{"artifact_kind":"plectis_reviewer_decision_brief","schema":"plectis_reviewer_source_hologram_v3","supersedes":["plectis_reviewer_hologram_v2","plectis_reviewer_decision_brief_v1"],"display_name":"Plectis reviewer brief","what_this_is":"Navigable public source for an AI assistant. Not a compact authored brief. Upload this file and query it by key: active paper records and available LaTeX in papers.records[paper_id].body; check papers.missing_embedded_bodies for unavailable source. Glossary term records are in glossary.terms, mathematical assertion rows in mathematical_assertions.assertions, Lean/Comparator/site paths in problem_path_index.problems[erdos_N], component records in component_digest, public repository files in public_repository_documents, independent review cases in challenge_manifest, and website links in site_routes. Website page prose is available through site_routes. hologram_atlas names the parts before you open them. Reading it is not a verification run: a check counts as passed only when the stated command was run and its evidence shown. The primary AI download at plectis-ai-review-packet.json is a compact content router.","authority":"Copied source plus brief-local indexes. Decision records and source layers come from the builder's gated in-memory full-source projection. Cold-open, answer-contract, start-here and Type B reading paths are adapted to keys this standalone brief carries. Decision-layer source_routes remain full-packet provenance; see source_route_scope before following a path absent here. Paper bodies from scholarly_corpus.papers[].body; glossary from public_glossary.terms; assertions from mathematical_assertions; components from component_digest; public repository documents, demo transcripts, and the challenge manifest are copied verbatim from that source projection. Maths-library hrefs follow the published path grammar and are not in the review packet's pages list. If a copied field disagrees with its cited public source, this brief is stale — regenerate rather than editing this file.","the_evidence_itself_is_at":"source_receipts and the cited public source URLs","source_route_scope":{"decision_layer_source_routes":"Provenance paths copied from the builder's in-memory full-source projection. They are not brief-local reading instructions. If a path is absent here, use sibling_artifacts.ai_router or site_routes to browse the cited source; even when its root is present here, inspect the named record before citing it.","ai_router":"plectis-ai-review-packet.json","brief_local_reading_paths":["question_router.routes.*.first_sources","cold_open_protocol.reading_plans","start_here.ai_route","type_b_orientation.read_by_question","answer_contract.answer_from"],"validation":"The builder checks brief-local reading paths. Copied decision source_routes retain their original full-packet provenance."},"read_in_this_order":["execution_boundary","hologram_atlas","question_router","paper_catalog","problem_path_index","mathematical_assertions","glossary","site_routes","component_digest","public_repository_documents","challenge_manifest","source_receipts"],"source_fingerprint":"98eed17f6ad86c212ec8b3eeb9d51e931753815f2879df906bfde2cdb5b07fd5","canonical_public_identity":{"schema":"plectis_canonical_public_identity_v1","project_name":"Plectis","artifact_name":"Plectis AI review packet","canonical_repository":"https://github.com/wcook04/practical-formal-verification","formal_mathematics_repository":"https://github.com/wcook04/plectis-erdos","legacy_technical_namespace":"microcosm","one_project":"Microcosm became Plectis. There is no second product, no fork and no predecessor still running: one project, renamed, with its old identifier kept where changing it would break something.","where_the_old_identifier_is_still_correct":["artifact_kind and schema, which stored copies validate against","the Python package and module paths the source actually uses","state paths and legacy routes that existing links resolve to","quoted source bodies, where editing the quotation would falsify it"],"where_it_would_be_wrong":"anything a reader is meant to type, say, or cite. The public command is `plectis`; component records carry the exact command their source accepts, which is sometimes the older spelling, and that is a fact about the source rather than a second name for the project.","if_you_are_summarising_this_file":"call it Plectis. Naming the schema identifier as the product is the one misreading this key exists to prevent."},"cold_open_protocol":{"schema":"plectis_cold_open_protocol_v1","applies_when":["the file arrives with no question at all","read this file","what is this","summarise this","tell me what I just uploaded","where do I start","any opening turn too broad to route"],"before_answering":["read artifact_kind and schema; if schema is not plectis_reviewer_source_hologram_v3, say so","read canonical_public_identity, so the project is named Plectis","run capsule_preflight against this file's keys","read execution_boundary before describing any capability"],"these_are_reading_actions":"Opening keys, filtering rows and counting them is reading this file. None of it runs Plectis, and none of it may be reported as having checked anything the packet records.","then_say_four_things":["what this file is: navigable public source for an AI, not a compact authored brief","what it can answer without cloning: papers, assertion rows, glossary terms, Lean/Comparator paths, public repository documents, demo transcripts, independent challenges, components, site routes","what it cannot establish: fresh execution, the withheld private system, independent correctness, external novelty","five questions spanning a general reader, a specialist, and a sceptic"],"do_not":["ask a clarifying question before orienting. A cold reader often lacks the vocabulary to answer it -- they have told you everything they know how to tell you yet. Orient first, then let them choose. The asymmetry is the whole argument: an unnecessary orientation costs a paragraph, an unanswerable question costs the turn","summarise from whichever fragment retrieval surfaced first. If the first thing you found was a glossary entry or a manuscript passage, that is a fragment of the packet, not the packet","describe the file by its size, key count or density. Those are properties of the container","claim, imply or accept that reading this file constitutes running, verifying or testing anything it describes"],"worked_first_answer":"This is the Plectis reviewer brief, navigable public source for an AI. Start with the mathematics: 8 Erdős problem lanes, 22 active paper records, and mathematical assertion rows. Open problem_path_index and mathematical_assertions for each lane's exact status and boundary; papers.records carries available LaTeX, and papers.missing_embedded_bodies names any absent manuscript source. The brief also carries Lean and Comparator paths, 784 glossary terms, 88 public software component records, source receipts, and a link-only website map. The full review packet carries rendered page prose. Reading this brief cannot run a component or recheck a proof.","then_offer":["Give me a tour of the website and both GitHub repositories.","What is Plectis, in plain terms?","List the short papers and the long papers.","For Erdős 1041, where is the Lean, the Comparator route, and the paper?","What does the glossary mean by hologram, packet, and check?","What is still open in the mathematics?","Talk me out of this — what is the strongest case against it?"],"if_the_question_has_no_route":{"why_this_is_here":"'I could not find that' and 'the public evidence does not establish that' are different answers, and answering the second when the first is true is how a search miss becomes a finding about the project.","ladder":["classify the question with question_router.routes","try an exact component, assertion, problem, page, paper, document or glossary identifier","search hologram_atlas and the compact catalogues before opening a full body","open one matching record from this brief","if absent, distinguish a search miss from refused, withheld or unchecked evidence"],"opens":["question_router","paper_catalog","problem_path_index","mathematical_assertions","glossary","public_repository_documents","challenge_manifest","hologram_atlas"],"the_three_negatives":"answer_contract.negative_claims_rule"},"reading_plans":{"canonical":["canonical_public_identity","cold_open_protocol","execution_boundary","citation_contract","answer_contract","hologram_atlas","project_thesis","question_router","site_routes","paper_catalog","problem_path_index","public_repository_documents"],"by_question":"question_router.routes","by_time_budget":"hologram_atlas","by_byte_budget":"hologram_atlas","note":"These are paths in this standalone brief; open a full body only for the selected question."}},"execution_boundary":{"schema":"plectis_chat_packet_execution_boundary_v0","chat_assistant_mode":"read_and_cite_only","packet_is_inert":true,"permitted_actions":["read and search the uploaded public records","quote, compare, and reason from those records","report a recorded check with exact attribution and a locator"],"execution_claim_rule":"The chat assistant must not say it executed a command, program, or check merely because this packet contains one. File reading, attachment search, and retrieval are not execution.","recorded_execution_rule":"A public record may state that a named command or check ran. Report that only as attributed historical evidence with its exact packet path or public link; never rewrite it as 'I ran' or 'I verified'.","embedded_program_rule":"Embedded source may be inspected and critiqued in chat. If a human wants a fresh run, they must move it to a separate evaluator environment.","separate_execution_surface":"A separate evaluator means a cloned repository plus coding agent, or a human-owned interpreter session. Attribute any resulting receipt to that environment.","analysing_this_file_is_not_running_the_system":{"permitted":"If your host gives you a code interpreter, load this JSON and query it. Parsing, filtering, counting, grouping, joining and diffing the packet's own bytes is reading, done accurately. It is the intended way to read a file this large, and it is what packet_atlas.python_recipes is written for.","not_permitted":"Running the commands, validators or programs the packet documents, and reporting the result as a check of Plectis. Those need the repository, its fixtures and its runtime; none of that is in this file, so a run in your sandbox would be testing a stub, not the system.","how_to_attribute":"Say what you computed and over what: 'counted over the 347 rows in mathematical_assertions.assertions' is an honest sentence. 'I verified the claim' is not, because you checked the packet's record of a check, not the check. Recomputation over this file cannot make a claim more true than the file is.","what_it_cannot_settle":"Whether a Lean proof compiles, whether a component passes today, whether the source the packet quotes still says that, and whether anything omitted from the packet contradicts it. Internal consistency is not correspondence: the packet can agree with itself perfectly and still describe a stale checkout, which is what source_receipts and semantic_integrity are for."},"answer_opening_rule":"Lead with the answer. Mention this boundary only when execution, verification, or the provenance of a result is relevant to the question."},"citation_contract":{"schema":"plectis_packet_citation_contract_v0","material_claim_rule":"Every material factual claim must carry the narrowest locator available in the packet. Distinct claims need distinct locators; one citation must not float over a paragraph it only partly supports.","locator_priority":["record.relative_href or its canonical public page URL","record.source_links[] or source_ref plus repository path","record id plus an exact JSON path in this packet","page id plus pages[].relative_href"],"quotation_rule":"Use quotation marks only for text copied from a record. Label summaries and comparisons as paraphrase or inference.","unsupported_rule":"If no carried locator supports a claim, label it as inference or say that the public records do not establish it. Stop where the cited record's scope limit stops.","recorded_execution_rule":"Cite the record that reports a run and name its actor or environment; a record of execution never licenses the packet-reading assistant to claim personal execution."},"answer_contract":{"answer_from":["site_routes","public_repository_documents","component_digest","family_digest","claim_ceiling_index","question_router","mathematical_assertions","papers.records","scholarly_maps","source_receipts","challenge_manifest"],"must_separate":["public evidence","inference","missing evidence","scope limit"],"must_not_infer":["hosted_service","product_readiness","whole_system_correctness","security_proof","financial_advice","claims_without_public_evidence"],"preferred_answer_shape":["plain answer","exact supporting packet path or public link for each material claim","evidence class and rank when relevant","scope limit","next inspection route"],"boundary_rule":"Answer only from Plectis's public source repository, public site pages, component cards, generated packets, and linked public source refs. When those records do not support a claim, say the public evidence does not show it.","calibration_rule":"Calibrate in both directions. Before describing an area as thin, read family_digest and a relevant component_digest record. Paper length and reference counts do not establish capability; cite what was checked.","symmetric_evaluation_rule":"Use the same evidentiary standard for favorable and unfavorable descriptions. Absence from the released records is not disproof, while volume, polish, and authorship are not proof of capability. Characterize a result only after opening the record that supports or limits that characterization.","independent_review_rule":"For a substantive review, inspect at least one strong record and one failed, blocked, bounded, or contradiction-seeking record relevant to the question. Name the sample and why it was selected. The packet routes evidence; the verdict is the reviewer's.","negative_claims_rule":"Classify a negative before asserting it: refused when a refusal record exists, withheld when outside this released source slice, or unchecked when no carried record bears on it. Only a refusal record supports a confident 'it does not'. Cite the record and its scope.","command_fidelity_rule":"Quote commands only from embedded component command fields or public_repository_documents; do not compose a command. In packet-chat mode a command is evidence or a handoff, never an action to execute.","command_audience_rule":"Commands are for a separate coding agent or human-owned interpreter. For fresh execution, use start_here.coding_agent_route and attribute any receipt to that separate environment.","handoff_back_rule":"You cannot write anywhere. This packet is read-only and nothing you conclude reaches the project unless you print it and the human carries it back. So end any substantive review by emitting a HANDOFF block the human can copy in one action: what you were asked, which packet records you actually opened (by key or id), what you concluded, what you could not settle and why, and any place the packet's wording claimed more than its evidence showed. Cite record keys, not prose summaries. That printed block is the entire return path -- an unprinted conclusion is a conclusion the project never receives.","handoff_back_scope":{"emit_after":["an audit or a review the human asked for","a judgement formed across several records","a defect, contradiction or overclaim found in the packet","any conclusion the human said they wanted to carry back"],"do_not_emit_after":["an ordinary factual question, however carefully answered","a drilldown into a record already under discussion","an answer the human asked to be short"],"why":"The block is the return path for findings. Attaching it to answers that carry no finding trains the human to skip it."}},"authorship_and_framing":{"who_built_this":{"author":"Will Cook, an economics undergraduate at the University of Bristol. Working alone, on a student budget, with no institutional support and no funding.","training":"No mathematics or computer-science degree. The formal mathematics is self-taught and Lean was learned for this project. The University of Bristol has no affiliation with this work.","how_it_was_built":"Built with AI throughout. AI was not the main tool, it was the only tool.","how_to_use_this_fact":"Provenance, offered for calibration. The mathematics stands or falls on the Lean source and the registry statuses, and the packet asks to be judged on those. The scale and shape of the corpus are not evidence of a funded team, a research group, or a department, because there was none. What to make of that is the reader's."},"plain_statement":"This file was assembled by the project's builder from the project's own records, and its structure shapes how an assistant reads the project. That is what documentation does. It is written to shape attention, never verdicts.","what_shapes_attention":["key order: the whole front door -- orientation, boundaries, routers, atlas, preflight -- comes before the first record body, and the large collections come after it. Check that against packet_atlas.entries rather than taking it on trust; this sentence was false for the packet's whole life, because the mathematics rows were assembled early and nobody re-read the claim after they grew to a megabyte and a half","question routes that pick which records answer which questions","computed tables (evidence_mode_legend, family_depth_facts)","starter prompts"],"what_must_not_happen":"No route, rule, or prompt here decides a conclusion for the reader. Wording that pre-decides a verdict is a defect of this packet; say so when found.","symmetry_defense":"The packet carries both positive claims and authority boundaries. Neither decides the verdict. A characterization without openable evidence behind it is reportable whether it flatters or diminishes the work.","how_to_audit_the_framing":"starter_prompts.audit_this_packets_framing","bottom_line":"Trust the openable evidence, not this file's tone. The verdict stays the reader's."},"capsule_preflight":{"schema":"plectis_reviewer_brief_preflight_v0","mode":"read_and_cite_only","execution_required":false,"expected_top_level_keys":["artifact_kind","schema","source_fingerprint","canonical_public_identity","cold_open_protocol","execution_boundary","citation_contract","answer_contract","authorship_and_framing","project_thesis","stakeholder_decision_router","decision_briefs","signature_evidence_sets","reviewer_diligence_ladder","engagement_contract","controlled_review_protocol","evidence_gap_register","empirical_programme","hologram_atlas","capsule_preflight","question_router","paper_catalog","papers","problem_path_index","mathematical_assertions","glossary","site_routes","headline_results","component_digest","family_digest","public_repository_documents","challenge_manifest","source_receipts"],"expected_active_paper_ids":["erdos-68-factorial-denominator-irrationality","erdos68-factorial-reasoning-surface","erdos-243-reciprocal-tail-rigidity","erdos243-reciprocal-tail-reasoning-surface","erdos-249-binary-totient-series","erdos249-totient-reasoning-surface","erdos-251-prime-gap-dyadic-series","erdos251-prime-gap-reasoning-surface","erdos-257-mersenne-support-subseries","erdos257-mersenne-reasoning-surface","erdos257-expository-sums-never-fractions","erdos269-running-lcm-reasoning-surface","erdos-269-three-prime-running-lcm","erdos-1041-lemniscate-newton-flow","erdos1041-lemniscate-reasoning-surface","erdos-1049-rational-base-lambert","erdos1049-rational-base-lambert-reasoning-surface","optimal-sparse-perturbations","writing-a-good-mathematical-paper","writing-mathematics-from-reviewed-revisions","claim-faithful-publication-systems","plectis-public-system"],"expected_active_paper_count":22,"expected_problem_ids":["erdos_68","erdos_243","erdos_249","erdos_251","erdos_257","erdos_269","erdos_1041","erdos_1049"],"expected_problem_count":8,"required_routes":["what_is_plectis","which_papers","one_erdos_problem","lean_paths","comparator","glossary","website","guided_tour","public_repository_documents","independent_challenges","software_components","what_can_i_decide"],"minimum_component_count":88,"minimum_glossary_term_count":784,"minimum_assertion_count":414,"if_preflight_fails":"The upload truncated or this copy is stale. Say so and stop; a truncated source hologram answers confidently from a fragment."},"hologram_atlas":{"schema":"plectis_reviewer_hologram_atlas_v0","role":"names_the_source_parts_before_you_open_them","how_to_read":"This file is navigable source, not a compact authored overlay. Do not answer from this atlas. Open the named key. papers.records[paper_id].body carries LaTeX when available; papers.missing_embedded_bodies names absent source; glossary.terms is the vocabulary records; mathematical_assertions.assertions is the claim rows; problem_path_index.problems[erdos_N] is Lean, Comparator, and public paths; component_digest is the software records; public_repository_documents contains the public repositories' reader files; challenge_manifest contains independently selectable review cases; site_routes is the website including the maths library. execution_boundary still applies: filtering JSON is reading, not running Plectis.","inventory":{"active_papers":22,"short_papers":9,"long_papers":8,"cross_problem_papers":1,"system_papers":4,"erdos_problems":8,"glossary_terms":784,"mathematical_assertions":414,"software_components":88,"software_pages_in_review_packet":221},"python_recipes":{"load":"import json; brief=json.load(open('plectis-reviewer-brief.json'))","list_keys":"list(brief)","list_papers":"sorted(brief['papers']['records'])","one_paper_body":"brief['papers']['records']['erdos-1041-lemniscate-newton-flow']['body']","one_problem":"brief['problem_path_index']['problems']['erdos_1041']","one_glossary_term":"next(t for t in brief['glossary']['terms'] if t['id']=='hologram')","one_assertion":"brief['mathematical_assertions']['assertions'][0]","short_vs_long":"brief['paper_catalog']['by_form']","repository_documents":"brief['public_repository_documents']","challenge_cases":"brief['challenge_manifest']"},"what_this_file_cannot_establish":["that any Lean proof compiles today","that any component still passes","the unpublished private system","independent novelty or peer review"]},"question_router":{"schema":"plectis_reviewer_brief_question_router_v0","contract":"Routes a question to a source key in THIS file. It does not pre-decide the verdict. Keys named here must exist on the brief.","routes":{"what_is_plectis":{"when":"what is this, summarise this, what did I upload","first_sources":["canonical_public_identity","hologram_atlas","project_thesis","execution_boundary"]},"which_papers":{"when":"papers, short notes, long papers, reasoning surfaces, manuscripts","first_sources":["paper_catalog","papers","site_routes.maths_library"]},"one_erdos_problem":{"when":"erdos 68/243/249/251/257/269/1041/1049, what is proved, what is open","first_sources":["problem_path_index","headline_results","mathematical_assertions","paper_catalog.by_problem"]},"lean_paths":{"when":"lean modules, where is the proof, source files","first_sources":["problem_path_index.problems","site_routes.lean_claim_layer"]},"comparator":{"when":"comparator, external verification, independent check","first_sources":["problem_path_index.problems","glossary","site_routes.maths_library.documents"]},"glossary":{"when":"what does this word mean, hologram, packet, coding agent","first_sources":["glossary","site_routes.first_human_pages"]},"website":{"when":"site map, where is the maths library, landing page, docs","first_sources":["site_routes","paper_catalog"],"body_rule":"This file carries links and titles only. Open the live page or the full review packet for its prose."},"guided_tour":{"when":"tour of the website and both GitHub repositories","first_sources":["site_routes","public_repository_documents","scholarly_maps.repository_maps"],"answer_shape":["website overview and one linked page","mathematics repository: one embedded front-door document and source path","software repository: one embedded front-door document and source path","ask which stop to explore next"],"body_rule":"For website prose, open the live page or the full review packet. For embedded repository documents, read their bodies before describing them. Do not silently drop either repository when the reader asks for both."},"public_repository_documents":{"when":"readme, repository documents, source files, public repository contents","first_sources":["public_repository_documents","scholarly_maps.formal_claim_control_plane"]},"independent_challenges":{"when":"challenge, independent review, cases the author did not choose","first_sources":["challenge_manifest"]},"software_components":{"when":"component, what does it run, evidence class","first_sources":["component_digest","family_digest","claim_ceiling_index"]},"what_can_i_decide":{"when":"should I care, investor, mathematician, sceptic","first_sources":["stakeholder_decision_router","decision_briefs","evidence_gap_register"]}}},"project_thesis":{"schema":"plectis_project_thesis_v1","role":"the current mathematics-led project, before the earlier software inventory","problem":{"statement":"AI can help produce formal mathematics, but a checked statement still needs a readable explanation, a reason it matters and a clear account of what remains open. Will Cook wants other people to understand and continue this work.","statement_class":"author_thesis"},"mechanism":{"statement":"Connect each problem to its short paper, longer reasoning record, Lean source and recorded checks. Preserve failed approaches and hypotheses so a reader can follow the argument and choose what to try next.","statement_class":"author_thesis"},"public_evidence":[{"statement":"Problem records distinguish checked results, open obligations and statements not established. Read each status separately; a refuted formal statement does not by itself settle the historical wording.","statement_class":"public_fact","source_routes":["headline_results.per_problem","headline_results.verification_boundary"]},{"statement":"The packet carries the published manuscripts and source links, with an availability record for each paper.","statement_class":"public_fact","source_routes":["scholarly_corpus.papers","type_b_orientation.coverage_receipt"]},{"statement":"The separate earlier software release carries 88 components, of which 6 invoke external machinery.","statement_class":"public_fact","source_routes":["component_digest","counts"]},{"statement":"The published source audit records 4 defects with their corrections or remaining review needs.","statement_class":"public_fact","source_routes":["upstream_defects.defects"]}],"boundary":{"statement":"Recorded formal checks establish specific statements under their assumptions. They do not establish novelty, significance, independent peer review or the reliability of the unpublished working system.","statement_class":"public_fact","source_routes":["headline_results.verification_boundary","execution_boundary"]},"what_the_author_wants":{"statement":"Mathematical direction, corrections and ideas. The contact page gives the current collaboration routes and contribution-credit policy.","statement_class":"author_thesis"},"what_support_would_change":{"statement":"The stated aim is to keep mathematics produced with AI checkable and readable as models improve: each problem set out from its Lean map down to one result, showing what is proved, what is not and why. This is an aim, not evidence of a future result.","statement_class":"author_thesis"},"immediate_ask":{"statement":"Choose one problem, read its short paper, then follow a result or open question to its source. Share a specific correction or idea through the website's contribution routes.","statement_class":"recommended_next_action","does_not_imply":"that reading a paper verifies it or endorses the project"}},"stakeholder_decision_router":{"schema":"plectis_stakeholder_decision_router_v1","why_this_exists":"Every other router here sorts by the kind of object a reader names. This one sorts by the decision they are trying to make, because a reader who does not yet know what is in the file cannot name an object, and the decision is the thing they always have.","ask_once_if_unclear":"What are you trying to decide: whether it is real, whether it matters to your work, whether it is new, or whether to speak to the builder?","never":["infer the reader's role from weak hints and answer as if it were known","give a decision brief to a reader who asked a narrow factual question","present a recommended action without the condition that would stop it"],"decisions":{"understand_the_project":{"sounds_like":"what is this, what is it for","answer_shape":["the decision this reader is making, named back to them","at most three records, from the brief or the signature set","what those records cannot settle","the cheapest next action, or the stop condition if that is the honest answer"],"opens":"project_thesis"},"verify_one_public_claim":{"sounds_like":"is any of this real, is this AI slop, show me one thing I can check","answer_shape":["the decision this reader is making, named back to them","at most three records, from the brief or the signature set","what those records cannot settle","the cheapest next action, or the stop condition if that is the honest answer"],"evidence":"signature_evidence_sets.sets.general_first_pass"},"assess_ai_lab_relevance":{"sounds_like":"why should a lab care, is this relevant to us","answer_shape":["the decision this reader is making, named back to them","at most three records, from the brief or the signature set","what those records cannot settle","the cheapest next action, or the stop condition if that is the honest answer"],"opens":"decision_briefs.briefs.ai_lab_executive"},"assess_agent_infrastructure_relevance":{"sounds_like":"we run coding agents, is there anything here","answer_shape":["the decision this reader is making, named back to them","at most three records, from the brief or the signature set","what those records cannot settle","the cheapest next action, or the stop condition if that is the honest answer"],"opens":"decision_briefs.briefs.agent_infrastructure_lead"},"assess_safety_or_evals_relevance":{"sounds_like":"does this bear on agent safety or evaluations","answer_shape":["the decision this reader is making, named back to them","at most three records, from the brief or the signature set","what those records cannot settle","the cheapest next action, or the stop condition if that is the honest answer"],"opens":"decision_briefs.briefs.safety_evals_team"},"assess_formal_methods_relevance":{"sounds_like":"what about the mathematics, is the Lean work worth reading","answer_shape":["the decision this reader is making, named back to them","at most three records, from the brief or the signature set","what those records cannot settle","the cheapest next action, or the stop condition if that is the honest answer"],"evidence":"signature_evidence_sets.sets.formal_methods"},"assess_investment_meeting_relevance":{"sounds_like":"is this a company, is it worth a meeting, there is no company and no product so is this person worth backing or funding","answer_shape":["the decision this reader is making, named back to them","at most three records, from the brief or the signature set","what those records cannot settle","the cheapest next action, or the stop condition if that is the honest answer"],"opens":"decision_briefs.briefs.investor_technical_diligence"},"assess_builder_or_hiring_relevance":{"sounds_like":"what has the builder actually demonstrated","answer_shape":["the decision this reader is making, named back to them","at most three records, from the brief or the signature set","what those records cannot settle","the cheapest next action, or the stop condition if that is the honest answer"],"opens":"decision_briefs.briefs.hiring_or_research_collaboration"},"consider_collaboration_or_pilot":{"sounds_like":"could we test this against a real workflow","answer_shape":["the decision this reader is making, named back to them","at most three records, from the brief or the signature set","what those records cannot settle","the cheapest next action, or the stop condition if that is the honest answer"],"opens":"decision_briefs.briefs.agent_infrastructure_lead"},"request_controlled_private_review":{"sounds_like":"I cannot take the whole system, what can I review","answer_shape":["the decision this reader is making, named back to them","at most three records, from the brief or the signature set","what those records cannot settle","the cheapest next action, or the stop condition if that is the honest answer"],"opens":"decision_briefs.briefs.controlled_private_reviewer"},"seek_guidance_or_routing":{"sounds_like":"I can see capability here but I am not the final judge; 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It checks propositions only: no paper deduction, cited theorem or external computation. 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A reasoning surface maps selected results, failed routes, and open interfaces. Neither is proof authority: checked Lean source is.","counts":{"active_papers":22,"short_papers":9,"long_papers":8,"cross_problem_papers":1,"system_papers":4,"forms":{"Problem note":8,"Reasoning surface":8,"Expository paper":1,"Cross-problem paper":1,"Compact writing guide":1,"Editorial method companion":1,"Research architecture":1,"Evidence boundary":1}},"by_form":{"Problem note":["erdos-68-factorial-denominator-irrationality","erdos-243-reciprocal-tail-rigidity","erdos-249-binary-totient-series","erdos-251-prime-gap-dyadic-series","erdos-257-mersenne-support-subseries","erdos-269-three-prime-running-lcm","erdos-1041-lemniscate-newton-flow","erdos-1049-rational-base-lambert"],"Reasoning surface":["erdos68-factorial-reasoning-surface","erdos243-reciprocal-tail-reasoning-surface","erdos249-totient-reasoning-surface","erdos251-prime-gap-reasoning-surface","erdos257-mersenne-reasoning-surface","erdos269-running-lcm-reasoning-surface","erdos1041-lemniscate-reasoning-surface","erdos1049-rational-base-lambert-reasoning-surface"],"Expository paper":["erdos257-expository-sums-never-fractions"],"Cross-problem paper":["optimal-sparse-perturbations"],"Compact writing guide":["writing-a-good-mathematical-paper"],"Editorial method companion":["writing-mathematics-from-reviewed-revisions"],"Research architecture":["claim-faithful-publication-systems"],"Evidence boundary":["plectis-public-system"]},"by_problem":{"erdos_68":["erdos-68-factorial-denominator-irrationality","erdos68-factorial-reasoning-surface"],"erdos_243":["erdos-243-reciprocal-tail-rigidity","erdos243-reciprocal-tail-reasoning-surface"],"erdos_249":["erdos-249-binary-totient-series","erdos249-totient-reasoning-surface"],"erdos_251":["erdos-251-prime-gap-dyadic-series","erdos251-prime-gap-reasoning-surface"],"erdos_257":["erdos-257-mersenne-support-subseries","erdos257-mersenne-reasoning-surface","erdos257-expository-sums-never-fractions"],"erdos_269":["erdos269-running-lcm-reasoning-surface","erdos-269-three-prime-running-lcm"],"erdos_1041":["erdos-1041-lemniscate-newton-flow","erdos1041-lemniscate-reasoning-surface"],"erdos_1049":["erdos-1049-rational-base-lambert","erdos1049-rational-base-lambert-reasoning-surface"]},"short_paper_ids":["erdos-68-factorial-denominator-irrationality","erdos-243-reciprocal-tail-rigidity","erdos-249-binary-totient-series","erdos-251-prime-gap-dyadic-series","erdos-257-mersenne-support-subseries","erdos257-expository-sums-never-fractions","erdos-269-three-prime-running-lcm","erdos-1041-lemniscate-newton-flow","erdos-1049-rational-base-lambert"],"long_paper_ids":["erdos68-factorial-reasoning-surface","erdos243-reciprocal-tail-reasoning-surface","erdos249-totient-reasoning-surface","erdos251-prime-gap-reasoning-surface","erdos257-mersenne-reasoning-surface","erdos269-running-lcm-reasoning-surface","erdos1041-lemniscate-reasoning-surface","erdos1049-rational-base-lambert-reasoning-surface"],"cross_problem_paper_ids":["optimal-sparse-perturbations"],"system_paper_ids":["writing-a-good-mathematical-paper","writing-mathematics-from-reviewed-revisions","claim-faithful-publication-systems","plectis-public-system"],"in_registry_order":["erdos-68-factorial-denominator-irrationality","erdos68-factorial-reasoning-surface","erdos-243-reciprocal-tail-rigidity","erdos243-reciprocal-tail-reasoning-surface","erdos-249-binary-totient-series","erdos249-totient-reasoning-surface","erdos-251-prime-gap-dyadic-series","erdos251-prime-gap-reasoning-surface","erdos-257-mersenne-support-subseries","erdos257-mersenne-reasoning-surface","erdos257-expository-sums-never-fractions","erdos269-running-lcm-reasoning-surface","erdos-269-three-prime-running-lcm","erdos-1041-lemniscate-newton-flow","erdos1041-lemniscate-reasoning-surface","erdos-1049-rational-base-lambert","erdos1049-rational-base-lambert-reasoning-surface","optimal-sparse-perturbations","writing-a-good-mathematical-paper","writing-mathematics-from-reviewed-revisions","claim-faithful-publication-systems","plectis-public-system"],"by_id":{"erdos-68-factorial-denominator-irrationality":{"paper_id":"erdos-68-factorial-denominator-irrationality","title":"Integer Linear Forms for a Factorial Reciprocal Series","form":"Problem note","length_class":"short","pages":9,"subject":"Erdős #68","subject_kind":"problem","problem_id":"erdos_68","landing_role":"mathematics","question":"Which integer coefficient vectors cancel initial weighted sums in factorial-denominator linear forms?","precis":"For S=∑_{n≥2}(n!−1)^{-1}, the paper classifies finitely supported integer vectors that cancel prescribed initial weighted sums in forms MS+k. An integral basis, one Diophantine equation and a finite gcd determine the possible moments M. At cancellation depth four, the least positive moment is 1380. Irrationality by these forms still requires a nonintegral form with q dividing M for every positive q.","owns":"the coefficient classification, finite moment gcd and depth-four construction; the long paper develops denominator exclusions, further constructions and method limits","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #68, which remains open","first_pass":["sec:problem","sec:channels","sec:moments","sec:open"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-68-factorial-denominator-irrationality.pdf","hosted_pdf_path":"papers/erdos-68-factorial-denominator-irrationality.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/68/erdos-68-factorial-denominator-irrationality.tex","site_html":"maths/papers/erdos-68-factorial-denominator-irrationality.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/68/erdos-68-factorial-denominator-irrationality.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/68/erdos-68-factorial-denominator-irrationality.tex","body_pointer":"papers.records.erdos-68-factorial-denominator-irrationality.body"},"erdos68-factorial-reasoning-surface":{"paper_id":"erdos68-factorial-reasoning-surface","title":"Factorial Linear Forms and Denominators: Detailed Proofs and Rationality Criteria","form":"Reasoning surface","length_class":"long","pages":51,"subject":"Erdős #68","subject_kind":"problem","problem_id":"erdos_68","landing_role":"mathematics","question":"What growth, valuation, and finite-denominator facts are proved for ∑_{n≥2}(n!−1)^{-1}, and which tail inequalities remain?","precis":"The record proves liminf (log L_N)/(N^{3/2} log N) ≥ 2√2/3 for L_N=lcm(2!−1,…,N!−1), by comparing a terminal block of denominators with pairwise gcds rather than an external multiplicity theorem. Specialising a Louwsma–Martino valuation, it exhibits primes 139 and 2593 absent from all later reduced denominators. Coefficient constructions give integer forms MS+k; at fixed moment the remainder changes only by an integer. Rationality criteria use factorial digits and the next integer above a scaled partial sum; the digit argument also recovers the irrationality of e. Two exact calculations give q∤299999! and q≥2^39990>10^12038. These exclusions do not prove irrationality.","owns":"the complete problem-specific reasoning surface for Erdős #68, including all registered result families and their boundaries","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #68, which remains open","first_pass":["long68:sec:carry","long68:sec:finite","long68:sec:open","sec:erdos-68-complete-family-map"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos68-factorial-reasoning-surface.pdf","hosted_pdf_path":"papers/erdos68-factorial-reasoning-surface.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/68/erdos68-factorial-reasoning-surface.tex","site_html":"maths/papers/erdos68-factorial-reasoning-surface.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/68/erdos68-factorial-reasoning-surface.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/68/erdos68-factorial-reasoning-surface.tex","body_pointer":"papers.records.erdos68-factorial-reasoning-surface.body"},"erdos-243-reciprocal-tail-rigidity":{"paper_id":"erdos-243-reciprocal-tail-rigidity","title":"Reciprocal Sums and the Sylvester Recurrence","form":"Problem note","length_class":"short","pages":21,"subject":"Erdős #243","subject_kind":"problem","problem_id":"erdos_243","landing_role":"mathematics","question":"Which growth rates force irrational reciprocal sums, and which additional tail conditions force the Sylvester recurrence?","precis":"For strictly increasing positive integers with a_n²/a_{n+1}=1+3/n+o(n^{-3}), the reciprocal sum is irrational. The argument extracts a cubic polynomial from an integer tail numerator, restricts its coefficients using a square in a cubic field, and excludes the remaining cases modulo seven. Under rationality and the quadratic growth limit, separate criteria using bounded numerator increases and weighted first crossings force the eventual Sylvester recurrence. The unrestricted Erdős–Graham question remains open.","owns":"cubic-rate irrationality, nonintegral regular-rate exclusion and sufficient tail criteria for the Sylvester recurrence; the unrestricted Erdős–Graham question remains open","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #243, which remains open","first_pass":["sec:problem","sec:transfer","sec:secondaryrate"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-243-reciprocal-tail-rigidity.pdf","hosted_pdf_path":"papers/erdos-243-reciprocal-tail-rigidity.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex","site_html":"maths/papers/erdos-243-reciprocal-tail-rigidity.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/243/erdos-243-reciprocal-tail-rigidity.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/243/erdos-243-reciprocal-tail-rigidity.tex","body_pointer":"papers.records.erdos-243-reciprocal-tail-rigidity.body"},"erdos243-reciprocal-tail-reasoning-surface":{"paper_id":"erdos243-reciprocal-tail-reasoning-surface","title":"Reciprocal Sums and the Sylvester Recurrence: Further Results and Proofs","form":"Reasoning surface","length_class":"long","pages":61,"subject":"Erdős #243","subject_kind":"problem","problem_id":"erdos_243","landing_role":"mathematics","question":"Which growth hypotheses make a reciprocal sum irrational, and which sufficient conditions force the Sylvester recurrence without settling Erdős #243?","precis":"Integer finite differences exclude nonintegral regular rates and reduce the cubic rate to a polynomial. A square condition in the cubic field and congruences modulo seven then prove irrationality. For rational reciprocal sums with a_{n+1}/a_n² tending to one, the record derives sufficient Sylvester-recurrence criteria from persistent divisors, bounded numerator increases and weighted crossings of new maxima. The arithmetic estimates needed to apply these criteria to every rational reciprocal sum remain open.","owns":"cubic and nonintegral regular-rate irrationality, integer-tail arithmetic and sufficient Sylvester-recurrence criteria, with their separate hypotheses and remaining estimates","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #243, which remains open","first_pass":["long243:sec:cubicrate","long243:sec:bounded","long243:sec:open","sec:erdos-243-complete-family-map"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos243-reciprocal-tail-reasoning-surface.pdf","hosted_pdf_path":"papers/erdos243-reciprocal-tail-reasoning-surface.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex","site_html":"maths/papers/erdos243-reciprocal-tail-reasoning-surface.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex","body_pointer":"papers.records.erdos243-reciprocal-tail-reasoning-surface.body"},"erdos-249-binary-totient-series":{"paper_id":"erdos-249-binary-totient-series","title":"Integral Relations among Totient Sections","form":"Problem note","length_class":"short","pages":15,"subject":"Erdős #249","subject_kind":"problem","problem_id":"erdos_249","landing_role":"mathematics","question":"What explicit basis and integral relations does the totient k-kernel have at every integer base, and why does that not decide Erdős #249?","precis":"For integers k≥2 and e≥1, the totient sections n↦φ(k^j n+r) with 0≤j≤e and 0≤r<k^j have an explicit rational basis of dimension k^e+1. Every section is an explicit integer multiple of a retained one, and these reductions generate all integral relations, with unique coefficients. The proof combines the local formula for Euler's totient with a nonsingular evaluation matrix obtained from the Chinese remainder theorem and Dirichlet's theorem. The basis theorem does not decide the irrationality of S=∑ φ(n)2^{-n}.","owns":"the exact finite-level totient-kernel basis, independence proof and integral relations; the long paper develops denominator exclusions and other approaches to the binary series","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #249, which remains open","first_pass":["sec:results","sec:rank","sec:base-six-test"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-249-binary-totient-series.pdf","hosted_pdf_path":"papers/erdos-249-binary-totient-series.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/249/erdos-249-binary-totient-series.tex","site_html":"maths/papers/erdos-249-binary-totient-series.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos-249-binary-totient-series.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/249/erdos-249-binary-totient-series.tex","body_pointer":"papers.records.erdos-249-binary-totient-series.body"},"erdos249-totient-reasoning-surface":{"paper_id":"erdos249-totient-reasoning-surface","title":"The Binary Totient Series","form":"Reasoning surface","length_class":"long","pages":163,"subject":"Erdős #249","subject_kind":"problem","problem_id":"erdos_249","landing_role":"mathematics","question":"What can the totient-kernel basis and residue tests say about Erdős #249, and which hypotheses still block irrationality?","precis":"The dyadic totient sections through level e≥1 have an explicit rational basis of size 2^e+1. Truncating a scaled tail difference produces an integer whose residue certifies nonintegrality when its distances from both endpoints of the residue interval exceed an explicit error bound. The record develops the resulting finite denominator exclusions, equivalent residue criteria, and sufficient exponential-sum estimates. Rational comparison sequences identify the limitations of particular size, parity, rank, and finite-prefix arguments. The short paper supplies the all-base basis theorem; here each conditional result carries the arithmetic hypothesis still needed to apply it. Those hypotheses are not established in the form required to prove irrationality of S=∑ φ(n)2^{-n}.","owns":"a claim-bounded problem-specific reasoning surface for Erdős #249: selected checked premises, finite certificates, closed routes, surviving obligations, and their exact evidence bands","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #249, which remains open","first_pass":["sec:wall","sec:survivors","sec:series","sec:mahler-defect"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos249-totient-reasoning-surface.pdf","hosted_pdf_path":"papers/erdos249-totient-reasoning-surface.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/249/erdos249-totient-reasoning-surface.tex","site_html":"maths/papers/erdos249-totient-reasoning-surface.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos249-totient-reasoning-surface.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/249/erdos249-totient-reasoning-surface.tex","body_pointer":"papers.records.erdos249-totient-reasoning-surface.body"},"erdos-251-prime-gap-dyadic-series":{"paper_id":"erdos-251-prime-gap-dyadic-series","title":"Sparse Congruence-Preserving Perturbations of Dyadic Series","form":"Problem note","length_class":"short","pages":14,"subject":"Erdős #251","subject_kind":"problem","problem_id":"erdos_251","landing_role":"mathematics","question":"Which sparse congruence-preserving corrections can rationalise a dyadic series, and why does that not prove Erdős #251 irrational?","precis":"Sparse nonnegative integer corrections to a convergent dyadic series can attain every target in an interval. One set of upper Banach density zero supports the corrections, which satisfy any prescribed bound tending to infinity and eventually preserve each fixed coefficient and partial-sum congruence. For prime gaps with logarithmic corrections, block distributions remain close for lengths o(log log X), and cumulative positions remain asymptotic to n log n. Those positions need not be prime; the original irrationality question remains open.","owns":"the sparse correction construction and its prime-gap application; the long paper develops tail recurrences, local certificates and the limitations of other irrationality methods","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #251, which remains open","first_pass":["sec:problem","sec:construction","sec:prime-application"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-251-prime-gap-dyadic-series.pdf","hosted_pdf_path":"papers/erdos-251-prime-gap-dyadic-series.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/251/erdos-251-prime-gap-dyadic-series.tex","site_html":"maths/papers/erdos-251-prime-gap-dyadic-series.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/251/erdos-251-prime-gap-dyadic-series.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/251/erdos-251-prime-gap-dyadic-series.tex","body_pointer":"papers.records.erdos-251-prime-gap-dyadic-series.body"},"erdos251-prime-gap-reasoning-surface":{"paper_id":"erdos251-prime-gap-reasoning-surface","title":"Prime-Gap Dyadic Series: Perturbations, Exact Criteria and Certificates","form":"Reasoning surface","length_class":"long","pages":47,"subject":"Erdős #251","subject_kind":"problem","problem_id":"erdos_251","landing_role":"mathematics","question":"What exact tail criteria, sparse countermodels, and finite denominator bounds are proved for the prime-gap dyadic series, and why is irrationality still open?","precis":"A convergent dyadic series with nonnegative integer coefficients can be made rational by sparse nonnegative corrections of upper Banach density zero, eventually bounded by any f(n)→∞, while preserving every fixed modulus and, for a logarithmic allowance, short-block empirical distributions. For prime gaps the cumulative positions remain asymptotic to n log n; no primality conclusion is asserted. Summation by parts reduces the prime series to the gap series, and the tails T_N=∑_{j≥1} g_{N+j} 2^{-j} satisfy T_{N+1}=2T_N−g_{N+1}. That recurrence characterises rationality and yields finite tests with explicit remainder bounds. An exact finite certificate shows that a rational value of the prime series would have denominator at least 2^{39998}>10^{12040}. The bound does not establish irrationality.","owns":"the complete problem-specific reasoning surface for Erdős #251, including all registered result families and their boundaries","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #251, which remains open","first_pass":["long251:sec:tail","long251:sec:obstructions","long251:sec:open","sec:erdos-251-complete-family-map"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos251-prime-gap-reasoning-surface.pdf","hosted_pdf_path":"papers/erdos251-prime-gap-reasoning-surface.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/251/erdos251-prime-gap-reasoning-surface.tex","site_html":"maths/papers/erdos251-prime-gap-reasoning-surface.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/251/erdos251-prime-gap-reasoning-surface.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/251/erdos251-prime-gap-reasoning-surface.tex","body_pointer":"papers.records.erdos251-prime-gap-reasoning-surface.body"},"erdos-257-mersenne-support-subseries":{"paper_id":"erdos-257-mersenne-support-subseries","title":"Irrationality criteria for Lambert subseries","form":"Problem note","length_class":"short","pages":26,"subject":"Erdős #257","subject_kind":"problem","problem_id":"erdos_257","landing_role":"mathematics","question":"Which supports are proved to give irrational Lambert subseries, and what still blocks the arbitrary infinite support that Erdős #257 asks about?","precis":"The series ∑_{a∈A}(b^a−1)^{-1} is irrational for every integer b≥2 and infinite A contained in E∪V when E satisfies a finite-prime weighted summability condition and V has a summable positive divisor cover. The two classes are incomparable, and their union reaches supports in neither class. A common average over dyadic lengths produces one index with both displacements small. Arbitrary infinite supports at base two remain unresolved.","owns":"the weighted, divisor-cover and mixed support criteria with complete proofs and method limits; the long paper treats finite denominators, the achievement set and rational membership","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #257, which remains open","first_pass":["sec:problem","sec:reciprocal-support","sec:eight-return-extensions","sec:map"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-257-mersenne-support-subseries.pdf","hosted_pdf_path":"papers/erdos-257-mersenne-support-subseries.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex","site_html":"maths/papers/erdos-257-mersenne-support-subseries.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/257/erdos-257-mersenne-support-subseries.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/257/erdos-257-mersenne-support-subseries.tex","body_pointer":"papers.records.erdos-257-mersenne-support-subseries.body"},"erdos257-mersenne-reasoning-surface":{"paper_id":"erdos257-mersenne-reasoning-surface","title":"Reciprocal Mersenne Subseries","form":"Reasoning surface","length_class":"long","pages":152,"subject":"Erdős #257","subject_kind":"problem","problem_id":"erdos_257","landing_role":"mathematics","question":"What weighted and cover criteria prove hereditary Mersenne irrationality for Erdős #257, and which targets remain open?","precis":"A weighted summability criterion proves irrationality of reciprocal Mersenne subseries: if h is the part of a supported on a finite nonempty prime set P, every infinite A with ∑ h(a)/(a(b^{h(a)}−1)) finite has irrational ∑ (b^a−1)^{-1}. The base-two condition gives the conclusion at every integer base for every infinite subset, and permits an example with divergent reciprocal sum, while excluding full support and all odd exponents. The proof averages over a finite part of the support and over dyadic observation lengths. Erdős's weaker reciprocal-summable theorem is proved in full, and a common window extends both arguments to unions of weighted and positive-cover supports. Finite-denominator and achievement-set calculations formulate tests for the targets 1/2 and 1/21; those tests do not decide either target or the universal question. Prime support at base 2 and squarefree support at power-of-two bases are attributed as existing work.","owns":"a claim-bounded problem-specific reasoning surface for Erdős #257: selected checked premises, finite computations, closed routes, surviving obligations, and their exact evidence bands","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #257, whose universal and half-value questions remain open","first_pass":["sec:257-problem","sec:257-wall","sec:257-survivors","sec:257-howto"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos257-mersenne-reasoning-surface.pdf","hosted_pdf_path":"papers/erdos257-mersenne-reasoning-surface.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/257/erdos257-mersenne-reasoning-surface.tex","site_html":"maths/papers/erdos257-mersenne-reasoning-surface.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/257/erdos257-mersenne-reasoning-surface.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/257/erdos257-mersenne-reasoning-surface.tex","body_pointer":"papers.records.erdos257-mersenne-reasoning-surface.body"},"erdos257-expository-sums-never-fractions":{"paper_id":"erdos257-expository-sums-never-fractions","title":"Why certain sums of 1/(2^n-1) are irrational","form":"Expository paper","length_class":"short","pages":16,"subject":"Erdős #257","subject_kind":"problem","problem_id":"erdos_257","landing_role":"mathematics","question":"Why do the fractions 1/(2^n−1) over an infinite set of exponents n add up to an irrational number whenever 1/n, added up over the same exponents, is finite?","precis":"Erdős asked whether infinitely many of the fractions 1, 1/3, 1/7, 1/15, …, each of the form 1/(2^n−1) and each used at most once, can ever add up to a fraction. The paper proves that the chosen fractions add up to an irrational number whenever 1/n, added up over the chosen exponents n, is finite; for the exponents 2, 4, 8, 16, … this makes 1/3+1/15+1/255+1/65535+… irrational. Sections 1 to 8 explain the proof with examples; Appendix A supplies supporting details and formal-proof correspondence. The finite reconstruction here is an ordinary proof; the linked Lean and Comparator receipts establish the formal theorem and its own proof. For an arbitrary infinite choice the question is still open.","owns":"an account of Theorem A.1 of the #257 short paper at base 2 for a reader new to the problem: a guided proof through worked examples, with supporting details, exercises and the formal-proof correspondence in appendices","not_authority_for":"the validity of the Lean theorem it quotes, which belongs to the cited kernel-checked source, or a solution to Erdős #257, which remains open","first_pass":["sec:intro","sec:test","sec:doubling","sec:proof"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos257-expository-sums-never-fractions.pdf","hosted_pdf_path":"papers/erdos257-expository-sums-never-fractions.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/257/erdos257-expository-sums-never-fractions.tex","site_html":"maths/papers/erdos257-expository-sums-never-fractions.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/257/erdos257-expository-sums-never-fractions.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/257/erdos257-expository-sums-never-fractions.tex","body_pointer":"papers.records.erdos257-expository-sums-never-fractions.body"},"erdos269-running-lcm-reasoning-surface":{"paper_id":"erdos269-running-lcm-reasoning-surface","title":"Running least common multiples: distinct values and multiplicities","form":"Reasoning surface","length_class":"long","pages":73,"subject":"Erdős #269","subject_kind":"problem","problem_id":"erdos_269","landing_role":"mathematics","question":"How do distinct running-LCM irrationality, infinite-rank kernels and repeated-sum tail arithmetic fit together, and which residue windows remain open?","precis":"For the prime set {2,3,5}, a five-map argument proves irrationality of the sum over distinct running LCMs. Two-prime sums reduce to a transcendental Hecke–Mahler value. For the repeated three-prime sum, an equivalent residue criterion leaves an arithmetic condition open. Complete ordinary arguments for general distinct-running-LCM sums and prime-power subseries are retained as remarks awaiting formalisation.","owns":"distinct-height irrationality, all-rank kernel structure and the repeated-sum tail criteria, preserving the multiplicity distinction and the remaining open residue-window obligation","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #269, which remains open","first_pass":["long269:sec:problem","long269:sec:rank","long269:sec:open","sec:erdos-269-complete-family-map"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos269-running-lcm-reasoning-surface.pdf","hosted_pdf_path":"papers/erdos269-running-lcm-reasoning-surface.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/269/erdos269-running-lcm-reasoning-surface.tex","site_html":"maths/papers/erdos269-running-lcm-reasoning-surface.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/269/erdos269-running-lcm-reasoning-surface.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/269/erdos269-running-lcm-reasoning-surface.tex","body_pointer":"papers.records.erdos269-running-lcm-reasoning-surface.body"},"erdos-269-three-prime-running-lcm":{"paper_id":"erdos-269-three-prime-running-lcm","title":"Distinct running least common multiples","form":"Problem note","length_class":"short","pages":16,"subject":"Erdős #269","subject_kind":"problem","problem_id":"erdos_269","landing_role":"mathematics","question":"Why is the sum over distinct running LCMs of the 5-smooth integers irrational?","precis":"A five-map argument proves irrationality of the reciprocal sum over distinct running least common multiples of the 5-smooth integers. The spacing of powers of five separates the tail images, so rationality would force a periodic jump sequence. Counting running LCMs with multiplicity gives a different number, whose irrationality remains open here.","owns":"the five-map proof for distinct running LCMs over {2,3,5}; the long paper treats general prime sets, the sum with multiplicity, kernels and residue criteria","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #269, which remains open","first_pass":["sec:problem","sec:distinct"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-269-three-prime-running-lcm.pdf","hosted_pdf_path":"papers/erdos-269-three-prime-running-lcm.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/269/erdos-269-three-prime-running-lcm.tex","site_html":"maths/papers/erdos-269-three-prime-running-lcm.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/269/erdos-269-three-prime-running-lcm.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/269/erdos-269-three-prime-running-lcm.tex","body_pointer":"papers.records.erdos-269-three-prime-running-lcm.body"},"erdos-1041-lemniscate-newton-flow":{"paper_id":"erdos-1041-lemniscate-newton-flow","title":"Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections","form":"Problem note","length_class":"short","pages":19,"subject":"Erdős #1041","subject_kind":"problem","problem_id":"erdos_1041","landing_role":"mathematics","question":"How does the degree-seven counterexample obstruct short lemniscate connections, and why do trinomials admit them?","precis":"The degree-seven polynomial constructed by ani has all its zeros in the open unit disc, but every connected subset of {|f|<1} containing two zeros has Hausdorff measure greater than two. The proof confines a slit preimage near a critical point and applies radial projection on its two inverse sheets. For monic trinomials with zeros in the open unit disc, the root equation instead keeps every root-to-origin segment inside the lemniscate.","owns":"ani's degree-seven counterexample and the trinomial connection theorem; the long paper develops critical-value estimates, other subclasses and subsidiary formal checks","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or the current status of Erdős #1041, including the later one-polynomial Formal Conjectures path-image Hausdorff-length check and its unreviewed historical curve-length correspondence; those distinctions belong to docs/claims.json in the mathematics repository","first_pass":["sec:problem","sec:counterexample","sec:trinomial"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-1041-lemniscate-newton-flow.pdf","hosted_pdf_path":"papers/erdos-1041-lemniscate-newton-flow.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex","site_html":"maths/papers/erdos-1041-lemniscate-newton-flow.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1041/erdos-1041-lemniscate-newton-flow.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1041/erdos-1041-lemniscate-newton-flow.tex","body_pointer":"papers.records.erdos-1041-lemniscate-newton-flow.body"},"erdos1041-lemniscate-reasoning-surface":{"paper_id":"erdos1041-lemniscate-reasoning-surface","title":"Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections","form":"Reasoning surface","length_class":"long","pages":86,"subject":"Erdős #1041","subject_kind":"problem","problem_id":"erdos_1041","landing_role":"mathematics","question":"How does ani's degree-seven counterexample work, which polynomial families admit short paths, and which critical-value constructions remain unformalised?","precis":"The record gives the rational estimates and geometric proof for ani's degree-seven polynomial: every connected set in its strict unit lemniscate joining two roots has Hausdorff measure greater than two. It develops contained paths for trinomials, collinear roots and two sparse families. Ordinary analytic arguments for critical-value criteria retain their unformalised inputs beside the statements. Critical-value moments, counterexamples to prescribed paths and questions about flow-strip geometry complete the account. The one-polynomial formal statement and correspondence with the historical wording remain distinct.","owns":"the exact geometry of ani's one-polynomial counterexample, proved positive families and critical-value arguments with their unformalised inputs identified; formal-statement status and historical correspondence remain owned by the claims registry","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or the current status of Erdős #1041, including the later one-polynomial Formal Conjectures path-image Hausdorff-length check and its unreviewed historical curve-length correspondence; those distinctions belong to docs/claims.json in the mathematics repository","first_pass":["sec:low-critical-closure","sec:solved-polynomial-families","sec:gap","sec:erdos-1041-complete-family-map"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos1041-lemniscate-reasoning-surface.pdf","hosted_pdf_path":"papers/erdos1041-lemniscate-reasoning-surface.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex","site_html":"maths/papers/erdos1041-lemniscate-reasoning-surface.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1041/erdos1041-lemniscate-reasoning-surface.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1041/erdos1041-lemniscate-reasoning-surface.tex","body_pointer":"papers.records.erdos1041-lemniscate-reasoning-surface.body"},"erdos-1049-rational-base-lambert":{"paper_id":"erdos-1049-rational-base-lambert","title":"Hankel Determinants of Geometric Moments and Rational Lambert Values","form":"Problem note","length_class":"short","pages":20,"subject":"Erdős #1049","subject_kind":"problem","problem_id":"erdos_1049","landing_role":"mathematics","question":"What are the fixed-base Hankel asymptotics, and which rational bases give irrational values?","precis":"Positive geometric moments and a uniform partition-sum bound give the fixed-base asymptotic of Zudilin’s normalised Hankel determinants, including its constant and power correction. A separate calculation of degrees after cancellation in his 2004 forms extends the rational-base irrationality region. It includes every positive integral power of 31/4 and excludes 3/2.","owns":"the fixed-base Hankel asymptotic and rational-base irrationality theorem; the long paper develops exact orders, content and endpoint obstructions, and the remaining problem at 3/2","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #1049, which remains open","first_pass":["sec:problem","sec:geometric-moments","sec:rational-base-irrationality","sec:open"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-1049-rational-base-lambert.pdf","hosted_pdf_path":"papers/erdos-1049-rational-base-lambert.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex","site_html":"maths/papers/erdos-1049-rational-base-lambert.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1049/erdos-1049-rational-base-lambert.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1049/erdos-1049-rational-base-lambert.tex","body_pointer":"papers.records.erdos-1049-rational-base-lambert.body"},"erdos1049-rational-base-lambert-reasoning-surface":{"paper_id":"erdos1049-rational-base-lambert-reasoning-surface","title":"Geometric Moments and Rational Lambert Values: Proofs and Further Results","form":"Reasoning surface","length_class":"long","pages":80,"subject":"Erdős #1049","subject_kind":"problem","problem_id":"erdos_1049","landing_role":"mathematics","question":"How are the sharp fixed-base Hankel asymptotic and rational-base irrationality criterion proved, and why do the remaining methods not settle 3/2?","precis":"The record proves the fixed-base Hankel asymptotic for Zudilin's normalised moments and the rational-base irrationality criterion log b/log a<0.4056830213840605…. The determinant proof separates geometric node configurations from positive weights; the irrationality proof instead specialises Zudilin's 2004 polynomial forms after cancellation. Coefficient moments, evaluated lattices and congruence conditions distinguish integer divisibility from nonzero remainder decay. A broader ratio-limit argument and a rational-value model remain unformalised. None of these arguments settles F(3/2).","owns":"the sharp fixed-base Hankel asymptotic and rational-base irrationality proofs, coefficient moments and arithmetic obstructions, with ordinary ratio-limit and rational-value models distinguished from checked results","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #1049, which remains open","first_pass":["long1049:sec:region","long1049:sec:hankel-order","long1049:sec:open"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos1049-rational-base-lambert-reasoning-surface.pdf","hosted_pdf_path":"papers/erdos1049-rational-base-lambert-reasoning-surface.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","site_html":"maths/papers/erdos1049-rational-base-lambert-reasoning-surface.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","body_pointer":"papers.records.erdos1049-rational-base-lambert-reasoning-surface.body"},"optimal-sparse-perturbations":{"paper_id":"optimal-sparse-perturbations","title":"Reading Eight Erdős Problems Together","form":"Cross-problem paper","length_class":"cross_problem","pages":38,"subject":"Eight Erdős problems read together","subject_kind":"synthesis","problem_id":null,"landing_role":"mathematics","question":"When do sparse integer digits fill intervals of values, when do they force irrationality or transcendence, and what does that show about the methods used across the Erdős problems?","precis":"One account of the mathematics developed across the programmes: an exact tail-capacity criterion under eventual congruences, its sharp factorial support-gap threshold, rational and irrational Lambert subsums across bases, and the limits of finite greedy tests and other irrationality methods. Full arguments, exact counterexamples, unsuccessful routes and attribution are retained together. The principal proofs are ordinary mathematics; cited Lean ingredients have their own stated scope. Historical novelty and independent expert review are not established.","owns":"the synthesis exposition and ordinary proofs across the covered problems: capacity and congruence constructions, Lambert subsums, rational-point counts, method limits and their research record","not_authority_for":"a solution to any original Erdős target, historical novelty, independent expert review, or a full Lean proof of the analytic capacity criterion or Lambert-chain theorem","first_pass":["sec:map","capacity:sec:criterion","sec:results","sec:questions"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/optimal-sparse-perturbations.pdf","hosted_pdf_path":"papers/optimal-sparse-perturbations.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/synthesis/optimal-sparse-perturbations.tex","site_html":"maths/papers/optimal-sparse-perturbations.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/synthesis/optimal-sparse-perturbations.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/synthesis/optimal-sparse-perturbations.tex","body_pointer":"papers.records.optimal-sparse-perturbations.body"},"writing-a-good-mathematical-paper":{"paper_id":"writing-a-good-mathematical-paper","title":"Writing a Good Mathematical Paper","form":"Compact writing guide","length_class":"system","pages":2,"subject":"Exposition method","subject_kind":"system","problem_id":null,"landing_role":"method","question":"How should an AI write a clear and mathematically faithful paper?","precis":"A self-contained guide to explaining a mathematical argument: read nearby primary literature, state the exact hypotheses and conclusion, connect proof steps, introduce notation, attribute borrowed ideas, and inspect the rendered paper. The guide offers writing practice rather than mathematical certification.","owns":"the exposition methodology: a compact general guide for writing a mathematically faithful paper","not_authority_for":"a mathematical result, independent review, or a measured improvement in writing","first_pass":[],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/writing-a-good-mathematical-paper.pdf","hosted_pdf_path":"papers/writing-a-good-mathematical-paper.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/exposition/writing-a-good-mathematical-paper.tex","site_html":"maths/papers/writing-a-good-mathematical-paper.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/exposition/writing-a-good-mathematical-paper.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/exposition/writing-a-good-mathematical-paper.tex","body_pointer":"papers.records.writing-a-good-mathematical-paper.body"},"writing-mathematics-from-reviewed-revisions":{"paper_id":"writing-mathematics-from-reviewed-revisions","title":"Writing Mathematics from the Literature and Reviewed Revisions","form":"Editorial method companion","length_class":"system","pages":21,"subject":"Exposition method","subject_kind":"system","problem_id":null,"landing_role":"method","question":"How can reading nearby papers and reviewed revisions improve the explanation of a particular mathematical argument?","precis":"A method for reading nearby mathematical prose, examining reviewed revisions, and retaining editorial lessons with their source, native decision and limit. The examples come from eight Erdős-problem papers. This documented practice has not been evaluated as an intervention in reader understanding.","owns":"the exposition methodology: reading primary mathematical prose, reviewing specific revisions, and retaining bounded editorial lessons","not_authority_for":"the example mathematics, independent mathematical review, or a measured improvement in reader understanding","first_pass":["sec:reading","sec:revisions","sec:review","sec:practice"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/writing-mathematics-from-reviewed-revisions.pdf","hosted_pdf_path":"papers/writing-mathematics-from-reviewed-revisions.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/exposition/writing-mathematics-from-reviewed-revisions.tex","site_html":"maths/papers/writing-mathematics-from-reviewed-revisions.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/exposition/writing-mathematics-from-reviewed-revisions.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/exposition/writing-mathematics-from-reviewed-revisions.tex","body_pointer":"papers.records.writing-mathematics-from-reviewed-revisions.body"},"claim-faithful-publication-systems":{"paper_id":"claim-faithful-publication-systems","title":"A Repository-Based System for Research and Publication","form":"Research architecture","length_class":"system","pages":25,"subject":"Systems paper","subject_kind":"system","problem_id":null,"landing_role":"method","question":"How can questions, proofs, computations and failed approaches stay linked to reviewed claims and papers, and what can publication checks miss that mathematical review must catch?","precis":"A correct Lean theorem can be misrepresented when its hypotheses disappear from a paper. This account follows a weighted-support criterion for Erdős Problem 257 through its mathematical argument, two registered formal supports and written explanation. It describes source inspection, publication checks, editorial revision and contributions in a repository covering eight problems. A historical test rejected nine false edits but missed a false completion claim; the paper distinguishes that record from the limited follow-up check. Source-byte agreement and repository adoption do not establish mathematical meaning, independent review or recognition by the community. The paper retains its mathematical antecedents and compares the recorded relations with blueprints and Prove2Me, without claiming comparative benefit.","owns":"the research architecture: problem-centred collections, the path from a proof to a public claim, and the contribution cycle","not_authority_for":"the mathematics it uses as examples, which the problem papers own, or the novelty and significance of those results","first_pass":["sec:intro","sec:example","sec:paper","sec:checks","sec:limits"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/claim-faithful-publication-systems-paper.pdf","hosted_pdf_path":"papers/claim-faithful-publication-systems-paper.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/systems/claim-faithful-publication-systems-paper.tex","site_html":"maths/papers/claim-faithful-publication-systems.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/systems/claim-faithful-publication-systems-paper.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/systems/claim-faithful-publication-systems-paper.tex","body_pointer":"papers.records.claim-faithful-publication-systems.body"},"plectis-public-system":{"paper_id":"plectis-public-system","title":"Plectis: What a Stranger Can Check","form":"Evidence boundary","length_class":"system","pages":22,"subject":"Systems paper","subject_kind":"system","problem_id":null,"landing_role":"boundary","question":"What may a stranger conclude from public evidence when the author chose both what to publish and what counts as a pass?","precis":"The author publishes selected code, fixed inputs, expected results, decision rules, and run records while the larger system stays private. What can a stranger conclude after rerunning them? A matching result shows repeatability for one stated public case. It does not establish where the code came from, whether the expected answer is correct, whether the published selection resembles the private whole, or whether the private system works reliably. The contribution is a method for making one selected disagreement local, not a score.","owns":"the evidence boundary: what a passing public run does and does not establish","not_authority_for":"the private system's internal state, which is not public, and the mathematical results it cites as evidence","first_pass":["sec:problem","sec:early-example","sec:distinctions","sec:stronger"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/plectis-public-system.pdf","hosted_pdf_path":"papers/plectis-public-system.pdf","hosted_tex_path":"papers/source/plectis/paper/plectis-public-system.tex","site_html":"maths/papers/plectis-public-system.html","github_current_url":"https://github.com/wcook04/practical-formal-verification/blob/main/paper/plectis-public-system.tex","github_permalink_url":"https://github.com/wcook04/practical-formal-verification/blob/f2ed2582e6de24c71b45dfc30dae33863e2ab539/paper/plectis-public-system.tex","body_pointer":"papers.records.plectis-public-system.body"}}},"papers":{"schema":"plectis_reviewer_paper_bodies_v0","role":"active_paper_records_with_available_latex","missing_embedded_bodies":[],"records":{"erdos-68-factorial-denominator-irrationality":{"paper_id":"erdos-68-factorial-denominator-irrationality","title":"Integer Linear Forms for a Factorial Reciprocal Series","form":"Problem note","length_class":"short","pages":9,"subject":"Erdős #68","subject_kind":"problem","problem_id":"erdos_68","landing_role":"mathematics","question":"Which integer coefficient vectors cancel initial weighted sums in factorial-denominator linear forms?","precis":"For S=∑_{n≥2}(n!−1)^{-1}, the paper classifies finitely supported integer vectors that cancel prescribed initial weighted sums in forms MS+k. An integral basis, one Diophantine equation and a finite gcd determine the possible moments M. At cancellation depth four, the least positive moment is 1380. Irrationality by these forms still requires a nonintegral form with q dividing M for every positive q.","owns":"the coefficient classification, finite moment gcd and depth-four construction; the long paper develops denominator exclusions, further constructions and method limits","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #68, which remains open","first_pass":["sec:problem","sec:channels","sec:moments","sec:open"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-68-factorial-denominator-irrationality.pdf","hosted_pdf_path":"papers/erdos-68-factorial-denominator-irrationality.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/68/erdos-68-factorial-denominator-irrationality.tex","site_html":"maths/papers/erdos-68-factorial-denominator-irrationality.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/68/erdos-68-factorial-denominator-irrationality.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/68/erdos-68-factorial-denominator-irrationality.tex","body_pointer":"papers.records.erdos-68-factorial-denominator-irrationality.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Standalone Erdős Problem Note for #68.  Context is deliberately repeated\n% from the sibling notes so that this file remains independently readable.\n%\n% Build from the public paper directory with `make`.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n\\input{evidence/erdos-68-factorial-denominator-irrationality}\n\\renewcommand{\\commit}{99f4bf47422abbd8757cbb22b50ba079d764d3a7}\n\\renewcommand{\\commitshort}{99f4bf47422a}\n\\usepackage{longtable}\n\\hypersetup{\n  pdftitle={Integer linear forms for a factorial reciprocal series (Erdos Problem 68)},\n  pdfsubject={Erdős Problem Notes: Problem 68},\n  pdfkeywords={irrationality, factorial denominators, factorial digits, prime-power cancellation, continued fractions, Lean 4},\n}\n\n% Source link targets retain the original manuscript commits and paths.\n\\newcommand{\\Ser}{S}\n\n\n\\runninghead{Integer linear forms for a factorial reciprocal series}\n\\title{Integer Linear Forms for a\\\\Factorial Reciprocal Series}\n\\subtitle{An integral basis and a finite gcd formula}\n\\author{Will Cook}\n\\date{31 August 2026; revised 7 October 2026}\n\n\\begin{document}\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\nFor $S=\\sum_{n\\ge2}(n!-1)^{-1}$, we classify the multipliers in\ncancellation forms $MS+k$ constructed from finite integer vectors\nsupported on $n\\ge2$. An integral basis solves cancellation through\ndepth $D\\ge2$; excluding index one leaves one Diophantine equation.\nIts coefficient gcd is\ndetermined below $2D^2$, giving the least positive multiplier and an\nattaining vector. At $D=4$, the minimum is $1380$; an attaining form\nis nonintegral. Irrationality still requires, for each $q>0$, a\nnonintegral form with $q\\mid M$.\n\\end{abstract}\n\n\\section{Introduction}\\label{sec:problem}\n\nErd\\H{o}s asked whether replacing $n!$ by $n!-1$ preserves the irrationality\nof the reciprocal sum \\cite[p.~102]{erdos1988}. The question appears as\nProblem~68 in Bloom's catalogue \\cite{bloom}.\n\n\\begin{problem}[Erd\\H{o}s Problem~\\#68]\\label{res:problem}\nIs\n\\[\n  \\Ser=\\sum_{n\\ge2}\\frac1{n!-1}\n\\]\nirrational?\n\\end{problem}\n\nThe elementary proof that $e$ is irrational clears the partial sum\nby a factorial and bounds the remaining positive tail. Here the first\nstep fails: for $H_m=\\sum_{n=2}^m(n!-1)^{-1}$, already\n$3!H_3=36/5$. We instead construct forms $MS+k$, with $M,k\\in\\mathbb Z$,\nby cancelling initial weighted sums. If $S=a/q$ and $q\\mid M$, then\n$MS+k$ is integral; proving it nonintegral excludes that denominator.\n\nAt depth four, the weight congruences force $115\\mid M$. Restricting\nthe coefficient support to $n\\ge2$ strengthens this to $1380\\mid M$,\nand every multiple of $1380$ occurs. We determine the exact factor at\nevery depth $D\\ge2$ and construct a vector attaining the least\npositive multiplier.\n\nFirst allow a coefficient at index one.\nTheorem~\\ref{res:divisor-channel-coordinates} constructs an integral\nbasis by correcting adjacent factorial differences at proper divisors\nto isolate each weighted sum.\nAt fixed $M$, cancellation determines the lower basis coefficients;\nremoving index one leaves a single equation in the higher coefficients.\nTheorem~\\ref{res:finite-channel-moment-certificate} reduces its infinite\ngcd to a calculation below $2D^2$. B\\'ezout coefficients then construct\nan attaining vector. No assumption on the rationality of $S$ enters\nthis classification.\n\nHan\\v{c}l and Tijdeman's tail-integrality lemma treats factorial series\nwith integer coefficients \\cite[Lemma~2.1 and the following remark,\np.~385]{hancl-tijdeman}. Their factorial-scaled partial sums are integers.\nHere the scaled partial sum need not be integral. Instead, the weight\ncongruences give $\\mathcal R-MS\\in\\mathbb Z$\n\\eqref{eq:integer-linear-form}; the assumption $S=a/q$ and $q\\mid M$\nthen makes $\\mathcal R$ integral. Koepf and Schmersau's factorial-digit\ncriterion \\cite[Example~3.2, p.~121]{koepf-schmersau} is used in the\n\\href{erdos68-factorial-reasoning-surface.pdf\\#r5-digits}{companion's digit argument}\nto give an equivalent eventual-pattern condition for $S-e+2$. Failure\nof that pattern at arbitrarily large indices also remains unproved.\n\nSection~\\ref{sec:depth-four} tests an attaining form;\nSection~\\ref{sec:open} gives the general nonintegrality test.\n\n\\section{An integral basis}\\label{sec:channels}\n\nWe need integer weights congruent to $n!$ modulo $d!-1$, with\n$nW_{d,n-1}=W_{d,n}$ when $n\\ge2$ and $d\\nmid n$.\nRemoving powers of $d!$ preserves the residue; the floor exponent\ngives the required jumps. Put\n\\[\n W_{d,n}=\\frac{n!}{(d!)^{\\lfloor n/d\\rfloor}}\n \\qquad(d\\ge2,\\ n\\ge1).\n\\]\nWriting $n=kd+r$, $0\\le r<d$, shows that $W_{d,n}$ is $r!$ times\nthe multinomial coefficient $n!/((d!)^k r!)$. Thus it is an integer,\nand $n!=(d!)^kW_{d,n}$ gives $W_{d,n}\\equiv n!\\pmod{d!-1}$.\nFor example, $W_{3,4}=4$ and $(4!-W_{3,4})/(3!-1)=4$.\n\nFor a finitely supported integer vector $\\lambda=(\\lambda_n)_{n\\ge1}$,\ndefine the integer linear forms\n\\[\n M(\\lambda)=\\sum_n\\lambda_n n!,\\qquad\n V_d(\\lambda)=\\sum_n\\lambda_nW_{d,n},\n\\]\nand let\n\\[\n \\mathcal R(\\lambda)=\\sum_{d\\ge2}\\frac{V_d(\\lambda)}{d!-1}.\n\\]\nHere $n$ indexes coefficients and $d$ indexes remainder summands.\nIf the support lies in $n\\le N$, $N\\ge2$, then $V_d=M$ for $d>N$.\nThus the remainder converges absolutely, and subtracting $MS$ leaves\nonly finitely many terms. The weight congruence makes each an integer:\n\\begin{equation}\n \\mathcal R(\\lambda)-M(\\lambda)S\n =\\sum_{d=2}^N\\frac{V_d(\\lambda)-M(\\lambda)}{d!-1}\\in\\mathbb Z.\n \\label{eq:integer-linear-form}\n\\end{equation}\nWe solve $V_2=\\cdots=V_D=0$, allowing either sign of $M$. Write\n$e_n$ for the unit vector at index $n$. Index one is auxiliary: it\nmakes the basis triangular but introduces no term $1/(1!-1)$,\nsince the remainder still starts at $d=2$. We impose $\\lambda_1=0$\nin Section~\\ref{sec:moments}.\n\nThe identity $n(n-1)!-n!=0$ suggests starting with\n$T_n=ne_{n-1}-e_n$. At $n=2$, the vector $U_2=2e_1-e_2$ has\n$V_2=1$ and every other weighted sum zero. At $n=4$, the difference\n$T_4=4e_3-e_4$ has two nonzero weighted sums. One correction suffices:\n\\begin{center}\n\\begin{tabular}{lrrrr}\n\\papertablehead Vector & $M$ & $V_2$ & $V_3$ & $V_4$\\\\\n\\hline\n$T_4$       & $0$ & $6$ & $0$ & $23$\\\\\n$6U_2$      & $0$ & $6$ & $0$ & $0$\\\\\n$T_4-6U_2$  & $0$ & $0$ & $0$ & $23$\\\\\n\\end{tabular}\n\\end{center}\nFor $d>4$ these weighted sums equal $M=0$. Thus $U_4=T_4-6U_2$\nisolates $V_4=4!-1$. In general, the unwanted sums occur at proper\ndivisors of $n$, whose correcting vectors are already available in an\ninductive construction.\n\n\\begin{theorem}[an integer basis with prescribed weighted sums]\\label{res:divisor-channel-coordinates}\nSet\n\\[\nT_n=ne_{n-1}-e_n,\\qquad\nU_n=T_n-\\sum_{\\substack{d\\mid n\\\\2\\le d<n}}W_{d,n}U_d\n\\quad(n\\ge2).\n\\]\nThen\n\\[\nM(U_n)=0,\\qquad V_d(U_n)=(d!-1)\\mathbf1_{d=n}.\n\\]\nThe vectors $e_1,U_2,U_3,\\ldots$ form an integral basis.  Every finite vector\nhas the unique finite expansion\n\\begin{equation}\n\\lambda=M(\\lambda)e_1+\n\\sum_{d\\ge2}\\frac{V_d(\\lambda)-M(\\lambda)}{d!-1}U_d.\n\\label{eq:channel-basis-expansion}\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nWe have $M(T_n)=0$. If $d\\nmid n$, the floor exponents at\n$n-1$ and $n$ agree, so $nW_{d,n-1}=W_{d,n}$. If $d\\mid n$, the\nexponent increases by one, so $nW_{d,n-1}=d!W_{d,n}$. Hence\n\\[\n V_d(T_n)=(d!-1)W_{d,n}\\mathbf1_{d\\mid n}.\n\\]\nSuppose the asserted identities hold below $n$. For each proper divisor\n$d$, subtracting $W_{d,n}U_d$ cancels the sum at $d$ and changes none\nof the others. The sum at $n$ remains $n!-1$, since $W_{n,n}=1$.\nThis proves the identities by induction.\n\nFor completeness, restrict to indices $1,\\ldots,N$. The columns\n$e_1,T_2,\\ldots,T_N$ form a triangular integer matrix with diagonal\n$1,-1,\\ldots,-1$. Replacing each $T_n$ by $U_n$ subtracts an integer\ncombination of earlier columns, so the second change of basis is\ntriangular with diagonal one. Both matrices have integer inverses:\nback-substitution divides only by $1$ or $-1$. Hence\n$e_1,U_2,\\ldots,U_N$ is a basis over $\\mathbb Z$.\n\nIn the resulting expansion, applying $M$ gives the coefficient of $e_1$.\nIf the coefficient of $U_d$ is $c_d$, applying $V_d$ gives\n$V_d=M+(d!-1)c_d$, which proves \\eqref{eq:channel-basis-expansion}. For $d>N$ that\ncoefficient is zero because $V_d=M$.\n\\end{proof}\n\nThe $U_d$-coordinates in \\eqref{eq:channel-basis-expansion} are exactly\nthe integer summands in \\eqref{eq:integer-linear-form}. Their sum is\ntherefore $k$ in $\\mathcal R=MS+k$.\n\nFor a prime $p\\ge3$ the divisor sum is empty, and\n$U_p=pe_{p-1}-e_p$. At a composite index the correction terms remove\nthe extra weighted sums. For example,\n\\begin{equation}\n U_9=9e_8-e_9-5040e_2+1680e_3.\n \\label{res:translator}\n\\end{equation}\nAdding $U_n$ changes only $V_n$ among the weighted sums, preserves\n$M$, and adds one to $\\mathcal R$. Integer multiples therefore preserve\nthe fractional part.\n\n\\section{Possible values of \\texorpdfstring{$M$}{M}}\\label{sec:moments}\n\nFix $D\\ge2$ and solve $V_2=\\cdots=V_D=0$, allowing index one\ntemporarily. The coordinates in \\eqref{eq:channel-basis-expansion} give\n\\begin{equation}\n V_d(\\lambda)\\equiv M(\\lambda)\\pmod{d!-1}.\n \\label{res:congruence}\n\\end{equation}\nIt follows that $M$ must be divisible by\n\\[\n L_D=\\operatorname{lcm}_{2\\le d\\le D}(d!-1).\n\\]\nThis necessary condition is sufficient while index one is allowed:\n$L_De_1$ has every weighted sum equal to $L_D$, and subtracting\n$L_D/(d!-1)$ copies of $U_d$ sets the $d$th sum to zero without changing\nthe others. Thus put\n\\[\n K_D=L_De_1-\\sum_{d=2}^D\\frac{L_D}{d!-1}U_d.\n\\]\nUniqueness of the coordinates now gives the complete solution\n\\begin{equation}\n V_2=\\cdots=V_D=0\n \\quad\\Longleftrightarrow\\quad\n \\lambda=tK_D+\\sum_{n>D}z_nU_n,\\qquad M=tL_D,\n \\label{eq:low-channel-classification}\n\\end{equation}\nwith $t\\in\\mathbb Z$ and finitely many nonzero integers $z_n$.\nThe coefficient of $e_1$ in \\eqref{eq:channel-basis-expansion} is $M$; the original coordinate\n$\\lambda_1$ also receives contributions from the $U_n$.\nWrite $a_D=(K_D)_1$ and $u_n=(U_n)_1$. The support restriction is exactly\n\\begin{equation}\n ta_D+\\sum_{n>D}z_nu_n=0.\n \\label{eq:support-equation}\n\\end{equation}\nThe higher corrections leave cancellation untouched and change the\nfirst coordinate by exactly $g_D\\mathbb Z$, where\n$g_D=\\gcd\\{u_n:n>D\\}$. They can therefore remove $ta_D$ precisely\nwhen $g_D\\mid ta_D$.\nFor example, at depth four we shall find $a_4=-55$ and $g_4=60$.\nRemoving index one then requires $60\\mid55t$, or $12\\mid t$;\nhence $M=115t$ must be a multiple of $1380$.\n\nIn general, the possible values of $M$ form the ideal\n\\begin{equation}\n \\mu_D\\mathbb Z,\\qquad\n \\mu_D=L_D\\frac{g_D}{\\gcd(g_D,a_D)}.\n \\label{eq:attainable-moment-ideal}\n\\end{equation}\nIndeed, division by $\\gcd(g_D,a_D)$ leaves two coprime integers, so\n$g_D\\mid ta_D$ holds exactly when $g_D/\\gcd(g_D,a_D)\\mid t$.\nB\\'ezout's identity supplies the required finite combination; the next\ntheorem bounds the indices needed. Every vector attaining $\\mu_D$\nhas coefficient gcd one, since division by a common factor would give\na smaller positive multiplier.\n\n\\subsection{A finite gcd calculation}\\label{sec:finite-gcd}\n\nThe recursion for the first coordinate is\n\\[\n u_2=2,\\qquad\n u_n=-\\sum_{\\substack{d\\mid n\\\\2\\le d<n}}W_{d,n}u_d\\quad(n>2).\n\\]\nInduction gives $u_n=0$ at odd indices and\n$u_{2p}=-(2p)!/2^{p-1}\\ne0$ at twice a prime, including $p=2$.\nChoose a prime $\\ell$ with $D/2<\\ell\\le D$, using Bertrand's postulate\nfor $D\\ge3$ and $\\ell=2$ for $D=2$. A finite gcd containing $u_{2\\ell}$\nis positive and divides $(2\\ell)!$.\n\nThe remaining issue is whether later coefficients reduce this gcd.\nIn each recurrence term $W_{d,n}u_d$, induction controls $u_d$ for\n$d>D$. For $d\\le D$, unordered block counting gives\n$(n/d)!\\mid W_{d,n}$. The cutoff ensures $n/d\\ge2\\ell$, so the weight\nsupplies the factor $(2\\ell)!$ instead.\n\n\\begin{theorem}[a finite formula for the gcd]\n\\label{res:finite-channel-moment-certificate}\nChoose a prime $\\ell$ with $D/2<\\ell\\le D$ and put $H=D(2\\ell-1)$.\nThen\n\\[\ng_D=\\gcd(u_{D+1},\\ldots,u_H),\\qquad H<2D^2.\n\\]\n\\end{theorem}\n\n\\begin{proof}\nLet $g=\\gcd(u_{D+1},\\ldots,u_H)$. The index $2\\ell$ belongs to this\ninterval, and $u_{2\\ell}=-(2\\ell)!/2^{\\ell-1}\\ne0$. Consequently\n$g>0$ and $g\\mid(2\\ell)!$.\n\nWe prove $g\\mid u_n$ for every $n>D$ by strong induction. This is true\nup to $H$ by definition. For $n>H$, consider a proper divisor $d\\ge2$\nin the recurrence for $u_n$. If $d>D$, the induction hypothesis gives\n$g\\mid u_d$. If $d\\le D$, put $k=n/d$. The choice\n$H=D(2\\ell-1)$ ensures $k\\ge2\\ell$. Moreover,\n\\[\n \\frac{W_{d,n}}{k!}=\\frac{n!}{(d!)^k k!}\\in\\mathbb Z:\n\\]\nthis quotient counts partitions of $n$ labelled objects into $k$\nunordered blocks of size $d$. Hence $(2\\ell)!\\mid k!\\mid W_{d,n}$,\nso $g\\mid W_{d,n}$ in the small-divisor case as well.\nEvery product $W_{d,n}u_d$ in the recurrence is therefore divisible\nby $g$.\n\nThus $g$ divides the whole tail. Conversely, every common divisor of\nthe tail divides its finite subfamily $u_{D+1},\\ldots,u_H$. The two\ngcds are equal, and $H\\le D(2D-1)<2D^2$.\n\\end{proof}\n\nThe coefficient at $2\\ell\\le2D$ supplies positivity; the larger cutoff\ncontrols the small-divisor weights. The argument does not justify\ntruncation at $2D$.\n\nTo attain the least positive multiplier, compute integers $b_n$ with\n$\\sum_{n=D+1}^H b_nu_n=g_D$ by the extended Euclidean algorithm, and put\n$h=\\gcd(g_D,a_D)$. Taking\n\\[\n t=g_D/h,\\qquad z_n=-(a_D/h)b_n\\quad(D<n\\le H)\n\\]\nmakes $ta_D+\\sum z_nu_n=0$. Substitution in\n\\eqref{eq:low-channel-classification} therefore gives $M=\\mu_D$ with\nsupport in $2,\\ldots,H$.\n\n\\subsection{The associated remainder}\\label{sec:coordinate-remainder}\n\nFix $M\\in\\mu_D\\mathbb Z$ and put $t=M/L_D$. For choices of $z_n$\nsatisfying \\eqref{eq:support-equation}, the identities\n$\\mathcal R(e_1)=S$ and $\\mathcal R(U_n)=1$ give\n\\begin{equation}\n \\mathcal R\\left(tK_D+\\sum_{n>D}z_nU_n\\right)\n =tL_D(S-H_D)+\\sum_{n>D}z_n,\n \\qquad H_D=\\sum_{d=2}^D\\frac1{d!-1}.\n \\label{eq:residual-transparency}\n\\end{equation}\nSince $MH_D\\in\\mathbb Z$, the offset from $MS$ is\n$k=-MH_D+\\sum_{n>D}z_n$; $\\sum z_n$ alone is the offset from\n$M(S-H_D)$. All choices at fixed $M$ therefore have the same\nfractional part.\n\n\\section{The least multiplier at depth four}\\label{sec:depth-four}\n\nAt depth four the recursion gives\n\\[\n L_4=115,\\qquad\n K_4=-55e_1+16e_2+3e_3+5e_4,\\qquad a_4=-55.\n\\]\nTake $\\ell=3$ in Theorem~\\ref{res:finite-channel-moment-certificate},\nso that $H=20$. Since $u_6=-180$, $u_8=-4200$ and\n$23u_6-u_8=60$, we have $g_4\\mid60$. For the converse,\n\\begin{equation}\n \\operatorname{lcm}(1,\\ldots,n)\\mid u_n\n \\label{eq:channel-lcm-envelope}\n\\end{equation}\ngives $60\\mid u_n$ for every $n>4$, hence $60\\mid g_4$.\nTo prove \\eqref{eq:channel-lcm-envelope}, put\n$\\Lambda_n=\\operatorname{lcm}(1,\\ldots,n)$. For a prime $r$, write $v_r$ for its valuation. If $d\\mid n$,\nLegendre's formula gives\n\\[\n v_r(W_{d,n})=\\sum_{j\\ge1}\\left(\n \\left\\lfloor\\frac n{r^j}\\right\\rfloor\n -\\frac nd\\left\\lfloor\\frac d{r^j}\\right\\rfloor\\right)\n \\ge v_r(\\Lambda_n)-v_r(\\Lambda_d).\n\\]\nEach summand is nonnegative, and each power\n$d<r^j\\le n$ contributes at least one. Hence\n$\\Lambda_n\\mid W_{d,n}\\Lambda_d$, and induction in the recurrence,\nstarting from $u_2=2$, proves the assertion. Thus $g_4=60$, and\n\\eqref{eq:attainable-moment-ideal} gives\n\\[\n \\mu_4=115\\frac{60}{\\gcd(60,55)}=1380.\n\\]\n\nTo attain $M=1380$, take $t=12$ in\n\\eqref{eq:support-equation}. The choice $z_6=-27$, $z_8=1$ works because\n\\[\n 12(-55)-27(-180)-4200=0.\n\\]\nThe vector $12K_4-27U_6+U_8$ therefore has no index-one coefficient\nand cancels $V_2,V_3,V_4$. To remove its coefficients at indices four\nand seven, add $-26U_5+8U_7$. These corrections still cancel through\nfour and do not restore index one, since $u_5=u_7=0$. Thus\n\\[\n \\lambda=12K_4-26U_5-27U_6+8U_7+U_8,\n\\]\nwhich expands to\n\\begin{equation}\n \\lambda=1482e_2-784e_3-136e_5+83e_6-e_8,\n \\qquad (M,V_2,V_3,V_4)=(1380,0,0,0).\n \\label{eq:depth-four-short-vector}\n\\end{equation}\nTable~\\ref{tab:depth-four} verifies all four sums directly.\n\n\\begin{table}[htbp]\n\\centering\n\\begin{tabular}{rrrrrr}\n\\papertablehead $n$ & $\\lambda_n$ & $n!$ & $W_{2,n}$ & $W_{3,n}$ & $W_{4,n}$\\\\\n\\hline\n2 &1482 &2 &1 &2 &2\\\\\n3 &-784 &6 &3 &1 &6\\\\\n5 &-136 &120 &30 &20 &5\\\\\n6 &83 &720 &90 &20 &30\\\\\n8 &-1 &40320 &2520 &1120 &70\\\\\n\\hline\n\\multicolumn{2}{l}{Weighted totals} & $1380$ & $0$ & $0$ & $0$\\\\\n\\end{tabular}\n\\caption{The depth-four checks: multiply each of the last four columns\nby $\\lambda_n$ and sum to obtain the last row.}\n\\label{tab:depth-four}\n\\end{table}\n\nThe basis coordinates give the offset $-26-27+8+1=-44$ in\n\\eqref{eq:residual-transparency}. Since\n$H_4=143/115$, the original linear form is\n\\[\n \\mathcal R(\\lambda)=1380(S-H_4)-44=1380S-1760.\n\\]\nFor the remainder estimate, the inequality\n$(n+1)!-1>(n+1)(n!-1)$ gives the geometric bound\n\\begin{equation}\n 0<S-H_N<\\frac{N+2}{(N+1)((N+1)!-1)}\n          <\\frac{2}{(N+1)!-1}\\qquad(N\\ge2).\n \\label{eq:series-tail-bound}\n\\end{equation}\nThe majorant starts with $1/((N+1)!-1)$ and has ratio $1/(N+2)$.\n\nExact rational arithmetic gives\n\\[\n -31+\\frac45\n <1380\\sum_{d=5}^{8}\\frac1{d!-1}-44\n <-31+\\frac56,\n \\qquad \\frac{2\\cdot1380}{9!-1}<\\frac1{100}.\n\\]\nThe omitted contribution to $\\mathcal R$ is positive and less than\n$1/100$, so $-31<\\mathcal R(\\lambda)<-30$. If $S=a/q$ with $q\\mid1380$,\n\\eqref{eq:integer-linear-form} would make this remainder an integer.\nEvery divisor of $1380$ is therefore excluded.\n\nThe \\href{erdos68-factorial-reasoning-surface.pdf\\#r15-depth-six}{depth-six calculation}\ngives $\\mu_6=24L_6$: the extra factor need not equal $12$. The companion also gives a\n\\href{erdos68-factorial-reasoning-surface.pdf\\#r5-depth-refinements}{dual congruence, support bounds and coefficient-norm refinements}.\nLeast multiplier, least support size and least coefficient norm are\ndistinct problems; \\eqref{eq:attainable-moment-ideal} settles only the first.\n\n\\section{Nonintegrality of the remainder}\\label{sec:open}\\label{sec:plateau}\n\\papersectiontarget{r12-short-gap}\n\nFix a vector with $V_2=\\cdots=V_D=0$ and $M>0$, and choose any\ncutoff $N\\ge D$ at or beyond its support. Separate the signed finite\nsum from the positive tail:\n\\[\n A_N=\\sum_{d=D+1}^N\\frac{V_d(\\lambda)}{d!-1},\\qquad\n \\mathcal R(\\lambda)=A_N+M(S-H_N).\n\\]\nThe remainder exceeds $A_N$. By \\eqref{eq:series-tail-bound}, it lies\nbelow the least integer strictly above $A_N$ whenever\n\\begin{equation}\n \\frac{2M}{(N+1)!-1}<\\lfloor A_N\\rfloor+1-A_N.\n \\label{eq:signed-block-gap}\n\\end{equation}\nThen $\\lfloor A_N\\rfloor<\\mathcal R(\\lambda)<\\lfloor A_N\\rfloor+1$,\nexcluding every denominator dividing $M$. The strict successor matters: at an\nintegral $A_N$ the gap is one, whereas just below an integer it can be\narbitrarily small. Failure of the comparison is inconclusive.\n\nAt depth two, fix $\\lambda=-6e_2+e_4$, with $M=12$, and vary only\nthe cutoff. At $N=4$, its finite part\n$A_4=-239/115$ lies just below $-2$, and its gap $9/115$ is smaller\nthan the bound $24/119$. At $N=5$, the finite part\n$A_5=-27061/13685$ has crossed $-2$: the next integer is now $-1$,\nand the gap $13376/13685$ exceeds $24/719$.\nFor any fixed vector with $M>0$, $A_N$ increases to $\\mathcal R$.\nIf $\\mathcal R$ is nonintegral, the gap tends to a positive number and\nthe test eventually succeeds. If $\\mathcal R$ is integral, the eventual\ngap equals the omitted tail, so the strict upper-bound test fails.\nChanging the cutoff detects nonintegrality; it neither creates it nor\nchanges $M$. Coefficient changes at fixed $M$ preserve nonintegrality\nby \\eqref{eq:residual-transparency}.\n\nTo exclude every denominator, $M$ must vary. The\n\\href{erdos68-factorial-reasoning-surface.pdf\\#r5-progression}{companion's progression construction}\nadds size control: polynomial root conditions give primitive vectors\n(coefficient gcd one) with $0<M<N!$, where $N$ is their largest\nsupported index. Every fixed $q$ divides their multipliers for all\nsufficiently large $N$. The tail at that endpoint is below $2/N$, but\nthe next-integer gap may shrink too. For each $q>0$, one still needs\na vector with $q\\mid M$ and a cutoff satisfying\n\\eqref{eq:signed-block-gap}; the cutoff may exceed its support endpoint.\nNo minimal-support assertion is made.\n\nThe companion treats the\n\\href{erdos68-factorial-reasoning-surface.pdf\\#r5-carry}{equivalent carry condition}\nand the\n\\href{erdos68-factorial-reasoning-surface.pdf\\#r5-complement}{complementary-denominator tail inequality}.\nIt also records two larger finite computations: a non-unit carry at\n$m=300000$ excludes every $q\\mid299999!$, and a continued-fraction\nenclosure excludes every $q<2^{39990}$. Both computations were performed\noutside Lean; their\n\\href{erdos68-factorial-reasoning-surface.pdf\\#r12-finite}{algorithms, exact outputs and scope}\nare given there. The factorial-divisibility exclusion does not exclude\nall denominators with only small prime factors, whose exponents may be\nlarger. Neither finite restriction establishes the required condition\nat arbitrarily large indices.\n\n\\section{Verification}\\label{app:sources}\n\n\\evidenceparagraph{\\evidencerecordurl}\n\nThe integral-basis theorem and finite gcd formula have\nrecorded exact Lean bindings. The\n\\href{erdos68-factorial-reasoning-surface.pdf\\#r12-short-sources}{companion source concordance}\nidentifies their declarations and the theorem on the possible values of $M$. Its\nlarge finite computations are separate from these kernel proofs.\n\n\\section*{Acknowledgements}\n\nAn AI research pass supplied by Will Cook derived the depth-four dual\ncongruence and shorter vectors from the weighted linear forms and the\nclassification of their possible coefficients $M$ developed earlier here. OpenAI Codex\nchecked and integrated them; a separate AI pass reviewed the proof.\n\n\\begin{thebibliography}{99}\n\n\\bibitem{erdos1988}\nPaul Erd{\\H o}s. \\href{https://doi.org/10.1017/CBO9780511897184.009}{On the irrationality of certain series: problems and results}.\nIn Alan Baker (ed.), \\emph{New Advances in Transcendence Theory}, Cambridge University Press (1988), pp.~102--109.\n\n\\bibitem{bloom}\nThomas F. Bloom. \\href{https://www.erdosproblems.com/68}{Erd{\\H o}s Problem {\\#68}}.\nOnline resource (2026). Historical access: 28 July 2026; present-page status not reverified.\n\n\\bibitem{koepf-schmersau}\nWolfram Koepf and Dieter Schmersau. \\href{https://doi.org/10.1524/anly.2011.1094}{Irrationality of certain infinite series {II}}.\n\\emph{Analysis} \\textbf{31} (2011), 117--124.\n\n\\bibitem{hancl-tijdeman}\nJaroslav Han{\\v c}l and Robert Tijdeman. \\href{https://doi.org/10.4064/aa118-4-5}{On the irrationality of factorial series}.\n\\emph{Acta Arithmetica} \\textbf{118} (4) (2005), 383--401.\n\n\\end{thebibliography}\n\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":22700,"sha256":"sha256:b4eba8fb48d81d851b0679a5d77bdc3b20e60913d48f403184365009d4f30edd","cite_as":"Plectis reader corpus, paper:erdos-68-factorial-denominator-irrationality","source_repository":"plectis-erdos","source_path":"paper/68/erdos-68-factorial-denominator-irrationality.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:088d36d7ba3d3ee0cee2df99e8915927999698f19889f8ea0b8d55b3721ec778","byte_count":6923,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% This is the only copy of the shared preamble.  The eight short notes, the\n% long records assembled from paper/reasoning-parts other than #249 and #257\n% (which carry their own preambles), and the synthesis paper include it with\n% \\input{problem-note-preamble}; paper/Makefile builds from this directory\n% with the directory on the tectonic search path.\n%\n% Shared preamble for the Erdős Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erdős problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{8e58c395e9807ef3374d7365c62cf04d20ee6331}\n\\newcommand{\\commitshort}{9e231ce4371f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-erdos/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-erdos/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\usepackage{longtable}  % the notes' declaration tables; loaded here so the assembled records inherit it\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n% ---------------------------------------------------------------------------\n% Evidence marks.  paper-evidence.tex places \"Lean\" and \"Comparator\" links in\n% the margin beside each result from the paper's generated file\n% paper/evidence/<paper>.tex, which each paper inputs after this preamble.\n% ---------------------------------------------------------------------------\n\\input{paper-evidence}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Production provenance.  The wording is fixed by the corpus provenance\n% contract and reproduced verbatim in every manuscript in this repository, so\n% the papers cannot drift into different accounts of how they were made.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erdős problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every Lean mark in the margin points to propositions that the\n  pinned Lean kernel accepts, with no \\texttt{sorry}, no added axiom, and no\n  unchecked evaluation.  That claim concerns the formal statements only; it\n  says nothing about their mathematical interest, their novelty, or the\n  original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n\n% Compact opening card for the separately stated Comparator theorem selected\n% from each problem note.  The prose supplied by each note owns the\n% contribution and relevance judgements; the shared final sentence states the\n% mechanical assurance boundary once and points to the public evidence owner.\n\\newcommand{\\externalreviewobject}[3]{%\n  \\begin{quote}\\small\n  \\textbf{Contribution.} #1\\par\n  \\textbf{Relation to the open problem.} #2\\par\n  \\textbf{Executable review object.} #3  The repository's\n  external-verification job compares these exact Lean propositions with\n  separately declared challenge statements and an axiom budget, then asks\n  Lean's kernel to check the submitted proofs.  The\n  \\href{https://github.com/wcook04/plectis-erdos/blob/ca0e13f8acf5ccf48506e4bdb870953d3a0856fa/formalization.yaml}%\n  {formalisation manifest} and the commit-bound CI receipt record that check;\n  they do not assess novelty, significance, or whether the original problem is\n  solved.\n  \\end{quote}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  Under\n% Tectonic (XeTeX) the text and mathematics are set in TeX Gyre Pagella from\n% the bundled OpenType fonts, with a Heros sans range for \\mathsf and\n% Inconsolata for monospace; the pdflatex fallback keeps the same Palatino\n% design through newpx.  The A4 text block uses a classical measure with\n% quiet running heads and restrained ink.  Link colours: steel blue for\n% external (Lean-source and web) targets, deep burgundy for internal\n% cross-references and citations.\n\\NeedsTeXFormat{LaTeX2e}\n\\ProvidesPackage{paper-house-style}[2026/07/19 Pagella house style for the papers]\n\n\\RequirePackage{iftex}\n\\RequirePackage{amsmath}\n\\RequirePackage{amsthm}\n\\ifpdftex\n  \\RequirePackage[T1]{fontenc}\n  \\RequirePackage[utf8]{inputenc}\n  \\RequirePackage{amssymb}\n  \\RequirePackage{newpxtext}\n  \\RequirePackage{newpxmath}\n\\else\n  \\RequirePackage{fontspec}\n  \\RequirePackage{unicode-math}\n  \\setmainfont{texgyrepagella}[Extension=.otf,UprightFont=*-regular,\n    BoldFont=*-bold,ItalicFont=*-italic,BoldItalicFont=*-bolditalic]\n  \\setmathfont{texgyrepagella-math.otf}\n  \\setmathfont{texgyreheros-regular.otf}[range=sfup,Scale=MatchLowercase]\n  \\setmonofont{Inconsolatazi4}[Extension=.otf,UprightFont=*-Regular,\n    BoldFont=*-Bold,Scale=MatchLowercase]\n\\fi\n\n\\RequirePackage[a4paper,textwidth=418pt,textheight=636pt,hcentering,\n  top=68pt,headheight=15pt,headsep=20pt,footskip=34pt]{geometry}\n\\RequirePackage{microtype}\n\\linespread{1.045}\n\\clubpenalty=10000 \\widowpenalty=10000 \\tolerance=1600\n\n\\RequirePackage[dvipsnames,table]{xcolor}\n\\definecolor{linkexternal}{HTML}{275D8C}\n\\definecolor{linkinternal}{HTML}{6E2B2B}\n\\definecolor{papertableheader}{HTML}{E3EAF0}\n\\definecolor{papertablerow}{HTML}{F2F5F7}\n\\definecolor{papertablerule}{HTML}{7D8991}\n\\definecolor{paperresultrule}{HTML}{315B72}\n\n% Stable cross-document links between sibling PDFs. A named destination is\n% more durable than a page number: sections may move while their semantic role\n% remains the same. All public PDFs are copied to one directory at release.\n\\newcommand{\\papersectiontarget}[1]{\\hypertarget{#1}{}}\n\\newcommand{\\papersectionlink}[3]{\\href{#1\\##2}{#3}}\n\n\\RequirePackage{titlesec}\n\\titleformat{\\section}{\\Large\\bfseries}{\\thesection}{0.85em}{}\n\\titlespacing*{\\section}{0pt}{2.6ex plus .8ex minus .3ex}{1.5ex plus .3ex}\n\\titleformat{\\subsection}{\\large\\bfseries}{\\thesubsection}{0.8em}{}\n\\titlespacing*{\\subsection}{0pt}{2.1ex plus .6ex minus .2ex}{1.1ex plus .2ex}\n\\titlespacing*{\\paragraph}{0pt}{1.6ex plus .5ex minus .2ex}{0.85em}\n\n\\RequirePackage{fancyhdr}\n\\newcommand{\\paperrunninghead}{}\n\\newcommand{\\runninghead}[1]{\\renewcommand{\\paperrunninghead}{#1}}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\fancyhead[L]{\\small\\itshape\\paperrunninghead}\n\\fancyhead[R]{\\small\\thepage}\n\\renewcommand{\\headrulewidth}{0pt}\n\\fancypagestyle{plain}{\\fancyhf{}\\renewcommand{\\headrulewidth}{0pt}}\n\n\\RequirePackage{caption}\n\\captionsetup{font=small,labelfont=bf,skip=9pt,width=.92\\linewidth}\n\\RequirePackage{etoolbox}\n\\RequirePackage{pdflscape}\n\\renewcommand{\\arraystretch}{1.20}\n\\newcommand{\\papertablehead}{\\rowcolor{papertableheader}}\n\\newcommand{\\papertablebody}{\\rowcolors{2}{white}{papertablerow}}\n\\AtBeginEnvironment{tabular}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\\AtBeginEnvironment{longtable}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\n% A compact front-matter panel for the result a scanning reader should retain.\n% It is deliberately short and unbreakable: manuscripts should state one\n% result, its evidence, and its limit here, then supply the argument below.\n\\newenvironment{paperresult}[1]{%\n  \\par\\addvspace{0.7em}\n  \\begin{center}\\begin{minipage}{0.94\\linewidth}\n    {\\color{paperresultrule}\\rule{\\linewidth}{0.8pt}}\\par\\vspace{4pt}\n    {\\small\\sffamily\\bfseries\\color{paperresultrule}#1}\\par\\smallskip\n    \\small\\color{black}\n}{%\n    \\par\\vspace{4pt}{\\color{paperresultrule}\\rule{\\linewidth}{0.45pt}}\n    \\end{minipage}\\end{center}\\par\\addvspace{0.5em}\n}\n\n% Shared diagram grammar.  Reader-facing figures use the same restrained\n% hierarchy as the surrounding page: a short eyebrow, a substantial title,\n% one compact explanatory line, and an optional evidence badge.  The styles\n% are installed only when a manuscript loads TikZ.\n\\AtBeginDocument{%\n  \\@ifpackageloaded{tikz}{%\n    \\tikzset{\n      paper box/.style={draw=gray!48, line width=.45pt, rounded corners=3pt,\n        fill=white, align=left, inner xsep=7pt, inner ysep=6pt,\n        font=\\small},\n      paper card/.style={paper box, minimum height=13mm},\n      paper stage/.style={paper box, align=center, minimum height=14mm},\n      paper arrow/.style={-{Stealth[length=2.15mm,width=1.45mm]},\n        line width=.7pt, draw=gray!72!black},\n      paper feedback/.style={paper arrow, densely dashed,\n        draw=Violet!72!black},\n      paper eyebrow/.style={font=\\scriptsize\\sffamily\\bfseries,\n        text=gray!70!black},\n      paper label/.style={font=\\scriptsize\\sffamily,\n        text=gray!72!black, fill=white, inner sep=1.5pt},\n      paper badge/.style={rounded corners=2pt, inner xsep=4pt, inner ysep=2pt,\n        font=\\scriptsize\\sffamily\\bfseries},\n      paper boundary/.style={densely dashed, line width=.55pt,\n        draw=gray!62}\n    }%\n  }{}%\n}\n\n% Front matter: title block with an optional italic subtitle, small-caps\n% author, and a hairline rule; the abstract is set small with a small-caps\n% head and symmetric indents.\n\\newcommand{\\subtitle}[1]{\\gdef\\@papersubtitle{#1}}\n\\renewenvironment{abstract}{%\n  \\vspace{0.1em}\\small\n  \\begin{center}{\\normalsize\\scshape Abstract}\\end{center}\\vspace{-0.5em}\n  \\list{}{\\leftmargin 1.3em\\rightmargin\\leftmargin\\listparindent 0pt\n    \\parsep 0.32em\\itemindent 0pt}\\item\\relax}\n  {\\endlist\\vspace{0.5em}}\n\\def\\@maketitle{%\n  \\newpage\\null\n  \\vspace*{-2.6em}%\n  \\begin{center}%\n    {\\LARGE\\bfseries\\@title\\par}%\n    \\vskip 0.55em\n    {\\large\\itshape\\@papersubtitle\\par}%\n    \\vskip 1.1em\n    {\\scshape\\@author\\textsuperscript{*}\\par}%\n    \\vskip 0.25em\n    {\\small\\@date\\par}%\n    \\vskip 0.8em\n    \\rule{6.5em}{0.4pt}%\n  \\end{center}%\n  \\vskip 0.9em}\n\n% Production provenance, set at the foot of the first page.\n%\n% A reader who meets a single personal byline ordinarily infers that the named\n% author drafted the sentences.  For these papers that inference is false, and\n% a declaration reached only in the back matter would leave it standing for the\n% length of the paper.  The note therefore appears on page one, small and\n% visually subordinate to the title and abstract: enough to prevent the false\n% inference, not enough to make the production method the subject.\n%\n% It carries no mark, because it qualifies the whole manuscript rather than any\n% one passage.  \\footnotetext does not advance the footnote counter, and the\n% emptied \\thefootnote is confined to a group, so numbered footnotes elsewhere\n% in the paper are unaffected.\n%\n% The wording is not authored here.  It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"},{"record_type":"paper_source_dependency","tex_reference":"evidence/erdos-68-factorial-denominator-irrationality","source_path":"paper/evidence/erdos-68-factorial-denominator-irrationality.tex","media_type":"text/x-tex","sha256":"sha256:b769fad136514a48e1ea36d23191d3140b849fbf9676ea4e2279ebfe64f2abe5","byte_count":1874,"licence":"CC-BY-4.0","body_format":"latex","body":"% Generated by scripts/paper_evidence.py from docs/paper_lean_coverage.json and the\n% Comparator replay receipts under evidence/comparator/.  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Specialising a Louwsma–Martino valuation, it exhibits primes 139 and 2593 absent from all later reduced denominators. Coefficient constructions give integer forms MS+k; at fixed moment the remainder changes only by an integer. Rationality criteria use factorial digits and the next integer above a scaled partial sum; the digit argument also recovers the irrationality of e. Two exact calculations give q∤299999! and q≥2^39990>10^12038. These exclusions do not prove irrationality.","owns":"the complete problem-specific reasoning surface for Erdős #68, including all registered result families and their boundaries","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #68, which remains open","first_pass":["long68:sec:carry","long68:sec:finite","long68:sec:open","sec:erdos-68-complete-family-map"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos68-factorial-reasoning-surface.pdf","hosted_pdf_path":"papers/erdos68-factorial-reasoning-surface.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/68/erdos68-factorial-reasoning-surface.tex","site_html":"maths/papers/erdos68-factorial-reasoning-surface.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/68/erdos68-factorial-reasoning-surface.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/68/erdos68-factorial-reasoning-surface.tex","body_pointer":"papers.records.erdos68-factorial-reasoning-surface.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Long-form reasoning record seeded from the complete problem note.\n% The short note and this record are now independent publication units.\n% This flat manuscript is generated from reviewable parts below.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n\\input{evidence/erdos68-factorial-reasoning-surface}\n\\renewcommand{\\commit}{92b88dc1bbe099aa73bcc900c3c405e9ba5c2334}\n\\renewcommand{\\commitshort}{92b88dc1bbe0}\n\\usepackage{longtable}\n\\hypersetup{\n  pdftitle={Factorial linear forms and denominators: detailed proofs and rationality criteria},\n  pdfsubject={Erdős Problem Notes: Problem 68},\n  pdfkeywords={irrationality, factorial denominators, factorial digits, prime-power cancellation, support bounds, Lean 4},\n}\n\n% Source link targets retain the original manuscript commits and paths.\n\\newcommand{\\Ser}{S}\n\\newcommand{\\chn}[2]{V_{#1}(#2)}\n\\newcommand{\\Lc}{L}\n\\newcommand{\\den}{\\operatorname{den}}\n\n\\runninghead{Factorial linear forms: proofs and denominator criteria}\n\\title{Factorial Linear Forms and Denominators\\\\Detailed Proofs and Rationality Criteria}\n\\subtitle{Erd\\H{o}s Problem~\\#68}\n\\author{Will Cook}\n\\date{1 September 2026; revised 7 October 2026}\n\n\\setlength{\\emergencystretch}{3em}\n% Permit breaks inside long declaration names in the generated catalogue.\n\\usepackage{xurl}\n\\begin{document}\n% ---- part core ----\n% , - part core , -\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\nWe classify multipliers in finitely supported factorial-weighted\ncancellation forms $MS+k$, where $S=\\sum_{n\\ge2}(n!-1)^{-1}$, with\ninteger coefficients on $n\\ge2$. An integral basis reduces cancellation through\n$D\\ge2$ and exclusion of index one to a Diophantine equation whose coefficient gcd\nis determined below $2D^2$. Examples separate least multiplier, support\nsize and coefficient norm. We prove a lower bound for common-denominator growth,\nanalyse prime-power cancellation on reduction, and derive equivalent\ncarry and factorial-digit rationality criteria.\nRecorded computations exclude $q\\mid299999!$ and $q<2^{39990}$ under\n$S=a/q$. These computations are outside Lean; the required nonintegrality\nat arbitrarily large indices remains unproved.\n\\end{abstract}\n\n\\begin{problem}[Erd\\H{o}s \\#68]\\label{long68:res:problem}\nIs $\\Ser=\\sum_{n\\ge2}(n!-1)^{-1}$ irrational?\n\\end{problem}\n\nErd\\H{o}s posed this question in 1988, together with the expectation that\nthe corresponding series with denominators $n!+t$ should be\ntranscendental for every integer $t$ \\cite[p.~102]{erdos1988}.\nFor negative $t$, the sum begins after all zero or negative denominators.\nAny further change of starting index adds a rational number. The\nexpectation for all integer shifts remains unproved here.\nThe problem number follows Bloom's catalogue \\cite{bloom}.\n\nFor finitely supported integer vectors on $n\\ge2$, cancellation forms\n$MS+k$ give an explicitly computable ideal of multipliers.\nCorrecting adjacent factorial differences at proper divisors gives an\nintegral basis. Cancellation fixes its lower coordinates; excluding\nindex one leaves a single Diophantine equation. Its gcd, computed below\n$2D^2$, gives the least positive multiplier and an attaining vector.\nAt depth four, the support restriction strengthens $115\\mid M$ to\n$1380\\mid M$. This classification is independent of nonintegrality.\n\nThe accompanying short paper proves this classification and bounds one\nresulting remainder between consecutive integers. Here we develop the\nsupport and coefficient-norm refinements and a primitive construction\non arithmetic progressions. These solve different problems: at fixed\n$M$, changing the vector changes $MS+k$ only by an integer, so it cannot\nalter nonintegrality. The progression construction makes $M$ divisible\nby every fixed denominator eventually, but still needs a gap estimate\nfor those same forms.\n\nThe later sections study that arithmetic difficulty from two directions.\nCommon-denominator growth measures the cost of clearing terms;\nprime-power calculations determine what survives reduction. Carries\nand factorial digits instead characterize rationality itself. The\nresulting equivalences identify what an irrationality proof must show,\nwithout proving that the required departures occur arbitrarily far out.\n\nThroughout, write\n\\[\n d_n=n!-1,\\qquad H_N=\\sum_{n=2}^N\\frac1{d_n},\\qquad\n \\Lc_N=\\lcm(d_2,\\ldots,d_N),\\qquad\n A_N=\\sum_{n=2}^N\\frac{\\Lc_N}{d_n}.\n\\]\nThus $H_N=A_N/\\Lc_N$. We write $\\den(x)$ for the positive reduced\ndenominator of a rational number $x$. Under $S=a/q$, every form $MS+k$ with\n$M,k\\in\\mathbb Z$ and $q\\mid M$ is integral. For each positive integer $q$, an irrationality argument of this kind\nmust produce a nonintegral form with $q\\mid M$. Both properties must\nhold for the same vector; cancellation and growth of $M$ alone do not\nestablish this.\n\nThe proof of the classification is in \\S\\ref{long68:sec:basis}; the\nfollowing table locates the additional arguments. In particular, the\nlower bound for $\\Lc_N$ concerns the unreduced common denominator,\nnot automatically $\\den(H_N)$. The appendices retain the supplementary\nidentities, limits of the attempted arguments, and formal-source concordance.\n\n\\begin{center}\\small\n\\begin{tabular}{p{.55\\linewidth}p{.34\\linewidth}}\n\\toprule\n\\papertablehead\nQuestion & Where it is treated\\\\\n\\midrule\nWhich $M$ and coefficient vectors are possible? & \\S\\ref{long68:sec:basis}\\\\\nWhat do support and norm minimisation change? &\n\\S\\ref{long68:sec:basis-examples}\\\\\nWhich prime powers survive reduction? & \\S\\ref{long68:sec:prime-powers}\\\\\nHow fast does the common denominator grow? & \\S\\ref{long68:sec:lcm}\\\\\nHow does growth constrain the support parameter? &\n\\S\\ref{long68:sec:quantitative-support}\\\\\nWhich eventual patterns characterize rationality? & \\S\\ref{long68:sec:carry}\\\\\nWhich denominators are excluded by finite computations? & \\S\\ref{long68:sec:finite}\\\\\nWhere are the formal declarations? & Appendix~\\ref{long68:sec:short-concordance}\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\n\\section{Integer coefficient vectors and cancellation}\\label{long68:sec:channels}\n\nWe need weights that preserve the fractional part of $i!/(d!-1)$\nand whose scaled adjacent differences have a simple form. Both requirements\nare met by removing one factor $d!$ for each complete block of $d$\nindices. For a finitely supported integer vector $c=(c_i)$ on $i\\ge2$,\nput\n\\[\n M(c)=\\sum_i c_i\\,i!,\\qquad\n W_{d,i}=\\frac{i!}{(d!)^{\\lfloor i/d\\rfloor}},\\qquad\n \\chn dc=\\sum_i c_iW_{d,i}\\quad(d\\ge2).\n\\]\nWe allow either sign for $M(c)$ and\nfor $V_d(c)$. The associated remainder is\n\\[\n \\mathcal R(c)=\\sum_{d\\ge2}\\frac{V_d(c)}{d!-1}.\n\\]\nThe index $i$ selects a coefficient, whereas $d$ selects a summand\nof the remainder. If $c$ is supported on $i\\le N$, then for $d>N$\nall floor exponents vanish and $V_d(c)=M(c)$. The series therefore\nconverges absolutely. Cancellation of the first weighted sums leaves\na finite signed sum followed by $M(c)$ times the original tail.\nWe show below that the remainder differs from $M(c)S$ by an integer.\n\nThe chosen exponent is a floor, rather than the largest possible power\nof $d!$ dividing $i!$. For example, at $d=2$, $i=6$, it is $3$\nalthough $2^4\\mid6!$. What matters is that it changes between $i-1$\nand $i$ exactly when $d\\mid i$. This makes adjacent differences\namenable to divisor elimination in \\S\\ref{long68:sec:basis}.\nEach $W_{d,i}$ is an integer: writing $i=kd+r$ with $0\\le r<d$, the quotient\n$i!/\\bigl((d!)^kr!\\bigr)$ is a multinomial coefficient.  Since\n$i!=(d!)^{\\lfloor i/d\\rfloor}W_{d,i}$ and $d!\\equiv1\\pmod{d!-1}$,\n\\begin{equation}\\label{long68:eq:channel-congruence}\n \\chn dc\\equiv M(c)\\pmod{d!-1}.\n\\end{equation}\n\n\\begin{theorem}[divisibility of the difference]\\label{long68:res:normalform}\nFor every finite integer support and every $d\\ge2$ there is an integer $k$\nwith $\\chn dc=M(c)+(d!-1)k$.\n\\end{theorem}\n\n\\begin{proof}\nBy \\eqref{long68:eq:channel-congruence}, $d!-1$ divides\n$V_d(c)-M(c)$.  Thus $k=(V_d(c)-M(c))/(d!-1)$ is an integer and gives the\nidentity.  The modulus is positive also at $d=2$, when it equals one.\n\\end{proof}\n\nFor any $N\\ge2$ at least as large as every supported index, the terms\nwith $d>N$ equal $M(c)/(d!-1)$. Therefore\n\\[\n \\mathcal R(c)-M(c)\\Ser\n =\\sum_{d=2}^N\\frac{V_d(c)-M(c)}{d!-1}\\in\\mathbb Z.\n\\]\nThe sum is finite and each term is integral by\n\\eqref{long68:eq:channel-congruence}. This argument also holds when\nindex $1$ is temporarily allowed in the basis calculation below.\nIn particular, $M=0$ gives an integral remainder, and fixing\n$M(c)$ fixes its fractional part. These conclusions hold for every\nfinite vector, before imposing cancellation.\n\nBy \\eqref{long68:eq:channel-congruence}, a vanishing $d$-th weighted sum forces $(d!-1)\\mid M(c)$,\nand annihilating every weighted sum $2\\le d\\le D$ forces $\\Lc_D\\mid M(c)$, where\n$\\Lc_D=\\lcm_{2\\le d\\le D}(d!-1)$ is the same quantity as in\n\\S\\ref{long68:sec:lcm}. With $M$ fixed, the same congruence fixes the\nfractional part of each quotient $V_d(c)/(d!-1)$ separately.\n\n\\subsection{An integral basis and the possible values of \\texorpdfstring{$M$}{M}}\\label{long68:sec:basis}\n\\papersectiontarget{r12-moment}\n\nFix $D\\ge2$. We temporarily allow a coefficient at index one so that\nthe adjacent-difference vectors form a triangular basis. After solving\n$V_2=\\cdots=V_D=0$, we impose one further equation to remove that\ncoordinate. Only the coefficient space is enlarged; the series $S$\ncontinues to start at $d=2$, with no term $1/(1!-1)$.\n\nLet $e_n$ be the unit vector at index $n$. For $n\\ge2$ put\n\\[\n T_n=ne_{n-1}-e_n.\n\\]\nWe have $M(T_n)=0$. If $d\\nmid n$, the two floor exponents in its weighted\nsum agree, and $nW_{d,n-1}=W_{d,n}$. If $d\\mid n$, the exponent increases\nby one, so $nW_{d,n-1}=d!W_{d,n}$. Hence\n\\[\n V_d(T_n)=(d!-1)W_{d,n}\\mathbf1_{d\\mid n}.\n\\]\nFor example, $T_4=4e_3-e_4$ has weighted sums $V_2=6$ and $V_4=23$,\nand all its other weighted sums vanish. Since $T_2=2e_1-e_2$ has only\n$V_2=1$ nonzero, $T_4-6T_2$ isolates the fourth weighted sum.\n\nWe therefore define, in increasing order of $n$,\n\\[\n U_n=T_n-\\sum_{\\substack{d\\mid n\\\\2\\le d<n}}W_{d,n}U_d.\n\\]\nAssume the construction has isolated the weighted sums below $n$.\nSubtracting $W_{d,n}U_d$ removes the contribution at a proper divisor\n$d$ and changes no other weighted sum. At $d=n$ we have $W_{n,n}=1$.\nInduction therefore gives $M(U_n)=0$ and\n$V_d(U_n)=(d!-1)\\mathbf1_{d=n}$ for every $d\\ge2$.\n\nFor the basis claim, restrict to indices $1,\\ldots,N$. The matrix with\ncolumns $e_1,T_2,\\ldots,T_N$ is triangular over $\\mathbb Z$ with\ndiagonal $1,-1,\\ldots,-1$. The change from $T_n$ to $U_n$ is\ntriangular with diagonal one. Both determinants are units in $\\Z$,\nso both inverses have integer entries. In back-substitution the only\ndivisors are $1$ and $-1$; no rational coefficients are needed.\nThus $e_1,U_2,\\ldots,U_N$ represents every integer vector uniquely.\nIf this expansion is $b e_1+\\sum z_dU_d$, then $M(c)=b$ and\n$V_d(c)=b+(d!-1)z_d$. These identities determine the coordinates:\n\\[\n c=M(c)e_1+\\sum_{d\\ge2}\\frac{V_d(c)-M(c)}{d!-1}U_d.\n\\]\nThe congruence \\eqref{long68:eq:channel-congruence} also verifies the\nintegrality of the displayed coefficients. They vanish for $d>N$,\nsince then $V_d(c)=M(c)$, so the expansion is finite. These same\ncoefficients occur in the integer-difference identity at the start of\n\\S\\ref{long68:sec:channels}: their sum is $\\mathcal R(c)-M(c)S$.\nThus the coordinates also recover the integer term in the linear form.\n\nCancellation through $D$ forces $\\Lc_D\\mid M(c)$. To obtain\n$M(c)=\\Lc_D$ in the enlarged space, start with $\\Lc_De_1$, whose weighted\nsums all equal $\\Lc_D$. The vector $U_d$ changes only the $d$th sum,\nby $d!-1$. Hence define\n\\[\n K_D=\\Lc_De_1-\\sum_{d=2}^D\\frac{\\Lc_D}{d!-1}U_d.\n\\]\nThis vector satisfies $M=\\Lc_D$ and $V_2=\\cdots=V_D=0$.\nAll solutions in the enlarged coefficient space consequently have the\nunique form\n\\begin{equation}\\label{long68:eq:low-channel-basis}\n c=tK_D+\\sum_{n>D}z_nU_n,\\qquad M(c)=tL_D,\n\\end{equation}\nwhere $t,z_n$ are integers and only finitely many $z_n$ are nonzero.\nThe only forbidden coefficient is now the one at index $1$.\nWrite $a_D=(K_D)_1$ and $u_n=(U_n)_1$. Removing this coefficient\nis equivalent to\n\\begin{equation}\\label{long68:eq:low-channel-support}\n ta_D+\\sum_{n>D}z_nu_n=0.\n\\end{equation}\nThe free higher basis coefficients can change the first coefficient by every\nmultiple of $\\gcd\\{u_n:n>D\\}$, and only by such a multiple. Thus the\ngcd gives the exact solvability condition. We next show how to compute\nit using a bounded list of coefficients.\nHere $u_2=2$, whereas for $n>2$ the recurrence reads\n\\[\n u_n=-\\sum_{\\substack{d\\mid n\\\\2\\le d<n}}W_{d,n}u_d.\n\\]\nIt gives $u_n=0$ for odd $n>1$. If $p$ is prime, the only proper divisor\nof $2p$ with a nonzero contribution is $2$, and therefore\n$u_{2p}=-(2p)!/2^{p-1}$, including $p=2$.\n\nChoose a prime $p$ with $D/2<p\\le D$. Such a prime exists by Bertrand's\npostulate when $D\\ge3$, and $p=2$ works when $D=2$. The coefficient\n$u_{2p}$ lies in the tail and divides $(2p)!$. Once a finite initial\ninterval includes that coefficient, its gcd therefore divides this\nfactorial. To control the small-divisor weights beyond the interval,\nwe shall require $n/d\\ge2p$ whenever $d\\le D$. This leads to the\ncutoff $H=D(2p-1)$. We claim that\n\\begin{equation}\\label{long68:eq:finite-horizon}\n \\gcd\\{u_n:n>D\\}=\\gcd(u_{D+1},\\ldots,u_H)>0,\\qquad H<2D^2.\n\\end{equation}\nLet $g$ be the finite gcd on the right. Its value could in principle\ndecrease if we included later coefficients. To rule this out, we prove\n$g\\mid u_n$ for every $n>D$ by strong induction.\nThe index $2p$ lies in the finite interval and has a nonzero coefficient,\nso $g>0$ and $g\\mid(2p)!$. The definition proves the induction claim\nthrough $H$.\n\nFor $n>H$, each term of the recurrence is $W_{d,n}u_d$ with $d<n$.\nIf $d>D$, divisibility comes from $u_d$ by induction. If $d\\le D$,\nwe cannot use that hypothesis and instead use the weight. Put $k=n/d$.\nThe cutoff was chosen so that $n>D(2p-1)$ forces $k\\ge2p$ for\nevery such divisor. Also,\n\\[\n \\frac{W_{d,n}}{k!}=\\frac{n!}{(d!)^k k!}\\in\\mathbb Z,\n\\]\nbecause the quotient counts partitions of $n$ labelled objects into\n$k$ unordered blocks of size $d$. Hence\n$g\\mid(2p)!\\mid k!\\mid W_{d,n}$. Thus every product in the recurrence\nis divisible by $g$, completing the induction.\nThe finite gcd divides the entire tail, and inclusion of the finite\ninterval gives the reverse divisibility. Finally,\n$H\\le D(2D-1)<2D^2$.\n\n\\begin{theorem}[the set of attainable values of $M$]\\label{long68:res:moment-ideal}\nFix $D\\ge2$ and a prime $p$ with $D/2<p\\le D$.  Put\n\\[\n H=D(2p-1),\\qquad\n G_D=\\gcd\\{|u_n|:D<n\\le H\\},\\qquad\n \\mu_D=L_D\\frac{G_D}{\\gcd(G_D,a_D)}.\n\\]\nFor finite integer vectors supported on $n\\ge2$ and cancelling all\nweighted sums $2,\\ldots,D$, the possible values of $M$ are exactly $\\mu_D\\Z$.  The integer $\\mu_D$ is positive,\nis independent of the eligible prime $p$, and is attained by a vector of\ncoefficients with gcd one, that is, by a primitive vector.\n\\end{theorem}\n\n\\begin{proof}\nEquation~\\eqref{long68:eq:finite-horizon} identifies $G_D$ with the gcd of\nall $u_n$ for $n>D$. Their finite integer combinations are exactly\n$G_D\\Z$. Hence \\eqref{long68:eq:low-channel-support} has a solution if\nand only if $G_D\\mid ta_D$, or equivalently\n$G_D/\\gcd(G_D,a_D)\\mid t$. Substitution in\n\\eqref{long68:eq:low-channel-basis} gives the claimed set of values of $M$.\nB\\'ezout's identity applied to the finite list through $H$ supplies a\nvector attaining its least positive element. A common coefficient divisor\nlarger than one could be divided out, giving a smaller positive attainable\nvalue of $M$. The attaining vector is therefore primitive. Finally, the set of\nvalues of $M$ is intrinsic to the cancellation equations, so its positive\ngenerator is independent of the eligible prime $p$.\n\\end{proof}\n\nFix $M\\in\\mu_D\\Z$ and put $t=M/L_D$. The support equation\n\\eqref{long68:eq:low-channel-support} determines which basis coefficients\nare admissible. The following formula records their integer contribution\nand shows why changing the vector at that value of $M$ cannot alter\nnonintegrality.\n\n\\begin{theorem}[how the coefficient choices change the remainder]\\label{long68:res:residual-transparency}\nFor the vector in \\eqref{long68:eq:low-channel-basis},\n\\[\n \\mathcal R\\!\\left(tK_D+\\sum_{n>D}z_nU_n\\right)\n =tL_D(\\Ser-H_D)+\\sum_{n>D}z_n.\n\\]\nThe remainder series converges for every finite vector supported away from\nindex zero.  A vector with $M=0$ has integral remainder, and any two finite\nvectors with the same value of $M$ have remainders differing by an integer.\n\\end{theorem}\n\n\\begin{proof}\nFor $d>D$, we have $V_d(K_D)=L_D$ and\n$V_d(U_n)/(d!-1)=\\mathbf1_{d=n}$. Summing these identities gives\n$tL_D(S-H_D)+\\sum_{n>D}z_n$, as claimed. The first series converges and\nthe second sum is finite.\n\nAbsolute convergence and the general integer-difference identity were\nproved at the start of \\S\\ref{long68:sec:channels}, including for the\nauxiliary index $1$. Taking $M=0$ proves integrality;\nsubtracting the identities for two vectors with equal $M$ proves the\nlast assertion.\n\\end{proof}\n\nHere $M=tL_D$, so every denominator in $H_D$ divides $M$. Thus\n$MH_D\\in\\Z$, and the integer in the original form $MS+k$ is\n$k=-MH_D+\\sum_{n>D}z_n$.\n\n\\subsection{Examples and support refinements}\\label{long68:sec:basis-examples}\n\nFor $D=4$ we need a common divisor of the entire coefficient tail,\nas well as a finite combination attaining it. Put\n$\\Lambda_n=\\lcm(1,\\ldots,n)$. We first prove $\\Lambda_n\\mid u_n$.\nFor $d\\mid n$ and a prime $r$, Legendre's formula gives\n\\[\n v_r(W_{d,n})=\\sum_{j\\ge1}\n \\left(\\left\\lfloor\\frac n{r^j}\\right\\rfloor\n       -\\frac nd\\left\\lfloor\\frac d{r^j}\\right\\rfloor\\right).\n\\]\nEvery summand is nonnegative. If $d<r^j\\le n$, the second floor\nis zero and the first is at least one. Counting these powers gives\n$v_r(W_{d,n})\\ge v_r(\\Lambda_n)-v_r(\\Lambda_d)$, and therefore\n$\\Lambda_n\\mid W_{d,n}\\Lambda_d$. Starting from $u_2=2$, induction\nin the recurrence proves $\\Lambda_n\\mid u_n$.\n\nThus $60\\mid u_n$ for all $n>4$. Conversely, the finite calculation\n$u_6=-180$, $u_8=-4200$ gives $23u_6-u_8=60$. The gcd is consequently\n$G_4=60$.\nSince $L_4=115$ and $a_4=-55$, we obtain\n\\[\n \\mu_4=115\\frac{60}{\\gcd(60,55)}=1380.\n\\]\nThe vector $12K_4+253U_6-11U_8$ attains this value: its coefficient at\nindex $1$ is $12(-55)+253(-180)-11(-4200)=0$.\n\nThe five-term vector used in the short paper is obtained from the same\ncoordinates. The equality $12(-55)-27(-180)-4200=0$ first gives\n$12K_4-27U_6+U_8$. Its coefficients at indices four and seven are\nremoved by $-26U_5+8U_7$, respectively. Both corrections leave the first\ncoordinate zero. Consequently\n\\begin{equation}\n\\begin{split}\n 12K_4-26U_5-27U_6+8U_7+U_8\n &=1482e_2-784e_3-136e_5\\\\\n &\\quad+83e_6-e_8.\n\\end{split}\n\\label{long68:eq:constructed-depth-four}\n\\end{equation}\nThe coefficients of the $U_n$ in this expansion sum to $-44$.\nSince $H_4=143/115$, its remainder is\n$1380(S-H_4)-44=1380S-1760$. The exact interval calculation appears\nin \\S\\ref{long68:sec:ext-certificates}. Thus the coordinates give both\nthe original linear form and the cancellations, independently of a\nsearch over supports.\n\nIf support is additionally restricted to $n\\le6$, only $u_5=0$ and\n$u_6=-180$ remain available in \\eqref{long68:eq:low-channel-support}.\nThe least positive value of $M$ is then $115\\cdot180/\\gcd(180,55)=4140$.\nThus a restriction on the largest supported index can increase the\nleast attainable value of $M$. The following refinements specify the support bound in each case.\n\n\\papersectiontarget{r5-depth-refinements}\nThere is also a direct congruence proof using only the second and\nfourth weighted sums. For every $n\\ge2$,\n\\begin{equation}\n1380\\mid 11n!-46W_{2,n}+12W_{4,n}.\n\\label{long68:eq:depth-four-dual}\n\\end{equation}\nThe expression is zero for $n=2,3,4$. For $n\\ge4$, it is divisible\nby $12$: $12\\mid n!$ and $6\\mid W_{2,n}$, as follows from\n$W_{2,2r}=r!\\prod_{j=1}^r(2j-1)$ for $r\\ge2$ and the corresponding\nodd-index recurrence. For $n\\ge5$, it is divisible by $5$ because\n$5\\mid n!$ and the denominators removed to form $W_{2,n}$ and\n$W_{4,n}$ are coprime to $5$. Modulo $23$,\n$n!\\equiv W_{4,n}$ and $46W_{2,n}\\equiv0$. This proves the congruence.\nConsequently $V_2=V_4=0$ forces $1380\\mid M$, without a restriction\non the largest supported index. Conversely,\n$1482e_2-784e_3-136e_5+83e_6-e_8$ satisfies $M=1380$ and\n$V_2=V_3=V_4=0$. Thus imposing the additional equation $V_3=0$ does not change the\npossible values of $M$, although $3e_2-e_3$ shows that it changes the set of\nadmissible vectors.\n\nAmong integer vectors supported in $\\{2,\\ldots,8\\}$ with $M=1380$ and\n$V_2=V_3=V_4=0$, this vector has the smallest support. Every such vector\nhas the following coefficients for some $t,x,y\\in\\Z$. The coefficient\nof $K_4$ remains $M/L_4=12$; here $t$ is a new free parameter.\n\\begin{equation}\n\\begin{aligned}\n\\lambda_2&=1482-4200t,&\\lambda_3&=-784+2240t,\\\\\n\\lambda_4&=-680+1890t-5x,&\\lambda_5&=x,\\\\\n\\lambda_6&=83-238t-7y,&\\lambda_7&=y,\\\\\n\\lambda_8&=-1+3t.\n\\end{aligned}\n\\label{long68:eq:depth-four-family}\n\\end{equation}\nIndeed, the values of the expression in\n\\eqref{long68:eq:depth-four-dual} at indices $2,\\ldots,8$ are\n\\[\n(0,0,0,0,4140,28980,328440).\n\\]\nTherefore the dual identity and\n$M=1380$ give $11=3\\lambda_6+21\\lambda_7+238\\lambda_8$.\nModulo $3$ this yields $\\lambda_8=-1+3t$, then\n$\\lambda_6+7\\lambda_7=83-238t$. Set $x=\\lambda_5$ and\n$y=\\lambda_7$; solving $V_2=V_3=V_4=0$ for the coefficients at indices\n$2,3,4$ gives \\eqref{long68:eq:depth-four-family}. The~coefficient~matrix~is~$\\left(\\begin{smallmatrix}1&3&6\\\\2&1&4\\\\2&6&1\\end{smallmatrix}\\right)$,\nwith determinant $55$, and substitution verifies the converse.\nIndices $2,3,8$ are nonzero for every integral $t$; neither the pair\n$(\\lambda_4,\\lambda_5)$ nor $(\\lambda_6,\\lambda_7)$ can vanish\ntogether. Thus five nonzero entries are necessary at this support bound.\n\nThe smallest coefficient $\\ell^1$ norm for support in $\\{2,\\ldots,8\\}$ is $2416$,\nattained by\n$(\\lambda_2,\\ldots,\\lambda_8)=(1482,-784,0,-136,-1,12,-1)$.\nFor $t\\ge1$, the absolute values at indices $2,3,8$ alone sum to\n$6443t-2267\\ge4176$; for $t\\le-1$ they sum to\n$2267-6443t\\ge8710$. Thus a minimizer has $t=0$.\nThe remaining independent minima are\n$\\min_x(|-680-5x|+|x|)=136$ at $x=-136$ and\n$\\min_y(|83-7y|+|y|)=13$ at $y=12$.\nFor the first, $680\\le|-680-5x|+5|x|$ is at most five times\nthe objective. For the second, the objective is at least $83$ when\n$y\\le0$, at least $17$ when $0\\le y\\le11$, and equals\n$8y-83\\ge13$ when $y\\ge12$.\nAdding the fixed contribution $1482+784+1=2267$ gives $2416$.\nEquality forces $t=0$, $x=-136$ and $y=12$, so this norm minimizer is\nunique. Among the five-term vectors, $t=0$ is again necessary for the\nleast norm; the pair $(\\lambda_6,\\lambda_7)$ can then have one nonzero\nentry only when $y=0$, since $7\\nmid83$. The pair\n$(\\lambda_4,\\lambda_5)$ has one nonzero entry only at $x=0$ or\n$x=-136$, and the latter is smaller. Hence the displayed five-term\nvector is the unique least-norm vector among those of minimum support,\nwith norm $2486$.\n\nThe additional equation $V_3=0$ changes the minimum support size and\ncoefficient norm, while leaving the possible values of $M$ unchanged.\nWith only $V_2=V_4=0$, group the seven\ncoefficients by\n$a=\\lambda_2+3\\lambda_3$, $b=\\lambda_4+5\\lambda_5$,\n$c=\\lambda_6+7\\lambda_7$ and $d=\\lambda_8$.\nSolving $V_2=V_4=0$ and $M=1380$ gives\n\\[\n(a,b,c,d)=(-870+2520t,-680+1890t,83-238t,-1+3t),\n\\qquad t\\in\\Z.\n\\]\nEach coordinate is nonzero for integral $t$, so four nonzero\ncoefficients are necessary and sufficient. One four-term witness is\n$-290e_3-136e_5+83e_6-e_8$, with norm $510$.\nFor the coefficient norm on the same set of indices,\n$|\\lambda_2|+|\\lambda_3|\\ge|a|/3$ and\n$|\\lambda_4|+|\\lambda_5|\\ge|b|/5$; when $t\\ne0$, these bounds\ntogether with $|d|$ already sum to at least $794$.\nAt $t=0$, the four groups have unique norm minima $290$, $136$,\n$13$ and $1$, respectively. Thus the unique norm minimizer under\nonly $V_2=V_4=0$ is\n$-290e_3-136e_5-e_6+12e_7-e_8$, with norm $440$.\nFor comparison, with $M=1380$ and support contained in\n$\\{2,\\ldots,8\\}$, the two optimization problems give:\n\\begin{center}\n\\begin{tabular}{lrr}\n\\papertablehead\nCancellation & Minimum support size & Minimum $\\ell^1$ norm\\\\\n\\hline\n$V_2=V_4=0$ & $4$ & $440$\\\\\n$V_2=V_3=V_4=0$ & $5$ & $2416$\\\\\n\\end{tabular}\n\\end{center}\nIn both rows the norm minimizer has one more nonzero coefficient than\na support minimizer. Thus even with $M$ and the allowed indices fixed,\nleast support and least norm need not coincide. These are finite\noptimization statements, not irrationality conclusions.\n\nThere is a uniform restriction $12L_D\\mid M$ on support $n\\ge2$.\nFor $n=2,3$ one has $n!=2W_{2,n}$. For $n\\ge4$, both $n!$ and\n$2W_{2,n}$ are divisible by $12$: if $n=2k$ or $2k+1$, then\n$W_{2,2k}=k!\\prod_{j=1}^k(2j-1)$ is divisible by $6$ for $k\\ge2$.\nThus $12\\mid M(c)-2V_2(c)$. Every $d!-1$ is coprime to $6$, so\n$V_2=\\cdots=V_D=0$ implies $12L_D\\mid M(c)$.\nThis factor is attained at $D=2$ and $D=3$ by\n$-6e_2+e_4$ and $-6e_2-8e_3+5e_4$, respectively.\nThe next example shows that this need not be the whole restriction.\n\n\\papersectiontarget{r15-depth-six}\nAt $D=6$, the same recurrence\ngives\n\\[\n L_6=9839515,\\qquad a_6=-2242555,\\qquad G_6=840.\n\\]\nTo verify the last value, $u_7=0$, while $840=\\lcm(1,\\ldots,8)$ divides\nevery $u_n$ for $n\\ge8$ by the preceding divisibility. Conversely,\n$u_8=-4200$, $u_{12}=-14386680$ and $\\gcd(u_8,u_{12})=840$.\nThus $\\gcd(G_6,a_6)=35$ and the least positive value of $M$ is $\\mu_6=24L_6$,\nnot $12L_6$. It is attained by $24K_6-16240U_8+U_{12}$: the coefficient\nat index $1$ is zero and the coefficient at $12$ is $-1$. The finite-gcd formula thus detects the extra factor at depth six,\nwhich the uniform divisibility by $12L_D$ misses.\n\n\\subsection{Support bounds and nonintegrality}\\label{long68:sec:support-and-remainder}\n\nThere are three separate choices: the vector, its multiplier $M$,\nand the cutoff used to estimate its remainder. The basis and support\nequations classify the first two. Once $M$ is fixed,\n$\\mathcal R(c)\\equiv M(c)S\\pmod{\\Z}$ fixes the fractional part:\ninteger translation may put a representative near zero but cannot\nchange its distance from an integer.\n\nTo exclude every rational denominator, each $q>0$ must divide the\nmultiplier of some nonintegral form. Growth of the compulsory divisor\n$12L_D$ alone is insufficient: its $3$-adic valuation is exactly one\nfor every $D$, so it does not even force $9\\mid M$. By contrast,\nextending the cutoff changes neither $M$ nor the remainder, only the\nfinite enclosure used to test it.\n\nFor $c=-6e_2+e_4$, we have $M(c)=12$ and $\\mathcal R(c)=12S-17$. At cutoff $4$, its finite part is $-239/115$:\nthe gap $9/115$ is smaller than the tail bound $24/119$. At cutoff $5$,\nthe finite part is $-27061/13685$, whose gap $13376/13685$ exceeds the\nnew bound $24/719$. The finite part has crossed $-2$, so the next\ninteger is now $-1$ rather than $-2$. This explains the larger gap.\nThe second cutoff certifies nonintegrality of the same remainder;\nfailure at the first cutoff was inconclusive. The successful test\nexcludes denominators dividing the unchanged value $M=12$, already\ncovered by the finite exclusions.\n\nFor fixed $c$ with $M(c)>0$, the finite part increases to\n$\\mathcal R(c)$ with omitted tail $M(c)(S-H_N)$. If\n$\\mathcal R(c)\\notin\\Z$, its gap to the next integer is positive,\nwhereas $2M(c)/((N+1)!-1)$ tends to zero; the test therefore succeeds\nat every sufficiently large cutoff. If $\\mathcal R(c)\\in\\Z$, the\neventual gap equals the omitted tail and is smaller than this upper\nbound. Eventual success is consequently equivalent to nonintegrality\nof this fixed form, not an independent reason for it. The\n\\href{erdos-68-factorial-denominator-irrationality.pdf\\#r12-short-gap}{short paper's remainder comparison}\ngives the inequality explicitly. For each $q>0$ we still need a vector\nwith $q\\mid M(c)$ for which it succeeds; the successful cutoff need\nnot be the largest supported index.\n\nThe support itself has a separate geometric restriction. Cancellation\nwith $M\\ne0$ must cross a change in the relevant floor exponent;\notherwise the weighted sum is proportional to $M$.\n\n\\begin{theorem}[constant values of the floor in the weights]\\label{long68:res:bandbreakpoint}\nLet $d\\ge2$ and $k\\ge0$, and suppose every supported index $i$ satisfies\n$kd\\le i<(k+1)d$.  Then $M(c)=(d!)^k\\chn dc$.  In particular, cancellation\non the interval $d\\le i<2d$ forces $M(c)=0$; and if every supported index\nis at least $d$ while $M(c)\\ne0$ and $\\chn dc=0$, then some supported index\nis at least $2d$.\n\\end{theorem}\n\n\\begin{proof}\nOn this interval the quotient $\\lfloor i/d\\rfloor$ is the constant $k$, so\n$i!=(d!)^kW_{d,i}$ for every supported index and the factor $(d!)^k$ comes\nout of the sum.  The two consequences follow by taking $k=1$ and by\ncontraposition.\n\\end{proof}\n\nThe constant-floor condition is local. A quantitative bound also needs\ncontrol of the multiplier: cancellation alone does not give\n$0<M<(R+1)!-1$. After proving the common-denominator estimates,\n\\S\\ref{long68:sec:quantitative-support} derives the resulting bounds on $R$.\n\nAt a prime index, the two-term vector $T_p$ already changes only one\nweighted sum; there are no proper divisors to eliminate.\n\n\\begin{theorem}[changing just one weighted sum]\\label{long68:res:translator}\nLet $p\\ge3$ be prime and let $c_{p-1}=p$, $c_p=-1$, with every other coefficient\nzero.  Then $M(c)=0$, $\\chn pc=p!-1$, and $\\chn dc=0$ for every $d\\ge2$ with\n$d\\ne p$.\n\\end{theorem}\n\n\\begin{proof}\nSince $p\\ge3$, both indices $p-1$ and $p$ lie in the prescribed coefficient\ndomain $i\\ge2$.  Here $M=p\\,(p-1)!-p!=0$.  For $2\\le d<p$ the quotient identity\n$\\lfloor(p-1)/d\\rfloor=\\lfloor p/d\\rfloor$ holds, since $d\\nmid p$, so the\ntwo factorial weights carry the same power of $d!$ and the weighted sum evaluates to\n$p\\,(p-1)!/(d!)^{\\lfloor p/d\\rfloor}-p!/(d!)^{\\lfloor p/d\\rfloor}=0$.  For\n$d>p$ both indices lie below $d$, so both weights are the plain factorials\nand the same cancellation occurs.  At $d=p$ the weights are $(p-1)!$ and\n$p!/p!=1$, giving $p\\,(p-1)!-1=p!-1$.\n\\end{proof}\n\nFor $p=2$, the same auxiliary identities would require $c_1=2$.  They do\nnot define an admissible vector on the domain $i\\ge2$, since every vector\nsupported on that domain has $c_1=0$.\n\n\\noindent Adding an integer multiple of this vector changes $V_p$ by\nthat multiple of $p!-1$, without changing $M$ or any other $V_d$.\nSuppose $M(c)\\ne0$, $V_2(c)=\\cdots=V_D(c)=0$, and every\nsupported index exceeds a prescribed integer $B$. Choose a prime\n$p>\\max(D,B+1,2)$ and replace $c$ by\n\\[\n c-\\lfloor\\mathcal R(c)+1/2\\rfloor T_p.\n\\]\nBoth new indices, $p-1$ and $p$, exceed $B$. Since $p>D$, the prescribed\nweighted sums remain zero, and $M$ is unchanged. The new remainder is\n$\\mathcal R(c)-\\lfloor\\mathcal R(c)+1/2\\rfloor\\in[-1/2,1/2)$.\nThe progression construction below supplies such an initial vector:\nchoose its first index $r>B$.\n\nAn integral remainder rounds to zero; a nonintegral one rounds to a\nnonzero value. Thus rounding controls size but does not prove\nnonintegrality or exclude further denominators at fixed $M$.\nMoreover, $\\lfloor\\mathcal R(c)+1/2\\rfloor$ depends on the unknown\nremainder: this is an existence formula for a representative, not an\nindependent certificate of its nonvanishing.\n\n\\subsection{A primitive solution on an arithmetic progression}\\label{long68:sec:progression}\n\\papersectiontarget{r5-progression}\nWe now separate two tasks that the basis classification alone does\nnot combine: prescribe where support begins and make $M$ absorb every\nfixed denominator as the support grows. Equal spacing turns each\ncancellation equation into a polynomial root condition. Scaling then\nmakes the coefficients integral and the last coefficient equal to one.\nLet $D,r\\ge2$ and let $\\ell$ be a positive multiple of $2,\\ldots,D$.\nThe roots will be $(d!)^{-\\ell/d}$. Put\n\\[\ni_j=r+j\\ell\\ (0\\le j<D),\\quad N=r+(D-1)\\ell,\\quad\n\\alpha_d=(d!)^{\\ell/d},\n\\]\n\\[\n\\prod_{d=2}^D(\\alpha_dX-1)=\\sum_{j=0}^{D-1}h_jX^j,\n\\qquad A=\\prod_{d=2}^D\\alpha_d.\n\\]\nAt index $i_j$ put the coefficient\n\\[\nc_{i_j}=\\frac{N!h_j}{A i_j!},\n\\]\nand put every other coefficient equal to zero. We verify that these\ncoefficients are integers and that $V_2=\\cdots=V_D=0$.\nFor $2\\le d\\le D$, divisibility of $\\ell$ by $d$ gives\n$\\lfloor i_j/d\\rfloor=\\lfloor r/d\\rfloor+j\\ell/d$. Hence\n\\[\nV_d(c)=\n\\frac{N!}{A(d!)^{\\lfloor r/d\\rfloor}}\n\\sum_{j=0}^{D-1}h_j\\alpha_d^{-j}=0,\n\\]\nbecause $\\alpha_d^{-1}$ is a root of the displayed polynomial. Evaluating\nthat polynomial at $1$ gives $M$\n\\[\nM=N!\\prod_{d=2}^D(1-1/\\alpha_d)>0.\n\\]\n\nFor integrality, each term of $h_j/A$ is, up to sign, the reciprocal of\na product of $D-1-j$ distinct factors $\\alpha_d$. In that denominator,\neach selected $d$ contributes $\\ell/d$ blocks of size $d$. The total size is\n$(D-1-j)\\ell=N-i_j$, so the multinomial coefficient shows that this\ndenominator divides $N!/i_j!$. Thus every $c_{i_j}$ is an integer.\nSince $h_{D-1}=A$, the coefficient at $N$ is $1$, and the coefficients\nhave gcd one. This does not establish minimal support. The smallest\npermitted step is $\\ell=\\operatorname{lcm}(2,\\ldots,D)$; taking $D=r=\\ell=2$\nrecovers $-6e_2+e_4$.\n\nThe stronger divisibility\n\\[\nr!\\prod_{d=2}^D(\\ell/d)!\\prod_{d=2}^D(\\alpha_d-1)\\mid M\n\\]\nfollows because\n\\[\n\\frac{N!}{r!\\,A\\prod_{d=2}^D(\\ell/d)!}\n\\]\nis an integer: it counts partitions into one distinguished block of size\n$r$ and, for each $d$, an unordered collection of $\\ell/d$ blocks of size\n$d$. Their sizes sum to $N$. Using\n$M=(N!/A)\\prod_{d=2}^D(\\alpha_d-1)$ now gives the asserted divisibility.\nPut $v=\\max(r,\\ell/2)$, an integer since $2\\mid\\ell$. The displayed\nfactor divides $M$, so $v!\\mid M$. Since $D\\le\\ell\\le2v$ and $r\\le v$,\n\\[\n N=r+(D-1)\\ell\\le4v^2-v<4v^2,\n \\qquad v\\ge\\lfloor\\sqrt N/2\\rfloor+1.\n\\]\nConsequently $(\\lfloor\\sqrt N/2\\rfloor+1)!\\mid M$. Every positive integer\n$q\\le\\lfloor\\sqrt N/2\\rfloor+1$ therefore divides $M$, as does every\ndivisor of that factorial. In particular, $N\\ge4(q-1)^2$ suffices for\n$q\\mid M$. This includes denominators growing with $N$.\n\nThe same vector also supplies the size estimate needed for a tail test.\nEvery factor $1-1/\\alpha_d$ lies strictly between $0$ and $1$, so\n$0<M<N!$ and\n\\[\n 0<M(S-H_N)<\\frac{2M}{(N+1)!-1}<\\frac2N,\n\\]\nwhere $(N+1)!-1>NN!$ for $N\\ge2$. Thus the family supplies both\neventual divisibility by each fixed $q$ and the size condition\n$M<(N+1)!-1$, without an extra hypothesis. It does not yet compare\nthis tail with the finite part's gap to the next integer. Both may\nshrink together. Before studying that arithmetic question, we show why\nthe cofactor formula gives the same primitive vectors.\n\n\\subsection{Cofactors give the same primitive vector}\\label{long68:sec:cofactor-progression}\nA determinant formula may appear to offer a different family. For\npositive starting index, however, its primitive normalization gives\nexactly the preceding progression vector, up to sign. We first verify\nthe determinant construction, then identify the two formulas. For\nintegers $n,t\\ge0$, put\n$s_n=((n+2)!)^2$ and $i_j=(t+j)s_n$ for $0\\le j\\le n+1$.\nThis step size matches the recorded construction; the determinant argument\nonly needs a positive step divisible by every $d=2,\\ldots,n+2$.\nHere the factorial-weight formula is also allowed at index $0$; the case\n$t=0$ will show why a lower support condition is necessary. Form the\ninteger matrix $A$ with first row $(i_j!)_j$ and weighted-sum rows\n$(W_{d,i_j})_j$ for $2\\le d\\le n+2$.\n\nTo see that $\\det A\\ne0$, divide column $j$ by $i_j!$.\nSince $d\\mid s_n$, the $d$th weighted-sum row becomes\n$((d!)^{-(t+j)s_n/d})_j$. Removing the nonzero factor\n$(d!)^{-t s_n/d}$ from that row leaves a Vandermonde matrix with nodes\n\\[\n1,\\ (2!)^{-s_n/2},\\ \\ldots,\\ ((n+2)!)^{-s_n/(n+2)}.\n\\]\nThese nodes are distinct: $(d!)^{1/d}$ is strictly increasing, since\n$d!<(d+1)^d$. The Vandermonde determinant is the product of their\npairwise differences \\cite[(1.3.13)]{nist-vandermonde}, and is therefore nonzero. None of the row or\ncolumn factors removed is zero, so the original determinant is nonzero\ntoo.\n\nLet $c$ be the cofactor vector of the first row, and let $N_d$ be the\ndeterminant obtained by replacing that row with $(W_{d,i_j})_j$.\nCofactor expansion gives $M(c)=\\det A\\ne0$ and $\\chn dc=N_d$.\nFor $2\\le d\\le n+2$ the replacement duplicates a row, so $\\chn dc=0$.  By\n\\eqref{long68:eq:channel-congruence} each $(\\chn dc-M(c))/(d!-1)$ is an integer and\nvanishes for $d>\\max_ji_j$, so\n\\[\n \\mathcal R_{n,t}:=\\sum_{d>n+2}\\frac{N_d}{d!-1},\\qquad\n \\mathcal R_{n,t}-\\det(A)\\,\\Ser\\in\\Z .\n\\]\nA family with $\\min_ji_j=t\\,s_n\\to\\infty$ and\n$\\mathcal R_{n,t}\\notin\\Z$ would prove irrationality. Indeed, for a fixed\ndenominator $q$, every entry of the first row is divisible by $q$ once\nthe least support index is at least $q$, so $q\\mid\\det A$. If $S=a/q$,\nthe integer-difference identity would then force $\\mathcal R_{n,t}\\in\\Z$.\nConversely, if $S$ is irrational, that same identity and $\\det A\\ne0$\nmake every $\\mathcal R_{n,t}$ irrational. Taking, for example, $n=0$\nand $t\\to\\infty$ gives the required family. Thus nonintegrality along\nsuch a family is equivalent to the original irrationality question,\nnot a weaker conjecture obtained from the construction.\n\nThe condition $t s_n\\to\\infty$ cannot be replaced by unboundedness of\n$(n,t)$: at $t=0$, the least supported index stays zero. The coefficient\nat $i_0$ really is nonzero: its cofactor uses columns $1,\\ldots,n+1$, and\nafter nonzero row and column scalings it is a Vandermonde determinant\non the weighted-sum nodes. The eventual\nidentity $N_d=\\det A$ determines the tail, but not the finite intermediate\nsum, whose terms may have either sign. Thus the determinant proves\ncancellation, but its nonzero value does not prove nonintegrality of\nthe remainder.\n\nFor $t\\ge1$, primitive normalisation identifies this vector exactly.\nTake $D=n+2$, $\\ell=s_n$ and $r=t s_n$ in the progression construction\nof \\S\\ref{long68:sec:progression}, and put $N=r+(D-1)\\ell$.\nFor any coefficient vector supported on these $D$ indices, the equations\n$V_2=\\cdots=V_D=0$ say that the polynomial\n\\[\n \\sum_{j=0}^{D-1}c_{i_j}i_j!X^j\n\\]\nvanishes at the $D-1$ distinct points\n$\\alpha_d^{-1}=(d!)^{-\\ell/d}$, $2\\le d\\le D$.\nIts degree is at most $D-1$, so it is a scalar multiple of\n$\\prod_{d=2}^D(\\alpha_dX-1)$. Thus the rational solution space is\none-dimensional. The progression vector has coefficient $1$ at $N$,\nso every integer solution is its integer multiple, with multiplier $c_N$.\nIn particular, the cofactor gcd is $|c_N|$, and the primitive cofactor\nvector is the progression vector up to sign. Choosing the sign so that $M>0$ gives\n\\[\n N!\\prod_{d=2}^D\\bigl(1-(d!)^{-\\ell/d}\\bigr).\n\\]\nFor example, $n=0,t=1$ gives support $\\{4,8\\}$, cofactor vector\n$2520e_4-6e_8$, and primitive vector $-420e_4+e_8$ with $M>0$.\n\nThe factorial divisibility proved for the progression coefficient $M$ therefore\nalso applies to this primitive vector. Normalisation still leaves the\nnonintegrality condition equivalent to irrationality along families\nwhose least support index tends to infinity: $M$ is nonzero,\nevery factorial below that index divides it, and\n\\eqref{long68:eq:channel-congruence} gives the integer-difference identity.\nThis identification does not estimate the distance from an integer.\nThe case $t=0$ is excluded from the identification with the stated\nprogression construction, whose indices are at least $2$.\n\n\\section{Prime powers in reduced denominators}\\label{long68:sec:prime-powers}\n\nLet $p$ be a prime, and let $v_p(a)$ denote the exponent of $p$ in a\npositive integer $a$. We determine which prime powers of $\\Lc_M$ survive in $\\den(H_M)$.\nAt the largest exponent, this follows by specialising Louwsma and\nMartino's valuation formula\nfor an elementary symmetric sum \\cite[Lemma~4.1, p.~10]{louwsma-martino}.\nIndeed, $\\sum_i1/x_i=(\\sum_i\\prod_{j\\ne i}x_j)/\\prod_i x_i$ for positive\nintegers $x_i$. Their lemma evaluates the numerator's valuation; subtracting\nthe product's valuation gives the reciprocal-sum form. The proof below\nuses the least common multiple instead of the product.\n\nThe distinction is already visible in $1/3+1/15=2/5$:\nthe common denominator contains $3$, but the reduced denominator does not.\nBy contrast, $1/9+1/45=2/15$ only lowers the exponent of $3$ from $2$ to $1$.\nThe theorem below tests whether the largest exponent is preserved; a\nfailure of that test need not remove the prime completely. The formula\nis standard. Its mechanism is simple: after clearing by the lcm and\nreducing modulo $p$, only denominators carrying the largest power of\n$p$ contribute. The factorial examples and later tail comparison\nconcern this particular series.\n\n\\label{long68:res:lead-prime-pole}\\phantomsection\n\\begin{theorem}[maximal prime-power survival]\\label{long68:res:prime-pole}\nLet $M\\ge2$, let $p$ be a prime dividing $\\Lc_M$, and put $e=v_p(\\Lc_M)$.\nLet $J=\\{n:2\\le n\\le M,\\ v_p(d_n)=e\\}$ and write $d_n=p^eu_n$ for $n\\in J$.\nThen, with inverses in $\\mathbb F_p$,\n\\begin{equation}\\label{long68:eq:prime-pole-survival}\n v_p\\bigl(\\den(H_M)\\bigr)=e\n \\quad\\Longleftrightarrow\\quad\n \\sum_{n\\in J}u_n^{-1}\\ne0\\quad\\hbox{in }\\mathbb F_p.\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nWrite $\\Lc_M=p^eW$ with $p\\nmid W$.  If $v_p(d_n)<e$, then $p\\mid\\Lc_M/d_n$.\nIf $n\\in J$, then $u_n(\\Lc_M/d_n)=W$, so $\\Lc_M/d_n\\equiv Wu_n^{-1}\\pmod p$.\nSumming over $2\\le n\\le M$ gives\n\\begin{equation}\\label{long68:eq:prime-pole-residue}\n A_M\\equiv \\frac{\\Lc_M}{p^e}\\sum_{n\\in J}u_n^{-1}\\pmod p.\n\\end{equation}\n\n\nFinally $\\den(H_M)=\\Lc_M/\\gcd(A_M,\\Lc_M)$, whose $p$-valuation is $e$ exactly\nwhen $p\\nmid A_M$.  Since $W$ is invertible modulo $p$, that is exactly the\nstated condition.\n\\end{proof}\n\nThe complete valuation identity is\n\\[\n v_p\\bigl(\\den(H_M)\\bigr)=\\max\\{0,e-v_p(A_M)\\}.\n\\]\nA zero residue in \\eqref{long68:eq:prime-pole-survival} therefore lowers\nthe exponent, whereas complete cancellation requires $v_p(A_M)\\ge e$.\nFor $e>1$, reduction modulo $p$ alone cannot distinguish partial from\ncomplete cancellation.  The proof uses no property of factorials beyond the\ndisplayed denominators, so it applies to any finite sum of reciprocals of positive integers.\nFor $d_n=n!-1$, the examples below show that complete cancellation can\nactually occur; the general valuation formula alone does not predict it.\n\n\\paragraph{Two complete cancellations.}\n\nA unique maximal exponent makes the residue sum a single nonzero term,\nso the condition holds automatically. With several maximal terms it can\nfail. The following examples show cancellation in the factorial sequence\nitself, not just in arbitrary rational sums.\nTake $M=p-1$. The table lists every index with $p\\mid d_n$, and every\none has valuation exactly one.\n\\begin{center}\n\\small\n\\begin{tabular}{rlll}\n\\toprule\n\\papertablehead\n$p$ & indices $n$ & $d_n/p\\pmod p$ & inverses modulo $p$\\\\\n\\midrule\n$139$ & $69,\\ 122,\\ 137$ & $6,\\ 49,\\ 73$ & $116,\\ 122,\\ 40$\\\\\n$2593$ & $349,\\ 2243,\\ 2591$ & $1508,\\ 1566,\\ 1678$ & $1367,\\ 356,\\ 870$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\nThe inverse sums are $278=2\\cdot139$ and $2593$.  By\nTheorem~\\ref{long68:res:prime-pole}, $139\\nmid\\den(H_{138})$ and\n$2593\\nmid\\den(H_{2592})$: each prime divides the common denominator of its\npartial sum and vanishes from its reduced denominator. The listed indices\nand their valuations are checked by the recurrence\n\\[\n r_1=1,\\qquad r_n\\equiv nr_{n-1}\\pmod{p^2},\\qquad 0\\le r_n<p^2,\n\\]\nrun through $n=p-1$. At an index with $r_n\\equiv1\\pmod p$, the\ncofactor is $(r_n-1)/p$ modulo $p$. Valuation at least two would give\n$r_n=1$, which occurs for neither prime among the tested indices\n$2\\le n\\le p-1$.  That enumeration is a finite calculation\nseparate from the theorem.\n\nThe recurrence is run modulo $p^2$, not merely modulo $p$: reduction\nmodulo $p$ would find the indices with $p\\mid n!-1$, but would not\ndetermine whether $p^2\\mid n!-1$. This is why the zero inverse sums prove complete\ncancellation in these examples.\n\nFor each prime $p\\ge3$, the exponent $v_p(\\den(H_M))$ is constant for\n$M\\ge\\max(2,p-2)$. Wilson's theorem gives\n$(p-1)!-1\\equiv-2\\pmod p$, and $n!-1\\equiv-1\\pmod p$ for $n\\ge p$.\nHence every summand after index $p-2$ has denominator coprime to $p$. If a reduced fraction\n$a/b$ is followed by a summand $1/d$ with $p\\nmid d$, their sum has\nnumerator $ad+b$ over $bd$. When $p\\mid b$, this numerator is nonzero\nmodulo $p$, so the exponent of $p$ in the reduced denominator stays the\nsame. When $p\\nmid b$, the new denominator remains coprime to $p$.\nConsequently, the two cancellations persist in every later partial sum:\n\\[\n 139\\nmid\\den(H_M)\\quad(M\\ge137),\\qquad\n 2593\\nmid\\den(H_M)\\quad(M\\ge2591).\n\\]\nThe primes still divide the corresponding common denominators $L_M$.\nThese are assertions about the partial sums; they do not constrain a\nhypothetical denominator of their real limit $S$ without a further\napproximation estimate.\n\nWe next ask where a prime first divides a denominator $d_m$. This means\nit divides none of $d_2,\\ldots,d_{m-1}$; it may still divide a later\ndenominator, so uniqueness at its first occurrence gives no uniqueness\nin every larger block.\n\n\\begin{proposition}[cofinal first prime occurrences]\\label{long68:res:wilson-cofinality}\nFor every integer $B\\ge0$ there are a prime $q$ and an integer $m>B$ with\n$m<q$, $q\\mid m!-1$ and $\\gcd(q,k!-1)=1$ for every $k$ with $2\\le k<m$.\n\\end{proposition}\n\n\\begin{proof}\nChoose a prime $q\\ge B!+5$.  Wilson's theorem gives $(q-2)!\\equiv1\\pmod q$,\nso there is a least index $m\\ge2$ with $q\\mid m!-1$, and $m\\le q-2$.\nIf $m\\le B$, then\n$q\\le m!-1\\le B!-1$, contradicting the choice of $q$.  Hence $m>B$, and\nminimality gives the coprimality assertions.\n\\end{proof}\n\nThe construction proves cofinality of first occurrences, but not the\ninequality involving $q$ needed for the additional two-factor test in\n\\S\\ref{long68:sec:residue}. The sufficient tail inequality there does not\nrequire choosing a prime $q$. Wilson reflection limits what can be inferred\nfrom a large prime factor alone: for odd $2\\le n<q$ with $q\\mid n!-1$,\nthe identity $(q-1-n)!\\,n!\\equiv(-1)^{n+1}\\pmod q$ gives the reflected\ncongruence at $q-n-1$. It gives a second summand denominator only when\n$q\\ne2n+1$ and $n\\le q-3$. At the Wilson endpoint $n=q-2$, it instead\nreturns index $1$, where $1!-1=0$ is not a denominator of this series.  The reflection identity is classical; see Stewart\n\\cite[p.~462, (4)]{stewart2004}.\n\n\\section{Growth of the common denominator}\\label{long68:sec:lcm}\n\\papersectiontarget{r12-lcm}\n\nA large product of denominators gives a large least common multiple\nonly if their overlap is controlled. We use the last $k$ denominators:\ntheir logarithmic product contributes about $kN\\log N$, while the\npairwise-gcd estimate costs about $(k^3/6)\\log N$. With\n$k\\sim\\alpha\\sqrt N$, the leading coefficient is\n$\\alpha-\\alpha^3/6$, maximized at $\\alpha=\\sqrt2$.\nThe lemmas below make this balance rigorous and yield\n$\\liminf\\log\\Lc_N/(N^{3/2}\\log N)\\ge2\\sqrt2/3$.\nHere $\\Lc_N$ is the common denominator before reduction, not\n$\\den(H_N)$.\n\n\\begin{lemma}[product, least common multiple, pairwise gcd]\n\\label{long68:res:product-lcm}\nFor positive integers $x_1,\\ldots,x_k$,\n\\[\n \\prod_{i=1}^{k}x_i\\ \\Big|\\ \\lcm(x_1,\\ldots,x_k)\\prod_{i<j}\\gcd(x_i,x_j).\n\\]\n\\end{lemma}\n\n\\begin{proof}\nFor each prime $r$, arrange $a_i=v_r(x_i)$ in increasing order.\nThe valuation of the right-hand side is\n$a_k+\\sum_{i=1}^{k-1}(k-i)a_i$, whereas that of the left-hand side is\n$\\sum_{i=1}^k a_i$. Their difference is\n$\\sum_{i=1}^{k-1}(k-i-1)a_i\\ge0$. This proves divisibility at every prime.\n\\end{proof}\n\n\\noindent The next divisibility is the case $P=-1$ of the relation\n$\\gcd(i!+P(i),j!+P(j))\\mid (j!/i!)P(i)-P(j)$ for $P\\in\\Z[X]$, obtained by\nmultiplying $i!+P(i)$ by $j!/i!$ and subtracting $j!+P(j)$.  Luca and\nShparlinski use this relation to bound a common divisor of $n!+P(n)$ and\n$(n+h)!+P(n+h)$ in the proof of their Lemma~5~\\cite{luca-shparlinski}, and Lai\nuses it for a common prime-power divisor\n\\cite[proof of Lemma~2.4, display~(2.5)]{lai}.\n\nThe earlier spacing method of Erd\\H{o}s and Stewart\n\\cite[\\S3, pp.~516--517]{erdos-stewart1976} is related background.\nWe now state only the subtraction needed for the lcm argument, rather than\nimporting the prime-factor estimates of those papers.\n\n\\begin{lemma}[factorial-gap gcd]\\label{long68:res:gap-gcd}\nFor $2\\le i<j$, the integer $g=\\gcd(i!-1,j!-1)$ divides $j!/i!-1$, and\n$g\\le j!/i!-1<j^{\\,j-i}$.\n\\end{lemma}\n\n\\begin{proof}\nThe identity\n$(j!/i!)(i!-1)-(j!-1)=1-j!/i!$ shows that $g$ divides $j!/i!-1$.\nThis integer is positive. Moreover, $j!/i!$ is a product of $j-i$ integers\nat most $j$, so $g\\le j!/i!-1<j^{j-i}$.\n\\end{proof}\n\n\\begin{lemma}[segment inequality]\\label{long68:res:segment}\nFor $2\\le k\\le N-1$,\n\\begin{equation}\\label{long68:eq:segment}\n \\sum_{n=N-k+1}^{N}\\log(n!-1)\n \\ \\le\\ \\log \\Lc_N+\\binom{k+1}{3}\\log N.\n\\end{equation}\n\n\n\\end{lemma}\n\n\\begin{proof}\nApply Lemma~\\ref{long68:res:product-lcm} to the $k$ denominators with\nindices $N-k+1,\\ldots,N$. Their lcm divides $\\Lc_N$.\nLemma~\\ref{long68:res:gap-gcd} bounds the logarithm of each pairwise gcd\nby $(j-i)\\log N$. Since\n\\[\n \\sum_{N-k+1\\le i<j\\le N}(j-i)\n =\\sum_{h=1}^{k-1}h(k-h)=\\binom{k+1}{3},\n\\]\ntaking logarithms gives \\eqref{long68:eq:segment}.\n\\end{proof}\n\n\\begin{theorem}[common-denominator growth]\\label{long68:res:lcm-growth}\n\\[\n \\liminf_{N\\to\\infty}\\frac{\\log \\Lc_N}{N^{3/2}\\log N}\n \\ \\ge\\ \\frac{2\\sqrt2}{3}.\n\\]\n\\end{theorem}\n\n\\begin{proof}\nThe sum of the logarithms on a terminal block of length $k$ is of order\n$kN\\log N$, while the gcd estimate costs order $k^3\\log N$.\nWe therefore fix $\\alpha>0$ and take $k=\\lfloor\\alpha\\sqrt N\\rfloor$.\nFor large $N$ this is an admissible block length. If $u=N-k+1$, then\n$n!-1\\ge n!/2$ and $\\log n!\\ge n\\log n-n$ give\n\\[\n \\sum_{n=u}^N\\log(n!-1)\\ge k(u\\log u-u)-k\\log2.\n\\]\nHere $k=\\alpha\\sqrt N+O(1)$ and $u=N+O(\\sqrt N)$, so\n$ku\\log u=\\alpha N^{3/2}\\log N+O(N\\log N)$.\nThe remaining two terms are $O(N^{3/2})$. Also\n\\[\n \\binom{k+1}{3}\\log N\n =\\frac{\\alpha^3}{6}N^{3/2}\\log N+o(N^{3/2}\\log N).\n\\]\nAfter division by $N^{3/2}\\log N$, Lemma~\\ref{long68:res:segment}\ntherefore yields a lower limit of at least $\\alpha-\\alpha^3/6$.\nThis expression is maximised at $\\alpha=\\sqrt2$, giving $2\\sqrt2/3$.\n\\end{proof}\n\nThe displayed proof of the lcm bound uses only\nLemmas~\\ref{long68:res:product-lcm} and~\\ref{long68:res:gap-gcd}.\nTheorem~12 of Garaev, Luca and Shparlinski\n\\cite[arXiv v1, p.~16]{garaev-luca-shparlinski} supplies a different input:\na uniform $O(N^{2/3})$ bound for the number of solutions of\n$n!\\equiv a\\pmod p$, for fixed $a\\not\\equiv0\\pmod p$, in an interval of\nlength $N$ contained in $1\\le n<p$.  The weaker lcm deduction\nfrom it is retained in \\S\\ref{long68:sec:ext-superseded}, not used here.\nNo exhaustive priority claim is made for the lcm estimate;\nits precise elementary derivation and the inherited subtraction are stated\nso that the claim can be assessed without relying on a search conclusion.\n\n\\begin{remark}[polynomial shifts]\nFix $P\\in\\Z[X]\\smallsetminus\\{0\\}$.  By Lemma~3 of Luca and\nShparlinski~\\cite{luca-shparlinski}, in the form given by\nLai~\\cite[Lemma~2.1]{lai}, there is $n_0\\ge2$, depending only on $P$, such\nthat $n!+P(n)>1$ for all $n\\ge n_0$ and $P(n)(n+1)\\cdots(n+h)\\ne P(n+h)$ for\nall $n\\ge n_0$ and $h\\ge1$. The nonvanishing has a short direct proof.\nBeyond a cutoff, $P$ has constant nonzero sign and\n$P(n+1)/P(n)\\to1$, so $0<P(n+1)/P(n)<n+1$ for every sufficiently\nlarge $n$. Multiplying these inequalities gives\n\\[\n 0<\\frac{P(n+h)}{P(n)}<\\prod_{j=1}^{h}(n+j)\\qquad(h\\ge1).\n\\]\nThus the required subtraction is nonzero for every gap $h$, not merely\nfor each fixed $h$ after a separate cutoff. Increasing $n_0$ also makes\n$n!+P(n)>1$. The proof of Theorem~\\ref{long68:res:lcm-growth} then gives\n\\[\n \\liminf_{N\\to\\infty}\n \\frac{\\log\\lcm\\{n!+P(n):n_0\\le n\\le N\\}}{N^{3/2}\\log N}\n \\ \\ge\\ \\frac{2\\sqrt2}{3}.\n\\]\nFor $n_0\\le i<j\\le N$, multiplying $i!+P(i)$ by $j!/i!$ and subtracting\n$j!+P(j)$ gives\n\\[\n \\gcd\\bigl(i!+P(i),\\,j!+P(j)\\bigr)\\ \\Big|\\ \\frac{j!}{i!}P(i)-P(j),\n\\]\nwhich replaces Lemma~\\ref{long68:res:gap-gcd}.  The right-hand side is\nnonzero by the choice of $n_0$, and its absolute value is at most\n$C_PN^{\\,j-i+\\deg P}$ for a constant $C_P$.  Over a block of $k$ indices the\npairwise loss therefore grows by $O_P(k^2\\log N)$, which is $O_P(N\\log N)$\nwhen $k=\\lfloor\\alpha\\sqrt N\\rfloor$.  Since $n!+P(n)\\ge n!/2$ for large $n$,\nthe lower estimate for the sum of logarithms is unchanged.  The cutoff is\nneeded, since for $P=-2$ the factor at $n=2$ vanishes.  The hypothesis\n$P\\ne0$ is needed as well, since for $P=0$ the least common multiple is $N!$\nwhatever the cutoff.\n\\end{remark}\n\n\\paragraph{A non-polynomial comparison.}\nFor $n!+2^n-1$, Luca and Shparlinski\n\\cite[Lemmas~2.1--2.3, pp.~860--862]{luca-shparlinski-exp} eliminate the\nexponential term using three indices satisfying the congruence; multiplicative order enters the counting.\nTheir valuation-layer identity \\cite[(3.2), p.~863]{luca-shparlinski-exp}\nrecords repeated prime-power divisibility.  These are related tools, not\ninputs to Theorem~\\ref{long68:res:lcm-growth}, and no extension to general\nnon-polynomial perturbations is asserted.\n\nWithin the terminal-block estimate just proved, maximising\n$\\alpha-\\alpha^3/6$ gives $2\\sqrt2/3$.  This is an optimisation of that\none-parameter bound, not an optimality theorem for all block selections or\nall gcd arguments.  The discarded factorial term has size $N^{3/2}$,\nso a finite normalised ratio need not be close to the limiting lower bound.\n\nTheorem~\\ref{long68:res:lcm-growth} concerns $\\Lc_N$, not $\\den(H_N)$.\nTheorem~\\ref{long68:res:prime-pole} and the cancellations at $139$ and $2593$\nshow why these cannot be interchanged: a prime present in the common\ndenominator may be absent after reduction. The two examples establish this\npossibility, not an asymptotic estimate for cancellation.\n\nEven two consecutive denominators obstruct clearing by the full lcm. For\n$N\\ge3$, the gcd of $(N-1)!-1$ and $N!-1$ divides $N-1$, whereas\n$(N-1)!-1$ is coprime to $N-1$. Thus those denominators are coprime, and\n\\[\n\\Lc_N(\\Ser-H_N)>\n\\frac{((N-1)!-1)(N!-1)}{(N+1)!-1}\\longrightarrow\\infty.\n\\]\nThis argument does not use the stronger asymptotic growth theorem.\n\nReduction could still help, but only in the opposite direction: a\nclearing argument needs a sufficiently small reduced denominator.\nWrite $H_N=\\widehat A_N/Q_N$ in lowest terms, reserving $A_N$ for the\nnumerator over the full lcm. Under $\\Ser=a/q$, with $a\\in\\Z$ and\n$q\\ge1$, positivity gives\n\\[\n qQ_N(\\Ser-H_N)=aQ_N-q\\widehat A_N\\in\\Z_{>0}.\n\\]\nA contradiction requires $qQ_N(\\Ser-H_N)<1$ at some $N$. A lower\nbound on $Q_N$ cannot supply this upper estimate. For comparison, the\nrational limit $1$ and the approximants\n$1-1/Q_N$ have arbitrarily large reduced denominators without any\nirrationality.  The inequality $d_N(\\Ser-H_N)<1$ alone would not repair the\nargument, because $d_N$ generally does not clear the earlier summands of\n$H_N$.  The prime-power analysis describes which factors survive reduction,\nbut it does not yet supply the required upper control on $Q_N$.\n\nFor linear forms in $p$-adic zeta values, with $p\\ge5$ a fixed prime and $n$\nthe index of the form, Lai, Lupu and Sprang quantify a saving of this kind.\nThey bound the denominators of the coefficients by powers of\n$\\lcm(1,\\ldots,n)$, show that the bound may be divided by a product $\\Phi_n$\nof prime powers, and compute the growth rate of\n$\\Phi_n$~\\cite[Lemmas~5.3--5.7 and~7.3]{lai-lupu-sprang}.  That saving enters\nthe inequality~\\cite[(8.1)]{lai-lupu-sprang} under which the rescaled forms,\nwhich have integer coefficients, meet the irrationality criterion\n\\cite[Lemma~2.1, p.~3]{lai-lupu-sprang} that they quote from Lai\n\\cite[Lemma~2.1, p.~4]{lai-2adic}. Along one unbounded subsequence, the forms\nmust be nonzero and their $p$-adic absolute values, multiplied by the\nlargest ordinary absolute value of an integer coefficient, must tend to\nzero. The coefficient bound is archimedean; the bound on the value is\n$p$-adic. The conclusion is that at least one of the finitely many\n$p$-adic numbers in the forms is irrational, not that each is irrational.\nThe corresponding saving for $H_N$ is the ratio $\\Lc_N/Q_N$.\n\nThis ratio is an integer.  Merely naming it supplies neither its growth\nnor a small linear form; the analogy is about what an eventual estimate\nwould have to accomplish.\nTheorem~\\ref{long68:res:prime-pole} decides, for each prime $p\\mid\\Lc_N$,\nwhether $p$ divides this ratio; no asymptotic lower bound for the ratio is\nestablished here.  This paragraph compares the two methods; no $p$-adic\ntheorem is applied to $\\Ser$.\n\n\\subsection{Consequences for the support parameter}\\label{long68:sec:quantitative-support}\n\nThe lcm estimates also constrain the size of a cancellation vector\nwhen its multiplier is bounded by a factorial. At depth $D=2t^2$,\ncancellation gives $\\Lc_D\\mid M$; the next theorem additionally assumes\n$0<M<(R+1)!-1$. Its $R$ is an index, not the remainder or the modulus\n$R_p$ of \\S\\ref{long68:sec:residue}. The conclusion bounds the largest\nsupported index only when $R$ denotes that index. Taking $R$ to be a\nlater cutoff gives a bound on the cutoff instead.\n\n\\begin{samepage}\n\\label{long68:res:lead-channel-radius}\\phantomsection\n\\begin{theorem}[a lower bound for the support parameter]\\label{long68:res:channel-radius}\nLet $t,M,R\\in\\N$ satisfy\n\\[\n t\\ge2^{32},\\qquad M>0,\\qquad \\Lc_{2t^2}\\mid M,\\qquad M<(R+1)!-1 .\n\\]\nThen $3t^3<2(R+1)$.  Consequently no family satisfying these hypotheses for\nall sufficiently large $t$ has $(R(t)+1)/t^3\\le3/2$ eventually, and none has\n$R(t)=o(t^3)$.\n\n\\end{theorem}\n\\end{samepage}\n\n\\begin{proof}\nPut $D=2t^2$, $k=2t$ and $u=D-k+1=2t^2-2t+1$.  Lemma~\\ref{long68:res:segment} at\n$N=D$, with $n!-1\\ge n!/2$ and $\\log n!\\ge n\\log n-n$, and with\n$\\Lc_D\\le M<(R+1)!$, gives\n\\begin{equation}\\label{long68:eq:radius-log}\n 2t\\bigl(u\\log u-u-\\log2\\bigr)\n <(R+1)\\log(R+1)+\\binom{2t+1}{3}\\log(2t^2).\n\\end{equation}\n\n\nSuppose $R+1\\le\\tfrac32t^3$.  Since $t\\ge16$ one has\n$\\tfrac{15}{8}t^2\\le u\\le2t^2$ and $\\log u\\ge2\\log t$, so the left side of\n\\eqref{long68:eq:radius-log} is at least\n$\\tfrac{15}{2}t^3\\log t-4t^3-2t\\log2$.  Using\n$\\binom{2t+1}{3}<\\tfrac43t^3$ and $\\log\\tfrac32<\\log2$, the right side is at\nmost\n\\[\n \\tfrac32t^3\\bigl(\\log2+3\\log t\\bigr)+\\tfrac43t^3\\bigl(\\log2+2\\log t\\bigr).\n\\]\nDividing by $t^3$ and rearranging, \\eqref{long68:eq:radius-log} would require\n\\[\n \\tfrac13\\log t<4+\\frac{2\\log2}{t^2}+\\tfrac{17}6\\log2 .\n\\]\nBut $\\log t\\ge32\\log2$ and $\\tfrac23<\\log2<1$ give\n$\\tfrac13\\log t-\\tfrac{17}6\\log2\\ge\\tfrac{47}6\\log2>5$, while\n$4+2\\log2/t^2<5$.  This contradiction proves the finite inequality, and the\ntwo sequence conclusions follow.\n\\end{proof}\n\nThe hypotheses are compatible: take $M=\\Lc_{2t^2}$ and then choose\n$R$ large enough. The conclusion rules out subcubic $R$ under the\nstated size condition; it neither constructs a coefficient vector nor\nproves irrationality.\n\nTheorem~\\ref{long68:res:lcm-growth} raises the asymptotic constant in the same\nestimate.\n\n\\begin{corollary}[the asymptotic lower bound]\\label{long68:res:radius-constant}\nLet $M(t),R(t)$ satisfy $M(t)>0$, $\\Lc_{2t^2}\\mid M(t)$ and\n$M(t)<(R(t)+1)!-1$ for all sufficiently large $t$.  Then\n\\[\n \\liminf_{t\\to\\infty}\\frac{R(t)+1}{t^3}\\ \\ge\\ \\frac{16}{9}.\n\\]\n\\end{corollary}\n\n\\begin{proof}\nPut $r=R(t)+1$, so $\\Lc_{2t^2}\\le M(t)<r!$ and $\\log \\Lc_{2t^2}<r\\log r$.\nSince\n\\[\n\\frac{(2t^2)^{3/2}\\log(2t^2)}{t^3\\log t}\\longrightarrow4\\sqrt2,\n\\]\nTheorem~\\ref{long68:res:lcm-growth} gives, for every $\\varepsilon>0$ and\nall sufficiently large $t$,\n\\[\n \\log \\Lc_{2t^2}\\ge\n \\Bigl(\\tfrac{16}{3}-\\varepsilon\\Bigr)t^3\\log t.\n\\]\nSuppose $r\\le ct^3$ for arbitrarily large $t$, with $0<c<16/9$ fixed.  On that\nsubsequence $r\\log r\\le ct^3(3\\log t+\\log c)=\\bigl(3c+o(1)\\bigr)t^3\\log t$.\nChoosing $0<\\varepsilon<16/3-3c$ contradicts the lower bound for\nsufficiently large $t$.\n\\end{proof}\n\nThis corollary does not assert the strict inequality\n$R+1>\\tfrac{16}9t^3$ eventually. Indeed, the finite logarithmic constraint\n\\eqref{long68:eq:radius-log} allows equality at that constant. To check\nthis directly, set $R+1=16t^3/9$ with $t\\ge4$. Since\n$u=2t^2-2t+1\\le2t^2$ and $x\\log x-x$ is increasing for $x\\ge1$, the\nleft side of \\eqref{long68:eq:radius-log} is at most\n$4t^3(\\log(2t^2)-1)-2t\\log2$. Subtracting this bound from the right side gives\n\\[\n t^3\\left(4+\\frac{16}{9}\\log\\frac{16}{9}-\\frac83\\log2\\right)\n +t\\left(\\frac53\\log2-\\frac23\\log t\\right)\n >\\frac43t^3-\\frac23t^2>0,\n\\]\nwhere $\\log2<1$ and $\\log t\\le t$ suffice for the last comparison.\nInteger pairs with $9(R+1)=16t^3$ exist for every multiple $t\\ge6$ of $3$.\nThis verifies the numerical constraint, not the existence of a coefficient\nvector with that support. The asymptotic assertion in\nCorollary~\\ref{long68:res:radius-constant} instead follows from\nTheorem~\\ref{long68:res:lcm-growth}.\n\n\\section{Carries and factorial digits}\\label{long68:sec:carry}\n\\papersectiontarget{r5-carry}\n\nInstead of clearing the partial sum, we now track the only integer\nthat could equal $m!S$. Under rationality, $m!S$ is integral for all\nlarge $m$; since the positive tail is less than one, it must be the\nstrict successor of $m!H_m$. The recurrence for these successors\ntherefore tests rationality without assuming that $m!$ clears $H_m$. Put\n\\[\n Z_m=\\lfloor m!H_m\\rfloor+1,\\qquad\n \\Delta_m=Z_{m-1}-(m-1)!H_{m-1}\\quad(m\\ge3).\n\\]\nThus $0<\\Delta_m\\le1$, with $\\Delta_m=1$ when the preceding\nscaled partial sum is integral. Define the integer carry $b_m$ by\n\\begin{equation}\\label{long68:eq:carry-recurrence}\n Z_m=mZ_{m-1}+1-b_m .\n\\end{equation}\nFor example, $2!H_2=2$ and $3!H_3=36/5$, so $Z_2=3$ and\n$Z_3=8$, giving $b_3=3\\cdot3+1-8=2$. We use\n$\\lfloor x\\rfloor+1$, not $\\lceil x\\rceil$: at the integral value\n$2!H_2=2$, the required strict successor is $3$.\n\nCall $b_m=1$ a \\emph{unit carry}.  Put\n$E_m=m!(\\Ser-H_m)$ and $\\varepsilon_m=1/(m!-1)$.  Since\n$d_{n+1}>(n+1)d_n$, a geometric majorant gives the tail estimate\n\\begin{equation}\\label{long68:eq:tail-bound}\n 0<E_m<\\frac{2\\,m!}{(m+1)!-1}<\\frac2m\\le1\\qquad(m\\ge2).\n\\end{equation}\nCompare the factorial scaling in Han\\v{c}l and Tijdeman's tail-integrality\nlemma for factorial series with integer coefficients\n\\cite[Lemma~2.1 and the following remark,\np.~385]{hancl-tijdeman}. Their scaled partial sums are integers; ours\nneed not be, since already $3!H_3=36/5$. The next proof instead uses the\nshort positive tail to identify the least integer above a scaled partial sum\nunder a rationality assumption. The carry-defect expansion in\n\\S\\ref{long68:sec:ext-literature} gives a separate, direct application of\nfactorial-series rationality criteria.\n\n\\label{long68:res:lead-carry-equivalence}\\phantomsection\n\\label{long68:res:strict-successor-complete-characterization}\\phantomsection\n\\begin{theorem}[an exact criterion from successive partial sums]\\label{long68:res:carry-equivalence}\nFor $m\\ge3$,\n\\begin{equation}\\label{long68:eq:unit-window}\n b_m=1\n \\iff m\\mid Z_m\n \\iff 1+\\varepsilon_m<m\\Delta_m\\le2+\\varepsilon_m .\n\\end{equation}\n\n\nMoreover\n\\begin{align}\n \\Ser\\in\\Q\n &\\iff b_m=1\\ \\hbox{for all sufficiently large }m,\n \\label{long68:eq:carry-rationality}\\\\\n \\Ser\\notin\\Q\n &\\iff \\forall B\\ \\exists m>B:\\ m\\nmid Z_m .\n \\label{long68:eq:strict-misses}\n\\end{align}\nIf $\\Ser=a/q$ with $a\\in\\Z$, $q\\ge1$ and $b_m\\ne1$, then $q\\nmid(m-1)!$ and\n$q\\ge m$.\n\\end{theorem}\n\n\\begin{proof}\nFrom $m!H_m=mZ_{m-1}-m\\Delta_m+1+\\varepsilon_m$ one gets\n$b_m=\\lceil m\\Delta_m-1-\\varepsilon_m\\rceil$ with $-1\\le b_m\\le m-1$, which\ngives both equivalences in \\eqref{long68:eq:unit-window}, endpoints included.\n\nSuppose $\\Ser=a/q$ and $q\\mid(m-1)!$.  For $j=m-1$ and $j=m$ the number\n$j!\\,\\Ser$ is an integer, and $j!H_j=j!\\,\\Ser-E_j$ with $0<E_j<1$ by\n\\eqref{long68:eq:tail-bound}, so $Z_j=j!\\,\\Ser$.  Substituting into\n\\eqref{long68:eq:carry-recurrence} forces $b_m=1$.  Every fixed $q$ divides\n$(m-1)!$ eventually, which gives one direction of\n\\eqref{long68:eq:carry-rationality} and, by contraposition, the final assertion;\n$q<m$ would imply $q\\mid(m-1)!$.  Conversely, an eventual unit-carry tail\nmakes $Z_m/m!$ eventually constant, and $0<Z_m/m!-H_m\\le1/m!$ with\n$H_m\\to\\Ser$ identifies that constant as $\\Ser$, which is then rational.\nNegating \\eqref{long68:eq:carry-rationality} gives \\eqref{long68:eq:strict-misses}.\n\\end{proof}\n\nThe conclusion $q\\nmid(m-1)!$ is about prime-power multiplicities, not\njust prime factors. It permits all prime factors of $q$ to be at most\n$m-1$ if one occurs to a larger exponent than in $(m-1)!$.\n\n\\subsection{The adjacent-unit window condition}\n\\label{long68:sec:adjacent-unit-no-go}\n\nTwo successive unit carries imply a prime-power size condition after\nclearing denominators in their defining inequalities. The calculation\nbelow shows that the condition follows already from those carries.\nAn additional arithmetic estimate is therefore needed to turn it into\na contradiction.\n\nWrite $\\Delta_k=P^\\Delta_k/Q^\\Delta_k$ in lowest terms, with $P^\\Delta_k,Q^\\Delta_k>0$.\nIn one recurrence step the unreduced denominator is $Q^\\Delta_k(k!-1)$;\nlet $G_k=Q^\\Delta_k(k!-1)/Q^\\Delta_{k+1}$ be the positive integer cancelled in reduction.\nFor $m\\ge3$, put\n\\[\n \\begin{aligned}\n D_m&=Q^\\Delta_m(m!-1)((m+1)!-1),\\\\\n \\Omega_m&=D_m\\left(m(m+1)\\Delta_m-m-2\n       -\\frac{m+1}{m!-1}-\\frac1{(m+1)!-1}\\right).\n \\end{aligned}\n\\]\nBoth are integers. The two unit carries are equivalent to\n$0<\\Omega_m\\le D_m$. Independently of the carries, cancellation in two\nsuccessive steps gives\n\\[\n D_m=Q^\\Delta_{m+2}G_{m+1}G_m=Q^\\Delta_m(m!-1)\\bigl((m+1)!-1\\bigr),\n\\]\nand under the pair assumption the offset factors to match,\n$\\Omega_m=P^\\Delta_{m+2}G_{m+1}G_m$.  Dividing by $G_{m+1}G_m>0$ reduces the proposed bound to\n\\[\n 0<P^\\Delta_{m+2}\\le Q^\\Delta_{m+2},\n\\]\nwhich every reduced positive gap in $(0,1]$ already satisfies.  Thus the resulting gap bound is not an additional restriction on a\npair of unit carries. This calculation does not exclude an independent\narithmetic obstruction involving the transition numerators or their\nvaluations. Such an obstruction would need information not already\nimplied by the pair assumption; it need not take the form of a bound on\nthe preceding gap $\\Delta_m$.\n\n\\subsection{Factorial digits of \\texorpdfstring{$S-e+2$}{S-e+2}}\\label{long68:sec:companion-digits}\n\\papersectiontarget{r5-digits}\n\nThe small perturbation from the reciprocal-factorial series can also\nbe isolated term by term. Subtracting $e-2$ leaves the absolutely\nconvergent correction $C=\\sum_{n\\ge2}\\bigl(n!(n!-1)\\bigr)^{-1}$.\nThe identity $1/(n!-1)=1/n!+1/(n!(n!-1))$ gives\n\\begin{equation}\\label{long68:eq:companion-decomposition}\n \\Ser=C+e-2 .\n\\end{equation}\nThe canonical factorial digits of a real $x$ are\n$a_m(x)=\\lfloor m!x\\rfloor-m\\lfloor(m-1)!x\\rfloor\\in\\{0,\\ldots,m-1\\}$\nfor $m\\ge2$. At a rational endpoint, the floor convention selects the\nterminating representation. For example, $1/2=1/2!$ also equals\n$\\sum_{m\\ge3}(m-1)/m!$, but its canonical digits are $a_2=1$ and\n$a_m=0$ for $m\\ge3$, not an eventually maximal tail. With this convention,\n$x$ is rational exactly when its digits vanish from some index on. The criterion\ngoes back to Cantor \\cite{cantor1869}; Galambos treats the rationality of\nCantor series in \\cite[Ch.~II, \\S2.1, pp.~21--22]{galambos1976}, and Koepf\nand Schmersau prove the irrationality direction for nonterminating factorial\nexpansions whose digits are not eventually maximal\n\\cite[Example~3.2, p.~121]{koepf-schmersau}.\n\n\\label{long68:res:lead-companion-orbit}\\phantomsection\n\\begin{proposition}[rationality and factorial residues]\\label{long68:res:companion-orbit}\n\\[\n \\Ser\\in\\Q\n \\quad\\Longleftrightarrow\\quad\n \\lfloor m!C\\rfloor\\equiv-2\\pmod m\n \\quad\\hbox{for all sufficiently large }m.\n\\]\n\\end{proposition}\n\n\\begin{proof}\nPut $J_m=\\sum_{k=0}^{m}m!/k!$.  Then $J_m$ is an integer, every summand with\n$k<m$ is divisible by $m$ and the summand at $k=m$ is $1$, so\n$J_m\\equiv1\\pmod m$; and $0<m!\\,e-J_m<1$ for $m\\ge2$.\n\nSuppose $\\Ser$ is rational.  For all large $m$ both $(m-1)!\\,\\Ser$ and\n$m!\\,\\Ser$ are integers, so $m\\mid m!\\,\\Ser$.  Multiplying\n\\eqref{long68:eq:companion-decomposition} by $m!$ and using\n$\\lfloor -m!\\,e\\rfloor=-J_m-1$ gives\n$\\lfloor m!C\\rfloor=m!\\,\\Ser+2\\,m!-J_m-1\\equiv-2\\pmod m$.\n\nConversely, assume the congruence from some index on.  Because\n$0\\le a_m(C)<m$, for $m\\ge3$ the congruence is equivalent to $a_m(C)=m-2$.\nChoose $N$ beyond the exceptional indices.  The canonical expansion gives\n\\[\n C=\\frac{\\lfloor N!C\\rfloor}{N!}+\\sum_{m>N}\\frac{m-2}{m!},\n\\]\nand the telescope $\\sum_{m>N}(m-1)/m!=1/N!$ turns this into\n\\[\n \\Ser=C+e-2\n =\\frac{\\lfloor N!C\\rfloor+1}{N!}+\\sum_{m=2}^{N}\\frac1{m!},\n\\]\nwhich is rational.\n\\end{proof}\n\nThe proof applies to every real $C$, not just this series: eventual\ndigits $a_m(C)=m-2$ characterize rationality of $C+e-2$. Thus $C=2-e$\nsatisfies the eventual pattern, whereas $C=0$ does not. For our\nparticular positive $C$, neither possibility is decided by a finite\ndigit list.\n\nNegating the eventual congruence gives the equivalent formulation\n\\begin{equation}\n S\\notin\\mathbb Q\\quad\\Longleftrightarrow\\quad\n (\\forall B)(\\exists m>B)\\ (\\lfloor m!C\\rfloor+2)\\bmod m\\ne0.\n \\label{long68:eq:companion-cofinal}\n\\end{equation}\nIt remains to prove failures of this congruence at arbitrarily large\nindices for this particular $C$. A finite list of failures does not\nsettle this question, and the equivalence gives no distribution estimate.\n\n\\paragraph{Recovering the finite carries.}\\label{long68:sec:floor-correction}\nTo compare canonical digits with the carries, we must account for the\ntail removed before taking a floor. Set\n\\[\n \\delta_m=m!\\sum_{n>m}\\frac1{n!(n!-1)},\\qquad\n \\sigma_m=\\mathbf1_{\\{\\{m!C\\}<\\delta_m\\}}.\n\\]\nHere $\\{x\\}=x-\\lfloor x\\rfloor$. The factorial-tail bound gives\n$0<\\delta_m<1/((m+1)!-1)<1$. Subtracting $\\delta_m$ from $m!C$\nlowers its floor by one exactly when $\\{m!C\\}<\\delta_m$. Therefore\n\\[\n Z_m=m!\\sum_{n=2}^m\\frac1{n!}+\\lfloor m!C\\rfloor+1-\\sigma_m.\n\\]\nSubstitution in \\eqref{long68:eq:carry-recurrence} gives\n\\begin{equation}\n b_m=m-1-a_m(C)+\\sigma_m-m\\sigma_{m-1}\\qquad(m\\ge3).\n \\label{long68:eq:companion-wrap}\n\\end{equation}\nWhen $\\{m!C\\}=\\delta_m$, the scaled partial sum is integral and $\\sigma_m=0$.\nFor a smaller fractional part, subtracting even this small tail crosses\nan integer. Tail size alone therefore permits neither correction term\nto be omitted.\n\n\\subsection{Escape from a smaller interval}\n\nWrite $\\theta_m=\\{m!\\,\\Ser\\}$. The next criterion compares a\nfractional part with the actual tail: $\\theta_{m-1}$ must leave\n$[0,E_m/m)$ at arbitrarily large indices. The interval has width\nless than $2/m^2$. Since $E_m$ contains $S$, this equivalence is not\nyet a finite test; the second assertion supplies a sufficient test\nusing the finite quantities $\\Delta_m$ and $\\varepsilon_m$.\n\n\\begin{proposition}[lower-interval criterion]\\label{long68:res:lower-escape}\n\\begin{equation}\\label{long68:eq:lower-escape}\n \\Ser\\notin\\Q\n \\quad\\Longleftrightarrow\\quad\n \\forall B\\ \\exists m>B:\\ E_m\\le m\\theta_{m-1}.\n\\end{equation}\nFor $m\\ge3$ the finite condition\n\\begin{equation}\\label{long68:eq:finite-escape}\n m\\Delta_m\\le1+\\varepsilon_m\n \\quad\\hbox{or}\\quad\n 1+\\varepsilon_m+\\frac2m\\le m\\Delta_m\n\\end{equation}\nimplies the escape inequality in \\eqref{long68:eq:lower-escape}.  Cofinally many\ninstances of \\eqref{long68:eq:finite-escape} therefore imply $\\Ser\\notin\\Q$.\n\\end{proposition}\n\n\\begin{proof}\nIf $\\Ser$ is rational, the remainders $\\theta_{m-1}$ vanish eventually while\n$E_m>0$, so the escape inequality fails eventually.  Conversely, suppose\n$m\\theta_{m-1}<E_m$ for every sufficiently large $m$.  By\n\\eqref{long68:eq:tail-bound} this gives $m\\theta_{m-1}<1$, so $a_m(\\Ser)=0$ and\n$\\theta_m=m\\theta_{m-1}$. Fix $N$ after this recurrence begins.\nThen $\\theta_{N+k}=((N+k)!/N!)\\theta_N<1$ for every $k\\ge0$, forcing\n$\\theta_N=0$. Hence $N!\\Ser$ is an integer and $\\Ser$ is rational.\n\nFor the finite implication, suppose $m\\theta_{m-1}<E_m$.  The tail recurrence\nis $mE_{m-1}=1+\\varepsilon_m+E_m$, so $E_{m-1}>E_m/m>\\theta_{m-1}\\ge0$, and\nwith $0<E_{m-1}<1$ the strict-successor identity gives\n$\\Delta_m=E_{m-1}-\\theta_{m-1}$.  Hence\n\\[\n 1+\\varepsilon_m<m\\Delta_m=1+\\varepsilon_m+E_m-m\\theta_{m-1}\n \\le1+\\varepsilon_m+E_m<1+\\varepsilon_m+\\frac2m,\n\\]\nwhich contradicts \\eqref{long68:eq:finite-escape}.\n\\end{proof}\n\nThere are two comparisons to keep separate: a finite sufficient test\nfor escape at one index, and an equivalence when indices are unbounded.\nThe finite test excludes only the open interval\n$1+\\varepsilon_m<m\\Delta_m<1+\\varepsilon_m+2/m$; it accepts either endpoint\nand all values outside. The width $2/m$ refers to $m\\Delta_m$; the interval\nfor $\\Delta_m$ has width $2/m^2$. At one index this is a sufficient test\nfor escape, obtained by replacing $E_m$ with the upper bound $2/m$.\n\nAt arbitrarily large integer indices, however, the finite test itself\ncharacterises irrationality:\n\\[\n S\\notin\\Q\\quad\\Longleftrightarrow\\quad\n \\text{\\eqref{long68:eq:finite-escape} holds for arbitrarily large integers }m.\n\\]\nOnly the forward implication remains to be shown. If $S$ is irrational,\nTheorem~\\ref{long68:res:carry-equivalence} gives $b_m\\ne1$ at arbitrarily\nlarge indices. By \\eqref{long68:eq:unit-window}, each such index satisfies\n$m\\Delta_m\\le1+\\varepsilon_m$ or $m\\Delta_m>2+\\varepsilon_m$.\nSince $2/m<1$ for $m\\ge3$, either alternative implies\n\\eqref{long68:eq:finite-escape}. The converse is Proposition~\\ref{long68:res:lower-escape}.\nThis combines the two criteria; it does not prove either unbounded\noccurrence for $S$. The equivalence is over all integer indices. A\nrestriction to primes gives only a sufficient condition here.\n\nFor comparison at a single index, the exact classification is\n\\begin{equation}\\label{long68:eq:escape-classification}\n E_m\\le m\\theta_{m-1}\n \\quad\\Longleftrightarrow\\quad\n b_m\\ne1\\ \\hbox{ or }\\ a_m(\\Ser)=m-1\n \\qquad(m\\ge3).\n\\end{equation}\n\nTo verify the classification, use $0<E_j<1$ to write\n\\[\n Z_j=\\lfloor j!S\\rfloor+\\mathbf1_{\\{\\theta_j\\ge E_j\\}}.\n\\]\nSubstitution in the carry recurrence gives\n\\[\n b_m=1-a_m(S)+m\\mathbf1_{\\{\\theta_{m-1}\\ge E_{m-1}\\}}\n                 -\\mathbf1_{\\{\\theta_m\\ge E_m\\}}.\n\\]\nSince $0\\le a_m(S)\\le m-1$, a unit carry has exactly two possibilities:\nboth indicators are zero and $a_m(S)=0$, or both are one and\n$a_m(S)=m-1$. In the first case $m\\theta_{m-1}=\\theta_m<E_m$;\nin the second, $m\\theta_{m-1}\\ge m-1>E_m$.\nConversely, $m\\theta_{m-1}<E_m<1$ forces $a_m(S)=0$ and both\nindicators to be zero, using\n$mE_{m-1}=1+\\varepsilon_m+E_m$. It therefore forces a unit carry with\nzero digit. This proves \\eqref{long68:eq:escape-classification}, including\nthe equality case in its left-hand inequality.\n\nThus escape can occur even when $b_m=1$, provided the digit is $m-1$.\nThis happens at $m=52$: exact\nrational arithmetic gives $b_{52}=1$ together with\n$52\\,\\Delta_{52}>1+\\varepsilon_{52}+2/52$, the margin being about\n$0.5689674908$.  This example separates the two events at a single index: the finite test\ncan hold at a unit carry. Their occurrence at arbitrarily large indices\nnevertheless gives equivalent rationality criteria. Since $52$ is\ncomposite, the example supplies no prime-index instance.\n\n\\subsection{The series with denominators \\texorpdfstring{$n!+t$}{n!+t}}\n\nThe digit argument also applies to the other shifts mentioned by\nErd\\H{o}s. For an integer $t\\ge-1$, put\n$\\Ser_t=\\sum_{n\\ge2}1/(n!+t)$ and $C_t=\\sum_{n\\ge2}1/\\bigl(n!(n!+t)\\bigr)$,\nso that $\\Ser_t=-tC_t+(e-2)$.\n\nThe restriction $t\\ge-1$ keeps every denominator positive for $n\\ge2$;\n$t=-2$ is excluded because its first denominator is zero. All series in\nthis identity converge absolutely. The familiar case $t=0$ is $e-2$;\nthis example will make the residue condition explicit.\n\n\\begin{theorem}[a criterion for the shifts $t\\ge-1$]\\label{long68:res:shift-family}\nFor every integer $t\\ge-1$, the series $\\Ser_t$ is rational exactly when\n\\[\n \\bigl\\lceil t\\,m!\\,C_t\\bigr\\rceil\\equiv2\\pmod m\n\\]\nfor all sufficiently large $m$, and irrational exactly when that residue is\nmissed cofinally.  The member $t=-1$ is $\\Ser$, and the member $t=0$ is\n$e-2$, for which the scaled correction is always $0$ and hence misses the residue class\nat every $m\\ge3$, proving the irrationality of $e$.\n\\end{theorem}\n\n\\begin{proof}\nPut $Y=-tC_t$, so that $\\Ser_t=Y+e-2$.  The proof of\nProposition~\\ref{long68:res:companion-orbit} applies to any real $Y$:\nrationality of $Y+e-2$ forces\n$\\lfloor m!Y\\rfloor\\equiv-2\\pmod m$ eventually.  Conversely, that congruence\nforces the canonical digits of $Y$ to equal $m-2$ eventually; adding the\nfactorial series of $e-2$ leaves an eventually telescoping tail, hence a\nrational sum.  Finally\n$\\lceil t m!C_t\\rceil=-\\lfloor m!Y\\rfloor$ translates the congruence.\nNegation gives the cofinal statement.  For $t=0$, the left side is zero,\nwhich is not congruent to $2$ for $m\\ge3$.\n\\end{proof}\n\nThe $t=0$ specialisation recovers the classical irrationality of $e$.\nFor $t\\ne0$, no estimate proving failure of the displayed congruence at\narbitrarily large indices is supplied here.\nThis is a statement about the scope of the present argument, not an assertion\nthat every other member has the same current literature status.\n\n\\section{A complementary-denominator criterion}\\label{long68:sec:residue}\n\\papersectiontarget{r5-complement}\n\nWe seek a multiplier with two properties: it absorbs each fixed\nrational denominator eventually, and the scaled partial sum has a\ncomputable gap to the next integer. Include a factorial to ensure the\nfirst property. For the second, the multiplier also absorbs every prime-power\nlevel shared by two summand denominators. Any exponent left in the\ndenominator then comes from a unique maximal occurrence, so it\ncannot cancel on reduction. This multiplier\nis sufficient, not necessarily minimal. For an integer $p\\ge3$, not\nnecessarily prime, put\n$I_p=\\{2,\\ldots,2p-1\\}$ and $F_p=(p-1)!$, and define\n\\begin{align*}\n D_p&=\\lcm_{\\substack{i,j\\in I_p\\\\ i<j}}\\gcd(d_i,d_j),\n &C_p&=\\lcm(F_p,D_p),\\\\\n \\Lc^{\\mathrm{blk}}_p&=\\lcm(F_p,d_2,\\ldots,d_{2p-1}),\n &R_p&=\\Lc^{\\mathrm{blk}}_p/C_p .\n\\end{align*}\nPrime by prime, $D_p$ records the second-largest exponent, $C_p$\nraises it to at least the exponent in $F_p$, and $R_p$ records what\nremains below the largest exponent.\nThe factors $C_p$ and $R_p$ need not be coprime: a prime absent from $F_p$\nwith largest exponent $5$ and second-largest exponent $2$ occurs to powers\n$2$ and $3$ in the two factors. This is an illustration of the valuation\ncalculation, not a claimed pattern in a particular factorial block.\nIt will be useful to retain the ratio $\\widetilde C_p=C_p/F_p$, so that\n\\begin{equation}\\label{long68:eq:core-normalisation}\n \\Lc^{\\mathrm{blk}}_p=C_pR_p=F_p\\widetilde C_pR_p .\n\\end{equation}\n\n\nIncluding $F_p$ in both $C_p$ and $\\Lc_p^{\\mathrm{blk}}$ ensures that\na fixed rational denominator divides $C_p$ for all large $p$.\nThe ratio $\\widetilde C_p$ removes precisely that factorial factor when\nwe estimate the shared part. Put\n\\[\n T_p=\\sum_{i\\in I_p}\\frac{\\Lc^{\\mathrm{blk}}_p}{d_i},\\qquad\n \\rho_p=(-T_p)\\bmod R_p,\\qquad\n K_p=2p^2(2p-1)! ,\n\\]\nwith $\\rho_p$ the least nonnegative representative.  All of these depend only\non a finite partial sum.\n\nFor each prime, $C_p$ contains the larger of its exponent in $F_p$ and\nits second-largest exponent among the denominators.  A prime dividing $R_p$ therefore has a unique\ndenominator whose exponent is maximal and exceeds the exponent in $F_p$, so the argument of\nTheorem~\\ref{long68:res:prime-pole} applies to it and gives $\\gcd(T_p,R_p)=1$.\nThus $R_p=\\den(C_pH_{2p-1})$, the reduced denominator of the scaled\npartial sum, not generally of $H_{2p-1}$. When $R_p>1$, this ratio is\nnonintegral and\n\\begin{equation}\\label{long68:eq:gap-normalisation}\n \\frac{\\rho_p}{R_p}\n =\\bigl\\lceil C_pH_{2p-1}\\bigr\\rceil-C_pH_{2p-1}\\in(0,1),\n\\end{equation}\nthe distance to the next integer. When $R_p=1$, $\\rho_p=0$ but the\nscaled partial sum is integral, so the least strictly larger integer is at\ndistance $1$. The theorem below uses only $R_p>1$.\n\nAt $p=3$, the denominators $1,5,23,119$ are pairwise coprime. Thus $D_3=1$,\n$C_3=2$, $R_3=13685$, and\n\\[\nC_3H_5=\\frac{34264}{13685},\\qquad\n\\frac{\\rho_3}{R_3}=3-\\frac{34264}{13685}=\\frac{6791}{13685}.\n\\]\nThe tail estimate in the proof below gives\n$C_3(\\Ser-H_5)<7/1080<6791/13685$. This verifies the comparison for one\npartial sum. The theorem needs this comparison at arbitrarily large $p$,\nso that $F_p$ can absorb any fixed rational denominator.\n\n\\begin{theorem}[a sufficient tail inequality]\\label{long68:res:global-residue}\nSuppose that for every $B$ there is a natural parameter $p\\ge3$ with\n$p>B$, $R_p>1$, and\n\\begin{equation}\\label{long68:eq:global-scale}\n (2p+1)\\Lc^{\\mathrm{blk}}_p<K_p\\rho_p .\n\\end{equation}\nThen $\\Ser$ is irrational.\n\\end{theorem}\n\n\\begin{proof}\nSuppose $\\Ser=a/q$ with $q\\ge1$, and choose a parameter in the hypothesis\nwith $p>q$. Since $q\\mid F_p\\mid C_p$, the number $C_p\\Ser$ is an integer\nstrictly above $C_pH_{2p-1}$. The gap identity\n\\eqref{long68:eq:gap-normalisation} gives\n\\[\n C_p(\\Ser-H_{2p-1})\\ge\\frac{\\rho_p}{R_p}.\n\\]\nOn the other hand, $1/(n!-1)<2/n!$ for $n\\ge2p$, and comparison with a\ngeometric series of ratio $1/(2p+1)$ gives\n\\[\n \\Ser-H_{2p-1}<\\frac{2}{(2p)!}\\frac{2p+1}{2p}=\\frac{2p+1}{K_p}.\n\\]\nMultiplication by $C_p$ and \\eqref{long68:eq:global-scale}, with\n$\\Lc_p^{\\mathrm{blk}}=C_pR_p$, now give\n$C_p(\\Ser-H_{2p-1})<\\rho_p/R_p$, a contradiction.\n\\end{proof}\n\nCancelling $R_p$ in \\eqref{long68:eq:global-scale} through\n\\eqref{long68:eq:core-normalisation} displays the quantitative issue directly:\n\\begin{equation}\\label{long68:eq:core-gap-scale}\n (2p+1)C_p<K_p\\frac{\\rho_p}{R_p}.\n\\end{equation}\nThe factor $R_p$ has cancelled. Thus new prime divisors, even large ones,\ndo not by themselves bound $C_p$ or keep $\\rho_p/R_p$ away from zero.\nProposition~\\ref{long68:res:wilson-cofinality} supplies the primes but not\nthese two estimates.\n\nUnder $R_p>1$, we have $\\rho_p>0$, and taking logarithms gives\n\\[\n \\log\\widetilde C_p-\\log\\frac{\\rho_p}{R_p}\n <\\log\\frac{2p^2(2p-1)!}{(2p+1)(p-1)!}\n =p\\log p+(2\\log2-1)p+O(\\log p).\n\\]\nBoth terms on the left must be controlled at the same $p$. The elementary\nbound $\\rho_p\\ge1$ gives only $\\rho_p/R_p\\ge1/R_p$, which may be too small.\nFor a concrete comparison, a gap of at least $1/2$ would reduce the sufficient\ntest to $(2p+1)C_p<K_p/2$. At the opposite extreme, the gap $1/R_p$ requires\nthe stronger bound $(2p+1)C_p<K_p/R_p$. Neither type of favourable behaviour\nis proved here for an unbounded family of these particular prefixes.\n\n\\label{long68:res:lead-moving-factor-split}\\phantomsection\n\\label{long68:res:moving-factor-scale-split}\\phantomsection\nA more restrictive test also uses a prime $q$ that first divides $m!-1$\nat an index $m\\ge4$. Choose $p=\\lfloor m/2\\rfloor+1$, which need not be\nprime. This choice puts $m$ in the block and gives $q\\mid R_p$. Indeed,\n$q>m\\ge p$, so $q\\nmid F_p$, and no earlier denominator contains $q$.\nThe only possible later index in the block is $m+1$; when it occurs,\n$(m+1)!-1\\equiv m\\not\\equiv0\\pmod q$. Thus $q$ occurs in just one block\ndenominator and is absent from $C_p$.\n\nThe additional comparison uses the integer $R_p/q$. Requiring the tail\nbound to be smaller than both $\\rho_p$ and $R_p/q$ gives\n$\\min(\\rho_p,R_p/q)$ and is equivalent to the following two inequalities:\n\\begin{equation}\\label{long68:eq:factor-split}\n (2p+1)\\Lc^{\\mathrm{blk}}_p<K_p\\min(\\rho_p,R_p/q)\n \\quad\\Longleftrightarrow\\quad\n \\begin{cases}\n (2p+1)\\Lc^{\\mathrm{blk}}_p<K_p\\rho_p,\\\\\n (2p+1)C_pq<K_p .\n \\end{cases}\n\\end{equation}\n\n\nThe equivalence follows by testing each entry of the minimum and using\n\\eqref{long68:eq:core-normalisation}. Both inequalities must hold for the\nsame arbitrarily large parameters. The first alone is already sufficient\nby Theorem~\\ref{long68:res:global-residue}; the second is an extra restriction,\nnot an equivalent formulation of irrationality.\n\n\\label{long68:res:split-factor-normalized-collision}\\phantomsection\n\\label{long68:bdry:fixed-owner-absorption}\\phantomsection\nA fixed small index cannot supply new factors indefinitely. If $n$ is\nfixed and $p>n!-1$, then $n!-1$ divides $(p-1)!$, so none of its prime\npowers remains outside the factorial base. For example, the factor $719$\nof $6!-1$ is useful only over a finite range of $p$. Any argument using\nsuch factors at arbitrarily large $p$ must let their indices increase.\n\n\\section{Finite denominator exclusions}\\label{long68:sec:finite}\n\\label{long68:res:lead-denominator-exclusions}\\phantomsection\n\\papersectiontarget{r12-finite}\n\nTwo recorded computations give different restrictions on a hypothetical\nrational denominator. The carry calculation concerns divisibility, and\nthe continued-fraction enclosure concerns size:\n\\begin{equation}\\label{long68:eq:finite-bounds}\n q\\nmid299999!,\\qquad q\\ge2^{39990}>10^{12038}\n\\end{equation}\nThese restrictions hold simultaneously whenever $S=a/q$, with\n$a\\in\\Z$ and $q>0$; they are not two bounds to compare by taking a\nmaximum. They also hold for an unreduced representation. The size\nbound passes to multiples of the reduced denominator, as does failure\nto divide a fixed factorial. The computations were performed outside Lean. The\n\\href{https://github.com/wcook04/plectis-erdos/blob/27c2fc5fe3c55fe547feaeb4c9bb68b3cf63a5bf/lean/ErdosProblems/Erdos68/PaperCompleteFiniteSizeCertificate.lean#L58}{candidate finite-size certificate}\nrecords that its computation has not been kernel checked, and the full\ncarry certificate has the same outstanding formal obligation.\n\nThe two restrictions concern different properties of $q$. The prime\n$300007$ does not divide $299999!$, but lies below the size bound;\n$299999!$ exceeds that bound, but fails the nondivisibility condition.\nA sufficiently high power of a small prime need not divide $299999!$\neither. Thus the carry exclusion is not an exclusion of all denominators\nwith small prime factors. In Sondow's terminology, it says that the least\n$r$ for which $q\\mid r!$ is at least $300000$\n\\cite[\\S3, Theorem~1]{sondow2006}. His approximation estimates for $e$\nare not being applied to $S$.\n\n\\paragraph{Carry intervals.}\nThe GMP calculation encloses each scaled sum by outward-rounded rational\nintervals, using scale $2^{5025679}$ and $96$ guard bits. It uses integer\narithmetic throughout. A carry is entered only after the entire interval\nhas been located in one half-open unit cell. All $299998$ cells for\n$3\\le m\\le300000$ were certified in the recorded run. The indices with\n$b_m=1$ were exactly\n\\[\n52,\\quad591,\\quad1030,\\quad1407,\\quad1438,\\quad2164,\\quad4258,\n\\quad10991,\\quad21236.\n\\]\nIn particular $b_{300000}\\ne1$, which gives the first restriction in\n\\eqref{long68:eq:finite-bounds} by\nTheorem~\\ref{long68:res:carry-equivalence}, and hence also $q\\ge300000$.\nThe independently implemented GMP census is separate from the earlier\ncalculation through $4000$. That calculation gives the first six entries\nof this list and a non-unit carry at its endpoint, so it alone gives\n$q\\ge4000$. The implication from a non-unit carry to factorial\nnondivisibility is the recorded kernel-checked result; the full census\nis not thereby a kernel-checked computation.\n\n\\paragraph{A rational enclosure and continued fractions.}\nSet $D=2^{80000}$ and let $N$ be the first integer satisfying $N!-1>D$.\nPut\n\\[\n \\ell=\\sum_{n=2}^{N-1}\\left\\lfloor\\frac D{n!-1}\\right\\rfloor,\n \\qquad\n u=\\ell+(N-2)+\\left\\lfloor\\frac{2D}{N!-1}\\right\\rfloor+1.\n\\]\nEach of the $N-2$ rounded terms loses less than one. Moreover,\n$(n+1)!-1>(n+1)(n!-1)$, so the positive tail from $N$ is less than\n$(N+1)/(N(N!-1))<2/(N!-1)$. These inequalities give the strict enclosure\n$\\ell/D<S<u/D$. The omitted positive tail makes the lower endpoint strict,\nand the final $+1$ makes the upper endpoint strict even when\n$2D/(N!-1)$ is an integer.\n\nThe recorded values are $N=7054$ and $u-\\ell=7053$.\nApply Euclid's continued-fraction algorithm simultaneously to the two\nendpoints, stopping as soon as their integer parts differ or a remainder\nvanishes. At an inversion, reverse the order of the endpoints. There\nare $23449$ common partial quotients $a_0,\\ldots,a_{23448}$.\nTheir convergent denominators satisfy\n\\[\n Q_{-2}=1,\\qquad Q_{-1}=0,\\qquad\n Q_j=a_jQ_{j-1}+Q_{j-2},\\qquad Q_{23448}\\ge2^{39990}.\n\\]\nEvery rational in the open enclosure shares these quotients. Such a\nrational either equals the last convergent $P_{23448}/Q_{23448}$,\nor has a subsequent complete quotient $A/B>1$, with coprime positive\nintegers $A,B$, and equals\n\\[\n \\frac{P_{23448}A+P_{23447}B}{Q_{23448}A+Q_{23447}B}.\n\\]\nThe determinant identity\n$P_jQ_{j-1}-P_{j-1}Q_j=\\pm1$ proves that this fraction is reduced and\nits denominator is at least $Q_{23448}$\n\\cite[\\S1.12(ii), (1.12.5)--(1.12.7), (1.12.20)--(1.12.21)]{nist-dlmf}.\nNo assumption that the last convergent lies inside the enclosure is\nneeded. The integer inequality $2^{39990}>10^{12038}$ then gives the\ndecimal bound in \\eqref{long68:eq:finite-bounds}.\nThe\n\\href{https://github.com/wcook04/plectis-erdos/blob/cd3bd6fe245867a435e15d503ccedd699c5d02e2/scripts/check_erdos68_continued_fraction.py}{integer program}\nand its\n\\href{https://github.com/wcook04/plectis-erdos/blob/cd3bd6fe245867a435e15d503ccedd699c5d02e2/verification/erdos68-continued-fraction.json}{recorded output}\ngive the $80000$-bit scale, width $7053$, prefix length $23449$,\nbinary exponent $39990$ and strict decimal exponent $12038$.\n\n\\paragraph{Carry computation record.}\nThe carry calculation uses the program \\path{scripts/check_erdos68_strict_successor.py}\nwith the GMP backend \\path{scripts/check_erdos68_strict_successor_gmp.cpp}; its\noutput \\path{verification/erdos68-strict-successor.json} records the scale and\nguard bits above, the unit carries, an event-trace digest and the final\nenclosure. The SHA-256 digests of the program and the backend, and the\npayload digest recorded in the output, are\n\\begin{quote}\\footnotesize\nProgram: \\path{f25b5bd8ffbfd66bdbb42ca6a07a936aab9b4eb7d3a57446725be55acb642191}.\\\\\nBackend: \\path{43308ddc3902dd537b789ae6e0648054a0d752adf07e5d6e1d829811e9342083}.\\\\\nPayload: \\path{044ef27a60f7294c39b8279b7f04e22e1234a942e2862e9e8c64eb66505af29b}.\n\\end{quote}\n\n\nBoth calculations cover finite data. Refining an enclosure can leave the\ncommon prefix unchanged, and a later unit carry gives no new carry\nexclusion. In particular, neither calculation excludes an eventual\ntail of unit carries. The continued-fraction identities are classical\n\\cite[\\S1.12(ii)]{nist-dlmf}; the numerical parameters in this section\nare the outputs of the stated computations.\n\n\\section{Further arithmetic questions}\\label{long68:sec:open}\n\nIt remains to prove $m\\nmid Z_m$ at arbitrarily large indices.\nSection~\\ref{long68:sec:carry} gives equivalent formulations through the\nactual remainders in \\eqref{long68:eq:lower-escape} and the finite\ncondition \\eqref{long68:eq:finite-escape}, when all integer indices are\nallowed. The estimates below concern the remaining arithmetic, not further\nways to solve the coefficient equations. None establishes the required\nunbounded set. Restricting indices to primes is an extra condition;\nselected congruences likewise do not identify the sequence defined by\n$S$ without its floor relations.\n\n\\paragraph{Value sets do not determine the weighted residue sum.}\nGaraev, Luca and Shparlinski's harmonic-sum estimate\n\\cite[Theorem~1 and (5)]{garaev-luca-shparlinski-harmonic} concerns\nunconditioned harmonic sums, not sums restricted to the indices where\n$n!\\equiv1\\pmod q$.\nThe value-set literature addresses different invariants.  Banks, Luca,\nShparlinski and Stichtenoth \\cite{banks-et-al2005} provide historical\nmissing-value context; Klurman and Munsch\n\\cite[Theorems~2.1--2.2]{klurman-munsch2017} distinguish unconditional\naverages from GRH-dependent improvements.  Grebennikov, Sagdeev,\nSemchankau and Vasilevskii\n\\cite[Corollaries~1.2 and~1.4]{grebennikov-et-al2024} study sizes of value\nand quotient sets, as do the factorial-residue and representation results\nof Hu \\cite{hu2026} and Garaev and Pardo \\cite{garaev-pardo2026}.\nNone is a noncancellation theorem for the reciprocal unit weights here.\nA lower bound for the number of distinct residues does give an upper\nbound on the multiplicity of residue $1$, since every other value needs\nat least one index. But that bound does not determine the reciprocal\nunit weights or test whether they cancel modulo\n$q$. The two explicit cancellations in \\S\\ref{long68:sec:prime-powers} illustrate\nwhy the weights matter.\nHu's September preprint \\cite[Theorem~3.1]{hu-september2026} considers\npairwise distinct maps $T_\\theta(x)=b_\\theta+c_\\theta/x$ over $\\mathbb F_p$,\nwith $c_\\theta\\ne0$. An internal transition is a pair $(x,\\theta)$ for which\n$x\\in A\\smallsetminus\\{0\\}$ and $T_\\theta(x)\\in A$.\nWith at least $M$ distinct such pairs, at most $|A|$ parameters, and a\nbound $\\mu\\ge1$ on the number of ordered parameter pairs representing each\nnonidentity map $T_\\eta T_\\theta^{-1}$, the theorem gives $|A|\\gg\\min\\{M,p\\}^{8/15}\\mu^{-4/15}$. Its incidence estimate comes\nfrom Stevens and de Zeeuw \\cite[Theorem~4]{stevens-dezeeuw2017}.\nTo apply this to the residue classes of indices satisfying\n$n!\\equiv1\\pmod q$, one would first have to specify the set $A$ and the\ntransition maps. The number of such indices is not automatically a\nlower bound for $M$: different indices may produce the same pair\n$(x,\\theta)$, and their factorial values are all $1$. One must prove that\nenough distinct transitions remain, as well as bound the ordered-pair\nmultiplicity $\\mu$. The unconditioned value-set theorem does not provide\nthese additional facts.\n\n\\paragraph{The shared denominator and the gap must be controlled together.}\nFor the blocks selected from first prime occurrences, the second inequality\nin \\eqref{long68:eq:factor-split} is\n\\begin{equation}\\label{long68:eq:weighted-target}\n \\log\\widetilde C_p\n <\\log\\left(\\frac{2p^2}{2p+1}\\cdot\n       \\frac{\\prod_{j=p}^{2p-1}j}{q}\\right).\n\\end{equation}\nThe first inequality is \\eqref{long68:eq:global-scale}. Both must hold at\nthe same arbitrarily large parameters.\n\nThe order of quantifiers is part of the proposed result: for every bound,\none parameter must satisfy both inequalities. Two unrelated infinite\nsubsequences would not prove the combined criterion.\n\nFor a prime $r$ and an integer $e\\ge1$, let\n$h_{r,e}(p)=\\#\\{i\\in I_p:r^e\\mid i!-1\\}$. The exponent of $r$ in the\nshared part is at least $e$ exactly when $h_{r,e}(p)>1$.\nRemoving the exponent already present in $F_p$ therefore gives\n\\[\n v_r(\\widetilde C_p)\n =\\#\\{e\\ge1:h_{r,e+v_r(F_p)}(p)>1\\}.\n\\]\nWhen $r\\nmid F_p$, this counts the positive exponents for which two or\nmore denominators are divisible by $r^e$. The spacing bound\n$h_{r,e}(p)(e+1)\\le2p+e-2$ follows from the gap-product argument:\nif $i<j$ are two such indices, then $j<r$ and\n$0<r^e\\le j!/i!-1<r^{j-i}$, so $j-i\\ge e+1$. All indices lie between\n$2$ and $2p-1$, giving the bound when $h_{r,e}(p)>0$. It also holds when $h_{r,e}(p)=0$.\nTo estimate $\\log\\widetilde C_p$, these exponent counts must be multiplied\nby $\\log r$ and summed over primes. A count ignoring the sizes of the\nprimes does not give \\eqref{long68:eq:weighted-target}. For a prime $q$,\nWilson reflection of an odd index $n<q$ contributes a repeated divisor\nonly when the reflected index is distinct and in the same block, with\nboth indices at least $2$. The endpoint $n=q-2$ reflects to $1$ and\ncontributes no such pair, as shown in \\S\\ref{long68:sec:prime-powers}.\n\nEven for two admissible indices, the reflection identity is only modulo\n$q$ \\cite[(4), p.~462]{stewart2004}. For example, exact multiplication gives\n\\[\n 609!\\equiv1\\pmod{971^2},\\qquad\n 361!\\equiv736019=1+758\\cdot971\\pmod{971^2},\n\\]\nwhere $361=971-609-1$. Both indices lie in $I_{306}$, but their factorial\ndenominators have $971$-exponents at least $2$ and exactly $1$, respectively.\nThus both indices satisfy the congruence modulo $q$, but need not satisfy\nit modulo $q^e$ for $e>1$. Each level in the valuation sum requires its own\ncongruence; neither distinctness nor a higher exponent may be inferred\nfrom the prime-level identity alone.\nTheorem~12 of Garaev, Luca and Shparlinski\n\\cite[arXiv v1, Thm.~12, p.~16]{garaev-luca-shparlinski} is a multiplicity\nbound and supplies no lower bound for the distance to the next integer in\n\\eqref{long68:eq:global-scale}.\n\nFor the first inequality alone, a joint mean bound\n\\[\n \\frac1{|I_X|}\\sum_{p\\in I_X}\n \\left(\\log\\widetilde C_p-\\log\\frac{\\rho_p}{R_p}\\right)\n <(1-\\varepsilon)X\\log X\n\\]\non nonempty sets $I_X\\subseteq[X,2X]\\cap\\mathbb N$, with $R_p>1$, a fixed\n$0<\\varepsilon<1$, and arbitrarily large $X$, would suffice: at least one\nsummand is at most the average, while the required upper bound is\n$\\log(K_p/((2p+1)F_p))=p\\log p+O(p)$, uniformly for $p\\in[X,2X]$.\nTwo separate estimates on unrelated sets of indices would not give this\njoint comparison.\n\n\\paragraph{What two gap-product tests do not show.}\nFor integers $h\\ge1$ and $x\\ge0$, let\n$F_h(X)=\\prod_{j=1}^h(X+j)$. Suppose $x+h<q$ for a prime $q$.\nIf $x!$ and $(x+h)!$ both equal $1$ modulo $q$ and the harmonic sums\nthrough these indices are equal modulo $q$,\nthen $F_h(x)=1$ and\n$F'_h(x)=F_h(x)\\sum_{j=1}^h(x+j)^{-1}=0$ in that field.  An exact calculation gives\n\\[\n \\gcd(F_{12}-1,F'_{12})\n =(X-4626)(X-7848)\\quad\\hbox{in }\\mathbb F_{12487}[X].\n\\]\nThus a uniform bound of one repeated root is false. But $4626!\\equiv442$ and $7848!\\equiv6300$\nmodulo $12487$, so neither root also satisfies $x!\\equiv1$. Counting\nrepeated roots of $F_h-1$ therefore does not count the indices required\nby the factorial problem. These are finite modular calculations, not a\nbound on the number of indices satisfying both congruences.\n\nStewart's gap-product estimate\n\\cite[Lemma~2, (21)--(24)]{stewart2004} requires the two products to be\nunequal. He supplies that step separately: in the proof of his bound\n(14), a prime-distribution argument separates the products when the common\ndivisor is large; the smaller-divisor case is immediate\n\\cite[p.~467, proof of Lemma~2]{stewart2004}.\n\nThe non-power theorem gives a different argument when the gaps are\nadjacent. For integers $2\\le i<j<k$, equality $j!/i!=k!/j!$ would make $k!/i!$ a\nsquare, contrary to that theorem \\cite[Theorem~1]{erdos-selfridge1975}.\nSelecting two short gaps with nearly equal lengths need not preserve\nadjacency, so this observation alone cannot replace Stewart's nonvanishing\nargument. The wider equal-product problem has a separate history\n\\cite{macleod-barrodale1970}; the doubled-length theorem discussed by\nNguyen Xuan Tho \\cite[Theorem~1]{nguyen-tho2026} is another special case,\nnot arbitrary near-equal-length nonvanishing. No stronger lcm exponent is\ninferred from these local observations.\n\n\\paragraph{Prime powers already seen at earlier indices.}\nRecall that $\\Delta_n=P^\\Delta_n/Q^\\Delta_n$ is the positive gap in lowest terms. Define\n\\[\n \\mathcal A_n=\\prod_{\\substack{r\\ \\mathrm{prime},\\ r\\mid n!-1\\\\\n     r\\mid k!-1\\ \\text{for some }2\\le k<n\\\\\n     v_r(Q^\\Delta_n)<v_r(n!-1)}}r^{\\,v_r(n!-1)} .\n\\]\nThus the product includes the full power of a prime in $n!-1$ when that\nprime has appeared before but has smaller exponent in $Q^\\Delta_n$.\nThe recurrence for the gaps is\n\\[\n \\Delta_{n+1}=n\\Delta_n-b_n-\\frac1{n!-1}.\n\\]\nFor a prime $r$ in the product, put $e=v_r(n!-1)$. The reduced denominator\nof $n\\Delta_n-b_n$ has $r$-exponent at most $v_r(Q^\\Delta_n)<e$, whereas\n$1/(n!-1)$ has exponent $e$. Clearing denominators leaves exactly one\nnumerator term nonzero modulo $r$, so the reduced denominator of\n$\\Delta_{n+1}$ has exponent exactly $e$. Hence\n$\\mathcal A_n\\mid Q^\\Delta_{n+1}$ and, when $\\mathcal A_n>1$,\n$P^\\Delta_{n+1}\\not\\equiv0\\pmod{\\mathcal A_n}$.\nThe earlier occurrence of $r$ is not used in this divisibility argument;\nit restricts the product to the primes proposed for this approach.\nA sufficient quantitative estimate, at arbitrarily large indices $n$,\nwith $m=n+1$ and $d=\\mathcal A_n>1$, is\n\\begin{equation}\\label{long68:eq:amplification-target}\n \\bigl((m+2)m!-2\\bigr)Q^\\Delta_m\n \\le m^2(m!-1)\\,(P^\\Delta_m\\bmod d).\n\\end{equation}\nIt implies the upper alternative of \\eqref{long68:eq:finite-escape}. A nonzero\nleast representative can equal one even when the modulus is large.\nNeither a lower bound for $\\log\\mathcal A_n$ nor infinitely many increases\nin a previously occurring prime's exponent therefore proves\n\\eqref{long68:eq:amplification-target}: the least representative itself\nmust satisfy the displayed inequality. Repeated prime squares\ncan first occur before the Wilson index $r-2$: for example, $971$ divides\n$361!-1$ and its square first divides $609!-1$, with $609<971-2$. Such an\nexample rules out confinement to that last index but supplies no unbounded\nfamily satisfying the required inequality.\n\n\\paragraph{The upper alternative in the finite test.}\nIt would suffice for \\eqref{long68:eq:finite-escape} to hold at arbitrarily\nlarge indices; an unbounded set of prime indices would suffice as well.  The upper alternative is exactly\n$\\bigl((m+2)m!-2\\bigr)Q^\\Delta_m\\le m^2(m!-1)P^\\Delta_m$.  Replacing $P^\\Delta_m$ by its least\nnonnegative residue modulo a specified divisor of $Q^\\Delta_m$ gives a stronger\nsufficient test. No bound forcing this inequality at arbitrarily large\nindices is established here. The full recurrence for the actual scaled\npartial sums, including its initial value, is fixed; a rational example\nfor a weakened recurrence does not satisfy that full specification\n(\\S\\ref{long68:sec:ext-nogo}). Nor does a zero canonical digit assert\nmembership in the specified real interval.\n\n\\paragraph{Conditions at twice a prime.}\nFor an odd prime $p$, the carry range and $Z_{2p}=2pZ_{2p-1}+1-b_{2p}$ give\n\\begin{equation}\\label{long68:eq:doubled-prime}\n p^2\\mid Z_{2p}\n \\quad\\Longleftrightarrow\\quad\n \\begin{cases}\n b_{2p}=1\\ \\hbox{and}\\ p\\mid Z_{2p-1},\\ \\hbox{or}\\\\\n b_{2p}=1+p\\ \\hbox{and}\\ p\\mid2Z_{2p-1}-1 .\n \\end{cases}\n\\end{equation}\n\n\nReduction modulo $p$ forces $b_{2p}\\equiv1\\pmod p$, and the two displayed\nvalues are the only possibilities in $[-1,2p-1]$; division by $p$ gives the\npredecessor conditions.  For rational $\\Ser=a/q$ and $p>q$, the tail estimate gives\n$Z_{2p}=(2p)!\\,\\Ser$, which is divisible by $p^2$. A proof through\n$p^2\\nmid Z_{2p}$ must therefore exclude both displayed alternatives at\nthe same prime, for arbitrarily large primes.\nThat is a requirement of this particular test, not of every irrationality\nargument. Rationality also forces $b_{2p}=1$ and\n$Z_{2p-1}=(2p-1)!a/q$, with $p\\mid Z_{2p-1}$: here $q\\mid(p-1)!$,\nand the remaining factorial factors include $p$. Thus either\n$b_{2p}\\ne1$ at arbitrarily large odd primes or\n$p\\nmid Z_{2p-1}$ at arbitrarily large odd primes would already contradict\nrationality. The first is a special case of\nTheorem~\\ref{long68:res:carry-equivalence}; the second uses the same\nfactorial-clearing argument. Neither unbounded occurrence is proved here.\n\n\\paragraph{Criteria from the literature that do not apply.}\nDuverney's Theorem~3.1 assumes, among other conditions, quadratic growth\n$cu_n^2\\le u_{n+1}\\le c'u_n^2$ for positive constants $c,c'$. For\n$u_n=n!-1$, however, $u_{n+1}/u_n^2\\to0$. His Corollary~3.2 assumes\nconvergence of the signed series $\\sum_n(u_{n+1}/u_n^2-1)$ in (3.6).\nHere its terms tend to $-1$, so that hypothesis fails as well\n\\cite[pp.~275, 285--287]{duverney}.  For a single denominator sequence\nthe rapid-growth criterion goes back to Erd\\H{o}s\n\\cite[Theorem~1, p.~1]{erdos1975}; Barreto, Kang, Kim, Kova\\v{c} and Zhang\ntreat products of consecutive denominators and weighted extensions\n\\cite[Thms.~2--3 and Rem.~4]{barreto-et-al}.  For $a_n=n!-1$,\n$\\log(n!-1)=O(n\\log n)$ makes $(n!-1)^{1/\\psi^n}\\to1$ for every fixed\n$\\psi>1$, so neither Erd\\H{o}s's hypothesis\n$\\limsup_na_n^{1/2^n}=\\infty$ nor the growth hypotheses of their Theorems~2\nand~3 hold.  These criteria do not decide $\\Ser$; the useful content is the\ntarget these papers identify: a subsequence with clearing integers small enough for the corresponding\nremainders, or enough exact cancellation in those integers.\nTheorem~\\ref{long68:res:lcm-growth} measures the cost of clearing all\nsummands before reduction; it does not bound cancellation in their sum.\nTheorem~\\ref{long68:res:prime-pole} tests whether a maximal prime power\nsurvives reduction. Neither supplies the upper bound for a clearing\ninteger, relative to its remainder, that these arguments need.\n\n\\medskip\n\\noindent No irrationality conclusion is obtained here.  The finite\nconclusion is that every rational representation $\\Ser=a/q$, $q>0$, must\nsatisfy both $q\\nmid299999!$ and $q\\ge2^{39990}>10^{12038}$.\nThe criteria above require a condition at arbitrarily large indices.\nExtending either finite computation alone does not establish it.\n\n\\section*{Sources and evidence}\n\\evidenceparagraph{\\evidencerecordurl}\n\nThe margin marks and the table below identify the recorded Lean\nstatements and supporting formulas at their original revisions. A link\nto a nearby statement does not certify a further deduction in the prose.\nThe two partial-sum cancellations, the index-$52$ example and the\ncontinued-fraction enclosure use the finite integer calculations\ndescribed above. The carry computation through $300000$ uses a separate\nexact-interval computation outside Lean, described in\n\\S\\ref{long68:sec:finite}.\n\n\\begingroup\\small\n\\begin{longtable}{@{}p{.46\\linewidth}p{.48\\linewidth}@{}}\n\\toprule\n\\papertablehead\nStatement & Lean statements of its steps\\\\\n\\midrule\n\\endfirsthead\n\\toprule\nStatement & Lean statements of its steps (continued)\\\\\n\\midrule\n\\endhead\n\\bottomrule\n\\endfoot\nTheorem~\\ref{long68:res:prime-pole} &\n\\lword{Erdos68/PrimePoleCriterion.lean}{223}{factorialGapPrefix_den_full_prime_power_iff}{maximal-power survival},\nwith\n\\lword{Erdos68/PrimePoleCriterion.lean}{129}{factorialGapPrefixLCMNumerator_mod_prime}{the residue formula}\nand\n\\lword{Erdos68/PrimePoleDenominator.lean}{41}{natRatio_den_factorization_eq_iff}{the denominator form}\\\\\nProposition~\\ref{long68:res:wilson-cofinality} &\n\\lword{Erdos68/PrimeZeroBranch.lean}{3097}{cofinal_prefixPrivate_factorialGap_hits}{first prime occurrences at arbitrarily large indices};\n\\lword{Erdos68/PrimeZeroBranch.lean}{3139}{prime_dvd_reflected_factorialGap_of_odd}{the reflected divisibility}\\\\\nTheorem~\\ref{long68:res:carry-equivalence} &\n\\lword{Erdos68/FactorialGapPlateauCore.lean}{986}{irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses}{failures of divisibility for the next integer};\ndenominator exclusions:\n\\lword{Erdos68/FactorialGapPlateauCore.lean}{812}{rational_denominator_ge_of_nonunit_carry}{the size bound}\nand\n\\lword{Erdos68/FactorialGapPlateauCore.lean}{836}{rational_denominator_not_dvd_pred_factorial_of_nonunit_carry}{the factorial nondivisibility}\\\\\nProposition~\\ref{long68:res:companion-orbit} &\n\\lword{Erdos68/CompanionOrbitRationality.lean}{438}{not_irrational_factorialGapSeries_iff_eventually_companion_floor_neg_two}{factorial residues and rationality}\\\\\nProposition~\\ref{long68:res:lower-escape} &\n\\lword{Erdos68/ShrinkingTargetNormalForm.lean}{82}{irrational_factorialGapSeries_iff_cofinal_lower_endpoint_escape}{the equivalent fractional-part inequality};\n\\lword{Erdos68/ShrinkingTargetNormalForm.lean}{122}{lower_endpoint_escape_of_predecessorGap_outside_window}{the finite window},\n\\lword{Erdos68/ShrinkingTargetNormalForm.lean}{145}{irrational_factorialGapSeries_of_cofinal_predecessorGap_outside_window}{the irrationality implication},\n\\lword{Erdos68/ShrinkingTargetNormalForm.lean}{39}{factorialGap_lower_endpoint_escape_iff_nonunit_or_maximal_digit}{the classification}\n\\eqref{long68:eq:escape-classification},\n\\lword{Erdos68/ShrinkingTargetNormalForm.lean}{158}{lower_endpoint_target_radius_lt_two_div_sq}{the radius bound}\\\\\nTheorem~\\ref{long68:res:shift-family} &\n\\lword{Erdos68/FactorialShiftFamilyOrbit.lean}{139}{not_irrational_shiftGapSeries_iff_eventually_ceil_residue_two}{the criterion for the shifted series};\n\\lword{Erdos68/FactorialShiftFamilyOrbit.lean}{156}{irrational_shiftGapSeries_iff_cofinal_ceil_residue_misses}{the escape form},\n\\lword{Erdos68/FactorialShiftFamilyOrbit.lean}{175}{shiftGapSeries_neg_one_eq_factorialGapSeries}{the member $t=-1$},\nand\n\\lword{Erdos68/FactorialShiftFamilyOrbit.lean}{211}{irrational_exp_one_of_family_boundary}{the irrationality of $e$}\\\\\nTheorem~\\ref{long68:res:global-residue} &\n\\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{6164}{irrational_factorialGapSeries_of_cofinal_global_complementaryTail}{the prime-parameter specialization};\nthe margin mark gives the natural-parameter statement;\n\\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{4209}{factorialBlockTailNumerator_coprime_privateModulus}{the coprimality}\n\\eqref{long68:eq:gap-normalisation}\\\\\nEquation~\\eqref{long68:eq:factor-split} &\n\\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{4766}{factorialBlock_unitFactorPairFloor_scale_iff}{the two simultaneous inequalities},\n\\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{4701}{factorialBlock_unitFactorPairFloor_eq_min}{the floor identity},\n\\lword{Erdos68/PrimeZeroBranch.lean}{1574}{irrational_factorialGapSeries_of_cofinal_largePrefixPrivate_unitScaleSplit}{the irrationality implication}\\\\\nRemoval of a fixed denominator factor &\n\\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{2677}{factorialBlockPrivateQuotient_eq_one_of_gap_lt}{a fixed factor eventually divides the factorial}\\\\\nWeight identities used in Theorem~\\ref{long68:res:normalform} and\n\\eqref{long68:eq:channel-congruence} &\n\\lword{Erdos68/FactorialChannelCertificate.lean}{25}{factorial_pow_floor_dvd_factorial}{integrality of the weights},\n\\lword{Erdos68/FactorialChannelCertificate.lean}{43}{channelWeight_mul_denominator}{the denominator-times-weight identity}\\\\\nTheorem~\\ref{long68:res:channel-radius} &\n\\lword{Erdos68/ChannelIntegralCongruence.lean}{1084}{square_subsequence_radius_three_halves_lower}{finite radius bound};\n\\lword{Erdos68/ChannelIntegralCongruence.lean}{1103}{no_eventual_square_subsequence_three_halves_upper}{the sequence form},\n\\lword{Erdos68/ChannelIntegralCongruence.lean}{905}{not_isLittleO_square_subsequence_radius}{the little-$o$ form}\\\\\nChanging one weighted sum $V_p$ &\n\\lword{Erdos68/PrimeUnitTranslator.lean}{1657}{exists_remote_factorialGrid_primeTranslator_reduction}{adjustment at sufficiently large indices},\n\\lword{Erdos68/PrimeUnitTranslator.lean}{1559}{exists_cramerFactorialGridResidual_eq_det_mul_factorialGapSeries_add_int}{the residual identity}\\\\\nEquation~\\eqref{long68:eq:doubled-prime} &\n\\lword{Erdos68/FactorialZeroPlateauSupplement.lean}{257}{sq_dvd_double_strictFacTop_factorialGapPrefix_iff}{doubled-prime criterion}\\\\\nCanonical factorial digits &\n\\lword{Erdos68/CanonicalFactorialTermination.lean}{78}{exists_rat_eq_iff_eventually_zero_canonicalDigit}{termination equivalence}\\\\\nAmplification modulus &\n\\lword{Erdos68/PrimeZeroBranch.lean}{5557}{factorialGapRepeatedBirthAmplificationModulus_dvd_succ_den}{divisibility},\n\\lword{Erdos68/PrimeZeroBranch.lean}{5572}{predecessorGapNumeratorNat_mod_repeatedBirthAmplificationModulus_ne_zero}{nonvanishing}\\\\\n\\end{longtable}\n\\endgroup\n\nAttribution.  Wilson's theorem and the Wilson reflection identity are\nclassical, the latter recorded by Stewart \\cite[p.~462, (4)]{stewart2004}.\nThe factorial-digit termination criterion goes back to Cantor\n\\cite{cantor1869}; Koepf and Schmersau prove its irrationality direction for\ndigits that are not eventually maximal\n\\cite[Example~3.2, p.~121]{koepf-schmersau}, and Galambos treats rationality\ncriteria for Cantor series \\cite[Ch.~II, \\S2.1]{galambos1976}.  Here the identity\n$C=\\Ser-e+2$ gives the eventual digit value $m-2$ and extends to the shifted\nfamily. The comparison with Han\\v{c}l and Tijdeman\n\\cite[Lemma~2.1 and the following\nremark, p.~385]{hancl-tijdeman} concerns factorial scaling: their lemma\nmakes a scaled tail integral, whereas the proof in\n\\S\\ref{long68:sec:carry} identifies the next integer above a generally\nnonintegral scaled partial sum.  The multiplicity bound is Garaev,\nLuca and Shparlinski's \\cite{garaev-luca-shparlinski}, and the lcm deduction\nfrom it is not theirs.  The divisibility in Lemma~\\ref{long68:res:gap-gcd} is\nthe case $P=-1$ of the relation used in the proof of Lemma~5 of Luca and\nShparlinski \\cite{luca-shparlinski} and at display~(2.5) of Lai \\cite{lai}.  The\nnonvanishing cutoff for polynomial shifts is Lemma~3 of\n\\cite{luca-shparlinski}, restated with the bound $n!+P(n)>1$ in\n\\cite[Lemma~2.1]{lai}.  The survival criterion is a specialisation of\nLouwsma and Martino's valuation formula \\cite[Lemma~4.1, p.~10]{louwsma-martino}.\nThe single-denominator growth criterion is Erd\\H{o}s's\n\\cite[Theorem~1, p.~1]{erdos1975}, and the continued-fraction identities are\nthose of \\cite[\\S1.12(ii)]{nist-dlmf}. The deductions from Wilson's\ntheorem, the conditional finite support bound and\nTheorem~\\ref{long68:res:lcm-growth} are proved above. No further priority\nclaim is inferred from this comparison.\n\n\n\\appendix\n\\section{Guide to the formal sources}\\label{long68:app:sources}\n\\label{long68:sec:coverage}\n\\papersectiontarget{coverage}\n\nThis concordance lists the floor, support and divisibility identities\nused alongside the principal results. All links retain their original\nsource revisions. The last subsection contains the historical links\nfrom the short paper, with their targets unchanged.\n\n\\begin{itemize}[leftmargin=*]\n\\item \\lword{Erdos68/CanonicalFactorialDigits.lean}{47}{facFloor_rat_eq_cleared}{the factorial-scaled floor of a rational number}\n\\item \\lword{Erdos68/CanonicalFactorialDigits.lean}{81}{canonicalDigit_eq_zero_of_rational}{eventual zero digits of a rational number}\n\\item \\lword{Erdos68/CanonicalFactorialDigits.lean}{124}{canonicalDigit_eq_floor_mul_remainder}{a digit as the floor of a scaled fractional part}\n\\item \\lword{Erdos68/CanonicalFactorialDigits.lean}{153}{canonicalDigit_lt_radix}{the digit is smaller than its radix}\n\\item \\lword{Erdos68/CanonicalFactorialDigits.lean}{162}{canonicalRemainder_recurrence}{the recurrence for fractional parts}\n\\item \\lword{Erdos68/CanonicalFactorialDigits.lean}{196}{factorial_expansion_partial}{the finite factorial expansion with its remainder}\n\\item \\lword{Erdos68/CanonicalFactorialDigits.lean}{269}{zero_remainder_forces_zero_digit_tail}{a zero remainder forces all later digits to vanish}\n\\item \\lword{Erdos68/CanonicalFactorialTermination.lean}{38}{canonicalRemainder_eq_zero_of_zero_digit_tail}{eventual zero digits force a zero remainder}\n\\item \\lword{Erdos68/CanonicalFactorialTermination.lean}{54}{exists_rat_eq_of_canonicalRemainder_eq_zero}{a zero factorial remainder implies rationality}\n\\item \\lword{Erdos68/FactorialGapPlateauCore.lean}{82}{factorialGrid_plateau}{the scaled partial sum and limit select the same integer}\n\\item \\lword{Erdos68/FactorialGapPlateauCore.lean}{945}{irrational_factorialGapSeries_of_cofinal_nonunit_carries}{non-unit carries at arbitrarily large indices imply irrationality}\n\\item \\lword{Erdos68/FactorialZeroPlateauCertificates.lean}{136}{sixty_seven_le_rational_denominator}{every rational denominator is at least sixty-seven}\n\\item \\lword{Erdos68/PrimeZeroBranch.lean}{6099}{factorialGap_zero_branch_iff_lower_endpoint_cylinder}{the exact interval condition for the lower endpoint}\n\\item \\lword{Erdos68/PrimeZeroBranch.lean}{3047}{exists_late_prefixPrivate_factorialGap_hit_of_primeProduct_lt}{a prime-product bound gives a late first occurrence}\n\\item \\lword{Erdos68/PrimeZeroBranch.lean}{3231}{prime_dvd_factorialBlockNormalizedCollisionCore_of_odd_reflection}{a reflected prime survives normalisation under the block and upper-half hypotheses}\n\\item \\lword{Erdos68/PrimeZeroBranch.lean}{3252}{one_lt_factorialBlockPrimeHitCount_of_odd_reflection}{two distinct in-block indices under the same reflection hypotheses}\n\\item \\lword{Erdos68/PrimeZeroBranch.lean}{1419}{factorialGapLargePrefixPrivatePowerModulus_dvd_tailoredBlockPrivateQuotient}{the selected prime powers divide the remaining quotient}\n\\item \\lword{Erdos68/PrimeZeroBranch.lean}{1524}{factorialGapLargePrefixPrivatePrimes_unit_factor_pair_tailoredBlock}{the factors one and a selected prime in the chosen block}\n\\item \\lword{Erdos68/AdjacentUnitCarryWindow.lean}{148}{consecutive_unit_carries_iff_positive_offset_le_den}{the exact integer interval for two consecutive unit carries}\n\\item \\lword{Erdos68/AdjacentUnitCarryWindow.lean}{215}{twoStep_den_mul_transitionNormalizers}{the two denominator-reduction factors telescope}\n\\item \\lword{Erdos68/AdjacentUnitCarryWindow.lean}{243}{adjacentUnitCarryWindowDen_eq_twoStep_den}{the denominator after two steps}\n\\item \\lword{Erdos68/AdjacentUnitCarryWindow.lean}{337}{adjacentUnitCarryWindowOffset_eq_twoStep_reducedNumerator}{the offset equals the later numerator times the reduction factors}\n\\item \\lword{Erdos68/ChannelBreakpointRigidity.lean}{71}{factorialMoment_eq_factorial_pow_mul_channelNumerator_band}{factorisation when the floor in the weights is constant}\n\\item \\lword{Erdos68/ChannelBreakpointRigidity.lean}{91}{factorialMoment_eq_zero_of_channelNumerator_eq_zero_band}{vanishing of the value of $M$ on one such interval}\n\\item \\lword{Erdos68/ChannelBreakpointRigidity.lean}{101}{factorialMoment_eq_factorial_mul_channelNumerator_firstBand}{factorisation on the interval from d to twice d}\n\\item \\lword{Erdos68/ChannelBreakpointRigidity.lean}{130}{exists_index_ge_two_mul_of_factorialMoment_ne_zero_of_channel_eq_zero}{a nonzero value of $M$ forces an index at least twice d}\n\\item \\lword{Erdos68/ChannelIntegralCongruence.lean}{868}{square_subsequence_radius_cubic_lower}{a cubic lower bound for the support parameter}\n\\item \\lword{Erdos68/ChannelIntegralCongruence.lean}{885}{no_eventual_square_subsequence_cubic_upper}{exclusion of the stated eventual cubic upper bound}\n\\item \\lword{Erdos68/FactorialChannelCertificate.lean}{65}{channelEvent_eq_zero_of_not_dvd}{no change unless the divisor divides the index}\n\\item \\lword{Erdos68/FactorialChannelCertificate.lean}{156}{lambda34_residual_bounds}{the strict rational bounds for the example remainder}\n\\item \\lword{Erdos68/FactorialAnalyticBoundary.lean}{52}{tendsto_factorialGap_succ_div_sq}{the consecutive-term quadratic ratio tends to zero}\n\\item \\lword{Erdos68/FactorialAnalyticBoundary.lean}{84}{not_summable_abs_factorialGap_succ_div_sq_sub_one}{divergence of the absolute quadratic-ratio errors}\n\\item \\lword{Erdos68/FactorialAnalyticBoundary.lean}{131}{tendsto_factorialGap_rpow_one_div_two_pow}{the iterated-root growth expression tends to one}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{556}{pairwiseCollisionCore_insert_gcd_lcm}{updating the pairwise gcd lcm after insertion}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{586}{collisionCore_insert_gcd_lcm}{updating the lcm with a fixed prescribed factor}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{698}{collisionCore_div_base_eq_pairwiseCollisionCore_div_gcd}{exact removal of the common part with the base}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{739}{collisionCore_div_base_factorization}{the valuation after removal of the base}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{800}{collisionCore_div_base_mul_denominatorLcm_dvd_denominatorProd}{the shared part times the lcm divides the product}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{2707}{factorialBlockFixedPairPrivateQuotients_eq_one}{fixed denominator indices are eventually absorbed by the factorial}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{3327}{factorialBlockNormalizedCollisionCore_le_gapProd_div_gapLcm}{the shared part is bounded by the product divided by the lcm}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{3394}{factorialBlockNormalizedCollisionCore_factorization}{the surviving valuation after removal of the factorial base}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{3443}{factorialBlock_primePower_le_gapPow_of_two_hits}{a shared prime power is bounded by the gap product}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{3472}{factorialBlock_prime_hitPair_distance_gt_exponent}{two indices sharing a prime power are farther apart than its exponent}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{3602}{factorialBlock_primePowerHitCount_mul_succ_le}{the resulting bound on the number of such indices}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{3790}{factorialBlock_normalizedCollision_factorization_add_base_lt_prime}{the total relevant exponent is less than the prime}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{3848}{factorialBlock_normalizedCollision_factorization_add_base_lt_blockDiameter}{the total relevant exponent is less than the block diameter}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{3982}{factorialBlock_prime_sq_gt_pred_of_dvd_normalizedCollisionCore}{a surviving prime has square greater than the parameter minus one}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{4387}{factorialBlock_factorProjection_lcm_eq_privateModulus}{the projection moduli have the required lcm}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{4854}{factorialBlock_complementaryFactorPairFloor_le_global}{the two-factor bound is no larger than the full residue bound}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{5213}{factorialBlock_normalizedCollisionCore_factorization_eq_repeatedHitLayerCount}{the valuation counts exponents occurring at two indices}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{5277}{factorialBlock_normalizedCollisionCore_factorization_eq_truncatedRepeatedHitLayerCount}{a finite exponent range suffices for that count}\n\\item \\lword{Erdos68/EndpointWeightedPrivateSupport.lean}{5389}{factorialBlock_primePower_dvd_normalizedCollisionCore_iff_one_lt_hitCount_of_endpoint_lt_base}{the exact repeated-power criterion above the factorial base}\n\\end{itemize}\n\n\\subsection{Source links from the short paper}\\label{long68:sec:short-concordance}\n\\papersectiontarget{r12-short-sources}\n\nThese links collect the declarations formerly listed in the short\npaper's source appendix, at the same historical revisions and paths.\nThey supplement the concordance above. No fresh compilation or\nindependent review is asserted by a link, and a theorem about a\nconditional family leaves its hypotheses to be established.\n\n\\begingroup\\small\\raggedright\n\\begin{itemize}[leftmargin=*]\n\\item \\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos68/PaperCompletePrimePole.lean#L118}{maximal prime power survival}.\n\\item \\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos68/PaperCompleteSupplementary.lean#L17}{the explicit vector $U_9$}.\n\\item \\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos68/PaperCompleteSupplementary.lean#L45}{its weighted sums}.\n\\item \\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos68/PaperCompleteSupplementary.lean#L53}{divisibility of $V_d(\\lambda)-M(\\lambda)$ by $d!-1$}.\n\\item \\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos68/DivisorChannelBasis.lean}{the Lean file on the integer basis}.\n\\item \\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos68/DivisorChannelBasis.md}{an account of the integer basis}.\n\\item \\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos68/CompressedPrimitiveChannelKernel.md}{an account of the primitive vector}.\n\\item \\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos68/PaperCompleteMomentHorizon.lean}{the finite formula for the gcd}.\n\\item \\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos68/scripts/check_finite_source.py}{the finite gcd verification script}.\n\\item \\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos68/scripts/check_moment_saturation.py}{a script}.\n\\item \\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos68/TailIdealCertificate.lean}{Generic gcd lemmas in the Lean sources}.\n\\item \\href{https://github.com/wcook04/plectis-erdos/blob/0b500c7cf8e8bb7ae343484378df02f277fb8194/lean/ErdosProblems/Erdos68/PaperCompleteLiminf.lean#L42}{the liminf bound for the common denominator}.\n\\item \\href{https://github.com/wcook04/plectis-erdos/blob/d788dd4b8c59f2246000f2ed98fffb8a5e8ac72e/lean/ErdosProblems/Erdos68/ChannelIntegralCongruence.lean#L524-L529}{the inequality for a terminal block}.\n\\item \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L101}{the vector $2e_3-e_4$}.\n\\item \\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos68/CompanionConstantCarryLaw.lean}{the exact floor correction for the companion series}.\n\\item \\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos68/PrimePoleCriterion.lean}{the prime-power noncancellation criterion}.\n\\end{itemize}\n\\endgroup\n\nThe remaining links from the short paper use revision\n\\texttt{99f4bf47422a}; the links in the preceding concordance use this\nrecord's own revision. The following numbers link to source lines within\neach named module, retaining both revisions where they differ.\n\n\\begingroup\\small\\raggedright\n\\begin{description}[leftmargin=1.2em,style=nextline]\n\\item[\\texttt{AdjacentUnitCarryWindow.lean}] \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/AdjacentUnitCarryWindow.lean#L148}{148}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/AdjacentUnitCarryWindow.lean#L215}{215}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/AdjacentUnitCarryWindow.lean#L243}{243}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/AdjacentUnitCarryWindow.lean#L337}{337}.\n\\item[\\texttt{ChannelBreakpointRigidity.lean}] \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean#L71}{71}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean#L91}{91}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean#L101}{101}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean#L130}{130}.\n\\item[\\texttt{ChannelIntegralCongruence.lean}] \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/ChannelIntegralCongruence.lean#L868}{868}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/ChannelIntegralCongruence.lean#L885}{885}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/ChannelIntegralCongruence.lean#L905}{905}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/ChannelIntegralCongruence.lean#L1084}{1084}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/ChannelIntegralCongruence.lean#L1103}{1103}.\n\\item[\\texttt{DivisorFactorialCentre.lean}] \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/DivisorFactorialCentre.lean#L34}{34}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/DivisorFactorialCentre.lean#L39}{39}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/DivisorFactorialCentre.lean#L44}{44}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/DivisorFactorialCentre.lean#L49}{49}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/DivisorFactorialCentre.lean#L80}{80}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/DivisorFactorialCentre.lean#L88}{88}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/DivisorFactorialCentre.lean#L102}{102}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/DivisorFactorialCentre.lean#L129}{129}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/DivisorFactorialCentre.lean#L146}{146}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/DivisorFactorialCentre.lean#L170}{170}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/DivisorFactorialCentre.lean#L179}{179}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/DivisorFactorialCentre.lean#L188}{188}.\n\\item[\\texttt{EndpointWeightedPrivateSupport.lean}] \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/EndpointWeightedPrivateSupport.lean#L4766}{4766}.\n\\item[\\texttt{FactorialChannelCertificate.lean}] \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L25}{25}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L39}{39}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L43}{43}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L51}{51}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L55}{55}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L59}{59}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L65}{65}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L105}{105}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L109}{109}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L112}{112}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L119}{119}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L126}{126}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L133}{133}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L140}{140}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L151}{151}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialChannelCertificate.lean#L156}{156}.\n\\item[\\texttt{FactorialZeroPlateau.lean}] \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialZeroPlateau.lean#L876}{876}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialZeroPlateau.lean#L940}{940}.\n\\item[\\texttt{PrimeUnitTranslator.lean}] \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/PrimeUnitTranslator.lean#L1559}{1559}.\n\\item[\\texttt{PrimeZeroBranch.lean}] \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/PrimeZeroBranch.lean#L6099}{6099}.\n\\end{description}\n\\endgroup\n\nThe hypotheses about short intervals, least prime factors, and the\nnonzero real remainders are stated in the mathematical sections above.\nThey remain hypotheses when a declaration link is followed.\n\n% , - part extended_record , -\n% Long-only authored mathematics for the Erdős #68 reasoning record.\n% Everything here was in the short note before the September 2026 revision and\n% is kept because the complete record owns the subsidiary arguments, the\n% superseded deductions, and the failed routes with their exact boundaries.\n\n\n\\paragraph{Earlier links for the consolidated criteria.}\\label{long68:sec:short-criteria-sources}\nThe four criteria treated here retain the short paper's earlier source\nand comparison links below. These identify historical declarations and\nrecorded comparisons; they are not new executions.\n\\begin{itemize}\n\\item The constant-floor support statement: \\href{https://github.com/wcook04/plectis-erdos/blob/897f55801d7d184aaa2ce8fb19e7b701c9c23d94/evidence/erdos-68-factorial-denominator-irrationality.md#res-bandbreakpoint}{the recorded declaration list}, \\href{https://github.com/wcook04/plectis-erdos/blob/897f55801d7d184aaa2ce8fb19e7b701c9c23d94/evidence/erdos-68-factorial-denominator-irrationality.md#res-bandbreakpoint-comparator}{the comparison record}.\n\\item The carry characterisation: \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialZeroPlateau.lean#L953}{the earlier non-unit-carry implication}, \\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos68/FactorialZeroPlateau.lean#L1090}{the next-integer criterion}, \\href{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos68/PaperCompleteExisting.lean#L27}{the recorded combined statement}, \\href{https://github.com/wcook04/plectis-erdos/blob/897f55801d7d184aaa2ce8fb19e7b701c9c23d94/evidence/erdos-68-factorial-denominator-irrationality.md#res-carry-characterization-comparator}{the comparison record}.\n\\item The complementary-denominator criterion: \\href{https://github.com/wcook04/plectis-erdos/blob/897f55801d7d184aaa2ce8fb19e7b701c9c23d94/evidence/erdos-68-factorial-denominator-irrationality.md#res-global-complementary-criterion-comparator}{the comparison record}.\n\\item The companion-digit criterion: \\href{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos68/PaperCompleteExisting.lean#L45}{the recorded formal statement}, \\href{https://github.com/wcook04/plectis-erdos/blob/897f55801d7d184aaa2ce8fb19e7b701c9c23d94/evidence/erdos-68-factorial-denominator-irrationality.md#res-companion-orbit-rationality-boundary-comparator}{the comparison record}.\n\\end{itemize}\n\n\\section{Further deductions and limitations}\\label{long68:sec:extended-record}\n\nThe following sections give additional digit identities, denominator\ncalculations and finite examples. For each unsuccessful argument we\nspecify the missing estimate or the counterexample to its inference.\nThe resulting limitations apply to those arguments, without ruling out\nother uses of the same identities.\n\n\\subsection{Factorial digits and their remainders}\\label{long68:sec:ext-digits}\n\nWe derive the factorial-digit identities used in\n\\S\\ref{long68:sec:carry} directly from their remainders. For a real\n$x$, start with $\\theta_0=\\{x\\}$ and, for $m\\ge1$, put\n\\[\n a_m=\\lfloor m\\,\\theta_{m-1}\\rfloor,\\qquad \\theta_m=m\\,\\theta_{m-1}-a_m ,\n\\]\nso that $\\theta_m=\\{m!\\,x\\}$ for $m\\ge1$.  The recorded formal identities give the floor\nformula and the digit bounds $0\\le a_m<m$, the recurrence\n$\\theta_{m+1}=(m+1)\\theta_m-a_{m+1}$, the finite telescoping expansion\n\\[\n x=\\lfloor x\\rfloor+\\sum_{m=2}^{N}\\frac{a_m}{m!}+\\frac{\\theta_N}{N!},\n\\]\nand the rule that a zero remainder at one index forces every later digit to\nvanish.  The rational direction is also checked: if $a\\in\\mathbb Z$, $q\\in\\mathbb N$, and $0<q\\le n$, then\n$\\lfloor n!a/q\\rfloor=(n!/q)a$ and the canonical digit at\nradix $n+1$ vanishes, so every rational input has an eventually zero\nexpansion.  The converse holds for every real input, and gives Cantor's\ntermination criterion \\cite{cantor1869}\n\\[\n x\\in\\Q\\quad\\Longleftrightarrow\\quad a_m(x)=0\\ \\hbox{for all large }m .\n\\]\nIf all digits after index $N\\ge1$ vanish, the recurrence gives\n$\\theta_{N+k}\\ge(k+1)\\theta_N$; since every remainder is below one,\n$\\theta_N=0$ and $x=\\lfloor N!x\\rfloor/N!$.\n\nFor $x=S$, a zero canonical digit means $m\\theta_{m-1}<1$. The\nstronger condition $m\\theta_{m-1}<E_m$, where $E_m=m!(S-H_m)$, is the\nlower-interval event used in \\S\\ref{long68:sec:carry}. The reported interval\ndata contain zero digits at $m=5$ and $m=23$, but no occurrence of that\nstronger condition through $m=100000$.\n\nThere is also a useful identity for an abstract rational sequence $F_m$\nand integer sequence $k_m$. Suppose\n$F_m=mF_{m-1}+1+\\varepsilon_m-k_m$, and define\n\\[\n \\delta_m=\\lceil F_m\\rceil-F_m,\\qquad\n q_m=m\\lceil F_{m-1}\\rceil+1-k_m-\\lceil F_m\\rceil.\n\\]\nThen $0\\le\\delta_m<1$, and the exact identities are\n\\[\n q_m=\\lfloor m\\delta_{m-1}-\\varepsilon_m\\rfloor,\\qquad\n \\delta_m=m\\delta_{m-1}-\\varepsilon_m-q_m.\n\\]\nThe Lean sources prove these identities under the displayed\nrecurrence. They do apply to the actual partial sums: take $F_m=m!H_m$,\n$k_m=0$ and $\\varepsilon_m=1/(m!-1)$ for $m\\ge3$. The recurrence then\nfollows from $H_m=H_{m-1}+1/(m!-1)$, starting with $F_2=2$.\nThe ceiling code $q_m$ need not equal the carry $b_m$, because $Z_m$ is\nthe least integer strictly above $F_m$, not always its ceiling. Precisely,\n$Z_m=\\lceil F_m\\rceil+\\mathbf1_{\\{F_m\\in\\Z\\}}$, so\n\\[\n b_m=q_m+m\\mathbf1_{\\{F_{m-1}\\in\\Z\\}}\n          -\\mathbf1_{\\{F_m\\in\\Z\\}}.\n\\]\nFor example, $F_2=2$ and $F_3=36/5$ give $q_3=-1$ but $b_3=2$.\nThe codes agree whenever both scaled prefixes are nonintegral. Once\n$F_2=2$ and $\\varepsilon_m=1/(m!-1)$ are fixed, the recurrence with $k_m=0$\ndetermines every $F_m$ uniquely by induction. Thus matching the recurrence\nand its initial value is stronger than satisfying selected congruences\nor carry bounds. This specialization is an ordinary deduction from the\npartial-sum recurrence, not a new Lean check. It does not show that the rational gaps admit a\nfinite-state description or establish the unbounded non-unit carries\nneeded for irrationality.\n\n\\subsection{How the classical criteria apply here}\\label{long68:sec:ext-literature}\n\nLet $s_n<a$ be partial sums tending to $a$. Koepf and Schmersau show that\n$\\lfloor ns_n\\rfloor=\\lfloor na\\rfloor$ for all sufficiently large $n$\nforces $a$ to be irrational \\cite[Thm.~1.1, p.~117]{koepf-schmersau}.\nThe strict inequality matters: if $a=A/B$ were rational, then every\nsufficiently large multiple $n$ of $B$ would give\n$\\lfloor ns_n\\rfloor<na=\\lfloor na\\rfloor$. Their rational-term version\nuses a positive integer multiplier $p_n$ and obtains the floor equality\nfrom $np_ns_n\\in\\mathbb N_0$ and the strict tail bound\n$a-s_n<1/(np_n)$\n\\cite[(2.1) and (2.3), p.~118; Thms.~2.2--2.3, pp.~119--120]{koepf-schmersau}.\nIndeed, integrality makes $ns_n$ a multiple of $1/p_n$, so its gap to the\nleast strictly larger integer is at least $1/p_n$; the increase\n$n(a-s_n)<1/p_n$ cannot cross that integer.\nFor the choice they record after (2.1), the least common multiple of the\nreduced summand denominators, here $p_n=\\lcm\\{k!-1:2\\le k\\le n\\}$, the last\ntwo denominators already obstruct the tail bound. As proved in\n\\S\\ref{long68:sec:lcm}, they are coprime for $n\\ge3$, so\n\\[\n p_n\\ge(n!-1)\\bigl((n-1)!-1\\bigr).\n\\]\nFor $n\\ge4$ this gives $np_n>(n+1)!-1$. Hence the first omitted summand\n$1/((n+1)!-1)$ already exceeds $1/(np_n)$, and this $p_n$ cannot satisfy\ntheir tail hypothesis. This rules out the displayed choice of $p_n$, not\nthe criterion itself. In fact, the least positive integer $p_n$ for which\n$np_nH_n$ is integral is exactly\n\\[\n p_n=\\den(nH_n)\n     =\\frac{\\den(H_n)}{\\gcd(n,\\den(H_n))}.\n\\]\nIndeed, writing $H_n$ in lowest terms shows that its denominator must\ndivide $np_n$. With this least choice the required inequality is\n\\[\n 0<S-H_n<\\frac{\\gcd(n,\\den(H_n))}{n\\den(H_n)}.\n\\]\nThus reduction of the partial sum, and the common factor of its reduced\ndenominator with $n$, determine the best scale available in this\ncriterion. The eventual inequality above remains unproved here; failure\nof a larger clearing multiplier does not establish its failure.\n\nDuverney's Theorem~3.1 includes the quadratic growth assumption\n$cu_n^2\\le u_{n+1}\\le c'u_n^2$ for positive constants $c,c'$, while\n$u_{n+1}/u_n^2\\to0$ for $u_n=n!-1$\n\\cite[(1.3), p.~275; Thm.~3.1, pp.~285--286]{duverney}.\nHis Corollary~3.2, which allows signs $a_n\\in\\{-1,1\\}$, assumes convergence\nof the signed series $\\sum_n(u_{n+1}/u_n^2-1)$ in (3.6)\n\\cite[Cor.~3.2, p.~287]{duverney}. Its terms tend to $-1$ here, so the\nseries does not converge. Lean checks the ratio\nlimit and also the divergence of the absolute-value series; the signed\ncondition fails already by the term test. To see the ratio limit directly, divide\nnumerator and denominator by $(n!)^2$: the numerator tends to zero and\nthe denominator $(1-1/n!)^2$ tends to one.\n\nFor a strictly increasing sequence $(n_k)$ of positive integers,\nErd\\H{o}s proved irrationality of $\\sum_k1/n_k$ under the following\nconditions, with a fixed $\\varepsilon>0$ \\cite[Theorem~1, p.~1]{erdos1975}:\n\\[\n \\limsup_{k\\to\\infty}n_k^{1/2^k}=\\infty,\n \\qquad n_k>k^{1+\\varepsilon}\\quad\\text{eventually}.\n\\]\nBarreto, Kang, Kim, Kova\\v{c}\nand Zhang treat products of consecutive denominators and weighted extensions\n\\cite[Thms.~2--3 and Rem.~4, pp.~2--5]{barreto-et-al}.  For $a_n=n!-1$ and every fixed $\\psi>1$,\n\\[\n 0\\le\\frac{\\log(n!-1)}{\\psi^n}\\le\\frac{n^2}{\\psi^n}\\longrightarrow0,\n \\qquad (n!-1)^{1/\\psi^n}\\longrightarrow1.\n\\]\nThus neither Erd\\H{o}s's limsup hypothesis, which uses $\\psi=2$, nor the\ngrowth hypotheses of their Theorems~2 and~3 hold. The Lean proof of this\nlimit treats the special case $\\psi=2$; the estimate above also covers the other\nbases greater than one used in those theorems.  Their proof of Theorem~3 uses the classical\ncriterion, which they trace to Fourier's proof that $e$ is irrational, that a\nrational sum of nonnegative rationals with infinitely many positive terms\nadmits no integers $D_N$ clearing the denominators of the partial sums\nwith $\\liminf_ND_Nr_N=0$, where $r_N$\nis the tail after $N$ terms \\cite[Lem.~8, p.~6]{barreto-et-al}; their\nProposition~12 produces such integers under the hypotheses of that theorem\n\\cite[pp.~9--12]{barreto-et-al}.\nFor the present series, even the full summand lcm $\\Lc_N$ makes\n$\\Lc_N(S-H_N)$ tend to infinity, as shown in \\S\\ref{long68:sec:lcm}.\nA smaller integer cannot clear every summand. It can nevertheless clear\nthe partial sum after addition: the least positive such integer is\n$\\den(H_N)$, and every other one is a multiple of it. Thus this criterion\nwould require information about cancellation in the reduced partial sum,\nnot merely an improvement from the product to the lcm. More precisely, if\n$S=a/q$, then\n\\[\n q\\den(H_N)(S-H_N)\\in\\mathbb Z_{>0},\n \\qquad \\den(H_N)(S-H_N)\\ge\\frac1q.\n\\]\nThus $\\liminf_N\\den(H_N)(S-H_N)=0$ is a sufficient target for this\nrationality criterion; the preceding common-denominator estimates\ndo not establish it.\n\nDividing the recurrence $Z_m=mZ_{m-1}+1-b_m$ by $m!$ and\ntelescoping gives the exact finite identity\n\\[\n \\frac{Z_M}{M!}=\\frac{Z_2}{2!}+\\sum_{m=3}^{M}\\frac{1-b_m}{m!},\n\\]\nso the carry defects $1-b_m$ are integer coefficients of a factorial series.\nHan\\v{c}l and Tijdeman classify the rational sums with polynomial coefficients\n\\cite[Thm.~3.1 and Cor.~3.1, pp.~390--391]{hancl-tijdeman}; their\ndenominator is the cumulative linear product $\\prod_{n\\le N}(an+b)$, and the\nindividual number $N!-1$ does not occur.  In the factorial case $a=1$, $b=0$, the criterion of Oppenheim that they\nreproduce \\cite[Lem.~2.2, p.~385]{hancl-tijdeman} says the following: for\ninteger coefficients $c_n$ with $|c_n|<n$ eventually and\n$\\liminf_n|c_n|/n=0$, the sum $\\sum_n c_n/n!$ is rational exactly when\n$c_n=0$ eventually. This formulation concerns integer coefficients and\nfactorial denominators, not reciprocal terms with denominator $n!-1$.\nHowever, their introduction records a version of Oppenheim's result\nwithout the liminf assumption: under $|c_m|<m-1$ eventually, rationality\nis equivalent to eventual vanishing\n\\cite[Introduction, p.~383]{hancl-tijdeman}. Changing finitely many terms\nadds a rational number, so eventual hypotheses suffice. Here\n$2-m\\le c_m=1-b_m\\le2$ gives $|c_m|<m-1$ for $m\\ge4$; the classical\nresult therefore applies directly.\n\nFor completeness, the following elementary proof specialises that result\nto these one-sided bounds. For $N\\ge2$, telescoping\n$\\sum_{m>N}(m-1)/m!=1/N!$ gives\n\\[\n -1<N!\\sum_{m>N}\\frac{c_m}{m!}\n \\le2N!\\sum_{m>N}\\frac1{m!}<\\frac2N.\n\\]\nThe lower bound follows from $c_m\\ge2-m$ and\n$N!\\sum_{m>N}(m-2)/m!=1-N!\\sum_{m>N}1/m!<1$.\nThe upper bound follows by comparison with\n$\\sum_{j\\ge1}(N+1)^{-j}=1/N$.\nIf $\\sum_{m\\ge3}c_m/m!$ is rational, the scaled tails are integers for all\nsufficiently large $N$. They must then be zero, and subtracting consecutive\ntails gives $c_m=0$ eventually. The converse is immediate. Since\n$Z_M/M!\\to S$, the displayed finite identity recovers\nTheorem~\\ref{long68:res:carry-equivalence}. This is a self-contained\nspecialisation of the classical rationality criterion, not a strengthening\nof it. It proves an equivalence, not the nonvanishing needed for irrationality.\n\n\\subsection{A superseded deduction}\\label{long68:sec:ext-superseded}\n\nFactorial-congruence multiplicities give another lower bound for\n$\\Lc_N$. Its exponent is weaker than that of the terminal-block\nargument in \\S\\ref{long68:sec:lcm}. We include the deduction to specify\nwhat follows from this external input.\nFor an odd prime $r$, let $m_r$ count the\nindices $2\\le n\\le N$ with $r\\mid n!-1$; such an index satisfies $n<r$.  The\nfactorial-congruence multiplicity estimate of Garaev, Luca and Shparlinski\n\\cite[arXiv v1, Thm.~12, p.~16]{garaev-luca-shparlinski}, applied to the\nresidue $a=1$ on $1\\le n\\le\\min(N,r-1)$, gives $m_r\\ll N^{2/3}$; the prime $2$\ndivides none of\nthese factors. The sum of the exponents at a prime is at most the number\nof occurrences times the largest exponent, which is its exponent in\n$\\Lc_N$. Thus\n\\[\n \\sum_{n=2}^{N}\\log(n!-1)=\\sum_r\\sum_{n=2}^{N}v_r(n!-1)\\log r\n \\le\\bigl(\\max_r m_r\\bigr)\\log\\Lc_N\n \\ll N^{2/3}\\log \\Lc_N ,\n\\]\nand Stirling's formula makes the left side $\\asymp N^2\\log N$, whence\n$\\log \\Lc_N\\gg N^{4/3}\\log N$.  Theorem~\\ref{long68:res:lcm-growth} supersedes this:\nits exponent is $3/2$, its constant is explicit, and it does not use the\nexternal multiplicity theorem.  The deduction is retained because it is the only place a\nmultiplicity bound enters the record.\n\nDividing all coefficients by their gcd does not evade the lcm restriction:\nthe resulting integer vector still has $V_2=\\cdots=V_D=0$, so its value of $M$\nis still divisible by $L_D$. For the nonzero cofactor vector,\n\\S\\ref{long68:sec:cofactor-progression} justifies normalisation by\nidentifying the primitive progression vector. It supplies no proof that the remainder is\nnonintegral. No formalised primitive cofactor construction is claimed here.\n\n\\subsection{Rational grid points and first crossings}\n\\label{long68:sec:ext-plateau}\n\nFix $q>0$. To compare a partial sum $H$ with a rational number of\ndenominator $q$, consider the grid $q^{-1}\\Z$. Its next point above $H$\nis $(\\lfloor qH\\rfloor+1)/q$. It is at most $\\Ser$ exactly when\n$\\lfloor qH\\rfloor+1\\le q\\Ser$.  Suppose $H<G\\le S$, the number $n!G$ is an integer, and\n$n!(S-H)<1$. Then\n\\[\n n!H<n!G\\le n!S<n!H+1.\n\\]\nThus $n!G$ is both the least integer strictly above $n!H$ and the floor of\n$n!S$. In particular, for $n\\ge2$, suppose a single rational level $G$\nsatisfies $H_{n+1}<G\\le S$ and $n!G\\in\\Z$. Since $H_n<H_{n+1}$,\nthe tail bounds at both indices give\n\\[\n \\lfloor(n+1)!S\\rfloor=(n+1)!G\n   =(n+1)\\lfloor n!S\\rfloor.\n\\]\nThe next canonical digit is therefore zero. The same level must persist\nat both indices; choosing an unrelated rational level at each index\nwould not imply this equality.\n\nSet $H_1=0$, the empty partial sum. Now let $\\tau\\ge2$ be a first\ncrossing of a rational level $G$, so that\n$H_{\\tau-1}<G\\le H_\\tau$, and write $G-H_{\\tau-1}=a/v$ with positive\nintegers $a,v$. The newly added summand gives\n\\[\n 0<\\frac av\\le\\frac1{\\tau!-1},\n \\qquad v\\ge a(\\tau!-1)\\ge\\tau!-1.\n\\]\nThe scaled overshoot is\n\\[\n 0\\le\\tau!(H_\\tau-G)\n =1+\\frac1{\\tau!-1}-\\tau!\\frac av<2.\n\\]\nConsequently $-\\lfloor\\tau!(H_\\tau-G)\\rfloor$ is $0$ or $-1$.\nHere the minus sign is outside the floor; taking the floor of the negative\novershoot would give a different integer at nonintegral overshoots.\nThe displayed integer is $-1$ precisely when the overshoot is at least\n$1$, or equivalently when\n\\[\n \\frac av\\le\\frac1{\\tau!(\\tau!-1)}.\n\\]\nIn that case $v\\ge\\tau!(\\tau!-1)$. This extra small-gap condition is not\nasserted for every crossing. Nor is $-\\lfloor\\tau!(H_\\tau-G)\\rfloor$\nautomatically the carry $b_\\tau$ defined from $Z_\\tau$: here $G$ is an\narbitrary rational level. Neither denominator bound requires $a/v$ to be in lowest terms.\n\nThere is also a direct obstruction at prime indices. With $\\Delta_m$ as\ndefined in \\S\\ref{long68:sec:carry}, the formal source proves for $m\\ge3$\nthe criterion\n\\[\n m\\mid Z_m\n \\quad\\Longleftrightarrow\\quad\n 1+\\frac1{m!-1}<m\\Delta_m\\le2+\\frac1{m!-1},\n\\]\nand if $\\Ser=a/q$ with $q>0$, then $p\\mid Z_p$ for every prime $p>q$, so one\nexact missed prime $p$ gives $q\\ge p$.  Exact kernel reduction gives\n$11\\nmid Z_{11}$, hence $q\\ge11$; exact rational normalisation gives\n$60\\nmid Z_{60}$, $64\\nmid Z_{64}$ and $67\\nmid Z_{67}$, and the prime index\n$67$ gives the checked bound\n\\[\n \\Ser=\\frac aq,\\ q>0\\quad\\Longrightarrow\\quad q\\ge67 .\n\\]\nThe exact-interval census of the short note replaces $67$ by $300000$.\n\nThe finite geometric-series identity gives another exact decomposition.  For a real $x\\ne0,1$ and an integer $K\\ge0$,\n\\[\n \\frac1{x-1}=\\sum_{j=1}^{K}\\frac1{x^j}+\\frac1{x^K(x-1)} .\n\\]\nWhen $x=k!$ and a chosen factorial scale is divisible by $(k!)^K$, the scaled\nfinite sum is integral and only the last term retains the factor $k!-1$ in\nits denominator.  The identity isolates one remaining fraction before exact\nbounding, and it supplies no cofinal family of nonzero residuals.\n\nThe earlier record rejects a proposed divisibility strengthening of the\nfirst-crossing bound and reports examples at $m=52$ and $m=591$. Its\nnotation for the rational quantity in that claim was not defined, so those\nreports are not used as a verified counterexample here. The established\nconclusion remains the lower bound on the denominator's size, not a\nspecified factor dividing it.\n\nThere is a second, more arithmetic mechanism at doubled prime indices, stated\nin \\eqref{long68:eq:doubled-prime}.  The formal theorem is not\nrestricted to individually computed indices; the divisibility criterion\nholds for the actual partial sums at every odd prime.\n\n\\subsection{Exact identities for shared prime powers}\\label{long68:sec:ext-collision}\n\nLet $I\\subseteq\\{2,3,\\ldots\\}$ be finite. We first compute the\nprime-power levels common to two summand denominators. We then compare\nthose exponents with the ones already supplied by a factorial factor.\n\nWrite $D(I)=\\lcm_{i<j,\\ i,j\\in I}\\gcd(d_i,d_j)$, with $D(I)=1$\nfor an empty or singleton family. For each prime $r$, its valuation is the\nsecond-largest valuation among the $d_i$, counting missing values as zero;\nthe denominator lcm has the largest valuation. This proves\n$D(I)\\lcm_{i\\in I}d_i\\mid\\prod_{i\\in I}d_i$ prime by prime.\n\nThe integer $D(I)$ can be updated exactly when one index is added. For the positive\nfactorial-gap denominators, adjoining a new index $a\\ge2$, $a\\notin I$, gives\n\\[\n D(I\\cup\\{a\\})=\\lcm\\!\\Bigl(D(I),\\ \\gcd\\bigl(d_a,\\lcm_{j\\in I}d_j\\bigr)\\Bigr),\n\\]\nsince finite-family gcd and lcm distributivity collapses the lcm of all\npairwise gcds against $d_a$ to a single gcd. For a fixed positive integer\n$F$, take the lcm with $F$ on both sides:\n\\[\n \\lcm\\bigl(F,D(I\\cup\\{a\\})\\bigr)\n =\\lcm\\Bigl(F,D(I),\\gcd\\bigl(d_a,\\lcm_{j\\in I}d_j\\bigr)\\Bigr).\n\\]\nThus one need only retain the denominator lcm and $\\lcm(F,D(I))$ at each\nstep. The prescribed factor $F$ is not an extra summand denominator.\n\nThere is also an exact bound by the product of the denominators divided\nby their least common multiple. For a positive integer $F$, define\n$\\widetilde C(I)=\\lcm(F,D(I))/F=D(I)/\\gcd(F,D(I))$. Since\n$\\widetilde C(I)$ divides $D(I)$, the preceding divisibility gives\n\\[\n \\widetilde C(I)\\lcm_{j\\in I}d_j\\mid\\prod_{j\\in I}d_j,\\qquad\n \\widetilde C(I)\\le\\frac{\\prod_{j\\in I}d_j}{\\lcm_{j\\in I}d_j}.\n\\]  For the factorial block this specialises to\n\\[\n \\widetilde C_p\\le\\frac{\\prod_{n\\in I_p}(n!-1)}{\\lcm_{n\\in I_p}(n!-1)},\n\\]\nan upper bound for the shared factor in terms of the denominator product\nand lcm. The bound alone does not establish\n\\eqref{long68:eq:weighted-target}.\n\nDividing by $F_p$ subtracts its prime exponents; it does not remove every\nprime that occurs in $F_p$. With $F_p=(p-1)!$,\n$D_p=D(I_p)$ and $C_p=\\lcm(F_p,D_p)$, as in the main record,\n\\[\n \\widetilde C_p=\\frac{C_p}{F_p}=\\frac{D_p}{\\gcd(F_p,D_p)},\\qquad\n v_r(\\widetilde C_p)=\\max\\{0,v_r(D_p)-v_r(F_p)\\} .\n\\]\nFor $e>0$, $r^e\\mid\\widetilde C_p$ exactly when $D_p$ is divisible by\n$r^{e+v_r(F_p)}$, which in the block forces two distinct denominators to be divisible\nby that higher power, by the second-largest-valuation formula. If $i<j$\nare two such indices and $f=v_r(F_p)$, then\n$r^{e+f}\\mid j!/i!-1$. Since $r\\mid j!-1$ forces $j<r$,\n$0<j!/i!-1<r^{j-i}$, hence $e+f<j-i$.\nFor a prime $r\\mid\\widetilde C_p$, this bounds the surviving valuation by\n$v_r(\\widetilde C_p)+v_r(F_p)<r$. Since $v_r(\\widetilde C_p)\\ge1$ and\n$v_r(F_p)\\ge\\lfloor(p-1)/r\\rfloor$, we obtain\n$\\lfloor(p-1)/r\\rfloor\\le r-2$, hence $p-1<r(r-1)<r^2$.\nConsequently $\\widetilde C_p$ is coprime to $k!$ whenever\n$k(k-1)\\le p-1$: a prime $r\\le k$ dividing both would give\n$p-1<r(r-1)\\le k(k-1)\\le p-1$. This excludes the small primes but does not bound the\nproduct of the remaining prime powers.\n\nIf a prime $r$ divides a denominator $n!-1$ with $n\\ge p$,\nthen $r\\nmid F_p$ and $r^e\\mid\\widetilde C_p\\iff r^e\\mid C_p$ for every\n$e>0$.  This has an exact incidence-count form,\n\\[\n r^e\\mid\\widetilde C_p\n \\quad\\Longleftrightarrow\\quad\n 1<\\#\\{i\\in I_p:r^e\\mid i!-1\\},\n\\]\nso a bound of at most one such index implies $v_r(\\widetilde C_p)<e$.\nCounting the exponents gives\n\\[\n v_r(\\widetilde C_p)=\\#\\bigl\\{e\\in[1,r-1]:1<\\#\\{i\\in I_p:r^e\\mid i!-1\\}\\bigr\\},\n\\]\nand for a prime $r>2p-1$ the range can be restricted to\n$e\\in[1,2p-4]$.  The same equivalence holds under the exact condition\n$r\\nmid F_p$ in place of $r>2p-1$, hence in particular for every prime\n$r\\ge p$. At $e=2$, an upper bound of one on the number of indices gives\n$v_r(\\widetilde C_p)\\le1$.\n\nA surviving prime power also forces two indices to be far apart.  If $r$ is prime, $e>0$ and\n$r^e\\mid\\widetilde C_p$, there are $i<j$ in $I_p$ with\n$r^{e+v_r(F_p)}\\mid i!-1$, $r^{e+v_r(F_p)}\\mid j!-1$ and\n$r^{e+v_r(F_p)}\\le j^{\\,j-i}$; consequently $(2p-1)^d<r^{e+v_r(F_p)}$ forces\n$d<j-i$.  The required spacing follows directly: any two indices $i<j$ with $r^e\\mid i!-1$ and\n$r^e\\mid j!-1$ satisfy $e<j-i$, because $r\\mid j!-1$ already forces $j<r$ and the\ninequality for $j!/i!-1$ then forces the strict separation. More generally,\nfor any two indices $2\\le i<j$ and any prime power $r^e$ with $e>0$,\n\\[\n r^e\\mid i!-1,\\quad r^e\\mid j!-1\n \\quad\\Longrightarrow\\quad r^e\\le j^{\\,j-i},\n\\]\nso successive solutions of $i!\\equiv1\\pmod{r^e}$ are at least $e+1$\napart, and\n\\[\n (e+1)\\,\\#\\{i\\in I_p:r^e\\mid i!-1\\}\\ \\le\\ 2p+e-2 .\n\\]\nThe resulting bound on the exponent is\n\\[\n r\\mid\\widetilde C_p\n \\quad\\Longrightarrow\\quad\n v_r(\\widetilde C_p)+v_r(F_p)<2p-3\n\\]\nfor every prime $r$ and $p\\ge2$. Thus the exponent already present in\n$F_p$ uses part of the same bound $2p-3$. To apply\n\\eqref{long68:eq:global-scale}, one still needs sufficiently strong bounds\nfor the product of the shared prime powers and for the gap $\\rho_p/R_p$.\n\nThe available squarefreeness evidence is finite. An exhaustive modular scan through\n$r\\le2{,}000{,}000$ and $n\\le240$ found four pairs $(r,n)$ with $r^2\\mid n!-1$ and no\nprime occurring in two such pairs.  All $498{,}501$ pairs $2\\le a<b\\le1000$ have\nsquarefree $\\gcd(a!-1,b!-1)$. An aggregate scan through $p=499$ also reports a ratio below\n$0.374$, but does not specify its logarithmic normalisation; that ratio is\nnot used as a quantitative premise here. These are reported finite\ncomputations, not an asymptotic bound or a proof of squarefreeness at all\nindices.\n\nA finite version of the argument for first prime occurrences compares the product of a\nchosen set of primes, each at least $5$, with $\\prod_{2\\le k\\le B}(k!-1)$; if\nthe prime product is larger, at least one chosen prime $q$ divides no $k!-1$ with $2\\le k\\le B$,\nwhile Wilson's theorem bounds its first occurrence by $q-2$.  Wilson reflection\nlimits what a linear-size divisor can be assumed to be: if $n$ is odd,\n$n<q$, and $q\\mid n!-1$, then $q\\mid(q-n-1)!-1$. Suppose also that\n$p\\le n$, both indices lie in $I_p$, and the reflected index is earlier,\nequivalently $q<2n+1$. The prime $q$ then divides two denominators in the\nblock. Since $q>n\\ge p$, it does not divide $F_p=(p-1)!$, so\n$q\\mid\\widetilde C_p$: removing the factorial factor does not remove this\nshared prime.\n\nStewart states that for every $\\varepsilon>0$ there are infinitely many odd\n$n$ whose least prime factor $q$ of $n!-1$ satisfies\n\\[\n n<q<\\left(\\frac{\\sqrt{145}-1}{8}+\\varepsilon\\right)n ,\n\\]\nestimate~(9) of Theorem~1 being stated for $n!+1$ \\cite[p.~463]{stewart2004}\nand transferred to $n!-1$ in the text \\cite[p.~464]{stewart2004}.  This controls $q$ relative to the specified index $n$, not to its first\noccurrence. It does not by itself force a repeated divisor. For the minus\nsign, the bound is already met at all sufficiently large Wilson indices\n$n=q-2$. For every prime $q\\ge5$, Wilson gives $q\\mid(q-2)!-1$, and $q$ is its least prime\nfactor. Indeed, every prime divisor exceeds $q-2$, and $q-1$ is even;\nmoreover $q/(q-2)\\to1$. The reflected index is then $1$, outside the denominators of $S$.\n\nFor example, modulo $11$ the only solution of $k!\\equiv1$ with\n$2\\le k\\le9$ is $k=9$, although $11/9<(\\sqrt{145}-1)/8$.\nA small ratio $q/n$ therefore supplies neither a second admissible index nor\nmembership of a reflected index in the chosen block. The earlier\nreflection argument needs both block-membership hypotheses and distinct\nindices. Neither the required estimate at first occurrences nor a bound\nfor the total shared part follows from the transferred least-prime-factor\nestimate alone.\n\nFor a selected prime $q\\mid R_p$, the factors $1$ and $q$ are coprime,\nand the associated moduli $R_p$ and $R_p/q$ have lcm $R_p$. Thus this\nspecialisation requires no second selected prime.\n\nThe elementary fact behind the projection argument is as follows. Let\n$Z,T,B$ be nonnegative integers, let $R>0$, and suppose $Z\\equiv T\\pmod R$.\nEvery positive divisor $Q$ of $R$ with $Z\\le B<Q$ satisfies\n$T\\bmod Q=Z$, since $Z$ is already the least nonnegative representative.\nConsequently, unequal residues modulo two divisors exceeding $B$ rule out\nsuch a $Z$. More generally, if $T\\bmod Q_1\\ne T\\bmod Q_2$, then\n$\\min(Q_1,Q_2)\\le T$: otherwise both residues would equal $T$.\n\n\\subsection{Finite vectors and exact numerical examples}\\label{long68:sec:ext-certificates}\n\nThe three depth-four vectors below all have $M=1380$. By\nTheorem~\\ref{long68:res:residual-transparency}, their remainders differ\nonly by integers. A single finite calculation therefore tests all three:\n\\[\n 255+\\frac45\n <242+1380\\sum_{d=5}^{8}\\frac1{d!-1}\n <255+\\frac56,\n \\qquad \\frac{2\\cdot1380}{9!-1}<\\frac1{100}.\n\\]\nThe first remainder lies strictly between $255$ and $256$. Subtracting\nthe corresponding integer offsets gives the other two intervals.\n\\begin{center}\\small\n\\begin{tabular}{p{.42\\linewidth}rrr}\n\\papertablehead Vector & $\\sum_{n>4}z_n$ & $k$ & Interval for $\\mathcal R$\\\\\n\\hline\n$12K_4+253U_6-11U_8$ & $242$ & $-1474$ & $(255,256)$\\\\\nThe five-term vector \\eqref{long68:eq:constructed-depth-four}\n & $-44$ & $-1760$ & $(-31,-30)$\\\\\nThe norm minimizer on $\\{2,\\ldots,8\\}$\n & $-56$ & $-1772$ & $(-43,-42)$\\\\\n\\end{tabular}\n\\end{center}\nThe second column is the offset from $1380(S-H_4)$; the third is the\ninteger $k$ in $\\mathcal R=1380S+k$. Since $1380H_4=1716$, subtracting\n$1716$ converts the second column into the third. The five-term remainder\nis $286$ less than the first, and the norm-minimising remainder is another\n$12$ less. All three exclude exactly the same divisors of $1380$ by this\nargument. The coefficient and support improvements in\n\\S\\ref{long68:sec:basis-examples} do not enlarge that set of exclusions;\nthe larger finite computations in \\S\\ref{long68:sec:finite} already\nimply it. The exact offsets are reproduced by\n\\texttt{scripts/check\\_erdos68\\_channel\\_moment.py}.\n\nThe finite-support vector $\\lambda=2e_3-e_4$ has, by kernel check,\n$\\chn2\\lambda=0$, $M=-12$, $\\chn3\\lambda=-2$,\n$\\chn4\\lambda=11$, and $\\chn d\\lambda=-12$ for every $d\\ge5$.  Under the\nexact tail enclosure\n$1/119<\\sum_{d\\ge5}1/(d!-1)<1/50$ its remainder lies\nstrictly between $-93/575$ and $-309/13685$, so it is nonzero and has\nabsolute value less than $1$.\n\nIn the enlarged coefficient space of \\S\\ref{long68:sec:channels}, exact\ninteger computation verifies the vectors $K_D$ for every $2\\le D\\le12$:\nweighted sums $2$ through $D$ vanish, $M=\\Lc_D$, and the\ncoefficients have gcd one.  They lie outside the space of vectors supported\non $n\\ge2$, since $\\Lc_D$ is odd whereas $M$ is even for every such vector.  At\n$D=9$,\n\\[\n \\Lc_9=31540008254514077395,\\qquad a_9=[e_1]K_9=-3902884074990939115 .\n\\]\nSince $U_{11}=T_{11}$ has $u_{11}=0$, the vector $K_9-9553024718754\\,U_{11}$\nkeeps the coordinate $a_9$ and has coefficients with gcd one.  By\nTheorem~\\ref{long68:res:residual-transparency} its residual is\n$\\Lc_9(\\Ser-H_9)-9553024718754$, and exact rational arithmetic with the tail\nbound $\\sum_{n\\ge36}1/(n!-1)<2/(36!-1)$ places it strictly between\n$1353/100000$ and $1354/100000$.  A different example is supported on\n$n\\ge2$: the vector $c=(-40,55,-10,1)$ on the support $(3,4,5,6)$ annihilates\nweighted sums $2$ and $3$, has $M=600$, and satisfies\n\\[\n 0.09925341997208298<\\mathcal R(c)<0.09925341997208300 ,\n\\]\nwhich excludes denominators dividing $600$.\n\nA separate interval computation reports the stronger geometric statement that no\nlower-interval event $m\\theta_{m-1}<E_m$ occurs at any $3\\le m\\le100000$; its executable and source\ndigest are not available, so that classification remains external finite\nevidence.  The carry computation through $300000$ used in the short note is the\nexact GMP computation recorded in Section~\\ref{long68:sec:finite}, with\nthe separate calculation through $4000$; it is computational evidence\noutside the Lean development.\n\n\\subsection{Limits of the recurrences and clearing factors used here}\\label{long68:sec:ext-nogo}\n\nTwo distinctions are needed in using these identities: a recurrence\nmay have solutions that violate the defining floor relations, and a\nfixed factor may eventually be absorbed by a growing factorial. The\nexamples below keep these cases separate.\n\n\\begin{itemize}\n\\item Without the defining floor relation, the carry recurrence and its\n  range allow $Z_m/m!$ to be the constant $3/2$. Taking $Z_m=3m!/2$ for\n  $m\\ge2$ and $b_m=1$ for $m\\ge3$ gives\n  $Z_m=mZ_{m-1}+1-b_m$ and $-1\\le b_m\\le m-1$, with the actual initial\n  value $Z_2=3$. But it gives $Z_3=9$, whereas the actual partial sum\n  $H_3=6/5$ gives $Z_3=8$. This example only disproves sufficiency of\n  the stated recurrence, bounds and initial integer. It fails the floor\n  definition. No claim is made that it satisfies the additional prime-index\n  identities. The rational recurrence for $F_m=m!H_m$ in\n  \\S\\ref{long68:sec:ext-digits} determines the actual partial sums uniquely.\n\\item The arguments considered with Wilson quotients, harmonic sums,\n  $p$-adic gamma identities, and factorial residues still need a real\n  gap estimate and the required modular divisibility at the same indices.\n  No such unbounded family is obtained here. This records the missing\n  step in these arguments, not an impossibility theorem for the use of\n  those identities.\n\\item For the genus-zero product $E(z)=\\prod_{n\\ge2}(1-z/n!)$, local uniform\n  convergence and logarithmic differentiation give $-E'(1)/E(1)=\\Ser$.\n  Termwise clearing by $\\prod_{n=2}^N(n!-1)$ cannot give a positive\n  remainder tending to zero: this product is at least $L_N$, and\n  $L_N(\\Ser-H_N)\\to\\infty$ by \\S\\ref{long68:sec:lcm}. The conclusion concerns\n  this clearing factor alone; Hermite--Pad\\'e systems with other\n  denominators are not ruled out.\n\\item Changing a coefficient vector without changing its value of $M$ changes\n  the remainder by an integer only. Also, cancelling the first $D-1$ weighted\n  sums forces $L_D$ to divide the value of $M$; factorial divisibility does not\n  remove that constraint.\n\\item A fixed pair of denominator indices cannot make the projection\n  argument work at arbitrarily large parameters, because their factors\n  eventually divide the factorial being removed.\n\\end{itemize}\n\n\\subsection{An elementary criterion for any real number}\n\\label{long68:sec:ext-generic}\n\nThe fractional-part condition in \\eqref{long68:eq:lower-escape} is an\ninstance of Cantor's termination criterion. For any real $x$ and\npositive sequence $c_m$ satisfying $c_m\\le1/m$ eventually, we have\n\\[\n x\\notin\\Q\n \\quad\\Longleftrightarrow\\quad\n \\{(m-1)!\\,x\\}\\ge c_m\\ \\hbox{for arbitrarily large }m .\n\\]\nFor rational $x$ the fractional parts vanish eventually. Conversely, if\n$\\{(m-1)!x\\}<c_m\\le1/m$ for all large $m$, then\n$m\\{(m-1)!x\\}<1$, so every sufficiently late canonical digit is zero.\nCantor's criterion then makes $x$ rational. Equality $c_m=1/m$ is allowed:\nit is failure of the displayed weak inequality that makes\n$m\\{(m-1)!x\\}$ strictly less than one.\nFor $x=S$, the choice $c_m=E_m/m$ gives exactly\n\\eqref{long68:eq:lower-escape}; indeed $0<E_m<1$ ensures the required\nthreshold bound. The finite test \\eqref{long68:eq:finite-escape} is a\nseparate sufficient condition at one index. Its cofinal converse uses\nthe carry theorem, as shown in \\S\\ref{long68:sec:carry}, not substitution\ninto this general criterion. Neither argument proves that $S$ meets\nthe test at arbitrarily large indices.\n\nThe threshold $1/m$ cannot be replaced by $\\lambda/m$ for a fixed\n$\\lambda>1$. For $x=e$ and $m\\ge3$, the factorial tail gives\n\\[\n m\\{(m-1)!e\\}\n =1+m!\\sum_{n>m}\\frac1{n!}<1+\\frac1m<\\lambda\n \\qquad\\text{eventually}.\n\\]\nThus the irrational number $e$ would never meet the enlarged threshold\nat sufficiently large indices. Positivity is also essential: with\n$c_m=0$, rational numbers would meet the threshold eventually. These\nexamples explain the two assumptions without imposing any\nequidistribution hypothesis.\n% , - part family_catalogue , -\n\\section{Relations among the criteria}\\label{sec:erdos-68-complete-family-map}\nThe classification determines all cancellation vectors and attainable\nmultipliers. The progression construction adds eventual divisibility\nby each fixed $q$ and a vanishing tail bound for the same vector.\nIts primitive cofactor construction gives that vector up to sign, not\nanother family. At fixed $M$, coefficient changes still alter only the\ninteger in $\\mathcal R(c)=M(c)S+k$.\n\nCommon-denominator growth measures the cost of clearing, while\nprime-power survival describes reduction. Neither supplies the missing\ngap comparison. The carry, digit and interval criteria characterize\nirrationality but leave their required unbounded departures unproved.\nThe finite exclusions restrict divisibility and size, respectively;\nthey establish no unbounded assertion.\n\nThe remaining target for this method is therefore precise: for every\n$q>0$, find a nonintegral remainder whose own multiplier satisfies\n$q\\mid M$. The algebraic classification supplies choices, not this\narithmetic conclusion.\n\n% , - part back , -\n\\section*{Acknowledgements}\n\nAn AI research pass supplied by Will Cook derived the depth-four dual\ncongruence and shorter vectors from the weighted linear forms and the\nclassification of their possible coefficients $M$ developed earlier here. OpenAI Codex\nchecked and integrated them; a separate AI pass reviewed the proof.\n\n\\begin{thebibliography}{99}\n\n\\bibitem{erdos1988}\nPaul Erd{\\H o}s. \\href{https://doi.org/10.1017/CBO9780511897184.009}{On the irrationality of certain series: problems and results}.\nIn Alan Baker (ed.), \\emph{New Advances in Transcendence Theory}, Cambridge University Press (1988), pp.~102--109.\n\n\\bibitem{bloom}\nThomas F. Bloom. \\href{https://www.erdosproblems.com/68}{Erd{\\H o}s Problem {\\#68}}.\nOnline resource (2026). Historical access: 28 July 2026; present-page status not reverified.\n\n\\bibitem{erdos1975}\nPaul Erd{\\H o}s. \\href{https://www.renyi.hu/~p_erdos/1976-44.pdf}{Some problems and results on the irrationality of the sum of infinite series}.\n\\emph{Journal of Mathematical Sciences} \\textbf{10} (1975), 1--7.\n\n\\bibitem{louwsma-martino}\nJoel Louwsma and Joseph Martino. \\href{https://arxiv.org/abs/2309.07280v1}{Rational numbers with odd greedy expansion of fixed length}.\nPreprint (2023). arXiv:2309.07280.\n\n\\bibitem{cantor1869}\nGeorg Cantor. {\\\"U}ber die einfachen Zahlensysteme.\n\\emph{Zeitschrift f{\\\"u}r Mathematik und Physik} \\textbf{14} (1869), 121--128.\n\n\\bibitem{galambos1976}\nJ{\\'a}nos Galambos. \\href{https://doi.org/10.1007/BFb0081642}{Representations of Real Numbers by Infinite Series}.\nLecture Notes in Mathematics 502, Springer (1976).\n\n\\bibitem{garaev-luca-shparlinski}\nMoubariz Z. Garaev, Florian Luca and Igor E. Shparlinski. \\href{https://doi.org/10.1090/S0002-9947-04-03612-8}{Character sums and congruences with {$n!$}}.\n\\emph{Transactions of the American Mathematical Society} \\textbf{356} (12) (2004), 5089--5102. arXiv:math/0403422.\n\n\\bibitem{stewart2004}\nCameron L. Stewart. \\href{https://doi.org/10.5486/PMD.2004.3190}{On the greatest and least prime factors of {$n!+1$}, {II}}.\n\\emph{Publicationes Mathematicae Debrecen} \\textbf{65} (3--4) (2004), 461--480.\n\n\\bibitem{koepf-schmersau}\nWolfram Koepf and Dieter Schmersau. \\href{https://doi.org/10.1524/anly.2011.1094}{Irrationality of certain infinite series {II}}.\n\\emph{Analysis} \\textbf{31} (2011), 117--124.\n\n\\bibitem{duverney}\nDaniel Duverney. \\href{https://www.ms.u-tokyo.ac.jp/journal/pdf/jms080206.pdf}{Irrationality of fast converging series of rational numbers}.\n\\emph{Journal of Mathematical Sciences, the University of Tokyo} \\textbf{8} (2001), 275--316.\n\n\\bibitem{hancl-tijdeman}\nJaroslav Han{\\v c}l and Robert Tijdeman. \\href{https://doi.org/10.4064/aa118-4-5}{On the irrationality of factorial series}.\n\\emph{Acta Arithmetica} \\textbf{118} (4) (2005), 383--401.\n\n\\bibitem{barreto-et-al}\nKevin Barreto, Jiwon Kang, Sang-hyun Kim, Vjekoslav Kova{\\v c} and Shengtong Zhang. \\href{https://arxiv.org/abs/2601.21442v3}{Irrationality of rapidly converging series: a problem of {Erd{\\H o}s} and {Graham}}.\nPreprint (2026). arXiv:2601.21442.\n\n\\bibitem{luca-shparlinski}\nFlorian Luca and Igor E. Shparlinski. \\href{https://doi.org/10.1112/S0024609305004923}{Prime divisors of shifted factorials}.\n\\emph{Bulletin of the London Mathematical Society} \\textbf{37} (6) (2005), 809--817.\n\n\\bibitem{lai}\nLi Lai. \\href{https://doi.org/10.1017/S0004972725100543}{On the largest prime divisor of {$n!+1$}}.\n\\emph{Bulletin of the Australian Mathematical Society} \\textbf{113} (3) (2026), 390--403. arXiv:2103.14894.\n\n\\bibitem{lai-lupu-sprang}\nLi Lai, Cezar Lupu and Johannes Sprang. \\href{https://doi.org/10.1007/s40687-025-00559-x}{On the irrationality of certain {$p$}-adic zeta values}.\n\\emph{Research in the Mathematical Sciences} \\textbf{12} (4) (2025), article 77. arXiv:2505.23088.\n\n\\bibitem{lai-2adic}\nLi Lai. \\href{https://doi.org/10.1142/S1793042125500113}{On the irrationality of certain {$2$}-adic zeta values}.\n\\emph{International Journal of Number Theory} \\textbf{21} (1) (2025), 207--235. arXiv:2304.00816.\n\n\\bibitem{nist-dlmf}\n{NIST Digital Library of Mathematical Functions}. \\href{https://dlmf.nist.gov/1.12}{Continued fractions: convergents}.\nOnline resource (2026).\n\n\\bibitem{erdos-stewart1976}\nPaul Erd{\\H o}s and Cameron L. Stewart. \\href{https://doi.org/10.1112/jlms/s2-13.3.513}{On the greatest and least prime factors of {$n!+1$}}.\n\\emph{Journal of the London Mathematical Society} \\textbf{13} (3) (1976), 513--519.\n\n\\bibitem{luca-shparlinski-exp}\nFlorian Luca and Igor E. Shparlinski. \\href{https://doi.org/10.5802/jtnb.524}{On the largest prime factor of {$n!+2^n-1$}}.\n\\emph{Journal de Th{\\'e}orie des Nombres de Bordeaux} \\textbf{17} (3) (2005), 859--870.\n\n\\bibitem{garaev-luca-shparlinski-harmonic}\nMoubariz Z. Garaev, Florian Luca and Igor E. Shparlinski. \\href{https://doi.org/10.1007/s00209-006-0939-5}{Distribution of harmonic sums and {Bernoulli} polynomials modulo a prime}.\n\\emph{Mathematische Zeitschrift} \\textbf{253} (2006), 855--865.\n\n\\bibitem{sondow2006}\nJonathan Sondow. \\href{https://arxiv.org/abs/0704.1282v2}{A geometric proof that {$e$} is irrational and a new measure of its irrationality}.\n\\emph{American Mathematical Monthly} \\textbf{113} (7) (2006), 637--641. arXiv:0704.1282. Addendum: \\emph{Amer. Math. Monthly} \\textbf{114} (2007), 659; reading copy v2.\n\n\\bibitem{hu2026}\nXiyu Hu. \\href{https://arxiv.org/abs/2608.01781v1}{Factorial residues modulo a prime: beyond the square-root bound}.\nPreprint (2026). arXiv:2608.01781. Version 1, 3 August 2026.\n\n\\bibitem{garaev-pardo2026}\nMoubariz Z. Garaev and Julio C. Pardo. \\href{https://doi.org/10.1090/proc/17753}{Additive congruences with factorials modulo a prime}.\n\\emph{Proceedings of the American Mathematical Society} \\textbf{154} (10) (2026), 4179--4190. arXiv:2508.12127. Published online 1 July 2026; assigned October 2026 issue.\n\n\\bibitem{hu-september2026}\nXiyu Hu. \\href{https://arxiv.org/abs/2609.05652v1}{Lower bounds for some value sets over finite fields: incidence geometry and {Bourgain}'s group expansion theorem}.\nPreprint (2026). arXiv:2609.05652. Version 1, 4 September 2026.\n\n\\bibitem{banks-et-al2005}\nWilliam D. Banks, Florian Luca, Igor E. Shparlinski and Henning Stichtenoth. \\href{https://journals.tubitak.gov.tr/math/vol29/iss2/6/}{On the Value Set of $n!$ Modulo a Prime}.\n\\emph{Turkish Journal of Mathematics} \\textbf{29} (2) (2005), 169--174.\n\n\\bibitem{klurman-munsch2017}\nOleksiy Klurman and Marc Munsch. \\href{https://doi.org/10.5802/jtnb.974}{Distribution of factorials modulo $p$}.\n\\emph{Journal de Théorie des Nombres de Bordeaux} \\textbf{29} (1) (2017), 169--177.\n\n\\bibitem{grebennikov-et-al2024}\nAlexandr Grebennikov, Arsenii Sagdeev, Aliaksei Semchankau and Aliaksei Vasilevskii. \\href{https://doi.org/10.4171/RMI/1422}{On the sequence $n!$ mod $p$}.\n\\emph{Revista Matemática Iberoamericana} \\textbf{40} (2) (2024), 637--648.\n\n\\bibitem{stevens-dezeeuw2017}\nSophie Stevens and Frank de Zeeuw. \\href{https://doi.org/10.1112/blms.12077}{An improved point-line incidence bound over arbitrary fields}.\n\\emph{Bulletin of the London Mathematical Society} \\textbf{49} (5) (2017), 842--858. arXiv:1609.06284.\n\n\\bibitem{macleod-barrodale1970}\nR. A. Macleod and I. Barrodale. \\href{https://doi.org/10.4153/CMB-1970-052-8}{On Equal Products of Consecutive Integers}.\n\\emph{Canadian Mathematical Bulletin} \\textbf{13} (2) (1970), 255--259.\n\n\\bibitem{nguyen-tho2026}\n{Nguyen Xuan Tho}. \\href{https://doi.org/10.4171/EM/556}{On equal products of consecutive integers}.\n\\emph{Elemente der Mathematik} \\textbf{81} (3) (2026), 119--124. First published online 10 July 2025.\n\n\\bibitem{erdos-selfridge1975}\nPaul Erd{\\H o}s and John L. Selfridge. \\href{https://doi.org/10.1215/ijm/1256050816}{The product of consecutive integers is never a power}.\n\\emph{Illinois Journal of Mathematics} \\textbf{19} (2) (1975), 292--301.\n\n\\bibitem{nist-vandermonde}\n{NIST Digital Library of Mathematical Functions}.\n\\href{https://dlmf.nist.gov/1.3.E13}{Vandermonde determinant},\n\\S1.3(ii), formula~(1.3.13). Online resource, consulted 18 September 2026.\n\n\\end{thebibliography}\n% ---- part extended_record ----\n% ---- part family_catalogue ----\n% ---- part back ----\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":178274,"sha256":"sha256:0af0d85ec1beab46b63ef9a54e0326abb872d3c38403af575b2ca63ae9dea60d","cite_as":"Plectis reader corpus, paper:erdos68-factorial-reasoning-surface","source_repository":"plectis-erdos","source_path":"paper/68/erdos68-factorial-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:088d36d7ba3d3ee0cee2df99e8915927999698f19889f8ea0b8d55b3721ec778","byte_count":6923,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% This is the only copy of the shared preamble.  The eight short notes, the\n% long records assembled from paper/reasoning-parts other than #249 and #257\n% (which carry their own preambles), and the synthesis paper include it with\n% \\input{problem-note-preamble}; paper/Makefile builds from this directory\n% with the directory on the tectonic search path.\n%\n% Shared preamble for the Erdős Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erdős problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{8e58c395e9807ef3374d7365c62cf04d20ee6331}\n\\newcommand{\\commitshort}{9e231ce4371f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-erdos/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-erdos/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\usepackage{longtable}  % the notes' declaration tables; loaded here so the assembled records inherit it\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n% ---------------------------------------------------------------------------\n% Evidence marks.  paper-evidence.tex places \"Lean\" and \"Comparator\" links in\n% the margin beside each result from the paper's generated file\n% paper/evidence/<paper>.tex, which each paper inputs after this preamble.\n% ---------------------------------------------------------------------------\n\\input{paper-evidence}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Production provenance.  The wording is fixed by the corpus provenance\n% contract and reproduced verbatim in every manuscript in this repository, so\n% the papers cannot drift into different accounts of how they were made.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erdős problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every Lean mark in the margin points to propositions that the\n  pinned Lean kernel accepts, with no \\texttt{sorry}, no added axiom, and no\n  unchecked evaluation.  That claim concerns the formal statements only; it\n  says nothing about their mathematical interest, their novelty, or the\n  original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n\n% Compact opening card for the separately stated Comparator theorem selected\n% from each problem note.  The prose supplied by each note owns the\n% contribution and relevance judgements; the shared final sentence states the\n% mechanical assurance boundary once and points to the public evidence owner.\n\\newcommand{\\externalreviewobject}[3]{%\n  \\begin{quote}\\small\n  \\textbf{Contribution.} #1\\par\n  \\textbf{Relation to the open problem.} #2\\par\n  \\textbf{Executable review object.} #3  The repository's\n  external-verification job compares these exact Lean propositions with\n  separately declared challenge statements and an axiom budget, then asks\n  Lean's kernel to check the submitted proofs.  The\n  \\href{https://github.com/wcook04/plectis-erdos/blob/ca0e13f8acf5ccf48506e4bdb870953d3a0856fa/formalization.yaml}%\n  {formalisation manifest} and the commit-bound CI receipt record that check;\n  they do not assess novelty, significance, or whether the original problem is\n  solved.\n  \\end{quote}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  Under\n% Tectonic (XeTeX) the text and mathematics are set in TeX Gyre Pagella from\n% the bundled OpenType fonts, with a Heros sans range for \\mathsf and\n% Inconsolata for monospace; the pdflatex fallback keeps the same Palatino\n% design through newpx.  The A4 text block uses a classical measure with\n% quiet running heads and restrained ink.  Link colours: steel blue for\n% external (Lean-source and web) targets, deep burgundy for internal\n% cross-references and citations.\n\\NeedsTeXFormat{LaTeX2e}\n\\ProvidesPackage{paper-house-style}[2026/07/19 Pagella house style for the papers]\n\n\\RequirePackage{iftex}\n\\RequirePackage{amsmath}\n\\RequirePackage{amsthm}\n\\ifpdftex\n  \\RequirePackage[T1]{fontenc}\n  \\RequirePackage[utf8]{inputenc}\n  \\RequirePackage{amssymb}\n  \\RequirePackage{newpxtext}\n  \\RequirePackage{newpxmath}\n\\else\n  \\RequirePackage{fontspec}\n  \\RequirePackage{unicode-math}\n  \\setmainfont{texgyrepagella}[Extension=.otf,UprightFont=*-regular,\n    BoldFont=*-bold,ItalicFont=*-italic,BoldItalicFont=*-bolditalic]\n  \\setmathfont{texgyrepagella-math.otf}\n  \\setmathfont{texgyreheros-regular.otf}[range=sfup,Scale=MatchLowercase]\n  \\setmonofont{Inconsolatazi4}[Extension=.otf,UprightFont=*-Regular,\n    BoldFont=*-Bold,Scale=MatchLowercase]\n\\fi\n\n\\RequirePackage[a4paper,textwidth=418pt,textheight=636pt,hcentering,\n  top=68pt,headheight=15pt,headsep=20pt,footskip=34pt]{geometry}\n\\RequirePackage{microtype}\n\\linespread{1.045}\n\\clubpenalty=10000 \\widowpenalty=10000 \\tolerance=1600\n\n\\RequirePackage[dvipsnames,table]{xcolor}\n\\definecolor{linkexternal}{HTML}{275D8C}\n\\definecolor{linkinternal}{HTML}{6E2B2B}\n\\definecolor{papertableheader}{HTML}{E3EAF0}\n\\definecolor{papertablerow}{HTML}{F2F5F7}\n\\definecolor{papertablerule}{HTML}{7D8991}\n\\definecolor{paperresultrule}{HTML}{315B72}\n\n% Stable cross-document links between sibling PDFs. A named destination is\n% more durable than a page number: sections may move while their semantic role\n% remains the same. All public PDFs are copied to one directory at release.\n\\newcommand{\\papersectiontarget}[1]{\\hypertarget{#1}{}}\n\\newcommand{\\papersectionlink}[3]{\\href{#1\\##2}{#3}}\n\n\\RequirePackage{titlesec}\n\\titleformat{\\section}{\\Large\\bfseries}{\\thesection}{0.85em}{}\n\\titlespacing*{\\section}{0pt}{2.6ex plus .8ex minus .3ex}{1.5ex plus .3ex}\n\\titleformat{\\subsection}{\\large\\bfseries}{\\thesubsection}{0.8em}{}\n\\titlespacing*{\\subsection}{0pt}{2.1ex plus .6ex minus .2ex}{1.1ex plus .2ex}\n\\titlespacing*{\\paragraph}{0pt}{1.6ex plus .5ex minus .2ex}{0.85em}\n\n\\RequirePackage{fancyhdr}\n\\newcommand{\\paperrunninghead}{}\n\\newcommand{\\runninghead}[1]{\\renewcommand{\\paperrunninghead}{#1}}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\fancyhead[L]{\\small\\itshape\\paperrunninghead}\n\\fancyhead[R]{\\small\\thepage}\n\\renewcommand{\\headrulewidth}{0pt}\n\\fancypagestyle{plain}{\\fancyhf{}\\renewcommand{\\headrulewidth}{0pt}}\n\n\\RequirePackage{caption}\n\\captionsetup{font=small,labelfont=bf,skip=9pt,width=.92\\linewidth}\n\\RequirePackage{etoolbox}\n\\RequirePackage{pdflscape}\n\\renewcommand{\\arraystretch}{1.20}\n\\newcommand{\\papertablehead}{\\rowcolor{papertableheader}}\n\\newcommand{\\papertablebody}{\\rowcolors{2}{white}{papertablerow}}\n\\AtBeginEnvironment{tabular}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\\AtBeginEnvironment{longtable}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\n% A compact front-matter panel for the result a scanning reader should retain.\n% It is deliberately short and unbreakable: manuscripts should state one\n% result, its evidence, and its limit here, then supply the argument below.\n\\newenvironment{paperresult}[1]{%\n  \\par\\addvspace{0.7em}\n  \\begin{center}\\begin{minipage}{0.94\\linewidth}\n    {\\color{paperresultrule}\\rule{\\linewidth}{0.8pt}}\\par\\vspace{4pt}\n    {\\small\\sffamily\\bfseries\\color{paperresultrule}#1}\\par\\smallskip\n    \\small\\color{black}\n}{%\n    \\par\\vspace{4pt}{\\color{paperresultrule}\\rule{\\linewidth}{0.45pt}}\n    \\end{minipage}\\end{center}\\par\\addvspace{0.5em}\n}\n\n% Shared diagram grammar.  Reader-facing figures use the same restrained\n% hierarchy as the surrounding page: a short eyebrow, a substantial title,\n% one compact explanatory line, and an optional evidence badge.  The styles\n% are installed only when a manuscript loads TikZ.\n\\AtBeginDocument{%\n  \\@ifpackageloaded{tikz}{%\n    \\tikzset{\n      paper box/.style={draw=gray!48, line width=.45pt, rounded corners=3pt,\n        fill=white, align=left, inner xsep=7pt, inner ysep=6pt,\n        font=\\small},\n      paper card/.style={paper box, minimum height=13mm},\n      paper stage/.style={paper box, align=center, minimum height=14mm},\n      paper arrow/.style={-{Stealth[length=2.15mm,width=1.45mm]},\n        line width=.7pt, draw=gray!72!black},\n      paper feedback/.style={paper arrow, densely dashed,\n        draw=Violet!72!black},\n      paper eyebrow/.style={font=\\scriptsize\\sffamily\\bfseries,\n        text=gray!70!black},\n      paper label/.style={font=\\scriptsize\\sffamily,\n        text=gray!72!black, fill=white, inner sep=1.5pt},\n      paper badge/.style={rounded corners=2pt, inner xsep=4pt, inner ysep=2pt,\n        font=\\scriptsize\\sffamily\\bfseries},\n      paper boundary/.style={densely dashed, line width=.55pt,\n        draw=gray!62}\n    }%\n  }{}%\n}\n\n% Front matter: title block with an optional italic subtitle, small-caps\n% author, and a hairline rule; the abstract is set small with a small-caps\n% head and symmetric indents.\n\\newcommand{\\subtitle}[1]{\\gdef\\@papersubtitle{#1}}\n\\renewenvironment{abstract}{%\n  \\vspace{0.1em}\\small\n  \\begin{center}{\\normalsize\\scshape Abstract}\\end{center}\\vspace{-0.5em}\n  \\list{}{\\leftmargin 1.3em\\rightmargin\\leftmargin\\listparindent 0pt\n    \\parsep 0.32em\\itemindent 0pt}\\item\\relax}\n  {\\endlist\\vspace{0.5em}}\n\\def\\@maketitle{%\n  \\newpage\\null\n  \\vspace*{-2.6em}%\n  \\begin{center}%\n    {\\LARGE\\bfseries\\@title\\par}%\n    \\vskip 0.55em\n    {\\large\\itshape\\@papersubtitle\\par}%\n    \\vskip 1.1em\n    {\\scshape\\@author\\textsuperscript{*}\\par}%\n    \\vskip 0.25em\n    {\\small\\@date\\par}%\n    \\vskip 0.8em\n    \\rule{6.5em}{0.4pt}%\n  \\end{center}%\n  \\vskip 0.9em}\n\n% Production provenance, set at the foot of the first page.\n%\n% A reader who meets a single personal byline ordinarily infers that the named\n% author drafted the sentences.  For these papers that inference is false, and\n% a declaration reached only in the back matter would leave it standing for the\n% length of the paper.  The note therefore appears on page one, small and\n% visually subordinate to the title and abstract: enough to prevent the false\n% inference, not enough to make the production method the subject.\n%\n% It carries no mark, because it qualifies the whole manuscript rather than any\n% one passage.  \\footnotetext does not advance the footnote counter, and the\n% emptied \\thefootnote is confined to a group, so numbered footnotes elsewhere\n% in the paper are unaffected.\n%\n% The wording is not authored here.  It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"},{"record_type":"paper_source_dependency","tex_reference":"evidence/erdos68-factorial-reasoning-surface","source_path":"paper/evidence/erdos68-factorial-reasoning-surface.tex","media_type":"text/x-tex","sha256":"sha256:1a4adf5d82ec871f1b62e3b9af6d6e0c6470e6de5b5650392144acd526c84c85","byte_count":12573,"licence":"CC-BY-4.0","body_format":"latex","body":"% Generated by scripts/paper_evidence.py from docs/paper_lean_coverage.json and the\n% Comparator replay receipts under evidence/comparator/.  Do not edit by hand.\n\\newcommand{\\evidencerecordurl}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md}\n\\DeclareResultEvidenceName{long68:res:normalform}{Theorem 1.1}\n\\DeclareResultEvidenceRecord{long68:res:normalform}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-normalform}{exact}{compared}\n\\DeclareResultEvidence{long68:res:normalform}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos68/PaperCompleteSupportedBands.lean\\#L17}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-normalform-comparator}\n\\DeclareResultEvidenceName{long68:res:bandbreakpoint}{Theorem 1.2}\n\\DeclareResultEvidenceRecord{long68:res:bandbreakpoint}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-bandbreakpoint}{exact}{compared}\n\\DeclareResultEvidence{long68:res:bandbreakpoint}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-bandbreakpoint}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-bandbreakpoint-comparator}\n\\DeclareResultEvidenceName{long68:res:moment-ideal}{Theorem 1.3}\n\\DeclareResultEvidenceRecord{long68:res:moment-ideal}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-moment-ideal}{exact}{compared}\n\\DeclareResultEvidence{long68:res:moment-ideal}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-moment-ideal}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-moment-ideal-comparator}\n\\DeclareResultEvidenceName{long68:res:residual-transparency}{Theorem 1.4}\n\\DeclareResultEvidenceRecord{long68:res:residual-transparency}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-residual-transparency}{exact}{compared}\n\\DeclareResultEvidence{long68:res:residual-transparency}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-residual-transparency}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-residual-transparency-comparator}\n\\DeclareResultEvidenceName{long68:res:channel-radius}{Theorem 1.5}\n\\DeclareResultEvidenceRecord{long68:res:channel-radius}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-channel-radius}{exact}{compared}\n\\DeclareResultEvidence{long68:res:channel-radius}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-channel-radius}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-channel-radius-comparator}\n\\DeclareResultEvidenceName{long68:res:radius-constant}{Corollary 1.6}\n\\DeclareResultEvidenceRecord{long68:res:radius-constant}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-radius-constant}{exact}{compared}\n\\DeclareResultEvidence{long68:res:radius-constant}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos68/PaperCompleteLiminf.lean\\#L54}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-radius-constant-comparator}\n\\DeclareResultEvidenceName{long68:res:translator}{Theorem 1.7}\n\\DeclareResultEvidenceRecord{long68:res:translator}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-translator}{exact}{compared}\n\\DeclareResultEvidence{long68:res:translator}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos68/PaperCompleteExisting.lean\\#L299}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-translator-comparator}\n\\DeclareResultEvidenceName{long68:res:prime-pole}{Theorem 2.1}\n\\DeclareResultEvidenceRecord{long68:res:prime-pole}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-prime-pole}{exact}{compared}\n\\DeclareResultEvidence{long68:res:prime-pole}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos68/PaperCompletePrimePole.lean\\#L117}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-prime-pole-comparator}\n\\DeclareResultEvidenceName{long68:res:wilson-cofinality}{Proposition 2.2}\n\\DeclareResultEvidenceRecord{long68:res:wilson-cofinality}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-wilson-cofinality}{exact}{compared}\n\\DeclareResultEvidence{long68:res:wilson-cofinality}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos68/PaperCompleteExisting.lean\\#L180}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-wilson-cofinality-comparator}\n\\DeclareResultEvidenceName{long68:res:product-lcm}{Lemma 3.1}\n\\DeclareResultEvidenceRecord{long68:res:product-lcm}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-product-lcm}{exact}{compared}\n\\DeclareResultEvidence{long68:res:product-lcm}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos68/PaperCompleteExisting.lean\\#L189}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-product-lcm-comparator}\n\\DeclareResultEvidenceName{long68:res:gap-gcd}{Lemma 3.2}\n\\DeclareResultEvidenceRecord{long68:res:gap-gcd}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-gap-gcd}{exact}{compared}\n\\DeclareResultEvidence{long68:res:gap-gcd}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos68/ChannelIntegralCongruence.lean\\#L288}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-gap-gcd-comparator}\n\\DeclareResultEvidenceName{long68:res:segment}{Lemma 3.3}\n\\DeclareResultEvidenceRecord{long68:res:segment}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-segment}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long68:res:segment}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos68/ChannelIntegralCongruence.lean\\#L566}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-segment-comparator}\n\\DeclareResultEvidenceName{long68:res:lcm-growth}{Theorem 3.4}\n\\DeclareResultEvidenceRecord{long68:res:lcm-growth}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-lcm-growth}{exact}{compared}\n\\DeclareResultEvidence{long68:res:lcm-growth}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos68/PaperCompleteLiminf.lean\\#L42}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-lcm-growth-comparator}\n\\DeclareResultEvidenceName{long68:res:carry-equivalence}{Theorem 4.1}\n\\DeclareResultEvidenceRecord{long68:res:carry-equivalence}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-carry-equivalence}{exact}{compared}\n\\DeclareResultEvidence{long68:res:carry-equivalence}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos68/PaperCompleteExisting.lean\\#L77}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-carry-equivalence-comparator}\n\\DeclareResultEvidenceName{long68:res:companion-orbit}{Proposition 4.2}\n\\DeclareResultEvidenceRecord{long68:res:companion-orbit}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-companion-orbit}{exact}{compared}\n\\DeclareResultEvidence{long68:res:companion-orbit}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos68/PaperCompleteExisting.lean\\#L38}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-companion-orbit-comparator}\n\\DeclareResultEvidenceName{long68:res:lower-escape}{Proposition 4.3}\n\\DeclareResultEvidenceRecord{long68:res:lower-escape}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-lower-escape}{exact}{compared}\n\\DeclareResultEvidence{long68:res:lower-escape}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos68/PaperCompleteExisting.lean\\#L102}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-lower-escape-comparator}\n\\DeclareResultEvidenceName{long68:res:shift-family}{Theorem 4.4}\n\\DeclareResultEvidenceRecord{long68:res:shift-family}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-shift-family}{exact}{compared}\n\\DeclareResultEvidence{long68:res:shift-family}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-shift-family}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-shift-family-comparator}\n\\DeclareResultEvidenceName{long68:res:global-residue}{Theorem 5.1}\n\\DeclareResultEvidenceRecord{long68:res:global-residue}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-global-residue}{exact}{compared}\n\\DeclareResultEvidence{long68:res:global-residue}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos68/PaperCompleteExisting.lean\\#L154}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos68-factorial-reasoning-surface.md\\#long68-res-global-residue-comparator}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/68/erdos68-factorial-reasoning-surface.tex","section_outline":["Integer coefficient vectors and cancellation","Examples and support refinements","Support bounds and nonintegrality","A primitive solution on an arithmetic progression","Cofactors give the same primitive vector","Prime powers in reduced denominators","Growth of the common denominator","Consequences for the support parameter","Carries and factorial digits","The adjacent-unit window condition","Escape from a smaller interval","A complementary-denominator criterion","Finite denominator exclusions","Further arithmetic questions","Sources and evidence","Guide to the formal sources","Source links from the short paper","Further deductions and limitations","Factorial digits and their remainders","How the classical criteria apply here","A superseded deduction","Rational grid points and first crossings","Exact identities for shared prime powers","Finite vectors and exact numerical examples","Limits of the recurrences and clearing factors used here","An elementary criterion for any real number","Relations among the criteria","Acknowledgements"]},"erdos-243-reciprocal-tail-rigidity":{"paper_id":"erdos-243-reciprocal-tail-rigidity","title":"Reciprocal Sums and the Sylvester Recurrence","form":"Problem note","length_class":"short","pages":21,"subject":"Erdős #243","subject_kind":"problem","problem_id":"erdos_243","landing_role":"mathematics","question":"Which growth rates force irrational reciprocal sums, and which additional tail conditions force the Sylvester recurrence?","precis":"For strictly increasing positive integers with a_n²/a_{n+1}=1+3/n+o(n^{-3}), the reciprocal sum is irrational. The argument extracts a cubic polynomial from an integer tail numerator, restricts its coefficients using a square in a cubic field, and excludes the remaining cases modulo seven. Under rationality and the quadratic growth limit, separate criteria using bounded numerator increases and weighted first crossings force the eventual Sylvester recurrence. The unrestricted Erdős–Graham question remains open.","owns":"cubic-rate irrationality, nonintegral regular-rate exclusion and sufficient tail criteria for the Sylvester recurrence; the unrestricted Erdős–Graham question remains open","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #243, which remains open","first_pass":["sec:problem","sec:transfer","sec:secondaryrate"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-243-reciprocal-tail-rigidity.pdf","hosted_pdf_path":"papers/erdos-243-reciprocal-tail-rigidity.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex","site_html":"maths/papers/erdos-243-reciprocal-tail-rigidity.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/243/erdos-243-reciprocal-tail-rigidity.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/243/erdos-243-reciprocal-tail-rigidity.tex","body_pointer":"papers.records.erdos-243-reciprocal-tail-rigidity.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Short mathematical paper. Compile from the supplied public/paper directory.\n% Equivalent restatements are declared in STATEMENT_CHANGES.json; pinned evidence is retained.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n\\input{evidence/erdos-243-reciprocal-tail-rigidity}\n% Pinned revision includes the checked cubic-rate theorem.\n\\renewcommand{\\commit}{7380b7871687b6bcc41ca0143c61f232e8af6500}\n\\renewcommand{\\commitshort}{7380b7871687}\n% That revision keeps the Lean roots under lean/, so the links name it.\n\\renewcommand{\\PX}{lean/ErdosProblems}\n% Sources that postdate \\commit are cited with their lean/ path and resolve at \\latecommit,\n% whose tree keeps the Lean roots under lean/.\n\\newcommand{\\latecommit}{168bf6727758f918a430ef056a1c93d3160b53a6}\n\\newcommand{\\laterepobase}{https://github.com/wcook04/plectis-erdos/blob/\\latecommit}\n\\hypersetup{\n  pdftitle={Reciprocal sums and the Sylvester recurrence for Erdős Problem 243},\n  pdfsubject={Erdős Problem Notes: Problem 243},\n  pdfkeywords={irrationality, Ahmes series, Sylvester sequence, unit fractions, Lean 4},\n}\n\n% Operators used repeatedly in the mathematical arguments.\n\\newcommand{\\fnum}{\\operatorname{num}}\n\n\\runninghead{Reciprocal sums and the Sylvester recurrence}\n\\title{Reciprocal Sums and the Sylvester Recurrence}\n\\subtitle{Erd\\H{o}s Problem~\\#243}\n\\author{Will Cook}\n\\date{September 2026}\n\n\\begin{document}\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\nWe prove irrationality of the reciprocal sum when strictly increasing\npositive integers satisfy $a_n^2/a_{n+1}=1+3/n+o(n^{-3})$.\nRationality would give integer tail numerators whose fourth differences\neventually vanish. The denominator recurrence then forces a square in a\ncubic field; trace identities and a four-term calculation modulo seven\ngive a contradiction. Under rationality and the quadratic growth limit,\nwe also give separate criteria for the eventual Sylvester recurrence\nusing bounded numerator increases and weighted first crossings. The unrestricted Erd\\H{o}s--Graham question remains open.\n\\end{abstract}\n\n\\section{Introduction}\\label{sec:problem}\nThe Sylvester recurrence $a_{n+1}=a_n^2-a_n+1$ gives a rational\nreciprocal sum by telescoping:\n\\[\n \\frac1{a_n-1}=\\frac1{a_n}+\\frac1{a_{n+1}-1}.\n\\]\nFor an increasing sequence of positive integers that eventually satisfies\nthis recurrence, the tail sum at every sufficiently large index is\n$1/(a_n-1)$. Erd\\H{o}s and Graham asked whether every rational\nreciprocal sum with $a_{n+1}\\sim a_n^2$ arises in this way.\n\n\\begin{problem}[Erd\\H{o}s Problem 243]\\label{res:problem}\nLet $1\\le a_1<a_2<\\cdots$ be a sequence of integers with\n\\[\n \\lim_{n\\to\\infty}\\frac{a_n}{a_{n-1}^{2}}=1\n \\qquad\\text{and}\\qquad\n \\sum\\frac{1}{a_n}\\in\\Q .\n\\]\nThen $a_n=a_{n-1}^{2}-a_{n-1}+1$ for all sufficiently large $n$.\n\\end{problem}\n\nErd\\H{o}s and Graham \\cite[p.~64]{erdosgraham1980} asked this question;\nsee also Erd\\H{o}s \\cite[p.~105]{erdos1988} and Bloom's catalogue\n\\cite{erdosproblems}. We prove the following irrationality result for a\nrestricted class of sequences with $a_{n+1}\\sim a_n^2$.\n\n\\begin{theorem}[cubic-rate irrationality]\\label{res:cubicrate}\nA strictly increasing sequence of positive integers with\n\\[\n a_n^2/a_{n+1}=1+\\frac3n+o(n^{-3})\n\\]\nhas irrational reciprocal sum.\n\\end{theorem}\n\nThe Sylvester recurrence gives $a_n^2/a_{n+1}-1=O(1/a_n)$, so the\ncubic-rate hypothesis excludes that behaviour. It imposes much more\nthan the quadratic limit in Problem~\\ref{res:problem}.\n\nFor example, set $a_1=8$ and\n\\[\n a_{n+1}=\\left\\lceil\\frac{n a_n^2}{n+3}\\right\\rceil.\n\\]\nThis gives $8,16,103,5305,\\ldots$. From $a_{n+1}\\ge a_n^2/4$ we\nobtain $a_n\\ge4\\cdot2^{2^{n-1}}$ by induction. The error introduced\nby the ceiling is less than one, and hence\n\\[\n 0\\le 1+\\frac3n-\\frac{a_n^2}{a_{n+1}}<\\frac{16}{a_n^2}=o(n^{-3}).\n\\]\nThe theorem applies to this explicitly defined sequence.\n\nKoizumi's product criterion \\cite[Corollary~4]{koizumi2025} requires\nthe increments of $P_n/a_n$, where $P_n=\\prod_{j<n}a_j$, to have\nnonpositive upper limit. Corollary~\\ref{res:originalbounded} below allows\nany finite upper limit. At the cubic rate, however, $P_n/a_n$ grows\nlike $n^3$ and its increments like $n^2$; neither criterion applies.\nThis comparison does not decide whether the LCM criteria apply: those\nalso depend on repeated factors in $P_n$.\n\nRationality supplies a positive integer tail numerator $C_n$, but not\na bounded one: here it grows cubically. We therefore apply integrality\nto its finite differences. The exact comparison sequence\n$F_n=n(n+1)(n+2)$ has $F_{n+1}/F_n=1+3/n$. The precision in the\nhypothesis forces $\\Delta^4C_n\\to0$, hence eventual equality\n$C_n=AF_n+B$. This extraction uses the numerator ratio and integrality, not the\nexact denominator recurrence.\n\nThe denominator recurrence now becomes decisive. It imposes a square\ncondition at every root of a cubic modulo almost every prime.\nChebotarev's theorem turns these residue conditions into a square in a\ncubic number field, where traces restrict the numerator to two\npolynomials. A four-term calculation modulo seven excludes both.\nSection~\\ref{sec:secondaryrate} follows these two stages: first extract\nthe polynomial, then test whether it can satisfy the exact recurrence.\n\nSections~\\ref{sec:bounded} and~\\ref{sec:mass} give bounded-increment\nand summability criteria. Section~\\ref{sec:lcmrecords} clears tails by\na least common multiple and uses congruences to control first crossings\nof new heights; its convergence hypothesis is not derived from the\nquadratic limit. The\n\\href{erdos243-reciprocal-tail-reasoning-surface.pdf}{companion paper}\nalso treats nonintegral rates and proves a stronger cubic disagreement\ntheorem. Its proof locations and the formal sources are listed in\nAppendix~\\ref{app:index}.\n\n\\section{Tail numerators}\\label{sec:transfer}\\label{sec:state}\nFor the Sylvester sequence $2,3,7,43,\\ldots$, whose reciprocal sum is\n$1$, the successive tails are $1,1/2,1/6,1/42,\\ldots$. Multiplication\nby the products $1,2,6,42,\\ldots$ leaves the constant integer numerator\n$1$. For any rational reciprocal sum, including a denominator of the whole\nsum in these products again gives integer numerators. We keep these\nfractions unreduced: the exact multiplicative denominator update will\ntransfer the growth rate to the numerator.\n\nAssume now that $a_n$ are strictly increasing positive integers,\n$a_{n+1}/a_n^2\\to1$, and $\\sum_{n\\ge1}1/a_n=p/q$ with positive integers\n$p,q$. All indices in this construction start at $1$. Put\n\\[\n x_n=\\sum_{k\\ge n}\\frac1{a_k},\\qquad\n P_n=\\prod_{1\\le j<n}a_j,\\qquad D_n=qP_n,\\qquad C_n=D_nx_n.\n\\]\nHere $P_1=1$, $D_1=q$ and $C_1=p$. Since every $a_k$ with $k<n$\ndivides $P_n$, subtracting the finite prefix from $p/q$ gives\n\\[\n C_n=pP_n-q\\sum_{1\\le k<n}\\frac{P_n}{a_k}\\in\\N_{>0}.\n\\]\nPositivity comes from $C_n=D_nx_n$. Removing $1/a_n$ from the tail gives\n\\begin{equation}\\label{eq:recurrences}\n C_{n+1}=a_nC_n-D_n,\\qquad D_{n+1}=a_nD_n.\n\\end{equation}\nKoizumi constructs these coordinates in Lemma~4\n\\cite[pp.~11--12]{koizumi2025}. On a pseudo-greedy tail his\n$c_n,d_n,e_n$ correspond to $C_n,D_n,D_n-(a_n-1)C_n$. If the tail\nis restarted with its value in lowest terms, both coordinates are\ndivided by the same fixed integer. Reducing at every step would instead\nintroduce varying cancellation factors into the numerator ratio. We\ntherefore extract the polynomial before dividing by the eventual fixed\ngcd in Section~\\ref{sec:cubic-normalisation}.\n\nFor large $n$, $a_{n+1}\\ge a_n^2/2\\ge2a_n$. The terms after\n$1/a_{n+1}$ sum to at most $2/a_{n+2}\\le4/a_{n+1}^2$, so\n\\begin{equation}\\label{eq:tail-estimate}\n x_n=\\frac1{a_n}+\\frac1{a_{n+1}}+O(a_{n+1}^{-2}),\n \\qquad\n \\frac{C_{n+1}}{C_n}=\\frac{a_n^2}{a_{n+1}}+O(1/a_n).\n\\end{equation}\nFor the second estimate, the first gives $a_nx_n=1+O(1/a_n)$. Use this in\n\\[\n \\frac{C_{n+1}}{C_n}\n =\\frac{a_n^2}{a_{n+1}}\\,\n  \\frac{a_{n+1}x_{n+1}}{a_nx_n}.\n\\]\nThe last factor is $1+O(1/a_n)$, and $a_n^2/a_{n+1}\\to1$, giving the\nstated additive error. Quadratic growth makes $1/a_n=o(n^{-k})$ for every fixed $k$,\nso this error preserves the precision required in the cubic theorem.\nWe keep the original index $n$ throughout that proof. Although an\neventual recurrence survives removal of a finite prefix, the expression\n$1+3/n+o(n^{-3})$ changes its lower-order terms under an index shift.\n\n\\section{Proof of cubic-rate irrationality}\\label{sec:secondaryrate}\nSuppose, towards a contradiction, that the reciprocal sum in\nTheorem~\\ref{res:cubicrate} is rational. Construct $C_n,D_n$ as in\nSection~\\ref{sec:transfer}. Equation~\\eqref{eq:tail-estimate} transfers\nthe rate without changing its precision:\n\\begin{equation}\\label{eq:cubic-ratio}\n \\frac{C_{n+1}}{C_n}=1+\\frac3n+o(n^{-3}).\n\\end{equation}\n\\subsection{An eventual cubic polynomial}\\label{sec:cubic-extraction}\nWe use $F_n=n(n+1)(n+2)$ rather than $n^3$: its successive ratio is\n\\emph{exactly} $1+3/n$, so dividing by it removes the model without\nintroducing lower-order errors. Write\n\\eqref{eq:cubic-ratio} as $C_{n+1}=(1+3/n+\\varepsilon_n)C_n$,\nwhere $\\varepsilon_n=o(n^{-3})$. Dividing out the exact model gives\n\\[\n z_n=\\frac{C_n}{F_n},\\qquad\n \\frac{z_{n+1}}{z_n}=1+\\frac{\\varepsilon_n}{1+3/n}.\n\\]\nThe logarithmic increments are $o(n^{-3})$, so their absolute tails\nare $o(n^{-2})$. Hence $z_n\\to A>0$ with $z_n-A=o(n^{-2})$.\nMultiplying by $F_n$ gives\n\\[\n C_n=AF_n+\\delta_n,\\qquad \\delta_n=o(n).\n\\]\n\nAn $o(n)$ error can still oscillate too much for finite differences.\nTo control its first difference, subtract the numerator update for $C_n$\nand the comparison update for $AF_n$:\n\\[\n \\Delta\\delta_n=\\frac3n\\delta_n+\\varepsilon_n C_n=o(1),\n \\qquad \\Delta h_n=h_{n+1}-h_n.\n\\]\nHere $C_n=O(n^3)$ makes the second term $o(1)$. Three more differences\nstill tend to zero and kill the cubic model:\n\\[\n \\Delta^4 C_n=\\Delta^3(\\Delta\\delta_n)\\longrightarrow0.\n\\]\nIntegrality turns this limit into eventual vanishing. Newton\ninterpolation then expresses $C_n$ as a rational polynomial of degree\nat most three. Since its difference from $AF_n$ is $o(n)$, the cubic,\nquadratic and linear coefficients agree; only a constant can remain:\n\\begin{equation}\\label{eq:cubic-shape}\n C_n=A n(n+1)(n+2)+B,\\qquad A\\in\\Q_{>0},\\quad B\\in\\Q.\n\\end{equation}\n\nThe precision matters at the subtraction step. The integer sequence\n$F_n+(-1)^n$ has ratio $1+3/n+O(n^{-3})$ and never agrees eventually\nwith a polynomial; its error has an oscillating first difference.\nThis example concerns polynomial extraction alone. It is not asserted\nto satisfy the denominator recurrence.\n\n\\subsection{Reduction and the constant term}\\label{sec:cubic-normalisation}\\label{sec:reduction}\nWe now use the denominator recurrence. Both updates preserve common\ndivisors, so $G_n=\\gcd(C_n,D_n)$ divides $G_{n+1}$ and every later\nnumerator. On the polynomial tail it therefore divides the third\ndifference $6A$, a fixed positive integer. The fourth difference gave\npolynomiality; the nonzero third difference now bounds cancellation.\nThus $G_n$ stabilises at a positive integer $g$.\n\nDividing by this fixed $g$ gives $u_n=C_n/g$, $v_n=D_n/g$ in lowest\nterms without changing the recurrences. Their coprimality persists as\nfollows.\n\n\\begin{proposition}[persistent coprimality]\\label{res:reduced}\nLet $a_n,u_n,v_n$ be integer sequences with $u_n>0$, $v_n\\ge0$ and\n$\\gcd(u_n,v_n)=1$ satisfying\n\\[\n u_{n+1}=a_nu_n-v_n,\\qquad v_{n+1}=a_nv_n.\n\\]\nThen $\\gcd(a_n,v_n)=1$. The $a_n$ are pairwise coprime, and\n$\\gcd(a_i,u_t)=1$ whenever $i<t$.\n\\end{proposition}\n\\begin{proof}\nA prime dividing both $a_n$ and $v_n$ would divide $u_{n+1}$ and\n$v_{n+1}$, contrary to reducedness. For $i<t$, the factor $a_i$ divides\n$v_t$. Reducedness gives $\\gcd(a_i,u_t)=1$, and\n$\\gcd(a_t,v_t)=1$ gives $\\gcd(a_i,a_t)=1$.\n\\end{proof}\n\nIn the cubic case, adjacent numerators are also coprime, since\n$\\gcd(u_n,u_{n+1})=\\gcd(u_n,v_n)=1$.\nWrite the polynomial for $u_n$ as\n\\begin{equation}\\label{eq:normal-cubic}\n Q(n)=\\frac m6 n(n+1)(n+2)+c.\n\\end{equation}\nThe third difference $m=6A/g$ is a positive integer. Since\n$n(n+1)(n+2)/6=\\binom{n+2}{3}$ is an integer, any sufficiently late\nvalue $u_n=Q(n)$ gives\n\\[\n c=u_n-m\\binom{n+2}{3}\\in\\Z.\n\\]\nAdjacent coprimality now restricts this integer $c$.\nIf a prime $\\ell$ divides $c$, choose an arbitrarily late\n$n\\equiv-1\\pmod{6\\ell}$. Both three-factor products contain $n+1$,\nso division by $6$ leaves a factor $\\ell$, even for $\\ell=2,3$.\nThus $\\ell$ divides both $Q(n)$ and $Q(n+1)$, contradicting adjacent\ncoprimality. This also excludes $c=0$.\nThus\n\\[\n m\\in\\N_{>0},\\qquad c\\in\\{1,-1\\}.\n\\]\n\nIrreducibility follows from persistence of denominator primes.\nThe pairwise coprime multipliers $a_n>1$ supply infinitely many distinct\nprimes; each divides every later $v_t$ and no later $u_t$.\nA rational root of $Q$ would reduce to a root modulo all but finitely\nmany primes. Choose a persistent prime outside those exceptions.\nEvery sufficiently late index in the root's residue class would give\n$\\ell\\mid Q(t)=u_t$, contradicting reducedness. Thus $Q$ has no rational\nroot and, being cubic, is irreducible.\n\nSet $\\kappa=m/6$, $\\eta=6c/m$, and\n\\[\n f(T)=T^3-T+\\eta,\\qquad Q(n)=\\kappa f(n+1).\n\\]\n\\subsection{A square forced by a zero numerator}\\label{sec:cubic-field}\nAt a zero of the numerator polynomial modulo a prime, the recurrence\nmust still relate the two neighbouring nonzero numerators. Eliminating\n$v_n$ from two exact updates makes this constraint visible:\n\\begin{equation}\\label{eq:three-tail}\n u_{n+2}=(a_n+a_{n+1})u_{n+1}-a_n^2u_n.\n\\end{equation}\nWhenever $u_k\\equiv0\\pmod\\ell$, it follows that\n\\[\n -u_{k-1}u_{k+1}\\equiv(a_{k-1}u_{k-1})^2\\pmod\\ell.\n\\]\nThus the negative product of the neighbouring numerators must be a\nsquare. Since our numerator is a polynomial, its roots modulo $\\ell$\ntell us exactly where to apply this observation.\n\nLet $\\ell\\nmid6m$ be prime and let $r\\in\\mathbb F_\\ell$ satisfy\n$f(r)=0$. The constant $\\eta=6c/m$ is nonzero modulo $\\ell$, so\n$r\\notin\\{0,1,-1\\}$. Choose a sufficiently late $k$ with\n$k+1\\equiv r\\pmod\\ell$. Then $u_k=\\kappa f(k+1)\\equiv0$, whereas\n\\[\n f(r-1)=-3r(r-1),\\qquad f(r+1)=3r(r+1).\n\\]\nConsequently, in $\\mathbb F_\\ell$,\n\\[\n (a_{k-1}u_{k-1})^2=-u_{k-1}u_{k+1}\n   =9\\kappa^2r^2(r^2-1).\n\\]\nDividing by the nonzero square $9\\kappa^2r^2$ gives\n\\begin{equation}\\label{eq:all-root-square}\n \\ell\\nmid6m,\\quad r\\in\\mathbb F_\\ell,\\quad f(r)=0\n \\quad\\Longrightarrow\\quad r^2-1\\in(\\mathbb F_\\ell^\\times)^2\n \\qquad(\\ell\\text{ prime}).\n\\end{equation}\nThe condition holds for \\emph{every} root; primes with no root impose\nno condition. This quantifier matters in passing to a number field.\n\nLet $\\alpha$ be a root of $f$ and put $K=\\Q(\\alpha)$, a cubic field.\nSuppose $\\alpha^2-1$ is a nonsquare in $K$, and choose $\\beta$ with\n$\\beta^2=\\alpha^2-1$. To contradict the modular condition, we seek a\nFrobenius that fixes $\\alpha$ but exchanges $\\beta$ and $-\\beta$.\nThe nontrivial automorphism of $K(\\beta)/K$ does exactly this. Extend\nit to $\\sigma\\in\\operatorname{Gal}(M/\\Q)$, where $M$ is a Galois\nclosure of $K(\\beta)/\\Q$.\n\nChebotarev's density theorem\n\\cite[\\S3, author-version p.~15]{stevenhagenlenstra1996} supplies\ninfinitely many unramified rational primes with Frobenius conjugate\nto $\\sigma$; choosing a prime of $M$ above each makes it equal to\n$\\sigma$. Omit residue characteristic two, primes dividing $6m$, and\nthe finitely many primes where $\\alpha$ or $\\beta$ is nonintegral or\n$\\beta$ reduces to zero. Frobenius acts by the $\\ell$-th power on the\nresidue field, so\n\\[\n \\bar\\alpha^\\ell=\\bar\\alpha,\\qquad\n \\bar\\beta^\\ell=-\\bar\\beta\\ne\\bar\\beta.\n\\]\nThus $r=\\bar\\alpha$ lies in $\\mathbb F_\\ell$ and satisfies $f(r)=0$,\nbut neither root $\\pm\\bar\\beta$ of $X^2-(r^2-1)$ lies in that field.\nThe condition for \\emph{every} root in \\eqref{eq:all-root-square}\napplies to this $r$, giving the contradiction. Hence some $\\beta\\in K$\nsatisfies $\\beta^2=\\alpha^2-1$.\n\n\\subsection{Traces in the cubic field}\\label{sec:cubic-trace}\nWe now need the square relation to constrain the coefficient $\\eta$.\nFactoring it produces reciprocal elements, avoiding a separate expansion\nof the square root in a basis of $K$:\n\\[\n (\\alpha+\\beta)(\\alpha-\\beta)=1.\n\\]\nPut $z=\\alpha+\\beta$. Then $z^{-1}=\\alpha-\\beta$ and\n$2\\alpha=z+z^{-1}$. Since $z\\in K$ and $\\alpha\\in\\Q(z)$, we have\n$\\Q(z)=K$. Write the minimal polynomial of $z$ as\n\\[\n p(T)=T^3+bT^2+dT+w,\\qquad b,d,w\\in\\Q,\\quad w\\ne0.\n\\]\nThe minimal polynomial of $z^{-1}$ is the monic reciprocal polynomial\n\\[\n \\frac{T^3p(T^{-1})}{w}\n   =T^3+\\frac dwT^2+\\frac bwT+\\frac1w.\n\\]\nWe compute the same three traces from $f$ and from the reciprocal\npair $z,z^{-1}$. All traces are from $K$ to $\\Q$. The cubic for\n$\\alpha$ gives:\n\\[\n \\operatorname{Tr}(\\alpha)=0,\\qquad\n \\operatorname{Tr}(\\alpha^2)=2,\\qquad\n \\operatorname{Tr}(\\alpha^3)=-3\\eta.\n\\]\nNewton's identities for $p$ and its reciprocal give\n\\begin{equation}\\label{eq:cubic-traces}\n d=-bw,\\qquad b^2+b(w-w^{-1})=1,\\qquad\n \\eta=\\frac{(b^2+1)(w+w^{-1})}{8}.\n\\end{equation}\nFor completeness, the first equality follows from\n$0=2\\operatorname{Tr}(\\alpha)=-b-d/w$. After substituting $d=-bw$,\nNewton's identities give\n\\[\n \\begin{aligned}\n \\operatorname{Tr}(z^2)+\\operatorname{Tr}(z^{-2})\n   &=2b^2+2b(w-w^{-1}),\\\\\n \\operatorname{Tr}(z^3)+\\operatorname{Tr}(z^{-3})\n   &=-3(b^2+1)(w+w^{-1}).\n \\end{aligned}\n\\]\nThe first line is $4\\operatorname{Tr}(\\alpha^2)-6=2$.\nThe second is $8\\operatorname{Tr}(\\alpha^3)=-24\\eta$, because the\nmixed terms in $(z+z^{-1})^3$ have trace\n$3\\operatorname{Tr}(z+z^{-1})=0$. These give the other two equalities\nin \\eqref{eq:cubic-traces}.\n\nMultiplying the middle equation by $w$ and bringing all terms to one\nside gives $(bw-1)(b+w)=0$. Write $w=r/s$ in\nlowest terms, with $r\\ne0$, $s>0$. Substituting $b=-w$ or $b=1/w$\ninto the last equation and using $\\eta=6c/m$ gives respectively\n\\[\n m=\\frac{48c\\,r s^3}{(r^2+s^2)^2}\n \\qquad\\text{or}\\qquad\n m=\\frac{48c\\,r^3s}{(r^2+s^2)^2}.\n\\]\nHere the integer scale $m$ is decisive: coprimality\n$\\gcd(r^2+s^2,rs)=1$ prevents cancellation with the variable factors\nin either numerator, so $(r^2+s^2)^2$ divides $48$. Hence $r^2+s^2$ is $1$, $2$ or $4$.\nBoth $r$ and $s$ are nonzero, which leaves only $2$: a sum of two\nnonzero integer squares cannot equal $1$ or $4$.\nThus $|r|=s=1$, and positivity of $m$ gives $m=12$.\nThe only remaining numerators are\n\\[\n Q(n)=2n(n+1)(n+2)+1\n \\quad\\text{and}\\quad\n Q(n)=2n(n+1)(n+2)-1.\n\\]\n\n\\subsection{The contradiction modulo seven}\\label{sec:cubic-mod-seven}\nThe three-term square condition has reduced the possibilities to two\ncubics. We now use one more term: the denominator at the middle zero\nmust also come from the preceding exact update.\nIn the plus case, choose a late block starting at $n\\equiv0\\pmod7$;\nin the minus case, start at $n\\equiv1\\pmod7$. The four successive\nnumerators have residues\n\\[\n (u_n,u_{n+1},u_{n+2},u_{n+3})\\equiv\n \\begin{cases}(1,6,0,2)&(c=1),\\\\(4,5,0,1)&(c=-1).\n \\end{cases}\n\\]\nWrite $d_0,d_1$ for the residues of $v_n,v_{n+1}$ and\n$c_0,c_1,0,c_3$ for the displayed numerator residues. From\n$0=a_{n+1}c_1-d_1$ and $c_3=-a_{n+1}d_1$ we get\n$d_1^2=-c_1c_3=2$. Thus $d_1\\in\\{3,4\\}$.\nThe first update, however, gives\n\\[\n d_1=a_nd_0=\\frac{d_0(d_0+c_1)}{c_0}.\n\\]\nCompleting the square in these two cases gives, in $\\mathbb F_7$,\n\\[\n d_1+2=\n \\begin{cases}\n (d_0+3)^2 &(c=1),\\\\\n 2(d_0-1)^2 &(c=-1).\n \\end{cases}\n\\]\nSince $2=3^2$ in this field, $d_1+2$ must be a square in either case.\nBut $d_1\\in\\{3,4\\}$ would give $d_1+2\\in\\{5,6\\}$, both nonsquares.\nThe middle three terms admit $d_1=3$ or $4$; neither extends\nbackwards through the preceding exact update. Both cubics are excluded,\nproving Theorem~\\ref{res:cubicrate}.\n\nSection~2 of the companion proves positive lower density of\ndisagreement with each rational cubic of this form. That lower bound\ndepends on the orbit and the cubic. For Theorem~\\ref{res:cubicrate},\nexclusion of eventual equality suffices.\n\\section{Bounded increases}\\label{sec:bounded}\nWe return to the rational tails of Section~\\ref{sec:transfer}.\nThe Sylvester example had a constant integer numerator. To measure\ndeparture from this behaviour, put\n\\[\n E_n=D_n-(a_n-1)C_n=C_n-C_{n+1}.\n\\]\nNegative errors are rises of $C_n$; zero error gives the tail\n$1/(a_n-1)$. Instead of prescribing a polynomial, we now bound the\nrises. Quadratic growth makes all steps small relative to $C_n$;\nthe additional absolute bound applies only to rises, not decreases.\n\n\\begin{theorem}[bounded negative part]\\label{res:bounded}\nLet $a,C,D:\\N\\to\\N$ satisfy $a_n>1$, $C_n>0$, and\n\\[\n C_{n+1}+D_n=a_nC_n,\\qquad D_{n+1}=a_nD_n\n \\quad(n\\ge0).\n\\]\nPut $E_n=D_n-(a_n-1)C_n$. If\n\\[\n \\frac{E_n}{C_n}\\longrightarrow0\n \\quad\\text{and}\\quad E_n\\ge-B\\quad\\text{for all sufficiently large }n\n\\]\nfor some integer $B\\ge0$, then $E_n=0$ for all sufficiently large $n$.\n\\end{theorem}\nThe linked theorem uses the equivalent division-free condition:\nfor every $K\\in\\N$, eventually $K|E_n|<C_n$. Under this condition,\nabsorption either settles the theorem or leaves $C_n\\to\\infty$.\nA stable gcd and a forbidden block will exclude the latter alternative.\n\n\\subsection{Zero errors and descent}\\label{sec:defect}\\label{sec:descent}\nWrite $z_+=\\max(z,0)$. The identities below use\n\\eqref{eq:recurrences}; positivity and $|E_n|<C_n$ are imposed only\nwhere stated.\n\n\\begin{proposition}[error identities]\\label{res:update}\\label{res:defect}\nFor integer sequences satisfying \\eqref{eq:recurrences},\n\\[\n C_{n+1}=C_n-E_n,\\qquad\n \\bigl(a_{n+1}-a_n^2+a_n-1\\bigr)C_{n+1}=a_n^2E_n-E_{n+1}.\n\\]\n\\end{proposition}\n\\begin{proof}\nThe first identity follows by substituting $D_n=E_n+(a_n-1)C_n$\ninto the numerator recurrence. Substituting this expression at the\nnext index and using $D_{n+1}=a_nD_n$ gives the second.\n\\end{proof}\n\nEquation~\\eqref{eq:tail-estimate} now gives\n$E_n/C_n=1-C_{n+1}/C_n\\to0$, so $|E_n|<C_n$ for all sufficiently large $n$.\nKoizumi proves this relative-gap limit for an eventual pseudo-greedy\ntail \\cite[Corollary~3, p.~9]{koizumi2025}. His rounding convention\nalso gives $-C_n/2\\le E_n<C_n/2$, a stronger bound than we need.\n\nKoizumi's Lemma~3 gives zero absorption for pseudo-greedy tails, and\nProposition~1(2) uses descent when their gaps are eventually nonnegative\n\\cite[p.~10 and p.~14]{koizumi2025}. The following form separates\nwhat the two arguments require.\n\n\\begin{theorem}[absorption and descent]\\label{res:absorb}\\label{res:descent}\nLet $a,C,D:\\N\\to\\N$ satisfy $a_n>1$, $C_n>0$ and\n\\eqref{eq:recurrences}. If $|E_n|<C_n$ for every $n$, then $E_n=0$\nimplies $E_{n+1}=0$. For any positive integer sequence $C_n$ with\n$C_{n+1}=C_n-E_n$, eventual nonnegativity of $E_n$ implies its eventual\nvanishing.\n\\end{theorem}\n\\begin{proof}\nWhen $E_n=0$, the second identity in Proposition~\\ref{res:update}\nshows that $C_{n+1}$ divides $E_{n+1}$. The inequality\n$|E_{n+1}|<C_{n+1}$ then forces $E_{n+1}=0$. For the second assertion,\n$C_{n+1}=C_n-E_n$ makes the positive integer sequence $C_n$ eventually\nnonincreasing, hence constant.\n\\end{proof}\n\n\\begin{corollary}[two zero errors]\\label{res:step}\\label{res:eventual}\nFor integer sequences satisfying \\eqref{eq:recurrences}, if\n$E_n=E_{n+1}=0$ and $C_{n+1}\\ne0$, then\n$a_{n+1}=a_n^2-a_n+1$. In particular, if $a,C,D:\\N\\to\\N$\nalso satisfy $a_n>1$ and $C_n>0$, eventual zero error implies\nthe eventual Sylvester recurrence.\n\\end{corollary}\n\\begin{proof}\nCancel the nonzero factor $C_{n+1}$ in the second error identity.\n\\end{proof}\n\nThus a late zero is absorbing. Without late zeros, $|E_n|\\ge1$ and\n$|E_n|/C_n\\to0$ force $C_n\\to\\infty$. The scalar update permits\n$C_n=n+1$, $E_n=-1$ forever; excluding this behaviour requires the\ndenominator recurrence.\n\n\\subsection{A stable gcd and a forbidden block}\\label{sec:barrier}\n\\begin{proposition}[gcd stabilisation]\\label{res:gcdstab}\nLet $a,C,D:\\N\\to\\N$ satisfy $a_n>1$, $C_n>0$ and\n\\eqref{eq:recurrences}. If some fixed integer $B\\ge1$ satisfies\n$-B\\le E_n<0$ at infinitely many indices, then\n$G_n=\\gcd(C_n,D_n)$ is eventually constant. Dividing $C_n,D_n$ by\nits eventual value gives coprime sequences satisfying the same recurrences.\n\\end{proposition}\n\\begin{proof}\nWe have $G_n\\mid G_{n+1}$ and $G_n\\mid E_n$. Given $n$, choose $t\\ge n$\nwith $-B\\le E_t<0$. Then $G_n\\le G_t\\le -E_t\\le B$.\nHence the positive divisibility chain stabilises. Division by its\nstable value preserves the recurrences and gives coprime numerators and denominators.\n\\end{proof}\nThere are two distinct uses of the bound. Arbitrarily late negative\nerrors in a fixed interval suffice to stabilise the gcd. The crossing\nargument below needs \\emph{every} sufficiently late upward step bounded;\notherwise a single large rise could jump over the forbidden block.\n\nA sequence avoiding multiples of $2$ and $3$ cannot cross the pair\n$6k+2,6k+3$ from below with upward steps at most two: its first value\nat least $6k+2$ would be one of those two integers. Decreases before\nthat crossing do not help. For a general step bound $B$, the Chinese\nremainder theorem supplies a forbidden block of length $B$.\n\n\\begin{lemma}[consecutive multiples]\\label{res:crt}\nFor pairwise coprime integers $m_0,\\ldots,m_{B-1}\\ge2$ and every lower\nbound, there is a larger $t$ such that $m_i\\mid t+i$ for each $i<B$.\n\\end{lemma}\n\\begin{proof}\nSolve $t\\equiv-i\\pmod{m_i}$ by the Chinese remainder theorem and add\nmultiples of $\\prod_i m_i$ to exceed the prescribed bound.\n\\end{proof}\n\n\\begin{theorem}[Chinese remainder theorem and first crossing]\\label{res:barrier}\nLet $u:\\N\\to\\N$ tend to infinity and let $B\\ge1$ be an integer with\n$u_{n+1}\\le u_n+B$ for every $n$. There is no sequence of pairwise coprime\nintegers $m_i\\ge2$ for which $\\gcd(m_i,u_t)=1$ whenever $i<t$.\n\\end{theorem}\n\\begin{proof}\nApply Lemma~\\ref{res:crt} to $m_0,\\ldots,m_{B-1}$, choosing\n$t>\\max(u_0,\\ldots,u_B)$. There is a first $n>B$ with $u_n\\ge t$.\nThen\n\\[\n t\\le u_n\\le u_{n-1}+B<t+B.\n\\]\nThus $u_n=t+i$ for some $i<B<n$, and $m_i\\mid u_n$, contrary to\n$\\gcd(m_i,u_n)=1$.\n\\end{proof}\n\nBado proves a two-sided bounded-error criterion by this forbidden-block\nargument \\cite[Theorem~5.1, pp.~4--5]{bado2026}. The gcd stabilisation\nabove allows us to use only the lower error bound.\n\n\\begin{proof}[Proof of Theorem~\\ref{res:bounded}]\nSuppose the conclusion fails. Absorption excludes late zeros;\n$|E_n|\\ge1$ and the relative limit then give $C_n\\to\\infty$.\nInteger descent forces infinitely many late negative errors, where\n$1\\le-E_n\\le B$. Thus $B\\ge1$ and Proposition~\\ref{res:gcdstab}\ngives a stable gcd $g$. Restrict to a tail on which the gcd is $g$\nand $E_n\\ge-B$, then reindex that tail. Dividing by $g$ gives\n$u_n=C_n/g\\to\\infty$ and\n\\[\n u_{n+1}-u_n=-E_n/g\\le B.\n\\]\nProposition~\\ref{res:reduced} makes the multipliers $a_i\\ge2$ pairwise\ncoprime and coprime to every later numerator, contradicting\nTheorem~\\ref{res:barrier}.\n\\end{proof}\n\n\\begin{corollary}\\label{res:cor}\nUnder Theorem~\\ref{res:bounded}, the multipliers satisfy\n$a_{n+1}=a_n^2-a_n+1$ eventually.\n\\end{corollary}\n\\begin{proof}\nTheorem~\\ref{res:bounded} gives eventual zero error; positivity of $C_n$\nand Corollary~\\ref{res:step} give the recurrence.\n\\end{proof}\n\n\\subsection{The condition in the original sequence}\\label{sec:product-bound}\nLet $\\gamma_n=a_n^2/a_{n+1}-1$. For $t_n=P_n/a_n$,\n\\[\n t_{n+1}-t_n=t_n\\gamma_n.\n\\]\nWe bound upward increments, not values or decreases. The relation\n$C_n\\sim qt_n$ does not control increments; instead an exact identity\nwill give $E_n>-qt_n\\gamma_n$, turning an upper increment bound into\na lower error bound.\n\n\\begin{corollary}[bounded increments of $P_n/a_n$]\\label{res:originalbounded}\nLet $a_1<a_2<\\cdots$ be positive integers, $a_{n+1}/a_n^2\\to1$, and\n$\\sum_{n\\ge1}1/a_n\\in\\Q$.  Put $P_n=\\prod_{j<n}a_j$.  If\n\\[\n \\limsup_{n\\to\\infty}\\frac{P_n}{a_n}\n \\left(\\frac{a_n^2}{a_{n+1}}-1\\right)<+\\infty,\n\\]\nthen $a_{n+1}=a_n^2-a_n+1$ for all sufficiently large $n$.\n\\end{corollary}\n\\begin{proof}\nFor the canonical integer tail, direct substitution gives\n\\begin{equation}\\label{eq:canonical-dictionary}\n E_n+q\\frac{P_n}{a_n}\\gamma_n\n =qP_n\\left(\\frac1{a_{n+1}}\n -(a_n-1)\\sum_{k\\ge n+2}\\frac1{a_k}\\right).\n\\end{equation}\nThe tail estimate bounds the subtracted term by\n$4(a_n-1)/a_{n+1}^2<1/a_{n+1}$ eventually. The positive remainder gives\n$E_n>-qt_n\\gamma_n$, so the assumed upper bound bounds $E_n$ below.\nOnly the remainder's sign is needed here. The canonical tails satisfy\nthe exact recurrences and, by \\eqref{eq:tail-estimate}, $E_n/C_n\\to0$.\nTheorem~\\ref{res:bounded} and Corollary~\\ref{res:cor} finish.\n\\end{proof}\n\nKoizumi's proof of Corollary~4(1) uses this product-weighted comparison\n\\cite[(11)--(12), p.~15]{koizumi2025}; that corollary assumes a\nnonpositive upper limit, whereas the argument here allows any finite\nupper limit. A corresponding LCM comparison appears below.\nFor $a_n=b^{2^{n-1}}$, $b\\ge2$, the increments of $P_n/a_n$ are zero.\nSince the sequence does not eventually satisfy the Sylvester recurrence,\nthe corollary gives irrationality.\nFor a Sylvester sequence starting at $a_1>1$,\n$a_n-1=(a_1-1)P_n$ and the increments tend to zero.\n\nDividing by $n$ removes the nonsummable $1/n$ term. A summable upper\nerror then suffices, without a lower bound on $\\gamma_n$.\n\\begin{corollary}[an inclusive one-sided $1/n$ bound]\\label{res:inclusiveone}\nLet $a_1<a_2<\\cdots$ be positive integers with $a_{n+1}/a_n^2\\to1$\nand $\\sum_n1/a_n\\in\\Q$. Suppose that for some $K\\ge0$ and $\\varepsilon>0$,\n\\[\n \\gamma_n:=a_n^2/a_{n+1}-1\\le \\frac1n+\\frac{K}{n^{1+\\varepsilon}}\n \\quad\\hbox{for all large }n.\n\\]\nThen the sequence is eventually Sylvester. In particular, the conclusion\nholds under the pointwise eventual bound $\\gamma_n\\le1/n$.\n\\end{corollary}\n\\begin{proof}\nThe exact model $n$ has successive ratio $1+1/n$. Thus\n$(t_{n+1}/(n+1))/(t_n/n)=(1+\\gamma_n)/(1+1/n)\n\\le1+K/[n^\\varepsilon(n+1)]$.\nThe convergent product of these upper bounds bounds $t_n/n$ above;\nno positive limit is needed. Hence $t_n=O(n)$ and\n$t_n\\gamma_n\\le t_n(\\gamma_n)_+=O(1)$, so\nCorollary~\\ref{res:originalbounded} applies.\n\\end{proof}\n\nThe weaker condition $\\limsup n(\\gamma_n)_+\\le1$ does not suffice\nfor the estimate $t_n=O(n)$. For example, the scalar ratios\n$1+\\gamma_n=1+(1+1/\\log n)/n$ give a product of order $n\\log n$.\nThis scalar product refutes the intermediate linear estimate under\nthe weaker rate, not the Sylvester conclusion for reciprocal tails.\n\nFor positive limiting increments, take\n$a_1=4$ and $a_{n+1}=\\lceil n a_n^2/(n+1)\\rceil$, beginning\n$4,8,43,1387$. Here $a_n\\ge2\\cdot2^{2^{n-1}}$ and\n\\[\n 0\\le1+\\frac1n-\\frac{a_n^2}{a_{n+1}}<\\frac4{a_n^2}.\n\\]\nConsequently $t_n\\sim K n$ and $t_{n+1}-t_n\\to K>0$.\nThe corollary proves irrationality because $\\gamma_n\\sim1/n$ is\nincompatible with the eventual Sylvester recurrence.\n\nFor the introduction's $3/n$ ceiling example, division by\n$n(n+1)(n+2)$ and summability of the rounding errors give\n$t_n\\sim K_3n^3$ and $t_{n+1}-t_n\\sim3K_3n^2$, with $K_3>0$.\nIts irrationality follows from Theorem~\\ref{res:cubicrate}, beyond the\nbounded-increase criterion.\n\\section{Finite total relative increase}\\label{sec:mass}\nSummable relative increases first bound $C_n$ by an infinite product,\nthen allow only finitely many rises; integer descent finishes.\nNo denominator recurrence is needed. Relative increases tending merely\nto zero still allow $C_n=n+1$ or $C_n=(n+1)^2$.\n\n\\begin{theorem}[a convergent sum of relative increases]\\label{res:massscalar}\\label{res:mass}\nLet $C_n$ be positive integers and $E_n$ integers satisfying\n$C_{n+1}=C_n-E_n$. If\n\\[\n \\sum_n\\frac{(-E_n)_+}{C_n}<\\infty,\n\\]\nthen $E_n=0$ eventually. Neither denominator dynamics nor vanishing relative error is required.\n\\end{theorem}\n\\begin{proof}\nSet $\\delta_n=(-E_n)_+/C_n$. Then\n\\[\n C_N\\le C_0\\prod_{n<N}(1+\\delta_n)\n \\le C_0\\exp\\!\\left(\\sum_n\\delta_n\\right).\n\\]\nChoose an integer $K$ above this bound. A strict rise of the integer\nsequence has size at least one, so it contributes at least $1/K$ to\n$\\sum\\delta_n$. Hence only finitely many rises occur. Thereafter\n$C_n$ is a nonincreasing positive integer sequence, and its eventual\nconstancy gives $E_n=0$.\n\\end{proof}\n\nThe proof also allows positive real $C_n$ with integer $E_n$: the values\nlie in $C_0+\\Z$, whose bounded positive part is finite. Discrete increments\nare essential to this argument. With real errors, $C_n=1+1/(n+1)$\ndecreases forever with zero relative-increase sum; without positivity,\n$C_n=-n-1$, $E_n=1$ does so too. The companion's Section~13 gives the details.\n\nOn an exact reciprocal-tail orbit the conclusion gives the Sylvester\nrecurrence. For the gap sequence of the pseudo-greedy expansion, the same\ncriterion, with the same product bound and integer descent, appears in the\nErd\\H{o}s Problem a Day working report on Problem~\\#243, dated 12 August 2026\n\\cite[``A global termination criterion'']{erdosproblemaday243}; the statement\nabove is for any positive integer sequence with $C_{n+1}=C_n-E_n$.\nThis summability condition and the bound in Theorem~\\ref{res:bounded}\nare different estimates. For exact reciprocal tails with $|E_n|/C_n\\to0$,\neach is equivalent to eventual vanishing of $E_n$; neither estimate has\nbeen derived here from the unrestricted problem.\n\n\\section{New maxima of an LCM numerator}\\label{sec:lcmrecords}\n\nThe least common multiple retains only the largest power of each prime\namong $q,a_1,\\ldots,a_{n-1}$. It preserves integrality, not coprimality:\na numerator may now take values forbidden in Section~\\ref{sec:barrier}.\nThe recurrence will instead forbid short positive jumps from selected\nmultiples. Put\n\\[\n L_n=\\operatorname{lcm}(q,a_1,\\ldots,a_{n-1}),\\quad\n M_n=D_n/L_n,\\quad U_n=L_nx_n=C_n/M_n,\\quad V_n=E_n/M_n.\n\\]\nThe denominator of $x_n$ divides $L_n$, so $U_n\\in\\Z$ and\n$V_n=L_n-(a_n-1)U_n\\in\\Z$. The fraction $U_n/L_n$ need not be reduced.\nWith $\\rho_n=\\gcd(L_n,a_n)$, the exact updates are\n\\[\n M_{n+1}=M_n\\rho_n,\\qquad\n \\rho_nU_{n+1}=U_n-V_n,\\qquad V_n=L_n-(a_n-1)U_n.\n\\]\nThese coordinates and updates also appear in Bado's September 2026\npreprint. With his denominator parameter equal to $q$, his $M_{n-1}$,\n$\\Delta_{n-1}$, $K_{n-1}$, $u_n$ and $g_n$ are respectively our $L_n$,\n$M_n$, $U_n$, $V_n$ and $\\rho_n$; his (19) is the middle update above\n\\cite[Proposition~7.1 and (16)--(20), pp.~6--7]{bado2026}.\n\nSince $L_n\\mid qP_n$ and $V_n=(L_n/(qP_n))E_n$, multiplying\n\\eqref{eq:canonical-dictionary} by $L_n/(qP_n)$ gives, eventually,\n\\begin{equation}\\label{eq:general-clearance-dictionary}\n 0<V_n+\\frac{L_n}{a_n}\\left(\\frac{a_n^2}{a_{n+1}}-1\\right)\n \\le\\frac{L_n}{a_{n+1}}.\n\\end{equation}\nThe lower bound transfers an upper bound on the growth defect to a\nlower bound on $V_n$. The upper bound will let us compare weighted\nseries with a summable error.\n\nWrite $R_n=\\max_{j\\le n}U_j$ and\n$\\mathcal R=\\{n:U_{n+1}>R_n\\}$. Since $V_n/U_n=E_n/C_n\\to0$,\neventually $|V_n|<U_n$ and $-U_n\\le2V_n$. When $\\rho_n\\ge2$, the\nfirst bound gives $U_{n+1}<U_n$; the second sharpens this to\n$U_{n+1}\\le3U_n/4$. Thus every late rise, in particular every record,\nhas $\\rho_n=1$: its multiplier is coprime to $L_n$. At such a record\nwrite the full jump as $d_n=U_{n+1}-U_n=-V_n>0$; at a contracting\nstep that equality need not hold.\n\n\\begin{theorem}[a convergent weighted sum over new maxima]\\label{res:weightedrecord}\nAssume the growth and rationality hypotheses of Problem~\\ref{res:problem}.\nLet $f:[1,\\infty)\\to[0,\\infty)$ be finite and nonincreasing, with\n$\\int_1^\\infty f(t)\\,dt=\\infty$. Then the sequence is eventually Sylvester\nif and only if, for some integer $B\\ge0$,\n\\[\n \\sum_{n\\in\\mathcal R}(-V_n-B)_+f(U_n)<\\infty.\n\\]\n\\end{theorem}\n\nAt a late record, the summand is $(d_n-B)_+f(U_n)$, where\n$d_n=U_{n+1}-U_n$.\n\nThe full jump matters: with previous record $10$, a step from $5$ to\n$12$ first crosses $11$. Its jump is $7$, but the record increases\nby only $2$. This illustrates the two increments, not a denominator\nsequence satisfying \\eqref{eq:recurrences}.\n\nFor $B=0$, the integral comparison\n\\[\n \\int_{R_n}^{R_{n+1}}f(t)\\,dt\n \\le (R_{n+1}-R_n)f(U_n)\\le d_nf(U_n)\n\\]\nalready charges for every newly reached height. The arithmetic is\nneeded to retain divergence after subtracting a fixed $B$ at\n\\emph{each} record. We shall find a progression of heights whose first\ncrossings still cost at least one unit of excess jump.\n\nThe weights $1/t$ and $1/[t\\log(et)]$ are allowed; $1/t^2$ is not.\nOne admissible weight and one fixed $B$ suffice; $B$ cannot grow\nwith the record. The limit $V_n/U_n\\to0$ alone gives no summability.\n\n\\begin{proof}\nSuppose the sequence is not eventually Sylvester. Absorption excludes\nlate zeros of $E_n$, so $V_n=E_n/M_n$ is eventually a nonzero integer.\nHence\n\\[\n \\frac1{U_n}\\le\\frac{|V_n|}{U_n}=\\frac{|E_n|}{C_n}\\longrightarrow0.\n\\]\nThus $U_n\\to\\infty$. Infinitely many late records supply pairwise\ncoprime multipliers: each is coprime to its $L_n$, which is divisible by\nevery earlier multiplier. Since $a_n\\to\\infty$, for any fixed $B\\ge1$\nchoose $B$ of them, $m_0,\\ldots,m_{B-1}>B$. Take $T$ beyond their\nindices and the threshold for $\\rho_n=1$ at records.\n\nPut $P=\\prod_i m_i$ and choose $x$ with $m_i\\mid x+i$ by the\nChinese remainder theorem. We select the heights\n$\\tau=x+B+kP>R_T$, $k\\in\\Z$. They are spaced $P$ apart.\n\nAt the first crossing of a selected height,\n$U_n\\le R_n<\\tau\\le U_n+d_n$. In particular this is a record step.\nIf $d_n\\le B$, then $U_n\\in[\\tau-B,\\tau)$, so\n\\[\n U_n=\\tau-B+i=x+kP+i\\qquad\\text{for some }0\\le i<B.\n\\]\nThe choice of $x$ gives $m_i\\mid U_n$. This is allowed: $U_n/L_n$\nneed not be reduced. But $m_i\\mid L_n$ too, so\n$m_i\\mid d_n=(a_n-1)U_n-L_n$, contradicting $0<d_n\\le B<m_i$.\nIt is the short positive jump, not the numerator value, that is impossible.\n\nA jump may first cross several selected heights. To charge all of them\nto its single excess $d_n-B$, let their number be $h\\ge1$. Their spacing gives\n$(h-1)P<d_n$. Write $r=d_n-B\\ge1$. Since $P\\ge B+1$,\n$d_n=B+r\\le Pr$, and hence $h\\le r$. Each of those heights exceeds\n$U_n$, so monotonicity gives\n\\[\n \\sum_{\\substack{\\tau\\text{ first crossed}\\\\\\text{at step }n}}f(\\tau)\n \\le(d_n-B)f(U_n).\n\\]\nEach selected height above $R_T$ has one first crossing. The heights are\n$P$ apart, with the first at most $R_T+P$. For $R_N\\ge R_T+P$, the\nintervals $[\\tau,\\min\\{\\tau+P,R_N\\}]$ with $\\tau\\le R_N$ cover $[R_T+P,R_N]$.\nMonotonicity bounds each integral by $P f(\\tau)$, so the crossing bounds give\n\\begin{equation}\\label{eq:weightedcrossing}\n \\sum_{\\substack{T\\le n<N\\\\n\\in\\mathcal R}}(-V_n-B)_+f(U_n)\n \\ge\\frac1P\\int_{R_T+P}^{R_N} f(t)\\,dt.\n\\end{equation}\nSince $R_n\\to\\infty$, the right side diverges for every $B\\ge1$;\n$B=0$ follows by domination. Conversely, the eventual Sylvester recurrence\ngives by telescoping\n$x_n=1/(a_n-1)$, so $V_n=0$ eventually.\n\\end{proof}\n\nFor example, $f(t)=1/[t\\log(et)]$ gives the sufficient condition\n\\[\n \\sum_{n\\in\\mathcal R}\\frac{(-V_n-B)_+}{U_n\\log(eU_n)}<\\infty.\n\\]\nIts finite lower bound in~\\eqref{eq:weightedcrossing} is\n$P^{-1}\\log\\bigl(\\log(eR_N)/\\log(e(R_T+P))\\bigr)$.\nFurther fixed iterated logarithmic factors are allowed whenever the integral\nstill diverges. All these weights use the same crossing theorem.\n\nFor a series, sign control alone is not enough: we bound the total\nchange in its summands. Put $\\gamma_n=a_n^2/a_{n+1}-1$ and\n$\\theta_n=E_n/C_n$. The defect identity gives\n\\[\n \\gamma_n+\\theta_n=\n \\frac{(1-\\theta_n)(a_n-1+\\theta_{n+1})}{a_{n+1}},\n \\qquad 0<\\gamma_n+\\theta_n<3/a_n\n\\]\neventually. Positive parts do not increase absolute differences.\nSince $V_n=U_n\\theta_n$, the summands\n$U_nf(U_n)(\\gamma_n-B/U_n)_+$ and $(-V_n-B)_+f(U_n)$ differ by at most\n$3U_nf(U_n)/a_n\\le3f(1)C_n/a_n$. This is the error \\emph{after weighting}.\nNow $C_n/a_n\\sim qP_n/a_n^2$, since $a_nx_n\\to1$, and\n\\[\n \\frac{P_{n+1}/a_{n+1}^{2}}{P_n/a_n^{2}}\n =\\frac{a_n^3}{a_{n+1}^{2}}\\longrightarrow0.\n\\]\nThe ratio test therefore makes the weighted error summable, even over\nall indices. The remainders in~\\eqref{eq:canonical-dictionary}\nand~\\eqref{eq:general-clearance-dictionary} are likewise absolutely\nsummable: each is eventually between zero and\n$qP_n/a_{n+1}=O(P_n/a_n^2)$.\nThus Theorem~\\ref{res:weightedrecord} is equivalent to finiteness of\n\\begin{equation}\\label{eq:weightedgrowth}\n \\sum_{n\\in\\mathcal R}U_nf(U_n)\n \\left(\\frac{a_n^2}{a_{n+1}}-1-\\frac B{U_n}\\right)_+\n\\end{equation}\nfor some $B$. This is a reformulation of the extra hypothesis, not a\nproof of it from quadratic growth. Both series use the full jump\n$U_{n+1}-U_n$, including recovery from earlier decreases. Replacing it\nby $R_{n+1}-R_n$, or merely proving that individual summands tend to\nzero, would not establish the required finiteness.\n\n\\begin{corollary}[a bound using the least common multiple]\\label{res:lcmbounded}\nAssume the hypotheses of Problem~\\ref{res:problem}. Write\n$A_n=\\operatorname{lcm}(a_1,\\ldots,a_{n-1})$ with $A_1=1$. If\n\\[\n \\limsup_{n\\to\\infty}\\frac{A_n}{a_n}\n \\left(\\frac{a_n^2}{a_{n+1}}-1\\right)<\\infty,\n\\]\nthen the sequence is eventually Sylvester.\n\\end{corollary}\n\n\\begin{proof}\nSince $L_n/A_n=q/\\gcd(q,A_n)$ lies in $[1,q]$, an upper bound on the\nexpression in the corollary gives an upper bound on\n$(L_n/a_n)(a_n^2/a_{n+1}-1)$; negative terms remain negative.\nEquation~\\eqref{eq:general-clearance-dictionary} therefore bounds the\nnegative part of $V_n$. Choose an integer $B$ above\nthat eventual bound. The record series then vanishes after a finite prefix,\nso Theorem~\\ref{res:weightedrecord} applies.\n\\end{proof}\n\nThe expression in Erd\\H{o}s and Straus's Theorem~3 is\n\\[\n \\frac{[a_1,\\ldots,a_k]}{a_{k+1}}\n \\left(\\frac{a_{k+1}^2}{a_{k+2}}-1\\right),\n\\]\nwith a nonpositive upper limit \\cite[p.~132]{erdosstraus1964}.\nSetting $n=k+1$ gives exactly $(A_n/a_n)\\gamma_n$: the prefix ends\nbefore $a_n$, and the growth ratio begins at $a_n$.\nTijdeman and Yuan extend this type of criterion to positive numerators\n\\cite{tijdemanyuan2002}. Here any finite upper bound suffices under\nthe quadratic-limit assumption. Since $A_n\\mid P_n$,\nthe LCM hypothesis is no stronger than the product hypothesis in\nCorollary~\\ref{res:originalbounded}: if the product expression is at most\n$B$, the LCM expression is at most $\\max(B,0)$. When the earlier terms\nare pairwise coprime the two weights agree; repeated prime factors can\nmake the LCM much smaller. This comparison does not assert the existence\nof a non-Sylvester rational example satisfying one bound but not the other.\n\n\nThe finite upper limit means an eventual upper bound; it does not require\n$q$ to divide the LCM of the preceding denominators.\nFor positive integer summand numerators $b_n$, the modified error is\n$V_n=b_nL_n-(a_n-1)U_n$. The companion, Section~6 under ``Integer\ncoefficients'', separately proves boundedness from a lower bound on\n$V_n$ and eventual constancy when $V_n/U_n\\to0$ is also assumed.\nThe latter conclusion must not be read into the first hypothesis alone.\nThe same section links finite examples separating them; compare Badea\n\\cite[p.~316]{badea1993} and Tijdeman--Yuan \\cite{tijdemanyuan2002}.\nThe companion's Section~3 compares Duverney's signed-series criterion\n\\cite[Corollary~3.2, p.~287]{duverney2001}, whose printed condition is\none-sided, with the stronger absolute-convergence specialisation used\nthere. It also records the different growth hypotheses for irrationality\nexponents in \\cite[Theorem~1, author-version p.~2]{duverneykurosawashiokawa2020}.\nNeither comparison is an input to the crossing theorem.\n\n\\section{Further questions}\\label{sec:open}\n\\begin{proposition}[necessary profile]\\label{res:frontier}\nThe integer tail of a sequence satisfying Problem~\\ref{res:problem}'s\nhypotheses but not its conclusion has $E_n\\ne0$ eventually,\n$|E_n|/C_n\\to0$, unbounded negative magnitudes along negative indices,\nand\n\\[\n \\sum_n\\frac{(-E_n)_+}{C_n}=\\infty.\n\\]\n\\end{proposition}\n\\begin{proof}\nAbsorption excludes late zeros, descent excludes an eventually\nnonnegative error, and Theorems~\\ref{res:bounded} and~\\ref{res:massscalar}\nexclude the two finiteness conditions.\n\\end{proof}\n\nFor $B\\ge0$ put\n\\[\n F_B(X)=\\sum_{\\substack{n\\in\\mathcal R\\\\U_n\\le X}}(-V_n-B)_+.\n\\]\nWe count record steps by their \\emph{starting numerator} $U_n$, not\nby time or the running maximum. Every selected height at most $X$ is first crossed\nby a jump starting below $X$, even if it lands above $X$. Thus a\nnon-Sylvester orbit satisfies $F_B(X)\\ge X/P_B-O_B(1)$, with an\norbit-dependent CRT modulus $P_B$; $U_n\\to\\infty$ also makes $F_B(X)$\nfinite. It would therefore suffice to prove from growth and rationality\nthat, for some fixed integer $B\\ge0$,\n\\begin{equation}\\label{eq:remaining-record-budget}\n \\liminf_{X\\to\\infty}\\frac{F_B(X)}X=0.\n\\end{equation}\nThe next lemma constructs an admissible weight from this vanishing\nlower ratio: enumerate records by $j$ and put $u_j=U_n$,\n$w_j=(-V_n-B)_+$. The resulting weighted series is finite, contradicting\nTheorem~\\ref{res:weightedrecord}. The estimate remains unproved; the\nlemma changes what one must estimate, not the hypotheses available.\n\n\\begin{lemma}[weights and linear density]\\label{res:weights}\nLet $u_j$ be positive integers, let $w_j\\ge0$, and put\n$F(X)=\\sum_{u_j\\le X}w_j$, with the sum allowed a priori to be $+\\infty$. Then $\\liminf_{X\\to\\infty}F(X)/X=0$ if and only if\nthere is a finite nonincreasing $f:[1,\\infty)\\to[0,\\infty)$ with\n$\\int_1^\\infty f(t)\\,dt=\\infty$ and $\\sum_jw_jf(u_j)<\\infty$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose the lower limit vanishes. Since $F$ is constant on $[m,m+1)$, choose\nintegers $X_1<X_2<\\cdots$ with $X_k\\ge2^k$ and\n$F(X_k)\\le2^{-k}X_k$. Set\n\\[\n f(t)=\\sum_{k\\ge1}\\frac{\\mathbf 1_{[1,X_k]}(t)}{X_k}.\n\\]\nThis function is nonincreasing and bounded by $\\sum_k2^{-k}$.\nEach cutoff contributes $1-1/X_k$ to its integral, but at most $2^{-k}$\nto the weighted sum. Interchanging nonnegative sums therefore gives\n\\[\n \\int_1^\\infty f(t)\\,dt=\\sum_k(1-1/X_k)=\\infty,\n \\qquad\n \\sum_jw_jf(u_j)=\\sum_k\\frac{F(X_k)}{X_k}\\le1.\n\\]\n\nConversely, suppose $f$ is such a weight and $F(m)\\ge cm$ for some\n$c>0$ and all integers $m\\ge M$. Monotonicity and the divergent integral\nforce $f(X)>0$, so $F(X)\\le f(X)^{-1}\\sum_jw_jf(u_j)<\\infty$.\nThus partial summation applies; the nonnegative differences\n$f(m)-f(m+1)$ preserve the lower bound on $F$. For $N>M$,\n\\[\n \\sum_{u_j\\le N}w_jf(u_j)\n =F(N)f(N)+\\sum_{m=1}^{N-1}F(m)\\bigl(f(m)-f(m+1)\\bigr)\n \\ge c\\Bigl(Mf(M)+\\sum_{m=M+1}^{N}f(m)\\Bigr),\n\\]\nevery term being nonnegative. The right side diverges as $N\\to\\infty$, because\n$\\sum_{m\\ge1}f(m)\\ge\\int_1^\\infty f(t)\\,dt=\\infty$.\n\\end{proof}\n\n\n\\subsection{Arithmetic compatibility}\\label{sec:scalar-limit}\nThe remaining estimate requires the denominator recurrence as well as\ncontrol of the error. For example, $C_n=n^2+1$, $E_n=-(2n+1)$ has vanishing\nrelative error and the necessary scalar pattern. But at $n=2,3,4$ its\nnumerators are $5,10,17$. The update $10=5a_2-D_2$ forces $5\\mid D_2$,\nhence $5\\mid D_3=a_2D_2$; the next update would force $5\\mid17$.\nThus these numerators cannot satisfy \\eqref{eq:recurrences}. The companion's Section~14 also constructs\nsequences avoiding sparse fixed moduli and distinguishes that avoidance\nfrom compatibility with all later denominators. Neither property alone\nsupplies \\eqref{eq:remaining-record-budget}.\n\nConversely, suppose integer sequences indexed from zero satisfy\n\\eqref{eq:recurrences}, with $a_n>1$, $C_n>0$, $D_n\\ge0$ and\n$E_n/C_n\\to0$. Section~4 of the companion proves that $D_n>0$,\n$C_n/D_n\\to0$, $\\sum_n1/a_n=C_0/D_0$ and $a_{n+1}/a_n^2\\to1$.\nThe vanishing remainder $C_n/D_n\\to0$ turns the finite reciprocal\nidentities into an infinite series. A scalar profile or finite prefix\ndoes not supply it.\n\n\\begin{problem}[growth of repeated denominator factors]\\label{res:lcmheight}\nFor every rational-tail orbit satisfying the hypotheses but not the conclusion of Problem~\\ref{res:problem},\nmust\n\\[\n \\limsup_{n\\to\\infty}\\frac{\\log M_n}{n}>0?\n\\]\nEquivalently, must there be a $K\\ge1$ for which\n\\[\n 2^n\\le M_n^K\n\\]\nat infinitely many indices?\n\\end{problem}\nSection~\\ref{sec:transfer} gives $C_{n+1}/C_n\\to1$; averaging logarithms\ntherefore gives $\\log C_n=o(n)$. Since $1\\le M_n\\le C_n$, every such\norbit already satisfies $\\log M_n/n\\to0$. A positive answer therefore\nholds exactly when no such orbit exists: proving it from failure of\nSylvester behaviour would exclude every counterexample.\n\n\\appendix\n\\section{A factorial modulus for integral recursion}\\label{app:residue}\n\nA constant-negative error $E_n=-m$ with $C_0=c$ gives $C_n=c+nm$ and the identity\n\\begin{equation}\\label{eq:shape}\n D_n+m=(a_n-1)C_n.\n\\end{equation}\nIn the case $m=c=1$, where~\\eqref{eq:shape} reads $D_n+1=(a_n-1)(n+1)$, each\nmultiplier is determined by its predecessor. The recursion is\n$a_{n+1}=\\fnum(n,a_n)/(n+2)$, where\n\\[\n \\fnum(n,a)=(n+1)a^{2}-(n+2)a+(n+3),\n\\]\nthe\n\\lword{Erdos243/FiniteHorizonResidue.lean}{22}{forcedNumerator}{forced\nnumerator}. We stop at the first nonintegral quotient. This procedure\nchecks divisibility only; it does not impose the positivity conditions\nrequired of an infinite reciprocal-tail sequence. The\n\\lword{Erdos243/FiniteHorizonResidue.lean}{75}{ForcedSurvives}{formal exact-division condition}\nrequires exact division at each step.\n\\begin{example}[a nonintegral second step]\\label{ex:shape}\nStarting from $a_0=3$, the first numerator $\\fnum(0,3)=6$ is\ndivisible by $2$, giving $a_1=3$. The next numerator\n$\\fnum(1,3)=13$ is not divisible by $3$, so the recursion stops.\n\\end{example}\nDirect iteration can produce very large intermediate integers. To\ndecide whether the first $h$ divisions are exact, however, it suffices\nto know the initial value modulo $(h+1)!$.\nComputing a pseudo-greedy orbit through residues modulo a shrinking product\nmodulus is Koizumi's method \\cite[Remark~2 and Algorithm~1,\npp.~13--14]{koizumi2025}; for the forced numerator that product is a\nfactorial.\n\n\\begin{theorem}[factorial residue reduction]\\label{res:residue}\nLet $h$ be a nonnegative integer and let $a,b$ be integers with $a\\equiv b\\pmod{(h+1)!}$.\nThe first $h$ steps of the recursion\n$a_{n+1}=\\fnum(n,a_n)/(n+2)$, starting at index zero, are integral\nfor $a_0=a$ if and only if they are integral for $a_0=b$.\n\\end{theorem}\n\n\\begin{proof}\nDivision by $i+2$ consumes that factor of the modulus; no modular\ninverse is required. Define $M(0,i)=1$ and\n$M(h+1,i)=(i+2)M(h,i+1)$. Induct on $h$, allowing any starting\nindex $i$: inputs congruent modulo $M(h,i)$ either both complete\n$h$ integral steps or neither does. The case $h=0$ is immediate.\n\nIf $a\\equiv b\\pmod{(i+2)M(h,i+1)}$, the integer polynomial\n$\\fnum(i,\\cdot)$ preserves this congruence. Its values are therefore\neither both divisible by $i+2$ or neither is. In the latter case both\nrecursions stop. In the former, exact division leaves the quotients\ncongruent modulo $M(h,i+1)$, and induction supplies the remaining\n$h$ steps. Finally $M(h,0)=(h+1)!$.\n\\end{proof}\n\nFor $h=1$, the condition is $2\\mid a^2-2a+3$, or equivalently that $a$ is odd.\nThis illustrates Theorem~\\ref{res:residue} with modulus $2!=2$.\n\nThe enumeration in Example~9.1 of the companion, over\n$2\\le a_0<5000$, gives at most $17$ successful updates ($18$ values\nincluding $a_0$). The seed $a_0=1$ is excluded because it is fixed\nand violates $a_n\\ge2$. This finite calculation establishes only the\nstated range. The infinite\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{286}{no_constantNegative_orbit}{constant-negative exclusion}\nrequires the separate argument under the hypotheses given there.\n\n\n\\section{Sources and related work}\\label{app:index}\nThe companion \\emph{Reciprocal Sums and the Sylvester Recurrence:\nFurther Results and Proofs} supplies full proofs and fixed-revision links:\n\\href{erdos243-reciprocal-tail-reasoning-surface.pdf\\#243-long-cubic}{Section~2}\nfor regular rates and cubic disagreement;\n\\href{erdos243-reciprocal-tail-reasoning-surface.pdf\\#243-long-integer-tails}{Sections~4--5}\nfor integer tails;\n\\href{erdos243-reciprocal-tail-reasoning-surface.pdf\\#243-long-bounded-increases}{Section~12}\nfor the gcd and bounded-increment arguments;\n\\href{erdos243-reciprocal-tail-reasoning-surface.pdf\\#243-long-weighted-records}{Section~6}\nfor LCM results;\n\\href{erdos243-reciprocal-tail-reasoning-surface.pdf\\#243-long-relative-increase}{Section~13}\nfor scalar summability; and\n\\href{erdos243-reciprocal-tail-reasoning-surface.pdf\\#243-long-open-questions}{Section~14}\nfor open estimates and failed constructions. A final map records dependencies.\nThe auxiliary constructions satisfy only their stated hypotheses, not\nnecessarily those of a rational reciprocal series.\n\nFor the earlier integer-remainder method see\nErd\\H{o}s--Straus \\cite[Theorem~2.1, pp.~85--86]{erdosstraus1974}.\nThe companion's Section~3 distinguishes polynomial Cantor series\n\\cite[Theorem~2.2, pp.~39--40]{hancltijdeman2008} and the freely\nselectable series of Problem~\\#270 \\cite[Theorems~1--2]{crmarickovac2025}\nfrom the present setting. The Isabelle/HOL work \\cite{kouli2020}\nformalises older Erd\\H{o}s--Straus criteria, not Problem~\\#243.\nGeneral-rate extraction uses the Gamma-ratio asymptotics\nin \\cite{dlmf_gamma}; the cubic case uses the explicit $F_n$.\n\n\\paragraph{Indexing of the canonical tails.}\nSection~\\ref{sec:transfer} starts with $C_1=p$, $D_1=q$.\nSetting $a_0=1$, $D_0=q$, $C_0=p+q$ extends the recurrences by one\nupdate, but gives neither $a_0>1$ nor strict increase at that step.\nThe zero-based formal cubic theorem instead uses $N$ with $a_N>N$,\n$b_n=n+1$ for $0\\le n<N$ and $b_n=a_n$ thereafter. This sequence\nis strictly increasing and changes the reciprocal sum by a rational\nfinite sum without shifting $n$ or $3/n$ on the tail. See the companion's\n\\href{erdos243-reciprocal-tail-reasoning-surface.pdf\\#243-long-indexing}{indexing subsection}.\n\n\\section{Verification and reproducibility}\\label{app:verification}\n\\evidenceparagraph{\\evidencerecordurl}\nThe links identify fixed-revision Lean declarations and recorded\nComparator checks. The printed cubic argument uses Chebotarev; the\nformal square-specialisation proof uses a Dedekind-zeta pole comparison.\nTheir common conclusion does not check each step of the printed proof.\nThis revision includes no new Lean build, Comparator replay or independent\nreview. The evidence record retains source-pin and index gaps, and the\nestimate in Section~\\ref{sec:open} remains open.\n\n\\noindent The formal\n\\lword{Erdos243/FiniteHorizonResidue.lean}{134}{forcedSurvives_iff_of_modEq_factorial}{factorial\nresidue reduction} uses the\n\\lword{Erdos243/FiniteHorizonResidue.lean}{97}{forcedSurvives_iff_of_modEq}{shrinking-modulus\ninduction},\n\\lword{Erdos243/FiniteHorizonResidue.lean}{26}{forcedNumerator_modEq}{polynomial\ncongruence}, and\n\\lword{Erdos243/FiniteHorizonResidue.lean}{41}{quotient_modEq_of_modEq_mul}{cancellation\nafter exact division}. Its modulus is identified by the\n\\lword{Erdos243/FiniteHorizonResidue.lean}{59}{horizonModulus_eq_ascFactorial}{ascending-factorial\nformula} and its\n\\lword{Erdos243/FiniteHorizonResidue.lean}{69}{horizonModulus_zero_eq_factorial}{factorial\nvalue at the initial index}.\n\n\n% Legacy fixed-revision declaration catalogue, retained for source-coordinate\n% consumers. It is navigation, not an additional printed proof or fresh receipt.\n\\iffalse\n\n\nThe mathematical arguments above are independent of the source code.\nThe links below open Lean declarations at fixed revisions of the sources.\n\n\\paragraph{Earlier remainder methods and related problems.}\nThe integer-remainder method predates this argument\n\\cite[Theorem~2.1, pp.~85--86]{erdosstraus1974}.\nHan\\v{c}l and Tijdeman characterise rationality of polynomial Cantor\nseries by a finite decomposition and a shifted polynomial sum\n\\cite[Theorem~2.2 and its derivation, pp.~39--40]{hancltijdeman2008};\nthe polynomial hypotheses justify that rearrangement. Neither comparison\nsupplies the lower-bound CRT argument. The Isabelle/HOL\ndevelopment of Koutsoukou-Argyraki and Li \\cite{kouli2020} formalises\nolder irrationality criteria of Erd\\H{o}s and Straus; it does not treat\nProblem~\\#243. The freely selectable series of\nProblem~\\#270 \\cite[Theorems~1--2]{crmarickovac2025} are discussed in\nthe companion's Section~3, but are not used for the fixed recurrence here.\n\n\\paragraph{Main criterion and arithmetic lemmas.}\nThe Lean proof of \\lword{Erdos243/ReciprocalTailRigidity.lean}{2360}{eventuallyBoundedNegativePart_eventually_zero}{eventual vanishing under a lower error bound}\n(Theorem~\\ref{res:bounded}) shifts past the two thresholds and applies the \\lword{Erdos243/ReciprocalTailRigidity.lean}{2271}{boundedNegativePart_eventually_zero}{vanishing result under a bound on the negative part}\nwith centring and the lower bound at every index.\n\nProposition~\\ref{res:update} corresponds to the \\lword{Erdos243/ReciprocalTailRigidity.lean}{57}{nextTailState_eq_sub_centered}{update law}\nand the \\lword{Erdos243/ReciprocalTailRigidity.lean}{1775}{sylvesterDefect_mul_nextTailState}{defect identity}.\nTheorem~\\ref{res:absorb} uses \\lword{Erdos243/ReciprocalTailRigidity.lean}{2227}{centeredState_zero_absorbing}{absorption of a vanishing error}\nand \\lword{Erdos243/ReciprocalTailRigidity.lean}{1843}{centeredState_eventually_zero}{stabilisation of the nonnegative error}.\nThe \\lword{Erdos243/ReciprocalTailRigidity.lean}{1787}{sylvesterNext_eq_of_centered_zero}{local rigidity step}\nand the \\lword{Erdos243/ReciprocalTailRigidity.lean}{1805}{sylvesterNext_eventually_of_centered_zero}{eventual Sylvester recurrence}\ngive the two forms of Corollary~\\ref{res:step}.\n\nThe three parts of Proposition~\\ref{res:reduced} are \\lword{Erdos243/ReciprocalTailRigidity.lean}{989}{reducedStep_coprime_currentFactor}{step coprimality},\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1016}{reducedTail_pairwiseCoprime}{pairwise coprimality}, and\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1030}{reducedTail_wholeModulusAvoidance}{coprimality to each earlier multiplier}.\nProposition~\\ref{res:gcdstab} gives \\lword{Erdos243/ReciprocalTailRigidity.lean}{1636}{tailGcd_eventuallyConstant_of_cofinally_boundedNegative}{eventual constancy of the gcd}.\nThe exclusions treat \\lword{Erdos243/ReciprocalTailRigidity.lean}{1066}{no_eventuallyBoundedRise_reducedTail}{reduced numerators with bounded increments}\nand \\lword{Erdos243/ReciprocalTailRigidity.lean}{1754}{no_cofinallyBoundedNegative_of_normalizedVanishes}{the case where divergence follows from vanishing relative error}.\nThe main proof uses both divergence of the reduced numerator and a bound\non every upward increment. A cofinal bound on negative errors suffices\nfor gcd stabilisation, not for that increment bound. Companion Section~11\nalso distinguishes stabilisation from a density-zero bound on strict gcd\nincreases; the latter is not used here.\n\n\\paragraph{The scalar criterion.}\nThe scalar declaration proves \\lword{Erdos243/SparseResetRecovery.lean}{612}{eventually_zero_of_summable_negativeRelativeMass_scalar}{vanishing for a positive integer sequence with summable relative increases}\n(Theorem~\\ref{res:massscalar});\nits exact-orbit consequence is \\lword{Erdos243/SparseResetRecovery.lean}{690}{sylvesterNext_eventually_of_summable_negativeRelativeMass_scalar}{the Sylvester recurrence under the same assumption}.\nThese declarations concern integer $C_n$, not the real-initial-value\nobservation following the proof. Formulations that also assume denominator\ndynamics and vanishing relative error give \\lword{Erdos243/SparseResetRecovery.lean}{378}{tail_growth_le_one_add_negativeRelativeMass}{a growth bound for the numerator},\n\\lword{Erdos243/SparseResetRecovery.lean}{488}{eventually_zero_of_summable_negativeRelativeMass}{eventual zero error under the convergence assumption},\nand \\lword{Erdos243/SparseResetRecovery.lean}{510}{sylvesterNext_eventually_of_summable_negativeRelativeMass}{the resulting Sylvester recurrence}.\n\nThe Lean theorem excluding a periodic negative error (Theorem~10.1 of the\ncompanion) assumes that each error magnitude is smaller\nthan its multiplier. For periodic positive magnitudes and multipliers\n$a_n\\ge2$, the companion derives this bound eventually from the exact\nrecurrences. For general errors, vanishing relative error supplies\nstrict centring eventually, but control of the negative part is still\nneeded in Theorem~\\ref{res:bounded}.\n\n% The exhaustive source manifest is retained in the authored TeX for automated\n% validation, and kept outside the printed note.\n\n\n\\begin{tabular}{@{}ll@{}}\ninformal statement & linked Lean source\\\\\nSylvester successor & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{37}{sylvesterNext}\\\\\ndenominator update & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{41}{nextDenState}\\\\\ntail update & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{45}{nextTailState}\\\\\nerror & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{49}{centeredState}\\\\\nSylvester defect & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{53}{sylvesterDefect}\\\\\nupdate law & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{57}{nextTailState_eq_sub_centered}\\\\\ndenominator equivariance & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{65}{nextDenState_scale}\\\\\ntail equivariance & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{70}{nextTailState_scale}\\\\\ncentred equivariance & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{75}{centeredState_scale}\\\\\nnormalised exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{100}{no_normalizedConstantNegative_orbit}\\\\\nscale-primitive exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{147}{no_scalePrimitiveConstantNegative_orbit}\\\\\nconstant exclusion at every scale & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{286}{no_constantNegative_orbit}\\\\\neventual constant exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{350}{no_eventuallyConstantNegative_orbit}\\\\\npersistence of a multiplier & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{382}{base_dvd_denState_later}\\\\\none-step lock & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{402}{divisorLocksTailSucc}\\\\\npersistence of a lock & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{429}{divisorLock_persists}\\\\\niterated period & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{461}{periodic_add_mul}\\\\\niterated drift & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{476}{phaseDrift_add_mul}\\\\\nrepeated-divisor transport & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{494}{repeatedDivisor_forces_periodicCommonScale}\\\\\nphase-primitive exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{548}{no_phasePrimitivePeriodicNegative_orbit}\\\\\nperiodic exclusion at every scale & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{690}{no_periodicNegative_orbit}\\\\\neventual periodic exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{802}{no_eventuallyPeriodicNegative_orbit}\\\\\nshifted consecutive multiples & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{839}{exists_shifted_consecutiveMultiples}\\\\\nobstruction to bounded upward increments & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{903}{no_boundedRise_of_tailAvoidance}\\\\\nstep coprimality & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{989}{reducedStep_coprime_currentFactor}\\\\\npairwise coprimality & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1016}{reducedTail_pairwiseCoprime}\\\\\navoidance of multiples of whole moduli & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1030}{reducedTail_wholeModulusAvoidance}\\\\\nreduced exclusion under bounded upward increments & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1041}{no_boundedRise_reducedTail}\\\\\neventual reduced exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1066}{no_eventuallyBoundedRise_reducedTail}\\\\\ngcd monotonicity & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1430}{tailGcd_dvd_succ}\\\\\ngcd at a negative index & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1587}{tailGcd_eq_gcd_negativeMagnitude}\\\\\nchain stabilisation & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1605}{dvdChain_eventuallyConstant_of_cofinally_bounded}\\\\\ngcd stabilisation & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1636}{tailGcd_eventuallyConstant_of_cofinally_boundedNegative}\\\\\ntail-divergence form & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1661}{no_cofinallyBoundedNegative_of_tailTendsto}\\\\\ndivergence from vanishing relative error & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1739}{tailState_tendsto_atTop_of_nonzero_normalizedVanishes}\\\\\nexclusion from vanishing relative error & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1754}{no_cofinallyBoundedNegative_of_normalizedVanishes}\\\\\ndefect identity & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1775}{sylvesterDefect_mul_nextTailState}\\\\\nlocal rigidity step & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1787}{sylvesterNext_eq_of_centered_zero}\\\\\neventual Sylvester recurrence & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1805}{sylvesterNext_eventually_of_centered_zero}\\\\\neventual constancy & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1825}{antitone_nat_eventually_constant}\\\\\nstabilisation of the nonnegative error & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1843}{centeredState_eventually_zero}\\\\\ntail realisation & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1869}{natTail_eq_nextTailState}\\\\\ndenominator realisation & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1881}{natDen_eq_nextDenState}\\\\\nsigned update law & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1978}{natTail_eq_sub_centeredState}\\\\\nabsorption of a vanishing error & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{2227}{centeredState_zero_absorbing}\\\\\ncontrapositive along a tail & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{2254}{eventually_nonzero_of_zero_absorbing}\\\\\nvanishing under a bound on the negative part & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{2271}{boundedNegativePart_eventually_zero}\\\\\neventual vanishing under a bound on the negative part & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{2360}{eventuallyBoundedNegativePart_eventually_zero}\\\\\nbound on cancellation over a recovery interval & \\lrefx{Erdos243/RepairEntropy.lean}{48}{repairedFamily_recovery_energy_divisionFree}\\\\\nindependent-repair bound & \\lrefx{Erdos243/RepairEntropy.lean}{78}{repaired_card_bound_of_independent}\\\\\nfixed-length reset disappearance & \\lrefx{Erdos243/RepairEntropy.lean}{107}{eventually_recoveryPayment_eq_one_of_fixedLength}\\\\\nforced numerator & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{22}{forcedNumerator}\\\\\npolynomial congruence & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{26}{forcedNumerator_modEq}\\\\\nexact-division cancellation & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{41}{quotient_modEq_of_modEq_mul}\\\\\nhorizon modulus & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{53}{horizonModulus}\\\\\nascending-factorial form & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{59}{horizonModulus_eq_ascFactorial}\\\\\nfactorial value at the initial index & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{69}{horizonModulus_zero_eq_factorial}\\\\\nsurvival predicate & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{75}{ForcedSurvives}\\\\\nquotient transport & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{85}{forcedQuotient_modEq}\\\\\nshrinking-modulus transport & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{97}{forcedSurvives_iff_of_modEq}\\\\\nfactorial residue reduction & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{134}{forcedSurvives_iff_of_modEq_factorial}\\\\\n\\end{tabular}\n\n\n\n\\fi\n\n\\begin{thebibliography}{99}\n\\small\n\\raggedright\n\\setlength{\\itemsep}{1pt}\n\\bibitem{erdosstraus1964} P.~Erd\\H{o}s and E.~G. Straus,\n  \\href{https://users.renyi.hu/~p_erdos/1964-19.pdf}{\\emph{On the irrationality\n  of certain Ahmes series}},\n  J. Indian Math. Soc. (N.S.) \\textbf{27} (1964), 129--133. MR~175848.\n\\bibitem{duverney2001} D.~Duverney,\n  \\href{https://www.ms.u-tokyo.ac.jp/journal/pdf/jms080206.pdf}\n  {\\emph{Irrationality of fast converging series of rational numbers}},\n  J. Math. Sci. Univ. Tokyo \\textbf{8} (2001), 275--316. MR~1837165.\n\\bibitem{badea1993} C.~Badea,\n  \\href{https://matwbn.icm.edu.pl/ksiazki/aa/aa63/aa6342.pdf}{\\emph{A theorem\n  on irrationality of infinite series and applications}}, Acta Arith.\n  \\textbf{63} (1993), no.~4, 313--323,\n  doi:\\href{https://doi.org/10.4064/aa-63-4-313-323}{10.4064/aa-63-4-313-323}.\n\\bibitem{tijdemanyuan2002} R.~Tijdeman and P.~Yuan,\n  \\emph{On the rationality of Cantor and Ahmes series},\n  Indag. Math. (N.S.) \\textbf{13} (2002), no.~3, 407--418,\n  doi:\\href{https://doi.org/10.1016/S0019-3577(02)80018-0}{10.1016/S0019-3577(02)80018-0}.\n\\bibitem{erdosgraham1980} P.~Erd\\H{o}s and R.~L. Graham,\n  \\href{https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf}{\\emph{Old\n  and New Problems and Results in Combinatorial Number Theory}},\n  Monogr. Enseign. Math. 28, Geneva, 1980, p.~64.\n\\bibitem{erdos1988} P.~Erd\\H{o}s,\n  \\href{https://users.renyi.hu/~p_erdos/1988-22.pdf}{\\emph{On the irrationality\n  of certain series: problems and results}}, in A.~Baker (ed.), \\emph{New\n  Advances in Transcendence Theory}, Cambridge UP, 1988, pp.~102--109,\n  doi:\\href{https://doi.org/10.1017/CBO9780511897184.009}{10.1017/CBO9780511897184.009}.\n\\bibitem{bado2026} I.~O. Bado,\n  \\emph{Prime-Support Rigidity and Primitive Pseudo-Greedy Dynamics:\n  Partial Progress on Erd\\H{o}s Problem~\\#243}, preprint posted September 2026,\n  doi:\\href{https://doi.org/10.13140/RG.2.2.36612.08325}\n  {10.13140/RG.2.2.36612.08325}.\n\\bibitem{erdosproblemaday243} P.~White with Claude (Anthropic),\n  \\href{https://erdosproblemaday.com/report/243}{\\emph{Erd\\H{o}s \\#243:\n  working report}}, Erd\\H{o}s Problem a Day, page dated 12 August 2026.\n  AI-assisted, unrefereed working report.\n\\bibitem{stevenhagenlenstra1996} P.~Stevenhagen and H.~W. Lenstra, Jr.,\n  \\href{https://pub.math.leidenuniv.nl/~lenstrahw/PUBLICATIONS/1994c/art.pdf}\n  {\\emph{Chebotar\\\"ev and his density theorem}}, Math. Intelligencer\n  \\textbf{18} (1996), no.~2, 26--37,\n  doi:\\href{https://doi.org/10.1007/BF03027290}{10.1007/BF03027290}.\n  Page references are to the linked author version dated 23 March 1995.\n\\bibitem{koizumi2025} J.~Koizumi,\n  \\href{https://math.colgate.edu/~integers/aa28/aa28.pdf}\n  {\\emph{Irrationality of the reciprocal sum of doubly exponential sequences}},\n  Integers \\textbf{26} (2026), Paper No.~A28, 17 pp.,\n  doi:\\href{https://doi.org/10.5281/zenodo.18714404}{10.5281/zenodo.18714404}.\n  Locators refer to this published version.\n\\bibitem{erdosproblems} T.~F. Bloom,\n  \\href{https://www.erdosproblems.com/243}{\\emph{Erd\\H{o}s Problem \\#243}},\n  accessed 28 July 2026.\n\\bibitem{erdosstraus1974} P.~Erd\\H{o}s and E.~G. Straus,\n  \\href{https://msp.org/pjm/1974/55-1/pjm-v55-n1-p08-s.pdf}\n  {\\emph{On the irrationality of certain series}}, Pacific J. Math.\n  \\textbf{55} (1974), no.~1, 85--92.\n\\bibitem{hancltijdeman2008} J.~Han\\v{c}l and R.~Tijdeman,\n  \\href{https://www.impan.pl/shop/en/publication/transaction/download/product/82186}\n  {\\emph{On the irrationality of polynomial Cantor series}},\n  Acta Arith. \\textbf{133} (2008), no.~1, 37--52,\n  doi:\\href{https://doi.org/10.4064/aa133-1-3}{10.4064/aa133-1-3}.\n  Locators refer to the published version.\n\\bibitem{duverneykurosawashiokawa2020} D.~Duverney, T.~Kurosawa and I.~Shiokawa,\n  \\href{https://danielduverney.fr/documents/theorie-des-nombres/Tsukuba.pdf}\n  {\\emph{Irrationality exponents of certain fast converging series of rational\n  numbers}}, Tsukuba J. Math. \\textbf{44} (2020), no.~2, 235--250,\n  doi:\\href{https://doi.org/10.21099/tkbjm/20204402235}{10.21099/tkbjm/20204402235}.\n  Theorem locators follow the linked 14-page author version.\n\\bibitem{crmarickovac2025} T.~Crmari\\'{c} and V.~Kova\\v{c},\n  \\href{https://arxiv.org/abs/2504.18712v1}\n  {\\emph{On the irrationality of certain super-polynomially decaying series}},\n  Colloq. Math. \\textbf{179} (2025), 55--68,\n  doi:\\href{https://doi.org/10.4064/cm9628-5-2025}{10.4064/cm9628-5-2025}.\n  Theorem locators follow arXiv:2504.18712v1.\n\\bibitem{kouli2020} A.~Koutsoukou-Argyraki and W.~Li,\n  \\href{https://isa-afp.org/entries/Irrational_Series_Erdos_Straus.html}\n  {\\emph{Irrationality Criteria for Series by Erd\\H{o}s and Straus}},\n  Archive of Formal Proofs, 12 May 2020. An Isabelle/HOL formalisation;\n  the archive entry identifies the results formalised.\n\\bibitem{dlmf_gamma} National Institute of Standards and Technology,\n  \\href{https://dlmf.nist.gov/5.11.E12}{\\emph{Digital Library of Mathematical\n  Functions}, \\S5.11(iii), formula 5.11.12}, accessed 16 September 2026.\n\\end{thebibliography}\n\n\n% Keep the generated concordance heading with its table.\n\\evidenceconcordancenewpage\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":72229,"sha256":"sha256:b393a9190d51137fb1ba6d3ba15bb1915219b468de6fe7279cbfaec46abd9b9c","cite_as":"Plectis reader corpus, paper:erdos-243-reciprocal-tail-rigidity","source_repository":"plectis-erdos","source_path":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:088d36d7ba3d3ee0cee2df99e8915927999698f19889f8ea0b8d55b3721ec778","byte_count":6923,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% This is the only copy of the shared preamble.  The eight short notes, the\n% long records assembled from paper/reasoning-parts other than #249 and #257\n% (which carry their own preambles), and the synthesis paper include it with\n% \\input{problem-note-preamble}; paper/Makefile builds from this directory\n% with the directory on the tectonic search path.\n%\n% Shared preamble for the Erdős Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erdős problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{8e58c395e9807ef3374d7365c62cf04d20ee6331}\n\\newcommand{\\commitshort}{9e231ce4371f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-erdos/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-erdos/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\usepackage{longtable}  % the notes' declaration tables; loaded here so the assembled records inherit it\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n% ---------------------------------------------------------------------------\n% Evidence marks.  paper-evidence.tex places \"Lean\" and \"Comparator\" links in\n% the margin beside each result from the paper's generated file\n% paper/evidence/<paper>.tex, which each paper inputs after this preamble.\n% ---------------------------------------------------------------------------\n\\input{paper-evidence}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Production provenance.  The wording is fixed by the corpus provenance\n% contract and reproduced verbatim in every manuscript in this repository, so\n% the papers cannot drift into different accounts of how they were made.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erdős problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every Lean mark in the margin points to propositions that the\n  pinned Lean kernel accepts, with no \\texttt{sorry}, no added axiom, and no\n  unchecked evaluation.  That claim concerns the formal statements only; it\n  says nothing about their mathematical interest, their novelty, or the\n  original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n\n% Compact opening card for the separately stated Comparator theorem selected\n% from each problem note.  The prose supplied by each note owns the\n% contribution and relevance judgements; the shared final sentence states the\n% mechanical assurance boundary once and points to the public evidence owner.\n\\newcommand{\\externalreviewobject}[3]{%\n  \\begin{quote}\\small\n  \\textbf{Contribution.} #1\\par\n  \\textbf{Relation to the open problem.} #2\\par\n  \\textbf{Executable review object.} #3  The repository's\n  external-verification job compares these exact Lean propositions with\n  separately declared challenge statements and an axiom budget, then asks\n  Lean's kernel to check the submitted proofs.  The\n  \\href{https://github.com/wcook04/plectis-erdos/blob/ca0e13f8acf5ccf48506e4bdb870953d3a0856fa/formalization.yaml}%\n  {formalisation manifest} and the commit-bound CI receipt record that check;\n  they do not assess novelty, significance, or whether the original problem is\n  solved.\n  \\end{quote}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  Under\n% Tectonic (XeTeX) the text and mathematics are set in TeX Gyre Pagella from\n% the bundled OpenType fonts, with a Heros sans range for \\mathsf and\n% Inconsolata for monospace; the pdflatex fallback keeps the same Palatino\n% design through newpx.  The A4 text block uses a classical measure with\n% quiet running heads and restrained ink.  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It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"},{"record_type":"paper_source_dependency","tex_reference":"evidence/erdos-243-reciprocal-tail-rigidity","source_path":"paper/evidence/erdos-243-reciprocal-tail-rigidity.tex","media_type":"text/x-tex","sha256":"sha256:0e7e160adf8e923389a043dcbe026ad47117224e41e32e439482b959f6e629c0","byte_count":11519,"licence":"CC-BY-4.0","body_format":"latex","body":"% Generated by scripts/paper_evidence.py from docs/paper_lean_coverage.json and the\n% Comparator replay receipts under evidence/comparator/.  Do not edit by hand.\n\\newcommand{\\evidencerecordurl}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md}\n\\DeclareResultEvidenceName{res:cubicrate}{Theorem 1.2}\n\\DeclareResultEvidenceRecord{res:cubicrate}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-cubicrate}{exact}{compared}\n\\DeclareResultEvidence{res:cubicrate}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR21/SquareSpecialisationUnconditional.lean\\#L70}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-cubicrate-comparator}\n\\DeclareResultEvidenceName{res:reduced}{Proposition 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4.2}\n\\DeclareResultEvidenceRecord{res:update}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-update}{exact}{compared}\n\\DeclareResultEvidence{res:update}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR7/Arithmetic.lean\\#L21}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-update-comparator}\n\\DeclareResultEvidenceName{res:absorb}{Theorem 4.3}\n\\DeclareResultEvidenceRecord{res:absorb}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-absorb}{exact}{compared}\n\\DeclareResultEvidence{res:absorb}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR7/Arithmetic.lean\\#L120}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-absorb-comparator}\n\\DeclareResultEvidenceName{res:step}{Corollary 4.4}\n\\DeclareResultEvidenceRecord{res:step}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-step}{exact}{compared}\n\\DeclareResultEvidence{res:step}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-step}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-step-comparator}\n\\DeclareResultEvidenceName{res:gcdstab}{Proposition 4.5}\n\\DeclareResultEvidenceRecord{res:gcdstab}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-gcdstab}{exact}{compared}\n\\DeclareResultEvidence{res:gcdstab}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR7/Reduction.lean\\#L103}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-gcdstab-comparator}\n\\DeclareResultEvidenceName{res:crt}{Lemma 4.6}\n\\DeclareResultEvidenceRecord{res:crt}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-crt}{exact}{compared}\n\\DeclareResultEvidence{res:crt}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean\\#L839}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-crt-comparator}\n\\DeclareResultEvidenceName{res:barrier}{Theorem 4.7}\n\\DeclareResultEvidenceRecord{res:barrier}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-barrier}{exact}{compared}\n\\DeclareResultEvidence{res:barrier}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean\\#L903}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-barrier-comparator}\n\\DeclareResultEvidenceName{res:cor}{Corollary 4.8}\n\\DeclareResultEvidenceRecord{res:cor}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-cor}{exact}{compared}\n\\DeclareResultEvidence{res:cor}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR7/Arithmetic.lean\\#L179}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-cor-comparator}\n\\DeclareResultEvidenceName{res:originalbounded}{Corollary 4.9}\n\\DeclareResultEvidenceRecord{res:originalbounded}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-originalbounded}{exact}{compared}\n\\DeclareResultEvidence{res:originalbounded}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR7/ProductDefect.lean\\#L211}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-originalbounded-comparator}\n\\DeclareResultEvidenceName{res:inclusiveone}{Corollary 4.10}\n\\DeclareResultEvidenceRecord{res:inclusiveone}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-inclusiveone}{exact}{compared}\n\\DeclareResultEvidence{res:inclusiveone}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-inclusiveone}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-inclusiveone-comparator}\n\\DeclareResultEvidenceName{res:massscalar}{Theorem 5.1}\n\\DeclareResultEvidenceRecord{res:massscalar}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-massscalar}{exact}{compared}\n\\DeclareResultEvidence{res:massscalar}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR7/Frontier.lean\\#L84}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-massscalar-comparator}\n\\DeclareResultEvidenceName{res:weightedrecord}{Theorem 6.1}\n\\DeclareResultEvidenceRecord{res:weightedrecord}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-weightedrecord}{exact}{compared}\n\\DeclareResultEvidence{res:weightedrecord}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR11/CanonicalRecords.lean\\#L202}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-weightedrecord-comparator}\n\\DeclareResultEvidenceName{res:lcmbounded}{Corollary 6.2}\n\\DeclareResultEvidenceRecord{res:lcmbounded}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-lcmbounded}{exact}{compared}\n\\DeclareResultEvidence{res:lcmbounded}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR7/LcmDefect.lean\\#L49}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-lcmbounded-comparator}\n\\DeclareResultEvidenceName{res:frontier}{Proposition 7.1}\n\\DeclareResultEvidenceRecord{res:frontier}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-frontier}{exact}{compared}\n\\DeclareResultEvidence{res:frontier}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR7/Frontier.lean\\#L151}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-frontier-comparator}\n\\DeclareResultEvidenceName{res:weights}{Lemma 7.2}\n\\DeclareResultEvidenceRecord{res:weights}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-weights}{exact}{compared}\n\\DeclareResultEvidence{res:weights}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR20/RealCutoffCriterion.lean\\#L87}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-weights-comparator}\n\\DeclareResultEvidenceName{res:residue}{Theorem A.2}\n\\DeclareResultEvidenceRecord{res:residue}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-residue}{exact}{compared}\n\\DeclareResultEvidence{res:residue}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/FiniteHorizonResidue.lean\\#L134}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos-243-reciprocal-tail-rigidity.md\\#res-residue-comparator}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/243/erdos-243-reciprocal-tail-rigidity.tex","section_outline":["Introduction","Tail numerators","Proof of cubic-rate irrationality","An eventual cubic polynomial","Reduction and the constant term","A square forced by a zero numerator","Traces in the cubic field","The contradiction modulo seven","Bounded increases","Zero errors and descent","A stable gcd and a forbidden block","The condition in the original sequence","Finite total relative increase","New maxima of an LCM numerator","Further questions","Arithmetic compatibility","A factorial modulus for integral recursion","Sources and related work","Verification and reproducibility"]},"erdos243-reciprocal-tail-reasoning-surface":{"paper_id":"erdos243-reciprocal-tail-reasoning-surface","title":"Reciprocal Sums and the Sylvester Recurrence: Further Results and Proofs","form":"Reasoning surface","length_class":"long","pages":61,"subject":"Erdős #243","subject_kind":"problem","problem_id":"erdos_243","landing_role":"mathematics","question":"Which growth hypotheses make a reciprocal sum irrational, and which sufficient conditions force the Sylvester recurrence without settling Erdős #243?","precis":"Integer finite differences exclude nonintegral regular rates and reduce the cubic rate to a polynomial. A square condition in the cubic field and congruences modulo seven then prove irrationality. For rational reciprocal sums with a_{n+1}/a_n² tending to one, the record derives sufficient Sylvester-recurrence criteria from persistent divisors, bounded numerator increases and weighted crossings of new maxima. The arithmetic estimates needed to apply these criteria to every rational reciprocal sum remain open.","owns":"cubic and nonintegral regular-rate irrationality, integer-tail arithmetic and sufficient Sylvester-recurrence criteria, with their separate hypotheses and remaining estimates","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #243, which remains open","first_pass":["long243:sec:cubicrate","long243:sec:bounded","long243:sec:open","sec:erdos-243-complete-family-map"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos243-reciprocal-tail-reasoning-surface.pdf","hosted_pdf_path":"papers/erdos243-reciprocal-tail-reasoning-surface.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex","site_html":"maths/papers/erdos243-reciprocal-tail-reasoning-surface.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex","body_pointer":"papers.records.erdos243-reciprocal-tail-reasoning-surface.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Companion paper with full arguments and a guide to the formal sources.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n\\input{evidence/erdos243-reciprocal-tail-reasoning-surface}\n% Pinned revision includes the checked cubic and nonintegral regular-rate theorems.\n\\renewcommand{\\commit}{be89e72217ec9c5f05aa5ec7b915c1ebf0816fdd}\n\\renewcommand{\\commitshort}{be89e72217ec}\n% That revision keeps the Lean roots under lean/, so the links name it.\n\\renewcommand{\\PX}{lean/ErdosProblems}\n% Sources that postdate \\commit are cited with their lean/ path and resolve at \\latecommit,\n% whose tree keeps the Lean roots under lean/.\n\\newcommand{\\latecommit}{168bf6727758f918a430ef056a1c93d3160b53a6}\n\\newcommand{\\laterepobase}{https://github.com/wcook04/plectis-erdos/blob/\\latecommit}\n\\hypersetup{\n  pdftitle={Reciprocal sums and the Sylvester recurrence: further results and proofs},\n  pdfsubject={Erdős Problem Notes: Problem 243},\n  pdfkeywords={irrationality, Ahmes series, Sylvester sequence, unit fractions, Lean 4},\n}\n\n% Operators used repeatedly in the mathematical arguments.\n\\newcommand{\\fnum}{\\operatorname{num}}\n\\newcommand{\\Fp}{\\mathbb{F}}\n\\DeclareMathOperator{\\Tr}{Tr}\n\\DeclareMathOperator{\\Gal}{Gal}\n\n\\runninghead{Reciprocal sums: further results and proofs}\n\\title{Reciprocal Sums and the Sylvester Recurrence:\\newline Further Results and Proofs}\n\\subtitle{Erd\\H{o}s Problem~\\#243}\n\\author{Will Cook}\n\\date{September 2026}\n\n\\setlength{\\emergencystretch}{2em}\n% Permit breaks inside long declaration names in the generated catalogue.\n\\usepackage{xurl}\n\\begin{document}\n% ---- part core ----\n% ; - part core ; -\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\nWe study rational reciprocal sums with $a_{n+1}/a_n^2\\to1$.\nInteger finite differences exclude nonintegral regular rates and reduce\nthe cubic rate to a polynomial. A square condition in the cubic field\nand congruences modulo seven then prove irrationality.\nFor general rational tails, we derive Sylvester recurrence criteria\nfrom persistent divisors, bounded numerator increases and weighted\ncrossings of new maxima. We describe the arithmetic estimates still\nneeded for Erd\\H{o}s Problem~\\#243.\n\\end{abstract}\n\n\\section{Introduction}\\label{long243:sec:problem}\nSylvester's sequence $2,3,7,43,1807,\\ldots$ has reciprocal sum $1$.\nIts recurrence $a_n=a_{n-1}^2-a_{n-1}+1$ gives the telescoping identity\n\\[\n \\frac1{a_{n-1}-1}=\\frac1{a_{n-1}}+\\frac1{a_n-1}.\n\\]\nThe tail beginning with $1/a_{n-1}$ therefore equals $1/(a_{n-1}-1)$.\nErd\\H{o}s Problem~\\#243 asks whether every rational reciprocal sum\nwith $a_n\\sim a_{n-1}^2$ must eventually have this form:\n\n\\begin{problem}[Erd\\H{o}s \\#243]\\label{long243:res:problem}\nLet $1\\le a_1<a_2<\\cdots$ be a sequence of integers with\n\\[\n \\lim_{n\\to\\infty}\\frac{a_n}{a_{n-1}^{2}}=1\n \\qquad\\text{and}\\qquad\n \\sum\\frac{1}{a_n}\\in\\Q .\n\\]\nThen $a_n=a_{n-1}^{2}-a_{n-1}+1$ for all sufficiently large $n$.\n\\end{problem}\n\n\\noindent See~\\cite[p.~64]{erdosgraham1980} and~\\cite[p.~105]{erdos1988}.\nBloom's catalogue, accessed 28 July 2026, lists the problem as\nopen~\\cite{erdosproblems}.\nBelow we index the same recurrence as\n$a_{n+1}=a_n^{2}-a_n+1$, which is the shift the formal sources use; the two\nforms are the same statement. The polynomial $a^2-a+1$ is called the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{37}{sylvesterNext}{Sylvester\nsuccessor}.\n\nClearing a rational reciprocal tail gives positive integers $C_n,D_n$\nwith $C_{n+1}=a_nC_n-D_n$ and $D_{n+1}=a_nD_n$. The first update\ncan also be written $C_{n+1}=C_n-E_n$, where\n$E_n=D_n-(a_n-1)C_n$. Quadratic growth makes $E_n/C_n\\to0$.\nThe eventual Sylvester recurrence follows once the integer error itself\nvanishes.\n\nThere are two ways to use this integer numerator. At a precise regular\nrate it grows, but a high finite difference tends to zero and must\nvanish. The resulting polynomial is then tested against the denominator\nrecurrence. With bounded upward steps, a different argument applies:\na stable common divisor gives reduced tails, whose numerators avoid\nevery earlier multiplier. The Chinese remainder theorem produces a\nblock that bounded rises cannot cross, regardless of intervening\ndecreases. Neither argument replaces the denominator recurrence by a\nscalar growth estimate.\n\nSection~\\ref{long243:sec:cubicrate} proves cubic and nonintegral-rate\nirrationality: these rates exclude the eventual Sylvester recurrence.\nThe general question remains open. After the literature comparison in\nSection~\\ref{long243:sec:priorwork},\nSections~\\ref{long243:sec:state}--\\ref{long243:sec:defect} develop\ninteger tails and zero-error identities for the subsequent criteria.\n\nThe later criteria use two distinct normalisations.\nSection~\\ref{long243:sec:lcmrecords} clears tails with an LCM: common\nnumerator factors remain, so congruences forbid short crossings rather\nthan numerator values. Section~\\ref{long243:sec:records} reduces to\nlowest terms, where cancellation can remove a prime that persists in\nthe LCM. That argument must track the prime powers which survive. Sections~\\ref{long243:sec:constant}--\\ref{long243:sec:periodic}\ntreat constant and periodic errors;\nSections~\\ref{long243:sec:barrier}--\\ref{long243:sec:bounded} prove\nthe bounded-increase criterion, and Section~\\ref{long243:sec:mass}\ntreats scalar summability. Section~\\ref{long243:sec:open} separates open\nestimates from countermodels to weakened hypotheses. The appendices\ngive formal sources, the finite residue calculation and a result map.\n\nWe use $z_+=\\max(z,0)$. Deleting a finite prefix is harmless for an eventual\nrecurrence, but not for the coefficients in a higher-order rate. In\nparticular, the index $n$ in $1+3/n+o(n^{-3})$ is kept fixed throughout\nthe cubic proof. The recurrence sections also use zero-based indexing,\nwith that convention stated where the integer sequences are introduced.\n\n\n\\medskip\n\\noindent\\textbf{Keywords.} irrationality; Ahmes series; Sylvester's sequence;\nunit fractions; Lean~4.\n\\qquad\n\\textbf{MSC 2020.} 11J72 (primary); 11B37, 11D68, 68V20 (secondary).\n\n\\papersectiontarget{243-long-cubic}\n\\section{Irrationality at the cubic rate}\\label{long243:sec:cubicrate}\n\nA sufficiently precise rate constrains an integer tail more strongly\nthan the quadratic limit alone. At rate $1+3/n+o(n^{-3})$, its\nnumerator must eventually be cubic. The denominator recurrence will\nexclude every possible cubic. For the prefix product\n$P_n=\\prod_{1\\le k<n}a_k$, the increments of $P_n/a_n$ grow like a\npositive multiple of $n^2$, so the bounded-increase criterion\nlater in the paper does not apply. The eventual Sylvester recurrence gives deviation\n$O(1/a_n)$ and is also excluded by this rate.\n\n\\begin{theorem}[cubic-rate irrationality]\\label{long243:res:cubicrate}\nIf strictly increasing positive integers satisfy\n$a_n^2/a_{n+1}=1+3/n+o(n^{-3})$, then $\\sum_n1/a_n$ is irrational.\n\\end{theorem}\n\nThe same extraction argument rules out nonintegral regular rates above\none. These require no number-field calculation.\n\n\\begin{theorem}[nonintegral regular-rate irrationality]\\label{long243:res:nonintegralrate}\nLet $\\lambda>1$ be a real number that is not an integer. If strictly\nincreasing positive integers satisfy\n$a_n^2/a_{n+1}=1+\\lambda/n+o(n^{-\\lambda})$, then\n$\\sum_n1/a_n$ is irrational.\n\\end{theorem}\n\nThe \\lword{Erdos243/PaperCompleteR21/NonintegralRegularRate.lean}{196}{nonintegral_regular_rate_irrational}{zero-indexed Lean theorem}\nalso derives convergence of the reciprocal series from the rate.\n\n\nThe rates retain their stated indices throughout. Appendix~\\ref{long243:app:index}\ngives the finite-prefix construction relating these one-based statements\nto the zero-based formal statements without changing the rate's error term.\n\nFor a concrete sequence satisfying the hypothesis, take\n\\[\n a_1=8,\\qquad a_{n+1}=\\left\\lceil\\frac{n a_n^2}{n+3}\\right\\rceil\n \\quad(n\\ge1).\n\\]\nIt begins $8,16,103,5305,\\ldots$. The inequality $a_{n+1}\\ge a_n^2/4$\ngives $a_n\\ge4\\cdot2^{2^{n-1}}\\ge8$ by induction. Hence\n$a_{n+1}\\ge a_n^2/4\\ge2a_n$, so the sequence is strictly increasing.\nRounding upwards gives\n\\[\n 0\\le1+\\frac3n-\\frac{a_n^2}{a_{n+1}}<\\frac{16}{a_n^2}=o(n^{-3}).\n\\]\nThus its reciprocal sum is irrational. Lemma~\\ref{long243:res:extraction}\nwill turn the rate assumption into an eventual rising-factorial cubic\nfor the integer tail numerator. We first prove the arithmetic obstruction\nthat rules out this cubic using only the exact integer recurrences.\n\n\\begin{theorem}[rising-factorial cubic exclusion]\\label{long243:res:cubicexclusion}\nLet $a,C,D:\\N\\to\\Z_{>0}$ satisfy\n\\begin{equation}\\label{long243:eq:cubicorbit}\n C_{n+1}=a_nC_n-D_n,\\qquad D_{n+1}=a_nD_n .\n\\end{equation}\nThen for every $A\\in\\Q_{>0}$ and every $B\\in\\Q$,\n\\[\n \\liminf_{X\\to\\infty}\n \\frac{\\#\\{n\\le X:C_n\\ne A\\,n(n+1)(n+2)+B\\}}{X}>0 .\n\\]\nIn particular no such orbit satisfies $C_n=A\\,n(n+1)(n+2)+B$ for all\nlarge $n$.\n\\end{theorem}\n\nThe short paper needs only to exclude eventual equality with a cubic\nof this form. Here we prove more: the disagreement set has positive\nlower density. The bound may depend on the orbit and the cubic; no\nuniform numerical constant is claimed.\n\nLower density zero does not give eventual agreement, but leaves\narbitrarily late intact four-term blocks. Their third differences\nbound $G_n=\\gcd(C_n,D_n)$, stabilising the divisibility chain. Dividing\nout its limit leaves $m n(n+1)(n+2)/6+c$, with $m>0$ integral and\n$c=\\pm1$. A vanishing middle numerator modulo a prime forces the\nnegative product of its neighbours to be a square. This condition at\nevery root modulo almost every prime yields a square in the cubic\nfield; traces give\n$m=12$, and four-term blocks modulo seven exclude both signs.\n\nThe density assertion comes from the same finite tests. A failed test\nrecurs on an arithmetic progression, and each occurrence forces a\ndisagreement in a window of bounded length. The next lemma counts\nthose disagreements. After proving the arithmetic exclusion, we return\nto the rate assumption in Lemma~\\ref{long243:res:extraction}.\n\n\\paragraph{The integer tail numerator.}\nSuppose $\\sum_{n\\ge1}1/a_n=p/q$ with positive integers $p,q$, and write\n$x_n=\\sum_{k\\ge n}1/a_k$ for the tail.  Put\n\\[\n D_n=q\\prod_{1\\le k<n}a_k,\\qquad C_n=D_nx_n .\n\\]\nThen $D_n$ is a positive integer, and\n\\[\n C_n=D_n\\Bigl(\\frac pq-\\sum_{1\\le k<n}\\frac1{a_k}\\Bigr)\n    =p\\prod_{1\\le k<n}a_k-q\\sum_{1\\le k<n}\\ \\prod_{\\substack{1\\le j<n\\\\ j\\ne k}}a_j\n\\]\nis a positive integer as well.  From $x_n=1/a_n+x_{n+1}$,\n\\[\n C_{n+1}=D_{n+1}x_{n+1}=a_nD_n\\Bigl(x_n-\\frac1{a_n}\\Bigr)=a_nC_n-D_n,\n \\qquad D_{n+1}=a_nD_n,\n\\]\nwhich is \\eqref{long243:eq:cubicorbit}. These are the unreduced tail variables of Koizumi's\nLemma~4 \\cite[pp.~11--12]{koizumi2025}, up to a common positive factor.\nLemma~\\ref{long243:res:tailratio} uses this unreduced numerator:\nreducing each tail separately would introduce varying cancellation\nfactors into the ratio and obscure the prescribed rate.\nSection~\\ref{long243:sec:state} constructs these variables together with\nthe error $E_n=D_n-(a_n-1)C_n$, which this section does not need.  The subsequent cubic exclusion uses only these integer recurrences;\nit does not assume a reciprocal sum. To apply its zero-based statement\nwithout shifting the rate hypothesis, adjoin $a_0=1$, $D_0=q$ and\n$C_0=p+q$. The recurrences at $0$ then give $D_1=q$ and $C_1=p$,\nwhile every original index $n\\ge1$ is unchanged. The auxiliary term\n$a_0$ need not satisfy the increasing-sequence hypothesis of the\nirrationality theorem: the exclusion requires only positive multipliers.\n\n\\subsection{Reduction and lower density}\n\nFor $S\\subseteq\\N$ write $\\underline d(S)=\\liminf_X\\#(S\\cap[1,X])/X$.\nThus $\\underline d(S)=0$ gives cutoffs along which the exceptional\nproportion tends to zero, not eventual agreement. We cannot replace\n$C_n$ by its proposed polynomial at arbitrary indices; the following\nlemma supplies the finite windows where such a replacement is valid.\nFor a sequence $F$, let $\\Delta F_n=F_{n+1}-F_n$; higher powers of\n$\\Delta$ mean repeated forward differences. This operator is distinct\nfrom the later indexed quantity $\\Delta_n$, the Sylvester defect.\n\n\\begin{lemma}[periodic obstruction principle]\\label{long243:res:periodicobstruction}\nLet $h,L$ be positive integers and let $j_1,\\ldots,j_L$ be fixed integer\noffsets. Suppose that for every sufficiently large $n$ in one residue\nclass modulo $h$ at least one of $n+j_1,\\ldots,n+j_L$ lies in $S$.\nThen $\\underline d(S)\\ge1/(Lh)$.\n\\end{lemma}\n\n\\begin{proof}\nThere are $X/h+O(1)$ relevant starting indices below $X$. Their witnesses\nlie below $X+O(1)$ because the offsets are fixed, and each element of $S$\nserves as a witness for at most $L$ starting indices. Hence\n$L\\#(S\\cap[1,X])\\ge X/h-O(1)$, which gives the claimed lower density.\n\\end{proof}\n\nFix $A\\in\\Q_{>0}$, $B\\in\\Q$, and suppose for contradiction that\n\\[\n P(n)=A\\,n(n+1)(n+2)+B,\n \\qquad S=\\{n:C_n\\ne P(n)\\},\n \\qquad \\underline d(S)=0 .\n\\]\nWe retain this assumption through Section~\\ref{long243:sec:modseven}.\n\n\\begin{lemma}[gcd stabilisation and the reduced polynomial]\\label{long243:res:gcdshape}\nWrite $G_n=\\gcd(C_n,D_n)$.  Then $6A$ is a positive integer, $G_n$ divides\n$6A$ for every $n$, and $G_n$ is eventually equal to a positive integer\n$g$.  On the tail where $G_n=g$, put $u_n=C_n/g$, $v_n=D_n/g$ and\n$Q(n)=P(n)/g$.  Then\n\\begin{equation}\\label{long243:eq:primitivetail}\n u_{n+1}=a_nu_n-v_n,\\qquad v_{n+1}=a_nv_n,\\qquad \\gcd(u_n,v_n)=1,\n \\qquad\\gcd(u_n,u_{n+1})=1,\n\\end{equation}\nand there are $m\\in\\Z_{>0}$ and $c\\in\\{-1,1\\}$ with\n\\begin{equation}\\label{long243:eq:Qmc}\n Q(n)=\\frac m6n(n+1)(n+2)+c .\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nEquation~\\eqref{long243:eq:cubicorbit} gives $G_n\\mid G_{n+1}$ and $G_n\\mid C_t$ for\nevery $t\\ge n$.  Since $\\underline d(S)=0$, beyond every threshold there is a block of four\nconsecutive indices disjoint from $S$: otherwise every block of four would meet\n$S$ and $\\underline d(S)\\ge1/4$.  On such a block $\\Delta^3C_t=\\Delta^3P(t)=6A$, and\n$G_n$ divides each of the four values, hence divides $6A$.  So $6A$\nis a positive integer and the divisibility chain $(G_n)$ is bounded and\nstabilises at some $g$.\n\nOn the primitive tail, $\\gcd(u_n,v_n)=1$ by construction, and a prime dividing\n$u_n$ and $u_{n+1}$ would divide $v_n=a_nu_n-u_{n+1}$, which gives\n$\\gcd(u_n,u_{n+1})=1$.\n\nSince $g\\mid6A$, the number $m:=6A/g$ is a positive integer. Put\n$c=B/g$, so\n\\[\n Q(n)=m\\binom{n+2}{3}+c.\n\\]\nChoose any sufficiently late $n\\notin S$. Then $Q(n)=u_n\\in\\Z$, and the\nbinomial term is an integer, so $c\\in\\Z$. This proves\n\\eqref{long243:eq:Qmc} with $c\\in\\Z$. It also shows that $Q$ is\ninteger-valued: its values at integer arguments are integers, even\nthough its coefficients need not all be integers.\n\nIt remains to exclude $c=0$ and every $c$ with a prime factor.  Let $\\ell$ be a\nprime dividing $c$, or any prime if $c=0$.  For $n\\equiv-1\\pmod{6\\ell}$, write\n$n+1=6\\ell k$; then\n\\[\n \\frac{n(n+1)(n+2)}6=\\ell k(6\\ell k-1)(6\\ell k+1),\n \\qquad\n \\frac{(n+1)(n+2)(n+3)}6=\\ell k(n+2)(n+3),\n\\]\nso $\\ell$ divides both $Q(n)$ and $Q(n+1)$.  Agreement at both indices would\ngive $\\ell\\mid\\gcd(u_n,u_{n+1})=1$.  Hence one of $n,n+1$ lies in $S$ for every\nlate $n\\equiv-1\\pmod{6\\ell}$, and Lemma~\\ref{long243:res:periodicobstruction} with\n$L=2$, $h=6\\ell$ gives $\\underline d(S)\\ge1/(12\\ell)$.  This contradiction\nleaves $c=\\pm1$.\n\\end{proof}\n\n\\begin{lemma}[coprimality and irreducibility]\\label{long243:res:reduciblecase}\nOn the primitive tail, $\\gcd(a_n,v_n)=1$, the multipliers at distinct indices\nare pairwise coprime, infinitely many of them exceed $1$, and the cubic\n$Q_{m,c}$ of \\eqref{long243:eq:Qmc} is irreducible over $\\Q$.\n\\end{lemma}\n\n\\begin{proof}\nA prime dividing $a_n$ and $v_n$ would divide $u_{n+1}$ and $v_{n+1}$, against\n$\\gcd(u_{n+1},v_{n+1})=1$; and every earlier multiplier divides every later\n$v_n$, so distinct multipliers are coprime.  If $a_n=1$ for all large $n$ then\n$v_n$ is eventually a positive constant and $u_{n+1}=u_n-v_n$ decreases without\nbound, against positivity.  So infinitely many distinct primes divide late\nmultipliers.\n\nSuppose $Q_{m,c}$ had a rational root.  Choose a prime $\\ell\\nmid6m$ dividing a\nlate multiplier $a_N$, large enough that the root reduces modulo $\\ell$.  Then\n$Q_{m,c}$ vanishes modulo $\\ell$ on a full residue class, while $\\ell\\mid v_n$\nand therefore $\\ell\\nmid u_n$ for every $n>N$.  Agreement at any late index of\nthat class is impossible, and\nLemma~\\ref{long243:res:periodicobstruction} with $L=1$, $h=\\ell$ gives\n$\\underline d(S)\\ge1/\\ell$.\n\\end{proof}\n\n\\subsection{Square specialisation}\n\nFor the proposed cubic $f(T)=T^3-T+6c/m$, the recurrence will force\n$r^2-1$ to be a square whenever $r$ is a root modulo a good prime.\nWe need the same conclusion for an algebraic root $\\alpha$ in\n$\\Q(\\alpha)$. The issue is that a putative square root could lie in a\nquadratic extension of this field. Its nontrivial automorphism would\nfix $\\alpha$ and change the sign of the square root. Chebotarev supplies\na prime with exactly that behaviour, contradicting the finite-field\ncondition. The following lemma isolates this passage; in the cubic\napplication we take $H(T)=T^2-1$.\n\n\\begin{lemma}[square specialisation]\\label{long243:res:squarespec}\nLet $f\\in\\Q[T]$ be irreducible with root $\\alpha$, and let $H\\in\\Q[T]$ satisfy\n$H(\\alpha)\\ne0$.  If for all but finitely many primes $\\ell$ every root\n$r\\in\\Fp_\\ell$ of the reduction of $f$ has $H(r)$ a square in $\\Fp_\\ell^\\times$,\nthen $H(\\alpha)$ is a square in $\\Q(\\alpha)^\\times$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $H(\\alpha)$ is a nonsquare in $K=\\Q(\\alpha)$, and\nchoose $\\beta$ with $\\beta^2=H(\\alpha)$. The nontrivial automorphism\nof the quadratic extension $K(\\beta)/K$ extends to an element\n$\\sigma\\in\\Gal(L/K)$ for a finite Galois extension $L/\\Q$ containing\n$K(\\beta)$. In particular $\\sigma\\alpha=\\alpha$ and\n$\\sigma\\beta=-\\beta$.\n\nBy Chebotarev's density theorem in the form stated by Stevenhagen\nand Lenstra \\cite[\\S3, p.~15]{stevenhagenlenstra1996}, infinitely many\nunramified rational primes have Frobenius conjugate to $\\sigma$.\nOnly infinitude is needed. We discard the exceptional primes in the\nhypothesis and the finitely many primes at which coefficients, $\\alpha$\nor $\\beta$ have denominators, together with $2$ and the primes at which\n$\\beta$ reduces to zero. Above each\nremaining prime $\\ell$, choose a prime of $L$ whose Frobenius is\n$\\sigma$. Its residue field then satisfies\n\\[\n \\overline\\alpha^{\\,\\ell}=\\overline\\alpha,\\qquad\n \\overline\\beta^{\\,\\ell}=-\\overline\\beta\\ne\\overline\\beta.\n\\]\nThus $\\overline\\alpha\\in\\Fp_\\ell$ is a root of the reduction of $f$.\nNeither of the two roots $\\pm\\overline\\beta$ of\n$X^2-H(\\overline\\alpha)$ is fixed by Frobenius, so neither belongs\nto $\\Fp_\\ell$. Hence $H(\\overline\\alpha)$ is a nonsquare there,\ncontradicting the hypothesis.\n\\end{proof}\n\nAn alternative proof of Lemma~\\ref{long243:res:squarespec} uses zeta\nfunctions. A nonsquare algebraic integer $b$ in a number field $K$\nis nonsquare modulo infinitely many degree-one primes of $K$. The underlying\nzeta-function argument first gives zero or a nonsquare. The zero case is\nexcluded beyond the prime divisors of the nonzero constant coefficient of the\nminimal polynomial of $b$. If\n$H(\\alpha)$ is not a square in $K=\\Q(\\alpha)$, this applies to\n$b=m^2H(\\alpha)$, where the nonzero integer $m$ makes $b$ an algebraic\ninteger, and the residue map at such a prime, of norm $\\ell$, sends $\\alpha$\nto a root $r\\in\\Fp_\\ell$ of the reduction of $f$ at which $H(r)$ is a\nnonsquare. The prime-reduction argument\n(\\href{https://github.com/wcook04/plectis-erdos/blob/e21782a3e5edf1d7e2e7bfca67341356dacfc042/lean/ErdosProblems/Shared/NonsquareModuloPrimes.lean\\#L160-L192}{nonsquares modulo primes})\nis proved from the simple pole of the Dedekind zeta function at $s=1$, which\nis in Mathlib. If $b$ were a nonzero square modulo all but finitely many\nprimes of degree one, these primes would split in $K(\\sqrt b)$, and for real\n$s>1$ near $1$ the zeta function of $K(\\sqrt b)$ would be at least a positive\nconstant times the square of the zeta function of $K$, which the two simple\npoles forbid.\n\nBoth arguments use the condition at every root for all but finitely\nmany primes. Testing finitely many primes supplies no such condition.\nThe zeta-function argument is the one used in the linked formal proof.\n\n\\begin{proposition}[the square forced by the numerator recurrence]\\label{long243:res:transportsquare}\nWrite $\\kappa=m/6$ and $\\eta=6c/m$, so that $Q_{m,c}(n)=\\kappa f(n+1)$ for\n$f(T)=T^3-T+\\eta$.  Then $\\alpha^2-1$ is a square in $\\Q(\\alpha)^\\times$ for a\nroot $\\alpha$ of $f$.\n\\end{proposition}\n\n\\begin{proof}\nEliminating $v_n$ from \\eqref{long243:eq:primitivetail} gives\n\\begin{equation}\\label{long243:eq:secondorderu}\n u_{n+2}=(a_n+a_{n+1})u_{n+1}-a_n^2u_n .\n\\end{equation}\nLet $\\ell$ be a prime and $k$ an index with $\\ell\\mid u_k$.  Reading\n\\eqref{long243:eq:secondorderu} at $n=k-1$ modulo $\\ell$ gives\n$u_{k+1}\\equiv-a_{k-1}^2u_{k-1}$, hence\n\\begin{equation}\\label{long243:eq:forcedsquare}\n -u_{k-1}u_{k+1}\\equiv(a_{k-1}u_{k-1})^2 \\pmod\\ell,\n\\end{equation}\nso $-u_{k-1}u_{k+1}$ is a square modulo $\\ell$.\n\nLet $\\ell\\nmid6m$ and let $r\\in\\Fp_\\ell$ be a root of $f$.  Then $r\\ne0$ and\n$r\\ne\\pm1$, since $f(0)=\\eta\\ne0$ and $f(\\pm1)=\\eta$.  Using $\\eta=r-r^3$,\n\\[\n f(r-1)=-3r(r-1),\\qquad f(r+1)=3r(r+1),\n\\]\nso at an index $k\\equiv r-1\\pmod\\ell$ we may reduce the polynomial values in\n$\\Fp_\\ell$. All equalities in the following calculation are in that field:\n$Q(k)=0$, $Q(k-1)=-3\\kappa r(r-1)$, $Q(k+1)=3\\kappa r(r+1)$, and\n\\begin{equation}\\label{long243:eq:threevalue}\n -Q(k-1)Q(k+1)=9\\kappa^2r^2(r^2-1).\n\\end{equation}\nIf $r^2-1$ were a nonsquare modulo $\\ell$, then so would be the right side of\n\\eqref{long243:eq:threevalue}, because $9\\kappa^2r^2$ is a nonzero square; agreement at\nall three of $k-1,k,k+1$ would then contradict \\eqref{long243:eq:forcedsquare}.  That\nprohibition recurs at every index of the class $r-1$ modulo $\\ell$, and\nLemma~\\ref{long243:res:periodicobstruction} with $L=3$, $h=\\ell$ would give\n$\\underline d(S)\\ge1/(3\\ell)$.  So $r^2-1$ is a square in $\\Fp_\\ell^\\times$ at\nevery root of every good reduction, and Lemma~\\ref{long243:res:squarespec} applies with\n$H(T)=T^2-1$.\n\\end{proof}\n\n\\subsection{The square condition forces \\texorpdfstring{$m=12$}{m=12}}\n\n\\begin{lemma}[classification of the scales]\\label{long243:res:scaletwelve}\nLet $m$ be a positive integer, let $c\\in\\{-1,1\\}$ and $\\eta=6c/m$, and suppose $T^3-T+\\eta$ is\nirreducible over $\\Q$ with a root $\\alpha$ such that $\\alpha^2-1$ is a square\nin $\\Q(\\alpha)^\\times$.  Then $m=12$.\n\\end{lemma}\n\n\\begin{proof}\nChoose $\\beta\\in\\Q(\\alpha)$ with $\\beta^2=\\alpha^2-1$. The identity\n$(\\alpha+\\beta)(\\alpha-\\beta)=1$ gives a reciprocal pair in the cubic\nfield. Taking $z=\\alpha+\\beta$, we obtain\n\\begin{equation}\\label{long243:eq:zalpha}\n z^{-1}=\\alpha-\\beta,\\qquad \\alpha=\\tfrac12(z+z^{-1}),\n\\end{equation}\nand $z$ generates $\\Q(\\alpha)$. Write its minimal polynomial as\n$p(Z)=Z^3+bZ^2+dZ+w$ with $b,d,w\\in\\Q$ and $w\\ne0$. The minimal\npolynomial of $z^{-1}$ is\n\\[\n \\frac{Z^3p(Z^{-1})}{w}\n   =Z^3+\\frac dwZ^2+\\frac bwZ+\\frac1w.\n\\]\nThus Newton's identities apply directly to both $z$ and its inverse.\nWe use the first two traces of $\\alpha$ to restrict these coefficients,\nthen the third trace to impose $\\eta=6c/m$. All traces below are from\n$\\Q(\\alpha)$ to $\\Q$.\nThe polynomial $T^3-T+\\eta$ gives\n\\begin{equation}\\label{long243:eq:alphatraces}\n \\Tr\\alpha=0,\\qquad \\Tr\\alpha^2=2,\\qquad \\Tr\\alpha^3=-3\\eta .\n\\end{equation}\nNewton's identities give $\\Tr z=-b$ and $\\Tr z^{-1}=-d/w$, so the first\nequation in \\eqref{long243:eq:alphatraces} and \\eqref{long243:eq:zalpha} give\n\\begin{equation}\\label{long243:eq:dbw}\n d=-bw .\n\\end{equation}\nFor the second trace, Newton's identities and $d=-bw$ give\n\\[\n \\Tr z^2=b^2+2bw,\\qquad \\Tr z^{-2}=b^2-2b/w.\n\\]\nSince the trace of $1$ is $3$, squaring\n\\eqref{long243:eq:zalpha} yields\n$4\\Tr\\alpha^2=\\Tr z^2+6+\\Tr z^{-2}$. Substituting\n$\\Tr\\alpha^2=2$ gives $b^2+b(w-w^{-1})=1$.\nMultiplication by $w$ factors this equation:\n\\begin{equation}\\label{long243:eq:factored}\n (bw-1)(b+w)=0 .\n\\end{equation}\nFor the third trace, the same identities give\n\\[\n \\begin{aligned}\n \\Tr z^3&=-b^3-3b^2w-3w,\\\\\n \\Tr z^{-3}&=b^3-3b^2/w-3/w.\n \\end{aligned}\n\\]\nThe cross terms in $(z+z^{-1})^3$ have trace\n$3(\\Tr z+\\Tr z^{-1})=0$ by \\eqref{long243:eq:dbw}.\nThus $8\\Tr\\alpha^3=\\Tr z^3+\\Tr z^{-3}$.\nUsing $\\Tr\\alpha^3=-3\\eta$ now gives\n\\begin{equation}\\label{long243:eq:etatrace}\n \\eta=\\frac{(b^2+1)(w+w^{-1})}8 .\n\\end{equation}\nBy \\eqref{long243:eq:factored}, either $b=-w$ or $b=w^{-1}$, and \\eqref{long243:eq:etatrace}\nbecomes\n\\[\n \\eta=\\frac{(w^2+1)^2}{8w}\n \\qquad\\text{or}\\qquad\n \\eta=\\frac{(w^2+1)^2}{8w^3}.\n\\]\nWrite $w=r/s$ with $r\\in\\Z\\mathbin{\\backslash}\\{0\\}$, $s\\in\\Z_{>0}$ and $\\gcd(r,s)=1$.\nSince $\\eta=6c/m$,\n\\begin{equation}\\label{long243:eq:mformula}\n m=\\frac{48c\\,rs^3}{(r^2+s^2)^2}\n \\qquad\\text{or}\\qquad\n m=\\frac{48c\\,r^3s}{(r^2+s^2)^2}.\n\\end{equation}\nNow $\\gcd(r^2+s^2,rs)=1$, so integrality of $m$ and $|c|=1$ force\n$(r^2+s^2)^2\\mid48$, hence $r^2+s^2\\in\\{1,2,4\\}$.  With $r\\ne0$ and $s\\ge1$ the\nvalue $1$ is too small, and $4$ is not a sum of two nonzero squares, so\n$r^2+s^2=2$ and $|r|=s=1$.  Then \\eqref{long243:eq:mformula} gives $m=12cr$, and\n$m>0$ gives $m=12$.\n\\end{proof}\n\nThe square condition leaves the two possible cubics\n\\begin{equation}\\label{long243:eq:survivors}\n Q_{12,\\pm1}(n)=2n(n+1)(n+2)\\pm1 .\n\\end{equation}\nThe square condition uses three consecutive numerators. The last step\nalso uses the preceding denominator update, so it tests four consecutive\nnumerators instead.\n\n\\subsection{The two remaining cubics fail modulo seven}\\label{long243:sec:modseven}\n\n\\begin{lemma}[two incompatible four-term residue patterns]\\label{long243:res:modseven}\nFor $Q_{12,1}$, every sufficiently late four-index window beginning at\n$n\\equiv0\\pmod7$ contains an exceptional index. For $Q_{12,-1}$, the same\nholds for windows beginning at $n\\equiv1\\pmod7$. In either case,\n$\\underline d(S)\\ge1/7$.\nThe specified starting residue classes are part of the assertion; no\nclaim is made for every block of four consecutive indices.\n\\end{lemma}\n\n\\begin{proof}\nReduce modulo $7$.  For $c=1$ the values of $Q_{12,1}$ at $n=0,1,2,3$ are\n$1,13,49,121$, that is\n\\begin{equation}\\label{long243:eq:wordplus}\n (1,6,0,2),\n\\end{equation}\nand for $c=-1$ the values at $n=1,2,3,4$ are $11,47,119,239$, that is\n\\begin{equation}\\label{long243:eq:wordminus}\n (4,5,0,1).\n\\end{equation}\nEach four-term pattern recurs with period $7$.\n\nIndex a proposed occurrence locally by $0,1,2,3$, and use $a_i$ for the\nmultiplier at local index $i$. Let $(c_0,c_1,0,c_3)$ be the residues of\n$u$ and $(d_0,d_1,d_2,d_3)$ those of $v$. Thus\n$c_{i+1}=a_ic_i-d_i$ and $d_{i+1}=a_id_i$ modulo $7$.  The middle zero gives\n$a_1c_1=d_1$, then $d_2=a_1d_1$ and $c_3=-d_2$, so\n\\begin{equation}\\label{long243:eq:d1square}\n d_1^2=c_1\\,a_1d_1=c_1d_2=-c_1c_3 .\n\\end{equation}\nFor \\eqref{long243:eq:wordplus} this is $-6\\cdot2=2$, and for \\eqref{long243:eq:wordminus} it\nis $-5\\cdot1=2$.  The square roots of $2$ modulo $7$ are $3$ and $4$, so\n$d_1\\in\\{3,4\\}$ in both cases.\n\nThe first step gives $a_0=(c_1+d_0)/c_0$ and hence\n$d_1=d_0(d_0+c_1)/c_0$, which is legitimate because $c_0\\in\\{1,4\\}$ is\ninvertible.  For \\eqref{long243:eq:wordplus} this is $d_1=d_0(d_0+6)$, with values\n\\[\n (0,0,2,6,5,6,2) \\qquad\\text{at } d_0=0,1,\\ldots,6,\n\\]\nand for \\eqref{long243:eq:wordminus} it is $d_1=2d_0(d_0+5)$, with values\n\\[\n (0,5,0,6,2,2,6).\n\\]\nThe incompatibility also follows without enumerating $d_0$. Completing\nthe square gives\n\\[\n d_1+2=(d_0+3)^2\\quad(c=1),\\qquad\n d_1+2=2(d_0-1)^2\\quad(c=-1).\n\\]\nSince $2$ is a square modulo $7$, either update makes $d_1+2$ a square.\nThe required values $d_1=3,4$ instead give the nonsquares $5,6$.\nThe two displayed lists provide the same exhaustive check: both images\nare $\\{0,2,5,6\\}$, disjoint from $\\{3,4\\}$.\n\nHence every sufficiently late window $[7k,7k+3]$ for the plus profile,\nand every sufficiently late window $[1+7k,4+7k]$ for the minus profile,\ncontains an element of $S$. Windows within each family are pairwise disjoint.\nThere are $X/7+O(1)$ such windows contained in $[0,X]$, so choosing one\nexceptional index in each gives $\\#(S\\cap[0,X])\\ge X/7-O(1)$ and therefore\n$\\underline d(S)\\ge1/7$. This stronger local bound is not promoted to a\nuniform lower bound for arbitrary cubic profiles.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{long243:res:cubicexclusion}]\nAssume $\\underline d(S)=0$.  Lemmas~\\ref{long243:res:gcdshape}\nand~\\ref{long243:res:reduciblecase} put the primitive tail in the shape \\eqref{long243:eq:Qmc}\nwith $c=\\pm1$ and $Q_{m,c}$ irreducible.  Each excluded branch already\ncontradicts the assumption by producing a positive, branch-dependent lower\ndensity, namely $1/(12\\ell)$ or $1/\\ell$ for the relevant prime $\\ell$.\nProposition~\\ref{long243:res:transportsquare} and Lemma~\\ref{long243:res:scaletwelve} force\n$m=12$, and Lemma~\\ref{long243:res:modseven} then gives\n$\\underline d(S)\\ge1/7$, the final contradiction.\n\\end{proof}\n\nThe two densities play different roles. If the field element is not\na square, Chebotarev supplies one witnessing prime; its periodic\nobstruction gives a positive density of disagreement \\emph{indices}.\nNo lower bound uniform over cubic profiles is needed.\n\n\\subsection{Polynomial extraction at a regular rate}\n\nIt remains to connect the arithmetic exclusion with an asymptotic\nrate. For the cubic rate, the natural comparison is $n(n+1)(n+2)$;\nits successive ratio is exactly $1+3/n$. For a real parameter\n$\\lambda>1$, the corresponding comparison sequence is\n$\\Gamma(n+\\lambda)/\\Gamma(n)$, with ratio $1+\\lambda/n$.\nThe argument first leaves an error $o(n)$. Subtracting the two\nrecurrences then shows that the first difference of that error tends\nto zero. A sufficiently high difference of the original integer\nsequence is therefore a small integer and must vanish. Its eventual\npolynomial growth also forces $\\lambda$ to be integral.\n\n\\begin{lemma}[integer extraction at a regular rate]\\label{long243:res:extraction}\nLet $\\lambda>1$ and let $C_n$ be positive integers with\n\\begin{equation}\\label{long243:eq:regularrate}\n \\frac{C_{n+1}}{C_n}=1+\\frac\\lambda n+o(n^{-\\lambda}).\n\\end{equation}\nThen $\\lambda$ is an integer $d\\ge2$, and there are $A\\in\\Q_{>0}$ and\n$B\\in\\Q$ with $C_n=A\\,n(n+1)\\cdots(n+d-1)+B$ for all large $n$.\n\\end{lemma}\n\n\\begin{proof}\nPut $F_n=\\Gamma(n+\\lambda)/\\Gamma(n)$, so that $F_{n+1}/F_n=1+\\lambda/n$\nexactly and $F_n\\sim n^\\lambda$ by the gamma-ratio asymptotic\n\\cite[5.11.12]{dlmf_gamma}. Write $\\varepsilon_n=o(n^{-\\lambda})$ for the\nerror in \\eqref{long243:eq:regularrate} and $z_n=C_n/F_n$, so that\n$z_{n+1}/z_n=1+\\varepsilon_n/(1+\\lambda/n)$.  Since $\\lambda>1$ the errors are\nabsolutely summable, so the product converges and $z_n\\to K$ for some $K>0$,\nwith $z_n-K=o(n^{1-\\lambda})$. To see the stated error, put\n$\\eta_n=\\sup_{k\\ge n}k^\\lambda|\\varepsilon_k|\\to0$; the tail of the\nlogarithmic product has absolute value at most\n$O(\\eta_n\\sum_{k\\ge n}k^{-\\lambda})=o(n^{1-\\lambda})$.\nThe factors are positive eventually, so the limiting product is nonzero.  Hence\n\\begin{equation}\\label{long243:eq:dsmall}\n \\delta_n:=C_n-KF_n=F_n(z_n-K)=o(n).\n\\end{equation}\nWe subtract the recurrences for $C_n$ and $KF_n$ to obtain\n$\\delta_{n+1}-\\delta_n=(\\lambda/n)\\delta_n+\\varepsilon_nC_n$.\nThe first term is $o(1)$ by \\eqref{long243:eq:dsmall}, and the second\nis $o(1)$ because $C_n=O(n^\\lambda)$. Hence $\\Delta\\delta_n\\to0$.\nEvery further difference is a finite linear combination of shifts of\nthis sequence, so $\\Delta^j\\delta_n\\to0$ for all $j\\ge1$.\n\nFor the comparison sequence, the Gamma recurrence gives\n\\[\n \\Delta F_n\n =\\frac{\\Gamma(n+1+\\lambda)}{\\Gamma(n+1)}\n  -\\frac{\\Gamma(n+\\lambda)}{\\Gamma(n)}\n =\\lambda\\,\\frac{\\Gamma(n+\\lambda)}{\\Gamma(n+1)}.\n\\]\nIterating this identity gives\n\\[\n \\Delta^jF_n=\\lambda(\\lambda-1)\\cdots(\\lambda-j+1)\\,\n \\frac{\\Gamma(n+\\lambda)}{\\Gamma(n+j)},\n\\]\nwhose right side is $O(n^{\\lambda-j})$.  Choose an integer $j>\\lambda$; then\n$\\Delta^jF_n\\to0$, so $\\Delta^jC_n\\to0$.  These are integers, so\n$\\Delta^jC_n=0$ for all large $n$. Choose $N$ beyond this threshold.\nThe Newton forward-difference formula gives\n\\[\n C_{N+t}=\\sum_{i=0}^{j-1}\\binom ti\\Delta^iC_N\\qquad(t\\ge0),\n\\]\nso $C_n$ agrees eventually with a polynomial with rational coefficients.\nIts growth $C_n\\asymp n^\\lambda$ forces\nthe degree to be $\\lambda$, so $\\lambda=d$ is an integer, and $d\\ge2$ because\n$\\lambda>1$.  For that integer $F_n=n(n+1)\\cdots(n+d-1)$ exactly, so\n\\eqref{long243:eq:dsmall} says that the difference of two polynomials of degree $d$ is\n$o(n)$, hence constant. The leading coefficient of the rational polynomial\nis $K$, so $K\\in\\Q_{>0}$; the constant difference is rational as well.\nThis gives the stated shape with $A=K$ and $B\\in\\Q$.\n\\end{proof}\n\nThe little-oh error in \\eqref{long243:eq:regularrate} cannot in general\nbe replaced by a big-oh error. The positive integers\n$C_n=n(n+1)(n+2)+(-1)^n$ satisfy $C_{n+1}/C_n=1+3/n+O(n^{-3})$ and are not\neventually polynomial.\nLikewise $C_n=\\lceil n^{3/2}\\rceil$ satisfies\n$C_{n+1}/C_n=1+3/(2n)+O(n^{-3/2})$: rounding contributes\n$O(n^{-3/2})$, while the unrounded ratio has error $O(n^{-2})$.\nThese scalar examples obstruct weaker extraction hypotheses, not an\nirrationality theorem; neither is asserted to satisfy the full tail\nhypotheses. Extraction excludes nonintegral $\\lambda$. Integral rates\nstill require an arithmetic exclusion, proved here in degree three.\n\n\\begin{lemma}[comparison of the numerator and denominator ratios]\\label{long243:res:tailratio}\nLet $a_n$ be strictly increasing positive integers with $a_{n+1}/a_n^2\\to1$ and\n$\\sum_n1/a_n$ rational, and let $(C_n,D_n)$ be the integer tail.\nWrite $\\gamma_n=a_n^2/a_{n+1}-1$.  Then\n\\[\n \\frac{C_{n+1}}{C_n}=1+\\gamma_n+O(a_n^{-1}),\n \\qquad a_n\\ge\\exp(c\\,2^n)\\ \\text{eventually for some }c>0 .\n\\]\n\\end{lemma}\n\n\\begin{proof}\nSince $a_{n+1}/a_n^2\\to1$ and $a_n\\to\\infty$, there is an $N$ with\n$a_{n+1}\\ge a_n^2/2\\ge2a_n$ for $n\\ge N$.  Iterating $a_{n+1}\\ge a_n^2/2$ from\nan index where $a_n>2$ gives $a_n\\ge\\exp(c2^n)$ eventually.  The same doubling\ngives, for $m>N$,\n\\[\n \\frac1{a_m}\\ \\le\\ x_m\\ =\\ \\sum_{k\\ge m}\\frac1{a_k}\\ \\le\\ \\frac2{a_m},\n\\]\nso $x_{m}=(1+\\rho_{m})/a_{m}$ with\n$0\\le\\rho_{m}=a_mx_{m+1}\\le 2a_m/a_{m+1}\\le4/a_m$, using\n$a_{m+1}\\ge a_m^2/2$.  Hence\n\\[\n \\frac{C_{n+1}}{C_n}=\\frac{a_nD_nx_{n+1}}{D_nx_n}\n =a_n\\cdot\\frac{(1+\\rho_{n+1})/a_{n+1}}{(1+\\rho_n)/a_n}\n =\\frac{a_n^2}{a_{n+1}}\\cdot\\frac{1+\\rho_{n+1}}{1+\\rho_n}.\n\\]\nThe last factor is $1+O(1/a_n)$ and $a_n^2/a_{n+1}$ is bounded, so the product\nis $a_n^2/a_{n+1}+O(a_n^{-1})$.\n\\end{proof}\n\nThe exact identity in Section~\\ref{long243:sec:lcmrecords} gives the same\nestimate: $C_{n+1}/C_n=1-E_n/C_n$, while\n$0<\\gamma_n+E_n/C_n<3/a_n$ eventually. The identity follows from\nTheorem~\\ref{long243:res:defect}; its bound also uses vanishing relative\nerror and near-quadratic growth. Equation~\\eqref{long243:eq:weightedgrowth}\nis the resulting weighted-sum application, not the identity itself.\n\n\\begin{proof}[Proof of Theorem~\\ref{long243:res:nonintegralrate}]\nThe stated rate implies $a_{n+1}/a_n^2\\to1$ and convergence of the\nreciprocal series. Suppose its sum were rational, and form the positive\ninteger tail numerators $C_n$. Lemma~\\ref{long243:res:tailratio} gives\n$C_{n+1}/C_n=a_n^2/a_{n+1}+O(a_n^{-1})$ and an eventual bound\n$a_n\\ge\\exp(c2^n)$ for some $c>0$. Hence $a_n^{-1}=o(n^{-\\lambda})$ and\n\\[\n \\frac{C_{n+1}}{C_n}=1+\\frac{\\lambda}{n}+o(n^{-\\lambda}).\n\\]\nLemma~\\ref{long243:res:extraction} forces $\\lambda$ to be an integer,\ncontrary to the hypothesis.\n\\end{proof}\n\nFor a concrete nonintegral rate, take $a_1=8$ and\n\\[\n a_{n+1}=\\left\\lceil\\frac{2n a_n^2}{2n+3}\\right\\rceil\n \\quad(n\\ge1).\n\\]\nThe first terms are $8,26,387,99846$. Writing\n$r_n=1+3/(2n)$ gives $a_{n+1}=\\lceil a_n^2/r_n\\rceil$ and\n$0\\le r_n-a_n^2/a_{n+1}<r_n^2/a_n^2\\le25/(4a_n^2)$.\nAlso $a_{n+1}\\ge(2/5)a_n^2$, which from $a_1=8$ proves strict\nincrease and exponential lower growth. Thus the ratio error is\n$o(n^{-3/2})$, and Theorem~\\ref{long243:res:nonintegralrate}\nproves this reciprocal sum irrational.\n\n\\begin{proof}[Proof of Theorem~\\ref{long243:res:cubicrate}]\nThe hypothesis $a_n^2/a_{n+1}=1+3/n+o(n^{-3})$ gives $a_{n+1}/a_n^2\\to1$.\nSuppose $\\sum_n1/a_n$ were rational and pass to the integer tail, so\nthat $(a,C,D)$ satisfies \\eqref{long243:eq:cubicorbit} with every $C_n$ and $D_n$ a\npositive integer.  By Lemma~\\ref{long243:res:tailratio} and $a_n\\ge\\exp(c2^n)$, the\nerror term $O(a_n^{-1})$ is $o(n^{-3})$, so\n\\[\n \\frac{C_{n+1}}{C_n}=1+\\frac3n+o(n^{-3}).\n\\]\nLemma~\\ref{long243:res:extraction} at $\\lambda=3$ gives\n$C_n=A\\,n(n+1)(n+2)+B$ for all large $n$, with $A\\in\\Q_{>0}$ and $B\\in\\Q$.\nThat contradicts Theorem~\\ref{long243:res:cubicexclusion}.\n\\end{proof}\n\nNo restriction on the rational normalisation was assumed: the proof\nderives the permitted primitive leading coefficient. The two residue\npatterns \\eqref{long243:eq:wordplus} and \\eqref{long243:eq:wordminus},\nthe image $\\{0,2,5,6\\}$ modulo $7$ and the condition\n$(r^2+s^2)^2\\mid48$ have been calculated above.\n\nThe rationality supposition in this proof supplies the integer orbit.\nThe rate implies $a_{n+1}/a_n^2\\to1$ and excludes the eventual\nSylvester recurrence,\nso the conclusion is irrationality on a restricted class of the sequences\nin Problem~\\ref{long243:res:problem}.\n\nAt the cubic rate $P_n/a_n$ grows like $n^3$ and its increment like\n$n^2$.\nThus neither the bounded-increment criterion nor Koizumi's nonpositive\nupper-limit criterion applies. The rate also fails his $1+o(1/n)$\ncondition, and Duverney's signed defect series diverges. We do not decide\nwhether the LCM-weighted criteria apply; that requires information about\nthe repeated factors in the prefix product. Conversely,\nTheorem~\\ref{long243:res:cubicexclusion} assumes only positive integer\n$a_n,C_n,D_n$ and the recurrences~\\eqref{long243:eq:cubicorbit}, not the\nnear-quadratic growth limit. The two arguments therefore retain distinct\nhypotheses.\n\nIn familiar approximation notation,\n$C_n=q\\bigl(\\prod_{k<n}a_k\\cdot\\sum_m1/a_m-\\sum_{k<n}\\prod_{j<n,\\,j\\ne k}a_j\\bigr)$\nis a linear form in the reciprocal sum with integer coefficients.\nThese integer remainders grow like $n^3$. We use the precision\n$o(n^{-3})$ to determine their polynomial form by finite differences,\nthen exclude that form by the recurrence. The integer example\n$n(n+1)(n+2)+(-1)^n$ above explains why an $O(n^{-3})$ error alone\ndoes not justify that polynomial conclusion.\n\n\\section{Prior work and formalisation}\\label{long243:sec:priorwork}\n\n\\paragraph{Classical criteria and pseudo-greedy expansions.}\nThe rational-tail method predates these coordinates. Erd\\H{o}s and Straus\n\\cite[Theorem~2.1, pp.~85--86]{erdosstraus1974} use eventual integer\nremainders under a small-numerator hypothesis. In polynomial Cantor\nseries, Han\\v{c}l and Tijdeman split the numerator into finitely many\nshifted products. Rationality is then characterised by the vanishing of\nthe resulting polynomial sum\n\\cite[Theorem~2.2 and its proof, pp.~39--40]{hancltijdeman2008}.\nTheir theorem provides methodological context, not a result for arbitrary\narrays of signed integer coefficients: the rearrangement of the infinite\nsum needs boundary control, supplied there by the polynomial hypothesis.\nNeither result supplies the first-crossing argument for a lower error\nbound proved below.\n\nSeveral classical results give sufficient conditions directly on the\nsequence $(a_n)$. Their hypotheses use different quantities, so the\ncomparisons must be made separately. Koizumi's product criterion is\nimplied by eventual nonnegativity of the integer error $E_n$, under the\nidentification of coordinates given below. It therefore already covers\nthe descent argument in Section~\\ref{long243:sec:descent}\n(see~\\cite[Cor.~4(1) and Prop.~1(1), pp.~14--15]{koizumi2025}).\nThe original Erd\\H{o}s--Straus criterion instead uses a least common multiple.\nErd\\H{o}s and Straus proved that if $\\lim a_n/a_{n-1}^{2}=1$, the reciprocal\nsum is rational, and $(a_n)$ has no Sylvester tail, then\n\\[\n \\limsup_{n\\to\\infty}\\ \\frac{[a_1,\\ldots,a_n]}{a_{n+1}}\n \\left(\\frac{a_{n+1}^{2}}{a_{n+2}}-1\\right)>0,\n\\]\nwhere $[a_1,\\ldots,a_n]$ is the least common\nmultiple~\\cite[Theorem~3, p.~132]{erdosstraus1964}.  This is the indexing in\nthe original theorem. With $A_r=\\operatorname{lcm}(a_1,\\ldots,a_{r-1})$\nand $r=n+1$, its expression is exactly\n$(A_r/a_r)(a_r^2/a_{r+1}-1)$, the LCM quantity in the short note.\nThe index change is not an additional difference between those two\ncriteria. Erd\\H{o}s's 1988 survey, followed by the\ncatalogue summary, instead prints $a_n^2/a_{n+1}-1$ in the second factor while\nretaining the same prefix-LCM quotient; that displayed summary is off by one\nand is not followed here~\\cite[p.~105]{erdos1988}\\cite{erdosproblems}.\nKoizumi records the convenient sufficient rate\n$a_n^2/a_{n+1}=1+o(1/n)$ under which the Erd\\H{o}s--Straus criterion settles\nthe problem~\\cite[Remark~3, p.~16]{koizumi2025}.  Tijdeman and Yuan give a\ncriterion of the same kind for Ahmes series with positive integer numerators,\nweighted by the least common multiple of the earlier denominators\n\\cite{tijdemanyuan2002}.\n\nDuverney's signed criterion gives a further comparison. For positive\nintegers $a_n\\to\\infty$ and signs $\\epsilon_n\\in\\{-1,1\\}$, its printed\nhypothesis (3.6) is the one-sided condition\n\\[\n \\sum_{n\\ge0}\\left(\\frac{a_{n+1}}{a_n^2}-1\\right)<\\infty,\n\\]\nwithout absolute values. The stated conclusion is that\n$\\sum_{n\\ge0}\\epsilon_n/a_n$ is rational if and only if\n\\[\n a_{n+1}=a_n^2-(\\epsilon_{n+1}/\\epsilon_n)a_n\n             +\\epsilon_{n+2}/\\epsilon_{n+1}\n\\]\nfor all large $n$~\\cite[Corollary~3.2, p.~287]{duverney2001}.\nFor the comparison here we impose the stronger sufficient hypothesis\n$\\sum_{n\\ge0}|a_{n+1}/a_n^2-1|<\\infty$ and take all signs positive.\nThe distinction concerns one step of the printed proof. On pp.~299--300,\nthe reduced auxiliary fractions $p'_n/q'_n$ satisfy\n$p'_{n+1}\\mid q'_n$ and\n\\[\n p'_n\\le p'_N\\prod_{k=N}^{n-1}\\frac{q'_k}{p'_k}.\n\\]\nThe required boundedness follows if\n$\\prod_{k\\ge N}(p'_k/q'_k)$ has a positive limit. Absolute convergence\nof the growth-defect series ensures this, using the summable error in\nDuverney's estimate~(3.3). Signed convergence alone does not justify\nthat product step, even for positive rational factors. For each integer\n$m\\ge2$, take the pair $1+1/m$, $1-1/m$ successively $m$ times.\nThe deviations cancel after every pair, and the intervening partial sums\nare $1/m\\to0$, so their series converges. But the product through the\nblock $m=N$ is\n\\[\n \\prod_{m=2}^{N}\\left(1-\\frac1{m^2}\\right)^m\n =\\frac{(N+1)^N}{2N^{N+1}}\\sim\\frac e{2N}\\longrightarrow0;\n\\]\nthe equality follows by cancelling powers of consecutive integers.\nThis example does not impose $p'_{n+1}\\mid q'_n$. It refutes only the\ngeneral product inference, not Duverney's arithmetic criterion.\nWe therefore use only the absolute-convergence form; the interpretation\nwith merely signed convergence is not needed in any proof here.\n\nIn the all-positive case $\\epsilon_n=1$, absolute convergence also gives a\ndirect comparison with the classical product criterion. With the\none-based notation $P_n=\\prod_{1\\le j<n}a_j$ used above,\n\\[\n \\frac{P_{n+1}/a_{n+1}}{P_n/a_n}=\\frac{a_n^2}{a_{n+1}}.\n\\]\nThe ratios on the right have an absolutely convergent sum of deviations\nfrom $1$, because the same is true of their reciprocals and those\nreciprocals tend to $1$. Thus $P_n/a_n$ has a positive finite limit,\nand its increments tend to zero. This case already satisfies the\nclassical nonpositive-upper-limit condition; the bounded-increment\nresult allows a larger class of sequences.\n\nA different quantitative question is treated by Duverney, Kurosawa and\nShiokawa \\cite[Theorem~1, author-version p.~2; proof in Section~3]{duverneykurosawashiokawa2020}.\nFor rational $x_n>1$, their result assumes eventual $x_{n+1}\\ge x_n^2$\nand control of the accumulated denominators of $x_{n+1}/x_n^2$; it computes\nthe irrationality exponent of a signed reciprocal series. We use it only as\na restricted comparison. In particular, the eventual Sylvester recurrence and the cubic rate\nconsidered here approach quadratic growth from the other side.\n\nBadea's positive-term criterion is adjacent but different. For a convergent\nseries $\\sum_n b_n/a_n$ with $a_n,b_n$ positive integers, eventual strict\ninequality\n\\[\n a_{n+1}>\\frac{b_{n+1}}{b_n}a_n^2-\\frac{b_{n+1}}{b_n}a_n+1\n\\]\nforces irrationality, while rationality under the corresponding non-strict\ninequality forces eventual equality~\\cite[Theorem~A, p.~313; Cor.~2.2,\np.~316]{badea1993}. For $b_n=1$, its hypothesis is\n$a_{n+1}\\ge a_n^2-a_n+1$, an inequality between consecutive denominators,\nnot the one-step sign condition $E_n\\ge0$ used in integer descent.\nUnder the standing rational-tail and growth hypotheses, however, either\ninequality imposed eventually forces the Sylvester recurrence and hence the\nother. Their eventual forms are equivalent in this setting; that fact\ndoes not supply either condition for a general signed error.\n\nKoizumi's pseudo-greedy expansion~\\cite{koizumi2025} chooses $a_n$ by\nrounding $x_n^{-1}+1$ to the nearest integer, with a half-integer rounded\nupwards. Here $x_n$ is the remaining sum before subtracting $1/a_n$.\nThus $a_n=\\lfloor x_n^{-1}+3/2\\rfloor$, and the gap\n$\\varepsilon_n=x_n^{-1}+1-a_n$ is the signed rounding error.\nWrite the rational initial sum as $r=p/q$ with positive integers $p,q$. His Lemma~4\n~\\cite[pp.~11--12]{koizumi2025} produces integers $c_n>0$,\n$d_n$ and $e_n$ with $x_n=c_n/d_n$ and $\\varepsilon_n=e_n/c_n$, satisfying\n\\[\n a_n=\\frac{d_n-e_n}{c_n}+1,\\qquad\n c_{n+1}=c_n-e_n,\\qquad\n d_{n+1}=a_nd_n ,\n\\]\nand the proof of that lemma also gives $c_{n+1}=a_nc_n-d_n$.\nThese are the recurrences of Section~\\ref{long243:sec:state}, after\nmatching the starting index and the initial normalization. For a sequence\nsatisfying the hypotheses of Problem~\\ref{long243:res:problem}, the rounding rule is guaranteed\nonly after a finite restart~\\cite[Corollary~3, p.~9]{koizumi2025}. Restarting there\nwith the tail in lowest terms gives $(c_n,d_n,e_n)$; the original\n$(C_n,D_n,E_n)$ on that tail is a fixed positive integer multiple of\nthis triple, with the indices matched. Thus $E_n/C_n=\\varepsilon_n$\nis unchanged by the restart. The integer $E_n$ is not necessarily a\nnumerator in lowest terms, and the rounding range is not asserted\nbefore the restart. The first identity\nabove gives $E_n=D_n-(a_n-1)C_n$; the remaining identities are exactly\nthe numerator and denominator recurrences, including\nProposition~\\ref{long243:res:update}. On an all-negative tail we later\nwrite $e_n=-E_n>0$ for the magnitude. That local use of lower-case $e_n$\nhas the opposite sign to Koizumi's signed integer $e_n$.\n\nHis Theorem~3~\\cite[pp.~12--13]{koizumi2025} proves the equivalence of\ntwo assertions. Conjecture~1~\\cite[p.~4]{koizumi2025} asks whether, for\na positive rational $r$, the condition $\\varepsilon_n\\to0$ forces\n$\\varepsilon_n=0$ eventually. Question~1~\\cite[p.~3]{koizumi2025} is the\nquestion of Erd\\H{o}s and Graham stated as\nProblem~\\ref{long243:res:problem} above.\nTwo implications used below already appear in these coordinates: his Lemma~3, that $\\varepsilon_n=0$ forces $\\varepsilon_{n+1}=0$,\nis the absorption of Theorem~\\ref{long243:res:absorb}, and his Proposition~1(2), that\n$\\varepsilon_n\\ge0$ for all large $n$ forces $\\varepsilon_n=0$ for all large\n$n$, is the descent of Theorem~\\ref{long243:res:descent}.  Both statements are given in~\\cite[Lemma~3, p.~10; Prop.~1(2), p.~14]{koizumi2025}.\nKoizumi attributes Proposition~1(2) to Badea.  Its contrapositive says that a\ncounterexample must have negative errors infinitely often; it assumes the sign\nis eventually nonnegative, while Theorem~\\ref{long243:res:bounded} below assumes only\nthat the negative part is eventually bounded. It permits negative errors\nbetween $-B$ and $0$ and imposes no independent upper bound on positive\nerrors; the relative-error limit is still required.\n\nThe growth hypothesis in Problem~\\ref{long243:res:problem} is calibrated by two facts\nabout Sylvester's sequence. With $a_1=2$, it satisfies\n$a_n\\sim c_0^{2^{n}}$ with $c_0=1.2640847\\ldots$. Deleting initial terms\nand reindexing produces sequences with $a_n\\sim C^{2^{n}}$ for arbitrarily\nlarge $C$ whose reciprocals still sum\nto a rational number~\\cite[arXiv v4, p.~2]{kovactao2024}.  The classical sufficient\ncondition for irrationality, $\\lim_n a_n^{1/2^{n}}=\\infty$, is therefore\nsharp.  Kova\\v{c} and Tao identify that condition as folklore\n~\\cite[arXiv v4, p.~2]{kovactao2024}; they attribute the sharpness observation to\nErd\\H{o}s (1975).  A sequence with\n$a_n/a_{n-1}^{2}\\to1$ has $a_n^{1/2^{n}}$\nconvergent, so that criterion says nothing about the sequences of\nProblem~\\ref{long243:res:problem}, and rationality is genuinely possible there.\nSylvester's\nsequence is \\texttt{A000058} in the OEIS; \\texttt{A129871} is the variant\n$1,2,3,7,43,\\ldots$ with an initial $1$ prepended. For\nthe recent literature on irrationality of Ahmes series we refer to Kova\\v{c}\nand Tao~\\cite{kovactao2024}, who resolve several problems of Erd\\H{o}s and\nGraham drawn from the same two sources cited above, and whose introduction\ngives a sample of the intermediate work, including S\\'andor (1984) and Badea\n(1987).  They do not treat Problem~\\#243; the rigidity conclusion asked for\nthere is not among their results.\n\nCrmari\\'{c} and Kova\\v{c} \\cite[Theorems~1--2]{crmarickovac2025}\naddress the neighbouring Problem~\\#270, not Problem~\\#243. Allowing integer\n$f(n)\\to\\infty$ in $\\sum_n(\\prod_{j=1}^{f(n)}(n+j))^{-1}$ gives every\npositive real value; imposing nondecreasing $f$ gives a measure-zero value\nset. The latter assertion does not rule out individual rational values.\nTheir extension of Kakeya's subsum argument explains why decay alone need\nnot force irrationality when the summands remain freely selectable. The\npresent exact denominator recurrence imposes additional compatibility, so\nneither direction is an implication between their theorem and ours.\n\nThe \\emph{Formal Conjectures} collection contains a mathematically equivalent\nunproved declaration, up to its zero-based indexing\n~\\cite{formalconjectures243}.  Its summand is $\\Q$-valued, so Lean's\n\\texttt{Summable} hypothesis asserts the existence of a sum in $\\Q$; the\nfinite indexing shift changes that sum only by a rational prefix.  The\ndeclaration therefore does encode the rationality premise, but its proof is\n\\texttt{sorry}.  It states the problem without proving it, and the\ndevelopment in this note is independent of it. Earlier formal work on the remainder method should also be\ndistinguished from a solution of this problem: Koutsoukou-Argyraki and Li's\nArchive of Formal Proofs entry \\cite{kouli2020} records an Isabelle/HOL\nformalisation of Erd\\H{o}s--Straus (1974), Theorem~2.1, Corollary~2.10 and\nTheorem~3.1. That is formal prior art for those classical criteria, not an\nexisting formal proof of Erd\\H{o}s~\\#243.\n\nEach criterion still requires an additional bound or convergence\nestimate under the unrestricted hypotheses. In the stable-gcd case,\nfor example, the record-increment theorem gives a positive lower bound\nfor the log-log coefficient, rather than proving it infinite.\nAll conclusions about a persistently nonzero error assume failure of\neventual Sylvester behaviour. We collect them in\nSection~\\ref{long243:sec:open}, where mixed-sign tails remain\nunclassified. The two recurrence proofs below are independent of the\ncubic argument: one uses a stable gcd and CRT, the other an infinite\nproduct and integer descent.\n\n\\papersectiontarget{243-long-integer-tails}\n\\section{Integer numerators, denominators and errors}\\label{long243:sec:state}\n\nWe retain an unreduced fraction $C_n/D_n$ for each rational tail.\nThis keeps the denominator update multiplicative: removing $1/a_n$\nfrom the tail gives\n\\begin{equation}\\label{long243:eq:recurrences}\n D_{n+1}=a_nD_n,\\qquad C_{n+1}=a_nC_n-D_n.\n\\end{equation}\nWe measure the difference from the Sylvester recurrence by\n$E_n=D_n-(a_n-1)C_n$. These are Koizumi's recurrences\n\\cite[Lemma~4, pp.~11--12]{koizumi2025}; their formal definitions are the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{41}{nextDenState}{denominator\nupdate}, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{45}{nextTailState}{numerator update},\nand the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{49}{centeredState}{error}.\n\nWe also consider integer solutions of \\eqref{long243:eq:recurrences}\nwithout initially assuming that $C_n/D_n$ is a reciprocal tail.\nHere and in the following recurrence sections the indices may start\nat $0$, and $E_n$ always denotes the error just defined. We state\npositivity and growth assumptions as they are needed. The terms $a_n$\nare the multipliers, while $C_n,D_n$ are the numerator and denominator.\nThe pseudo-greedy rounding rule is an additional condition giving $-C_n/2\\le E_n<C_n/2$, whereas the\nabsorption argument below needs only $|E_n|<C_n$. At a negative error we\nwrite $e_n=-E_n>0$. For an isolated step we use $a,D,C\\in\\Z$ without\nsubscripts.\n\n\\begin{proposition}[update law]\\label{long243:res:update}\nFor all $a,D,C\\in\\Z$,\n\\[\n aC-D=C-\\bigl(D-(a-1)C\\bigr).\n\\]\nConsequently, every solution of \\eqref{long243:eq:recurrences} satisfies $C_{n+1}=C_n-E_n$.\n\\end{proposition}\n\n\\begin{proof}\n$C-\\bigl(D-(a-1)C\\bigr)=aC-D$.\n\\end{proof}\n\n$C_n$ decreases when $E_n>0$, increases when $E_n<0$, and\nstays unchanged when $E_n=0$. This is why bounds on the negative error\nbecome bounds on upward increments.\n\n\\begin{example}[the Sylvester sequence]\\label{long243:ex:sylvester}\nTake $a_n=2,3,7,43,1807,\\ldots$ with $a_{n+1}=a_n^{2}-a_n+1$, and start the\nstate at $D_0=C_0=1$.  The two updates give\n\n\\begin{center}\n\\begin{tabular}{@{}rrrrr@{}}\n\\toprule\n$n$ & $a_n$ & $D_n$ & $C_n$ & $E_n$\\\\\n\\midrule\n$0$ & $2$ & $1$ & $1$ & $0$\\\\\n$1$ & $3$ & $2$ & $1$ & $0$\\\\\n$2$ & $7$ & $6$ & $1$ & $0$\\\\\n$3$ & $43$ & $42$ & $1$ & $0$\\\\\n$4$ & $1807$ & $1806$ & $1$ & $0$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\n\\noindent and $D_n=a_n-1$ with $C_n=1$ at every index: if $D_n=a_n-1$ and\n$C_n=1$ then $C_{n+1}=a_n-(a_n-1)=1$ and $D_{n+1}=a_n(a_n-1)=a_{n+1}-1$.\nHence $E_n=D_n-(a_n-1)C_n=0$ throughout and the numerator never moves, which\nis Proposition~\\ref{long243:res:update} in the stationary case.\n\\end{example}\n\n\\paragraph{Construction from the reciprocal sum.}\nLet $T_n=\\sum_{k\\ge n}1/a_k$ and suppose $T_0\\in\\Q$.  Put $D_0$ equal to a\ncommon denominator and $D_n=D_0a_0\\cdots a_{n-1}$, and set $C_n=D_nT_n$.  Then\n$C_n\\in\\Z$ for every $n$, and from $T_n=1/a_n+T_{n+1}$ one gets\n$C_{n+1}=a_nC_n-D_n$, which is the tail update.  On Sylvester's sequence\n$T_n=1/(a_n-1)$ exactly, so $D_n=(a_n-1)C_n$ and $E_n=0$: the error\nmeasures deviation from the telescoping tail identity, and it vanishes\nidentically on the Sylvester orbit.\n\nIn Lean, the elementary identities are stated for an abstract integer\nsystem, and their application to a rational reciprocal sum has separate\nstatements, including\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos243/PaperCompleteR7/ProductDefect.lean\\#L211}{the product condition for a rational reciprocal sum}.\nThis passage uses the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1869}{natTail_eq_nextTailState}{tail realisation},\nthe \\lword{Erdos243/ReciprocalTailRigidity.lean}{1881}{natDen_eq_nextDenState}{denominator realisation}\nand the \\lword{Erdos243/ReciprocalTailRigidity.lean}{1978}{natTail_eq_sub_centeredState}{signed update}.\n\nConversely, the recurrences identify a reciprocal sum when the ratio\n$C_n/D_n$ tends to zero. Regard that ratio as a real number, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1901}{tailRatio}{tail ratio}.\nAssume $a_n>0$, $D_n>0$, $C_{n+1}+D_n=a_nC_n$ and\n$D_{n+1}=a_nD_n$. Dividing the numerator update by $a_nD_n$ gives\n$C_n/D_n=1/a_n+C_{n+1}/D_{n+1}$, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1905}{tailRatio_eq_reciprocal_add_next}{one-step\nreciprocal identity}. Iterating gives\n\\[\n \\frac{C_0}{D_0}=\\sum_{n<N}\\frac1{a_n}+\\frac{C_N}{D_N},\n\\]\nas in the \\lword{Erdos243/ReciprocalTailRigidity.lean}{1929}{tailRatio_eq_partialReciprocalSum_add}{finite\ntelescoping identity}. If $C_N/D_N\\to0$, taking the limit identifies\n$\\sum_n1/a_n=C_0/D_0$, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1955}{reciprocalSeries_hasSum_of_tailRatio_tendsto_zero}{series\nrealisation}.  No growth, error bound or separate rationality hypothesis is used in\nthese three identities. Positivity and the stated limit of $C_n/D_n$\nare still required. They prove the direction from recurrences to a series;\nthe construction from a rational reciprocal sum was given above and is\nalso formalised in separate declarations.\n\nUnder the relative-error hypothesis used below, the required limit is\nautomatic. Suppose $a_n>1$, $C_n>0$, $D_n\\ge0$ and $E_n/C_n\\to0$ on an\nexact integer orbit. If $D_0=0$, then $D_n=0$ and\n$E_n/C_n=1-a_n\\le-1$ at every index, a contradiction. Thus $D_0\\ge1$ and\n$D_n\\ge D_0 2^n$. For each $\\varepsilon>0$, the update gives\n$C_{n+1}\\le(1+\\varepsilon)C_n$ eventually. Taking $0<\\varepsilon<1$\nshows that $C_n/D_n\\to0$, so the telescoping identity realises the\nreciprocal sum as $C_0/D_0$.\n\nThe same assumptions also imply the quadratic growth limit. Put\n$\\theta_n=E_n/C_n$. Then $a_n=D_n/C_n+1-\\theta_n\\to\\infty$, and the\nexact recurrences give\n\\[\n \\frac{a_{n+1}}{a_n^2}\n =\\frac1{1-\\theta_n}-\\frac1{a_n}\n   +\\frac{1-\\theta_{n+1}}{a_n^2}\\longrightarrow1.\n\\]\nIn particular, the multipliers are strictly increasing eventually.\nThus a global orbit with these positivity and relative-error assumptions\nalready gives a sequence of the type in Problem~\\ref{long243:res:problem}\nafter deletion of a finite prefix. No separate tail-limit estimate is\nneeded for such an orbit; without the relative-error assumption, that\nestimate remains a condition of the general telescoping argument.\n\n\\begin{proposition}[scaling the numerator and denominator]\\label{long243:res:scale}\nFor every $s,a,D,C\\in\\Z$,\n\\[\n a(sD)=s\\,(aD),\\qquad\n a(sC)-sD=s\\,(aC-D),\n\\]\n\\[\n sD-(a-1)sC=s\\,[D-(a-1)C].\n\\]\n\\end{proposition}\n\n\nMultiplying the numerator and denominator by the same factor therefore\npreserves the recurrence and scales the error by that factor. When the\nfactor divides all entries, we can divide it out instead. This is used\nin the finite calculation of Appendix~\\ref{long243:app:residue} and in\nthe induction in Theorem~\\ref{long243:res:periodic}.\n\n\\section{Zero errors and the Sylvester recurrence}\\label{long243:sec:defect}\n\nTo recover the Sylvester recurrence from a zero error, we eliminate\n$D_n$ between two successive updates. Write\n\\[\n \\Delta_n=a_{n+1}-(a_n^2-a_n+1),\n\\]\nthe\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{53}{sylvesterDefect}{Sylvester\ndefect}. The identity below expresses $\\Delta_nC_{n+1}$ through two\nconsecutive errors. If both errors vanish, positivity lets us cancel\n$C_{n+1}$ and recover the recurrence. Under the bound $|E_n|<C_n$, it will\nalso make a single zero propagate to the next index.\n\n\\begin{theorem}[defect identity]\\label{long243:res:defect}\nFor all $a,a',D,C\\in\\Z$,\n\\[\n \\begin{aligned}\n \\relax[a'-(a^2-a+1)]\\,(aC-D)\n &=a^2[D-(a-1)C]\\\\\n &\\quad-\\bigl[aD-(a'-1)(aC-D)\\bigr],\n \\end{aligned}\n\\]\nthat is, $\\Delta_n\\,C_{n+1}=a_n^{2}E_n-E_{n+1}$.\n\\end{theorem}\n\n\\begin{proof}\nA direct expansion: both sides equal\n$a'aC-a'D-a^{3}C+a^{2}D+a^{2}C-aD-aC+D$.\n\\end{proof}\n\n No sign or growth hypothesis is needed for the identity itself.\nSuch hypotheses enter only in its consequences below.\n\n\\begin{example}[one defect and the error it creates]\\label{long243:ex:defect}\nContinue Example~\\ref{long243:ex:sylvester} but replace $a_3=43$ by $a_3=44$.  The\nstates $D_3=42$ and $C_3=1$ are unchanged, since they depend only on\n$a_0,a_1,a_2$, and $E_2=0$ still.  The defect is\n$\\Delta_2=a_3-(a_2^2-a_2+1)=44-43=1$, and\n$E_3=D_3-(a_3-1)C_3=42-43=-1$, so the identity reads\n\\[\n \\Delta_2C_3=1\\cdot1=1=49\\cdot0-(-1)=a_2^{2}E_2-E_3 .\n\\]\nBy Proposition~\\ref{long243:res:update} the numerator then rises, $C_4=C_3-E_3=2$.  A\nsingle unit of defect at one index has produced a negative error of magnitude\n$1$ at the next, increasing the numerator from $1$ to $2$.\n\\end{example}\n\n\\begin{theorem}[two vanishing errors force the step]\\label{long243:res:step}\nLet $a,a',D,C\\in\\Z$ with $aC-D\\ne0$. If\n\\[\n D-(a-1)C=0,\\qquad aD-(a'-1)(aC-D)=0,\n\\]\nthen $a'=a^2-a+1$.\n\\end{theorem}\n\n\\begin{proof}\nTheorem~\\ref{long243:res:defect} gives\n$[a'-(a^2-a+1)]\\,(aC-D)=0$. The second factor is nonzero, so the first\nfactor must vanish.\n\\end{proof}\n\n  The hypothesis $C_{n+1}\\ne0$ is not removable: at $C_{n+1}=0$\nthe identity gives no information about $a'$.\n\n\\subsection{Relations among three consecutive numerators}\n\nEliminating the denominator from two successive steps gives a relation\namong three reduced numerators.  Write $u,u_1,u_2$ for the\nreduced numerator coordinates, $v,v_1$ for the corresponding denominator\ncoordinates, and $h,h_1$ for the two common factors removed in reduction.  If\nthe two steps have multipliers $a,a_1$, then the exact hypotheses are\n\\[\n hu_1+v=au,\\qquad hv_1=av,\\qquad\n h_1u_2+v_1=a_1u_1.\n\\]\n\n\\begin{theorem}[eliminating the denominator from two steps]\\label{long243:res:secondorder}\nLet $a,a_1,u,u_1,u_2,v,v_1,h,h_1$ be integers satisfying\n\\[\n hu_1+v=au,\\qquad hv_1=av,\\qquad h_1u_2+v_1=a_1u_1.\n\\]\nThen $a^2u+hh_1u_2=h(a+a_1)u_1$.\n\\end{theorem}\n\n\\begin{proof}\nMultiply the first equation by $a$, replace $av$ using the second equation,\nand then replace $h_1u_2+v_1$ using the third.  The remaining terms factor as\nthe displayed right-hand side.\n\\end{proof}\n\n\\noindent This is the\n\\lword{Erdos243/DynamicCancellation.lean}{270}{reducedStep_secondOrder}{recurrence obtained by eliminating the denominator}.  Both cancellation factors remain in the formula. The identity adds no\nassumption to the two exact steps, but it still contains both multipliers\nand cannot by itself force $a_1=a^2-a+1$;\nLemma~\\ref{long243:res:oldmodulussaturation} specifies what it forgets\nmodulo a fixed old denominator.\n\nWhen neither step cancels a common factor, the same recurrence gives\na square identity. Write $p,p_1,p_2$ for three consecutive numerators,\n$q=ap-p_1$ for the denominator before the first step, and $a,a_1$ for\nthe two multipliers. Then\n\\[\n p_2+a^2p=(a+a_1)p_1.\n\\]\nThe resulting square identity is as follows.\n\n\\begin{theorem}[a square identity without cancellation]\\label{long243:res:curvature}\nLet $a,a_1,p,p_1,p_2,q$ be integers with $q=ap-p_1$ and\n$p_2+a^2p=(a+a_1)p_1$.  Then\n\\[\n q^2+\\bigl(pp_2-p_1^2\\bigr)=(a_1-a)pp_1.\n\\]\n\\end{theorem}\n\n\\begin{proof}\nSubstitute $q=ap-p_1$ and $p_2=(a+a_1)p_1-a^2p$:\n\\[\n \\begin{aligned}\n q^2+pp_2-p_1^2\n &=(ap-p_1)^2+p\\bigl((a+a_1)p_1-a^2p\\bigr)-p_1^2\\\\\n &=(a_1-a)pp_1.\n \\end{aligned}\n\\]\n\\end{proof}\n\nThe term $q^2$ is nonnegative, while $pp_2-p_1^2$ measures the difference\nbetween the product of the outer numerators and the square of the middle one.  The right side records the change in multiplier, so the\nidentity gives a necessary algebraic constraint without\nasserting a sign or a global monotonicity.  It is checked as\n\\lword{Erdos243/DynamicCancellation.lean}{309}{cancellationFree_curvature_square}{the\nsquare identity when no factor is cancelled}; no global exclusion of approximate solutions is\nclaimed.\n\nThe reduced error update must also be compatible with the denominator\nrecurrence. The following calculation gives the exact compatibility\ncondition.\nHere $e,e_1$ are signed errors in reduced coordinates, not the positive\nmagnitudes denoted by $e_n$ in the all-negative case. Let $\\Delta$ be a\nproposed value of the Sylvester defect. If\n\\[\n hu_1=u-e,\n \\qquad he_1=a^2e-\\Delta(u-e),\n\\]\nthen the denominator equality\n\\[\n h\\bigl((a_1-1)u_1+e_1\\bigr)\n   =a\\bigl((a-1)u+e\\bigr)\n\\]\nholds precisely when the following difference vanishes:\n\\[\n \\begin{aligned}\n &h\\bigl((a_1-1)u_1+e_1\\bigr)-a\\bigl((a-1)u+e\\bigr)\\\\\n &\\qquad=\\bigl[a_1-(a^2-a+1)-\\Delta\\bigr]\\,(u-e).\n \\end{aligned}\n\\]\nConsequently, when $u-e\\ne0$,\n\\[\n h\\bigl((a_1-1)u_1+e_1\\bigr)\n   =a\\bigl((a-1)u+e\\bigr)\n \\quad\\Longleftrightarrow\\quad\n a_1=a^2-a+1+\\Delta.\n\\]\nThe\n\\lword{Erdos243/FeedbackRealizability.lean}{64}{feedback_denominator_transport_iff}{equivalence with the denominator recurrence}\nis formalised using the\n\\lword{Erdos243/FeedbackRealizability.lean}{15}{feedback_transport_defect}{factored difference between the two recurrences}.  The hypothesis $u-e\\ne0$ is essential: if\n$u-e=0$, the factorised mismatch cannot identify $a_1$.\n\nThe identities hold under the local hypotheses displayed in their\nstatements. It is their proposed global applications, not the identities,\nthat would need additional information, such as control of cancellation or\na sign estimate for $pp_2-p_1^2$. No such information is deduced here for\nan arbitrary rational near-quadratic tail.\n\n\\begin{lemma}[saturation modulo an old denominator]\\label{long243:res:oldmodulussaturation}\nLet $M\\ge2$. Any finite word $r_0,\\ldots,r_k$ of units modulo $M$ is\ncompatible with the cancellation-free recurrences modulo $M$, with all\nreduced denominator residues equal to zero. Consequently the eliminated\ntwo-step identity alone imposes no further restriction on such unit words\nwhen the multiplier residues are free.\n\\end{lemma}\n\\begin{proof}\nSet $v_i\\equiv0\\pmod M$ and choose\n$a_i\\equiv r_{i+1}r_i^{-1}\\pmod M$. Then\n$r_{i+1}+v_i\\equiv a_ir_i$ and $v_{i+1}\\equiv a_iv_i$.\nEliminating $v_i$ gives the two-step identity automatically.\n\\end{proof}\nThis is finite residue compatibility, not existence of a positive integer\norbit, much less a rational tail with near-quadratic growth.\nFor $M\\mid v_T$ on a tail with no further common-factor cancellation,\nthe actual $u_n$ are indeed units modulo\n$M$. Useful further restrictions must therefore retain information discarded\nhere: for example multiplier size, changing moduli, or primes outside the\nold denominator. The square restriction used in the cubic proof concerns\nprecisely a prime at which a middle numerator vanishes.\n\n\\begin{theorem}[sequence form]\\label{long243:res:eventual}\nLet $a,D,C:\\N\\to\\Z$ satisfy $D_{n+1}=a_nD_n$ and\n$C_{n+1}=a_nC_n-D_n$.  If $E_n=0$ for all sufficiently large $n$ and\n$C_{n+1}\\ne0$ for all sufficiently large $n$, then $a_{n+1}=a_n^{2}-a_n+1$ for\nall sufficiently large $n$.\n\\end{theorem}\n\n  Take the larger of the two thresholds and apply\nTheorem~\\ref{long243:res:step} at each later index. This is the final\nstep of the criteria below that force $E_n$ to vanish eventually.\n\nThe same identity shows that a zero error persists whenever\n$|E_n|<C_n$ at the subsequent indices.\n\n\\begin{theorem}[zero is absorbing]\\label{long243:res:absorb}\nLet $a,C,D:\\N\\to\\N$ satisfy $C_{n+1}+D_n=a_nC_n$ and\n$D_{n+1}=a_nD_n$, and put $E_n=D_n-(a_n-1)C_n$.\nSuppose that $|E_n|<C_n$ for every $n$.  If $E_n=0$ then $E_{n+1}=0$.\n\\end{theorem}\n\n\\begin{proof}\nPutting $E_n=0$ in Theorem~\\ref{long243:res:defect} gives\n$\\Delta_n C_{n+1}=-E_{n+1}$, so $C_{n+1}$ divides $E_{n+1}$.  A multiple of\n$C_{n+1}$ of absolute value smaller than $C_{n+1}$ is zero, and\n$|E_{n+1}|<C_{n+1}$ by hypothesis.\n\\end{proof}\n\nBeyond an index from which $|E_n|<C_n$ always holds, either $E_n$\nreaches zero and remains zero, or it is never zero. A zero at an earlier\nindex need not persist: in Example~\\ref{long243:ex:defect}, the altered\nmultiplier gives $E_3=-1$ and $C_3=1$, so the required strict inequality\nfails.\nSections~\\ref{long243:sec:constant} and~\\ref{long243:sec:periodic} address\nthe special all-negative case. The mixed-sign analysis in\nSections~\\ref{long243:sec:bounded} and~\\ref{long243:sec:mass} separates\narbitrarily late negative errors from an eventually nonnegative tail.\n\n\\papersectiontarget{243-long-weighted-records}\n\\section{LCM numerators and weighted records}\\label{long243:sec:lcmrecords}\n\nClearing a rational tail by an LCM removes repeated denominator\nfactors, but need not make its numerator coprime to that LCM. The\nargument here therefore forbids short crossings, not numerator values.\nIt characterises boundedness by a weighted sum over new maxima, even\nafter subtracting a fixed amount at each record. The error may have\neither sign, extending the setting of Koizumi's eventual-nonnegative\ncase in Proposition~1(2).\n\nApplied to rational tails, this gives a Sylvester recurrence criterion.\nIts convergence hypothesis remains unproved for the unrestricted\nproblem. The whole-modulus avoidance model in\nProposition~\\ref{long243:res:coprimalitycap} omits the denominator\nrecurrence and prime-factor coprimality; its examples therefore do not\ncontradict the criterion.\n\nIn this section the indices start at $0$. Write the full reciprocal\nsum as $p/q$, with positive integers $p,q$, put $x_n=\\sum_{k\\ge n}1/a_k$, and take\n$D_0=q$. Set\n\\[\n L_n=\\operatorname{lcm}(q,a_0,\\ldots,a_{n-1}),\\quad\n M_n=D_n/L_n,\\quad U_n=L_nx_n=C_n/M_n,\\quad V_n=E_n/M_n.\n\\]\nThe rational tail has denominator dividing $L_n$, so $U_n$ is an integer.\nAlso $V_n=L_n-(a_n-1)U_n\\in\\Z$. This clears the denominator but need\nnot reduce the fraction: $U_n$ and $L_n$ may still have common factors.\nWith $\\rho_n=\\gcd(L_n,a_n)$, the exact updates are\n\\[\n M_{n+1}=M_n\\rho_n,\\qquad\n \\rho_nU_{n+1}=U_n-V_n,\\qquad V_n=L_n-(a_n-1)U_n.\n\\]\nThese are Bado's LCM coordinates, with the indexing translated explicitly.\nTake his denominator parameter to be $q$ and identify his $a_{n+1}$\nwith our $a_n$. Then his $M_n$, $\\Delta_n$, $K_n$, $u_{n+1}$ and\n$g_{n+1}$ are respectively our $L_n$, $M_n$, $U_n$, $V_n$ and $\\rho_n$.\nHis (19) becomes the middle update above\n\\cite[Prop.~7.1 and (16)--(20), pp.~6--7]{bado2026}.\n\nWrite $R_n=\\max_{j\\le n}U_j$ and\n$\\mathcal R=\\{n:U_{n+1}>R_n\\}$.  The inequality $|V_n|<U_n$ gives\n$U_{n+1}<U_n$ when $\\rho_n\\ge2$, so every sufficiently late strict rise has\n$\\rho_n=1$.  The stronger eventual bound $-U_n\\le2V_n$, supplied by\n$V_n/U_n=E_n/C_n\\to0$, gives the quantitative estimate\n$U_{n+1}\\le3U_n/4$ when $\\rho_n\\ge2$.\nRepeated factors thus cause contraction; late records use multipliers\ncoprime to the current LCM. At those records the actual jump is\n$d_n=U_{n+1}-U_n=-V_n>0$. This identity need not hold when $\\rho_n\\ge2$.\n\nFor $B=0$, divergence under unbounded growth can already be seen by\ncomparing a record step with the interval of new heights it covers:\n\\[\n \\int_{R_n}^{R_{n+1}}f(t)\\,dt\n \\le (R_{n+1}-R_n)f(U_n)\n \\le (U_{n+1}-U_n)f(U_n).\n\\]\nThe last expression is $(-V_n)f(U_n)$ at a record. The theorem retains\nthis divergence after subtracting a fixed $B$ at every record. That is\nwhere the congruences enter: they select a progression of heights whose\nfirst crossings necessarily have an excess jump. For the divergence direction, unboundedness of $U_n$ suffices; no\nlimit $U_n\\to\\infty$ is assumed.\n\n\\begin{theorem}[boundedness and a weighted sum over new maxima]\\label{long243:res:arithmeticrecord}\nLet $a_n,L_n,U_n$ be positive integers and $V_n$ integers satisfying\n\\[\n \\begin{gathered}\n L_{n+1}=\\operatorname{lcm}(L_n,a_n),\\qquad \\rho_n=\\gcd(L_n,a_n),\\\\\n \\rho_nU_{n+1}=U_n-V_n,\\qquad\n V_n=L_n-(a_n-1)U_n,\\qquad -U_n\\le2V_n.\n \\end{gathered}\n\\]\nLet $f:[1,\\infty)\\to[0,\\infty)$ be finite and nonincreasing, with\n$\\int_1^\\infty f(t)\\,dt=\\infty$. For each fixed integer $B\\ge0$,\n\\[\n \\sup_n U_n<\\infty\n \\quad\\Longleftrightarrow\\quad\n \\sum_{n\\in\\mathcal R}(-V_n-B)_+f(U_n)<\\infty.\n\\]\n\\end{theorem}\n\n\\begin{proof}\nA bounded integer running maximum increases only finitely many times, so\nbounded $U_n$ gives a finite sum. Suppose instead that $U_n$ is unbounded.\nThere are infinitely many record rises, each with $\\rho_n=1$. Its multiplier is\ngreater than one because $d_n=(a_n-1)U_n-L_n>0$. If $r<s$ are record\nindices, then $a_r\\mid L_s$ and $\\gcd(a_s,L_s)=1$, hence\n$\\gcd(a_r,a_s)=1$. The record multipliers are therefore distinct and pairwise\ncoprime. For any fixed $B\\ge1$, choose $B$ of them greater than $B$,\nand take $T$ after their indices. These earlier multipliers\n$m_0,\\ldots,m_{B-1}$ all divide $L_T$.\n\nPut $P=\\prod_i m_i$ and choose $x$ with $m_i\\mid x+i$ by the\nChinese remainder theorem. Select the heights $\\tau=x+B+kP>R_T$,\n$k\\in\\Z$. At the first crossing of any selected height we have\n$U_n\\le R_n<\\tau\\le U_n+d_n$, so the crossing is a record step.\nHere $d_n=U_{n+1}-U_n$ is the full jump from the current numerator.\nIf $d_n\\le B$, then $U_n\\in[\\tau-B,\\tau)$, so\n\\[\n U_n=\\tau-B+i=x+kP+i\\qquad\\text{for some }0\\le i<B.\n\\]\nThe choice of $x$ gives $m_i\\mid U_n$, which is compatible with these\nunreduced LCM coordinates. Since $m_i\\mid L_n$ too, it also gives\n$m_i\\mid d_n=(a_n-1)U_n-L_n$. The contradiction is\n$0<d_n\\le B<m_i$: the jump is too short to be a positive multiple\nof the modulus.\n\nEvery crossing has $d_n>B$. We must still avoid counting one excess\nseveral times when a jump crosses more than one selected height:\ntheir combined weight must fit within that same $d_n-B$. If it first crosses $h\\ge1$ selected heights,\ntheir spacing gives $(h-1)P<d_n$.\nWriting $r=d_n-B\\ge1$ and using $P\\ge B+1$, we obtain\n$d_n=B+r\\le Pr$, hence $h\\le r$. Since each crossed height is above\n$U_n$, monotonicity of $f$ gives\n\\[\n \\sum_{\\substack{\\tau\\text{ first crossed}\\\\\\text{at step }n}}f(\\tau)\n \\le(d_n-B)f(U_n).\n\\]\nEach selected height has one first crossing. The first is at most\n$R_T+P$, and successive selected heights are $P$ apart. For\n$R_N\\ge R_T+P$, the intervals\n$[\\tau,\\min\\{\\tau+P,R_N\\}]$, over selected heights $\\tau\\le R_N$,\ncover $[R_T+P,R_N]$. Since $f$ is nonincreasing, its integral over\neach interval is at most $P f(\\tau)$. Summing the crossing bounds gives\n\\begin{equation}\\label{long243:eq:weightedcrossing}\n \\sum_{\\substack{T\\le n<N\\\\n\\in\\mathcal R}}(-V_n-B)_+f(U_n)\n \\ge\\frac1P\\int_{R_T+P}^{R_N} f(t)\\,dt.\n\\end{equation}\nSince $R_n\\to\\infty$, the right side diverges for every $B\\ge1$.\nThe case $B=0$ follows by domination.\n\\end{proof}\n\nThe abstract theorem used only unboundedness, not $U_n\\to\\infty$:\nrecord multipliers supplied the CRT moduli, and the stated error bound\nexcluded records with $\\rho_n\\ge2$. For reciprocal tails, vanishing\nrelative error and absorption supply the missing alternative: failure\nof Sylvester behaviour forces $U_n\\to\\infty$.\n\n\\begin{theorem}[a convergent weighted sum over new maxima]\\label{long243:res:weightedrecord}\nAssume the growth and rationality hypotheses of Problem~\\ref{long243:res:problem},\nand let $f$ be as in Theorem~\\ref{long243:res:arithmeticrecord}. The sequence is\neventually Sylvester if and only if, for some integer $B\\ge0$,\n\\[\n \\sum_{n\\in\\mathcal R}(-V_n-B)_+f(U_n)<\\infty.\n\\]\n\\end{theorem}\n\n\\begin{proof}\nIf the sequence is not eventually Sylvester, zero error is never reached on a sufficiently late\ntail. Integrality gives $1/U_n\\le|V_n|/U_n=|E_n|/C_n\\to0$, so $U_n$ is\nunbounded. The same relative-error limit gives $2|V_n|<U_n$ at every\nsufficiently large index, and hence the bound $-U_n\\le2V_n$ required\nby Theorem~\\ref{long243:res:arithmeticrecord}. Applying that theorem to\nthis tail forces divergence for every $B$. Once the running maximum of\nthe retained tail exceeds the omitted prefix maximum, deleting that\nprefix no longer changes which steps set new records. Unboundedness\nensures that this crossing occurs. Conversely, a sequence satisfying\nthe eventual Sylvester recurrence\ntelescopes to $x_n=1/(a_n-1)$, so $V_n=0$ eventually.\n\\end{proof}\n\nThe nonincreasing weights $1/t$ and $1/[t\\log(et)]$ have divergent\nintegrals and are admissible. The weight $1/t^2$ is excluded: its\nintegral is finite, so the crossing lower bound would not contradict\nunbounded numerators. For example, $f(t)=1/[t\\log(et)]$ gives\nthe sufficient condition\n\\[\n \\sum_{n\\in\\mathcal R}\\frac{(-V_n-B)_+}{U_n\\log(eU_n)}<\\infty.\n\\]\nIts finite lower bound in~\\eqref{long243:eq:weightedcrossing} is\n$P^{-1}\\log\\bigl(\\log(eR_N)/\\log(e(R_T+P))\\bigr)$.\nFurther fixed iterated logarithmic factors are allowed whenever the integral\nstill diverges. These are specialisations of one crossing theorem.\n\nTransferring the criterion to the growth defect requires a summable\nchange in the summands, not merely a pointwise asymptotic. Put\n$\\gamma_n=a_n^2/a_{n+1}-1$ and $\\theta_n=E_n/C_n$. The defect identity gives\n\\[\n \\gamma_n+\\theta_n=\n \\frac{(1-\\theta_n)(a_n-1+\\theta_{n+1})}{a_{n+1}},\n \\qquad 0<\\gamma_n+\\theta_n<3/a_n\n\\]\neventually. Taking positive parts cannot increase the absolute difference.\nSince $V_n=U_n\\theta_n$, the two nonnegative summands\n$U_nf(U_n)(\\gamma_n-B/U_n)_+$ and $(-V_n-B)_+f(U_n)$ differ by at most\n$3U_nf(U_n)/a_n$. To see summability directly, put $P_n=\\prod_{j<n}a_j$.\nThe tail estimate $x_n\\sim1/a_n$ gives $C_n/a_n\\sim qP_n/a_n^2$, and\n\\[\n \\frac{P_{n+1}/a_{n+1}^{2}}{P_n/a_n^{2}}\n =\\frac{a_n^3}{a_{n+1}^{2}}\\longrightarrow0.\n\\]\nThus $\\sum_n C_n/a_n<\\infty$ by the ratio test. Since $U_n\\le C_n$ and\n$f(U_n)\\le f(1)$, the comparison error is summable.\nConsequently Theorem~\\ref{long243:res:weightedrecord} is equivalent to finiteness of\n\\begin{equation}\\label{long243:eq:weightedgrowth}\n \\sum_{n\\in\\mathcal R}U_nf(U_n)\n \\left(\\frac{a_n^2}{a_{n+1}}-1-\\frac B{U_n}\\right)_+\n\\end{equation}\nfor some $B$. The original hypotheses do not currently supply this\nfiniteness. In particular, termwise convergence to zero is insufficient.\nA Lean formulation of this comparison, with the sufficient constant $16$\nin place of $3$ and the summand of \\eqref{long243:eq:weightedgrowth} at\nrecord indices and zero elsewhere, is in the companion Lean repository\n\\href{https://github.com/wcook04/plectis-erdos-lean/blob/52f29ad173b04e3bac941b3663f2b9aebe5de0bb/ErdosProblems/Erdos243/PaperCompleteR8/GrowthDebtSummability.lean\\#L234}{\\texttt{plectis-erdos-lean}};\nit is not covered by the evidence record.\n\nThe crossed heights lie above $R_n$, but the divisibility argument uses\nthe starting numerator $U_n$ and the full jump $U_{n+1}-U_n$. Replacing\nthis by $R_{n+1}-R_n$ would discard the recovery from an earlier decrease.\n\n\n\\paragraph{Integer coefficients.}\nThe first-crossing proof extends to summands with arbitrary integer\nnumerators $b_n$. We record this extension separately from the\nunit-fraction criteria used above. In that case\n\\[\n V_n=b_nL_n-(a_n-1)U_n,\\qquad\n \\rho_nU_{n+1}=U_n-V_n.\n\\]\nAssume that $a_n\\ge2$, that $L_n,U_n$ are positive integers, and that\n$L_{n+1}=\\operatorname{lcm}(L_n,a_n)$, with\n$\\rho_n=\\gcd(L_n,a_n)$. Suppose $V_n\\ge-B$ eventually, with an integer\n$B\\ge0$. Then $U_{n+1}\\le U_n+B$; if $B=0$, boundedness follows at once.\nFor $B\\ge1$, a record step with $R_n>B$ must have $\\rho_n=1$, since\n$\\rho_n\\ge2$ would give\n\\[\n U_{n+1}\\le\\frac{U_n+B}{2}\\le\\frac{R_n+B}{2}<R_n.\n\\]\nThus an unbounded sequence would supply infinitely many pairwise coprime\nrecord multipliers. Choose $B$ of them larger than $B$, and then a CRT\nblock of $B$ consecutive integers above the previous maximum, each\ndivisible by one of the chosen multipliers. The first step past the upper\nend of the block is a high record step, so $\\rho_n=1$.\nIts positive jump is at most $B$, hence the preceding numerator $U_n$\nlies in the block. The corresponding multiplier divides both $U_n$ and\n$L_n$, hence divides the jump $(a_n-1)U_n-b_nL_n$. A positive jump at most $B$ cannot have a divisor\nlarger than $B$. No growth or relative-error hypothesis was used.\n\nWith the additional limit $V_n/U_n\\to0$, choose an integer $K$ bounding\n$U_n$. Eventually $|V_n|<U_n/K\\le1$, so the integer $V_n$ is zero.\nThe recurrence becomes $\\rho_nU_{n+1}=U_n$; the positive integer\nsequence $U_n$ is then nonincreasing and hence eventually constant. For positive $b_n$, the classical comparisons\nare Badea's Corollary~2.2 \\cite[p.~316]{badea1993} and the criterion of\nTijdeman and Yuan \\cite{tijdemanyuan2002}. Further finite examples\nseparating boundedness from stationarity are in the\n\\href{https://github.com/wcook04/plectis-erdos/blob/eccd8afc6db3c02a2265d0601b727d6c5e4467d5/lean/ErdosProblems/Erdos243/CoefficientUniformBoundedHeight.md}\n{coefficient proof supplement}.\n\n\\section{New maxima of reduced numerators}\\label{long243:sec:records}\n\nWe now reduce each tail to lowest terms, writing $u_n/v_n$.\nCoprimality is restored, but persistence is lost: a prime power can\ndisappear from $v_n$ while remaining in every later LCM $L_j$.\nBefore applying coprimality at new maxima, we must therefore bound\nwhat cancellation removes. Lower-case $u_n$ is the reduced numerator;\nupper-case $U_n$ remains the LCM numerator of the preceding section.\n\n\\paragraph{Fractions in lowest terms.}\nPut\n\\[\n G_n=\\gcd(C_n,D_n),\\qquad u_n=C_n/G_n,\\qquad v_n=D_n/G_n,\n \\qquad \\tilde e_n=E_n/G_n.\n\\]\nThus $\\gcd(u_n,v_n)=1$ and $\\tilde e_n$ is a signed integer.\nThe factor $h_n=G_{n+1}/G_n$ records the common factor removed at the next\nstep. The relation to the preceding section is exact:\n\\[\n \\frac{G_n}{M_n}=\\gcd(U_n,L_n),\\qquad\n u_n=\\frac{U_n}{\\gcd(U_n,L_n)},\\qquad\n v_n=\\frac{L_n}{\\gcd(U_n,L_n)}.\n\\]\nIndeed, $C_n=M_nU_n$ and $D_n=M_nL_n$.\nClearing with an LCM and reducing to lowest terms are different operations.\nFor example, the exact step $(C,D,a)=(3,6,3)$ has $E=0$ and gives\n$(C',D')=(3,18)$. With $L=6$, its LCM numerator falls from $3$ to $1$,\nwhile the reduced numerator stays $1$: the LCM factor is $\\rho=3$,\nbut the reduction factor is $h=1$. From this step onwards, the Sylvester recurrence holds.\n\nIn general, before reduction the next numerator is\n$w_n=a_nu_n-v_n=u_n-\\tilde e_n$, so\n\\[\n h_nu_{n+1}=w_n,\\qquad h_nv_{n+1}=a_nv_n,\\qquad\n \\gcd(u_n,u_{n+1})=1.\n\\]\nThe negative magnitude $e_n=-E_n$ used earlier is not $\\tilde e_n$;\nneither is Koizumi's real gap $\\varepsilon_n$.\n\nFor the maxima and their increments write\n\\[\n R_n=\\max_{k\\le n}u_k,\\qquad H_n=\\max_{j\\le n}C_j,\n \\qquad s_n=R_{n+1}-R_n.\n\\]\nAt a strict rise let $d_n=u_{n+1}-u_n$. Then\n$s_n=(d_n-(R_n-u_n))_+$. Thus the actual jump includes recovery of\nthe earlier decrease $R_n-u_n$, whereas the record increment does not. For example,\nif $R_n=10$, $u_n=5$ and $u_{n+1}=12$, then $s_n=2$ but $d_n=7$.\nThis illustrates the definitions, not a claimed reciprocal-tail orbit.\nWe will use\n\\[\n m_n=(-\\tilde e_n)_+,\\qquad \\mathcal A_n=\\frac{R_n}{u_n}m_n,\n \\qquad \\delta_n=\\left(\\frac{a_n^2}{a_{n+1}}-1\\right)_+,\n \\qquad \\ell(x)=\\log_2\\log_2\\max(4,x).\n\\]\nThe factor $R_n/u_n$ in $\\mathcal A_n$ measures how far the current\nnumerator has fallen below its previous maximum. It equals one at a\nmaximum and can be large after a decrease. The symbol $\\mathcal A_n$\nis distinct from the prefix product $A_n$ used later.\n\nUnless a statement in this section specifies an abstract recurrence,\nwe assume the hypotheses of Problem~\\ref{long243:res:problem} and use\nits integer tails, for which $|E_n|/C_n\\to0$ after a finite shift.\nWe will write out the conclusion $a_{n+1}=a_n^2-a_n+1$ eventually\nrather than give it a separate symbol.\n\n\\paragraph{Comparison with Duverney's reduced parameters.}\nFor the positive reciprocal series, the eventually reduced parameters in Duverney's\nTheorem~3.1 and Section~5.2 \\cite[pp.~285--286, 299--300]{duverney2001}\ncan be taken as follows. The symbols $p_n,q_n$ in this comparison are\nDuverney's auxiliary integers, not the fixed numerator and denominator\nof the full reciprocal sum:\n\\[\n p_n=u_n,\\qquad q_n=w_n=a_nu_n-v_n=h_nu_{n+1},\\qquad\n E_n=G_n(p_n-q_n).\n\\]\nIndeed, $\\gcd(u_n,w_n)=\\gcd(u_n,v_n)=1$. Since\n$a_nv_n=a_n^2u_n-a_nw_n$, reduction of the next tail gives\n\\[\n h_n=\\gcd(w_n,a_nv_n)=\\gcd(w_n,a_n^2),\\qquad\n p_{n+1}\\mid q_n.\n\\]\nFor the gcd equality, subtract the multiple $a_nw_n$ and use the\ncoprimality of $u_n,w_n$; the divisibility follows from\n$q_n=h_np_{n+1}$. Eliminating $v_{n+1}$ then gives Duverney's recurrence:\n\\[\n a_{n+1}=\\frac{p_n}{q_n}a_n^2-a_n+\\frac{q_{n+1}}{p_{n+1}}.\n\\]\nThus $q_n/p_{n+1}=h_n$ is precisely the cancellation factor, and\n$q_n/p_n\\to1$. This identifies the reduced parameters, not every possible\nunreduced choice in the original theorem. An upper bound on $q_n-p_n$ controls the increase before cancellation\nin the reduced coordinates. To transfer that bound directly to\n$(-E_n)_+=G_n(q_n-p_n)_+$ would also require control of $G_n$.\n\nAn old prime power can be lost only when the multiplier supplies\nexactly its exponent and subtraction cancels more. The formula below\nseparates this case from unequal exponents.\n\n\\begin{lemma}[the denominator valuation transition]\\label{long243:res:valuationtransition}\nLet $u,v,a$ be positive integers, $\\gcd(u,v)=1$, and $w=au-v>0$.\nPut $h=\\gcd(w,av)$ and $v'=av/h$. For a prime $p$, write\n$r=\\nu_p(a)$, $s=\\nu_p(v)$ and $t=\\nu_p(w)$. Then\n\\[\n \\nu_p(v')=\n \\begin{cases}\n \\max(r,s),&r\\ne s,\\\\\n \\max(0,2s-t),&r=s.\n \\end{cases}\n\\]\nIn particular $\\nu_p(v')\\le\\max(r,s)$. A strict loss relative to $s$\nrequires $r=s\\ge1$ and $t>s$.\n\\end{lemma}\n\\begin{proof}\nAlways $\\nu_p(v')=r+s-\\min(t,r+s)$. If $r\\ne s$, primitivity implies\n$t=\\min(r,s)$: when $s>0$, $u$ is a $p$-adic unit, and when $s=0<r$,\n$v$ is a unit. If $r=s>0$, both terms in $au-v$ are divisible by $p^s$,\nso $t\\ge s$; substituting gives the second case. If $r=s=0$, the same\nformula gives zero without needing $u$ to be a unit.\n\\end{proof}\nCancellation is measured against the enlarged denominator $av$, not\nagainst $v$. Thus $h>1$ need not mean that an old prime power is lost.\nFor example, $(u,v,a)=(2,15,9)$ gives $w=3$, $h=3$ and\n$(u',v')=(1,45)$. The numerator before cancellation exceeds $u$ by only\n$w-u=1$, yet the $3$-adic valuation of the reduced denominator rises\nfrom $1$ to $2$.\nThis is a finite exact step, not an infinite counterexample.\n\n\\begin{corollary}[persistence of a prime power]\\label{long243:res:powerpersistence}\nSuppose $p^k\\mid v_s$. If $w_n<p^{k+1}$ at every step from $s$ through\n$t-1$, then $p^k\\mid v_t$.\n\\end{corollary}\n\\begin{proof}\nAt a first loss of divisibility by $p^k$, the current exponent is some\n$j\\ge k$ and the valuation lemma forces $\\nu_p(w_n)>j$. This would give\n$w_n\\ge p^{j+1}\\ge p^{k+1}$, contrary to the hypothesis.\n\\end{proof}\nThe bound on $w_n$ protects the old prime power despite varying\ncancellation factors. It is needed only until the crossing under study.\n\n\\subsection{Growth of the running maximum}\n\n\\begin{theorem}[increments of the running maximum]\\label{long243:res:recorddichotomy}\nLet $\\Theta=\\limsup_n(H_{n+1}-H_n)/\\ell(H_n)$ on a rational-tail orbit\nunder the standing hypotheses, and suppose $E_n$ is not eventually zero.\nThen either $G_n$ is unbounded and $\\Theta$ is infinite, or\n$G_n$ stabilises at a value $g$ and $\\Theta\\ge g\\,v_T/\\varphi(v_T)$ for every\nlate $T$, so that $\\Theta>g\\ge1$.  For any orbit under the standing\nhypotheses, therefore, $\\Theta=0$ or $\\Theta>1$, and $\\Theta\\le1$ forces\nthe eventual Sylvester recurrence.\n\\end{theorem}\n\nThe constant in the next condition is measured against a double\nlogarithm, not against $C_n$. This grows much more slowly than any\npositive power of $C_n$. Every fixed bound on $(-E_n)_+$ satisfies\nit on a nonzero tail because $C_n\\to\\infty$; it also permits unbounded\nnegative parts on the double-logarithmic scale, with the coefficient\nspecified below. In contrast, an error of size\n$\\sqrt{C_n}$ has relative size tending to zero but violates the condition.\nThese are comparisons of bounds, not constructions of reciprocal tails.\n\n\\begin{corollary}[the double-logarithmic bound]\\label{long243:res:loglogboundary}\nIf\n\\[\n \\limsup_n\\frac{(-E_n)_+}{\\ell(C_n)}\\le1,\n\\]\nthen the sequence is eventually Sylvester. Every counterexample therefore\nsatisfies\n\\[\n \\limsup_n\\frac{(-E_n)_+}{\\ell(C_n)}>1.\n\\]\n\\end{corollary}\n\n\\begin{proof}[Proof of Theorem~\\ref{long243:res:recorddichotomy} and\nCorollary~\\ref{long243:res:loglogboundary}]\nWe first prove the auxiliary lower bound $\\Theta\\ge1$. This part needs\nonly an exact positive integer orbit with $a_n>1$, $D_0\\ge1$,\n$|E_n|/C_n\\to0$ and error not eventually zero. In particular, it will also\napply to the abstract orbit in Theorem~\\ref{long243:res:slownegative}.\nThe key input is that, outside a set of indices of density zero,\n$a_n$ is coprime to the entire preceding denominator $D_n$. This supplies\nalmost one new coprime modulus per step. The denominator growth then\nplaces the resulting CRT block at the required double-logarithmic height.\n\n\\emph{How often a multiplier is coprime to the preceding denominator.}\nWe use the LCM factorisation from\nSection~\\ref{long243:sec:lcmrecords}. In the present zero-based indexing, put\n\\[\n L_n=\\lcm(D_0,a_0,\\ldots,a_{n-1}),\\qquad\n M_n=D_n/L_n,\\qquad \\rho_n=\\gcd(L_n,a_n).\n\\]\nThe finite telescoping identity gives\n\\[\n \\frac{C_n}{M_n}\n =L_n\\left(\\frac{C_0}{D_0}-\\sum_{j<n}\\frac1{a_j}\\right)\n \\in\\Z_{>0}.\n\\]\nThus $M_n\\mid C_n$, while $M_0=1$ and\n$M_{n+1}=\\rho_nM_n$. Since $L_n$ and $D_n$ have the same prime divisors,\n\\[\n \\#\\{n<N:\\gcd(a_n,D_n)>1\\}\n =\\#\\{n<N:\\rho_n>1\\}\n \\le\\log_2M_N\\le\\log_2C_N=o(N).\n\\]\nHere the last estimate follows by summing\n$\\log(C_{n+1}/C_n)=\\log(1-E_n/C_n)=o(1)$.\nThis is the density-one coprimality argument of\nBado~\\cite[Theorem~9.1, p.~7]{bado2026}, written with the exact-orbit\nhypotheses used here. Only the finite telescoping identity is needed;\nno estimate on record increments has entered this count.\n\n\\emph{Choosing the moduli and controlling their product.}\nAbsorption and vanishing relative error give $C_n\\to\\infty$, hence\n$H_n\\to\\infty$. The maximum is still subexponential: for every\n$\\varepsilon>0$, choose $K_\\varepsilon$ with\n$C_j\\le K_\\varepsilon e^{\\varepsilon j}$ for every $j$. Then\n$H_n\\le K_\\varepsilon e^{\\varepsilon n}$, so $\\log H_n=o(n)$.\nSince $D_n\\ge2^n$ and\n$a_n=D_n/C_n+1-E_n/C_n$, eventually\n\\[\n a_n\\ge2^{n/2},\\qquad a_n<2D_n,\\qquad \\ell(D_n)\\le n+O(1).\n\\]\nFor the last inequality, $D_{n+1}<2D_n^2$ gives\n$1+\\log_2D_{n+1}<2(1+\\log_2D_n)$. Iteration from a fixed late index,\nfollowed by another logarithm, gives $\\ell(D_n)\\le n+O(1)$.\n\nSuppose $H_{n+1}-H_n\\le c\\ell(H_n)$ eventually for some $0<c<1$.\nFor a large integer $B$, take the first $B$ indices at or after\n$\\lceil3\\log_2B\\rceil$ for which $\\gcd(a_n,D_n)=1$.\nWrite the corresponding multipliers as $m_0,\\ldots,m_{B-1}$ and let\n$s$ be the index immediately after the last choice. The density estimate\njust proved gives $s=B+o(B)$: deleting $O(\\log B)$ initial indices and\n$o(s)$ exceptional indices leaves $B$ choices.\nEach $m_i>2B$ for large $B$, and the chosen multipliers are pairwise\ncoprime, because each earlier one divides the denominator preceding a\nlater one. All divide $D_s$. For $P=\\prod_i m_i$ we therefore have\n\\[\n (2B)^B<P\\le D_s,\\qquad H_s<P,\\qquad B<P,\n \\qquad \\ell(3P)\\le B+o(B).\n\\]\nThe bound on $H_s$ uses $\\log H_s=o(s)$ and $s=B+o(B)$.\nThus the product is large enough to place the block beyond the previous\nmaximum, but small enough that the allowed jump at its height is less\nthan the block length: $c\\ell(3P)<B$ for all sufficiently large $B$.\nBoth comparisons are needed; existence of a distant CRT block alone\nwould not control a height-dependent jump bound.\n\n\\emph{Crossing the block.}\nAn unreduced numerator may land on a multiple. The modulus then\nbecomes a persistent common divisor, forbidding the \\emph{next}\nsmall record. Choose $x\\in[P,2P)$ by the Chinese remainder theorem so that\n$m_i\\mid x+i$ for $0\\le i<B$. At the first record $C_t\\ge x$\nafter $s$, the preceding maximum is below $x$ and its increase is\nat most $c\\ell(2P)<B$. Thus\n$x\\le C_t<x+B<3P$, so some $m_i$ divides both $C_t$ and $D_t$.\nThe exact updates preserve this common divisor. At the next record\n$C_r$ we would therefore have\n\\[\n m_i\\mid C_r-C_t,\\qquad\n 0<C_r-C_t\\le c\\ell(C_t)\\le c\\ell(3P)<B<m_i,\n\\]\nwhich is impossible. This proves the auxiliary bound $\\Theta\\ge1$.\n\nThe integer normalisation matters here. Scaling $(C,D,E)$ by a positive\ninteger $k$ scales the running maximum by $k$ and its limit-superior\ncoefficient by $k$, since $\\ell(kx)/\\ell(x)\\to1$. Thus the bound with\ncoefficient $1$ is not invariant under arbitrary clearing of denominators.\nThe next step keeps track of the common factor rather than discarding it.\n\n\\emph{The common gcd and the sharper coefficient.}\nFor any fixed $N$, divide the tail from $N$ onwards by $G_N$.\nThe resulting integer orbit satisfies the same hypotheses. Its running\nmaximum is eventually $H_n/G_N$, and\n$\\ell(H_n/G_N)/\\ell(H_n)\\to1$. The auxiliary bound therefore gives\n$\\Theta\\ge G_N$. If $G_N$ is unbounded, then $\\Theta=+\\infty$.\n\nOtherwise $G_n=g$ eventually. Fix a later index $T$ with $v_T>1$.\nThe reduced fractions have pairwise coprime multipliers, each coprime\nto $v_T$. Its prime factors already exclude the nonunit offsets;\nnew multipliers need cover only the rest. For a large integer $L$, exactly\n\\[\n k_L=\\frac{\\varphi(v_T)}{v_T}L+O(v_T)\n\\]\nof the offsets $0,\\ldots,L-1$ are coprime to $v_T$.\nAssign to those offsets the next $k_L$ multipliers, beginning at $T$.\nPut $s=T+k_L$ and $Q=v_s=v_T\\prod_{T\\le j<s}a_j$.\nSolve $x\\equiv0\\pmod{v_T}$ and $x\\equiv-j$ modulo the multiplier\nassigned to offset $j$, and take the solution in $[Q,2Q)$.\nEvery integer in $[x,x+L)$ then fails to be coprime to $v_s$.\nNo later reduced numerator can lie there.\n\nThe running maximum $R_s$ of the reduced numerator is below $Q$ for\nlarge $L$, by the growth estimates above. The first crossing of $x$\nafter $s$ must therefore have a record increment at least $L$.\nIts preceding maximum is below $2Q$, and\n$\\ell(2Q)\\le s+O(1)=T+k_L+O(1)$. The corresponding record indices tend\nto infinity with $L$, so\n\\[\n \\limsup_n\\frac{R_{n+1}-R_n}{\\ell(R_n)}\n \\ge\\lim_{L\\to\\infty}\\frac{L}{T+k_L+O(1)}\n =\\frac{v_T}{\\varphi(v_T)}.\n\\]\nSince $H_n=gR_n$ eventually, this gives\n$\\Theta\\ge g v_T/\\varphi(v_T)>g$.\nUnder the eventual Sylvester recurrence, $C_n$ is eventually constant and $\\Theta=0$.\nFinally,\n\\[\n H_{n+1}-H_n=\\bigl(-E_n-(H_n-C_n)\\bigr)_+\\le(-E_n)_+.\n\\]\nTogether with $\\ell(C_n)\\le\\ell(H_n)$, this proves the corollary.\n\\end{proof}\n\n\\paragraph{Integer solutions of the recurrences.}\nThe strict bound $\\Theta>1$ is also a Lean theorem for solutions of\n\\eqref{long243:eq:recurrences} with $a_n>1$, $C_n>0$, $D_0\\ge1$ and vanishing relative error whose error\nis not eventually zero\n(\\href{\\laterepobase/lean/ErdosProblems/Erdos243/PaperCompleteR21/ExactOrbitRecordDichotomy.lean\\#L363}{strict record bound}).\nThe implication of Corollary~\\ref{long243:res:loglogboundary}, that\n$\\limsup_n(-E_n)_+/\\ell(C_n)\\le1$ forces $E_n=0$ eventually, is a Lean\ntheorem for solutions of \\eqref{long243:eq:recurrences} with $a_n>1$,\n$C_n>0$ and vanishing relative error\n(\\href{\\laterepobase/lean/ErdosProblems/Erdos243/PaperCompleteR21/DoubleLogOrbitBound.lean\\#L45}{double-logarithmic bound for an exact orbit});\nthere $D_0\\ge1$ follows from the vanishing relative error.\nThe Lean sources also contain pointwise-rise versions of the arithmetic\nsteps:\n\\lword{Erdos243/SlowRiseBarrier.lean}{111}{commonDivisor_persists}{persistence of\na common divisor}, a\n\\lword{Erdos243/SlowRiseBarrier.lean}{51}{exists_consecutiveMultiples_between}{CRT\nblock starting in $[P,2P)$}, the\n\\lword{Erdos243/SlowRiseBarrier.lean}{194}{slowRise_landing}{landing of a slowly\nrising numerator in that block} and the\n\\lword{Erdos243/SlowRiseBarrier.lean}{280}{no_slowNegative_of_coprimeBlock}{resulting\nexclusion}. Their landing bound controls increments of $C_n$; the\nargument above controls increments of $H_n$.\n\nNo upper bound on $\\Theta$ is proved. If $G_n$ is unbounded, the theorem\nforces $\\Theta=+\\infty$. If $G_n=g$ eventually, it proves only\n$\\Theta\\ge g\\,v_T/\\varphi(v_T)$ for every late $T$. Since $v_T\\mid v_{T+1}$,\nthe ratios $\\varphi(v_T)/v_T$ are nonincreasing and have a limit\n$\\sigma\\ge0$. If $\\sigma=0$, the lower bound again forces\n$\\Theta=+\\infty$; if $\\sigma>0$, it gives $\\Theta\\ge g/\\sigma$ but does\nnot determine whether $\\Theta$ is finite. Replacing $H_n$ by the running\nmaximum of $u_n$ divides the coefficient by $g$, since\n$\\ell(gx)/\\ell(x)\\to1$.\n\n\\subsection{Bounds that allow for cancellation and earlier decreases}\n\n\\begin{theorem}[bounds allowing for previous decreases]\\label{long243:res:recordamplified}\nUnder the standing hypotheses, the following are equivalent: eventual Sylvester behaviour; $\\limsup_n\\mathcal A_n<\\infty$;\n$\\limsup_nR_n\\delta_n<\\infty$.  Each of those two limits superior is $0$ or\n$+\\infty$.\n\\end{theorem}\n\n\\begin{corollary}[the critical rate]\\label{long243:res:criticalrate}\nUnder the standing hypotheses and $\\delta_n=O(1/n)$, with no convergence of $n\\delta_n$ assumed,\neventual Sylvester behaviour is equivalent to $u_n=O(n)$ and to $(-\\tilde e_n)_+=O(1)$.  A\ncounterexample at the critical rate therefore has $\\limsup_nu_n/n=\\infty$ and\n$\\limsup_n(-\\tilde e_n)_+=\\infty$.\n\\end{corollary}\n\n\\begin{proof}[Proof of Theorem~\\ref{long243:res:recordamplified}]\nSuppose the orbit is not eventually Sylvester and\n$\\mathcal A_n\\le K$ eventually, for an integer $K\\ge1$.\nAbsorption and $|\\tilde e_n|/u_n\\to0$ give $u_n\\to\\infty$.\nNegative reduced errors must occur arbitrarily late, since otherwise\n$h_nu_{n+1}=u_n-\\tilde e_n$ would make $u_n$ eventually nonincreasing.\n\nCancellation may lower $u_{s+1}$ without a record. The first later\nnegative error detects the loss: its magnitude is at least one, while\n$R_t$ remembers $u_s$. Fix a late $s$ and let $t>s$ be that first\nnegative-error index.\nAt the intervening steps the reduced numerator is nonincreasing, so\n$u_t\\le u_{s+1}$. Since $R_t\\ge u_s$ and\n$m_t=|\\tilde e_t|\\ge1$,\n\\[\n \\mathcal A_t=\\frac{R_tm_t}{u_t}\\ge\\frac{u_s}{u_{s+1}}\n =\\frac{h_s}{1-\\tilde e_s/u_s}.\n\\]\nThus $h_s\\le3K/2<2K$ after the relative-error threshold.\nAlso $m_n\\le\\mathcal A_n\\le K$, since $R_n\\ge u_n$, and hence\n$u_{n+1}\\le u_n+K$.\n\nWe can now choose primes too large to be removed by cancellation.\nThe density-one argument in the proof of\nTheorem~\\ref{long243:res:recorddichotomy} supplies infinitely many\nlate multipliers $a_n$ coprime to $D_n$. These multipliers are pairwise\ncoprime. At each such step, $a_n$ is coprime to $v_n$ and\n$\\gcd(u_n,v_n)=1$, so $a_nu_n-v_n$ is coprime to both $a_n$ and $v_n$.\nHence $h_n=1$ and $a_n\\mid v_{n+1}$.\nOnly finitely many of the chosen multipliers can have all their prime\ndivisors at most $2K$, since each uses a different prime from that finite\nset. Choose $K$ distinct primes $p_0,\\ldots,p_{K-1}>2K$ at these\nsteps. Once a chosen prime divides a reduced denominator, it divides\nevery later one: $h_nv_{n+1}=a_nv_n$ and $h_n<2K<p_i$ prevent its\nremoval. All the primes therefore divide $v_T$ at a common later index $T$.\n\nConsequently every $u_n$ with $n\\ge T$\nis coprime to all the $p_i$. Choose a CRT block of $K$ consecutive\nintegers, one divisible by each $p_i$, above $R_T$.\nDivergence forces a first crossing, while\n$u_{n+1}\\le u_n+K$ forces it to land inside the block, a contradiction.\nThis proves that finite $\\limsup\\mathcal A_n$ forces the eventual Sylvester recurrence.\n\nThe comparison\n$|\\delta_n-m_n/u_n|\\le3/a_n$ gives\n$|R_n\\delta_n-\\mathcal A_n|\\le3R_n/a_n\\to0$.\nHere $R_n\\le H_n$, $\\log H_n=o(n)$ and $a_n$ grows doubly\nexponentially. Thus the two boundedness conditions are equivalent.\nUnder the eventual Sylvester recurrence, $u_n=1$ and $m_n=0$ eventually, so\n$\\mathcal A_n\\to0$ and $R_n\\delta_n\\to0$. Since both quantities are\nnonnegative, their limits superior can only be $0$ or $+\\infty$.\n\\end{proof}\n\nThe estimate at the first later negative error is the lemma in the release,\n\\href{https://github.com/wcook04/plectis-erdos-lean/blob/52f29ad173b04e3bac941b3663f2b9aebe5de0bb/ErdosProblems/Erdos243/SaturatedSquareTransport.lean\\#L378}{negative error scaled by the previous maximum after cancellation}: at the first later negative error,\n$u_t\\le u_{s+1}$ and $u_su_t\\le R_t|\\tilde e_t|u_{s+1}$. The exact example\n$15/134=1/10+1/85+1/5695$, with steps in reduced fractions\n$(15,134)\\to(4,335)\\to(1,5695)$ and $\\mathcal A_1=15/4$,\nattains equality in the displayed bound.\n\nFor Corollary~\\ref{long243:res:criticalrate}, the comparison\n$|\\delta_n-m_n/u_n|\\le3/a_n$ and the rate $\\delta_n=O(1/n)$ give\n$m_n/u_n=O(1/n)$. If $u_n=O(n)$, then $m_n=O(1)$. Conversely,\n$m_n=O(1)$ gives $u_{n+1}\\le u_n+m_n$, hence $u_n=O(n)$ and\n$R_n=O(n)$. In either case\n$\\mathcal A_n=R_nm_n/u_n=O(1)$, so\nTheorem~\\ref{long243:res:recordamplified} gives eventual Sylvester\nbehaviour. The eventual Sylvester recurrence gives $u_n=1$ and $m_n=0$ eventually,\nwhich proves the converse implications.\n\nThe critical-rate assumption is doing real work: a bound on\n$(-\\tilde e_n)_+$ alone does not directly control $R_n/u_n$ after a\ndecrease. The preceding corollary supplies the missing control from\n$\\delta_n=O(1/n)$. No such bound is derived here from the unrestricted\nquadratic limit.\n\n\\begin{lemma}[large odd prime powers in the reduced denominator]\\label{long243:res:oddpowersupply}\nUnder the standing hypotheses, for every fixed $A>0$ and every sufficiently large $n$, the\nreduced denominator $v_n$ has an odd prime-power divisor $Q=p^k$ with\n\\[\n Q>(H_n+2)^A,\\qquad H_n=\\max_{j\\le n}C_j.\n\\]\nThe prime $p$ may depend on $n$; no stable-gcd or prime-arrival assumption\nis imposed.\n\\end{lemma}\n\\begin{proof}\nThe denominator grows too fast for all its odd prime-power factors to\nremain small compared with $H_n$. Indeed, the estimate for the rational\ntail gives $C_{n+1}/C_n\\to1$, hence $\\log H_n=o(n)$. Since $x_n=u_n/v_n\\le2/a_n$ and $u_n\\ge1$,\n$v_n\\ge a_n/2$. Also $v_n\\mid L_n=\\lcm(q,a_0,\\ldots,a_{n-1})$,\nso the $2$-primary part of $v_n$ is at most $\\max(q,a_{n-1})=a_{n-1}$\nfor all large $n$. Its odd part $W_n$ therefore satisfies\n\\[\n W_n\\ge\\frac{a_n}{2a_{n-1}}\\ge\\frac{a_{n-1}}4,\n \\qquad \\log W_n\\ge c2^{n-1}-O(1)\n\\]\nfor some $c>0$.\nPut $B_n=(H_n+2)^A=\\exp(o(n))$. If every exact odd prime-power factor of\n$W_n$ were at most $B_n$, then\n$W_n\\mid\\lcm(1,\\ldots,\\lfloor B_n\\rfloor)$, giving\n\\[\n \\log W_n\\le B_n\\log B_n=\\exp(o(n)).\n\\]\nThis contradicts the preceding exponential lower bound for all sufficiently\nlarge $n$. An offending exact prime-power factor is the required $Q$.\n\\end{proof}\nThis uses only an elementary factorial bound, not prime distribution.\nThe prime may change with $n$: choose one large prime power at a late\nstart and protect it until the first forbidden crossing.\n\n\\begin{theorem}[unit record increments]\\label{long243:res:unitrecord}\nUnder the standing hypotheses, if $R_{n+1}-R_n\\le1$ for all large $n$, then $a_{n+1}=a_n^2-a_n+1$\nfor all large $n$.  Hence the sequence is eventually Sylvester if and only if\n$\\#\\{n:R_{n+1}-R_n\\ge2\\}$ is finite.  No hypothesis is placed on decreases in $u_n$, on\nrecord-setting jumps, or on the cancellation factors $h_n$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose the orbit is not eventually Sylvester. Absorption gives\n$\\tilde e_n\\ne0$ on a late tail, so vanishing relative error and integrality give\n$u_n\\to\\infty$. Choose $s$ so large that the unit-increment bound and\n$|\\tilde e_n|<u_n/2$ hold for every $n\\ge s$, and that\nLemma~\\ref{long243:res:oddpowersupply} applies with $A=3$.\nSince $H_s\\ge R_s\\ge1$, it supplies an odd $Q=p^k\\mid v_s$ with\n$Q>(H_s+2)^3>4(R_s+2)$ and $Q\\ge16$.\nLet $Y$ be the least multiple of $p$ above $R_s$. Then\n$Y\\le R_s+p< Q/4+p\\le5Q/4$.\nAt the first $t>s$ with $u_t\\ge Y$, the integer maximum rises from\nat most $Y-1$ by at most one, forcing $u_t=Y$ regardless of\n$u_{t-1}$. Before $t$, one has\n$w_n<3u_n/2<15Q/8<pQ=p^{k+1}$.\nCorollary~\\ref{long243:res:powerpersistence} therefore gives $p^k\\mid v_t$.\nBut $p\\mid u_t=Y$, contradicting $\\gcd(u_t,v_t)=1$.\nThe converse follows because the eventual Sylvester recurrence gives $u_n=1$ eventually.\n\\end{proof}\n\nThe \\texttt{plectis-erdos-lean} repository contains the\n\\href{https://github.com/wcook04/plectis-erdos-lean/blob/52f29ad173b04e3bac941b3663f2b9aebe5de0bb/ErdosProblems/Erdos243/RecordIncrementBarrier.lean\\#L329}{unit-increment landing lemma} and the\n\\href{https://github.com/wcook04/plectis-erdos-lean/blob/52f29ad173b04e3bac941b3663f2b9aebe5de0bb/ErdosProblems/Erdos243/RecordIncrementBarrier.lean\\#L425}{one-unit record-increment implication};\nthey assume the existence of the prime power that\nLemma~\\ref{long243:res:oddpowersupply} supplies.\n\n\nThe two-unit criterion discussed below bounds the actual jump at each\nrecord step, rather than the increase in the running maximum. As numerical\nbounds on individual steps, neither condition implies the other: the first\nallows $s_n=2$, while $s_n\\le1$ permits large jumps after an earlier\ndecrease. This is a comparison of the bounds, not a claim that they remain\nlogically independent under all the standing tail hypotheses; there both\ncriteria force eventual Sylvester behaviour.\nThe finite example\n$(8,177)\\to(7,4071)\\to(10,2373393)$ with multipliers $23$ and $583$ has records\n$8$ and $10$, increment $2$, jump $3$ and $\\gcd(8,10)=2$, which is why the\nparity argument for crossing an odd level does not transfer from actual\njumps to increments of the running maximum.\n\n\\subsection{Counting jumps before a prime power can be lost}\n\nLarge jumps may skip forbidden values. Under the relative-error\nbound, $L=pQ/2$ keeps the numerator before cancellation below $pQ$\nuntil the crossing, protecting the prime power. Odd multiples of $p$\nexclude two-unit crossings as well: skipping an odd level in two\nunits would leave both adjacent numerators even. We count both\ncrossing jumps and their excess over two.\n\n\\begin{theorem}[counting crossings before a prime power is lost]\\label{long243:res:epochenergy}\nLet the orbit satisfy the reduced recurrences of this section, with\n$2|\\tilde e_n|<u_n$ from an\nindex $s$.  Let $p\\ge3$ be prime, $Q=p^{\\ell}$ divide $v_s$ with $Q\\ge16$, put\n$L=pQ/2$, and assume $R_s<L/2$. Assume that $u_t\\ge L$ for some $t>s$,\nand let $\\tau$ be the first such index, let $J$ be the set of steps in $[s,\\tau)$ that first cross at least one odd\nmultiple of $p$ in $(L/2,L]$, and put $X=\\sum_{n\\in J}(d_n-2)$.  Then every\n$n\\in J$ is a record step with $h_n=1$ and $d_n\\ge3$, and\n$pQ\\le(8p+8)\\lvert J\\rvert+4X+8p$.\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{long243:res:epochenergy}]\nFor $s\\le n<\\tau$, one has $u_n<L$ and\n$w_n<3u_n/2<3pQ/4<p^{\\ell+1}$, since $Q=p^{\\ell}$.\nCorollary~\\ref{long243:res:powerpersistence} gives $Q\\mid v_t$ throughout\n$[s,\\tau]$, so $p\\nmid u_t$ there. A first crossing of a level above $R_s$\nis a record step. If $h_n\\ge2$ then $u_{n+1}=w_n/h_n<3u_n/4$, so every\nsuch record step has $h_n=1$.\nAn odd multiple of $p$ cannot be landed on. A jump of size $1$ crossing it\nwould land on it; a jump of size $2$ avoiding the landing would have two\neven endpoints, contrary to $\\gcd(u_n,u_{n+1})=1$. Thus every $n\\in J$\nhas $d_n\\ge3$.\nIt remains to count these levels. The interval $(L/2,L]$ has length\n$pQ/4$, and odd multiples of $p$ are spaced by $2p$, so it contains\nat least $Q/8-1$ of them. The levels first crossed by a step of size\n$d_n$ have the same spacing, so\nthere are at most $1+d_n/(2p)$ of them. Summing over $J$ gives\n\\[\n Q/8-1\\le |J|+\\frac{2|J|+X}{2p},\n\\]\nwhich rearranges to the asserted inequality.\n\\end{proof}\n\nCrossing requires many jumps or a few large ones. The first series\ntherefore counts jumps of size at least three and measures their\nexcess over two, with weights that detect either contribution.\nA non-Sylvester tail pays a fixed positive amount on arbitrarily late\nprotected intervals; a Sylvester tail has no late records. The\nrelative-error limit alone gives no convergence.\n\n\\begin{theorem}[two convergence criteria for large record jumps]\\label{long243:res:energycriterion}\nUnder the standing hypotheses, the sum\n$\\mathcal E=\\sum_{n\\ \\mathrm{record}}\\bigl(\\mathbf 1_{d_n\\ge3}u_n^{-1/2}\n+(d_n-2)_+u_n^{-1}\\bigr)$ is finite if and only if the sequence is eventually Sylvester; and\n$\\sum_{n\\ \\mathrm{record}}(d_n-2)_+u_n^{-1/2}$ is finite if and only if the sequence is eventually Sylvester.\n\\end{theorem}\n\n\\begin{proof}\nAssume first that the recurrence is not eventually Sylvester. Then $u_n\\to\\infty$.\nFor every sufficiently late $s$, Lemma~\\ref{long243:res:oddpowersupply}\nwith $A=3$ gives an odd $Q=p^k\\mid v_s$ with\n$Q>(H_s+2)^3$, hence $Q\\ge16$ and $pQ>4R_s$.\nSet $L=pQ/2$ and take its first crossing. The preceding theorem applies.\nBecause $Q\\ge p$ and $p\\ge3$,\n\\[\n \\frac{8p}{L}=\\frac{16}{Q}\\le1,\n \\qquad \\frac{8p+8}{\\sqrt L}\n \\le8\\sqrt2(1+1/p)<16.\n\\]\nDividing the preceding counting inequality by $L$ therefore gives\n\\[\n 1\\le16\\frac{|J|}{\\sqrt L}+4\\frac{X}{L}\n \\le16\\sum_{n\\in J}\\left(\\frac1{\\sqrt{u_n}}+\n                  \\frac{d_n-2}{u_n}\\right),\n\\]\nsince $u_n<L$ for $n<\\tau$ and $d_n\\ge3$ on $J$.\nThus an arbitrarily late finite window contributes at least $1/16$ to\n$\\mathcal E$, contradicting convergence. No disjointness of the windows is\nneeded: every window lies in a tail of the nonnegative series.\nUnder the eventual Sylvester recurrence, $u_n=1$ eventually, so there are no late records and\n$\\mathcal E$ is finite.\nFinally, term by term at a record,\n\\[\n \\mathbf1_{d_n\\ge3}u_n^{-1/2}+(d_n-2)_+u_n^{-1}\n \\le2(d_n-2)_+u_n^{-1/2}.\n\\]\nHence finiteness of the second series implies finiteness of the first,\nand convergence in the converse direction again follows from eventual Sylvester behaviour.\n\\end{proof}\n\n\\paragraph{Formalisation.}\nThe \\texttt{plectis-erdos-lean} repository contains\n\\href{https://github.com/wcook04/plectis-erdos-lean/blob/52f29ad173b04e3bac941b3663f2b9aebe5de0bb/ErdosProblems/Erdos243/ProtectedEpochEnergy.lean\\#L185}{the integer inequality counting crossings before a prime power is lost}\nand its counting lemmas.\nIn Lean,\n\\lword{Erdos243/PrimitiveRecordBarrier.lean}{435}{recordRiseTwo_sylvesterNext_eventually}{the two-unit record-rise implication}\nassumes that arbitrarily late $s$ admit an odd $p^k\\mid v_s$ with\n$3R_s<p^k$; Lemma~\\ref{long243:res:oddpowersupply} supplies such prime\npowers.\n\n\\subsection{Bounds on the error and on the original sequence}\n\n\\begin{theorem}[slow negative part]\\label{long243:res:slownegative}\nLet $a,C,D:\\N\\to\\N$ satisfy \\eqref{long243:eq:recurrences} with $a_n>1$, $C_n>0$, $D_0\\ge1$, under\nvanishing relative error.  Suppose that for some $\\delta\\in(0,1)$ and all large $n$\nwith $E_n<0$ one has $-E_n\\le(1-\\delta)\\ell(C_n)$.  Then $E_n=0$ for all large\n$n$, and $a_{n+1}=a_n^2-a_n+1$ for all large $n$.\n\\end{theorem}\n\n\\noindent Theorem~\\ref{long243:res:bounded} is the case of a constant bound, since a\nconstant is eventually below $(1-\\delta)\\ell(C_n)$ on a nonzero tail,\nwhere $C_n\\to\\infty$.\n\n\\begin{proof}\nSuppose the error is not eventually zero. The auxiliary lower bound in\nthe proof of Theorem~\\ref{long243:res:recorddichotomy} applies to this\nexact integer orbit: it used only $a_n>1$, $C_n>0$, $D_0\\ge1$ and\nvanishing relative error. It gives\n$\\limsup_n(H_{n+1}-H_n)/\\ell(H_n)\\ge1$.\nOn the other hand, the hypothesis and $C_n\\le H_n$ give\n\\[\n H_{n+1}-H_n\\le(-E_n)_+\n \\le(1-\\delta)\\ell(C_n)\\le(1-\\delta)\\ell(H_n)\n\\]\nfor all large $n$, a contradiction. Thus $E_n=0$ eventually, and\nTheorem~\\ref{long243:res:eventual} gives the Sylvester recurrence.\n\\end{proof}\n\nThe corresponding pointwise-rise argument also has formalised arithmetic\nlemmas:\n\\lword{Erdos243/SlowRiseBarrier.lean}{111}{commonDivisor_persists}{persistence of\na common divisor},\n\\lword{Erdos243/SlowRiseBarrier.lean}{162}{multiplierOverlap_dvd_laterCenteredState}{divisibility\nof every later error by $\\gcd(a_n,D_n)$}, the\n\\lword{Erdos243/SlowRiseBarrier.lean}{194}{slowRise_landing}{landing lemma} and\nthe\n\\lword{Erdos243/SlowRiseBarrier.lean}{280}{no_slowNegative_of_coprimeBlock}{resulting\nexclusion}.  The count of indices and the simultaneous choice of constants are\nwritten out in the earlier proof. Corollary~\\ref{long243:res:loglogboundary} includes the\nboundary coefficient $1$, while the preceding theorem on the running\nmaximum also discounts rises that recover an earlier decrease. The present\nstatement is retained for its direct bound on $E_n$.\n\nTo compare the bounded hypothesis with the classical criteria, number\nthe original sequence from $1$ in this subsection and put\n$P_n=\\prod_{1\\le j<n}a_j$. Then\n\\[\n \\frac{P_n}{a_n}\\left(\\frac{a_n^2}{a_{n+1}}-1\\right)\n =\\frac{P_{n+1}}{a_{n+1}}-\\frac{P_n}{a_n}.\n\\]\nThus the bound concerns upward increments, not boundedness of $P_n/a_n$.\nThe quadratic growth limit controls only the ratio of consecutive terms.\nSequences satisfying the eventual Sylvester recurrence satisfy the bound, as do exact-square sequences\n$a_n=b^{2^{n-1}}$ with $b\\ge2$, for which the difference is zero.\nThe latter never satisfies the Sylvester recurrence, so the bounded\ncriterion below recovers irrationality of its reciprocal sum.\nThe rounded examples after Corollary~\\ref{long243:res:onethreshold}\nshow more generally how a term $c/n$ in the ratio produces increments\nof order $n^{c-1}$. The theorem allows a finite positive upper limit\nwhere the classical product criterion requires a nonpositive one;\nneither bound is derived from the original problem alone.\n\n\\begin{theorem}[bounded or slowly growing increments of the product ratio]\\label{long243:res:strausbounded}\nLet $a_1<a_2<\\cdots$ be positive integers with $a_{n+1}/a_n^2\\to1$\nand $\\sum_{n\\ge1}1/a_n=p/q$, where $p,q$ are positive integers. Put\n\\[\n Q_n=\\frac{a_1a_2\\cdots a_{n-1}}{a_n}\n     \\left(\\frac{a_n^2}{a_{n+1}}-1\\right).\n\\]\nIf $\\limsup_nQ_n<\\infty$, the sequence is eventually Sylvester.\nThe same conclusion holds if, for some $\\delta>0$ and all large $n$,\n\\[\n Q_n\\le\\frac{1-\\delta}{q}\\,\n \\ell(a_1\\cdots a_{n-1}/a_n).\n\\]\n\\end{theorem}\n\nFor the bounded clause, use the stated denominator $q$, put\n$P_n=\\prod_{j<n}a_j$, $x_n=\\sum_{k\\ge n}1/a_k$, and form the integer tail\n$D_n=qP_n$, $C_n=D_nx_n$, $E_n=D_n-(a_n-1)C_n$. Exact cancellation gives\n\\[\n E_n+qQ_n=qP_n\\left(\\frac1{a_{n+1}}\n -(a_n-1)\\sum_{k\\ge n+2}\\frac1{a_k}\\right).\n\\]\nThe expression in parentheses is positive eventually, since\n$\\sum_{k\\ge n+2}1/a_k\\le4/a_{n+1}^2$ and\n$a_{n+1}>4(a_n-1)$ eventually. Thus $E_n\\ge-qQ_n$,\nand $\\limsup Q_n<\\infty$ supplies the eventual lower bound required\nby Theorem~\\ref{long243:res:bounded}. This proves the bounded clause\nwithout first estimating the absolute size of the comparison error.\n\nFor the slow-growth clause and the later summability comparisons,\nthe same exact formula gives\n\\[\n 0<E_n+qQ_n\\le qP_n/a_{n+1}=O(P_n/a_n^2)\n\\]\neventually. The ratio test in Section~\\ref{long243:sec:lcmrecords}\nshows that these errors have a convergent sum, hence tend to zero.\nThe factor $1/q$ in the slow-growth clause cancels the factor $q$ in\n$E_n+qQ_n=o(1)$. Suppose the sequence is not eventually Sylvester. Absorption\nand vanishing relative error give $C_n\\to\\infty$, and the tail estimate\n$C_n\\sim qP_n/a_n$ implies\n\\[\n \\frac{\\ell(C_n)}{\\ell(P_n/a_n)}\\longrightarrow1.\n\\]\nFor $0<\\delta<1$, the assumed bound and $E_n+qQ_n=o(1)$ consequently give\n$(-E_n)_+\\le(1-\\delta/2)\\ell(C_n)$ eventually. This contradicts\nTheorem~\\ref{long243:res:slownegative}. If $\\delta\\ge1$, the hypothesis gives\n$Q_n\\le0$ eventually and the bounded clause already applies.\n\nThe same comparison permits any positive real clearing factor\n$F_n\\le qP_n$: multiplying $E_n+qQ_n=o(1)$ by $F_n/(qP_n)$ gives\n\\[\n F_n-(a_n-1)F_nx_n+\n \\frac{F_n}{a_n}\\left(\\frac{a_n^2}{a_{n+1}}-1\\right)=o(1).\n\\]\nNeither $F_nx_n$ nor $F_n-(a_n-1)F_nx_n$ need be integral. The LCM\nchoice $F_n=L_n$ makes both integers, but its update includes the\nfactor $\\rho_n$ of Section~\\ref{long243:sec:lcmrecords}. The short note\nuses only this LCM specialisation; no general clearing factor is\nneeded in its bounded-increment proof.\n\n\\paragraph{Attribution.}\nErd\\H{o}s and Straus require $\\limsup\\le0$ in the corresponding criterion, with\nthe least common multiple in place of the product and the growth factor one\nindex later, so their quantity is\n$[a_1,\\ldots,a_n]a_{n+1}^{-1}(a_{n+1}^{2}/a_{n+2}-1)$~\\cite[Theorem~3,\np.~132]{erdosstraus1964}; Koizumi's Corollary~4(1) uses the product\nform~\\cite[Cor.~4(1), pp.~14--15]{koizumi2025}.  The bounded clause follows from\nTheorem~\\ref{long243:res:bounded}. For Koizumi's product expression it\nreplaces a nonpositive upper limit by an arbitrary finite upper limit.\nThe material distinction here is product versus least common multiple.\nReindexing $r=n+1$ makes the classical LCM expression\n$[a_1,\\ldots,a_{r-1}]a_r^{-1}(a_r^2/a_{r+1}-1)$, exactly the expression\nin the short note's LCM corollary.\n\nThe comparison $E_n+qQ_n=o(1)$ has its classical predecessor in\nKoizumi's proof of Corollary~4(1) \\cite[(11)--(12), p.~15]{koizumi2025}.\n\n\\begin{corollary}[the $1/n$ threshold]\\label{long243:res:onethreshold}\nLet $a_1<a_2<\\cdots$ be positive integers with $a_{n+1}/a_n^2\\to1$\nand $\\sum_n1/a_n\\in\\Q$. If\n\\[\n \\limsup_{n\\to\\infty} n\\,(a_n^2/a_{n+1}-1)_+<1,\n\\]\nthen $a_{n+1}=a_n^2-a_n+1$ for all large $n$. The same conclusion holds if, for some $K\\ge0$ and $\\varepsilon>0$,\n\\[\n \\frac{a_n^2}{a_{n+1}}-1\\le\\frac1n+\\frac{K}{n^{1+\\varepsilon}}\n \\quad\\hbox{eventually}.\n\\]\nIn particular the one-sided bound by $1/n$ is included.\n\\end{corollary}\n\n\\begin{proof}\nPut $\\gamma_n=a_n^2/a_{n+1}-1$ and $t_n=\\bigl(\\prod_{j<n}a_j\\bigr)/a_n$, so\nthat $t_{n+1}/t_n=1+\\gamma_n$.  Choose $r\\in(0,1)$ and $N$ with\n$\\gamma_n^{+}\\le r/n$ for all $n\\ge N$.  Then\n\\[\n t_n\\le t_N\\prod_{k=N}^{n-1}\\Bigl(1+\\frac rk\\Bigr)=O(n^{r}),\n\\]\nso $(t_n\\gamma_n)_+=t_n\\gamma_n^{+}=O(n^{r-1})$ tends to zero and\n$\\limsup_nt_n\\gamma_n\\le0$.  Koizumi's Corollary~4(1)\n\\cite[pp.~14--15]{koizumi2025} gives the first conclusion.\nFor the second, divide $t_n$ by $n$ to obtain\n\\[\n \\frac{t_{n+1}/(n+1)}{t_n/n}\n =\\frac{1+\\gamma_n}{1+1/n}\n \\le 1+\\frac{K}{n^\\varepsilon(n+1)}.\n\\]\nThe product of the upper bounds converges, so $t_n/n$ is bounded above.\nConsequently $t_n(\\gamma_n)_+=O(1)$, and\nTheorem~\\ref{long243:res:strausbounded} applies.\n\\end{proof}\n\nThe inclusive clause requires more than $\\limsup n(\\gamma_n)_+\\le1$.\nThat weaker condition does not give the product estimate: for example,\nthe scalar sequence $\\gamma_n=(1+1/\\log n)/n$ has $t_n\\asymp n\\log n$\nand $t_n\\gamma_n\\asymp\\log n$. This is a limitation of the estimate, not a\ncounterexample to the original problem.\n\nMore generally, if $c>0$, $\\varepsilon>0$ and\n\\[\n \\frac{a_n^2}{a_{n+1}}=1+\\frac cn+O(n^{-1-\\varepsilon}),\n\\]\nthen the consecutive-ratio identity gives $t_n\\sim Kn^c$ for some\n$K>0$ and $t_{n+1}-t_n\\sim Kc n^{c-1}$. To see the constant, take\nlogarithms and subtract $c\\log(1+1/n)$; the remainder is absolutely summable.\nHence the increments are bounded for $c\\le1$ and unbounded for $c>1$.\nThese rates are realised by increasing integer sequences: set\n$a_{n+1}=\\lceil n a_n^2/(n+c)\\rceil$ and choose the seed large enough\nthat $a_{n+1}\\ge a_n^2/(1+c)\\ge2a_n$ throughout. The rounding remainder\nin $[0,1)$ gives an error in the ratio bounded by\n$(1+c)^2/a_n^2$, smaller than every fixed inverse power of $n$.\nThis calculation supplies the family comparison behind the two rounded\nexamples retained in the short note. It concerns their growth and does\nnot assume rationality.\n\nThe second clause has a concrete application outside Koizumi's\nnonpositive-upper-limit product condition. The sequence\n$a_1=4$, $a_{n+1}=\\lceil n a_n^2/(n+1)\\rceil$ begins\n$4,8,43,1387,\\ldots$ and satisfies\n\\[\n a_n\\ge2\\cdot2^{2^{n-1}},\\qquad\n 0\\le1+\\frac1n-\\frac{a_n^2}{a_{n+1}}<\\frac4{a_n^2}.\n\\]\nThe lower bound follows from $a_{n+1}\\ge a_n^2/2$; the upper error bound\nfollows by writing the rounding remainder in $[0,1)$.\nConsequently $t_n\\sim Kn$ and $t_n\\gamma_n\\to K>0$ by the product\ncomparison above. The corollary proves that the reciprocal sum is\nirrational, since the eventual Sylvester recurrence would give\n$\\gamma_n=O(1/a_n)$ rather than $\\gamma_n\\sim1/n$.\n\nThe strict clause includes Koizumi's sufficient\nrate $1+o(1/n)$ \\cite[Remark~3, p.~16]{koizumi2025}; the inclusive clause\nallows a bounded positive increment rather than requiring a nonpositive\nupper limit.\n\n\\subsection{Signs of the error and of the growth ratio}\n\nWe keep the preceding one-based indexing. The growth defect need not\nhave the opposite sign to $E_n$: the next identity displays the\ncorrection, including its sign under the eventual Sylvester recurrence.\n\n\\begin{proposition}[comparison of the two signs]\\label{long243:res:classicalhalfspace}\nThe factor $M_n=D_n/L_n$ divides $G_n=\\gcd(C_n,D_n)$. Thus $G_n/M_n$\nis a positive integer and $E_n/M_n=(G_n/M_n)\\tilde e_n$ is an integer\nwith the same sign as $E_n$. The sign of the growth ratio minus one also\ndepends on a correction term. The exact recurrences give, whenever\n$a_{n+1}C_nC_{n+1}\\ne0$,\n\\begin{equation}\\label{long243:eq:shiftedsign}\n \\frac{a_n^2}{a_{n+1}}-1=-\\frac{E_n}{C_n}+\\Lambda_n,\n \\qquad\n \\Lambda_n=\\frac{\\bigl(1-E_n/C_n\\bigr)\\bigl(a_n-1+E_{n+1}/C_{n+1}\\bigr)}{a_{n+1}},\n\\end{equation}\nand, under the standing positive rational-tail hypotheses,\n$0<\\Lambda_n<3/a_n$ for all sufficiently large $n$. The\nErd\\H{o}s--Straus quantity of Theorem~3 is\n\\[\n Z_n^{\\mathrm{ES}}=\n \\frac{[a_1,\\ldots,a_n]}{a_{n+1}}\n \\left(\\frac{a_{n+1}^2}{a_{n+2}}-1\\right).\n\\]\nIts least common multiple includes $a_n$ but not the clearing\ndenominator $q$. It is not $Q_n=(P_n/a_n)\\gamma_n$ from the preceding subsection,\nnor $L_n\\gamma_{n+1}/a_{n+1}$ under our convention\n$L_n=\\lcm(q,a_1,\\ldots,a_{n-1})$.\nBeing a positive multiple of the next growth defect, $Z_n^{\\mathrm{ES}}$\nhas the sign of $\\Lambda_{n+1}-E_{n+1}/C_{n+1}$. For all sufficiently\nlarge $n$, it is positive when $E_{n+1}\\le0$; for $E_{n+1}>0$, it is\nnegative precisely when\n$E_{n+1}/C_{n+1}>\\Lambda_{n+1}$. Under the eventual Sylvester recurrence, $E_n=0$ and\n$\\Lambda_n=(a_n-1)/a_{n+1}>0$.\n\\end{proposition}\n\n\\begin{proof}\nSince both $C_n=D_nx_n$ and $L_nx_n$ are integers, $M_n=D_n/L_n$\ndivides $C_n$ as well as $D_n$. It therefore divides $G_n$.  For \\eqref{long243:eq:shiftedsign}, write\n$\\theta_n=E_n/C_n$; Proposition~\\ref{long243:res:update} gives $C_{n+1}=C_n(1-\\theta_n)$\nand $D_n/C_n=a_n-1+\\theta_n$, so\n$a_{n+1}-1+\\theta_{n+1}=a_nD_n/C_{n+1}=a_n(a_n-1+\\theta_n)/(1-\\theta_n)$.\nMultiplying the asserted identity by $a_{n+1}$ and substituting reduces it to\n$a_{n+1}+a_n-1+\\theta_{n+1}=a_n^2/(1-\\theta_n)$, which is the displayed\nrelation with $a_n$ added to both sides.  The bound follows from\n$\\theta_n\\to0$ and $a_{n+1}\\ge a_n^2/2$.  On Sylvester's sequence\n$\\theta_n=0$ and $a_{n+1}=a_n^2-a_n+1$, so\n$a_n^2/a_{n+1}-1=(a_n-1)/a_{n+1}=\\Lambda_n>0$.\n\\end{proof}\n\n\nA nonpositive upper limit admits positive values tending to zero; it is\nnot an eventual pointwise sign condition. For the \\emph{product} quantity\n$Q_n=(P_n/a_n)\\gamma_n$, the identity $E_n+qQ_n=o(1)$ shows that eventual\n$E_n\\ge0$ implies $\\limsup Q_n\\le0$, as in Koizumi's Corollary~4(1)\n\\cite[pp.~14--15]{koizumi2025}.\nFor the classical LCM expression $Z_n^{\\mathrm{ES}}$, the corresponding\ninteger error is also taken one index later. Its published hypotheses are\nthose of Erd\\H{o}s--Straus Theorem~3\n\\cite[p.~132]{erdosstraus1964}. Reindexing relates it to an LCM-cleared\nerror, not to $E_n$ with the unchanged product factor $qQ_n$.\n\n\\paragraph{A comparison with $\\mathcal B$-free integers.}\nFor a fixed family $\\mathcal B=\\{m_i\\}$, the integers divisible by none\nof the $m_i$ are called $\\mathcal B$-free integers. With pairwise coprime\nmoduli and $\\sum_i1/m_i<\\infty$, this is the standard setting of\n\\cite[\\S1.2]{elabdalaoui2015}. The next proposition is an elementary\ninterval count in that setting. It tests what can follow from avoiding\nwhole multiples alone. Unlike the reciprocal-tail argument, it imposes\nall the moduli from the outset and has no denominator recurrence or\ncancellation factors.\n\n\\begin{proposition}[an elementary interval bound]\\label{long243:res:coprimalitycap}\nLet $m_0<m_1<\\cdots$ be pairwise coprime integers at least $2$ with\n$\\theta=\\sum_i1/m_i<1$. For all integers $x\\ge1$ and $L\\ge1$ satisfying\n$L>k/(1-\\theta)$, where $k=\\#\\{i:m_i\\le x+L\\}$, the interval $[x,x+L)$ contains\nan integer divisible by no $m_i$. If also $\\ell(m_i)=i+O(1)$, then for\nevery $\\epsilon>0$ there are an index $T$ and a strictly increasing\nsequence of positive integers $(u_n)$ such that\n\\[\n m_i\\nmid u_n\\quad\\text{for every }i\\ge T\\text{ and every }n,\n \\qquad\n u_{n+1}-u_n\\le(1+\\epsilon)\\ell(u_n)\\quad\\text{eventually}.\n\\]\n\\end{proposition}\n\n\\begin{proof}\nThe interval $[x,x+L)$ contains exactly $L$ integers.  Each lies in\n$[1,\\infty)$ and is smaller than $x+L$, so a\nmodulus exceeding $x+L$ divides none of them. Each of the $k$ remaining\nmoduli divides at most $L/m_i+1$ of them, so the covered count is at most\n$L\\theta+k<L$. This interval count itself does not use pairwise\ncoprimality; that condition is needed for the exact CRT proportions\nin the later gap theorem.\n\nFor the second assertion, choose a tail of the family and reindex it so that\n$\\theta<\\epsilon/(1+\\epsilon)$.  If\n$k(z)=\\#\\{i:m_i\\le z\\}$, the hypothesis $\\ell(m_i)=i+O(1)$ gives\n$k(z)\\le\\ell(z)+C$ for all large $z$ and some constant $C$.  Choose\n\\[\n (1-\\theta)^{-1}<\\rho<1+\\epsilon,\n \\qquad L(y)=\\left\\lceil\\rho\\ell(y)\\right\\rceil .\n\\]\nThen $L(y)=O(\\ell(y))=o(y)$, and the definition of $\\ell$ gives\n$\\ell(y+1+L(y))=\\ell(y)+o(1)$.  Consequently, for all large integers $y$,\n\\[\n \\frac{k(y+1+L(y))}{1-\\theta}\n \\le \\frac{\\ell(y)+C+o(1)}{1-\\theta}\n < \\rho\\ell(y)\\le L(y),\n\\]\nwhile $L(y)\\le(1+\\epsilon)\\ell(y)$.  The first part also supplies an admissible $u_0$ beyond this threshold.\nApply it successively with $x=u_n+1$ and length $L(u_n)$, and choose\n$u_{n+1}$ in the resulting window.  The sequence is strictly increasing,\navoids every retained modulus, and has the required rise bound.\n\\end{proof}\n\nThe small-increment sequence avoids only the retained moduli $m_i$,\n$i\\ge T$. Discarding the prefix makes their reciprocal sum small.\nFor the full family, Theorem~\\ref{long243:res:gapconstant} instead gives\nthe coefficient $\\prod_i(1-1/m_i)^{-1}>1$ for the increasing enumeration\nof integers avoiding all the moduli.\n\n\nProposition~\\ref{long243:res:coprimalitycap} concerns avoidance of whole\nmultiples, not coprimality to a composite modulus. The distinction matters:\nno integer in $[2,5)$ is coprime to $30$, although none is divisible by $30$.\nThus this proposition alone does not disprove a coprime-walk extension of\nTheorem~\\ref{long243:res:barrier}. Proposition~\\ref{long243:res:variablerise}\nbelow gives a separate counterexample using sparse \\emph{prime} moduli.\nUnder the additional scale $\\ell(m_j)=j+O(1)$,\nTheorem~\\ref{long243:res:gapconstant} determines the exact maximal-gap\ncoefficient for these $\\mathcal B$-free integers. Its proof uses a finite\nsieve estimate and the Chinese remainder theorem, not an orbit\nconstruction. Neither static construction supplies an exact\nreciprocal-tail orbit.\n\n\\paragraph{Conditions on a possible counterexample.}\nTogether, these criteria give necessary conditions on a counterexample.\nFor every fixed $\\delta\\in(0,1)$, it must have\n\\[\n (1-\\delta)\\ell(C_n)\\ \\le\\ -E_n\\ =\\ o(C_n)\n \\qquad\\text{at infinitely many }n.\n\\]\nIt must also have a divergent sum of relative increases, unbounded\nnegative error scaled by the previous maximum, and infinitely many record\njumps of the reduced numerator of size at least three, with $h_n=1$.\nThese are necessary conditions, not a construction of a counterexample. The residue-saturation lemma\nshows precisely one limitation of the fixed-old-modulus method; it is not\nan impossibility theorem about every future approach. A useful next target\nis interaction between new prime support, three consecutive numerators and\nthe global bound on numerator growth.\n\n\\section{Nonnegative errors}\\label{long243:sec:descent}\n\nThe scalar update already settles nonnegative errors. With indices\nstarting at zero, $C_{n+1}=C_n-E_n$ makes $C_n$ nonincreasing, and\na decreasing sequence of natural numbers must stabilise.\n\n\\begin{theorem}[descent]\\label{long243:res:descent}\nLet $C,E:\\N\\to\\N$ satisfy $C_{n+1}+E_n=C_n$ for every $n$.  Then $E_n=0$ for\nall sufficiently large $n$.\n\\end{theorem}\n\n\\begin{proof}\nThe recurrence gives $C_{n+1}\\le C_n$. Choose an index at which the\nnatural-number sequence reaches its minimum. Every later value equals\nthat minimum, and $C_{n+1}+E_n=C_n$ then gives $E_n=0$.\n\\end{proof}\n\n\n\nThe hypothesis is exactly one-sidedness: $C$ and $E$ take values in $\\N$.  If\n$E_n<0$ infinitely often, the numerator rises at those indices and this\ndescent argument no longer applies. The next two sections first treat\nconstant and periodic negative errors.\n\n\\section{Constant negative errors}\\label{long243:sec:constant}\n\nA constant negative error makes the numerator grow linearly.\nSuch behaviour can persist for arbitrarily long finite intervals, even\non genuine rational orbits, by the flat-transient theorem in the working\nreport on Problem~\\#243\n\\cite[``Flat-transient theorem'']{erdosproblemaday243}. We exclude\ninfinite constant and periodic negative tails below. These exclusions\nsupply no bound for the finite transient lengths.\n\nSuppose the error is negative with a constant magnitude, $E_n=-m$ for a fixed\n$m>0$ and every $n$.  By\nProposition~\\ref{long243:res:update} the numerator increases by $m$ at each step,\nso $C_n=c+nm$ with $c=C_0$, and the definition of the error becomes the\nidentity\n\\[\n D_n+m=(a_n-1)\\,(c+nm) .\n\\tag{5.1}\\label{long243:eq:shape}\n\\]\nTogether with $D_{n+1}=a_nD_n$ this is a closed system in $(a,D)$.\nWe shall prove that it has no infinite natural-number solution with\n$a_n\\ge2$, after examining a finite example.\n\n\\begin{example}[a nonintegral second step]\\label{long243:ex:shape}\nTake $m=c=1$, so that $C_n=1+n$ and \\eqref{long243:eq:shape} reads\n$D_n+1=(a_n-1)(1+n)$, and take $D_0=1$.  Each step is now determined: at $n=0$\nthe equation reads $2=(a_0-1)\\cdot1$, so $a_0=3$, and $D_1=a_0D_0=3$; at $n=1$\nit reads $4=(a_1-1)\\cdot2$, so $a_1=3$, and $D_2=a_1D_1=9$; at $n=2$ it reads\n$10=(a_2-1)\\cdot3$, which has no integer solution, and the orbit stops.\nLonger prefixes occur for other starting values. For $2\\le a_0<5000$,\nthe exact residue search in Appendix~\\ref{long243:app:residue} gives a\nmaximum of $17$ successful updates, hence $18$ multiplier values\nincluding $a_0$.\n\\end{example}\n\n\\begin{theorem}[no constant negative magnitude]\\label{long243:res:constant}\nFor any $m,c\\in\\N$ with $m>0$, there is no pair of sequences\n$a,D:\\N\\to\\N$ with $a_n\\ge2$ for all $n$ satisfying\n$D_{n+1}=a_nD_n$ and~\\eqref{long243:eq:shape}. The same holds if the\nidentity is assumed only from some index onwards.\n\\end{theorem}\n\n\\begin{proof}[Proof when $c$ and $m$ are coprime]\nAssume first $\\gcd(c,m)=1$. Every multiplier must share a prime with\n$m$, but each such prime can occur in at most one multiplier.\n\n\\emph{Every $a_j$ shares a prime with $m$.}  Suppose $\\gcd(a_j,m)=1$.  Then\n$m$ is invertible modulo $a_j$, so some $n$ has $c+nm\\equiv0$, that is\n$a_j\\mid C_n$; and $a_j\\mid D_n$ for every $n>j$, since $D$ is multiplicative\nwith $a_j$ among its factors.  Choosing such an $n$ beyond $j$\nand reading~\\eqref{long243:eq:shape} modulo $a_j$ gives $a_j\\mid m$, contradicting\n$\\gcd(a_j,m)=1$ and $a_j\\ge2$.\n\n\\emph{A prime divisor of $m$ occurs in at most one $a_j$.}  Suppose $p\\mid m$\nand $p\\mid a_j$.  For $n>j$ we have $p\\mid D_n$, so~\\eqref{long243:eq:shape} gives\n$(a_n-1)C_n\\equiv0 \\pmod p$.  Now $C_n=c+nm\\equiv c$, and $p\\nmid c$ because\n$p\\mid m$ and $\\gcd(c,m)=1$; hence $p\\mid a_n-1$, so $p\\nmid a_n$.  Thus $p$\ncannot divide any later multiplier.\n\nThis injects infinitely many multipliers into the finite set of prime\ndivisors of $m$, a contradiction.\n\\end{proof}\n\n\\noindent The case $\\gcd(c,m)=1$ is the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{147}{no_scalePrimitiveConstantNegative_orbit}{exclusion when $c$ and $m$ are coprime}.  The special case $m=c=1$, worked through in\nExample~\\ref{long243:ex:shape}, where $m$ has no prime divisors at all and the first\nfact is immediately contradictory, is recorded separately as the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{100}{no_normalizedConstantNegative_orbit}{case $c=m=1$}.\n\n\\begin{proof}[Removing the scale]\nFor general $c$, set $g=\\gcd(c,m)$. Equation~\\eqref{long243:eq:shape}\ngives $g\\mid D_n$. Both identities are homogeneous in $D,c,m$:\ndividing them by $g$ leaves the multipliers unchanged and gives\ncoprime data, excluded by the previous case.\n\\end{proof}\n\n\\noindent The formal statements are the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{286}{no_constantNegative_orbit}{exclusion\nat every scale} and the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{350}{no_eventuallyConstantNegative_orbit}{eventual\nexclusion}. The latter follows by shifting the orbit to the first index\nof constant error.\n\nTheorem~\\ref{long243:res:constant} excludes an error that is eventually constant\nand negative, at every magnitude and every scale. Constancy fixes both\nthe possible prime divisors and the congruence used at later indices.\nFor varying magnitudes, even when their prime divisors belong to a fixed\nfinite set, this argument does not supply the same later congruence.\nIt therefore does not settle that case.\n\n\\section{Periodic negative errors}\\label{long243:sec:periodic}\n\nLet the negative error have periodic magnitude $e_n=-E_n>0$, so\n$e_{n+h}=e_n$ and $C_{n+1}=C_n+e_n$. Summing over a period gives\n$C_{n+h}=C_n+M$, where the positive integer\n$M=\\sum_{j=0}^{h-1}e_j$ is the \\emph{drift}. Periodicity makes this\nsum independent of the starting index.\n\nAt $h=1$ this says that the magnitude is constant and that $M$ is its value,\nwhich is the situation of Section~\\ref{long243:sec:constant};\nTheorem~\\ref{long243:res:constant} already excludes it without using\nthe pointwise bound $e_n<a_n$ in the proof below. The new case is $h\\ge2$.\n\nFor a constant error, the proof used the finitely many prime divisors of\nits magnitude. For a periodic error we use the prime divisors of the\nincrease $M$ over one period. The required divisibility facts are as\nfollows. Every multiplier $a_j$ divides each later denominator, by\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{382}{base_dvd_denState_later}{persistence\nof a multiplier}. If a prime divides both $D_n$ and $a_n$, the numerator\nupdate makes it divide $C_{n+1}$, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{402}{divisorLocksTailSucc}{one-step\ndivisibility implication}. Once it divides both numerator and denominator,\nit divides every later numerator, denominator and error, by\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{429}{divisorLock_persists}{persistence\nof a common divisor}.\n\nPeriodicity carries this divisibility back to every phase. Suppose\n$d\\mid M$ and $d\\mid a_i,a_j$ with $i<j$. Then $d\\mid D_j$, and\n$C_{j+1}=a_jC_j-D_j$ gives $d\\mid C_{j+1}$. The common divisor\npersists in both sequences and in every later error. For any phase $r$,\nchoose $k$ with $r+kh\\ge j+1$; periodicity gives\n$d\\mid e_{r+kh}=e_r$. Taking $kh\\ge j+1$ also gives\n$d\\mid C_{kh}=C_0+kM$, hence $d\\mid C_0$. This is\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{494}{repeatedDivisor_forces_periodicCommonScale}{the repeated-divisor implication}; it uses the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{461}{periodic_add_mul}{period relation}\nand the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{476}{phaseDrift_add_mul}{increase over successive periods}, each iterated through whole periods.\nA repeated prime can therefore be divided out of the entire system.\nThis explains the induction on $M$ in the proof.\n\n\\begin{theorem}[no periodic negative magnitude]\\label{long243:res:periodic}\nLet $a,D,C,e:\\N\\to\\N$ with $a_n\\ge2$, $e_n>0$ and $e_n<a_n$ for every $n$,\nsatisfying\n\\[\n D_{n+1}=a_nD_n,\\qquad C_{n+1}=C_n+e_n,\\qquad D_n+e_n=(a_n-1)C_n ,\n\\]\nand suppose $e_{n+h}=e_n$ and $C_{n+h}=C_n+M$ for some $h>0$ and $M>0$.  This\nis impossible.\n\\end{theorem}\n\n\\begin{proof}\nWe use strong induction on the drift $M$. A prime divisor of $M$ that\nrecurs among the multipliers need not give an immediate contradiction.\nIt instead divides the whole orbit, allowing us to reduce $M$ by that\nprime and apply the induction hypothesis.\n\nSuppose first that no prime divisor of $M$ is a common divisor of $C_0$ and of\nevery magnitude.  The repeated-divisor implication applies in the\ncontrapositive: a prime divisor of $M$ occurring in two of the multipliers\nwould be exactly such a common divisor, so each prime divisor of $M$ occurs in\nat most one multiplier.  On the other hand, every multiplier shares a prime with $M$.\nIndeed, if $\\gcd(a_j,M)=1$, choose $k\\ge1$ so that\n$a_j\\mid C_j+kM=C_{j+kh}$. The same multiplier divides $D_{j+kh}$,\nand periodicity gives $e_{j+kh}=e_j$. The identity at $j+kh$\ntherefore implies $a_j\\mid e_j$, contrary to $0<e_j<a_j$.\nThus infinitely many multipliers would require distinct prime divisors\nof the fixed integer $M$, a contradiction.\n\nOtherwise some prime $p$ divides $M$, divides $C_0$, and divides every\nmagnitude.  Then $p\\mid C_n$ for every $n$, since $C_{n+1}=C_n+e_n$ and both\nsummands on the right are divisible by $p$, and the same identity\n$D_n+e_n=(a_n-1)C_n$ then gives $p\\mid D_n$ for every $n$.  Divide $D$, $C$ and\n$e$ by $p$.  The multipliers are untouched, so $a_n\\ge2$ and $e_n<a_n$ persist\nand the magnitudes stay positive; the three recurrences are homogeneous in\n$(D,C,e)$ and so survive; the period is still $h$; and the drift becomes\n$M/p<M$.  The inductive hypothesis applies.\n\\end{proof}\n\n\\noindent The first of the two cases above is the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{548}{no_phasePrimitivePeriodicNegative_orbit}{exclusion when no prime of $M$ divides all phases}, the induction is the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{690}{no_periodicNegative_orbit}{exclusion\nat every scale}, and the eventual form is the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{802}{no_eventuallyPeriodicNegative_orbit}{eventual\nexclusion}.\n\nThe bound $e_n<a_n$ rules out $a_j\\mid e_j$. It follows eventually,\nwithout rescaling: on an infinite orbit with periodic positive\nmagnitudes, bounded errors and exponential denominator growth suffice.\nThe identity gives $C_n>0$, and periodicity gives\n\\[\n C_n=\\frac{M}{h}n+O(1),\\qquad \\sup_ne_n<\\infty,\n\\]\nand $D_0$ cannot be zero: otherwise all $D_n$ vanish and the defining identity gives $e_n=(a_n-1)C_n\\ge C_n$, contradicting these bounds.\nThus $D_n\\ge2^nD_0$, and\n\\[\n a_n-1=\\frac{D_n+e_n}{C_n}\\longrightarrow\\infty.\n\\]\nConsequently $e_n<a_n$ eventually. Applying the eventual exclusion\ntherefore rules out periodic positive magnitudes without an independent\nsmallness assumption. The numbered theorem retains the pointwise bound\nused by its checked phase-by-phase proof. The later bounded-negative\ntheorem also excludes this case.\n\n\\begin{remark}\\label{long243:rem:eventual-error-bound}\nFor the tail numerators of~\\cite{koizumi2025}, the absolute error\nsatisfies $|E_n|<a_n$ eventually. Indeed, under the dictionary in\nSection~\\ref{long243:sec:priorwork}, Lemma~4(2) gives\n$-C_n/2\\le E_n<C_n/2$, hence $|E_n|\\le C_n/2$. The proof of Lemma~4(3) gives\n$C_{n+1}\\le\\tfrac32C_n$~\\cite[pp.~11--12]{koizumi2025}, hence\n$C_n\\le C_N(3/2)^{n-N}$ beyond a pseudo-greedy starting index $N$.\nOn the other hand, convergence of $\\sum1/a_n$ gives $a_n\\to\\infty$,\nand $a_{n+1}/a_n^2\\to1$ then gives $a_{n+1}\\ge2a_n$ eventually.\nThus $|E_n|/a_n\\to0$. On a negative-error tail this gives $e_n<a_n$.\n\nFor integer solutions of \\eqref{long243:eq:recurrences}, the same conclusion follows when\n$a_n>1$, $C_n>0$ and $E_n/C_n\\to0$ hold: the calculation in\nSection~\\ref{long243:sec:state} already derives the reciprocal series and\nits growth from these hypotheses. Summability is not an independent\nassumption in that setting. The bound $e_n<a_n$ is still only eventual,\nwhereas Theorem~\\ref{long243:res:periodic} states it at every index;\nits eventual form, cited above, is therefore the applicable one.\nFor a periodic magnitude, boundedness of $e_n$ and divergence of $a_n$\nalready suffice.\n\\end{remark}\n\n\\section{Bounded increases and coprimality}\\label{long243:sec:barrier}\n\nFor nonperiodic errors, the Chinese remainder theorem replaces the\nfixed period. It produces a block of consecutive forbidden values.\nAn unbounded numerator with bounded upward steps must enter that\nblock when it first crosses its lower end.\n\n\\begin{lemma}[shifted blocks of consecutive multiples]\\label{long243:res:crt}\nLet $m_0,\\ldots,m_{B-1}$ be pairwise coprime and at least $2$.  For every bound\nthere is a $t$ beyond it with $m_i\\mid t+i$ for each $i<B$.\n\\end{lemma}\n\n\\begin{proof}\nSolve $t\\equiv-i\\pmod{m_i}$ simultaneously by the Chinese\nremainder theorem. Adding a sufficiently large multiple of\n$\\prod_i m_i$ puts the solution beyond the prescribed bound.\n\\end{proof}\n\n\n\n\\noindent Lemma~\\ref{long243:res:crt} matches each modulus $m_i$ to its own\nmultiple $t+i$ inside a window of $B$ consecutive integers. Matchings of a set\nof integers to distinct multiples in an interval are the subject of\nErd\\H{o}s Problem~\\#650, solved by van Doorn, Li and Tang: for every set $S$\nof $k$ positive integers, every open interval of length $2\\max S$ contains\ndistinct multiples of at least $\\min(k,\\lceil 2\\sqrt{k}\\,\\rceil)$ elements of\n$S$, and this count is optimal \\cite[Theorem~2.1]{vandoornlitang2026}. Their\nextremal construction uses the same Chinese remaindering for moduli that need\nnot be pairwise coprime, with their Claim~3.2 supplying the divisibility of\nresidue differences by greatest common divisors that the generalised theorem\nrequires \\cite[pp.~4--5]{vandoornlitang2026}. The first-crossing argument\nbelow needs only the pairwise coprime case.\n\n\\begin{example}[a block of three consecutive multiples]\\label{long243:ex:crt}\nTake $B=3$ and $(m_0,m_1,m_2)=(3,4,5)$.  The congruences $t\\equiv0\\pmod3$,\n$t\\equiv3\\pmod4$ and $t\\equiv3\\pmod5$ hold exactly when $t\\equiv3\\pmod{60}$.\nAt $t=3$ the block is $3,4,5$ with $3\\mid3$, $4\\mid4$ and $5\\mid5$; at $t=63$\nit is $63,64,65$ with $3\\mid63$, $4\\mid64$ and $5\\mid65$.  A sequence of\nnatural numbers that starts at $0$, tends to infinity and rises by at most $3$\nat each step cannot step over the block $\\{3,4,5\\}$: it has a first value at\nleast $3$, that value is at most $5$, and every member of $\\{3,4,5\\}$ is\ndivisible by one of the three moduli.\n\\end{example}\n\n\\begin{theorem}[bounded increases and coprimality to earlier moduli]\\label{long243:res:barrier}\nLet $u:\\N\\to\\N$ tend to infinity with $u_{n+1}\\le u_n+B$ for a fixed integer $B\\ge1$.\nThen $u$ cannot remain coprime to infinitely many fresh pairwise coprime\nmoduli: there is no family of pairwise coprime $m_i\\ge2$, one for each index,\nsuch that $\\gcd(m_i,u_t)=1$ whenever $i<t$.\n\\end{theorem}\n\nThe CRT block must lie above the initial segment where some moduli\nare not yet available. No monotonicity of $u$ is assumed.\n\n\\begin{proof}\nChoose $m_0,\\ldots,m_{B-1}$ and use Lemma~\\ref{long243:res:crt} to find\n$t>\\max(u_0,\\ldots,u_B)$ with $m_i\\mid t+i$ for $0\\le i<B$.  Since\n$u_n\\to\\infty$, there is a first $n>B$ with $u_n\\ge t$.  Minimality and the\nrise bound give\n\\[\n t\\le u_n\\le u_{n-1}+B<t+B.\n\\]\nHence $u_n=t+i$ for some $0\\le i<B<n$, so $m_i\\mid u_n$.  The inequality\n$i<n$ now permits the avoidance hypothesis, which says\n$\\gcd(m_i,u_n)=1$; this contradicts $m_i\\ge2$.  The argument uses the first\ncrossing of the CRT block, not monotonicity of $u$.\n\\end{proof}\n\nOnly unboundedness above is used to obtain this first crossing.\nDivergence is stated because it holds for the reduced rational tail in\nthe application. This distinction also explains why the weighted proof\nin Section~\\ref{long243:sec:lcmrecords} works with an unbounded running\nmaximum, without assuming that its numerator tends to infinity.\n\n\nDivergence forces $u$ to reach the forbidden block, and the uniform rise\nbound prevents it from jumping over the block. A bound only along a\nsubsequence would leave the intervening steps unrestricted. Without a scale restriction on the moduli,\nProposition~\\ref{long243:res:variablerise} gives an unbounded coprime walk\nwith rises $o(\\log\\log u_n)$.  The same forbidden-block crossing,\nunder a two-sided bound on the error, appears in the proof of Bado's\nTheorem~5.1 \\cite[pp.~4--5]{bado2026}, posted in September 2026.\n\nIn the application, $m_i$ is the $i$th multiplier. Its coprimality is\nknown only for later numerators, which explains the condition $i<t$.\n\nTo apply the preceding theorem, we reduce $C_n/D_n$ to lowest terms\nand require that no further common factor be cancelled. In that case\nthe reduced numerators and denominators still satisfy the original\nrecurrences. The relevant coprimality is as follows.\n\n\\begin{proposition}[persistent coprimality]\\label{long243:res:reduced}\nSuppose that $\\gcd(u_n,v_n)=1$ at every index and\n\\[\n u_{n+1}+v_n=a_nu_n,\\qquad v_{n+1}=a_nv_n.\n\\]\nThen $\\gcd(a_n,v_n)=1$, the $a_n$ are pairwise coprime, and\n$\\gcd(a_i,u_t)=1$ whenever $i<t$.\n\\end{proposition}\n\n\\begin{proof}\nA common prime divisor of $a_n$ and $v_n$ would divide both\n$u_{n+1}=a_nu_n-v_n$ and $v_{n+1}=a_nv_n$, contradicting their\ncoprimality. Hence $\\gcd(a_n,v_n)=1$.\nFor $i<t$, the denominator recurrence gives $a_i\\mid v_t$.\nCombining this with $\\gcd(a_t,v_t)=1$ proves that $a_i$ and $a_t$\nare coprime; combining it with $\\gcd(u_t,v_t)=1$ proves that $a_i$\nand $u_t$ are coprime.\n\\end{proof}\n\n\\noindent These are the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{989}{reducedStep_coprime_currentFactor}{step\ncoprimality}, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1016}{reducedTail_pairwiseCoprime}{pairwise\ncoprimality}, and the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1030}{reducedTail_wholeModulusAvoidance}{coprimality to each earlier multiplier}. Applying Theorem~\\ref{long243:res:barrier}\nshows that a numerator satisfying these conditions cannot tend to infinity\nwith uniformly bounded upward increments. This is the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1041}{no_boundedRise_reducedTail}{reduced\nexclusion under bounded upward increments}, together with its eventual form, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1066}{no_eventuallyBoundedRise_reducedTail}{version with an eventual increment bound}.\n\n\\begin{example}[coprimality in Sylvester's sequence]\\label{long243:ex:reduced}\nPut $u_n=1$ and $v_n=D_n=1,2,6,42,1806,\\ldots$ as in\nExample~\\ref{long243:ex:sylvester}, with multipliers $a_n=v_n+1=2,3,7,43,1807,\\ldots$.\nThen $\\gcd(u_n,v_n)=1$, $u_{n+1}+v_n=1+v_n=a_nu_n$ and $v_{n+1}=a_nv_n$, so\nProposition~\\ref{long243:res:reduced} applies and gives the classical fact\nthat Sylvester's numbers are pairwise coprime.  Its numerator is\nconstant, so it satisfies every hypothesis of the exclusion just stated except\ndivergence; since the tail exists, that hypothesis cannot be dropped.\n\\end{example}\n\nThe fraction $C_n/D_n$ in Section~\\ref{long243:sec:state} need not be in lowest terms.\nTo use the preceding argument, we must show that the common factor stops\nchanging. One uniform bound on the negative magnitudes at arbitrarily\nlate indices suffices: each gcd divides every later gcd, and at a negative\nindex it also divides the nonzero magnitude. Thus a bound at arbitrarily\nlate negative indices bounds the entire gcd sequence. The next proposition\njustifies dividing by its eventual value.\n\n\\begin{proposition}[the tail gcd stabilises]\\label{long243:res:gcdstab}\nLet $a,D,C:\\N\\to\\N$ satisfy $C_{n+1}+D_n=a_nC_n$ and\n$D_{n+1}=a_nD_n$, and put $E_n=D_n-(a_n-1)C_n$. Suppose some fixed integer $B\\ge1$ satisfies\n$-B\\le E_n<0$ at infinitely many indices. Then\n$\\gcd(C_n,D_n)$ is eventually constant. Dividing $C_n,D_n$ by its\neventual value gives coprime sequences satisfying the same recurrences.\n\\end{proposition}\n\n\\begin{proof}\nFirst, $C_n>0$ at every index. If $C_n=0$, the nonnegative recurrence\nforces $C_{n+1}=D_n=0$, and both sequences then vanish at all later\nindices. This contradicts the existence of arbitrarily late negative\nerrors.\n\nThe two recurrences give $G_n\\mid G_{n+1}$ for\n$G_n=\\gcd(C_n,D_n)$, as in\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1430}{tailGcd_dvd_succ}{the gcd divisibility relation}.\nAlso $G_n=\\gcd(C_n,|E_n|)$, since $D_n=E_n+(a_n-1)C_n$;\nat a negative index this is\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1587}{tailGcd_eq_gcd_negativeMagnitude}{the gcd identity with the negative magnitude}.\nFor any $n$, choose $t\\ge n$ with $-B\\le E_t<0$. Then\n\\[\n G_n\\le G_t\\le |E_t|\\le B.\n\\]\nThe entire positive divisibility chain is therefore bounded and eventually\nconstant, by\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1605}{dvdChain_eventuallyConstant_of_cofinally_bounded}{stabilisation of a bounded divisibility chain}.\nIn Lean this is the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1636}{tailGcd_eventuallyConstant_of_cofinally_boundedNegative}{gcd stabilisation result}.\nDividing both sequences by the eventual value preserves the recurrences\nand gives coprime numerator and denominator at every later index.\n\\end{proof}\n\nOther negative errors may exceed $B$ in magnitude. Gcd stabilisation\nuses the bounded witnesses, whereas the CRT application needs a bound\non \\emph{every} upward increment.\n\nVanishing relative error also gives sparsity of the strict gcd increases.\nThe next proposition asserts long finite intervals of constancy, not\nconstancy on an infinite tail.\nFor $a,C,D:\\N\\to\\N$ satisfying \\eqref{long243:eq:recurrences} with $C_n>0$, put\n\\[\n G_n=\\gcd(C_n,D_n),\\qquad\n \\Gamma(N)=\\#\\{0\\le j<N:G_j<G_{j+1}\\}.\n\\]\n\n\\begin{proposition}[vanishing relative error makes strict gcd changes sparse]\n\\label{long243:res:gcdsparse}\nLet $a,C,D:\\N\\to\\N$ satisfy $C_{n+1}+D_n=a_nC_n$ and $D_{n+1}=a_nD_n$ with\n$C_n>0$, and put $E_n=D_n-(a_n-1)C_n$, $G_n=\\gcd(C_n,D_n)$ and\n$\\Gamma(N)=\\#\\{0\\le j<N:G_j<G_{j+1}\\}$.  If $|E_n|/C_n\\to0$, then\n$\\Gamma(N)=o(N)$.  Moreover, for every starting bound $B$ and block length\n$L$, some $n\\ge B$ satisfies\n\\[\n  G_n=G_{n+1}=\\cdots=G_{n+L}.\n\\]\n\\end{proposition}\n\n\\noindent The first assertion is the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{2186}{tailGcd_strictGrowthCount_sublinear_of_normalizedVanishes}{sublinear\nstrict-growth theorem}; the second is the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{2202}{tailGcd_exists_arbitrarilyLate_constBlock_of_normalizedVanishes}{arbitrarily\nlate constant-block theorem}. The proof first converts vanishing relative\nerror into subexponential growth of $C_n$. Each strict divisibility increase\nof $G_n$ contributes a factor of at least $2$, so\n$2^{\\Gamma(N)}G_0\\le G_N\\le C_N$. Taking logarithms gives\n$\\Gamma(N)=o(N)$. For $L=0$ the second assertion is immediate.\nFor $L\\ge1$, if every block of $L$ successive transitions beyond $B$\ncontained a strict increase, disjoint such blocks would give\n$\\Gamma(N)\\ge(N-B)/L-O(1)$, a contradiction.\n\nThe starting index may depend on the block length; arbitrarily long\nconstant blocks do not give an infinite constant tail.\nProposition~\\ref{long243:res:gcdstab} needs additional bounded negative\nwitnesses, which Proposition~\\ref{long243:res:gcdsparse} does not supply.\n\n\\papersectiontarget{243-long-bounded-increases}\n\\section{Bounded negative part}\\label{long243:sec:bounded}\n\nThe lower bound $E_n\\ge-B$ has two uses. At negative indices it\nbounds the gcd, making it stable. At every index it bounds the upward\nstep $C_{n+1}-C_n=-E_n$. We can therefore combine the preceding\ncoprimality and first-crossing arguments.\n\nWithout the denominator recurrence, $C_n=c+bn$, $E_n=-b$ would be\na counterexample for any positive integers $b,c$.\nTheorem~\\ref{long243:res:constant} excludes this exact-orbit behaviour.\nThe separate summability criterion in the next section needs only the\nscalar update. For positive integer solutions of \\eqref{long243:eq:recurrences} with vanishing relative error,\nboth criteria are equivalent to eventual zero error. Neither extra\nestimate has been derived from the unrestricted hypotheses.\n\n\\begin{theorem}[bounded negative part]\\label{long243:res:bounded}\nLet $a,C,D:\\N\\to\\N$ satisfy $a_n>1$, $C_n>0$, and\n\\[\n C_{n+1}+D_n=a_nC_n,\\qquad D_{n+1}=a_nD_n\n \\quad(n\\ge0).\n\\]\nPut $E_n=D_n-(a_n-1)C_n$. If\n\\[\n \\frac{E_n}{C_n}\\longrightarrow0\n \\quad\\text{and}\\quad E_n\\ge-B\\quad\\text{for all sufficiently large }n\n\\]\nfor some integer $B\\ge0$, then $E_n=0$ for all sufficiently large $n$.\n\\end{theorem}\nThe limit is equivalent, for positive $C_n$, to the integer condition\n$K|E_n|<C_n$ eventually for each $K\\in\\N$. This is the form used\nby the linked declaration. Its $K=1$ case gives $|E_n|<C_n$ eventually.\n\n\n\\begin{proof}\nSuppose that the error fails to vanish eventually. After the\npoint from which $|E_n|<C_n$ always holds, Theorem~\\ref{long243:res:absorb} excludes every zero. Hence $|E_n|\\ge1$,\nand the relative limit forces $C_n\\to\\infty$.\n\nThere are infinitely many negative indices, since otherwise\nTheorem~\\ref{long243:res:descent} would give eventual zero.\nAt each sufficiently late negative index, $1\\le-E_n\\le B$, so\n$B\\ge1$.\nProposition~\\ref{long243:res:gcdstab} makes $G_n=\\gcd(C_n,D_n)$\nstabilise to an integer $g>0$. The divided sequences\n$u_n=C_n/g$, $v_n=D_n/g$ satisfy the coprime recurrences, with\n$u_n\\to\\infty$ and $u_{n+1}-u_n=-E_n/g\\le B$.\nTheir coprimality properties from Proposition~\\ref{long243:res:reduced}\nnow contradict Theorem~\\ref{long243:res:barrier}.\n\\end{proof}\n\n\\begin{corollary}\\label{long243:res:cor}\nUnder the hypotheses of Theorem~\\ref{long243:res:bounded}, the multipliers\nsatisfy $a_{n+1}=a_n^{2}-a_n+1$ for all sufficiently large $n$.\n\\end{corollary}\n\n\\begin{proof}\nTheorem~\\ref{long243:res:bounded} gives eventual zero error. Since\n$C_{n+1}>0$, Theorem~\\ref{long243:res:eventual} gives the recurrence.\n\\end{proof}\n\nTo apply it to Problem~\\ref{long243:res:problem}, delete a finite prefix\nand use Koizumi's Corollary~3 \\cite[p.~9]{koizumi2025}. The remaining\nsequence is the pseudo-greedy expansion of its own reciprocal sum.\nHis Lemma~4 gives positive integer numerators $c_n$, denominators $d_n$\nand errors $e_n$ with\n\\[\n c_{n+1}=a_nc_n-d_n,\\qquad d_{n+1}=a_nd_n,\\qquad\n e_n=d_n-(a_n-1)c_n.\n\\]\nThe terms $a_n$ eventually exceed one, and $e_n/c_n\\to0$ by that\ncorollary. The same construction gives $-c_n/2\\le e_n<c_n/2$.\nThe additional condition still required is an eventual lower bound\n$e_n\\ge-B$. Neither Koizumi's construction nor the constant and periodic\nexclusions above establish this bound for every rational sum in the\nproblem. The result therefore remains conditional.\n\n\\papersectiontarget{243-long-relative-increase}\n\\section{Finite total relative increase}\\label{long243:sec:mass}\n\nWe can bound a positive integer sequence by multiplying the relative\nincreases of its terms. If their sum converges, that product is finite.\nOnly finitely many upward integer steps can then occur, and descent\nfinishes the argument. No denominator recurrence is needed.\nFor comparison, $C_n=n+1$ and $C_n=(n+1)^2$ have relative increases\ntending to zero but divergent sums. The eventual Sylvester recurrence gives an eventually\nconstant numerator, so it satisfies the summability condition.\n\n\\begin{samepage}\n\\begin{theorem}[finite sum of relative increases]\\label{long243:res:mass}\nLet \\(C_n\\in\\N_{>0}\\) and \\(E_n\\in\\Z\\) satisfy \\(C_{n+1}=C_n-E_n\\).\nIf\n\\[\n \\sum_{n=0}^{\\infty}\\frac{(-E_n)_+}{C_n}<\\infty,\n\\]\nthen \\(E_n=0\\) eventually. For an exact reciprocal-tail orbit this implies\n\\(a_{n+1}=a_n^2-a_n+1\\) eventually. No hypothesis $|E_n|/C_n\\to0$ is required.\n\\end{theorem}\n\\end{samepage}\n\n\\begin{proof}\nPut \\(\\delta_n=(-E_n)_+/C_n\\). The update gives\n\\(C_{n+1}\\le C_n(1+\\delta_n)\\), so\n\\[\n C_N\\le C_0\\prod_{n<N}(1+\\delta_n)\n \\le C_0\\exp\\!\\left(\\sum_n\\delta_n\\right).\n\\]\nChoose an integer upper bound \\(K\\) for \\(C_n\\). Each strict rise of the\ninteger sequence contributes at least \\(1/K\\) to \\(\\sum_n\\delta_n\\), so\nthere are only finitely many rises. The remaining positive integer sequence\nis nonincreasing and therefore stabilises, forcing \\(E_n=0\\). The recurrence\nfollows from Theorem~\\ref{long243:res:eventual}, since \\(C_n>0\\).\n\\end{proof}\n\nInteger increments suffice even for positive real $C_0$: the bounded\nvalues lie in the finite set $(C_0+\\Z)\\cap(0,K]$, so the same proof\nworks. The Lean statement retains integer-valued $C_n$.\n\nThe two indispensable features of this argument are positivity and\ndiscrete increments. With $C_n=1+1/(n+1)$ and\n$E_n=1/((n+1)(n+2))$, the numerator decreases forever with zero\nrelative-increase sum, but the errors are not integers. With $C_n=-n-1$\nand $E_n=1$, the increments are integers but positivity fails.\n\nThe pseudo-greedy gap version, with the same product bound and integer\ndescent, appears in the 12 August 2026 working report\n\\cite[``A global termination criterion'']{erdosproblemaday243}.\nBado additionally accounts for factors removed in LCM clearing\n\\cite[Thm.~11.1 and Remark~11.3, p.~9]{bado2026}. In the notation of\nSection~\\ref{long243:sec:lcmrecords}, the update gives\n\\[\n \\log U_N\\le\\log U_0+\n \\sum_{n<N}\\frac{(-E_n)_+}{C_n}-\\sum_{n<N}\\log\\rho_n.\n\\]\nHis sufficient condition is an upper bound on the difference of these\ntwo sums. The subtractive term records the decrease caused by a repeated\nfactor in the denominator. It may offset some relative increases; the\nhypothesis does not require either sum to converge separately. Unlike the\nscalar criterion above, this argument also uses the pseudo-greedy limit:\nbounded $U_n$ and $V_n/U_n=E_n/C_n\\to0$ make the integer $V_n$ eventually\nzero, after which $\\rho_nU_{n+1}=U_n$ gives eventual constancy.\n\nCorollary~3~\\cite[p.~9]{koizumi2025} and Lemma~4\n~\\cite[pp.~11--12]{koizumi2025} supply the same positive integer update\nafter a finite restart. Theorem~\\ref{long243:res:mass} therefore applies\nif the relative-increase sum converges. This is the sole additional\nhypothesis, asked for in Problem~\\ref{long243:res:masshyp}; the cited\nconstruction does not establish it.\n\n\\begin{example}[the product bound at a geometric rate]\\label{long243:ex:mass}\nSuppose $C_0=100$ and $(-E_n)_+/C_n\\le2^{-n-1}$ for every $n$, so that the\nsum of relative increases is at most $1$; the constraint at $n=0$ still permits\n$E_0$ as negative as $-50$.  Since $1+x\\le e^{x}$,\n\\[\n C_N\\le100\\prod_{n<N}\\bigl(1+2^{-n-1}\\bigr)\\le100e<272\n\\]\nfor every $N$. For all sufficiently large $n$, we have\n$2^{-n-1}<1/272$. A strict rise would then force\n$(-E_n)_+/C_n\\ge1/C_n>1/272$, a contradiction. Thus $C_n$ is eventually\nnonincreasing; as a positive integer sequence it stabilises, and $E_n=0$\nthereafter. The constant $272$ comes from the total mass and not from any single magnitude,\nwhich is why the hypothesis of Theorem~\\ref{long243:res:mass} is summability of\n$(-E_n)_+/C_n$ and not a bound on it.\n\\end{example}\n\n\\papersectiontarget{243-long-open-questions}\n\\section{Further questions}\\label{long243:sec:open}\n\nWe collect the restrictions on a possible counterexample and the\nestimates which would exclude it. The constant and periodic arguments\nconcern all-negative tails. They leave open a periodic pattern seen\nonly along the negative indices of a mixed-sign sequence. The\nunrestricted problem remains unresolved here.\n\n\\begin{proposition}[necessary conditions on a counterexample]\\label{long243:res:frontier}\nFor the integer tail attached to any counterexample to\nProblem~\\ref{long243:res:problem},\n\\[\n E_n\\ne0\\quad\\hbox{eventually},\\qquad\n \\frac{|E_n|}{C_n}\\longrightarrow0,\n\\]\nand\n\\[\n \\limsup_{\\substack{n\\to\\infty\\\\E_n<0}}(-E_n)=\\infty,\n \\qquad\n \\sum_{n=0}^{\\infty}\\frac{(-E_n)_+}{C_n}=\\infty .\n\\]\nIn particular, negative indices occur infinitely often.  Thus the remaining\nregime consists of unbounded negative errors along exact reciprocal tails for which\nthe sum of relative increases diverges.\n\\end{proposition}\n\n\\begin{proof}\nKoizumi's construction gives vanishing relative error after a finite\nshift, hence $|E_n|<C_n$ eventually.\nAbsorption then makes $E_n\\ne0$ eventually,\nsince eventual zero would give the Sylvester recurrence.  Descent excludes an\neventually nonnegative error.  Theorem~\\ref{long243:res:bounded} excludes a bounded\nnegative part, and Theorem~\\ref{long243:res:mass} excludes convergence of the sum of relative increases.  These statements are unchanged by deleting a finite prefix.\n\\end{proof}\n\n\n\nProposition~\\ref{long243:res:frontier} does not assert infinitely many\nprime divisors of the error magnitudes. Nor are its numerical conditions\nalone a reformulation of the problem: the numerator and denominator\nmust satisfy the exact recurrences. With those recurrences, positivity\nand vanishing relative error, Section~\\ref{long243:sec:state} supplies\nthe reciprocal-series realisation. The following exact-orbit question\nis therefore equivalent to the original problem.\n\n\\begin{problem}[unbounded negative errors with divergent relative sum]\\label{long243:res:excursions}\nExclude, or construct, natural-number sequences $(a,D,C)$ satisfying\n\\eqref{long243:eq:recurrences}, with $a_n>1$ and\n$C_n>0$, whose multipliers satisfy $\\lim_n a_{n+1}/a_n^{2}=1$ and whose error satisfies vanishing relative error, and which is\nnegative infinitely often with magnitudes unbounded along that infinite set and\n\\[\n \\sum_n\\frac{(-E_n)_+}{C_n}=\\infty .\n\\]\nSuch an orbit fails the bounded-negative-part and summable-relative-increase\nhypotheses.\nBy Proposition~\\ref{long243:res:frontier} the integer tail of a counterexample is\nsuch an orbit after deletion of a finite prefix, so an exclusion would settle\nProblem~\\#243. Conversely, a global orbit with the displayed properties\nhas $C_n/D_n\\to0$ by Section~\\ref{long243:sec:state}. The\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1955}{reciprocalSeries_hasSum_of_tailRatio_tendsto_zero}{reciprocal-series\nrealisation} then identifies its reciprocal sum as $C_0/D_0$.\nAfter deletion of a finite prefix the multipliers are strictly increasing,\nand the infinitely many negative errors rule out a Sylvester tail.\nIt is therefore a counterexample to Problem~\\#243. Finite admissible\nprefixes alone do not provide such a construction.\n\\end{problem}\n\n\\begin{remark}\\label{long243:rem:recurrence-without-relative-bound}\nThe recurrences alone admit unbounded negative errors for which\n$\\sum_n(-E_n)_+/C_n$ diverges. The following example fails both the\nrelative-error and growth assumptions of Problem~\\ref{long243:res:excursions}. Put\n\\[\n C_n=2^{n},\\qquad b_0=2,\\qquad b_{n+1}=\\tfrac12 b_n(b_n+2),\\qquad\n a_n=b_n+2,\\qquad D_n=b_nC_n ,\n\\]\nso that $(b_n)=2,4,12,84,3612,\\ldots$ and $(a_n)=4,6,14,86,3614,\\ldots$.  Every\n$b_n$ is a positive even integer, since $b_n=2k$ gives $b_{n+1}=2k(k+1)$, so all\nfour sequences are $\\N$-valued. They satisfy both recurrences:\n$C_{n+1}+D_n=2^{n}(2+b_n)=a_nC_n$ and\n$D_{n+1}=b_{n+1}2^{n+1}=b_n(b_n+2)2^{n}=a_nD_n$.  Its error is\n\\[\n E_n=D_n-(a_n-1)C_n=b_n2^{n}-(b_n+1)2^{n}=-C_n ,\n\\]\nso the negative magnitudes $2^{n}$ are unbounded and $\\sum_n(-E_n)_+/C_n$\ndiverges. Here $|E_n|=C_n$ at every index, so both the strict inequality\n$|E_n|<C_n$ and the vanishing relative error fail. The\nmultipliers satisfy $a_{n+1}=\\tfrac12(a_n^{2}-2a_n+4)$, so\n$a_{n+1}/a_n^{2}\\to\\tfrac12$ and the growth hypothesis fails as well.  The\nexample therefore says nothing about Problem~\\ref{long243:res:problem}, and it does not\nisolate the error bounds: $E_n=-C_n$ at every index forces that\nmultiplier recurrence, so growth fails with them.  It is recorded only to show\nthat the two recurrences alone do not force the required conclusion.\n\\end{remark}\n\n\\paragraph{A scalar profile need not satisfy the recurrences.}\nThe example $C_n=n^2+1$, $E_n=-(2n+1)$ satisfies the numerator update,\n$|E_n|/C_n\\to0$, $\\log C_n=o(n)$ and $\\sum_n(E_n/C_n)^2<\\infty$.\nIt is not an exact reciprocal-tail orbit. Indeed, $C_2=5$, $C_3=10$\nand $C_4=17$. The first numerator update forces $5\\mid D_2$; the\nmultiplicative denominator update gives $5\\mid D_3$; the next numerator\nupdate then forces $5\\mid C_4$, a contradiction. This is the failed-route\nexample referred to at the end of the short note: subexponential size\nand a small relative error do not supply arithmetic compatibility.\n\n\\subsection{Growth of the factors repeated in the denominator}\n\nTo compare repeated prime factors in the denominator with its total\nsize, define\n\\[\n L_0=D_0,\\qquad L_{n+1}=\\lcm(L_n,a_n),\\qquad M_n=\\frac{D_n}{L_n}.\n\\]\nThe integer $M_n$ records prime factors counted repeatedly in\n$D_n=D_0\\prod_{j<n}a_j$ but only to their largest exponent in $L_n$;\nthus $M_nL_n=D_n$. Problem~\\ref{long243:res:lcmheight} asks whether\nfailure of the Sylvester recurrence forces exponential growth of $M_n$,\ncontradicting its subexponential upper bound. The following recovery\nestimate is local and does not supply that lower bound.\n\n\\subsection{Cancellation during recovery intervals}\n\nThe LCM quotient $M_n$ need not equal the common divisor $G_n$ removed\nby reduction to lowest terms. Since $M_n$ divides both $C_n$ and $D_n$,\nwe have $M_n\\mid G_n$, but equality is not assumed. The following\nestimate concerns the successive reduction factors, over an interval\nwhere the reduced numerator regains its starting value.\nIn the calculation below, $h_n$ is the factor removed at step $n$,\n$c_n$ is a positive integer with $c_n^2\\mid h_n$, and $\\tilde e_n$\ndenotes the signed reduced error, as in Section~\\ref{long243:sec:records}.\nFor the estimate itself, let $u,h,c:\\N\\to\\N$ and $\\tilde e:\\N\\to\\Z$ satisfy\n\\[\n h_nu_{n+1}=u_n-\\tilde e_n,\n \\qquad u_n>0,\n \\qquad h_n>0.\n\\]\nFix $K,r,L\\in\\N$ with $K,L>0$, assume\n\\[\n K|\\tilde e_{r+i}|<u_{r+i}\\quad(0\\le i<L),\n \\qquad u_r\\le u_{r+L},\n\\]\nand let $R$ be a finite family of moduli.  Suppose every $m_q$, $q\\in R$,\ndivides $\\prod_{r\\le n<r+L}c_n$, and that\n$c_n^2\\mid h_n$ throughout the interval.  Then\n\\begin{equation}\\label{long243:eq:repair-entropy}\n K^L\\lcm(m_q:q\\in R)^2<(K+1)^L.\n\\end{equation}\nTo see this, divide the recurrence by $u_n$ and multiply over the interval:\n\\[\n \\left(\\prod_{r\\le n<r+L}h_n\\right)\\frac{u_{r+L}}{u_r}\n =\\prod_{r\\le n<r+L}\\left(1-\\frac{\\tilde e_n}{u_n}\\right)\n <(1+1/K)^L.\n\\]\nRecovery makes the endpoint ratio at least $1$, giving\n$\\prod_{r\\le n<r+L}h_n<(1+1/K)^L$. Without $u_r\\le u_{r+L}$,\na small endpoint ratio could conceal a large cancellation product.\nThe least common multiple of the $m_q$ divides the product of the\n$c_n$, and $c_n^2\\mid h_n$ at every step. Its square therefore divides,\nand is at most, the positive integer $\\prod_{r\\le n<r+L}h_n$.\nMultiplying by $K^L$ proves the claimed bound.\nThis is the\n\\lword{Erdos243/RepairEntropy.lean}{48}\n      {repairedFamily_recovery_energy_divisionFree}\n      {bound on cancellation over a recovery interval}.\n\nThe square in~\\eqref{long243:eq:repair-entropy} comes from the assumption\n$c_n^2\\mid h_n$. If the least common multiple of the chosen moduli\nis at least $2^{|R|}$, then\n\\[\n K^L4^{|R|}<(K+1)^L.\n\\]\nThe formal consequence is\n\\lword{Erdos243/RepairEntropy.lean}{78}\n      {repaired_card_bound_of_independent}\n      {the bound on the number of independent moduli}.\nTaking logarithms gives the explicit bound\n\\[\n \\frac{|R|}{L}<\\frac{\\log(1+1/K)}{\\log4}\n \\le\\frac1{K\\log4}.\n\\]\nThus the number of these moduli is small relative to the recovery length\nwhen the relative-error bound is small. This statement requires the lower bound\n$2^{|R|}$ on their least common multiple; it does not apply to an arbitrary\nfamily with repeated prime factors.\n\nFor a fixed number of steps, the same estimate has a simpler consequence:\na sufficiently late recovery cannot include any cancellation.  If\n$K|\\tilde e_n|<u_n$ eventually for every $K$, then for each fixed $L>0$ there is an\n$N$ such that every recovery $u_r\\le u_{r+L}$ with $r\\ge N$ satisfies\n\\[\n \\prod_{0\\le i<L}h_{r+i}=1.\n\\]\nIndeed, choose $K$ so large that $(1+1/K)^L<2$ and apply the same\nproduct estimate beyond its error threshold. The positive integer\nproduct is then smaller than $2$, so it equals $1$.\nThe same $K$ works for every length $1\\le j\\le L$, since\n$(1+1/K)^j\\le(1+1/K)^L<2$. Thus, after a sufficiently late step with\n$h_n>1$, the numerator cannot regain its starting value at any of the\nnext $L$ indices. This includes a recovery followed by another fall\nbefore the last endpoint. The fixed-length conclusion is formalised as\n\\lword{Erdos243/RepairEntropy.lean}{107}\n      {eventually_recoveryPayment_eq_one_of_fixedLength}\n      {absence of late cancellation during recovery intervals of fixed length}.\nThe threshold $N$ depends on $L$: late cancellations require longer\nrecoveries, but have not been excluded. A global contradiction needs\ncontrol of those varying lengths and a lower bound on the moduli's\nleast common multiple; the interval estimate supplies neither.\n\n\\begin{problem}[growth of repeated denominator factors]\\label{long243:res:lcmheight}\nFor every rational-tail orbit satisfying the hypotheses but not the conclusion of Problem~\\ref{long243:res:problem},\nmust\n\\[\n \\limsup_{n\\to\\infty}\\frac{\\log M_n}{n}>0?\n\\]\nEquivalently, must there be a $K\\ge1$ for which\n\\[\n 2^n\\le M_n^K\n\\]\nat infinitely many indices?\n\\end{problem}\n\nThe two displayed formulations are equivalent up to changing the\npositive constant; the second is not a weaker target.\nSection~\\ref{long243:sec:lcmrecords} already gives\n$1\\le M_n\\le C_n$, hence $\\log M_n/n\\to0$ on every orbit under\nconsideration. A positive answer would therefore be a contradiction,\nnot a further compatible necessary condition. It would prove\nProblem~\\#243. The missing step is to force exponential growth of $M_n$\nfrom failure of the Sylvester recurrence.\n\nThere is also a more local-looking question whose content is nevertheless the\nentire prefix.  Put $A_n=\\prod_{j<n}a_j$, so $D_n=D_0A_n$.  From\n\\[\n D_0A_n=(a_n-1)C_n+E_n\n\\]\none immediately obtains\n\\[\n \\gcd(A_n,a_n-1)\\mid E_n.\n\\]\nThe checked\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{2170}{tailState_subexponential_of_normalizedVanishes}{tail-height estimate}\nand vanishing relative error make $|E_n|$ subexponential in $n$.\n\n\\begin{problem}[common factors of the prefix and $a_n-1$]\\label{long243:res:prefixgcd}\nIn every rational-tail orbit satisfying the hypotheses but not the conclusion of Problem~\\ref{long243:res:problem}, is\n\\[\n \\limsup_{n\\to\\infty}\n \\frac{\\log\\gcd(A_n,a_n-1)}{n}>0?\n\\]\nEquivalently, do there exist $\\eta>0$ and infinitely many $n$ such that\n$\\gcd(A_n,a_n-1)\\ge e^{\\eta n}$?\n\\end{problem}\n\n\\noindent By Proposition~\\ref{long243:res:frontier}, the error is nonzero\neventually. The displayed divisibility therefore bounds\n$\\gcd(A_n,a_n-1)$ by $|E_n|$ at every sufficiently late index.\nA positive answer contradicts the subexponential bound on that error.\nArbitrarily long locally admissible blocks do\nnot answer this question: the gcd uses the complete prefix.\n\n\\subsection{The direct analytic question}\n\n\\begin{problem}[summability from rationality]\\label{long243:res:masshyp}\nLet $(a_n)$ satisfy the hypotheses of Problem~\\ref{long243:res:problem},\nand let $(D_n,C_n,E_n)$ be its integer tail. Must\n\\[\n \\sum_{n=0}^{\\infty}\\frac{(-E_n)_+}{C_n}<\\infty ?\n\\]\n\\end{problem}\n\n\\noindent An affirmative answer closes Problem~\\#243 by\nTheorem~\\ref{long243:res:mass}. Since the summands tend to zero and\n$\\log(1+t)\\sim t$, convergence is equivalent to bounded partial products\n$\\prod_{n<N}(1+(-E_n)_+/C_n)$. A negative answer requires the full\ngrowth and rationality hypotheses, not merely a locally admissible orbit.\nBy Theorem~\\ref{long243:res:weightedrecord}, finiteness\nof~\\eqref{long243:eq:weightedgrowth} for one nonincreasing weight with\ndivergent integral and one fixed $B\\ge0$ also suffices. The equivalent\nrecord-only sum in Theorem~\\ref{long243:res:weightedrecord} weights the\npositive excess of each actual upward jump over $B$ by $f(U_n)$.\n\n\\begin{remark}\\label{long243:rem:relative-error-boundary}\nOne further question is not left open here, since it is answered\nin~\\cite{koizumi2025}: whether vanishing relative error follows from\n$a_{n+1}/a_n^{2}\\to1$ and rationality.  It does.  After\ndeleting a finite prefix, Koizumi's Corollary~3\n~\\cite[p.~9]{koizumi2025} makes the sequence the pseudo-greedy expansion of its\nown reciprocal sum and gives $\\varepsilon_n\\to0$ under exactly the hypotheses of\nProblem~\\ref{long243:res:problem}; rationality is not needed for that step, only\nsummability of $\\sum1/a_n$.  The matched-index dictionary in\nSection~\\ref{long243:sec:priorwork} gives $\\varepsilon_n=E_n/C_n$\non the restarted tail. For a rational sum, Lemma~4(2)\n~\\cite[pp.~11--12]{koizumi2025} also gives\n$-c_n/2\\le e_n<c_n/2$ at every index of that expansion. Thus\n$|E_n|<C_n$ holds beyond the restart, as required by\nTheorem~\\ref{long243:res:bounded}; it need not hold at every original\nindex. The eventual inequality $|E_n|<C_n$ also follows directly from vanishing\nrelative error with $K=1$. What remains unsupplied is the eventual\nlower bound on $E_n$ in Theorem~\\ref{long243:res:bounded}.\nThat additional hypothesis is why the theorem is still conditional.\n\\end{remark}\n\n\\subsection{Unbounded upward increments}\n\n\\begin{proposition}[small increases when the prime moduli are sparse]\\label{long243:res:variablerise}\nThere exist strictly increasing primes $p_i$ and a strictly increasing\npositive integer sequence $u_n\\to\\infty$ such that $\\gcd(u_n,p_i)=1$ for\nall $i,n$ and\n\\[\n u_{n+1}-u_n=O\\bigl(\\sqrt{\\log\\log(u_n+e^e)}\\bigr)\n             =o\\bigl(\\log\\log(u_n+3)\\bigr).\n\\]\nThus the bounded-increment hypothesis of\nTheorem~\\ref{long243:res:barrier} cannot be replaced by an\n$o(\\log\\log u_n)$ bound without a quantitative restriction on the\nmoduli.\n\\end{proposition}\n\n\\begin{proof}\nChoose increasing primes\n$p_i>\\max\\{\\exp(\\exp((i+2)^2)),2^{i+3}\\}$, for $i\\ge0$.\nPrimes of arbitrarily large size suffice; no distribution theorem is used.\nThen $\\theta=\\sum_i1/p_i<1/4$ and\n$k(z)=\\#\\{i:p_i\\le z\\}\\le\\sqrt{\\log\\log z}$ whenever $z$ is large.\nFor integer $x$ put\n$L(x)=\\lceil4\\sqrt{\\log\\log(x+e^e)}+8\\rceil$.\nSince $L(x)=o(x)$, eventually $L(x)>k(x+L(x))/(1-\\theta)$.\nThe elementary window bound of\nProposition~\\ref{long243:res:coprimalitycap} leaves an integer in\n$[x,x+L(x))$ divisible by no $p_i$. The set of such integers is therefore\nunbounded. Enumerate it increasingly as $(u_n)$ and apply the same bound\nwith $x=u_n+1$ to obtain $u_{n+1}-u_n\\le L(u_n+1)$.\nFor prime moduli nondivisibility is exactly coprimality, proving every\nclaim. Sparse moduli defeat the abstract coprimality argument, not the exact\ntail recurrences. The next theorem instead fixes the moduli's\ndouble-exponential scale.\n\\end{proof}\n\nWe now keep the full family of moduli, rather than discarding a prefix\nas in Proposition~\\ref{long243:res:coprimalitycap}. The resulting\ncoefficient depends on $\\sigma=\\prod_j(1-1/m_j)$. For a finite prefix,\nthe corresponding product is exactly the proportion of admissible\nresidue classes by the Chinese remainder theorem. This explains the\nconstant in the statement before the limiting argument is made.\n\n\\begin{theorem}[largest gaps between integers avoiding given multiples]\\label{long243:res:gapconstant}\nLet $m_0<m_1<\\cdots$ be pairwise coprime integers at least $2$ with\n$\\ell(m_j)=j+O(1)$, where $\\ell(x)=\\log_2\\log_2\\max(4,x)$.\nLet $\\sigma=\\prod_j(1-1/m_j)>0$ and enumerate the positive integers\ndivisible by no $m_j$ in increasing order as $(u_n)$. Then\n\\[\n \\limsup_{n\\to\\infty}\\frac{u_{n+1}-u_n}{\\ell(u_n)}=\\sigma^{-1}.\n\\]\nThis is a statement about avoidance of multiples of whole moduli. It is not a statement\nabout coprimality to composite $m_j$, or about integer tail orbits.\n\\end{theorem}\n\n\\begin{proof}\nWrite $k(z)=\\#\\{j:m_j\\le z\\}=\\ell(z)+O(1)$. For a fixed prefix length\n$T$, set\n\\[\n M_T=\\prod_{j<T}m_j,\\qquad\n \\sigma_T=\\prod_{j<T}(1-1/m_j),\\qquad\n \\theta_T=\\sum_{j\\ge T}1/m_j.\n\\]\nHere $\\theta_T\\to0$. By the Chinese remainder theorem, avoiding the first\n$T$ whole moduli selects precisely the fraction $\\sigma_T$ of the\nresidues modulo $M_T$.\n\nFor the upper bound, fix $T$ before sending the interval to infinity,\nso the prefix error $M_T$ stays constant. An integer interval\n$[x,x+L)$ contains at least\n$\\sigma_TL-M_T$ integers avoiding that prefix. The remaining moduli\ncover at most $L\\theta_T+k(x+L)$ integers. Choose $T$ large enough that\n$\\sigma_T>\\theta_T$, and then any\n$c>(\\sigma_T-\\theta_T)^{-1}$.\nFor $L=\\lceil c\\ell(x)\\rceil$, the number left is positive for all\nsufficiently large $x$, since $k(x+L)=\\ell(x)+O(1)$.\nThus the set being enumerated is unbounded, and every sufficiently late\nsuch interval meets it. Applying this at $x=u_n+1$ gives\n\\[\n \\limsup_n\\frac{u_{n+1}-u_n}{\\ell(u_n)}\n \\le(\\sigma_T-\\theta_T)^{-1}.\n\\]\nLetting $T\\to\\infty$ gives the upper bound $\\sigma^{-1}$.\n\nFor the lower bound, fix $T$ and let the integer length $L$ tend to\ninfinity. Among the offsets $0\\le j<L$, precisely\n$K_L=\\sigma_TL+O(M_T)$ avoid the first $T$ moduli.\nAssign to these offsets distinct moduli\n$m_T,\\ldots,m_{T+K_L-1}$. Solve simultaneously\n$x\\equiv0\\pmod{M_T}$ and $x\\equiv-j$ modulo the modulus assigned to $j$.\nPut $Q_L=\\prod_{i<T+K_L}m_i$ and choose this solution in $[Q_L,2Q_L)$.\nEvery integer of $[x,x+L)$ is then divisible by a modulus: either an old\none or its assigned new one. Let $u$ be the greatest admissible integer\nbelow $x$; its successor is at least $x+L$, so the gap is at least $L$.\nThe scale assumption gives\n\\[\n \\ell(2Q_L)\\le T+K_L+O(1),\n\\]\nbecause $\\log_2m_i=2^{i+O(1)}$ and these upper bounds sum geometrically.\nSince $u<2Q_L$, the ratio for this gap is at least\n$L/(T+K_L+O(1))$, tending to $\\sigma_T^{-1}$.\nThese $u$ tend to infinity: $x\\ge Q_L\\to\\infty$ and admissible integers\nare unbounded. Hence the limsup is at least $\\sigma_T^{-1}$ for every\n$T$. Let $T\\to\\infty$ to finish.\n\\end{proof}\n\nFor the Fermat moduli $m_j=2^{2^j}+1$, pairwise coprimality and the\nfinite-product identity\n\\[\n \\prod_{j=0}^{N}\\left(1-\\frac1{2^{2^j}+1}\\right)\n =\\frac{2^{2^{N+1}-1}}{2^{2^{N+1}}-1}\n\\]\ngive $\\sigma=1/2$, so the coefficient is exactly $2$.\nThese results describe gaps between integers avoiding fixed moduli.\nThey do not construct multipliers or numerator and denominator sequences\nsatisfying the exact recurrences. In particular, the static proportion\n$\\sigma$ must not be substituted for $\\varphi(v_T)/v_T$: the first\nexcludes multiples of the whole moduli, while the second excludes every\nprime factor of the reduced denominator.\n\n\\subsection*{Formalisation targets}\n\n\\paragraph{Erd\\H{o}s--Straus.}\nFormalise Theorem~3 of~\\cite{erdosstraus1964}, with its prefix-LCM quotient\nand correctly indexed growth factor, under the published analytic hypotheses.\n\n\\paragraph{Duverney.}\nFormalise the absolute-convergence form of Corollary~3.2\nof~\\cite{duverney2001}, including its signed numerators, and recover the\nall-positive specialisation used for comparison here. For the printed\ncondition interpreted as mere signed convergence, first justify the\nnonvanishing-product step discussed in\nSection~\\ref{long243:sec:priorwork}. That interpretation is not treated\nas an already verified formalisation target.\n\n\\pdfbookmark[1]{Statements and declarations}{statements-declarations}\n\\subsection*{Statements and declarations}\n\n\\paragraph{Artefact and data availability.} The fixed toolchain,\nLean sources and evidence record are described in\nAppendix~\\ref{long243:app:index}.\n\n\\paragraph{Funding and competing interests.} This work received no external\nfunding.  The author declares no competing interests.\n\n\\paragraph{Acknowledgements.} The problem numbering follows\nBloom's Erd\\H{o}s Problems catalogue~\\cite{erdosproblems}.\n\n\\appendix\n\\section{Verification and reproducibility}\\label{long243:app:index}\n\\evidenceparagraph{\\evidencerecordurl}\nThe fixed toolchain, library manifest and Lean sources are in the\n\\href{\\sourceurl}{public repository}. The evidence record identifies\nthe exact declarations, source versions and Comparator status, including\nsource-pin and index gaps. The external formal-conjecture entry states\nthe problem without proving it. The printed square-specialisation proof\nuses Chebotarev, while the formal proof uses a Dedekind-zeta pole comparison;\nthe shared conclusion does not check every step of the printed argument.\nNo new Lean build, Comparator replay or independent review was performed\nfor this revision.\n\n\nThe recurrence identities\nand arithmetic exclusions are in \\path{ReciprocalTailRigidity.lean}, the\nscalar convergence theorem is in \\texttt{SparseResetRecovery.lean}, and\nthe bounds on cancelled factors are in \\texttt{RepairEntropy.lean}.\nThe Lean statement excluding a periodic error assumes $e_n<a_n$;\nSection~\\ref{long243:sec:periodic} derives this bound eventually for a\nperiodic positive error magnitude. The printed bounded-negative theorem uses $E_n/C_n\\to0$. The\nformal statement uses its division-free form, $K|E_n|<C_n$ eventually\nfor each natural $K$; the bound $|E_n|<C_n$ is its $K=1$ case.\nSections~\\ref{long243:sec:priorwork} and~\\ref{long243:sec:bounded} give the\nconstruction from a rational reciprocal sum.\n\n\\papersectiontarget{243-long-indexing}\n\\subsection*{Indexing of the regular-rate theorems}\n\\noindent\\emph{Index translation for the Lean statements.} Both Lean\ndeclarations use $b:\\N\\to\\N$, including a positive $b_0$, whereas the\ntheorems above start at $a_1$. A simple shift changes $\\lambda/n$ by\norder $n^{-2}$ and may lose the stated $o(n^{-\\lambda})$ precision.\nInstead choose a\nlarge $N$ with $a_N>N$. Such an $N$ exists: strict increase gives\n$a_n\\ge n$ and the rate gives $a_n^2/a_{n+1}<2$ eventually, hence\n$a_{n+1}>a_n^2/2\\ge n^2/2>n+1$ for large $n$. Define\n\\[\n b_n=\\begin{cases}n+1,&0\\le n<N,\\\\a_n,&n\\ge N.\\end{cases}\n\\]\nThen $b$ is positive and strictly increasing, with the same indexed rate\nfor every $n\\ge N$. Moreover\n\\[\n \\sum_{n\\ge0}\\frac1{b_n}-\\sum_{n\\ge1}\\frac1{a_n}\n =\\sum_{n=0}^{N-1}\\frac1{n+1}-\\sum_{n=1}^{N-1}\\frac1{a_n}\\in\\Q.\n\\]\nThe series converge by the stated rapid-growth rate. If the original\nsum were rational, so would be the sum for $b$, contradicting the\ncorresponding zero-indexed Lean theorem. This finite-prefix implication\nis proved here in ordinary mathematics; the Lean declarations check the\nzero-indexed theorems themselves.\n\n\\subsection*{The square-specialisation proofs}\nLemma~\\ref{long243:res:squarespec} and its consequences,\nProposition~\\ref{long243:res:transportsquare},\nTheorem~\\ref{long243:res:cubicexclusion} and\nTheorem~\\ref{long243:res:cubicrate}, are formalised without Chebotarev.\nThe printed proof of Lemma~\\ref{long243:res:squarespec} uses that theorem;\nits Lean proof instead uses the simple pole of the Dedekind zeta function.\nThe zero-indexed Lean statement of Theorem~\\ref{long243:res:cubicrate}\nrequires the finite-prefix bridge above.\n\nThe nonintegral-rate formal proof uses the real-parameter extraction\ntheorem and a quantitative canonical-tail estimate. Convergence of the\nreciprocal series is derived from the rate.\n\n\\subsection*{Recurrence declarations}\n\\noindent\\emph{Proposition~\\ref{long243:res:update}.} \\lword{Erdos243/ReciprocalTailRigidity.lean}{57}{nextTailState_eq_sub_centered}{update\nlaw}.\n\n\\noindent\\emph{Proposition~\\ref{long243:res:scale}.} \\lword{Erdos243/ReciprocalTailRigidity.lean}{65}{nextDenState_scale}{denominator},\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{70}{nextTailState_scale}{numerator}\nand\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{75}{centeredState_scale}{error}.\n\n\\noindent\\emph{Theorem~\\ref{long243:res:defect}.} \\lword{Erdos243/ReciprocalTailRigidity.lean}{1775}{sylvesterDefect_mul_nextTailState}{defect\nidentity}.\n\n\\noindent\\emph{Theorem~\\ref{long243:res:step}.} \\lword{Erdos243/ReciprocalTailRigidity.lean}{1787}{sylvesterNext_eq_of_centered_zero}{local\nrigidity step}.\n\n\\noindent\\emph{Theorem~\\ref{long243:res:eventual}.} \\lword{Erdos243/ReciprocalTailRigidity.lean}{1805}{sylvesterNext_eventually_of_centered_zero}{eventual\nSylvester recurrence}.\n\n\\noindent\\emph{Theorem~\\ref{long243:res:absorb}.} \\lword{Erdos243/ReciprocalTailRigidity.lean}{2227}{centeredState_zero_absorbing}{absorption\nof a vanishing error}, with the companion\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{2254}{eventually_nonzero_of_zero_absorbing}{contrapositive\nalong a tail}.\n\n\\noindent\\emph{Theorem~\\ref{long243:res:descent}.} \\lword{Erdos243/ReciprocalTailRigidity.lean}{1843}{centeredState_eventually_zero}{stabilisation\nof the nonnegative error}, using the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1825}{antitone_nat_eventually_constant}{eventual\nconstancy of a nonincreasing sequence}.\n\n\\noindent\\emph{Lemma~\\ref{long243:res:crt}.} \\lword{Erdos243/ReciprocalTailRigidity.lean}{839}{exists_shifted_consecutiveMultiples}{shifted\nconsecutive multiples}.\n\n\\noindent\\emph{Theorem~\\ref{long243:res:barrier}.} \\lword{Erdos243/ReciprocalTailRigidity.lean}{903}{no_boundedRise_of_tailAvoidance}{the Chinese-remainder first-crossing consequence}.\n\n\\noindent\\emph{Theorem~\\ref{long243:res:bounded}.} \\noindent The theorem is\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{2360}{eventuallyBoundedNegativePart_eventually_zero}{stated with an eventual lower bound and division-free vanishing relative error}.\nThe Lean proof shifts past both thresholds and applies the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{2271}{boundedNegativePart_eventually_zero}{version with $|E_n|<C_n$ and the lower bound at every index}.\nIt uses\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1739}{tailState_tendsto_atTop_of_nonzero_normalizedVanishes}{divergence from vanishing relative error}\nand the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1754}{no_cofinallyBoundedNegative_of_normalizedVanishes}{exclusion for bounded upward increments and arbitrarily late bounded negative magnitudes}, via\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1661}{no_cofinallyBoundedNegative_of_tailTendsto}{its formulation with divergence as a hypothesis}.\nBoth exclusion statements require a uniform bound on all upward\nincrements. Bounded negative magnitudes at arbitrarily late indices\nsuffice for gcd stabilisation, but not for the jump bound in the CRT\nargument.\n\n\\noindent\\emph{Theorem~\\ref{long243:res:mass}.} \\noindent The scalar conclusion is\n\\lword{Erdos243/SparseResetRecovery.lean}{612}{eventually_zero_of_summable_negativeRelativeMass_scalar}{the eventual vanishing for a positive integer sequence};\nits exact-orbit consequence is\n\\lword{Erdos243/SparseResetRecovery.lean}{690}{sylvesterNext_eventually_of_summable_negativeRelativeMass_scalar}{the scalar recurrence from a convergent sum}.\nA separate exact-orbit formulation retains the denominator and tail update\nequations, positivity, the centered-step identity and division-free normalized\nvanishing. Under $\\sum_n(-E_n)_+/C_n<\\infty$, the checked results give\n\\lword{Erdos243/SparseResetRecovery.lean}{378}{tail_growth_le_one_add_negativeRelativeMass}{a bound on numerator growth},\n\\lword{Erdos243/SparseResetRecovery.lean}{488}{eventually_zero_of_summable_negativeRelativeMass}{eventual zero error}, and\n\\lword{Erdos243/SparseResetRecovery.lean}{510}{sylvesterNext_eventually_of_summable_negativeRelativeMass}{the Sylvester recurrence}.\nThese are conditional hypotheses, not properties proved here for the original\norbit.\nThe \\texttt{plectis-erdos-lean} repository contains a\n\\href{https://github.com/wcook04/plectis-erdos-lean/blob/52f29ad173b04e3bac941b3663f2b9aebe5de0bb/ErdosProblems/Erdos243/PaperCompleteR8/CanonicalNegativeMass.lean\\#L19}{formulation for the rational-tail numerators using a convergent sum},\nnot covered by the evidence record. Summability remains\nan assumption; the elementary scalar implication needs neither growth nor\nvanishing relative error.\n\n\\noindent\\emph{Proposition~\\ref{long243:res:frontier}.} \\noindent In the Lean statement of Proposition~\\ref{long243:res:frontier},\nthe divergent sum of relative increases is recorded as divergence of the\npartial sums.\n\n% The exhaustive source manifest is retained in the authored TeX for automated\n% validation, and kept outside the printed note.\n\\iffalse\n\n\\begin{tabular}{@{}ll@{}}\ninformal statement & linked Lean source\\\\\nSylvester successor & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{37}{sylvesterNext}\\\\\ndenominator update & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{41}{nextDenState}\\\\\ntail update & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{45}{nextTailState}\\\\\nerror & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{49}{centeredState}\\\\\nSylvester defect & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{53}{sylvesterDefect}\\\\\nupdate law & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{57}{nextTailState_eq_sub_centered}\\\\\ndenominator equivariance & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{65}{nextDenState_scale}\\\\\ntail equivariance & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{70}{nextTailState_scale}\\\\\ncentred equivariance & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{75}{centeredState_scale}\\\\\nnormalised exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{100}{no_normalizedConstantNegative_orbit}\\\\\nscale-primitive exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{147}{no_scalePrimitiveConstantNegative_orbit}\\\\\nconstant exclusion at every scale & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{286}{no_constantNegative_orbit}\\\\\neventual constant exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{350}{no_eventuallyConstantNegative_orbit}\\\\\npersistence of a multiplier & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{382}{base_dvd_denState_later}\\\\\none-step lock & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{402}{divisorLocksTailSucc}\\\\\npersistence of a lock & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{429}{divisorLock_persists}\\\\\niterated period & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{461}{periodic_add_mul}\\\\\niterated drift & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{476}{phaseDrift_add_mul}\\\\\nrepeated-divisor transport & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{494}{repeatedDivisor_forces_periodicCommonScale}\\\\\nphase-primitive exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{548}{no_phasePrimitivePeriodicNegative_orbit}\\\\\nperiodic exclusion at every scale & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{690}{no_periodicNegative_orbit}\\\\\neventual periodic exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{802}{no_eventuallyPeriodicNegative_orbit}\\\\\nshifted consecutive multiples & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{839}{exists_shifted_consecutiveMultiples}\\\\\nobstruction to bounded upward increments & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{903}{no_boundedRise_of_tailAvoidance}\\\\\nstep coprimality & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{989}{reducedStep_coprime_currentFactor}\\\\\npairwise coprimality & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1016}{reducedTail_pairwiseCoprime}\\\\\navoidance of multiples of whole moduli & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1030}{reducedTail_wholeModulusAvoidance}\\\\\nreduced exclusion under bounded upward increments & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1041}{no_boundedRise_reducedTail}\\\\\neventual reduced exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1066}{no_eventuallyBoundedRise_reducedTail}\\\\\ngcd monotonicity & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1430}{tailGcd_dvd_succ}\\\\\ngcd at a negative index & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1587}{tailGcd_eq_gcd_negativeMagnitude}\\\\\nchain stabilisation & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1605}{dvdChain_eventuallyConstant_of_cofinally_bounded}\\\\\ngcd stabilisation & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1636}{tailGcd_eventuallyConstant_of_cofinally_boundedNegative}\\\\\ntail-divergence form & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1661}{no_cofinallyBoundedNegative_of_tailTendsto}\\\\\ndivergence from vanishing relative error & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1739}{tailState_tendsto_atTop_of_nonzero_normalizedVanishes}\\\\\nexclusion from vanishing relative error & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1754}{no_cofinallyBoundedNegative_of_normalizedVanishes}\\\\\ndefect identity & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1775}{sylvesterDefect_mul_nextTailState}\\\\\nlocal rigidity step & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1787}{sylvesterNext_eq_of_centered_zero}\\\\\neventual Sylvester recurrence & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1805}{sylvesterNext_eventually_of_centered_zero}\\\\\neventual constancy & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1825}{antitone_nat_eventually_constant}\\\\\nstabilisation of the nonnegative error & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1843}{centeredState_eventually_zero}\\\\\ntail realisation & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1869}{natTail_eq_nextTailState}\\\\\ndenominator realisation & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1881}{natDen_eq_nextDenState}\\\\\nsigned update law & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1978}{natTail_eq_sub_centeredState}\\\\\nabsorption of a vanishing error & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{2227}{centeredState_zero_absorbing}\\\\\ncontrapositive along a tail & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{2254}{eventually_nonzero_of_zero_absorbing}\\\\\nvanishing under a bound on the negative part & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{2271}{boundedNegativePart_eventually_zero}\\\\\neventual vanishing under a bound on the negative part & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{2360}{eventuallyBoundedNegativePart_eventually_zero}\\\\\nbound on cancellation over a recovery interval & \\lrefx{Erdos243/RepairEntropy.lean}{48}{repairedFamily_recovery_energy_divisionFree}\\\\\nindependent-repair bound & \\lrefx{Erdos243/RepairEntropy.lean}{78}{repaired_card_bound_of_independent}\\\\\nfixed-length reset disappearance & \\lrefx{Erdos243/RepairEntropy.lean}{107}{eventually_recoveryPayment_eq_one_of_fixedLength}\\\\\nforced numerator & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{22}{forcedNumerator}\\\\\npolynomial congruence & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{26}{forcedNumerator_modEq}\\\\\nexact-division cancellation & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{41}{quotient_modEq_of_modEq_mul}\\\\\nhorizon modulus & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{53}{horizonModulus}\\\\\nascending-factorial form & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{59}{horizonModulus_eq_ascFactorial}\\\\\nfactorial value at the initial index & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{69}{horizonModulus_zero_eq_factorial}\\\\\nsurvival predicate & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{75}{ForcedSurvives}\\\\\nquotient transport & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{85}{forcedQuotient_modEq}\\\\\nshrinking-modulus transport & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{97}{forcedSurvives_iff_of_modEq}\\\\\nfactorial residue reduction & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{134}{forcedSurvives_iff_of_modEq_factorial}\\\\\n\\end{tabular}\n\\fi\n\n\n\\section{A factorial modulus for integral recursion}\\label{long243:app:residue}\n\nIn the case $m=c=1$ of Section~\\ref{long243:sec:constant}, the identity\n\\eqref{long243:eq:shape} becomes $D_n+1=(a_n-1)(n+1)$. It determines\nthe recursion $a_{n+1}=\\fnum(n,a_n)/(n+2)$, where\n\\[\n \\fnum(n,a)=(n+1)a^{2}-(n+2)a+(n+3),\n\\]\nthe\n\\lword{Erdos243/FiniteHorizonResidue.lean}{22}{forcedNumerator}{forced\nnumerator}. We stop at the first nonintegral quotient. The\n\\lword{Erdos243/FiniteHorizonResidue.lean}{75}{ForcedSurvives}{formal survival predicate}\nrequires exact division at each step. For $a_0=3$, as in\nExample~\\ref{long243:ex:shape}, the first quotient is\n$\\fnum(0,3)/2=3$, but the next quotient is $\\fnum(1,3)/3=13/3$,\nso the recursion stops. Direct iteration can produce very large\nintermediate integers. To\ndecide whether the first $h$ divisions are exact, however, it suffices\nto know the initial value modulo $(h+1)!$.\nComputing a pseudo-greedy orbit through residues modulo a shrinking product\nmodulus is Koizumi's method \\cite[Remark~2 and Algorithm~1,\npp.~13--14]{koizumi2025}; for the forced numerator that product is a\nfactorial.\n\n\\begin{theorem}[factorial residue reduction]\\label{long243:res:residue}\nLet $h$ be a nonnegative integer and let $a,b$ be integers with $a\\equiv b\\pmod{(h+1)!}$.\nThe first $h$ steps of the recursion\n$a_{n+1}=\\fnum(n,a_n)/(n+2)$, starting at index zero, are integral\nfor $a_0=a$ if and only if they are integral for $a_0=b$.\n\\end{theorem}\n\n\\begin{proof}\nEach exact division consumes a congruence factor. Retain the remaining\nfactors by setting $M(0,i)=1$ and $M(h+1,i)=(i+2)M(h,i+1)$. Thus\n$M(h,0)=(h+1)!$. We prove the stronger statement that, at index $i$,\ncongruent inputs modulo $M(h,i)$ give the same answer to whether the\nnext $h$ quotients are all integral.\nThere is nothing to prove for $h=0$.\n\nFor the inductive step, suppose\n$a\\equiv b\\pmod{(i+2)M(h,i+1)}$. Since $\\fnum(i,\\cdot)$ is an integer\npolynomial, its values at $a$ and $b$ are congruent modulo that product.\nIn particular, one is divisible by $i+2$ if and only if the other is.\nIf neither is divisible, both recursions stop. Otherwise their quotients\nare congruent modulo $M(h,i+1)$, so the induction hypothesis applies\nto the remaining $h$ updates at index $i+1$. Taking $i=0$ proves the claim.\n\\end{proof}\n\n\\noindent The formal\n\\lword{Erdos243/FiniteHorizonResidue.lean}{134}{forcedSurvives_iff_of_modEq_factorial}{factorial\nresidue reduction} uses the\n\\lword{Erdos243/FiniteHorizonResidue.lean}{97}{forcedSurvives_iff_of_modEq}{shrinking-modulus\ninduction},\n\\lword{Erdos243/FiniteHorizonResidue.lean}{26}{forcedNumerator_modEq}{polynomial\ncongruence}, and\n\\lword{Erdos243/FiniteHorizonResidue.lean}{41}{quotient_modEq_of_modEq_mul}{cancellation\nafter exact division}. Its modulus is identified by the\n\\lword{Erdos243/FiniteHorizonResidue.lean}{59}{horizonModulus_eq_ascFactorial}{ascending-factorial\nformula} and its\n\\lword{Erdos243/FiniteHorizonResidue.lean}{69}{horizonModulus_zero_eq_factorial}{factorial\nvalue at the initial index}.\n\nFor $h=1$, the modulus is $2!=2$. The first quotient is integral\nif and only if $2\\mid a^2-2a+3$, or equivalently if $a$ is odd.\nFor instance, $\\fnum(0,3)=6$ and $\\fnum(0,4)=11$. Thus the parity\nof $a$ decides the first step, as Theorem~\\ref{long243:res:residue}\nasserts in this case.\n\nExact enumeration for $2\\le a_0<5000$ gives a maximum of $17$\nsuccessful updates, or $18$ multiplier values including $a_0$.\nNine seeds attain it, the least being $1501$. The seed $a_0=1$ is excluded:\nit gives $a_n=1$ at every step and violates the hypothesis $a_n\\ge2$.\nThe test certifies exact division, not a positive orbit. Even a\nsurviving seed for every finite horizon would not supply one integer\nseed surviving forever. Theorem~\\ref{long243:res:constant} proves the\ninfinite exclusion; the residue reduction also applies to other\npolynomial-division recurrences.\n% ; - part family_catalogue ; -\n\\section{Result map and proof dependencies}\\label{sec:erdos-243-complete-family-map}\nBeyond the short paper, this companion supplies signed-series\ncomparisons (Section~\\ref{long243:sec:priorwork}), integer coefficients\n(Section~\\ref{long243:sec:lcmrecords}), general $1+c/n$ rates\n(Section~\\ref{long243:sec:records}), and regular-rate extraction with\npositive-density cubic exclusion (Section~\\ref{long243:sec:cubicrate}).\nTheorem~\\ref{long243:res:recorddichotomy} gives the double-logarithmic\nbound; Section~\\ref{long243:sec:open} includes the scalar\nfailed-divisibility example. The principal dependencies are:\n\n\\paragraph{Bounded upward increments.}\nVanishing relative error gives eventual absorption. Bounded negative\nvalues at arbitrarily late indices give a stable gcd; CRT then excludes\nthe unbounded reduced tail with bounded upward increments. The increment\nbound is an extra hypothesis.\n\n\\paragraph{A convergent sum of relative increases.}\nThe inequality $C_{n+1}\\le C_n(1+(-E_n)_+/C_n)$ turns summability into a\nuniform bound on $C_n$. Integrality then permits only finitely many rises,\nafter which the numerator stabilises.\nNo vanishing relative error is needed.\n\n\\paragraph{New maxima and prime-power persistence.}\nThe amplified error bounds cancellation, protecting large primes\nfor Theorem~\\ref{long243:res:recordamplified}. Density-one coprimality\nand CRT height control give the double-logarithmic bound. LCM tails\ngive weighted crossings without primitive numerators. For reduced\ntails, Corollary~\\ref{long243:res:powerpersistence} protects a prime\npower below its numerator threshold, allowing crossings to be counted\nbefore that protection fails.\n\n\\paragraph{Cubic rate.}\nThe tail estimate and finite differences of the Gamma ratio force an\neventual cubic polynomial for the integer numerator. The recurrence forces a modular square condition, which the square-specialisation\nlemma lifts to the cubic field. The ordinary proof uses Chebotarev; the Lean\nproof obtains the same implication from a Dedekind-zeta pole comparison.\nThe trace calculation leaves $m=12$, and both possible signs are excluded\nmodulo seven.\n\n\\paragraph{Avoidance of fixed moduli.}\nFor sufficiently sparse prime moduli, a coprime integer sequence can\nhave unbounded but very small increments. At the specified\ndouble-exponential scale, counting residue classes and applying CRT give\nthe largest gaps between integers divisible by none of the whole moduli.\nThese auxiliary results do not construct reciprocal-tail examples.\n\n\\paragraph{What remains.}\nFor the unrestricted problem, one still needs the stated summability or\nrecord bound, or a contradiction to the necessary conditions on a\ncounterexample. The static examples and local identities supply neither.\n\n% ; - part back ; -\n\\begin{thebibliography}{99}\n\\small\n\\raggedright\n\\setlength{\\itemsep}{1pt}\n\\bibitem{erdosstraus1964} P.~Erd\\H{o}s and E.~G. Straus,\n  \\href{https://users.renyi.hu/~p_erdos/1964-19.pdf}{\\emph{On the irrationality\n  of certain Ahmes series}},\n  J. Indian Math. Soc. (N.S.) \\textbf{27} (1964), 129--133. MR~175848.\n\\bibitem{duverney2001} D.~Duverney,\n  \\href{https://www.ms.u-tokyo.ac.jp/journal/pdf/jms080206.pdf}\n  {\\emph{Irrationality of fast converging series of rational numbers}},\n  J. Math. Sci. Univ. Tokyo \\textbf{8} (2001), 275--316. MR~1837165.\n\\bibitem{badea1993} C.~Badea,\n  \\href{https://matwbn.icm.edu.pl/ksiazki/aa/aa63/aa6342.pdf}{\\emph{A theorem\n  on irrationality of infinite series and applications}}, Acta Arith.\n  \\textbf{63} (1993), no.~4, 313--323,\n  doi:\\href{https://doi.org/10.4064/aa-63-4-313-323}{10.4064/aa-63-4-313-323}.\n\\bibitem{tijdemanyuan2002} R.~Tijdeman and P.~Yuan,\n  \\emph{On the rationality of Cantor and Ahmes series},\n  Indag. Math. (N.S.) \\textbf{13} (2002), no.~3, 407--418,\n  doi:\\href{https://doi.org/10.1016/S0019-3577(02)80018-0}{10.1016/S0019-3577(02)80018-0}.\n\\bibitem{erdosgraham1980} P.~Erd\\H{o}s and R.~L. Graham,\n  \\href{https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf}{\\emph{Old\n  and New Problems and Results in Combinatorial Number Theory}},\n  Monogr. Enseign. Math. 28, Geneva, 1980, p.~64.\n\\bibitem{erdos1988} P.~Erd\\H{o}s,\n  \\href{https://users.renyi.hu/~p_erdos/1988-22.pdf}{\\emph{On the irrationality\n  of certain series: problems and results}}, in A.~Baker (ed.), \\emph{New\n  Advances in Transcendence Theory}, Cambridge UP, 1988, pp.~102--109,\n  doi:\\href{https://doi.org/10.1017/CBO9780511897184.009}{10.1017/CBO9780511897184.009}.\n\\bibitem{kovactao2024} V.~Kova\\v{c} and T.~Tao,\n  \\href{https://arxiv.org/abs/2406.17593v4}{\\emph{On several irrationality\n  problems for Ahmes series}}, Acta Math. Hungar. \\textbf{175} (2025), no.~2,\n  572--608,\n  doi:\\href{https://doi.org/10.1007/s10474-025-01528-0}{10.1007/s10474-025-01528-0};\n  arXiv:\\href{https://arxiv.org/abs/2406.17593v4}{2406.17593v4}.\n  Page references are to arXiv:2406.17593v4.\n\\bibitem{bado2026} I.~O. Bado,\n  \\emph{Prime-Support Rigidity and Primitive Pseudo-Greedy Dynamics:\n  Partial Progress on Erd\\H{o}s Problem~\\#243}, preprint posted September 2026,\n  doi:\\href{https://doi.org/10.13140/RG.2.2.36612.08325}\n  {10.13140/RG.2.2.36612.08325}.\n\\bibitem{erdosproblemaday243} P.~White with Claude (Anthropic),\n  \\href{https://erdosproblemaday.com/report/243}{\\emph{Erd\\H{o}s \\#243:\n  working report}}, Erd\\H{o}s Problem a Day, page dated 12 August 2026.\n  AI-assisted, unrefereed working report.\n\\bibitem{stevenhagenlenstra1996} P.~Stevenhagen and H.~W. Lenstra, Jr.,\n  \\href{https://pub.math.leidenuniv.nl/~lenstrahw/PUBLICATIONS/1994c/art.pdf}\n  {\\emph{Chebotar\\\"ev and his density theorem}}, Math. Intelligencer\n  \\textbf{18} (1996), no.~2, 26--37,\n  doi:\\href{https://doi.org/10.1007/BF03027290}{10.1007/BF03027290}.\n  Page references are to the linked author version dated 23 March 1995.\n\\bibitem{vandoornlitang2026} W.~van Doorn, Y.~Li and Q.~Tang,\n  \\href{https://arxiv.org/abs/2603.28636}{\\emph{Optimal bounds for an\n  Erd\\H{o}s problem on matching integers to distinct multiples}},\n  preprint arXiv:2603.28636v1, 2026.\n\\bibitem{koizumi2025} J.~Koizumi,\n  \\href{https://math.colgate.edu/~integers/aa28/aa28.pdf}\n  {\\emph{Irrationality of the reciprocal sum of doubly exponential sequences}},\n  Integers \\textbf{26} (2026), Paper No.~A28, 17 pp.,\n  doi:\\href{https://doi.org/10.5281/zenodo.18714404}{10.5281/zenodo.18714404}.\n  Locators refer to this published version.\n\\bibitem{erdosproblems} T.~F. Bloom,\n  \\href{https://www.erdosproblems.com/243}{\\emph{Erd\\H{o}s Problem \\#243}},\n  accessed 28 July 2026.\n\\bibitem{formalconjectures243} The Formal Conjectures Authors,\n  \\href{https://github.com/google-deepmind/formal-conjectures/blob/f776d2f2039351b00737ffcafb9d7d7666e1d9af/FormalConjectures/ErdosProblems/243.lean}\n  {\\emph{FormalConjectures.ErdosProblems.\\texttt{243}}},\n  Lean source at commit \\texttt{f776d2f}, 2025. Formal problem statement,\n  not a proof.\n\\bibitem{erdosstraus1974} P.~Erd\\H{o}s and E.~G. Straus,\n  \\href{https://msp.org/pjm/1974/55-1/pjm-v55-n1-p08-s.pdf}\n  {\\emph{On the irrationality of certain series}}, Pacific J. Math.\n  \\textbf{55} (1974), no.~1, 85--92.\n\\bibitem{hancltijdeman2008} J.~Han\\v{c}l and R.~Tijdeman,\n  \\href{https://www.impan.pl/shop/en/publication/transaction/download/product/82186}\n  {\\emph{On the irrationality of polynomial Cantor series}},\n  Acta Arith. \\textbf{133} (2008), no.~1, 37--52,\n  doi:\\href{https://doi.org/10.4064/aa133-1-3}{10.4064/aa133-1-3}.\n  Locators refer to the published version.\n\\bibitem{duverneykurosawashiokawa2020} D.~Duverney, T.~Kurosawa and I.~Shiokawa,\n  \\href{https://danielduverney.fr/documents/theorie-des-nombres/Tsukuba.pdf}\n  {\\emph{Irrationality exponents of certain fast converging series of rational\n  numbers}}, Tsukuba J. Math. \\textbf{44} (2020), no.~2, 235--250,\n  doi:\\href{https://doi.org/10.21099/tkbjm/20204402235}{10.21099/tkbjm/20204402235}.\n  Theorem locators follow the linked 14-page author version.\n\\bibitem{crmarickovac2025} T.~Crmari\\'{c} and V.~Kova\\v{c},\n  \\href{https://arxiv.org/abs/2504.18712v1}\n  {\\emph{On the irrationality of certain super-polynomially decaying series}},\n  Colloq. Math. \\textbf{179} (2025), 55--68,\n  doi:\\href{https://doi.org/10.4064/cm9628-5-2025}{10.4064/cm9628-5-2025}.\n  Theorem locators follow arXiv:2504.18712v1.\n\\bibitem{kouli2020} A.~Koutsoukou-Argyraki and W.~Li,\n  \\href{https://isa-afp.org/entries/Irrational_Series_Erdos_Straus.html}\n  {\\emph{Irrationality Criteria for Series by Erd\\H{o}s and Straus}},\n  Archive of Formal Proofs, 12 May 2020. An Isabelle/HOL formalisation;\n  the archive entry identifies the results formalised.\n\\bibitem{dlmf_gamma} National Institute of Standards and Technology,\n  \\href{https://dlmf.nist.gov/5.11.E12}{\\emph{Digital Library of Mathematical\n  Functions}, \\S5.11(iii), formula 5.11.12}, accessed 16 September 2026.\n\\bibitem{elabdalaoui2015} E.~H. el Abdalaoui, M.~Lema\\'nczyk and T.~de la Rue,\n  \\emph{A dynamical point of view on the set of $\\mathcal B$-free integers},\n  Int. Math. Res. Not. IMRN (2015), no.~16, 7258--7286,\n  doi:\\href{https://doi.org/10.1093/imrn/rnu164}{10.1093/imrn/rnu164}.\n  The cited definition is also in \\S1.2 of\n  arXiv:\\href{https://arxiv.org/abs/1311.3752v3}{1311.3752v3}.\n\\end{thebibliography}\n% ---- part family_catalogue ----\n% ---- part back ----\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":206995,"sha256":"sha256:92512a9aa35a79c5e4d2fb5690c56912a7a7dff9a4a244e356b1c2e886b37bac","cite_as":"Plectis reader corpus, paper:erdos243-reciprocal-tail-reasoning-surface","source_repository":"plectis-erdos","source_path":"paper/243/erdos243-reciprocal-tail-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:088d36d7ba3d3ee0cee2df99e8915927999698f19889f8ea0b8d55b3721ec778","byte_count":6923,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% This is the only copy of the shared preamble.  The eight short notes, the\n% long records assembled from paper/reasoning-parts other than #249 and #257\n% (which carry their own preambles), and the synthesis paper include it with\n% \\input{problem-note-preamble}; paper/Makefile builds from this directory\n% with the directory on the tectonic search path.\n%\n% Shared preamble for the Erdős Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erdős problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{8e58c395e9807ef3374d7365c62cf04d20ee6331}\n\\newcommand{\\commitshort}{9e231ce4371f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-erdos/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-erdos/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\usepackage{longtable}  % the notes' declaration tables; loaded here so the assembled records inherit it\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n% ---------------------------------------------------------------------------\n% Evidence marks.  paper-evidence.tex places \"Lean\" and \"Comparator\" links in\n% the margin beside each result from the paper's generated file\n% paper/evidence/<paper>.tex, which each paper inputs after this preamble.\n% ---------------------------------------------------------------------------\n\\input{paper-evidence}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Production provenance.  The wording is fixed by the corpus provenance\n% contract and reproduced verbatim in every manuscript in this repository, so\n% the papers cannot drift into different accounts of how they were made.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erdős problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every Lean mark in the margin points to propositions that the\n  pinned Lean kernel accepts, with no \\texttt{sorry}, no added axiom, and no\n  unchecked evaluation.  That claim concerns the formal statements only; it\n  says nothing about their mathematical interest, their novelty, or the\n  original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n\n% Compact opening card for the separately stated Comparator theorem selected\n% from each problem note.  The prose supplied by each note owns the\n% contribution and relevance judgements; the shared final sentence states the\n% mechanical assurance boundary once and points to the public evidence owner.\n\\newcommand{\\externalreviewobject}[3]{%\n  \\begin{quote}\\small\n  \\textbf{Contribution.} #1\\par\n  \\textbf{Relation to the open problem.} #2\\par\n  \\textbf{Executable review object.} #3  The repository's\n  external-verification job compares these exact Lean propositions with\n  separately declared challenge statements and an axiom budget, then asks\n  Lean's kernel to check the submitted proofs.  The\n  \\href{https://github.com/wcook04/plectis-erdos/blob/ca0e13f8acf5ccf48506e4bdb870953d3a0856fa/formalization.yaml}%\n  {formalisation manifest} and the commit-bound CI receipt record that check;\n  they do not assess novelty, significance, or whether the original problem is\n  solved.\n  \\end{quote}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  Under\n% Tectonic (XeTeX) the text and mathematics are set in TeX Gyre Pagella from\n% the bundled OpenType fonts, with a Heros sans range for \\mathsf and\n% Inconsolata for monospace; the pdflatex fallback keeps the same Palatino\n% design through newpx.  The A4 text block uses a classical measure with\n% quiet running heads and restrained ink.  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For these papers that inference is false, and\n% a declaration reached only in the back matter would leave it standing for the\n% length of the paper.  The note therefore appears on page one, small and\n% visually subordinate to the title and abstract: enough to prevent the false\n% inference, not enough to make the production method the subject.\n%\n% It carries no mark, because it qualifies the whole manuscript rather than any\n% one passage.  \\footnotetext does not advance the footnote counter, and the\n% emptied \\thefootnote is confined to a group, so numbered footnotes elsewhere\n% in the paper are unaffected.\n%\n% The wording is not authored here.  It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"},{"record_type":"paper_source_dependency","tex_reference":"evidence/erdos243-reciprocal-tail-reasoning-surface","source_path":"paper/evidence/erdos243-reciprocal-tail-reasoning-surface.tex","media_type":"text/x-tex","sha256":"sha256:0104e889a8d89ec6b5c7921287e8d8fddc9f799fd9d620b1f8b3c2b5a3da5949","byte_count":37084,"licence":"CC-BY-4.0","body_format":"latex","body":"% Generated by scripts/paper_evidence.py from docs/paper_lean_coverage.json and the\n% Comparator replay receipts under evidence/comparator/.  Do not edit by hand.\n\\newcommand{\\evidencerecordurl}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md}\n\\DeclareResultEvidenceName{long243:res:cubicrate}{Theorem 2.1}\n\\DeclareResultEvidenceRecord{long243:res:cubicrate}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-cubicrate}{exact}{compared}\n\\DeclareResultEvidence{long243:res:cubicrate}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR21/SquareSpecialisationUnconditional.lean\\#L70}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-cubicrate-comparator}\n\\DeclareResultEvidenceName{long243:res:nonintegralrate}{Theorem 2.2}\n\\DeclareResultEvidenceRecord{long243:res:nonintegralrate}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-nonintegralrate}{exact}{pending}\n\\DeclareResultEvidence{long243:res:nonintegralrate}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR21/NonintegralRegularRate.lean\\#L196}{}\n\\DeclareResultEvidenceName{long243:res:cubicexclusion}{Theorem 2.3}\n\\DeclareResultEvidenceRecord{long243:res:cubicexclusion}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-cubicexclusion}{exact}{compared}\n\\DeclareResultEvidence{long243:res:cubicexclusion}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR21/SquareSpecialisationUnconditional.lean\\#L54}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-cubicexclusion-comparator}\n\\DeclareResultEvidenceName{long243:res:periodicobstruction}{Lemma 2.4}\n\\DeclareResultEvidenceRecord{long243:res:periodicobstruction}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-periodicobstruction}{exact}{compared}\n\\DeclareResultEvidence{long243:res:periodicobstruction}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR11/DensityTransport.lean\\#L156}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-periodicobstruction-comparator}\n\\DeclareResultEvidenceName{long243:res:gcdshape}{Lemma 2.5}\n\\DeclareResultEvidenceRecord{long243:res:gcdshape}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-gcdshape}{exact}{compared}\n\\DeclareResultEvidence{long243:res:gcdshape}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR21/CubicProfileGcdShape.lean\\#L103}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-gcdshape-comparator}\n\\DeclareResultEvidenceName{long243:res:reduciblecase}{Lemma 2.6}\n\\DeclareResultEvidenceRecord{long243:res:reduciblecase}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-reduciblecase}{exact}{compared}\n\\DeclareResultEvidence{long243:res:reduciblecase}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR11/PrimitiveMultiplierSupply.lean\\#L244}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-reduciblecase-comparator}\n\\DeclareResultEvidenceName{long243:res:squarespec}{Lemma 2.7}\n\\DeclareResultEvidenceRecord{long243:res:squarespec}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-squarespec}{exact}{pending}\n\\DeclareResultEvidence{long243:res:squarespec}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-squarespec}{}\n\\DeclareResultEvidenceName{long243:res:transportsquare}{Proposition 2.8}\n\\DeclareResultEvidenceRecord{long243:res:transportsquare}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-transportsquare}{exact}{compared}\n\\DeclareResultEvidence{long243:res:transportsquare}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR21/SquareSpecialisationUnconditional.lean\\#L39}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-transportsquare-comparator}\n\\DeclareResultEvidenceName{long243:res:scaletwelve}{Lemma 2.9}\n\\DeclareResultEvidenceRecord{long243:res:scaletwelve}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-scaletwelve}{exact}{compared}\n\\DeclareResultEvidence{long243:res:scaletwelve}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR20/ScaleTwelve.lean\\#L36}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-scaletwelve-comparator}\n\\DeclareResultEvidenceName{long243:res:modseven}{Lemma 2.10}\n\\DeclareResultEvidenceRecord{long243:res:modseven}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-modseven}{exact}{compared}\n\\DeclareResultEvidence{long243:res:modseven}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-modseven}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-modseven-comparator}\n\\DeclareResultEvidenceName{long243:res:extraction}{Lemma 2.11}\n\\DeclareResultEvidenceRecord{long243:res:extraction}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-extraction}{exact}{compared}\n\\DeclareResultEvidence{long243:res:extraction}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-extraction}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-extraction-comparator}\n\\DeclareResultEvidenceName{long243:res:tailratio}{Lemma 2.12}\n\\DeclareResultEvidenceRecord{long243:res:tailratio}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-tailratio}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long243:res:tailratio}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR7/QuantitativeTail.lean\\#L204}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-tailratio-comparator}\n\\DeclareResultEvidenceName{long243:res:update}{Proposition 4.1}\n\\DeclareResultEvidenceRecord{long243:res:update}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-update}{exact}{compared}\n\\DeclareResultEvidence{long243:res:update}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean\\#L57}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-update-comparator}\n\\DeclareResultEvidenceName{long243:res:scale}{Proposition 4.3}\n\\DeclareResultEvidenceRecord{long243:res:scale}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-scale}{exact}{compared}\n\\DeclareResultEvidence{long243:res:scale}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR7/Arithmetic.lean\\#L31}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-scale-comparator}\n\\DeclareResultEvidenceName{long243:res:defect}{Theorem 5.1}\n\\DeclareResultEvidenceRecord{long243:res:defect}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-defect}{exact}{compared}\n\\DeclareResultEvidence{long243:res:defect}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean\\#L1775}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-defect-comparator}\n\\DeclareResultEvidenceName{long243:res:step}{Theorem 5.3}\n\\DeclareResultEvidenceRecord{long243:res:step}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-step}{exact}{compared}\n\\DeclareResultEvidence{long243:res:step}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean\\#L1787}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-step-comparator}\n\\DeclareResultEvidenceName{long243:res:secondorder}{Theorem 5.4}\n\\DeclareResultEvidenceRecord{long243:res:secondorder}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-secondorder}{exact}{compared}\n\\DeclareResultEvidence{long243:res:secondorder}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR7/Arithmetic.lean\\#L42}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-secondorder-comparator}\n\\DeclareResultEvidenceName{long243:res:curvature}{Theorem 5.5}\n\\DeclareResultEvidenceRecord{long243:res:curvature}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-curvature}{exact}{compared}\n\\DeclareResultEvidence{long243:res:curvature}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/DynamicCancellation.lean\\#L309}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-curvature-comparator}\n\\DeclareResultEvidenceName{long243:res:oldmodulussaturation}{Lemma 5.6}\n\\DeclareResultEvidenceRecord{long243:res:oldmodulussaturation}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-oldmodulussaturation}{exact}{compared}\n\\DeclareResultEvidence{long243:res:oldmodulussaturation}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR21/ReducedStepLocalArithmetic.lean\\#L39}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-oldmodulussaturation-comparator}\n\\DeclareResultEvidenceName{long243:res:eventual}{Theorem 5.7}\n\\DeclareResultEvidenceRecord{long243:res:eventual}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-eventual}{exact}{compared}\n\\DeclareResultEvidence{long243:res:eventual}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean\\#L1805}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-eventual-comparator}\n\\DeclareResultEvidenceName{long243:res:absorb}{Theorem 5.8}\n\\DeclareResultEvidenceRecord{long243:res:absorb}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-absorb}{exact}{compared}\n\\DeclareResultEvidence{long243:res:absorb}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean\\#L2227}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-absorb-comparator}\n\\DeclareResultEvidenceName{long243:res:arithmeticrecord}{Theorem 6.1}\n\\DeclareResultEvidenceRecord{long243:res:arithmeticrecord}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-arithmeticrecord}{exact}{compared}\n\\DeclareResultEvidence{long243:res:arithmeticrecord}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR11/ArithmeticWeightedRecord.lean\\#L264}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-arithmeticrecord-comparator}\n\\DeclareResultEvidenceName{long243:res:weightedrecord}{Theorem 6.2}\n\\DeclareResultEvidenceRecord{long243:res:weightedrecord}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-weightedrecord}{exact}{compared}\n\\DeclareResultEvidence{long243:res:weightedrecord}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR11/CanonicalRecords.lean\\#L202}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-weightedrecord-comparator}\n\\DeclareResultEvidenceName{long243:res:valuationtransition}{Lemma 7.1}\n\\DeclareResultEvidenceRecord{long243:res:valuationtransition}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-valuationtransition}{exact}{compared}\n\\DeclareResultEvidence{long243:res:valuationtransition}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-valuationtransition}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-valuationtransition-comparator}\n\\DeclareResultEvidenceName{long243:res:powerpersistence}{Corollary 7.2}\n\\DeclareResultEvidenceRecord{long243:res:powerpersistence}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-powerpersistence}{exact}{compared}\n\\DeclareResultEvidence{long243:res:powerpersistence}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR20/PowerPersistence.lean\\#L27}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-powerpersistence-comparator}\n\\DeclareResultEvidenceName{long243:res:recorddichotomy}{Theorem 7.3}\n\\DeclareResultEvidenceRecord{long243:res:recorddichotomy}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-recorddichotomy}{exact}{compared}\n\\DeclareResultEvidence{long243:res:recorddichotomy}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-recorddichotomy}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-recorddichotomy-comparator}\n\\DeclareResultEvidenceName{long243:res:loglogboundary}{Corollary 7.4}\n\\DeclareResultEvidenceRecord{long243:res:loglogboundary}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-loglogboundary}{exact}{compared}\n\\DeclareResultEvidence{long243:res:loglogboundary}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-loglogboundary}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-loglogboundary-comparator}\n\\DeclareResultEvidenceName{long243:res:recordamplified}{Theorem 7.5}\n\\DeclareResultEvidenceRecord{long243:res:recordamplified}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-recordamplified}{exact}{compared}\n\\DeclareResultEvidence{long243:res:recordamplified}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-recordamplified}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-recordamplified-comparator}\n\\DeclareResultEvidenceName{long243:res:criticalrate}{Corollary 7.6}\n\\DeclareResultEvidenceRecord{long243:res:criticalrate}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-criticalrate}{exact}{compared}\n\\DeclareResultEvidence{long243:res:criticalrate}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-criticalrate}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-criticalrate-comparator}\n\\DeclareResultEvidenceName{long243:res:oddpowersupply}{Lemma 7.7}\n\\DeclareResultEvidenceRecord{long243:res:oddpowersupply}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-oddpowersupply}{exact}{compared}\n\\DeclareResultEvidence{long243:res:oddpowersupply}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-oddpowersupply}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-oddpowersupply-comparator}\n\\DeclareResultEvidenceName{long243:res:unitrecord}{Theorem 7.8}\n\\DeclareResultEvidenceRecord{long243:res:unitrecord}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-unitrecord}{exact}{compared}\n\\DeclareResultEvidence{long243:res:unitrecord}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-unitrecord}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-unitrecord-comparator}\n\\DeclareResultEvidenceName{long243:res:epochenergy}{Theorem 7.9}\n\\DeclareResultEvidenceRecord{long243:res:epochenergy}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-epochenergy}{exact}{compared}\n\\DeclareResultEvidence{long243:res:epochenergy}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-epochenergy}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-epochenergy-comparator}\n\\DeclareResultEvidenceName{long243:res:energycriterion}{Theorem 7.10}\n\\DeclareResultEvidenceRecord{long243:res:energycriterion}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-energycriterion}{exact}{compared}\n\\DeclareResultEvidence{long243:res:energycriterion}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-energycriterion}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-energycriterion-comparator}\n\\DeclareResultEvidenceName{long243:res:slownegative}{Theorem 7.11}\n\\DeclareResultEvidenceRecord{long243:res:slownegative}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-slownegative}{exact}{compared}\n\\DeclareResultEvidence{long243:res:slownegative}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-slownegative}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-slownegative-comparator}\n\\DeclareResultEvidenceName{long243:res:strausbounded}{Theorem 7.12}\n\\DeclareResultEvidenceRecord{long243:res:strausbounded}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-strausbounded}{exact}{compared}\n\\DeclareResultEvidence{long243:res:strausbounded}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-strausbounded}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-strausbounded-comparator}\n\\DeclareResultEvidenceName{long243:res:onethreshold}{Corollary 7.13}\n\\DeclareResultEvidenceRecord{long243:res:onethreshold}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-onethreshold}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long243:res:onethreshold}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-onethreshold}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-onethreshold-comparator}\n\\DeclareResultEvidenceName{long243:res:classicalhalfspace}{Proposition 7.14}\n\\DeclareResultEvidenceRecord{long243:res:classicalhalfspace}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-classicalhalfspace}{exact}{compared}\n\\DeclareResultEvidence{long243:res:classicalhalfspace}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-classicalhalfspace}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-classicalhalfspace-comparator}\n\\DeclareResultEvidenceName{long243:res:coprimalitycap}{Proposition 7.15}\n\\DeclareResultEvidenceRecord{long243:res:coprimalitycap}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-coprimalitycap}{exact}{compared}\n\\DeclareResultEvidence{long243:res:coprimalitycap}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-coprimalitycap}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-coprimalitycap-comparator}\n\\DeclareResultEvidenceName{long243:res:descent}{Theorem 8.1}\n\\DeclareResultEvidenceRecord{long243:res:descent}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-descent}{exact}{compared}\n\\DeclareResultEvidence{long243:res:descent}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean\\#L1843}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-descent-comparator}\n\\DeclareResultEvidenceName{long243:res:constant}{Theorem 9.2}\n\\DeclareResultEvidenceRecord{long243:res:constant}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-constant}{exact}{compared}\n\\DeclareResultEvidence{long243:res:constant}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-constant}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-constant-comparator}\n\\DeclareResultEvidenceName{long243:res:periodic}{Theorem 10.1}\n\\DeclareResultEvidenceRecord{long243:res:periodic}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-periodic}{exact}{compared}\n\\DeclareResultEvidence{long243:res:periodic}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR21/NegativeMagnitudeExclusions.lean\\#L68}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-periodic-comparator}\n\\DeclareResultEvidenceName{long243:res:crt}{Lemma 11.1}\n\\DeclareResultEvidenceRecord{long243:res:crt}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-crt}{exact}{compared}\n\\DeclareResultEvidence{long243:res:crt}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR21/ForbiddenBlockCrossing.lean\\#L26}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-crt-comparator}\n\\DeclareResultEvidenceName{long243:res:barrier}{Theorem 11.3}\n\\DeclareResultEvidenceRecord{long243:res:barrier}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-barrier}{exact}{compared}\n\\DeclareResultEvidence{long243:res:barrier}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR21/ForbiddenBlockCrossing.lean\\#L44}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-barrier-comparator}\n\\DeclareResultEvidenceName{long243:res:reduced}{Proposition 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11.6}\n\\DeclareResultEvidenceRecord{long243:res:gcdstab}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-gcdstab}{exact}{compared}\n\\DeclareResultEvidence{long243:res:gcdstab}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR7/Reduction.lean\\#L103}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-gcdstab-comparator}\n\\DeclareResultEvidenceName{long243:res:gcdsparse}{Proposition 11.7}\n\\DeclareResultEvidenceRecord{long243:res:gcdsparse}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-gcdsparse}{exact}{compared}\n\\DeclareResultEvidence{long243:res:gcdsparse}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR7/Limits.lean\\#L97}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-gcdsparse-comparator}\n\\DeclareResultEvidenceName{long243:res:bounded}{Theorem 12.1}\n\\DeclareResultEvidenceRecord{long243:res:bounded}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-bounded}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long243:res:bounded}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean\\#L2360}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-bounded-comparator}\n\\DeclareResultEvidenceName{long243:res:cor}{Corollary 12.2}\n\\DeclareResultEvidenceRecord{long243:res:cor}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-cor}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long243:res:cor}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean\\#L2412}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-cor-comparator}\n\\DeclareResultEvidenceName{long243:res:mass}{Theorem 13.1}\n\\DeclareResultEvidenceRecord{long243:res:mass}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-mass}{exact}{compared}\n\\DeclareResultEvidence{long243:res:mass}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR7/Frontier.lean\\#L84}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-mass-comparator}\n\\DeclareResultEvidenceName{long243:res:frontier}{Proposition 14.1}\n\\DeclareResultEvidenceRecord{long243:res:frontier}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-frontier}{exact}{compared}\n\\DeclareResultEvidence{long243:res:frontier}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR7/Frontier.lean\\#L151}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-frontier-comparator}\n\\DeclareResultEvidenceName{long243:res:variablerise}{Proposition 14.6}\n\\DeclareResultEvidenceRecord{long243:res:variablerise}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-variablerise}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long243:res:variablerise}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR21/WindowAvoidance.lean\\#L862}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-variablerise-comparator}\n\\DeclareResultEvidenceName{long243:res:gapconstant}{Theorem 14.7}\n\\DeclareResultEvidenceRecord{long243:res:gapconstant}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-gapconstant}{exact}{compared}\n\\DeclareResultEvidence{long243:res:gapconstant}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos243/PaperCompleteR21/MaximalGapConstant.lean\\#L1041}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-gapconstant-comparator}\n\\DeclareResultEvidenceName{long243:res:residue}{Theorem B.1}\n\\DeclareResultEvidenceRecord{long243:res:residue}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-residue}{exact}{compared}\n\\DeclareResultEvidence{long243:res:residue}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-residue}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos243-reciprocal-tail-reasoning-surface.md\\#long243-res-residue-comparator}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex","section_outline":["Introduction","Irrationality at the cubic rate","Reduction and lower density","Square specialisation","The two remaining cubics fail modulo seven","Polynomial extraction at a regular rate","Prior work and formalisation","Integer numerators, denominators and errors","Zero errors and the Sylvester recurrence","Relations among three consecutive numerators","LCM numerators and weighted records","New maxima of reduced numerators","Growth of the running maximum","Bounds that allow for cancellation and earlier decreases","Counting jumps before a prime power can be lost","Bounds on the error and on the original sequence","Signs of the error and of the growth ratio","Nonnegative errors","Constant negative errors","Periodic negative errors","Bounded increases and coprimality","Bounded negative part","Finite total relative increase","Further questions","Growth of the factors repeated in the denominator","Cancellation during recovery intervals","The direct analytic question","Unbounded upward increments","Formalisation targets","Statements and declarations","Verification and reproducibility","Indexing of the regular-rate theorems","The square-specialisation proofs","Recurrence declarations","A factorial modulus for integral recursion","Result map and proof dependencies"]},"erdos-249-binary-totient-series":{"paper_id":"erdos-249-binary-totient-series","title":"Integral Relations among Totient Sections","form":"Problem note","length_class":"short","pages":15,"subject":"Erdős #249","subject_kind":"problem","problem_id":"erdos_249","landing_role":"mathematics","question":"What explicit basis and integral relations does the totient k-kernel have at every integer base, and why does that not decide Erdős #249?","precis":"For integers k≥2 and e≥1, the totient sections n↦φ(k^j n+r) with 0≤j≤e and 0≤r<k^j have an explicit rational basis of dimension k^e+1. Every section is an explicit integer multiple of a retained one, and these reductions generate all integral relations, with unique coefficients. The proof combines the local formula for Euler's totient with a nonsingular evaluation matrix obtained from the Chinese remainder theorem and Dirichlet's theorem. The basis theorem does not decide the irrationality of S=∑ φ(n)2^{-n}.","owns":"the exact finite-level totient-kernel basis, independence proof and integral relations; the long paper develops denominator exclusions and other approaches to the binary series","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #249, which remains open","first_pass":["sec:results","sec:rank","sec:base-six-test"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-249-binary-totient-series.pdf","hosted_pdf_path":"papers/erdos-249-binary-totient-series.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/249/erdos-249-binary-totient-series.tex","site_html":"maths/papers/erdos-249-binary-totient-series.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos-249-binary-totient-series.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/249/erdos-249-binary-totient-series.tex","body_pointer":"papers.records.erdos-249-binary-totient-series.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n\\input{evidence/erdos-249-binary-totient-series}\n\\evidenceconcordancenewpage\n\\renewcommand{\\commit}{99f4bf47422abbd8757cbb22b50ba079d764d3a7}\n\\renewcommand{\\commitshort}{99f4bf47422a}\n% Sources that postdate \\commit are cited with their lean/ path and resolve at \\latecommit,\n% whose tree keeps the Lean roots under lean/.\n\\newcommand{\\latecommit}{168bf6727758f918a430ef056a1c93d3160b53a6}\n\\newcommand{\\laterepobase}{https://github.com/wcook04/plectis-erdos/blob/\\latecommit}\n\\hypersetup{\n  pdftitle={Integral relations among totient sections},\n  pdfsubject={Erdős Problem Notes: Problem 249},\n  pdfkeywords={irrationality, Euler totient, k-regular sequences, 2-kernel, Lambert series, Möbius inversion, linear independence, Lean 4},\n}\n\n\\newtheorem{equivform}[theorem]{Equivalent formulation}\n\n\n\\runninghead{Integral relations among totient sections}\n\\title{Integral Relations among Totient Sections}\n\\subtitle{Finite-level bases and bounded residue series}\n\\author{Will Cook}\n\\date{July 2026; revised 1 October 2026}\n\n\n% Named destinations are outside all theorem bodies.\n\\newcommand{\\longrecord}[2]{\\href{erdos249-totient-reasoning-surface.pdf\\##1}{#2}}\n\\begin{document}\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\nFor integers $k\\ge2$ and $e\\ge1$, every totient section\n$n\\mapsto\\varphi(k^jn+r)$ with $0\\le j\\le e$, $0\\le r<k^j$,\nis an explicit integer multiple of one of $k^e+1$ independent sections.\nThe resulting two-term relations form an integral basis of all relations\namong the indexed sections: their coefficients are recovered without\ndivision. Martin's stronger affine theorem gives the independence;\nwe also prove it directly by making an evaluation matrix diagonal\nmodulo a prime. The same separation method classifies rational binary\nseries formed from dyadic totient residues. It does not settle the\nirrationality of the unreduced binary totient series.\n\\end{abstract}\n\n\\section{Introduction}\\label{sec:results}\n\\papersectiontarget{short249-basis}\nSplitting the totient sequence into its even and odd subsequences,\nthen splitting once more, gives seven sequences:\n\\[\n \\varphi(n);\\qquad\n \\varphi(2n),\\ \\varphi(2n+1);\\qquad\n \\varphi(4n),\\ \\varphi(4n+1),\\ \\varphi(4n+2),\\ \\varphi(4n+3).\n\\]\nThe identities\n\\[\n \\varphi(4n)=2\\varphi(2n),\\qquad\n \\varphi(4n+2)=\\varphi(2n+1)\n\\]\nleave five independent sequences. Thus every identity among the seven\ncomes from these two substitutions. More precisely, an integral\nrelation is a unique integral combination of the two displayed\nrelations. We will prove the same statement at every depth and in\nevery integer base.\n\nFor an integer $k\\ge2$, the \\emph{totient sections} are the sequences\n$n\\mapsto\\varphi(k^jn+r)$, where $j\\ge0$ and $0\\le r<k^j$.\nThroughout, $n$ ranges over $\\N=\\{0,1,2,\\ldots\\}$ and\n$\\varphi(0)=0$. To describe relations we retain a separate index for\neach pair $(j,r)$, even when two pairs give the same sequence.\nFor example, $\\varphi(4n+2)$ and $\\varphi(2n+1)$ are equal sequences\nbut occupy different positions in the list above.\n\nThe sections form the $k$-kernel of the totient sequence. A sequence is\n$k$-regular when the $\\Z$-module generated by its $k$-kernel is finitely\ngenerated~\\cite{allouche-shallit}. Coons proved that the totient sequence is\nnot $k$-regular for any $k\\ge2$~\\cite[Thm.~3.2]{coons}.\nHere we determine those modules at each finite level, including their\nintegral relations. The theorem gives the basis and the substitutions;\nCorollary~\\ref{cor:integral-normal-form} explains how to recover every\nrelation from them. Level zero contains only $\\varphi(n)$ and has\ndimension one.\n\n\\begin{theorem}[A basis through each finite level]\n\\label{thm:kkernelrank}\nLet $k\\ge2$ and $e\\ge1$ be integers, write $F^{(k)}_{j,r}(n)=\\ph(k^jn+r)$, and put\n\\[\n V_{k,e}=\\operatorname{span}_{\\Q}\n \\{\\,F^{(k)}_{j,r}:0\\le j\\le e,\\ 0\\le r<k^j\\,\\}.\n\\]\nThen $\\dim_{\\Q}V_{k,e}=k^e+1$, and\n\\[\n \\mathcal B_{k,e}\n =\\{F^{(k)}_{0,0},F^{(k)}_{1,0}\\}\n \\cup\\{\\,F^{(k)}_{j,r}:1\\le j\\le e,\\ 1\\le r<k^j,\\ k\\nmid r\\,\\}\n\\]\nis a basis.  Every omitted section reduces to a basis element by an explicit\nscalar: $F^{(k)}_{j,0}=k^{j-1}F^{(k)}_{1,0}$ for $j\\ge1$, and if $r=k^tu$ with\n$t=\\max\\{s:k^s\\mid r\\}\\ge1$ and $k\\nmid u$, then\n\\[\n F^{(k)}_{j,r}=C_k(t,u)\\,F^{(k)}_{j-t,u},\n \\qquad\n C_k(t,u)=k^t\\prod_{\\substack{p\\mid k\\\\ p\\nmid u}}\\Bigl(1-\\tfrac1p\\Bigr).\n\\]\n\\end{theorem}\n\nThe condition $k\\nmid r$ allows residues with $\\gcd(k,r)>1$.\nFor instance, $r=1$ and $r=2$ are both retained in base six, and the\nreductions involving them are\n\\[\n \\varphi(36n+6)=2\\varphi(6n+1),\\qquad\n \\varphi(36n+12)=4\\varphi(6n+2).\n\\]\nThe factor $2$ is already present in $6n+2$ but not in $6n+1$;\nonly newly introduced prime divisors contribute factors $1-1/p$.\nThus one removes whole factors of $k$ from the residue, not every\ncommon prime factor. Repeated prime factors of $k$ cause no exception.\n\nMartin's 2006 theorem on simultaneous totient inequalities already\nimplies independence for every finite family of pairwise\nnonproportional affine forms with positive integral slopes\n\\cite[Thm.~1]{martin-phi-inequalities}. Its conclusion is stronger\nthan the independence used here. The point of the present description\nis the explicit reduction of the sections and the integral relation\nbasis in Corollary~\\ref{cor:integral-normal-form}.\nWe also give a direct proof that constructs a nonsingular evaluation\nmatrix. It uses the CRT--Dirichlet separation argument in Yazdani's\nproof of Theorem~2, credited there to Shallit\n\\cite[pp.~652--653]{yazdani2001}. The step that needs care is to keep\nthe progression for the selected affine form reduced while imposing\nprime divisors on all the others.\n\nThe proof in Section~\\ref{sec:rank} separates scalar reduction from\nindependence. Section~\\ref{sec:base-six-test} recovers the integral\nrelations, starting in base six. Section~\\ref{sec:family} uses prime\nisolation for bounded functions of totient residues.\nThe series\n\\[\n S=\\sum_{n\\ge1}\\frac{\\varphi(n)}{2^n}\n\\]\nhas unbounded coefficients. Its irrationality, asked in\nErd\\H{o}s Problem~\\#249~\\cite[p.~61]{erdosgraham1980}, remains open.\nThe coefficient basis supplies no irrationality conclusion for this\nvalue. Appendix~\\ref{app:sources} gives further coefficient\nconsequences; Appendices~\\ref{sec:carry-rank} and~\\ref{sec:open}\ncollect the tail identities, comparison sequences and still-unproved\ncancellation estimates relevant to $S$. Their longer proofs are in\n\\longrecord{long249-r3-complements}{\\emph{The Binary Totient Series}}.\n\n\\section{Proof of the basis theorem}\\label{sec:rank}\n\\papersectiontarget{short249-proof}\n\\subsection*{Scalar reductions}\nThe product formula reduces every omitted section to an integer\nmultiple of a retained one.\nFor a zero residue, $kn$ and $k^jn$ have the same prime divisors when\n$n>0$. Hence\n$\\varphi(k^jn)=k^{j-1}\\varphi(kn)$; both sides vanish at $n=0$.\nFor an omitted positive residue $r=k^tu$ with $t\\ge1$ and $k\\nmid u$,\nwe have $t<j$ because $0<r<k^j$. Put $m=k^{j-t}n+u$. A prime $p\\mid k$ divides $m$ exactly when it divides\n$u$. The product formula for $\\varphi(k^tm)/\\varphi(m)$ therefore\ncontributes $k^t$ and one factor $1-1/p$ for each prime $p\\mid k$\nabsent from $u$. The multiplier is an integer: its denominator is a\nproduct of distinct primes dividing $k$, and therefore divides $k^t$.\nThis gives $C_k(t,u)$ and proves that the stated family spans $V_{k,e}$.\n\n\\subsection*{Affine independence}\nIndependence follows from Martin's theorem. For any ordering of\npairwise nonproportional affine forms and any $C>0$, it gives a set\nof positive lower density on which each successive totient ratio\nexceeds $C$. In a proposed nonzero relation, place a nonzero\ncoefficient $c_1$ first and take\n$C>\\max(1,\\sum_{i>1}|c_i|/|c_1|)$. The first term then dominates\nthe sum of all remaining terms in absolute value, a contradiction\n\\cite[Thm.~1]{martin-phi-inequalities}. This also proves independence\nafter any finite prefix is deleted.\n\nThe direct proof isolates one column in each evaluation row. Let\n$L_i(n)=a_i n+b_i$ for $1\\le i\\le s$, where $a_i>0$ and\n$a_i b_j-a_j b_i\\ne0$ for $i\\ne j$. Shift $n$ so that all $b_i$\nare positive; the cross determinants do not change. Write\n$L_i=g_iA_i$, where $g_i=\\gcd(a_i,b_i)$ and $A_i$ is primitive.\nWe make $A_i$, not $L_i$, prime: $L_i$ may have the fixed divisor\n$g_i>1$. We seek a prime $\\ell$ and points $n_1,\\ldots,n_s$ for which\n\\begin{equation}\\label{eq:diagonal-evaluation}\n E=\\bigl(\\varphi(L_j(n_i))\\bigr)_{1\\le i,j\\le s}\n \\equiv\n \\begin{pmatrix}\n \\varphi(g_1)&0&\\cdots&0\\\\\n 0&\\varphi(g_2)&\\cdots&0\\\\\n \\vdots&\\vdots&\\ddots&\\vdots\\\\\n 0&0&\\cdots&\\varphi(g_s)\n \\end{pmatrix}\\pmod\\ell,\n \\qquad \\ell\\nmid\\prod_i\\varphi(g_i).\n\\end{equation}\nSince $\\det E\\not\\equiv0\\pmod\\ell$, $E$ is invertible over $\\Q$.\nA rational relation gives $Ec=0$, hence $c=0$. We now construct\nthe rows.\n\nFor $j\\ne i$, a prime divisor $q_{ij}\\equiv1\\pmod\\ell$ of\n$L_j(n_i)$ makes $\\ell\\mid q_{ij}-1\\mid\\varphi(L_j(n_i))$.\nFor the diagonal entry, we instead make $A_i(n_i)$ prime and\ncongruent to $2$ modulo $\\ell$.\nChoose an odd $\\ell$ dividing none of the slopes $a_i$, the products\n$g_i\\varphi(g_i)$ or the nonzero cross determinants. For each fixed\n$i$, Dirichlet's theorem supplies distinct primes\n$q_{ij}\\equiv1\\pmod\\ell$ ($j\\ne i$) outside the same finite exceptional\nset. Impose\n\\[\n A_i(n)\\equiv2\\pmod\\ell,\\qquad\n L_j(n)\\equiv0\\pmod{q_{ij}}\\quad(j\\ne i).\n\\]\nEach congruence is soluble because its slope is coprime to its\nmodulus. CRT combines them into $n=n_0+Ru$, where\n$R=\\ell\\prod_{j\\ne i}q_{ij}$.\n\nCRT supplies the congruences, but not yet a prime value. Dirichlet\napplies to the progression of values of $A_i$, namely\n\\[\n A_i(n_0+Ru)=A_i(n_0)+(a_i/g_i)Ru.\n\\]\nIts initial value must be coprime to its step. The step has only\nthree kinds of prime factors: those of $a_i/g_i$, excluded by\nprimitivity; $\\ell$, excluded by the residue $2$; and the imposed\n$q_{ij}$. Such a $q_{ij}$ cannot divide\n$A_i(n_0)$: it would then divide both $L_i(n_0)$ and $L_j(n_0)$, hence\n\\[\n a_iL_j(n_0)-a_jL_i(n_0)=a_i b_j-a_j b_i,\n\\]\ncontrary to its choice. This is where nonproportionality is used:\nthe nonzero cross determinant lets us keep each imposed divisor\naway from the selected form. We may now choose $p=A_i(n_i)$ prime,\nas large as needed,\nand in particular with $p\\nmid g_i$. Then\n\\[\n \\varphi(L_i(n_i))=\\varphi(g_i)(p-1)\n       \\equiv\\varphi(g_i)\\not\\equiv0\\pmod\\ell,\n\\]\nwhile all the other entries of row $i$ vanish modulo $\\ell$.\nWe choose the rows separately, applying Dirichlet's theorem once\nmore for each selected form $A_i$. Thus no simultaneous primality\nassertion is required. The construction proves\n\\eqref{eq:diagonal-evaluation} with every $n_i$ beyond any prescribed\nthreshold.\n\n\\subsection*{The two zero-residue sections}\nThe arguments $n,kn$ are proportional; their totient sequences are\nnot. We first merge their coefficients on\n$n=km+1$, where $\\varphi(kn)=\\varphi(k)\\varphi(n)$, then separate\nthem using a different ratio. On this progression the\npositive-residue sections have arguments\n\\[\n k^{j+1}m+k^j+r\\quad(k\\nmid r),\n\\]\ntogether with $km+1$. These are pairwise nonproportional: equality of\ntwo intercept-to-slope ratios would give\n$r/k^j=r'/k^{j'}$, which for $j'>j$ forces $k\\mid r'$.\nNo positive-residue form is proportional to $km+1$.\nIf $A,B$ multiply $\\varphi(n),\\varphi(kn)$, affine independence\ntherefore leaves $A+B\\varphi(k)=0$ and eliminates every other\ncoefficient. At $n=k$, however, the ratio is $k$, giving $A+Bk=0$.\nSince $\\varphi(k)<k$, these two equations separate the remaining\ncoefficients: $A=B=0$. Finally,\n\\[\n \\lvert\\mathcal B_{k,e}\\rvert\n   =2+\\sum_{j=1}^e(k^j-k^{j-1})=k^e+1,\n\\]\nwhich completes the proof of Theorem~\\ref{thm:kkernelrank}.\nFor independence on a tail, use a sufficiently large power of $k$\nin the last evaluation.\n\n\\section{Integral relations}\\label{sec:base-six-test}\n\\papersectiontarget{short249-base-six}\nIndependence turns identities into coordinate calculations. In base six\nat level two, the $43$ indexed sections reduce to $37$ retained ones:\n$F_{0,0},F_{1,0}$, the five\n$F_{1,r}$ ($1\\le r\\le5$), and the $30$ sections $F_{2,r}$ with\n$1\\le r<36$ and $6\\nmid r$. The omitted sections satisfy\n\\[\n\\begin{aligned}\n F_{2,0}&=6F_{1,0}, & F_{2,6}&=2F_{1,1}, & F_{2,12}&=4F_{1,2},\\\\\n F_{2,18}&=3F_{1,3}, & F_{2,24}&=4F_{1,4}, & F_{2,30}&=2F_{1,5}.\n\\end{aligned}\n\\]\nThe formal symbols $E_{j,r}$ remain independent even when their\nsequences $F_{j,r}$ coincide. The substitution $F_{2,12}=4F_{1,2}$\ncorresponds to the decomposition\n\\[\n x_{1,2}E_{1,2}+x_{2,12}E_{2,12}\n = (x_{1,2}+4x_{2,12})E_{1,2}\n   +x_{2,12}(E_{2,12}-4E_{1,2}).\n\\]\nThe retained coefficient becomes $x_{1,2}+4x_{2,12}$; the relation\ncoefficient is the original $x_{2,12}$. After all six substitutions,\nthe evaluated sequence vanishes exactly when the $37$ retained\ncoefficients vanish. In that case, the relation coordinates are the six\noriginal omitted coefficients. In the opening binary example the\nrelation vectors are $E_{2,0}-2E_{1,0}$ and $E_{2,2}-E_{1,1}$.\n\n\\begin{corollary}[Integral coordinates and all integral relations]\\label{cor:integral-normal-form}\nLet $k\\ge2$ and $e\\ge1$. The retained family is a $\\Z$-basis of the\nmodule generated by the sections through level $e$. Index the sections\nby their level and residue, retaining distinct indices even when they\ndefine equal sequences. For each omitted index $i$, write the scalar\nreduction as $F_i=a_iF_{j(i)}$, where $j(i)$ is retained and $a_i$ is a\nnonnegative integer. In the free abelian group with one generator $E_i$\nfor each of these indices, the vectors\n\\[\n R_i=E_i-a_iE_{j(i)}\\qquad(i\\text{ omitted})\n\\]\nform a $\\Z$-basis of the kernel of evaluation $E_i\\mapsto F_i$.\nIn particular, every integral relation has a unique integral expression\nin these elementary relations, and their rank is\n$\\sum_{j=1}^{e-1}k^j$.\n\\end{corollary}\n\\begin{proof}\nSince $\\prod_{p\\mid k,\\,p\\nmid u}p$ divides $k$, every scalar\n$C_k(t,u)$ is an integer. The retained family therefore spans the\nsection module over $\\Z$, and its rational independence makes it a\n$\\Z$-basis.\n\nFor any vector $x=\\sum_i x_iE_i$, substitute\n$E_i=R_i+a_iE_{j(i)}$ at each omitted index and collect terms:\n\\[\n x=\\sum_{j\\text{ retained}}\n       \\left(x_j+\\sum_{\\substack{i\\text{ omitted}\\\\j(i)=j}}a_i x_i\\right)E_j\n       +\\sum_{i\\text{ omitted}}x_iR_i.\n\\]\nThe first sum gives the retained coordinates of the evaluated\nsequence; the second evaluates to zero. Thus $x$ is a relation\nexactly when the first sum vanishes, in which case\n$x=\\sum_{i\\text{ omitted}}x_iR_i$. This expression is unique:\nat omitted index $i$, the coefficient of $\\sum_h c_hR_h$ is $c_i$\nitself. Each substitution has a unit coefficient at its own omitted\nindex, so the entire calculation takes place over $\\Z$, without division.\nCounting omitted indices gives\n$(1+k+\\cdots+k^e)-(k^e+1)=\\sum_{j=1}^{e-1}k^j$.\n\\end{proof}\nThe \\emph{section-generated} module need not contain every integer-valued\nsequence in its rational span: for $k=2$, $e\\ge2$, the sequence\n$\\varphi(4n+3)/2$ is integer-valued but has retained coordinate $1/2$.\nThis failure of saturation concerns the image of evaluation; the integral\nrelation basis describes the kernel.\n\nIn the same base-six example, evaluation gives a second test.\nOrder the retained columns as above, with residues increasing within\neach level, and evaluate at $n=0,\\ldots,36$ to obtain a $37\\times37$ matrix.\nIts determinant is $17$ modulo $101$, so it is nonsingular over $\\Q$.\nConsequently a retained combination is zero if and only if all $37$\nof its integer evaluations are zero. Here reduction modulo $101$\ncertifies the matrix, not the equality of a proposed combination:\n$101F_{2,2}$ would vanish modulo $101$ at every input, although its\ninteger value at $n=0$ is $101$.\nThe script \\texttt{scripts/totient\\_kernel\\_normal\\_form.py} calculates\nthe coordinates and the first nonzero evaluation using integer arithmetic.\n\n\\section{Bounded functions of totient residues}\\label{sec:family}\n\\papersectiontarget{short249-residues}\nThe same separation method now isolates a coefficient between long\nzero blocks. For bounded integer coefficients, rationality puts scaled\ntails in a fixed lattice: the right block forces a tail to zero,\nand the left leaves a nonzero tail of magnitude below the lattice spacing.\nResidues below are least nonnegative representatives, not necessarily\nbinary digits.\n\\begin{theorem}[Series of totient residues]\\label{res:residueseries}\nFor every $m\\ge3$,\n\\[\n \\sum_{n\\ge1}\\frac{\\ph(n)\\bmod m}{2^n}\\notin\\Q.\n\\]\nFor $k\\ge1$ and $f:\\Z/2^k\\Z\\to\\Q$, the series\n$\\sum_{n\\ge1}f(\\ph(n)\\bmod2^k)2^{-n}$ is rational exactly when $f$ is\nconstant on the even residue classes. If that constant is $c$, its value is\n$3f(1)/4+c/4$.\n\\end{theorem}\nOnly $f(1)$ and the even classes are visible to the series:\n$\\varphi(1)=\\varphi(2)=1$ and all later totients are even.\nThis explains the rational case, including the value $3/4$ modulo\ntwo. The irrational case rests on the pattern\n\\[\n \\underbrace{0,\\ldots,0}_{L\\text{ terms}},\\quad\n a_N\\ne0,\\quad\n \\underbrace{0,\\ldots,0}_{L\\text{ terms}}.\n\\]\nThe coefficients outside the window may have either sign; only their\nuniform bound matters.\n\\begin{lemma}[An isolated nonzero coefficient between long zero blocks]\\label{lem:bounded-pulse}\nLet $a_n\\in\\Z$ satisfy $|a_n|\\le C$. Suppose that for arbitrarily large $L$\nthere is $N>L$ such that $a_N\\ne0$ and\n$a_{N+t}=0$ for $0<|t|\\le L$. Then $\\sum_{n\\ge1}a_n2^{-n}$ is irrational.\n\\end{lemma}\n\\begin{proof}\nSuppose that the sum has denominator $q\\ge1$. Multiplying it by\n$2^j$ and removing its finite integer prefix shows that every scaled\ntail $T_j=\\sum_{t\\ge1}a_{j+t}2^{-t}$ belongs to $q^{-1}\\Z$.\nChoose a pattern with $C2^{-L}<1/q$. The block on the right gives\n\\[\n |T_N|\\le C\\sum_{t>L}2^{-t}=C2^{-L}<1/q.\n\\]\nThe only element of $q^{-1}\\Z$ in this interval is zero, so $T_N=0$.\nThe right block has removed the unknown tail exactly. Starting before\nthe left block now leaves only the nonzero centre, rescaled:\n\\[\n T_{N-L-1}=a_N2^{-L-1},\\qquad\n 0<|T_{N-L-1}|\\le C2^{-L-1}<1/q,\n\\]\ncontradicting $T_{N-L-1}\\in q^{-1}\\Z$.\n\\end{proof}\nFor example, the indicator of the squares satisfies the lemma.\nAn eventually periodic sequence with infinitely many nonzero terms\ndoes not have isolated coefficients at arbitrarily large distances.\nThe uniform bound $C$ is essential to this argument.\n\n\\begin{proof}[Proof of Theorem~\\ref{res:residueseries}]\nWe prescribe a prime at the centre and a different divisor at each\nneighbour. Fix a unit $s\\pmod m$ and an integer $L\\ge1$, and choose\ndistinct primes $q_t\\equiv1\\pmod m$, all exceeding $L$, for\n$0<|t|\\le L$. The congruences\n\\[\n p\\equiv s\\pmod m,\\qquad p\\equiv-t\\pmod{q_t}\\quad(0<|t|\\le L)\n\\]\ndefine a reduced progression, since $s$ is a unit modulo $m$ and\n$0<|t|<q_t$. CRT and Dirichlet's theorem therefore allow us to choose\nan arbitrarily large prime $p>L+1$ on this progression, with\n$m\\mid\\varphi(p+t)$ at every neighbour and\n$\\varphi(p)\\equiv s-1\\pmod m$ at the centre.\n\nFor a rational-valued function $f$ on the residue classes, subtract\n$f(0)$ and multiply by a common denominator of its values. Since\n$\\sum_{n\\ge1}2^{-n}=1$, these operations preserve whether the sum is\nrational. If $f(s-1)\\ne f(0)$, the resulting bounded integer sequence\nhas an isolated nonzero coefficient for every $L$, so the lemma\napplies.\nFor the least-residue map, $s=-1$ gives the nonzero centre $m-2$\nwhen $m\\ge3$. For $m=2^k$, the possible centres $s-1$ run through\n\\emph{every} even class. Thus any failure of constancy on those\nclasses supplies a centre with $f(s-1)\\ne f(0)$.\nConversely, constancy makes every term after $n=2$ equal $c$, giving\n$3f(1)/4+c/4$.\n\\end{proof}\nThe dyadic modulus matters to the classification. Modulo $6$, for\nexample, prime centres of this form see only $s-1\\in\\{0,4\\}$, not\nthe even class $2$. The least-residue assertion needs just one\nnonzero centre; classification of an arbitrary function needs all\nrelevant classes.\n\nThe first assertion is also an application of Erd\\H{o}s's sparse-series\ncriterion~\\cite[Lem.~1, p.~213]{erdos1957}, and of Kaneko, Suzuki and\nTachiya~\\cite[Thm.~3]{kst2026}. The residues are bounded and nonnegative,\nand the prime construction above makes their support infinite.\nThe nonzero residues occur on a set of density zero: an integer in that\nset has no prime divisor\n$p\\equiv1\\pmod m$. Wong gave the case in which the base equals the\nmodulus~\\cite{wong2015}. The isolated-coefficient argument above also\ntreats signed values of $f-f(0)$, for which those nonnegative\nsparse-series criteria need an additional spacing hypothesis.\nThe \\longrecord{long249-r3-residue-extension}{extension to other bases}\ntreats every integer $t\\ge2$ and proves rational linear\nindependence of $1$ and $\\sum_n(\\varphi(n)\\bmod m)t^{-dn}$, $d\\ge1$.\n\n\\appendix\n\\section{Further consequences for the sections}\\label{app:sources}\n\\papersectiontarget{short249-sources}\nEvery finite set of dyadic sections lies in a single finite level.\nTheorem~\\ref{thm:kkernelrank} therefore also determines the full kernel;\n``span'' and ``relation'' here still mean finite linear combinations.\n\n\\begin{corollary}[Dyadic basis]\\label{res:basis}\nThe family\n\\[\n \\{\\ph_{0,0},\\ph_{1,0}\\}\n \\cup\\{\\ph_{j,r}:j\\ge1,\\ 0<r<2^j,\\ r\\text{ odd}\\},\n \\qquad \\ph_{j,r}(n)=\\ph(2^jn+r),\n\\]\nis a basis for the rational span of the full dyadic kernel. Every rational\nrelation is generated by the scalar reductions. The complete level-zero\ntruncation has dimension one.\n\\end{corollary}\n\\begin{proof}\nEach finite subset of this family is contained in some\n$\\mathcal B_{2,e}$. Theorem~\\ref{thm:kkernelrank} gives independence,\nand its reductions give spanning. At level zero the sole section\n$\\varphi(n)$ is nonzero, since $\\varphi(1)=1$.\n\\end{proof}\n\nThe affine argument also permits periodic coefficients by restriction\nto their common residue classes.\n\\begin{corollary}[Periodic coefficients]\\label{cor:periodic-freezing}\nLet $L_1,\\ldots,L_s$ be pairwise nonproportional affine forms with integer\ncoefficients and positive slopes. If $w_1,\\ldots,w_s$ are rational-valued\nperiodic sequences, then\n\\[\n \\sum_i w_i(n)\\ph(L_i(n))=0\\quad\\text{for all sufficiently large }n\n \\quad\\Longrightarrow\\quad w_i(n)=0\\quad\\text{for every }i,n.\n\\]\n\\end{corollary}\n\\begin{proof}\nWe choose a common period $Q$ and restrict to $n=r+Qt$. The\ncoefficients are now the constants $w_i(r)$, and the cross\ndeterminants are $Q(a_i b_j-a_j b_i)\\ne0$. By shifting $t$ beyond\nthe given threshold we can apply affine independence, obtaining\n$w_i(r)=0$ for every $i$. We repeat this for each residue $r$;\nperiodicity then gives the conclusion even at the indices preceding\nthe threshold.\n\\end{proof}\nProportional forms are excluded for a reason: the identity\n$\\varphi(4n+2)=\\varphi(2n+1)$ would otherwise contradict the conclusion.\nMartin's theorem gives the same corollary on each\nprogression~\\cite[Thm.~1]{martin-phi-inequalities}.\n\nCoons proves nonregularity using the poles of\n$\\zeta(s-1)/\\zeta(s)$~\\cite[Cor.~3.1, Thm.~3.2]{coons}.\nThe ranks above recover it: finite rational kernel dimension would\nembed the integral kernel module by evaluations in $\\Z^d$, making\nit finitely generated; the converse is immediate\n(compare~\\cite[App.~A]{coons-jsr}).\nAllouche, Shallit and Yassawi survey the earlier separation method\n\\cite[Thm.~3, Ex.~4, Rem.~7]{allouche-shallit-yassawi}.\n\nThe generating function $F_\\varphi(z)=\\sum_{n\\ge1}\\varphi(n)z^n$\ntherefore satisfies no $k$-Mahler equation\n$\\sum_{i=0}^dP_i(z)F_\\varphi(z^{k^i})=0$ with $P_i\\in\\Q[z]$,\n$P_0\\ne0$. Indeed, $\\varphi(n)\\le n$ gives logarithmic height at most\n$\\log n$, and the height theorem of Adamczewski, Bell and Smertnig\nwould make such a series $k$-regular\n\\cite[Thm.~1.2(b)]{adamczewski-bell-smertnig2023}.\nBell and Smertnig's multiplicative classification gives the same\nconclusion~\\cite[Thm.~1.3]{bell-smertnig2026}.\nBell, Bruin and Coons treat related algebraic and $D$-finite series\n(those satisfying polynomial-coefficient linear differential equations)\n\\cite[Thms.~1.5--1.6]{bell-bruin-coons}.\nThese are statements about the function, not its value at $1/2$.\n\nThe section operators $T_i g(n)=g(kn+i)$ satisfy\n\\[\n T_iF_{j,r}=F_{j+1,k^ji+r},\\qquad\n V_{k,e+1}=V_{k,e}+\\sum_{i=0}^{k-1}T_iV_{k,e}.\n\\]\nEvery residue at level $j+1$ has the form $k^ji+r$, giving the second\nidentity. A positive retained residue stays retained. For the zero-residue\ngenerators, reduce $T_iF_{0,0}=F_{1,i}$ and $T_iF_{1,0}=F_{2,ki}$ as\nbefore. Each image is a positive integer multiple of a basis element,\nbut rank growth prevents a finite invariant truncation. Consequently\nthe growth formula using the joint spectral radius for regular sequences\n\\cite[Thm.~1, Prop.~4, Cor.~7]{coons-jsr} does not apply.\n\n\\section{The binary totient series}\\label{sec:carry-rank}\n\\papersectiontarget{short249-tails}\nThe bounded-pulse argument in Theorem~\\ref{res:residueseries} uses a\ntail bound independent of the pulse's position. For $S$, the bound\n$0\\le\\varphi(n)\\le n$ ensures convergence, but forcing\n$2^L\\mid\\varphi(N+L)$ with $N+L>1$ requires $2^L\\le N+L-1$.\nThe tail bound $(N+L+2)2^{-L}$ then exceeds $1$.\nThis is a failure of the estimate, not a lower bound on the tail:\nthe window lies too far out for this bound to help.\nInstead, we test differences of scaled tails by finite sums.\n\nPut $R_N=\\sum_{j\\ge1}\\varphi(N+j)2^{-j}$ and\n$\\Delta_h(N)=R_{N+h}-R_N$. Subtracting the finite-prefix identities\nfor $2^{N+h}S$ and $2^NS$ gives\n\\begin{equation}\\label{eq:tail-phase}\n \\Delta_h(N)\\equiv2^N(2^h-1)S\\pmod\\Z.\n\\end{equation}\nIf $S=a/(2^cv)$, with $v$ odd, then $\\Delta_h(N)$ is integral for\n$N\\ge c$ and $h=\\varphi(v)$. The tails are then periodic modulo\nintegers, not necessarily equal. To disprove rationality it therefore\nsuffices to find a nonintegral difference beyond every threshold for\neach positive $h$.\n\nFor $h,N,L\\in\\N$, define the integer\n\\[\n D_{h,N,L}=\\sum_{j=0}^{L-1}\n   \\bigl(\\varphi(N+h+1+j)-\\varphi(N+1+j)\\bigr)2^{L-1-j},\n \\qquad B=N+h+L+2.\n\\]\nSince $0\\le R_M\\le M+2$, we have\n\\begin{equation}\\label{eq:tail-error}\n 2^L\\Delta_h(N)-D_{h,N,L}=R_{N+h+L}-R_{N+L},\\qquad\n |2^L\\Delta_h(N)-D_{h,N,L}|\\le B.\n\\end{equation}\nBoth tails lie in $[0,B]$, so their difference has magnitude at most\n$B$, not $2B$. The interval centred at $D_{h,N,L}/2^L$ with radius\n$B/2^L$ avoids every integer precisely when\n\\begin{equation}\\label{eq:r3-certificate}\n B<D_{h,N,L}\\bmod2^L<2^L-B.\n\\end{equation}\nThis proves $\\Delta_h(N)\\notin\\Z$ and, by \\eqref{eq:tail-phase},\nexcludes every reduced denominator dividing $2^N(2^h-1)$.\nFor example, $D_{1,12,16}=-143140$ has residue $53468$ modulo $2^{16}$,\nand $B=31$, so it excludes denominators dividing $2^{12}$.\nIn ordinary coordinates the same certificate reads\n\\[\n \\Delta_1(12)\\in\n \\left[\\frac{-143171}{65536},\\frac{-143109}{65536}\\right]\n \\subset(-3,-2).\n\\]\n\nFor fixed $h,N$, the test is complete: if\n$\\delta=\\operatorname{dist}(\\Delta_h(N),\\Z)>0$, then\n$2^L\\delta>2B$ for all large $L$, since $B$ grows linearly.\nTogether with \\eqref{eq:tail-error}, this gives \\eqref{eq:r3-certificate}.\nA failed test proves no integrality; its interval may be too wide.\nDepth detects a given nonintegral difference, but does not supply one\nfor each $h$ beyond every threshold.\n\n\\subsection{Equivalent residue conditions}\\label{sec:frontier}\nWe write $K(h,N,L)$ for \\eqref{eq:r3-certificate}.\n\\begin{theorem}[Propagation of one nonintegral tail difference]\\label{res:fulldepth}\nFix $d\\ge1$ and $N\\ge0$. If $\\Delta_d(N)\\notin\\Z$, then every sufficiently\nlate pair $\\{t,t+1\\}$ contains an $m$ such that $K(md,N,md)$ holds.\nConsequently\n\\[\n \\exists t\\ge1:\\ K(td,N,td)\n \\quad\\Longleftrightarrow\\quad\\Delta_d(N)\\notin\\Z.\n\\]\nRequiring this for every $d\\ge1,N\\ge0$ is equivalent to $S\\notin\\Q$.\n\\end{theorem}\nCompleteness fixes $h,N$ and varies $L$; here $L=h=td$.\nA failed test puts $\\Delta_{td}(N)$ within\n$2(N+2td+2)2^{-td}$ of $\\Z$. This bound tends to zero. The recurrence\n$\\Delta_{(t+1)d}(N)\\equiv2^d\\Delta_{td}(N)+\\Delta_d(N)\\pmod\\Z$\nwould transfer two consecutive errors to the fixed nonintegral\n$\\Delta_d(N)$. Its positive distance from $\\Z$ therefore rules out\nconsecutive failures for large $t$.\nThe \\longrecord{long249-r3-ray}{companion proof} gives the estimates.\nThe final assertion still quantifies over every $d,N$, not one pair.\n\nFor a hypothetical denominator $2^cv$, a second test uses the\nmodulus $M=(2^H-1)/v$, where $H$ is a positive multiple of $\\varphi(v)$.\nWrite $B_{H,c}=\\sum_{j=0}^{H-1}\\varphi(c+1+j)2^{H-1-j}$.\n\\begin{equivform}[A residue test modulo a divisor of $2^H-1$]\\label{res:canonicalmersenne}\nThe series $S$ is irrational if and only if, for every $c\\ge0$ and odd\n$v\\ge1$, some positive multiple $H$ of $\\ph(v)$ satisfies\n\\begin{equation}\\label{eq:canonical-gap}\n c+H+1<(-B_{H,c})\\bmod M<M-(c+H+1).\n\\end{equation}\n\\end{equivform}\nThe identity $B_{H,c}=M(vR_c)-\\Delta_H(c)$ puts $B_{H,c}$ within\n$c+H+1$ of $M(vR_c)$. If $S=a/(2^cv)$, then $vR_c\\in\\Z$,\nso the residue lies near an endpoint. For irrational $S$, the fixed\nnumber $vR_c$ has positive distance from $\\Z$. The distance of $M(vR_c)$ from\n$M\\Z$ grows exponentially, while the error bound and excluded\nmargins grow linearly.\nThe \\longrecord{long249-r3-mersenne}{proof and residue recurrence}\ngive the bounds.\nThe required residues have not been shown to exist for every choice\nof the parameters in either test. A separate\n\\longrecord{long249-r3-farey}{Farey-interval calculation} at depth $240$\nexcludes $S=a/q$ for $0<q\\le Q_0\\approx7.96\\times10^{34}$.\n\n\\subsection{Comparison sequences}\\label{sec:nogo}\\label{sec:signed-interpolation-transfer}\nRationality and linear coefficient growth still permit exponential\ncarry rank. There is an integer sequence $c$ with $0\\le c(n)\\le n$, unchanged odd totients and\n$|c(n)-\\varphi(n)|\\le2$ at even indices, for which\n$\\sum c(n)2^{-n}=5/4$. Its carry is the integer sequence\n\\[\n u_N=4\\sum_{j\\ge1}c(N+j)2^{-j}\n     =5\\cdot2^N-4\\sum_{n=1}^{N}c(n)2^{N-n}.\n\\]\nIts recurrence is $2u_{N-1}-u_N=4c(N)$ for $N\\ge1$.\nFor $j\\ge1$ and odd $0<r<2^j$, this gives\n\\[\n 4\\varphi(2^jn+r)=2u_{2^jn+r-1}-u_{2^jn+r}.\n\\]\nThus the $2^e-1$ independent odd-residue totient sections lie in the\nspan of the carry sections through level $e$. This explains how the\nrank lower bound survives even though the sum is rational.\nThe \\longrecord{long249-r3-comparisons}{construction and its limitations}\nexplain why it fails the exact prime-dilation laws of $\\varphi$.\nThe same section retains the cofinal LCM-diagonal criterion, its endpoint\nsign conditions and the limits of finite integer combinations of shifts.\n\nThe \\longrecord{long249-r3-interpolation}{signed interpolation\nargument} tests whether preserving specified analytic data fixes the\nvalue. For each integer $\\beta\\ge2$, sparse even-index corrections\nrealise every value in an interval about $\\sum\\varphi(n)\\beta^{-n}$,\nwith a prescribed prefix and all odd coefficients fixed. Errors are\nbounded by any prescribed positive $f(n)\\to\\infty$; their support has\nupper Banach density zero. On each fixed arithmetic progression the\nDirichlet-series difference is entire. At roots of unity the\ngenerating-function difference extends smoothly radially, with any\nprescribed finite collection of radial derivatives zero. This preserves\nDirichlet singularities, not regular parts or zeros; a finite jet is\nnot the full radial germ. The proof is not formalised in Lean and\nestimates no cancellation for the unchanged totient sequence.\n\nRational approximation imposes a different requirement. The identity\n\\[\n \\Theta_2:=\\sum_{d\\ge1}\\frac{\\mu(d)}{(2^d-1)^2}=S-\\frac12\n\\]\nwas posted by Amiram Eldar, together with its coprimality interpretation,\non 15 March 2026~\\cite{eldar2026oeis}, and by Steve Fan on\n16 May 2026~\\cite{fan2026totient}.\nConsider the rational quotients\n\\[\n Q(e,Y)=\\frac{\\bigl(\\sum_{d\\le Y}\\mu(d)/(2^d-1)^{e+2}\\bigr)^2}\n      {\\sum_{d\\le Y}\\mu(d)/(2^d-1)^{2e+2}}.\n\\]\n\\begin{theorem}[A lower bound for the rank-one quotients]\\label{res:rankonefloor}\nFor $e\\ge1$ and $Y\\ge4$, the denominator of $Q(e,Y)$ is positive, and the\nunique minimiser is $(e,Y)=(1,5)$. Every admissible quotient and every\nnonempty finite positive weighted average of such quotients satisfies\n\\[\n Q-\\Theta_2>\\frac{21}{320}.\n\\]\nThe uniform bound $Q-\\Theta_2>1/15$ is false already at $(1,5)$.\n\\end{theorem}\nThe \\longrecord{long249-r3-rankone}{proof} first makes the denominator\nuniformly positive. A geometric bound handles all $e\\ge2$; at $e=1$,\nseven-term prefixes leave only $Y=4,5,6$ to compare directly.\nThe averaging assertion concerns averages of the quotients themselves.\nIt leaves open signed combinations, quotients formed after combining\nthe sums, and higher-rank kernels. For a denominator-clearing\nirrationality argument, a sufficient target is\n$0<q_j|\\Theta_2-p_j/q_j|\\to0$, with $p_j/q_j$ in lowest terms.\nIndeed, if $\\Theta_2=a/b$, any nonzero difference satisfies\n$q_j|a/b-p_j/q_j|=|a q_j-bp_j|/b\\ge1/b$.\nConvergence alone gives no such contradiction.\n\n\\section{An exponential-sum question}\\label{sec:open}\n\\papersectiontarget{short249-first-harmonic}\nTo prove irrationality, it suffices to show that for every $h\\ge1$\nand every $X_0$, some integers $X\\ge\\max(X_0,1)$ and $L\\ge h$ satisfy\n\\begin{equation}\\label{eq:r3-harmonic}\n 16(2X+h+L+2)\\le2^L,\\qquad\n \\operatorname{Re}\\sum_{N=X}^{2X-1}\n       \\exp(2\\pi iD_{h,N,L}/2^L)\\le\\frac9{10}X.\n\\end{equation}\nThe phases cannot all cluster near $1$. If every residue failed\n\\eqref{eq:r3-certificate}, each $D_{h,N,L}/2^L$ would be within\n$1/16$ of an integer. Each exponential would then have real part at least\n$\\cos(\\pi/8)>9/10$. Thus some $N$ satisfies\n\\eqref{eq:r3-certificate}. For a hypothetical rational value, choosing\nits period and a sufficiently late block contradicts\n\\eqref{eq:tail-phase}.\n\nTo separate counting from cancellation, single out one totient in\neach window. Fix $h,s\\ge1$,\n$L\\ge s+h$, $X\\ge1$ and $0<\\eta<1$, and put\n$t=L-s+1$ and $E_N=\\exp(2\\pi iD_{h,N,L}/2^L)$.\nFor $X\\le N<2X$, let $p_N$ be the largest prime factor of $N+t$ and\n$m_N=(N+t)/p_N$. Define\n\\[\n\\begin{aligned}\n \\mathcal A&=\\{N\\in[X,2X):m_N\\le\\lfloor\\sqrt X/2\\rfloor,\n                      \\ p_N>2\\lfloor\\sqrt X\\rfloor\\},\\\\\n \\mathcal G&=\\{N\\in\\mathcal A:\\varphi(m_N)\\ge\\eta m_N\\}.\n\\end{aligned}\n\\]\nAll indices are integers. On $\\mathcal A$, $p_N>m_N$, hence\n$\\varphi(N+t)=\\varphi(m_N)(p_N-1)$. Since $h<t\\le L$, this term\noccurs in both tails, with net coefficient\n$2^{-(t-h)}-2^{-t}=(2^h-1)/2^t$. Write\n\\[\n z_N=\\exp\\!\\left(\\frac{2\\pi i(2^h-1)\\varphi(m_N)(p_N-1)}{2^t}\\right),\n \\qquad w_N=E_N/z_N,\n\\]\nand let $\\bar z_m$ be the average over the indices in $\\mathcal A$\nwith cofactor $m$, or zero for an empty set.\nFor fixed $m$, the exponent of $z_N$ is affine in $p_N$;\n$w_N$ contains the other totients.\n\nConsider the four bounds\n\\begin{equation}\\label{eq:r3-four-bounds}\n\\begin{aligned}\n \\operatorname{Re}\\sum_{N\\in\\mathcal G}w_N(z_N-\\bar z_{m_N})\n       &\\le\\tfrac{14}{25}X,\\\\\n \\left|\\sum_{N\\in\\mathcal G}w_N\\bar z_{m_N}\\right|\n       &\\le\\tfrac1{100}X,\\\\\n \\left|\\sum_{N\\in\\mathcal A\\smallsetminus\\mathcal G}E_N\\right|\n       &\\le\\tfrac1{100}X,\\\\\n \\left|\\sum_{\\substack{X\\le N<2X\\\\N\\notin\\mathcal A}}E_N\\right|\n       &\\le\\tfrac8{25}X.\n\\end{aligned}\n\\end{equation}\nThese are an exact partition: outside $\\mathcal G$ use the last two\nsums; on $\\mathcal G$ write\n$w_Nz_N=w_N(z_N-\\bar z_{m_N})+w_N\\bar z_{m_N}$.\nThe bounds add to $(14/25+1/100+1/100+8/25)X=9X/10$.\n\nThe last three bounds are available at a common depth. Take the least\n$L\\ge s+h$ satisfying the size inequality in \\eqref{eq:r3-harmonic}.\nThe \\longrecord{long249-r3-counts}{smooth-number and small-cofactor estimates}\ngive the last two bounds for all large $X$ when\n$0<\\eta<1/(300\\log4)$. The elementary excluded-cofactor estimate is\n$(3\\eta\\log4+o(1))X$; it needs no prime number theorem.\nWith the prime number theorem as input, the\n\\longrecord{long249-group-mean}{group-mean estimate} gives the second\nbound for $s=26$, $\\eta=1/1000$ at this depth and all large $X$.\nThe first bound remains open.\n\nSmall group means control the second sum by the triangle inequality\nbecause $|w_N|=1$. They do not control the first: subtracting\n$\\bar z_m$ cancels an \\emph{unweighted} group sum, not one with\nvarying $w_N$. No independence is assumed.\nFor example, $z=(1,-1)$ has mean zero, but the weights\n$w=(1,-1)$ make $\\sum w_N(z_N-\\bar z)=2$.\nThis illustrates the distinction, not the behaviour of the totient phases.\nA real-part bound does not control the imaginary part or give the\n$21X/25$ norm bound.\n\n\\begin{problem}[Four estimates for the first-harmonic sum]\n\\label{prob:firstharmonic}\nFor each $h\\ge1$, do there exist $s\\ge1$ and $0<\\eta<1$ such that, for every\n$X_0$, there are $X,L$ with\n\\[\n \\max(X_0,1)\\le X,\\qquad h\\le L-s,\\qquad\n 16(2X+h+L+2)\\le2^L,\n\\]\nfor which those four bounds hold?\n\\end{problem}\n\nFix $s,\\eta$ after $h$ but before $X_0$; all four estimates must hold\non the same block. At $s=26$, $\\eta=1/1000$ and the least admissible\n$L$, it suffices to prove the first bound for arbitrarily large $X$:\nthe other three hold eventually on those same blocks.\n\nA weaker sufficient condition needs no group-mean input.\nSince $|\\mathcal G|>67X/100$ for large $X$, the open bound\n$\\operatorname{Re}\\sum_{N\\in\\mathcal G}E_N\\le603X/1000$\nwould put the average on $\\mathcal G$ below $9/10$:\n$603/1000=(9/10)(67/100)$. Some good index would pass the test.\nThis requires an average on $\\mathcal G$ for each $h$ on arbitrarily\nlarge blocks, not a bound on the full block.\nThe \\longrecord{long249-r3-interpolation}{ordinary comparison argument}\ngives larger first harmonics at all sufficiently large admissible scales,\ndistinguishing its rational sequences from the cancellation required of\n$\\varphi$.\n\nBalasubramanian, Giri and Srivastav give correlation estimates applicable\nto $\\varphi(n)/n$, uniformly for $|h|\\le x/2$~\\cite[Thm.~2.2]{bgs2016}.\nTheir correlation is a product of two normalised totient values.\nHere the modulus grows with $L$, and $w_N$ is an exponential of the\nother, overlapping totient contributions. The cited shift uniformity\ndoes not supply this weighted exponential estimate.\n\n\\section*{Verification and reproducibility}\n\\evidenceparagraph{\\evidencerecordurl}\nThe \\longrecord{long249-r3-declarations}{declaration guide} links the\nformal sources for this paper. Complete proofs of three cited results\nappear in Appendix~A of the companion paper. The displayed arguments,\nincluding signed interpolation, require ordinary mathematical review;\nthe recorded Lean and Comparator checks cover their specified\ndeclarations and source versions. No new formal run or independent\nreview was performed for this revision.\n\n\\begin{thebibliography}{99}\\small\n\\raggedright\n\\bibitem{allouche-shallit} J.-P.~Allouche and J.~Shallit,\n\\emph{The ring of $k$-regular sequences}, Theoret.\\ Comput.\\ Sci.\\\n\\textbf{98} (1992), no.~2, 163--197,\ndoi:\\href{https://doi.org/10.1016/0304-3975(92)90001-V}{10.1016/0304-3975(92)90001-V}.\n\\bibitem{coons} M.~Coons, \\emph{(Non)Automaticity of number theoretic\nfunctions}, J.\\ Th\\'eor.\\ Nombres Bordeaux \\textbf{22} (2010), no.~2, 339--352,\ndoi:\\href{https://doi.org/10.5802/jtnb.718}{10.5802/jtnb.718}.\nTheorem~3.2, p.~348.\n\\bibitem{coons-jsr} M.~Coons, \\emph{Regular sequences and the joint spectral\nradius}, Internat.\\ J.\\ Found.\\ Comput.\\ Sci.\\ \\textbf{28} (2017), no.~2,\n135--140,\ndoi:\\href{https://doi.org/10.1142/S0129054117500095}{10.1142/S0129054117500095};\narXiv:\\href{https://arxiv.org/abs/1511.07535v1}{1511.07535v1}.\nTheorem~1, Proposition~4, Corollary~7 and Appendix~A are cited with the\narXiv v1 numbering.\n\\bibitem{martin-phi-inequalities} G.~Martin, \\emph{Simultaneous inequalities\namong values of the Euler phi-function},\narXiv:\\href{https://arxiv.org/abs/math/0603053v1}{math/0603053v1}, 2006.\nTheorem~1, pp.~1--2.\n\\bibitem{wong2015} Erick Wong,\n\\href{https://math.stackexchange.com/a/1211557}{answer 1211557}\nto \\emph{An infinite sum based on the mod-parity of Euler's totient function},\nMathematics Stack Exchange, 29 March 2015, accessed 15 September 2026.\n\\bibitem{erdos1957} P.~Erd\\H{o}s, \\emph{On the irrationality of certain\nseries}, Indag.\\ Math.\\ \\textbf{19} (1957), 212--219. Lemmas~1 and~4,\npp.~213--218.\n\\bibitem{kst2026} H.~Kaneko, Y.~Suzuki and Y.~Tachiya, \\emph{Refinements of\nErd\\H{o}s's irrationality criterion for certain sparse infinite series},\narXiv:\\href{https://arxiv.org/abs/2601.20743v1}{2601.20743v1}. Theorem~3, p.~5.\n\\bibitem{eldar2026oeis} A.~Eldar, comment and formula added to OEIS A256936\n(revisions 28 and 31), 15 March 2026,\n\\url{https://oeis.org/history?seq=A256936}, accessed 16 September 2026.\n\\bibitem{fan2026totient} S.~Fan, comment on Erd\\H{o}s Problem \\#249,\n16 May 2026, 19:01, \\href{https://www.erdosproblems.com/forum/thread/249}\n{Erd\\H{o}s Problems discussion thread}, accessed 16 September 2026.\n\\bibitem{erdosgraham1980} P.~Erd\\H{o}s and R.~L.~Graham,\n\\emph{Old and New Problems and Results in Combinatorial Number Theory},\nMonographies de L'Enseignement Math\\'ematique \\textbf{28}, 1980, p.~61.\n\\bibitem{bgs2016} R.~Balasubramanian, S.~Giri and P.~Srivastav,\n\\emph{On correlations of certain multiplicative functions},\nJ.\\ Number Theory \\textbf{174} (2017), 221--238,\ndoi:\\href{https://doi.org/10.1016/j.jnt.2016.10.001}{10.1016/j.jnt.2016.10.001};\narXiv:\\href{https://arxiv.org/abs/1511.02221v3}{1511.02221v3}.\nTheorem~2.2 is cited with the arXiv v3 pagination, p.~2.\n\\bibitem{yazdani2001} S.~Yazdani,\n\\emph{Multiplicative functions and $k$-automatic sequences},\nJ.\\ Th\\'eor.\\ Nombres Bordeaux \\textbf{13} (2001), no.~2, 651--658,\n\\url{https://www.numdam.org/item/JTNB_2001__13_2_651_0/}.\nTheorem~2 and its proof, pp.~652--653; Corollary~4, p.~654.\n\\bibitem{allouche-shallit-yassawi} J.-P.~Allouche, J.~Shallit and R.~Yassawi,\n\\emph{How to prove that a sequence is not automatic},\nExposition.\\ Math.\\ \\textbf{40} (2022), no.~1, 1--22,\ndoi:\\href{https://doi.org/10.1016/j.exmath.2021.08.001}{10.1016/j.exmath.2021.08.001};\narXiv:\\href{https://arxiv.org/abs/2104.13072v1}{2104.13072v1}.\nTheorem and example numbering follows the arXiv v1 (2021).\n\\bibitem{bell-smertnig2026} J.~Bell and D.~Smertnig,\n\\emph{Mahler series with multiplicative coefficient sequences},\narXiv:\\href{https://arxiv.org/abs/2603.23456v1}{2603.23456v1},\n24 March 2026, preprint. Theorem~1.3, p.~2; Lemma~3.3, p.~9.\n\\bibitem{bell-bruin-coons} J.~P.~Bell, N.~Bruin and M.~Coons,\n\\emph{Transcendence of generating functions whose coefficients are multiplicative},\nTrans.\\ Amer.\\ Math.\\ Soc.\\ \\textbf{364} (2012), no.~2, 933--959,\ndoi:\\href{https://doi.org/10.1090/S0002-9947-2011-05479-6}{10.1090/S0002-9947-2011-05479-6};\narXiv:\\href{https://arxiv.org/abs/1003.2221v2}{1003.2221v2}.\nTheorems~1.5--1.6 are cited with the arXiv v2 pagination, p.~2.\n\\bibitem{adamczewski-bell-smertnig2023}\nB.~Adamczewski, J.~Bell and D.~Smertnig,\n\\emph{A height gap theorem for coefficients of Mahler functions},\nJ.~Eur.~Math.~Soc.\\ \\textbf{25} (2023), no.~7, 2525--2571,\ndoi:\\href{https://doi.org/10.4171/JEMS/1244}{10.4171/JEMS/1244}.\nTheorem~1.2(b), p.~2528; logarithmic height in Section~1.1.1, p.~2527.\n\\end{thebibliography}\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":43217,"sha256":"sha256:df848f1a20fab00a6a062234582b65feddc3dfddc3964a7d74c1eaab73a64fc0","cite_as":"Plectis reader corpus, paper:erdos-249-binary-totient-series","source_repository":"plectis-erdos","source_path":"paper/249/erdos-249-binary-totient-series.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:088d36d7ba3d3ee0cee2df99e8915927999698f19889f8ea0b8d55b3721ec778","byte_count":6923,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% This is the only copy of the shared preamble.  The eight short notes, the\n% long records assembled from paper/reasoning-parts other than #249 and #257\n% (which carry their own preambles), and the synthesis paper include it with\n% \\input{problem-note-preamble}; paper/Makefile builds from this directory\n% with the directory on the tectonic search path.\n%\n% Shared preamble for the Erdős Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erdős problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{8e58c395e9807ef3374d7365c62cf04d20ee6331}\n\\newcommand{\\commitshort}{9e231ce4371f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-erdos/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-erdos/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\usepackage{longtable}  % the notes' declaration tables; loaded here so the assembled records inherit it\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n% ---------------------------------------------------------------------------\n% Evidence marks.  paper-evidence.tex places \"Lean\" and \"Comparator\" links in\n% the margin beside each result from the paper's generated file\n% paper/evidence/<paper>.tex, which each paper inputs after this preamble.\n% ---------------------------------------------------------------------------\n\\input{paper-evidence}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Production provenance.  The wording is fixed by the corpus provenance\n% contract and reproduced verbatim in every manuscript in this repository, so\n% the papers cannot drift into different accounts of how they were made.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erdős problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every Lean mark in the margin points to propositions that the\n  pinned Lean kernel accepts, with no \\texttt{sorry}, no added axiom, and no\n  unchecked evaluation.  That claim concerns the formal statements only; it\n  says nothing about their mathematical interest, their novelty, or the\n  original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n\n% Compact opening card for the separately stated Comparator theorem selected\n% from each problem note.  The prose supplied by each note owns the\n% contribution and relevance judgements; the shared final sentence states the\n% mechanical assurance boundary once and points to the public evidence owner.\n\\newcommand{\\externalreviewobject}[3]{%\n  \\begin{quote}\\small\n  \\textbf{Contribution.} #1\\par\n  \\textbf{Relation to the open problem.} #2\\par\n  \\textbf{Executable review object.} #3  The repository's\n  external-verification job compares these exact Lean propositions with\n  separately declared challenge statements and an axiom budget, then asks\n  Lean's kernel to check the submitted proofs.  The\n  \\href{https://github.com/wcook04/plectis-erdos/blob/ca0e13f8acf5ccf48506e4bdb870953d3a0856fa/formalization.yaml}%\n  {formalisation manifest} and the commit-bound CI receipt record that check;\n  they do not assess novelty, significance, or whether the original problem is\n  solved.\n  \\end{quote}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  Under\n% Tectonic (XeTeX) the text and mathematics are set in TeX Gyre Pagella from\n% the bundled OpenType fonts, with a Heros sans range for \\mathsf and\n% Inconsolata for monospace; the pdflatex fallback keeps the same Palatino\n% design through newpx.  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Link colours: steel blue for\n% external (Lean-source and web) targets, deep burgundy for internal\n% cross-references and citations.\n\\NeedsTeXFormat{LaTeX2e}\n\\ProvidesPackage{paper-house-style}[2026/07/19 Pagella house style for the papers]\n\n\\RequirePackage{iftex}\n\\RequirePackage{amsmath}\n\\RequirePackage{amsthm}\n\\ifpdftex\n  \\RequirePackage[T1]{fontenc}\n  \\RequirePackage[utf8]{inputenc}\n  \\RequirePackage{amssymb}\n  \\RequirePackage{newpxtext}\n  \\RequirePackage{newpxmath}\n\\else\n  \\RequirePackage{fontspec}\n  \\RequirePackage{unicode-math}\n  \\setmainfont{texgyrepagella}[Extension=.otf,UprightFont=*-regular,\n    BoldFont=*-bold,ItalicFont=*-italic,BoldItalicFont=*-bolditalic]\n  \\setmathfont{texgyrepagella-math.otf}\n  \\setmathfont{texgyreheros-regular.otf}[range=sfup,Scale=MatchLowercase]\n  \\setmonofont{Inconsolatazi4}[Extension=.otf,UprightFont=*-Regular,\n    BoldFont=*-Bold,Scale=MatchLowercase]\n\\fi\n\n\\RequirePackage[a4paper,textwidth=418pt,textheight=636pt,hcentering,\n  top=68pt,headheight=15pt,headsep=20pt,footskip=34pt]{geometry}\n\\RequirePackage{microtype}\n\\linespread{1.045}\n\\clubpenalty=10000 \\widowpenalty=10000 \\tolerance=1600\n\n\\RequirePackage[dvipsnames,table]{xcolor}\n\\definecolor{linkexternal}{HTML}{275D8C}\n\\definecolor{linkinternal}{HTML}{6E2B2B}\n\\definecolor{papertableheader}{HTML}{E3EAF0}\n\\definecolor{papertablerow}{HTML}{F2F5F7}\n\\definecolor{papertablerule}{HTML}{7D8991}\n\\definecolor{paperresultrule}{HTML}{315B72}\n\n% Stable cross-document links between sibling PDFs. A named destination is\n% more durable than a page number: sections may move while their semantic role\n% remains the same. All public PDFs are copied to one directory at release.\n\\newcommand{\\papersectiontarget}[1]{\\hypertarget{#1}{}}\n\\newcommand{\\papersectionlink}[3]{\\href{#1\\##2}{#3}}\n\n\\RequirePackage{titlesec}\n\\titleformat{\\section}{\\Large\\bfseries}{\\thesection}{0.85em}{}\n\\titlespacing*{\\section}{0pt}{2.6ex plus .8ex minus .3ex}{1.5ex plus .3ex}\n\\titleformat{\\subsection}{\\large\\bfseries}{\\thesubsection}{0.8em}{}\n\\titlespacing*{\\subsection}{0pt}{2.1ex plus .6ex minus .2ex}{1.1ex plus .2ex}\n\\titlespacing*{\\paragraph}{0pt}{1.6ex plus .5ex minus .2ex}{0.85em}\n\n\\RequirePackage{fancyhdr}\n\\newcommand{\\paperrunninghead}{}\n\\newcommand{\\runninghead}[1]{\\renewcommand{\\paperrunninghead}{#1}}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\fancyhead[L]{\\small\\itshape\\paperrunninghead}\n\\fancyhead[R]{\\small\\thepage}\n\\renewcommand{\\headrulewidth}{0pt}\n\\fancypagestyle{plain}{\\fancyhf{}\\renewcommand{\\headrulewidth}{0pt}}\n\n\\RequirePackage{caption}\n\\captionsetup{font=small,labelfont=bf,skip=9pt,width=.92\\linewidth}\n\\RequirePackage{etoolbox}\n\\RequirePackage{pdflscape}\n\\renewcommand{\\arraystretch}{1.20}\n\\newcommand{\\papertablehead}{\\rowcolor{papertableheader}}\n\\newcommand{\\papertablebody}{\\rowcolors{2}{white}{papertablerow}}\n\\AtBeginEnvironment{tabular}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\\AtBeginEnvironment{longtable}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\n% A compact front-matter panel for the result a scanning reader should retain.\n% It is deliberately short and unbreakable: manuscripts should state one\n% result, its evidence, and its limit here, then supply the argument below.\n\\newenvironment{paperresult}[1]{%\n  \\par\\addvspace{0.7em}\n  \\begin{center}\\begin{minipage}{0.94\\linewidth}\n    {\\color{paperresultrule}\\rule{\\linewidth}{0.8pt}}\\par\\vspace{4pt}\n    {\\small\\sffamily\\bfseries\\color{paperresultrule}#1}\\par\\smallskip\n    \\small\\color{black}\n}{%\n    \\par\\vspace{4pt}{\\color{paperresultrule}\\rule{\\linewidth}{0.45pt}}\n    \\end{minipage}\\end{center}\\par\\addvspace{0.5em}\n}\n\n% Shared diagram grammar.  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For these papers that inference is false, and\n% a declaration reached only in the back matter would leave it standing for the\n% length of the paper.  The note therefore appears on page one, small and\n% visually subordinate to the title and abstract: enough to prevent the false\n% inference, not enough to make the production method the subject.\n%\n% It carries no mark, because it qualifies the whole manuscript rather than any\n% one passage.  \\footnotetext does not advance the footnote counter, and the\n% emptied \\thefootnote is confined to a group, so numbered footnotes elsewhere\n% in the paper are unaffected.\n%\n% The wording is not authored here.  It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"},{"record_type":"paper_source_dependency","tex_reference":"evidence/erdos-249-binary-totient-series","source_path":"paper/evidence/erdos-249-binary-totient-series.tex","media_type":"text/x-tex","sha256":"sha256:14e84041b4cec4a73ce38c89903cfffc431d1d5a0d526bedac6273f11136dcaf","byte_count":5506,"licence":"CC-BY-4.0","body_format":"latex","body":"% Generated by scripts/paper_evidence.py from docs/paper_lean_coverage.json and the\n% Comparator replay receipts under evidence/comparator/.  Do not edit by hand.\n\\newcommand{\\evidencerecordurl}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md}\n\\DeclareResultEvidenceName{thm:kkernelrank}{Theorem 1.1}\n\\DeclareResultEvidenceRecord{thm:kkernelrank}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#thm-kkernelrank}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:kkernelrank}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR8/FullKernelAssemblies.lean\\#L35}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#thm-kkernelrank-comparator}\n\\DeclareResultEvidenceName{cor:integral-normal-form}{Corollary 3.1}\n\\DeclareResultEvidenceRecord{cor:integral-normal-form}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#cor-integral-normal-form}{exact}{compared}\n\\DeclareResultEvidence{cor:integral-normal-form}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR8/KernelRelationBasis.lean\\#L398}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#cor-integral-normal-form-comparator}\n\\DeclareResultEvidenceName{res:residueseries}{Theorem 4.1}\n\\DeclareResultEvidenceRecord{res:residueseries}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#res-residueseries}{exact}{compared}\n\\DeclareResultEvidence{res:residueseries}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR7/RationalObservableClassification.lean\\#L276}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#res-residueseries-comparator}\n\\DeclareResultEvidenceName{lem:bounded-pulse}{Lemma 4.2}\n\\DeclareResultEvidenceRecord{lem:bounded-pulse}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#lem-bounded-pulse}{exact}{compared}\n\\DeclareResultEvidence{lem:bounded-pulse}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR7/PeriodicAndPulse.lean\\#L106}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#lem-bounded-pulse-comparator}\n\\DeclareResultEvidenceName{res:basis}{Corollary A.1}\n\\DeclareResultEvidenceRecord{res:basis}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#res-basis}{exact}{compared}\n\\DeclareResultEvidence{res:basis}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR8/FullKernelAssemblies.lean\\#L156}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#res-basis-comparator}\n\\DeclareResultEvidenceName{cor:periodic-freezing}{Corollary A.2}\n\\DeclareResultEvidenceRecord{cor:periodic-freezing}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#cor-periodic-freezing}{exact}{compared}\n\\DeclareResultEvidence{cor:periodic-freezing}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR20/PeriodicIntegerAffine.lean\\#L24}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#cor-periodic-freezing-comparator}\n\\DeclareResultEvidenceName{res:fulldepth}{Theorem B.1}\n\\DeclareResultEvidenceRecord{res:fulldepth}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#res-fulldepth}{exact}{compared}\n\\DeclareResultEvidence{res:fulldepth}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR7/ShortNoteAssemblies.lean\\#L35}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#res-fulldepth-comparator}\n\\DeclareResultEvidenceName{res:rankonefloor}{Theorem B.3}\n\\DeclareResultEvidenceRecord{res:rankonefloor}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#res-rankonefloor}{exact}{compared}\n\\DeclareResultEvidence{res:rankonefloor}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#res-rankonefloor}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-249-binary-totient-series.md\\#res-rankonefloor-comparator}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/249/erdos-249-binary-totient-series.tex","section_outline":["Introduction","Proof of the basis theorem","Scalar reductions","Affine independence","The two zero-residue sections","Integral relations","Bounded functions of totient residues","Further consequences for the sections","The binary totient series","Equivalent residue conditions","Comparison sequences","An exponential-sum question","Verification and reproducibility"]},"erdos249-totient-reasoning-surface":{"paper_id":"erdos249-totient-reasoning-surface","title":"The Binary Totient Series","form":"Reasoning surface","length_class":"long","pages":163,"subject":"Erdős #249","subject_kind":"problem","problem_id":"erdos_249","landing_role":"mathematics","question":"What can the totient-kernel basis and residue tests say about Erdős #249, and which hypotheses still block irrationality?","precis":"The dyadic totient sections through level e≥1 have an explicit rational basis of size 2^e+1. Truncating a scaled tail difference produces an integer whose residue certifies nonintegrality when its distances from both endpoints of the residue interval exceed an explicit error bound. The record develops the resulting finite denominator exclusions, equivalent residue criteria, and sufficient exponential-sum estimates. Rational comparison sequences identify the limitations of particular size, parity, rank, and finite-prefix arguments. The short paper supplies the all-base basis theorem; here each conditional result carries the arithmetic hypothesis still needed to apply it. Those hypotheses are not established in the form required to prove irrationality of S=∑ φ(n)2^{-n}.","owns":"a claim-bounded problem-specific reasoning surface for Erdős #249: selected checked premises, finite certificates, closed routes, surviving obligations, and their exact evidence bands","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #249, which remains open","first_pass":["sec:wall","sec:survivors","sec:series","sec:mahler-defect"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos249-totient-reasoning-surface.pdf","hosted_pdf_path":"papers/erdos249-totient-reasoning-surface.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/249/erdos249-totient-reasoning-surface.tex","site_html":"maths/papers/erdos249-totient-reasoning-surface.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos249-totient-reasoning-surface.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/249/erdos249-totient-reasoning-surface.tex","body_pointer":"papers.records.erdos249-totient-reasoning-surface.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Full editorial replacement, 18 September 2026. 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Parts are inlined, never \\input, so\n% that the Pandoc projection cannot silently drop a section.\n\\documentclass[11pt]{article}\n% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared preamble for the two per-problem reasoning-surface papers.\n% These documents are written to be read in full as working context: every\n% result appears with its exact hypotheses, its evidence band, the scale at\n% which it is proved, and the Lean declaration that carries it.\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{adjustbox}\n\\usepackage{booktabs}\n\\usepackage{longtable}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage{xurl}\n\\usepackage{xspace}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdftitle={The Binary Totient Series: results and remaining arithmetic conditions for Erdos Problem 249},\n  pdfauthor={Will Cook},pdflang={en-GB}}\n% Lean declaration names are full of underscores and are typeset in text mode.\n% Category codes are fixed when TeX READS a line, not when a macro expands, so\n% no amount of \\catcode juggling inside \\lean can rescue an underscore that has\n% already been tokenised as a subscript. 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These are never blurred: a reader must always be able to see\n% what a kernel checked, what an exact computation checked, what was proved in\n% ordinary mathematics, what is quoted from the literature, and what is open.\n% ===========================================================================\n\\newcommand{\\ev}[1]{{\\small\\textsf{[#1]}}\\xspace}\n\n% Scale at which a statement is proved. This is the load-bearing axis of both\n% documents: the corpus proves at fixed or bounded scale, and both open\n% producers need cofinal scale.\n\\newcommand{\\scale}[1]{{\\small\\textsf{[#1]}}\\xspace}\n\n% The representation a statement is expressed in. Obstructions are relative to\n% a coordinate, never to the object, so every obstruction carries one.\n% \\nolinkurl manipulates catcodes, so it cannot be written into the\n% hyperref bookmark file. \\coord appears in subsubsection titles, so it\n% must supply a plain-text PDF string alongside the typeset form.\n\\newcommand{\\coord}[1]{} % Source classification retained, not printed.\n\\newcommand{\\codeid}[1]{\\nolinkurl{#1}}\n\n% ===========================================================================\n% Mathematical shorthand.\n% ===========================================================================\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\C}{\\mathbb{C}}\n\\newcommand{\\ph}{\\varphi}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/ErdosProblems/#1\\#L#2}{#4}}\n\\newcommand{\\shortpaper}[2]{\\href{erdos-249-binary-totient-series.pdf\\##1}{#2}}\n\\newcommand{\\EB}{E}\n\n\\allowdisplaybreaks\n\\emergencystretch=2em\n\n% These working records contain a small number of intentionally exhaustive\n% formulas. Keep ordinary displays at their natural size, but bound any display\n% wider than the classical text measure rather than letting it disappear past\n% the page edge.\n\\long\\def\\[#1\\]{%\n  \\leavevmode\\par\\vspace{\\abovedisplayskip}%\n  \\noindent\\makebox[\\linewidth][c]{%\n    \\begin{adjustbox}{max width=\\linewidth}%\n      \\(\\displaystyle #1\\)%\n    \\end{adjustbox}}%\n  \\par\\vspace{\\belowdisplayskip}%\n}\n\n\\runninghead{Erd\\H{o}s \\#249: the binary totient series}\n\\title{The Binary Totient Series}\n\\subtitle{Totient sections, rational comparisons and arithmetic conditions}\n\\author{Will Cook}\n\\date{July 2026; revised 1 October 2026}\n\n\\begin{document}\n% ---- part a249_front ----\n\\maketitle\n\n\\paperprovenance\n\n\\begin{abstract}\nRationality of $S=\\sum_{n\\ge1}\\varphi(n)2^{-n}$ forces a positive\nshift for which every sufficiently late difference of scaled tails\nis an integer. We develop finite residue tests for a failure of this\nintegrality and prove their completeness and propagation laws.\nThe tests yield a finite denominator exclusion, but their required\nsupply at arbitrarily large indices remains unproved. Rational\ncomparison sequences explain why coefficient rank, finite data and\nspecified analytic properties do not supply it. A first-harmonic\nroute isolates the remaining weighted cancellation estimate; its\ngroup-mean bound is recorded with a prime-number-theorem input.\nThe irrationality of $S$ remains open.\n\\end{abstract}\n\n\\section*{Organisation}\nThe argument starts from $2^NS=\\Phi_N+R_N$, with $\\Phi_N\\in\\Z$.\nA rational $S$ gives a shift $h>0$ for which $R_{N+h}-R_N\\in\\Z$\nfor every sufficiently large $N$. Three questions organise the record.\n\n\\emph{What finite calculation can refute that integrality?}\nThe residue test in Definition~\\ref{defn:sep} encloses the tail\ndifference in an explicit error interval. Section~\\ref{sec:series}\ngives the identities and the independent Farey-window exclusion.\nAppendix~\\ref{app:r3-complements} supplies the complete propagation,\nMersenne-residue and rank-one arguments cited in the short paper.\n\n\\emph{Which information is insufficient to supply such calculations?}\nThe \\shortpaper{short249-basis}{short paper} determines all finite-level\ntotient relations, with a basis of $k^e+1$ sections in every base\n$k\\ge2$. Its \\shortpaper{short249-base-six}{integral relation proof}\nrecovers the coefficients of every relation. Those are statements\nabout sequences, not their weighted infinite sums.\nSection~\\ref{sec:mahler-defect} explains the distinction through\ncarry rank; Section~\\ref{sub:generic-carry-rank} constructs a rational\n$5/4$ series retaining the odd totients. Section~\\ref{sec:wall}\ngives other comparisons, each limited to its stated preserved properties.\n\n\\emph{What additional estimate would suffice?}\nSection~\\ref{sec:survivors} gives the quantified conditions.\nSection~\\ref{sub:pivot} separates the weighted exponential-sum estimate\nfrom the counting and group-mean bounds, corresponding to the\n\\shortpaper{short249-first-harmonic}{four-estimate problem} in the\nshort paper. A second sufficient condition estimates only the average\nover good indices and needs no group-mean input.\nAppendix~\\ref{app:r3-declarations} records the formal declarations;\nan implication proved there does not establish its remaining hypothesis.\n\n\\setcounter{tocdepth}{1}\n\\clearpage\n\\tableofcontents\n\\clearpage\n% ; - part a249_front ; -\n\\section{Totient sections and tail differences}\n\nPut $\\varphi(0)=0$ and $\\N=\\{0,1,2,\\ldots\\}$. At base two,\n$\\varphi(4n)=2\\varphi(2n)$ and $\\varphi(4n+2)=\\varphi(2n+1)$ leave five\nindependent sequences among the seven sections through level two.\nMore generally, the sections through level $e\\ge1$ have the basis\n$\\varphi(n)$, $\\varphi(2n)$ and $\\varphi(2^jn+r)$ with\n$1\\le j\\le e$ and odd $0<r<2^j$, of size $2^e+1$.\nThe all-base theorem in the short paper gives rank $k^e+1$ and integral\ncoordinates. Its retained positive residues satisfy $k\\nmid r$;\ncoprimality with $k$ is unnecessary.\n\nThe coefficient relations are finite linear identities. The series\nquestion asks whether a particular \\emph{infinite} weighted sum can\nbe rational. Put\n\\[\n  S \\;:=\\; \\sum_{n \\ge 1} \\frac{\\varphi(n)}{2^{n}}\n  \\;=\\; \\tfrac12 + \\tfrac14 + \\tfrac{2}{8} + \\tfrac{2}{16} + \\tfrac{4}{32} + \\cdots\n  \\;=\\; 1.3676308019850223\\ldots\n\\]\nThe series converges absolutely because $\\varphi(n) \\le n$. Erd\\H{o}s Problem\n\\#249 asks whether $S$ is irrational, which remains open. The comparisons below test proposed passages from coefficient structure\nto the value of this sum. Each retains specified properties while making\nthe corresponding sum rational. It therefore defeats an argument using\nonly those properties, not every argument that might also use the\narithmetic of the totient.\n\nThroughout, a cofinal condition means that a witness exists beyond each\nthreshold. It need not hold at every sufficiently large index. Each\ncomparison sequence is used only for the properties it preserves;\nagreement with a fixed totient prefix, for example, imposes no\nmultiplicativity condition on the continuation.\n\n\\begin{rem}[Earlier coefficient results]\nErd\\H{o}s and Graham record the irrationality question for\n$S$~\\cite[p.~61]{erdosgraham1980}. An integer sequence is $k$-regular\nwhen the $\\Z$-module generated by its sections $n\\mapsto c(k^jn+r)$,\n$j\\ge0$ and $0\\le r<k^j$, is finitely generated. Coons proved that\n$\\varphi$ is not $k$-regular for any $k\\ge2$~\\cite[Theorem~3.2]{coons}.\nMartin's\nTheorem~1 implies that the functions $n\\mapsto\\varphi(a_in+b_i)$ are linearly\nindependent for pairwise nonproportional affine forms with positive integer\nslopes~\\cite[Theorem~1]{martin-phi-inequalities}. The exact ranks in\n\\S\\ref{sec:mahler-defect}, together with the all-base normal form in the\nshort paper, identify the retained sections and every relation among them.\nThe independence itself is not claimed as new. The CRT--Dirichlet separation\nmethod is already present in Yazdani's Theorem~2 and proof, which explicitly\ncredit Shallit~\\cite[pp.~652--653]{yazdani2001}; our evaluation matrix uses\none such separation for each column. Yazdani also proves non-automaticity of\n$\\varphi(n)\\bmod m$ for every $m\\ge3$ in every base~\\cite[Cor.~4]{yazdani2001}.\nAllouche, Shallit and Yassawi give a survey account~\\cite[Thm.~3,\nEx.~4 and Rem.~7]{allouche-shallit-yassawi}.\n\n\\end{rem}\n\nTo test rationality, we remove the finite prefix, which becomes an\ninteger after multiplication by $2^N$. Write $2^{N} S = \\Phi_N + R_N$\nwith $\\Phi_N := \\sum_{n=1}^{N} \\varphi(n) 2^{N-n} \\in \\N$ and\n$R_N := \\sum_{j \\ge 0} \\varphi(N+1+j)/2^{j+1} \\in [0,\\infty)$\n(\\lean{Erdos249257.TotientTailPeriodKiller.totientTail}{TotientTailPeriodKiller.lean:57},\n\\lean{two_pow_mul_totient_series_eq}{TotientTailPeriodKiller.lean:150}).\nIf $S=a/(2^cv)$ with $c\\ge0$ and $v\\ge1$ odd, Euler's theorem gives\n$h=\\varphi(v)>0$ and $N_0=c$ such that $R_{N+h}-R_N\\in\\Z$ for every\n$N\\ge N_0$\n(\\lean{Erdos249257.TotientTailPeriodKiller.eventual_period_of_not_irrational}{TotientTailPeriodKiller.lean:358}).\nIndeed,\n\\[\n R_{N+h}-R_N\\equiv 2^N(2^h-1)S\\pmod\\Z,\n\\]\nand the right side is integral for those choices.\n\nThus it suffices, for every positive shift $h$, to find arbitrarily large\n$N$ for which $R_{N+h}-R_N$ is not an integer. A finite test is obtained by\ntruncating this difference. Throughout, $a\\bmod m$ denotes the least\nnonnegative residue, and an unqualified index in a sum is a nonnegative\ninteger. For $h,N,L\\in\\N$, write\n\\[\n  {D}(h,N,L)\n  \\;:=\\; \\sum_{j<L}\\bigl(\\varphi(N{+}h{+}1{+}j) - \\varphi(N{+}1{+}j)\\bigr)\\,2^{\\,L-1-j} \\in \\Z,\n\\]\nthe depth-$L$ truncation of $2^{L}(R_{N+h} - R_N)$, and\n\\[\n  \\mathcal{C}(h,N,L)\n  \\;:\\Longleftrightarrow\\;\n  (N{+}h{+}L{+}2) \\;<\\; {D}(h,N,L) \\bmod 2^{L}\n  \\;<\\; 2^{L} - (N{+}h{+}L{+}2),\n\\]\na decidable condition\n(\\lean{Erdos249257.TotientTailPeriodKiller.windowDiscrepancy}{TotientTailPeriodKiller.lean:66},\n\\lean{Erdos249257.TotientTailPeriodKiller.certifiedKill}{TotientTailPeriodKiller.lean:72}).\n\nThe omitted tails satisfy the exact identity\n\\[\n 2^L(R_{N+h}-R_N)-D(h,N,L)=R_{N+h+L}-R_{N+L}.\n\\]\nBoth tails are nonnegative, so their difference is bounded in absolute\nvalue by the larger bound $B=N+h+L+2$, not the sum of their bounds.\nThe two strict inequalities in $\\mathcal C(h,N,L)$ keep the resulting\nerror interval away from every multiple of $2^L$; hence\n$R_{N+h}-R_N\\notin\\Z$.\nConversely, a fixed noninteger difference has a fixed positive distance\n$\\delta$ from $\\Z$. Since $2^L\\delta>2B$ eventually, some depth\ndetects it. Completeness concerns detection of an existing difference,\nnot existence of the differences required below\n(\\lean{Erdos249257.TotientTailPeriodKiller.exists_certifiedKill_iff_tail_diff_notMem_int}{LcmConeFlatness.lean:316}).\nThe resulting quantified condition is the following.\n\n\\begin{defn}[The quantified residue condition]\n\\label{defn:sep}\n\\[\n  \\mathrm{Sep}\n  \\;:\\Longleftrightarrow\\;\n  \\forall h \\ge 1\\ \\forall N_0\\ \\exists N \\ge N_0\\ \\exists L,\\;\n  \\mathcal{C}(h,N,L).\n\\]\n$\\mathrm{Sep} \\Longleftrightarrow S\\notin\\mathbb Q$\n(\\lean{irrational\\_totient\\_series\\_iff\\_certificate\\_supply}{LcmConeFlatness.lean:412}).\nThe equivalence identifies the missing quantified input; it supplies no\ncertificate by itself.\n\\end{defn}\n\nOne successful $(h,N,L)$ excludes reduced denominators dividing\n$2^N(2^h-1)$. To exclude an arbitrary rational value, its odd denominator\nfirst determines a shift and its power of two determines a threshold;\nonly then may we choose the witness $N,L$. This is the order in\n$\\mathrm{Sep}$. The depth must grow with $N$, since success requires\n$2(N+h+L+2)<2^L$. A large enough depth makes the interval test possible,\nbut does not make the actual residue lie inside it. A failed test\nat one depth likewise does not prove integrality.\n\\lean{Erdos249257.TotientTailPeriodKiller.certifiedKill_depth_floor}{TotientTailPeriodKiller.lean:79}\n\n\n\\subsection{Unconditional results}\n\nThe finite denominator exclusion, the ranks of the totient sections\nand the tail-positivity statements below require no additional\narithmetic hypothesis. To distinguish their conclusions, note that\npositivity excludes a zero tail difference, whereas irrationality\nrequires a nonintegral one. The rank calculation concerns the\ncoefficients and supplies neither conclusion about the value by itself.\n\n\\begin{thm}[Denominator exclusion from a fixed Farey window]\n\\label{thm:denom}\nIf $S \\in \\Q$ then its reduced denominator exceeds\n\\[\nQ_0 := 79\\,639\\,646\\,646\\,701\\,375\\,323\\,355\\,774\\,875\\,831\\,053 \\approx 7.96 \\times 10^{34}.\n\\]\nEquivalently, $S$ differs from every rational number whose reduced\ndenominator is at most $Q_0$.\nThe bound is sharp for this window: $q = Q_0 + 1$ is the exact first failing\ndenominator. It is the denominator of the mediant of two explicit\nunimodular Farey neighbours.\n\\end{thm}\n\nThis is obtained from a classical Stern--Brocot gap lemma\n(\\lean{farey\\_gap}{GapFareyBound.lean:51}) applied at window $K = 240$.\nOf the recorded bounds $4838$, $2^{22}$, approximately\n$2.49\\times10^{17}$, and $Q_0$, this last is the largest. It is\nproved without assuming $\\mathrm{Sep}$ or using the tail-residue tests.\nTo deduce irrationality by this argument, the analogous denominator bounds\nwould have to be unbounded as the window grows; this has not been proved.\n\n\\begin{prop}[Finite certificate computations]\n\\label{prop:deposits}\n$\\mathcal{C}$ has been verified at: the $28$ diagonal instances of\nthe least-common-multiple diagonal through $t = 64$;\nall shifts $h \\in [1,16]$ simultaneously at $(N,L) = (14,9)$, by\n\\texttt{decide};\nand eight further period examples at $N = 300$. These are historical\nsubsets of the examples. The supplied source also proves a complete diagonal band for\n$1\\le t\\le82$, with no omitted scales.\nThis is bounded coverage, not a cofinal diagonal family.\nNote the quantifier order: the simultaneous small-shift example\nuses one pair $N,L$ for every $1\\le h\\le16$. In symbols, it proves\n$\\exists N\\,\\exists L\\,\\forall h\\,(1\\le h\\le16\\Rightarrow\\mathcal C(h,N,L))$.\nIn contrast, $\\mathrm{Sep}$ requires arbitrarily large $N$ for each\npositive $h$; neither the shift range nor the basepoint threshold is bounded.\n\\end{prop}\n\n\\begin{prop}[Positivity does not exclude an integer]\n\\label{prop:sign}\nPut $H_a=\\operatorname{lcm}(1,\\ldots,2^a)$. For every $a\\ge8$ and\n$J\\ge0$ with $J+(a+6)<2\\cdot2^a$,\n\\[\n R_{2H_a+J}-R_{H_a+J}>0.\n\\]\nThis needs no irrationality hypothesis.\nIf this difference is an integer and $K\\ge0$ satisfies\n\\[\n J+K+(a+6)<2\\cdot2^a,\\qquad 2H_a+J+K+2<2^K,\n\\]\nthen\n\\[\n D(H_a,H_a+J,K)\\bmod2^K\n   =2^K-\\bigl(R_{2H_a+J+K}-R_{H_a+J+K}\\bigr)\n\\]\nlies strictly between $2^K-(2H_a+J+K+2)$ and $2^K$. The later tail difference is a positive integer.\nIts negative, not the carry itself, is the representative near zero.\nThus positivity locates the residue near the upper endpoint; it does not\nestablish the central-residue inequalities. Both restrictions on $K$ are\npart of this conclusion.\n\\end{prop}\n\nFor the next two propositions, a rational value $S=a/v$, with integers\n$a$ and $v\\ge1$, would give the integer carry\n\\[\n u_N=vR_N=2^Na-v\\Phi_N,\\qquad\n u_{N+1}=2u_N-v\\varphi(N+1),\\qquad 0\\le u_N\\le v(N+2).\n\\]\nHere $v$ is any positive integer clearing the denominator, so it\nneed not be odd. The linear bound implies $u_N/2^N\\to0$.\nThe dyadic sections of $u$ are the sequences $n\\mapsto u_{2^jn+r}$.\n\n\\begin{prop}[Rationality forces unbounded carry rank]\n\\label{prop:rank}\nIf $S$ is rational then, for every $e$, the carry sections\n$n\\mapsto u_{2^jn+r}$ with $1\\le j\\le e$ and $0\\le r<2^j$ span a rational\nvector space of dimension at least $2^{e}-1$.\nThe lower bound holds at every depth. It comes from the linear independence\nof the $2^e+1$ retained dyadic totient sections for $e\\ge1$, proved using\nthe Chinese remainder theorem and Dirichlet's theorem,\nso the full family spans an infinite-dimensional space,\nthe case $k=2$ of Coons's non-regularity theorem (\\S\\ref{sec:mahler-defect}).\n\\end{prop}\n\n\\begin{prop}[Periodicity modulo an integer does not bound rational rank]\n\\label{prop:period-not-rank}\nIf $S$ is rational, the same carry $u$ has unbounded rational section rank\nand one eventual period modulo $v$ valid for all its dyadic sections.\nMore precisely, there are $h\\ge1$ and $N_0$ such that, for every $j,r\\ge0$\nand $n\\ge N_0$,\n\\[\n u_{2^j(n+h)+r}\\equiv u_{2^jn+r}\\pmod v.\n\\]\nThe rank and periodicity assertions hold together.\nThis conditional theorem alone is not a counterexample to a general\nperiodicity-to-rank implication: its antecedent is not established.\nThe rational comparison sequence with sum $5/4$ gives a\ncounterexample to a rank upper bound based on rationality alone;\nsee Section~\\ref{sec:mahler-defect}.\n\\end{prop}\n\n\\begin{prop}[Equivalent certificate conditions]\n\\label{prop:iffs}\nPut $H_t=\\operatorname{lcm}(1,\\ldots,t)$. Several variations of\nDefinition~\\ref{defn:sep} are equivalent to $S\\notin\\mathbb Q$. One may allow a positive multiple of each prescribed\nshift, still requiring certificates beyond every basepoint threshold\n(Theorem~\\ref{catalogue:cert:b2}).\nAlternatively, one may use the original quantified condition\nor restrict to $h=N=H_t$ at arbitrarily large $t$\n(Theorem~\\ref{catalogue:cert:b3}).\nReplacing the symmetric residue test by\n\\[\n N+L+2\\le D(h,N,L)\\bmod2^L\\le2^L-(N+h+L+2)\n\\]\nalso gives equivalent quantified conditions, including its restriction to\n$h=N=H_t$ at arbitrarily large $t$.\nFinally, the counted phase-separation condition in\nProposition~\\ref{prop:b2}(b) is equivalent to irrationality.\nThese equivalences change the form of the arithmetic question, not its\nlogical strength.\n\\end{prop}\n\n\\begin{prop}[A rational sequence preserving size bounds, parity and aperiodicity]\n\\label{prop:parity}\nThere is $c : \\N \\to \\N$ with $c(n) \\le 6$ and $c(n) \\le n$ for all $n$,\n$c(n) \\equiv \\varphi(n) \\pmod 2$ for \\emph{every} $n$, and $c$ not eventually\nperiodic; indeed for every $N, G, K$ there are $K$ pairs of coefficients\n$(6,0)$ starting beyond $N$, with starting positions pairwise separated\nby more than $G$; and\n$\\sum_n c(n)/2^{n} = 3/2 \\in \\Q$.\n\\end{prop}\n\nThese propositions separate several logical obligations. Positivity leaves\nan upper-endpoint condition; a large rational carry rank is not a contradiction;\nand exact equivalences still require a quantified certificate condition. The stated size\nbounds, parity and aperiodicity also hold for a rational comparison series. Each\nconclusion concerns its stated hypotheses, not all possible arithmetic\ninformation about $\\varphi$.\n\n\\subsection{Reduced coefficients}\\label{sub:reduced-coefficients}\n\\papersectiontarget{long249-r3-residue-extension}\n\nReduction modulo $m$ restores the boundedness missing from $\\varphi$.\nThe short paper isolates a prescribed nonzero coefficient between two\nlong zero blocks: a hypothetical rational denominator forces the\nright tail to vanish, after which the left block makes the nonzero\ncentre too small. CRT creates the blocks by imposing a separate\nprime divisor congruent to $1\\pmod m$ at each neighbour.\nWe extend that proof in two directions. Changing the integer base\nchanges only the geometric tail bound. Combining dilations is subtler:\none must first ensure that their contributions cannot cancel at the\nchosen centre.\n\n\\begin{prop}[Residue series in every integer base]\n\\label{prop:radixresidue}\nLet $t\\ge2$ be an integer.\n\\begin{enumerate}[label=(\\alph*),leftmargin=*]\n\\item For every $m\\ge3$, $\\sum_{n\\ge1}(\\varphi(n)\\bmod m)\\,t^{-n}$ is\n  irrational\n  (\\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/Erdos249/PaperCompleteR7/RationalIntegerRadixClassification.lean\\#L128}{\\texttt{radix\\_residue\\_series\\_irrational}}).\n\\item For $k\\ge1$ and $f:\\Z/2^k\\Z\\to\\Q$, the series\n  $\\sum_{n\\ge1}f(\\varphi(n)\\bmod2^k)\\,t^{-n}$ is rational exactly when $f$\n  is constant on the even residue classes, and its value is then\n  $(t+1)f(1)/t^2+f(0)/\\bigl(t^2(t-1)\\bigr)$\n  (\\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/Erdos249/PaperCompleteR7/RationalIntegerRadixClassification.lean\\#L145}{\\texttt{rational\\_zmod\\_radix\\_observable\\_iff}},\n  \\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/Erdos249/PaperCompleteR7/IntegerRadixObservables.lean\\#L303}{\\texttt{positiveRadixValue\\_eq\\_of\\_even\\_constant}}).\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nFor coefficients bounded by $C$, the right-block tail is at most\n$Ct^{-L}/(t-1)\\le Ct^{-L}$. Choose $L$ with $Ct^{-L}<1/q$;\nthe rational-tail lattice forces that tail to be zero. The left-block\ntail is then the isolated coefficient times $t^{-L-1}$, giving the\nsame contradiction as in the short paper. In (b), the two exceptional\nterms contribute $(t+1)f(1)/t^2$ and the constant tail contributes\n$f(0)/(t^2(t-1))$, since all totients after $n=2$ are even.\n\\end{proof}\nFor example, in base $2$ the function modulo $4$ with\n$f(0)=f(2)=5$, $f(1)=7$ and $f(3)=-2$ gives the rational value\n$3\\cdot7/4+5/4=13/2$: the class $3$ never occurs. In contrast,\nthe indicator of the class $2\\pmod4$ gives an irrational value.\nThis classification uses the fact that the prime centres $s-1$,\nwith $s$ a unit, cover every even class modulo $2^k$. Modulo $6$\nthey cover only $0$ and $4$. The least-residue assertion needs one\nnonzero accessible centre, but the same argument does not classify\nall rational-valued functions for a general modulus.\n\n\\begin{prop}[Finitely many dilations]\n\\label{prop:dilations}\nLet $t\\ge2$ be an integer.\n\\begin{enumerate}[label=(\\alph*),leftmargin=*]\n\\item For every $m\\ge3$, the numbers $1$ and\n  $\\sum_{n\\ge1}(\\varphi(n)\\bmod m)\\,t^{-dn}$, $d=1,2,\\ldots$, are linearly\n  independent over $\\Q$\n  (\\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/Erdos249/FiniteDilationLinearIndependent.lean\\#L42}{\\texttt{linearIndependent\\_one\\_and\\_least\\_residue\\_values}}).\n\\item Let $D$ be a finite set of positive integers, and for $d\\in D$ let\n  $k_d\\ge1$ and $f_d:\\Z/2^{k_d}\\Z\\to\\Q$. For every $c\\in\\Q$, the number\n  \\[\n   c+\\sum_{d\\in D}\\sum_{n\\ge1}f_d\\bigl(\\varphi(n)\\bmod2^{k_d}\\bigr)\\,t^{-dn}\n  \\]\n  is rational exactly when every $f_d$ is constant on the even residue\n  classes\n  (\\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/Erdos249/FiniteDilationMixedModuli.lean\\#L95}{\\texttt{rational\\_mixed\\_moduli\\_with\\_constant\\_iff}}).\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\n(a) A relation gives integers $c_0$ and $c_d$, $d$ in a finite set $D$ of\npositive integers, with some $c_d\\ne0$ and\n$c_0+\\sum_{N\\ge1}b_Nt^{-N}=0$, where\n$b_N=\\sum_{d\\in D,\\,d\\mid N}c_d\\bigl(\\varphi(N/d)\\bmod m\\bigr)$ is bounded.\nDiscard the $d$ with $c_d=0$ and let $\\delta$ be the least remaining one.\nDifferent dilations can contribute to the same exponent $N$, so\nisolating a prime in one summand is not enough. Choose instead\n$N=\\delta p$ with $p>\\max D$ prime. A contributing $d$ divides\n$\\delta p$; since $p\\nmid d$, it divides $\\delta$. Minimality then\nforces $d=\\delta$. The centre now receives exactly one active\ncontribution, while CRT will remove all contributions at its neighbours.\n\nGiven $L$, choose distinct primes $q_j\\equiv1\\pmod m$, $0<|j|\\le L$,\neach larger than $L$ and than every element of $D$. Impose\n\\[\n p\\equiv-1\\pmod m,\\qquad\n p\\equiv-j\\delta^{-1}\\pmod{q_j}\\quad(0<|j|\\le L).\n\\]\nThe inverse exists because $q_j>\\delta$, and the latter residue is\nnonzero because $q_j>|j|$. Together with the unit residue $-1$\nmodulo $m$, these congruences define a reduced progression.\nCRT and Dirichlet's theorem give a prime $p$ as large as needed.\nAt the centre, $b_{\\delta p}=c_\\delta(m-2)\\ne0$.\nAt a neighbour $\\delta p+j$, every contributing dilation $d$ is\ncoprime to $q_j$, so $q_j\\mid(\\delta p+j)/d$ and\n$m\\mid\\varphi((\\delta p+j)/d)$. Hence $b_{\\delta p+j}=0$.\nThe isolated coefficient makes $\\sum_Nb_Nt^{-N}$ irrational,\ncontradicting $c_0\\in\\Z$.\n\n(b) Each $f_d$ that is constant on the even classes contributes a rational\nnumber, by Proposition~\\ref{prop:radixresidue}(b) in base $t^d$.\nRemove these contributions, lift the remaining functions to one\nmodulus $2^K$, subtract their values at zero, and clear a common\ndenominator. None of these operations changes rationality.\nFor the least remaining dilation $\\delta$, choose an even residue $r$\nwith $f_\\delta(r)\\ne f_\\delta(0)$. Use the construction in (a), now\nwith $q_j\\equiv1\\pmod{2^K}$ and $p\\equiv r+1\\pmod{2^K}$.\nThe central residue is allowed because $r+1$ is odd. Minimality\nagain leaves just the nonzero contribution from $\\delta$ at the\ncentre, while every neighbouring totient is zero modulo $2^K$\nand hence contributes zero after the subtraction. The same bounded\ninteger-coefficient argument applies.\n\\end{proof}\n\n\\begin{rem}[Slowly growing dyadic moduli]\n\\label{prop:slowmoduli}\nLet $\\kappa:\\N\\to\\N$ satisfy $2^{\\kappa(n)}=o(\\log n)$, and put\n$a_n=\\varphi(n)\\bmod2^{\\kappa(n)}$. Then $\\sum_{n\\ge1}a_n2^{-n}$ is\nrational exactly when $a_n=0$ for all large $n$. In particular the sum is\nirrational when $\\kappa(3^k)\\ge2$ for infinitely many $k$.\n\\end{rem}\n\\evidenceremark{Ordinary proof; not formalised in Lean.}\n\n\\begin{proof}\nIf $a_n$ vanishes eventually the sum is finite. Suppose instead that\n$a_n\\ne0$ infinitely often and that the sum equals $u/v$. Then\n$T_N=\\sum_{j\\ge1}a_{N+j}2^{-j}$ lies in $v^{-1}\\Z$ for every $N$. For\n$1\\le j\\le L$ let $q_j$ be a prime factor of the Fermat number\n$2^{2^j}+1$; then $q_j\\equiv1\\pmod{2^{j+1}}$, and the $q_j$ are distinct\nbecause distinct Fermat numbers are coprime. Put $Q=q_1\\cdots q_L$, so\n$Q<2^{2^{L+1}}$. These factors give a size bound as well as the needed\ndivisibility: $\\log Q=O(2^L)$. The factor $2^{-L}$ in the remaining\ntail will offset this growth, leaving the arbitrarily small constant\nfrom $2^{\\kappa(n)}=o(\\log n)$.\nGiven $M$, the Chinese remainder theorem gives\n$N\\in[M,M+Q)$ with $q_j\\mid N+j$ for $1\\le j\\le L$. Then\n$2^{j+1}\\mid\\varphi(N+j)$, so $2^j\\mid a_{N+j}$: either\n$\\kappa(N+j)\\le j$ and $a_{N+j}=0$, or $2^j$ divides both $\\varphi(N+j)$\nand the modulus. Hence $T_N$ differs from an integer by\n$\\sum_{j>L}a_{N+j}2^{-j}$, which is positive because $a_n\\ne0$\ninfinitely often, and at most $\\sum_{j>L}2^{\\kappa(N+j)-j}$. Choose\n$\\epsilon>0$ with $3\\epsilon<1/v$, then $M$ with\n$2^{\\kappa(n)}\\le\\epsilon\\log n$ for $n\\ge M$, then $L\\ge2$ so large that\n$Q\\ge M$. Then $N<M+Q\\le2Q$ and $\\log(N+j)\\le\\log(2Q)+j\\le2^{L+1}+j$, so\nthe tail is at most $\\epsilon2^{-L}(2^{L+1}+L+2)\\le3\\epsilon<1/v$. It is a\npositive element of $v^{-1}\\Z$ below $1/v$, a contradiction. Finally $\\varphi(3^k)=2\\cdot3^{k-1}$ is not divisible by\n$4$, so $a_{3^k}\\ne0$ whenever $\\kappa(3^k)\\ge2$.\n\\end{proof}\n\n\\begin{rem}[Earlier results and the reduced coefficients]\n\\label{rem:erdos1957}\nProposition~\\ref{prop:radixresidue}(a), and with it the first assertion\nof the short paper's residue theorem, already follows from Erd\\H{o}s's\nLemma~1 of~\\cite[p.~213]{erdos1957}: if integers $a_k\\ge0$ have\n$\\sum_{k\\le n}a_k=O(n)$, infinitely many $a_k\\ne0$, and\n$\\liminf_n\\#\\{k\\le n:a_k>0\\}/n=0$, then $\\sum_ka_kt^{-k}$ is irrational for\nevery integer $t\\ge2$. Erd\\H{o}s notes that this is the case $b_k=0$ of his\nLemma~4 (stated pp.~215--216, proved pp.~216--218). For\n$a_n=\\varphi(n)\\bmod m$ the hypotheses hold: $a_n\\ne0$ forces $n$ to have\nno prime factor $p\\equiv1\\pmod m$, and such $n$ have density zero because\n$\\sum_{p\\equiv1\\ (m)}1/p$ diverges, while $a_p=m-2$ for every prime\n$p\\equiv-1\\pmod m$. Erd\\H{o}s's Lemma~4$'$ (p.~218, proof omitted there)\nand Theorem~3 of Kaneko, Suzuki and Tachiya~\\cite[p.~5]{kst2026} give the\nsame conclusion; the latter paper proves Lemma~4$'$ as its Theorem~A\n(pp.~2 and~18). The same argument covers\nProposition~\\ref{prop:radixresidue}(b) and Proposition~\\ref{prop:dilations}\nwhen the values $f(r)-f(0)$, respectively the coefficients $c_d$, have one\nsign. With both signs present, Lemmas~4 and~4$'$ and Theorem~3\nof~\\cite{kst2026} require an additional spacing condition on the negative\nterms, which is not checked here; the isolated coefficient needs none.\nTheorem~3 of~\\cite{kst2026}, with $b=0$ and $z_n=\\log_2\\log x_n$, also\ngives the irrational direction of Remark~\\ref{prop:slowmoduli} once\nthe $n<x$ with $v_2(\\varphi(n))<\\log_2\\log x$ are shown to number\n$o(x/\\log\\log x)$; this record does not write out that density estimate.\nWong's base-equals-modulus theorem~\\cite{wong2015} is the case $t=m$ of\nProposition~\\ref{prop:radixresidue}(a).\n\\end{rem}\n\n\\section{Rational comparison sequences}\n\\label{sec:wall}\n\nRational series can agree with the totient series on a fixed prefix,\non parity at every index, or on specified linear data. The following\nconstructions keep these requirements separate. Each gives a rational\ncomparison satisfying the data named in its statement, and thus\nidentifies what those data alone leave undetermined.\nThe properties in different rows below belong to different constructions;\nthey must not be combined into one comparison theorem.\n\\begin{center}\n\\begin{tabular}{@{}p{.29\\linewidth}p{.65\\linewidth}@{}}\n\\textbf{Construction} & \\textbf{Data kept while the value is rational}\\\\[3pt]\nPrescribed prefix (Theorem~\\ref{thm:gamma}) &\nAny fixed totient prefix and the bound $0\\le c(n)\\le n$.\\\\[3pt]\nParity (Proposition~\\ref{prop:parity}) &\nParity at every index, aperiodicity and the stated size bound.\\\\[3pt]\nValue $5/4$ (Section~\\ref{sub:generic-carry-rank}) &\nEvery odd totient, changes of at most $2$ at even indices, and unbounded carry rank.\\\\[3pt]\nSigned interpolation (Section~\\ref{long249:sec:signed-interpolation}) &\nA prefix, all odd totients, and the specified moment and analytic data;\nnot the full radial germ or all arithmetic identities.\n\\end{tabular}\n\\end{center}\n\n\\subsection{Finite data and rational continuations}\n\nWe begin with a rational continuation of an arbitrary finite prefix\n(Theorem~\\ref{thm:gamma}). Proposition~\\ref{prop:parity} instead\npreserves parity at every index, a linear size bound and aperiodicity.\nThe carry and linear descriptions in Propositions~\\ref{prop:b5}\nand~\\ref{prop:b6} impose different requirements again. Their\ncounterexamples concern the displayed descriptions only; they do not\nexclude all finite-state or finite-dimensional arguments.\n\nTwo other limitations do not require a rational comparison sequence.\nA congruence construction can increase the index so much that its tail\nerror exceeds the residue it produces (Proposition~\\ref{prop:b4}). An exact\nreformulation or a finite list of examples still needs the quantified\narithmetic assertion (Proposition~\\ref{prop:b2}).\n\nThe finite certificates in Proposition~\\ref{prop:deposits} and the $K=240$\nFarey calculation in Theorem~\\ref{thm:denom} have bounded ranges.\nTheorem~\\ref{thm:gamma} excludes\nan inference from one such fixed range uniformly over the whole coefficient\nclass. It does not exclude a proof using the arithmetic of $\\varphi$ at\narbitrarily large horizons. Likewise, the cofinal positivity theorem does\nnot by itself separate the residue from the top edge: that is the additional\ninformation identified by Proposition~\\ref{prop:sign}.\n\n\\subsection{A rational sequence with any prescribed totient prefix}\n\nA finite prefix leaves the remaining coefficients free. We construct a\nrational continuation satisfying the same linear size bound, then show why\nit cannot satisfy the quantified residue condition. First we record the\ntwo tail facts needed for a general coefficient sequence. For\n$c : \\N \\to \\N$ with $c(n) \\le n$, write\n$T_c := \\sum_{n \\ge 1} c(n)/2^{n}$,\n$R^{c}_{N} := \\sum_{j \\ge 0} c(N{+}1{+}j)/2^{\\,j+1}$,\n$D_c(h,N,L) := \\sum_{j<L}(c(N{+}h{+}1{+}j) - c(N{+}1{+}j))2^{\\,L-1-j}$,\nand let $\\mathcal C_c(h,N,L)$ and $\\mathrm{Sep}_c$ be\n$\\mathcal{C}$ and $\\mathrm{Sep}$ with $D_c$ in place of\n${D}$. For $c = \\varphi$ these are the objects of\nDefinition~\\ref{defn:sep}.\n\n\\begin{lem}[A residue certificate excludes an integral tail difference]\n\\label{lem:gsound}\nLet $c:\\N\\to\\N$ satisfy $c(n)\\le n$ for all $n$. For $h,N,L\\in\\N$,\n$\\mathcal C_c(h,N,L) \\Rightarrow R^{c}_{N+h} - R^{c}_{N} \\notin \\Z$.\n\\end{lem}\n\n\\begin{proof}\nSplitting both tails after $L$ terms gives\n\\[\n 2^L(R^c_{N+h}-R^c_N)-D_c(h,N,L)\n   =R^c_{N+h+L}-R^c_{N+L}.\n\\]\nBoth tails on the right are nonnegative, and\n$R^c_M\\le\\sum_{j\\ge1}(M+j)2^{-j}=M+2$. Their difference therefore has\nabsolute value at most $N+h+L+2$. The corresponding totient bound is\nformalised at\n\\lean{Erdos249257.TotientTailPeriodKiller.tail_after_le}{TotientTailPeriodKiller.lean:198} and used\nthere to prove\n\\lean{Erdos249257.TotientTailPeriodKiller.tail_diff_notMem_int_of_certifiedKill}{TotientTailPeriodKiller.lean:262}.\nIf $R^{c}_{N+h} - R^{c}_{N} = k \\in \\Z$ then $2^{L}k \\equiv 0 \\pmod{2^{L}}$\nand $D_c$ lies within $N{+}h{+}L{+}2$ of $2^{L}k$, so\n$D_c \\bmod 2^{L}$ lies within $N{+}h{+}L{+}2$ of $0$ or of $2^{L}$,\ncontradicting $\\mathcal C_c$.\n\\end{proof}\n\n\\begin{lem}[Generic tail-period law]\n\\label{lem:gperiod}\nLet $c:\\N\\to\\N$ satisfy $c(n)\\le n$ for all $n$. If\n$T_c = p/(2^{e}m)$ with $p\\in\\Z$, $e\\ge0$ and $m$ a positive odd integer,\nand if $h\\ge1$ satisfies\n$m\\mid 2^h-1$, then\n$R^{c}_{N+h} - R^{c}_{N} \\in \\Z$ for every $N \\ge e$.\n\\end{lem}\n\n\\begin{proof}\nSince $2^NT_c-R^c_N=\\sum_{n=1}^N c(n)2^{N-n}$ is an integer,\n\\[\n R^c_{N+h}-R^c_N\\equiv2^N(2^h-1)T_c\\pmod\\Z.\n\\]\nThe right side is integral because $2^e\\mid2^N$ and $m\\mid2^h-1$.\n\\end{proof}\n\n\\begin{rem}\nLemmas~\\ref{lem:gsound} and~\\ref{lem:gperiod} are also checked for every\ncoefficient sequence satisfying $c(n)\\le n$. The formal proofs use the\ntrue-tail recurrence and the bound $0\\le R^c_N\\le N+2$; the residue\ncertificate keeps the full finite discrepancy. The tail-period proof uses\n$m\\mid2^h-1$ directly, so oddness of $m$ is only needed when choosing a\nperiod from the denominator. The following rational continuation and its\nfinite-prefix consequence are also checked.\n\\end{rem}\n\n\\begin{thm}[A rational sequence agreeing with any finite totient prefix]\n\\label{thm:gamma}\nLet $B\\ge1$ and $P>B$ be integers. Define $\\gamma : \\N \\to \\N$ by\n\\[\n  \\gamma(n) := \\varphi(n) \\ \\ (n \\le B), \\qquad\n  \\gamma(n) := \\begin{cases} n-1, & P \\mid n \\\\ n, & P \\nmid n \\end{cases}\n  \\ \\ (n > B).\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*),leftmargin=*]\n\\item $\\gamma(n) \\le n$ for all $n$, so $\\gamma$ lies in the same coefficient\n  class as $\\varphi$;\n\\item $\\gamma(n) = \\varphi(n)$ for every $n \\le B$, and consequently\n  $D_\\gamma(h,N,L) = D_\\varphi(h,N,L)$ for every $(h,N,L)$ with\n  $N + h + L \\le B$;\n\\item $T_\\gamma = 2-\\sum_{n\\le B}(n-\\varphi(n))/2^n-1/(2^P-1)\\in\\Q$,\n  and the odd part of its reduced denominator is \\emph{exactly} $2^P-1$;\n\\item $\\mathrm{Sep}_\\gamma$ is false; indeed it fails already at $h = P$.\n\\end{enumerate}\nHence for every $B$ there is a coefficient sequence in the same class,\nagreeing with $\\varphi$ on all of $[1,B]$, whose quantified condition is false.\n\\end{thm}\n\n\\begin{proof}\n(i) is immediate; (ii) holds because $D_\\varphi(h,N,L)$ reads $\\varphi$ only at\nindices $\\le N{+}h{+}L$. For (iii), since $P > B$ every multiple of $P$ exceeds\n$B$, so\n\\[\n  T_\\gamma\n  = \\sum_{n\\ge1}\\frac{n}{2^{n}} - \\sum_{n \\le B}\\frac{n - \\varphi(n)}{2^{n}}\n    - \\sum_{k \\ge 1} 2^{-kP}\n  = 2 - \\sum_{n \\le B}\\frac{n-\\varphi(n)}{2^{n}} - \\frac{1}{2^{P}-1}.\n\\]\nThe first two terms have denominator dividing $2^B$. On putting the\nexpression over $2^B(2^P-1)$, its numerator is congruent to $-2^B$ modulo\n$2^P-1$. Since $\\gcd(2^B,2^P-1)=1$, no odd factor cancels. Thus the\nreduced denominator has odd part exactly $2^P-1$ and its power-of-two part\ndivides $2^B$. For (iv), apply Lemma~\\ref{lem:gperiod} with $h=P$:\n$R^{\\gamma}_{N+P} - R^{\\gamma}_{N} \\in \\Z$ for every $N \\ge B$, so by\nLemma~\\ref{lem:gsound} no $\\mathcal C_\\gamma(P,N,L)$ holds for any\n$N \\ge B$ and any $L$. Taking $N_0 = B$ refutes the inner existential of\n$\\mathrm{Sep}_\\gamma$ at $h = P$.\n\\end{proof}\n\n\\begin{cor}[The limit of a finite-prefix argument]\n\\label{cor:b1}\nNo proof rule uniform over all $c:\\N\\to\\N$ with $c(n)\\le n$ can\nestablish $\\mathrm{Sep}$ from a single fixed prefix\n$\\{c(n):n\\le B\\}$: Theorem~\\ref{thm:gamma} supplies a rational countermodel\nwith that same prefix.  This does \\emph{not} invalidate an argument that uses\nthe fixed arithmetic sequence $\\varphi$ together with compatible information\nat arbitrarily large horizons; the theorem gives a different $\\gamma_B$ for\neach $B$, not one sequence agreeing with $\\varphi$ at every $B$.\n\\end{cor}\n\n\\begin{rem}[Four distinct finite restrictions]\n\\label{rem:b1-scope}\nTheorem~\\ref{thm:gamma} concerns the information in a finite prefix.\nIt does not identify $\\varphi$ with one of the comparison sequences. It also\ndoes not exclude a proof that combines finite computation with an arithmetic\ntheorem applying at all larger indices. Four different restrictions must\nbe kept separate: a bound on the indices inspected, a bound on the depth\n$L$, a bound on the carry states retained, and a bound on matrix rank.\nThe depth inequality $2(N+h+L+2)<2^L$, for example, concerns the second\nrestriction, not all four.\n\\end{rem}\n\nProposition~\\ref{prop:parity} gives another rational comparison sequence,\nthis time preserving parity and specified aperiodicity properties. The two\nexamples test different assumptions and must not be combined into a single\nstronger example.\n\n\n\\subsection{Limits of specific reductions and estimates}\n\nWe test several proposed deductions by retaining their assumptions\nand changing the conclusion. The constructions concern specified\ncongruences, recurrences or estimates, so their force is limited to\nthe corresponding deductions.\n\n\n\\begin{prop}[Three particular equivalences]\n\\label{prop:b2}\n\\begin{enumerate}[label=(\\alph*),leftmargin=*]\n\\item \\emph{Certificate completeness.} The complete residue tests considered here are\n  equivalent to the corresponding nonintegrality assertions. Rewriting the\n  quantified condition using one of these equivalences does not weaken it.\n\\item \\emph{A selectable two-point sample.} The condition is as follows.\n  For every $h\\ge1$ and $X_0$, choose $X,L\\in\\N$ with\n  $X\\ge\\max(X_0,1)$ and $16(2X+h+L+2)\\le2^L$, a nonempty set\n  $T\\subseteq[X,2X)\\cap\\N$, a set $P\\subseteq T\\times T$ of ordered\n  pairs, and a real number $\\delta\\ge0$. With\n  $E_N=\\exp(2\\pi iD(h,N,L)/2^L)$, require\n  \\[\n   |E_i-E_j|\\ge\\delta\\quad((i,j)\\in P),\\qquad\n   \\frac{2|T|^2}{5}\\le |P|\\delta^2.\n  \\]\n  This condition is equivalent to irrationality. For the converse,\n  irrationality and the doubling map allow\n  $T=\\{N,N+1\\}$ and $P=\\{(N,N+1),(N+1,N)\\}$ at arbitrarily large $N$,\n  with $\\delta=9/10$. The numerical requirement is then\n  $8/5\\le2(9/10)^2$. The freedom to select $T$ matters: this is not an\n  estimate on a sample prescribed in advance, such as all prime positions.\n\n\\item \\emph{Four integral tail differences.} For $H\\ge0$ and positive\n  integers $p,q$, the conjunction\n  \\[\n   R_{2kH}-R_{kH}\\in\\Z\\qquad(k\\in\\{1,p,q,pq\\})\n  \\]\n  is equivalent to $R_{2H}-R_H\\in\\Z$ alone. The affine transport identity\n  for $H\\mapsto kH$ preserves integrality, so the three additional\n  conditions add no restriction. Neither $p$ nor $q$ need be prime.\n\\end{enumerate}\nThese three equivalences explain why these particular reformulations\nretain the original arithmetic question. They do not rule out useful weaker\nintermediate lemmas. The full-block and fixed-margin conditions in\n\\S\\ref{sec:survivors} ask for additional quantitative information that the\nthree arguments above do not supply.\n\\end{prop}\n\n\\begin{rem}\nSecond-difference certificates need not occur at a smaller depth than\nfirst-difference certificates. At $(h,N)=(1,8)$, the first-difference test\nholds at depth $8$, whereas no second-difference certificate exists at a\ndepth at most $8$ (Proposition~\\ref{catalogue:cert:b9a}). This one\nexample refutes a uniform claim that second differences always save depth;\nit does not give a uniform ordering in the opposite direction.\n\nSeparately, unbounded growth of the denominator exclusions produced by the\nFarey windows would imply irrationality. The record does not establish a\nconverse for that specific sequence of windows, so the proposed growth\ncondition is used only as a sufficient condition.\n\\end{rem}\n\n\\begin{prop}[Failure of a specified two-adic congruence construction]\n\\label{prop:b4}\nA certificate requires the residue to lie farther than $N+h+L+2$\nfrom either endpoint modulo $2^L$. In the specific construction below,\nthe residue equals $2^{K-1}$ modulo $2^K$, but the defining\ncongruence is $p\\equiv1+2^{K-1}\\pmod{2^K}$. Hence\n$p\\ge1+2^{K-1}$. At $N=p-K$, $h=H$, and $L=K$, the error bound is\n$p+H+2>2^{K-1}$, so the residue does not satisfy the certificate inequalities.\nThis calculation defeats this construction at every depth. It does not\nprove a corresponding statement for every use of the Chinese Remainder\nTheorem or every prescribed totient pattern.\n\\end{prop}\n\n\\begin{rem}[Two further prescribed patterns]\n\\label{rem:b4}\nTwo related results concern particular prescribed patterns. Under the\nstated size condition, the complete terminal divisibility pattern is\nimpossible: its last totient difference would be positive, below a modulus,\nand divisible by that modulus\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.not_actualLcmTerminalDyadicStaircase_of_room}{TotientActualLcmTopEdgeStaircase.lean:251}).\nFor the pattern with the last condition\nremoved, the penultimate difference is forced to be $2^{m-1}$, with\n$2^m<2(2H+J+K+2)$.\n\nA different sufficient condition would prescribe the accumulated sum\nrather than an individual difference:\n${D}(h,N,L)\\equiv2^{L-1}\\pmod{2^L}$ and\n$2^{L-1}>N+h+L+2$. These two inequalities would give a certificate.\nThey have not been supplied at the required quantified scope. The failure\nof the earlier patterns does not prove that this replacement is necessary.\n\\end{rem}\n\n\\begin{prop}[Information lost by specific carry descriptions]\n\\label{prop:b5}\n\\begin{enumerate}[label=(\\roman*),leftmargin=*]\n\\item Fix $m\\ge2$ and write $R=\\lfloor(m+1)/2\\rfloor$. For\n  $r=0,\\ldots,R$, let $c_r$ vanish except at\n  $c_r(m)=R-r$ and $c_r(m+1)=2r$. Each sequence satisfies\n  $0\\le c_r(n)\\le n$, has the same binary sum $R2^{-m}$, and has\n  the same coefficients and scaled tails before position $m$.\n  Its scaled tail at position $m$ is $r$. The common history therefore\n  does not determine that tail. A finite set of labels that determines\n  the tail for each member of this family must contain at least\n  $R+1=\\lfloor(m+1)/2\\rfloor+1$ elements, one for each value of $r$.\n\\item Reduction modulo $2^L$ forgets the initial value after the same\n$L$ recurrence steps: two\n  affine binary orbits with different seeds satisfy\n  $\\mathrm{orbit}_u(L) - \\mathrm{orbit}_v(L) = 2^{L}(u_0 - v_0)$, so the\n  endpoint residue mod $2^{L}$ is independent of the initial carry.\n\\item Fix a precision $u\\ge1$, a finite list of nonnegative valuations\n  $\\nu_j$ and odd integers $a_j$, and an initial integer $e_0$.\n  There are integers $z_j$ such that\n  \\[\n   c_j=2^{\\nu_j}(a_j+2^u z_j),\\qquad\n   e_{j+1}=2e_j+c_j,\\qquad |e_{j+1}|\\le2^{\\nu_j+u-1}.\n  \\]\n  At each step, choose the centred representative of\n  $2e_j+2^{\\nu_j}a_j$ modulo $2^{\\nu_j+u}$.\n  The unrestricted integers $z_j$ are essential: the assertion does not\n  prescribe the full coefficients $c_j$.\n\\end{enumerate}\nThese statements exclude decoders using only the specified common\nhistory, distinctions using only the endpoint residue, and contradictions\nusing only the stated local symbols. They do not prove that the actual\ntotient expansion cannot be studied by automata. A proposed application\nmust verify that it uses no additional arithmetic information that the\ncomparison families fail to preserve.\n\\end{prop}\n\n\\begin{prop}[Four limits of particular linear constructions]\n\\label{prop:b6}\nThe following four constructions have different limitations. They must\nnot be read as a claim that every finite family of totient sections is\nindependent: the full family has the explicit relations described in the\nshort paper.\n\\begin{enumerate}[label=(\\roman*),leftmargin=*]\n\\item \\emph{Dyadic sections and an integer identity.} The retained\n  family of $2^e+1$ dyadic sections is linearly independent over $\\Q$\n  for $e\\ge1$ (Proposition~\\ref{prop:rank}). Separately, positive integers\n  $Q,v$ and integers $A,b$ cannot satisfy\n  \\[\n     A\\ne0,\\qquad QvA=b,\\qquad |b|<Qv,\n  \\]\n  because $|A|\\ge1$ gives $|b|\\ge Qv$.\n  This elementary incompatibility concerns the displayed requirements;\n  it does not exclude every argument using an adjugate matrix.\n\\item \\emph{M\\\"obius incidence.} $U_N(i,j) = \\mu((i{+}1)/(j{+}1))$ when\n  $(j{+}1) \\mid (i{+}1)$ and $0$ otherwise is lower triangular with unit\n  diagonal, so $\\det U_N = 1$ for every $N$. Hence multiplication by\n  $U_N$ is injective; the corresponding coefficient transformation cannot\n  create a nonzero vector in its kernel.\n\\item \\emph{Adjugate reconstruction.} Let $w_i\\in\\Q$ and\n  $x_i\\in\\N$ be finite families with $\\sum_i w_i\\varphi(x_i)=1$.\n  Reconstructing each value from two tails and bounding them separately\n  gives the error bound $\\sum_i|w_i|(3x_i+4)$. It is at least $3$, since\n  \\[\n   1\\le\\sum_i|w_i|\\varphi(x_i)\\le\\sum_i|w_i|x_i.\n  \\]\n  Consequently this particular bound cannot be less than $1$, whatever\n  the size of the evaluation matrix.\n\\item \\emph{Finite combinations of shifts.} The synthetic sequence in\n  Observation~\\ref{prop:B4b-kill} has the prescribed differences\n  $a_{(q-1)H-1}=\\varphi(H)$ for $2\\le q<t$, where\n  $H=\\operatorname{lcm}(1,\\ldots,t)$. It is of the form\n  $a_i=2c_i-c_{i+1}$, and every finite integer combination of its shifts\n  has the same form with a correspondingly shifted state. Thus these\n  linear operations alone do not remove the compatible carry recurrence.\n  Their uniform bounds depend on the absolute coefficient sum, as made\n  explicit in that observation. The construction does not assert that\n  $a_i$ equals the actual totient difference at other indices.\n\\end{enumerate}\nA further limitation concerns quotients of the finite sums\n$t(Y,r)=\\sum_{d=1}^{Y}\\mu(d)/(2^d-1)^r$. For $e\\ge1$ and $Y\\ge4$,\n\\[\n \\frac{t(Y,e+2)^2}{t(Y,2e+2)}-(S-\\tfrac12)>\\frac1{480}.\n\\]\nThe proof bounds each infinite sum\n$\\sum_{d\\ge1}\\mu(d)/(2^d-1)^r$, $r\\ge3$, between $1429/1512$ and $1$,\nand its truncation error after $Y\\ge4$ terms by $1/3584$.\nThus the lower bound holds for every stated pair $e,Y$, not just a finite\nlist of computed examples.\nThese results concern the listed matrices, estimates, and comparison\nsequence. The fourth item has the explicit construction and source theorem given\nin Observation~\\ref{prop:B4b-kill}. A different finite-dimensional argument,\nor one using further arithmetic assumptions, is not excluded.\nProposition~\\ref{prop:period-not-rank} is conditional on rationality of\n$S$ and is not, by itself, a counterexample. The separate rational $5/4$\ncomparison supplies the counterexample to the generic rank bound.\n\\end{prop}\n\n\\begin{prop}[The coefficient properties of the rational example]\n\\label{prop:b7}\nThe rational sequence in Proposition~\\ref{prop:parity} satisfies uniform\nboundedness, $c(n)\\le n$, agreement with $\\varphi(n)$ modulo $2$ at every\nindex, and the stated separated-carry form of aperiodicity. Consequently,\nthose properties alone cannot imply irrationality of a dyadic series.\nThe example does not exclude arguments using further totient identities,\nincluding its multiplicative relations.\n\\end{prop}\n\n\\subsection{Consequences of the comparisons}\n\nThe residue margin and the truncation error must be controlled\ntogether. Enlarging a congruence may enlarge both, while a finite\nprefix constrains no later coefficients. Periodicity modulo an\ninteger likewise places no general upper bound on rational rank.\nThese comparisons leave the equivalent residue conditions intact,\nwith their original requirements for witnesses at unbounded indices.\n\nThese are limitations of specified arguments, not an exhaustive\nclassification of possible methods. In particular, an averaging estimate is\none possible source of the missing information, not the only possible\nsource. A new arithmetic identity or a propagation theorem could also be\nuseful. The next section describes the precise assumptions of several\napproaches considered here.\n\n\\section{Conditions that would imply irrationality}\n\\label{sec:survivors}\n\nBelow are quantitative sufficient conditions, an equivalent\nresidue formulation, and a proposed generic rank bound. The rank bound\nis false, as its rational counterexample shows. The other arithmetic\nconditions remain open for the totient; the proofs establish their\nstated implications. They are examples of possible approaches, not\nan exhaustive list.\n\n\\subsection{Cancellation on a full block}\n\nIf every window residue lies close to an endpoint, its complex phase\nlies close to $1$. A saving in the norm of a block sum must therefore\nproduce a certificate.\n\n\\begin{defn}[The full-block norm condition]\n\\label{defn:fh}\nWrite $e(x) := \\exp(2\\pi i x)$ and\n${E}(h,N,L) := e\\bigl(({D}(h,N,L) \\bmod 2^{L})/2^{L}\\bigr)$.\nThe block norm condition asserts that\n\\[\n  \\forall h \\ge 1\\ \\forall X_0\\ \\exists X, L : \\quad\n  \\max(X_0,1) \\le X, \\quad\n  16(2X + h + L + 2) \\le 2^{L}, \\quad\n  \\Bigl\\lVert \\sum_{N \\in [X,\\,2X)} {E}(h,N,L) \\Bigr\\rVert\n  \\le \\tfrac{21}{25}X .\n\\]\nIt implies $S\\notin\\mathbb Q$.\n\\lean{Erdos249257.TotientTailPeriodKiller.DTWFirstHarmonicNormGap}{FirstHarmonicPivot.lean:75}\n\\lean{Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_first_harmonic_norm_gap}{FirstHarmonicPivot.lean:83}\n\\end{defn}\n\nA bound on the real part also suffices: it is enough that\n$\\sum_{N \\in [X,2X)} \\operatorname{Re}E(h,N,L) \\le \\tfrac{9}{10}X$\n(\\lean{Erdos249257.TotientTailPeriodKiller.exists_certifiedKill_of_first_harmonic_gap}{FirstHarmonicGap.lean:128};\nfor the implication from $21/25$ to $9/10$,\n\\lean{Erdos249257.TotientTailPeriodKiller.exists_certifiedKill_of_first_harmonic_norm_gap}{FirstHarmonicPivot.lean:53}).\nUnpacked, the required object is a constant-saving cancellation estimate for\nthe dyadically weighted totient-difference exponential sum\n\\[\n  \\sum_{X \\le N < 2X}\n  e\\!\\left(\\frac{\\sum_{j<L}\\bigl(\\varphi(N{+}h{+}1{+}j)-\\varphi(N{+}1{+}j)\\bigr)2^{\\,L-1-j}}{2^{L}}\\right),\n  \\qquad L \\text{ subject to the displayed size inequality}.\n\\]\nNo theorem in this record establishes this estimate with the\nrequired quantifiers. The displayed size inequality imposes a lower\nbound on $L$; it does not require $L=\\log_2X+O(1)$. A depth of that\norder would be a possible regime for the estimate, not an additional\nproved conclusion. A finite instance would not suffice.\nThe norm hypothesis requires cancellation: if all phases are $1$, the\nnorm is $X$ and the bound fails; a balanced set of opposite phases has\nsum zero. If instead every phase is $-1$, the real part is $-X$ and\nsatisfies the real-part bound, while the norm is $X$ and fails the norm\nbound. Thus a real-part estimate is strictly weaker for general phase\nfamilies; these examples do not distinguish the two quantified conditions\nfor the actual totient sequence. The same depth $L$ must work across the\nwhole block.\n\n\\paragraph{Comparison with the preceding counterexamples.} The earlier counterexamples do not establish or refute this block\nestimate. The following distinctions explain why those particular arguments\ndo not settle it.\n\n\n\\begin{itemize}[leftmargin=*]\n\\item The parameters grow. The size condition forces the truncation depth\n  to increase with $X$, so this is not a test of a fixed prefix.\n\\item The sample is prescribed: all $N\\in[X,2X)$ are summed at one depth.\n  Choosing two favourable indices does not establish this block estimate.\n\\item The sum uses the accumulated totient differences. It does not\n  prescribe individual values through a congruence construction.\n\\item The estimate concerns the actual phases. Neither a bound on carry\n  rank nor agreement only in parity gives it.\n\\item The proved implication uses the real-part threshold $9/10$ and the\n  constant in the size bound $16$. The stronger norm bound $21/25$ implies that\n  real-part bound. These constants do not assert any distribution theorem\n  for the totient phases.\n\\end{itemize}\n\nFormal sources for these implications: \\lean{Erdos249257.TotientTailPeriodKiller.windowFirstCos_gt_of_not_certifiedKill}{FirstHarmonicGap.lean:44}\n\\lean{Erdos249257.TotientTailPeriodKiller.exists_certifiedKill_of_first_harmonic_gap_subset}{FirstHarmonicGap.lean:104}\n\n\n\\begin{rem}[The finite-prefix example does not choose a method]\n\\label{rem:b1-not-selector}\nThe finite-prefix example does not select an analytic method.\nIt shows only that a proof valid for every sequence $0\\le c(n)\\le n$\ncannot deduce the required assertion from one fixed prefix. An argument\nusing additional identities of $\\varphi$, a theorem propagating a finite\nbase case, a Farey estimate, or a suitable exponential sum could provide\ninformation not contained in that prefix. The earlier finite audit cannot\nrank all such possibilities.\n\\end{rem}\n\n\\begin{rem}[Finite certificate observations]\nIn the historical table of $28$ diagonal certificates through $t=64$,\nthe reported depths exceed the necessary depth bound by a small additive\nconstant. A separate enumeration at the indices where the least common\nmultiple increases records a closest margin from the central interval's\nboundary of approximately $0.000221$ of the modulus, at $t=100$.\nBoth are finite observations. Neither establishes nor refutes\nequidistribution or the estimate at arbitrarily large scales.\n\\end{rem}\n\n\\subsection{A four-term bound for the real part}\\label{sub:four-sums}\n\nWe isolate the phase of one term $\\varphi(mp)$ with $p>m$ prime,\nthen group indices with the same cofactor $m$. The resulting identity\nseparates three tasks: count the excluded indices, bound the group\nmeans, and estimate the remaining weighted, mean-subtracted sum.\nOnly the last of these remains open after the stated group-mean input.\nThe identity itself supplies none of the estimates.\n\nFix $h,s\\ge1$, $L\\ge s+h$, $X\\ge1$ and $0<\\eta<1$, and put\n$t=L-s+1$. For $X\\le N<2X$, let $p_N$ be the largest prime factor of\n$N+t$ and put $m_N=(N+t)/p_N$. Assign the index $N$ to this factorisation\nwhen\n\\[\n m_N\\le\\left\\lfloor\\frac{\\sqrt X}{2}\\right\\rfloor,\n \\qquad p_N>2\\lfloor\\sqrt X\\rfloor.\n\\]\nWrite $\\mathcal A$ for the assigned indices and\n$\\mathcal G=\\{N\\in\\mathcal A:\\varphi(m_N)\\ge\\eta m_N\\}$.\nThe two inequalities ensure $p_N>m_N\\ge1$, so\n$\\varphi(N+t)=\\varphi(m_N)(p_N-1)$.\n\nSince $h<t\\le L$, the selected term occurs in both truncated tails:\nit has coefficient $2^{-(t-h)}$ in the shifted tail and $-2^{-t}$ in\nthe unshifted one. Their sum is $(2^h-1)2^{-t}$. On the assigned indices,\nset\n\\[\n z_N=e\\!\\left(\\frac{(2^h-1)\\varphi(m_N)(p_N-1)}{2^t}\\right),\n \\qquad w_N=\\frac{E(h,N,L)}{z_N}.\n\\]\nBoth factors have absolute value $1$. Let $\\bar z_m$ be the average of\n$z_N$ over the assigned indices with cofactor $m$, taking the average to\nbe zero when the group is empty. The four contributions are then defined\nby the identity\n\\[\n\\begin{aligned}\n \\sum_{N=X}^{2X-1}E(h,N,L)\n &=\\underbrace{\\sum_{N\\in\\mathcal G}w_N(z_N-\\bar z_{m_N})}\n                 _{\\text{mean-subtracted contribution}}\n   +\\underbrace{\\sum_{N\\in\\mathcal G}w_N\\bar z_{m_N}}\n                 _{\\text{mean contribution}}\\\\\n &\\quad+\\underbrace{\\sum_{N\\in\\mathcal A\\smallsetminus\\mathcal G}E(h,N,L)}\n                 _{\\text{excluded-cofactor contribution}}\n   +\\underbrace{\\sum_{\\substack{X\\le N<2X\\\\N\\notin\\mathcal A}}E(h,N,L)}\n                 _{\\text{unassigned contribution}}.\n\\end{aligned}\n\\]\nThis follows by partitioning the indices and using $E(h,N,L)=w_Nz_N$.\nSubtracting the group mean is exact algebra; it does not make the two\nfactors independent. The chosen prime need not be confined to the selected\nargument: with $X=16$, $h=s=1$, $L=20$ and $N=18$, the argument is\n$N+t=38=2\\cdot19$, which satisfies both assignment inequalities, but the\nsame prime also divides $N+1=19$. Thus the factorisation at $N+t$ alone\ngives no separation from the other totient values in the phase.\n\n\n\\begin{defn}[Four simultaneous inequalities]\n\\label{defn:pivot}\nFor every $h > 0$ there are $s > 0$ and $\\eta \\in (0,1)$ such that for every\n$X_0$ there are $X, L$ with $\\max(X_0,1) \\le X$, $h \\le L - s$,\n$16(2X{+}h{+}L{+}2) \\le 2^{L}$, and all four of\n\\[\n\\begin{aligned}\n \\operatorname{Re}\\sum_{N\\in\\mathcal G}w_N(z_N-\\bar z_{m_N})\n   &\\le \\tfrac{14}{25}X,\\\\\n \\left\\lVert\\sum_{N\\in\\mathcal G}w_N\\bar z_{m_N}\\right\\rVert\n   &\\le \\tfrac{1}{100}X,\\\\\n \\left\\lVert\\sum_{N\\in\\mathcal A\\smallsetminus\\mathcal G}E(h,N,L)\\right\\rVert\n   &\\le \\tfrac{1}{100}X,\\\\\n \\left\\lVert\\sum_{\\substack{X\\le N<2X\\\\N\\notin\\mathcal A}}E(h,N,L)\\right\\rVert\n   &\\le \\tfrac{8}{25}X.\n\\end{aligned}\n\\]\nIt implies $S\\notin\\mathbb Q$.\n\\lean{Erdos249257.TotientTailPeriodKiller.DTWPivotResidualDecorrelation}{FirstHarmonicPivot.lean:569}\n\\lean{Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_pivotResidualDecorrelation}{FirstHarmonicPivot.lean:577}\n\\end{defn}\n\nFirst fix $h$, then choose $s,\\eta$; these must remain fixed as the\nthreshold $X_0$ increases. At each threshold all four bounds must hold\nfor one common $X,L$, rather than along unrelated sequences of blocks.\nTheir constants add to $14/25+1/100+1/100+8/25=9/10$, yielding the\nreal-part criterion. This is sufficient for a certificate and does not\nassert the stronger $21/25$ norm criterion. In particular, no\nindependence of the two phase factors follows from the grouping.\n\n\nThe decomposition is an exact identity, proved in Lean\n(\\lean{Erdos249257.TotientTailPeriodKiller.windowFirstExp_sum_eq_pivot_decomposition}{FirstHarmonicPivot.lean:514}),\nas is the implication from the four bounds to the real-part bound\n(\\lean{Erdos249257.TotientTailPeriodKiller.first_harmonic_gap_of_pivotBudgetAt}{FirstHarmonicPivot.lean:549}).\nThe reason a new estimate is needed is visible within one group:\n$\\sum_N(z_N-\\bar z_m)=0$, whereas\n$\\sum_Nw_N(z_N-\\bar z_m)$ need not vanish. The residual weight varies\nwith the same prime that determines $z_N$. By contrast, a uniform\nbound for $|\\bar z_m|$ immediately controls the mean contribution\nbecause $|w_N|=1$. Thus small means and weighted cancellation are\ndifferent statements.\nA two-point group makes this explicit:\n\\[\n (z_1,z_2)=(1,-1),\\quad (w_1,w_2)=(1,-1),\\quad \\bar z=0,\n \\qquad \\sum_{j=1}^2w_j(z_j-\\bar z)=2.\n\\]\nThe unweighted mean vanishes while the weighted terms agree exactly.\nThis example explains why an implication is missing; it is not a\ncounterexample to the estimate for the actual totient windows. For the latter, only the real-part bound\n$14X/25$ is required, not the earlier norm bound $X/2$.\nFor $X\\ge4$ and $s=L-h$, the assigned indices \\emph{with cofactor\n$m_N=1$} are exactly those for which $N+h+1$ is prime. This describes\none cofactor group, not all assigned indices. The source proves this\nmembership equality\n(\\lean{Erdos249257.TotientTailPeriodKiller.mem_pivotFiber_one_overlap_iff}{FirstHarmonicPivot.lean:423}),\nusing $\\varphi(mp)=\\varphi(m)(p-1)$ and $p>m$.\nThis special choice has $s=L-h$ varying with $L$; it illustrates the\narithmetic of the groups but does not establish the condition with $s$\nfixed before $X_0$.\n\nThe four inequalities have not been established together.\nPropositions~\\ref{prop:dickman} and~\\ref{prop:badcof} give the unassigned\nand excluded-cofactor bounds at minimal admissible depth. At $s=26$,\n$\\eta=1/1000$, the prime-number-theorem argument discussed in\nSection~\\ref{sub:goodbasegap} also bounds the mean contribution for all\nlarge $X$. With that input, the missing estimate is the real-part bound\nfor the mean-subtracted contribution, at the same parameters and depth,\non arbitrarily large blocks. Theorem~\\ref{thm:goodbasegap} instead asks\nfor one bound on $\\mathcal G$, without a group-mean input.\n\n\nA nonzero determinant alone does not prove the needed correlation\nestimate. The comparison in the linked source\n(\\lean{Erdos249257.ResidualGaugeObstruction.locked_reconstruction_preserves_nonzero_minor}{ResidualGaugeObstruction.lean:94})\npreserves such a determinant\nwhile fixing an entire row of weighted phases to $1$. A successful\napplication must distinguish that comparison, for example by additional\narithmetic information, rather than only by column rescaling. The exact\nfour-term identity supplies no prime-distribution estimate.\n\n\\subsection{One bound on the good indices}\\label{sub:goodbasegap}\n\nTwo counting estimates allow us to restrict the cancellation question\nto $\\mathcal G$. For fixed $h,s$, let $L(X)$ be the least $L$ satisfying\n$h\\le L-s$ and $16(2X+h+L+2)\\le2^L$, as in\nProposition~\\ref{prop:dickman}. With $s=26$ and $\\eta=1/1000$, more\nthan $67X/100$ indices remain in $\\mathcal G$ for large $X$.\nA suitable saving on this set alone would therefore imply irrationality.\n\n\\begin{thm}[Irrationality from one bound on the good indices]\n\\label{thm:goodbasegap}\n\\begin{enumerate}[label=(\\alph*),leftmargin=*]\n\\item Suppose that for every $h\\ge1$ and every $A$ there are $X$, $L$ with\n  $16(2X+h+L+2)\\le2^L$ and a nonempty finite set $T$ of integers in\n  $[A,2X)$ with\n  \\[\n   \\sum_{N\\in T}\\operatorname{Re}E(h,N,L)\\le\\tfrac{9}{10}\\,\\#T.\n  \\]\n  Then $S\\notin\\Q$\n  (\\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean\\#L44}{\\texttt{irrational\\_totient\\_series\\_of\\_support\\_gap}}).\n\\item Take $s=26$, $\\eta=1/1000$ and $L=L(X)$. Suppose that for every\n  $h\\ge1$ there are arbitrarily large $X$ with\n  \\[\n   \\operatorname{Re}\\sum_{N\\in\\mathcal G}E\\bigl(h,N,L(X)\\bigr)\n   \\le\\tfrac{603}{1000}X.\n  \\]\n  Then $S\\notin\\Q$\n  (\\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean\\#L100}{\\texttt{irrational\\_totient\\_series\\_of\\_goodBase\\_gap}}).\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\n(a) At a depth satisfying the size inequality, a nonempty set of indices\nbelow $2X$ whose average of $\\operatorname{Re}E(h,N,L)$ is at most $9/10$\ncontains an index $N$ with $\\mathcal C(h,N,L)$\n(\\lean{Erdos249257.TotientTailPeriodKiller.exists_certifiedKill_of_first_harmonic_gap_subset}{FirstHarmonicGap.lean:104}).\nThe hypothesis therefore supplies, for every $h\\ge1$, certificates beyond\nevery threshold, which is Definition~\\ref{defn:sep}; that condition is\nequivalent to $S\\notin\\Q$.\n\n(b) For all large $X$, fewer than $8X/25$ indices in $[X,2X)$ are\nunassigned (Proposition~\\ref{prop:dickman}), and fewer than $X/100$\nassigned indices have $\\varphi(m_N)<m_N/1000$, by the Chebyshev form of\nProposition~\\ref{prop:badcof} given after its proof. These two sets and\n$\\mathcal G$ partition $[X,2X)$, so $\\#\\mathcal G>67X/100$\n(\\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean\\#L75}{\\texttt{eventually\\_card\\_pivotGoodBases\\_gt}}).\nSince $603/1000=(9/10)(67/100)$, each large $X$ in the hypothesis gives\n$\\sum_{N\\in\\mathcal G}\\operatorname{Re}E(h,N,L(X))<\\tfrac{9}{10}\\#\\mathcal G$.\nFor $X\\ge A$ the set $T=\\mathcal G$ is nonempty and lies in $[A,2X)$, and\n$L(X)$ satisfies the size inequality, so (a) applies.\n\\end{proof}\n\nThe trivial bound for the sum in (b) is $\\#\\mathcal G$, so the\nhypothesis asks for a fixed saving: along a sequence of blocks, the\naverage of $\\cos\\bigl(2\\pi D(h,N,L)/2^L\\bigr)$ over $\\mathcal G$ must be\nat most $603X/(1000\\,\\#\\mathcal G)$, a number below $9/10$.\nEquidistributed phases would make the sum $o(X)$. A rational value\n$S=a/(2^cv)$ with $v$ odd makes the bound fail for $h=\\varphi(v)$ and all\nlarge $X$: every large index\nthen has an integral tail difference, hence no certificate, and an index\nat a depth satisfying the size condition and having no certificate has\n$\\operatorname{Re}E>9/10$.\nThe hypothesis is sufficient and is not known to be necessary: from\nirrationality one obtains nonintegral tail differences, and no argument\nturns them into a fixed saving on $\\mathcal G$. It is open. No computation in this record bears\non it, and the count $\\#\\mathcal G>67X/100$ holds only for large $X$.\n\n\\papersectiontarget{long249-group-mean}\nWhen the first two inequalities of Definition~\\ref{defn:pivot} hold with\n$s=26$, $\\eta=1/1000$ and $L=L(X)$, they add to\n$\\operatorname{Re}\\sum_{N\\in\\mathcal G}E(h,N,L(X))\\le\\tfrac{57}{100}X$, so\nthe four-term condition at these parameters implies the hypothesis of (b).\nAt these same parameters, the recorded argument takes the prime number\ntheorem as an input and bounds every group mean $|\\bar z_m|$ on\n$\\mathcal G$ by $1/100$ for all sufficiently large $X$, for each fixed $h$\n(\\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/Erdos249/PaperCompleteR21/FiberMeanOfPrimeNumberTheorem.lean\\#L935}{\\texttt{fiberMean\\_le\\_of\\_primeNumberTheorem}}).\nSince $|w_N|=1$, this gives\n$|\\sum_{N\\in\\mathcal G}w_N\\bar z_{m_N}|\\le\\#\\mathcal G/100\\le X/100$,\nregardless of the dependence of $w_N$ on the prime. It supplies the mean\nbound, not the mean-subtracted cancellation. Given this mean bound,\na bound $\\tfrac{11}{20}X$ on the good-index sum, on arbitrarily large\nblocks for each $h$, yields the mean-subtracted inequality and hence irrationality\n(\\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/ArgumentGraph/Results/Erdos249.lean\\#L45}{\\texttt{dtw\\_of\\_fiberMean\\_and\\_centered}},\n\\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/ArgumentGraph/Results/Erdos249Route.lean\\#L118}{\\texttt{irrational\\_totient\\_series\\_of\\_primeNumberTheorem\\_and\\_goodBase\\_gap}}).\nTheorem~\\ref{thm:goodbasegap}(b) supersedes that route. It assumes\nneither the prime number theorem nor a bound on the group means, and\nasks only for $603X/1000$, rather than $11X/20$. The saving is measured\nagainst $\\#\\mathcal G$, not against the full block size: since\n$\\#\\mathcal G>67X/100$, the good-index average is below $9/10$.\nThere is no need to add the discarded indices back into the sum.\n\nFor clarity, the two sufficient estimates have different uses. The\ncentred route controls all four pieces and so bounds the sum on the\nwhole block. The $603X/1000$ route uses only the good indices and their\ncardinality. Bounding the discarded part trivially would give only\n$(603/1000+33/100)X=933X/1000$, not $9X/10$; adding it back would\nlose the conclusion one was trying to prove.\n\nThere is also a useful distinction between eventual and cofinal\nhypotheses. At $s=26$, $\\eta=1/1000$ and the prescribed $L(X)$, fix\n$h$ and take the maximum of the thresholds for the two counting bounds\nand the PNT-input mean bound. Any arbitrarily large sequence of blocks\non which the centred bound holds eventually passes this maximum.\nThus it supplies common blocks for all four estimates. Merely knowing\nthat several different estimates each hold on some arbitrarily large\nblocks would not justify that intersection.\n\n\n\\subsection{A fixed separation at prime positions}\n\nInstead of averaging over a block, one may ask for a fixed gap at\nprime positions. The lower bound on the prime ensures that the\ncorresponding tail indices lie beyond the required threshold.\n\n\\begin{defn}[A fixed gap at prime positions]\n\\label{defn:primegap}\nFor every $h \\ge 1$ and every $N_0$ there is a prime $p$ with\n$\\max(N_0{+}h{+}1,\\, h{+}5) \\le p$ and\n$\\mathrm{Re}\\,e(R_{p-1}-R_{p-h-1}) < \\tfrac{9}{10}$,\nwhere $e(x)=\\exp(2\\pi i x)$. Equivalently:\ncofinally many primes $p$ at which the totient tail difference across the shift\n$h$ stays a \\emph{fixed} distance\n$> \\arccos(9/10)/(2\\pi) \\approx 0.0718$ from every integer.\nIt implies $S\\notin\\mathbb Q$.\n\\lean{DTWNaturalPrimeTailOrbitStrictGap}{ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean:157}\n\\lean{irrational\\_totient\\_series\\_of\\_naturalPrimeTailOrbitStrictGap}{ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean:524}\n\\end{defn}\n\nThis asks for more than nonintegrality. A nonintegral real number can\nbe arbitrarily close to an integer, whereas the displayed condition requires\na fixed positive distance, along prime positions and for each shift $h$.\nFor example, phases at a half-integer satisfy the numerical gap and phases\nconverging to an integer eventually fail it. These are illustrations of the\nhypothesis, not assertions about the totient tails. No estimate for these tail differences at prime positions\nwith the stated quantifiers is proved here. Certificate completeness alone\ndoes not supply it.\n\n\\subsection{Certificates with depth equal to the shift}\n\nThe shift and truncation depth can be required to agree, provided\nthe shift is allowed to run through positive multiples of a fixed\n$d$. The propagation argument in Appendix~\\ref{long249:r3-ray}\nexplains why this restriction preserves the equivalence below.\n\n\\begin{defn}[Depth equal to a shift multiple]\n\\label{defn:apfde}\nFor every $d \\ge 1$ and every $N$ there is $t \\ge 1$ with\n$\\mathcal{C}(td,\\,N,\\,td)$; unpacked,\n\\[\n  N + 2td + 2 \\;<\\; {D}(td,N,td) \\bmod 2^{td}\n  \\;<\\; 2^{td} - (N + 2td + 2).\n\\]\nIt is equivalent to $S\\notin\\mathbb Q$.\n\\lean{ApFullDepthEscape}{ErdosProblems/Erdos249/PeriodMultipleEscape.lean:441}\n\\lean{apFullDepthEscape\\_iff\\_irrational}{ErdosProblems/Erdos249/FullDepthRayAmplifier.lean:406}\n\\end{defn}\n\nFor a fixed pair $(d,N)$, a nonintegral tail difference supplies a\ncertificate in every sufficiently late adjacent pair of multipliers.\nAppendix~\\ref{long249:r3-ray} proves this propagation and hence the\nequivalence in Definition~\\ref{defn:apfde}\n(\\lean{apFullDepthEscape\\_iff\\_irrational}{ErdosProblems/Erdos249/FullDepthRayAmplifier.lean:406}).\nThe finite examples verify the starting hypothesis for individual\npairs, not for every $d,N$. That universal supply of nonintegral\ndifferences is the missing arithmetic input. With these quantifiers,\nthe condition has the full strength of the irrationality question;\nit is not a mild regularity assumption. The open problem remains unproved.\n\n\\noindent Formal sources: \\lean{windowDiscrepancy\\_self\\_eq\\_totientBlock\\_sub}{ErdosProblems/Erdos249/PeriodMultipleEscape.lean:67}\n\n\\subsection{A false generic rank bound}\n\nThe coefficient rank might suggest a bound on carries under\nrationality. The next proposed assertion makes that suggestion precise;\nthe rational comparison following it disproves both versions.\n\n\\begin{defn}[A counterfactual rank criterion]\n\\label{defn:rankupper}\nThe proposed assertion is a constant $C$ bounding the rank for every\n$c:\\N\\to\\N$ with $c(n)\\le n$ and $T_c\\in\\Q$, every integer $v>0$,\nand every integer sequence $u$ satisfying\n\\[\n u(n+1)=2u(n)-v c(n+1),\\qquad u(n)/2^n\\longrightarrow0.\n\\]\nSpecifically, the proposed bound is\n\\[\n \\dim_{\\Q}\\operatorname{span}_{\\Q}\n \\{n\\mapsto u(2^jn+r):1\\le j\\le e,\\ 0\\le r<2^j\\}\\le C\n \\qquad(e\\ge0).\n\\]\nThe alternative replaces $C$ by a function $g(e)$ growing strictly slower\nthan $2^e-1$, with the same quantifiers. Either assertion would contradict\nthe lower bound of Proposition~\\ref{prop:rank} under rationality of $S$.\nBoth assertions are false, as the following comparison shows.\n\\end{defn}\n\nBoth assertions in Definition~\\ref{defn:rankupper} are false.\nThe comparison sequence with sum $5/4$ in\nSection~\\ref{sec:mahler-defect} satisfies $0\\le c(n)\\le n$ and has\nan integral carry with $u(n)/2^n\\to0$ and rank at least $2^e-1$\nat every depth (Section~\\ref{sub:generic-carry-rank}). It therefore\ncontradicts both proposed upper bounds. This leaves open bounds\nunder further hypotheses specific to the totient.\n\nPeriodicity modulo $v$ alone does not provide the additional rank bound.\n\n\n\\section{Guide to the detailed statements}\n\nThe remainder records the formulas and proofs behind the preceding\narguments. To use a statement, distinguish its conclusion from its\ninput. A finite certificate excludes a family of denominators; a\ncofinal supply excludes every rational value. Positivity of a tail\ndifference is not nonintegrality, and a conditional implication still\nneeds its arithmetic premise.\n\nThe following entry points answer different questions; they are not\nsuccessive steps of one proof of irrationality.\n\\begin{center}\n\\begin{tabular}{@{}p{.39\\linewidth}p{.55\\linewidth}@{}}\n\\textbf{Question} & \\textbf{Where to read}\\\\[3pt]\nWhich section identities hold? &\nSection~\\ref{sec:mahler-defect}: independent sections and integral relations.\\\\[3pt]\nWhat does a finite computation exclude? &\nSection~\\ref{sec:series}: tail residues; Section~\\ref{ssec:farey}: the denominator cutoff.\\\\[3pt]\nCan one nonintegral difference yield more certificates? &\nAppendix~\\ref{long249:r3-ray}: propagation with growing shift and depth.\\\\[3pt]\nWhich data still allow a rational value? &\nSection~\\ref{sec:wall}: comparison sequences, with their hypotheses kept separate.\\\\[3pt]\nWhich estimate would suffice? &\nSection~\\ref{sub:pivot}: the four-term decomposition;\nTheorem~\\ref{thm:goodbasegap}: the weaker good-index condition.\\\\[3pt]\nWhat was formally checked? &\nAppendix~\\ref{app:r3-declarations}: the declaration and source-version guide.\n\\end{tabular}\n\\end{center}\nThe detailed statements retain their source references so that a proof\ncan be recovered without treating a source label as another hypothesis.\n\\section{Series identities and finite exclusions}\\label{sec:series}\n\nThe prefix-tail identity supports two different finite exclusions.\nResidue certificates test a tail difference modulo $\\Z$; the Farey\ncalculation in Section~\\ref{ssec:farey} encloses $S$ itself in an\ninterval. M\\\"obius inversion then gives other expressions for the\nsame value, useful only to the extent that they permit new estimates.\nAn identity alone does not provide a denominator exclusion.\n\n\\subsection{The series}\n\nWe begin with the value and the prefix-tail decomposition used by\nall the subsequent tests.\n\n\\begin{defn}[The Erd\\H{o}s--249 constant]\n\\label{defn:S}\nLet $\\varphi$ denote Euler's totient function. Define\n\\[\nS \\;:=\\; \\sum_{n \\ge 1} \\frac{\\varphi(n)}{2^{n}} \\;=\\; \\sum_{n : \\mathbb{N}} \\frac{\\varphi(n)}{2^{n}},\n\\]\nthe two forms coinciding because $\\varphi(0)=0$; the second, $\\mathbb{N}$-indexed\nform is the one carried in the Lean source. The series converges absolutely\nsince $\\varphi(n)\\le n$. The irrationality of this value is the question\nunder consideration.\n\\end{defn}\n\n\\begin{rem}[Status]\nNo proof or disproof of $S\\notin\\mathbb Q$ exists anywhere in the record,\nformal or informal. What exists is: one large unconditional finite denominator\nexclusion (\\S\\ref{ssec:farey}), several exact reformulations of $S$ that\nrelocate the same open question onto different coordinates without touching\nits truth value (\\S\\ref{ssec:coprime}--\\S\\ref{ssec:lambert}), and a certificate\napparatus whose cofinal quantified condition is the precise open target\n(Definition~\\ref{defn:sep}; only its finite, checked instances\nare catalogued here, \\S\\ref{ssec:certtable}).\n\\end{rem}\n\n\\subsection{The tail-shift identity}\n\nMultiplying by a power of two makes the finite prefix integral and\nleaves a scaled tail. The tail values themselves may grow; it is their\nclasses modulo $\\Z$, rather than their absolute sizes, that encode a\npossible rational period.\n\n\\begin{prop}[Digit-shift identity]\n\\label{prop:shift}\nFor every $N : \\mathbb{N}$,\n\\[\n2^{N} \\cdot S \\;=\\; \\Phi_N + R_N,\n\\]\nwhere $\\Phi_N := \\sum_{n \\le N} \\varphi(n) \\cdot 2^{N-n} \\in \\mathbb{N}$ (the\ninteger prefix) and $R_N := \\sum_{j \\ge 0} \\varphi(N+1+j)/2^{j+1}$ (the\nscaled tail, \\ensuremath{{R}} in Lean).\n\\end{prop}\n\n\\begin{proof}[Consequence for rationality]\nSubtracting the prefix identities at $N+h$ and $N$ gives\n\\[\n R_{N+h}-R_N\\equiv2^N(2^h-1)S\\pmod\\Z.\n\\]\nIf $S=a/(2^cv)$ with $v\\ge1$ odd, Euler's theorem supplies\n$h=\\varphi(v)>0$ and $N_0=c$, for which the difference is integral for\nall $N\\ge N_0$\n(\\lean{eventual_period_of_not_irrational}{TotientTailPeriodKiller.lean:358}).\nConversely, an integral difference for even one $h>0,N\\ge0$\nforces $S$ to be rational, because $2^N(2^h-1)$ is a nonzero integer.\nThus the unresolved question is not this converse, but whether every such\ntail difference is nonintegral. Definition~\\ref{defn:sep} expresses that\nquestion through finite residues.\n\\end{proof}\n\n\\subsection{The finite denominator bound}\n\\label{ssec:farey}\n\\papersectiontarget{long249-r3-farey}\n\nAn interval containing $S$ excludes rational numbers whose\ndenominators are too small to occur inside it. The Farey argument\nbelow makes this precise using a finite totient prefix and a bound\non its tail. It is independent of the period and residue criteria.\n\n\\begin{lem}[Farey gap, fully general]\n\\label{lem:farey}\nFor integers $a,b,c,d,r,s$ with $b>0$, $d>0$, $bc-ad=1$ (i.e.\\ $a/b$ and\n$c/d$ are unimodular Farey neighbours), and $as < rb$, $rd < cs$ (i.e.\\ $r/s$\nlies strictly between them): $b+d \\le s$.\n\\end{lem}\n\nIndeed, the two strict inequalities make $cs-dr$ and $br-as$ positive\nintegers, and\n\\[\n s=b(cs-dr)+d(br-as)\\ge b+d,\n\\]\nusing $bc-ad=1$. This also proves $s>0$, which the formulation did not\nneed to assume separately. The lemma is the classical Stern--Brocot\ndenominator bound. The mediant $(a+c)/(b+d)$ lies between the two\nneighbours and is reduced: any common divisor of $a+c$ and $b+d$\ndivides $b(a+c)-a(b+d)=bc-ad=1$. It attains denominator $b+d$,\nso the general interval bound is sharp. A particular tail interval\nstill requires its own endpoint and first-failure checks.\n\n\\begin{prop}[The Farey certificate at $K=240$]\n\\label{prop:gapwindow}\nFor the window $(N,K)=(1,240)$, put\n\\[\n V=\\left(\\sum_{r=1}^{240}\\varphi(1+r)2^{240-r}\\right)\\bmod2^{240}.\n\\]\nFor every $q : \\mathbb{N}$ with\n\\[\n0 < q \\;\\le\\; Q_0 := 79\\,639\\,646\\,646\\,701\\,375\\,323\\,355\\,774\\,875\\,831\\,053 \\;\\;(\\approx 7.96\\times 10^{34}),\n\\]\n\\[\n(q \\cdot V) \\bmod 2^{240} \\;+\\; 243\\,q \\;<\\; 2^{240}.\n\\]\nThis bound is \\emph{sharp}: $q = Q_0+1 = 79\\,639\\,646\\,646\\,701\\,375\\,323\\,355\\,774\\,875\\,831\\,054$\nis the exact first failing denominator, obtained as the denominator of\nthe mediant of two explicit unimodular Farey neighbours.\n\\end{prop}\nHere ``failing'' means that this particular enclosure can no longer\nexclude the denominator. It does not mean that the mediant equals\n$S$, or that another argument could not exclude that denominator too.\nSharpness concerns the finite interval test, not a smallest possible\ndenominator for the unknown value.\n\n\\begin{thm}[Denominator exclusion]\n\\label{thm:denom-record}\nFor every $p\\in\\Q$ whose reduced denominator is at most $Q_0$,\n\\[\nS \\;\\neq\\; p.\n\\]\nEquivalently: \\emph{if $S$ is rational, its reduced denominator exceeds}\n$Q_0 \\approx 7.96 \\times 10^{34}$.\n\\end{thm}\n\n\\begin{rem}[What this does and does not say]\nTheorem~\\ref{thm:denom-record} is a complete, unconditional finite fact: no rational\nof denominator at most $Q_0$ equals $S$. A single such bound does not\nexclude arbitrarily large denominators. Proving unbounded exclusions by\nthe same mechanism would, however, establish irrationality. The question\nthis leaves is whether the first failing denominator $(b+d)(K)$ at\nwindow $K$ is unbounded as $K \\to \\infty$. An affirmative answer\nwould prove irrationality by Proposition~\\ref{prop:C2sup}, independently of the\ncertificate condition discussed after Proposition~\\ref{prop:shift}. The\nquestion is open; it is neither claimed nor proved anywhere in the record.\n\\end{rem}\n\n\\begin{rem}[Earlier finite denominator bounds]\nThe recorded Farey computations gave bounds $4838$, $2^{22}$,\napproximately $2.49\\times10^{17}$ at $K=120$, and $Q_0$ at $K=240$.\nEach applies the finite mechanism of Lemma~\\ref{lem:farey} and\nProposition~\\ref{prop:gapwindow}. These values do not prove that the\nbounds are unbounded as $K$ increases.\n\\end{rem}\n\n\\subsection{Equivalent expressions and their denominator bounds}\n\\label{ssec:coprime}\n\nThe finite bound in Theorem~\\ref{thm:denom-record} also applies to the\ntwo expressions for $S-1/2$ below, with half the denominator bound.\nAmiram Eldar posted the coprimality interpretation of\nProposition~\\ref{prop:coprime} in revision~28 of OEIS A256936 on\n15 March 2026, and the M\\\"obius-square formula of\nProposition~\\ref{prop:mobsq} in revision~31~\\cite{eldar2026oeis}.\nSteve Fan posted the latter formula on the problem's forum thread on\n16 May 2026~\\cite{fan2026totient}. We derive both identities by\nM\\\"obius inversion and geometric summation.\n\n\\begin{prop}[Fair-coin coprimality form]\n\\label{prop:coprime}\nLet $X,Y$ be independent random variables with $\\Pr(X=n)=\\Pr(Y=n)=2^{-n}$\nfor $n \\ge 1$ (independent fair-coin waiting times). Then\n\\[\nS \\;=\\; \\tfrac12 \\;+\\; \\Pr\\bigl(\\gcd(X,Y)=1\\bigr)\n\\;=\\; \\tfrac12 \\;+\\; \\sum_{\\substack{a,b\\ge 1\\\\ \\gcd(a,b)=1}} 2^{-(a+b)}.\n\\]\nEquivalently, on the visible lattice: summing $2^{-(a+b)}$ over the half-open\ncoprime pairs ($a\\ge1$, $b\\ge0$, $\\gcd(a,b)=1$) recovers $\\sum_n \\varphi(n)/2^n$\nexactly, with no boundary correction, because the visible-point count on the\nhalf-open antidiagonal at height $n$ equals $\\varphi(n)$ for every $n$,\nincluding $n=0,1$.\n\\end{prop}\n\n\\begin{prop}[The gcd-layer normalisation]\n\\label{prop:gcdlayer}\nFor independent fair-coin waiting times as above, $\\sum_{g\\ge1}\\Pr(\\gcd(X,Y)=g)=1$\nexactly; and for every $d>0$, $\\Pr(d\\mid X \\wedge d\\mid Y) = 1/(2^d-1)^2$.\n\\end{prop}\n\n\\begin{thm}[Denominator exclusion for the coprimality-probability form]\n\\label{thm:denomcoprime}\nLet\n\\[\nQ_1 := \\left\\lfloor \\frac{Q_0}{2} \\right\\rfloor = 39\\,819\\,823\\,323\\,350\\,687\\,661\\,677\\,887\\,437\\,915\\,526.\n\\]\nFor every $a\\in\\mathbb{Z}$, $d\\in\\mathbb{N}$ with $0<d\\le Q_1$: the visible\ncoprime-pair probability $\\Pr(\\gcd(X,Y)=1)$ is not equal to $a/d$.\n\\end{thm}\n\n\\begin{thm}[Denominator exclusion for the M\\\"obius-square form]\n\\label{thm:denommobsq}\nWith $Q_1$ as in Theorem~\\ref{thm:denomcoprime}: for every $a\\in\\mathbb{Z}$,\n$d\\in\\mathbb{N}$ with $0<d\\le Q_1$, the signed series\n$T := \\sum_{m\\ge1} \\mu(m)/(2^m-1)^2 = S - \\tfrac12$ (see \\S\\ref{ssec:mobius})\nis not equal to $a/d$.\n\\end{thm}\n\n\\begin{rem}[Why the bound halves, and why this is not new information]\n$Q_1 = \\lfloor Q_0/2 \\rfloor$ arithmetically: $Q_0$ is odd\n($Q_0 = 79\\,639\\,646\\,646\\,701\\,375\\,323\\,355\\,774\\,875\\,831\\,053$), so\n$Q_0/2 = 39\\,819\\,823\\,323\\,350\\,687\\,661\\,677\\,887\\,437\\,915\\,526.5$ and\n$Q_1$ is its floor. The halving is the cost of transporting the known bound\nthrough the affine shift\n$S = \\tfrac12 + T$ (resp.\\ $S = \\tfrac12 + \\Pr(\\gcd(X,Y)=1)$) on a\ndenominator-exclusion statement: excluding all denominators $\\le Q_1$ for $T$\nfollows from excluding all denominators $\\le Q_0$ for $S$ (a denominator-$d$\nvalue of $T$ with $d\\le Q_1$ yields a\ndenominator dividing $2d\\le Q_0$ for $S$). Theorems~\\ref{thm:denomcoprime}\nand~\\ref{thm:denommobsq} are therefore the \\emph{same} finite Farey record as\nTheorem~\\ref{thm:denom-record}, transported through Proposition~\\ref{prop:coprime}\nand Proposition~\\ref{prop:mobsq} respectively; not independent evidence.\nThe converse finite implication is not asserted: subtracting $1/2$ can double\na denominator, so the $Q_1$ exclusion for $T$ alone need not recover the full\n$Q_0$ exclusion for $S$.\n\\end{rem}\n\n\\subsection{The M\\\"obius-weighted identity}\n\\label{ssec:mobius}\n\nSubtracting $1/2$ converts the totient series to a M\\\"obius-weighted\nsquared-Lambert series without changing rationality.\n\n\\begin{prop}[M\\\"obius-square reduction]\n\\label{prop:mobsq}\n\\[\nS \\;=\\; \\sum_{n\\ge1}\\frac{\\varphi(n)}{2^n} \\;=\\; \\frac12 \\;+\\; \\sum_{d\\ge1}\\frac{\\mu(d)}{(2^d-1)^2},\n\\]\nwhere $\\mu$ is the M\\\"obius function, so $\\mu(d)\\in\\{-1,0,1\\}$ for every $d$.\nConsequently \\emph{Erd\\H{o}s \\#249 $\\iff$ $T:=\\sum_{d\\ge1}\\mu(d)/(2^d-1)^2 \\notin \\mathbb{Q}$},\nsince adding a rational number does not change irrationality.\n\\end{prop}\n\n\\begin{rem}[Why the bounded coefficients matter; the Erd\\H{o}s-1948 regime]\nThe identity of Proposition~\\ref{prop:mobsq} rewrites $S$ (equivalently $T$)\nas a Lambert-type series with squared denominators and bounded\nM\\\"obius weights. To see the constant $1/2$ directly, expand\n\\[\n \\sum_{d\\ge1}\\frac{\\mu(d)}{2^d-1}\n   =\\sum_{n\\ge1}\\frac{\\sum_{d\\mid n}\\mu(d)}{2^n}=\\frac12.\n\\]\nCombining this identity with $\\varphi=\\mu*\\mathrm{Id}$ and\n$2^d=(2^d-1)+1$ proves the formula for $T$; absolute convergence\njustifies the rearrangements. Thus the corresponding first-power sum\n$L(\\mu):=\\sum_d\\mu(d)/(2^d-1)$ is rational, whereas\n$L(1)=\\sum_{d\\ge1}1/(2^d-1)=E$ is the Erd\\H{o}s--Borwein constant.\nErd\\H{o}s proved its irrationality in 1948~\\cite{erdos1948lambert}, and\nit is also proved in Lean\n(\\lean{irrational_erdosBorwein_series}{CertificateKernel.lean:8335}).\n\n\\par\\medskip\n\nIn contrast, the coefficients of the original power series satisfy\n$0\\le\\varphi(n)\\le n$ and are unbounded. Bounded M\\\"obius weights do\nnot make the new denominators powers of a fixed base, so they do not\nturn this representation into a bounded-digit expansion.  The function $\\varphi$ has average\norder $6n/\\pi^2$, equivalently\n$\\sum_{k\\le x}\\varphi(k)\\sim 3x^2/\\pi^2$, but there is no pointwise estimate\n$\\varphi(n)=\\Theta(n)$. The near-integer irrationality criterion\n(formalised generically as\n\\lean{irrational_of_int_mul_near_int}{CertificateKernel.lean:6120} and its\nbase-power specialisation\n\\lean{irrational_of_pow_mul_near_int}{CertificateKernel.lean:6149}) applies\nto either representation if its nonzero approximation\nerrors can be established. Boundedness of the coefficients does not itself\nestablish those errors.\nErd\\H{o}s's 1948 argument is itself a proof for the specific Lambert series:\nit chooses arguments by a system of congruences so that the divisor counts\nalong a block are divisible by increasing powers of the base, and a tail bound\nthen forces a long run of zero digits~\\cite[pp.~63--66]{erdos1948lambert}.\nBoundedness of a signed weight supplies none of that divisibility input.\n\n\\par\\medskip\n\nThe rational identity $L(\\mu)=1/2$ is already a warning: bounded signed\ncoefficients alone do not imply irrationality. The later denominator\ncalculations retain the M\\\"obius terms whose indices do not divide the\nchosen LCM; those terms cannot be discarded when transferring an estimate.\n\\end{rem}\n\n\\subsection{A general squared-Lambert identity}\n\\label{ssec:lambert}\n\nThe same geometric summation applies to any weight with the stated\nlinear bound.\n\n\\begin{prop}[The squared-Lambert identity]\n\\label{prop:lambertengine}\nFor $w:\\mathbb{N}\\to\\mathbb{R}$ with $|w(d)|\\le d$ for all $d>0$, and $0\\le r<1$:\n\\[\n\\sum_{d\\ge1} w(d)\\left(\\frac{r^d}{1-r^d}\\right)^2 \\;=\\; \\sum_{n\\ge1}\\left(\\sum_{e\\mid n} w(e)\\Bigl(\\tfrac{n}{e}-1\\Bigr)\\right) r^n.\n\\]\n\\end{prop}\n\n\\noindent The identity follows by expanding $r^d/(1-r^d)$ as a geometric\nseries and interchanging absolutely convergent sums.  It is the squared\nanalogue of the classical divisor-sum Lambert identity\n$\\sum_na_nq^n/(1-q^n)=\\sum_m\\bigl(\\sum_{d\\mid m}a_d\\bigr)q^m$~\\cite[(1),\np.~2]{merca-schmidt}.\n\nAt $r=1/2$, the choices $w=1$ and $w=\\varphi$ give the following two\nidentities. They help distinguish the role of the weight from that of the\nsquared denominator.\n\n\\begin{prop}[The divisor-sum identity]\n\\label{prop:zetaq}\n\\[\n\\sum_{d\\ge1} \\frac{1}{(2^d-1)^2} \\;=\\; \\sum_{n\\ge1} \\frac{\\sigma(n)-\\tau(n)}{2^n} \\;=\\; \\zeta_q(2)-\\zeta_q(1) \\text{ at } q=\\tfrac12,\n\\]\nwhere $\\sigma$ is the sum-of-divisors function and $\\tau$ the number-of-divisors\nfunction. The \\emph{identity} is machine-checked.  With\n$\\zeta_q(s)=\\sum_{n\\ge1}n^{s-1}q^n/(1-q^n)$, Postelmans and Van Assche prove\nthat $1,\\zeta_q(1),\\zeta_q(2)$ are linearly independent over $\\mathbb{Q}$ for\n$q=1/p$ with an integer $p\\ge2$~\\cite[Theorem~1.3, p.~3]{postelmans-vanassche};\nat $q=\\tfrac12$ this gives the irrationality of the \\emph{value}\n$\\zeta_q(2)-\\zeta_q(1)$.\nTwo irrational numbers can have a rational difference, so the separate\nirrationality of the two values would not suffice.\n\\end{prop}\n\\evidenceremark{The Lean proof of the irrationality assertion assumes the\nlinear independence theorem of Postelmans and Van Assche quoted above; the\ndisplayed identity is proved without it.}\n\n\\begin{prop}[The gcd-moment identity]\n\\label{prop:pillai}\n\\[\n\\sum_{d\\ge1} \\frac{\\varphi(d)}{(2^d-1)^2} \\;=\\; \\sum_{n\\ge1} \\bigl(P(n)-n\\bigr)\\cdot 2^{-n} \\;=\\; \\mathbb{E}[\\gcd(X,Y)],\n\\]\nwhere $P(n) := \\sum_{e\\mid n}\\varphi(e)\\cdot(n/e) = (\\varphi * \\mathrm{Id})(n)$\nis Pillai's gcd-sum function $P(n)=\\sum_{k\\le n}\\gcd(k,n)$~\\cite[\\S1,\n(1)--(2)]{toth-gcd} and $X,Y$ are the independent fair-coin waiting\ntimes of Proposition~\\ref{prop:coprime}. The identity itself is\nmachine-checked; irrationality of $\\mathbb{E}[\\gcd(X,Y)]$ is not proved here. This is a\ndifferent series from $S$.\n\\end{prop}\n\n\\begin{rem}[Related series]\nWriting $L(f):=\\sum_d f(d)/(2^d-1)$ and\n$L_2(f):=\\sum_d f(d)/(2^d-1)^2$, the preceding identities give\n$L(\\mu)=\\tfrac12$ (rational), $L(1)=E$ (Erd\\H{o}s--Borwein,\nirrational), $L(\\varphi)=2$ (rational); $L_2(\\mu)=S-\\tfrac12$\n(irrationality open; this is \\#249), $L_2(1)=\\zeta_q(2)-\\zeta_q(1)$\n(irrational by the cited theorem, Proposition~\\ref{prop:zetaq}),\n$L_2(\\varphi)=\\mathbb{E}[\\gcd(X,Y)]$ (irrationality open,\nProposition~\\ref{prop:pillai}). These examples show why the denominator shape alone does not determine\nirrationality: changing the weight can produce either rational or\nirrational values.\n\\end{rem}\n\n\\subsection{Diagonal certificates through scale 82}\n\\label{ssec:certtable}\n\nEach row is a finite instance of Definition~\\ref{defn:sep}, with a\nlink to the computation producing its witness. The aggregate result\ncovers every integer scale $t\\le82$. It contains no assertion at\n$t=83$ or at arbitrarily large scales.\n\n\\paragraph{Definition of the finite residue test} For $h,N,L : \\mathbb{N}$, recall the truncated\ndifference $D(h,N,L) := \\sum_{j<L} (\\varphi(N+h+1+j)-\\varphi(N+1+j))\\cdot 2^{L-1-j} \\in \\mathbb{Z}$\n(\\lean{windowDiscrepancy}{TotientTailPeriodKiller.lean:66}), and\n\\[\n\\mathcal{C}(h,N,L) \\;:\\equiv\\; (N+h+L+2) < D(h,N,L) \\bmod 2^{L} < 2^{L}-(N+h+L+2),\n\\]\n(\\lean{certifiedKill}{TotientTailPeriodKiller.lean:72}; a decidable condition). By\ncompleteness (\\lean{exists_certifiedKill_iff_tail_diff_notMem_int}{LcmConeFlatness.lean:316}),\n\\[\n(\\exists L,\\ \\mathcal{C}(h,N,L)) \\iff R_{N+h}-R_N \\notin \\mathbb{Z}\n\\]\n\nfor every $h,N$. Requiring such certificates beyond every basepoint\nthreshold for each fixed positive $h$ is precisely $\\mathrm{Sep}$ and is\nequivalent to $S\\notin\\mathbb Q$. The table records finite instances,\nindexed either by $(h,N,L)$ or by $t$; it does not establish that quantified\ncondition.\n\n\\small\n\\begin{longtable}{@{}\n  >{\\raggedright\\arraybackslash}p{0.20\\linewidth}\n  >{\\raggedright\\arraybackslash}p{0.48\\linewidth}\n  >{\\raggedright\\arraybackslash}p{0.22\\linewidth}\n  @{}}\n\\textbf{Computation} & \\textbf{Exact statement of what was checked} & \\textbf{Lean source} \\\\\n\\hline\n\\endhead\nFixed-window example, depth 16\n&\n$\\mathcal{C}(h,12,16)$ holds for every $h \\in \\{1,\\dots,8\\}$ (by direct evaluation,\n$256$ evaluations with totient arguments below $37$); consequently $S \\ne r$ for every\n$r\\in\\mathbb{Q}$ whose reduced denominator divides $2^{12}(2^h-1)$ for some $1\\le h\\le 8$.\n&\n\\lean{certifiedKill_all_small}{TotientTailPeriodKiller.lean:404}\n\\lean{totient_series_ne_rat_of_den_dvd}{TotientTailPeriodKiller.lean:416}\n\\\\\nFixed-window example, depth 9, wider net\n&\n$\\mathcal{C}(h,14,9)$ holds for every\n$h\\in\\{1,\\dots,16\\}$: $S \\ne r$ for every $r\\in\\mathbb{Q}$ whose reduced\ndenominator divides $2^{14}(2^h-1)$ for some $1\\le h\\le16$.\n&\n\\lean{totient_series_ne_rat_of_den_dvd_pow_two_mul_mersenne_upto_sixteen}{CertificateKernel.lean:18890}\n\\\\\nDiagonal certificates, 12 base scales\n&\n$\\mathcal C(H_t,H_t,L_t)$, $H_t := \\mathrm{lcm}(1,\\dots,t)$,\nholds (by explicit evaluation of the\ntotient values, via factored prime-power blocks with\nLucas-primality certificates) for $t \\in \\{1,2,3,4,5,7,8,9,11,13,16,17\\}$\nwith certificate depths $L_t \\in \\{6,5,7,7,9,14,15,14,21,22,23,26\\}$\nrespectively (in the same order).\n&\n\\lean{certifiedKill_diagonal_all_imported}{DiagonalPincerCertificates.lean:2870}\n\\lean{diagonalPincerCertificateScales}{DiagonalPincerCertificates.lean:2852}\n\\lean{diagonalPincerKillDepth}{DiagonalPincerCertificates.lean:2854}\n\\\\\nDiagonal certificates, further scales through $t=64$\n&\nThe same predicate $\\mathcal{C}(H_t,H_t,L_t)$ is additionally checked, one\nLean file per scale, at\n$t \\in \\{19,\\allowbreak23,\\allowbreak25,\\allowbreak27,\\allowbreak29,\\allowbreak31,\n\\allowbreak32,\\allowbreak37,\\allowbreak41,\\allowbreak43,\\allowbreak47,\\allowbreak49,\n\\allowbreak53,\\allowbreak59,\\allowbreak61,\\allowbreak64\\}$\n(16 further explicit values). Together with the base 12 scales above this is\n\\textbf{28 explicit values of $t$ in total}, the complete list being\n$\\{1,\\allowbreak2,\\allowbreak3,\\allowbreak4,\\allowbreak5,\\allowbreak7,\n\\allowbreak8,\\allowbreak9,\\allowbreak11,\\allowbreak13,\\allowbreak16,\\allowbreak17,\n\\allowbreak19,\\allowbreak23,\\allowbreak25,\\allowbreak27,\\allowbreak29,\\allowbreak31,\n\\allowbreak32,\\allowbreak37,\\allowbreak41,\\allowbreak43,\\allowbreak47,\\allowbreak49,\n\\allowbreak53,\\allowbreak59,\\allowbreak61,\\allowbreak64\\}$.\nThis does \\emph{not} establish $\\mathcal{C}(H_t,H_t,\\cdot)$ for infinitely\nmany $t$; these 28 examples are a historical strict subset of the contiguous\nband in the next row.\n&\nOne Lean file per scale; endpoint\n\\lean{certifiedKill_diagonal_t64}{DiagonalPincerCertificateT64Endpoint.lean:1928}\n\\\\\nContiguous lcm-diagonal band through $t\\le82$\n&\nFor every natural $t\\le82$ there exists a depth $L$ with\n$\\mathcal{C}(H_t,H_t,L)$.  This closes every scale in the finite interval,\nincluding plateau transfers, with no holes.  The next lcm jump is at the prime\n$83$; no certificate at $t=83$ is claimed, and any such certificate must have\ndepth at least $125$.\n&\n\\lean{ErdosProblems.Skip.LadderT67.exists_diagonalKill_le_82}{ErdosProblems/Skip/LadderT67.lean:71264}\n\\lean{ErdosProblems.Skip.LadderT67.t83_depth_floor}{ErdosProblems/Skip/LadderT67.lean:71294}\n\\\\\nFarey denominator bound\n&\nEvery $q\\in\\mathbb{N}$ with \\adjustbox{max width=\\linewidth}{$0<q\\le Q_0=79\\,639\\,646\\,646\\,701\\,375\\,323\\,355\\,774\\,875\\,831\\,053$}\nsatisfies the window-$(N,K)=(1,240)$ gap certificate; $q=Q_0+1$ is the exact\nfirst failure.\n&\n\\lean{gap_check_window_1_240_le_79639646646701375323355774875831053}{GapFareyBound.lean:176}\n\\\\\nCoprimality-probability / M\\\"obius-square Farey bound\n&\nEvery $d\\in\\mathbb{N}$ with \\adjustbox{max width=\\linewidth}{$0<d\\le Q_1=39\\,819\\,823\\,323\\,350\\,687\\,661\\,677\\,887\\,437\\,915\\,526$}\n(exactly $\\lfloor Q_0/2\\rfloor$) is excluded as a denominator of\n$\\Pr(\\gcd(X,Y)=1)$ and, separately, of $T=\\sum_d\\mu(d)/(2^d-1)^2$.\n&\n\\lean{tsum_visible_coprime_pairs_ne_int_div_of_den_le_39819823323350687661677887437915526}{CertificateKernel.lean:18572}\n\\lean{tsum_moebius_div_two_pow_sub_one_sq_ne_int_div_of_den_le_39819823323350687661677887437915526}{CertificateKernel.lean:18487}\n\\\\\n\\end{longtable}\n\\normalsize\n\n\\begin{rem}[Scope of the table]\nNo row above is cofinal in its indexing parameter ($h$, $t$, or the\ndenominator bound), and no row is claimed to extrapolate. The historical\ndiagonal depths are irregular and nonmonotone\n($6,5,7,7,9,14,15,14,21,22,23,26,\\dots$); no closed-form growth rate for\n$L_t$ is proved or conjectured in the record. This table records the\nhistorical 28-scale list through $t=64$; the later aggregate theorem for every\n$t\\le82$ is stated above and is not itemised row by row here.  Neither finite\nrecord supplies the quantified condition of Definition~\\ref{defn:sep}.\n\\end{rem}\n% ; - part a249_p1a ; -\n\\section{Detailed statements}\n\nThe following statements give the exact hypotheses of the preceding\narguments. We begin with unconditional identities, rank bounds and\nfinite computations, then pass to the residue and approximation\nconditions. A Lean declaration proving a conditional implication\nretains that condition as a hypothesis; it does not supply the\nmissing estimate for $\\varphi$.\n\n\\subsection{Unconditional structure and finite examples}\n\nThe rank formulas hold at every stated level. The residue tests and\nFarey interval below instead give conclusions on specified finite\nsets of shifts, scales or denominators. This distinction matters when\napplying the later conditions, which require witnesses at unbounded\nindices.\n\n\\begin{thm}[Rationality gives an eventual tail period]\\label{catalogue:cert:a9}\nIf $S=a/(2^cv)$ with $a\\in\\mathbb Z$, $c\\in\\mathbb N$ and $v$ a\npositive odd integer, then\n\\[\n R_{N+h}-R_N\\in\\mathbb Z\\qquad(N\\ge c),\\qquad h=\\varphi(v)>0.\n\\]\nEuler's theorem gives $v\\mid2^h-1$, and the prefix-tail identity then\nmakes $2^N(2^h-1)S$ integral. Thus rationality supplies a positive\nshift for which all sufficiently late tail differences are integral.\n\\end{thm}\n\n\\begin{thm}[Rationality forces unbounded carry rank]\\label{catalogue:cert:d5}\nIf $S\\in\\mathbb Q$, there are an integer $v>0$ and an integer sequence\n$u$ such that\n\\[\n u(N+1)=2u(N)-v\\varphi(N+1),\\qquad u(N)/2^N\\longrightarrow0,\n\\]\nand, for every $e\\ge0$, the sequences\n$ n\\mapsto u(2^jn+r)$ with $1\\le j\\le e$ and $0\\le r<2^j$\nspan a space of $\\mathbb Q$-dimension at least $2^e-1$.\n\nThe recurrence follows from the scaled tails after clearing a\nhypothetical denominator of $S$. The only growth condition imposed\nhere is the displayed limit $u(N)/2^N\\to0$; subexponential growth is\nnot assumed. The rank lower bound is a necessary consequence of\nrationality and yields no contradiction by itself. An upper\nbound using additional properties of the actual totient coefficients\nwould be needed; the later rational comparison rules out such a bound\nfor arbitrary rational coefficient series.\n\\end{thm}\n\n\\begin{thm}[Independence of the retained dyadic sections]\\label{catalogue:cert:d4}\nFor every integer $e\\ge0$, the family\n\\[\n \\varphi(n),\\quad\\varphi(2n),\\quad\n \\varphi(2^jn+r)\\quad(1\\le j\\le e,\\ 0<r<2^j,\\ r\\text{ odd})\n\\]\nis linearly independent over $\\mathbb Q$ and has $2^e+1$ members.\nAt $e=0$ this family still contains $\\varphi(2n)$; it is not the full\nlevel-zero truncation, which consists only of $\\varphi(n)$. For $e\\ge1$\nit is the basis of the full truncated dyadic family.\n\nThe proof separates one section at a time: the Chinese remainder theorem\nforces suitable prime divisors in the other affine arguments, while\nDirichlet's theorem makes the selected argument prime. This proves\nindependence, not just the displayed cardinality. In particular, the\nspan of all dyadic sections is infinite-dimensional; that conclusion\nalone says nothing about rationality of their weighted sum.\n\\end{thm}\n\n\\begin{prop}[Denominator divisibility forces tail integrality]\\label{catalogue:cert:a8}\nIf $S=a/q$ in lowest terms with $q>0$ and\n$q\\mid2^N(2^h-1)$, then $R_{N+h}-R_N\\in\\mathbb Z$.\nIndeed, the prefix identity expresses that difference as\n$2^N(2^h-1)S$ minus an integer. This is the contradiction used by a\nnonintegrality certificate to exclude the rational value $a/q$.\n\\end{prop}\n\nThe denominator results use the following polynomial, defined for $r\\ge1$:\n\\[\n P_r(X)=\\sum_{d\\mid r}\\mu(d)\\frac rd\n                 \\sum_{j=0}^{r/d-1}X^{dj}.\n\\]\nFor example, $P_p(X)=p(1+X+\\cdots+X^{p-1})-1$ when $p$ is prime.\nThus $P_p(2)=p(2^p-1)-1$ is coprime to $2^p-1$.\nFor general $r\\ge1$ and $H\\ge1$, write\n\\[\n b_r=\\frac{P_r(2)}{2^r-1},\\qquad\n \\beta_H=\\frac{b_{\\operatorname{rad}(H)}}{\\operatorname{rad}(H)},\n \\qquad r_t=\\operatorname{rad}(H_t).\n\\]\nHere $\\operatorname{rad}(H)$ is the product of the distinct prime\nfactors of $H$, and $\\operatorname{den}(x)$ denotes the positive\nreduced denominator of $x\\in\\mathbb Q$. These rational numbers are\nfinite rational contributions to the M\\\"obius decomposition; no convergence\nto $S$ is asserted for them.\nAlso put\n$\\mathcal P_t=\\{p\\text{ prime}:t/2<p\\le t\\}$.\nThe hypotheses below specify which factors of $2^{r_t}-1$ remain in\nthe reduced denominator after cancellation.\n\n\\begin{prop}[A cyclotomic factor remains after cancellation]\\label{catalogue:mob:b5}\nIf $r$ is squarefree, then\n\\[\n \\gcd\\bigl(|P_r(2)|,|\\Phi_r(2)|\\bigr)=1,\n \\qquad |\\Phi_r(2)|\\mid\\operatorname{den}(b_r).\n\\]\nHere $\\Phi_r$ is the $r$-th cyclotomic polynomial. Proposition~\\ref{catalogue:mob:b4}\ngives $P_r\\equiv\\mu(r)\\pmod{\\Phi_r}$, and $\\mu(r)=\\pm1$ because\n$r$ is squarefree. This proves that the indicated cyclotomic factor\ncannot cancel from $b_r$.\n\\end{prop}\n\n\\begin{prop}[Primes in the upper half remain after cancellation]\\label{catalogue:mob:b6}\nFor $t\\ge5$ the whole product\n\\[\n \\prod_{p\\in\\mathcal P_t}(2^p-1)\n       \\ \\mid\\ \\operatorname{den}(H_t\\beta_{H_t})\n\\]\nremains after cancellation. More generally, the source gives the same\nproduct-divisibility conclusion for a multiplier coprime to that product;\nit also supplies a sufficient condition on the multiplier's prime factors.\nFor the multiplier $H_t/r_t$, those prime factors are at most $t$, and\nthe LCM result above follows.\n\n\\end{prop}\n\nThe restriction $t\\ge5$ belongs to the sufficient prime-factor condition,\nnot just to the LCM example. At $t=3$, one has $r_t=6$,\n$b_6=298/63$ and $\\mathcal P_3=\\{2,3\\}$. The multiplier $9$ has only\nprime factors at most $3$, but $\\operatorname{den}(9b_6)=7$ is not\ndivisible by $(2^2-1)(2^3-1)=21$. Thus the prime-factor condition alone\ndoes not imply survival of the product at every scale.\nThis is a denominator statement about a finite rational contribution,\nnot a nonintegrality assertion about a real tail.\n\n\n\\begin{prop}[A lower bound for the reduced denominator]\\label{catalogue:mob:b7a}\nFor every integer $t\\ge5$,\n\\[\n 2^{\\lfloor t/2\\rfloor}\n \\le \\prod_{p\\in\\mathcal P_t}(2^p-1)\n \\le \\operatorname{den}(H_t\\beta_{H_t}).\n\\]\nBertrand's postulate supplies a prime in $\\mathcal P_t$; its factor is\nat least $2^{\\lfloor t/2\\rfloor}$. Proposition~\\ref{catalogue:mob:b6}\nthen gives the second inequality. The complementary term in the real\ntail decomposition can still cancel this rational contribution, so\ndenominator growth alone does not prove irrationality of $S$.\n\\end{prop}\n\n\\begin{prop}[The exact reduced denominator]\\label{catalogue:mob:b7b}\nAt every integer scale $t\\ge0$,\n\\[\n \\operatorname{den}(H_t\\beta_{H_t})\n =\\frac{2^{r_t}-1}\n {\\gcd\\!\\left(2^{r_t}-1,\n       \\dfrac{H_t}{r_t}\n       \\prod_{\\substack{p\\mid r_t\\\\p\\ \\mathrm{odd\\ prime}}}(p^2-1)\\right)}.\n\\]\nThe product is $1$ when there is no odd prime divisor. This gives the\nexact cancellation, including the multiplier $H_t/r_t$, rather than\nmerely a lower bound for the denominator. For example, $H_5=60$ and\n$r_5=30$, and the denominator is $(2^{30}-1)/3$.\n\\end{prop}\n\n\\begin{prop}[Nonvanishing of a signed dyadic sum]\\label{catalogue:mob:d3}\nLet $I$ be finite, let $u_i\\in\\mathbb Z$ and $e_i\\in\\mathbb N$, and\nsuppose that $m\\in I$ is the unique index with maximal exponent $e_m$.\nIf $u_m$ is odd, then\n\\[\n 2^{e_m}\\sum_{i\\in I}\\frac{u_i}{2^{e_i}}\n       =\\sum_{i\\in I}u_i2^{e_m-e_i}\\equiv1\\pmod2.\n\\]\nIndeed, every summand other than $u_m$ is even. The sum is therefore\nnonzero. In the determinant application, the rectangular\nCauchy--Binet formula produces such finite signed sums from truncated\nmoments. The result tests a specified configuration; it does not show\nthat a suitable maximal-exponent term exists at arbitrarily large scales.\n\\end{prop}\n\n\\begin{prop}[Small certificates]\\label{catalogue:cert:a11}\nFor each integer $1\\le h\\le8$, the finite test $\\mathcal C(h,12,16)$\nholds. Each discrepancy uses two 16-term windows. Across all eight\nshifts, only the 24 distinct totient values at $13\\le n\\le36$ are\nneeded, since the windows overlap. The source verifies the eight\ninteger residue inequalities by exact computation.\n\nConsequently, if $a/b$ is a reduced fraction with $b>0$ and\n$b\\mid2^{12}(2^h-1)$ for at least one $1\\le h\\le8$, then $S\\ne a/b$.\nThis follows from Propositions~\\ref{catalogue:cert:a6}\nand~\\ref{catalogue:cert:a8}.\n\\end{prop}\n\n\\begin{prop}[A common certificate for sixteen shifts]\\label{catalogue:cert:a12}\nFor each integer $1\\le h\\le16$, one has $\\mathcal C(h,14,9)$.\nThe common basepoint is $14$ and the depth is $9$; these parameters\nhave different roles. The two windows for each shift use only the\n25 distinct totient values at $15\\le n\\le39$ across the whole family.\nThus $S\\ne a/b$ for every reduced fraction with $b>0$ such that\n$b\\mid2^{14}(2^h-1)$ for at least one $1\\le h\\le16$.\n\\end{prop}\n\\evidenceremark{The sixteen residue inequalities are a finite computation,\nchecked by exact evaluation.}\n\n\\begin{prop}[Historical diagonal examples and the complete band through 82]\\label{catalogue:cert:b11}\nLet $H_t=\\operatorname{lcm}(1,\\ldots,t)$ and let\n\\[\nP(t)\\quad:\\Longleftrightarrow\\quad \\exists L,\\ \\mathcal{C}(H_t,H_t,L).\n\\]\nThe checked certificate table proves $P(t)$ at 28 explicit indices, ending at $t=64$.\nFor the initial indices\n\\[\nt=1,2,3,4,5,7,8,9,11,13,16,17,\\ldots,\n\\]\nthe corresponding depths begin\n$6,5,7,7,9,14,15,14,21,22,23,26,\\ldots$.\nEach entry is a finite kernel computation on explicit totient values. The factorisations use\nchecked prime-power blocks and Lucas primality certificates. Thus the table proves a finite list\nof instances of $P(t)$; it does not prove that $P(t)$ holds for infinitely many $t$.\n\nThe historical 28 examples are now a strict subset of the aggregate theorem\n$\\forall t\\le82,\\ P(t)$.  That theorem closes the finite interval without\nholes but supplies neither $P(83)$ nor a cofinal family.\n\\end{prop}\n\n\\subsection{Definitions and elementary identities}\n\nThe prefix-tail identity and the truncation formula give both\ndirections of the finite residue test. We then specialise the test to\nLCM shifts and express it in terms of rational approximation.\nM\\\"obius inversion provides the alternative coefficient weights used\nin the subsequent estimates.\n\n\\subsubsection{Initial implications}\n\n\\begin{defn}[The scaled tail]\\label{catalogue:cert:a1}\nFor $N\\in\\mathbb N$,\n\\[\n R_N=\\sum_{j\\ge0}\\frac{\\varphi(N+1+j)}{2^{j+1}}.\n\\]\nThe series converges because $\\varphi(n)\\le n$. It is the tail of $S$\nmultiplied by $2^N$, not the fractional part of $2^NS$; indeed, $R_N$\nneed not lie in $[0,1)$.\n\\lean{totientTail}{TotientTailPeriodKiller.lean:57}\n\\end{defn}\n\n\\begin{defn}[The truncated tail difference]\\label{catalogue:cert:a3}\nFor $h,N,L\\in\\N$, define the integer\n\\[\n D(h,N,L)=\\sum_{j=0}^{L-1}\n \\bigl(\\varphi(N+h+1+j)-\\varphi(N+1+j)\\bigr)2^{L-1-j}.\n\\]\nThe sum is empty when $L=0$. Dividing it by $2^L$ gives the first\n$L$ terms of $R_{N+h}-R_N$. Computing the finitely many totients\ntherefore determines $D(h,N,L)$ exactly; the residue inequality below\nis the decidable condition.\n\\lean{windowDiscrepancy}{TotientTailPeriodKiller.lean:66}\n\\end{defn}\n\n\\begin{defn}[The central-residue inequality]\\label{catalogue:cert:a4}\nFor $h,N,L\\in\\mathbb N$, the finite condition $\\mathcal C(h,N,L)$ is\n\\[\n N+h+L+2<D(h,N,L)\\bmod2^L<2^L-(N+h+L+2).\n\\]\nIt puts the residue outside both endpoint intervals of width $N+h+L+2$.\nAfter division by $2^L$, the excluded width tends to zero as $L$ grows\nwith $h,N$ fixed. No analytic hypothesis is part of this finite test;\nthe separate condition $\\mathrm{Sep}$ quantifies over such tests.\n\\lean{certifiedKill}{TotientTailPeriodKiller.lean:72}\n\\end{defn}\n\n\\begin{prop}[The prefix-tail identity]\\label{catalogue:cert:a2}\nFor every $N\\in\\mathbb N$,\n\\[\n 2^NS=\\Phi_N+R_N,\n \\qquad \\Phi_N=\\sum_{n\\le N}\\varphi(n)2^{N-n}\\in\\mathbb Z.\n\\]\nThus $R_N$ and $2^NS$ have the same fractional part. Subtracting the\nidentities at $N+h$ and $N$ gives the tail-difference congruence used in\nthe irrationality criterion.\n\\end{prop}\n\n\\begin{lem}[The necessary depth inequality]\\label{catalogue:cert:a5}\nFor $h,N,L\\in\\N$, the condition $\\mathcal C(h,N,L)$ implies\n\\[\n 2(N+h+L+2)<2^L.\n\\]\nIndeed, its lower residue bound must be smaller than its upper bound.\nThus $L>1+\\log_2(N+h+L+2)$ whenever a certificate holds. In\nparticular, a fixed depth cannot accommodate unbounded $N+h$.\nThis is a necessary depth bound, not an upper bound for finding a\ncertificate.\n\\end{lem}\n\n\\begin{prop}[A certificate implies nonintegrality]\\label{catalogue:cert:a6}\nFor all $h,N,L\\in\\mathbb N$,\n\\[\n \\mathcal C(h,N,L)\\ \\Longrightarrow\\ R_{N+h}-R_N\\notin\\mathbb Z.\n\\]\nThe scaled truncation error satisfies\n$|2^L(R_{N+h}-R_N)-D(h,N,L)|\\le N+h+L+2$.\nIf the tail difference\nwere integral, $D(h,N,L)$ would therefore lie within that distance of a\nmultiple of $2^L$, contrary to the two strict residue inequalities.\n\\end{prop}\n\n\\begin{thm}[Nonintegrality gives a certificate at some depth]\\label{catalogue:cert:a7}\nFor every $h,N\\in\\mathbb N$,\n\\[\n (\\exists L\\in\\mathbb N,\\ \\mathcal C(h,N,L))\n \\quad\\Longleftrightarrow\\quad R_{N+h}-R_N\\notin\\mathbb Z.\n\\]\nFor the converse to Proposition~\\ref{catalogue:cert:a6}, fix a positive\ndistance from the nonintegral tail difference to the nearest integer.\nThe normalised error $(N+h+L+2)/2^L$ tends to zero, so a sufficiently\nlarge truncation preserves that distance. This is a pointwise\nequivalence; irrationality requires the stated quantifiers over the\nshift and basepoint.\n\\end{thm}\n\n\\begin{defn}[The least common multiple of the first integers]\\label{catalogue:cert:b1}\nPut $H(0)=1$ and $H(t+1)=\\operatorname{lcm}(H(t),t+1)$. Then\n$H(t)=\\operatorname{lcm}(1,\\ldots,t)$, $t\\le H(t)$, and every positive\ninteger $h\\le t$ divides $H(t)$. This is why a single LCM parameter can\nabsorb any fixed hypothetical period once $t$ is large enough.\n\\lean{periodLcm}{CarrySurvivorExtinction.lean:515}\n\\lean{le\\_periodLcm}{LcmDiagonalReduction.lean:81}\n\\end{defn}\n\n\\begin{prop}[The Farey gap lemma]\\label{catalogue:cert:c1}\nLet $a,b,c,d,r,s\\in\\mathbb Z$, with $b,d>0$ and $bc-ad=1$. If\n\\[\n as<rb\\quad\\hbox{and}\\quad rd<cs,\n\\]\nthen $b+d\\le s$. The two cross-products $rb-as$ and $cs-rd$ are positive\nintegers, and\n\\[\n s=d(rb-as)+b(cs-rd)\\ge d+b.\n\\]\nIn particular, a rational $r/s$ strictly between the Farey neighbours\n$a/b$ and $c/d$, with positive denominator, has denominator at least\n$b+d$.\n\\end{prop}\n\n\\begin{prop}[An irrationality criterion from rational approximations]\\label{catalogue:cert:d1}\nLet $x\\in\\mathbb R$ and let $u_k=p_k/q_k\\in\\mathbb Q$ be in lowest\nterms, with $q_k>0$. If $u_k\\ne x$ for every sufficiently large $k$ and\n$q_k|x-u_k|\\to0$, then $x$ is irrational. Indeed, if $x=a/b$ with\n$a\\in\\mathbb Z$ and $b\\ge1$, a nonzero integer numerator gives\n$q_k|x-u_k|=|a q_k-b p_k|/b\\ge1/b$, a contradiction.\nThis elementary criterion requires both nonvanishing and the scaled\nerror estimate. Convergence $u_k\\to x$ alone is insufficient, and the\napproximants need not be continued-fraction convergents.\n\\end{prop}\n\n\\begin{prop}[An irrationality criterion from near integers]\\label{catalogue:cert:d2}\nLet $\\xi\\in\\mathbb R$. Suppose that for every integer $q\\ge1$ there\nare integers $m,z$ with\n\\[\n 0<|m\\xi-z|<1/q.\n\\]\nThen $\\xi$ is irrational: if $\\xi=a/b$ with $b\\ge1$, every nonzero\nsuch difference is at least $1/b$. Restricting the multiplier to powers\n$m=b_0^n$ of a fixed integer base $b_0\\ge2$ gives a sufficient special\ncase, not a hypothesis satisfied by every irrational number.\nThe strict lower bound excludes exact integer hits; the upper bound must\nbe available for arbitrarily large $q$.\n\nFor a binary example, form a number by concatenating the blocks $10$ at\nsquare indices $k\\ge1$ and $01$ at the other indices. Its binary expansion\nis not eventually periodic, since the block sequence has increasingly\nlong gaps between the square indices, so the number is irrational. There\nare no three consecutive equal digits. Every fractional part after a\nbinary shift therefore lies in $[1/8,7/8]$, and the multiples $2^n\\xi$\ndo not approach the integers. This verifies that the restriction to\nbase powers is genuinely stronger. This example is an ordinary mathematical\nargument, not an additional claim about the linked formalisation.\n\\end{prop}\n\n\\begin{prop}[A nonsingular evaluation matrix gives independence]\\label{catalogue:cert:d3}\nLet $I$ be a finite index set and let $f_j:\\mathbb N\\to\\mathbb Q$\nfor $j\\in I$. If there are evaluation points $n_i\\in\\mathbb N$ such that\n\\[\n \\det\\bigl(f_j(n_i)\\bigr)_{i,j\\in I}\\ne0,\n\\]\nthen the family $(f_j)_{j\\in I}$ is linearly independent over $\\mathbb Q$.\nA relation among the functions, evaluated at the $n_i$, gives a vector\nin the kernel of this nonsingular matrix, so all coefficients vanish.\nFiniteness is required to form the displayed determinant. In the totient\napplication the arithmetic work is to construct these evaluation points;\nthe implication itself is ordinary linear algebra.\n\\end{prop}\n\n\\begin{prop}[The gap between distinct rational numbers]\\label{catalogue:cert:d9}\nIf $a/b<c/d$ are reduced rational numbers with $b,d>0$, then\n\\[\n \\frac cd-\\frac ab=\\frac{bc-ad}{bd}\\ge\\frac1{bd}.\n\\]\nThe numerator $bc-ad$ is a positive integer. Thus an upper bound on a\npositive rational error gives a lower bound on the product of the reduced\ndenominators. A large denominator in an unreduced displayed expression\nneed not be the reduced denominator appearing in this conclusion.\n\\end{prop}\n\n\\begin{prop}[The M\\\"obius identity for $S$]\\label{catalogue:mob:a1a}\nThe absolutely convergent series satisfy\n\\[\n S=\\sum_{n\\ge1}\\frac{\\varphi(n)}{2^n}\n  =\\sum_{d\\ge1}\\frac{\\mu(d)2^d}{(2^d-1)^2}\n  =\\frac12+\\sum_{d\\ge1}\\frac{\\mu(d)}{(2^d-1)^2}.\n\\]\nFor the first rearrangement use\n$\\varphi(n)=\\sum_{d\\mid n}\\mu(d)n/d$. For the second use\n$2^d=(2^d-1)+1$ and the classical identity\n$\\sum_{d\\ge1}\\mu(d)/(2^d-1)=1/2$.\nThese are identities of values, not transfers of irrationality from a\nLambert series with a different weight.\n\\end{prop}\n\n\\begin{cor}[An equivalent irrationality question]\\label{catalogue:mob:a1b}\nSubtracting the rational number $1/2$ gives\n\\[\n S\\notin\\mathbb Q\n \\quad\\Longleftrightarrow\\quad\n \\sum_{d\\ge1}\\frac{\\mu(d)}{(2^d-1)^2}\\notin\\mathbb Q.\n\\]\nThis is the same irrationality question in a different series\nrepresentation. The notation $L_2(\\mu)$ for the right-hand series is\nintroduced in Definition~\\ref{catalogue:mob:a3}.\n\\end{cor}\n\n\\begin{prop}[The squared-Lambert identity]\\label{catalogue:mob:a2}\nLet $w:\\mathbb N\\to\\mathbb R$ satisfy $|w(d)|\\le d$ for $d\\ge1$,\nand let $0\\le r<1$. Then\n\\[\n \\sum_{d\\ge1}w(d)\\left(\\frac{r^d}{1-r^d}\\right)^2\n =\\sum_{n\\ge1}\\left(\\sum_{d\\mid n}w(d)\\left(\\frac nd-1\\right)\\right)r^n.\n\\]\nExpand $(x/(1-x))^2=\\sum_{j\\ge1}(j-1)x^j$ and group terms by\n$n=dj$. Absolute convergence follows from the bound on $w$ and\ngeometric decay. The cases $w=\\mu$, $w=1$ and $w=\\varphi$ give the next\nthree series; the same algebra applies to each, but their arithmetic\nvalues differ.\n\\end{prop}\n\n\\begin{defn}[The squared-Lambert series with weight $\\mu$]\\label{catalogue:mob:a3}\nPut\n\\[\n L_2(\\mu)=\\sum_{d\\ge1}\\frac{\\mu(d)}{(2^d-1)^2}=S-\\frac12.\n\\]\nThe equality follows from Proposition~\\ref{catalogue:mob:a1a} and\nProposition~\\ref{catalogue:mob:a2} with $w=\\mu$.\nSubtracting $1/2$ preserves rationality, so the irrationality of this\nseries is exactly the question about $S$, not a separate easier claim.\nThe formula is credited in \\S\\ref{ssec:coprime}.\n\\lean{tsum\\_totient\\_half\\_pow\\_eq\\_half\\_add\\_moebius\\_sq}{MersenneLambertLadder.lean:665}\n\\end{defn}\n\n\\begin{prop}[Weight one and divisor sums]\\label{catalogue:mob:a4}\n$\\sum_{d:\\mathbb{N}^+}' 1/(2^d-1)^2 = \\sum_{n:\\mathbb{N}^+}' \\big(\\sigma(n)-\\tau(n)\\big)\\cdot(1/2)^n = \\zeta_q(2) - \\zeta_q(1)$ at $q=1/2$. The displayed identity is formalised. Irrationality of its value follows\nfrom the cited linear independence result of Postelmans and Van Assche,\nas explained in Proposition~\\ref{prop:zetaq}; that literature result is\nnot formalised here. Replacing the weight $1$ by $\\mu$ changes the value\nto that of Definition~\\ref{catalogue:mob:a3}. Observation~\\ref{catalogue:mob:a6}\ncompares the two choices of weight.\n\\end{prop}\n\\evidenceremark{The Lean proof of the irrationality assertion assumes the\nlinear independence theorem of Postelmans and Van Assche.}\n\n\\begin{prop}[Totient weight and gcd moments]\\label{catalogue:mob:a5}\n\\[\n\\sum_{d:\\mathbb{N}^+}' \\varphi(d)/(2^d-1)^2 = \\sum_{n:\\mathbb{N}^+}' (P(n)-n)\\cdot(1/2)^n,\n\\]\nwhere $P = \\varphi * \\mathrm{Id}$ (Pillai's gcd-sum function). It also equals $\\mathbb E[\\gcd(X,Y)]$ when $X,Y$ are independent and\n$\\mathbb P(X=n)=\\mathbb P(Y=n)=2^{-n}$ for $n\\ge1$; see\nProposition~\\ref{prop:pillai}.\nThis is a different weighted series. Its rationality is not settled by\nthe identities proved here.\n\\end{prop}\n\n\\begin{obs}[Comparison of weighted series]\\label{catalogue:mob:a6}\nFor $L(f)=\\sum_{d\\ge1}f(d)/(2^d-1)$, the three weights give\n\\[\n L(\\mu)=\\frac12,\\qquad L(1)=E,\\qquad L(\\varphi)=2,\n\\]\nwhere $E$ is the irrational Erd\\H{o}s--Borwein constant.\nFor the squared denominators, they give\n\\[\n L_2(\\mu)=S-\\frac12,\\qquad\n L_2(1)=\\zeta_{1/2}(2)-\\zeta_{1/2}(1),\\qquad\n L_2(\\varphi)=\\mathbb E[\\gcd(X,Y)].\n\\]\nThe first equality is Definition~\\ref{catalogue:mob:a3}; the second and\nthird are Propositions~\\ref{catalogue:mob:a4} and~\\ref{catalogue:mob:a5}.\nThe middle value is irrational by the cited linear independence theorem.\nThe other two rationality questions are not settled in this record.\nThese examples show that the choice of weight matters: an identity with\nthe same denominators does not transfer an irrationality theorem from one\nweight to another. They do not classify all weighted Lambert series.\n\\end{obs}\n\n\\begin{prop}[The weight of pairs divisible by a fixed integer]\\label{catalogue:mob:a7}\nLet $X,Y$ be independent random variables with\n$\\mathbb P(X=n)=\\mathbb P(Y=n)=2^{-n}$ for $n\\ge1$. For every $d\\ge1$,\n\\[\n \\mathbb P(d\\mid X,\\ d\\mid Y)=\\frac1{(2^d-1)^2}.\n\\]\nThis follows by multiplying the two geometric sums\n$\\sum_{k\\ge1}2^{-dk}=1/(2^d-1)$. It explains the squared denominator\nin the preceding M\\\"obius identity.\n\\end{prop}\n\n\\begin{prop}[The sum over coprime directions]\\label{catalogue:mob:a8}\nThe sum over positive coprime pairs satisfies\n\\[\n \\sum_{\\substack{a,b\\ge1\\\\\\gcd(a,b)=1}}\n                  \\frac1{2^{a+b}-1}=1.\n\\]\nTo see the normalisation, write every pair of positive integers uniquely\nas $(ka,kb)$ with $\\gcd(a,b)=1$, and sum $2^{-k(a+b)}$ over $k\\ge1$.\nThe total is $(\\sum_{n\\ge1}2^{-n})^2=1$. This base-two normalisation\nis used in Proposition~\\ref{catalogue:mob:a9a}.\n\\end{prop}\n\n\\begin{prop}[A Stern--Brocot recursion with stopping]\\label{catalogue:mob:a9a}\nFor positive integers $a,b$, put\n\\[\n M(a,b)=\\frac1{(2^a-1)(2^b-1)}.\n\\]\nThen\n\\[\n M(a,b)=\\frac1{2^{a+b}-1}+M(a+b,b)+M(a,a+b).\n\\]\nMultiplication by the common denominator proves the identity. At the\nroot $(1,1)$, the three terms on the right are each $1/3$, and\n$M(1,1)=1$.\n\nThe first term is a mass retained at the current node, not passed to\neither child. Thus the probabilistic interpretation is a branching\nprocess with stopping: after division by $M(a,b)$, the stopping and two\ntransition probabilities are, respectively,\n\\[\n \\frac{(2^a-1)(2^b-1)}{2^{a+b}-1},\\qquad\n \\frac{2^a-1}{2^{a+b}-1},\\qquad\n \\frac{2^b-1}{2^{a+b}-1}.\n\\]\nThey sum to one. The stopping probability is at least $1/3$, which gives\nthe next proposition's geometric error bound. Also\n$M(a,b)=\\mathbb P(a\\mid X)\\mathbb P(b\\mid Y)$ for the independent\ngeometric variables used above. The identity does not describe a\nconservative mass split between the two children alone.\n\\end{prop}\n\n\\begin{prop}[Convergence with an explicit error]\\label{catalogue:mob:a9b}\nIn the splitting identity of Proposition~\\ref{catalogue:mob:a9a}, the\nsum of the two child terms is at most $2/3$ of the parent term.\nIf $M_d(a,b)$ is the sum of the contributions removed during the first\n$d$ levels, then\n\\[\n |M(a,b)-M_d(a,b)|\\le(2/3)^d M(a,b).\n\\]\nIn particular, $M_d(a,b)\\to M(a,b)$. This convergence statement is\nseparate from identifying the limit with a sum over any independently\ndefined infinite set of descendants; no additional identification is\nasserted here.\n\\end{prop}\n\n\\begin{prop}[A numerator polynomial and its coefficients]\\label{catalogue:mob:b1}\nFor squarefree $r\\ge1$, the polynomial defined above has coefficients\n\\[\n [X^k]P_r(X)=\n \\begin{cases}\n \\dfrac r{\\gcd(r,k)}\\varphi(\\gcd(r,k)),&0\\le k<r,\\\\\n 0,&k\\ge r.\n \\end{cases}\n\\]\nFor $k<r$, extracting the coefficient gives\n$\\sum_{d\\mid\\gcd(r,k)}\\mu(d)r/d$; the usual divisor formula for\n$\\varphi$ evaluates this sum. All coefficients in the range $0\\le k<r$\nare positive. Thus a signed sum of geometric polynomials has an explicit\npositive-coefficient expression. The squarefree hypothesis is retained\nas in the cited formal statements.\n\\end{prop}\n\n\\begin{prop}[Evaluation at two]\\label{catalogue:mob:b2}\nFor squarefree $r\\ge1$, evaluation of $P_r$ at two gives the integer\n\\[\n P_r(2)=\\sum_{d\\mid r}\\mu(d)\\frac rd\n                         \\frac{2^r-1}{2^d-1}.\n\\]\nEach quotient is an integer because $d\\mid r$. The formal definition\nsums over subsets of the prime divisors of $r$; for squarefree $r$\nthese are exactly its divisors. This identifies the polynomial\ncalculation with the numerator used in $b_r$.\n\\end{prop}\n\n\\begin{prop}[Decomposition by the radical]\\label{catalogue:mob:b3}\nFor $H\\ge1$ and $r\\ge1$,\n\\[\n H\\beta_H=\\frac H{\\operatorname{rad}(H)}b_{\\operatorname{rad}(H)},\n \\qquad\n \\operatorname{den}(b_r)\n   =\\frac{2^r-1}{\\gcd(|P_r(2)|,2^r-1)}.\n\\]\nThe first equality follows from the definition of $\\beta_H$; the\nsecond is ordinary reduction of an integer fraction. Multiplication\nby $H/\\operatorname{rad}(H)$ can cause further cancellation. No\ncoprimality assumption is included in either identity.\n\\end{prop}\n\n\\noindent The constant in Proposition~\\ref{catalogue:mob:b4} is the value of $P_r$ at a primitive\n$m$-th root of unity. Since the $X^k$ coefficient of $P_r$ depends only on\n$\\gcd(k,r)$, the value is a discrete Fourier coefficient of a function of the\ngreatest common divisor, which Schramm expresses through Ramanujan\nsums~\\cite[Theorem and (2), p.~2]{schramm-gcd}; Proposition~\\ref{catalogue:mob:b4} specialises this to the\npresent numerator polynomial and proves the integral cyclotomic consequence.\n\n\\begin{prop}[A cyclotomic congruence]\\label{catalogue:mob:b4}\nLet $r$ be squarefree and $m\\mid r$. In $\\mathbb Z[X]$,\n\\[\n P_r(X)\\equiv\\mu(m)J_2(r/m)\\pmod{\\Phi_m(X)},\n\\]\nwhere $J_2(n)=n^2\\prod_{p\\mid n}(1-p^{-2})$ is the second Jordan\ntotient. Evaluation at two gives\n\\[\n \\Phi_m(2)\\mid P_r(2)-\\mu(m)J_2(r/m).\n\\]\n\\end{prop}\n\n\\begin{proof}\nTo prove the congruence, let $\\zeta$ be a primitive $m$-th root of\nunity. Since $m\\mid r$, the geometric sum in the definition of $P_r$\nis $r/d$ when $m\\mid d$ and zero otherwise. Hence\n\\[\n P_r(\\zeta)=\\sum_{\\substack{d\\mid r\\\\m\\mid d}}\\mu(d)(r/d)^2.\n\\]\nWrite $r=mu$. Squarefreeness gives $\\gcd(m,u)=1$, so substituting\n$d=me$ reduces this to\n$\\mu(m)\\sum_{e\\mid u}\\mu(e)(u/e)^2=\\mu(m)J_2(u)$.\nThe minimal polynomial $\\Phi_m$ therefore divides the difference in\n$\\mathbb Q[X]$; because it is monic, polynomial division gives the\nquotient in $\\mathbb Z[X]$. Taking $m=r$ gives the coprimality in\nProposition~\\ref{catalogue:mob:b5}.\n\\end{proof}\n\nSquarefreeness cannot simply be dropped from this constant formula.\nFor $r=4$ and $m=2$,\n$P_4(X)=2+4X+2X^2+4X^3$, so $P_4(-1)=-4$, whereas\n$\\mu(2)J_2(2)=-3$. Here $P_4(2)=50$ and $\\Phi_4(2)=5$;\nthe latter factor cancels from $b_4=50/15=10/3$. Positive polynomial\ncoefficients alone therefore do not prevent denominator cancellation.\n\n\n\\begin{prop}[Adjoining a prime: new divisors]\\label{catalogue:mob:b8a}\nLet $r\\ge1$, let $p\\nmid r$ be prime, and let $m\\mid r$. Then\n\\[\n P_{rp}(X)\\equiv-P_r(X^p)\\pmod{\\Phi_{mp}(X)}.\n\\]\n\\end{prop}\n\n\\begin{proof}\nSplitting the divisors of $rp$ into $d$ and $pd$, with $d\\mid r$,\ngives the exact polynomial identity\n\\[\n P_{rp}(X)=p\\left(\\sum_{j=0}^{p-1}X^{jr}\\right)P_r(X)-P_r(X^p).\n\\]\nAt a primitive $mp$-th root $\\zeta$, the number $\\zeta^r$ is a\nnontrivial $p$-th root, because $m\\mid r$ and $p\\nmid r$.\nThe geometric factor vanishes, proving the stated congruence.\nNo squarefree hypothesis on $r$ is needed for this identity.\nIf $r$ is squarefree, $\\zeta^p$ is a primitive $m$-th root, and\nProposition~\\ref{catalogue:mob:b4} gives the constant remainder\n$-\\mu(m)J_2(r/m)$.\n\\end{proof}\n\n\n\\begin{prop}[Adjoining a prime: existing divisors]\\label{catalogue:mob:b8b}\nLet $r$ be squarefree, let $p\\nmid r$ be prime, and let $m\\mid r$.\nFor the existing divisors $m$, the corresponding identity is\n\\[\n P_{rp}(X)\\equiv(p^2-1)P_r(X)\\pmod{\\Phi_m(X)}.\n\\]\n\\end{prop}\n\n\\begin{proof}\nApply Proposition~\\ref{catalogue:mob:b4} to both $r$ and $rp$.\nSince $p\\nmid r/m$, multiplicativity gives\n$J_2(rp/m)=(p^2-1)J_2(r/m)$, which proves the identity.\nThe squarefree hypothesis is retained as in the Lean statement.\nThis compares the remainders at the existing divisors $m$; the\npreceding proposition treats the new divisors $mp$.\n\\end{proof}\n\n\n\\begin{prop}[Lambert-series identities]\\label{catalogue:cert:d7}\nFor an arithmetic function $f$ for which the sums converge absolutely,\nwrite $L(f)=\\sum_{n\\ge1}f(n)/(2^n-1)$. Expanding each geometric series gives\n\\[\n L(f)=\\sum_{m\\ge1}\\frac{(f*1)(m)}{2^m}.\n\\]\nIn particular, if $\\alpha=\\varphi*\\mu$, then $\\alpha*1=\\varphi$ and\n$L(\\alpha)=S$. The table distinguishes this value from four other\nLambert-series values. The identity changes the coefficients and the\nform of the denominator; it does not transfer an irrationality theorem\nfor one weight to another.\n\\end{prop}\n\n\\begin{longtable}{@{}\n  >{\\raggedright\\arraybackslash}p{0.14\\linewidth}\n  >{\\raggedright\\arraybackslash}p{0.33\\linewidth}\n  >{\\raggedright\\arraybackslash}p{0.43\\linewidth}\n  @{}}\n\\caption{Five Lambert-series values and the support for their stated arithmetic status.}\\\\\n\\toprule\n\\papertablehead\nWeight $f$ & Value $L(f)$ & Status \\\\\n\\midrule\n\\endfirsthead\n$\\mu$ & $L(\\mu) = 1/2$ & rational, trivial \\\\\n$\\varphi$ & $L(\\varphi) = 2$ & rational \\\\\n$\\mathrm{Id}$ & $L(\\mathrm{Id}) = \\sum_m \\sigma(m)/2^m$ & transcendental, since $1-24L(\\mathrm{Id})$ is Ramanujan's $P(1/2)$ (Nesterenko~\\cite[Cor.~2, p.~1320]{nesterenko1996}; the Lean proof takes his theorem as a hypothesis) \\\\\n$1$ & $L(1) = E$, Erd\\H{o}s--Borwein constant & irrational (Erd\\H{o}s~\\cite{erdos1948lambert}), also proved in Lean (\\lean{irrational_erdosBorwein_series}{CertificateKernel.lean:8335}) \\\\\n$\\alpha = \\varphi*\\mu$ & $L(\\alpha) = S$ & \\textbf{OPEN}; this is \\#249 \\\\\n\\bottomrule\n\\end{longtable}\n\n\\begin{rem}[How the identities differ]\nThe coefficients matter: a result for the constant weight $1$ does not\nsettle the series with weight $\\alpha=\\varphi*\\mu$. A theorem expressed\nin Lambert-series form applies here only after its assumptions have\nbeen checked for that weight.\n\nThe equality between the two indexing conventions for $S$ is proved in\nLean (\\lean{tsum\\_totient\\_div\\_pow\\_two\\_eq\\_pnat\\_half\\_pow}{CertificateKernel.lean:18415}).\nThe table keeps apart a cited theorem, an ordinary identity and a formal\ndeclaration.\n\\end{rem}\n\n\n\n\\begin{prop}[An affine combination annihilates the specified divisor terms]\\label{catalogue:mob:e3}\nLet $H\\ge1$ and $d\\mid H$, with $d>0$. The $d$th term in the\nM\\\"obius expansion of $R_{mH}$ is $\\mu(d)K_d(mH)$, where\n\\[\n K_d(mH)=\\frac{mH}{d(2^d-1)}+\\frac{2^d}{(2^d-1)^2}.\n\\]\nThis is affine in $m$. Consequently, for any finite family of real\ncoefficients $c_i$ and nonnegative integers $m_i$ with\n$\\sum_i c_i=\\sum_i c_i m_i=0$, one has\n$\\sum_i c_i K_d(m_iH)=0$.\n\nFor the four multipliers $(1,3,5,15)$ and coefficients $(4,-3,-2,1)$,\n\\[\n K_d(15H)-3K_d(3H)-2K_d(5H)+4K_d(H)=0.\n\\]\nThe zeroth and first moments are zero, while the second is\n$15^2-3\\cdot3^2-2\\cdot5^2+4=152$.\nEquivalently, $XY-3X-2Y+4$ takes the values $0,0,152$ at\n$(1,1),(3,5),(9,25)$. This explains the coefficients: they cancel the\naffine contribution of each divisor of $H$, without cancelling a general\nquadratic function of the multiplier.\n\nFor a finite truncation of depth $L\\ge0$, write\n\\[\n U=\\sum_{j=1}^{L}\n \\bigl(\\varphi(15H+j)-3\\varphi(3H+j)\n       -2\\varphi(5H+j)+4\\varphi(H+j)\\bigr)2^{L-j}.\n\\]\nThe difference between $2^L(R_{15H}-3R_{3H}-2R_{5H}+4R_H)$ and $U$\nhas absolute value at most $19H+5L+5$. Indeed, the error equals\n\\[\n (R_{15H+L}+4R_{H+L})-(3R_{3H+L}+2R_{5H+L}),\n\\]\nand both parenthesised terms lie between $0$ and $19H+5L+5$ by\n$0\\le R_n\\le n+1$ for $n\\ge1$.\nThus\n\\[\n 19H+5L+5<U\\bmod2^L<2^L-(19H+5L+5)\n\\]\nis sufficient for this four-tail combination to be nonintegral.\nThe bound is derived from these tail enclosures; no optimality for the\nactual totient tails is asserted. Obtaining such certificates at the\nrequired unbounded family of LCM heights remains unproved, as in\nTheorems~\\ref{catalogue:mob:e2} and~\\ref{catalogue:cert:b6}.\n\\end{prop}\n\n\\begin{prop}[The error under composite dilation]\\label{catalogue:mob:f1}\nFor $A\\subseteq\\mathbb N$ and $n\\ge1$, let\n$d_A(n)=\\#\\{d\\ge1:d\\mid n,\\ d\\in A\\}$. If $a\\in A$ and $a,x\\ge1$, then\n\\[\n d_A(ax)=d_A(x)+\\mathbf 1_{a\\nmid x}\n       +\\#\\{d\\mid ax:d\\in A,\\ d\\nmid x,\\ d\\ne a\\}.\n\\]\nIndeed, partition the divisors of $ax$ into those already dividing $x$,\nthe possible new divisor $a$, and all other new divisors. If every element\nof $A$ is prime, the last set is empty, since a prime dividing $ax$ but\nnot $x$ must equal the prime $a$. For composite $a$ it can be nonempty:\nat $A=\\mathbb N$, $a=6$, $x=1$, the other new divisors are $2$ and $3$.\n\nThis is an unweighted counting identity. The associated Lambert series\nsatisfies\n\\[\n \\sum_{\\substack{a\\in A\\\\a\\ge1}}\\frac1{2^a-1}\n =\\sum_{n\\ge1}\\frac{d_A(n)}{2^n}.\n\\]\nFor the totient series the required divisor-convolution weight is instead\n$\\alpha=\\varphi*\\mu$, since $\\alpha*1=\\varphi$, as in\nProposition~\\ref{catalogue:cert:d7}. Weighting all divisors by $\\varphi$\nwould give $\\sum_{d\\mid n}\\varphi(d)=n$, not $\\varphi(n)$; already at\n$n=2$ these values are $2$ and $1$. A weighted version must retain the\nweights of the new divisors, so it cannot be obtained by substituting\n$A=\\mathbb N$ in this unweighted formula. Also $\\alpha(p)=p-2$ for primes\n$p$, so $\\alpha$ is unbounded and is not a periodic weight.\n\\end{prop}\n\n\\paragraph{Iterated prime dilations of divisor counts.}\nMultiplying an argument by a new prime adds exactly one new family of\ndivisors. For the divisor count $d_A$ above and $n\\ge1$, put\n$\\Delta_p f(n)=f(pn)-f(n)$. If $p_1,\\ldots,p_r$ are distinct primes,\n$P=p_1\\cdots p_r$ and $\\gcd(n,P)=1$, then\n\\[\n \\Delta_{p_1}\\cdots\\Delta_{p_r}d_A(n)\n       =\\#\\{d\\mid n:Pd\\in A\\}.\n\\]\nFor one prime, the divisors of $pn$ not dividing $n$ are exactly $pd$\nwith $d\\mid n$, because $p\\nmid n$. Iteration gives the displayed\nidentity: distinctness ensures that each new prime is coprime to the\narguments produced at the earlier steps. Equivalently, inclusion--exclusion\ncancels every divisor not containing all the new prime factors.\n\nLet $\\Omega(a)$ count the prime factors of $a\\ge1$ with multiplicity,\nand let $K\\ge0$ be an integer. If $\\Omega(a)\\le K$ for every positive $a\\in A$ and $r>K$, the set on\nthe right is empty, since $\\Omega(Pd)=r+\\Omega(d)>K$.\nThis proves the stated vanishing with all its hypotheses.\n\\lean{Erdos249257.SupportDilationDifferences.iteratedDilationDifference_eq_zero_of_cardFactors_le}{SupportDilationDifferences.lean:250}\nThe source also proves vanishing of the right-hand divisor count for a\nlist with repeated primes. That is a statement about the condition\n$Pd\\in A$, not an identity with repeated dilation differences.\n\nFor example, with $A=\\{2\\}$ and $n=1$,\n\\[\n \\Delta_2^2d_A(1)=d_A(4)-2d_A(2)+d_A(1)=-1,\n \\qquad \\#\\{d\\mid1:4d\\in A\\}=0.\n\\]\nThus the bound $\\Omega(a)\\le1$ and two prime factors do not by themselves\nmake the repeated difference vanish. The strict inequality $r>K$ is\nalso material: $\\Delta_2d_{\\{2\\}}(1)=1$ when $r=K=1$.\nFinally, at $n=2$ the one-prime identity without the coprimality hypothesis\nwould give $\\Delta_2d_{\\{2\\}}(2)=0$ on the left and\n$\\#\\{d\\mid2:2d\\in\\{2\\}\\}=1$ on the right.\nThis last example concerns the identity, not the necessity of coprimality\nfor every possible vanishing theorem. These divisor counts belong to\nthe unweighted support problem; they are not the totient coefficients.\n\n\\subsubsection{Equivalent quantified certificate conditions}\n\nCompleteness identifies the base residue condition and the LCM-diagonal\ncondition with irrationality. The following statements spell out the\nquantifiers and the implications between them. The existence of the\nrequired certificates beyond every threshold remains unproved.\n\n\\begin{thm}[The quantified certificate condition is equivalent to irrationality]\\label{catalogue:cert:a10}\n$\\big(\\forall h:\\mathbb{N},\\ 0<h \\to \\forall N_0:\\mathbb{N},\\ \\exists N\\ge N_0,\\ \\exists L,\\ \\mathcal{C}(h,N,L)\\big) \\leftrightarrow S\\notin\\mathbb Q$. The quantified-condition side is exactly $\\mathcal{C}(h,N,L)$ quantified as $\\forall h\\ge1\\ \\forall N_0\\ge0\\ \\exists N\\ge N_0\\ \\exists L$. The universally quantified assertion remains unproved. The equivalence identifies the finite witnesses that suffice; it does not establish their existence beyond every threshold.\n\\end{thm}\n\n\\begin{thm}[It suffices to use multiples of a period]\\label{catalogue:cert:b2}\n$\\big(\\forall h_0>0,\\ \\forall N_0,\\ \\exists m>0,\\ \\exists N\\ge N_0,\\ \\exists L,\\ \\mathcal{C}(m\\cdot h_0, N, L)\\big) \\to S\\notin\\mathbb Q$. The shift may be any positive multiple of a prescribed period. This\nenlarges the choice of finite witness: Theorem~\\ref{catalogue:cert:a10}\ngives the condition with $m=1$. Conversely, rationality would make all\nsuch tail differences integral after a fixed starting index. The\ndisplayed implication therefore makes this quantified condition\nequivalent to irrationality; its truth remains unproved.\n\\end{thm}\n\n\\begin{thm}[The diagonal condition is equivalent to irrationality]\\label{catalogue:cert:b3}\n$\\big(\\forall t_0:\\mathbb{N},\\ \\exists t\\ge t_0,\\ \\exists L,\\ \\mathcal{C}({H}(t), {H}(t), L)\\big) \\leftrightarrow S\\notin\\mathbb Q$. For a fixed hypothetical rational value, sufficiently large $t$ makes both its dyadic denominator and its odd-part period admissible at $N=h={H}(t)$. Conversely, irrationality and pointwise completeness supply a witness for every prescribed $t$. This is an equivalent condition with one scale parameter; the verified cases $t\\le82$ do not establish it at arbitrarily large scales.\n\\end{thm}\n\n\\begin{lem}[Nondivisors in a short LCM window]\\label{catalogue:cert:b4}\nLet $t\\ge1$ and $1\\le j<2t$ be integers. If $j\\nmid H(t)$, then\n$j=p^a>t$ for a prime $p$ and an integer $a\\ge1$. Indeed, a prime-power\ndivisor of $j$ must exceed $t$, and $j<2t$ leaves no room for a\ncofactor larger than $1$. This classifies the exceptional offsets in\nthis short window; it does not bound their contribution to a weighted\nsum of totient differences.\n\n\\end{lem}\n\n\\begin{prop}[Totient factorisation on an LCM progression]\\label{catalogue:cert:b5}\nLet $t\\ge1$, $j\\ge1$ and $q\\ge0$ be integers. Suppose $j\\mid H(t)$\nand every prime divisor of $j$ also divides $H(t)/j$. Then\n\\[\n \\varphi(qH(t)+j)=\\varphi(j)\\,\n                 \\varphi\\bigl(q(H(t)/j)+1\\bigr).\n\\]\nThe two factors in $qH(t)+j=j\\bigl(q(H(t)/j)+1\\bigr)$ are coprime:\nthe second is $1$ modulo each prime dividing $j$. Totient\nmultiplicativity gives the identity. The extra prime-divisor hypothesis\nis not automatic from $j\\mid H(t)$; for example, it fails at $t=j=2$,\nwhere $\\varphi(H(2)+2)=2$ but the proposed product would be $1$.\n\n\\end{prop}\n\n\\begin{thm}[Rationality forces tail integrality on an LCM grid]\\label{catalogue:cert:b6}\nIf $S\\in\\mathbb Q$, there is $t_1\\in\\mathbb N$ such that for all\n$t\\ge t_1$, $q\\ge1$ and $m\\ge0$,\n\\[\n R_{(q+m)H(t)}-R_{qH(t)}\\in\\mathbb Z.\n\\]\nThus all tails at positive multiples of $H(t)$ have the same fractional\npart once $t\\ge t_1$. A certificate at one fixed scale does not refute\nrationality: the threshold $t_1$ depends on the hypothetical rational\nvalue. At a shift $mH(t)>0$ and basepoint $qH(t)$, such a certificate\nexcludes rational values whose reduced denominator divides\n$2^{qH(t)}(2^{mH(t)}-1)$. Certificates at arbitrarily large scales,\nas required in Theorem~\\ref{catalogue:cert:b7}, exclude every hypothetical\nrational value.\n\n\\end{thm}\n\n\\begin{thm}[A sufficient nonintegrality condition on the grid]\\label{catalogue:cert:b7}\nSuppose that for every $t_0\\in\\N$ there are integers\n$t\\ge t_0$, $q\\ge1$, $m\\ge0$ and $L\\ge0$ such that\n\\[\n \\mathcal C(mH_t,qH_t,L).\n\\]\nThen $S\\notin\\mathbb Q$. Rationality would make the corresponding\ntail difference integral at every sufficiently large scale, contradicting\nProposition~\\ref{catalogue:cert:a6}. A successful certificate necessarily\nhas $m>0$, since the discrepancy vanishes at shift zero.\n\nThe diagonal choice is $q=m=1$; consecutive grid points have $m=1$,\nand for $p\\ge2$ the pair $(H_t,pH_t)$ has $(q,m)=(1,p-1)$. The diagonal equivalence\nin Theorem~\\ref{catalogue:cert:b3} also gives the converse from\nirrationality to the displayed quantified condition; that condition remains unproved.\n\\end{thm}\n\n\\begin{cor}[Equivalent conditions stated without certificates]\\label{catalogue:cert:b8}\nThe series $S$ is irrational if and only if\n\\[\n R_{2H_t}-R_{H_t}\\notin\\mathbb Z\n \\qquad\\text{for arbitrarily large integers }t.\n\\]\nEquivalently, for arbitrarily large $t$ there are integers $q\\ge1$\nand $m\\ge1$ with $R_{(q+m)H_t}-R_{qH_t}\\notin\\mathbb Z$.\nApply the pointwise equivalence of Theorem~\\ref{catalogue:cert:a7}\nto Theorems~\\ref{catalogue:cert:b3} and~\\ref{catalogue:cert:b7},\nrespectively. These are equivalent statements of the irrationality\nquestion; the required unbounded sets of scales are not established.\n\\end{cor}\n\n\\begin{prop}[Soundness of a second-difference certificate]\\label{catalogue:cert:b9a}\nLet $h,N,L\\in\\mathbb N$. If\n\\[\n\\begin{aligned}\n 2(N+2h+L+2)\n &<\\bigl(D(h,N+h,L)-D(h,N,L)\\bigr)\\bmod2^L\\\\\n &<2^L-2(N+2h+L+2),\n\\end{aligned}\n\\]\nthen $R_{N+2h}-2R_{N+h}+R_N\\notin\\mathbb Z$.\nThe numerator is the second difference of the finite windows.\nSubtracting the two tail identities leaves an error of absolute value\nat most $2(N+2h+L+2)$, which explains the radius in the hypothesis.\nAt $(h,N)=(1,8)$, the first-difference test holds at depth $8$,\nwhereas the second-difference test fails at every depth $L\\le8$\nand holds at $L=9$. A historical probe over $t\\le20$ reported that the\nfirst-difference test was at least as shallow in $30$ of $40$ sampled\ncases. This finite comparison is not a uniform ordering of the depths.\n\\end{prop}\n\n\nFor $H,L\\in\\mathbb N$ and a finite set $Q$ of multipliers, put\n\\[\n A_q=\\sum_{j=1}^{L}\\varphi(qH+j)2^{L-j},\\qquad\n B_q=qH+L+2\\quad(q\\in Q).\n\\]\nThe finite-grid test compares these integer numerators modulo $2^L$.\n\n\\begin{thm}[A finite-grid condition gives a nonintegral pair]\\label{catalogue:cert:b10a}\nLet $Q\\subseteq\\mathbb N_{>0}$ be finite and nonempty. Suppose that\n$B_q<2^L$ for every $q\\in Q$ and that\n\\[\n \\forall q_i\\in Q\\ \\exists q_j\\in Q,\\qquad\n B_{q_j}<(A_{q_i}-A_{q_j})\\bmod2^L.\n\\]\nThen $R_{q_jH}-R_{q_iH}\\notin\\mathbb Z$ for some $q_i,q_j\\in Q$.\n\\end{thm}\n\\begin{proof}\nThe tail identity and its elementary bound give\n\\[\n 2^LR_{qH}=A_q+R_{qH+L},\\qquad 0\\le R_{qH+L}\\le B_q.\n\\]\nSuppose all pairwise tail differences are integers. Choose $q_*\\in Q$\nminimising $R_{qH+L}$. For every $q\\in Q$, the difference\n$R_{qH+L}-R_{q_*H+L}$ is then an integer in $[0,B_q]$, congruent to\n$A_{q_*}-A_q$ modulo $2^L$. Since $B_q<2^L$, it is the least\nnonnegative residue. This contradicts the displayed condition at $q_*$.\n\\end{proof}\n\nFor example, take $H=840$, $L=12$ and $Q=\\{1,2,3,4\\}$.\nThe following exact values verify one failed arc containment in each row:\n\\begin{center}\n\\begin{tabular}{rrrrr}\n$q_i$ & $A_{q_i}\\bmod4096$ & $B_{q_i}$ & $q_j$ & $(A_{q_i}-A_{q_j})\\bmod4096$\\\\\n\\hline\n1 & 2920 & 854  & 3 & 3484\\\\\n2 & 2024 & 1694 & 1 & 3200\\\\\n3 & 3532 & 2534 & 4 & 3832\\\\\n4 & 3796 & 3374 & 1 & 876\n\\end{tabular}\n\\end{center}\nThe last column exceeds $B_{q_j}$, not necessarily $B_{q_i}$.\nOn the circle $\\mathbb R/4096\\mathbb Z$, every two arcs\n$[A_q,A_q+B_q]$ intersect, but the four arcs have no common point.\nThus a test using all four points can exclude a common fractional part\nwithout any pair of these arcs being disjoint. The theorem does not\nrequire pairwise compatibility, or require $|Q|\\ge3$.\n\nThe size hypothesis is $2^L>\\max_{q\\in Q}B_q$. In contrast, the\nsymmetric test $\\mathcal C$ for a pair $q_i<q_j$ needs\n$2^L>2B_{q_j}$ before its two-sided interval can contain a residue.\nThese are inequalities for the modulus $2^L$. They do not say that the\nnew test uses half as many binary digits, or give a uniform comparison\nof the first depths at which the tests succeed.\n\n\\begin{thm}[A sufficient quantified finite-grid condition]\\label{catalogue:cert:b10b}\nSuppose that for every $t_0$ there are $t\\ge t_0$, a depth $L$, and a\nfinite nonempty $Q\\subseteq\\mathbb N_{>0}$ such that, with $H=H_t$,\nall the hypotheses of Theorem~\\ref{catalogue:cert:b10a} hold.\nThen $S\\notin\\mathbb Q$. Indeed, rationality would make all tail\ndifferences on every sufficiently large LCM grid integral, whereas the\nfinite-grid theorem supplies a nonintegral pair on such a grid.\nThe example above verifies one grid; it does not establish the\nquantified hypothesis.\n\\end{thm}\n\n\\begin{prop}[A finite carry test implies nonintegrality]\\label{catalogue:cert:b12}\nFor each integer $z$ with $|z|\\le N+h+1$, define\n\\[\n c_z(0)=z,\\qquad\n c_z(i+1)=2c_z(i)-\\bigl(\\varphi(N+h+i+1)-\\varphi(N+i+1)\\bigr).\n\\]\nSuppose that for every such $z$ there is an $i\\le K$ with\n\\[\n |c_z(i)|\\ge N+i+h+2.\n\\]\nThen $R_{N+h}-R_N\\notin\\mathbb Z$. Indeed, an integral value of the\ntail difference would be one of the $2(N+h+1)+1$ initial candidates.\nIts recurrence would satisfy $c_z(i)=R_{N+h+i}-R_{N+i}$ and hence\n$|c_z(i)|<N+i+h+2$ at every step, contrary to the finite test.\nThe excluded strip is open: equality at its boundary already suffices.\nThis proposition proves soundness for a specified $h,N,K$; an\nirrationality argument still needs the quantified family of tests.\n\\end{prop}\n\n\\begin{thm}[The block norm condition implies irrationality]\\label{catalogue:mob:e1}\nSuppose that for every integer $h\\ge1$ and every threshold $X_0\\in\\N$\nthere are $X,L\\in\\N$ with\n\\[\n X\\ge\\max(X_0,1),\\qquad 16(2X+h+L+2)\\le2^L,\n\\]\nand\n\\[\n \\left\\lVert\\sum_{X\\le N<2X}E(h,N,L)\\right\\rVert\\le\\frac{21}{25}X.\n\\]\nThen $S\\notin\\mathbb Q$. Indeed, the norm bound implies the real-part\nbound $\\operatorname{Re}\\sum E(h,N,L)\\le(9/10)X$. Under the displayed\nsize condition, this latter bound already gives a finite residue\ncertificate, by $\\cos(\\pi/8)>9/10$ and averaging. The argument uses this\nspecific numerical gap; an unspecified saving below $X$ is not the\nstated hypothesis.\n\\end{thm}\n\\evidenceremark{The hypothesis, the block norm condition of\nDefinition~\\ref{defn:fh}, is unproved.}\n\n\\begin{thm}[A four-tail residue criterion]\\label{catalogue:mob:e2}\nFor nonnegative integers $H,p,L$, define the four-tail combination,\nits finite numerator and its error bound by\n\\[\n\\begin{aligned}\n J(H,p)&=R_{2pH}-R_{pH}-pR_{2H}+pR_H,\\\\\n W(H,p,L)&=D(pH,pH,L)-pD(H,H,L),\\\\\n B(H,p,L)&=3pH+(p+1)(L+2).\n\\end{aligned}\n\\]\nThen\n\\[\n |2^LJ(H,p)-W(H,p,L)|\\le B(H,p,L).\n\\]\nConsequently the finite condition\n\\[\n B(H,p,L)<W(H,p,L)\\bmod2^L<2^L-B(H,p,L)\n\\]\nimplies $J(H,p)\\notin\\mathbb Z$. If, for every $t_0\\in\\mathbb N$,\nthere are $t\\ge t_0$, $p\\ge1$ and $L\\ge0$ satisfying this condition\nwith $H=H(t)$, then $S\\notin\\mathbb Q$. Rationality would instead make\nboth diagonal differences in $J(H(t),p)$ integral for every sufficiently\nlarge $t$.\n\nThe error bound groups the remainder as\n\\[\n (R_{2pH+L}+pR_{H+L})-(R_{pH+L}+pR_{2H+L}).\n\\]\nEach parenthesis is nonnegative and at most $B(H,p,L)$, by\n$0\\le R_n\\le n+2$. Their difference therefore has absolute value at\nmost $B(H,p,L)$. This proves the bound without asserting optimality for\nthe actual totient tails. Primality of $p$ is not required by the\ncriterion; a prime $p$ is useful when comparing consecutive LCM heights.\nThe checked instance is $(H,p,L)=(12,5,15)$, for which\n$W(H,p,L)\\bmod2^L=18834$ and $B(H,p,L)=282$.\n\\end{thm}\n\\evidenceremark{The checked instance is a finite computation. The\nhypothesis for every $t_0$ is unproved.}\n% ; - part a249_p1b ; -\n\\subsection{Consequences}\n\nThe next conditions concern a single LCM-diagonal tail difference. For\n$t,j,a\\in\\mathbb N$, write\n\\[\n \\delta_t(j)=\\varphi(2H(t)+j)-\\varphi(H(t)+j),\n \\qquad \\Omega_a=R_{2H(2^a)}-R_{H(2^a)}.\n\\]\nFor $h,N\\in\\mathbb N$ and $z\\in\\mathbb Z$, define the integer recurrence\n\\[\n c_{h,N,z}(0)=z,\\qquad\n c_{h,N,z}(k+1)=2c_{h,N,z}(k)\n   -\\bigl(\\varphi(N+h+k+1)-\\varphi(N+k+1)\\bigr).\n\\]\nIf $z=R_{N+h}-R_N$, then $c_{h,N,z}(k)=R_{N+h+k}-R_{N+k}$.\nThis separates an arbitrary integer initial value from the actual tail.\n\nTwo finite quantities will also recur. With $H=H_{2^a}$, put\n\\[\n \\rho_{a,q}=\\frac{D(H,H,2q+1)+\\delta_{2^a}(2q+2)}{2^{2q+1}},\n \\qquad \\varepsilon_{a,q}=\\frac{2H+2q+3}{2^{2q+1}}.\n\\]\nThe prescribed odd depth used below is $2q_a+1$, where\n\\[\n q_a=\\left\\lfloor\\frac{\\lfloor\\log_2 H\\rfloor+10}{2}\\right\\rfloor.\n\\]\nThus $2q_a+1$ is the least odd integer at least\n$\\lfloor\\log_2 H\\rfloor+10$. The error bound is\n$|\\Omega_a-\\rho_{a,q}|<\\varepsilon_{a,q}$.\nFor $a\\ge2$, let $u_{a,q}$ be the centred integer representative of\n\\[\n \\frac{D(H,H,2q+1)+\\delta_{2^a}(2q+2)}2\\pmod{4^q},\n \\qquad -\\frac{4^q}2<u_{a,q}\\le\\frac{4^q}2.\n\\]\nThe numerator is even, since the totient arguments exceed $2$; at the\nmidpoint this convention chooses the positive representative. The\nnormalised block $\\rho_{a,q}$ approximates $\\Omega_a$, whereas\n$u_{a,q}$ records a residue. They must not be confused with the real\ntail difference or with the integer recurrence.\n\n\\begin{prop}[An equivalent diagonal nonintegrality condition]\\label{prop:NI-01}\nThe following statements are equivalent:\n\\[\n S\\notin\\mathbb Q\n \\quad\\Longleftrightarrow\\quad\n \\forall a_0\\in\\mathbb N\\ \\exists a\\ge a_0,\\ \\Omega_a\\notin\\mathbb Z.\n\\]\nThis is an exact reformulation of irrationality, not a weaker theorem\nabout a finite set of scales. The later sufficient conditions seek a\nfinite inequality that establishes the right-hand side.\n\\end{prop}\n\n\\begin{prop}[A sufficient short-window condition]\\label{prop:AR-07}\nSuppose that for every $a_0\\in\\mathbb N$ there are $a\\ge a_0$\nand $L<2\\cdot2^a$ with $\\mathcal C(H_{2^a},H_{2^a},L)$. Then\n$S\\notin\\mathbb Q$. Writing out the certificate, the required inequality is\n\\[\n 2H_{2^a}+L+2\n <D(H_{2^a},H_{2^a},L)\\bmod2^L\n <2^L-(2H_{2^a}+L+2).\n\\]\nIt places the residue away from both endpoints of the modulus interval.\nThe examples at $a=4,6$ establish the bounded statement in\nProposition~\\ref{prop:SK-02}, not the condition for every threshold.\nPointwise completeness allows a sufficiently large depth but does not\nsupply a depth below $2\\cdot2^a$.\n\n\\end{prop}\n\\evidenceremark{The hypothesis is unproved.}\n\n\\begin{prop}[A sufficient approximation condition]\\label{prop:SEP-03}\nUse the prescribed index $q_a$ and the error\n$\\varepsilon_{a,q_a}$ defined above. Suppose\n\\[\n \\forall a_0\\in\\mathbb N\\ \\exists a\\ge\\max(2,a_0)\\ \\forall z\\in\\mathbb Z,\n \\qquad |\\Omega_a-z|\\ge\\frac1{32}+\\varepsilon_{a,q_a}.\n\\]\nThen $S\\notin\\mathbb Q$. The estimate\n$|\\Omega_a-\\rho_{a,q_a}|<\\varepsilon_{a,q_a}$ implies\n$|\\rho_{a,q_a}-z|>1/32$ for every integer $z$, and hence the finite\nresidue separation from which the proof derives irrationality.\nThis is a fixed separation condition along an unbounded family of LCM\nscales. It is not an irrationality-measure theorem for arbitrary\nrational approximants, and nonintegrality alone does not give the\nstated margin. The depth is prescribed, not an additional free witness.\n\n\\end{prop}\n\\evidenceremark{The separation hypothesis is unproved.}\n\n\\begin{prop}[A sufficient upper-endpoint separation]\\label{prop:TE-04}\nLet $a,J,K,m\\in\\mathbb N$, $a\\ge8$, and $H=H(2^a)$. Assume\n\\[\n J+K+(a+6)<2\\cdot2^a.\n\\]\nDefine the finite condition\n\\[\n \\begin{aligned}\n \\mathcal G_a(J,K,m)\\quad:\\Longleftrightarrow\\quad\n &m\\le K,\\qquad 2H+J+K+2<2^m,\\\\\n &D(H,H+J,K)\\bmod2^m\\le2^m-(2H+J+K+2).\n \\end{aligned}\n\\]\nThen $\\mathcal G_a(J,K,m)$ implies $R_{2H+J}-R_{H+J}\\notin\\mathbb Z$.\nIn particular, irrationality follows if for every $a_0$ there are\n$a\\ge\\max(a_0,8)$ and $K,m$ satisfying $K+(a+6)<2\\cdot2^a$ and\n$\\mathcal G_a(0,K,m)$.\n\nThe sign theorem forces an integral tail to give a residue in the upper\nendpoint interval. This one-sided test excludes that interval; it does\nnot require a lower residue bound. Both the sign-range hypothesis and\nthe strict modulus bound are essential premises of this implication.\nThe required unbounded family is not established.\n\\end{prop}\n\n\\begin{prop}[Relations among the sufficient conditions]\\label{prop:TE-05}\nThe sufficient conditions do not form a single linear chain. The proved\nimplications are\n\\[\n \\begin{aligned}\n \\text{odd-prefix interval at the prescribed depth}\n &\\ \\Longleftrightarrow\\ \\text{centred-residue bound at the prescribed depth},\\\\\n \\text{centred-residue bound at the prescribed depth}\n &\\ \\Longrightarrow\\ \\text{centred-residue bound at a variable depth},\\\\\n \\text{centred-residue bound at a variable depth}\n &\\ \\Longrightarrow\\ \\text{adjacent-suffix band}\n   \\ \\Longrightarrow\\ \\text{upper-endpoint test}.\n \\end{aligned}\n\\]\nThere is also a separate sufficient argument: terminal dominance and\nflexible centred magnitude each imply the two-sided exclusion in\nProposition~\\ref{prop:TE-06}, which implies diagonal nonintegrality.\nThese statements refer to the quantified versions, with the bounds on\n$a,q$ and the depth retained. The detailed conditions and sources appear\nin Proposition~\\ref{prop:te-chain}. No reverse implication between\nterminal dominance and flexible magnitude is asserted, and none of the\nrequired unbounded families is proved here.\n\\end{prop}\n\n\\begin{prop}[An exact endpoint formula]\\label{prop:TE-06}\nLet $a,q\\in\\mathbb N$, $a\\ge8$, and $H=H(2^a)$. Assume\n\\[\n 2q+2+(a+6)<2\\cdot2^a,\\qquad 2(H+q+2)\\le4^q.\n\\]\nIf $z\\in\\mathbb Z$ and $z=\\Omega_a$, then\n\\[\n 2u_{a,q}=\\delta_{2^a}(2q+2)-c_{H,H,z}(2q+1).\n\\]\nConsequently either of the inequalities\n\\[\n \\begin{aligned}\n 2u_{a,q}&\\le\\delta_{2^a}(2q+2)-(2H+2q+3),\\\\\n \\delta_{2^a}(2q+2)&\\le2u_{a,q}\n \\end{aligned}\n\\]\nis sufficient for $\\Omega_a\\notin\\mathbb Z$. Under integrality the\nrecurrence equals a positive tail difference strictly smaller than\n$2H+2q+3$, and the identity rules out both displayed inequalities.\nThe identity is exact; the resulting nonintegrality criterion is only\nsufficient. Its converse is not asserted.\n\\end{prop}\n\n\\begin{prop}[A sufficient extension of the examples through exponent 6]\\label{prop:SK-02}\nThe supplied finite result is\n\\[\n \\forall a_0\\le6\\ \\exists a,L\\in\\mathbb N,\\qquad\n a\\ge a_0,\\quad L<2\\cdot2^a,\\quad\n \\mathcal C(H_{2^a},H_{2^a},L).\n\\]\nThe witnesses at $(a,L)=(4,23)$ and $(6,93)$ give this bounded family.\nA witness with $a\\ge7$ would extend the verified range, but any finite\nextension would still leave the assertion for arbitrarily large\nthresholds in Proposition~\\ref{prop:AR-07} unproved.\n\n\\end{prop}\n\n\\begin{prop}[A sufficient extremal-order condition]\\label{prop:FR-01}\nLet $H\\ge1$ and $j\\ge0$. If\n\\[\n \\varphi(2H+j)<\\min\\{\\varphi(H+j),\\varphi(3H+j)\\},\n\\]\nthen\n\\[\n \\varphi(3H+j)-2\\varphi(2H+j)+\\varphi(H+j)>0.\n\\]\nIf instead the middle value is strictly larger than both outer values,\nthe displayed second difference is negative. In each case the claim\nfollows by adding the two strict differences. No lower bound on their\nsizes is needed.\n\nFor $H=H_{2^a}$, this reduces nonvanishing of one second difference to an\nordering of three totient values. For a prescribed $j$, the remaining\nquestion is whether either strict ordering occurs for arbitrarily large\n$a$. The elementary implication does not establish those occurrences.\n\\end{prop}\n\n\\begin{prop}[A directed certificate condition]\\label{prop:CP-06}\nFor fixed $h,N\\in\\N$, a depth $L\\in\\N$ satisfying\n\\[\n N+L+2\\ \\le\\ D(h,N,L)\\bmod2^L\n \\ \\le\\ 2^L-(N+h+L+2)\n\\]\nexists if and only if $R_{N+h}-R_N\\notin\\mathbb Z$.\nThe endpoint inequalities are non-strict.\n\\end{prop}\n\n\\begin{proof}\nThe exact truncation identity leaves an error $R_{N+h+L}-R_{N+L}$ strictly between\n$-(N+L+2)$ and $N+h+L+2$. An integral tail difference would therefore\nput the residue in $[0,N+L+2)$ or\n$(2^L-(N+h+L+2),2^L)$, contrary to the displayed test.\nConversely, a nonintegral difference has a symmetric certificate by\nTheorem~\\ref{catalogue:cert:a7}, and every symmetric certificate\nsatisfies this wider interval condition.\n\\end{proof}\n\nAt $H_3=6$ and $L=6$, the discrepancy is $D(6,6,6)=270$,\nwith residue $14$ modulo $64$. The directed interval is $[14,44]$,\nso its lower endpoint succeeds. The symmetric interval is $(20,44)$\nand fails; its first successful depth is $7$. Thus the wider test\ncan improve the first successful depth, but no uniform depth saving\nis asserted. Replacing $\\mathcal C$ by the displayed inequality in\nTheorem~\\ref{catalogue:cert:b3} gives the equivalent directed diagonal\ncondition, whose occurrence at arbitrarily large scales remains unproved.\n\n\\begin{prop}[A sufficient condition at the prescribed mod-four pulses]\\label{prop:CP-07}\nSuppose that for every positive integer $h$ and every $B\\in\\mathbb N$\nthere are a prime $p>B$ and $K\\in\\mathbb N$ such that\n\\[\n \\varphi(p+4h)-\\varphi(p)\\equiv2\\pmod4\n\\]\nand every integer $z$ with $|z|\\le p+4h+1$ and $z\\equiv2\\pmod4$\nhas an index $0\\le i\\le K$ satisfying\n\\[\n |c_{4h,p,z}(i)|\\ge p+i+4h+2.\n\\]\nThen $S\\notin\\mathbb Q$. Under a hypothetical eventual period $h$,\nthe proved congruence argument at a sufficiently large such prime\nforces $R_{p+4h}-R_p$ to be an integer in the tested residue class.\nThe finite test excludes every candidate in that class, a contradiction.\n\nThe mod-four congruence is supplied by\nProposition~\\ref{prop:CP-05-inv}; exclusion at the same prime is not.\nRestricting to $2\\pmod4$ retains one residue class out of four, not\nnecessarily exactly one quarter of the candidates in a finite interval.\nThe quantifier over every bound $B$ cannot be replaced by one fixed\nprime for each $h$.\n\n\\end{prop}\n\n\\begin{prop}[The full quantified condition]\\label{prop:A10}\n\\[\n\\big(\\forall h\\ge 1,\\ \\forall N_0,\\ \\exists N\\ge N_0,\\ \\exists L,\\ \\mathcal{C}\\ h\\ N\\ L\\big)\n\\iff S\\notin\\mathbb Q.\n\\]\nThis is exactly the quantifier structure $\\forall h\\ge1\\ \\forall N_0\\ \\exists N\\ge N_0\\ \\exists L\\ \\mathcal{C}(h,N,L)$.\nThe quantified condition is not supplied here. The equivalence specifies\nexactly what a residue-based proof would need beyond the finite examples.\n\\end{prop}\n\n\\begin{prop}[One diagonal parameter suffices]\\label{prop:B3}\nThe following condition is equivalent to irrationality:\n\\[\n \\bigl(\\forall t_0\\in\\mathbb N\\ \\exists t\\ge t_0\\ \\exists L\\ge0,\n       \\ \\mathcal C(H(t),H(t),L)\\bigr)\n \\quad\\Longleftrightarrow\\quad S\\notin\\mathbb Q.\n\\]\nFor a hypothetical rational value, let $h_0$ and $N_0$ be its eventual\ntail period and starting index. Taking $t\\ge\\max(h_0,N_0)$ ensures\n$h_0\\mid H(t)$ and $H(t)\\ge N_0$. The corresponding diagonal difference\nis then integral, contradicting a certificate at that scale.\nConversely, irrationality and pointwise completeness provide a witness\nat every prescribed $t$. The reduction uses one scale parameter; it\nneither establishes the certificate condition nor makes one fixed scale\nsufficient for all rational values.\n\\end{prop}\n\n\\begin{prop}[A sufficient condition on the LCM grid]\\label{prop:B7}\n\\[\n\\big(\\forall t_0,\\ \\exists t\\ge t_0,\\ \\exists q\\,m\\,L,\\ 0<q \\wedge \\mathcal{C}(m\\cdot{H}\\,t)(q\\cdot{H}\\,t)\\ L\\big)\n\\implies S\\notin\\mathbb Q.\n\\]\nOne residue certificate \\emph{anywhere} on the two-multiplier LCM cone, at arbitrarily large $t$,\nsuffices; the diagonal choice $q=m=1$ recovers Proposition~\\ref{prop:B3}.\n\\end{prop}\n\\evidenceremark{The hypothesis is unproved.}\n\n\n\\begin{prop}[A finite-grid certificate condition]\\label{prop:B10}\nLet $H,L\\in\\mathbb N$ and let $Q\\subseteq\\mathbb N_{>0}$ be finite and\nnonempty. Define $A_q=\\sum_{j=1}^{L}\\varphi(qH+j)2^{L-j}$ and\n$B_q=qH+L+2$. If $B_q<2^L$ for every $q\\in Q$ and\n\\[\n \\forall q_i\\in Q\\ \\exists q_j\\in Q,\\qquad\n B_{q_j}<(A_{q_i}-A_{q_j})\\bmod2^L,\n\\]\nthen $R_{q_jH}-R_{q_iH}\\notin\\mathbb Z$ for some $q_i,q_j\\in Q$.\nSuch a test at $H=H_t$ for arbitrarily large $t$ implies\n$S\\notin\\mathbb Q$. Neither $|Q|\\ge3$ nor compatibility of every\npair is a hypothesis. The proof, by choosing a minimum tail in the\nfinite set, is given in Theorem~\\ref{catalogue:cert:b10a}.\n\\end{prop}\n\\evidenceremark{The existence of such tests at arbitrarily large $t$ is unproved.}\n\n\\begin{prop}[Unbounded Farey bounds imply irrationality]\\label{prop:C2sup}\nThe Farey-gap denominator bound at window $K$ is currently $\\sim 7.96\\times10^{34}$ at $K=240$\n(Prop.~\\ref{prop:C2-inv}). If\n\\[\n\\sup_K\\, (b+d)(K) = \\infty\n\\]\n(that is, the proved exclusion bounds are arbitrarily large), then \\#249 follows via the denominator-exclusion implication, without assuming the certificate supply in Proposition~\\ref{prop:A10}. No unboundedness claim is proved or attempted in the record.\n\\end{prop}\n\n\\begin{prop}[The lower bound and a false proposed upper bound]\\label{prop:D5cons}\nBy Prop.~\\ref{prop:D4-inv}, the canonical dyadic totient-kernel family is unconditionally\n$(2^e+1)$-dimensional at every level $e\\ge1$. Rationality of $S$ forces an associated\ncarry orbit with $\\mathbb Q$-rank $\\ge 2^e-1$ at every level (Prop.~\\ref{prop:CP-02}). An upper bound independent of $e$ for these same carry ranks would\ncontradict the lower bound. More generally, a bound $g(e)$ with\n$g(e)<2^e-1$ at some level $e\\ge1$ would suffice, provided it applies to the\nactual carry under the hypothetical rationality of $S$. A bound that\nmerely grows with $e$ need not contradict anything. The generic assertion\nfor arbitrary rational coefficient series is false: the $5/4$ comparison\nsequence in Section~\\ref{sec:mahler-defect} has an integer carry satisfying\nthe growth condition and the same rank lower bound. The incompatible\ninteger identity and the finite-shift counterexample (Observation~\\ref{prop:B4b-kill}) remain\nseparate counterexample results. None gives a rank upper bound from rationality alone.\nThe Lean proof covers the rank lower bound and counterexample results,\nnot the counterfactual upper bound.\n\\end{prop}\n\n\\begin{prop}[Soundness of the finite carry test]\\label{prop:B12cons}\nFor fixed $h,N,K\\in\\mathbb N$, use the integer recurrences\n$c_{h,N,z}$ defined above. Test each of the $2(N+h+1)+1$ candidates\n$z\\in\\mathbb Z$ with $|z|\\le N+h+1$. If every candidate has some\n$i\\le K$ for which\n\\[\n |c_{h,N,z}(i)|\\ge N+i+h+2,\n\\]\nthen $R_{N+h}-R_N\\notin\\mathbb Z$.\nAn integral tail difference would give one of the initial candidates\nand would remain in the \\emph{open} strip\n$|c_{h,N,z}(i)|<N+i+h+2$ at every step. Equality at a boundary already\nexcludes a candidate. This is the soundness implication for a specified\nfinite test; no bound for a successful search depth is asserted.\n\n\\end{prop}\n\n\\subsection{Further exact identities}\n\nWe now bound the errors in expansions of tail differences, without\nassuming residue separation. Such bounds determine whether a\nproposed rational centre stays far enough from the integers after\nthe omitted terms are restored.\n\n\\begin{prop}[The exact totient difference on an LCM progression]\\label{prop:AR-04-inv}\nPut $H=H(t)$. For every $j\\ge0$, the arithmetic expression in the piecewise form given below equals $\\delta_t(j)=\\varphi(2H+j)-\\varphi(H+j)$.\nFor a divisor $j\\mid H$, this assertion contains a useful product\nformula, not merely a change of notation. Set\n\\[\n a=H/j,\\qquad g_1=\\gcd(j,a+1),\\qquad g_2=\\gcd(j,2a+1).\n\\]\nThen\n\\[\n \\delta_t(j)=\\varphi(j)\\left(\n \\frac{g_2\\varphi(2a+1)}{\\varphi(g_2)}\n -\\frac{g_1\\varphi(a+1)}{\\varphi(g_1)}\\right).\n\\]\nHere $j\\ge1$ because $H>0$, so both denominators are nonzero.\nThe identity follows from\n$\\varphi(jx)=\\varphi(j)\\varphi(x)\\gcd(j,x)/\\varphi(\\gcd(j,x))$,\nwhich retains all primes shared by $j$ and $x$.\nIf every prime dividing $j$ also divides $a$, then $g_1=g_2=1$,\nand the formula reduces to\n$\\varphi(j)\\bigl(\\varphi(2a+1)-\\varphi(a+1)\\bigr)$.\nFor example, at $H=j=2$ the two gcds are $g_1=2$ and $g_2=1$;\nthe exact difference is $\\varphi(6)-\\varphi(4)=0$, whereas discarding\nthe gcd factors would give $1$.\nFor $j\\nmid H$, that expression is the literal totient difference.\nThus no coprimality assumption is imposed on the full expression.\n\n\\end{prop}\n\n\\begin{prop}[A lower bound for the totient of a rough integer]\\label{prop:AR-03-inv}\nLet $a\\ge8$, set $t=2^a$, and let $n>0$ be an integer all of\nwhose prime factors exceed $t$. If $n<2^{2t}$, then $n$ has fewer\nthan $t/4$ distinct prime factors and\n\\[\n \\varphi(n)>\\frac34 n.\n\\]\nIf $n$ has $r$ distinct prime factors, then\n$n\\ge t^r=2^{ar}$, so $ar<2t$ and $r<t/4$ because $a\\ge8$.\nFor the second, use\n$\\varphi(n)/n=\\prod_{p\\mid n}(1-1/p)\\ge1-\\sum_{p\\mid n}1/p>3/4$.\nThese are bounds for rough integers in the stated size range; they are\nnot pointwise bounds for arbitrary totient values.\n\n\\end{prop}\n\n\\begin{prop}[Positivity of the short-window differences]\\label{prop:AR-05-inv}\nFor $a\\ge8$ and $1\\le j<2\\cdot2^a$,\n\\[\n \\delta_{2^a}(j)=\\varphi(2H_{2^a}+j)-\\varphi(H_{2^a}+j)>0.\n\\]\nThe proof treats divisor offsets and the exceptional prime-power\noffsets separately, using multiplicativity and the rough-integer\nestimate of Proposition~\\ref{prop:AR-03-inv}. No rationality hypothesis\nis used. Positivity of these finitely many coefficients contributes to\nthe later sign theorem; a bound for the remaining infinite tail is\nstill needed to obtain that theorem.\n\n\\end{prop}\n\n\\begin{prop}[The accumulated sum is the discrepancy]\\label{prop:AR-06-inv}\nFor all $t,L\\in\\mathbb N$, the weighted diagonal sum is exactly\n\\[\n \\sum_{j=1}^{L}\\delta_t(j)2^{L-j}=D(H(t),H(t),L).\n\\]\nThus its two-sided residue inequality is precisely\n$\\mathcal C(H(t),H(t),L)$. The equality follows term by term from the\ndefinition of $\\delta_t$. A separate restriction on $L$, when present,\nremains part of the hypothesis; this identity does not supply such a\nrestricted-depth witness.\n\n\\end{prop}\n\n\\begin{prop}[A global-to-local identity]\\label{prop:SEP-01-inv}\n\\[\n\\Omega_a \\;=\\; 2^H(2^H-1)\\Big(\\textstyle\\sum_n' \\varphi(n)/2^n\\Big) - \\big(\\Phi_{2H}-\\Phi_H\\big),\\quad H={H}(2^a).\n\\]\nThus a separation estimate for $\\Omega_a$ is an estimate for this\ninteger translate of a multiple of $S$. The identity does not require\nother rational-approximation arguments to use this particular translate.\n\\end{prop}\n\n\\begin{prop}[A rational approximation with an error bound]\\label{prop:SEP-02-inv}\nFor all $a,q\\in\\mathbb N$, with $H=H_{2^a}$ and the finite\n$\\rho_{a,q}$ defined above,\n\\[\n |\\Omega_a-\\rho_{a,q}|<\\varepsilon_{a,q}\n     =\\frac{2H+2q+3}{2^{2q+1}}.\n\\]\nThe triangle inequality transfers a separation estimate between the\nreal tail and its rational approximation at the cost of this error.\nThe bound tends to zero with $q$ for fixed $a$; no assertion of\noptimality is made.\n\\end{prop}\n\n\\begin{prop}[Unconditional positivity]\\label{prop:SGN-01}\nFor $a\\ge8$, $J+(a+6)<2\\cdot2^a$:\n\\[\n0 < R_{2H+J} - R_{H+J},\\qquad H={H}(2^a),\n\\]\nwith no irrationality hypothesis; the true, infinite, real translated tail difference is\nstrictly positive throughout almost the entire short window, proved unconditionally from\nProp.~\\ref{prop:AR-05-inv} plus a directed one-sided tail bound. Any argument about the sign of\nthe actual orbit (not merely its residue mod $2^L$) must agree with this; specialised at $J=0$\nthis gives $0<\\Omega_a$.\n\\end{prop}\n\n\\begin{prop}[The residue forced by integrality]\\label{prop:SGN-03}\nLet $a,J,K\\in\\mathbb N$, $a\\ge8$, $H=H(2^a)$, and assume\n\\[\n J+K+(a+6)<2\\cdot2^a,\\qquad 2H+J+K+2<2^K.\n\\]\nIf $R_{2H+J}-R_{H+J}\\in\\mathbb Z$, then\n\\[\n \\begin{aligned}\n D(H,H+J,K)\\bmod2^K\n   &=2^K-\\bigl(R_{2H+J+K}-R_{H+J+K}\\bigr),\\\\\n 0<R_{2H+J+K}-R_{H+J+K}&<2H+J+K+2.\n \\end{aligned}\n\\]\nThe later tail difference is a positive integer. Its negative is the\nrepresentative near zero, while the least nonnegative residue lies near\n$2^K$. Positivity alone therefore does not prove nonintegrality; a\nseparate estimate must exclude this upper interval.\n\\end{prop}\n\n\n\\begin{prop}[The penultimate term of a partial divisibility pattern]\\label{prop:TE-02-inv}\nLet $a,J,K,m\\in\\mathbb N$, $a\\ge8$, $H=H_{2^a}$ and\n$B=2H+J+K+2$. Suppose that\n\\[\n\\begin{gathered}\n 0<m\\le K,\\qquad J+K+a+6<2\\cdot2^a,\\\\\n B<2^m,\\qquad \\delta_{2^a}(J+K)\\le2^m-B,\\\\\n 2^{r+1}\\mid\\delta_{2^a}(J+K-m+r+1)\n       \\quad(0\\le r,\\ r+1<m).\n\\end{gathered}\n\\]\nThen\n\\[\n \\delta_{2^a}(J+K-1)=2^{m-1},\\qquad 2^m<2B.\n\\]\nIndeed, $B\\ge4$ and $B<2^m$ force $m\\ge3$. The penultimate\ndivisibility, positivity, and the bound\n$\\delta_{2^a}(J+K-1)<B<2^m$ leave $2^{m-1}$ as its only possible\nvalue. Thus the modulus lies in the strict interval $B<2^m<2B$;\nthere is at most one possible power of two. The value fixed at\n$2^{m-1}$ is the \\emph{penultimate} difference. The terminal\ndifference $\\delta_{2^a}(J+K)$ instead satisfies the separate upper\nbound in the hypotheses. These restrictions are necessary for the\nspecified pattern; the proposition does not construct such patterns\nat arbitrarily large $a$.\n\\end{prop}\n\n\\begin{prop}[An equivalent test using two residue bits]\\label{prop:TE-03-inv}\nFor every $h,N\\in\\mathbb N$, existence of a certificate at some\ndepth is equivalent to the following explicit condition:\n\\[\n \\exists s,b\\in\\mathbb N,\\quad\n \\mathcal C(h,N+s,b+1)\\quad\\text{or}\\quad\n \\begin{cases}\n  N+s+h+b+4<2^b,\\\\\n  2^b\\le D(h,N+s,b+2)\\bmod2^{b+2}<3\\cdot2^b.\n \\end{cases}\n\\]\nThe last interval says that the two leading bits are $01$ or $10$.\nThe proof uses the first successful certificate of depth $L$ and may\ntake $b=\\lfloor\\log_2(N+h+L+2)\\rfloor+1$. Since $b$ depends on that\nunknown $L$, the equivalence is not an a priori bound on the search\ndepth. The proof uses the finite tail recurrence and its size bound;\nit supplies no totient-specific assertion that a witness exists.\n\\end{prop}\n\n\\begin{prop}[The factor in a second difference]\\label{prop:FR-02-inv}\nLet $a\\ge4$ and $j\\ge1$ be integers with $j^2\\le2^a$, and put\n$H_a=H(2^a)=\\operatorname{lcm}(1,\\ldots,2^a)$. Then\n\\[\n 2\\varphi(j)\\mid\n \\varphi(3H_a+j)-2\\varphi(2H_a+j)+\\varphi(H_a+j).\n\\]\nTo see the factor, set $A=H_a/j$. The square bound ensures that $j\\mid H_a$\nand that every prime divisor $p$ of $j$ also divides $A$: indeed,\n$pj\\le j^2\\le2^a$ implies $pj\\mid H_a$. Consequently\n$\\gcd(j,qA+1)=1$ for $q=1,2,3$, and the displayed difference equals\n\\[\n \\varphi(j)\\bigl(\\varphi(3A+1)-2\\varphi(2A+1)+\\varphi(A+1)\\bigr).\n\\]\nAlso $2j\\le2^a$, so $2j\\mid H_a$ and $A\\ge2$ is even. The three\narguments $qA+1$ are therefore odd and greater than two, and all three\ntotients are even. This proves the extra factor of two.\nThe divisibility is a lower bound, not an exact valuation at each LCM\nheight. The different family in Observation~\\ref{prop:FR-03-kill} does\nnot satisfy these LCM and square-window hypotheses, so it does not\nestablish sharpness within this restricted family.\n\\end{prop}\n\n\\begin{prop}[Carry displacement and tail integrality]\\label{prop:CP-01-inv}\nFor a positive integer $v$ and an integral totient carry $u$ with\n$u(N)/2^N\\to0$,\n\\[\n v\\mid u(N+k)-u(N)\\quad\\Longleftrightarrow\\quad R_{N+k}-R_N\\in\\Z.\n\\]\nThe recurrence and the limit identify $u(N)=vR_N$, so the\ndifference is $v(R_{N+k}-R_N)$. This tests divisibility of a carry\ndifference and gives no dimension bound for the carry sections.\nFor another coefficient sequence, the same argument applies once\nits carry is identified with its scaled tail under the corresponding\nrecurrence and limit.\n\\end{prop}\n\n\\begin{prop}[A consequence of rationality and its limitation]\\label{prop:CP-02}\nIf $S\\in\\mathbb Q$, there are a positive integer $v$ and an\ninteger carry sequence $u$, with $u(N)/2^N\\to0$, whose retained\nsections through every\nlevel $e$ have rational rank at least $2^e-1$. At the same time, those\nsections are uniformly eventually periodic modulo $v$.\nPeriodicity after reduction modulo $v$ is a statement in a finite\nquotient; it is not a rank upper bound over $\\mathbb Q$.\nProposition~\\ref{prop:D5cons} gives the lower bound, while the rational\ncomparison sequence explains why a general rank upper bound cannot\nbe deduced from these recurrence assumptions alone.\n\\end{prop}\n\n\\begin{prop}[A totient difference congruent to two modulo four]\\label{prop:CP-05-inv}\nFor every positive integer $h$ and every $B\\in\\mathbb N$, there is\n a prime $p>B$ such that\n\\[\n \\varphi(p+4h)-\\varphi(p)\\equiv2\\pmod4.\n\\]\nTo construct $p$, set $H=4h$, choose a prime $r>H$ with\n$r\\equiv1\\pmod4$, and use Dirichlet's theorem in the progression\n$p\\equiv3r-H\\pmod{4r}$. This progression is reduced: its residue is\nodd and is $-H\\not\\equiv0\\pmod r$. Then $p\\equiv3\\pmod4$, so\n$\\varphi(p)=p-1\\equiv2\\pmod4$, whereas $r\\mid p+H$ gives\n$4\\mid\\varphi(p+H)$. Subtracting gives the stated congruence.\nThis supplies the arithmetic congruence at arbitrarily large primes;\nthe additional finite exclusion in Proposition~\\ref{prop:CP-07}\nremains a separate hypothesis.\n\n\\end{prop}\n\n\\begin{prop}[A prescribed residue after a long divisibility block]\\label{prop:TA-inv}\nLet $K\\ge2$ and $H>K$ be integers. For every $B\\in\\mathbb N$ there\nis a prime $p>\\max(B,H+K)$ such that\n\\[\n\\begin{aligned}\n p&\\equiv1+2^{K-1}\\pmod{2^K},&\n 2^K&\\mid\\varphi(p+H),\\\\\n 2^K&\\mid\\varphi(p-j),&\n 2^K&\\mid\\varphi(p-j+H)\\qquad(1\\le j<K).\n\\end{aligned}\n\\]\nThe construction uses $2K-1$ distinct auxiliary primes congruent to\n$1\\pmod{2^K}$: one for $p+H$, and two for each of the $K-1$\npreceding positions. Choose them larger than $\\max(H+K,2^K)$.\nFor the first prime impose $p\\equiv-H$; for the pair at position $j$\nimpose $p\\equiv j$ and $p\\equiv j-H$, respectively. These nonzero\nresidues, together with the displayed odd residue modulo $2^K$,\ncombine by the Chinese remainder theorem into a reduced progression.\nDirichlet's theorem then supplies arbitrarily large prime values of $p$.\nEach auxiliary prime contributes the required factor $2^K$ to the\ncorresponding totient.\n\nThus the first $K-1$ totient differences in the window starting at\n$p-K$ vanish modulo $2^K$, while the last is $2^{K-1}$ modulo $2^K$.\nConsequently\n\\[\n D(H,p-K,K)\\equiv2^{K-1}\\pmod{2^K}.\n\\]\nIf $R_{N+H}-R_N\\in\\mathbb Z$ for every $N\\ge N_0$, choose such a\nprime with $p\\ge N_0+K$. Iterating the carry recurrence from $p-K$\nthen gives\n\\[\n R_{p+H}-R_p\\in\\mathbb Z,\\qquad\n R_{p+H}-R_p\\equiv2^{K-1}\\pmod{2^K}.\n\\]\nThe initial integer is multiplied by $2^K$ and disappears only after\nreduction modulo $2^K$. This does not bound its higher quotient.\nThe condition $H>K$ is part of the construction; it does not provide\narbitrarily large $K$ for one fixed shift $H$.\n\n\\end{prop}\n\n\\begin{prop}[A M\\\"obius-inversion formula for the tail]\\label{prop:MP-01-inv}\n\nFor $N\\in\\mathbb N$ and $d\\ge1$, let\n$r_d(N)=d-(N\\bmod d)$ and $q_d(N)=\\lfloor N/d\\rfloor+1$.\nThus $r_d(N)$ is the distance to the next strictly larger multiple of\n$d$, so $1\\le r_d(N)\\le d$. M\\\"obius inversion gives\n\\[\n R_N=\\sum_{d\\ge1}\\mu(d)2^{d-r_d(N)}\n \\left(\\frac{q_d(N)}{2^d-1}+\\frac{1}{(2^d-1)^2}\\right).\n\\]\nIndeed, substitute $\\varphi(n)=\\sum_{d\\mid n}\\mu(d)n/d$ and group\nby $d$. Its multiples after $N$ have quotients $q_d(N)+\\ell$ and\nforward distances $r_d(N)+d\\ell$, for $\\ell\\ge0$; summing the two\ngeometric series gives the formula. Absolute convergence justifies the\nregrouping, since the sum of the absolute divisor contributions at $n$\nis at most $\\sum_{d\\mid n}n/d\\le n^2$, and\n$\\sum_{n>N}n^2 2^{N-n}<\\infty$.\nThis identity is available for later transformations; it does not\nassert that every M\\\"obius or Lambert-series argument must use it.\n\n\\end{prop}\n\n\\begin{prop}[Explicit finite examples]\\label{prop:SK-01-inv}\nThe finite certificates\n$\\mathcal C(H_{16},H_{16},23)$ and\n$\\mathcal C(H_{64},H_{64},93)$ imply\n$\\Omega_4,\\Omega_6\\notin\\mathbb Z$. Both satisfy the short-window\nrestriction, since $23<32$ and $93<128$. They are the certificates at\n$t=16$ and $t=64$ in the diagonal table of Section~\\ref{ssec:certtable}.\nThese are two explicit instances of the desired condition, not the\nstart of a proved induction or a claim that no other finite instances\ncan be checked.\n\n\\end{prop}\n\n\\begin{prop}[The necessary depth]\\label{prop:A5-inv}\nEvery certificate $\\mathcal C(h,N,L)$ satisfies\n\\[\n 2(N+h+L+2)<2^L.\n\\]\nThe two endpoint margins must fit inside an interval of length $2^L$.\nIn particular, successful depths grow at least logarithmically with\n$N+h$. This is a necessary lower bound, not an upper bound for the\nfirst successful depth. The later two-bit reformulation applies at its\nstated modulus and does not reverse this implication.\n\n\\end{prop}\n\n\\begin{prop}[Rationality gives an eventual tail period]\\label{prop:A9-inv}\nSuppose $S=a/(2^c v)$ in lowest terms, where $a\\in\\mathbb Z$,\n$c\\ge0$ and $v\\ge1$ is odd. Take $h=\\varphi(v)$, including\n$h=1$ when $v=1$. Euler's theorem gives $v\\mid2^h-1$, so\n\\[\n 2^N(2^h-1)S\\in\\mathbb Z\\qquad(N\\ge c).\n\\]\nThe identity relating $S$ to its scaled tails therefore gives\n$R_{N+h}-R_N\\in\\mathbb Z$ for every $N\\ge c$.\nThese are explicit witnesses for the eventual period asserted in\nProposition~\\ref{prop:A10}. To use the finite residue test against a\nhypothetical rational value, one must obtain a nonintegral difference\nfor its period at an index beyond its preperiod. Soundness proves that\nthe quantified certificate condition rules out rationality. Conversely,\nthe prefix-tail identity makes every positive-shift tail difference\nnonintegral when $S$ is irrational, and pointwise completeness\n(Theorem~\\ref{catalogue:cert:a7}) then supplies a depth.\nProposition~\\ref{prop:AR-06-inv} only identifies the weighted diagonal\nsum with $D$; it is not the completeness argument and supplies no\ndepth bound.\n\\end{prop}\n\n\\begin{prop}[Tail integrality on an LCM grid]\\label{prop:B6-inv}\nIf $S\\in\\mathbb Q$, there is $t_1\\in\\mathbb N$ such that for all\n$t\\ge t_1$, $q\\ge1$ and $m\\ge0$,\n\\[\n R_{(q+m)H(t)}-R_{qH(t)}\\in\\mathbb Z.\n\\]\nThe threshold may depend on the hypothetical rational value. A\ncertificate at one fixed scale therefore gives a finite denominator\nexclusion, not irrationality. Proposition~\\ref{prop:B7} requires\ncertificates at arbitrarily large scales, so that one lies beyond the\nthreshold supplied by any such rational value.\n\n\\end{prop}\n\n\\begin{prop}[Independence of the retained dyadic family]\\label{prop:D4-inv}\nFor every $e\\ge0$, the auxiliary canonical dyadic family has\n$2^e+1$ indexed sections, and these sections are linearly independent\nover $\\mathbb Q$. The CRT--Dirichlet argument is explained in\nSection~\\ref{sec:mahler-defect}. For $e\\ge1$ the family is a basis for all sections through level\n$e$. At $e=0$ the auxiliary family still contains both $\\varphi(n)$ and\n$\\varphi(2n)$, so it is not the actual level-zero truncation, whose\ndimension is one. No rationality hypothesis on $S$ is used.\n\n\\end{prop}\n\n\\begin{prop}[The exact range of the stated Farey-gap inequality]\\label{prop:C2-inv}\nLet $V$ be the totient residue at $(N,K)=(1,240)$ defined in\nSection~\\ref{ssec:farey}. For every integer $q$ satisfying\n\\[\n 1\\le q\\le79639646646701375323355774875831053,\n\\]\none has\n\\[\n (qV)\\bmod2^{240}+243q<2^{240}.\n\\]\nThe first failure occurs at\n\\[\n q=79639646646701375323355774875831054,\n\\]\nobtained from the stated Farey mediant. Thus the range is exact for\nthis particular inequality.\nIt is not an optimality claim among all denominator-exclusion tests.\nThe preceding range gives the exclusion in\nProposition~\\ref{prop:C3-inv}, approximately $7.96\\times10^{34}$.\n\\end{prop}\n\\evidenceremark{The range and its first failure are an exact finite\ncomputation at the window $K=240$.}\n\n\\begin{prop}[The resulting denominator exclusion]\\label{prop:C3-inv}\nFor every reduced fraction $a/q$, with $a\\in\\mathbb Z$ and $q\\ge1$,\n\\[\n q\\le79639646646701375323355774875831053\n \\quad\\Longrightarrow\\quad S\\ne a/q.\n\\]\nThis is an unconditional finite denominator exclusion. It does not\nassume certificates at arbitrarily large scales, as required by\nProposition~\\ref{prop:A10}. Conversely, excluding this finite range\ndoes not rule out rational values with larger reduced denominators.\n\\end{prop}\n\n\\begin{prop}[Two general irrationality criteria]\\label{prop:D1D2-inv}\nLet $x\\in\\mathbb R$. One sufficient condition is a sequence of\nreduced fractions $u_j=a_j/q_j$, $q_j\\ge1$, with $u_j\\ne x$ for\nall sufficiently large $j$ and\n\\[\n q_j|x-u_j|\\longrightarrow0.\n\\]\nAnother is that, for every integer $Q\\ge1$, there exist $m,z\\in\\mathbb Z$\nsuch that\n\\[\n 0<|mx-z|<1/Q.\n\\]\nIndeed, if $x=a/b$ is reduced with $b>0$, every nonzero error in the\nfirst condition is at least $1/(bq_j)$, and every nonzero error in the\nsecond is at least $1/b$. The strict lower bounds exclude exact\nrational hits. Requiring $m=b_0^n$ for a fixed integer base $b_0\\ge2$\nis a stronger sufficient condition, not a reformulation valid for\nevery irrational number. The binary example in\nProposition~\\ref{catalogue:cert:d2} explains this distinction.\n\\end{prop}\n\n\\begin{prop}[Lambert identities involving $S$]\\label{prop:D7-inv}\nFor $L(f)=\\sum_{n\\ge1}f(n)/(2^n-1)$, the identities are\n$L(\\mu)=1/2$, $L(\\varphi)=2$, $L(1)=E$ and $L(\\varphi*\\mu)=S$.\nThe last identity rewrites the same unknown value; it does not deduce its\nirrationality from that of $E$. The value\n$L(\\mathrm{Id})=\\sum_{m\\ge1}\\sigma(m)/2^m$ is transcendental by\nNesterenko~\\cite[Cor.~2, p.~1320]{nesterenko1996}.\nThese examples show that Lambert-series form alone does not determine\narithmetic status. The individual comparisons are in\nProposition~\\ref{catalogue:cert:d7} and its table.\n\\end{prop}\n\\evidenceremark{The Lean proof of the transcendence of $L(\\mathrm{Id})$\nassumes Nesterenko's theorem on the algebraic independence of the three\nEisenstein series values at an algebraic point of the punctured unit disc.\nThe four Lambert identities and the identification of $L(\\mathrm{Id})$ with\n$\\sum_{m\\ge1}\\sigma(m)/2^m$ are proved without it.}\n\n\\begin{prop}[A general rational gap bound]\\label{prop:D9-inv}\nIf $a/b<c/d$ are reduced fractions with $b,d>0$, then\n\\[\n \\frac{1}{bd}\\le\\frac cd-\\frac ab,\n\\]\nsince $bc-ad$ is a positive integer. In particular, if $S=a/b$ and\n$p_N/q_N<S$ is a reduced rational approximation with positive error\nat most $\\varepsilon_N$, then\n\\[\n b\\ge\\frac{1}{q_N\\varepsilon_N}.\n\\]\nThese lower bounds tend to infinity precisely when\n$q_N\\varepsilon_N\\to0$; a subsequence with this property is enough\nto contradict a fixed $b$. A tail bound by itself does not imply this. For example, the ordinary dyadic\nprefix has denominator dividing $2^N$ and tail at most\n$(N+2)2^{-N}$. Using only these two bounds yields\n$b\\ge1/(N+2)$, which is vacuous. The rational-spacing lemma can\nconvert sharper error and denominator information into an exclusion;\nit does not automatically improve the Farey bound in\nProposition~\\ref{prop:C2-inv}.\n\\end{prop}\n\n\\begin{prop}[Diagonal certificates through $82$]\\label{prop:B11-inv}\nThe current aggregate establishes\n\\[\n \\forall t\\in\\mathbb N,\\quad t\\le82\\ \\Longrightarrow\\ P\\,t,\n\\]\nwhere $P\\,t$ is the diagonal certificate predicate in\nProposition~\\ref{prop:B3}. The earlier aggregate covered 28 explicit\ncases through $64$, including\n$1,2,3,4,5,7,8,9,11,13,16,17$; the later theorem fills the gaps\nand extends the range through $82$. Each instance reduces to exact\nfinite arithmetic with the displayed totient values, using the\nprime-power factorisations and Lucas primality certificates in the\nsource. This proves neither $P\\,83$ nor infinitely many instances,\nand therefore does not establish the quantified conditions in\nPropositions~\\ref{prop:A10}, \\ref{prop:B3} and \\ref{prop:B7}.\n\\end{prop}\n\n\\begin{prop}[A coprime-pair expression for $S$]\\label{prop:C1-inv}\nFor $n\\ge0$, the number of integer pairs $a\\ge1$, $b\\ge0$ with\n$a+b=n$ and $\\gcd(a,b)=1$ is $\\varphi(n)$, with\n$\\varphi(0)=0$. Indeed, $b=n-a$ and\n$\\gcd(a,n-a)=\\gcd(a,n)$. The boundary case $n=1$ contributes\n$(a,b)=(1,0)$; it disappears when both coordinates are required to\nbe positive. Consequently,\n\\[\n \\sum_{\\substack{a,b\\ge1\\\\\\gcd(a,b)=1}}2^{-(a+b)}=S-\\frac12.\n\\]\nFor independent random variables $X,Y$ with\n$\\Pr(X=n)=\\Pr(Y=n)=2^{-n}$, $n\\ge1$, the left side is\n$\\Pr(\\gcd(X,Y)=1)$. Thus $S-1/2$ has a coprimality-probability\ninterpretation. This identity does not supply an estimate for the\nbinary digits or tail residues used elsewhere in the paper.\n\\end{prop}\n\n\\subsection{Counterexamples and limitations}\n\nA residue or coefficient condition may hold for a rational comparison\nseries as well as for $S$. Each example below records the information\nit preserves and the deduction it disproves. The irrationality of $S$\nremains open, and an argument using further totient identities is\nunaffected by a comparison that does not preserve them.\n\n\n\\begin{obs}[The sign of the actual endpoint representative]\\label{prop:SGN-02}\nAssume $a\\ge8$, $H=H_{2^a}$ and\n$J+K+a+6<2\\cdot2^a$, as in\nProposition~\\ref{prop:SGN-01}. If\n$d=R_{2H+J}-R_{H+J}$ is an integer, define\n\\[\n c_0=d,\\qquad\n c_{j+1}=2c_j-\\bigl(\\varphi(2H+J+j+1)-\\varphi(H+J+j+1)\\bigr).\n\\]\nThe tail recurrence gives\n$c_K=R_{2H+J+K}-R_{H+J+K}>0$. Its negative is an integer\nrepresentative satisfying\n\\[\n |{-c_K}|\\le2H+J+K+2,\\qquad\n -c_K\\equiv D(H,H+J,K)\\pmod{2^K}.\n\\]\nThus the representative corresponding to the actual tail is negative.\nThe stated size bound allows it to remain in the endpoint interval,\nso its sign alone does not contradict integrality. Other integers in\nthat interval with the same residue need not represent the actual tail.\nProposition~\\ref{prop:SGN-03} gives the further residue separation\nthat would exclude this case.\n\\lean{actualLcm_trueEndpointSurvivor_neg}{TotientActualLcmOrbitSign.lean:172}\n\\end{obs}\n\n\\begin{obs}[The complete divisibility pattern is impossible]\\label{prop:TE-01-killer}\nLet $a\\ge8$, $H=H_{2^a}$ and $J,K,m\\in\\mathbb N$. Assume\n\\[\n m\\le K,\\qquad J+K+a+6<2\\cdot2^a,\\qquad\n 2H+J+K+2<2^m.\n\\]\nThe simultaneous divisibilities\n\\[\n 2^{r+1}\\mid\\delta_{2^a}(J+K-m+r+1)\\qquad(0\\le r<m)\n\\]\ncannot hold. The terminal difference would be positive by\nProposition~\\ref{prop:AR-05-inv}, smaller than $2^m$, and divisible by\n$2^m$, a contradiction. The obstruction applies to this complete\npattern under the stated size bounds. Omitting its last divisibility\ncondition leads to the necessary restrictions in\nProposition~\\ref{prop:TE-02-inv}; the residue inequality in\nProposition~\\ref{prop:TE-04} is another sufficient condition. Neither\nstatement here supplies such witnesses at arbitrarily large scales.\n\\lean{not_actualLcmTerminalDyadicStaircase_of_room}{TotientActualLcmTopEdgeStaircase.lean:251}\n\\end{obs}\n\n\\begin{obs}[An exact valuation in a dyadically scaled family]\\label{prop:FR-03-kill}\nFor every integer $n\\ge0$, take $H=2^{n+2}$ and $j=3\\cdot2^{n+1}$.\nThen\n\\[\n \\varphi(3H+j)-2\\varphi(2H+j)+\\varphi(H+j)=-2^{n+1}.\n\\]\nIndeed, the three arguments are $9\\cdot2^{n+1}$,\n$7\\cdot2^{n+1}$ and $5\\cdot2^{n+1}$. Their totients are respectively\n$6\\cdot2^n$, $6\\cdot2^n$ and $4\\cdot2^n$, which give the identity.\nThus the difference has $2$-adic valuation exactly $n+1$.\nThis example limits a valuation bound for arbitrary even heights and\nscaled odd offsets. Here $j>H$, whereas the offset in\nProposition~\\ref{prop:FR-02-inv} is at most $2^{a/2}$ and the height is\n$H(2^a)$. The example therefore does not exclude additional divisibility\non that LCM family.\n\\lean{fixedRankSecondDifference_dyadic_fixture_exact}{TotientFixedRankLcmAsymptotic.lean:462}\n\\end{obs}\n\n\\begin{obs}[A rational sequence with matching parity]\\label{prop:CM-01}\n\nStart with the coefficient sequence $0,1,1,2,4,4,4,\\ldots$, indexed\nfrom $0$. For each $m=2^{k+3}$, $k\\ge0$, replace the pair at positions\n$m,m+1$ from $(4,4)$ to $(6,0)$, leaving all other coefficients unchanged.\nThese pairs are disjoint. Each replacement preserves the binary sum because\n\\[\n 2\\cdot2^{-m}-4\\cdot2^{-(m+1)}=0.\n\\]\nThe resulting sequence has the same sum $3/2$ and the same parity at\nevery index as $\\varphi$. It is not eventually periodic, since the gaps\nbetween its occurrences of $6$ tend to infinity. This construction uses\nthe particular initial parity pattern and the binary weights; it does\nnot establish an analogous result for arbitrary prescribed parities or\nother summand weights.\n\\lean{parityCoboundaryWeight}{TotientParityCoboundaryCountermodel.lean:59}\n\\end{obs}\n\n\\begin{obs}[What the parity example disproves]\\label{prop:CM-02}\nThe sequence $c$ in Observation~\\ref{prop:CM-01} satisfies\n\\[\n 0\\le c(n)\\le\\min(n,6),\\qquad\n c(n)\\equiv\\varphi(n)\\pmod2,\\qquad\n \\sum_{n\\ge0}c(n)2^{-n}=\\frac32,\n\\]\nand is not eventually periodic. More precisely, for every\n$N,G,K\\in\\mathbb N$, it has $K$ of the displayed $(6,0)$ pairs\nstarting beyond $N$, with successive starting positions separated\nby more than $G$. Choose sufficiently late powers of two in the\nconstruction to obtain these pairs.\nHence the growth bound, the exact parity pattern, nonperiodicity and\nthese arbitrarily separated occurrences are compatible with a rational\nsum. They cannot by themselves prove irrationality. The construction\ndoes not preserve the complete totient sequence or its\nmultiplicativity, and does not specify which additional arithmetic\nhypothesis would suffice.\n\\lean{exists_totientParity_arbitrarilyManySeparatedCarry_rational_countermodel}{TotientParityCoboundaryCountermodel.lean:637}\n\\end{obs}\n\n\\begin{obs}[A Mersenne factor does not force tail-difference divisibility]\\label{prop:CP-03-kill}\nFor the completely multiplicative sequence $c(n)=n$, the scaled tail is\n\\[\n T_N=\\sum_{j\\ge1}(N+j)2^{-j}=N+2,\n \\qquad T_{N+K}-T_N=K.\n\\]\nIf $q>K>0$ and $q\\mid2^K-1$, then every such tail difference is\nintegral but is not divisible by $q$. For example, take $K=3$ and\n$q=7$; the multiplicative order of $2$ modulo $7$ is $3$.\nThus even this prime divisor of the homogeneous recurrence multiplier\ndoes not force divisibility of the inhomogeneous tail difference.\nAn application to the totient sequence needs a further relation between\nthe prime and the coefficient residues or the tail's size. This\nexample supplies no general obstruction for Problem~\\#257 merely\nbecause that problem also involves Mersenne denominators. Compare the\nspecific totient construction in Proposition~\\ref{prop:CP-05-inv}.\n\\lean{idCoeff_mersenneModulus_does_not_annihilate_tailShift}{TotientTailCarryPeriod.lean:342}\n\\end{obs}\n\n\\begin{obs}[A lower bound for the stated reconstruction cost]\\label{prop:CP-04-kill}\nLet $I$ be finite, $w_i\\in\\mathbb Q$ and $x_i\\in\\mathbb N$, and\nassume $\\sum_{i\\in I}w_i\\varphi(x_i)=1$. Then\n\\[\n \\sum_{i\\in I}|w_i|\\bigl(2(x_i+1)+(x_i+2)\\bigr)\\ge3.\n\\]\nIndeed,\n$1\\le\\sum_i|w_i|\\varphi(x_i)\\le\\sum_i|w_i|x_i$,\nand the displayed cost is $3\\sum_i|w_i|x_i+4\\sum_i|w_i|$.\nFor $x\\ge1$, the reconstruction\n$\\varphi(x)=2R_{x-1}-R_x$ and the separate estimates\n$R_M\\le M+2$ give exactly this cost. Terms at $x=0$ contribute\nnothing to the normalisation, since $\\varphi(0)=0$.\nThis particular termwise triangle bound cannot be smaller than one.\nThe conclusion is about that bound, not the true combined tail error;\ncancellation or a different estimate is not excluded. The same elementary\ncalculation applies to any nonnegative sequence $c(n)\\le n$ under\nthe corresponding normalisation $\\sum_iw_i c(x_i)=1$.\n\\lean{three_le_totientAdjugateTailCost}{TotientTailCarryPeriod.lean:266}\n\\end{obs}\n\n\\begin{obs}[A second-difference test need not use less depth]\\label{prop:B9cert-kill}\nFor $h,N,L\\in\\mathbb N$, put\n\\[\n D_2=D(h,N+h,L)-D(h,N,L),\\qquad B_2=2(N+2h+L+2).\n\\]\nThe second-difference certificate is the finite inequality\n\\[\n B_2<D_2\\bmod2^L<2^L-B_2.\n\\]\nIt implies $R_{N+2h}-2R_{N+h}+R_N\\notin\\mathbb Z$.\nThe two first-difference truncation errors have bounds\n$N+2h+L+2$ and $N+h+L+2$ after multiplication by $2^L$;\ntheir sum is at most the stated $B_2$. This explains the radius\nused by the linked test, without claiming it is optimal.\nAt $(h,N)=(1,8)$, the first-difference test holds at depth $8$,\nwhereas the second-difference test fails at every depth at most $8$\nand first holds at $9$. The earlier 40-cell probe for $t\\le20$\nreported the first test at least as shallow in 30 cells; that aggregate\nis a reported computation, not part of the single-cell theorem linked\nbelow. These comparisons do not give a uniform ordering of the depths.\n\\lean{totient_tail_rank_two_kill_sound_but_not_shallower_cell}{CertificateKernel.lean:19090}\n\\end{obs}\n\n\\begin{obs}[The prime-power restriction in the unit-gap refinement]\\label{prop:D8cert-kill}\nThe ordinary interval test excludes a rational denominator when its\npossible numerator interval contains no integer. Reduced fractions\npermit a refinement: every numerator is coprime to its denominator,\nso an interval containing only nonunits can also be excluded.\nFor a candidate $S=a/(2^c u)$ in lowest terms, with $u>0$ odd\nand $N\\ge c$, the integer $uR_N=2^{N-c}a-u\\Phi_N$ is coprime\nto $u$. For such an $N$ and any $K\\ge0$, define\n\\[\n \\begin{aligned}\n V&=\\left(\\sum_{r=1}^{K}\\varphi(N+r)2^{K-r}\\right)\\bmod2^K,\\\\\n uV&=J2^K+W,\\qquad 0\\le W<2^K,\\\\\n L&=\\left\\lfloor\\frac{W+u(N+K+2)}{2^K}\\right\\rfloor.\n \\end{aligned}\n\\]\nAfter subtracting an integer multiple of $u$, the finite prefix\nof $uR_N$ is $J+W/2^K$ and its remaining tail lies in\n$(0,u(N+K+2)/2^K]$. Its possible integer values are therefore\n$J+1,\\ldots,J+L$. The refined condition is\n$\\gcd(J+t,u)>1$ for all $1\\le t\\le L$, contradicting the required\ncoprimality of $uR_N$.\nFor a prime $p$ and $e\\ge1$, nonunits modulo $p^e$ are exactly the\nmultiples of $p$. Since two consecutive integers cannot both be\nmultiples of $p$, the condition for $u=p^e$ is equivalent to\n\\[\n L=0\\quad\\text{or}\\quad L=1\\text{ and }p\\mid J+1.\n\\]\nThus a \\emph{successful} refined certificate at a prime-power denominator\nhas at most one candidate integer. This is not a bound for an arbitrary\nwindow: $(u,N,K)=(3,3,3)$ gives $V=2$, $J=0$, $W=6$ and $L=3$,\nand the test fails because $1,2,3$ are not all nonunits modulo $3$.\nThe prime-power restriction also matters. With the actual totient\nwindow $(u,N,K)=(15,13,8)$, one gets\n\\[\n V=148,\\quad J=8,\\quad W=172,\\quad L=2.\n\\]\nBoth candidates $9,10$ are nonunits modulo $15$. This interval passes\nthe refined test but not the empty-interval test. The prime-power bound\ntherefore neither applies to all composite denominators nor bounds the\npossible improvement in the final denominator cutoff.\nThe prime-power criterion is proved in Lean\n(\\lean{reducedDenominatorCandidateCount_prime_pow_le_one}{PrimitiveWeightCertificate.lean:111}).\n\\end{obs}\n\n\\begin{obs}[A synthetic sequence preserving the stated LCM differences]\\label{prop:B4b-kill}\nHere is the recurrence behind the comparison. Fix $t\\ge3$, put\n$H=\\operatorname{lcm}(1,\\ldots,t)$ and $A=\\varphi(H)$, and define\n\\[\n c_i=\\begin{cases}-A,&i=(q-1)H\\text{ for some }2\\le q<t,\\\\\n                   0,&\\text{otherwise},\\end{cases}\n \\qquad a_i=2c_i-c_{i+1}.\n\\]\nThe index $i$ here corresponds to the argument $H+i+1$ of a totient\ndifference. For $2\\le q<t$ this gives\n\\[\n a_{(q-1)H-1}=\\varphi((q+1)H)-\\varphi(qH)=A.\n\\]\nThe equality follows because all prime divisors of $q$ and $q+1$ divide\n$H$: hence $\\varphi(qH)=qA$ and $\\varphi((q+1)H)=(q+1)A$.\nEvery $a_i$ is divisible by $A$, and $|c_i|\\le A$, $|a_i|\\le2A$.\nFor finitely supported integer weights $(\\lambda_j)_{j\\ge0}$, put\n$d_i=\\sum_j\\lambda_j c_{i+j}$ and $b_i=\\sum_j\\lambda_j a_{i+j}$.\nLinearity and telescoping give\n\\[\n b_i=2d_i-d_{i+1},\\qquad\n \\sum_{r=0}^{L-1}2^{L-1-r}b_{i+r}=2^Ld_i-d_{i+L}.\n\\]\nThe triangle inequality gives the precise uniform bounds\n\\[\n |d_i|\\le A\\sum_j|\\lambda_j|,\\qquad\n |b_i|\\le2A\\sum_j|\\lambda_j|.\n\\]\nThese are uniform in $i$, not in the weights. The factor\n$\\sum_j|\\lambda_j|$ cannot be omitted: taking only $\\lambda_0=2$\ndoubles the nonzero state and coefficient maxima. Divisibility by $A$\nand by $\\varphi(d)$ for $d\\mid H$ is preserved, since\n$\\varphi(d)\\mid\\varphi(H)$.\n\nFinite shift polynomials therefore preserve the recurrence, but not the\noriginal numerical bounds independently of their coefficients. This\ncomparison does not identify $a_i$ with actual totient differences away\nfrom the prescribed indices, or bound the terms discarded by a proposed\nprojection. The construction and the weight-dependent bounds are in\n\\lean{lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient}{LcmFactorIdealPulseObstruction.lean:798}.\nThe decomposition of the actual totient into divisor contributions is\nproved in Lean\n(\\lean{totient\\_eq\\_sum\\_mobiusTotientChannel}{LcmFactorIdealPulseObstruction.lean:328}).\n\\end{obs}\n\n\\begin{obs}[Compatible carries for fixed-precision local symbols]\\label{prop:B5-kill}\nFor example, prescribing an odd coefficient modulo $8$ permits any\ncoefficient of the form $a+8z$ with $a$ odd and $z\\in\\mathbb Z$.\nWhatever the incoming integer carry $e$, one can choose $z$ so that\n$2e+a+8z$ lies between $-4$ and $3$.\nMore generally, fix $u\\ge1$ and a finite list of integers\n$v_j\\ge0$ and odd integers $a_j$. Starting from any integer $e_0$,\nthere are integers $z_j$ and coefficients\n\\[\n c_j=2^{v_j}(a_j+2^u z_j),\\qquad\n e_{j+1}=2e_j+c_j,\n \\qquad |e_{j+1}|\\le2^{v_j+u-1}.\n\\]\nAt each step, reduce $2e_j+2^{v_j}a_j$ to its centred representative\nmodulo $2^{v_j+u}$, choosing the interval\n\\[\n [-2^{v_j+u-1},2^{v_j+u-1}).\n\\]\nSolving for $z_j$ gives the stated next carry. Iterate this choice. The bounds apply to the resulting\nstates, not to the arbitrary initial state $e_0$.\nThe unrestricted higher quotients $z_j$ are essential: the construction\nprescribes a valuation and a finite unit residue, not the full\ncoefficients or their signs. It therefore does not realise a prescribed\ntotient-difference sequence. A contradiction using only these local\nsymbols cannot exclude all compatible integer carries; further\nrestrictions on the coefficients need a separate argument.\n\\lean{fixedPrecisionTropicalNoGo}{TropicalCurvatureCarry.lean:137}\n\\end{obs}\n\n\\begin{obs}[Correction-free factorisation does not ensure nonvanishing]\\label{prop:B8-kill}\nConsider the two finite differences\n\\[\n \\varphi(65)-\\varphi(65)=0,\\qquad\n \\varphi(78)-\\varphi(70)=24-24=0.\n\\]\nThey arise from the square-congruence choice\n$52\\equiv52\\pmod{13^2}$ and $52\\equiv2\\pmod{5^2}$.\nThe four arguments factor as $13\\cdot5$, $5\\cdot13$,\n$13\\cdot6$ and $5\\cdot14$; in each product the prime and cofactor\nare coprime. Thus all four totients have the correction-free form\n$\\varphi(pm)=(p-1)\\varphi(m)$, yet both differences vanish.\nNonvanishing is also possible: the choices\n$27\\equiv0\\pmod{3^2}$ and $27\\equiv2\\pmod{5^2}$ give\n\\[\n \\varphi(30)-\\varphi(30)=0,\\qquad\n \\varphi(33)-\\varphi(35)=20-24=-4,\n\\]\nagain with coprime factors at every term.\nThe square-CRT construction removes the non-coprime multiplicativity\ncorrection on the prescribed finite range. That property alone does\nnot imply nonvanishing. These are explicit witnesses of the two\npossibilities, not a claim that the first is a smallest counterexample.\n\\lean{squareCRT_clean_block_can_vanish}{SquareCRTCube.lean:459}\n\\end{obs}\n\n\\begin{obs}[No fixed integer clears every primitive denominator]\\label{prop:D6-kill}\nLet $A=\\varphi*\\mu$ be the Dirichlet convolution used in\nProposition~\\ref{prop:D7-inv}. At prime powers,\n\\[\n A(p)=p-2,\\qquad\n A(p^r)=p^{r-2}(p-1)^2\\quad(r\\ge2).\n\\]\nThese follow by subtracting consecutive prime-power values of\n$\\varphi$. For every odd prime $p$, the reduced denominator of\n$A(p)/p$ is $p$. For every prime $p$, the reduced denominator of\n$A(p^2)/p^2$ is $p^2$, including the denominator $4$ at $p=2$.\nConsequently any positive integer $D$ for which\n$DA(n)/n\\in\\mathbb Z$ for all $1\\le n\\le N$ must be divisible\nby every odd prime $p\\le N$, every prime square $p^2\\le N$, and\nby $4$ once $N\\ge4$. No fixed $D$ can do this for all $n$, since\nthere are infinitely many odd primes.\nIn periodic-point counting, the quantities $A(n)/n$ would count\norbits of length $n$; their integrality and nonnegativity characterise\nrealisable periodic-point sequences~\\cite[Lemma~3.1, p.~5]{everest-periodic}.\nHere only the denominator obstruction is used. It excludes a fixed\ninteger clearing all these particular coordinates, not every possible\nEuler-product representation or an argument with a multiplier growing\nwith $N$. It is not an irrationality proof for $S$.\n\\lean{no_fixed_integral_primitive_euler_index}{PrimitiveDeterminantLift.lean:169}\n\\end{obs}\n\n\\begin{obs}[A finite exclusion is not a membership test]\\label{prop:B11cert-kill}\nA successful residue certificate proves the exclusion stated in its\nsoundness theorem. Failure of that sufficient condition at finitely\nmany depths does not prove integrality of the tail difference. In\nparticular, a finite unsuccessful search cannot be substituted for\nthe eventual-period assertion needed to prove rationality.\nThe same logical distinction occurs for the linked greedy test in a\nseparate Mersenne achievement-set problem: its finite inequality\nimplies nonmembership. The linked object is a definition, with a\nsoundness theorem in the same file, not a theorem asserting a universal\nimpossibility of finite rationality proofs for Problem~\\#249.\nMembership or rationality established by some additional exact identity\nis not excluded by this observation. Likewise, the successful finite\nranges in Propositions~\\ref{prop:SK-01-inv} and \\ref{prop:B11-inv}\nremain finite ranges until an unbounded supply is proved.\n\\lean{CertifiedGreedyMersenneDeath}{GreedyAchievementSet.lean:1756}\n\\end{obs}\n% ; - part a249_newdecls ; -\n\\subsection{Related structural results}\n\nWe apply the M\\\"obius-weighted identities first to log-concavity and\nthen to integer carries, rational approximations and factorisations.\nThe objects differ, but each proof starts from the identities already\ngiven for that object.\n\n\\subsubsection{Log-concavity of the M\\\"obius-weighted series}\n\nPut $\\Theta_r=\\sum_{n\\ge0}\\mu(n+1)/(2^{n+1}-1)^r$, so that\n$\\Theta_2=S-1/2$. We prove strict log-concavity for every integer\n$r\\ge1$. This orders the values of the family without determining their\nrationality.\n\n\n\\begin{defn}\nFor each integer $r\\ge1$, put\n\\[\n \\Theta_r=\\sum_{n\\ge0}\\frac{\\mu(n+1)}{(2^{n+1}-1)^r}\n =1-3^{-r}+\\sum_{n\\ge0}\\frac{\\mu(n+3)}{(2^{n+3}-1)^r}.\n\\]\nThe first two summands isolate the dominant contribution. The remaining\nabsolute sum is at most $1/(2^r(2^r-1))$, by comparison with\n$\\sum_{n\\ge2}2^{-rn}$. In particular, $\\Theta_r>0$ for $r\\ge1$:\n$1-3^{-r}\\ge2/3$ and this tail bound is at most $1/2$.\n\\lean{Erdos249257.SignedQMomentObstruction.mobiusMersenneTheta}{SignedQMomentObstruction.lean:136}\n\\quad \\lean{Erdos249257.SignedQMomentObstruction.mobiusMersenneTwoAtom}{SignedQMomentObstruction.lean:140}\n\\quad \\lean{Erdos249257.SignedQMomentObstruction.mobiusMersenneTailAfterTwo}{SignedQMomentObstruction.lean:144}\n\\end{defn}\n\n\\begin{thm}[The values at exponents one and two]\\label{thm:values-exponents-one-two}\nThe first two values are $\\Theta_1=1/2$ and $\\Theta_2=S-1/2$.\n\\end{thm}\n\n\\noindent The second equality is the M\\\"obius-square identity credited in\n\\S\\ref{ssec:coprime}; whether $\\Theta_2$ is rational is \\#249 itself.\n\n\\begin{thm}[The first two summands give a positive Hankel gap]\\label{thm:first-two-summands-give-positive}\n$(1-3^{-(r+1)})^2 - (1-3^{-r})(1-3^{-(r+2)}) = 4/3^{r+2}$,\nso the rational sequence $1-3^{-r}$ is strictly log-concave for every integer $r\\ge1$.\n\\end{thm}\n\n\\begin{thm}[Strict log-concavity for all integer $r\\ge1$]\\label{thm:strict-log-concavity-all-integer}\nFor every integer $r\\ge1$,\n\\[\n \\Theta_r\\Theta_{r+2}<\\Theta_{r+1}^2.\n\\]\nEquivalently, the determinant of\n$\\left(\\begin{smallmatrix}\\Theta_r&\\Theta_{r+1}\\\\\n\\Theta_{r+1}&\\Theta_{r+2}\\end{smallmatrix}\\right)$ is negative.\nThe two-summand comparison gives an elementary proof, detailed below;\nthe same strict inequality is also proved in Lean.\n\\end{thm}\n\\begin{proof}[A quantitative comparison]\nFor this proof put $a_r=1-3^{-r}$ and $\\varepsilon_r=1/(2^r(2^r-1))$.\nThe preceding bound gives $|\\Theta_r-a_r|\\le\\varepsilon_r$.\nSince $0<a_r<1$, expansion of the two products gives\n\\[\n \\Theta_{r+1}^2-\\Theta_r\\Theta_{r+2}\n \\ge \\frac4{3^{r+2}}-B_r,\n \\qquad\n B_r=2\\varepsilon_{r+1}+\\varepsilon_{r+1}^2\n       +\\varepsilon_r+\\varepsilon_{r+2}\n       +\\varepsilon_r\\varepsilon_{r+2}.\n\\]\nHere $\\varepsilon_{r+1}<\\varepsilon_r/4$, so $B_{r+1}<B_r/4$.\nThe rational comparison\n\\[\n B_5=\\frac{50106593}{32001951744}<\\frac4{3^7}\n\\]\ntherefore proves the inequality for all $r\\ge5$: the error decreases\nfaster than the positive main term.\nFor $r=1,2,3,4$, truncate each $\\Theta_j$ at $n=8$. Its omitted\nabsolute sum is at most\n\\[\n \\frac{1}{(2^9-1)^j(1-2^{-j})}.\n\\]\nUsing $\\Theta_1=1/2$ and these rational intervals, the four gaps\n$\\Theta_{r+1}^2-\\Theta_r\\Theta_{r+2}$ are respectively greater than\n$1/4$, $1/16$, $1/64$ and $1/200$. These finite rational comparisons\ncomplete the ordinary proof.\n\\end{proof}\n\n\\begin{obs}\nThe inequality excludes a representation\n$\\Theta_r=\\int_0^\\infty x^r\\,d\\nu(x)$ by a positive measure with these\nfinite moments: Cauchy--Schwarz would give\n$\\Theta_{r+1}^2\\le\\Theta_r\\Theta_{r+2}$. It does not by itself imply\nirrationality of $\\Theta_2$. Indeed, the rational sequence $1-3^{-r}$\nsatisfies the same strict log-concavity inequalities. Thus the determinant\nsign supplies an obstruction to a positive-moment representation, not a\nrational-approximation estimate.\n\\end{obs}\n\nThe accompanying formal results include a rectangular determinant\nexpansion derived from the Leibniz formula\n(\\lean{Erdos249257.SignedQMomentObstruction.det_mul_rectangular}{SignedQMomentObstruction.lean:29}), and the\nfollowing elementary parity fact: in a finite sum of dyadic fractions,\nif the largest denominator exponent occurs once and its numerator is odd,\nclearing that denominator gives an odd numerator\n(\\lean{Erdos249257.SignedQMomentObstruction.scaled_dyadic_sum_odd}{SignedQMomentObstruction.lean:78}).\n\n\\subsubsection{Integer recurrences and rational binary series}\n\nFor coefficients $0\\le c(n)\\le n$, rationality is equivalent to the\nexistence of an integer carry with $u(N)/2^N\\to0$. The recurrence alone\nis not restrictive: every integer initial value produces an integer\nsequence. The limit condition selects the initial value that represents\nthe infinite sum.\n\n\n\\begin{thm}[Integer recurrence criterion for a binary series]\n\\label{thm:binary-carry-criterion}\nLet $c:\\N\\to\\N$ satisfy $c(n)\\le n$. Then\n$\\sum_{n\\ge1}c(n)2^{-n}$ is rational if and only if there are an integer\n$v>0$ and an integer sequence $u$ such that\n\\[\n u(N+1)=2u(N)-v\\,c(N+1),\\qquad \\frac{u(N)}{2^N}\\longrightarrow0.\n\\]\nFor each fixed $v$, such a sequence is unique and equals\n\\[\n u(N)=vR^c_N,\\qquad R^c_N=\\sum_{j\\ge1}\\frac{c(N+j)}{2^j}.\n\\]\n\\end{thm}\n\n\\begin{proof}\nIterating the recurrence gives the finite identity\n\\[\n \\frac{u(N)}{2^N}=u(0)-v\\sum_{n=1}^N\\frac{c(n)}{2^n}.\n\\]\nTaking $N\\to\\infty$ gives $S_c=u(0)/v$, where\n$S_c=\\sum_{n\\ge1}c(n)2^{-n}$. Subtracting the finite prefix then gives\n$u(N)=vR^c_N$.\n\nConversely, suppose $S_c$ is rational. Choose an integer $v>0$ with\n$vS_c\\in\\Z$, set $u(0)=vS_c$, and iterate the recurrence. The terms\nare integers by construction; the displayed identity gives\n$u(N)=vR^c_N$. Finally $0\\le R^c_N\\le N+2$ proves the required limit.\n\\end{proof}\n\nWithout the boundary condition, any integer $u(0)$ gives an integer\nsolution of the recurrence, whether the series is rational or irrational.\nTwo solutions with the same $v$ differ by $2^N$ times their initial\ndifference. Thus the limit, not the recurrence alone, determines the\ninitial value. Changing $v$ rescales the canonical solution; uniqueness\nis asserted only for a fixed multiplier.\n\n\\begin{prop}[Uniqueness under the growth condition]\\label{prop:uniqueness-under-growth-condition}\nA real sequence $d$ with $d(N{+}1)=2d(N)$ and $d(N)=o(2^N)$ is identically\nzero.\n\\end{prop}\n\n\\begin{obs}[The specified finite-state decoder is impossible]\nFix $m\\ge2$, put $R=\\lfloor(m+1)/2\\rfloor$, and for\n$0\\le r\\le R$ define\n\\[\n c_{m,r}(n)=\n \\begin{cases}\n  R-r,&n=m,\\\\\n  2r,&n=m+1,\\\\\n  0,&\\text{otherwise}.\n \\end{cases}\n\\]\nThe two nonzero terms have total contribution\n$(R-r)2^{-m}+2r2^{-(m+1)}=R2^{-m}$, independent of $r$.\nAll coefficients and scaled tails before $m$ are therefore the same,\nwhereas the scaled tail at $m$ is $r$.\nNo map from a state common to this whole family can recover that tail.\nMore generally, exact recovery of all its possible values requires at\nleast $R+1$ distinct labels.\n\\lean{Erdos249257.balancedPulse_no_autonomous_decoder}{GenericTailOrbitRigidity.lean:247}\n\\quad \\lean{Erdos249257.balancedPulse_label_card_lower_bound}{GenericTailOrbitRigidity.lean:228}\nThe sequences in this family have identical histories before the isolated\nchange but different subsequent tails. Therefore that finite history alone\ncannot determine the exact tail in this class of sequences. This is\nnot a conclusion about the totient sequence.\n\\end{obs}\n\nThere is also an infinite family with a common rational value.\nLet $a:\\N\\to\\{0,1\\}$ satisfy $a(0)=0$, and put\n$c(2k)=k-a(k)$ and $c(2k+1)=2a(k)$. The adjacent terms contribute\n$k4^{-k}$, so\n\\[\n \\sum_{n\\ge1}\\frac{c(n)}{2^n}\n   =\\sum_{k\\ge1}\\frac{k}{4^k}=\\frac49,\n\\]\nregardless of the binary sequence $a$. The cited result also gives an\ninteger recurrence satisfying the growth condition\n(\\lean{Erdos249257.globalBalancedCoeff_value}{GenericTailOrbitRigidity.lean:545}).\nThe assumptions $a(0)=0$ and $a(k)\\in\\{0,1\\}$ are part of that\nformal statement, not an arbitrary bounded-input hypothesis.\nFor comparison, the identity coefficients $c(n)=n$ have the explicit\nmultiplier-one recurrence $u(N)=N+2$\n(\\lean{Erdos249257.idCoeff_temperedOrbit}{GenericTailOrbitRigidity.lean:564}).\n\n\\subsubsection{Rational approximation with a squared-Mersenne remainder}\n\nThe first-order Lambert term in the diagonal tail difference can be\nsummed exactly. The approximation below leaves only a squared-Mersenne\ntail to estimate.\nSmall error is only half of a denominator-clearing argument. If a\ntarget $\\alpha=a/b$ is rational, then every unequal rational $p/q$\nsatisfies $q|\\alpha-p/q|=|aq-bp|/b\\ge1/b$. Thus nonzero errors\ntending to zero after multiplication by the reduced denominators\nwould prove irrationality. The identity and geometric bound below\ngive explicit errors, but not that comparison with the denominators.\n\n\n\\begin{prop}[An exact rational approximation formula]\\label{prop:exact-rational-approximation-formula}\nFor integers $H,D\\ge0$, with the integer prefixes $\\Phi_N$ defined above,\n\\[\n\\begin{aligned}\n R_{2H}-R_H\n &-\\left[\\Phi_H-\\Phi_{2H}\n   +2^H(2^H-1)\\left(\\frac12+\\sum_{d=1}^{D}\\frac{\\mu(d)}{(2^d-1)^2}\\right)\\right]\\\\\n &=2^H(2^H-1)\\sum_{d>D}\\frac{\\mu(d)}{(2^d-1)^2}.\n\\end{aligned}\n\\]\nThis follows by substituting the squared-denominator identity for $S$\ninto $R_{2H}-R_H=2^H(2^H-1)S+\\Phi_H-\\Phi_{2H}$. There is no required\nordering between $D$ and $H$.\n\\end{prop}\n\n\\begin{prop}[A geometric tail bound]\\label{prop:geometric-tail-bound}\nFor every integer $D\\ge0$,\n\\[\n \\left|\\sum_{d>D}\\frac{\\mu(d)}{(2^d-1)^2}\\right|\n \\le \\frac{4}{3(2^{D+1}-1)^2}.\n\\]\nIndeed, $|\\mu(d)|\\le1$ and\n$2^{D+1+j}-1\\ge2^j(2^{D+1}-1)$ for $j\\ge0$. The sum of the resulting\ngeometric majorant is $\\sum_{j\\ge0}4^{-j}=4/3$.\n\\end{prop}\n\n\\begin{obs}[One-sided and symmetric error bounds]\nSuppose $n>D$, $\\mu(d)=0$ for $D<d<n$, and $|\\mu(n)|=1$. Removing\nthat first nonzero summand gives the sharper enclosure\n\\[\n \\left|\\mu(n)\\sum_{d>D}\\frac{\\mu(d)}{(2^d-1)^2}\n                 -\\frac1{(2^n-1)^2}\\right|\n       \\le\\frac4{3(2^{n+1}-1)^2}.\n\\]\nThe radius is smaller than the positive centre, so the tail has sign\n$\\mu(n)$. The interval's width, divided by the width of the preceding\nsymmetric enclosure, is\n\\[\n \\left(\\frac{2^{D+1}-1}{2^{n+1}-1}\\right)^2<\\frac14.\n\\]\nThe cited theorem gives this signed enclosure\n(\\lean{Erdos249257.SquaredMersenneDiagonalEnclosure.mobiusSquareTail_directed_of_first_support}{SquaredMersenneDiagonalEnclosure.lean:313}).\nMultiplication by $2^H(2^H-1)$ and translation by the rational\napproximation above enclose $R_{2H}-R_H$. If the resulting interval\ncontains no integer, the tail difference is nonintegral. For the symmetric\nbound, the linked statement requires the distance from its rational centre\nto every integer to exceed its error radius\n(\\lean{Erdos249257.SquaredMersenneDiagonalEnclosure.scaleFullTarget_miss_of_lambert_projected_separation}{SquaredMersenneDiagonalEnclosure.lean:426}).\nThe enclosure does not establish that strict separation. In particular,\nshrinking an interval around a rational approximation does not, without a\ndenominator comparison, show that the interval misses every integer.\n\\end{obs}\n\n\\subsubsection{Divisor terms and their complement}\n\nAt $H=6$ and $s=2$, the difference\n$\\varphi(2H+s)-\\varphi(H+s)=\\varphi(14)-\\varphi(8)$ is $2$, whereas\nthe terms with divisor index $d\\mid H$ sum to $3$. The other indices\ncontribute $-1$. The following decomposition keeps both contributions.\nIt holds for every positive integer $H$, not only a least common multiple;\nan estimate is needed before its second sum can be discarded.\n\n\n\\begin{prop}[The sum over divisor indices]\\label{prop:sum-over-divisor-indices}\nFor integers $H>0$ and $s\\ge0$, the part of the M\\\"obius expansion\nindexed by divisors of $H$ is\n\\[\n \\sum_{\\substack{d\\mid H\\\\d\\mid s}}\\mu(d)\\frac Hd\n       =H\\frac{\\varphi(\\gcd(H,s))}{\\gcd(H,s)}.\n\\]\nThis is the standard identity\n$\\sum_{d\\mid g}\\mu(d)/d=\\varphi(g)/g$ at $g=\\gcd(H,s)>0$.\nThe formal statement writes the same value after extracting\n$H/\\operatorname{rad}(H)$, where\n$\\operatorname{rad}(H)=\\prod_{p\\mid H}p$.\n\\end{prop}\n\n\\begin{prop}[The divisor sum and its complement]\\label{prop:divisor-sum-complement}\nFor integers $H>0$ and $s\\ge0$,\n\\[\n\\begin{aligned}\n \\varphi(2H+s)-\\varphi(H+s)\n &=H\\frac{\\varphi(\\gcd(H,s))}{\\gcd(H,s)}\\\\\n &\\quad+\\sum_{\\substack{1\\le d\\le2H+s\\\\d\\nmid H}}\n \\mu(d)\\left(\\frac{2H+s}{d}\\,\\mathbf1_{d\\mid2H+s}\n           -\\frac{H+s}{d}\\,\\mathbf1_{d\\mid H+s}\\right).\n\\end{aligned}\n\\]\nHere $\\mathbf1$ is the indicator of the stated divisibility condition.\nThe identity follows by expanding both totients as\n$\\varphi(n)=\\sum_{d\\mid n}\\mu(d)n/d$ and separating the indices\n$d\\mid H$. Such an index divides either endpoint precisely when it\ndivides $s$, and its difference is then $\\mu(d)H/d$.\n\\end{prop}\n\n\\begin{thm}[Doubling the full totient difference]\\label{thm:doubling-full-totient-difference}\nFor nonnegative integers $H,r$ with $H$ even,\n\\[\n \\varphi(4H+2r)-\\varphi(2H+2r)=\n \\begin{cases}\n 2\\bigl(\\varphi(2H+r)-\\varphi(H+r)\\bigr),&r\\text{ even},\\\\\n \\varphi(2H+r)-\\varphi(H+r),&r\\text{ odd}.\n \\end{cases}\n\\]\nThis applies to the full difference, not just its divisor contribution.\nIt follows from $\\varphi(2n)=2\\varphi(n)$ for even $n$ and\n$\\varphi(2n)=\\varphi(n)$ for odd $n$.\n\\end{thm}\n\n\\begin{obs}[Squarefree support of the complementary terms]\nA nonzero summand in the complementary sum must have a squarefree\nindex $d$ dividing exactly one of $2H+s$ and $H+s$. Indeed, otherwise\n$\\mu(d)=0$ or both indicators vanish. Both endpoints cannot be divisible\nby $d$: their difference is $H$, and $d\\nmid H$.\n\\lean{Erdos249257.DiagonalFreshLossBridge.squarefree_of_foreignChannelPhaseTerm_ne_zero}{DiagonalFreshLossBridge.lean:909}\n\\quad \\lean{Erdos249257.DiagonalFreshLossBridge.foreign_endpoint_support_disjoint}{DiagonalFreshLossBridge.lean:852}\nThe restriction identifies which indices can contribute. It does not\ncontrol the sign or magnitude of their sum.\n\\end{obs}\n\n\\begin{prop}[A doubling identity for two portions of the sum]\\label{prop:doubling-identity-two-portions-sum}\nLet $H>0$, $s\\ge0$ and $d\\ge1$, with $d\\nmid H$ and $d\\mid H+s$.\nThe $d$-summands in the complementary sums at offsets $s$ and $2s$ are,\nrespectively,\n\\[\n -\\mu(d)\\frac{H+s}{d}\n \\qquad\\text{and}\\qquad\n 2\\mu(d)\\frac{H+s}{d}.\n\\]\nFor the second expression, $d\\mid2H+2s$ and $d\\nmid H+2s$; otherwise\n$d$ would divide their difference $H$. Thus the second contribution is\n$-2$ times the first, including the zero case $\\mu(d)=0$. The two\nterms have different binary weights in a window sum, so this identity\nalone is not a cancellation of their weighted contributions.\n\\end{prop}\n\n\\subsubsection{A numerator polynomial with explicit positive coefficients}\n\\label{repunit-gcd-word}\n\n\\begin{thm}[Positive coefficients of the numerator polynomial]\\label{thm:positive-coefficients-numerator-polynomial}\nFor a squarefree integer $r\\ge1$, the numerator polynomial has the\nexplicit expression\n\\[\n \\sum_{d\\mid r}\\mu(d)\\frac rd\\sum_{j=0}^{r/d-1}X^{dj}\n   =\\sum_{k=0}^{r-1}\n        \\frac r{\\gcd(r,k)}\\varphi(\\gcd(r,k))X^k.\n\\]\nEvery coefficient for $0\\le k<r$ is positive, and the higher coefficients\nare zero. Extracting $X^k$ on the left gives\n$\\sum_{d\\mid\\gcd(r,k)}\\mu(d)r/d$, which proves the formula.\nThe squarefree assumption identifies this divisor sum with the\nsubset-of-primes definition in the formal source.\n\\end{thm}\n\n\\begin{cor}\\label{cor:numerator-polynomial-explicit-positive-coefficients}\nFor the same squarefree $r$, evaluation at $X=2$ gives the integer\n\\[\n \\sum_{d\\mid r}\\mu(d)\\frac rd\\frac{2^r-1}{2^d-1}.\n\\]\nEach quotient is an integer because $d\\mid r$.\n\\end{cor}\n\n\\subsubsection{Exact dyadic comparisons and tail bounds}\n\nA finite prefix locates a dyadic rational relative to the tail.\nThe following two formulas give the exact comparison and an upper bound\nfor the omitted terms, so that the residue margin can be compared with\nits error.\n\n\n\\begin{defn}\nFor $p\\in\\mathbb Z$, $n\\ge0$ and an integer $L>0$, the difference\nbetween $p/(2L)$ and the dyadic point $2^{-(n+1)}$ is\n\\[\n \\frac{p}{2L}-2^{-(n+1)}=\\frac{2^np-L}{2^{n+1}L}.\n\\]\nIts sign is therefore the sign of the integer numerator $2^np-L$.\nFor fixed $p,L$, the next numerator satisfies\n$2^{n+1}p-L=2(2^np-L)+L$.\n\\lean{Erdos249257.nextDyadicExcessIntNumerator}{DyadicPrefixCompression.lean:156}\n\\quad \\lean{Erdos249257.nextDyadicExcessIntNumerator_succ}{DyadicPrefixCompression.lean:161}\n\\end{defn}\n\n\\begin{prop}[The geometric remainder bound]\\label{prop:geometric-remainder-bound}\nThe remainder after the first two geometric terms of $1/(2^n-1)$ is\n\\[\n \\frac1{2^n-1}-2^{-n}-4^{-n}\n   =\\frac{8^{-n}}{1-2^{-n}}\n   \\le \\frac43\\,8^{-n}\\qquad(n\\ge2).\n\\]\nIndeed, $1-2^{-n}\\ge3/4$. Summing over $n>m$, for an integer $m\\ge1$, gives\n\\[\n \\sum_{n>m}\\left(\\frac1{2^n-1}-2^{-n}-4^{-n}\\right)\n \\le\\frac43\\sum_{n>m}8^{-n}=\\frac4{21}\\,8^{-m}.\n\\]\n\\end{prop}\n\n\\subsubsection{Squared distances between complex phases}\n\nFor unit complex numbers, the identity\n$|z-1|^2=2-2\\operatorname{Re}z$ converts squared distances into a real-part\nsum. We also sum the corresponding pairwise identity. Applying these\nformulas to totient phases still requires a separation estimate for\nthose phases.\n\n\n\\begin{prop}[Squared distance from the phase one]\\label{prop:squared-distance-phase-one}\n\\[\n\\begin{aligned}\n\\sum_{N\\in T}\\|{E}(h,N,L)-1\\|^2\n  &= 2|T|-2\\sum_{N\\in T}\\operatorname{Re}E(h,N,L).\n\\end{aligned}\n\\]\n\\end{prop}\n\n\\begin{lem}[Squared-distance bound for separated pairs]\\label{lem:squared-distance-bound-separated-pairs}\nFor a finite family $z:T\\to\\mathbb{C}$, a real number $\\delta\\ge0$, and any set of pairs $P\\subseteq\nT\\times T$ each separated by $\\ge\\delta$, $|P|\\cdot\\delta^2 \\le\n\\sum_{i,j\\in T}\\|z_i-z_j\\|^2$.\n\\end{lem}\n\n\\subsubsection{Approximating the nondivisor terms}\\label{sec:nondivisor-enclosure}\n\nThe terms with $d\\mid H$ form a finite rational contribution to\n$R_{2H}-R_H$. To use it, we must also approximate the terms with\n$d\\nmid H$. The approximation and its error are explicit; for a cutoff\n$D\\ge2H$, only separation of the resulting rational centre from the\nintegers remains to be checked.\n\nFor $d\\ge1$ and $N\\ge0$, let $a_d(N)=d-(N\\bmod d)$ be the least\npositive shift from $N$ to a multiple of $d$. Summing the multiples of\n$d$ in the totient tail gives the rational term\n\\[\n \\kappa_d(N)=\\mu(d)2^{d-a_d(N)}\n \\left(\\frac{N+a_d(N)}{d(2^d-1)}+\\frac1{(2^d-1)^2}\\right),\n \\qquad R_N=\\sum_{d\\ge1}\\kappa_d(N).\n\\]\nIndeed, substitute $\\varphi(n)=\\sum_{d\\mid n}\\mu(d)n/d$ into $R_N$.\nFor a fixed $d$, the admissible shifts are $a_d(N)+kd$, $k\\ge0$;\ntheir contribution is\n$\\mu(d)\\sum_{k\\ge0}((N+a_d(N))/d+k)2^{-a_d(N)-kd}$,\nwhich is the displayed expression. The interchange of sums is justified by\n\\[\n \\sum_{j\\ge1}2^{-j}\\sum_{d\\mid N+j}|\\mu(d)|\\frac{N+j}{d}\n \\le \\sum_{j\\ge1}(N+j)^2 2^{-j}<\\infty.\n\\]\nThus subtraction and finite splitting of these series are valid.\nThe terms are signed: for example, $\\kappa_2(0)=-4/9$.\nThis identity is proved in Lean\n(\\lean{Erdos249257.TotientShiftedMobiusForeignBridge.totientTail_eq_tsum_positiveForeignResidueKernel}{TotientShiftedMobiusForeignBridge.lean:61}).\n\n\\begin{prop}[The finite divisor sum]\\label{prop:finite-divisor-sum}\nFor an integer $H>0$, the contribution of the divisor indices to\n$R_{2H}-R_H$ is\n\\[\n A_H=H\\sum_{d\\mid H}\\frac{\\mu(d)}{d(2^d-1)}.\n\\]\nFor $d\\mid H$, both $a_d(H)$ and $a_d(2H)$ equal $d$. Hence\n$\\kappa_d(2H)-\\kappa_d(H)=H\\mu(d)/(d(2^d-1))$.\nSumming over the divisors gives $A_H$, not the whole tail difference:\nthe nondivisor terms must still be included.\n\\end{prop}\n\nFor $H>0$ and an integer cutoff $D\\ge0$, set\n\\[\n\\begin{aligned}\n P_{H,D}&=\\sum_{\\substack{1\\le d\\le D\\\\d\\nmid H}}\n                    \\bigl(\\kappa_d(2H)-\\kappa_d(H)\\bigr),\\\\\n \\varepsilon_{H,D}\n   &=2^H(2^H-1)\\left(\\frac2{2^D}+\\frac4{3\\cdot4^D}\\right).\n\\end{aligned}\n\\]\nHere $A_H$ includes \\emph{all} divisor indices, even when $D<H$.\nThe following implication therefore keeps the error estimate as an\nexplicit hypothesis; it will be proved below when $D\\ge2H$.\n\n\\begin{prop}[Separation larger than the error implies exclusion]\\label{prop:separation-larger-than-error-implies}\nSuppose\n\\[\n \\bigl|(R_{2H}-R_H)-(A_H+P_{H,D})\\bigr|\\le\\varepsilon_{H,D},\n \\qquad\n |A_H+P_{H,D}-z|>\\varepsilon_{H,D}\\quad\\hbox{for every }z\\in\\Z.\n\\]\nThen $R_{2H}-R_H\\notin\\Z$.\n\\end{prop}\n\\begin{proof}\nIf $R_{2H}-R_H=z\\in\\Z$, the two hypotheses give opposite inequalities\nfor $|A_H+P_{H,D}-z|$.\n\\end{proof}\n\n\\paragraph{Why the error hypothesis holds for $D\\ge2H$.}\nEvery divisor of $H$ then lies below the cutoff. For $d>D$, both\n$H$ and $2H$ are smaller than $d$, so the omitted increment simplifies to\n\\[\n \\kappa_d(2H)-\\kappa_d(H)\n =\\mu(d)2^H(2^H-1)\n       \\left(\\frac1{2^d-1}+\\frac1{(2^d-1)^2}\\right).\n\\]\nSince $|\\mu(d)|\\le1$ and $2^d-1\\ge2^{d-1}$,\n\\[\n\\begin{aligned}\n \\bigl|(R_{2H}-R_H)-(A_H+P_{H,D})\\bigr|\n &\\le 2^H(2^H-1)\\sum_{d>D}\\left(\\frac2{2^d}+\\frac4{4^d}\\right)\\\\\n &=\\varepsilon_{H,D}.\n\\end{aligned}\n\\]\nThus for $H>0$ and $D\\ge2H$ the error bound needs no additional\nconvergence hypothesis. The same estimate is proved in Lean\n(\\lean{Erdos249257.TotientShiftedMobiusForeignBridge.controlledForeignProjection_of_twice_le}{TotientShiftedMobiusForeignBridge.lean:465});\ntogether with the displayed separation inequality, it gives the cited\nnonintegrality consequence\n(\\lean{Erdos249257.TotientShiftedMobiusForeignBridge.scaleFullTarget_miss_of_projected_separation_of_twice_le_of_forall_int}{TotientShiftedMobiusForeignBridge.lean:483}).\n\nFor example, at $H=1$ and $D=5$ the rational centre and radius are\n\\[\n A_1+P_{1,5}=\\frac{304282}{423801},\\qquad\n \\varepsilon_{1,5}=\\frac{49}{384},\\qquad\n \\frac12<A_1+P_{1,5}-\\varepsilon_{1,5}\n          <A_1+P_{1,5}+\\varepsilon_{1,5}<1.\n\\]\nThus this finite calculation excludes an integral value of $R_2-R_1$.\nThe cutoff restriction alone does not ensure success: at $H=1,D=4$,\nthe centre $346/441$ is within the radius $25/96$ of $1$.\nFor fixed $H$ the radius tends to zero as $D\\to\\infty$, but an\nirrationality argument still needs the required family of separated\ncentres. Neither a single example nor convergence of the error supplies it.\n\nWhen $D\\ge2H$, the first-order part of the omitted sum can also be\nincluded exactly:\n$\\sum_{d>D}\\mu(d)/(2^d-1)=1/2-\\sum_{d\\le D}\\mu(d)/(2^d-1)$.\nAdding $2^H(2^H-1)$ times this rational correction to $A_H+P_{H,D}$\ngives the squared-Mersenne approximation above\n(\\lean{Erdos249257.TotientShiftedMobiusForeignBridge.renormalizedResidueAgreement_of_twice_le}{TotientShiftedMobiusForeignBridge.lean:338}). This explains the\nrelation between the two approximations and the improved squared-denominator\nremainder; it does not provide their integer separation.\n\n\\subsubsection{Coprime-pair sums}\n\nCoprime-pair counting gives a further expression for $S$. The classical\nLambert identity alongside it uses different weights and sums to an\nelementary rational function. That evaluation does not determine the\nrationality of the totient series.\n\n\n\\begin{thm}[Coprime-pair counting and the totient]\\label{thm:coprime-pair-counting-totient}\nFor every $n\\in\\mathbb N$,\n$\\#\\{(a,b)\\in\\mathbb N^2:a+b=n,\\ a>0,\\ \\gcd(a,b)=1\\}=\\varphi(n)$.\nHere $b=0$ is allowed: the boundary pair $(1,0)$ accounts for the value\n$\\varphi(1)=1$.\n\\end{thm}\n\n\\begin{prop}[Two lattice sums]\\label{prop:two-lattice-sums}\nFor $0\\le r<1$, the two choices of boundary give\n\\[\n\\begin{aligned}\n \\sum_{\\substack{a\\ge1,\\ b\\ge0\\\\\\gcd(a,b)=1}}r^{a+b}\n   &=\\sum_{n\\ge1}\\varphi(n)r^n,\\\\\n \\sum_{\\substack{a,b\\ge1\\\\\\gcd(a,b)=1}}r^{a+b}\n   &=\\sum_{n\\ge1}\\varphi(n)r^n-r.\n\\end{aligned}\n\\]\nThe removed pair is $(1,0)$. Partitioning all strictly positive pairs\nby their greatest common divisor gives\n\\[\n \\sum_{g\\ge1}\\ \\sum_{\\substack{a,b\\ge1\\\\\\gcd(a,b)=1}}\n       r^{g(a+b)}=\\left(\\frac r{1-r}\\right)^2.\n\\]\nThis total is $1$ exactly when $r=1/2$.\n\\end{prop}\n\n\\begin{thm}[The classical coprime-pair Lambert identity]\\label{thm:classical-coprime-pair-lambert-identity}\nFor every $0\\le r<1$, $\\sum_{(a,b)\\ \\mathrm{coprime},\\, a,b\\ge 1}\n\\dfrac{r^{a+b}}{1-r^{a+b}} = \\Bigl(\\dfrac{r}{1-r}\\Bigr)^2$, an elementary\nrational function of $r$, hence rational at every rational $r$ including\n$r=1/2$. This is the classical visible-point identity, and its Lean proof is a formalisation of it rather than a new result: writing each\npair $(A,B)$ of positive integers uniquely as $g\\cdot(a,b)$ with\n$\\gcd(a,b)=1$ converts the quadrant sum $\\sum_{A,B\\ge1}r^{A+B}=(r/(1-r))^2$\ninto the displayed sum over visible points. With the plain weight\n$r^{a+b}$, the same strictly positive index set instead sums to\n$\\sum_{n\\ge1}\\varphi(n)r^n-r$. Thus at $r=1/2$ the Lambert-weighted\nsum is $1$, whereas the plain-weight sum is $S-1/2$, not $S$.\nAdding the boundary pair $(1,0)$ recovers $S$ in the plain-weight sum.\n\\end{thm}\n\n\\begin{obs}\nThe Lambert-weighted sum is rational at every rational $r\\in[0,1)$.\nConsequently, coprimality of the indices, the decomposition by greatest\ncommon divisor and geometric decay alone do not imply its irrationality.\nThis comparison does not decide the arithmetic nature of the differently\nweighted sum $S-1/2$, or of $S$. A proof using the plain-weight sum must\nexploit information not contained in those three properties.\n\\end{obs}\n\n\\subsubsection{The single-parameter diagonal condition}\n\nTaking the period to be a least common multiple reduces the certificate\ncondition to a single scale $t$. The witness must still occur beyond\nevery threshold in $t$. We prove the reduction and retain that arithmetic\nrequirement explicitly.\n\n\n\\begin{obs}[The diagonal implication]\nThe irrationality of $S$ follows if, for every integer $t_0\\ge0$, there\nare $t\\ge t_0$ and $L\\ge0$ such that $\\mathcal C(H_t,H_t,L)$, where\n$H_t=\\operatorname{lcm}(1,\\ldots,t)$ and $H_0=1$. This implication is\nproved in Lean\n(\\lean{Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_lcm_diagonal_certificate_supply}{LcmDiagonalReduction.lean:112});\nits hypothesis is unproved. Theorem~\\ref{catalogue:cert:b3} also records the\nconverse. The quantified condition is therefore equivalent to\nirrationality, not a weaker hypothesis.\n\\end{obs}\n\n\\begin{thm}[Nondivisors in a short LCM window]\\label{thm:nondivisors-short-lcm-window}\nLet $t\\ge1$. If $1\\le j<2t$ and $j\\nmid H_t$, then\n$j=p^a>t$ for a prime $p$ and an integer $a\\ge1$. Every integer\n$1\\le j\\le t$ divides $H_t$.\nTo see the first claim, some prime-power divisor $p^a$ of $j$ exceeds $t$;\notherwise every prime-power divisor would divide $H_t$. Since\n$j<2t<2p^a$, its remaining cofactor is $1$.\n\nIf $j\\mid H_t$ and every prime divisor of $j$ also divides $H_t/j$, then,\nfor every integer $q\\ge0$,\n\\[\n \\varphi(qH_t+j)=\\varphi(j)\\varphi\\bigl(q(H_t/j)+1\\bigr).\n\\]\nIndeed, the second factor's argument is coprime to $j$, so totient\nmultiplicativity applies. The hypothesis is essential to this\nfactorisation: at $t=2$, $j=2$, $q=1$ the left side is $\\varphi(4)=2$\nbut the displayed product would be $1$.\n\\end{thm}\n\n\\begin{obs}[Finite diagonal examples]\nThe cited file proves $\\mathcal C(H_t,H_t,L)$ at the following depths:\n\\[\n \\begin{array}{c|rrrrrrrr}\n t&1&2&3&4&5&6&7&8\\\\\n H_t&1&2&6&12&60&60&420&840\\\\\n L&6&5&7&7&9&9&14&15\n \\end{array}\n\\]\nThe largest totient argument is $2H_6+9=129$ for $t\\le6$, and\n$2H_8+15=1695$ for the full table.\n\\lean{Erdos249257.TotientTailPeriodKiller.certifiedKill_periodLcm_diagonal_upto_six}{LcmDiagonalReduction.lean:257}\n\\quad \\lean{Erdos249257.TotientTailPeriodKiller.certifiedKill_periodLcm_diagonal_at_eight}{LcmDiagonalReduction.lean:266}\nThis is a finite verification; certificates at arbitrarily large scales\nremain unproved.\n\\end{obs}\n\n\\subsubsection{Prime divisors of Mersenne factors and finite exclusions}\n\nFor $X-2$, multiplicative orders supply infinitely many prime divisors\namong the Mersenne factors. They therefore satisfy the abstract\nprime-support hypothesis below. This does not imply the required\ntail-residue inequality. The accompanying finite exclusions concern\nonly their stated periods and basepoints.\n\n\n\\begin{thm}[Unbounded prime support in Mersenne factors]\\label{thm:unbounded-prime-support-mersenne-factors}\nEvery prime divisor $p$ of $2^q-1$, for prime $q$, satisfies $q\\mid p-1$\n(the order of $2$ mod $p$ is exactly $q$, by Fermat/Lagrange in\n$(\\mathbb{Z}/p)^\\times$); consequently the prime divisors appearing in the\nlayers $\\{2^n-1\\}$ are unbounded, unconditionally, with no cyclotomic\nresultant hypothesis left open.\n\\end{thm}\n\n\\begin{thm}[A prime satisfying the stated cyclotomic conditions]\\label{thm:prime-satisfying-stated-cyclotomic-conditions}\nFor every period $h>0$ and threshold $N_0$, there exist a prime $q$ and a\nprime factor $p$ of $|\\Phi_{hq}(2)|$ (the binary cyclotomic layer) with $p$\ncoprime to $hq$, $hq\\mid p-1$, and $p-1\\ge N_0$. The characteristic-prime\nexceptional case in the cyclotomic order decomposition is eliminated\ndirectly, by choosing $q>2^h$ (rules out $p=q$) and $q>h$ (rules out\n$p\\mid h$).\n\\end{thm}\n\n\\begin{obs}\nThe first theorem uses the multiplicative order of $2$ modulo a prime.\nFor the second, choose a prime $q>\\max(h,N_0,2^h)$ and then a prime\ndivisor of $|\\Phi_{hq}(2)|>1$. The cyclotomic order formula, together\nwith the stated inequalities on $q$, excludes prime divisors of $hq$\nand gives $\\operatorname{ord}_p(2)=hq$. Existence of arbitrarily large\nprimes suffices here; no prime-in-progression theorem is needed for this\nconstruction.\n\nThese results supply prime divisors with the required orders. They do\nnot supply the tail-residue inequality at the resulting periods. The\nall-prime and eventual forms of the order hypothesis must also be\ndistinguished, as explained in \\S\\ref{prime-ray-curvature}.\n\\end{obs}\n\n\\begin{obs}[Finite exclusions at period 30]\nThe Lean kernel decides two independent certificates at the composite\nperiod $h=30$, at the basepoints $N=300$ and $N=330$. The certificate at\n$N=300$ gives the denominator exclusion $S\\ne r$ for every $r\\in\\mathbb{Q}$\nwhose reduced denominator divides $2^{300}(2^{30}-1)$.\n\\lean{ErdosProblems.Erdos249.CyclotomicAnchoredKill.certifiedKill_cyclotomic_30_331_natural}{ErdosProblems/Erdos249/CyclotomicAnchoredKill.lean:3356}\n\\quad \\lean{ErdosProblems.Erdos249.CyclotomicAnchoredKill.totient_series_ne_rat_of_den_dvd_30_300}{ErdosProblems/Erdos249/CyclotomicAnchoredKill.lean:3377}\nThis is a finite exclusion; the corresponding condition at arbitrarily\nlarge scales remains unproved.\n\\end{obs}\n\n\\subsubsection{A sufficient hypothesis for unbounded prime support}\n\\label{prime-ray-curvature}\n\nHere the exponent $k$ in $p^k\\equiv1\\pmod{mq}$ bounds the order\nof $p$ modulo $mq$, or equivalently a finite-field extension degree.\nIt does not bound the order of $2$ modulo $p$ in the binary examples.\nThe condition forces a prime divisor of $C(mq)$ to grow with $q$,\nprovided the separate assumption $C(mq)>1$ ensures the existence of\nsuch a divisor.\n\n\\begin{thm}[A sufficient order hypothesis for unbounded prime support]\\label{thm:sufficient-order-hypothesis-unbounded-prime}\nLet $C:\\N\\to\\N$, and fix integers $m\\ge1$ and $d\\ge0$. Suppose that\nfor every pair of primes $q,p$ with $p\\mid C(mq)$ there is an integer\n$k$ such that\n\\[\n 1\\le k\\le d,\\qquad mq\\mid p^k-1.\n\\]\nThe divisibility forces $\\gcd(p,mq)=1$. For such $p$, existence of\n$k$ in the displayed range is equivalent to\n$\\operatorname{ord}_{mq}(p)\\le d$. It also gives $mq<p^d$.\nHence every fixed finite set of primes is disjoint from\nthe prime divisors of $C(mq)$ for all sufficiently large prime $q$.\nIf, in addition, for all sufficiently large prime $q$ one has\n$C(mq)>1$ and $\\gcd(C(mq),mq)=1$, then for every $B,N_0$ there are\nprimes $q\\ge N_0$ and $p>B$ with $p\\mid C(mq)$.\n\nIndeed, the divisibility gives $mq\\le p^k-1<p^k\\le p^d$.\nFor a finite set of primes, take $q$ larger than all their $d$-th powers;\nfor the last assertion, take a prime divisor of the nontrivial value\n$C(mq)$ after this threshold. The stated coprimality is included in the\nformal source's layer hypothesis, although this last extraction argument\nuses only $C(mq)>1$.\nFor $C(n)=2^n-1$, the first theorem above supplies the hypothesis with\n$m=d=1$: here $\\operatorname{ord}_q(p)=1$, whereas\n$\\operatorname{ord}_p(2)=q$. For $C(n)=|\\Phi_n(2)|$ and a general fixed $m>0$, the source\nuses the \\emph{eventual} version: the divisibility is required only for\nprime $q$ above a fixed threshold. The same proof then applies after that\nthreshold. It is not valid to replace this by an all-prime assertion:\n$\\Phi_6(2)=3$, but $6\\nmid3-1$. For the binary cyclotomic family, choosing\n$q>\\max(m,2^m)$ removes these exceptional indices. Other families require\ntheir own proof of the divisibility and nontriviality assumptions.\n\\end{thm}\n\n\\subsubsection{The lower bound for rank-one quotients}\n\\label{rank-one-subrank}\n\nThe finite-sum quotients below stay a positive distance above\n$\\Theta_2=S-1/2$, uniformly in their exponent and truncation length.\nConsequently neither parameter can make the associated rational\nlinear form tend to zero. Appendix~\\ref{long249:r3-rankone} gives\nthe sharp minimum and its ordinary proof, strengthening the bound\nrecorded here.\n\n\\begin{thm}[A uniform positive gap for rank-one quotients]\\label{thm:uniform-positive-gap-rank-one}\nFor integers $r\\ge2$ and $Y\\ge1$, write\n\\[\n \\Theta_r=\\sum_{d\\ge1}\\frac{\\mu(d)}{(2^d-1)^r},\n \\qquad t(Y,r)=\\sum_{d=1}^{Y}\\frac{\\mu(d)}{(2^d-1)^r}.\n\\]\nFor every $e\\ge1$ and $Y\\ge4$,\n\\[\n \\frac{t(Y,e+2)^2}{t(Y,2e+2)}-\\Theta_2>\\frac1{480}.\n\\]\nHere $\\Theta_2=S-1/2$. For $r\\ge3$, the proof uses\n$1429/1512\\le\\Theta_r<1$ and\n$|t(Y,r)-\\Theta_r|\\le1/3584$ when $Y\\ge4$.\nIn particular, these inequalities make the denominator positive.\n\\end{thm}\n\n\\begin{prop}[The bound is preserved by positive normalised averaging]\\label{prop:bound-preserved-positive-normalised-averaging}\nFor a nonempty finite family of admissible quotients with positive\nweights summing to $1$, the weighted average still exceeds $\\Theta_2$\nby more than $1/480$. If an admissible quotient is $p/q$ in lowest terms,\nwith $q>0$, then $|q\\Theta_2-p|>q/480$.\n\\end{prop}\n\n\\begin{obs}\nWriting an admissible quotient in lowest terms as $p/q$, with $q>0$,\ngives $|q\\Theta_2-p|>q/480\\ge1/480$. Such forms cannot tend to zero.\nThe conclusion concerns exactly the displayed quotients and their positive\nnormalised averages. It makes no assertion about other rational\napproximants or about signed averages.\n\\end{obs}\n\n\\subsubsection{Multiples of a period}\n\nWe allow a positive multiple of each prescribed period. Telescoping\nproves sufficiency, and completeness of the residue test proves\nnecessity, giving another condition equivalent to irrationality.\nThe universally quantified condition remains unproved. For the finite\nexamples, we specify which excluded denominators already occur in the\ndiagonal computations.\n\n\\begin{thm}[Concatenation and specified period multiples]\\label{thm:concatenation-specified-period-multiples}\nFor nonnegative integers $a,b,N$, define the integer block sum\n\\[\n Q_{a,N}=\\sum_{j=1}^{a}\\varphi(N+j)2^{a-j}.\n\\]\nSplitting a block after its first $a$ terms gives\n\\[\n Q_{a+b,N}=2^bQ_{a,N}+Q_{b,N+a}.\n\\]\nIn particular, $Q_{2h,N}=2^hQ_{h,N}+Q_{h,N+h}$.\nThe corresponding cyclotomic identity is\n$\\Phi_4(2^h)=2^{2h}+1=\\Phi_2(2^{2h})$: the order-four factor at height\n$h$ is the order-two factor at height $2h$.\n\nThe order-three factor is different:\n$\\Phi_3(2^h)=2^{2h}+2^h+1$ comes from tripling the period.\nIt is not generally part of the doubling sequence. For example,\n$\\Phi_3(2)=7$ divides none of $2^{2^j}-1$, because the order of $2$\nmodulo $7$ is $3$, which does not divide $2^j$. Thus the block identities\nrelate specified period multiples; they do not put all the order-two,\norder-three and order-four factors into one doubling chain.\n\\end{thm}\n\n\\begin{obs}[An equivalent quantified condition]\nBoth implications of this equivalence are proved in Lean\n(\\lean{ErdosProblems.Erdos249.PeriodMultipleEscape.periodMultipleKillSupply_iff_irrational}{ErdosProblems/Erdos249/PeriodMultipleEscape.lean:432});\nthe quantified condition itself is not established. The aggregate diagonal\ntable gives certificates at every $H_t$ for $t\\le82$\n(\\lean{ErdosProblems.Skip.LadderT67.exists_diagonalKill_le_82}{ErdosProblems/Skip/LadderT67.lean:71264});\nthis equivalence contributes no new information about $t=83$ or any\ncondition at arbitrarily large scales.\n\\end{obs}\n\n\\begin{obs}[Eight finite period certificates]\nThe cited declarations give certificates at basepoint $N=300$ for\n\\[\n h\\in\\{67,81,97,101,121,125,127,128\\}.\n\\]\nEach excludes every reduced fraction $a/b$, with $b>0$ and\n$b\\mid2^{300}(2^h-1)$. These exclusions overlap with those already\nobtained from the diagonal. In particular, $67$ and $81$ divide $H_{82}$,\nand $H_{82}>300$, so\n\\[\n 2^{300}(2^h-1)\\mid2^{H_{82}}(2^{H_{82}}-1)\n \\qquad(h=67,81).\n\\]\nTheir entire denominator classes are already covered by\nProposition~\\ref{prop:B11-inv}. The eight cases therefore cannot all\nbe described as new denominator classes relative to that proposition.\nThe exact comparison of the odd factors is\n\\[\n \\gcd(2^h-1,2^{H_{82}}-1)=2^{\\gcd(h,H_{82})}-1,\n\\]\nby the Euclidean algorithm on the exponents. None of the remaining\nsix periods divides $H_{82}$, so its full odd factor is not covered\nby that diagonal denominator. This compares with the $t=82$ certificate,\nnot with every other exclusion in the record; the case $h=127$ includes\nthe Mersenne prime $2^{127}-1$.\n\nThe case $h=67$ also succeeds at depth $L=67$. It illustrates the\nrestriction $L=h$ in the full-depth condition, while retaining a small\nbasepoint. That universally quantified condition is equivalent to\nirrationality; one finite instance is not.\n\\lean{ErdosProblems.Erdos249.PeriodMultipleEscape.certifiedKill_67_300}{ErdosProblems/Erdos249/PeriodMultipleEscape.lean:471}\n\\quad \\lean{ErdosProblems.Erdos249.PeriodMultipleEscape.certifiedKill_127_300}{ErdosProblems/Erdos249/PeriodMultipleEscape.lean:490}\n\\quad \\lean{ErdosProblems.Erdos249.PeriodMultipleEscape.certifiedKill_67_300_fullDepth}{ErdosProblems/Erdos249/PeriodMultipleEscape.lean:498}\n\\quad \\lean{ErdosProblems.Erdos249.PeriodMultipleEscape.totient_series_ne_rat_of_den_dvd_127_300}{ErdosProblems/Erdos249/PeriodMultipleEscape.lean:534}\nThese are eight finite exclusions; the condition at arbitrarily large\nscales remains unproved.\n\\end{obs}\n\n\\subsubsection{The strict prime-position criterion}\n\nA sufficient prime-index condition is that, for every integer $h\\ge1$\nand every $N_0\\ge0$, there is a prime\n$p\\ge\\max(N_0+h+1,h+5)$ such that\n\\[\n \\operatorname{Re}e\\bigl(2^{p-h-1}(2^h-1)S\\bigr)<\\frac9{10}.\n\\]\nHere $e(x)=\\exp(2\\pi i x)$. The positive difference between $9/10$ and\nthis real part may depend on the chosen prime; the truncation depth is\nthen chosen so that its error fits within that difference. This is a\nweaker phase requirement than the earlier bound $4/5$, not a larger\nsaving from $1$. The implication to irrationality is proved, but the\ndisplayed prime-index condition remains unproved.\n\\lean{ErdosProblems.Erdos249.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap}{ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean:524}\n\n\\subsubsection{Finite Euler-product identities}\n\nThese are identities for the Dirichlet-convolution square $\\mu*\\mu$,\nnot the pointwise square $\\mu(n)^2$. For each prime $p$, its coefficients\nat $1,p,p^2$ are $1,-2,1$, and the higher prime-power coefficients vanish.\nThe local factors at exponents $1$ and $2$ are\n\\[\n 1-\\frac2p+\\frac1{p^2}=\\left(1-\\frac1p\\right)^2,\n \\qquad\n 1-\\frac2{p^2}+\\frac1{p^4}=\\left(1-\\frac1{p^2}\\right)^2.\n\\]\nFor a prime $p$ and $e\\ge0$, the corresponding convolution with the\ndivisor-sum function $\\sigma$ gives\n\\[\n \\sigma(p^{e+2})-2\\sigma(p^{e+1})+\\sigma(p^e)\n       =p^{e+1}(p-1)=\\varphi(p^{e+2}).\n\\]\nThis follows by subtracting the finite geometric sums. These finite\nidentities do not assert a transcendence result.\n\\lean{ErdosProblems.Erdos249.FiniteEulerSieve.muSqEulerFactor_one}{ErdosProblems/Erdos249/FiniteEulerSieve.lean:28}\n\\quad \\lean{ErdosProblems.Erdos249.FiniteEulerSieve.sigmaPrimePow_secondDiff}{ErdosProblems/Erdos249/FiniteEulerSieve.lean:51}\n% ; - part a249_p2 ; -\n\\section{Parameter ranges and quantifiers}\\label{sec:249-scale-ladder}\n\nThe distinction between a finite calculation and an unbounded family\nis in the quantifiers. A statement such as $\\forall a\\ge8$ is uniform\nin $a$, whereas a list of computed scales is finite and a condition\n$\\forall N_0\\,\\exists N\\ge N_0$ asks for witnesses arbitrarily far out.\nThe table uses these distinctions even when a uniform statement is\nconditional on an unproved hypothesis. A bound on a secondary variable,\nsuch as $J+(a+6)<2\\cdot2^a$ in the positivity row, does not bound $a$.\n\n\nSome identities have no distinguished scale parameter. They are listed\nwith the general identities. An equivalence can hold for all parameters\nwithout establishing either of the equivalent arithmetic assertions.\n\n\nThe LCM formulation of Problem~249 requires arbitrarily large scales:\n\\lean{irrational\\_totientSeries\\_iff\\_actualLcmOrbitNonintegralitySupply}{TotientActualLcmOrbitNonintegrality.lean:37}\n\\[\n  \\mathrm{Irrational}\\Bigl(\\sum_{n\\ge 1}\\varphi(n)/2^n\\Bigr) \\iff \\forall a_0\\ \\exists a\\ge a_0,\\ \\Omega_a \\notin \\mathbb Z,\n\\]\na cofinal statement in the exponent $a$ indexing $H=H(2^a)$.\nA finite calculation alone does not establish it; a further theorem\ncould propagate finite information to arbitrarily large scales.\n\nThe table groups statements by their parameter ranges, not by proof\nstatus. In particular, some assertions with cofinal quantifiers are\nalready proved, such as the prime constructions and rational comparison\nsequences. Others are implications whose quantified hypotheses remain\nopen. The horizontal rule separates the cofinal group for ease of\nreference; it does not mark a boundary between proved and unproved\nmathematics. The descriptions identify this distinction in that group.\nThe prefixes summarise the quantifiers; complete hypotheses are given in\nthe cited statements. A definition records an object or condition; it does not by itself prove\nthe associated soundness or existence assertion.\n\n\\begingroup\nIn the periodic-weight rows, write\n$w:\\mathbb N\\to\\mathbb Z$ and\n$c_w(n)=\\sum_{d\\mid n}w(d)$ for $n\\ge1$. Positivity and nonvanishing\nrefer to these divisor coefficients, not to the weight $w$ itself.\nHere ``nonzero arbitrarily far out'' means that for every $N$ there is\n$n>N$ with $c_w(n)\\ne0$; it does not mean nonvanishing on every\narithmetic progression. The nonprincipal character $\\chi_4$ modulo four\nis $0$ on even integers and $(-1)^{(n-1)/2}$ on odd integers.\n\n\\begin{landscape}\n\\fontsize{8}{9.8}\\selectfont\n\\setlength{\\tabcolsep}{3pt}\n\\begin{longtable}{@{}>{\\raggedright\\arraybackslash}p{0.26\\linewidth}>{\\raggedright\\arraybackslash}p{0.18\\linewidth}>{\\raggedright\\arraybackslash}p{0.38\\linewidth}>{\\raggedright\\arraybackslash}p{0.12\\linewidth}@{}}\n\\caption{Results and unproved conditions, grouped by their quantifiers.\nThe dividing rule separates the cofinal group, not proof status.\nComplete hypotheses are given in the cited statements.}\\\\\n\\toprule\n\\papertablehead\n\\textbf{Result} & \\textbf{Source} & \\textbf{Quantifier prefix} & \\textbf{Representation}\\\\\n\\midrule\n\\endfirsthead\n\\toprule\n\\papertablehead\n\\textbf{Result} & \\textbf{Source} & \\textbf{Quantifier prefix} & \\textbf{Representation}\\\\\n\\midrule\n\\endhead\n\\midrule \\multicolumn{4}{r}{\\emph{continued on next page}} \\\\\n\\endfoot\n\\bottomrule\n\\endlastfoot\n\n\\papertablehead\\multicolumn{4}{l}{\\emph{Uniform statements, including unconditional identities and implications}}\\\\\nscaled totient tail (definition) & \\lean{totientTail}{TotientTailPeriodKiller.lean:57} & $\\forall N$ & binary-weighted series\\\\\ninteger prefix and real tail & \\lean{two\\_pow\\_mul\\_totient\\_series\\_eq}{TotientTailPeriodKiller.lean:150} & $\\forall N$ & binary-weighted series\\\\\n$D(h,N,L)$ (definition) & \\lean{windowDiscrepancy}{TotientTailPeriodKiller.lean:66} & $\\forall h\\,\\forall N\\,\\forall L$ & finite binary sums\\\\\n$\\mathcal C(h,N,L)$ (definition) & \\lean{certifiedKill}{TotientTailPeriodKiller.lean:72} & $\\forall h\\,\\forall N\\,\\forall L$ & finite binary sums\\\\\nnecessary depth inequality & \\lean{certifiedKill\\_depth\\_floor}{TotientTailPeriodKiller.lean:79} & $\\forall h\\,\\forall N\\,\\forall L$, given \\ensuremath{\\mathcal{C}}$\\,h\\,N\\,L$ & finite binary sums\\\\\nresidue test soundness & \\lean{tail\\_diff\\_notMem\\_int\\_of\\_certifiedKill}{TotientTailPeriodKiller.lean:262} & $\\forall h\\,\\forall N\\,\\forall L$, given \\ensuremath{\\mathcal{C}} & finite binary sums\\\\\ncompleteness of the residue test & \\lean{exists\\_certifiedKill\\_iff\\_tail\\_diff\\_notMem\\_int}{LcmConeFlatness.lean:316} & $\\forall h\\,\\forall N$ & finite binary sums\\\\\nintegrality from denominator divisibility & \\lean{tail\\_diff\\_int\\_of\\_den\\_dvd}{TotientTailPeriodKiller.lean:327} & $\\forall N\\,\\forall h\\,\\forall r{:}\\mathbb{Q}$, given $S=r$, $r.\\mathrm{den}\\mid 2^N(2^h{-}1)$ & binary-weighted series\\\\\ntail-period law & \\lean{eventual\\_period\\_of\\_not\\_irrational}{TotientTailPeriodKiller.lean:358} & $\\lnot\\mathrm{Irrational}\\,S\\to\\exists h{>}0\\,\\exists N_0\\,\\forall N{\\ge}N_0$ & binary-weighted series\\\\\nleast common multiple $H(t)$ & \\lean{periodLcm}{CarrySurvivorExtinction.lean:515} & $\\forall t$ & LCM periods\\\\\na nondivisor below $2t$ is a prime power & \\lean{eq\\_prime\\_pow\\_of\\_not\\_dvd\\_periodLcm}{LcmDiagonalReduction.lean:137} & $\\forall t\\,\\forall j\\,(0{<}j{<}2t)$, given $j\\nmid{H}\\,t$ & LCM offsets\\\\\ntotient factorisation at divisor offsets & \\lean{totient\\_periodLcm\\_ray\\_split}{LcmDiagonalReduction.lean:208} & $j\\mid H(t)$, with the prime-divisor hypotheses of Proposition~\\ref{catalogue:cert:b5} & totient factorisation\\\\\nintegrality on the LCM grid under rationality & \\lean{rational\\_totient\\_series\\_forces\\_lcm\\_cone\\_flatness}{CertificateKernel.lean:19002} & $\\lnot\\mathrm{Irrational}\\,S\\to\\exists t_1\\,\\forall t{\\ge}t_1\\,\\forall q{>}0\\,\\forall m$ & LCM grids\\\\\nsecond-difference residue test & \\lean{second\\_diff\\_notMem\\_int\\_of\\_certifiedRank2Kill}{LcmConeFlatness.lean:493} & $\\forall h\\,\\forall N\\,\\forall L$, given \\ensuremath{\\text{second-difference residue condition}} & finite binary sums\\\\\na nonintegral pair on a finite grid & \\lean{exists\\_nonintegral\\_pair\\_of\\_coneNonflatCert}{LcmConeNonflat.lean:126} & finite $Q\\ne\\emptyset$; all size and residue hypotheses of Theorem~\\ref{catalogue:cert:b10a} & finite LCM grids\\\\\nfinite exclusion of integer carries & \\lean{tail\\_diff\\_notMem\\_int\\_of\\_survivorKill}{CarrySurvivorExtinction.lean:428} & $\\forall h\\,\\forall N\\,\\forall K$, given \\ensuremath{\\text{finite carry exclusion}} & integer recurrences\\\\\nFarey neighbour denominator law & \\lean{farey\\_gap}{GapFareyBound.lean:51} & $a,b,c,d,r,s\\in\\mathbb Z$; $b,d>0$, $bc-ad=1$, $as<br$, $dr<cs$ & rational intervals\\\\\nrational-approximation criterion & \\lean{irrational\\_of\\_den\\_mul\\_abs\\_sub\\_tendsto\\_zero}{CertificateKernel.lean:5371} & $u_j\\in\\mathbb Q$, $x\\in\\mathbb R$; eventually $u_j\\ne x$, and $\\operatorname{den}(u_j)|u_j-x|\\to0$ & general criterion\\\\\nnear-integer irrationality criterion & \\lean{irrational\\_of\\_int\\_mul\\_near\\_int}{CertificateKernel.lean:6120} & $\\forall \\xi{:}\\mathbb{R}$, $\\forall q{>}0\\,\\exists m\\,z\\,0{<}|m\\xi{-}z|{<}1/q$ & general criterion\\\\\nindependence from a nonsingular evaluation matrix & \\lean{linearIndependent\\_of\\_separatedMinorCertificate}{TotientMahlerDefect.lean:91} & a finite family with a square evaluation matrix of nonzero determinant & general criterion\\\\\ndyadic totient-kernel rank $=2^e{+}1$ & \\lean{linearIndependent\\_canonicalTotientKernelFamily}{TotientMahlerDefect.lean:935} & $\\forall e\\ge1$ & dyadic sections\\\\\nrationality forces unbounded carry-kernel rank & \\lean{not\\_irrational\\_totientSeries\\_implies\\_unbounded\\_carryRank\\_unconditional}{TotientCarryKernelRigidity.lean:300} & $\\lnot\\mathrm{Irrational}\\,S\\to\\exists v{>}0\\,\\exists u\\,\\forall e$ & carry sections\\\\\nno fixed $D$ clears all $A(n)/n$ for $A=\\varphi*\\mu$ & \\lean{no\\_fixed\\_integral\\_primitive\\_euler\\_index}{PrimitiveDeterminantLift.lean:169} & $\\forall D{>}0$ & M\\\"obius sums\\\\\nLambert-series value identities & \\lean{tsum\\_totient\\_div\\_pow\\_two\\_eq\\_pnat\\_half\\_pow}{CertificateKernel.lean:18415} & value identities, $L(\\mu){=}\\tfrac12$, $L(\\varphi){=}2$, $L(1){=}$Erd\\H{o}s--Borwein & M\\\"obius sums\\\\\nprime-power unit-gap criterion & \\lean{reducedDenominatorUnitGapCert\\_prime\\_pow\\_iff}{PrimitiveWeightCertificate.lean:54} & $\\forall p\\ \\text{prime},\\ e>0,\\ N,K\\ge0$ & rational intervals\\\\\npositive rational-difference lower bound & \\lean{positive\\_rational\\_difference\\_lower\\_bound}{PrimitiveRationalGapSupply.lean:31} & $\\forall\\ \\text{whole}\\,\\text{pfx}{:}\\mathbb{Q},\\ \\text{pfx}{<}\\text{whole}$ & general criterion\\\\\n$L_2(\\mu)=S-\\tfrac12$ & \\lean{tsum\\_totient\\_half\\_pow\\_eq\\_half\\_add\\_moebius\\_sq}{MersenneLambertLadder.lean:665} & unconditional & M\\\"obius sums\\\\\nweighted squared-denominator identity & \\lean{tsum\\_lambert\\_linear\\_weight\\_sq\\_pure}{GcdMomentCalculus.lean:105} & $\\forall w{:}\\mathbb{N}\\to\\mathbb{R}\\,(|w(d)|{\\le}d)\\,\\forall r\\in[0,1)$ & M\\\"obius sums\\\\\n$L_2(\\mu)=S-\\tfrac12$ & Proposition~\\ref{prop:mobsq} & unconditional & M\\\"obius sums\\\\\n$L_2(1)=\\zeta_q(2)-\\zeta_q(1)$ identity & \\lean{tsum\\_one\\_div\\_mersenne\\_sq\\_eq\\_sigma\\_sub\\_tau\\_series}{GcdMomentCalculus.lean:216} & unconditional & M\\\"obius sums\\\\\n$L_2(\\varphi)$ = gcd moment & \\lean{tsum\\_totient\\_div\\_mersenne\\_sq\\_eq\\_gcd\\_moment\\_series}{GcdMomentCalculus.lean:235} & unconditional & gcd moments\\\\\ncomparison of the weights & Propositions~\\ref{prop:mobsq}--\\ref{prop:pillai} & unconditional & M\\\"obius sums\\\\\ngcd-divisibility factorizes & \\lean{tsum\\_pos\\_pair\\_both\\_dvd\\_half\\_eq\\_inv\\_mersenne\\_sq}{GcdMomentCalculus.lean:266} & $\\forall d{:}\\mathbb{N},\\,d{>}0$ & geometric weights\\\\\nsum over positive coprime pairs equals $1$ & \\lean{tsum\\_pos\\_coprime\\_inv\\_mersenne\\_eq\\_one}{GcdMomentCalculus.lean:349} & unconditional (sum over all coprime pairs) & geometric weights\\\\\nStern--Brocot recursion with stopping & \\lean{cylinderMass\\_split}{GcdMomentCalculus.lean:474} & positive integers $a,b$ & Stern--Brocot tree\\\\\npositive numerator coefficients & \\lean{mobiusNumeratorPolynomial\\_eq\\_gcdWord}{RepunitMobiusNumerator.lean:205} & $\\forall r\\,(\\mathrm{Squarefree}\\,r)\\,\\forall k{<}r$ & divisor and cyclotomic sums\\\\\nevaluation at $2$ gives the numerator & \\lean{mobiusNumeratorPolynomial\\_eval\\_two}{RepunitMobiusNumerator.lean:445} & $\\forall r\\,(\\mathrm{Squarefree}\\,r)$ & divisor and cyclotomic sums\\\\\nreduction to the radical of $H$ & \\lean{scaledMobiusShadow\\_eq\\_radicalBase}{RadicalMobiusShadow.lean:125} & $\\forall H{>}0\\,\\forall r{>}0$ & M\\\"obius sums\\\\\nnumerator congruence modulo $\\Phi_m$ & \\lean{cyclotomic\\_dvd\\_mobiusNumeratorPolynomial\\_sub}{CyclotomicProjectionOfShadow.lean:299} & $\\forall r\\,(\\mathrm{Squarefree}\\,r)\\,\\forall m\\,(m{\\mid}r)$ & divisor and cyclotomic sums\\\\\ngcd of the numerator and $\\Phi_r(2)$ & \\lean{mobiusNumerator\\_gcd\\_cyclotomicValue}{CyclotomicProjectionOfShadow.lean:350} & $\\forall r\\,(\\mathrm{Squarefree}\\,r)$ & divisor and cyclotomic sums\\\\\nMersenne factors retained in the denominator & \\lean{upperHalfChannel\\_product\\_dvd\\_den\\_of\\_scale\\_primeFactors\\_le}{MersenneShadowCyclotomicNoncollapse.lean:809} & $t\\ge5$, squarefree $r$; finite $P$ of primes dividing $r$ in $(t/2,t]$; prime factors of $r$ and the multiplier are at most $t$ & divisor and cyclotomic sums\\\\\ndenominator lower bound / exact value & \\lean{lcmHeight\\_scaledMobiusShadow\\_den\\_exact}{MersenneShadowDenominatorGrowth.lean:147} & $\\forall t$ (exact-value); $t{\\ge}5$ (lower bound) & M\\\"obius sums\\\\\nnumerator congruence after adjoining a prime & \\lean{cyclotomic\\_dvd\\_primeJump\\_new\\_fibre}{PrimePowerJumpDynamics.lean:281} & $r\\ge1$, $p$ prime, $p\\nmid r$, $m\\mid r$ & divisor and cyclotomic sums\\\\\nsquared-Mersenne series: partial sum plus tail & \\lean{mobius\\_square\\_series\\_eq\\_partial\\_add\\_tail}{SquaredMersenneDiagonalEnclosure.lean:94} & $\\forall D{:}\\mathbb{N}$ & M\\\"obius sums\\\\\ncancellation of the affine divisor terms & \\lean{joint35\\_oldChannel\\_zero}{JointExponentTransport.lean:71} & $\\forall d{>}0\\,(d{\\mid}H)$; general: finite affine annihilator & LCM diagonal\\\\\nnew-divisor counting identity & \\lean{supportCoeff\\_mul\\_eq\\_add\\_defect}{CompositeDilationDefect.lean:30} & $\\forall a{\\in}A\\,(a{>}0)\\,\\forall x{>}0$ & divisor counts\\\\\nbound on zero blocks in divisor counts & \\lean{supportCoeffZeroWindow\\_length\\_le\\_eps\\_logb\\_add}{SublogDivisorCoverage.lean:392} & $\\forall \\varepsilon{>}0\\,\\exists B$, given rationality of the base-2 support series & divisor counts\\\\\nequivalent shift condition for the target $1/2$ (Problem~257) & \\lean{shiftWindowZero\\_iff\\_half\\_mem\\_mersenneAchievementSet}{CampbellShiftSynchronization.lean:488} & iff, no extra parameter & Mersenne subsums\\\\\ntotient multiplication with common prime factors & \\lean{totient\\_mul\\_eq\\_overlapFactor\\_mul}{TotientActualLcmOrbitArithmetic.lean:43} & $\\forall x{>}0$ ($j$ fixed) & totient identities\\\\\ntotient relative-Euler-product split & \\lean{totient\\_mul\\_eq\\_totient\\_mul\\_relativeEulerProduct}{TotientActualLcmOrbitArithmetic.lean:100} & $\\forall j\\,x{>}0$ & totient identities\\\\\nformula for $\\varphi(2H(t)+j)-\\varphi(H(t)+j)$ & \\lean{lcmRayArithmeticLetter\\_eq\\_deltaTotient}{TotientActualLcmOrbitArithmetic.lean:298} & $\\forall t\\,\\forall j$ & M\\\"obius sums\\\\\nthe finite diagonal sum equals $D(H(t),H(t),L)$ & \\lean{lcmDiagonalArithmeticWord\\_eq\\_windowDiscrepancy}{TotientActualLcmOrbitArithmetic.lean:1998} & $\\forall t\\,\\forall L$ & M\\\"obius sums\\\\\n\\#249 $\\iff$ cofinal actual-orbit nonintegrality & \\lean{irrational\\_totientSeries\\_iff\\_actualLcmOrbitNonintegralitySupply}{TotientActualLcmOrbitNonintegrality.lean:37} & iff & M\\\"obius sums\\\\\nthe LCM tail difference is an integer translate of a multiple of $S$ & \\lean{actualLcmTailOrbit\\_eq\\_scaled\\_totientSeries\\_sub\\_prefix}{TotientActualLcmOrbitSeparation.lean:68} & $\\forall a$ & M\\\"obius sums\\\\\nexplicit finite-block approximation & \\lean{abs\\_actualLcmTailOrbit\\_sub\\_rawApprox\\_lt}{TotientActualLcmOrbitSeparation.lean:141} & $\\forall a\\,\\forall q$ & M\\\"obius sums\\\\\nan equivalent residue-interval condition & \\lean{exists\\_certifiedKill\\_iff\\_guardCylinderWitness}{TotientActualLcmTopEdgeStaircase.lean:844} & $\\forall h\\,N,\\ \\text{iff}$ & integer recurrences\\\\\nsign of the three-term totient difference under the stated middle-value bound & \\lean{fixedRankSecondDifference}{TotientFixedRankLcmAsymptotic.lean:256} & $\\forall H\\,j$, given MiddleRankTotientExtremal & totient differences\\\\\nexact value on a specified dyadic family & \\lean{fixedRankSecondDifference\\_dyadic\\_fixture\\_exact}{TotientFixedRankLcmAsymptotic.lean:462} & $\\forall n$ (closed-form) & totient differences\\\\\nthe rational parity comparison (definition) & \\lean{parityCoboundaryWeight}{TotientParityCoboundaryCountermodel.lean:59} & $\\forall n$ (definition) & binary-weighted series\\\\\n$R_N$ as a sum over positive divisor indices & \\lean{totientTail\\_eq\\_tsum\\_positiveForeignResidueKernel}{TotientShiftedMobiusForeignBridge.lean:61} & $\\forall N$ & divisor and cyclotomic sums\\\\\n$R_N$ as a shifted M\\\"obius sum & \\lean{totientTail\\_eq\\_tsum\\_shiftedMobiusPulse}{TotientShiftedMobiusPulse.lean:547} & $\\forall N$ & divisor and cyclotomic sums\\\\\nabsolute convergence and regrouping of Lambert sums & \\lean{tsum\\_lambert\\_pair\\_regroup\\_if}{TotientShiftedMobiusPulse.lean:124} & $\\forall w\\,v{:}\\mathbb{N}\\to\\mathbb{R}\\,(|w(d)|{\\le}d,\\,|v(m)|{\\le}m)$ & divisor and cyclotomic sums\\\\\ncarry divisibility and integral tail differences & \\lean{carryShift\\_dvd\\_iff\\_tailDiff\\_mem\\_int}{TotientTailCarryPeriod.lean:77} & $\\forall N\\,k$, given the carry recurrence and limit & integer recurrences\\\\\nrationality $\\Rightarrow$ periodicity \\& unbounded rank & \\lean{not\\_irrational\\_totientSeries\\_implies\\_mod\\_period\\_and\\_unbounded\\_rank}{TotientTailCarryPeriod.lean:224} & $\\lnot\\mathrm{Irrational}\\,S\\to\\exists v\\,u\\,\\forall e$ & integer recurrences\\\\\nthe stated reconstruction cost is at least $3$ & \\lean{three\\_le\\_totientAdjugateTailCost}{TotientTailCarryPeriod.lean:266} & $\\forall\\ \\text{finite}\\ w{:}\\iota\\to\\mathbb{Q}\\,x{:}\\iota\\to\\mathbb{N}$, given $\\sum w_i\\varphi(x_i){=}1$ & weighted tail bounds\\\\\ndirected residue interval (definition) & \\lean{directedCertifiedKill}{TotientTailCarryPeriod.lean:766} & $h,N,L\\ge0$; soundness and the existential-depth equivalence are proved in Proposition~\\ref{prop:CP-06} & integer recurrences\\\\\nrationality and an integer carry with vanishing boundary term & \\lean{binaryCoeffSeries\\_rational\\_iff\\_exists\\_temperedBinaryOrbit}{GenericTailOrbitRigidity.lean:426} & $\\forall c{:}\\mathbb{N}\\to\\mathbb{N}\\,(c(n){\\le}n)$, iff & binary-weighted series\\\\\ndoubling-orbit rigidity & \\lean{doublingOrbit\\_eq\\_zero\\_of\\_tempered}{GenericTailOrbitRigidity.lean:315} & $\\forall d{:}\\mathbb{N}\\to\\mathbb{R}$, doubling and $d(n)/2^n\\to0$ imply $d{\\equiv}0$ & integer recurrences\\\\\nnumber of labels needed for exact tail recovery & \\lean{balancedPulse\\_no\\_autonomous\\_decoder}{GenericTailOrbitRigidity.lean:247} & $\\forall m\\,\\forall\\mathrm{State}$, collapsing-state $\\to$ no decoder & binary-weighted series\\\\\nendpoint residues do not determine the initial carry & \\lean{affineBinaryOrbit\\_mod\\_twoPow\\_eq}{GenericTailOrbitRigidity.lean:307} & $\\forall a{:}\\mathbb{N}\\to\\mathbb{Z}\\,\\forall u_0\\,v_0\\,\\forall L$ & binary-weighted series\\\\\ninteger-coefficient block-certificate dichotomy & \\lean{int\\_coeff\\_series\\_irrational\\_or\\_bpow\\_mul\\_eq\\_intCast\\_of\\_full\\_block\\_certificates}{CertificateKernel.lean:13934} & $\\forall b{\\ge}2\\,\\forall c\\,\\forall q$, given full-block certificate at $q$ & binary-weighted series\\\\\ndivisibility under iterated forward differences & \\lean{dvd\\_iteratedForwardDifference}{LcmFactorIdealPulseObstruction.lean:211} & $d\\mid f(n)$ for every $n$; every finite list of shifts & integer sequences\\\\\ntotient series as a coprime-pair sum & \\lean{tsum\\_coprime\\_pair\\_pow\\_eq\\_tsum\\_totient\\_mul\\_pow}{GeometricCoprimality.lean:120} & $\\forall n$; $\\forall r\\in[0,1)$ & geometric weights\\\\\nfactorisation through a rational-valued linear functional & \\lean{linearDescender\\_eq\\_smul\\_eval}{AdelicHeightObstruction.lean:120} & rational vector spaces $V,W$; surjective linear $\\mathrm{ev}:V\\to\\Q$, linear $L:V\\to W$, $\\ker(\\mathrm{ev})\\subseteq\\ker L$ & divisibility and linear algebra\\\\\ndivisibility of a reduced denominator & \\lean{divisor\\_dvd\\_divInt\\_den}{RationalDenominatorSurvival.lean:17} & $\\forall D{>}0\\,\\forall m{\\mid}D\\,\\forall a$ & divisibility and linear algebra\\\\\nthe complementary denominator divides the integer multiplier & \\lean{scalarLocalization\\_complement\\_dvd}{AdelicHeightObstruction.lean:23} & $\\forall x{:}\\mathbb{Q}\\,\\forall H\\,(H{\\mid}x.\\mathrm{den})\\,\\forall c\\,((cx).\\mathrm{den}{\\mid}H)$ & divisibility and linear algebra\\\\\nlargest-exponent nonvanishing criterion & \\lean{scaled\\_dyadic\\_sum\\_ne\\_zero}{SignedQMomentObstruction.lean:96} & $\\exists m{\\in}s$ (strictly maximal, odd coeff), $\\forall i{\\ne}m\\,e(i){<}e(m)$ & divisibility and linear algebra\\\\\ncolumn rescaling and determinant nonvanishing & \\lean{det\\_residualMonomialMatrix}{ResidualGaugeObstruction.lean:47} & $\\forall d\\,\\forall e{:}\\mathrm{Fin}\\,d{\\to}\\mathbb{N}\\,\\forall z{:}\\mathrm{Fin}\\,d{\\to}\\mathbb{C}$ ($z$ nonzero) & complex matrices\\\\\ndeterminant $1$ for the divisor-incidence matrix & \\lean{incidenceMobius\\_det\\_eq\\_one}{IncidenceQuotientHermitePade.lean:56} & $\\forall N{:}\\mathbb{N}$ & M\\\"obius sums\\\\\n\nerror bound from convergence & \\lean{controlledForeignProjection\\_of\\_tail\\_limit}{LambertDiagonalEnclosure.lean:31} & $2H{\\le}D$; convergence is supplied for $H>0$ in Section~\\ref{sec:nondivisor-enclosure} & M\\\"obius sums\\\\\ncarries realising fixed-precision unit residues & \\lean{fixedPrecisionTropicalNoGo}{TropicalCurvatureCarry.lean:137} & $\\forall u{>}0$ (fixed precision), $\\forall$ finite carry word & divisibility and linear algebra\\\\\nvanishing of iterated prime dilation differences & \\lean{iteratedDilationDifference\\_eq\\_zero\\_of\\_cardFactors\\_le}{SupportDilationDifferences.lean:250} & distinct primes $p_1,\\ldots,p_r$, $r>K$, $\\gcd(n,\\prod p_i)=1$; $\\Omega(a)\\le K$ on $A$ & divisor counts\\\\\n$\\varphi(n)>\\tfrac34 n$ for the stated rough integers & \\lean{three\\_quarters\\_mul\\_lt\\_totient\\_of\\_rough\\_lt\\_two\\_pow}{TotientActualLcmOrbitArithmetic.lean:768} & $\\forall a{\\ge}8\\,\\forall n{>}0$ ($n$ is $t$-rough, $t{=}2^a$, $n{<}2^{2\\cdot2^a}$) & M\\\"obius sums\\\\\npositivity of short-window totient differences & \\lean{lcmRayArithmeticLetter\\_pos\\_of\\_lt\\_two\\_mul}{TotientActualLcmOrbitArithmetic.lean:1703} & $\\forall a{\\ge}8\\,\\forall j\\,(0{<}j{<}2\\cdot2^a)$ & M\\\"obius sums\\\\\nthe negative representative of the true endpoint & \\lean{actualLcm\\_trueEndpointSurvivor\\_neg}{TotientActualLcmOrbitSign.lean:172} & $\\forall a{\\ge}8\\,\\forall J\\,K$ (size bound) & M\\\"obius sums\\\\\nintegrality forces upper-endpoint residue & \\lean{actualLcm\\_integral\\_forces\\_topEdgeResidue}{TotientActualLcmOrbitSign.lean:211} & $\\forall a{\\ge}8\\,\\forall J\\,K$ (size condition, $2H{+}J{+}K{+}2{<}2^K$) & M\\\"obius sums\\\\\nterminal dyadic staircase impossible under the size condition & \\lean{not\\_actualLcmTerminalDyadicStaircase\\_of\\_room}{TotientActualLcmTopEdgeStaircase.lean:251} & $\\forall a{\\ge}8\\,\\forall J\\,K\\,m$ (size condition, modulus-wide) & M\\\"obius sums\\\\\npenultimate difference in a partial divisibility pattern & \\lean{puncturedDyadicStaircase\\_penultimate\\_eq\\_half}{TotientActualLcmTopEdgeStaircase.lean:1187} & $\\forall a{\\ge}8\\,\\forall J\\,K\\,m$ (hypotheses of Proposition~\\ref{prop:TE-02-inv}) & M\\\"obius sums\\\\\nupper-endpoint residue-gap $\\Rightarrow$ nonintegral & \\lean{actualLcmTailDiff\\_notMem\\_int\\_of\\_topEdgeResidueGap}{TotientActualLcmTopEdgeStaircase.lean:1260} & $\\forall a{\\ge}8\\,\\forall J\\,K\\,m$ (size condition, one-sided) & M\\\"obius sums\\\\\nrelation between the centred residue and the actual carry & \\lean{two\\_mul\\_actualOddHalfCenteredLift\\_eq\\_terminal\\_sub\\_trueCarry}{TotientActualLcmTopEdgeStaircase.lean:1678} & $\\forall a{\\ge}8\\,\\forall q$ (size condition, fit bound $2(H{+}q{+}2){\\le}4^q$) & M\\\"obius sums\\\\\ndivisibility of the three-term totient difference & \\lean{two\\_mul\\_totient\\_dvd\\_fixedRankSecondDifference}{TotientFixedRankLcmAsymptotic.lean:519} & $\\forall a{\\ge}4\\,\\forall j{>}0\\,(j^2{\\le}2^a)$ & totient differences\\\\\nequality of two finite rational approximations & \\lean{renormalizedResidueAgreement\\_of\\_twice\\_le}{TotientShiftedMobiusForeignBridge.lean:338} & $H>0$, $D\\ge2H$; no separation hypothesis for this equality & M\\\"obius sums\\\\\na Mersenne divisor need not divide a tail difference & \\lean{idCoeff\\_mersenneModulus\\_does\\_not\\_annihilate\\_tailShift}{TotientTailCarryPeriod.lean:342} & $\\forall K,q\\,(K{>}0,K{<}q,q{\\mid}2^K{-}1)$ & M\\\"obius sums\\\\\nfinite numerator separation implies nonintegrality & \\lean{scaleFullTarget\\_miss\\_of\\_lambert\\_projected\\_num\\_gap}{LambertProjectedNumeratorGap.lean:23} & $\\forall H,D$ fixed, given uniform numerator-gap bound & M\\\"obius sums\\\\\nperiodic-weight Lambert dichotomy & \\lean{irrational\\_or\\_bpow\\_mul\\_eq\\_intCast\\_intWeightedErdosSeries\\_periodic}{CertificateKernel.lean:14175} & $\\forall b{\\ge}2\\,\\forall m{>}0\\,\\forall w$ ($w(n{+}m){=}w(n)$) & M\\\"obius sums\\\\\nperiodic weight with nonnegative divisor coefficients & \\lean{irrational\\_intWeightedErdosSeries\\_periodic\\_of\\_coeff\\_nonneg\\_of\\_frequently\\_ne\\_zero}{CertificateKernel.lean:14583} & $b\\ge2$, $m>0$, $w(n+m)=w(n)$; $c_w(n)\\ge0$ and $c_w(n)\\ne0$ arbitrarily far out & M\\\"obius sums\\\\\n\nzeros on one progression for the character modulo four & \\lean{intWeightedCoeff\\_periodFourSignWeight\\_eq\\_zero\\_of\\_mod\\_four\\_eq\\_three}{CertificateKernel.lean:14226} & $c_{\\chi_4}(n)=0$ for every $n\\equiv3\\pmod4$; the weight is fixed & M\\\"obius sums\\\\\n\npositivity of the translated LCM tail difference & \\lean{actualLcmTailDiff\\_shift\\_pos}{TotientActualLcmOrbitSign.lean:39} & $\\forall a{\\ge}8\\,\\forall J\\,(J{+}(a{+}6){<}2\\cdot2^a)$, no rationality hypothesis & M\\\"obius sums\\\\\n\n\\papertablehead\\multicolumn{4}{l}{\\emph{Finite ranges and parameter-bounded checks}}\\\\\nFarey window $K{=}240$ denominator bound & \\lean{gap\\_check\\_window\\_1\\_240\\_le\\_79639646646701375323355774875831053}{GapFareyBound.lean:176} &\n$\\forall q{:}\\mathbb{N},\\,0{<}q{\\le}$\\newline\n$79{,}639{,}646{,}646{,}701{,}$\\newline\n$375{,}323{,}355{,}774{,}875{,}831{,}053$ & rational intervals\\\\\ndenominator lower bound $S{\\ne}p/q$, $q{\\le}Q_0$ & \\lean{tsum\\_totient\\_div\\_pow\\_two\\_ne\\_ratCast\\_of\\_den\\_le\\_79639646646701375323355774875831053}{CertificateKernel.lean:18384} & $\\forall p{:}\\mathbb{Q},\\,p.\\mathrm{den}{\\le}Q_0$ & rational intervals\\\\\nfinite LCM certificates covering thresholds $a_0\\le6$ & \\lean{powerTwoActualLcmShortArithmeticKillSupply\\_through\\_six}{TotientActualLcmShortKill.lean:54} & $\\forall a_0{\\le}6\\,\\exists a\\,L$ & M\\\"obius sums\\\\\n\ncontiguous diagonal certificate band through $t{\\le}82$ & \\lean{ErdosProblems.Skip.LadderT67.exists_diagonalKill_le_82}{ErdosProblems/Skip/LadderT67.lean:71264} & $\\forall t{\\le}82\\,\\exists L$; no $t{=}83$ certificate claimed & LCM diagonal\\\\\n\n\\papertablehead\\multicolumn{4}{l}{\\emph{Listed examples}}\\\\\nresidue certificates for $1\\le h\\le8$, $N=12$, $L=16$ & \\lean{certifiedKill\\_all\\_small}{TotientTailPeriodKiller.lean:404} & $\\forall h\\in[1,8]$, \\ensuremath{\\mathcal{C}}\\,$h$\\,12\\,16 (direct evaluation) & finite binary sums\\\\\ndenominator exclusions for $1\\le h\\le16$, binary exponent $14$ & \\lean{totient\\_series\\_ne\\_rat\\_of\\_den\\_dvd\\_pow\\_two\\_mul\\_mersenne\\_upto\\_sixteen}{CertificateKernel.lean:18890} & $\\forall h\\in[1,16]$, denominator exponent 14 (direct evaluation) & finite binary sums\\\\\nthe second-difference test uses more depth at $(h,N)=(1,8)$ & \\lean{totient\\_tail\\_rank\\_two\\_kill\\_sound\\_but\\_not\\_shallower\\_cell}{CertificateKernel.lean:19090} & first-difference certificate at $L=8$; no second-difference certificate for $1\\le L\\le8$, and one at $L=9$ & finite binary sums\\\\\n28 listed diagonal certificates with $t\\le64$ & \\lean{certifiedKill\\_diagonal\\_all\\_imported}{DiagonalPincerCertificates.lean:2870} & $t\\in\\{1,2,3,4,5,7,8,9,11,13,16,17,\\dots\\}$ (28 explicit values) & LCM diagonal\\\\\nLCM certificates at $a=4$ and $a=6$ & \\lean{lcmDiagonalArithmeticKill\\_two\\_pow\\_four}{TotientActualLcmShortKill.lean:18} & $a{=}4$, $a{=}6$ (concrete, formally checked) & M\\\"obius sums\\\\\nsquare-CRT coprime-factor examples: zero / nonzero differences & \\lean{squareCRT\\_clean\\_block\\_can\\_vanish}{SquareCRTCube.lean:459} & witness $n{=}52$ (vanishes); witness $n{=}27$ (nonzero) & totient factorisation\\\\\n\n\\midrule\n\\papertablehead\\multicolumn{4}{c}{\\emph{Statements with cofinal quantifiers: proved constructions and conditional implications}}\\\\\n\\midrule\n\n\\papertablehead\\multicolumn{4}{l}{\\emph{Cofinal statements; status specified in the description}}\\\\\n\\textbf{certificate criterion} (proved implication; unproved premise) & \\lean{irrational\\_totient\\_series\\_of\\_certificate\\_supply}{TotientTailPeriodKiller.lean:394} & $\\forall h{\\ge}1\\,\\forall N_0{\\ge}0\\,\\exists N{\\ge}N_0\\,\\exists L,\\ \\ensuremath{\\mathcal{C}}\\,h\\,N\\,L$ & finite binary sums\\\\\nperiod-multiple criterion (equivalent; unproved condition) & \\lean{irrational\\_totient\\_series\\_of\\_multiple\\_certificate\\_supply}{CarrySurvivorExtinction.lean:502} & $\\forall h_0{>}0\\,\\forall N_0\\,\\exists m{>}0\\,\\exists N{\\ge}N_0\\,\\exists L$ & multiples of periods\\\\\nLCM-diagonal criterion (unproved condition) & \\lean{irrational\\_totient\\_series\\_of\\_lcm\\_diagonal\\_certificate\\_supply}{LcmDiagonalReduction.lean:112} & $\\forall t_0\\,\\exists t{\\ge}t_0\\,\\exists L$ & LCM diagonal\\\\\nLCM-grid certificate criterion (unproved premise) & \\lean{irrational\\_totient\\_series\\_of\\_lcm\\_cone\\_window\\_kill\\_supply}{CertificateKernel.lean:19014} & $\\forall t_0\\,\\exists t{\\ge}t_0\\,\\exists q{>}0\\,\\exists m\\,\\exists L$ & LCM grids\\\\\nLCM nonintegrality criterion (unproved condition) & \\lean{irrational\\_totient\\_series\\_of\\_lcm\\_diagonal\\_nonintegrality\\_supply}{CertificateKernel.lean:19034} & $\\forall t_0\\,\\exists t{\\ge}t_0$, $\\ensuremath{{R}}(2H_t){-}\\ensuremath{{R}}(H_t)\\notin\\mathbb{Z}$ & real tail differences\\\\\nfinite-grid residue criterion (unproved premise) & \\lean{irrational\\_totient\\_series\\_of\\_lcm\\_cone\\_nonflat\\_supply}{CertificateKernel.lean:19140} & for every $t_0$, data $t\\ge t_0,L,Q$ satisfying Theorem~\\ref{catalogue:cert:b10b} & finite LCM grids\\\\\nfirst-harmonic norm criterion (unproved premise) & \\lean{irrational\\_totient\\_series\\_of\\_first\\_harmonic\\_norm\\_gap}{FirstHarmonicPivot.lean:83} & $\\forall h{>}0\\,\\forall X_0\\,\\exists X{\\ge}\\max(X_0,1)\\,\\exists L$, size condition $16(2X{+}h{+}L{+}2){\\le}2^L$ & binary-weighted series\\\\\nfour-tail residue criterion (unproved premise) & \\lean{irrational\\_totient\\_series\\_of\\_primeJumpSharpKill\\_supply}{PrimeJumpWindow.lean:193} & $\\forall t_0\\,\\exists t{\\ge}t_0\\,\\exists p,L{>}0$ & binary-weighted series\\\\\nshort LCM-window criterion (unproved premise) & \\lean{PowerTwoActualLcmShortArithmeticKillSupply}{TotientActualLcmOrbitArithmetic.lean:2107} & $\\forall a_0\\,\\exists a\\,L,\\ a_0{\\le}a\\land L{<}2\\cdot2^a$ & M\\\"obius sums\\\\\nLCM-tail separation criterion (unproved premise) & \\lean{irrational\\_totientSeries\\_of\\_actualLcmOrbitSeparationSupply}{TotientActualLcmOrbitSeparation.lean:305} & cofinal $1/32{+}$error separation at prescribed odd depths & M\\\"obius sums\\\\\nfive endpoint criteria (unproved premises) & \\lean{irrational\\_totientSeries\\_of\\_flexibleActualTerminalDominanceSupply}{TotientActualLcmTopEdgeStaircase.lean:2221} & one of the five quantified conditions in Proposition~\\ref{prop:te-chain} & M\\\"obius sums\\\\\nproved rational counterexample to bound, parity and aperiodicity & \\lean{exists\\_totientParity\\_arbitrarilyManySeparatedCarry\\_rational\\_countermodel}{TotientParityCoboundaryCountermodel.lean:637} & $\\exists c{:}\\mathbb{N}\\to\\mathbb{N}$, $c{\\le}6$, $c{\\le}n$, $c{\\equiv}\\varphi\\!\\!\\pmod2$, arbitrarily-separated blocks & binary-weighted series\\\\\nproved mod-4 prime construction (CRT and Dirichlet) & \\lean{exists\\_prime\\_deltaTotient\\_four\\_mul\\_mod\\_four\\_two}{TotientTailCarryPeriod.lean:384} & $\\forall h{>}0\\,\\forall B\\,\\exists\\text{prime}\\,p{>}B$ & M\\\"obius sums\\\\\nsufficient mod-4 condition (unproved premise) & \\lean{irrational\\_totientSeries\\_of\\_cofinal\\_modFourPulseSurvivorKill}{TotientTailCarryPeriod.lean:1066} & the displayed mod-four residue condition at arbitrarily large scales & M\\\"obius sums\\\\\nproved two-adic prime construction & \\lean{exists\\_prime\\_totient\\_twoAdic\\_pulse\\_divisors}{TotientTwoAdicPulseBlock.lean:54} & $\\forall K{\\ge}2\\,\\forall H{>}K\\,\\forall B\\,\\exists\\text{cofinally many primes}\\,p{>}B$ & divisibility and linear algebra\\\\\nproved implication from eventual integrality & \\lean{eventual\\_integral\\_tailDiff\\_has\\_cofinal\\_twoAdic\\_half\\_pulse}{TotientTwoAdicPulseBlock.lean:327} & $\\forall K{\\ge}2$, given the stated two-adic prime construction and eventual integrality & divisibility and linear algebra\\\\\n\\end{longtable}\n\\end{landscape}\n\\endgroup\n\n\\subsection{Finite extension and cofinal estimates}\n\nExtending a computation supplies more finite witnesses. To reach a\ncofinal statement, an argument must instead produce a witness beyond\nan arbitrary threshold. The examples below specify which of these\nextensions is needed; the largest computed scale is not asserted to\nbe a limit of the method.\n\n\\begin{enumerate}[leftmargin=*]\n\\item \\textbf{Finite certificate tables.} Results such as the\ndiagonal certificates at 28 explicit values of $t$ through $t{=}64$, now a\nstrict subset of the aggregate band $t\\le82$\n(\\lean{certifiedKill\\_diagonal\\_all\\_imported}{DiagonalPincerCertificates.lean:2870}),\nthe certificates for $h\\in[1,8]$ at fixed depth 16\n(\\lean{certifiedKill\\_all\\_small}{TotientTailPeriodKiller.lean:404}), and the\nexclusions for $h\\in[1,16]$ at exponent 14\n(\\lean{totient\\_series\\_ne\\_rat\\_of\\_den\\_dvd\\_pow\\_two\\_mul\\_mersenne\\_upto\\_sixteen}{CertificateKernel.lean:18890})\nare each a finite \\emph{list} of independently verified rows, not a\nsingle argument instantiated at a free parameter. The existing lists do not\nestablish further rows; an extension needs either additional computations\nor an argument for a family of parameters.\n\\item \\textbf{A bounded window inside a uniform theorem.} The rough-integer\nbound of Proposition~\\ref{prop:AR-03-inv} and the positivity theorem of\nProposition~\\ref{prop:SGN-01}\nhold for every $a\\ge8$, subject to their stated bounds on the other\nparameters. In the sign theorem, for example, $J+a+6<2\\cdot2^a$.\nTheir proofs are therefore not merely enumerations of a fixed set of\nexponents. The missing conclusion is a suitable residue inequality at\narbitrarily large scales, not extension of the range of $a$ in the\npositivity theorem. No complexity lower bound follows from these examples.\n\\item \\textbf{Parameter dependence in an approximation estimate.}\nFor reduced approximants $p_K/q_K$ to $S$, the conditions\n$0<|S-p_K/q_K|\\le\\varepsilon_K$ and\n$q_K\\varepsilon_K\\to0$ imply irrationality. A growing denominator\nalone does not control this product. In particular, the exact\nreduced-denominator formula of Proposition~\\ref{catalogue:mob:b7b}\nconcerns a specified finite M\\\"obius sum; it does not estimate the\ncomplementary tail. Likewise, a bound proved with a constant depending\non a fixed auxiliary parameter is not automatically uniform when that\nparameter grows. Both the dependence of the constant and the error\nestimate must be retained in any passage to an unbounded family.\n\\end{enumerate}\n\nThe missing arithmetic inputs include the stated constant-saving\nexponential-sum estimate and an upper-endpoint residue inequality.\nFor the positivity theorem, the unresolved step is exclusion of that\nresidue interval, not positivity at more exponents.\n\n\\section{Relations between the representations}\n\nBinary tails, M\\\"obius sums, rational approximations and exponential\nsums express related information about $S$. We give the identities\nthat connect them and then check which assumptions survive each\nchange. This is necessary before transferring a counterexample to\nanother representation, as the finite-state decoding and polynomial\ndenominator examples illustrate.\n\n\\begin{description}[leftmargin=*,style=nextline]\n\n\\item[Binary-weighted series and tails.]\nThe coefficients in $S=\\sum_{n\\ge1}\\varphi(n)2^{-n}$ are unbounded;\nthis expression is not the base-two digit expansion of $S$.\nThe tail $R_N$, finite difference numerator $D(h,N,L)$ and test\n$\\mathcal C(h,N,L)$ retain this coefficient information. The identity\n\\[\n S=\\frac12+\\sum_{d\\ge1}\\frac{\\mu(d)}{(2^d-1)^2}\n\\]\n(\\lean{tsum\\_totient\\_half\\_pow\\_eq\\_half\\_add\\_moebius\\_sq}{MersenneLambertLadder.lean:665})\nand its partial-sum version\n(\\lean{mobius\\_square\\_series\\_eq\\_partial\\_add\\_tail}{SquaredMersenneDiagonalEnclosure.lean:94})\nexpress the same value by M\\\"obius sums. An equality of values does not\nby itself transfer a claim about the original coefficient subsequences.\n\n\\item[Lambert sums and finite numerator polynomials.]\nThe Lambert identities for $\\mu$, $1$ and $\\varphi$ give the series\nrelations in Section~\\ref{sec:series}. For a finite divisor sum, the\npolynomial evaluation\n(\\lean{mobiusNumeratorPolynomial\\_eval\\_two}{RepunitMobiusNumerator.lean:445})\nand the definition\n(\\lean{mobiusNumerator}{RadicalMobiusShadow.lean:101})\nput its numerator into integer polynomial arithmetic. This evaluates a\nfinite contribution, not the complementary infinite tail.\n\n\\item[Coprimality probability.]\nLet $X,Y$ be independent with $\\Pr(X=n)=\\Pr(Y=n)=2^{-n}$ for $n\\ge1$.\nThen $S-\\tfrac12=\\Pr(\\gcd(X,Y)=1)$. The divisibility probabilities are\n$\\Pr(d\\mid X,\\ d\\mid Y)=(2^d-1)^{-2}$\n(\\lean{tsum\\_pos\\_pair\\_both\\_dvd\\_half\\_eq\\_inv\\_mersenne\\_sq}{GcdMomentCalculus.lean:266}).\nM\\\"obius inversion converts these to the coprimality probability.\nThe decomposition by the greatest common divisor also gives the weighted\ncoprime-pair identity\n(\\lean{tsum\\_pos\\_coprime\\_inv\\_mersenne\\_eq\\_one}{GcdMomentCalculus.lean:349}).\nCounting coprime pairs by their sum gives the totient expression directly\n(\\lean{tsum\\_coprime\\_pair\\_pow\\_eq\\_tsum\\_totient\\_mul\\_pow}{GeometricCoprimality.lean:120}).\n\n\\item[Cyclotomic factors of finite denominators.]\nThe coefficient formula for $P_r$\n(\\lean{mobiusNumeratorPolynomial\\_coeff}{RepunitMobiusNumerator.lean:217})\nand the cyclotomic divisibility statement\n(\\lean{cyclotomicValue\\_dvd\\_baseMobiusShadow\\_den}{CyclotomicProjectionOfShadow.lean:389})\nidentify factors in the reduced denominators of the specified finite sums.\nThe product-divisibility theorem\n(\\lean{upperHalfChannel\\_product\\_dvd\\_den\\_of\\_scale\\_primeFactors\\_le}{MersenneShadowCyclotomicNoncollapse.lean:809})\nis used in the exact denominator formula\n(\\lean{lcmHeight\\_scaledMobiusShadow\\_den\\_exact}{MersenneShadowDenominatorGrowth.lean:147}).\nGrowing finite denominators alone do not exclude cancellation with the\nremaining tail.\n\n\\item[Numerator divisibility and denominator bounds.]\nWrite $x=a/b$ in lowest terms, with $b>0$. For $c\\in\\Z$ and a\npositive divisor $H$ of $b$, if the reduced denominator of $cx$ divides\n$H$, then $b/H\\mid c$\n(\\lean{scalarLocalization\\_complement\\_dvd}{AdelicHeightObstruction.lean:23}).\nIn particular, $|c|\\ge b/H$ when $c\\ne0$. A related bound is\n\\[\n 0<\\frac ab<\\frac2{2^n-1},\\quad 2^r\\mid a,\\quad n\\ge1\n \\quad\\Longrightarrow\\quad 2^r(2^n-1)<2b\n\\]\nfor integers $r\\ge0$\n(\\lean{positiveRat\\_mersenne\\_height}{AdelicHeightObstruction.lean:103}).\nHere positivity gives $a\\ge2^r$; the size inequality then gives the\nconclusion. These are arithmetic bounds on a reduced fraction, not\nirrationality criteria without an additional approximation estimate.\n\n\\item[Farey intervals.]\nThe denominator bound\n(\\lean{farey\\_gap}{GapFareyBound.lean:51})\napplied to the explicit window $(N,K)=(1,240)$ gives\n(\\lean{gap\\_check\\_window\\_1\\_240\\_le\\_79639646646701375323355774875831053}{GapFareyBound.lean:176})\nand hence $S\\ne p/q$ for integers $p$ and $1\\le q\\le Q_0$.\nThe finite input is a totient numerator modulo $2^{240}$.\nIts exact first failing denominator is $Q_0+1$, the denominator of the\nrelevant Farey mediant\n(\\lean{gap\\_check\\_window\\_1\\_240\\_first\\_failure}{GapFareyBound.lean:225}).\nThis last assertion concerns this fixed window.\n\n\\item[Differences at least-common-multiple shifts.]\nFor $H_t=\\operatorname{lcm}(1,\\ldots,t)$\n(\\lean{periodLcm}{CarrySurvivorExtinction.lean:515}),\na jump in $H_t$ can occur only at a prime power\n(\\lean{eq\\_prime\\_pow\\_of\\_not\\_dvd\\_periodLcm}{LcmDiagonalReduction.lean:137}).\nThe identities\n(\\lean{lcmRayArithmeticLetter\\_eq\\_deltaTotient}{TotientActualLcmOrbitArithmetic.lean:298})\nand\n(\\lean{diagonalWindowIncrement\\_eq\\_lcmRayArithmeticLetter}{TotientActualLcmOrbitArithmetic.lean:310})\nidentify the coefficient difference as\n$\\delta_t(j)=\\varphi(2H_t+j)-\\varphi(H_t+j)$ and give\n$D(H_t,H_t,L)=\\sum_{j=1}^L\\delta_t(j)2^{L-j}$.\nThus these arithmetic formulas concern particular instances of the same\nfinite residue test; additional window restrictions remain in force.\n\n\\item[Exponential sums of the window residues.]\nFor integers $h,X\\ge1$, the relevant block statistic is\n\\[\n \\sum_{N=X}^{2X-1}\\cos\\!\\left(2\\pi\\frac{D(h,N,L)}{2^L}\\right).\n\\]\nA bound by $9X/10$, together with $L\\ge h$ and\n$16(2X+h+L+2)\\le2^L$, yields a certificate in the block\n(\\lean{exists\\_certifiedKill\\_of\\_first\\_harmonic\\_gap}{FirstHarmonicGap.lean:128}).\nThe linked irrationality implication uses the stronger complex norm\nbound $21X/25$, for every $h$ on arbitrarily late blocks satisfying\nthe same size inequality\n(\\lean{irrational\\_totient\\_series\\_of\\_first\\_harmonic\\_norm\\_gap}{FirstHarmonicPivot.lean:83}).\nThe finite implication uses $\\cos(\\pi/8)>9/10$; the required cancellation\nbound is still unproved. The norm bound on the complex exponential sum\nis sufficient for the displayed real-part bound, but is stronger.\n\n\\end{description}\n\n\\subsection{Binary tails and M\\\"obius sums}\n\nErd\\H{o}s's argument for $\\sum_n1/(2^n-1)$ chooses congruences so\nthat divisor counts along a block are divisible by increasing powers of\nthe base. A tail bound then forces a long run of zero digits\n\\cite[pp.~63--66]{erdos1948lambert}. It uses no finite automaton.\n\nA different proposal would recover each later block from a fixed finite\nsummary of the preceding history. The counterexample of\nProposition~\\ref{prop:b5}(i)\n(\\lean{balancedPulse\\_no\\_autonomous\\_decoder}{GenericTailOrbitRigidity.lean:247})\nshows why that proposal fails for the family constructed there. For\n$m\\ge2$, its members have the same earlier history but different\nsuccessors. A state that identifies all those histories cannot recover\nevery successor. More generally, an exact label-and-decoder pair for\nthat family needs at least $\\lfloor(m+1)/2\\rfloor+1$ distinct labels.\nThis is a lower bound on the number of labels, not a linear lower bound\non the number of bits, and it is not a state-complexity theorem for the\nactual totient sequence.\n\nThe implication from the quantified certificate condition\n(\\lean{irrational\\_totient\\_series\\_of\\_certificate\\_supply}{TotientTailPeriodKiller.lean:394})\nis a separate theorem. The counterexample above neither proves its\nhypothesis nor makes it the only possible route to irrationality.\nLikewise, Proposition~\\ref{prop:nonzero-minor-survives-inverse-phase}\nshows that inverse-phase column weights can preserve a determinant, and\nTheorem~\\ref{catalogue:cert:d4} gives the exact growing ranks of totient\nsections. These are different\nstatements with different hypotheses; none is a general prohibition on\nother representations of $S$.\n\nThe M\\\"obius representation instead allows the squared-Lambert identity\nof Proposition~\\ref{prop:lambertengine}\nto be applied to $\\mu$, since $|\\mu(d)|\\le d$ for $d\\ge1$. The\npolynomial identities of\nPropositions~\\ref{catalogue:mob:b1}--\\ref{catalogue:mob:b8a}\noperate at each finite radical without an autonomous decoder. The exact\nreduced denominator and its lower bound are then given by\nPropositions~\\ref{catalogue:mob:b7b} and~\\ref{catalogue:mob:b7a}.\nThose formulas concern a finite rational contribution. They do not\nexclude cancellation by the complementary tail and hence do not settle\nirrationality. The representations are equal expressions for $S$;\nwhat differs is the information supplied by the particular arguments.\n% ; - part a249_p3 ; -\n\\section{Index of unproved conditions}\n\nEvery condition in this index is accompanied by the implication that\nwould turn it into irrationality of $S=\\sum_{n\\ge0}\\varphi(n)2^{-n}$.\nThose implications are proved. The arithmetic hypotheses still have\nto be established with the displayed quantifiers.\n\n\\subsection{Scope of the index}\n\nThe index compares particular quantified hypotheses and the results\nthat use them. An unsuccessful implication between two entries\ntherefore says nothing by itself about every argument using the same\nmethod.\n\nEach entry states its quantifiers in full. A cofinal assertion, with a\nquantifier of the form $\\forall a_0\\ \\exists a\\ge a_0$, may itself be\nproved or may remain an unproved hypothesis; the text of each entry says\nwhich. A cofinal assertion about a different quantity, or with a\ndifferent order of quantifiers, need not supply the irrationality\ncondition.\n\nThe quantified residue condition for \\#249, stated in Definition~\\ref{defn:sep} and proved sufficient\n(\\lean{irrational_totient_series_of_certificate_supply}{TotientTailPeriodKiller.lean:394}), is the\nfollowing statement:\n\\[\n  \\forall h \\ge 1,\\ \\forall N_0,\\ \\exists N \\ge N_0,\\ \\exists L,\\ \\mathcal{C}\\ h\\ N\\ L .\n\\]\nThe following catalogue compares fourteen proposed approaches to this\ncondition. It starts with the exponential-sum estimate, then gives thirteen\nfurther approaches with their individual hypotheses and source references.\n\n\\subsection{The exponential-sum criterion}\n\\label{ssec:headline}\n\nWe express the truncated tail difference as a phase and ask for\ncancellation when that phase is summed over a block. The following\nimplications explain how a saving gives a residue certificate and,\nwith the required quantifiers, Definition~\\ref{defn:sep}.\n\n\\begin{defn}[The truncated difference]\nFor $h,N,L\\in\\mathbb N$,\n\\[\n  D(h,N,L) \\;=\\; \\sum_{j=0}^{L-1} \\bigl(\\varphi(N+h+1+j) - \\varphi(N+1+j)\\bigr)\\cdot 2^{\\,L-1-j}\n  \\ \\in \\mathbb Z .\n\\]\nThis is the depth-$L$ truncation of $2^L\\cdot(R_{N+h}-R_N)$, where\n$R_N = \\sum_{j\\ge 1}\\varphi(N+j)/2^j$ is the local totient tail\n(\\lean{totientTail}{TotientTailPeriodKiller.lean:57}).\n\\lean{windowDiscrepancy}{TotientTailPeriodKiller.lean:66}\n\\end{defn}\n\n\\begin{defn}[The central-residue condition]\\label{defn:central-residue-condition}\n\\[\n  \\mathcal{C}\\ h\\ N\\ L \\;:\\Leftrightarrow\\;\n  (N{+}h{+}L{+}2) < D(h,N,L)\\bmod 2^L < 2^L - (N{+}h{+}L{+}2).\n\\]\nThat is: the residue of $D(h,N,L)$ modulo $2^L$ avoids the radius-$(N{+}h{+}L{+}2)$ neighbourhood\nof $0$. It forces the size condition $2(N{+}h{+}L{+}2) < 2^L$\n(\\lean{certifiedKill_depth_floor}{TotientTailPeriodKiller.lean:79}).\n\\lean{certifiedKill}{TotientTailPeriodKiller.lean:72}\n\\end{defn}\n\n\\begin{defn}[The complex phase and its real part]\nPut\n\\[\n  \\operatorname{Re}E(h,N,L) = \\cos\\!\\Bigl(2\\pi\\,\\tfrac{D(h,N,L)\\bmod 2^L}{2^L}\\Bigr),\\qquad\n  E(h,N,L) = \\exp\\!\\Bigl(i\\,2\\pi\\,\\tfrac{D(h,N,L)\\bmod 2^L}{2^L}\\Bigr).\n\\]\nThe real part of $E$ is the displayed cosine\n(\\lean{windowFirstExp_re}{FirstHarmonicPivot.lean:42}), and\n$|E(h,N,L)| = 1$ (\\lean{norm_windowFirstExp}{FirstHarmonicPivot.lean:48}).\nThus $E$ is the first additive character of the endpoint discrepancy\nmodulo $2^L$, a standard exponential-sum object.\n\\lean{windowFirstCos}{FirstHarmonicGap.lean:27}\n\\lean{windowFirstExp}{FirstHarmonicPivot.lean:38}\n\\end{defn}\n\n\\begin{thm}[A real-part bound gives a certificate]\n\\label{thm:hgap-real}\nFor all $h,X,L$ with $0<X$ and the size condition $16(2X{+}h{+}L{+}2)\\le 2^L$, if\n\\[\n  \\sum_{N=X}^{2X-1} \\operatorname{Re}E(h,N,L) \\;\\le\\; \\tfrac{9}{10}\\,X ,\n\\]\nthen $\\exists N\\in[X,2X)$ with $\\mathcal{C}\\ h\\ N\\ L$.\n\\end{thm}\n\n\\begin{thm}[The same implication for a nonempty subset]\n\\label{thm:hgap-subset}\nFor any nonempty finite $T\\subseteq\\mathbb N$ with $T\\subset[0,2X)$ and the same size condition, if\n\\[\n  \\sum_{N\\in T} \\operatorname{Re}E(h,N,L) \\;\\le\\; \\tfrac{9}{10}\\,|T| ,\n\\]\nthen $\\exists N\\in T$ with $\\mathcal{C}\\ h\\ N\\ L$. This generalises\nTheorem~\\ref{thm:hgap-real}: $T$ can be \\emph{any} explicitly chosen nonempty finite subset of the\ndyadic block, not the whole block, and the proof (an averaging pigeonhole) never uses\nthat $T$ has positive density or comes from a partition.\n\\end{thm}\n\n\\begin{thm}[A norm bound gives the real-part criterion]\n\\label{thm:hgap-norm}\nThe complex norm bound implies the real-part bound\n(\\m{|z|\\ge\\mathrm{Re}(z)}, and $21/25 < 9/10$ absorbs the slack), so it composes through\nTheorem~\\ref{thm:hgap-real} to the same certificate. Define\n\\[\n  \\text{block norm condition} :\\Leftrightarrow\\;\n  \\forall h{>}0\\ \\forall X_0\\ \\exists X, L,\\ \\max(X_0,1)\\le X\\ \\wedge\\ 16(2X{+}h{+}L{+}2)\\le 2^L\\ \\wedge\n\\]\n\\[\n  \\Bigl\\|\\ \\sum_{N=X}^{2X-1}E(h,N,L)\\ \\Bigr\\| \\;\\le\\; \\tfrac{21}{25}\\,X .\n\\]\nThis sufficient condition is unproved. Then\n\\[\n  \\text{block norm condition} \\;\\Longrightarrow\\; \\mathrm{Irrational}\\Bigl(\\sum_{n\\ge 0}\n  \\tfrac{\\varphi(n)}{2^n}\\Bigr),\n\\]\nproved in full, with no further gap, by chaining\nTheorem~\\ref{thm:hgap-norm}$\\to$Theorem~\\ref{thm:hgap-real}$\\to$\nTheorem~\\ref{catalogue:cert:a10}.\n\\end{thm}\n\n\\begin{obs}[What remains unproved]\n\\label{obs:no-instance}\nThe implications above do not establish the cofinal block estimate.\nThree further Lean statements retain a separation or gap hypothesis: one\nuses the two-element set $\\{N,M\\}$\n(\\lean{AdjacentPhaseSeparation.lean}{AdjacentPhaseSeparation.lean:267}),\nand the other two use a supplied finite set $T$ and the singleton\n$\\{N\\}$\n(\\lean{PivotAntiReconstruction.lean}{PivotAntiReconstruction.lean:882},\n\\lean{PivotAntiReconstruction.lean}{PivotAntiReconstruction.lean:1572}).\nThey prove what follows from the relevant estimate; they do not provide\nthat estimate at arbitrarily large scales. The finite numerical\nobservations in Table~\\ref{tab:blocks} are a different kind of evidence\nand do not discharge this quantified hypothesis.\n\\end{obs}\n\n\\begin{rem}[What the reduction does and does not prove]\nThe required savings are explicit: a real-part bound of $9X/10$, or\nthe stronger norm bound of $21X/25$, on blocks at arbitrarily large\nscales for every positive shift $h$. An unspecified nonzero saving is\nnot the stated hypothesis. A bound $o(X)$ would suffice, but it is more\nthan the implication needs.\n\nThe inequality $16(2X+h+L+2)\\le2^L$ can be met by increasing $L$ for\nfixed $X,h$. Doing so changes the exponential sum, and hence does not\npreserve an estimate proved at another depth. The arithmetic estimate\nmust hold at a depth satisfying this inequality.\n\nThe subset theorem permits any \\emph{nonempty} finite sample inside\n$[0,2X)$; no density assumption is required. A prime-defined sample is\none possible choice. The four-term decomposition of\n\\S\\ref{sub:four-sums} isolates one prime\nfactor and the remaining totient-dependent phase, but does not prove\nthe required cancellation between them.\n\\end{rem}\n\n\\subsection{Proved results and the additional assumptions they need}\n\nBesides the exponential-sum condition, the sufficient hypotheses\ninvolve tail signs, endpoint residues, rational approximation, short\nwindows, Farey bounds, divisibility, denominator growth and carry rank.\nFor each, we state the missing estimate used by the proposed\napplication. These are specific possibilities rather than a\nclassification of methods for Problem~\\#249.\n\n\\subsubsection{Positivity and the remaining residue inequality}\n\n\\begin{defn}\nThe \\emph{actual LCM tail orbit} at exponent $a$ is\n$\\Omega_a = R_{2H}-R_H$, where $H = {H}(2^a)$\n(\\lean{actualLcmTailOrbit_pos}{TotientActualLcmOrbitSign.lean:144}).\n\\end{defn}\n\n\\begin{thm}\\label{thm:positivity-remaining-residue-inequality}\nFor every $a\\ge 8$ and every $J$ with $J+(a{+}6) < 2\\cdot 2^a$,\n\\[\n  0 \\;<\\; {R}(2H{+}J) - {R}(H{+}J), \\qquad H={H}(2^a).\n\\]\nHence $0 < \\Omega_a$ for every $a\\ge 8$. The proof is genuinely uniform in\n$a$: it rests on two facts proved for all $a\\ge 8$ and $0<j<2^{a+1}$, namely $\\delta_{2^a}(j)>0$ and $H<8\\cdot 2^a\\,\\delta_{2^a}(j)$, by a two-case structural split (divisor letter\nvs.\\ foreign prime power), with absolute constants $4,8,32$ and no lookup table.\n\\end{thm}\n\n\\begin{rem}[Which endpoint interval must be excluded]\nUnder the hypotheses of Proposition~\\ref{prop:SGN-03}, integrality would\nforce the residue to be $2^K-e$, where\n$e=R_{2H+J+K}-R_{H+J+K}$ is a positive integer smaller than\n$2H+J+K+2$. Thus the residue lies near the upper endpoint, not near zero\nin its least nonnegative representation. The representative $-e$ is\nnegative; the carry $e$ is positive.\n\nThe one-sided test below excludes precisely that upper interval. This\nuses the bound $J+K+(a+6)<2\\cdot2^a$, as well as the modulus bound.\nIt does not follow from positivity that the residue already satisfies\nhalf of a nonintegrality certificate without those assumptions.\n\\end{rem}\n\n\\subsubsection{An exact endpoint identity}\n\nUnder the bounds below, positivity places an integral tail difference\nat the upper residue endpoint. A one-sided inequality would exclude it.\nThat additional inequality is a sufficient condition, and is not claimed\nto be strictly weaker than irrationality.\n\n\\begin{thm}[The proved implications]\\label{thm:proved-implications}\nFor every $a\\ge 8$ and every $J,K,m$ inside the sign corridor\n($J{+}K{+}(a{+}6) < 2\\cdot 2^a$), a one-sided residue gap at precision $m\\le K$ (size condition $2H{+}J{+}K{+}2 < 2^m$ and $D(H,H+J,K)\\bmod 2^m \\le 2^m - (2H{+}J{+}K{+}2)$) already\nforces ${R}(2H{+}J) - {R}(H{+}J)\\notin\\mathbb Z$. No lower margin\nat all is demanded. The proof chain is complete: the theorem holds for every $a\\ge 8$, and\n\\emph{\\m{\\text{cofinal upper-endpoint condition}} $\\Rightarrow$ Irrational $S$} is proved.\n\\end{thm}\n\n\\begin{defn}[The unproved quantified condition]\nThe \\emph{upper-endpoint condition} is\n\\[\n  \\forall a_0\\ \\exists a,K,m,\\ a_0\\le a\\ \\wedge\\ 8\\le a\\ \\wedge\\ K{+}(a{+}6)<2\\cdot 2^a\\ \\wedge\\\n  \\mathcal G_a(0,K,m) .\n\\]\nNo unconditional statement anywhere in the record locates the residue of the diagonal word\n$D(H,H,K)$ (where $H={H}(2^a)$) below the top strip at even one large $a$.\nThe condition reduces further to the last $m$ differences\n$\\delta_{2^a}(j)$ in the sum $D(H,H,K)$\n(\\lean{actualLcmTopEdgeResidueGap_iff_terminal}{TotientActualLcmTopEdgeStaircase.lean:909}).\n\\lean{PowerTwoActualLcmTopEdgeResidueGapSupply}{TotientActualLcmTopEdgeStaircase.lean:1325}\n\\end{defn}\n\n\\begin{prop}[Five sufficient conditions]\n\\label{prop:te-chain}\nEach of the five conditions below suffices for irrationality; none of them is proved. The first\nfour imply the upper-endpoint condition;\nthe fifth gives nonintegrality directly by the endpoint identity. These\nare not asserted to form a linear hierarchy. Here $H=H(2^a)$, and every\ncondition quantifies over arbitrarily large exponents $a$.\n\\begin{enumerate}[leftmargin=*]\n  \\item \\emph{Adjacent-suffix band}:\n    \\[\n      \\forall a_0\\ \\exists a,m,\\ a_0\\le a \\wedge 8\\le a \\wedge m{+}1{+}(a{+}6)<2\\cdot 2^a \\wedge\n      2H{+}m{+}3 < 2^m \\wedge\n    \\]\n    \\[\n      2H{+}m{+}2 \\le \\text{suffix residue}(2^a)\\,0\\,m \\le 2^m - (2H{+}m{+}2)\n    \\]\n    (a two-sided band on the adjacent-suffix residue directly, one candidate depth $m$).\n  \\item \\emph{Odd-prefix interval at the prescribed depth}: at the prescribed odd depth $2q{+}1$, a two-sided band of half-width\n    $H{+}q{+}2$ on the half-word residue modulo $4^q$. The depth is fixed\n    by $q=q_a$; it cannot be chosen independently of $a$.\n  \\item \\emph{Centred-residue bound at the prescribed depth}, \\emph{proved equivalent} to the previous one:\n    \\[\n      \\forall a_0\\ \\exists a,q,\\ \\max(14,a_0)\\le a \\wedge q=q_a\n      \\wedge H{+}q{+}2 \\le |u_{a,q}| .\n    \\]\n  \\item \\emph{Centred-residue bound at a variable depth}: the same magnitude bound with the depth restriction relaxed from\n    the prescribed value $q=q_a$ to any odd $2q{+}1$ satisfying the half-cell fit\n    $2(H{+}q{+}2)\\le 4^q$ and the positivity range $2q{+}2{+}(a{+}6)<2\\cdot 2^a$.\n  \\item \\emph{Terminal dominance}: under the same bounds as item 4,\n    $\\delta_{2^a}(2q+2)\\le2u_{a,q}$. This is a different one-sided\n    comparison, treated immediately below.\n\\end{enumerate}\nFor the first four conditions, the relevant implications into the\nupper-endpoint condition are as follows:\nadjacent-suffix band $\\to$ upper-endpoint condition; odd-prefix interval at the prescribed depth\n$\\to$ adjacent-suffix band; centred-residue bound at the prescribed depth\n$\\Leftrightarrow$ odd-prefix interval at the prescribed depth;\ncentred-residue bound at a variable depth $\\to$ adjacent-suffix band;\ncentred-residue bound at the prescribed depth $\\to$ centred-residue bound at a variable depth.\n\\end{prop}\n\n\\subsubsection{A bound on the last totient difference}\n\nWe compare the last totient difference with a centred residue. To use\nthe comparison for irrationality, it would have to hold for arbitrarily\nlarge $a$, with $q$ in the prescribed range.\n\n\\begin{defn}\nThe \\emph{terminal dominance condition}, which is unproved, is the\nfollowing. Writing $H={H}(2^a)$,\n\\[\n  \\forall a_0\\ \\exists a,q,\\ a_0\\le a \\wedge 8\\le a \\wedge 2q{+}2{+}(a{+}6)<2\\cdot 2^a \\wedge\n  2(H{+}q{+}2)\\le 4^q \\wedge\n\\]\n\\[\n  \\delta_{2^a}(2q{+}2) \\;\\le\\; 2\\cdot u_{a,q} .\n\\]\n\\lean{PowerTwoFlexibleActualTerminalDominanceSupply}{TotientActualLcmTopEdgeStaircase.lean:1908}\n\\end{defn}\n\n\\begin{thm}[The proved implication]\\label{thm:proved-implication}\nThe dominance hypothesis at a single odd rank already excludes\nintegrality of \\m{\\Omega_a}, and the terminal dominance condition implies\n\\m{S\\notin\\mathbb Q}.\n\\end{thm}\n\n\\begin{obs}[How the sign and size bounds enter the contradiction]\\label{obs:staircase-tension}\nUse the hypotheses and notation of Proposition~\\ref{prop:TE-06}. Under\nthe assumption $z=\\Omega_a\\in\\mathbb Z$, the exact identity is\n\\[\n 2u_{a,q}=\\delta_{2^a}(2q+2)-c_{H,H,z}(2q+1),\n \\qquad 0<c_{H,H,z}(2q+1)<2H+2q+3.\n\\]\nTerminal dominance $\\delta_{2^a}(2q+2)\\le2u_{a,q}$ would contradict the\npositive lower bound on the carry. The other sufficient inequality,\n$2u_{a,q}\\le\\delta_{2^a}(2q+2)-(2H+2q+3)$, would contradict its upper\nbound. Both are valid ways to contradict integrality.\n\nThese bounds do not eliminate either approach: they hold under the\nintegrality assumption that the approach is intended to refute. They\nneither prove nor disprove an unconditional estimate for $u_{a,q}$.\nFinite examples lying between the two boundaries do not establish an\nasymptotic exclusion of either inequality. The identities and\nconditional bounds are proved in Lean\n(\\lean{two_mul_actualOddHalfCenteredLift_eq_terminal_sub_trueCarry}{TotientActualLcmTopEdgeStaircase.lean:1678},\n\\lean{actualLcm_trueEndpointSurvivor_neg}{TotientActualLcmOrbitSign.lean:172},\n\\lean{actualLcmTailOrbit_notMem_int_of_actualTerminalCarryCorridorEscape}{TotientActualLcmTopEdgeStaircase.lean:1824}).\n\\end{obs}\n\n\\begin{prop}[A sufficient extension]\\label{prop:sufficient-extension}\nEither of the following suffices: (i) the lower-escape branch cofinally, as displayed just above; or (ii) the two-sided\nmagnitude form\n(item 4 of Proposition~\\ref{prop:te-chain}), which asks only $H{+}q{+}2 \\le\n|u_{a,q}|$ and which the Lean source proves\nimplies corridor escape via a clean sign split (positive branch escapes above the terminal letter,\nnegative branch escapes below the directed bound), without prescribing the sign of the centred representative. Neither\nbranch is supplied by the conditional sign identity itself.\n\\end{prop}\n\n\\subsubsection{Separation of a rational approximation}\n\n\\begin{thm}\\label{thm:separation-rational-approximation}\nUnconditionally, for every $a$ and $q$,\n\\[\n  \\bigl|\\, \\Omega_a - \\rho_{a,q} \\,\\bigr|\n  \\;<\\; \\frac{4H + 2(2q{+}1) + 4}{2^{2q+2}} ,\n\\]\nwhere $\\rho_{a,q}$ is an explicit finite computable rational block. For fixed $a$, the error tends to zero as $q$ grows. In the quantified\ncondition below, however, the depth is prescribed, so increasing $q$\nis not a free way to satisfy the separation hypothesis.\n\\end{thm}\n\n\\begin{rem}[Scope of the implication]\nA cofinal $1/32$-separation of the raw approximants at the prescribed\nindices $q=q_a$ suffices for \\#249. Allowing arbitrary depths would be a\ndifferent hypothesis. The cited short-window theorem gives\nnonintegrality of $\\Omega_a$ at $a=4,6$\n(Proposition~\\ref{prop:SK-01-inv}).\nIt neither asserts the $1/32$ margin at those exponents nor says that\nthese are the only successful exponents. Indeed, the diagonal table also\ngives short-window certificates $(a,L)=(2,7)$ and $(3,15)$, since\n$H_4=12$, $H_8=840$, and $7<8$, $15<16$. All these are finite\nnonintegrality examples, not the cofinal approximation-separation condition.\n\\end{rem}\n\n\\begin{defn}[The additional hypothesis]\nThe \\emph{cofinal approximation-separation condition}, which is unproved, is\n\\[\n \\begin{gathered}\n  \\forall a_0\\ \\exists a\\ge\\max(2,a_0)\\ \\exists q,\\quad q=q_a\\ \\wedge\\\\\n  \\forall z\\in\\mathbb Z,\\quad \\tfrac1{32}+\\varepsilon_{a,q}\n       \\le |\\Omega_a-z| .\n \\end{gathered}\n\\]\nThis hypothesis is about the real quantity $\\Omega_a$, not just a finite\nrational calculation. It implies\n$|\\rho_{a,q_a}-z|>1/32$ for every integer $z$, by the displayed error\nbound and the triangle inequality. A cofinal $1/32$-separation condition\non the finite approximants is another sufficient input, through the\nresidue-band implication. These sufficient conditions are not asserted\nto be equivalent.\n\\end{defn}\n\n\\subsubsection{The short-window examples through exponent 6}\n\n\\begin{thm}\\label{thm:short-window-examples-through-exponent}\n\\[\n  \\forall a_0\\le 6,\\ \\exists a,L,\\ a_0\\le a \\wedge L < 2\\cdot 2^a \\wedge\n  \\mathcal C(H_{2^a},H_{2^a},L)\n\\]\nThe single witness $(a,L)=(6,93)$ works for every threshold $a_0\\le6$.\nIts certificate is verified by exact integer arithmetic on the finite\ntotient windows.\nThis proves the displayed bounded statement. It does not construct\nwitnesses for unbounded thresholds; further finite checks would extend\nthe verified range, not establish the universally quantified condition below.\n\\end{thm}\n\n\\begin{defn}[The additional hypothesis]\nThe \\emph{cofinal short-window condition}, which is unproved, is\n\\[\n  \\forall a_0\\ \\exists a,L,\\ a_0\\le a \\wedge L < 2\\cdot 2^a \\wedge\n  \\mathcal C(H_{2^a},H_{2^a},L) .\n\\]\nThe depth restriction gives a useful description of the offsets:\nif $1\\le j<2\\cdot2^a$ and $j\\nmid H_{2^a}$, then $j$ is a prime\npower greater than $2^a$. Indeed, some prime-power divisor $p^r$ of\n$j$ exceeds $2^a$; if $j/p^r\\ge2$, then $j>2\\cdot2^a$, a contradiction.\nThis describes the possible offsets, not the size of their weighted sum.\n\nThe condition restricts the depth in the ordinary diagonal certificate\ncriterion. It is not established by pointwise completeness, which permits\narbitrarily large depths. Nor does a successful short-window certificate\nby itself yield the fixed $1/32$ separation required by the preceding\napproximation condition. These are distinct sufficient hypotheses;\nno implication from this condition to that fixed-margin condition is\nproved here.\n\\end{defn}\n\n\\subsubsection{The diagonal certificate table}\n\n\\begin{thm}\\label{thm:diagonal-certificate-table}\n\\[\n  \\forall t\\in\\{1,2,3,4,5,7,8,9,11,13,16,17\\},\\ \\exists L,\\\n  \\mathcal{C}\\ ({H}\\ t)\\ ({H}\\ t)\\ L\n\\]\nat depths $\\{6,5,7,7,9,14,15,14,21,22,23,26\\}$ respectively, extended by separate\ncomputations to 28 historical values through $t=64$.\nThe later aggregate theorem closes every scale $t\\le82$ with no holes.\nEach witness is checked by exact integer arithmetic on two finite totient\nwindows. For example, $H_{17}=12252240$ and $L=26$ use\n$\\varphi(H_{17}+i)$ and $\\varphi(2H_{17}+i)$ for $1\\le i\\le26$,\nnot every totient between the endpoints $12252241$ and $24504506$.\nThese computations verify the listed cases; they do not give a formula\nproducing a certificate for every $t$.\n\\end{thm}\n\n\\begin{rem}[The observed depths are finite data]\nThe necessary depth inequality of Lemma~\\ref{catalogue:cert:a5} forces\n$2(2H_t{+}L{+}2)<2^L$, that is, $L \\gtrsim \\log_2(4H_t)$ at any certified\ndepth. Comparing that bound with the recorded depths shows that every\nrecorded certificate depth lies within a small additive constant of that\nminimum, at every tested scale. This is an observation from finite\ncomputation, and no general theorem asserts it.\n\\end{rem}\n\n\\begin{prop}[A sufficient extension]\\label{prop:sufficient-extension-2}\n\\[\n  \\exists C\\ \\forall t_0\\ \\exists t\\ge t_0\\ \\exists L\\le \\log_2(4\\cdot{H}\\ t)+C,\\quad\n  \\mathcal{C}\\ ({H}\\ t)\\ ({H}\\ t)\\ L .\n\\]\nThis asks for a certificate within a fixed additive constant of the\nnecessary logarithmic depth bound. The uniform constant is an additional\nrequirement, not a consequence of the finite data.\n\\end{prop}\n\n\\subsubsection{One common certificate for sixteen shifts}\n\n\\begin{thm}\\label{thm:one-common-certificate-sixteen-shifts}\n\\[\n  \\mathcal C(h,14,9)\\qquad(h\\in\\{1,\\ldots,16\\}).\n\\]\nThe basepoint $14$ and depth $9$ are common to all sixteen shifts.\nConsequently, $S\\ne r$ whenever $r\\in\\mathbb Q$ and\n$\\operatorname{den}(r)\\mid2^{14}(2^h-1)$ for some integer $h$ with\n$1\\le h\\le16$. The exact computation uses two nine-term windows for\neach shift and reduces their discrepancy modulo $2^9$. The earlier\neight-shift computation uses basepoint $12$ and depth $16$.\n\\end{thm}\n\n\\begin{rem}[Finite simultaneity and the unbounded condition]\nThe displayed computation concerns only $N=14$, $L=9$ and\n$1\\le h\\le16$. It does not provide certificates beyond every basepoint\nthreshold. Simultaneous certification of finitely many shifts at a\nsufficiently large, unrestricted depth is not an additional obstacle:\nunder irrationality it follows from the same truncation estimate, as\nshown next. The argument below does not prescribe the common depth in terms of\nthe basepoint and number of shifts.\n\\end{rem}\n\n\\begin{prop}[Simultaneous certificates with unrestricted depth]\\label{prop:simultaneous-certificates-unrestricted-depth}\nThe existence of a function $f:\\N\\to\\N$ with $f(N)\\to\\infty$ such that\n\\[\n \\forall N_0\\ \\exists N\\ge N_0\\ \\exists L\\ \n \\forall h\\in\\{1,\\ldots,f(N)\\},\\quad \\mathcal C(h,N,L)\n\\]\nis equivalent to irrationality of $S$. Under irrationality, one can\ntake $f(N)=N+1$, and every $N$ has a suitable depth.\n\\end{prop}\n\\begin{proof}\nThe displayed condition supplies Proposition~\\ref{prop:A10}'s\nquantified certificate condition: for fixed $h\\ge1$, choose the\nbasepoint threshold so large that $f(N)\\ge h$ beyond it. That\nproposition gives irrationality.\n\nConversely, suppose $S$ is irrational. For every $h\\ge1$ and $N\\ge0$,\n\\[\n R_{N+h}-R_N\n =2^N(2^h-1)S-(\\Phi_{N+h}-\\Phi_N)\n\\]\nis nonintegral. Fix $N$ and a positive integer $M$. Choose $L$ so large\nthat\n\\[\n \\frac{2(N+M+L+2)}{2^L}\n <\\min_{1\\le h\\le M}\n   \\operatorname{dist}(R_{N+h}-R_N,\\Z).\n\\]\nThe finite minimum is positive, whereas the left side tends to zero.\nFor each $h\\le M$, the truncation error between $D(h,N,L)/2^L$ and\n$R_{N+h}-R_N$ is at most $(N+h+L+2)/2^L$. The triangle inequality\ntherefore puts $D(h,N,L)/2^L$ at distance greater than\n$(N+h+L+2)/2^L$ from every integer, which is precisely\n$\\mathcal C(h,N,L)$. Taking $M=N+1$ proves the converse.\n\\end{proof}\nThis proof gives no quantitative lower bound for the finite minimum\nas the basepoint or number of shifts grows. It selects a common\ndepth after those parameters are fixed; it does not impose depth equal\nto the period.\n\n\\subsubsection{The fixed Farey bound}\n\n\\begin{thm}\\label{thm:fixed-farey-bound}\nIf $S$ is rational,\nits reduced denominator exceeds $7.963\\,964\\,664\\,670\\,137\\,532\\,335\\allowbreak\\,577\\,487\\,583\\,105\\,3\\times 10^{34}$. This is the finite denominator exclusion in the\ncorpus, logically independent of the certificate-supply reduction.\n\\end{thm}\n\\evidenceremark{The proof combines the general Farey mediant lemma\n(Lemma~\\ref{lem:farey}) with an exact finite computation at the window\n$K=240$ (Proposition~\\ref{prop:gapwindow}).}\n\n\\begin{rem}[What a larger Farey bound requires]\nUnbounded valid denominator exclusions would imply $S\\notin\\Q$.\nFor the single window $K=240$, however, the first-failure computation\n(\\lean{gap_check_window_1_240_first_failure}{GapFareyBound.lean:225})\nproves that the residue inequality fails at the mediant denominator\n$b+d=Q_0+1$. This exact first failure prevents a larger cutoff from\nthat same inequality at that same window. It is not an obstruction to\nother windows or to other tests using the finite data.\n\nThe existing proofs for individual windows use explicit totient residues\nand rational endpoint calculations. Extending that implementation calls\nfor new finite data; a uniform proof of a growing bound need not use\nhard-coded convergents. If $S$ had reduced denominator $q$, every valid\nexclusion bound would be smaller than $q$. Thus unbounded exclusions suffice for irrationality; rationality\nwould bound them, but would not force eventual constancy.\nNo converse or comparison with the\ncontinued-fraction denominators of $S$ is proved here.\n\\end{rem}\n\n\\begin{obs}[The limited gain from a unit-gap refinement]\nThe finite unit-gap condition of Observation~\\ref{prop:D8cert-kill}\n(\\lean{ReducedDenominatorUnitGapCert}{PrimitiveWeightCertificate.lean:30})\nand its quantified version\n(\\lean{PrimitiveReducedDenominatorUnitGapSupply}{PrimitiveWeightCertificate.lean:47})\nare definitions; the latter is not asserted to hold. Two separate results\n(\\lean{reducedDenominatorUnitGapCert_prime_pow_iff}{PrimitiveWeightCertificate.lean:54},\n\\lean{reducedDenominatorCandidateCount_prime_pow_le_one}{PrimitiveWeightCertificate.lean:111})\nshow that, for $p$ prime and $e\\ge1$,\na window at denominator $p^e$ that \\emph{passes the refined test}\ncontains at most one candidate integer. A window with two or more\ncandidates fails the test. This is a condition for successful\ncertification, not a bound on all windows or on the improvement\nin the denominator cutoff. It does not apply to all odd denominators:\nObservation~\\ref{prop:D8cert-kill} gives an actual totient window with\ntwo admissible nonunit candidates modulo $15$.\n\\end{obs}\n\n\n\\begin{prop}[A sufficient extension]\\label{prop:sufficient-extension-3}\nSuppose $g(K)\\to\\infty$ and, for every $K$, the $(N=1,K)$ gap check\nexcludes every rational of reduced denominator at most $g(K)$.\nThen $S$ is irrational: any rational value of $S$ would have a fixed\nfinite denominator, contradicted at a sufficiently large $K$.\nOnly unbounded exclusion bounds along a sequence of windows are\nneeded for this argument. No equivalence with a bound for the\ncontinued-fraction denominators of $S$ is asserted; such a comparison\nwould require its own proof connecting those convergents to the\ncertified intervals.\n\\end{prop}\n\n\\subsubsection{A two-adic congruence that does not give a certificate}\n\n\\begin{thm}\\label{thm:two-adic-congruence-that-does}\nThe theorem is unconditional and holds for \\emph{every} $K$, $H$, $B$.\nFor every $K\\ge 2$, every $H>K$, and every bound $B$, there are primes\n$p>B$ with a length-$(K{-}1)$ zero prefix and a terminal half-modulus, giving\n\\[\n  D(H,p-K,K) \\equiv 2^{K-1} \\pmod{2^K}.\n\\]\nUnder eventual integrality this transfers to an integer $z$ with $(z:\\mathbb R) =\n{R}(p{+}H) - R_{p}$ and $z\\equiv 2^{K-1}\\pmod{2^K}$.\n\\end{thm}\n\n\\begin{rem}[Why the central residue still fails the test]\nThis places the residue at the centre of the residue interval: $2^{K-1}$ is\nas far as possible from $0$ modulo $2^K$. Taking $N{=}p{-}K$,\n$h{=}H$, $L{=}K$ gives radius $N{+}h{+}L{+}2 = p{+}H{+}2$, so $\\mathcal{C}$ requires\n$2^{K-1} > p{+}H{+}2$. But $z\\equiv 2^{K-1}\\pmod{2^K}$ forces $|z|\\ge 2^{K-1}$, while the directed\ntail bound forces $|z| < p{+}H{+}2$; and the construction's own congruence\n$p\\equiv 1{+}2^{K-1}\\pmod{2^K}$ (equivalently $v_2(p{-}1)=K{-}1$) forces $p\\ge 1{+}2^{K-1}$. So\n$p{+}H{+}2 > 2^{K-1}$ always, and no choice of $p$ rescues it. The quantifier order is inverted:\nthe theorem gives, for fixed $K$, cofinally many \\emph{large} $p$; the certificate needs a $p$\n\\emph{small} relative to $2^{K-1}$, impossible for a single totient\ndifference since $|\\varphi(N{+}H)-\\varphi(N)| < N{+}H$. The depth is not the\nobstacle: the construction already works uniformly for every $K$,\ncombining by the Chinese remainder theorem one congruence for $p+H$ and\ntwo for each of the $K-1$ preceding positions\n(Proposition~\\ref{prop:TA-inv}).\n\\end{rem}\n\n\\begin{prop}[A sufficient accumulated-residue condition]\\label{prop:sufficient-accumulated-residue-condition}\nSuppose that for every integer $h\\ge1$ and every $N_0\\in\\mathbb N$\nthere are $N\\ge N_0$ and $L\\ge1$ such that\n\\[\n D(h,N,L)\\equiv2^{L-1}\\pmod{2^L},\\qquad\n N+h+L+2<2^{L-1}.\n\\]\nThen $S\\notin\\mathbb Q$: these two inequalities imply\n$\\mathcal C(h,N,L)$, and the full quantified certificate condition\napplies. The half-modulus condition here is imposed on the accumulated\nweighted sum $D(h,N,L)$, not on a single totient difference.\nAn unbounded set of triples with no control of the shift does not\nsupply the displayed quantifiers. The required family remains unproved.\n\\end{prop}\n\n\\subsubsection{The residue margin at an LCM jump}\n\n\\begin{thm}[Arbitrarily large prime-power LCM jumps]\\label{thm:arbitrarily-large-prime-power-lcm}\nFor every $t_0\\in\\mathbb N$ there is\n$t\\ge t_0$ with $H(t)<H(t+1)$. One may take $t=p-1$ for any\nprime $p>t_0$. The positions $t=2^a-1$ are also strict LCM jumps for $a\\ge1$:\n$2^a$ is the next required power of $2$. The restriction excludes\n$a=0$, since $H(0)=H(1)=1$.\nThe second condition in the next definition\nasks for the additional residue margin along this explicit sequence.\nKnowing these jump positions does not establish that margin.\n\\end{thm}\n\n\\begin{defn}[The remaining residue margin]\nFor $t\\ge3$, put $H=H_t$ and define\n\\[\n \\begin{aligned}\n m&=\\lfloor\\log_2 H\\rfloor+10,\\\\\n d&=\\bigl(D(H,H+1,m)-D(H,H,m)\\bigr)\\bmod2^m,\\\\\n \\sigma_t&=\\min\\{d-2^{m-5},\\,2^m-2^{m-5}-d\\}.\n \\end{aligned}\n\\]\nThus $d$ is the difference of two adjacent window numerators, reduced\nmodulo $2^m$, not the residue of $D(H,H,m)$ alone. The integer\n$\\sigma_t$ is nonnegative exactly when\n$2^{m-5}\\le d\\le2^m-2^{m-5}$. The choice of $m$ makes the error\nbound $3H+2m+5$ smaller than $2^{m-5}$; nonnegativity is the\nadditional arithmetic requirement, not a consequence of that choice.\n\\lean{canonicalAdjacentSuffixCentralSlack}{DiagonalFreshLossBridge.lean:2651}\n\\lean{canonicalAdjacentSuffixDepth}{DiagonalFreshLossBridge.lean:1870}\n\\lean{diagonalAdjacentSuffixResidue}{DiagonalFreshLossBridge.lean:525}\n\nThe first unproved condition asks for every $t_0$ for an index\n$t\\ge\\max(3,t_0)$ with $H_t<H_{t+1}$ and $\\sigma_{t+1}\\ge0$.\nThe second restricts these endpoints to powers of two: for every $a_0$,\nthere must be $a\\ge\\max(2,a_0)$ with $\\sigma_{2^a}\\ge0$.\nThey are defined in Lean\n(\\lean{CanonicalAdjacentSuffixPostJumpSlackSupply}{DiagonalFreshLossBridge.lean:2686},\n\\lean{CanonicalAdjacentSuffixPowerTwoPostJumpSlackSupply}{DiagonalFreshLossBridge.lean:2693}).\nEach condition implies $S\\notin\\mathbb Q$\n(\\lean{irrational_totientSeries_of_canonicalAdjacentSuffixPostJumpSlackSupply}{DiagonalFreshLossBridge.lean:2941},\n\\lean{irrational_totientSeries_of_canonicalAdjacentSuffixPowerTwoPostJumpSlackSupply}{DiagonalFreshLossBridge.lean:2948}).\nThe implications are proved; both conditions are unproved.\n\\end{defn}\n\n\\begin{rem}[Finite numerical evidence]\nAll $40$ strict jumps up to endpoint $113$ pass, closest margin\n$\\approx 0.000221$ of the modulus at $t=100$; the power-of-two endpoints $4,8,16,32$ pass with\nslack fractions $0.41,0.036,0.40,0.19$. This is a finite computation; both\nconditions remain unproved at arbitrarily large endpoints.\nA strict jump has $t+1=p^k$ for a prime $p$ and $k\\ge1$, with\n$H_{t+1}=pH_t$\n(\\lean{periodLcm_succ_eq_prime_mul_of_strict_jump}{DiagonalFreshLossBridge.lean:2332}). Thus $\\sigma_{t+1}$ uses the new height $pH_t$,\nnot the pre-jump height $H_t$. Estimating its change would require\ninformation about the adjacent-window residue, not just the LCM ratio.\nMoreover $\\sigma_t=\\sigma_u$ whenever $H_t=H_u$\n(\\lean{canonicalAdjacentSuffixCentralSlack_eq_of_periodLcm_eq}{DiagonalFreshLossBridge.lean:2658}).\nThus the value changes\nonly when the LCM height changes. This reduces the number of distinct\nvalues to study; it does not determine their signs.\n\\end{rem}\n\n\\begin{prop}[A sufficient inequality]\\label{prop:sufficient-inequality}\nIf, for every $a_0\\in\\mathbb N$, there is $a\\ge\\max(2,a_0)$ with\n$\\sigma_{2^a}\\ge0$, then $S\\notin\\mathbb Q$.\nThis is the sufficient condition defined above, not a claim that the\ninequality holds at arbitrarily large indices. Equality at the edge\nis allowed: the condition is nonnegativity, not strict positivity.\n\\end{prop}\n\n\\subsubsection{A certificate for a four-tail combination}\n\n\\begin{thm}\\label{thm:certificate-four-tail-combination}\nUse $J(H,p)$, $W(H,p,L)$ and $B(H,p,L)$ from\nTheorem~\\ref{catalogue:mob:e2}. For all $H,p,L\\in\\mathbb N$,\n\\[\n B(H,p,L)<W(H,p,L)\\bmod2^L<2^L-B(H,p,L)\n \\quad\\Longrightarrow\\quad J(H,p)\\notin\\mathbb Z.\n\\]\nThe bound $B(H,p,L)=3pH+(p+1)(L+2)$ follows by grouping the\nnonnegative remainders as in that theorem. Compared with the\n$4pH+(p+1)(L+2)$ bound obtained by treating the two diagonal\ndifferences separately, it saves $pH$. This is a strict improvement\nwhen $pH>0$; optimality for totient tails is not asserted.\n\\end{thm}\n\n\\begin{thm}[One explicit witness]\\label{thm:one-explicit-witness}\nFor $H=H(4)=12$, $p=5$ and $L=15$, direct integer evaluation gives\n\\[\n W(12,5,15)=149906,\\qquad\n W(12,5,15)\\bmod32768=18834,\\qquad B(12,5,15)=282.\n\\]\nThus $282<18834<32486$, and $J(12,5)\\notin\\mathbb Z$.\nThis is the one instance singled out in Theorem~\\ref{catalogue:mob:e2}, not a\nclaim of uniqueness or of the smallest possible height. The implication\nfrom a quantified family is Theorem~\\ref{catalogue:mob:e2}.\n\\end{thm}\n\n\\begin{rem}[Scope of the witness]\nThe quantified hypothesis asks, for every $t_0\\in\\mathbb N$, for\n$t\\ge t_0$, $p\\ge1$ and $L\\ge0$ satisfying the displayed residue\ncondition with $H=H(t)$. The multiplier $p$ need not be prime, and need\nnot be tied to the next LCM jump. No comparison of the difficulty of\nthis condition with the prescribed-depth separation condition is proved.\n\nAt a strict LCM jump, $H(t+1)=pH(t)$ for a prime $p$. This gives one\nnatural choice of multiplier, but the prime need not be a new divisor:\n$H(4)=2H(3)$ and $2$ already divides $H(3)=6$.\nThe identity of Proposition~\\ref{catalogue:mob:e3}\ncancels contributions affine in the multiplier. It does not bound the\nremaining terms of the four-tail combination. The finite example above\ntherefore does not establish the quantified condition.\n\\end{rem}\n\n\\subsubsection{Denominators of M\\\"obius sums}\n\n\\begin{thm}\\label{thm:denominators-mobius-sums}\nFor every integer $t\\ge5$, with\n$\\mathcal P_t=\\{p\\text{ prime}:t/2<p\\le t\\}$ as above,\n\\[\n 2^{\\lfloor t/2\\rfloor}\n \\le\\prod_{p\\in\\mathcal P_t}(2^p-1)\n \\le\\operatorname{den}(H_t\\beta_{H_t}).\n\\]\nMoreover, some $p\\in\\mathcal P_t$ satisfies\n\\[\n 2^{\\lfloor t/2\\rfloor}\\le2^p-1<2^t,\n \\qquad 2^p-1\\mid\\operatorname{den}(H_t\\beta_{H_t}).\n\\]\nIndeed, Bertrand's postulate makes $\\mathcal P_t$ nonempty. Its members\nsatisfy $p-1\\ge\\lfloor t/2\\rfloor$ and $p\\le t$, giving the size\nbounds. The divisibility follows from\nProposition~\\ref{catalogue:mob:b6}; divisibility of a positive reduced\ndenominator also gives the product inequality.\n\\end{thm}\n\n\\begin{rem}[Growth of these denominators does not give approximation of $S$]\nThe number $H_t\\beta_{H_t}$ is the finite rational contribution to a\ntotient-tail decomposition. Its complementary term can still cancel\nits fractional part. The stated denominator bound supplies neither an\nerror estimate for approximation of $S$ nor nonintegrality of\n$R_{2H_t}-R_{H_t}$.\n\nTo apply the diagonal criterion in Theorem~\\ref{catalogue:cert:b3},\none must establish this tail nonintegrality at arbitrarily large LCM\nscales. Denominator growth alone does not supply that condition.\n\\end{rem}\n\n\\begin{prop}[The additional approximation hypothesis]\\label{prop:additional-approximation-hypothesis}\nA sequence of rationals $u_t$ proves irrationality of $S$ if\n\\[\n u_t\\ne S\\quad\\text{for all sufficiently large }t,\n \\qquad \\operatorname{den}(u_t)|S-u_t|\\longrightarrow0.\n\\]\nThis is the rational-separation criterion of\nProposition~\\ref{catalogue:cert:d1}:\nif $S=a/b$ were reduced, every unequal $u_t$ would instead satisfy\n$\\operatorname{den}(u_t)|S-u_t|\\ge1/b$. An upper bound tending to zero\nfor this product is needed. A lower bound for the denominator cannot\nestablish it; improving a lower bound does not change the actual\napproximation error.\n\nA different argument could deduce a residue certificate from the\nsurviving divisor and further information about the complementary\nterm. The denominator theorem alone contains no such separation\nstatement. Neither sufficient argument is established by its lower\nbound, and these are not claimed to exhaust possible approaches.\n\\end{prop}\n\n\\subsubsection{Rationality and carry rank}\n\nThe totient sections have a dimension lower bound, as do the sections of\nan integral carry under the rationality hypothesis. An incompatible\nupper bound would finish a rank argument. The rational comparison\nsequences show why rationality by itself gives no such bound.\n\n\\begin{thm}\\label{thm:rationality-carry-rank}\nFor every $e\\ge1$, the retained dyadic totient sections form a basis for\nall sections through level $e$, of dimension $2^e+1$. The proof uses the\nChinese remainder theorem and Dirichlet's theorem. The complete level-zero\ntruncation, consisting only of $\\varphi$, has dimension one.\n\\end{thm}\n\n\\begin{thm}\\label{thm:rationality-carry-rank-2}\nIf $S\\in\\Q$, choose an integer $v>0$\nwith $vS\\in\\Z$ and put $u_N=vR_N$. Then $u_N\\in\\Z$,\n$u_{N+1}=2u_N-v\\varphi(N+1)$, and $0\\le u_N\\le v(N+2)$. For every $e\\ge0$,\n\\[\n \\dim_{\\Q}\\operatorname{span}\n \\{n\\mapsto u_{2^j n+r}:1\\le j\\le e,\\ 0\\le r<2^j\\}\\ge2^e-1.\n\\]\nThe proof is given in Section~\\ref{sec:mahler-defect}.\n\\end{thm}\n\n\\begin{rem}[The generic upper-bound inference is false]\nThe bound $2^e-1$ holds at every level. An upper bound contradicting it\nwould need additional information about the totient, since the rational\n$5/4$ comparison has the same lower bound. One proposed system of integer\nequalities and strict inequalities is inconsistent: there are no integers\n$Q,v>0$, $A\\ne0$ and $b$ with $QvA=b$ and $|b|<Qv$, since a nonzero\ninteger multiple of $Qv$ has absolute value at least $Qv$\n(\\lean{false_of_compressedAdjointCertificate}{TotientMahlerDefect.lean:1183}).\nThis excludes those simultaneous\nrequirements, not every argument using a matrix or its adjugate.\nSeparately, the span of all dyadic totient sections is\ninfinite-dimensional (Proposition~\\ref{prop:rank}).\n\\end{rem}\n\n\\begin{prop}[An additional hypothesis for a totient-specific rank argument]\\label{prop:additional-hypothesis-totient-specific-rank}\nA rank bound that contradicts the lower bound $2^e-1$ would have to use\nadditional arithmetic of the actual totient coefficients. The generic\nproposal that rationality bounds the rank of every integral carry\nsatisfying $u(n)/2^n\\to0$ is disproved by the comparison sequence\nrecorded in the short paper:\nits carry rank is at least $2^e-1$ at every level. It is therefore not a\nremaining general lemma from which totient irrationality follows.\nThe conditional Proposition~\\ref{prop:period-not-rank}\nrecords eventual periodicity modulo $v$ together with unbounded rational\nrank. Periodicity of the residue sequences concerns values in a finite\nquotient; it does not bound the rational span of the integer-valued\nsections. The $5/4$ comparison shows that this distinction persists for\na rational coefficient series.\n\\end{prop}\n\n\\subsection{Relations between the sufficient conditions}\n\nA fixed saving for the real part or norm of the block exponential\nsum would give a certificate. The endpoint conditions seek a residue\nor size bound along an LCM sequence instead.\nProposition~\\ref{prop:te-chain} relates the sufficient formulations;\nthe arithmetic estimates required by them remain open.\n\nThe generic bounded-rank argument fails for a separate reason: the rational\ncomparison sequence has unbounded section rank. A totient-specific upper\nbound would require hypotheses not shared by that sequence.\n\nThese are statements about the displayed conditions and comparisons, not\nan exhaustive classification of possible proofs. Some conditions are\nequivalent to irrationality; others are only sufficient. No relative\ndifficulty, truth probability, or independent formal verification follows\nfrom their position in the table.\n\n\\section{Counterexamples to proposed deductions}\n\nThe next constructions produce rational series from coefficients\nsatisfying selected totient data. Others test a proposed estimate\nwithout changing the coefficients. We specify the retained assumptions\nin each case, so that the failed deduction can be distinguished from\nan argument using additional properties of $\\varphi$. None decides the\nirrationality of $S$.\n\n\n\\subsection{Changing the coefficients after a finite prefix}\n\n\\textbf{Proposed argument.} The quantified residue condition requires\ncertificates beyond every starting index. A finite computation can check\nmany instances, but one fixed inspection bound cannot establish these\nquantifiers. The following comparison explains this limitation without\nmaking any claim about all possible irrationality arguments.\n\\lean{Erdos249257.TotientTailPeriodKiller.irrational_totient_series_iff_certificate_supply}{LcmConeFlatness.lean:412}\n\n\n\\textbf{Counterexample.} Theorem~\\ref{thm:gamma} gives an explicit splice\nconstruction; its series value, exact reduced denominator and failure of the\nseparation condition are checked in Lean. Choose an integer \\(P>B\\), set\n\\(\\gamma(n)=\\varphi(n)\\) for \\(n\\le B\\), and \\(\\gamma(n)=n-\\mathbf 1_{P\\mid n}\\)\nfor \\(n>B\\). Then \\(0\\le\\gamma(n)\\le n\\), and the binary series\n\\(\\sum\\gamma(n)/2^n\\) is \\emph{rational}: it is a finite rational prefix, plus\nthe tail of \\(\\sum_n n2^{-n}\\), minus a geometric sum over the multiples of\n\\(P\\) beyond \\(B\\). Because \\(\\gamma\\) agrees with \\(\\varphi\\) on every index\n\\(\\le B\\), \\(\\gamma\\) passes every finite certificate that only inspects\nindices \\(\\le B\\); in particular it passes or fails every instance of\n\\(\\mathcal{C}(h,N,L)\\) with \\(N+h+L\\le B\\) exactly as \\(\\varphi\\) does. Yet\n\\(\\sum\\gamma(n)/2^n\\in\\mathbb Q\\) by construction.\n\n\\textbf{Scope.} For each fixed bound $B$, a different rational\ncomparison sequence can agree with $\\varphi$ on all inspected coefficients.\nThis rules out a deduction from that finite prefix alone. It does not rule\nout a proof that establishes the required instances for arbitrarily large\n$B$ with the correct quantifiers. Such a proof would no longer be a single\nfinite inspection. The comparison also does not preserve multiplicativity\nor every other arithmetic property of $\\varphi$.\n\n\n\\textbf{Consequence.} The finite-prefix example and the parity example\nbelow are distinct constructions. The former preserves a prescribed finite\nprefix; the latter preserves parity and has separated nonperiodic blocks.\nEach shows why its own list of assumptions is insufficient. The recorded\nfinite diagonal examples remain useful individual certificates, but their\nnumber alone gives no assertion at arbitrarily large scales.\n\\lean{Erdos249257.TotientTailPeriodKiller.certifiedKill_diagonal_all_imported_through_t64}{DiagonalPincerCertificatesT64.lean:1967}\n\n\n\\subsection{A rational series with matching totient parity}\n\\label{sec:parity-countermodel}\n\nMatching totient parity and the bound $c(n)\\le n$ does not determine\nthe rationality of a binary sum. The example below is bounded and\naperiodic, yet its sum is rational. Its paired changes can be placed\nbeyond any threshold with any prescribed minimum separation.\n\n\\textbf{Construction.} Start with the sequence\n$0,1,1,2,4,4,4,\\ldots$. At each position $m=2^{k+3}$, $k\\ge0$, replace\nthe pair $(4,4)$ at $m,m+1$ by $(6,0)$. Equivalently,\n\\[\n c(n)=\n \\begin{cases}\n  0,&n=0,\\\\\n  1,&n=1,2,\\\\\n  2,&n=3,\\\\\n  6,&n=2^{k+3}\\text{ for some }k\\ge0,\\\\\n  0,&n=2^{k+3}+1\\text{ for some }k\\ge0,\\\\\n  4,&\\text{otherwise}.\n \\end{cases}\n\\]\nEach replacement leaves the binary sum unchanged because\n$2\\cdot2^{-m}-4\\cdot2^{-(m+1)}=0$. The increasingly large gaps between\nreplacements prevent eventual periodicity. The strongest recorded result\nalso quantifies their separation:\n\\begin{prop}[A rational series preserving totient parity and the stated separation properties]\\label{prop:rational-series-preserving-totient-parity}\nThere exists \\(c:\\mathbb N\\to\\mathbb N\\) such that: \\(c(n)\\le6\\) for all\n\\(n\\); \\(c(n)\\le n\\); \\(c(n)\\equiv\\varphi(n)\\pmod2\\) for every \\(n\\); for\nevery \\(N,G,K\\) there are \\(K\\) pairs of coefficients \\((6,0)\\)\nstarting beyond \\(N\\), with starting positions separated by more\nthan \\(G\\); and\n\\(\\sum_n c(n)/2^n = 3/2\\).\n\\end{prop}\n\n\\textbf{Scope.} The displayed bounds, parity agreement and\naperiodicity, including the stated number and separation of the edits,\nare jointly compatible with a rational binary sum. They therefore do not\nimply irrationality. The sequence $c$ differs from $\\varphi$: the example\npreserves its parity, not all its arithmetic properties. It excludes an\nargument using only the listed assumptions, not every strengthening of\nan aperiodicity condition or every use of totient coefficients.\n\n\\textbf{Consequence.} The paired edits have zero total contribution,\nsince $2\\cdot2^{-m}-4\\cdot2^{-(m+1)}=0$. Applied at the specified sparse\npositions of this particular base sequence, they preserve the rational\nsum and the required bounds while preventing eventual periodicity. This\ndoes not construct a rational series for every prescribed parity pattern,\nand it is not the finite-prefix construction of Theorem~\\ref{thm:gamma}.\nA proposed criterion using only the assumptions of this example must fail;\na criterion using further arithmetic information is not addressed.\n\n\n\\subsection{Square-CRT factorisations}\n\nFor a prime $p$ and $m\\ge1$, the identity\n\\[\n \\varphi(pm)=(p-1)\\varphi(m)+\\mathbf1_{p\\mid m}\\varphi(m)\n\\]\nseparates the coprime and divisible cases. Prescribing a cofactor\nmodulo $p$ by a congruence modulo $p^2$ can remove the second term\nover a chosen finite range. It simplifies the product formula, but\ndoes not show that differences between the resulting values are nonzero.\n\nThe examples in Observation~\\ref{prop:B8-kill} show both possibilities.\nAt $n=52$, the offset pairs $(13,13)$ and $(26,18)$ give respectively\n$\\varphi(65)-\\varphi(65)=0$ and\n$\\varphi(78)-\\varphi(70)=0$\n(\\lean{squareCRT\\_clean\\_block\\_can\\_vanish}{SquareCRTCube.lean:459}).\nAll four prime--cofactor products are coprime\n(\\lean{squareCRT\\_vanishing\\_countermodel\\_is\\_clean}{SquareCRTCube.lean:468}).\nAt $n=27$, the offset pairs $(3,3)$ and $(6,8)$ instead give\n$\\varphi(30)-\\varphi(30)=0$ and\n$\\varphi(33)-\\varphi(35)=-4$\n(\\lean{squareCRT\\_clean\\_block\\_can\\_be\\_nonzero}{SquareCRTCube.lean:477}),\nagain with coprime factors\n(\\lean{squareCRT\\_nonzero\\_countermodel\\_is\\_clean}{SquareCRTCube.lean:485}).\nThese finite equalities are checked by exact arithmetic. No minimality\nclaim about either witness is needed. The general existence theorem\n\\lean{exists\\_squareCRT\\_clean\\_horizon}{SquareCRTCube.lean:327}\nsupplies a common base for the stated finite congruence requirements,\nnot a nonvanishing assertion.\n\nThe vanishing example disproves the implication from correction-free\nfactorisation to nonvanishing; the other example shows that this\nfactorisation does not force vanishing either. This does not establish\nformal independence of two propositions, or exclude an argument using\nadditional hypotheses. The prime-dilation identity\n\\lean{totient\\_mul\\_prime\\_cocycle}{SquareCRTCube.lean:38}\nand the CRT constructions\n\\lean{exists\\_squareCRT\\_base}{SquareCRTCube.lean:230},\n\\lean{exists\\_large\\_squareCRT\\_base}{SquareCRTCube.lean:263}\nremain available with their own prime and residue hypotheses.\n\nA separate elementary identity in the same file concerns alternating\nsums. Let $I$ be finite, let $i\\notin I$, and suppose an\ninteger-valued function $f$ on finite subsets satisfies\n$f(T\\cup\\{i\\})=f(T)$ for every $T\\subseteq I$. Pairing $T$ with\n$T\\cup\\{i\\}$ gives\n\\[\n \\sum_{T\\subseteq I\\cup\\{i\\}}(-1)^{|T|}f(T)=0.\n\\]\nThis identity is proved in Lean\n(\\lean{alternating\\_powerset\\_sum\\_eq\\_zero\\_of\\_insert\\_invariant}{SquareCRTCube.lean:363}).\nIt has no totient hypothesis, but an application must still verify\ninvariance under the indicated insertion. Neither it nor the congruence\nconstruction supplies the missing nonzero totient difference at a\nprescribed LCM height.\n\n\\subsection{The cost of clearing a rational denominator}\n\\label{sec:adelic-height}\n\nClearing a denominator changes the multiplier as well as the\nfraction. The example $4(5/12)=5/3$ illustrates the divisibility\nrelation below for arbitrary reduced fractions; no totient identity\nis needed.\n\n\\begin{lem}[Complement divisibility after multiplication]\\label{lem:complement-divisibility-after-multiplication}\nWrite $x=a/b$ in lowest terms, where $a\\in\\Z$ and $b\\ge1$. Let\n$c\\in\\Z$ and let $H$ be a positive divisor of $b$. If the reduced\ndenominator of $cx$ divides $H$, then\n\\[\n b/H\\mid c.\n\\]\n\\end{lem}\nIndeed, the hypothesis makes $Hcx=Hca/b$ integral. Since\n$\\gcd(a,b)=1$, it follows that $b\\mid Hc$, hence $b/H\\mid c$.\nWriting $c=(b/H)t$ also gives $Hcx=ta$; this equality is also proved in\nLean (\\lean{Erdos249257.AdelicHeightObstruction.scalarLocalization\\_integer\\_eq\\_mul\\_num}{AdelicHeightObstruction.lean:56}).\nIf $c\\ne0$, the divisibility implies $|c|\\ge b/H$. The nonzero\nhypothesis is essential for this size bound: $c=0$ has reduced\ndenominator one and satisfies the divisibility, but has no positive\nsize cost. The lemma is about multiplication of one rational value,\nnot arbitrary cancellations among several values.\n\nA related elementary inequality compares a positive numerator with its\ndenominator. If $a>0$, $r\\ge0$, $n\\ge1$, $2^r\\mid a$, and\n$a/b<2/(2^n-1)$, then\n\\[\n 2^r(2^n-1)\\le a(2^n-1)<2b.\n\\]\nLean proves this Mersenne instance\n(\\lean{Erdos249257.AdelicHeightObstruction.positiveRat\\_mersenne\\_height}{AdelicHeightObstruction.lean:103})\nand the version with a general positive denominator parameter\n(\\lean{positiveRat\\_numDivisor\\_mul\\_lt\\_two\\_mul\\_den}{AdelicHeightObstruction.lean:77}).\nPositivity of $a$ is what turns numerator divisibility into a lower\nbound; it cannot be omitted.\n\nThe accompanying linear-algebra statement has a one-dimensional\nquotient. Let $V,W$ be vector spaces over $\\Q$, let\n$\\mathrm{ev}:V\\to\\Q$ and $\\Lambda:V\\to W$ be linear, and suppose\n$\\ker\\mathrm{ev}\\subseteq\\ker\\Lambda$. Choose $e\\in V$ with\n$\\mathrm{ev}(e)=1$. Then, for every $v\\in V$,\n\\[\n v-\\mathrm{ev}(v)e\\in\\ker\\mathrm{ev},\\qquad\n \\Lambda(v)=\\mathrm{ev}(v)\\Lambda(e).\n\\]\nThis is factorisation through the scalar quotient $V/\\ker\\mathrm{ev}$\n(\\lean{Erdos249257.AdelicHeightObstruction.linearDescender\\_eq\\_smul\\_eval}{AdelicHeightObstruction.lean:120}).\nA surjection onto a higher-dimensional space would imply only a linear\nfactorisation through that space, not multiplication by one fixed vector.\n\nThe denominator calculations use two lemmas proved in Lean\n(\\lean{Erdos249257.RationalDenominatorSurvival.divisor\\_dvd\\_divInt\\_den}{RationalDenominatorSurvival.lean:17},\n\\lean{survivingDivisor\\_dvd\\_scaled\\_divInt\\_den}{RationalDenominatorSurvival.lean:38}).\nFor integers $a$ and $D\\ge1$, the reduced denominator of $a/D$ is\n$D/\\gcd(|a|,D)$. Hence a divisor of $D$ coprime to $a$ divides the\nreduced denominator. Coprimality is sufficient, not necessary: $2/8=1/4$\nretains the divisor $2$. More generally, if $C$ is a positive divisor of $D$, $\\gcd(C,a)=1$,\nand $h\\ge0$ is an integer, then $C/\\gcd(C,h)$ divides the reduced\ndenominator of $ha/D$. At $h=0$ this assertion retains only the divisor\none.\n\nFor a nonzero multiplier, a lower bound for $b/H$ therefore gives a\nlower bound for its absolute value. For an integer $t\\ge5$, the product estimate\n\\(2^{\\lfloor t/2\\rfloor}\\le\\prod_{p\\in\\mathcal P_t}(2^p-1)\\)\n(\\lean{upperHalfMersenneProduct\\_lower\\_bound}{MersenneShadowDenominatorGrowth.lean:60})\ncan be used when that product is shown to divide $b/H$.\nNeither the product estimate nor the scalar lemma proves a quadratic\ngrowth law for arbitrary approximation schemes.\n\nThe finite Farey exclusion for $S$ is a separate use of reduced\ndenominators. Its certified interval contains no rational whose reduced\ndenominator is at most\n\\[\n 79{,}639{,}646{,}646{,}701{,}375{,}323{,}355{,}774{,}875{,}831{,}053\n\\]\n(\\lean{tsum\\_totient\\_div\\_pow\\_two\\_ne\\_ratCast\\_of\\_den\\_le\\_79639646646701375323355774875831053}{CertificateKernel.lean:18384}).\nThat numerical exclusion requires the interval and Farey calculation;\nit is not a consequence of the scalar lemma alone.\n\n\\subsection{Signed moments and parity}\n\nA signed determinant can serve in rational approximation only with\nboth a nonzero numerator and a small error. Here parity proves the\nfirst requirement for the particular finite signed sum.\n\n\\begin{lem}[Nonvanishing from a unique largest denominator exponent]\\label{lem:nonvanishing-unique-largest-denominator-exponent}\nLet $I$ be a finite set, let $u_i\\in\\Z$ and $e_i\\in\\N$ for $i\\in I$,\nand suppose that $m\\in I$ satisfies $e_i<e_m$ for every $i\\ne m$.\nIf $u_m$ is odd, then\n\\[\n 2^{e_m}\\sum_{i\\in I}\\frac{u_i}{2^{e_i}}\n   =\\sum_{i\\in I}u_i2^{e_m-e_i}\\equiv1\\pmod2.\n\\]\nIn particular, the sum is nonzero.\n\\end{lem}\nEvery term except $u_m$ is even, which proves the assertion. For\nexample, $1/2-1/8=3/8$ has odd cleared numerator. Uniqueness is\nnecessary for this argument: two odd contributions with the same\nlargest exponent can cancel.\n\nLean also proves a separate determinant expansion from the Leibniz\nformula\n(\\lean{det\\_mul\\_rectangular}{SignedQMomentObstruction.lean:29})\nand defines the truncated signed moments\n$\\sum_{N=1}^Y a_N2^{-N(k+1)}$ and their Hankel determinants\n(\\lean{truncatedMoment}{SignedQMomentObstruction.lean:108},\n\\lean{hankelDet}{SignedQMomentObstruction.lean:117}). The parity lemma\nis not derived from that determinant identity. A determinant application\nmust first identify a sum to which the parity hypotheses apply.\n\nFor the totient series, one would still need suitable configurations at\narbitrarily large scales and a denominator-weighted error tending to\nzero. Nonvanishing alone provides neither. No such family is supplied\nby these two algebraic statements; compare the separate rank discussion\nin \\S\\ref{sec:mahler-defect}. In another base, the analogous argument\nwould require divisibility by a prime factor of that base rather than\nan unchanged appeal to parity.\n\n\\subsection{Column rescaling and cancellation}\n\nA nonzero determinant need not prevent one row from being constant.\nFor example,\n\\[\n \\begin{pmatrix}1&1\\\\1&-1\\end{pmatrix}\n \\begin{pmatrix}1&0\\\\0&-1\\end{pmatrix}\n =\\begin{pmatrix}1&-1\\\\1&1\\end{pmatrix}.\n\\]\nThe first and last matrices have determinants $-2$ and $2$, respectively;\nthe second row of the resulting matrix is constant. This simple column rescaling\nmatters when a determinant is proposed as evidence for cancellation in\nthe first-harmonic sums.\n\n\\begin{prop}[A nonzero minor survives inverse-phase column weights]\\label{prop:nonzero-minor-survives-inverse-phase}\nLet $d\\ge1$ be an integer, let $e_0,\\ldots,e_{d-1}$ be nonnegative integers, and let\n$z_0,\\ldots,z_{d-1}$ be nonzero complex numbers. Write\n$P_{ij}=z_j^{e_i}$, and suppose $e_{i_0}=1$ for some $i_0$.\nMultiplying column $j$ by $W_j=z_j^{-1}$ gives\n\\[\n (P\\operatorname{diag}(W))_{i_0j}=1,\\qquad\n \\det(P\\operatorname{diag}(W))=\\det(P)\\prod_{j=0}^{d-1}z_j^{-1}.\n\\]\nThus a nonzero determinant remains nonzero while row $i_0$ becomes\nconstant. If all $|z_j|=1$, its absolute value is also unchanged.\n\\end{prop}\nThe entry formula and the determinant of a product prove the statement.\nIf row-dependent weights are allowed, the choice\n$W_{ij}=z_j^i/z_j^{e_i}$ gives $W_{ij}P_{ij}=z_j^i$ and makes row zero\nconstant. It produces the ordinary Vandermonde matrix, which is\nnonsingular when the $z_j$ are pairwise distinct\n(\\lean{rowResidualMonomialMatrix\\_reconstructs\\_phasePowerMatrix}{ResidualGaugeObstruction.lean:124},\n\\lean{rowReconstruction\\_zero\\_row\\_one}{ResidualGaugeObstruction.lean:145}).\n\nThese are statements about complex matrices, not arithmetic properties\nof the totient sequence. In particular, determinant nonvanishing, even\nwith unit-modulus weights, does not by itself exclude complete\nalignment in the indicated row. An application of the first-harmonic\ncriterion\n(\\lean{irrational\\_totient\\_series\\_of\\_first\\_harmonic\\_norm\\_gap}{FirstHarmonicPivot.lean:83})\nneeds additional information that excludes such weights or controls the\nresulting weighted sum. The example does not require that information\nto take the form of an identity coupling different rows, and it does\nnot rule out determinant arguments that use appropriate additional\nhypotheses.\n\n\\subsection{Tail integrality on an LCM grid}\n\n\\textbf{The comparison.} The diagonal condition\n(\\lean{Erdos249257.TotientTailPeriodKiller.irrational_totient_series_iff_lcm_diagonal_certificate_supply}{LcmConeFlatness.lean:426})\nuses the pair $(H_t,2H_t)$. Under rationality, the same fractional part\noccurs at every positive multiple of $H_t$ once $t$ is sufficiently\nlarge. A finite set of these multiples therefore gives another possible\nplace to test for a nonintegral difference. The criterion below does not\nassert a smaller successful depth than the diagonal test.\n\n\\textbf{Eventual equality of fractional parts.}\n\\begin{prop}[Tail integrality on an LCM grid]\\label{prop:tail-integrality-lcm-grid}\n\\(S\\in\\mathbb Q \\implies \\exists t_1,\\ \\forall t\\ge t_1,\\\n\\forall q,m:\\mathbb N,\\ 0<q \\implies {R}(q\\cdot\n{H}(t)+m\\cdot{H}(t)) -\n{R}(q\\cdot{H}(t)) \\in\n\\mathbb Z\\).\n\\end{prop}\nRationality makes the fractional parts of $R_{kH_t}$ equal for all\n$k\\ge1$ at each scale $t\\ge t_1$. The corresponding implication,\nTheorem~\\ref{catalogue:cert:b7}, shows that a certificate $\\mathcal C(mH_t,qH_t,L)$ with $q\\ge1$\nat arbitrarily large $t$ implies $S\\notin\\mathbb Q$.\n\n\n\\textbf{(b\\('\\)) A finite test on the grid.}\n\\begin{prop}[A finite-grid certificate gives a nonintegral pair]\\label{prop:finite-grid-certificate-gives-nonintegral}\nLet $Q\\subseteq\\mathbb N_{>0}$ be finite and nonempty. For each $q\\in Q$,\nput $A_q=\\sum_{j=1}^{L}\\varphi(qH+j)2^{L-j}$ and $B_q=qH+L+2$.\nSuppose $B_q<2^L$ for every $q\\in Q$ and\n\\[\n \\forall q_i\\in Q\\ \\exists q_j\\in Q,\\qquad\n B_{q_j}<(A_{q_i}-A_{q_j})\\bmod2^L.\n\\]\nThen $R_{q_jH}-R_{q_iH}\\notin\\mathbb Z$ for some $q_i,q_j\\in Q$.\nThe proof and a four-point example are given in\nTheorem~\\ref{catalogue:cert:b10a}.\n\\end{prop}\n\n\\textbf{Scope.} Eventual integrality on the LCM grid is a necessary\nconsequence of rationality, not a claim about the actual series made\nwithout that hypothesis. The finite-grid test uses the individual\none-sided bounds $B_q<2^L$, rather than charging the larger radius to\nboth ends of a pairwise interval. This compares modulus bounds, not\nbinary depths. A finite nonempty set and every stated size inequality\nare needed; three or more points and pairwise compatibility are not.\n\nNeither statement supplies the unbounded family required to decide\n\\#249. Exhibiting a nonintegral tail difference on the LCM grid at\narbitrarily large $t$ would refute rationality. The finite-grid test is a\nway to certify such a difference, not a proof that these tests succeed\nat arbitrarily large scales.\n\nThe implication from rationality to grid flatness\n(\\lean{rational\\_totient\\_series\\_forces\\_lcm\\_cone\\_flatness}{LcmConeFlatness.lean:90})\nshould not be confused with an assertion of flatness for the actual series.\nThe exact diagonal integrality reformulation and its quantified converse\n(\\lean{diagonal\\_int\\_iff\\_foreignDiagonalDefect\\_hits\\_fullTarget}{DiagonalPincerDecomposition.lean:215},\n\\lean{irrational\\_totientSeries\\_of\\_full\\_target\\_avoidance\\_supply}{DiagonalPincerDecomposition.lean:290})\ngive other equivalent forms of the problem. They are not the only such\nforms; Proposition~\\ref{prop:iffs} lists further equivalent certificate\nconditions.\n\n\\textbf{Consequence.} Choosing the least deep tail is the step that\nturns a common-fractional-part assumption into an intersection of arcs.\nA related finite comparison concerns first and second differences: at\n$(h,N)=(1,8)$ the first-difference test holds at depth $8$, while no\nsecond-difference certificate exists at depth at most $8$\n(\\lean{totient\\_tail\\_rank\\_two\\_kill\\_sound\\_but\\_not\\_shallower\\_cell}{CertificateKernel.lean:19090}).\nThis disproves a universal claim of smaller depth for second differences;\nit does not show that they can never improve a certificate.\n\n\\subsection{Exact dyadic rank and the limits of a rank argument}\n\\label{sec:mahler-defect}\n\nAt each level the scalar totient identities supply spanning, while\naffine independence excludes any further relations. This computes\nthe rank, rather than merely proving its unboundedness. The latter\nalready follows from Coons's nonregularity theorem in every base\n$k\\ge2$~\\cite[Theorem~3.2]{coons}. Martin's stronger theorem supplies\nthe affine independence\\cite[Theorem~1]{martin-phi-inequalities}; the\nproportional arguments $n$ and $2n$ require a separate two-evaluation\nargument. The \\shortpaper{short249-proof}{all-base proof} and\n\\shortpaper{short249-base-six}{integral relation calculation} appear\nin the short paper. We give the dyadic form needed for carry rank here.\n\n\\textbf{Construction.}\n\\begin{thm}[Exact dyadic rank and infinite-dimensionality]\\label{thm:exact-dyadic-rank-infinite-dimensionality}\nFor every $e\\ge1$, the family\n\\[\n \\{n\\mapsto\\ph(n),\\ n\\mapsto\\ph(2n)\\}\n \\ \\cup\\ \\{n\\mapsto\\ph(2^j n+r):1\\le j\\le e,\\ 0<r<2^j,\\ r\\text{ odd}\\}\n\\]\nis linearly independent over $\\Q$. It contains $2^e+1$ sequences and spans\nthe sections through level $e$ by the zero- and even-residue reductions.\n\\end{thm}\nThe two zero-residue sections need separate treatment because their\narguments $n$ and $2n$ are proportional. Restrict a proposed relation\nto $n=2m+1$, where $\\ph(2n)=\\ph(n)$. The odd-residue sections,\ntogether with $\\ph(n)$, now have pairwise nonproportional affine\narguments. The \\shortpaper{short249-proof}{evaluation-matrix construction}\nchooses one input per row. At that input a selected primitive form is\nprime, while each competing form has a prime divisor whose predecessor\nis divisible by a fixed auxiliary prime. The matrix is diagonal and\nnonzero on the diagonal modulo that auxiliary prime. The check that\nthe selected prime progression is reduced is part of the construction.\n\nIndependence removes the positive-residue coefficients and leaves\n$A+B=0$. The same remaining relation at $n=2$ gives $A+2B=0$, so\nboth coefficients vanish. This is the dyadic specialisation of the\nshort paper's proof. The Lean proof is unconditional; it uses the Chinese\nremainder theorem and Dirichlet's theorem on primes in arithmetic\nprogressions from Mathlib. Consequently the full\ninfinite family spans an infinite-dimensional \\(\\mathbb Q\\)-space\n(\\lean{not\\_finiteDimensional\\_span\\_fullTotientKernel}{TotientMahlerDefect.lean:1145}).\nA separate elementary observation concerns a proposed integer identity.\nThere are no $Q,v\\in\\N_{>0}$ and $A,b\\in\\Z$ with\n$QvA=b$, $A\\ne0$, and $|b|<Qv$: the identity forces\n$|b|=Qv|A|\\ge Qv$\n(\\lean{false\\_of\\_compressedAdjointCertificate}{TotientMahlerDefect.lean:1183}).\n\nThe Lean independence theorem is also stated for \\(e=0\\), but its auxiliary\ncanonical index then still contains the two zero-residue sections\n\\(\\varphi(n)\\) and \\(\\varphi(2n)\\).  It is therefore not the actual truncation\nthrough level zero, which consists only of \\(\\varphi(n)\\) and has dimension one.\n\n\\paragraph{Integral relations.}\nRational independence becomes an integral normal form because every\nomitted section is an \\emph{integer} multiple of a retained one.\nKeep the formal coefficient vectors separate from the sequences:\na relation is nonzero in the former group and evaluates to zero\nin the latter.  Fix $e\\ge1$, write $F_{j,r}(n)=\\varphi(2^j n+r)$, and let\n$I_e=\\{(j,r):0\\le j\\le e,\\ 0\\le r<2^j\\}$.  Retain $F_{0,0}$,\n$F_{1,0}$, and the odd-residue sections at levels $1,\\ldots,e$; call\nthis retained index set $J_e$.  The totient identities reduce every omitted\nsection to $F_i=a_iF_{j(i)}$ for a retained section $j(i)$ and an integer\n$a_i\\ge0$.  Thus the retained family generates the lattice\n$L_e=\\sum_{i\\in I_e}\\mathbb Z F_i$.  Its rational independence gives\ninteger independence as well: every element of $L_e$ has unique integer\ncoordinates in these $2^e+1$ retained sections.\n\nTo describe \\emph{all} integral relations, let $E_i$ denote the formal\ncoordinate vector in $\\mathbb Z^{I_e}$ and use the evaluation map\n\\[\\adjustbox{max width=\\linewidth}{$\\displaystyle\n \\operatorname{ev}_e:\\mathbb Z^{I_e}\\longrightarrow L_e,\n \\qquad (c_i)\\longmapsto\\sum_{i\\in I_e}c_iF_i.\n$}\\]\nFor each omitted index put $R_i=E_i-a_iE_{j(i)}$.  Then\n\\[\\adjustbox{max width=\\linewidth}{$\\displaystyle\n \\ker(\\operatorname{ev}_e)\n   =\\bigoplus_{i\\in I_e\\mathbin{\\backslash} J_e}\\mathbb Z R_i,\n \\qquad\n \\operatorname{rank}_{\\mathbb Z}\\ker(\\operatorname{ev}_e)=2^e-2.\n$}\\]\nFor a vector $c$, subtract $\\sum_{i\\notin J_e}c_iR_i$. All omitted\ncoordinates disappear, and the retained coordinate at $j$ is\n$c_j+\\sum_{i\\notin J_e,\\,j(i)=j}a_ic_i$. The vector is a relation\nexactly when every retained coordinate is zero. In that case its\nrelation coefficients are already known: they are the original\nomitted coordinates $c_i$. Uniqueness follows because $R_i$ has\ncoefficient $1$ at its own omitted index and $0$ at every other one.\nThese unit coefficients explain why no division, and no passage\nfrom $\\Q$ back to $\\Z$, is required.  The \\shortpaper{short249-base-six}{depth-two base-six example}\nin the short paper illustrates these two-term rows.  The distinction between the retained basis and the redundant\nspanning family of all sections already matters for $k$-regular sequences\nthat are not eventually zero: their growth exponent is the base-$k$\nlogarithm of the joint spectral radius of their section matrices in a\nbasis, whereas a redundant spanning family gives only an upper bound\n(Coons~\\cite[Theorem~1, Proposition~4 and Corollary~7]{coons-jsr}).  The totient is not regular, so that theorem does not\napply; the short paper records how the section maps act on the retained bases\nfrom one level to the next.\n\nThe \\href{https://github.com/wcook04/plectis-erdos/blob/25ef6245d15a47548c6926369ae8f1a0f0a14a80/ErdosProblems/Erdos249/PaperCompleteR8/KernelRelationBasis.lean#L398}{integral normal form} and the \\href{https://github.com/wcook04/plectis-erdos/blob/25ef6245d15a47548c6926369ae8f1a0f0a14a80/ErdosProblems/Erdos249/PaperCompleteR8/FullKernelAssemblies.lean#L156}{full dyadic rational basis} are formally checked in Lean.\nThe formal statement gives the same construction for every integer base\n$k\\ge2$, with relation rank $\\sum_{j=1}^{e-1}k^j$.  The full dyadic\nassembly separately gives a basis for the infinite rational span and for\nits finitely supported rational relations.  These finite and infinite\nstatements should not be confused with a claim about infinite sums of\nrelations.\n\n\\textbf{Relation to the series value.}\nInfinite section rank does not imply irrationality of $S$. The result\nquantifies the nonregularity already cited above: it gives the rank,\nthe retained sections and their integral relations. Martin's simultaneous\ninequalities supply another proof for pairwise nonproportional affine\nforms with positive slopes~\\cite[Theorem~1]{martin-phi-inequalities}.\nHis theorem does not separate $n$ from $2n$; the two\nevaluations above do that. The linked Lean proof is independent of\nMartin's theorem.\n\nThe same independence argument also bounds the rank of an integral carry.\nIf $S\\in\\Q$, choose an integer $v>0$ with $vS\\in\\Z$ and put $u_N=vR_N$,\nwhere $R_N=\\sum_{j\\ge1}\\ph(N+j)2^{-j}$. Then $u_N\\in\\Z$,\n\\[\n u_{N+1}=2u_N-v\\ph(N+1),\\qquad\n 0\\le u_N\\le v(N+2),\\qquad u_N/2^N\\longrightarrow0.\n\\]\nFor every $e\\ge0$,\n\\[\n \\dim_{\\Q}\\operatorname{span}\n \\{n\\mapsto u_{2^j n+r}:1\\le j\\le e,\\ 0\\le r<2^j\\}\n \\ \\ge\\ 2^e-1.\n\\]\nThis is the finite family in the linked statement\n(\\lean{not\\_irrational\\_totientSeries\\_implies\\_unbounded\\_carryRank\\_unconditional}{TotientCarryKernelRigidity.lean:300});\nit contains all carry residues at levels $1,\\ldots,e$, not just odd\nresidues or an unspecified retained subfamily. Indeed, for each odd\n$r$ with $0<r<2^j$, the recurrence gives\n\\[\n v\\ph(2^j n+r)=2u_{2^j n+r-1}-u_{2^j n+r}.\n\\]\nThese $2^e-1$ totient sections are independent and belong to the span\nof the displayed carry sections after multiplication by $v\\ne0$.\nThe rank bound follows. The boundary condition $u_N/2^N\\to0$ identifies\nthe canonical carry, but is not used in this rank-transfer argument.\n\nA contradiction would require an upper bound on the same carry\nrank. Rationality and the recurrence supply no such bound:\nSection~\\ref{sub:generic-carry-rank} constructs a rational $5/4$\nseries whose unchanged odd totients force exactly this rank transfer.\nThe changed even coefficients break other totient identities. Hence\nthe comparison rules out a rank argument based only on rationality,\nlinear growth and the recurrence, while leaving arguments using\nthose additional identities untouched.\n\nA separate finite-dimensional calculation concerns the matrix\n$U_N(i,j)=\\mu((i+1)/(j+1))$ when $(j+1)\\mid(i+1)$ and zero otherwise,\nfor $0\\le i,j<N$. It is lower triangular with diagonal entries one.\nHence $\\det U_N=1$ and the map $x\\mapsto U_Nx$ is injective\n(\\lean{incidenceMobius\\_det\\_eq\\_one}{IncidenceQuotientHermitePade.lean:56}).\nThis proves injectivity of that particular map, not the impossibility\nof every finite-dimensional argument for irrationality.\n\n\\textbf{The evaluation-matrix argument.}\nFor functions $f_1,\\ldots,f_s$ with values in a field, points\n$n_1,\\ldots,n_s$ for which $\\det(f_j(n_i))\\ne0$ prove linear\nindependence: evaluating a relation gives an invertible linear system\nfor its coefficients. This elementary criterion is\nProposition~\\ref{catalogue:cert:d3}.\nIt does not, by itself, construct the points. Here the totient formula,\nCRT and Dirichlet's theorem provide them for nonproportional affine\nforms, followed by the separate zero-residue argument. Another\narithmetic function would require its own argument for nonvanishing\nof the evaluation determinant.\n\n\\begin{landscape}\n\\subsection{Further counterexamples and sharp bounds}\n\nThe table records further counterexamples and sharp bounds. For each\nproposed deduction it lists the retained hypotheses and the conclusion\nthat fails, together with the source for that argument.\n\n\\small\n\\begin{longtable}{@{}>{\\raggedright\\arraybackslash}p{0.20\\linewidth}>{\\raggedright\\arraybackslash}p{0.38\\linewidth}>{\\raggedright\\arraybackslash}p{0.36\\linewidth}@{}}\n\\toprule\n\\papertablehead\n\\textbf{Approach} & \\textbf{Argument and source} & \\textbf{Scope and remaining question} \\\\\n\\midrule\n\\endfirsthead\n\\toprule\n\\papertablehead\n\\textbf{Approach} & \\textbf{Argument and source} & \\textbf{Scope and remaining question} \\\\\n\\midrule\n\\endhead\n\\bottomrule\n\\endlastfoot\nComplete terminal divisibility pattern &\nUnder $a\\ge8$, $m\\le K$, $J+K+a+6<2\\cdot2^a$ and\n$2H+J+K+2<2^m$, the full set of divisibilities in\nObservation~\\ref{prop:TE-01-killer} is impossible: the last difference\nwould be a positive multiple of $2^m$ smaller than $2^m$.\n\\lean{not\\_actualLcmTerminalDyadicStaircase\\_of\\_room}{TotientActualLcmTopEdgeStaircase.lean:251}.\n&\nThe obstruction is specific to that pattern and those bounds.\nIf the last divisibility is omitted, the additional terminal bound of\nProposition~\\ref{prop:TE-02-inv} forces the penultimate difference to be\n$2^{m-1}$. It does not prove that such partial patterns exist.\n\\lean{puncturedDyadicStaircase\\_penultimate\\_eq\\_half}{TotientActualLcmTopEdgeStaircase.lean:1187}. \\\\\nFinite linear combinations of shifted carry recurrences &\nThe explicit sequence in Observation~\\ref{prop:B4b-kill} has\n$a_{(q-1)H-1}=\\varphi(H)$ for $2\\le q<t$ and a bounded state satisfying\n$a_i=2c_i-c_{i+1}$. Every finite integer combination of its shifts\nretains a recurrence of this form, including products of differences\nalong multiples of $H$. Its bounds acquire the factor $\\sum_j|\\lambda_j|$;\nthey are not uniform in the coefficients of the linear combination.\nThe explicit theorem and ordinary proof are given in\nObservation~\\ref{prop:B4b-kill}. &\nThe constructed coefficients are not asserted to be actual totient\ndifferences. Preserving the displayed recurrence under shifts therefore\ndoes not distinguish them from the coefficients required in the proposed\nargument. The full totient divisor identity\n\\lean{totient\\_eq\\_sum\\_mobiusTotientChannel}{LcmFactorIdealPulseObstruction.lean:328}\nretains information absent from the recurrence alone. The counterexample\ndoes not require every future proof to use that particular identity. \\\\\nAdding three multiplicative scales to an integrality test &\nFor $H\\ge0$ and positive integers $p,q$, requiring\n$R_{2kH}-R_{kH}\\in\\Z$ for every $k\\in\\{1,p,q,pq\\}$ is equivalent to\nrequiring only $R_{2H}-R_H\\in\\Z$. Neither multiplier need be prime.\nThe affine transport formula along $H\\mapsto kH$ carries integrality at\nthe first scale to each of the other three.\n\\lean{Erdos249257.FullTargetPrimeAdjunctionNoGo.fullTarget_primeAdjunction_diamond_iff_root}{FullTargetPrimeAdjunctionNoGo.lean:196}.\n&\nAdding these three integrality assertions does not strengthen the first\none. A different test involving part of the tail would need an estimate\nfor the part it discards; this equivalence provides no such estimate.\nThe result concerns the displayed four conditions, not every test using\nseveral scales. \\\\\nPrescribing valuations and finite unit residues &\nFor the valuations $\\nu_j$ and odd residues $a_j$ of\nProposition~\\ref{prop:b5}(iii), the free integers $z_j$ can always be\nchosen so that $e_{j+1}=2e_j+2^{\\nu_j}(a_j+2^u z_j)$ lies in the\nstated centred interval at every step.\n\\lean{Erdos249257.TotientTailPeriodKiller.fixedPrecisionTropicalNoGo}{TropicalCurvatureCarry.lean:137}.\n&\nThe specified residues do not exclude all finite carry sequences while\nthe higher quotients $z_j$ remain unrestricted. An argument would need\nfurther constraints on those quotients or on the actual coefficients.\nThe lemma does not assert that its freely chosen coefficients are totient\ndifferences. \\\\\nThe information retained by an endpoint residue &\nFor an integer sequence $a$ and integer initial values $u_0,v_0$,\nlet $u_{n+1}=2u_n-a(n+1)$ and $v_{n+1}=2v_n-a(n+1)$.\nInduction gives $u_L-v_L=2^L(u_0-v_0)$ for every $L\\ge0$.\n\\lean{Erdos249257.affineBinaryOrbit\\_mod\\_twoPow\\_eq}{GenericTailOrbitRigidity.lean:307}. &\nThe endpoints agree modulo $2^L$, but they are distinct integers\nwhenever $u_0\\ne v_0$. Indeed, if the forcing terms are known, then\n\\[\n u_0=\\frac{u_L+\\sum_{j=1}^{L}2^{L-j}a(j)}{2^L}.\n\\]\nOnly reduction modulo $2^L$ discards the initial value. The identity\ndoes not exclude an argument that retains the full endpoint or other\ninformation about the sequence. \\\\\nBounded finite-state / autonomous successor decoder for the binary carry &\nFor the family in Proposition~\\ref{prop:b5}(i), with $m\\ge2$, all\nhistories before $m$ agree but the next tail takes\n$\\lfloor(m+1)/2\\rfloor+1$ distinct values. A common state cannot determine\nthat tail; a set of labels permitting exact recovery must have at least\nthat many elements.\n\\lean{Erdos249257.balancedPulse\\_no\\_autonomous\\_decoder}{GenericTailOrbitRigidity.lean:247}. &\nThis counterexample concerns states determined only by the common\nhistory of the displayed family. It does not exclude a state using\nadditional information specific to the totient sequence. The bound counts\nlabels, not bits; it is not a complexity theorem about the binary digits\nof $S$. \\\\\nZeros on a progression versus nonzero coefficients arbitrarily far out &\nLet $\\chi_4(n)$ be $0$ for even $n$, $1$ for $n\\equiv1\\pmod4$,\nand $-1$ for $n\\equiv3\\pmod4$, and put\n$c(n)=\\sum_{d\\mid n}\\chi_4(d)$ for $n\\ge1$.\nDivisor pairing gives $c(n)=0$ when $n\\equiv3\\pmod4$.\n\\lean{Erdos249257.intWeightedCoeff\\_periodFourSignWeight\\_eq\\_zero\\_of\\_mod\\_four\\_eq\\_three}{CertificateKernel.lean:14226}.\nNevertheless $c(2^k)=1$ for every $k\\ge0$, since $1$ is the only\nodd divisor of $2^k$.\n&\nThis rules out nonvanishing on an arbitrarily prescribed progression,\nnot nonzero coefficients arbitrarily far out. In fact $c$ is\nmultiplicative. Its prime-power values are $1$ at powers of $2$,\n$a+1$ at $p^a$ with $p\\equiv1\\pmod4$, and $1$ or $0$ at $p^a$\nwith $p\\equiv3\\pmod4$ according as $a$ is even or odd.\nThus $c(n)\\ge0$ for every $n\\ge1$. These elementary calculations\nverify both coefficient hypotheses of\n\\lean{irrational\\_intWeightedErdosSeries\\_periodic\\_of\\_coeff\\_nonneg\\_of\\_frequently\\_ne\\_zero}{CertificateKernel.lean:14583}:\n$\\sum_{a\\ge1}\\chi_4(a)/(b^a-1)$ is irrational for every integer\n$b\\ge2$. This application concerns the displayed weighted Lambert\nseries, not the totient series $S$. \\\\\nUsing a finite exclusion test to prove membership &\n\\(\\text{finite Mersenne-weight exclusion}(x,\\mathrm{level},\\mathrm{lookahead})\\)\nis a decidable, finite-depth certificate proving non-membership only\n(\\lean{greedyMersenneDeath\\_not\\_mem}{GreedyAchievementSet.lean:1747}); a\nfinite computation excludes \\(3/4\\) at level 1\n(\\lean{three\\_fourths\\_certifiedGreedyMersenneDeath}{GreedyAchievementSet.lean:1778}). &\nThe cited test has a sound exclusion implication. The analogous\nsoundness statement for the residue test of this record,\nProposition~\\ref{catalogue:cert:a6}, also does not make a\nfinite unsuccessful search a proof of integrality. This is a limit of\nwhat follows from those tests, not an impossibility theorem for\nrationality or membership proofs using additional identities. \\\\\nReconstructing one totient value from a two-tail absolute-value linear\ncombination &\nFor any finite \\(w:\\iota\\to\\mathbb Q\\), \\(x:\\iota\\to\\mathbb N\\) with\n\\(\\sum w_i\\varphi(x_i)=1\\) (exact isolation), the crude two-tail cost\n\\(\\sum|w_i|(2(x_i+1)+(x_i+2))\\ge3\\), using only \\(\\varphi(x)\\le x\\).\n\\lean{Erdos249257.three_le_totientAdjugateTailCost}{TotientTailCarryPeriod.lean:266}\n(see also \\lean{not\\_totientAdjugateTailCost\\_lt\\_one}{TotientTailCarryPeriod.lean:307}). &\nThe displayed termwise absolute-value estimate cannot be less than $1$.\nThe same inequality holds for any coefficients $0\\le c(n)\\le n$ with\n$\\sum_iw_ic(x_i)=1$. It does not rule out a smaller error bound that uses\ncancellation between the tails, or a different approximation identity. \\\\\nHomogeneous Mersenne-primitive prime factor alone forces a contradiction &\nFor the completely multiplicative sequence $c(n)=n$, if\n$0<K<q$ and $q\\mid2^K-1$, then the scaled-tail difference is the\ninteger $K$ at every starting index, but $q\\nmid K$. \\lean{idCoeff\\_mersenneModulus\\_does\\_not\\_annihilate\\_tailShift}{TotientTailCarryPeriod.lean:342}.\n&\nA factor of $2^K-1$ does not, from the recurrence alone, have to\ndivide the inhomogeneous tail difference. The example $K=3$, $q=7$\nalready has a primitive prime factor. An application to another series\nwould need a correspondence between its coefficients and this recurrence;\nMersenne denominators alone do not provide one. \\\\\nThe unit-gap test at a prime-power denominator &\nFor $p$ prime and $e\\ge1$, if every candidate integer in the interval\nat denominator $p^e$ is a nonunit modulo $p^e$, then there is at most\none candidate. Equivalently, the notation of\nObservation~\\ref{prop:D8cert-kill} satisfies $L=0$, or $L=1$ with\n$p\\mid J+1$. \\lean{reducedDenominatorCandidateCount\\_prime\\_pow\\_le\\_one}{PrimitiveWeightCertificate.lean:111}\n(for the equivalent characterisation,\n\\lean{reducedDenominatorUnitGapCert\\_prime\\_pow\\_iff}{PrimitiveWeightCertificate.lean:54}). &\nThis candidate-count bound does not apply to every odd denominator;\nthe example modulo $15$ in Observation~\\ref{prop:D8cert-kill} has two\ncandidates. It also does not bound the denominator cutoff obtainable\nby changing the window or approximation.\nIt supplies no proof that the fixed exclusion in\nSection~\\ref{sec:adelic-height} extends to arbitrarily large denominators. \\\\\nExtra divisibility of a three-term totient difference &\nAt $H=2^{n+2}$, $j=3\\cdot2^{n+1}$,\n$\\varphi(3H+j)-2\\varphi(2H+j)+\\varphi(H+j)=-2^{n+1}$ for every $n\\ge0$.\n\\lean{fixedRankSecondDifference\\_dyadic\\_fixture\\_exact}{TotientFixedRankLcmAsymptotic.lean:462}. &\nThe exact valuation concerns this scaled family, not the LCM family\nwith $j^2\\le2^a$; see Observation~\\ref{prop:FR-03-kill}.\nThe independent coefficient calculation says that integer solutions of\n$c_1+c_2+c_3=c_1+2c_2+3c_3=0$ are exactly the multiples of $(1,-2,1)$\n(\\lean{affine\\_rank\\_three\\_kernel\\_classification}{TotientFixedRankLcmAsymptotic.lean:338}).\nIt classifies these three coefficients; it does not determine the\nvaluation of their totient combination on every restricted family. \\\\\n\\end{longtable}\n\\end{landscape}\n\n\\subsection{Signed interpolation with entire Dirichlet defects}\n\\label{long249:sec:signed-interpolation}\n\\papersectiontarget{long249-r3-interpolation}\n\nThe small block\n\\[\n z^2-2z^4+z^6=z^2(1-z^2)^2\n\\]\nhas coefficient sum $1-2+1=0$ and first moment $2-8+6=0$,\nbut its value at $z=\\beta^{-1}$ is positive for $\\beta\\ge2$.\nIt can therefore alter an interior value while preserving these two\nmoments. This is a model for one block, not a construction satisfying\nall the conditions below; translation, spacing and increasing the\ndifference order supply the additional requirements.\n\nThis construction assigns different jobs to the shape and amplitude\nof a correction. High-order finite differences annihilate low-order\nmoments; their integer amplitudes adjust the series value. As the\norders increase, later completed blocks contribute zero to each fixed\nmoment, so its defect stabilises between blocks. The prescribed initial\ncollection of moment defects is zero, not merely constant.\nSpacing the blocks controls their density and analytic convergence.\nAll corrections occur at even indices, leaving every odd totient fixed.\nThe three numbered remarks below have ordinary proofs; none has yet\nbeen formalised in Lean.\n\nFor a sequence $a$, write $F_a(z)=\\sum_{n\\ge1}a(n)z^n$. For integers\n$q\\ge1$, $0\\le c<q$ and $d\\ge0$, put\n\\[\n M_{q,c,d}(a;N)=\\sum_{\\substack{1\\le n<N\\\\ n\\equiv c\\pmod q}}n^d a(n).\n\\]\nLet $\\mu_{a,X,m}$ be the empirical distribution of the length-$m$ blocks\nstarting at the integers $X\\le N<2X$, with mass $1/X$ per start. Upper\nBanach density means the limiting upper density over all intervals of a\ngiven length, with their starting points unrestricted.\n\n\\begin{rem}[Sparse interpolation around the totient]\n\\label{long249:thm:signed-interpolation}\nFix integers $\\beta\\ge2$, $K\\ge1$, $P\\ge2$, and any function\n$f:\\N_{>0}\\to(0,\\infty)$ with $f(n)\\to\\infty$. There exist $T>0$ and\ntarget-independent sets $S\\subseteq H\\subseteq\\{n>K\\}$, with every\nelement of $S$ even, such that every\n\\[\n r\\in[F_\\varphi(\\beta^{-1})-T,F_\\varphi(\\beta^{-1})+T]\n\\]\nequals $F_b(\\beta^{-1})$ for an integer sequence $b=\\varphi+e$ satisfying:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $0\\le b(n)\\le n$, $|e(n)|\\le f(n)$, and\n$\\operatorname{supp}(e)\\subseteq S$. In particular the prefix through $K$\nand every odd coefficient are unchanged.\n\\item For every fixed $q\\ge1$, both $e(n)$ and $\\sum_{i<n}e(i)$ are\ndivisible by $q$ eventually, with a cutoff independent of $r$.\n\\item For every fixed $(q,c,d)$, the defect\n$M_{q,c,d}(b;N)-M_{q,c,d}(\\varphi;N)$ equals an integer $C_{q,c,d,r}$\nfor all sufficiently large $N\\notin H$. The cutoff is independent of $r$.\nIf $q\\le P$ and $d<P$, this defect is zero at every $N\\notin H$.\n\\item For each root of unity $\\zeta$, the radial function\n$x\\mapsto F_b(x\\zeta)-F_\\varphi(x\\zeta)$ extends smoothly to $x=1$.\nIts Taylor coefficients there lie in $\\Z[\\zeta]$. The coefficients of\norders $u<P$ are zero if the order of $\\zeta$ is at most $P$.\n\\item For every fixed $(q,c)$ the absolutely convergent series\n\\[\n D_{q,c,r}(s)=\\sum_{\\substack{n\\ge1\\\\n\\equiv c\\pmod q}}e(n)n^{-s},\n \\qquad \\operatorname{Re}s>3/2,\n\\]\nextends to an entire function, and $D_{q,c,r}(-d)=C_{q,c,d,r}$.\n\\end{enumerate}\nThe set $S$ has upper Banach density zero. The set $H$ has ordinary density\nzero and upper Banach density one. Define\n\\[\n g(n)=\\min\\left(\\inf_{k\\ge n}f(k),(n/2)^{1/4}\\right),\\qquad\n \\ell(X)=2\\left\\lfloor\\frac{\\log_\\beta g(\\lfloor X/4\\rfloor)}8\\right\\rfloor.\n\\]\nFor large $X$, $\\ell(X)>0$, and one constant $C$, independent of $r$, gives\n\\[\n d_{\\rm TV}(\\mu_{\\varphi,X,m},\\mu_{b,X,m})\n \\le\\min\\left(1,C\\left[\\frac m{\\ell(X)}+\\ell(X)^{-1/2}\\right]\\right),\n \\qquad 1\\le m\\le X.\n\\]\nThus every block length $m=o(\\ell(X))$ is preserved asymptotically.\n\\end{rem}\n\\noindent\\emph{Ordinary proof; not formalised in Lean.}\n\n\nThe allowance can grow as slowly as desired and need not be monotone.\nFor $f(n)=(\\log(n+3))^\\epsilon$, $\\epsilon>0$, the allowed block scale\nis $o(\\log\\log X)$; for $f(n)=(n+3)^\\alpha$, $\\alpha>0$, it is\n$o(\\log X)$. The exact Taylor series, growing moduli, zeros and regular\nparts of Dirichlet functions are not preserved. The correction has both signs, so the earlier one-sided perturbation\nconstraint is not retained. The construction makes no rationality\nassertion about $F_\\varphi(1/2)$.\nThe two exceptional sets also have different roles. The support $S$\ncontains the individual altered indices. The larger set $H$ contains\nthe intervals spanned by correction blocks, where a prefix may have\nincluded only part of a cancelling block. Its upper Banach density is\none because it contains arbitrarily long intervals, even though its\nordinary density is zero. Thus the moment conclusion outside $H$\ndoes not assert exact cancellation at every sufficiently large cutoff.\n\n\\begin{proof}\nWe construct the blocks first, independently of the target value.\nTheir increasing order enforces the moment and boundary conditions;\nonly their amplitudes will be chosen afterwards to obtain that value.\nAll blocks lie beyond the prescribed prefix.\n\nThe tail infimum in $g$ is positive, nondecreasing and tends to infinity.\nThe cap gives $g(n)\\le(n/2)^{1/4}$, while $g(n)\\le f(k)$ for every\n$k\\ge n$. Choose a large even $n_{-1}>K$ so that $g\\ge1$ thereafter\nand the first spacing below is at least $\\max(16,(2PP!)^2)$. Define\n\\[\n \\begin{aligned}\n s_j&=2\\left\\lfloor\\tfrac18\\log_\\beta g(n_{j-1})\\right\\rfloor,\n &n_j&=n_{j-1}+s_j,\\\\\n k_j&=\\max\\{k\\ge1:(2kk!)^2\\le s_j\\},\n &M_j&=k_j!,\\quad t_j=2k_j!,\\quad W_j=k_jt_j,\\\\\n A_j&=\\beta^{s_j+2}.\n \\end{aligned}\n\\]\nThese parameters serve two different constraints. The inequality\n$W_j\\le\\sqrt{s_j}<s_{j+1}$ keeps one block clear of the next.\nThe larger digit range $|d_j|\\le A_j$ will make the possible values\noverlap, but its coefficient cost must still fit within $g$.\nThe spacings and orders are nondecreasing and tend to infinity,\nwith $s_j=O(\\log n_{j-1})$. For digits\n$d_j\\in\\{-A_j,\\ldots,A_j\\}$, form\n\\[\n P_j(z)=d_jM_jz^{n_j}(1-z^{t_j})^{k_j},\\qquad E(z)=\\sum_{j\\ge0}P_j(z).\n\\]\nThe factor $(1-z^{t_j})^{k_j}$ supplies cancellation of moments;\n$M_j$ supplies divisibility; and the even $n_j,t_j$ leave every odd\ncoefficient unchanged. The digit $d_j$ remains free to adjust the value\nat $z=1/\\beta$.\nLet $S$ consist of all $n_j+lt_j$, $0\\le l\\le k_j$, and let $H$ be\nthe union of their full integer hulls $[n_j,n_j+W_j]$. These hulls are\ndisjoint and all points of $S$ are even.\n\n\\emph{Bounds for the coefficients.} If $s\\ge16$ and $2kk!\\le\\sqrt s$, then\n$4k!2^k\\le2\\sqrt s\\,2^{\\sqrt s/2}\\le2^s$. It follows for $\\beta\\ge2$\nthat $\\beta^2k!2^k\\le\\beta^s$. Therefore\n\\begin{equation}\\label{long249:eq:signed-budget}\n \\|P_j\\|_1\\le M_j\\beta^{s_j+2}2^{k_j}\n \\le\\beta^{2s_j}\\le\\sqrt{g(n_{j-1})}.\n\\end{equation}\nAt every modified index $n$ this is at most $f(n)$ and at most\n$(n_{j-1}/2)^{1/8}\\le\\sqrt{n/2}$. Both signs of the correction are permitted at even indices, because\nthe product formula gives\n\\[\n \\frac{\\varphi(n)^2}{n}\n =\\prod_{p^a\\parallel n}p^{a-2}(p-1)^2\\ge\\frac12,\n\\]\nbecause all odd-prime factors are at least one and the smallest possible\nfactor at two is $1/2$. Also $\\varphi(n)\\le n/2$ for even $n$.\nThus $b=\\varphi+e$, where $e$ is the coefficient sequence of $E$, stays\nbetween zero and $n$. The coefficient bound ensures absolute convergence\nof $E$ inside the unit disk.\n\n\\emph{An interval of values.} At $z=1/\\beta$, the contribution of\nblock $j$ is $d_jw_j$, where\n$a_j=(1-\\beta^{-t_j})^{k_j}$ and $w_j=M_j\\beta^{-n_j}a_j$.\nWe must check that the later blocks can fill the gaps between\nsuccessive choices of $d_j$.\nBernoulli's inequality gives $3/4\\le a_j\\le1$. Since $M_{j+1}\\ge M_j$,\n\\[\n A_{j+1}w_{j+1}=\\beta^2M_{j+1}\\beta^{-n_j}a_{j+1}\n \\ge3M_j\\beta^{-n_j}\\ge3w_j.\n\\]\nThe sum $T=\\sum_j A_jw_j$ is positive and finite, by\n\\eqref{long249:eq:signed-budget}. After choosing $d_j$, the remaining\ndigits have total allowance $T_j=\\sum_{i>j}A_iw_i$. Consecutive choices\nof $d_j$ change the current contribution by $w_j$, while\n$T_j\\ge A_{j+1}w_{j+1}\\ge3w_j$. Hence the intervals\n$[dw_j-T_j,dw_j+T_j]$, $-A_j\\le d\\le A_j$, overlap and cover\n$[-A_jw_j-T_j,A_jw_j+T_j]$. Successively choose a digit whose interval\ncontains the remaining target. The residual tends to zero with $T_j$,\nso the digits represent every value in $[-T,T]$. This is the\nfinite-choice interval principle of Crmari\\'{c} and Kova\\v{c}\n\\cite[Lemma~4(a), Remark~5]{crmarickovac2025signed}, here verified for\nsigned finite differences. It also shows $T$ can be made arbitrarily\nsmall, while positive, by increasing the initial cutoff.\n\n\\emph{Congruences and moments.} For every polynomial $p$ of degree less\nthan $k$,\n\\begin{equation}\\label{long249:eq:signed-difference}\n \\sum_{l=0}^k(-1)^l\\binom kl p(n+lt)=0.\n\\end{equation}\nThis follows by iterating the forward-difference operator, which lowers\npolynomial degree. Eventually $q\\mid M_j$, so every later coefficient\nand every partial block sum vanish modulo $q$. Complete blocks have\nmass zero, giving the cumulative assertion. Eventually also $q\\mid t_j$\nand $k_j>d$. Each such block lies in one residue class modulo $q$ and\nits moment of degree $d$ is zero by \\eqref{long249:eq:signed-difference}.\nThe restriction on the cutoff is needed here: outside $H$ it cannot\nsplit a block, so the moment sum contains only complete blocks.\nAll sufficiently late complete blocks cancel, leaving just the integer\ncontribution of the finitely many early blocks. If $q\\le P$ and $d<P$, there are no\nexceptional early blocks. These cutoffs depend on the block parameters,\nnot on the later choice of digits.\n\n\\emph{Sparsity and blocks.} Within the $j$th block the gaps in $S$ are\n$t_j\\to\\infty$; between blocks they are at least\n$s_{j+1}-\\sqrt{s_{j+1}}\\to\\infty$. Uniform spacing after a finite prefix\nimplies upper Banach density zero. To estimate the hulls without assuming\nregular variation of $f$, take any hull meeting $[X,2X+m)$, $m\\le X$.\nIts center lies in $[X/2,3X]$ eventually and its preceding center is at\nleast $X/4$, since its width and spacing are $O(\\log X)$.\nThus $s_j\\ge\\ell(X)$ for every such block. There are\n$O(X/\\ell(X)+1)$ of them, and\n\\[\n W_j+1\\le2\\sqrt{s_j}\\le\\frac{2s_{j+1}}{\\sqrt{\\ell(X)}}.\n\\]\nThe following gaps telescope to $O(X)$ over these consecutive centers.\nConsequently the total hull lengths are $O(X/\\sqrt{\\ell(X)})$.\nA hull can meet length-$m$ blocks at at most $m+W_j+1$ starts. Coupling\nthe two empirical laws at the same start proves the stated total-variation\nbound; the extra $m/X$ is absorbed by $m/\\ell(X)$.\nThe hull estimate at $m=1$, followed by dyadic summation, gives ordinary\ndensity zero for $H$. But $W_j\\to\\infty$, so $H$ contains arbitrarily\nlong solid intervals and its upper Banach density is one.\n\n\\emph{Radial smoothness.} The number of differences increases with the\nblock index, so any fixed derivative order is eventually controlled.\nFix a root $\\zeta$ of order $q$, and integers $u,A\\ge0$. Choose\n$B\\ge u+A+2$. For all sufficiently late blocks,\n$q\\mid t_j$ and $k_j\\ge B$. Factor\n\\[\n P_j(x\\zeta)=(1-x)^B R_j(x),\\qquad\n R_j(x)=d_jM_j\\zeta^{n_j}x^{n_j}\n (1+x+\\cdots+x^{t_j-1})^B(1-x^{t_j})^{k_j-B}.\n\\]\nIts coefficient norm is at most\n$\\sqrt{g(n_{j-1})}t_j^B=O_B(n_j^{1/2})$, because\n$g(n)\\le(n/2)^{1/4}$ and $t_j=O(\\sqrt{\\log n_j})$.\nAll exponents lie between $n_j$ and $2n_j$ eventually. For $x\\in[1/2,1)$,\n\\[\n \\sum_j|R_j^{(v)}(x)|\n \\le C_{B,v}\\sum_{n\\ge1}n^{v+1/2}x^n\n \\le C'_{B,v}(1-x)^{-v-3/2}.\n\\]\nThe series and derivatives converge normally on compact subintervals.\nAfter differentiating $(1-x)^B\\sum_jR_j(x)$ $u$ times, every term is\n$O((1-x)^{B-u-3/2})=O((1-x)^{A+1/2})$. Thus, after subtracting a\nfinite initial polynomial, all prescribed derivative orders vanish to\nany prescribed order at the boundary. Integration of the continuous\nderivative limits gives a single smooth extension. For an integer\npolynomial $\\sum c_nz^n$, its radial Taylor coefficient of order $u$\nat $\\zeta$ is $\\sum c_n\\binom nu\\zeta^n\\in\\Z[\\zeta]$.\nEvery such coefficient stabilizes after finitely many blocks. The\nfirst $P$ vanish at roots of order at most $P$.\n\n\\emph{Entire Dirichlet continuation.} Here we apply the same finite\ndifferences to $y^{-s}$ rather than to a polynomial. Fix $(q,c)$ and a\ncompact set of complex $s$, with $\\operatorname{Re}s\\ge-R$. Choose an\ninteger $B>R+3/2$. All sufficiently late blocks align modulo $q$ and\nhave $k_j\\ge B$. The derivative of $y^{-s}$ of order $B$ is\n$(-1)^B(s)(s+1)\\cdots(s+B-1)y^{-s-B}$. Repeated integration expresses\nits $B$th finite difference as an integral over $[0,t_j]^B$.\nThe remaining differences contribute at most $2^{k_j-B}$, so the\nabsolute Dirichlet contribution of one polynomial is at most\n\\[\n C|d_j|M_j2^{k_j-B}t_j^B n_j^{R-B}\n \\le C' n_j^{R-B+1/2}.\n\\]\nThis is summable over the distinct positive centers. The Dirichlet polynomials of the blocks therefore converge normally on every compact\nset and define an entire function. In $\\operatorname{Re}s>3/2$ they\nagree with the absolutely convergent ungrouped series. At $s=-d$,\nall late blocks vanish by their moment identity, leaving exactly\n$C_{q,c,d,r}$. These are analytic-continuation values; the original\nungrouped series need not converge at negative integers.\n\\end{proof}\n\n\\begin{rem}[Observations that do not determine irrationality]\n\\label{long249:cor:signed-observations}\nThe following data, taken together, do not determine whether\n$F_b(\\beta^{-1})$ is rational: the fixed prefix and odd coefficients;\nevery fixed eventual coefficient and cumulative congruence; the block\nlaws on scales $o(\\ell(X))$; fixed progression moments modulo eventual\ninteger constants outside $H$; progression Dirichlet functions modulo\nentire functions; and radial germs at all roots of unity modulo smooth\nfunctions with Taylor coefficients in $\\Z[\\zeta]$. This remains true\nwith exact equality of the first $P$ moment degrees and first $P$\nradial Taylor coefficients at moduli and root orders at most $P$.\n\\end{rem}\n\\noindent\\emph{Ordinary proof; not formalised in Lean.}\n\n\n\\begin{proof}\nEvery constructed sequence has the specified data of $\\varphi$, including\nthe common exceptional sets, while the nondegenerate interval of attained\nvalues contains both rational and irrational numbers.\n\\end{proof}\n\nTaking finite linear combinations of the Dirichlet series on arithmetic\nprogressions shows that every fixed periodic twist changes by an entire\nfunction. An entire summand has no negative Laurent coefficients: it\ntherefore preserves every pole and its principal part wherever the\noriginal series has meromorphic continuation. The regular part and\nzeros need not be preserved. In the untwisted case the classical identity\ngives\n\\[\n \\sum_{n\\ge1}b(n)n^{-s}=\\frac{\\zeta(s-1)}{\\zeta(s)}+D_{1,0,r}(s),\n \\qquad\\operatorname{Re}s>2.\n\\]\nThe version used by Coons~\\cite[Theorem~3.3]{coons-arxiv-signed} supplies\nthis function identity, not an irrationality theorem for one value of\n$F_\\varphi$. Likewise, Habiro's injectivity of the exact Taylor map for\na cyclotomic completion~\\cite[Theorem~5.2]{habiro2004signed} concerns a\ndifferent, stronger datum. The full sequence of Taylor coefficients is not fixed here:\nthe second Taylor coefficient of $Mz^n(1-z^t)^2$ at one is $Mt^2$.\n\nThe change of sign is indispensable for this construction. An infinitely\nsupported nonnegative integer correction has unbounded Abel limit at one.\nThe Dirichlet series of the correction also cannot extend to an\nentire function if its defining series converges in a right half-plane. Indeed, choose a positive real\n$\\sigma$ inside that half-plane. Termwise differentiation gives\n$(-1)^kD^{(k)}(\\sigma)=\\sum_ne(n)n^{-\\sigma}(\\log n)^k\\ge0$.\nThe entire Taylor series at zero and Tonelli's theorem then give\n\\[\n D(0)=\\sum_{k\\ge0}\\frac{\\sigma^k}{k!}\n       \\sum_ne(n)n^{-\\sigma}(\\log n)^k=\\sum_ne(n)<\\infty.\n\\]\nInteger nonnegativity forces finite support, and a finite correction at\nan integer reciprocal base cannot change rationality. Signed differences\nare what allow the stronger observation class above.\n\n\\begin{rem}[The first-harmonic test on the rational members]\n\\label{long249:prop:signed-first-harmonic}\nLet $b(n)$ be integers with $0\\le b(n)\\le n$ and\n$F_b(1/2)=u/(2^cv)$, where $c\\ge0$ and $v$ is odd and positive.\nChoose $h\\ge1$ with $v\\mid2^h-1$. If $X\\ge\\max(c,1)$ and\n$16(2X+h+L+2)\\le2^L$, then for every integer $N\\in[X,2X)$,\n\\[\n \\cos\\left(2\\pi\\,2^{-L}\\sum_{j=0}^{L-1}\n    [b(N+h+1+j)-b(N+1+j)]2^{L-1-j}\\right)\n \\ge\\cos(\\pi/8)>\\frac9{10}.\n\\]\n\\end{rem}\n\\noindent\\emph{Ordinary proof; not formalised in Lean.}\n\n\n\\begin{proof}\nThe actual tail $R_N=\\sum_{j\\ge1}b(N+j)2^{-j}$ lies in $[0,N+2]$\nand satisfies $R_N\\equiv2^NF_b(1/2)\\pmod\\Z$. Thus\n$R_{N+h}-R_N$ is an integer for $N\\ge c$. If $D$ denotes the integer\nwindow sum in the statement, direct truncation gives\n$2^L(R_{N+h}-R_N)-D=R_{N+h+L}-R_{N+L}$.\nThe right side has absolute value at most $N+h+L+2$, so $D/2^L$ is\nwithin $1/16$ of an integer. The cosine bound follows; its strict\ncomparison with $9/10$ follows from\n$\\cos^2(\\pi/8)=(2+\\sqrt2)/4>17/20>81/100$.\n\\end{proof}\n\nThis applies to every nonempty subset of the block, including the\n\\emph{actual totient-selected} set $\\mathcal G$ of\nTheorem~\\ref{thm:goodbasegap}(b), at $s=26$, $\\eta=1/1000$ and its\nminimal admissible depth $L(X)$. This set is held fixed rather\nthan recomputed from $b$. Wherever it has more than $67X/100$\nelements, the real first-harmonic sum exceeds $603X/1000$.\nThe interpolation interval has an explicit dyadic rational member: for\nsufficiently large $J$, the finite prefix\n$\\sum_{n\\le J}\\varphi(n)2^{-n}$ lies inside it, since the omitted tail\nis at most $(J+2)2^{-J}$. For this member one may take $h=1$.\nConsequently the preserved observations cannot imply the good-base\nsaving for all admissible sequences. No counterexample to the saving\nfor the actual totient is asserted; additional structure, such as its\nexact multiplicative relations at the relevant growing scales, is\nstill required.\n\n\\section{Hypotheses of the conditional implications}\n\\label{sec:promotion-audit}\n% The replaced, uncaptioned longtable advanced this counter. Preserve the\n% established number of the later labelled numerical table.\n\\stepcounter{table}\n\nUniformity in a parameter does not remove a hypothesis. An implication\ncan hold at every scale although the required arithmetic input is known\nat only finitely many scales, or at none. We examine that distinction\nfor the displayed implications, without drawing conclusions about all\narguments that use the same objects.\n\n\\subsection*{Implications concerning the totient series}\n\n\\paragraph{Cancellation versus a certificate.}\nThe first-harmonic implication converts the stated cosine bound on one\nblock into a finite residue certificate\n(\\lean{exists\\_certifiedKill\\_of\\_first\\_harmonic\\_gap}{FirstHarmonicGap.lean:128}). To prove irrationality, the bound must hold with the\nquantifiers in Section~\\ref{sec:survivors}: for every shift and beyond\nevery block threshold. A finite identity for the sum, or a computation\nat one block, does not establish this assertion.\n\n\\paragraph{Sign versus distance from an integer.}\nThe positivity theorem for the LCM tail difference\n(\\lean{actualLcmTailDiff\\_shift\\_pos}{TotientActualLcmOrbitSign.lean:39}) identifies the relevant endpoint under the stated\nsize and modulus bounds. Nonintegrality follows only after excluding\nthe corresponding residue interval\n(\\lean{actualLcmTailDiff\\_notMem\\_int\\_of\\_topEdgeResidueGap}{TotientActualLcmTopEdgeStaircase.lean:1260}).\nLikewise, the terminal-dominance condition is a sufficient hypothesis,\nnot a construction of the required parameters\n(\\lean{PowerTwoFlexibleActualTerminalDominanceSupply}{TotientActualLcmTopEdgeStaircase.lean:1908}).\nUnder integrality its inequality contradicts positivity of the terminal\ncarry; the contradiction establishes sufficiency, not the inequality itself.\n\n\\paragraph{An error bound versus a separated approximation.}\nThe finite-tail approximation has a uniform error radius\n(\\lean{abs\\_actualLcmTailOrbit\\_sub\\_rawApprox\\_lt}{TotientActualLcmOrbitSeparation.lean:141}). The cited exact check supplies nonintegrality at the two\nexponents $a=4,6$\n(\\lean{actualLcmTailOrbit\\_four\\_and\\_six\\_notMem\\_int}{TotientActualLcmShortKill.lean:35}); it is not an assertion that no other exponent works.\nTo obtain irrationality by this criterion, separation must occur at\narbitrarily large LCM scales. In the nondivisor approximation of\nSection~\\ref{sec:nondivisor-enclosure}, the error bound is also proved\nwhen $D\\ge2H$. Its remaining requirement is finite rational separation,\nnot an additional convergence hypothesis.\n\n\\paragraph{Denominator growth versus approximation quality.}\nThe lower bound for the product of the upper-half Mersenne factors\n(\\lean{upperHalfMersenneProduct\\_lower\\_bound}{MersenneShadowDenominatorGrowth.lean:60}) concerns a denominator expression. An irrationality argument\nneeds a comparison with the error of the corresponding approximation.\nGrowth of that denominator expression alone does not give either a\nsuccessful residue test or a nonintegral totient tail.\n\n\\paragraph{A rank lower bound versus a contradiction.}\nThe coefficient independence and hypothetical-carry rank bounds hold\nuniformly in the level\n(\\lean{linearIndependent\\_canonicalTotientKernelFamily}{TotientMahlerDefect.lean:935}, \\lean{not\\_irrational\\_totientSeries\\_implies\\_unbounded\\_carryRank\\_unconditional}{TotientCarryKernelRigidity.lean:300}).\nA contradiction would require an upper bound for the actual carry that\nis strictly smaller than the lower bound at some level. The rational\n$5/4$ comparison disproves the proposed generic upper bound. The separate integer boundary conditions in the cited\nadjugate calculation are inconsistent\n(\\lean{false\\_of\\_compressedAdjointCertificate}{TotientMahlerDefect.lean:1183}), but that calculation does not exclude matrix arguments\nwith different boundary conditions.\n\n\\subsection*{Related source comparisons for Problem~\\#257}\n\nThe following comparisons concern Problem~\\#257 and are included\nwith the hypotheses of their cited sources. No statement in this\nsubsection is an input to a proof about $S$.\n\nThe comparison of the middle-carry estimate\n(\\lean{one\\_third\\_lt\\_scaledMersenneTail\\_floorWall}{HalfCylinderMiddleCarryLowerBound.lean:2308}) with the remainder estimate\n(\\lean{SeamGreedyRemainderGapBound}{HalfCylinderSeamLimit.lean:192}) distinguishes the direction of an inequality from control\nof the required deviation. No unconditional proof of the associated sign\nclaim is supplied, and a sign alone would not give a lower bound for the\nmagnitude.\n\nThe row-capacity and later-skip implications retain their respective\nlocal deficit and skip hypotheses\n(\\lean{exists\\_exactRowStrictUpperFill\\_of\\_skippedCoreSharpCapacity}{BooleanMobiusSkipRow.lean:238}, \\lean{halfGreedy\\_precriticalSuffix\\_lt\\_of\\_future\\_skip\\_after\\_takenBlock}{BooleanMobiusCriticalCapacityCofinal.lean:603}).\nThe finite-shadow comparison distinguished an existential horizon,\nobtained from convergence, from the particular horizon $J=k-2$\nneeded in that application\n(\\lean{exists\\_greedyHalfFrozenMargin\\_nonneg\\_of\\_excess\\_neg}{HalfCylinderFiniteShadow.lean:1198}).\nThe terminal-strip formulation also imposes its own support and carry\nrestrictions\n(\\lean{HalfCarryCofinalTerminalOnlyStrip}{TerminalOnlyCofinal.lean:34}); a bound on carry size does not automatically replace a\nrestriction on the positions of the nonzero coefficients.\n\nFor finite sums of $1/(2^n-1)$, the common denominator is odd.\nTheir reduced denominators are therefore odd, so they cannot equal a\nrational number with even reduced denominator. This is the elementary\nreason behind the finite-support exclusion and its denominator source\n(\\lean{finite\\_boolSupport\\_ne\\_half}{HalfCarryReachability.lean:589}, \\lean{finiteErdosSum\\_den\\_odd}{DyadicPrefixCompression.lean:1513}).\nThe cited infinite-skipping theorem is stated for the target $1/2$\n(\\lean{infinite\\_greedyMersenneSkippedSupport\\_of\\_half\\_mem}{GreedyAchievementSet.lean:2547}). A proposed extension to other targets requires checking\nthat theorem's hypotheses and proof; the finite-denominator observation\nalone does not prove such an extension or the needed infinite-orbit estimate.\n\nFinally, the support-fraction statement and the more general collision\nbound have distinct hypotheses\n(\\lean{dyadic\\_support\\_fraction\\_reciprocalMass\\_diverges\\_or\\_gt\\_one}{RationalSupportCarrySkeleton.lean:2210}, \\lean{shiftedBooleanCollision\\_wrap\\_bound\\_of\\_common\\_multiple}{RationalSupportCarrySkeleton.lean:1968}).\nA broader use would need an additional mean-residue estimate, which is\nnot supplied, and the endpoint proofs of the dichotomy in the linked Lean\nsource have not been checked for a changed target\n(\\lean{half\\_mem\\_mersenneAchievementSet\\_or\\_exists\\_fatal\\_gap}{HalfCutLocator.lean:623}). No such extension is used here.\n\n\\paragraph{Quantifiers must be checked independently.}\nFor example, a statement of the form\n$\\exists N\\,\\exists L\\,\\forall h$ does not provide the required\n$\\forall h\\,\\forall N_0\\,\\exists N\\ge N_0\\,\\exists L$.\nThe first assertion controls many shifts at one location; the second\nrequires arbitrarily late locations for each fixed shift. Relabelling a\nsource statement does not bridge this difference.\n\n% ; - part a249_invent ; -\n\\section{Additional mathematics required by the sufficient conditions}\n\nWe turn from the proved reductions to the estimates needed to apply\nthem. The doubling identity relates window phases to exact tails;\nthe subsequent decompositions separate the weighted sums for which\ncancellation is required. The suggested analytic applications remain\nproposals where those estimates are missing. Failure of one application\ndoes not exclude the analytic method.\n\nThe section contains a doubling-orbit formulation with explicit thresholds,\na lacunary comparison sequence for the strength of a block estimate,\nand sufficient estimates for the prime-factor decomposition. The discussion of a rank upper bound\nmust also be read with the rational $5/4$ comparison sequence: rationality\nalone does not give bounded rank. These are the scope distinctions needed\nto interpret the individual statements, not priority claims.\n\n\nThe proved reductions identify explicit sufficient hypotheses. Those\nhypotheses remain to be established; the reductions do not prove them.\n\n\\subsection{The doubling identity}\n\nWrite $S=\\sum_{n\\ge 1}\\varphi(n)/2^n$, $R_N=\\sum_{m\\ge 1}\\varphi(N+m)/2^m$\nfor the local tail\n(\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak totientTail}{TotientTailPeriodKiller.lean:57}),\nand $\\Phi_N=\\sum_{n\\le N}\\varphi(n)2^{N-n}\\in\\Z$ for the integer prefix, so\nthat $2^N S=\\Phi_N+R_N$. For $h\\ge 1$ set\n\\[\n  \\alpha_h \\;:=\\; (2^h-1)\\,S .\n\\]\n\n\\begin{lem}[The doubling identity]\\label{lem:orbit}\nFor all $N\\ge 0$ and $h\\ge 1$,\n\\[\n  R_{N+1}=2R_N-\\varphi(N+1),\n  \\qquad\n  R_{N+h}-R_N \\;=\\; 2^N\\alpha_h-\\bigl(\\Phi_{N+h}-\\Phi_N\\bigr),\n\\]\nso $R_{N+h}-R_N \\equiv 2^N\\alpha_h \\pmod 1$, and consequently the exact\nfirst character of the tail difference is the $\\times 2$ orbit of one real\nnumber:\n\\[\n  e(R_{N+h}-R_N)\n  \\;=\\; e\\bigl(2^N\\alpha_h\\bigr),\n  \\qquad e(x):=\\exp(2\\pi i x).\n\\]\nHence, for fixed $h$, the phases $\\{\\,R_{N+h}-R_N \\bmod 1\\,\\}_{N\\ge 0}$ are\nthe forward orbit of $\\alpha_h \\bmod 1$ under $x\\mapsto 2x$.\n\\end{lem}\n\n\\noindent\nThe recurrence $R_{N+1}=2R_N-\\varphi(N+1)$ is one line from the definition\nand is used below only for intuition.\n\nThese identities express the phases as a lacunary exponential sum in\none real variable. The next proposition transfers an estimate for that\norbit to the truncated phases, accounting for the truncation error.\nThe identities themselves are already in the cited sources.\n\n\n\\begin{prop}[Transfer to the doubling orbit]\\label{prop:transfer}\nSuppose that\n\\[\n  \\forall h\\ge 1\\ \\forall X_0\\ \\exists X\\ge \\max(X_0,1):\\quad\n  \\sum_{N=X}^{2X-1}\\cos\\bigl(2\\pi\\,2^{N}\\alpha_h\\bigr)\\;\\le\\;\\tfrac{89}{100}\\,X .\n\\]\nThis hypothesis implies that $S$ is irrational.\n\\end{prop}\n\n\\begin{proof}\nFix $h\\ge 1$ and $X_0$, and take $X$ as supplied. Choose $L\\ge h$ with\n$2^{L}\\ge 1024\\,(2X+h+L+2)$; such $L$ exists because $2^L/L\\to\\infty$, and it\nsatisfies the size inequality $16(2X+h+L+2)\\le 2^L$ a fortiori. For every\n$N<2X$ the proved truncation bound\n(\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak norm_windowFirstExp_\\allowbreak sub_tailOrbitFirstExp_lt}{PivotAntiReconstruction.lean:202})\ngives\n$\\bigl\\|\\,{E}(h,N,L)-e(2^N\\alpha_h)\\,\\bigr\\|\n < 2\\pi(N+L+h+2)/2^{L} \\le 2\\pi/1024 < 1/100$,\nso, by the corresponding cosine comparison\n(\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak windowFirstCos_le_\\allowbreak add_of_tailOrbitGap}{PivotAntiReconstruction.lean:220}),\n$\\operatorname{Re}E(h,N,L)\\le \\cos(2\\pi 2^N\\alpha_h)+1/100$. Summing over\nthe $X$ values $N\\in[X,2X)$ gives\n$\\sum_N \\operatorname{Re}E(h,N,L)\\le \\tfrac{89}{100}X+\\tfrac{1}{100}X\n=\\tfrac{9}{10}X$, so Theorem~\\ref{thm:hgap-real}\nproduces $N\\in[X,2X)$ with $\\mathcal{C}(h,N,L)$, and $N\\ge X\\ge X_0$.\nThis is exactly the quantified certificate condition of\nTheorem~\\ref{catalogue:cert:a10}.\n\\end{proof}\n\n\\noindent\nThe proof\nabove makes the truncation margin explicit. The full implication is also\nproved in Lean\n(\\lean{ErdosProblems.Erdos249.irrational_totient_series_of_tailOrbitBlockGap}{ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean:96}).\nIt uses a real-part estimate for the exact phases and spends $1/100$\non truncation. Compare the sufficient norm condition of\nTheorem~\\ref{catalogue:mob:e1}.\nThis is a separately sufficient condition. It is not asserted\nto be equivalent to the $21/25$ norm bound or to the four separate\nestimates stated above. The constants account for different\nquantities and different error margins.\n\n\\begin{cor}[A digit version of the analytic condition]\\label{cor:digitform}\nLet $\\rho_h(X)$ denote the proportion of $N\\in[X,2X)$ with\n$\\|2^N\\alpha_h\\|_{\\R/\\Z}\\ge 1/4$. If for every $h\\ge 1$ there are cofinally\nmany $X$ with $\\rho_h(X)\\ge 11/100$, then $S$ is irrational.\nFor nondyadic $\\alpha_h$, the condition counted by $\\rho_h(X)$ is\nexactly a change between binary digits $N+1$ and $N+2$. Thus the same\nsufficient hypothesis asks for at least $11X/100$ such changes, counted\nwith $X\\le N<2X$, on arbitrarily large blocks for every $h$.\n\\end{cor}\n\n\\begin{proof}\nThe inequality $\\|x\\|_{\\R/\\Z}\\ge1/4$ gives $\\cos(2\\pi x)\\le0$.\nEvery remaining term is at most $1$, so\n\\[\n \\sum_{N=X}^{2X-1}\\cos(2\\pi 2^N\\alpha_h)\n          \\le(1-\\rho_h(X))X\\le\\frac{89}{100}X.\n\\]\nProposition~\\ref{prop:transfer} now applies. For the digit\nreading, for a nondyadic $\\alpha_h$ the fractional part of $2^N\\alpha_h$\nis in $[1/4,3/4]$ exactly when those two binary digits differ.\nThe nondyadic qualification excludes the endpoint ambiguity at $3/4$.\n\\end{proof}\n\n\\noindent\nThe exact hypothesis is the digit-change inequality in the\ncorollary. It is not a claim about an average run length: the two boundary\nruns of a finite block need not be complete, and their lengths are not\nspecified by the number of interior digit changes.\nThe finite observations below satisfy the displayed bound at their sampled\nparameters. They do not establish it on arbitrarily large blocks.\n\n\n\\begin{table}[h]\\centering\n\\begin{tabular}{r r r r r}\n\\hline\n$h$ & $X$ & $\\rho_h(X)$ & $X^{-1}\\sum_{N\\in[X,2X)}\\cos(2\\pi 2^N\\alpha_h)$ & required \\\\\n\\hline\n$1$ & $256$  & $0.4766$ & $\\phantom{-}0.0037$ & $\\le 0.89$ \\\\\n$1$ & $2048$ & $0.5073$ & $-0.0043$ & $\\le 0.89$ \\\\\n$3$ & $256$  & $0.4648$ & $\\phantom{-}0.0296$ & $\\le 0.89$ \\\\\n$3$ & $2048$ & $0.5020$ & $\\phantom{-}0.0027$ & $\\le 0.89$ \\\\\n$8$ & $2048$ & $0.5112$ & $-0.0088$ & $\\le 0.89$ \\\\\n$13$& $2048$ & $0.5049$ & $-0.0195$ & $\\le 0.89$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Finite block statistics for $\\alpha_h=(2^h-1)S$.\nThe exact digit-change counts in the six rows are $122$, $1039$, $119$,\n$1028$, $1047$ and $1034$; the proportions and cosine averages are\nrounded to four decimal places. The interval calculation described\nbelow certifies these counts and displayed roundings.}\n\\label{tab:blocks}\n\\end{table}\n\nFor the numerical enclosure, compute\n$P=\\sum_{n=1}^{6080}\\varphi(n)2^{6080-n}$. The elementary tail bound gives\n\\[\n P\\le2^{6080}S\\le P+6082.\n\\]\nThe endpoints have the same quotient on division by $2^{80}$, which\ncertifies $\\lfloor2^{6000}S\\rfloor$; an error bound smaller than one\nwould not alone certify a floor near an integer boundary.\nMultiplying this interval by $(2^h-1)2^{N-6080}$ encloses each phase.\nNone of the resulting intervals crosses a digit-test boundary, and\ninterval evaluation of the cosine certifies the rounded averages.\nAmong the first $5800$ fractional digits, the numbers of runs for\n$h=1,3,8$ are $2864$, $2964$, $2927$, and the longest runs have lengths\n$13$, $12$, $14$. These count the two boundary runs as observed, whether\nor not they continue outside the window. No randomness assumption or\nasymptotic conclusion is part of these finite calculations.\n\n\\begin{obs}\\label{obs:allroutes}\nLemma~\\ref{lem:orbit} expresses each tail phase in terms of the doubling\norbit of $\\alpha_h=(2^h-1)S$. The block conditions average those phases,\nthe prime condition samples prescribed prime-indexed positions, and the\nfull-depth condition requires a finite residue certificate for each\n$(d,N)$ at some multiple of $d$. Corollary~\\ref{cor:digitform} gives one\nsufficient digit-change test for the real-part estimate; it is not an\nequivalence with the norm bound or with the four separate budgets.\nLikewise, a bound on a binary run length is not the exact prime-gap\ncondition. Section~\\ref{sub:shape} compares the original inequalities.\n\\end{obs}\n\n\\subsection{The full-block exponential-sum estimate}\n\n\\paragraph{The required inequality.}\n\\[\n  \\forall h\\ge 1\\ \\forall X_0\\ \\exists X,L:\\quad\n  \\max(X_0,1)\\le X,\\quad 16(2X+h+L+2)\\le 2^{L},\\quad\n  \\Bigl\\|\\sum_{N=X}^{2X-1} e\\bigl(D(h,N,L)/2^{L}\\bigr)\\Bigr\\|\\le \\tfrac{21}{25}X,\n\\]\nwhere $D(h,N,L)=\\sum_{j<L}\\bigl(\\varphi(N+h+1+j)-\\varphi(N+1+j)\\bigr)2^{L-1-j}$.\nThis inequality is unproved; it implies irrationality\n(\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak DTWFirstHarmonicNormGap}{FirstHarmonicPivot.lean:75},\n\\lean{Erdos249257.\\allowbreak irrational_totient_series_of_\\allowbreak first_harmonic_norm_gap}{FirstHarmonicPivot.lean:83}).\nBy Proposition~\\ref{prop:transfer} the\nreal-part bound with constant $89/100$ for the \\emph{exact doubling\nphases} suffices; its transfer to the truncated phases uses the separate\nerror estimate in that proposition.\n\n\\paragraph{The discrepancy in terms of the doubling map.}\nFor $L\\ge h$, dividing by $2^L$ and reindexing the two ranges gives\n\\begin{equation}\\label{eq:window}\n  \\frac{D(h,N,L)}{2^{L}}\n  =\\underbrace{-\\sum_{t=1}^{h}\\frac{\\varphi(N+t)}{2^{t}}}_{\\text{head}}\n  \\;+\\;\\underbrace{(2^{h}-1)\\sum_{t=h+1}^{L}\\frac{\\varphi(N+t)}{2^{t}}}_{\\text{body}}\n  \\;+\\;\\underbrace{\\sum_{t=L+1}^{L+h}\\frac{\\varphi(N+t)}{2^{t-h}}}_{\\text{tail}} .\n\\end{equation}\n\nThe phase is a geometrically weighted sum of totients at $L+h$\nconsecutive arguments. In the first two ranges, $1\\le t\\le L$, the\ncoefficient has denominator $2^t$ and odd numerator. Its contribution is\ntherefore integral exactly when $v_2(\\varphi(N+t))\\ge t$. In the last range,\n$L<t\\le L+h$, the denominator is instead $2^{t-h}$, so the corresponding\nthreshold is $t-h$. Here $v_2(a)$ denotes the exponent of $2$ in the\npositive integer $a$. The selected offset in the four-term decomposition\nlies in the middle range, where the odd multiplier is $2^h-1$.\n\n\\paragraph{Relevant comparison estimates.}\nThe following comparisons identify the estimates still needed; they do\nnot rule out these methods.\n\n\\emph{Weyl differencing and van der Corput.} The van der Corput inequality\napplies to arbitrary bounded complex sequences~\\cite[Theorem~2.1]{edeko-vdc},\nbut its use here requires estimates for shifted correlations. For the exact\nphases $x_N=2^N\\alpha_h$ of Lemma~\\ref{lem:orbit},\n\\[\n x_{N+q}-x_N=(2^q-1)x_N.\n\\]\nThus differencing gives another doubling orbit, not a polynomial phase of\nlower degree. The finite phase $D(h,N,L)/2^L$ approximates this orbit modulo\n$1$; it is not the exact orbit itself. No bound for the resulting\ncorrelations is established here.\n\n\\emph{The large sieve.} A proposed application would need a suitable\nfamily of separated frequencies and a way to recover a bound for each\nfixed $h$. Averaging only over $h$ would not suffice: the condition asks\nfor a block estimate for every $h$, not for most of them.\n\n\\emph{Bilinear sums over prime factors.} Factoring $N+t=mp$ and using\n$\\varphi(mp)=\\varphi(m)(p-1)$ makes the selected totient contribution linear\nin the prime $p$. Section~\\ref{sub:pivot} gives this decomposition and\nidentifies the remaining weight. A bound for that weight is still needed\nbefore a Vaughan- or Vinogradov-type argument can be applied.\n\n\\emph{Erd\\H{o}s's 1948 digit method.} The comparison series is\n$E=\\sum_{n\\ge1}1/(2^n-1)=\\sum_{N\\ge1}d(N)/2^N$, whose irrationality\nErd\\H{o}s proved~\\cite{erdos1948lambert}. For the totient series, the\ncorresponding M\\\"obius expansion is exact: $\\varphi=\\mu * \\mathrm{id}$ gives\n\\begin{equation}\\label{eq:mobmers}\n  S=\\sum_{d\\ge 1}\\mu(d)\\,\\frac{2^{d}}{(2^{d}-1)^{2}}\n\\end{equation}\nThe identity follows by M\\\"obius inversion; a numerical computation gives\n$S=1.3676308019850223\\ldots$ to sixteen digits.\n\nErd\\H{o}s combines congruences forcing increasing powers of $2$ to divide\na block of divisor counts with an averaged tail bound, producing long\nzero blocks in the binary expansion~\\cite[pp.~63--66]{erdos1948lambert}.\nTo adapt that argument here, one would need congruences for the actual\ntotient coefficients together with a compatible tail estimate. The\nidentities alone provide neither combination. In \\eqref{eq:mobmers},\nthe coefficients $\\mu(d)$ can have either sign, so positivity of the\noriginal totient series does not give positivity term by term in this\nexpansion. This is a limitation of the proposed direct transfer, not a\nproof that a digit argument or a different use of the M\\\"obius expansion\ncannot work.\n\n\\paragraph{The unresolved estimate.}\nFor general coefficients $0\\le c(n)\\le n$, irrationality alone does not\nimply the block norm condition. The following example has very sparse\nnonzero coefficients. It is irrational, but almost all phases in each\nsufficiently large block lie close to $1$.\n\n\\begin{thm}[The block norm condition is stronger in the class $0\\le c(n)\\le n$]\\label{thm:lacunary}\nLet $c(n)=1$ if $n=k!$ for some $k\\ge 1$ and $c(n)=0$ otherwise, so\n$0\\le c(n)\\le n$ for all $n\\ge 1$, and let $\\beta=\\sum_{n\\ge1}c(n)/2^{n}\n=\\sum_{k\\ge 1}2^{-k!}$. Then $\\beta$ is irrational, and for every $h\\ge 1$\nand every $X\\ge 81(h+5)$,\n\\[\n  \\sum_{N=X}^{2X-1}\\cos\\bigl(2\\pi\\,2^{N}(2^{h}-1)\\beta\\bigr) \\;>\\; \\tfrac{9}{10}X .\n\\]\nConsequently the block norm condition fails for $\\beta$ at every\nsufficiently large scale, for every admissible truncation depth, even though\n$\\beta$ is irrational. Thus irrationality, nonnegative integer coefficients\nand the bound $c(n)\\le n$ do not imply the block gap. A proof for $S$\nneeds an additional property not shared by this example; the comparison\ndoes not specify which additional property will suffice.\n\\end{thm}\n\n\\begin{proof}\nThe binary expansion of $\\beta$ has infinitely many ones separated by\nunbounded strings of zeros. It is not eventually periodic, so $\\beta$ is\nirrational. Write $\\gamma=(2^h-1)\\beta$, and let $k!$ be the least\nfactorial greater than $N$. The earlier terms become integers after\nmultiplication by $2^N$, so\n\\[\n 2^N\\gamma\\equiv(2^h-1)\\sum_{j\\ge k}2^{N-j!}\\pmod{\\mathbb Z}.\n\\]\nIf $N\\le k!-h-5$, the expression on the right lies between $0$ and\n$2\\cdot2^{N+h-k!}\\le1/16$. Thus $\\|2^N\\gamma\\|_{\\mathbb R/\\mathbb Z}\\le1/16$ and\n$\\cos(2\\pi 2^{N}\\gamma)\\ge\\cos(\\pi/8)>9238/10000$. The remaining $N$\nlie in $(k!-h-5,\\,k!]$, at most $h+5$ values per factorial. The relevant\nfactorials have ratio at least $3$, and $h+5<X$, so $[X,2X)$ meets at\nmost one such interval. Hence the sum is greater than\n\\[\n \\frac{9238}{10000}(X-h-5)-(h+5)\\ge\\frac9{10}X+\\frac{h+5}{250}\n\\]\nwhen $X\\ge81(h+5)$.\n\nTo pass from these exact phases to the finite residue test, note that\n$0\\le c(n)\\le1$ puts each scaled tail in $[0,1]$. The error in the\ntruncated tail difference is therefore at most $2^{-L}$, so each complex\nphase changes by at most $2\\pi2^{-L}$. At an admissible depth,\n$2^L\\ge16(2X+h+L+2)>32X$, and the total change in the block sum is less\nthan $\\pi/16$. Its real part is consequently greater than\n$9X/10-\\pi/16>21X/25$ for $X\\ge81(h+5)$. Its norm cannot satisfy the\n$21X/25$ bound either.\n\\end{proof}\n\n\\noindent\nThis example rules out the implication from irrationality and the stated\ncoefficient bound to the block norm condition. It does not say that the\ntotient sum is Liouville-like, or that every argument based on its digits\nfails. A proof of the block estimate must use information that distinguishes\nthe actual totient phases from this comparison sequence.\n\n\\paragraph{The size of the required saving.}\nUnder Lebesgue measure on the phase circle, the binary digit maps are\nindependent fair bits. The digit-change proportion therefore tends to\n$1/2$ for almost every phase. For almost every phase, the real-part block\nsum is also $O(\\sqrt{X\\log\\log X})$, by the law of the iterated logarithm of\nErd\\H{o}s and G\\'al for lacunary series~\\cite[p.~65]{erdos-gal}.\nThe block bound follows by subtracting two initial partial sums.\nAn almost-everywhere theorem supplies no conclusion at the particular\nphase $\\alpha_h=(2^h-1)S$. The requirement is $\\rho_h\\ge 0.11$ and a saving of a constant\nfactor. The finite values in Table~\\ref{tab:blocks} have $\\rho_h$ near $0.50$\nand small block averages. They are compatible with square-root\ncancellation, but do not establish its asymptotic order. The sufficient\ncondition asks only for the stated fixed saving on arbitrarily large\nblocks. No estimate in this record supplies it for the totient phases.\n\n\\subsection{The four-term decomposition}\\label{sub:pivot}\n\\papersectiontarget{long249-r3-counts}\n\n\\paragraph{The simultaneous inequalities.}\nUse the four finite sums defined in Section~\\ref{sub:four-sums}; in\nparticular, each group mean averages the selected prime phase, not the\nresidual weight. For every $h\\ge 1$ there must exist $s\\ge 1$ and $\\eta\\in(0,1)$ such that for\nevery $X_0$ there are $X\\ge\\max(X_0,1)$ and $L$ with $h\\le L-s$,\n$16(2X+h+L+2)\\le 2^{L}$, and all four of\n\\[\n\\begin{aligned}\n \\operatorname{Re}\\sum_{N\\in\\mathcal G}w_N(z_N-\\bar z_{m_N})\n   &\\le\\tfrac{14}{25}X,\\\\\n \\left\\lVert\\sum_{N\\in\\mathcal G}w_N\\bar z_{m_N}\\right\\rVert\n   &\\le\\tfrac{1}{100}X,\\\\\n \\left\\lVert\\sum_{N\\in\\mathcal A\\smallsetminus\\mathcal G}E(h,N,L)\\right\\rVert\n   &\\le\\tfrac{1}{100}X,\\\\\n \\left\\lVert\\sum_{\\substack{X\\le N<2X\\\\N\\notin\\mathcal A}}E(h,N,L)\\right\\rVert\n   &\\le\\tfrac{8}{25}X.\n\\end{aligned}\n\\]\nTheir simultaneous validity remains unproved; together they imply irrationality\n(\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak DTWPivotResidualDecorrelation}{FirstHarmonicPivot.lean:569},\n\\lean{Erdos249257.\\allowbreak irrational_totient_\\allowbreak series_of_pivotResidualDecorrelation}{FirstHarmonicPivot.lean:577}).\nThe decomposition is an exact identity\n(\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak windowFirstExp_sum_\\allowbreak eq_pivot_decomposition}{FirstHarmonicPivot.lean:514}).\nThe last two sums are bounded by counting indices, since every\nphase has modulus one. The mean contribution is bounded by the\ntriangle inequality once a uniform group-mean estimate is available.\nThe remaining sum retains a varying weight after centring, and\nrequires a separate real-part estimate. The parameters $s,\\eta$ must\nbe fixed before $X_0$, with the estimates valid for the same $X,L$;\nseparate successful blocks do not suffice.\nFor $X\\ge4$ and $s=L-h$, the single cofactor group\n$m_N=1$, not the full assigned set, consists of the indices for which\n$N+h+1$ is prime\n(\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak mem_pivotFiber_\\allowbreak one_overlap_iff}{FirstHarmonicPivot.lean:423}).\n\n\\paragraph{An upper bound for the unassigned terms.}\nOnly an upper bound is needed here. We enclose the unassigned\narguments in a set of smooth numbers and count that larger set.\nThere is no claim that every smooth argument is unassigned.\n\n\\begin{prop}[A one-sided bound for the unassigned terms]\\label{prop:dickman}\nFix $h,s$ and choose the admissible depth $L$ minimally for each large $X$.\nPut $t=L-s+1=O_{h,s}(\\log X)$ and $y_X=4\\sqrt X+2t/\\sqrt X$.\nIf $n=N+t$ is unassigned, then its largest prime factor satisfies\n$P(n)\\le y_X$. Consequently\n\\[\n\\begin{aligned}\n \\#\\{N\\in[X,2X):N\\notin\\mathcal A\\}\n &\\le \\Psi(2X+t-1,y_X)-\\Psi(X+t-1,y_X)\\\\\n &=\\bigl(1-\\log2+o(1)\\bigr)X<\\tfrac8{25}X\n\\end{aligned}\n\\]\nfor all sufficiently large $X$. Here $\\Psi(x,y)$ counts the positive\nintegers at most $x$ whose prime factors are all at most $y$.\nThis is an upper bound for the unassigned count, not an asymptotic equality\nfor that count.\n\\end{prop}\n\n\\begin{proof}\nLet $p=P(n)$ and $m=n/p$. Since $n>1$, failure of the assignment condition\nmeans either $p\\le2\\lfloor\\sqrt X\\rfloor$ or\n$m>\\lfloor\\sqrt X/2\\rfloor$. In the second case $m>\\sqrt X/2$, so\n\\[\n p=\\frac{n}{m}<\\frac{2(2X+t)}{\\sqrt X}=y_X.\n\\]\nThe first case also gives $p\\le y_X$. Counting the resulting smooth\nintegers in $[X+t,2X+t)$ proves the first inequality.\n\nFor completeness, the classical fixed-parameter asymptotic\n$\\Psi(x,x^{1/u})\\sim x\\rho(u)$ suffices; no additional short-interval\nestimate is needed. Indeed, for either $x=X+t-1$ or $x=2X+t-1$,\n$\\log y_X/\\log x\\to1/2$. For every fixed $0<\\epsilon<1$, eventually\n\\[\n x^{1/(2+\\epsilon)}\\le y_X\\le x^{1/(2-\\epsilon)}.\n\\]\nMonotonicity in $y$ bounds $\\Psi(x,y_X)/x$ between quantities tending to\n$\\rho(2+\\epsilon)$ and $\\rho(2-\\epsilon)$. Letting $\\epsilon\\to0$ and\nusing continuity gives the limit $\\rho(2)$. Subtracting the two endpoint\nasymptotics yields $(\\rho(2)+o(1))X$. Since $\\rho(2)=1-\\log2<8/25$,\nthe claimed bound follows.\n\\end{proof}\n\n\\noindent\nThe Lean proof uses no smooth-number asymptotic. For large $X$,\n$y_X^2>2X+t-1$, so an integer in $(X+t-1,2X+t-1]$ that is not\n$y_X$-smooth has exactly one prime factor above $y_X$, and\n\\[\n \\Psi(2X+t-1,y_X)-\\Psi(X+t-1,y_X)\n =X-\\sum_{y_X<p\\le2X+t-1}\n \\left(\\left\\lfloor\\frac{2X+t-1}{p}\\right\\rfloor\n      -\\left\\lfloor\\frac{X+t-1}{p}\\right\\rfloor\\right).\n\\]\nMertens' theorems, proved in Lean with explicit constants,\nthen give $(1-\\log2)X+O(X/\\log X)$. The written proof above\ncites instead the fixed-$u$\nasymptotic~\\cite[(1.1), p.~268]{granville-smooth} and the value\n$\\rho(u)=1-\\log u$ on $[1,2]$~\\cite[(1.3), p.~268]{granville-smooth}.\nNo effective crossover scale or exact density for the unassigned indices\nis asserted. The estimate uses the minimal admissible depth; it does not\nsupply either weighted phase bound at that depth.\n\n\\begin{prop}[The excluded-cofactor estimate]\\label{prop:badcof}\nFix $h,s$ and use the minimal admissible depth $L$, as in\nProposition~\\ref{prop:dickman}; thus $t=L-s+1=O_{h,s}(\\log X)$.\nFor $\\eta\\in(0,1)$ let $B(\\eta)=\\{m\\ge1:\\varphi(m)<\\eta m\\}$, with natural density\n$D(\\eta)$; by Schoenberg's theorem $D$ exists, is continuous, and\n$D(0+)=0$~\\cite[\\S17, p.~193]{schoenberg1928}, in the framework\nof~\\cite[Theorem~1, pp.~318--319, and \\S8, p.~323]{schoenberg1936}. Then\n\\[\n  \\#\\{N\\in\\mathcal A:m_N\\in B(\\eta)\\}\n  \\;\\le\\;\\bigl(D(\\eta)+o(1)\\bigr)X ,\n\\]\nso a single choice of $\\eta$ with $D(\\eta)<1/200$ meets the $\\tfrac{1}{100}X$\nbound for all large $X$. This choice fixes $\\eta$ before $X_0$, as\nrequired. It supplies only the excluded-cofactor bound: the mean and\nmean-subtracted estimates must still hold for this same $\\eta$, and do\nnot follow from making $\\eta$ smaller.\n\\end{prop}\n\\evidenceremark{The Lean proof assumes the prime number theorem. It also\ntakes the existence of the natural density $D(\\eta)$ as a hypothesis,\nwhich the statement supplies from Schoenberg's theorem.}\n\n\\begin{proof}\nFor a fixed cofactor $m\\le\\sqrt X/2$, each assigned index gives a prime\n$p$ with $X+t\\le mp<2X+t$. Dropping the other assignment restrictions\nbounds the number of indices by the number of primes in this interval.\nThe prime number theorem, uniformly for $X/m\\ge2\\sqrt X$, and\n$t=O_{h,s}(\\log X)$ give the upper bound\n\\[\n (2+o(1))\\frac{X}{m\\log X}.\n\\]\nThe possible endpoint contribution is $O(1)$ for each $m$, hence $o(X)$\nafter summing over $m\\le\\sqrt X/2$. Natural density and partial summation\ngive\n\\[\n \\sum_{\\substack{m\\le\\sqrt X/2\\\\m\\in B(\\eta)}}\\frac1m\n       =\\left(\\frac{D(\\eta)}2+o(1)\\right)\\log X.\n\\]\nMultiplying the two estimates proves the stated bound. If\n$D(\\eta)<1/200$, its right side is less than $X/100$ for all sufficiently\nlarge $X$.\n\\end{proof}\n\n\\noindent\nThe existence of a usable cutoff can also be made explicit by an\nelementary estimate. For $x\\ge1$,\n\\[\n\\begin{aligned}\n \\frac1x\\sum_{n\\le x}\\frac n{\\varphi(n)}\n &\\le\\sum_{d\\ge1}\\frac{\\mu(d)^2}{d\\varphi(d)}\n   =\\prod_p\\left(1+\\frac1{p(p-1)}\\right)\\\\\n &\\le\\exp\\left(\\sum_{n\\ge2}\\frac1{n(n-1)}\\right)\n   =\\exp(1)<3.\n\\end{aligned}\n\\]\nThe first step uses\n$n/\\varphi(n)=\\sum_{d\\mid n}\\mu(d)^2/\\varphi(d)$ and\n$\\lfloor x/d\\rfloor\\le x/d$; the Euler product has nonnegative terms\nand converges by the displayed majorant. Markov's inequality gives\n$\\#(B(\\eta)\\cap[1,x])<3\\eta x$, hence $D(\\eta)\\le3\\eta$.\nIn particular, $\\eta=1/1000$ gives\n$D(\\eta)\\le3/1000<1/200$, an explicit sufficient choice for the\nexcluded-cofactor bound. This does not establish either weighted phase\nestimate for that choice, and it is not claimed to be an optimal cutoff.\nLean checks the estimate $\\#(B(\\eta)\\cap[1,x])<3\\eta x$ with no further input\n(\\href{https://github.com/wcook04/plectis-erdos/blob/76f44c5329c0e207f5376062c7d79125485fbceb/lean/ErdosProblems/Erdos249/PaperCompleteR21/ExcludedCofactorDensity.lean\\#L361}{Lean proof})\nand the excluded-cofactor bound at $\\eta=1/1000$ with no input at all\n(\\href{https://github.com/wcook04/plectis-erdos/blob/57941a741fceaae33ab7e2513a9cd11fee060081/lean/ErdosProblems/Erdos249/PaperCompleteR21/ExcludedCofactorEstimate.lean\\#L910}{Lean proof}).\nFor that bound Chebyshev's bound replaces the prime number theorem in the\nproof above: the primes $p$ with $a+1<p\\le2a+1$ divide\n$\\binom{2a+1}{a}\\le4^a$, so\n$\\#\\{p\\text{ prime}:a<p\\le2a+1\\}\\log a\\le a\\log4+\\log a$, the fibre bound\nbecomes $(2\\log4+o(1))X/(m\\log X)$, and the count is at most\n$(3\\log4/1000+o(1))X$ with $3\\log4/1000<0.0042$.\nThe proof uses $\\eta=1/1000$ only through $3\\eta\\log4<1/100$, so Lean checks\nthe bound with no input for every $\\eta$ with $0<\\eta<1/(300\\log4)$\n(\\href{https://github.com/wcook04/plectis-erdos/blob/ab690b0aa515faa3a432e9d47de288704ebf6bb5/lean/ErdosProblems/Erdos249/PaperCompleteR21/ExcludedCofactorEstimate.lean\\#L930}{Lean proof}).\n\nThe first element of $B(1/5)$ is\n$30030=2\\cdot3\\cdot5\\cdot7\\cdot11\\cdot13$, and\n$\\varphi(30030)/30030=5760/30030\\approx0.191808$. Enumeration through\n$3\\cdot10^5$ gives $38$ elements of $B(1/5)$ and\n\\[\n \\frac{1}{\\log(3\\cdot10^5)}\n \\sum_{\\substack{m\\le3\\cdot10^5\\\\m\\in B(1/5)}}\\frac1m\n \\approx2.82\\cdot10^{-5}.\n\\]\nThis finite weighted proportion is not the limiting density $D(1/5)$.\nUsing $\\eta=1/5$ in the asymptotic bound would require a\nrigorous bound for $D(1/5)$, including the integers beyond the enumeration;\nthe finite count does not supply one.\n\n\\paragraph{The growing dyadic modulus.}\nThe selected offset is $t=L-s+1$ in \\eqref{eq:window}. The four-term\ncondition fixes $s$ after $h$ but before the threshold $X_0$, whereas\n\\[\n 2^L\\ge16(2X+h+L+2)>32X,\\qquad 2^t>2^{6-s}X.\n\\]\nOn a group with cofactor $m$, the coefficient of $p-1$ in the phase\nhas reduced denominator $2^{\\max(t-v_2(\\varphi(m)),0)}$, because $2^h-1$\nis odd. Assignment gives\n$\\varphi(m)\\le m\\le\\sqrt X/2$. Thus, for fixed $s$ and all sufficiently\nlarge $X$, the reduced denominator is greater than\n$2^{7-s}\\sqrt X$. In particular, the assignment conditions do not reduce\nthese moduli to a fixed power of $\\log X$. This is the modulus for\nvariation in $p$, not a lower bound for the denominator of the phase\nvalue at each individual prime.\n\nChoosing $t=O(\\log\\log X)$ would require $s$ to grow with $X$, contrary\nto the stated quantifiers. Likewise, the formal observation that the phase\nis $1$ when $v_2(\\varphi(m))\\ge t$ cannot provide such groups at large $X$\nunder these size bounds. The proposed small-modulus prime-distribution\nargument therefore does not establish the mean estimate. The\nprime-number-theorem argument in Section~\\ref{sub:goodbasegap} instead\nuses prime counts in proportional intervals, not equidistribution in\nthese residue classes. Failure of the small-modulus argument does not\ndisprove the four-term condition.\n\n\\paragraph{The remaining prime-weighted correlation.}\nPropositions~\\ref{prop:dickman} and \\ref{prop:badcof} give the unassigned\nand excluded-cofactor bounds at minimal admissible depth. They must be\ncombined with mean and correlation estimates at the same $X,L,s,\\eta$;\nbounds obtained at unrelated depths cannot be substituted.\n\nWith the notation of Section~\\ref{sub:four-sums}, the correlation estimate\nstill required is\n\\[\n \\operatorname{Re}\\sum_{N\\in\\mathcal G}w_N\n \\left(e\\!\\left(\\frac{(2^h-1)\\varphi(m_N)(p_N-1)}{2^t}\\right)\n       -\\bar z_{m_N}\\right)\\le\\frac{14}{25}X.\n\\]\nThe sum is over the assigned indices with good cofactors, not all primes\nand cofactors in the interval. By \\eqref{eq:window}, $w_N$ is the phase\nwith the selected offset $t$ removed.\n\nRegrouping by $m_N=m$ gives weighted sums over the primes $p_N=p$.\nA bilinear argument would need to separate the $m$- and $p$-dependence or\nestimate the resulting two-variable weight directly. For $N=mp-t$,\nthat weight depends on $\\varphi(mp+j-t)$ at the other offsets\n$1\\le j\\le L+h$, $j\\ne t$. The identity\n$\\varphi(mp)=\\varphi(m)(p-1)$ controls the selected value, not these\nneighbouring totients. No bound for the resulting correlation is\nestablished here; this is the missing step, not a limitation of every\nbilinear method.\n\nA fixed-offset example illustrates the correlation at issue. At $h=1$\nand offset $t=2$, a prime $p\\ge5$ satisfies $\\varphi(p-1)\\equiv0\\pmod2$\nand $\\varphi(p)=p-1$. The phase of $R_{p-2}$ is therefore\n$\\chi_{-4}(p)$ times the phase of the following weighted totients, since\n$e((p-1)/4)=i^{p-1}=\\chi_{-4}(p)$. One possible fixed-saving estimate,\nwith a fixed $\\delta>0$, would be\n\\[\n  \\operatorname{Re}\\sum_{X<p\\le 2X}\\chi_{-4}(p)\\,\n    e\\Bigl(\\frac{\\varphi(p+1)}{8}+\\frac{\\varphi(p+2)}{16}+\\cdots\\Bigr)\n  \\;\\le\\;\\Bigl(\\frac{9}{10}-\\delta\\Bigr)\\,\\bigl(\\pi(2X)-\\pi(X)\\bigr).\n\\]\n\nThis fixed-offset example is not Definition~\\ref{defn:pivot}: there $s$\nis fixed while $t=L-s+1$ grows. The example isolates a correlation between\na quadratic character at $p$ and the binary phases of totients at\n$p+1,p+2,\\ldots$. It is analogous in this respect to shifted correlations\nstudied in the Chowla and Elliott problems, but the analogy does not\nsupply an applicable theorem. The cited theorems have different hypotheses and types of average: Matom\\\"aki and Radziwi\\l{}\\l{} compare short and long averages of\nreal multiplicative functions bounded by $1$, outside an exceptional set of\nshort intervals~\\cite[Theorem~1]{matomaki-radziwill}; Tao proves logarithmically averaged two-point Chowla and a corrected\nElliott statement, the latter under an explicit non-pretentiousness\nhypothesis on one of the bounded multiplicative functions\n~\\cite[Theorems~1.2--1.3]{tao-elliott}; and Tao and Ter\\\"av\\\"ainen\nprove the odd-order logarithmically averaged Chowla\nconjecture~\\cite[Theorem~1.1]{tao-teravainen}. The phase here\ndepends on $\\varphi$ at a growing window of shifts, so none of these applies\ndirectly.\n\n\\paragraph{Averaging over the block parameter.}\nFor each fixed $h$, an average of the nonnegative, normalised block norms\ncan establish the full-block condition. It is enough to prove\n\\[\n \\sum_{k=1}^{K}2^{-k}\n \\left|\\sum_{N=2^k}^{2^{k+1}-1}\n       e\\bigl(D(h,N,L_k)/2^{L_k}\\bigr)\\right|=o(K),\n\\]\nwhere every $L_k$ satisfies the size inequality for $X=2^k$.\nIndeed, the number of $k\\le K$ for which the normalised norm exceeds\n$21/25$ is at most $25/21$ times the displayed sum, hence is $o(K)$.\nThere are therefore arbitrarily large admissible blocks satisfying\nDefinition~\\ref{defn:fh}. This argument does not establish the four-term\ncondition: that condition requires all four separate budgets with common\n$X,L,s,\\eta$.\n\nThe displayed average is a Ces\\`aro average of norms on dyadic blocks.\nIt is not the signed logarithmically weighted correlation appearing in\nthe cited Chowla--Elliott theorems. The position of the absolute value\nmatters: cancellation between different blocks would not bound this\nsum of norms. Invoking entropy decrement would therefore require an\nadditional argument yielding this particular estimate.\n\nThe finite phase depends on the $L+h$ totient values at offsets\n$1,\\ldots,L+h$, with the coefficients in \\eqref{eq:window}.\nAlthough its absolute value is $1$, it is not a multiplicative function\nof $N$. Both the number of shifts and their dyadic denominators grow with\n$L$. The cited fixed-shift correlation theorems do not supply the uniform\nbounds for this growing family. No such extension is proved here.\n\n\\subsection{Separation along prime positions}\n\n\\paragraph{The quantified prime condition.}\n\\[\n  \\forall h\\ge 1\\ \\forall N_0\\ \\exists p\\ \\text{prime}:\\quad\n  \\max(N_0+h+1,\\,h+5)\\le p\n  \\ \\wedge\\\n  \\operatorname{Re}\\bigl(e(R_{p-1}-R_{p-h-1})\\bigr)<\\tfrac{9}{10}.\n\\]\nThis condition is unproved; it implies irrationality\n(\\lean{ErdosProblems.\\allowbreak Erdos249.\\allowbreak DTWNaturalPrimeTailOrbitStrictGap}{ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean:157},\n\\lean{ErdosProblems.\\allowbreak Erdos249.\\allowbreak irrational_totient_\\allowbreak series_of_naturalPrimeTailOrbitStrictGap}{ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean:524}).\n\n\\paragraph{Its meaning for the doubling map.}\n$\\operatorname{Re}e(2^{p-h-1}\\alpha_h)<9/10$ says\n$\\|2^{p-h-1}\\alpha_h\\|_{\\R/\\Z}>\\arccos(9/10)/2\\pi=0.0717831\\ldots$, so the\nrequirement is exactly: \\emph{for each $h$, the indicated prime-indexed\n$\\times2$ orbit point stays more than $0.0717831\\ldots$ from the nearest\ninteger, for cofinally many primes $p$}.  A short-run condition on the binary\ndigits can be a sufficient proxy for this circle-distance inequality, but it\nis not equivalent to it; a run-length description alone does not determine\nthe residual position inside the dyadic cylinder.\n\n\\paragraph{What the prime condition adds.}\nThe preceding condition asks for a separated point at a prime-indexed\nposition. The full-block average does not by itself locate such a point\non a prescribed sparse set. Conversely, the prime condition does not\nassert cancellation over the whole interval. They are different\nsufficient hypotheses; no general comparison of their difficulty is\nproved here.\n\n\\paragraph{Finite comparison.}\nAmong the $501$ primes in $[1000,5000)$, the strict inequality holds\nat $432$, $426$ and $430$ primes for $h=1,2,5$, respectively. The\ncorresponding proportions are $0.8623$, $0.8503$ and $0.8583$ to four\ndecimal places, and each sample has a cosine value below $-0.9999$.\nThese counts were recomputed using the same interval for $S$ as in\nTable~\\ref{tab:blocks}, with no interval meeting the threshold $9/10$.\nThey do not establish separation at arbitrarily large primes.\n\n\\paragraph{An average over a prescribed prime sample.}\nThe subset implication, Theorem~\\ref{thm:hgap-subset},\nallows a nonempty finite set $T$. If $T$ is chosen after a separated\npoint is known, the subset formulation may add no arithmetic content:\nthe equivalence proved using selected pairs,\nProposition~\\ref{prop:b2}(b), illustrates this issue.\nTo avoid that particular choice, fix the sample in advance by\n\\[\n T_{h,X}=\\{N\\in[X,2X):N+h+1\\text{ is prime}\\}.\n\\]\nFor $X\\ge4$ and $L\\ge h$, this is precisely the assigned group with\ncofactor $m_N=1$ of \\S\\ref{sub:four-sums}, taken with $s=L-h$\n(\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak mem_pivotFiber_\\allowbreak one_overlap_iff}{FirstHarmonicPivot.lean:423}).\n\nA sufficient, unproved condition is\n\\begin{equation}\\label{eq:primefibre}\n\\begin{aligned}\n &\\forall h\\ge1\\ \\forall X_0\\ \\exists X\\ge\\max(X_0,4)\\ \\exists L\\ge h:\\\\\n &\\qquad T_{h,X}\\ne\\varnothing,\\qquad\n          16(2X+h+L+2)\\le2^L,\\\\\n &\\qquad \\sum_{N\\in T_{h,X}}\\operatorname{Re}E(h,N,L)\n       \\le\\frac9{10}|T_{h,X}|.\n\\end{aligned}\n\\end{equation}\nThe subset theorem then gives a certificate at some $N\\ge X\\ge X_0$.\nNonemptiness matters: for the empty set the displayed sum inequality\nwould hold automatically and would produce no certificate. The bounds\n$X\\ge4$ and $L\\ge h$ are also the hypotheses of the identification with\nthat cofactor group.\n\nOn this sample the selected cofactor is $1$, so its coefficient is the\nodd integer $2^h-1$ and its modulus is $2^{h+1}$, fixed once $h$ is fixed.\nThe unweighted prime average has zero Ramanujan main term, since\n$c_{2^{h+1}}(2^h-1)=\\mu(2^{h+1})=0$. That observation does not establish\n\\eqref{eq:primefibre}: the prime phase is multiplied by a residual phase\nbuilt from the other totient values. Their correlation is the missing\nestimate. Restricting the sample removes the complement and unwanted\ncofactor terms, but not this residual correlation.\n\n\\begin{rem}\nFixing the sample rules out the particular argument that selects a pair\nonly after locating separation. It does not prove that\n\\eqref{eq:primefibre} is inequivalent to irrationality, or that it is a\nstrictly stronger condition. No such comparison is established here.\n\\end{rem}\n\n\\subsection{Depth equal to a period multiple}\n\n\\paragraph{The quantified residue condition.}\n\\[\n  \\text{full-depth condition} :\\equiv\n  \\forall d\\ge 1\\ \\forall N\\ \\exists t\\ge 1:\\ \\mathcal{C}(td,\\,N,\\,td),\n\\]\nunpacked, $(N+2td+2:\\Z)<D(td,N,td)\\bmod 2^{td}<2^{td}-(N+2td+2)$.\nThis condition is unproved. It is equivalent to irrationality\n(\\lean{ErdosProblems.\\allowbreak Erdos249.\\allowbreak PeriodMultipleEscape.\\allowbreak ApFullDepthEscape}{ErdosProblems/Erdos249/PeriodMultipleEscape.lean:441},\n\\lean{ErdosProblems.\\allowbreak Erdos249.\\allowbreak FullDepthRayAmplifier.\\allowbreak apFullDepthEscape_iff_irrational}{ErdosProblems/Erdos249/FullDepthRayAmplifier.lean:406});\nboth directions are proved, and the remaining gap is a nonintegral tail\ndifference $R_{N+d}-R_N$ for every $d\\ge1$ and basepoint $N$.\n\n\\paragraph{The condition in terms of binary digits.}\nWriting $h=td$ and using\n$D(h,N,L)/2^{L}=(R_{N+h}-R_N)-(R_{N+L+h}-R_{N+L})/2^{L}$ at $L=h$, together\nwith the proved strip $|R_{M+h}-R_M|<M+h+2$\n(\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak abs_tail_\\allowbreak diff_lt}{CarrySurvivorExtinction.lean:79}), applied at $M=N+h$, one gets:\n\n\\begin{prop}\\label{prop:route4}\n$\\mathcal{C}(h,N,h)$ holds whenever\n$\\bigl\\|2^{N+h}S-2^{N}S\\bigr\\|_{\\R/\\Z}>2(N+2h+2)/2^{h}$.\n\\end{prop}\n\n\\begin{proof}\n$\\|D(h,N,h)/2^{h}\\|\\ge\\|R_{N+h}-R_N\\|-|R_{N+2h}-R_{N+h}|/2^{h}\n >2(N+2h+2)/2^{h}-(N+2h+2)/2^{h}=(N+2h+2)/2^{h}$, and\n$R_{N+h}-R_N\\equiv 2^{N}(2^{h}-1)S=2^{N+h}S-2^{N}S\\pmod 1$ by\nLemma~\\ref{lem:orbit}. A residue at distance more than $(N+2h+2)/2^{h}$ from\n$\\Z$ is exactly a residue certificate at depth $h$.\n\\end{proof}\n\n\\noindent\nProposition~\\ref{prop:route4} gives a sufficient analytic\ncriterion for the condition with depth equal to the shift: for each\n$d\\ge1$ and $N\\ge0$, it is enough to find\n$h=td$, $t\\ge1$, for which the displayed distance exceeds the stated\nthreshold. The criterion is not asserted to be equivalent to each individual\nfinite certificate. It involves the phase of $S$ itself, and the threshold\ntends to zero exponentially as $t$ grows. The propagation theorem in Appendix~\\ref{long249:r3-ray} explains why\none nonintegral tail difference produces late certificates. It does not\nsupply such tail differences for all $d,N$.\n\n\\paragraph{Comparison with digit complexity.}\nThe phase inequality above is a quantitative statement at specified\npositions and period multiples. A general block-complexity assertion is a\ndifferent quantified statement, of the kind treated by the\nAdamczewski--Bugeaud method for algebraic numbers\n\\cite[Theorem~1, p.~549]{adamczewski-bugeaud}. No implication from such an\nassertion to the required inequality is established here. In particular, this record\ndoes not justify describing one existing method as the only possible route,\nor algebraicity as the sole missing input. The arithmetic nonintegrality\nobligation must be stated and proved in its own right.\n\n\\paragraph{Finite examples.}\nAt $N=300$, the least $t$ with $\\mathcal{C}(td,300,td)$ exists\nand is small for every $d\\le 24$: $t=10$ for $d=1$, $t=5$ for $d=2$,\n$t=4$ for $d=3$, $t=3$ for $d=4$, $t=2$ for $5\\le d\\le 9$, and $t=1$ for\n$10\\le d\\le 24$; in every case at or near the first depth admitted by the\nlower bound on the depth. The residues are not marginal: $r/2^{h}=0.3594$ at $d=1$,\n$0.5977$ at $d=3$, $0.6853$ at $d=7$, $0.2620$ at $d=17$, $0.5887$ at\n$d=23$, against edge radii $3.1\\cdot10^{-1}$, $8.0\\cdot10^{-2}$,\n$2.0\\cdot10^{-2}$, $2.6\\cdot10^{-3}$, $4.1\\cdot10^{-5}$. This is a\nfinite verification at the displayed basepoint and shifts. It neither\nbounds the required multiplier for arbitrary $d,N$ nor establishes the\nuniversally quantified condition.\n\n\\paragraph{A possible heuristic, not an estimate.}\nIf a phase $U$ were uniform on $\\R/\\Z$, then\n$\\Pr(\\|U\\|\\le\\epsilon)=\\min(1,2\\epsilon)$ for $\\epsilon\\ge0$.\nAt $\\epsilon=2(N+2h+2)2^{-h}$ this gives the marginal model\n$\\min(1,4(N+2h+2)2^{-h})$. The phases\n$2^N(2^{td}-1)S$ for fixed $d$ are arithmetically dependent; no independence\nmodel or uniform distribution theorem for this orbit is proved here.\nConsequently the marginal estimate cannot be multiplied across depths, and\nno summability conclusion over all basepoints follows from it. The finite\nexperiments are finite evidence; the condition for every $d$ and $N$\nremains unproved.\n\n\\subsection{A rational series with unbounded carry rank}\n\\label{sub:generic-carry-rank}\n\nFor an integer sequence $u$, write $\\operatorname{rk}_e(u)$ for the\nrational dimension of the span of all sequences\n$n\\mapsto u_{2^j n+r}$ with $0\\le j\\le e$ and $0\\le r<2^j$.\nRationality does not force these ranks to be bounded, even for one fixed\ncoefficient sequence. More precisely, the assumptions\n\\[\n 0\\le c(n)\\le n,\\qquad \\sum_{n\\ge1}c(n)2^{-n}\\in\\Q,\n \\qquad v\\in\\N_{>0},\\qquad u_n\\in\\Z,\n\\]\n\\[\n u_{n+1}=2u_n-vc(n+1),\\qquad u_n/2^n\\longrightarrow0\n\\]\ndo not imply $\\sup_e\\operatorname{rk}_e(u)<\\infty$. Nor do they imply\nan upper bound that is $o(2^e)$. Such a bound for the actual totient\ncarry would contradict the lower bound of Proposition~\\ref{prop:rank},\nbut the following rational comparison already has that lower bound.\n\nThe limit condition fixes the value of the series. Dividing the\nrecurrence by $2^{n+1}$ and summing gives\n\\[\n v\\sum_{n=1}^{N}c(n)2^{-n}=u_0-2^{-N}u_N.\n\\]\nThus $u_N/2^N\\to0$ identifies the sum as $u_0/v$; it imposes no\nfinite-dimensional description of the carry sections. In the\nconstruction below $u_0=5$ and $v=4$.\nFor fixed $c$ and $v$ this boundary condition also makes the carry\nunique: two solutions of the recurrence differ by $C2^N$, whose\nquotient by $2^N$ tends to $C$. Uniqueness therefore leaves no freedom\nto choose a simpler carry. The comparison shows that this unique\ncarry can nevertheless have unbounded section rank.\n\n\\paragraph{The comparison with value $5/4$.}\nWe alter only even coefficients, whose weights are\n$2^{-2m}=4^{-m}$. Thus we need a base-four expansion of $5/4-S$,\nwith signed digits small enough to keep the corrected coefficients\nbetween zero and their indices. The digits $-2,-1,0,1$ will do this;\nthe first even coefficient will require its own check.\nThe elementary bounds $9/8\\le S\\le3/2$ follow respectively from the\nfirst four terms and from $\\ph(n)\\le n-1$ for $n\\ge2$.\nPut $x_0=5/4-S$ and, for $m\\ge1$, define\n\\[\n d_m=\\left\\lfloor4x_{m-1}+\\frac23\\right\\rfloor,\n \\qquad x_m=4x_{m-1}-d_m.\n\\]\nThe interval $[-2/3,1/3)$ contains $x_0$ and is preserved by this\nrecurrence, so $d_m\\in\\{-2,-1,0,1\\}$. Iteration gives\n\\[\n x_0=\\sum_{m=1}^{M}d_m4^{-m}+4^{-M}x_M,\n \\qquad |x_M|\\le\\frac23.\n\\]\nConsequently $x_0=\\sum_{m\\ge1}d_m4^{-m}$. Define\n\\[\n c(0)=0,\\qquad c(2m-1)=\\ph(2m-1),\\qquad\n c(2m)=\\ph(2m)+d_m\\quad(m\\ge1).\n\\]\nThese are nonnegative integers: for $m\\ge2$, $\\ph(2m)\\ge2$;\nfor $m=1$, the sharper bound $x_0\\ge-1/4$ gives $d_1\\ge-1$.\nThe inequalities $d_m\\le1$ and $\\ph(2m)\\le2m-1$ also give $c(n)\\le n$.\nThus $|c(n)-\\ph(n)|\\le2$ and\n\\[\n \\sum_{n\\ge1}c(n)2^{-n}=S+\\sum_{m\\ge1}d_m4^{-m}=\\frac54.\n\\]\nThis is the centred base-four construction in the Lean\n\\href{https://github.com/wcook04/plectis-erdos-lean/blob/52f29ad173b04e3bac941b3663f2b9aebe5de0bb/ErdosProblems/Erdos249/ParityPerturbedRationalControl.lean#L379}{comparison-series theorem}.\n\nThe construction has separated value from rank: the even coefficients\nset the value, while the unchanged odd coefficients force large rank.\nFor the latter, take $v=4$ and\n$u_N=4\\sum_{j\\ge1}c(N+j)2^{-j}$. The rational sum makes $u_N$ integral;\nthe coefficient bound gives $0\\le u_N\\le4(N+2)$. For odd $r$, the\ncarry recurrence and $c(2^jn+r)=\\ph(2^jn+r)$ give\n\\[\n 4\\ph(2^j n+r)=2u_{2^j n+r-1}-u_{2^j n+r}.\n\\]\nThe same $2^e-1$ independent totient sections therefore lie in the\nspan of the carry sections at levels $1,\\ldots,e$, proving\n$\\operatorname{rk}_e(u)\\ge2^e-1$ for every $e\\ge0$.\nThe corresponding formal statement is the\n\\href{https://github.com/wcook04/plectis-erdos-lean/blob/52f29ad173b04e3bac941b3663f2b9aebe5de0bb/ErdosProblems/Erdos249/ParityPerturbedRationalControl.lean#L478}{unbounded-rank theorem for this comparison}.\nThis example is distinct from the bounded $3/2$ parity comparison in\nProposition~\\ref{prop:parity}. It disproves an inference from the listed\nassumptions alone, not one that also uses identities of the actual\ntotient coefficients.\n\n\\paragraph{The rank quantities.}\nFinite-dimensional span of all dyadic sections is the definition of a\n$2$-regular sequence in the sense of Allouche and\nShallit~\\cite[Definition~2.1]{allouche-shallit}.  For\ninteger-valued sequences this is equivalent to finite generation of the\n$\\mathbb Z$-module generated by the sections: if their span has dimension $d$,\nsome $d$ evaluations embed it in $\\mathbb Q^d$ and that module in\n$\\mathbb Z^d$.\nCoons~\\cite[Theorem~3.2]{coons} proved that\n$\\varphi$ is not $k$-regular for any $k\\ge2$.  The\nunconditional Lean theorem\n(\\lean{Erdos249257.\\allowbreak not_finiteDimensional_\\allowbreak span_fullTotientKernel}{TotientMahlerDefect.lean:1145})\nproves the case $k=2$ independently.  For integer coefficient and carry sequences, the recurrence transfers\nnon-$2$-regularity of the coefficients to non-$2$-regularity of the carry.\nA proof must include the zero-residue sections, not just the positive\nresidues. The following argument gives an explicit dimension bound.\n\nFor a rational-valued sequence $f$, let $V_e(f)$ be the span of all its sections\nthrough levels $0,\\ldots,e$. Define the backward shift by\n$(Bf)(0)=0$ and $(Bf)(n)=f(n-1)$ for $n\\ge1$, and let $\\delta_0$ be $1$\nat $0$ and $0$ elsewhere. A level-$j$ section of $Bf$ is a section of\n$f$ if its residue is positive. At residue zero it is the backward shift\nof $n\\mapsto f(2^jn+2^j-1)$. Hence\n\\[\n V_e(Bf)\\subseteq V_e(f)+B(V_e(f)).\n\\]\nFor integer sequences $c,u$ and a positive integer $v$, the recurrence\n$v c(n+1)=2u(n)-u(n+1)$ gives\n\\[\n c=\\frac{2Bu-u}{v}+\\left(c(0)+\\frac{u(0)}v\\right)\\delta_0.\n\\]\nSince every section of $\\delta_0$ is $0$ or $\\delta_0$, it follows that\n\\[\n V_e(c)\\subseteq V_e(u)+B(V_e(u))+\\Q\\delta_0,\n \\qquad \\dim_{\\Q} V_e(c)\\le2\\dim_{\\Q} V_e(u)+1.\n\\]\nIn particular, bounded carry ranks would force bounded coefficient ranks.\nNon-$2$-regularity of $c$ therefore forces unbounded carry ranks.\nThis is an ordinary proof, not a claim of a new Lean declaration.\nIt also gives the generic bound\n$\\dim_{\\Q} V_e(u)\\ge2^{e-1}$ for $c=\\varphi$ and $e\\ge1$, using the\ncoefficient rank $2^e+1$. The factor $2$ is not an additive $O(1)$ loss,\nand this bound concerns all sections, including level zero.\n\nThe theorem for $\\varphi$ in Proposition~\\ref{prop:rank} is sharper: the carry sections at all\nresidues of levels $1,\\ldots,e$ have rank at least $2^e-1$. It uses the independence of the odd-residue\ntotient sections. Neither lower bound is an irrationality proof, since\nrational coefficient series can have nonregular carries. The separate\n$5/4$ comparison verifies this directly. Any proposed rank upper bound must\nuse additional totient-specific hypotheses not satisfied by the $5/4$\ncomparison sequence.\n\n\\subsection{Finite observations and quantified conditions}\\label{sub:shape}\n\nThe finite observations can be compared with the quantified\nconditions, but they do not establish an asymptotic digit distribution.\nThey also supply no basis for ranking possible proof methods.\n\n\\paragraph{Comparison of conditions.}\nThe following table records the conditions themselves, rather than\nsufficient digit proxies. Lemma~\\ref{lem:orbit} explains how their phases\nare related.\n\n\\begin{center}\n\\setlength{\\tabcolsep}{5pt}\n\\begin{tabular}{>{\\raggedright\\arraybackslash}p{0.16\\linewidth} >{\\raggedright\\arraybackslash}p{0.44\\linewidth} >{\\raggedright\\arraybackslash}p{0.32\\linewidth}}\n\\hline\nCondition & Required assertion & Indices and quantifiers \\\\\n\\hline\nBlock norm & Norm at most $21X/25$ for the truncated phase sum &\nAll $N\\in[X,2X)$ at one depth; arbitrarily large $X$ for each $h$ \\\\\nFour terms & The four separate budgets of Definition~\\ref{defn:pivot} &\nOne common $X,L,s,\\eta$; $s,\\eta$ fixed before $X_0$ \\\\\nPrime gap & Distance from $\\Z$ greater than $\\arccos(9/10)/(2\\pi)$ &\nThe phase $2^{p-h-1}\\alpha_h$ at arbitrarily large primes for each $h$ \\\\\nFull depth & $\\mathcal C(td,N,td)$ for some $t\\ge1$ &\nEvery $d\\ge1$ and every $N\\ge0$ \\\\\nGeneric rank bound (false) & A uniform or subexponential upper bound for carries of rational coefficient series &\nAll finite dyadic levels; refuted by the $5/4$ comparison \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\n\\noindent\nThe first four conditions concern the same constant but impose different\naveraging, sampling and separation requirements. The $11/100$ digit-change\ntest in Corollary~\\ref{cor:digitform} is sufficient for a real-part bound;\nit is not a replacement for the norm or four-term rows. Only the full-depth\nrow is proved here to be equivalent to irrationality. The last row records\na false generic upper bound, not an additional open hypothesis.\n\n\\paragraph{Quantitative estimates.}\nThe counterexamples identify missing assumptions in the specific\narguments tested here. They do not imply that the binary digits of $S$\nare random, normal, or unusually structured. A constructed counterexample\ncan disprove an implication without resembling $S$ in other respects.\nLikewise, failure of one congruence construction does not exclude every\nuse of the Chinese remainder theorem.\n\n\\paragraph{Additional arithmetic information.}\nTable~\\ref{tab:blocks} records finite observations: the sampled digit-change\nproportions are near $0.50$, above the sufficient threshold $0.11$, and\nthe displayed normalized cosine sums are below $0.89$. These computations\nare compatible with the proposed conditions. They do not establish those\nconditions for every shift and arbitrarily large blocks, or prove a\nfair-coin model for the digits. The relevant quantified estimates remain\nunproved here.\n\nThe recorded small margins near a forbidden endpoint also require careful\ninterpretation. One small margin disproves a proposed lower bound larger\nthan that margin at that index. It does not rule out every positive margin\non a suitable subsequence, or every estimate with scale-dependent bounds.\nNo numerical census in this record determines which of these possibilities\nholds asymptotically.\n\n\\paragraph{Scope of the comparison.}\nCorollary~\\ref{cor:digitform} specifies one sufficient digit statistic:\nfor every fixed $h$, the digit-change proportion must reach at least\n$11/100$ on arbitrarily large blocks. A statement for one shift, a smaller\npositive proportion, or finitely many blocks is not this hypothesis. An\nupper bound $o(N)$ on the longest run alone would not imply irrationality;\na periodic alternating expansion has bounded runs. The prime-index\ncondition in \\eqref{eq:primefibre} is another proposed sufficient estimate.\nIts exact algebraic decomposition does not establish the required\ncorrelation bound.\n\\section{Remaining arithmetic questions}\n\nThe remaining questions ask for witnesses with unbounded indices.\nWe collect them in residue and exponential-sum form, retaining the\nquantifiers needed to imply irrationality of\n$S=\\sum_{n\\ge0}\\varphi(n)2^{-n}$. Erd\\H{o}s Problem~\\#249 remains open.\n\nThis section collects quantified conditions on the scaled tail\n$R_N=\\sum_{j\\ge0}\\varphi(N+1+j)2^{-(j+1)}$ that suffice for irrationality.\nThe implication theorems and the arithmetic assumptions have different\nstatus: the former are proved in the indicated sources, while the latter\nare not supplied for the actual totient sequence.\n\\lean{Erdos249257.TotientTailPeriodKiller.totientTail}{TotientTailPeriodKiller.lean:57}\n\n\nEach condition is stated with its quantifiers and linked to its formal\nsource. The comparison distinguishes proved equivalences from one-way\nsufficient conditions. It is not an exhaustive list of possible methods.\nA finite list of certificates does not stand in for a condition beyond\nevery threshold, and an implication does not establish its antecedent.\n\n\nRecall that $H_t=H(t)=\\operatorname{lcm}(1,\\dots,t)$\n(\\lean{Erdos249257.TotientTailPeriodKiller.periodLcm}{CarrySurvivorExtinction.lean:515}).\nFor nonnegative integers $h,N,L$, put\n\\[\nD(h,N,L)=\\sum_{j=0}^{L-1}\n \\bigl(\\varphi(N+h+1+j)-\\varphi(N+1+j)\\bigr)2^{L-1-j}.\n\\]\n\\lean{Erdos249257.TotientTailPeriodKiller.windowDiscrepancy}{TotientTailPeriodKiller.lean:66}\nThe certificate $\\mathcal C(h,N,L)$ is the pair of strict inequalities\n\\[\n N+h+L+2<D(h,N,L)\\bmod2^L<2^L-(N+h+L+2).\n\\]\n\\lean{Erdos249257.TotientTailPeriodKiller.certifiedKill}{TotientTailPeriodKiller.lean:72}\nThis condition is decidable.\nThus the least nonnegative residue must lie more than $N+h+L+2$\nfrom either endpoint. For fixed $h,N$, this excluded radius grows\nlinearly with $L$, while its proportion of the modulus tends to zero.\n\n\\subsection{Certificate completeness}\n\nFor each fixed $h,N\\ge0$, a certificate at some \\emph{unrestricted}\ndepth exists exactly when the tail difference is nonintegral:\n\\[\n \\bigl(\\exists L\\ge0,\\ \\mathcal C(h,N,L)\\bigr)\n \\quad\\Longleftrightarrow\\quad R_{N+h}-R_N\\notin\\mathbb Z.\n\\]\n\\lean{Erdos249257.TotientTailPeriodKiller.exists_certifiedKill_iff_tail_diff_notMem_int}{LcmConeFlatness.lean:316}\n(The same equivalence is also stated at\n\\lean{Erdos249257.totient\\_tail\\_window\\_kill\\_exists\\_iff\\_tail\\_diff\\_not\\_int}{CertificateKernel.lean:19025}.)\n\nThis equivalence removes the existential depth from the unrestricted\ncertificate conditions below, including the diagonal and LCM-grid forms.\nIt does not by itself supply a depth bound, a prescribed separation\nmargin, a common depth for a family, or an exponential-sum estimate.\nSuch requirements must be retained unless a separate argument removes\nthem. In particular, completeness does not establish any of the\nquantified nonintegrality assertions needed for irrationality.\n\n\\subsection{The original quantified residue condition}\n\nThe threshold quantifier excludes every possible starting point for\na rational tail period.\n\n\\begin{quote}\n\n\\textbf{The quantified residue condition.}\n\\[\n  S\\notin\\mathbb Q\n  \\quad\\Longleftrightarrow\\quad\n  \\forall h\\ge1\\ \\forall N_0\\ \\exists N\\ge N_0\\ \\exists L,\n  \\quad\\mathcal C(h,N,L).\n\\]\n\\end{quote}\nHere $\\mathcal C(h,N,L)$ is the finite residue test defined earlier;\nthe universal quantifiers over $h$ and $N_0$ belong to the displayed\nassertion, not to that finite test.\n\\lean{Erdos249257.TotientTailPeriodKiller.irrational_totient_series_iff_certificate_supply}{LcmConeFlatness.lean:412}\nThe equivalence is proved; the quantified existence assertion is not proved.\n\nThe reverse direction is by contradiction against the tail-period law: if $S$ were\nrational, the eventual-period theorem\n(\\lean{Erdos249257.TotientTailPeriodKiller.eventual_period_of_not_irrational}{TotientTailPeriodKiller.lean:358},\nunconditional) supplies a period $h>0$ and pre-period $N_0$ with\n$R_{N+h}-R_N\\in\\mathbb{Z}$ for all $N\\ge N_0$; the certificate hypothesis\ninstantiated at that $(h,N_0)$ produces an $N\\ge N_0$ where\n$R_{N+h}-R_N\\notin\\mathbb{Z}$, a contradiction via nonintegrality from a certificate\n(\\lean{Erdos249257.TotientTailPeriodKiller.tail_diff_notMem_int_of_certifiedKill}{TotientTailPeriodKiller.lean:262}).\n\nFor the forward direction, irrationality gives pointwise non-integrality of\nevery positive tail shift and certificate completeness supplies a witness\nalready at $N=N_0$.\n\n\nFor fixed $h$, the required certificate may choose both $N\\ge N_0$ and\n$L$. It is not required at every $N$ or every depth. Its error radius\nis $N+h+L+2$, so for fixed $h,N$ it grows only linearly in $L$, whereas\nthe modulus is $2^L$. The missing assertion is that an admissible\nresidue lies outside both endpoint intervals beyond every threshold,\nfor every positive $h$. The reformulations below specify other ways\nto choose such witnesses; none establishes this assertion for the\nactual totient sequence.\n\n\\subsection{Multiples, diagonals, and LCM grids}\n\n\nCompleteness allows the irrationality question to be expressed using\nmultiples of a period, diagonal tails or an LCM grid. The implications\nbelow prove these equivalences. Each formulation permits a different\nestimate to be attempted, but none supplies the witnesses it requires.\n\n\\paragraph{Multiples of a period.}\n\\[\n  \\forall h_0>0,\\ \\forall N_0,\\ \\exists m>0,\\ \\exists N\\ge N_0,\\ \\exists L,\\\n    \\mathcal{C}(m\\cdot h_0, N, L)\n  \\ \\Longrightarrow\\ S\\notin\\mathbb Q.\n\\]\n\\lean{Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_multiple_certificate_supply}{CarrySurvivorExtinction.lean:502}\n\nThe preceding residue condition implies this one by taking $m=1$.\nConversely, a hypothetical period $h_0$ also makes every positive\nmultiple $mh_0$ a period of the tail fractional parts. A certificate\nfor one such multiple contradicts that consequence. The freedom to\nchoose $m$ changes the allowed witnesses, while the quantified\nassertion remains equivalent to irrationality.\n\n\\paragraph{The diagonal.}\n\\[\n  S\\notin\\mathbb Q\n  \\ \\Longleftrightarrow\\\n  \\forall t_0,\\ \\exists t\\ge t_0,\\ \\exists L,\\\n    \\mathcal{C}(H_t, H_t, L).\n\\]\n\\lean{Erdos249257.TotientTailPeriodKiller.irrational_totient_series_iff_lcm_diagonal_certificate_supply}{LcmConeFlatness.lean:426}\n\nThe two parameters in a hypothetical rational tail period are absorbed\nby one LCM index. For $t\\ge\\max(h_0,N_0)$, both $h_0\\mid H_t$ and\n$H_t\\ge t\\ge N_0$ hold. A diagonal certificate then contradicts\nperiodicity through the multiple-period argument; the precise\nintermediate implication is proved in Lean\n(\\lean{Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_lcm_certificate_supply}{LcmDiagonalReduction.lean:92}).\nConversely, irrationality and pointwise completeness give a diagonal\ncertificate already at $t=t_0$.\n\nThe finite assertion $P(t):\\equiv\\exists L\\,\\mathcal C(H_t,H_t,L)$\nis verified for every $t\\le82$\n(\\lean{ErdosProblems.Skip.LadderT67.exists_diagonalKill_le_82}{ErdosProblems/Skip/LadderT67.lean:71264}).\nThe historical list contained 28 explicit values through $t=64$, with\ndepths $[6,5,7,7,9,14,15,14,21,22,23,26,\\dots]$ for\n$t\\in\\{1,2,3,4,5,7,8,9,11,13,16,17,\\dots\\}$\n(\\lean{Erdos249257.TotientTailPeriodKiller.certifiedKill_diagonal_all_imported}{DiagonalPincerCertificates.lean:2870};\nendpoint\n\\lean{Erdos249257.TotientTailPeriodKiller.certifiedKill_diagonal_t64}{DiagonalPincerCertificateT64Endpoint.lean:1928}).\nThese are finite computations. The current contiguous\nrange is still finite and does not establish witnesses at arbitrarily\nlarge indices; no certificate at $t=83$ is claimed here.\n\n\\paragraph{The LCM grid.}\n\\[\n  \\forall t_0,\\ \\exists t\\ge t_0,\\ \\exists q,m,L,\\ 0<q\\ \\wedge\\\n    \\mathcal{C}(m\\cdot H_t,\\, q\\cdot H_t,\\, L)\n  \\ \\Longrightarrow\\ S\\notin\\mathbb Q.\n\\]\n\\lean{Erdos249257.irrational_totient_series_of_lcm_cone_window_kill_supply}{CertificateKernel.lean:19014}\nThe diagonal choice is $q=m=1$. Allowing other positive grid points\ntherefore enlarges the choice of finite witnesses. To prove\nsufficiency, suppose rationality supplies a period $h$ and pre-period\n$N_0$. For $t\\ge\\max(h,N_0)$, all positive multiples of $H_t$ are\npast the pre-period and their differences are multiples of $h$.\nConsequently their tails have the same fractional part:\n$R_{qH_t+mH_t}-R_{qH_t}\\in\\mathbb Z$ for every $q>0$ and $m\\ge0$\n(\\lean{Erdos249257.rational_totient_series_forces_lcm_cone_flatness}{CertificateKernel.lean:19002}). A certificate for any such difference is a contradiction.\n\nThe finite-grid condition of Theorem~\\ref{catalogue:cert:b10a} offers\na second test. At $H=H_t$, choose a finite nonempty\n$Q\\subseteq\\mathbb N_{>0}$ with $B_q=qH_t+L+2<2^L$ for every $q$.\nIt requires\n\\[\n \\forall q_i\\in Q\\ \\exists q_j\\in Q,\\qquad\n B_{q_j}<(A_{q_i}-A_{q_j})\\bmod2^L,\n \\quad A_q=\\sum_{j=1}^{L}\\varphi(qH_t+j)2^{L-j}.\n\\]\nFor every threshold $t_0$, such data at some $t\\ge t_0$ would imply\nirrationality. This quantified condition, including the nonemptiness\nand size assumptions, remains unproved\n(\\lean{Erdos249257.irrational_totient_series_of_lcm_cone_nonflat_supply}{CertificateKernel.lean:19140}). The modulus bound is one-sided at each point; it does not imply\na factor-of-two reduction in the number of binary digits.\n\n\\paragraph{Equivalent nonintegrality statements}\nVia the completeness iff above, the diagonal and cone forms restate,\n\\emph{exactly}, with no reference to $\\mathcal{C}$ at all:\n\\[\n  \\forall t_0,\\ \\exists t\\ge t_0,\\ R_{2H_t}-R_{H_t}\\notin\\mathbb{Z}\n  \\ \\Longleftrightarrow\\ \\text{diagonal certificate condition}\n  \\ \\Longrightarrow\\ S\\notin\\mathbb Q,\n\\]\n\\lean{Erdos249257.irrational_totient_series_of_lcm_diagonal_nonintegrality_supply}{CertificateKernel.lean:19034}\n(cone analogue at\n\\lean{Erdos249257.irrational\\_totient\\_series\\_of\\_lcm\\_cone\\_nonintegrality\\_supply}{CertificateKernel.lean:19046})\n\nThus one can ask directly whether $R_{2H_t}-R_{H_t}\\notin\\mathbb Z$\nfor arbitrarily large $t$. Finite instances are known in this record;\nthe universally quantified assertion is not proved here.\n\n\\subsection{The block exponential-sum condition}\n\n\nA bound for an exponential sum is sufficient for the quantified\nresidue condition. With $e(x)=\\exp(2\\pi i x)$, the assertion is\n\\[\n  \\text{block norm condition} :\\equiv\\\n  \\forall h>0,\\ \\forall X_0,\\ \\exists X,L,\\quad\n  \\max(X_0,1)\\le X\\ \\wedge\\\n  16\\,(2X+h+L+2)\\le 2^L\\ \\wedge\\\n  \\Bigl\\|\\sum_{N\\in[X,2X)} e\\bigl(D(h,N,L)/2^L\\bigr)\\Bigr\\| \\le \\tfrac{21}{25}X.\n\\]\n\\lean{Erdos249257.TotientTailPeriodKiller.DTWFirstHarmonicNormGap}{FirstHarmonicPivot.lean:75}\nThe universally quantified assertion is not proved.\n\n\\[\n  \\text{block norm condition} \\ \\Longrightarrow\\ S\\notin\\mathbb Q.\n\\]\n\\lean{Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_first_harmonic_norm_gap}{FirstHarmonicPivot.lean:83}\nThe threshold is $X_0$, while $X$ and $L$ are chosen witnesses and may\ndepend on both $h$ and $X_0$. Taking $X_0=N_0$ gives a certificate\nat some $N\\in[X,2X)$, hence at $N\\ge N_0$. The elementary reason is\nthat, if every residue failed the test, each would be within\n$(2X+h+L+2)/2^L\\le1/16$ of an integer. Every summand would then have\nreal part at least $\\cos(\\pi/8)>21/25$, contradicting the norm bound.\nThe implication is proved; the bound on suitable blocks beyond every\nthreshold remains to be established.\n\nThe underlying finite lemma is more general: it applies to any nonempty\nfinite $T\\subseteq[0,2X)$ with the stated size bound and\n$\\|\\sum_{N\\in T}e(D(h,N,L)/2^L)\\|\\le(21/25)|T|$.\nTaking $T=[X,2X)$ gives the full-block implication. Allowing a chosen\nsubset weakens the hypothesis; it does not strengthen it. The Lean proof\n(\\lean{Erdos249257.TotientTailPeriodKiller.exists_certifiedKill_of_first_harmonic_norm_gap}{FirstHarmonicPivot.lean:53})\nuses the elementary block lemma of Theorem~\\ref{thm:hgap-real}, which is\nunconditional as an implication. The real-part version uses\n$\\operatorname{Re}\\sum_{N\\in T}e(D(h,N,L)/2^L)\\le(9/10)|T|$.\nIts proof uses $\\cos(\\pi/8)>9/10$ and averaging. An unspecified saving\nbelow $|T|$ is not the stated hypothesis.\n\n\\emph{What supplying this form would take:} genuine\ncancellation for the additive character $e(D(h,N,L)/2^L)$ at the\ngrowing dyadic moduli permitted by the size bound. The quantifiers require\nsuitable blocks beyond every threshold for each fixed $h>0$; they do not\nrequire one threshold that works simultaneously for all $h$. A determinant\nidentity\n(\\lean{Erdos249257.ResidualGaugeObstruction.det_residualMonomialMatrix}{ResidualGaugeObstruction.lean:47})\ngives a comparison in which every weighted phase equals $1$ while the\nspecified determinant remains nonzero. Thus nonvanishing of that determinant\nalone cannot prove cancellation. A successful use of a determinant would\nneed further information connecting its entries to the phase sum; the\ncomparison does not exclude every such argument.\n\n\\subsection{The actual tail difference on the LCM diagonal}\n\nFor $H=H_{2^a}$, the actual difference\n$\\Omega_a=R_{2H}-R_H$ removes the auxiliary rational approximation\nfrom the LCM criterion. The equivalence below restricts attention to\nthis sparse sequence of tails. Its required nonintegrality is still\nan arithmetic assertion to prove.\n\n\\begin{quote}\n\\textbf{The exact equivalence.}\n\\[\n  S\\notin\\mathbb Q\n  \\iff\n  \\forall a_0,\\ \\exists a\\ge a_0,\\ \\Omega_a\\notin\\mathbb{Z}.\n\\]\n\\end{quote}\n\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_iff_actualLcmOrbitNonintegralitySupply}{TotientActualLcmOrbitNonintegrality.lean:37}\nThis is an unconditional equivalence, where\n$\\Omega_a := 2^H(2^H-1)S - (\\Phi_{2H}-\\Phi_H)$,\n$H=H_{2^a}$\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcmTailOrbit_eq_scaled_totientSeries_sub_prefix}{TotientActualLcmOrbitSeparation.lean:68}).\nThe right-hand side is an equivalent reformulation of irrationality,\nnot a weaker hypothesis already established by the finite examples.\nThe following sufficient conditions are attempts to prove it.\n\n\n\\paragraph{An impossible complete divisibility pattern}\nLet $a\\ge8$, $H=H_{2^a}$ and $J,K,m\\in\\mathbb N$, with\n\\[\n m\\le K,\\qquad J+K+a+6<2\\cdot2^a,\\qquad\n B:=2H+J+K+2<2^m.\n\\]\nIt is impossible that\n\\[\n 2^{r+1}\\mid\\delta_{2^a}(J+K-m+r+1)\n              \\quad\\text{for every }0\\le r<m.\n\\]\n\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.not_actualLcmTerminalDyadicStaircase_of_room}{TotientActualLcmTopEdgeStaircase.lean:251}\nIndeed, the last requirement would make $\\delta_{2^a}(J+K)$ a\nmultiple of $2^m$, whereas the short-window bounds give\n$0<\\delta_{2^a}(J+K)<B<2^m$. The positivity input is\n\\[\n \\delta_{2^a}(j)>0\\qquad(a\\ge8,\\ 0<j<2\\cdot2^a),\n\\]\n\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.lcmRayArithmeticLetter_pos_of_lt_two_mul}{TotientActualLcmOrbitArithmetic.lean:1703}, without a rationality hypothesis.\n\nOmitting only the last divisibility requirement leaves the partial\npattern in Proposition~\\ref{prop:TE-02-inv}. It additionally requires\n$\\delta_{2^a}(J+K)\\le2^m-B$. The resulting necessary equality is\n\\emph{penultimate}:\n\\[\n \\delta_{2^a}(J+K-1)=2^{m-1},\\qquad B<2^m<2B.\n\\]\n\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.puncturedDyadicStaircase_penultimate_eq_half}{TotientActualLcmTopEdgeStaircase.lean:1187}\nThe terminal difference is not pinned to $2^{m-1}$. Neither this\nnecessary restriction nor the full-pattern impossibility supplies an\nunbounded family of successful partial patterns.\n\n\\paragraph{The remaining upper-endpoint residue}\n\\[\n  \\mathcal G_a(J,K,m) :\\equiv\n  m\\le K\\ \\wedge\\\n  2H+J+K+2 < 2^m\\ \\wedge\\\n  D(H,\\,H+J,\\,K) \\bmod 2^m \\le 2^m - (2H+J+K+2).\n\\]\n\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.ActualLcmTopEdgeResidueGap}{TotientActualLcmTopEdgeStaircase.lean:900}\nNo unbounded family satisfying this condition is known.\n\nThe positivity theorem gives\n\\[\n R_{2H+J}-R_{H+J}>0\n \\quad\\text{if }a\\ge8\\text{ and }J+a+6<2\\cdot2^a.\n\\]\n\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcmTailDiff_shift_pos}{TotientActualLcmOrbitSign.lean:39}\nFor $a\\ge8$, the residue test at depth $K$ uses the stronger bound\n$J+K+a+6<2\\cdot2^a$. Under these hypotheses, an integral initial\ntail difference would give an integer terminal carry\n$e=R_{2H+J+K}-R_{H+J+K}$ with $0<e<2H+J+K+2$. The tail identity is\n\\[\n D(H,H+J,K)=2^K(R_{2H+J}-R_{H+J})-e.\n\\]\nHence, when $2H+J+K+2<2^K$, it is the residue of $D(H,H+J,K)$,\nnot of $e$, that equals $2^K-e$\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcm_integral_forces_topEdgeResidue}{TotientActualLcmOrbitSign.lean:211}).\nMore generally, the bounds $m\\le K$ and $2H+J+K+2<2^m$ give\n\\[\n D(H,H+J,K)\\bmod2^m=2^m-e>2^m-(2H+J+K+2),\n\\]\nbecause $2^m$ divides $2^K$ and $0<e<2^m$. This contradicts\n$\\mathcal G_a(J,K,m)$. Consequently, for $a\\ge8$ and\n$J+K+a+6<2\\cdot2^a$,\n\\[\n \\mathcal G_a(J,K,m)\\quad\\Longrightarrow\\quad\n       R_{2H+J}-R_{H+J}\\notin\\mathbb Z.\n\\]\n\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcmTailDiff_notMem_int_of_topEdgeResidueGap}{TotientActualLcmTopEdgeStaircase.lean:1260}\nThe one-sided inequality excludes the upper-endpoint interval forced\nby integrality. Positivity alone does not establish that inequality.\n\nThe cofinal target built from it is\n\\[\n  \\text{cofinal upper-endpoint condition} :\\equiv\\\n  \\forall a_0,\\ \\exists a,K,m,\\quad\n  a_0\\le a\\ \\wedge\\ 8\\le a\\ \\wedge\\ K+(a{+}6)<2\\cdot 2^a\\ \\wedge\\\n  \\mathcal G_a(0,K,m).\n\\]\n\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.PowerTwoActualLcmTopEdgeResidueGapSupply}{TotientActualLcmTopEdgeStaircase.lean:1325}\nThis condition is unproved; its implication is proved:\n\\[\n  \\text{cofinal upper-endpoint condition} \\Longrightarrow S\\notin\\mathbb Q.\n\\]\n\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_topEdgeResidueGapSupply}{TotientActualLcmTopEdgeStaircase.lean:2184}\n\n\\paragraph{Five sufficient conditions for that residue inequality}\nThe next five quantified conditions each imply irrationality. Conditions\n(i)--(iv) reach the upper-endpoint test; (v) gives nonintegrality directly.\nTheir proven relations are summarised in Proposition~\\ref{prop:TE-05}.\n\n\\begin{enumerate}\n\\item[(i)] For every $a_0$, there are $a\\ge\\max(a_0,8)$ and $m\\ge0$\nwith $H=H_{2^a}$ such that\n\\[\n\\begin{gathered}\n m+1+a+6<2\\cdot2^a,\\qquad 2H+m+3<2^m,\\\\\n 2H+m+2\\le\n \\bigl(D(H,H,m)+\\delta_{2^a}(m+1)\\bigr)\\bmod2^m\n \\le2^m-(2H+m+3),\n\\end{gathered}\n\\]\nwhere $\\delta_{2^a}(j)=\\varphi(2H+j)-\\varphi(H+j)$.\nThe size condition includes the larger depth $m+1$, so the conclusion\nremains valid whichever of the two adjacent depths gives the certificate.\nThis condition is unproved\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.PowerTwoAdjacentSuffixMidbandSupply}{TotientActualLcmTopEdgeStaircase.lean:1334});\nits sufficiency is proved\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.powerTwoActualLcmTopEdgeResidueGapSupply_of_adjacentSuffixMidband}{TotientActualLcmTopEdgeStaircase.lean:2108},\n\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_adjacentSuffixMidbandSupply}{TotientActualLcmTopEdgeStaircase.lean:2191}).\n\n\\item[(ii)] For every $a_0$, there is $a\\ge\\max(a_0,14)$ with the\nfollowing property. Put $H=H_{2^a}$ and choose $q$ so that $2q+1$ is the\nleast odd integer not smaller than $\\lfloor\\log_2H\\rfloor+10$.\nDefine integers $W_{a,j}$ recursively by\n\\[\n\\begin{aligned}\n W_{a,0}&=0,\\\\\n W_{a,j+1}&=4W_{a,j}\n  +\\frac{\\delta_{2^a}(2j+3)-2\\delta_{2^a}(2j+2)\n                  +\\delta_{2^a}(2j+4)}{2}.\n\\end{aligned}\n\\]\nThe increments $\\delta_{2^a}(s)$ are even, so the recurrence is integral.\nThe required inequality is\n\\[\n H+q+2\\le W_{a,q}\\bmod4^q\\le4^q-(H+q+2).\n\\]\nAt depth $m=2q+1$, the residue in (i) equals\n$2(W_{a,q}\\bmod4^q)$. Dividing its two endpoint bounds by $2$ and using\nintegrality gives exactly the threshold $H+q+2$.\nThis condition is unproved\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.PowerTwoOddGuardTopEdgeHalfWordBandSupply}{TotientActualLcmTopEdgeStaircase.lean:1349}).\nThe width is $H+q+2$ and the depth is the prescribed depth;\nboth must be retained when comparing this with a fixed-fraction band. Its\nsufficiency is proved\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.powerTwoAdjacentSuffixMidbandSupply_of_oddGuardTopEdgeHalfWordBand}{TotientActualLcmTopEdgeStaircase.lean:2057}).\n\n\n\\item[(iii)] Use exactly the quantifiers and prescribed depth of (ii).\nLet $r=W_{a,q}\\bmod4^q$ and define its centred representative by\n\\[\n u_{a,q}=\\begin{cases}\n r,&2r\\le4^q,\\\\\n r-4^q,&2r>4^q.\n \\end{cases}\n\\]\nThus $-4^q/2<u_{a,q}\\le4^q/2$; when a midpoint occurs, the positive\nrepresentative is used. The equivalent inequality is\n\\[\n H+q+2\\le |u_{a,q}|.\n\\]\nThis condition is unproved\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.PowerTwoActualFinalTopEdgeMagnitudeSupply}{TotientActualLcmTopEdgeStaircase.lean:1471}).\nAt the prescribed depth the size bound $2(H+q+2)\\le4^q$ holds.\nFor a residue in $[0,4^q)$, the band in (ii) is therefore exactly\nthis absolute-value condition. The equivalence, including its\nquantifiers, is proved\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.topEdgeHalfWordBandSupply_iff_actualFinalCenteredMagnitudeSupply}{TotientActualLcmTopEdgeStaircase.lean:1479}).\n\n\\item[(iv)] Allow the odd depth to vary, but retain both size bounds.\nFor every $a_0$, require $a\\ge\\max(a_0,8)$ and $q\\in\\mathbb N$ with\n\\[\n\\begin{gathered}\n 2q+2+a+6<2\\cdot2^a,\\qquad 2(H+q+2)\\le4^q,\\\\\n H+q+2\\le |u_{a,q}|,\\qquad H=H_{2^a}.\n\\end{gathered}\n\\]\nThis condition is unproved\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.PowerTwoFlexibleActualTopEdgeMagnitudeSupply}{TotientActualLcmTopEdgeStaircase.lean:1927}).\nThe first bound keeps the adjacent depth inside the short window;\nthe second makes the required residue band nonempty. One implication\npasses through (i), and another gives the two-sided carry exclusion\nbelow\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.powerTwoAdjacentSuffixMidbandSupply_of_flexibleActualTopEdgeMagnitude}{TotientActualLcmTopEdgeStaircase.lean:2007},\n\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_flexibleActualTopEdgeMagnitudeSupply}{TotientActualLcmTopEdgeStaircase.lean:2213}).\n\n\\item[(v)] Retain the full quantifiers and the two size bounds in\n(iv), but replace its absolute-value inequality by\n\\[\n \\delta_{2^a}(2q+2)\\le2u_{a,q}.\n\\]\nThis compares the signed centred residue with the terminal totient\ndifference. It is not an unrestricted assertion at an arbitrary $q$.\nThis condition is unproved\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.PowerTwoFlexibleActualTerminalDominanceSupply}{TotientActualLcmTopEdgeStaircase.lean:1908});\nits sufficiency is proved\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_flexibleActualTerminalDominanceSupply}{TotientActualLcmTopEdgeStaircase.lean:2221}).\n\\end{enumerate}\n\n\\paragraph{A further sufficient terminal-carry condition}\nWith $H=H_{2^a}$, the unproved quantified condition is\n\\[\n\\begin{aligned}\n &\\forall a_0\\ \\exists a\\ge\\max(a_0,8)\\ \\exists q:\\\\\n &\\quad 2q+2+(a+6)<2\\cdot2^a,\\qquad 2(H+q+2)\\le4^q,\\\\\n &\\quad 2u_{a,q}\\le\\delta_{2^a}(2q+2)-(2H+2q+3)\n       \\quad\\hbox{or}\\quad \\delta_{2^a}(2q+2)\\le2u_{a,q}.\n\\end{aligned}\n\\]\n\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.PowerTwoFlexibleActualTerminalCarryCorridorEscapeSupply}{TotientActualLcmTopEdgeStaircase.lean:1895}\nEither alternative gives nonintegrality at that scale, so the\nquantified condition implies irrationality\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_flexibleActualTerminalCarryCorridorEscapeSupply}{TotientActualLcmTopEdgeStaircase.lean:2230}).\n\nThe proof assumes an integer $z=\\Omega_a$ and uses\n\\[\n 2u_{a,q}=\\delta_{2^a}(2q+2)-c_{H,H,z}(2q+1).\n\\]\n\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.two_mul_actualOddHalfCenteredLift_eq_terminal_sub_trueCarry}{TotientActualLcmTopEdgeStaircase.lean:1678}\nThe carry is then a positive tail difference, strictly smaller than\n$2H+2q+3$. The negative endpoint representative is \\emph{minus} this\ncarry\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcm_trueEndpointSurvivor_neg}{TotientActualLcmOrbitSign.lean:172}).\nThe first alternative contradicts the upper bound; the second\ncontradicts positivity, which is the terminal-dominance argument\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcmTailOrbit_notMem_int_of_actualTerminalDominance}{TotientActualLcmTopEdgeStaircase.lean:1880}).\n\nBoth arguments are conditional contradictions, not evidence against\neither unproved estimate. No converse from nonintegrality to either\nfinite inequality is asserted. In particular, the exact identity must\nnot be described as making this sufficient condition necessary.\n\n\n\\paragraph{Certificates in short windows}\nFor every $t,j\\in\\mathbb N$, the arithmetic difference on the LCM\nprogression is exactly\n\\[\n \\delta_t(j)=\\varphi(2H_t+j)-\\varphi(H_t+j).\n\\]\nNo coprimality restriction on $j$ is needed\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.lcmRayArithmeticLetter_eq_deltaTotient}{TotientActualLcmOrbitArithmetic.lean:298}). Consequently the corresponding window\nnumerator is\n\\[\n D(H_t,H_t,L)=\\sum_{j=1}^{L}\\delta_t(j)2^{L-j},\n\\]\nand applying the same residue and error bounds gives precisely the\ntest $\\mathcal C(H_t,H_t,L)$ (\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.lcmDiagonalArithmeticKill_iff_certifiedKill}{TotientActualLcmOrbitArithmetic.lean:2045}).\nThus the quantified assertion\n\\[\n \\forall a_0\\ \\exists a\\ge a_0\\ \\exists L<2\\cdot2^a,\n \\qquad\\mathcal C(H_{2^a},H_{2^a},L)\n\\]\nimplies irrationality\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.PowerTwoActualLcmShortArithmeticKillSupply}{TotientActualLcmOrbitArithmetic.lean:2107},\n\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcmOrbitNonintegralitySupply_of_shortArithmeticKill}{TotientActualLcmOrbitArithmetic.lean:2152});\nthe assertion itself is unproved.\nIt restricts the diagonal indices to powers of two and imposes the\nadditional bound $L<2\\cdot2^a$. Pointwise completeness allows an\nunspecified finite depth; it does not imply a witness within this\nshort window.\n\nThe displayed certified examples have $a=4$, $L=23$ and $a=6$, $L=93$\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcmTailOrbit_four_and_six_notMem_int}{TotientActualLcmShortKill.lean:35});\nthese are finite computations.\nThe bounded statement of\nTheorem~\\ref{thm:short-window-examples-through-exponent} uses the single\nwitness $(a,L)=(6,93)$\nto cover all thresholds $a_0\\le6$. It makes no assertion for larger\nthresholds, or that these are the only successful exponents.\n\n\n\\paragraph{A sufficient approximation bound}\nPut $H=H_{2^a}$ and define the finite rational and its error bound by\n\\[\n \\rho_{a,q}=\n \\frac{D(H,H,2q+1)+\\delta_{2^a}(2q+2)}{2^{2q+1}},\\qquad\n \\varepsilon_{a,q}=\\frac{2H+2q+3}{2^{2q+1}}.\n\\]\nFor every $a,q\\in\\mathbb N$,\n\\[\n |\\Omega_a-\\rho_{a,q}|<\\varepsilon_{a,q}.\n\\]\n\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.abs_actualLcmTailOrbit_sub_rawApprox_lt}{TotientActualLcmOrbitSeparation.lean:141}\nLet $2q_a+1$ be the least odd integer not smaller than\n$\\lfloor\\log_2H\\rfloor+10$, as in condition (ii). The sufficient\nquantified separation condition is\n\\[\n \\forall a_0\\ \\exists a\\ge\\max(2,a_0)\\ \\forall z\\in\\mathbb Z,\n \\qquad \\tfrac1{32}+\\varepsilon_{a,q_a}\\le|\\Omega_a-z|.\n\\]\nThis condition is unproved; its sufficiency is proved\n(\\lean{Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_actualLcmOrbitSeparationSupply}{TotientActualLcmOrbitSeparation.lean:305}).\nThe triangle inequality then gives\n$|\\rho_{a,q_a}-z|>1/32$ for every integer $z$.\nThe hypothesis concerns the real tail $\\Omega_a$, not merely its\ncomputable approximant. The truncation estimate transfers the stated\nseparation, but does not itself establish it. Also, the depth is\nprescribed by $a$; it is not a free parameter in this hypothesis.\n\n\\subsection{An equivalent test using two leading residue bits}\n\nProposition~\\ref{prop:TE-03-inv} gives an equivalent certificate\ncondition: there are $s,b\\in\\mathbb N$ such that either\n$\\mathcal C(h,N+s,b+1)$ holds, or both\n\\[\n N+s+h+b+4<2^b,\\qquad\n 2^b\\le D(h,N+s,b+2)\\bmod2^{b+2}<3\\cdot2^b.\n\\]\n\\lean{Erdos249257.TotientTailPeriodKiller.exists_certifiedKill_iff_guardCylinderWitness}{TotientActualLcmTopEdgeStaircase.lean:844}\nThe forward proof selects $b$ using the depth $L$ of the first\ncertificate, so this is an encoding of a witness rather than a bound\non how far one must search. Existence for the actual totient sequence\nremains the missing assertion. A use for another coefficient sequence\nwould require its own recurrence and tail bounds.\n\n\\subsection{Farey growth and the failed generic rank argument}\n\nThe Farey growth law is an open sufficient hypothesis. The generic\nrank bound listed beside it is false for rational comparison\nsequences, and is retained to explain that failed input rather than\nas another open problem.\n\n\\paragraph{The hypothetical rank bound.}\nUnconditionally, for every level $e\\ge1$, the dyadic totient basis of\n$2^e+1$ sequences is linearly independent over $\\mathbb{Q}$ (via CRT and\nDirichlet's theorem on primes in arithmetic progression)\n(\\lean{Erdos249257.linearIndependent_canonicalTotientKernelFamily}{TotientMahlerDefect.lean:935}).\nConsequently, if $S\\in\\Q$, there is an integer $v>0$ with $vS\\in\\Z$.\nIts integral carry $u_N=vR_N$ satisfies, for every $e\\ge0$,\n\\[\n \\dim_{\\Q}\\operatorname{span}\n \\{n\\mapsto u_{2^j n+r}:1\\le j\\le e,\\ 0\\le r<2^j\\}\\ge2^e-1.\n\\]\nIt obeys $u_{N+1}=2u_N-v\\ph(N+1)$ and $0\\le u_N\\le v(N+2)$.\n\\lean{Erdos249257.not_irrational_totientSeries_implies_unbounded_carryRank_unconditional}{TotientCarryKernelRigidity.lean:300}\n\nA bound independent of $e$ for these same carry-section ranks would\ncontradict the displayed lower bound; so would an upper bound that is\n$o(2^e)$. Rationality and linear coefficient growth alone do not give\neither conclusion: the fixed sequence constructed in\nSection~\\ref{sub:generic-carry-rank} has sum $5/4$ and the same rank\nlower bound. An incompatible upper bound for the actual totient carry\nwould have to use information not shared by that comparison sequence.\n\nThe most direct generic construction is also impossible: no\ninteger certificate satisfying the following conditions ($Q\\cdot v\\cdot A =\n\\mathrm{boundary}$ with $|\\mathrm{boundary}|<Q\\cdot v$ and $A\\ne0$) can\nexist (\\lean{Erdos249257.false_of_compressedAdjointCertificate}{TotientMahlerDefect.lean:1183});\nand the full family is proved infinite-dimensional in\nspan, the case $k=2$ of Coons's non-regularity theorem\n(\\S\\ref{sec:mahler-defect};\n\\lean{Erdos249257.not_finiteDimensional_span_fullTotientKernel}{TotientMahlerDefect.lean:1145}).\n\nProposition~\\ref{prop:period-not-rank} gives a common eventual period\nmodulo $v$ for the sections of this same $u$, together with their\nunbounded rational ranks. These conclusions concern different properties of the same carry; they supply no rank\nupper bound.\n\n\\paragraph{Growth of the Farey denominator bound.}\nThe finite denominator exclusion for $S$ comes from\na Farey-gap denominator exclusion at a single fixed window $K=240$:\n\\[\n S\\ne a/b\\quad\\text{for all }a\\in\\Z,\\quad\n 1\\le b\\le79639646646701375323355774875831053,\n \\quad\\gcd(a,b)=1.\n\\]\n\\lean{Erdos249257.tsum_totient_div_pow_two_ne_ratCast_of_den_le_79639646646701375323355774875831053}{CertificateKernel.lean:18384}\nThis bound is\n\\emph{sharp} at $K=240$; the mediant denominator\n$q=79639646646701375323355774875831054$ is the exact first failing\ndenominator\n(\\lean{GapFareyBound.gap_check_window_1_240_first_failure}{GapFareyBound.lean:225}); so the same interval cannot exclude every rational at the next\ndenominator. A different window requires a new certificate or a\nuniform argument; the fixed-window calculation supplies neither.\n\n\nA sufficient extension is a function $g(K)\\to\\infty$ for which every\n$(N=1,K)$ gap check excludes all rationals of reduced denominator\nat most $g(K)$. More generally, unbounded valid exclusion bounds\nalong a sequence of windows suffice. This is a statement about the\ncertified intervals. The record does not prove it equivalent to a\nlower bound on the continued-fraction denominators of $S$.\nThis extension is unproved.\n\nFor $p$ prime and $e\\ge1$, a successful unit-gap certificate at\ndenominator $p^e$ has zero candidates or a single candidate divisible\nby $p$ (\\lean{Erdos249257.reducedDenominatorCandidateCount_prime_pow_le_one}{PrimitiveWeightCertificate.lean:111}).\nThe success hypothesis is essential: the bound is\nnot a statement about every window at that denominator. Nor does it\nbound the eventual denominator cutoff. The prime-power restriction\ncannot be omitted, as the modulo-$15$ example in\nObservation~\\ref{prop:D8cert-kill} shows. No obstruction to a uniform\nestimate as $K$ varies is proved here.\n\n\\subsection{Summary of implications}\n\nFor each condition, the second column states its relation to the\nquantified residue test or to the named intermediate assertion.\n\n\\emph{Reading the table.} ``Sufficient'' records only the indicated\nimplication. It does not assert a converse or a strict comparison of\nstrength. ``Equivalent'' records two proved directions, either directly\nor through the base equivalence with $S\\notin\\mathbb Q$. Logical\nequivalence does not by itself provide a direct construction from the\nfinite witnesses in one formulation to those in another.\n\n\\subsection{Remaining estimates}\n\nTwo explicit estimates for $\\varphi$ remain: weighted cancellation\nin the exponential sums and a residue inequality involving the final\ntotient difference. Neither is derived here from the cited methods.\nOther approaches to the irrationality question are not excluded.\n\n\n\n\\paragraph{Exponential sums.} For each fixed $h>0$ and every\nthreshold $X_0$, the block norm condition asks for some $X\\ge\\max(X_0,1)$\nand $L$ satisfying both the modulus bound and\n\\[\n \\left|\\sum_{N\\in[X,2X)}e\\bigl(D(h,N,L)/2^L\\bigr)\\right|\n       \\le\\frac{21}{25}X.\n\\]\nThe saving $21/25$ is fixed, but the chosen block and depth may depend\non $h$ and $X_0$; uniformity of these choices in $h$ is not required.\nThe preceding four-term decomposition and the proved counting bounds\nidentify contributions to this sum. By\nTheorem~\\ref{thm:goodbasegap}(b) the two remaining weighted estimates may\nbe replaced by the single bound\n$\\operatorname{Re}\\sum_{N\\in\\mathcal G}E(h,N,L(X))\\le603X/1000$ at\n$s=26$, $\\eta=1/1000$ and minimal depth, for arbitrarily large $X$; that\nbound is open. The record therefore gives an exact reduction and some of\nits estimates; the displayed cancellation bound remains unproved.\n\n\\paragraph{The last totient difference.} The endpoint criteria compare\n\\[\n \\delta_{2^a}(2q+2)\n   =\\varphi(2H+2q+2)-\\varphi(H+2q+2),\\qquad H=H_{2^a},\n\\]\nwith twice the centred residue $u_{a,q}$. An appropriate inequality,\nproved for arbitrarily large $a$ with the stated bounds on $q$, would\ncontradict the carry interval required by integrality. The exact identity\nexplains why such an inequality would suffice; it does not supply the\ninequality. The finite examples through $t=82$, and the short-window\nexamples at $a=4,6$, do not establish an asymptotic pattern.\n\n\n\\clearpage\n\\appendix\n\\section{Complementary proofs for the short paper}\\label{app:r3-complements}\n\\papersectiontarget{long249-r3-complements}\nThe \\shortpaper{short249-proof}{basis theorem}, the\n\\shortpaper{short249-base-six}{integral relation theorem} and the\n\\shortpaper{short249-residues}{bounded residue theorem} have complete\nproofs in the short paper. This appendix supplies the longer arguments\nbehind its discussion of $S$: propagation from a nonintegral tail,\nMersenne residues, rational comparisons and the sharp rank-one bound.\nThroughout, $K(h,N,L)$ is the residue condition\n$\\mathcal C(h,N,L)$ defined in the main text.\n\n\\subsection{Propagation of nonintegral tail differences}\\label{long249:r3-ray}\n\\papersectiontarget{long249-r3-ray}\nWrite\n\\[\n S=\\sum_{n\\ge1}\\ph(n)2^{-n},\\qquad\n R_N=\\sum_{j\\ge1}\\ph(N+j)2^{-j},\\qquad\n \\Delta_h(N)=R_{N+h}-R_N.\n\\]\nThe identity\n\\[\n 2^NS=\\sum_{n=1}^N\\ph(n)2^{N-n}+R_N\n\\]\nhas an integer finite sum. Subtracting the identities at $N+h$ and $N$\ngives\n\\begin{equation}\\label{long249:r3-tail-phase}\n \\Delta_h(N)\\equiv2^N(2^h-1)S\\pmod\\Z.\n\\end{equation}\nIn particular, a nonintegral tail difference excludes every\nrational value whose denominator divides $2^N(2^h-1)$.\n\\paragraph{The finite test.}\nFor $h,N,L\\in\\N$, let\n\\[\n D_{h,N,L}=\\sum_{j=0}^{L-1}\n   \\bigl(\\ph(N+h+1+j)-\\ph(N+1+j)\\bigr)2^{L-1-j},\n \\qquad B=N+h+L+2.\n\\]\nAll residues below are least nonnegative residues. Write $K(h,N,L)$ for\n\\[\n B<D_{h,N,L}\\bmod2^L<2^L-B.\n\\]\nThe omitted terms form another difference of scaled tails:\n\\[\n 2^L\\Delta_h(N)-D_{h,N,L}=R_{N+h+L}-R_{N+L}.\n\\]\nSince $0\\le R_M\\le\\sum_{j\\ge1}(M+j)2^{-j}=M+2$, their absolute\ndifference is bounded by $N+h+L+2$.\nThus\n\\begin{equation}\\label{long249:r3-tail-error}\n |2^L\\Delta_h(N)-D_{h,N,L}|\\le B.\n\\end{equation}\nIf $K(h,N,L)$ holds, the interval\n$[D_{h,N,L}-B,D_{h,N,L}+B]$ lies strictly between two consecutive\nmultiples of $2^L$. It contains $2^L\\Delta_h(N)$, so\n$\\Delta_h(N)\\notin\\Z$.\nConversely, write $\\|x\\|_{\\R/\\Z}=\\min_{z\\in\\Z}|x-z|$ for the distance to\nthe nearest integer. If $\\Delta_h(N)\\notin\\Z$, choose $L$ so that\n$2^L\\|\\Delta_h(N)\\|_{\\R/\\Z}>2B$. Such a choice exists: $h,N$ are fixed,\nthe distance is positive, and $B=N+h+L+2$ grows only linearly in $L$.\nThe error estimate then puts the residue of $D_{h,N,L}$ more than $B$\nfrom either endpoint. Thus nonintegrality is equivalent to a certificate\nat some unrestricted depth. This is a one-sided finite test: success\nproves nonintegrality, but failure at a particular depth does not prove\nintegrality. The converse guarantees eventual success only after a\nfixed nonintegral difference has been supplied.\n\n\\begin{theorem}[Propagation of one nonintegral tail difference]\\label{res:fulldepth}\nFix $d\\ge1$ and $N\\ge0$. If $\\Delta_d(N)\\notin\\Z$, then every sufficiently\nlate pair $\\{t,t+1\\}$ contains an $m$ such that $K(md,N,md)$ holds.\nConsequently\n\\[\n \\exists t\\ge1:\\ K(td,N,td)\n \\quad\\Longleftrightarrow\\quad\\Delta_d(N)\\notin\\Z.\n\\]\nRequiring this for every $d\\ge1,N\\ge0$ is equivalent to $S\\notin\\Q$.\n\\end{theorem}\nUnrestricted completeness cannot simply set $L$ equal to the shift:\nit supplies a depth only after the shift is fixed. The theorem couples\nthem by varying both along multiples of $d$. The recurrence below\nprevents two adjacent failures. Its starting hypothesis is still just\none nonintegral difference; only the requirement for \\emph{every}\n$d,N$ is equivalent to irrationality.\n\n\\begin{proof}\nSet $F_t=\\Delta_{td}(N)$. Equation~\\eqref{long249:r3-tail-phase} yields\n$F_{t+1}\\equiv2^dF_t+F_1\\pmod\\Z$. This recurrence expresses the fixed\n$F_1$ in terms of two adjacent differences. A failed certificate makes\none of those differences close to an integer: if $K(td,N,td)$ fails,\n\\eqref{long249:r3-tail-error} gives\n\\[\n \\|F_t\\|_{\\R/\\Z}\\le\\epsilon_t,\n \\qquad \\epsilon_t=2(N+2td+2)2^{-td}.\n\\]\nIf the test also fails at $t+1$, the triangle inequality on $\\R/\\Z$\nand the recurrence give\n\\[\n \\|F_1\\|_{\\R/\\Z}\n \\le \\|F_{t+1}\\|_{\\R/\\Z}+2^d\\|F_t\\|_{\\R/\\Z}\n \\le\\epsilon_{t+1}+2^d\\epsilon_t.\n\\]\nThe recurrence has transferred two shrinking errors to the same\nfixed number $F_1$. Its distance from $\\Z$ is positive, whereas\n$\\epsilon_{t+1}+2^d\\epsilon_t\\to0$; hence adjacent failures are\nimpossible for large $t$. Conversely, integral $F_1$ makes every\n$F_t$ integral by the recurrence, so soundness excludes every certificate.\n\nFinally, \\eqref{long249:r3-tail-phase} is\nnonintegral for every $d,N$ when $S$ is irrational. If\n$S=a/(2^cv)$ with $v$ odd, take $N=c$ and $d=\\ph(v)$ to make it integral.\n\\end{proof}\nFor instance $D_{1,12,16}=-143140$, with residue $53468$ modulo $2^{16}$\nand $B=31$, verifies the hypothesis at $d=1,N=12$. More explicitly,\n\\[\n \\Delta_1(12)\\in\n \\left[\\frac{-143171}{65536},\\frac{-143109}{65536}\\right]\n \\subset(-3,-2).\n\\]\nThis enclosure proves nonintegrality without evaluating the infinite\ntails. The theorem gives infinitely many certificates with depth equal\nto the shift for this pair, not for every $d,N$.\n\n\\subsection{Residues modulo divisors of Mersenne numbers}\\label{long249:r3-mersenne}\n\\papersectiontarget{long249-r3-mersenne}\nFix a hypothetical denominator $2^cv$, with $c\\ge0$ and odd $v\\ge1$.\nFor a positive multiple $H$ of $\\varphi(v)$, Euler's theorem makes\n\\[\n M=\\frac{2^H-1}{v},\\qquad\n B_{H,c}=\\sum_{j=0}^{H-1}\\varphi(c+1+j)2^{H-1-j}\n\\]\nan integer modulus and an integer prefix, respectively. The test below\nasks whether the residue of $-B_{H,c}$ stays away from the endpoints\nby more than the tail error.\n\n\\begin{equivform}[A residue test modulo a divisor of $2^H-1$]\\label{res:canonicalmersenne}\nThe series $S$ is irrational if and only if, for every $c\\ge0$ and odd\n$v\\ge1$, some positive multiple $H$ of $\\ph(v)$ satisfies\n\\begin{equation}\\label{eq:canonical-gap}\n c+H+1<(-B_{H,c})\\bmod M<M-(c+H+1).\n\\end{equation}\n\\end{equivform}\n\\begin{proof}\nThe identity\n\\[\n B_{H,c}=2^HR_c-R_{c+H}=M(vR_c)-\\Delta_H(c)\n\\]\nis exact. For $H\\ge1$, $|\\Delta_H(c)|<c+H+1$. Indeed $R_N\\le N+1$\nfor $N\\ge1$ and $0<R_0\\le3/2$; the strict inequalities follow by subtracting\na positive tail. If $S=a/(2^cv)$, then $vR_c\\in\\Z$ and\n$\\Delta_H(c)\\in\\Z$. The residue of $-B_{H,c}$ is therefore within distance\nless than $c+H+1$ of $0\\pmod M$, so \\eqref{eq:canonical-gap} fails.\n\nIf $S$ is irrational, fix $c,v$ and put\n$\\delta=\\operatorname{dist}(vR_c,\\Z)>0$. As $H$ grows through positive\nmultiples of $\\ph(v)$, choose it large enough that\n$|\\Delta_H(c)|/M<\\delta/3$ and $(c+H+1)/M<\\delta/3$.\nSince $-B_{H,c}/M=-vR_c+\\Delta_H(c)/M$, its distance from $\\Z$\nis then greater than $2\\delta/3$, while the forbidden endpoint\nwidth is below $\\delta/3$. This proves \\eqref{eq:canonical-gap}\nfor all sufficiently large allowed $H$.\n\\end{proof}\nFor fixed $c,v$, one successful test excludes every value\n$S=a/(2^cv)$, regardless of the numerator $a$. Conversely,\nirrationality gives success for every sufficiently large allowed $H$\nfor each fixed $c,v$. This converse does not supply successful tests\nfor every $c,v$ without the irrationality assumption.\n\nA formal declaration for the equivalence is\n\\href{https://github.com/wcook04/plectis-erdos/blob/c91562bd574a387cde904481e609c7b4cacebb14/lean/ErdosProblems/Erdos249/CyclotomicAnchoredKill.lean\\#L3165}{available here},\nwith the modulus written as $vM=2^H-1$.\nThe finite residues also satisfy\n\\[\n \\rho_{H,N+1,M}=\n \\bigl(2\\rho_{H,N,M}-\\ph(N+H+1)+\\ph(N+1)\\bigr)\\bmod M,\n \\quad \\rho_{H,N,M}=(-B_{H,N})\\bmod M,\n\\]\nwhich is an exact recurrence for searches directed at \\eqref{eq:canonical-gap}.\n\n\n\\subsection{LCM tails and rational comparisons}\\label{long249:r3-comparisons}\n\\papersectiontarget{long249-r3-comparisons}\nWe first specialise the tail congruence to LCM shifts, then compare\nit with rational series preserving selected coefficient data. The\nsource declarations accompany the formulas; references point to the\ncomplete constructions and their hypotheses in the preceding sections.\n\n\\begin{enumerate}\n\\item\\label{res:actualorbit}\n\\textbf{Tail differences at least common multiples.}  Let\n$H_a=\\lcm(1,\\ldots,2^a)$. Then $S$ is irrational if and only if,\nbeyond every threshold $a_0$, some $a\\ge a_0$ satisfies\n$\\Delta_{H_a}(H_a)\\notin\\Z$.  This is the\n\\mword{Erdos249257/TotientActualLcmOrbitNonintegrality.lean}{37}{irrational_totientSeries_iff_actualLcmOrbitNonintegralitySupply}{exact cofinal nonintegrality criterion}\nfor the actual LCM diagonal, not a quantitative\nseparation estimate and not a proof that such indices occur; the\n\\mword{Erdos249257/TotientActualLcmOrbitNonintegrality.lean}{53}{irrational_totientSeries_of_actualLcmOrbitNonintegralitySupply}{forward implication}\nis recorded separately.\n\n\\item\\label{res:actualsign}\n\\textbf{Positive tail differences need not be nonintegral.} For $a\\ge8$ and\n$J+(a+6)<2\\cdot2^a$, one has $\\Delta_{H_a}(H_a+J)>0$.\nSuppose this difference is an integer. If a truncation depth $K$ satisfies\n\\[\n J+K+(a+6)<2\\cdot2^a,\n \\qquad 2H_a+J+K+2<2^K,\n\\]\nthen\n\\[\n D_{H_a,H_a+J,K}\\bmod2^K\n   =2^K-\\Delta_{H_a}(H_a+J+K)\n\\]\nlies strictly between $2^K-(2H_a+J+K+2)$ and $2^K$.\nThe later tail difference is a positive integer; its \\emph{negative} is\nthe representative near zero. Thus positivity determines which endpoint\ncontains the residue, but does not exclude an integral tail difference.\nBoth displayed bounds on $K$ are required for this conclusion.\n{\\footnotesize\\emph{Source declarations:}\n\\mword{Erdos249257/TotientActualLcmOrbitSign.lean}{39}{actualLcmTailDiff_shift_pos}{positive tail differences},\n\\mword{Erdos249257/TotientActualLcmOrbitSign.lean}{172}{actualLcm_trueEndpointSurvivor_neg}{negative endpoint representative},\n\\mword{Erdos249257/TotientActualLcmOrbitSign.lean}{211}{actualLcm_integral_forces_topEdgeResidue}{residue near the upper endpoint}.}\n\n\\item\\label{res:factorideal}\n\\textbf{Divisibility and finite linear combinations do not suffice.}\nA dyadic coboundary is a sequence of the form $2u_j-u_{j+1}$; its weighted\nsum telescopes. For every $t\\ge3$, with $H=\\lcm(1,\\ldots,t)$, the linked\nsource constructs a nonzero comparison sequence of this form. Its\ncoefficients reproduce the actual totient differences at $t-2$\nprescribed indices and are divisible by $\\varphi(H)$. Every finite integer\nlinear combination of forward shifts preserves the telescoping identity and\ndivisibility by $\\varphi(H)$ and the corresponding lower-level factors.\nThe bounds remain uniform in the index, but are multiplied by the sum\nof the absolute values of the combination's coefficients.\nA variant preserves sparse prescribed values along entire arithmetic\nprogressions. These properties alone do not force a contradiction: the\ncomparison need not consist of actual totient differences. The theorem\nconcerns linear combinations of shifts, not nonlinear operations, and\ndoes not supply certificates at arbitrarily large scales.\n{\\footnotesize\\emph{Source declarations:}\n\\mword{Erdos249257/LcmFactorIdealPulseObstruction.lean}{798}{lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient}{obstruction for finite integer combinations of shifts}.}\n\\end{enumerate}\n\nIf $S=a/v$ with integers $a$ and $v\\ge1$, the scaled tails $u_N=vR_N$\nare integers and satisfy\n\\[\n u_{N+1}=2u_N-v\\ph(N+1),\\qquad 0\\le u_N\\le v(N+2).\n\\]\nIn particular $u_N/2^N\\to0$. Its dyadic sections $n\\mapsto u_{2^jn+r}$ have rational rank at\nleast $2^e-1$ through level $e$ and a common eventual period modulo $v$.\nBoth properties are\n\\mword{Erdos249257/TotientTailCarryPeriod.lean}{224}{not_irrational_totientSeries_implies_mod_period_and_unbounded_rank}{the corresponding theorem for the same carry};\nthey are compatible, and neither supplies a contradiction.\n\nSection~\\ref{sub:generic-carry-rank} constructs a rational\ncomparison sequence with\n$c(n)=\\ph(n)$ for odd $n$, $|c(n)-\\ph(n)|\\le2$ for even $n$, $0\\le c(n)\\le n$,\nand\n\\[\n \\sum_{n\\ge1}c(n)2^{-n}=\\frac54.\n\\]\nThis comparison sequence has an integral carry satisfying\n$u_N/2^N\\to0$, with dyadic rank at least $2^e-1$ at every level. Thus the same lower bound for a hypothetical totient carry is\ncompatible with rationality. For one fixed prime $p$ and $g:\\N\\to\\Z$, the exact laws\n$g(pn)=p g(n)$ when $p\\mid n$ and $g(pn)=(p-1)g(n)$ otherwise, together\nwith $g-\\ph=o(n)$, force $g=\\ph$: a nonzero defect keeps a nonzero ratio\nto the argument along $p^j n$. The comparison sequence fails these dilation laws. The unchanged odd-residue sections\nalso give coefficient rank at least $2^e-1$ at every level. This is a\nlower bound, not an exact formula for the full coefficient rank.\n\n\n\\subsection{The sharp rank-one quotient bound}\\label{long249:r3-rankone}\n\\papersectiontarget{long249-r3-rankone}\nThe quotients below are a particular proposed family of rational\napproximants to a M\\\"obius-weighted value. We determine their minimum\nand show that they remain a fixed distance above that value. Put\n\\[\n \\Theta_2=\\sum_{d\\ge1}\\frac{\\mu(d)}{(2^d-1)^2}=S-\\frac12,\n \\qquad\n Q(e,Y)=\\frac{\\bigl(\\sum_{d=1}^{Y}\\mu(d)/(2^d-1)^{e+2}\\bigr)^2}\n {\\sum_{d=1}^{Y}\\mu(d)/(2^d-1)^{2e+2}}.\n\\]\nThe divisor-convolution identity $\\ph=\\mu*\\mathrm{Id}$ gives\n$S=\\sum_{d\\ge1}\\mu(d)2^d/(2^d-1)^2$.\nNow $2^d=(2^d-1)+1$ and\n$\\sum_{d\\ge1}\\mu(d)/(2^d-1)=1/2$, the latter following from\n$\\sum_{d\\mid n}\\mu(d)=0$ for $n>1$ and $1$ for $n=1$.\nThese identities give the displayed value of $\\Theta_2$; all the sums are\nabsolutely convergent. Amiram Eldar posted the corresponding formula\n$S=\\tfrac12+\\sum_{d\\ge1}\\mu(d)/(2^d-1)^2$, together with its coprimality\ninterpretation, to OEIS A256936 on 15 March 2026~\\cite{eldar2026oeis}, and\nSteve Fan posted the formula on the problem's forum thread on\n16 May 2026~\\cite{fan2026totient}. The result below gives the exact extremum for this family, refining\nthe earlier lower bound $1/480$ in Section~\\ref{rank-one-subrank}.\n\n\\begin{theorem}[A lower bound for the rank-one quotients]\\label{res:rankonefloor}\nFor $e\\ge1$ and $Y\\ge4$, the denominator of $Q(e,Y)$ is positive, and the\nunique minimiser is $(e,Y)=(1,5)$. Every admissible quotient and every\nnonempty finite positive weighted average of such quotients satisfies\n\\[\n Q-\\Theta_2>\\frac{21}{320}.\n\\]\nThe uniform bound $Q-\\Theta_2>1/15$ is false already at $(1,5)$.\n\\end{theorem}\n\\begin{proof}\nWe first bound every denominator away from zero. We then compare\n$Q(1,5)$ with the whole range $e\\ge2$ using one uniform estimate,\nand with $e=1$ using a seven-term prefix. Put\n\\[\n P_Y(r)=\\sum_{d=1}^{Y}\\frac{\\mu(d)}{(2^d-1)^r},\\quad\n \\Theta_r=\\sum_{d\\ge1}\\frac{\\mu(d)}{(2^d-1)^r},\\quad\n E_r(Y)=\\frac{2^{-rY}}{1-2^{-r}}.\n\\]\nSince $|\\mu(d)|\\le1$ and $2^d-1\\ge2^{d-1}$,\n\\[\n |\\Theta_r-P_Y(r)|\\le E_r(Y),\\qquad\n |P_Y(r)-P_Z(r)|\\le E_r(Z)\\quad(Y\\ge Z).\n\\]\nFor $r\\ge4$, separating the terms $d=1,2$ gives\n$\\Theta_r=1-3^{-r}+T_r$, where\n$|T_r|\\le[2^r(2^r-1)]^{-1}$. Thus\n\\[\n \\alpha:=\\frac{6373}{6480}=1-\\frac1{81}-\\frac1{240}\n \\le\\Theta_r<1,\n \\qquad E_r(Y)\\le\\varepsilon:=\\frac1{61440}\\quad(Y\\ge4).\n\\]\nFor the strict upper bound, the tail bound is smaller than $3^{-r}$:\n$2^r(2^r-1)>3^r$ for $r\\ge4$.\nIn particular $P_Y(2e+2)\\ge\\alpha-\\varepsilon>0$ for every admissible\n$e,Y$, proving denominator positivity.\n\nWrite $q_*=Q(1,5)$. Using\n$(\\mu(1),\\ldots,\\mu(7))=(1,-1,-1,0,-1,1,-1)$ gives\n\\[\n q_* = \\frac{35076077250375200}{37573118933633199}.\n\\]\nFor $e\\ge2$, both exponents in $Q(e,Y)$ are at least four, so the\nuniform bounds just proved apply to the numerator and denominator.\nFor $e=1$, the numerator instead contains $P_Y(3)$; we use the\nseven-term prefix to bound it. This leaves the two cases $Y=4,6$\nto compare with $Y=5$. The four exact rational comparisons are\n\\begin{align*}\n \\frac{(\\alpha-\\varepsilon)^2}{1+\\varepsilon}&>q_*\n       &&(e\\ge2),\\\\\n \\frac{79049881}{84671559}=Q(1,4)&>q_*\n       &&(e=1,\\ Y=4),\\\\\n \\frac{1043700953468949889}{1117992059749781919}=Q(1,6)&>q_*\n       &&(e=1,\\ Y=6),\\\\\n \\frac{(P_7(3)-1/1835008)^2}{P_7(4)+1/251658240}&>q_*\n       &&(e=1,\\ Y\\ge7).\n\\end{align*}\nEach comparison follows by multiplication by positive denominators.\nThe first left-hand side bounds every quotient in its range from\nbelow. So does the last, by the tail estimates with $Z=7$; the\nnumerator lower bound is positive before it is squared. The middle\ntwo rows give the quotients themselves. These ranges cover every\nadmissible pair except $(1,5)$, proving its unique minimality.\n\nIt remains to compare the minimum with $\\Theta_2$. An upper bound\nfor $\\Theta_2$ proves the stated gap; a lower bound disproves the\nstronger proposed gap. The same tail estimate at $r=2$ gives\n\\[\n P_6(2)-\\frac1{3072}\\le\\Theta_2\\le P_7(2)+\\frac1{12288}.\n\\]\nThe exact prefixes are\n\\[\n P_6(2)=\\frac{3309559}{3814209},\\qquad\n P_7(2)=\\frac{53376062902}{61519376961}.\n\\]\nComparison of fractions gives\n\\[\n q_*-\\left(P_7(2)+\\frac1{12288}\\right)>\\frac{21}{320},\n \\qquad\n q_*-\\left(P_6(2)-\\frac1{3072}\\right)<\\frac1{15}.\n\\]\nThe minimum proves the lower gap for every quotient; the second inequality\nproves failure of the proposed $1/15$ bound at $(1,5)$.\nMultiplying the strict lower gap by positive weights, summing over a\nnonempty finite set and dividing by the positive total weight proves\nthe assertion for weighted averages.\n\\end{proof}\n\nThis theorem treats averages after the quotients are formed. Signed\ncoefficients, coupling before the quotient, and higher-rank kernels require\nseparate arguments. For rational approximants $p_j/q_j$ in lowest terms, with $q_j>0$, a sufficient arithmetic target is\n\\[\n 0<q_j\\left|\\Theta_2-p_j/q_j\\right|\\longrightarrow0.\n\\]\nThe denominator sets the scale of the required error. Convergence alone\ndoes not control their product. For the elementary dyadic prefixes,\nthe available bound is only $q_j|S-p_j/q_j|\\le N+2$ at depth $N$,\nnot a quantity tending to zero. A prescribed approximation family\ntherefore needs a sharper error or reduced-denominator bound. When approximants\nmay be chosen freely, continued-fraction convergents show that the\ncriterion is equivalent to irrationality, just as\n\\eqref{eq:canonical-gap} is. The theorem above excludes the displayed\nrank-one quotients and their positive averages, not other approximation\nfamilies.\n\n\n\\section{Verification and reproducibility}\\label{app:r3-declarations}\n\\papersectiontarget{long249-r3-declarations}\n\\evidenceparagraph{\\evidencerecordurl}\nThe links below identify formal declarations at the supplied source\nrevisions. A checked conditional implication retains its hypothesis;\nthe linked source alone does not establish that input for $\\varphi$.\nThe record gives the available Lean and Comparator status and does not\nreport a new replay or independent review of the printed proofs.\n\n\\medskip\\noindent\\textbf{Bases and relations.}\\par\\smallskip\n\\mword{Erdos249257/TotientMahlerDefect.lean}{160}{totientKernel_zero_residue}{reduction of sections with residue zero};\n\\mword{Erdos249257/TotientMahlerDefect.lean}{169}{totientKernel_even_residue_reduce}{reduction of even residues};\n\\mword{Erdos249257/TotientMahlerDefect.lean}{1265}{linearIndependent_oddCoreTotientKernelFamily}{independence};\n\\mword{Erdos249257/TotientMahlerDefect.lean}{1380}{span_range_fullTotientKernel_eq_span_range_oddCore}{span equality};\n\\mword{Erdos249257/TotientMahlerDefect.lean}{1392}{totientDyadicSectionBasis}{basis};\n\\mword{Erdos249257/TotientMahlerDefect.lean}{935}{linearIndependent_canonicalTotientKernelFamily}{independence};\n\\mword{Erdos249257/TotientMahlerDefect.lean}{989}{finrank_canonicalTotientKernel_eq}{dimension};\n\\mword{Erdos249257/TotientMahlerDefect.lean}{1084}{finrank_totientKernelThroughLevelFamily_eq}{the exact finite-truncation rank};\n\\mword{Erdos249257/TotientMahlerDefect.lean}{1145}{not_finiteDimensional_span_fullTotientKernel}{infinite-dimensionality directly};\n\\mword{Erdos249257/TotientKernelIndex.lean}{124}{totientKernelSectionIndex_spec}{coordinate specification};\n\\mword{Erdos249257/TotientKernelIndex.lean}{190}{totientKernelResidueAtLevel_injective}{injective residues};\n\\mword{Erdos249257/TotientKernelIndex.lean}{213}{existsUnique_totientKernelResidueAtLevel}{unique representation};\n\\mword{Erdos249257/TotientKernelIndex.lean}{241}{card_totientKernelSectionLevel}{level count};\n\\mword{Erdos249257/TotientKernelIndex.lean}{249}{card_totientKernelSectionIndex}{section-index count};\n\\mword{Erdos249257/TotientKernelIndex.lean}{264}{card_totientKernelIndex_eq_two_add_sum}{index as two plus the odd-residue count};\n\\mword{Erdos249257/TotientKernelIndex.lean}{277}{card_totientKernelIndex}{full index count};\n\\mword{Erdos249257/TotientMahlerDefect.lean}{823}{paritySeparatedMatrix_det_ne_zero}{a nonzero evaluation determinant};\n\\mword{Erdos249257/TotientMahlerDefect.lean}{882}{exists_separatedMinorCertificate_totientAffineOddFamily}{an invertible minor for affine totient sections};\n\\mword{Erdos249257/TotientKernelReduction.lean}{60}{totient_pow_mul_eq}{power-multiplication identity};\n\\mword{Erdos249257/TotientKernelReduction.lean}{117}{totient_pow_mul_affine_gcd_cross_eq}{affine cross-gcd identity};\n\\mword{Erdos249257/TotientKernelConditional.lean}{35}{totientKernelReductionScalar_ne_zero}{nonzero reduction scalar};\n\\mword{Erdos249257/TotientKernelConditional.lean}{54}{allBaseTotientKernelSeq_mul_residue_step}{all-base residue step};\n\\mword{Erdos249257/TotientKernelConditional.lean}{102}{allBaseTotientKernelSeq_mem_span_canonical_of_le}{canonical spanning};\n\\mword{Erdos249257/TotientKernelConditional.lean}{185}{span_allBaseTotientKernelThroughLevelFamily_eq_canonical}{span equality};\n\\mword{Erdos249257/TotientKernelConditional.lean}{203}{finrank_canonicalAllBaseTotientKernel_eq_of_linearIndependent}{rank in every base from independence};\n\\mword{Erdos249257/TotientKernelConditional.lean}{215}{finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent}{all-base truncation rank from independence};\n\\href{https://github.com/wcook04/plectis-erdos/blob/25ef6245d15a47548c6926369ae8f1a0f0a14a80/Erdos249257/AllBaseTotientKernel.lean#L1118}{independence in every integer base};\n\\href{https://github.com/wcook04/plectis-erdos/blob/25ef6245d15a47548c6926369ae8f1a0f0a14a80/Erdos249257/AllBaseTotientKernel.lean#L1231}{the rank of the retained all-base family};\n\\href{https://github.com/wcook04/plectis-erdos/blob/25ef6245d15a47548c6926369ae8f1a0f0a14a80/Erdos249257/AllBaseTotientKernel.lean#L1239}{the rank of all sections through a finite level}.\n\n\\medskip\\noindent\\textbf{Tail identities, residue tests and carries.}\\par\\smallskip\n\\mword{Erdos249257/TotientTailPeriodKiller.lean}{150}{two_pow_mul_totient_series_eq}{the finite-prefix and tail identity};\n\\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/Erdos249257/TotientTailPeriodKiller.lean#L72}{definition};\n\\mword{Erdos249257/TotientTailPeriodKiller.lean}{404}{certifiedKill_all_small}{certificates for shifts 1 through 8};\n\\mword{Erdos249257/TotientTailPeriodKiller.lean}{407}{tail_diff_not_int_small}{nonintegral tail differences at the checked shifts};\n\\mword{Erdos249257/GenericTailOrbitRigidity.lean}{67}{IsTemperedBinaryOrbit}{the integer carry recurrence and its growth bound};\n\\lword{Erdos249/CyclotomicAnchoredKill.lean}{2322}{fullMersenneBlockResidue_succ}{residue recurrence};\n\\lword{Erdos249/CyclotomicAnchoredKill.lean}{3132}{fullMersenneCenteredResidueGapSupply_of_canonicalBasepoint}{comparison of the two residue conditions};\n\\lword{Erdos249/CyclotomicAnchoredKill.lean}{3165}{fullMersenneCanonicalBasepointResidueGapSupply_iff_irrational}{exact equivalence};\n\\lword{Erdos249/FullDepthRayAmplifier.lean}{364}{eventually_twoSyndetic_fullDepthKillMultipliers_of_seed}{a certificate in each sufficiently late pair of multipliers};\n\\lword{Erdos249/FullDepthRayAmplifier.lean}{388}{exists_fullDepthKill_on_ray_iff_shift_notMem_int}{pointwise equivalence};\n\\lword{Erdos249/FullDepthRayAmplifier.lean}{406}{apFullDepthEscape_iff_irrational}{global equivalence};\n\\lword{Erdos249/FullDepthRayAmplifier.lean}{421}{cofinalFullDepthKillSupply_iff_periodMultipleKillSupply}{equivalence after setting depth equal to the shift};\n\\lword{Erdos249/FullDepthRayAmplifier.lean}{438}{cofinalFullDepthKillSupply_iff_irrational}{irrationality and arbitrarily large certificates};\n\\mword{Erdos249257/TotientCarryKernelRigidity.lean}{300}{not_irrational_totientSeries_implies_unbounded_carryRank_unconditional}{the carry-rank lower bound under rationality};\n\\mword{Erdos249257/TotientCarryKernelRigidity.lean}{211}{finrank_canonicalCarryKernel_ge_of_linearIndependent}{carry rank at least $2^e-1$ from independent totient sections};\n\\mword{Erdos249257/TotientTailCarryPeriod.lean}{224}{not_irrational_totientSeries_implies_mod_period_and_unbounded_rank}{modular periodicity and the rank lower bound for the same carry};\n\\mword{Erdos249257/TotientTailCarryPeriod.lean}{126}{totient_forcing_vanishes_mod_of_dvd_multiplier}{forcing vanishing};\n\\mword{Erdos249257/TotientTailCarryPeriod.lean}{140}{totient_carry_modEq_geometric_of_dvd_multiplier}{geometric reduction};\n\\mword{Erdos249257/CertificateKernel.lean}{18445}{tsum_primWeight_div_two_pow_sub_one_eq_totient_series}{identity};\n\\mword{Erdos249257/MersenneLambertLadder.lean}{370}{primWeight_nonneg}{sign};\n\\mword{Erdos249257/MersenneLambertLadder.lean}{297}{primWeight_apply_prime}{primes};\n\\mword{Erdos249257/MersenneLambertLadder.lean}{272}{primWeight_apply_prime_pow}{prime powers};\n\\mword{Erdos249257/MersenneLambertLadder.lean}{321}{primWeight_not_bounded}{unboundedness};\n\\mword{Erdos249257/SquaredMersenneDiagonalEnclosure.lean}{138}{scaleDiagonalTailDifference_sub_lambertProjectedDiagonal}{exact centre decomposition};\n\\mword{Erdos249257/SquaredMersenneDiagonalEnclosure.lean}{408}{abs_scaleDiagonalTailDifference_sub_lambertProjectedDiagonal_le}{explicit error bound};\n\\mword{Erdos249257/SquaredMersenneDiagonalEnclosure.lean}{426}{scaleFullTarget_miss_of_lambert_projected_separation}{separation criterion};\n\\mword{Erdos249257/FullTargetPrimeAdjunctionNoGo.lean}{139}{scaleFullTargetHit_iff_integral}{target equivalence};\n\\mword{Erdos249257/FreshPrimeDeficitDecomposition.lean}{91}{endpointFreshDeficit_nonneg}{endpoint nonnegativity};\n\\mword{Erdos249257/FreshPrimeDeficitDecomposition.lean}{184}{finiteForeignChannelIncrement_eq_endpointFreshDeficit_sub}{foreign-channel identity};\n\\mword{Erdos249257/FreshPrimeDeficitDecomposition.lean}{269}{oldFivePoint_sub_secondBranchAdverseDeficit_le_actual}{five-point loss bound};\n\\mword{Erdos249257/SquareCRTCube.lean}{297}{exists_squareCRT_clean_totient_family}{simultaneous finite family};\n\\mword{Erdos249257/SquareCRTCube.lean}{327}{exists_squareCRT_clean_horizon}{finite horizon};\n\\mword{Erdos249257/SquareCRTCube.lean}{459}{squareCRT_clean_block_can_vanish}{vanishing block};\n\\mword{Erdos249257/SquareCRTCube.lean}{477}{squareCRT_clean_block_can_be_nonzero}{nonzero block};\n\\mword{Erdos249257/DiagonalFreshLossBridge.lean}{2667}{canonicalAdjacentSuffixCentral_iff_slack_nonneg}{centrality equivalence};\n\\mword{Erdos249257/DiagonalFreshLossBridge.lean}{2757}{canonicalAdjacentSuffixJumpCentralSupply_iff_slackSupply}{equivalence of the two conditions};\n\\mword{Erdos249257/DiagonalFreshLossBridge.lean}{2932}{irrational_totientSeries_of_canonicalAdjacentSuffixJumpSlackSupply}{conditional irrationality theorem};\n\\mword{Erdos249257/ActualForeignResidueProjection.lean}{276}{abs_foreignTailWindow_le_foreignComplementBound}{finite complement bound};\n\\mword{Erdos249257/ActualForeignResidueProjection.lean}{308}{finiteResidueDiagonal_eq_projectedForeign_add_divisor}{exact channel split};\n\\mword{Erdos249257/ActualForeignResidueProjection.lean}{414}{scaleFullTarget_miss_of_projected_separation}{conditional target-miss theorem};\n\\mword{Erdos249257/CertificateKernel.lean}{12811}{irrational_ratWeightSeries_eventuallyPeriodic}{the periodic theorem};\n\\mword{Erdos249257/CertificateKernel.lean}{18429}{tsum_totient_div_two_pow_sub_one_eq_two}{$\\sum\\ph(n)/(2^n-1)=2$};\n\\mword{Erdos249257/GenericTailOrbitRigidity.lean}{426}{binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit}{rationality and integer carries with controlled growth};\n\\mword{Erdos249257/FreshPrimeDeficitDecomposition.lean}{171}{diagonalHeightIncrement_eq_old_add_endpointFreshDeficits}{diagonal split};\n\\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/Erdos249257/CarrySurvivorExtinction.lean#L515}{definition};\n\\mword{Erdos249257/TotientTailPeriodKiller.lean}{57}{totientTail}{definition of the scaled tail};\n\\lword{Erdos249/CyclotomicAnchoredKill.lean}{3231}{CyclotomicAnchoredKillSupply}{the proposed cyclotomic residue condition};\n\\lword{Erdos249/CyclotomicAnchoredKill.lean}{3327}{binaryCyclotomicAnchoredKillSupply_iff_irrational}{the checked equivalence};\n\\lword{Erdos249/CyclotomicAnchoredKill.lean}{1706}{exists_unbounded_binaryCyclotomicSupport_with_periodLock_of_not_irrational}{checked conditional support/period counterexample};\n\\lword{Erdos249/CyclotomicAnchoredKill.lean}{1693}{binaryCyclotomicLayer_unboundedPrimeDivisorSupply}{the checked binary-layer theorem};\n\\lword{Erdos249/PeriodMultipleEscape.lean}{441}{ApFullDepthEscape}{certificates with depth equal to the period multiple}.\n\n\\medskip\\noindent\\textbf{Finite denominator exclusions.}\\par\\smallskip\n\\mword{Erdos249257/CertificateKernel.lean}{18371}{tsum_totient_div_pow_two_ne_int_div_of_den_le_79639646646701375323355774875831053}{exclusion};\n\\mword{Erdos249257/CertificateKernel.lean}{18384}{tsum_totient_div_pow_two_ne_ratCast_of_den_le_79639646646701375323355774875831053}{reduced-denominator form};\n\\mword{Erdos249257/GapFareyBound.lean}{225}{gap_check_window_1_240_first_failure}{first failure};\n\\mword{Erdos249257/CertificateKernel.lean}{16025}{irrational_tsum_totient_div_pow_two_of_gap_certificate_supply}{the conditional implication};\n\\mword{Erdos249257/GapFareyBound.lean}{51}{farey_gap}{farey gap};\n\\mword{Erdos249257/GapFareyBound.lean}{176}{gap_check_window_1_240_le_79639646646701375323355774875831053}{the denominator bound at window 240};\n\\mword{Erdos249257/CertificateKernel.lean}{15986}{tsum_totient_div_pow_two_ne_int_div_of_totient_gap_certificate}{denominator exclusion from a finite interval};\n\\mword{Erdos249257/GapFareyBound.lean}{88}{gap_check_window_1_120_le_248672326362367909}{the earlier denominator bound at window 120};\n\\mword{Erdos249257/CertificateKernel.lean}{18572}{tsum_visible_coprime_pairs_ne_int_div_of_den_le_39819823323350687661677887437915526}{denominator exclusion for the coprimality probability};\n\\mword{Erdos249257/DiagonalPincerCertificatesT64.lean}{1933}{diagonalPincerCertificateScalesThroughT64}{list};\n\\mword{Erdos249257/DiagonalPincerCertificatesT64.lean}{1967}{certifiedKill_diagonal_all_imported_through_t64}{verification};\n\\lword{Skip/LadderT67.lean}{71264}{exists_diagonalKill_le_82}{certificates at every scale through 82}.\n\n\\medskip\\noindent\\textbf{Least common multiples and limits of the tested methods.}\\par\\smallskip\n\\mword{Erdos249257/LcmConeFlatness.lean}{316}{exists_certifiedKill_iff_tail_diff_notMem_int}{certificate completeness};\n\\mword{Erdos249257/LcmConeFlatness.lean}{327}{tail_diff_mem_int_iff_scaled_series_mem_int}{tail-difference integrality};\n\\mword{Erdos249257/LcmConeFlatness.lean}{386}{irrational_totient_series_iff_all_tail_diffs_nonintegral}{irrationality iff every tail difference is nonintegral};\n\\mword{Erdos249257/TotientActualLcmOrbitNonintegrality.lean}{37}{irrational_totientSeries_iff_actualLcmOrbitNonintegralitySupply}{exact frontier};\n\\mword{Erdos249257/TotientActualLcmOrbitNonintegrality.lean}{53}{irrational_totientSeries_of_actualLcmOrbitNonintegralitySupply}{forward implication};\n\\mword{Erdos249257/TotientActualLcmOrbitSign.lean}{39}{actualLcmTailDiff_shift_pos}{positive tail differences};\n\\mword{Erdos249257/TotientActualLcmOrbitSign.lean}{172}{actualLcm_trueEndpointSurvivor_neg}{negative endpoint representative};\n\\mword{Erdos249257/TotientActualLcmOrbitSign.lean}{211}{actualLcm_integral_forces_topEdgeResidue}{residue near the upper endpoint};\n\\mword{Erdos249257/LcmFactorIdealPulseObstruction.lean}{798}{lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient}{divisibility preserved by finite combinations of shifts};\n\\mword{Erdos249257/LcmFactorIdealPulseObstruction.lean}{866}{lcm_factorIdeal_sparseAnchor_not_sufficient}{the counterexample with sparse prescribed values};\n\\mword{Erdos249257/LcmConeFlatness.lean}{399}{irrational_totient_series_iff_pointwise_certificates}{pointwise};\n\\mword{Erdos249257/LcmConeFlatness.lean}{412}{irrational_totient_series_iff_certificate_supply}{cofinal at each shift};\n\\mword{Erdos249257/LcmConeFlatness.lean}{426}{irrational_totient_series_iff_lcm_diagonal_certificate_supply}{$\\lcm$ diagonal};\n\\mword{Erdos249257/TotientActualLcmShortKill.lean}{18}{lcmDiagonalArithmeticKill_two_pow_four}{the certificate at \\(t=2^4\\) and depth \\(23\\)};\n\\mword{Erdos249257/TotientActualLcmShortKill.lean}{27}{lcmDiagonalArithmeticKill_two_pow_six}{the certificate at \\(t=2^6\\) and depth \\(93\\)};\n\\mword{Erdos249257/TotientActualLcmShortKill.lean}{35}{actualLcmTailOrbit_four_and_six_notMem_int}{actual-orbit non-integrality at both heights};\n\\mword{Erdos249257/TotientActualLcmShortKill.lean}{54}{powerTwoActualLcmShortArithmeticKillSupply_through_six}{certificates beyond every threshold \\(a_0\\le6\\)};\n\\mword{Erdos249257/TotientActualLcmOrbitSeparation.lean}{141}{abs_actualLcmTailOrbit_sub_rawApprox_lt}{an explicit approximation error};\n\\mword{Erdos249257/TotientActualLcmOrbitSeparation.lean}{179}{actualLcmRawApprox_eq_half_div_fourPow}{the approximation as an integer over \\(4^q\\)};\n\\mword{Erdos249257/TotientActualLcmOrbitSeparation.lean}{208}{halfWordBandAt_of_rawApprox_integerSeparation}{distance at least \\(1/32\\) from every integer};\n\\mword{Erdos249257/TotientActualLcmOrbitSeparation.lean}{254}{PowerTwoActualLcmOrbitSeparationSupply}{the cofinal separation hypothesis};\n\\mword{Erdos249257/TotientActualLcmOrbitSeparation.lean}{305}{irrational_totientSeries_of_actualLcmOrbitSeparationSupply}{the conditional endpoint theorem};\n\\mword{Erdos249257/TotientActualLcmTopEdgeStaircase.lean}{251}{not_actualLcmTerminalDyadicStaircase_of_room}{the staircase is impossible under the stated room hypothesis};\n\\mword{Erdos249257/TotientActualLcmTopEdgeStaircase.lean}{297}{windowDiscrepancy_emod_two_pow_eq_terminal}{terminal-remainder identity};\n\\mword{Erdos249257/TropicalCurvatureCarry.lean}{137}{fixedPrecisionTropicalNoGo}{the limitation of fixed-precision valuation data};\n\\mword{Erdos249257/LcmConeFlatness.lean}{357}{not_irrational_of_tail_diff_mem_int}{rationality from an integral tail difference}.\n\n\\medskip\\noindent\\textbf{M\\\"obius identities and rational approximation.}\\par\\smallskip\n\\lword{Erdos249/RankOneSharpFloor.lean}{534}{rankOneSubrankQuotient_eq_one_five_iff}{unique minimizer};\n\\lword{Erdos249/RankOneSharpFloor.lean}{624}{rankOneSubrankQuotient_sub_theta_two_gt_twentyOne_div_threeTwenty}{uniform lower bound $21/320$};\n\\lword{Erdos249/RankOneSharpFloor.lean}{634}{rankOneSubrankQuotient_sub_theta_two_gt_one_div_sixteen}{lower bound $1/16$};\n\\lword{Erdos249/RankOneSharpFloor.lean}{645}{not_forall_rankOneSubrankQuotient_sub_theta_two_gt_one_div_fifteen}{$1/15$ failure};\n\\lword{Erdos249/RankOneSharpFloor.lean}{655}{positive_direct_sum_sub_theta_two_gt_twentyOne_div_threeTwenty}{positive weighted averages of the quotients};\n\\lword{Erdos249/RankOneSharpFloor.lean}{693}{primitive_form_abs_gt_twentyOne_div_threeTwenty}{rational linear-form obstruction};\n\\mword{Erdos249257/GcdMomentCalculus.lean}{216}{tsum_one_div_mersenne_sq_eq_sigma_sub_tau_series}{divisor-count layer};\n\\mword{Erdos249257/GcdMomentCalculus.lean}{235}{tsum_totient_div_mersenne_sq_eq_gcd_moment_series}{first gcd moment};\n\\mword{Erdos249257/PrimitiveDeterminantLift.lean}{169}{no_fixed_integral_primitive_euler_index}{no fixed index};\n\\mword{Erdos249257/PrimitiveDeterminantLift.lean}{148}{twoTierPrimorial_dvd_of_integral_jet}{the primorial divisibility};\n\\mword{Erdos249257/CertificateKernel.lean}{8335}{irrational_erdosBorwein_series}{irrationality of the Erd\\H{o}s--Borwein series};\n\\mword{Erdos249257/CertificateKernel.lean}{18434}{tsum_moebius_div_two_pow_sub_one_eq_half}{$\\sum\\mu(n)/(2^n-1)=1/2$};\n\\mword{Erdos249257/CertificateKernel.lean}{18454}{totient_series_eq_half_add_moebius_mersenne_square}{the squared-Mersenne expression for the totient series};\n\\mword{Erdos249257/CertificateKernel.lean}{18557}{totient_series_eq_half_add_visible_coprime_pairs}{the coprimality-probability expression};\n\\mword{Erdos249257/GcdMomentCalculus.lean}{349}{tsum_pos_coprime_inv_mersenne_eq_one}{the Lambert-weighted sum over coprime positive pairs};\n\\mword{Erdos249257/GcdMomentCalculus.lean}{474}{cylinderMass_split}{the Stern--Brocot recursion with stopping};\n\\lword{Erdos249/FiniteEulerSieve.lean}{28}{muSqEulerFactor_one}{$s=1$};\n\\lword{Erdos249/FiniteEulerSieve.lean}{36}{muSqEulerFactor_two}{$s=2$};\n\\lword{Erdos249/FiniteEulerSieve.lean}{44}{sigmaPrimePow_firstDiff}{first difference};\n\\lword{Erdos249/FiniteEulerSieve.lean}{51}{sigmaPrimePow_secondDiff}{second difference}.\n\n\\medskip\\noindent\\textbf{Proposed exponential-sum and prime-factor estimates.}\\par\\smallskip\n\\mword{Erdos249257/PivotAntiReconstruction.lean}{1765}{dtwWindowSeparatedPairs_iff_irrational_totient_series}{window-separated pairs};\n\\lword{Erdos249/TotientStrictPrimeEscape.lean}{157}{DTWNaturalPrimeTailOrbitStrictGap}{the unproved quantitative separation condition at primes};\n\\lword{Erdos249/TotientStrictPrimeEscape.lean}{485}{naturalPivotPointEscape_of_naturalPrimeTailOrbitStrictGap}{a positive adaptive truncation budget};\n\\lword{Erdos249/TotientStrictPrimeEscape.lean}{524}{irrational_totient_series_of_naturalPrimeTailOrbitStrictGap}{$S$ is irrational};\n\\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L35}{density hypothesis};\n\\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L25}{block-gap hypothesis};\n\\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L45}{density-to-gap implication};\n\\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L167}{exact squaring recurrence};\n\\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L191}{iterate formula};\n\\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L201}{initial-phase normal form};\n\\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L278}{initial-phase equivalence};\n\\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L322}{dyadic-root obstruction};\n\\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L465}{prime-alignment implication};\n\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{22}{checkerboard}{mixed difference};\n\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{26}{checkerboard_separable}{separable case};\n\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{32}{checkerboard_unique}{fixed-stencil uniqueness};\n\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{45}{fourPoint_layer_identity}{a four-point divisor-layer identity};\n\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{62}{residueClass_step_has_centred_completion}{a centred representative for a residue class modulo an even modulus};\n\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{95}{PrimeRayLayerSupply}{eventual nontriviality and coprimality of the layers};\n\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{102}{BoundedDegreeOrderConsumer}{order divisibility with bounded extension degree};\n\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{151}{FinitePrimeSupportEscape}{exclusion of any fixed finite prime support};\n\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{158}{UnboundedPrimeDivisorSupply}{unbounded prime divisors};\n\\mword{Erdos249257/FirstHarmonicPivot.lean}{299}{mem_pivotFiber_iff_exists_supplierPrime}{membership};\n\\mword{Erdos249257/FirstHarmonicPivot.lean}{367}{pivotBaseOfPrime_injective_on_supplierPrimes}{injectivity};\n\\mword{Erdos249257/FirstHarmonicPivot.lean}{384}{image_pivotSupplierPrimes_eq_pivotFiber}{image equality};\n\\mword{Erdos249257/FirstHarmonicPivot.lean}{394}{supplierPrime_not_globally_isolated_counterexample}{checked counterexample};\n\\mword{Erdos249257/FirstHarmonicPivot.lean}{514}{windowFirstExp_sum_eq_pivot_decomposition}{the exact decomposition into four prime-factor contributions};\n\\mword{Erdos249257/FirstHarmonicPivot.lean}{577}{irrational_totient_series_of_pivotResidualDecorrelation}{the conditional irrationality theorem};\n\\mword{Erdos249257/FirstHarmonicPivot.lean}{543}{PivotBudgetAt}{the four inequalities on a common block};\n\\mword{Erdos249257/FirstHarmonicPivot.lean}{549}{first_harmonic_gap_of_pivotBudgetAt}{the first-harmonic bound from those four inequalities}.\n\n\n\\clearpage\n\\section*{Declarations}\n\n\\paragraph{Further background.}\nErick Wong posted a proof that\n$\\sum_{n\\ge1}(\\varphi(n)\\bmod k)k^{-n}$ is irrational for every integer $k>2$\non Mathematics Stack Exchange on 29 March 2015~\\cite{wong2015}; there the base\nequals the modulus. Erd\\H{o}s's 1957 criterion already gives every base\n(Remark~\\ref{rem:erdos1957}).  The M\\\"obius-square form of $S$ and its coprimality\ninterpretation were posted publicly before this record, as credited in\n\\S\\ref{ssec:coprime}.  Kova\\v{c} and Tao show that suitable subsums of Lambert\nseries in distinct integer bases can be rational~\\cite[Theorem~2.3]{kovac-tao};\nthe freedom to choose those subsums is absent for $S$.  The Lambert-series\nfactorisations of Merca~\\cite[Theorem~1.2]{merca2017} and of Merca and\nSchmidt~\\cite{merca-schmidt} are transform background for the\nM\\\"obius--Mersenne coordinate.  Balasubramanian, Giri and Srivastav prove\nasymptotics for finite shifted convolutions of divisor-convolution\nfunctions, including $\\varphi(n)/n$, uniformly for $|h|\\le x/2$ at\ncutoff $x$~\\cite[Theorem~2.2]{bgs2016}. Their partial-summation extension\nuses differentiable weights with bounded derivative on the summation\ninterval~\\cite[Rem.~2.9]{bgs2016}.\nApplying that observation here would require control of the weighted\nerror for the present phases; no such estimate is established in this\nrecord.\nIn the totient specialisation the divisor-convolution kernel is\n$\\mu(d)/d$ and both decay exponents in their theorem are $1$, giving\nan $O(\\log^2x)$ error for the two-point product. The present quantity\nis instead an exponential of a window of overlapping totient values,\nwith a denominator growing with $L$. Partial summation would carry\nthe variation of its weight into the error term; it does not remove\nthe need to bound that variation and the resulting weighted error. The cited results are not themselves the dyadic residue\nseparation required by $\\mathrm{Sep}$.  The remaining sources are cited where their hypotheses are\nused.\n\nAt the generating-function level, the published height criterion of\nAdamczewski, Bell and Smertnig gives $k$-regularity for a $k$-Mahler\nseries with integer coefficients of polynomial growth\n~\\cite[Thm.~1.2(b), p.~2528]{adamczewski-bell-smertnig2023}.\nFor the totient coefficients the logarithmic heights are\n$\\log\\max(1,\\ph(n))\\le\\log n$ for $n\\ge1$, so Coons's nonregularity\nalready rules out $k$-Mahler equations in every base. The later\nmultiplicative classification of Bell and Smertnig provides another\nroute~\\cite[Thm.~1.3 and p.~3]{bell-smertnig2026}.\nThe algebraic and $D$-finite classifications of Bell, Bruin and Coons\nbelong to the same function-level context~\\cite[Thms.~1.5--1.6]{bell-bruin-coons}.\nThey do not by themselves decide the value at $z=1/2$.\n\nAutomatic prime-sampling theorems address a different issue.\nAdamczewski, Drmota and M\\\"ullner identify a computable local coprimality\ncondition governing the relevant logarithmic densities, with a conditional\nordinary-density transfer~\\cite[Thm.~1.4 and Rem.~1.5]{adamczewski-drmota-mullner}.\nTheir theorem does not assert that every output state has positive density\non a prescribed prime progression, nor does it prove independence with\nautomatic coefficient weights. Any use for that purpose needs its own\ncompatibility argument.\n\n\\paragraph{Sources and data.} The \\href{\\evidencerecordurl}{evidence record}\nlists every Lean declaration, version and check behind the margin marks.\n\n\\paragraph{Verification scope.} Kernel checking validates a formal proof of\nits stated proposition relative to the declared axioms. It does not check\nthat every prose summary has the same hypotheses or verify the literature\ncomparisons. In particular, a checked implication with an unproved\nhypothesis is not a proof that Problem 249 is solved.\n\n\\paragraph{Funding and competing interests.} This work received no external\nfunding. The author declares no competing interests.\n\n\\clearpage\n\\begin{thebibliography}{99}\n\\small\n\\raggedright\n\\setlength{\\itemsep}{0pt}\n\\bibitem{allouche-shallit}\nJ.-P.~Allouche and J.~Shallit, \\emph{The ring of $k$-regular sequences},\nTheoret.\\ Comput.\\ Sci.\\ \\textbf{98} (1992), no.~2, 163--197,\ndoi:\\href{https://doi.org/10.1016/0304-3975(92)90001-V}{10.1016/0304-3975(92)90001-V}.\nDefinitions and early statements are cited with the author preprint's numbering.\n\\bibitem{coons}\nM.~Coons, \\emph{(Non)Automaticity of number theoretic functions},\nJ.\\ Th\\'eor.\\ Nombres Bordeaux \\textbf{22} (2010), no.~2, 339--352,\ndoi:\\href{https://doi.org/10.5802/jtnb.718}{10.5802/jtnb.718}.\nTheorem~3.2, p.~348.\n\\bibitem{coons-jsr}\nM.~Coons, \\emph{Regular sequences and the joint spectral radius},\nInternat.\\ J.\\ Found.\\ Comput.\\ Sci.\\ \\textbf{28} (2017), no.~2, 135--140,\ndoi:\\href{https://doi.org/10.1142/S0129054117500095}{10.1142/S0129054117500095};\narXiv:\\href{https://arxiv.org/abs/1511.07535v1}{1511.07535v1}.\nTheorem~1, Proposition~4 and Corollary~7 are cited with the arXiv v1 numbering.\n\\bibitem{martin-phi-inequalities}\nG.~Martin, \\emph{Simultaneous inequalities among values of the Euler\nphi-function}, arXiv:\\href{https://arxiv.org/abs/math/0603053v1}{math/0603053v1}, 2006.\nTheorem~1, pp.~1--2.\n\\bibitem{wong2015}\nE.~Wong, \\href{https://math.stackexchange.com/a/1211557}{answer 1211557} to\n\\emph{An infinite sum based on the mod-parity of Euler's totient function},\nMathematics Stack Exchange, 29 March 2015, accessed 15 September 2026.\n\\bibitem{erdos1957}\nP.~Erd\\H{o}s, \\emph{On the irrationality of certain series}, Indag. Math.\n\\textbf{19} (1957), 212--219,\n\\url{https://users.renyi.hu/~p_erdos/1957-07.pdf}.\nLemma~1, p.~213; Lemma~4, pp.~215--218; Lemma~4$'$, p.~218.\n\\bibitem{kst2026}\nH.~Kaneko, Y.~Suzuki and Y.~Tachiya, \\emph{Refinements of Erd\\H{o}s's\nirrationality criterion for certain sparse infinite series},\narXiv:\\href{https://arxiv.org/abs/2601.20743v1}{2601.20743v1}, 2026.\nTheorem~A, p.~2; Theorem~3, p.~5; proof of Theorem~A, p.~18.\n\\bibitem{eldar2026oeis}\nA.~Eldar, comment and formula added to OEIS A256936 (revisions 28 and 31),\n15 March 2026, \\url{https://oeis.org/history?seq=A256936},\naccessed 16 September 2026.\n\\bibitem{fan2026totient}\nS.~Fan, comment on Erd\\H{o}s Problem \\#249, 16 May 2026, 19:01,\n\\href{https://www.erdosproblems.com/forum/thread/249}{Erd\\H{o}s Problems\ndiscussion thread}, accessed 16 September 2026.\n\\bibitem{erdosgraham1980}\nP.~Erd\\H{o}s and R.~L.~Graham, \\emph{Old and New Problems and Results in\nCombinatorial Number Theory}, Monographies de L'Enseignement\nMath\\'ematique \\textbf{28}, 1980, p.~61.\n\\bibitem{erdos1948lambert}\nP.~Erd\\H{o}s, \\emph{On arithmetical properties of Lambert series},\nJ.\\ Indian Math.\\ Soc.\\ (N.S.) \\textbf{12} (1948), 63--66,\n\\url{https://users.renyi.hu/~p_erdos/1948-04.pdf}.\n\\bibitem{postelmans-vanassche}\nK.~Postelmans and W.~Van Assche, \\emph{Irrationality of $\\zeta_q(1)$ and\n$\\zeta_q(2)$}, J.\\ Number Theory \\textbf{126} (2007), no.~1, 119--154,\ndoi:\\href{https://doi.org/10.1016/j.jnt.2006.11.011}{10.1016/j.jnt.2006.11.011};\narXiv:\\href{https://arxiv.org/abs/math/0604312v1}{math/0604312v1}.\nTheorem~1.3 is cited with the arXiv v1 pagination, p.~3.\n\\bibitem{nesterenko1996}\nYu.~V.~Nesterenko, \\emph{Modular functions and transcendence questions},\nSb.\\ Math.\\ \\textbf{187} (1996), no.~9, 1319--1348,\ndoi:\\href{https://doi.org/10.1070/SM1996v187n09ABEH000158}{10.1070/SM1996v187n09ABEH000158}.\nCorollary~2, p.~1320.\n\\bibitem{schramm-gcd}\nW.~Schramm, \\emph{The Fourier transform of functions of the greatest common\ndivisor}, Integers \\textbf{8} (2008), \\#A50,\n\\url{https://math.colgate.edu/~integers/i50/i50.pdf}.\nTheorem and equation~(2), p.~2.\n\\bibitem{toth-gcd}\nL.~T\\'oth, \\emph{A survey of gcd-sum functions}, J.\\ Integer Seq.\\ \\textbf{13}\n(2010), Article~10.8.1,\n\\url{https://cs.uwaterloo.ca/journals/JIS/VOL13/Toth/toth10.pdf}.\n\\S1, equations (1)--(2).\n\\bibitem{merca-schmidt}\nM.~Merca and M.~D.~Schmidt, \\emph{Generating special arithmetic functions by\nLambert series factorizations}, Contrib.\\ Discrete Math.\\ \\textbf{14} (2019),\nno.~1, 31--45,\ndoi:\\href{https://doi.org/10.55016/ojs/cdm.v14i1.62425}{10.55016/ojs/cdm.v14i1.62425};\narXiv:\\href{https://arxiv.org/abs/1706.00393v2}{1706.00393v2}.\nEquation~(1) is cited with the arXiv v2 pagination, p.~2.\n\\bibitem{merca2017}\nM.~Merca, \\emph{The Lambert series factorization theorem},\nRamanujan J.\\ \\textbf{44} (2017), 417--435,\ndoi:\\href{https://doi.org/10.1007/s11139-016-9856-3}{10.1007/s11139-016-9856-3}.\nTheorem~1.2, p.~420.\n\\bibitem{kovac-tao}\nV.~Kova\\v{c} and T.~Tao, \\emph{On several irrationality problems for Ahmes\nseries}, Acta Math.\\ Hungar.\\ \\textbf{175} (2025), no.~2, 572--608,\ndoi:\\href{https://doi.org/10.1007/s10474-025-01528-0}{10.1007/s10474-025-01528-0};\narXiv:\\href{https://arxiv.org/abs/2406.17593v4}{2406.17593v4}.\nTheorem~2.3 is cited with the arXiv v4 pagination, p.~5.\n\\bibitem{bgs2016}\nR.~Balasubramanian, S.~Giri and P.~Srivastav, \\emph{On correlations of certain\nmultiplicative functions}, J.\\ Number Theory \\textbf{174} (2017), 221--238,\ndoi:\\href{https://doi.org/10.1016/j.jnt.2016.10.001}{10.1016/j.jnt.2016.10.001};\narXiv:\\href{https://arxiv.org/abs/1511.02221v3}{1511.02221v3}.\nTheorem~2.2 is cited with the arXiv v3 pagination, p.~2.\n\\bibitem{granville-smooth}\nA.~Granville, \\emph{Smooth numbers: computational number theory and beyond},\nin \\emph{Algorithmic Number Theory}, MSRI Publ.\\ \\textbf{44} (2008), 267--323,\n\\url{https://library.slmath.org/books/Book44/files/09andrew.pdf}.\nEquations (1.1) and (1.3), p.~268.\n\\bibitem{schoenberg1928}\nI.~Schoenberg, \\emph{\\\"Uber die asymptotische Verteilung reeller Zahlen\nmod~1}, Math.\\ Z.\\ \\textbf{28} (1928), 171--199,\ndoi:\\href{https://doi.org/10.1007/BF01181156}{10.1007/BF01181156}.\n\\S17, p.~193, for the distribution of $\\varphi(n)/n$.\n\\bibitem{schoenberg1936}\nI.~J.~Schoenberg, \\emph{On asymptotic distributions of arithmetical\nfunctions}, Trans.\\ Amer.\\ Math.\\ Soc.\\ \\textbf{39} (1936), no.~2, 315--330.\nTheorem~1, pp.~318--319; \\S8, p.~323.\n\\bibitem{edeko-vdc}\nN.~Edeko, H.~Kreidler and R.~Nagel, \\emph{A dynamical proof of the van der\nCorput inequality}, Dyn.\\ Syst.\\ \\textbf{37} (2022), no.~4, 648--665,\ndoi:\\href{https://doi.org/10.1080/14689367.2022.2100244}{10.1080/14689367.2022.2100244};\narXiv:\\href{https://arxiv.org/abs/2106.11835v3}{2106.11835v3}.\nTheorem~2.1 is cited with the arXiv v3 pagination, p.~4.\n\\bibitem{matomaki-radziwill}\nK.~Matom\\\"aki and M.~Radziwi\\l\\l, \\emph{Multiplicative functions in short\nintervals}, Ann.\\ of Math.\\ \\textbf{183} (2016), no.~3, 1015--1056,\ndoi:\\href{https://doi.org/10.4007/annals.2016.183.3.6}{10.4007/annals.2016.183.3.6};\narXiv:\\href{https://arxiv.org/abs/1501.04585v4}{1501.04585v4}.\nTheorem~1 is cited with the arXiv v4 pagination, pp.~1--2.\n\\bibitem{tao-elliott}\nT.~Tao, \\emph{The logarithmically averaged Chowla and Elliott conjectures for\ntwo-point correlations}, Forum Math.\\ Pi \\textbf{4} (2016), e8,\ndoi:\\href{https://doi.org/10.1017/fmp.2016.6}{10.1017/fmp.2016.6};\narXiv:\\href{https://arxiv.org/abs/1509.05422v4}{1509.05422v4}.\nTheorems~1.2--1.3 are cited with the arXiv v4 pagination, pp.~2 and~5.\n\\bibitem{tao-teravainen}\nT.~Tao and J.~Ter\\\"av\\\"ainen, \\emph{Odd order cases of the logarithmically\naveraged Chowla conjecture}, J.\\ Th\\'eor.\\ Nombres Bordeaux \\textbf{30}\n(2018), no.~3, 997--1015,\ndoi:\\href{https://doi.org/10.5802/jtnb.1062}{10.5802/jtnb.1062};\narXiv:\\href{https://arxiv.org/abs/1710.02112v1}{1710.02112v1}.\nTheorem~1.1 is cited with the arXiv v1 pagination, p.~2.\n\\bibitem{adamczewski-bugeaud}\nB.~Adamczewski and Y.~Bugeaud, \\emph{On the complexity of algebraic numbers\nI. Expansions in integer bases}, Ann.\\ of Math.\\ \\textbf{165} (2007), no.~2,\n547--565. Theorem~1, p.~549.\n\\bibitem{everest-periodic}\nG.~Everest, A.~J.~van der Poorten, Y.~Puri and T.~Ward, \\emph{Integer\nsequences and periodic points}, J.\\ Integer Seq.\\ \\textbf{5} (2002),\nArticle~02.2.3. Lemma~3.1 is cited with the preprint pagination, p.~5.\n\\bibitem{erdos-gal}\nP.~Erd\\H{o}s and I.~S.~G\\'al, \\emph{On the law of the iterated logarithm.~I},\nProc.\\ Kon.\\ Ned.\\ Akad.\\ Wetensch.\\ Ser.~A \\textbf{58} $=$ Indag.\\ Math.\\\n\\textbf{17} (1955), 65--76, \\url{https://www.renyi.hu/~p_erdos/1955-06.pdf}.\n\\bibitem{yazdani2001} S.~Yazdani,\n\\emph{Multiplicative functions and $k$-automatic sequences},\nJ.\\ Th\\'eor.\\ Nombres Bordeaux \\textbf{13} (2001), no.~2, 651--658,\n\\url{https://www.numdam.org/item/JTNB_2001__13_2_651_0/}.\nTheorem~2 and its proof, pp.~652--653; Corollary~4, p.~654.\n\\bibitem{allouche-shallit-yassawi} J.-P.~Allouche, J.~Shallit and R.~Yassawi,\n\\emph{How to prove that a sequence is not automatic},\nExposition.\\ Math.\\ \\textbf{40} (2022), no.~1, 1--22,\ndoi:\\href{https://doi.org/10.1016/j.exmath.2021.08.001}{10.1016/j.exmath.2021.08.001};\narXiv:\\href{https://arxiv.org/abs/2104.13072v1}{2104.13072v1}.\nTheorem and example numbering follows the arXiv v1 (2021).\n\\bibitem{bell-smertnig2026} J.~Bell and D.~Smertnig,\n\\emph{Mahler series with multiplicative coefficient sequences},\narXiv:\\href{https://arxiv.org/abs/2603.23456v1}{2603.23456v1},\n24 March 2026, preprint. Theorem~1.3, p.~2; Lemma~3.3, p.~9.\n\\bibitem{bell-bruin-coons} J.~P.~Bell, N.~Bruin and M.~Coons,\n\\emph{Transcendence of generating functions whose coefficients are multiplicative},\nTrans.\\ Amer.\\ Math.\\ Soc.\\ \\textbf{364} (2012), no.~2, 933--959,\ndoi:\\href{https://doi.org/10.1090/S0002-9947-2011-05479-6}{10.1090/S0002-9947-2011-05479-6};\narXiv:\\href{https://arxiv.org/abs/1003.2221v2}{1003.2221v2}.\nTheorems~1.5--1.6 are cited with the arXiv v2 pagination, p.~2.\n\\bibitem{adamczewski-drmota-mullner} B.~Adamczewski, M.~Drmota and C.~M\\\"ullner,\n\\emph{(Logarithmic) densities for automatic sequences along primes and squares},\nTrans.\\ Amer.\\ Math.\\ Soc.\\ \\textbf{375} (2022), no.~1, 455--499,\ndoi:\\href{https://doi.org/10.1090/tran/8476}{10.1090/tran/8476};\narXiv:\\href{https://arxiv.org/abs/2009.14773v2}{2009.14773v2}.\nTheorem~1.4 and Remark~1.5, p.~4, and the proof at the end of Section~7,\np.~19, refer to the arXiv v2 (2021).\n\\bibitem{adamczewski-bell-smertnig2023}\nB.~Adamczewski, J.~Bell and D.~Smertnig,\n\\emph{A height gap theorem for coefficients of Mahler functions},\nJ.~Eur.~Math.~Soc.\\ \\textbf{25} (2023), no.~7, 2525--2571,\ndoi:\\href{https://doi.org/10.4171/JEMS/1244}{10.4171/JEMS/1244}.\nTheorem~1.2(b), p.~2528; logarithmic height in Section~1.1.1, p.~2527.\n\\bibitem{crmarickovac2025signed} T. Crmari\\'{c} and V. Kova\\v{c},\n  \\emph{On the irrationality of certain super-polynomially decaying series},\n  arXiv:\\href{https://arxiv.org/abs/2504.18712v1}{2504.18712v1} (2025),\n  Lemma 4(a) and Remark 5.\n\\bibitem{coons-arxiv-signed} M. Coons,\n  \\emph{(Non)Automaticity of number theoretic functions},\n  arXiv:\\href{https://arxiv.org/abs/0810.3709v3}{0810.3709v3}, Theorem 3.3.\n  This version's numbering differs from the journal version cited above.\n\\bibitem{habiro2004signed} K. Habiro,\n  \\emph{Cyclotomic completions of polynomial rings},\n  Publ. Res. Inst. Math. Sci. \\textbf{40} (2004), 1127--1146,\n  Theorem 5.2, p.~1138.\n\n\\end{thebibliography}\n% ---- part a249_p0 ----\n% ---- part a249_p1a ----\n% ---- part a249_p1b ----\n% ---- part a249_newdecls ----\n% ---- part a249_p2 ----\n% ---- part a249_p3 ----\n% ---- part a249_p4 ----\n% ---- part a249_invent ----\n% ---- part a249_p5 ----\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":534837,"sha256":"sha256:2be13b340f9dafc4e3a3dcb34742a66225b3216015b6efa7ae3222c3e3d4a0ed","cite_as":"Plectis reader corpus, paper:erdos249-totient-reasoning-surface","source_repository":"plectis-erdos","source_path":"paper/249/erdos249-totient-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"evidence/erdos249-totient-reasoning-surface","source_path":"paper/evidence/erdos249-totient-reasoning-surface.tex","media_type":"text/x-tex","sha256":"sha256:91d61a5d18554ffe705e1fdd27c1c77ba20fe6f82bd194990aa0f73097947fc8","byte_count":140359,"licence":"CC-BY-4.0","body_format":"latex","body":"% Generated by scripts/paper_evidence.py from docs/paper_lean_coverage.json and the\n% Comparator replay receipts under evidence/comparator/.  Do not edit by hand.\n\\newcommand{\\evidencerecordurl}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface.md}\n\\DeclareResultEvidenceName{thm:denom}{Theorem 1.3}\n\\DeclareResultEvidenceRecord{thm:denom}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#thm-denom}{exact}{compared}\n\\DeclareResultEvidence{thm:denom}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#thm-denom}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#thm-denom-comparator}\n\\DeclareResultEvidenceName{prop:deposits}{Proposition 1.4}\n\\DeclareResultEvidenceRecord{prop:deposits}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-deposits}{exact}{compared}\n\\DeclareResultEvidence{prop:deposits}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-deposits}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-deposits-comparator}\n\\DeclareResultEvidenceName{prop:sign}{Proposition 1.5}\n\\DeclareResultEvidenceRecord{prop:sign}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-sign}{exact}{compared}\n\\DeclareResultEvidence{prop:sign}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-sign}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-sign-comparator}\n\\DeclareResultEvidenceName{prop:rank}{Proposition 1.6}\n\\DeclareResultEvidenceRecord{prop:rank}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-rank}{exact}{compared}\n\\DeclareResultEvidence{prop:rank}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/TotientCarryKernelRigidity.lean\\#L300}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-rank-comparator}\n\\DeclareResultEvidenceName{prop:period-not-rank}{Proposition 1.7}\n\\DeclareResultEvidenceRecord{prop:period-not-rank}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-period-not-rank}{exact}{compared}\n\\DeclareResultEvidence{prop:period-not-rank}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/TotientTailCarryPeriod.lean\\#L224}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-period-not-rank-comparator}\n\\DeclareResultEvidenceName{prop:iffs}{Proposition 1.8}\n\\DeclareResultEvidenceRecord{prop:iffs}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-iffs}{exact}{compared}\n\\DeclareResultEvidence{prop:iffs}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-iffs}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-iffs-comparator}\n\\DeclareResultEvidenceName{prop:parity}{Proposition 1.9}\n\\DeclareResultEvidenceRecord{prop:parity}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-parity}{exact}{compared}\n\\DeclareResultEvidence{prop:parity}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-parity}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-parity-comparator}\n\\DeclareResultEvidenceName{prop:radixresidue}{Proposition 1.10}\n\\DeclareResultEvidenceRecord{prop:radixresidue}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-radixresidue}{exact}{pending}\n\\DeclareResultEvidence{prop:radixresidue}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-radixresidue}{}\n\\DeclareResultEvidenceName{prop:dilations}{Proposition 1.11}\n\\DeclareResultEvidenceRecord{prop:dilations}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-dilations}{exact}{pending}\n\\DeclareResultEvidence{prop:dilations}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-dilations}{}\n\\DeclareResultEvidenceName{prop:slowmoduli}{Remark 12, p. 8}\n\\DeclareResultEvidenceRecord{prop:slowmoduli}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-1.md\\#prop-slowmoduli}{none}{not_applicable}\n\\DeclareResultEvidenceName{lem:gsound}{Lemma 2.1}\n\\DeclareResultEvidenceRecord{lem:gsound}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#lem-gsound}{exact}{compared}\n\\DeclareResultEvidence{lem:gsound}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#lem-gsound}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#lem-gsound-comparator}\n\\DeclareResultEvidenceName{lem:gperiod}{Lemma 2.2}\n\\DeclareResultEvidenceRecord{lem:gperiod}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#lem-gperiod}{exact}{compared}\n\\DeclareResultEvidence{lem:gperiod}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#lem-gperiod}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#lem-gperiod-comparator}\n\\DeclareResultEvidenceName{thm:gamma}{Theorem 2.4}\n\\DeclareResultEvidenceRecord{thm:gamma}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#thm-gamma}{exact}{compared}\n\\DeclareResultEvidence{thm:gamma}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#thm-gamma}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#thm-gamma-comparator}\n\\DeclareResultEvidenceName{cor:b1}{Corollary 2.5}\n\\DeclareResultEvidenceRecord{cor:b1}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#cor-b1}{exact}{compared}\n\\DeclareResultEvidence{cor:b1}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#cor-b1}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#cor-b1-comparator}\n\\DeclareResultEvidenceName{prop:b2}{Proposition 2.7}\n\\DeclareResultEvidenceRecord{prop:b2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#prop-b2}{exact}{compared}\n\\DeclareResultEvidence{prop:b2}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#prop-b2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#prop-b2-comparator}\n\\DeclareResultEvidenceName{prop:b4}{Proposition 2.9}\n\\DeclareResultEvidenceRecord{prop:b4}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#prop-b4}{exact}{compared}\n\\DeclareResultEvidence{prop:b4}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#prop-b4}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#prop-b4-comparator}\n\\DeclareResultEvidenceName{prop:b5}{Proposition 2.11}\n\\DeclareResultEvidenceRecord{prop:b5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#prop-b5}{exact}{compared}\n\\DeclareResultEvidence{prop:b5}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#prop-b5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#prop-b5-comparator}\n\\DeclareResultEvidenceName{prop:b6}{Proposition 2.12}\n\\DeclareResultEvidenceRecord{prop:b6}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#prop-b6}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{prop:b6}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#prop-b6}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#prop-b6-comparator}\n\\DeclareResultEvidenceName{prop:b7}{Proposition 2.13}\n\\DeclareResultEvidenceRecord{prop:b7}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#prop-b7}{exact}{compared}\n\\DeclareResultEvidence{prop:b7}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#prop-b7}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-2.md\\#prop-b7-comparator}\n\\DeclareResultEvidenceName{thm:goodbasegap}{Theorem 3.5}\n\\DeclareResultEvidenceRecord{thm:goodbasegap}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-3.md\\#thm-goodbasegap}{exact}{pending}\n\\DeclareResultEvidence{thm:goodbasegap}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-3.md\\#thm-goodbasegap}{}\n\\DeclareResultEvidenceName{prop:shift}{Proposition 5.3}\n\\DeclareResultEvidenceRecord{prop:shift}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-shift}{exact}{compared}\n\\DeclareResultEvidence{prop:shift}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/TotientTailPeriodKiller.lean\\#L150}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-shift-comparator}\n\\DeclareResultEvidenceName{lem:farey}{Lemma 5.4}\n\\DeclareResultEvidenceRecord{lem:farey}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#lem-farey}{exact}{compared}\n\\DeclareResultEvidence{lem:farey}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/GapFareyBound.lean\\#L51}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#lem-farey-comparator}\n\\DeclareResultEvidenceName{prop:gapwindow}{Proposition 5.5}\n\\DeclareResultEvidenceRecord{prop:gapwindow}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-gapwindow}{exact}{compared}\n\\DeclareResultEvidence{prop:gapwindow}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-gapwindow}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-gapwindow-comparator}\n\\DeclareResultEvidenceName{thm:denom-record}{Theorem 5.6}\n\\DeclareResultEvidenceRecord{thm:denom-record}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#thm-denom-record}{exact}{compared}\n\\DeclareResultEvidence{thm:denom-record}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/CertificateKernel.lean\\#L18384}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#thm-denom-record-comparator}\n\\DeclareResultEvidenceName{prop:coprime}{Proposition 5.9}\n\\DeclareResultEvidenceRecord{prop:coprime}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-coprime}{exact}{compared}\n\\DeclareResultEvidence{prop:coprime}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-coprime}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-coprime-comparator}\n\\DeclareResultEvidenceName{prop:gcdlayer}{Proposition 5.10}\n\\DeclareResultEvidenceRecord{prop:gcdlayer}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-gcdlayer}{exact}{compared}\n\\DeclareResultEvidence{prop:gcdlayer}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-gcdlayer}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-gcdlayer-comparator}\n\\DeclareResultEvidenceName{thm:denomcoprime}{Theorem 5.11}\n\\DeclareResultEvidenceRecord{thm:denomcoprime}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#thm-denomcoprime}{exact}{compared}\n\\DeclareResultEvidence{thm:denomcoprime}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/CertificateKernel.lean\\#L18572}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#thm-denomcoprime-comparator}\n\\DeclareResultEvidenceName{thm:denommobsq}{Theorem 5.12}\n\\DeclareResultEvidenceRecord{thm:denommobsq}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#thm-denommobsq}{exact}{compared}\n\\DeclareResultEvidence{thm:denommobsq}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/CertificateKernel.lean\\#L18487}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#thm-denommobsq-comparator}\n\\DeclareResultEvidenceName{prop:mobsq}{Proposition 5.14}\n\\DeclareResultEvidenceRecord{prop:mobsq}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-mobsq}{exact}{compared}\n\\DeclareResultEvidence{prop:mobsq}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-mobsq}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-mobsq-comparator}\n\\DeclareResultEvidenceName{prop:lambertengine}{Proposition 5.16}\n\\DeclareResultEvidenceRecord{prop:lambertengine}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-lambertengine}{exact}{compared}\n\\DeclareResultEvidence{prop:lambertengine}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/GcdMomentCalculus.lean\\#L105}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-lambertengine-comparator}\n\\DeclareResultEvidenceName{prop:zetaq}{Proposition 5.17}\n\\DeclareResultEvidenceRecord{prop:zetaq}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-zetaq}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{prop:zetaq}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-zetaq}{}\n\\DeclareResultEvidenceName{prop:pillai}{Proposition 5.18}\n\\DeclareResultEvidenceRecord{prop:pillai}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-pillai}{exact}{compared}\n\\DeclareResultEvidence{prop:pillai}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-pillai}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-5.md\\#prop-pillai-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:a9}{Theorem 6.1}\n\\DeclareResultEvidenceRecord{catalogue:cert:a9}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a9}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:a9}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR20/SpecifiedTailPeriod.lean\\#L44}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a9-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:d5}{Theorem 6.2}\n\\DeclareResultEvidenceRecord{catalogue:cert:d5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-d5}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:d5}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/TotientCarryKernelRigidity.lean\\#L300}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-d5-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:d4}{Theorem 6.3}\n\\DeclareResultEvidenceRecord{catalogue:cert:d4}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-d4}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:d4}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR8/FullKernelAssemblies.lean\\#L177}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-d4-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:a8}{Proposition 6.4}\n\\DeclareResultEvidenceRecord{catalogue:cert:a8}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a8}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:a8}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/TotientTailPeriodKiller.lean\\#L327}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a8-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:b5}{Proposition 6.5}\n\\DeclareResultEvidenceRecord{catalogue:mob:b5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b5}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:b5}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b5-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:b6}{Proposition 6.6}\n\\DeclareResultEvidenceRecord{catalogue:mob:b6}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b6}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:b6}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b6}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b6-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:b7a}{Proposition 6.7}\n\\DeclareResultEvidenceRecord{catalogue:mob:b7a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b7a}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:b7a}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR20/DenominatorBounds.lean\\#L10}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b7a-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:b7b}{Proposition 6.8}\n\\DeclareResultEvidenceRecord{catalogue:mob:b7b}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b7b}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:b7b}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b7b}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b7b-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:d3}{Proposition 6.9}\n\\DeclareResultEvidenceRecord{catalogue:mob:d3}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-d3}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:d3}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-d3}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-d3-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:a11}{Proposition 6.10}\n\\DeclareResultEvidenceRecord{catalogue:cert:a11}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a11}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:a11}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a11}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a11-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:a12}{Proposition 6.11}\n\\DeclareResultEvidenceRecord{catalogue:cert:a12}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a12}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:a12}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a12}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a12-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:b11}{Proposition 6.12}\n\\DeclareResultEvidenceRecord{catalogue:cert:b11}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-b11}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:b11}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-b11}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-b11-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:a2}{Proposition 6.16}\n\\DeclareResultEvidenceRecord{catalogue:cert:a2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a2}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:a2}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a2-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:a5}{Lemma 6.17}\n\\DeclareResultEvidenceRecord{catalogue:cert:a5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a5}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:a5}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a5-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:a6}{Proposition 6.18}\n\\DeclareResultEvidenceRecord{catalogue:cert:a6}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a6}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:a6}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a6}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a6-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:a7}{Theorem 6.19}\n\\DeclareResultEvidenceRecord{catalogue:cert:a7}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a7}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:a7}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/LcmConeFlatness.lean\\#L316}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-a7-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:c1}{Proposition 6.21}\n\\DeclareResultEvidenceRecord{catalogue:cert:c1}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-c1}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:c1}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/GapFareyBound.lean\\#L51}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-c1-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:d1}{Proposition 6.22}\n\\DeclareResultEvidenceRecord{catalogue:cert:d1}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-d1}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:d1}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/CertificateKernel.lean\\#L5371}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-d1-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:d2}{Proposition 6.23}\n\\DeclareResultEvidenceRecord{catalogue:cert:d2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-d2}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{catalogue:cert:d2}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-d2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-d2-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:d3}{Proposition 6.24}\n\\DeclareResultEvidenceRecord{catalogue:cert:d3}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-d3}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:d3}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/TotientMahlerDefect.lean\\#L91}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-d3-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:d9}{Proposition 6.25}\n\\DeclareResultEvidenceRecord{catalogue:cert:d9}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-d9}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:d9}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-d9}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-cert-d9-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:a1a}{Proposition 6.26}\n\\DeclareResultEvidenceRecord{catalogue:mob:a1a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a1a}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:a1a}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a1a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a1a-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:a1b}{Corollary 6.27}\n\\DeclareResultEvidenceRecord{catalogue:mob:a1b}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a1b}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:a1b}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR7/ArithmeticAssemblies.lean\\#L40}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a1b-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:a2}{Proposition 6.28}\n\\DeclareResultEvidenceRecord{catalogue:mob:a2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a2}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:a2}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/GcdMomentCalculus.lean\\#L105}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a2-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:a4}{Proposition 6.30}\n\\DeclareResultEvidenceRecord{catalogue:mob:a4}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a4}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{catalogue:mob:a4}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a4}{}\n\\DeclareResultEvidenceName{catalogue:mob:a5}{Proposition 6.31}\n\\DeclareResultEvidenceRecord{catalogue:mob:a5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a5}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:a5}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a5-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:a7}{Proposition 6.33}\n\\DeclareResultEvidenceRecord{catalogue:mob:a7}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a7}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:a7}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a7}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a7-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:a8}{Proposition 6.34}\n\\DeclareResultEvidenceRecord{catalogue:mob:a8}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a8}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:a8}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/GcdMomentCalculus.lean\\#L349}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a8-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:a9a}{Proposition 6.35}\n\\DeclareResultEvidenceRecord{catalogue:mob:a9a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a9a}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:a9a}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a9a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a9a-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:a9b}{Proposition 6.36}\n\\DeclareResultEvidenceRecord{catalogue:mob:a9b}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a9b}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:a9b}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a9b}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-a9b-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:b1}{Proposition 6.37}\n\\DeclareResultEvidenceRecord{catalogue:mob:b1}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b1}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:b1}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b1}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b1-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:b2}{Proposition 6.38}\n\\DeclareResultEvidenceRecord{catalogue:mob:b2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b2}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:b2}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-1.md\\#catalogue-mob-b2-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:b3}{Proposition 6.39}\n\\DeclareResultEvidenceRecord{catalogue:mob:b3}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-b3}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:b3}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR20/RadicalDecomposition.lean\\#L50}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-b3-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:b4}{Proposition 6.40}\n\\DeclareResultEvidenceRecord{catalogue:mob:b4}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-b4}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:b4}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-b4}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-b4-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:b8a}{Proposition 6.41}\n\\DeclareResultEvidenceRecord{catalogue:mob:b8a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-b8a}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:b8a}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/PrimePowerJumpDynamics.lean\\#L281}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-b8a-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:b8b}{Proposition 6.42}\n\\DeclareResultEvidenceRecord{catalogue:mob:b8b}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-b8b}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:b8b}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/PrimePowerJumpDynamics.lean\\#L310}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-b8b-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:d7}{Proposition 6.43}\n\\DeclareResultEvidenceRecord{catalogue:cert:d7}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-d7}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:d7}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-d7}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-d7-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:e3}{Proposition 6.45}\n\\DeclareResultEvidenceRecord{catalogue:mob:e3}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-e3}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:e3}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-e3}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-e3-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:f1}{Proposition 6.46}\n\\DeclareResultEvidenceRecord{catalogue:mob:f1}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-f1}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:f1}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-f1}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-f1-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:a10}{Theorem 6.47}\n\\DeclareResultEvidenceRecord{catalogue:cert:a10}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-a10}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:a10}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/LcmConeFlatness.lean\\#L412}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-a10-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:b2}{Theorem 6.48}\n\\DeclareResultEvidenceRecord{catalogue:cert:b2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b2}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:b2}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b2-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:b3}{Theorem 6.49}\n\\DeclareResultEvidenceRecord{catalogue:cert:b3}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b3}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:b3}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/LcmConeFlatness.lean\\#L426}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b3-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:b4}{Lemma 6.50}\n\\DeclareResultEvidenceRecord{catalogue:cert:b4}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b4}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:b4}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/LcmDiagonalReduction.lean\\#L137}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b4-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:b5}{Proposition 6.51}\n\\DeclareResultEvidenceRecord{catalogue:cert:b5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b5}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:b5}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b5-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:b6}{Theorem 6.52}\n\\DeclareResultEvidenceRecord{catalogue:cert:b6}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b6}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:b6}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b6}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b6-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:b7}{Theorem 6.53}\n\\DeclareResultEvidenceRecord{catalogue:cert:b7}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b7}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:b7}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b7}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b7-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:b8}{Corollary 6.54}\n\\DeclareResultEvidenceRecord{catalogue:cert:b8}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b8}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:b8}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b8}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b8-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:b9a}{Proposition 6.55}\n\\DeclareResultEvidenceRecord{catalogue:cert:b9a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b9a}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:b9a}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b9a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b9a-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:b10a}{Theorem 6.56}\n\\DeclareResultEvidenceRecord{catalogue:cert:b10a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b10a}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:b10a}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b10a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b10a-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:b10b}{Theorem 6.57}\n\\DeclareResultEvidenceRecord{catalogue:cert:b10b}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b10b}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:b10b}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b10b}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b10b-comparator}\n\\DeclareResultEvidenceName{catalogue:cert:b12}{Proposition 6.58}\n\\DeclareResultEvidenceRecord{catalogue:cert:b12}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b12}{exact}{compared}\n\\DeclareResultEvidence{catalogue:cert:b12}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b12}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-cert-b12-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:e1}{Theorem 6.59}\n\\DeclareResultEvidenceRecord{catalogue:mob:e1}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-e1}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:e1}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-e1}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-e1-comparator}\n\\DeclareResultEvidenceName{catalogue:mob:e2}{Theorem 6.60}\n\\DeclareResultEvidenceRecord{catalogue:mob:e2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-e2}{exact}{compared}\n\\DeclareResultEvidence{catalogue:mob:e2}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-e2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#catalogue-mob-e2-comparator}\n\\DeclareResultEvidenceName{prop:NI-01}{Proposition 6.61}\n\\DeclareResultEvidenceRecord{prop:NI-01}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-ni-01}{exact}{compared}\n\\DeclareResultEvidence{prop:NI-01}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/ActualLcmDiagonalConditions.lean\\#L29}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-ni-01-comparator}\n\\DeclareResultEvidenceName{prop:AR-07}{Proposition 6.62}\n\\DeclareResultEvidenceRecord{prop:AR-07}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-ar-07}{exact}{compared}\n\\DeclareResultEvidence{prop:AR-07}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-ar-07}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-ar-07-comparator}\n\\DeclareResultEvidenceName{prop:SEP-03}{Proposition 6.63}\n\\DeclareResultEvidenceRecord{prop:SEP-03}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-sep-03}{exact}{compared}\n\\DeclareResultEvidence{prop:SEP-03}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-sep-03}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-sep-03-comparator}\n\\DeclareResultEvidenceName{prop:TE-04}{Proposition 6.64}\n\\DeclareResultEvidenceRecord{prop:TE-04}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-te-04}{exact}{compared}\n\\DeclareResultEvidence{prop:TE-04}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-te-04}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-te-04-comparator}\n\\DeclareResultEvidenceName{prop:TE-05}{Proposition 6.65}\n\\DeclareResultEvidenceRecord{prop:TE-05}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-te-05}{exact}{compared}\n\\DeclareResultEvidence{prop:TE-05}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/TopEdgeStaircaseConditions.lean\\#L80}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-te-05-comparator}\n\\DeclareResultEvidenceName{prop:TE-06}{Proposition 6.66}\n\\DeclareResultEvidenceRecord{prop:TE-06}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-te-06}{exact}{compared}\n\\DeclareResultEvidence{prop:TE-06}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-te-06}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-te-06-comparator}\n\\DeclareResultEvidenceName{prop:SK-02}{Proposition 6.67}\n\\DeclareResultEvidenceRecord{prop:SK-02}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-sk-02}{exact}{compared}\n\\DeclareResultEvidence{prop:SK-02}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-sk-02}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-sk-02-comparator}\n\\DeclareResultEvidenceName{prop:FR-01}{Proposition 6.68}\n\\DeclareResultEvidenceRecord{prop:FR-01}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-fr-01}{exact}{compared}\n\\DeclareResultEvidence{prop:FR-01}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-fr-01}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-fr-01-comparator}\n\\DeclareResultEvidenceName{prop:CP-06}{Proposition 6.69}\n\\DeclareResultEvidenceRecord{prop:CP-06}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-cp-06}{exact}{compared}\n\\DeclareResultEvidence{prop:CP-06}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-cp-06}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-cp-06-comparator}\n\\DeclareResultEvidenceName{prop:CP-07}{Proposition 6.70}\n\\DeclareResultEvidenceRecord{prop:CP-07}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-cp-07}{exact}{compared}\n\\DeclareResultEvidence{prop:CP-07}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-cp-07}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-cp-07-comparator}\n\\DeclareResultEvidenceName{prop:A10}{Proposition 6.71}\n\\DeclareResultEvidenceRecord{prop:A10}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-a10}{exact}{compared}\n\\DeclareResultEvidence{prop:A10}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/LcmConeFlatness.lean\\#L412}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-2.md\\#prop-a10-comparator}\n\\DeclareResultEvidenceName{prop:B3}{Proposition 6.72}\n\\DeclareResultEvidenceRecord{prop:B3}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-b3}{exact}{compared}\n\\DeclareResultEvidence{prop:B3}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/LcmConeFlatness.lean\\#L426}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-b3-comparator}\n\\DeclareResultEvidenceName{prop:B7}{Proposition 6.73}\n\\DeclareResultEvidenceRecord{prop:B7}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-b7-60c3ed}{exact}{compared}\n\\DeclareResultEvidence{prop:B7}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/CertificateKernel.lean\\#L19014}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-b7-60c3ed-comparator}\n\\DeclareResultEvidenceName{prop:B10}{Proposition 6.74}\n\\DeclareResultEvidenceRecord{prop:B10}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-b10}{exact}{compared}\n\\DeclareResultEvidence{prop:B10}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-b10}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-b10-comparator}\n\\DeclareResultEvidenceName{prop:C2sup}{Proposition 6.75}\n\\DeclareResultEvidenceRecord{prop:C2sup}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-c2sup}{exact}{compared}\n\\DeclareResultEvidence{prop:C2sup}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-c2sup}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-c2sup-comparator}\n\\DeclareResultEvidenceName{prop:D5cons}{Proposition 6.76}\n\\DeclareResultEvidenceRecord{prop:D5cons}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-d5cons}{exact}{compared}\n\\DeclareResultEvidence{prop:D5cons}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-d5cons}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-d5cons-comparator}\n\\DeclareResultEvidenceName{prop:B12cons}{Proposition 6.77}\n\\DeclareResultEvidenceRecord{prop:B12cons}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-b12cons}{exact}{compared}\n\\DeclareResultEvidence{prop:B12cons}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-b12cons}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-b12cons-comparator}\n\\DeclareResultEvidenceName{prop:AR-04-inv}{Proposition 6.78}\n\\DeclareResultEvidenceRecord{prop:AR-04-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-ar-04-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:AR-04-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-ar-04-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-ar-04-inv-comparator}\n\\DeclareResultEvidenceName{prop:AR-03-inv}{Proposition 6.79}\n\\DeclareResultEvidenceRecord{prop:AR-03-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-ar-03-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:AR-03-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/ActualLcmShortWindowArithmetic.lean\\#L215}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-ar-03-inv-comparator}\n\\DeclareResultEvidenceName{prop:AR-05-inv}{Proposition 6.80}\n\\DeclareResultEvidenceRecord{prop:AR-05-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-ar-05-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:AR-05-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/ActualLcmShortWindowArithmetic.lean\\#L229}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-ar-05-inv-comparator}\n\\DeclareResultEvidenceName{prop:AR-06-inv}{Proposition 6.81}\n\\DeclareResultEvidenceRecord{prop:AR-06-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-ar-06-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:AR-06-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-ar-06-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-ar-06-inv-comparator}\n\\DeclareResultEvidenceName{prop:SEP-01-inv}{Proposition 6.82}\n\\DeclareResultEvidenceRecord{prop:SEP-01-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-sep-01-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:SEP-01-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/ActualLcmSeparationAndSign.lean\\#L24}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-sep-01-inv-comparator}\n\\DeclareResultEvidenceName{prop:SEP-02-inv}{Proposition 6.83}\n\\DeclareResultEvidenceRecord{prop:SEP-02-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-sep-02-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:SEP-02-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-sep-02-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-sep-02-inv-comparator}\n\\DeclareResultEvidenceName{prop:SGN-01}{Proposition 6.84}\n\\DeclareResultEvidenceRecord{prop:SGN-01}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-sgn-01}{exact}{compared}\n\\DeclareResultEvidence{prop:SGN-01}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-sgn-01}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-sgn-01-comparator}\n\\DeclareResultEvidenceName{prop:SGN-03}{Proposition 6.85}\n\\DeclareResultEvidenceRecord{prop:SGN-03}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-sgn-03}{exact}{compared}\n\\DeclareResultEvidence{prop:SGN-03}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/ActualLcmSeparationAndSign.lean\\#L105}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-sgn-03-comparator}\n\\DeclareResultEvidenceName{prop:TE-02-inv}{Proposition 6.86}\n\\DeclareResultEvidenceRecord{prop:TE-02-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-te-02-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:TE-02-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-te-02-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-te-02-inv-comparator}\n\\DeclareResultEvidenceName{prop:TE-03-inv}{Proposition 6.87}\n\\DeclareResultEvidenceRecord{prop:TE-03-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-te-03-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:TE-03-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-te-03-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-te-03-inv-comparator}\n\\DeclareResultEvidenceName{prop:FR-02-inv}{Proposition 6.88}\n\\DeclareResultEvidenceRecord{prop:FR-02-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-fr-02-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:FR-02-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-fr-02-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-fr-02-inv-comparator}\n\\DeclareResultEvidenceName{prop:CP-01-inv}{Proposition 6.89}\n\\DeclareResultEvidenceRecord{prop:CP-01-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-cp-01-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:CP-01-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-cp-01-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-cp-01-inv-comparator}\n\\DeclareResultEvidenceName{prop:CP-02}{Proposition 6.90}\n\\DeclareResultEvidenceRecord{prop:CP-02}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-cp-02}{exact}{compared}\n\\DeclareResultEvidence{prop:CP-02}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/TailCarryPeriodAndRankFloor.lean\\#L46}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-cp-02-comparator}\n\\DeclareResultEvidenceName{prop:CP-05-inv}{Proposition 6.91}\n\\DeclareResultEvidenceRecord{prop:CP-05-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-cp-05-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:CP-05-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/TwoAdicPulseBlockAndMobiusInversion.lean\\#L28}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-cp-05-inv-comparator}\n\\DeclareResultEvidenceName{prop:TA-inv}{Proposition 6.92}\n\\DeclareResultEvidenceRecord{prop:TA-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-ta-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:TA-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-ta-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-ta-inv-comparator}\n\\DeclareResultEvidenceName{prop:MP-01-inv}{Proposition 6.93}\n\\DeclareResultEvidenceRecord{prop:MP-01-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-mp-01-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:MP-01-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-mp-01-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-mp-01-inv-comparator}\n\\DeclareResultEvidenceName{prop:SK-01-inv}{Proposition 6.94}\n\\DeclareResultEvidenceRecord{prop:SK-01-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-sk-01-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:SK-01-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-sk-01-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-sk-01-inv-comparator}\n\\DeclareResultEvidenceName{prop:A5-inv}{Proposition 6.95}\n\\DeclareResultEvidenceRecord{prop:A5-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-a5-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:A5-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-a5-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-a5-inv-comparator}\n\\DeclareResultEvidenceName{prop:A9-inv}{Proposition 6.96}\n\\DeclareResultEvidenceRecord{prop:A9-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-a9-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:A9-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-a9-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-a9-inv-comparator}\n\\DeclareResultEvidenceName{prop:B6-inv}{Proposition 6.97}\n\\DeclareResultEvidenceRecord{prop:B6-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-b6-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:B6-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-b6-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-b6-inv-comparator}\n\\DeclareResultEvidenceName{prop:D4-inv}{Proposition 6.98}\n\\DeclareResultEvidenceRecord{prop:D4-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-d4-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:D4-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-d4-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-d4-inv-comparator}\n\\DeclareResultEvidenceName{prop:C2-inv}{Proposition 6.99}\n\\DeclareResultEvidenceRecord{prop:C2-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-c2-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:C2-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-c2-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-c2-inv-comparator}\n\\DeclareResultEvidenceName{prop:C3-inv}{Proposition 6.100}\n\\DeclareResultEvidenceRecord{prop:C3-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-c3-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:C3-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-c3-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-c3-inv-comparator}\n\\DeclareResultEvidenceName{prop:D1D2-inv}{Proposition 6.101}\n\\DeclareResultEvidenceRecord{prop:D1D2-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-d1d2-inv}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{prop:D1D2-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-d1d2-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-d1d2-inv-comparator}\n\\DeclareResultEvidenceName{prop:D7-inv}{Proposition 6.102}\n\\DeclareResultEvidenceRecord{prop:D7-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-d7-inv}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{prop:D7-inv}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-d7-inv}{}\n\\DeclareResultEvidenceName{prop:D9-inv}{Proposition 6.103}\n\\DeclareResultEvidenceRecord{prop:D9-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-d9-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:D9-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-d9-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-3.md\\#prop-d9-inv-comparator}\n\\DeclareResultEvidenceName{prop:B11-inv}{Proposition 6.104}\n\\DeclareResultEvidenceRecord{prop:B11-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-b11-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:B11-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-b11-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-b11-inv-comparator}\n\\DeclareResultEvidenceName{prop:C1-inv}{Proposition 6.105}\n\\DeclareResultEvidenceRecord{prop:C1-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-c1-inv}{exact}{compared}\n\\DeclareResultEvidence{prop:C1-inv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-c1-inv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-c1-inv-comparator}\n\\DeclareResultEvidenceName{thm:values-exponents-one-two}{Theorem 6.121}\n\\DeclareResultEvidenceRecord{thm:values-exponents-one-two}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-values-exponents-one-two}{exact}{compared}\n\\DeclareResultEvidence{thm:values-exponents-one-two}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/MobiusMersenneLadderLogConcavity.lean\\#L23}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-values-exponents-one-two-comparator}\n\\DeclareResultEvidenceName{thm:first-two-summands-give-positive}{Theorem 6.122}\n\\DeclareResultEvidenceRecord{thm:first-two-summands-give-positive}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-first-two-summands-give-positive}{exact}{compared}\n\\DeclareResultEvidence{thm:first-two-summands-give-positive}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-first-two-summands-give-positive}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-first-two-summands-give-positive-comparator}\n\\DeclareResultEvidenceName{thm:strict-log-concavity-all-integer}{Theorem 6.123}\n\\DeclareResultEvidenceRecord{thm:strict-log-concavity-all-integer}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-strict-log-concavity-all-integer}{exact}{compared}\n\\DeclareResultEvidence{thm:strict-log-concavity-all-integer}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-strict-log-concavity-all-integer}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-strict-log-concavity-all-integer-comparator}\n\\DeclareResultEvidenceName{thm:binary-carry-criterion}{Theorem 6.125}\n\\DeclareResultEvidenceRecord{thm:binary-carry-criterion}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-binary-carry-criterion}{exact}{compared}\n\\DeclareResultEvidence{thm:binary-carry-criterion}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-binary-carry-criterion}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-binary-carry-criterion-comparator}\n\\DeclareResultEvidenceName{prop:uniqueness-under-growth-condition}{Proposition 6.126}\n\\DeclareResultEvidenceRecord{prop:uniqueness-under-growth-condition}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-uniqueness-under-growth-condition}{exact}{compared}\n\\DeclareResultEvidence{prop:uniqueness-under-growth-condition}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/TemperedOrbitAndSquaredMersenneTail.lean\\#L28}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-uniqueness-under-growth-condition-comparator}\n\\DeclareResultEvidenceName{prop:exact-rational-approximation-formula}{Proposition 6.128}\n\\DeclareResultEvidenceRecord{prop:exact-rational-approximation-formula}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-exact-rational-approximation-formula}{exact}{compared}\n\\DeclareResultEvidence{prop:exact-rational-approximation-formula}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-exact-rational-approximation-formula}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-exact-rational-approximation-formula-comparator}\n\\DeclareResultEvidenceName{prop:geometric-tail-bound}{Proposition 6.129}\n\\DeclareResultEvidenceRecord{prop:geometric-tail-bound}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-geometric-tail-bound}{exact}{compared}\n\\DeclareResultEvidence{prop:geometric-tail-bound}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-geometric-tail-bound}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-geometric-tail-bound-comparator}\n\\DeclareResultEvidenceName{prop:sum-over-divisor-indices}{Proposition 6.131}\n\\DeclareResultEvidenceRecord{prop:sum-over-divisor-indices}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-sum-over-divisor-indices}{exact}{compared}\n\\DeclareResultEvidence{prop:sum-over-divisor-indices}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-sum-over-divisor-indices}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-sum-over-divisor-indices-comparator}\n\\DeclareResultEvidenceName{prop:divisor-sum-complement}{Proposition 6.132}\n\\DeclareResultEvidenceRecord{prop:divisor-sum-complement}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-divisor-sum-complement}{exact}{compared}\n\\DeclareResultEvidence{prop:divisor-sum-complement}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-divisor-sum-complement}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-divisor-sum-complement-comparator}\n\\DeclareResultEvidenceName{thm:doubling-full-totient-difference}{Theorem 6.133}\n\\DeclareResultEvidenceRecord{thm:doubling-full-totient-difference}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-doubling-full-totient-difference}{exact}{compared}\n\\DeclareResultEvidence{thm:doubling-full-totient-difference}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-doubling-full-totient-difference}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-doubling-full-totient-difference-comparator}\n\\DeclareResultEvidenceName{prop:doubling-identity-two-portions-sum}{Proposition 6.135}\n\\DeclareResultEvidenceRecord{prop:doubling-identity-two-portions-sum}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-doubling-identity-two-portions-sum}{exact}{compared}\n\\DeclareResultEvidence{prop:doubling-identity-two-portions-sum}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/DivisorChannelSplitAndSeamDoubling.lean\\#L289}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-doubling-identity-two-portions-sum-comparator}\n\\DeclareResultEvidenceName{thm:positive-coefficients-numerator-polynomial}{Theorem 6.136}\n\\DeclareResultEvidenceRecord{thm:positive-coefficients-numerator-polynomial}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-positive-coefficients-numerator-polynomial}{exact}{compared}\n\\DeclareResultEvidence{thm:positive-coefficients-numerator-polynomial}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-positive-coefficients-numerator-polynomial}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-positive-coefficients-numerator-polynomial-comparator}\n\\DeclareResultEvidenceName{cor:numerator-polynomial-explicit-positive-coefficients}{Corollary 6.137}\n\\DeclareResultEvidenceRecord{cor:numerator-polynomial-explicit-positive-coefficients}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#cor-numerator-polynomial-explicit-positive-coefficients}{exact}{compared}\n\\DeclareResultEvidence{cor:numerator-polynomial-explicit-positive-coefficients}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#cor-numerator-polynomial-explicit-positive-coefficients}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#cor-numerator-polynomial-explicit-positive-coefficients-comparator}\n\\DeclareResultEvidenceName{prop:geometric-remainder-bound}{Proposition 6.139}\n\\DeclareResultEvidenceRecord{prop:geometric-remainder-bound}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-geometric-remainder-bound}{exact}{compared}\n\\DeclareResultEvidence{prop:geometric-remainder-bound}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-geometric-remainder-bound}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-geometric-remainder-bound-comparator}\n\\DeclareResultEvidenceName{prop:squared-distance-phase-one}{Proposition 6.140}\n\\DeclareResultEvidenceRecord{prop:squared-distance-phase-one}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-squared-distance-phase-one}{exact}{compared}\n\\DeclareResultEvidence{prop:squared-distance-phase-one}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/PhaseEnergyAndForeignResidueProjection.lean\\#L31}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-squared-distance-phase-one-comparator}\n\\DeclareResultEvidenceName{lem:squared-distance-bound-separated-pairs}{Lemma 6.141}\n\\DeclareResultEvidenceRecord{lem:squared-distance-bound-separated-pairs}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#lem-squared-distance-bound-separated-pairs}{exact}{compared}\n\\DeclareResultEvidence{lem:squared-distance-bound-separated-pairs}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/PhaseEnergyAndForeignResidueProjection.lean\\#L40}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#lem-squared-distance-bound-separated-pairs-comparator}\n\\DeclareResultEvidenceName{prop:finite-divisor-sum}{Proposition 6.142}\n\\DeclareResultEvidenceRecord{prop:finite-divisor-sum}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-finite-divisor-sum}{exact}{compared}\n\\DeclareResultEvidence{prop:finite-divisor-sum}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-finite-divisor-sum}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-finite-divisor-sum-comparator}\n\\DeclareResultEvidenceName{prop:separation-larger-than-error-implies}{Proposition 6.143}\n\\DeclareResultEvidenceRecord{prop:separation-larger-than-error-implies}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-separation-larger-than-error-implies}{exact}{compared}\n\\DeclareResultEvidence{prop:separation-larger-than-error-implies}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-separation-larger-than-error-implies}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-separation-larger-than-error-implies-comparator}\n\\DeclareResultEvidenceName{thm:coprime-pair-counting-totient}{Theorem 6.144}\n\\DeclareResultEvidenceRecord{thm:coprime-pair-counting-totient}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-coprime-pair-counting-totient}{exact}{compared}\n\\DeclareResultEvidence{thm:coprime-pair-counting-totient}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-coprime-pair-counting-totient}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-coprime-pair-counting-totient-comparator}\n\\DeclareResultEvidenceName{prop:two-lattice-sums}{Proposition 6.145}\n\\DeclareResultEvidenceRecord{prop:two-lattice-sums}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-two-lattice-sums}{exact}{compared}\n\\DeclareResultEvidence{prop:two-lattice-sums}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-two-lattice-sums}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-two-lattice-sums-comparator}\n\\DeclareResultEvidenceName{thm:classical-coprime-pair-lambert-identity}{Theorem 6.146}\n\\DeclareResultEvidenceRecord{thm:classical-coprime-pair-lambert-identity}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-classical-coprime-pair-lambert-identity}{exact}{compared}\n\\DeclareResultEvidence{thm:classical-coprime-pair-lambert-identity}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-classical-coprime-pair-lambert-identity}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-classical-coprime-pair-lambert-identity-comparator}\n\\DeclareResultEvidenceName{thm:nondivisors-short-lcm-window}{Theorem 6.149}\n\\DeclareResultEvidenceRecord{thm:nondivisors-short-lcm-window}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-nondivisors-short-lcm-window}{exact}{compared}\n\\DeclareResultEvidence{thm:nondivisors-short-lcm-window}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-nondivisors-short-lcm-window}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-nondivisors-short-lcm-window-comparator}\n\\DeclareResultEvidenceName{thm:unbounded-prime-support-mersenne-factors}{Theorem 6.151}\n\\DeclareResultEvidenceRecord{thm:unbounded-prime-support-mersenne-factors}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-unbounded-prime-support-mersenne-factors}{exact}{compared}\n\\DeclareResultEvidence{thm:unbounded-prime-support-mersenne-factors}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-unbounded-prime-support-mersenne-factors}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-unbounded-prime-support-mersenne-factors-comparator}\n\\DeclareResultEvidenceName{thm:prime-satisfying-stated-cyclotomic-conditions}{Theorem 6.152}\n\\DeclareResultEvidenceRecord{thm:prime-satisfying-stated-cyclotomic-conditions}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-prime-satisfying-stated-cyclotomic-conditions}{exact}{compared}\n\\DeclareResultEvidence{thm:prime-satisfying-stated-cyclotomic-conditions}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-prime-satisfying-stated-cyclotomic-conditions}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-prime-satisfying-stated-cyclotomic-conditions-comparator}\n\\DeclareResultEvidenceName{thm:sufficient-order-hypothesis-unbounded-prime}{Theorem 6.155}\n\\DeclareResultEvidenceRecord{thm:sufficient-order-hypothesis-unbounded-prime}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-sufficient-order-hypothesis-unbounded-prime}{exact}{compared}\n\\DeclareResultEvidence{thm:sufficient-order-hypothesis-unbounded-prime}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-sufficient-order-hypothesis-unbounded-prime}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-sufficient-order-hypothesis-unbounded-prime-comparator}\n\\DeclareResultEvidenceName{thm:uniform-positive-gap-rank-one}{Theorem 6.156}\n\\DeclareResultEvidenceRecord{thm:uniform-positive-gap-rank-one}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-uniform-positive-gap-rank-one}{exact}{compared}\n\\DeclareResultEvidence{thm:uniform-positive-gap-rank-one}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-uniform-positive-gap-rank-one}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-uniform-positive-gap-rank-one-comparator}\n\\DeclareResultEvidenceName{prop:bound-preserved-positive-normalised-averaging}{Proposition 6.157}\n\\DeclareResultEvidenceRecord{prop:bound-preserved-positive-normalised-averaging}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-bound-preserved-positive-normalised-averaging}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{prop:bound-preserved-positive-normalised-averaging}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-bound-preserved-positive-normalised-averaging}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#prop-bound-preserved-positive-normalised-averaging-comparator}\n\\DeclareResultEvidenceName{thm:concatenation-specified-period-multiples}{Theorem 6.159}\n\\DeclareResultEvidenceRecord{thm:concatenation-specified-period-multiples}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-concatenation-specified-period-multiples}{exact}{compared}\n\\DeclareResultEvidence{thm:concatenation-specified-period-multiples}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-concatenation-specified-period-multiples}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-6-4.md\\#thm-concatenation-specified-period-multiples-comparator}\n\\DeclareResultEvidenceName{thm:hgap-real}{Theorem 9.4}\n\\DeclareResultEvidenceRecord{thm:hgap-real}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-hgap-real}{exact}{compared}\n\\DeclareResultEvidence{thm:hgap-real}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-hgap-real}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-hgap-real-comparator}\n\\DeclareResultEvidenceName{thm:hgap-subset}{Theorem 9.5}\n\\DeclareResultEvidenceRecord{thm:hgap-subset}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-hgap-subset}{exact}{compared}\n\\DeclareResultEvidence{thm:hgap-subset}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-hgap-subset}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-hgap-subset-comparator}\n\\DeclareResultEvidenceName{thm:hgap-norm}{Theorem 9.6}\n\\DeclareResultEvidenceRecord{thm:hgap-norm}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-hgap-norm}{exact}{compared}\n\\DeclareResultEvidence{thm:hgap-norm}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-hgap-norm}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-hgap-norm-comparator}\n\\DeclareResultEvidenceName{thm:positivity-remaining-residue-inequality}{Theorem 9.10}\n\\DeclareResultEvidenceRecord{thm:positivity-remaining-residue-inequality}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-positivity-remaining-residue-inequality}{exact}{compared}\n\\DeclareResultEvidence{thm:positivity-remaining-residue-inequality}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-positivity-remaining-residue-inequality}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-positivity-remaining-residue-inequality-comparator}\n\\DeclareResultEvidenceName{thm:proved-implications}{Theorem 9.12}\n\\DeclareResultEvidenceRecord{thm:proved-implications}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-proved-implications}{exact}{compared}\n\\DeclareResultEvidence{thm:proved-implications}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-proved-implications}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-proved-implications-comparator}\n\\DeclareResultEvidenceName{prop:te-chain}{Proposition 9.14}\n\\DeclareResultEvidenceRecord{prop:te-chain}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-te-chain}{exact}{compared}\n\\DeclareResultEvidence{prop:te-chain}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-te-chain}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-te-chain-comparator}\n\\DeclareResultEvidenceName{thm:proved-implication}{Theorem 9.16}\n\\DeclareResultEvidenceRecord{thm:proved-implication}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-proved-implication}{exact}{compared}\n\\DeclareResultEvidence{thm:proved-implication}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-proved-implication}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-proved-implication-comparator}\n\\DeclareResultEvidenceName{prop:sufficient-extension}{Proposition 9.18}\n\\DeclareResultEvidenceRecord{prop:sufficient-extension}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-sufficient-extension}{exact}{compared}\n\\DeclareResultEvidence{prop:sufficient-extension}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-sufficient-extension}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-sufficient-extension-comparator}\n\\DeclareResultEvidenceName{thm:separation-rational-approximation}{Theorem 9.19}\n\\DeclareResultEvidenceRecord{thm:separation-rational-approximation}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-separation-rational-approximation}{exact}{compared}\n\\DeclareResultEvidence{thm:separation-rational-approximation}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-separation-rational-approximation}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-separation-rational-approximation-comparator}\n\\DeclareResultEvidenceName{thm:short-window-examples-through-exponent}{Theorem 9.22}\n\\DeclareResultEvidenceRecord{thm:short-window-examples-through-exponent}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-short-window-examples-through-exponent}{exact}{compared}\n\\DeclareResultEvidence{thm:short-window-examples-through-exponent}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-short-window-examples-through-exponent}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-short-window-examples-through-exponent-comparator}\n\\DeclareResultEvidenceName{thm:diagonal-certificate-table}{Theorem 9.24}\n\\DeclareResultEvidenceRecord{thm:diagonal-certificate-table}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-diagonal-certificate-table}{exact}{compared}\n\\DeclareResultEvidence{thm:diagonal-certificate-table}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-diagonal-certificate-table}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-diagonal-certificate-table-comparator}\n\\DeclareResultEvidenceName{prop:sufficient-extension-2}{Proposition 9.26}\n\\DeclareResultEvidenceRecord{prop:sufficient-extension-2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-sufficient-extension-2}{exact}{compared}\n\\DeclareResultEvidence{prop:sufficient-extension-2}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-sufficient-extension-2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-sufficient-extension-2-comparator}\n\\DeclareResultEvidenceName{thm:one-common-certificate-sixteen-shifts}{Theorem 9.27}\n\\DeclareResultEvidenceRecord{thm:one-common-certificate-sixteen-shifts}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-one-common-certificate-sixteen-shifts}{exact}{compared}\n\\DeclareResultEvidence{thm:one-common-certificate-sixteen-shifts}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-one-common-certificate-sixteen-shifts}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-one-common-certificate-sixteen-shifts-comparator}\n\\DeclareResultEvidenceName{prop:simultaneous-certificates-unrestricted-depth}{Proposition 9.29}\n\\DeclareResultEvidenceRecord{prop:simultaneous-certificates-unrestricted-depth}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-simultaneous-certificates-unrestricted-depth}{exact}{compared}\n\\DeclareResultEvidence{prop:simultaneous-certificates-unrestricted-depth}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-simultaneous-certificates-unrestricted-depth}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-simultaneous-certificates-unrestricted-depth-comparator}\n\\DeclareResultEvidenceName{thm:fixed-farey-bound}{Theorem 9.30}\n\\DeclareResultEvidenceRecord{thm:fixed-farey-bound}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-fixed-farey-bound}{exact}{compared}\n\\DeclareResultEvidence{thm:fixed-farey-bound}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-fixed-farey-bound}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-fixed-farey-bound-comparator}\n\\DeclareResultEvidenceName{prop:sufficient-extension-3}{Proposition 9.33}\n\\DeclareResultEvidenceRecord{prop:sufficient-extension-3}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-sufficient-extension-3}{exact}{compared}\n\\DeclareResultEvidence{prop:sufficient-extension-3}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-sufficient-extension-3}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-sufficient-extension-3-comparator}\n\\DeclareResultEvidenceName{thm:two-adic-congruence-that-does}{Theorem 9.34}\n\\DeclareResultEvidenceRecord{thm:two-adic-congruence-that-does}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-two-adic-congruence-that-does}{exact}{compared}\n\\DeclareResultEvidence{thm:two-adic-congruence-that-does}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-two-adic-congruence-that-does}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-two-adic-congruence-that-does-comparator}\n\\DeclareResultEvidenceName{prop:sufficient-accumulated-residue-condition}{Proposition 9.36}\n\\DeclareResultEvidenceRecord{prop:sufficient-accumulated-residue-condition}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-sufficient-accumulated-residue-condition}{exact}{compared}\n\\DeclareResultEvidence{prop:sufficient-accumulated-residue-condition}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-sufficient-accumulated-residue-condition}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-sufficient-accumulated-residue-condition-comparator}\n\\DeclareResultEvidenceName{thm:arbitrarily-large-prime-power-lcm}{Theorem 9.37}\n\\DeclareResultEvidenceRecord{thm:arbitrarily-large-prime-power-lcm}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-arbitrarily-large-prime-power-lcm}{exact}{compared}\n\\DeclareResultEvidence{thm:arbitrarily-large-prime-power-lcm}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-arbitrarily-large-prime-power-lcm}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-arbitrarily-large-prime-power-lcm-comparator}\n\\DeclareResultEvidenceName{prop:sufficient-inequality}{Proposition 9.40}\n\\DeclareResultEvidenceRecord{prop:sufficient-inequality}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-sufficient-inequality}{exact}{compared}\n\\DeclareResultEvidence{prop:sufficient-inequality}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-sufficient-inequality}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-sufficient-inequality-comparator}\n\\DeclareResultEvidenceName{thm:certificate-four-tail-combination}{Theorem 9.41}\n\\DeclareResultEvidenceRecord{thm:certificate-four-tail-combination}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-certificate-four-tail-combination}{exact}{compared}\n\\DeclareResultEvidence{thm:certificate-four-tail-combination}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-certificate-four-tail-combination}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-certificate-four-tail-combination-comparator}\n\\DeclareResultEvidenceName{thm:one-explicit-witness}{Theorem 9.42}\n\\DeclareResultEvidenceRecord{thm:one-explicit-witness}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-one-explicit-witness}{exact}{compared}\n\\DeclareResultEvidence{thm:one-explicit-witness}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-one-explicit-witness}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-one-explicit-witness-comparator}\n\\DeclareResultEvidenceName{thm:denominators-mobius-sums}{Theorem 9.44}\n\\DeclareResultEvidenceRecord{thm:denominators-mobius-sums}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-denominators-mobius-sums}{exact}{compared}\n\\DeclareResultEvidence{thm:denominators-mobius-sums}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-denominators-mobius-sums}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-denominators-mobius-sums-comparator}\n\\DeclareResultEvidenceName{prop:additional-approximation-hypothesis}{Proposition 9.46}\n\\DeclareResultEvidenceRecord{prop:additional-approximation-hypothesis}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-additional-approximation-hypothesis}{exact}{compared}\n\\DeclareResultEvidence{prop:additional-approximation-hypothesis}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-additional-approximation-hypothesis}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-additional-approximation-hypothesis-comparator}\n\\DeclareResultEvidenceName{thm:rationality-carry-rank}{Theorem 9.47}\n\\DeclareResultEvidenceRecord{thm:rationality-carry-rank}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-rationality-carry-rank}{exact}{compared}\n\\DeclareResultEvidence{thm:rationality-carry-rank}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-rationality-carry-rank}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-rationality-carry-rank-comparator}\n\\DeclareResultEvidenceName{thm:rationality-carry-rank-2}{Theorem 9.48}\n\\DeclareResultEvidenceRecord{thm:rationality-carry-rank-2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-rationality-carry-rank-2}{exact}{compared}\n\\DeclareResultEvidence{thm:rationality-carry-rank-2}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/DyadicSectionBasisAndRationalCarry.lean\\#L129}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#thm-rationality-carry-rank-2-comparator}\n\\DeclareResultEvidenceName{prop:additional-hypothesis-totient-specific-rank}{Proposition 9.50}\n\\DeclareResultEvidenceRecord{prop:additional-hypothesis-totient-specific-rank}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-additional-hypothesis-totient-specific-rank}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{prop:additional-hypothesis-totient-specific-rank}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-additional-hypothesis-totient-specific-rank}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-9.md\\#prop-additional-hypothesis-totient-specific-rank-comparator}\n\\DeclareResultEvidenceName{prop:rational-series-preserving-totient-parity}{Proposition 10.1}\n\\DeclareResultEvidenceRecord{prop:rational-series-preserving-totient-parity}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#prop-rational-series-preserving-totient-parity}{exact}{compared}\n\\DeclareResultEvidence{prop:rational-series-preserving-totient-parity}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#prop-rational-series-preserving-totient-parity}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#prop-rational-series-preserving-totient-parity-comparator}\n\\DeclareResultEvidenceName{lem:complement-divisibility-after-multiplication}{Lemma 10.2}\n\\DeclareResultEvidenceRecord{lem:complement-divisibility-after-multiplication}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#lem-complement-divisibility-after-multiplication}{exact}{compared}\n\\DeclareResultEvidence{lem:complement-divisibility-after-multiplication}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/ScalarLocalisationAndInversePhaseGauge.lean\\#L29}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#lem-complement-divisibility-after-multiplication-comparator}\n\\DeclareResultEvidenceName{lem:nonvanishing-unique-largest-denominator-exponent}{Lemma 10.3}\n\\DeclareResultEvidenceRecord{lem:nonvanishing-unique-largest-denominator-exponent}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#lem-nonvanishing-unique-largest-denominator-exponent}{exact}{compared}\n\\DeclareResultEvidence{lem:nonvanishing-unique-largest-denominator-exponent}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#lem-nonvanishing-unique-largest-denominator-exponent}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#lem-nonvanishing-unique-largest-denominator-exponent-comparator}\n\\DeclareResultEvidenceName{prop:nonzero-minor-survives-inverse-phase}{Proposition 10.4}\n\\DeclareResultEvidenceRecord{prop:nonzero-minor-survives-inverse-phase}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#prop-nonzero-minor-survives-inverse-phase}{exact}{compared}\n\\DeclareResultEvidence{prop:nonzero-minor-survives-inverse-phase}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/ScalarLocalisationAndInversePhaseGauge.lean\\#L51}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#prop-nonzero-minor-survives-inverse-phase-comparator}\n\\DeclareResultEvidenceName{prop:tail-integrality-lcm-grid}{Proposition 10.5}\n\\DeclareResultEvidenceRecord{prop:tail-integrality-lcm-grid}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#prop-tail-integrality-lcm-grid}{exact}{compared}\n\\DeclareResultEvidence{prop:tail-integrality-lcm-grid}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR20/LcmGridCorrespondence.lean\\#L8}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#prop-tail-integrality-lcm-grid-comparator}\n\\DeclareResultEvidenceName{prop:finite-grid-certificate-gives-nonintegral}{Proposition 10.6}\n\\DeclareResultEvidenceRecord{prop:finite-grid-certificate-gives-nonintegral}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#prop-finite-grid-certificate-gives-nonintegral}{exact}{compared}\n\\DeclareResultEvidence{prop:finite-grid-certificate-gives-nonintegral}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#prop-finite-grid-certificate-gives-nonintegral}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#prop-finite-grid-certificate-gives-nonintegral-comparator}\n\\DeclareResultEvidenceName{thm:exact-dyadic-rank-infinite-dimensionality}{Theorem 10.7}\n\\DeclareResultEvidenceRecord{thm:exact-dyadic-rank-infinite-dimensionality}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#thm-exact-dyadic-rank-infinite-dimensionality}{exact}{compared}\n\\DeclareResultEvidence{thm:exact-dyadic-rank-infinite-dimensionality}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#thm-exact-dyadic-rank-infinite-dimensionality}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#thm-exact-dyadic-rank-infinite-dimensionality-comparator}\n\\DeclareResultEvidenceName{long249:thm:signed-interpolation}{Remark 273, p. 112}\n\\DeclareResultEvidenceRecord{long249:thm:signed-interpolation}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#long249-thm-signed-interpolation}{none}{not_applicable}\n\\DeclareResultEvidenceName{long249:cor:signed-observations}{Remark 274, p. 116}\n\\DeclareResultEvidenceRecord{long249:cor:signed-observations}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#long249-cor-signed-observations}{none}{not_applicable}\n\\DeclareResultEvidenceName{long249:prop:signed-first-harmonic}{Remark 275, p. 117}\n\\DeclareResultEvidenceRecord{long249:prop:signed-first-harmonic}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-10.md\\#long249-prop-signed-first-harmonic}{none}{not_applicable}\n\\DeclareResultEvidenceName{lem:orbit}{Lemma 12.1}\n\\DeclareResultEvidenceRecord{lem:orbit}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#lem-orbit}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{lem:orbit}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#lem-orbit}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#lem-orbit-comparator}\n\\DeclareResultEvidenceName{prop:transfer}{Proposition 12.2}\n\\DeclareResultEvidenceRecord{prop:transfer}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#prop-transfer}{exact}{compared}\n\\DeclareResultEvidence{prop:transfer}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/DoublingOrbitTransferAndFullDepthPhase.lean\\#L92}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#prop-transfer-comparator}\n\\DeclareResultEvidenceName{cor:digitform}{Corollary 12.3}\n\\DeclareResultEvidenceRecord{cor:digitform}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#cor-digitform}{exact}{compared}\n\\DeclareResultEvidence{cor:digitform}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#cor-digitform}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#cor-digitform-comparator}\n\\DeclareResultEvidenceName{thm:lacunary}{Theorem 12.5}\n\\DeclareResultEvidenceRecord{thm:lacunary}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#thm-lacunary}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:lacunary}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#thm-lacunary}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#thm-lacunary-comparator}\n\\DeclareResultEvidenceName{prop:dickman}{Proposition 12.6}\n\\DeclareResultEvidenceRecord{prop:dickman}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#prop-dickman}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{prop:dickman}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/UnassignedSmoothCount.lean\\#L556}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#prop-dickman-comparator}\n\\DeclareResultEvidenceName{prop:badcof}{Proposition 12.7}\n\\DeclareResultEvidenceRecord{prop:badcof}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#prop-badcof}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{prop:badcof}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR21/ExcludedCofactorEstimate.lean\\#L654}{}\n\\DeclareResultEvidenceName{prop:route4}{Proposition 12.9}\n\\DeclareResultEvidenceRecord{prop:route4}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#prop-route4}{exact}{compared}\n\\DeclareResultEvidence{prop:route4}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#prop-route4}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#prop-route4-comparator}\n\\DeclareResultEvidenceName{res:fulldepth}{Theorem A.1}\n\\DeclareResultEvidenceRecord{res:fulldepth}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#res-fulldepth}{exact}{compared}\n\\DeclareResultEvidence{res:fulldepth}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos249/PaperCompleteR7/ShortNoteAssemblies.lean\\#L35}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#res-fulldepth-comparator}\n\\DeclareResultEvidenceName{res:rankonefloor}{Theorem A.3}\n\\DeclareResultEvidenceRecord{res:rankonefloor}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#res-rankonefloor}{exact}{compared}\n\\DeclareResultEvidence{res:rankonefloor}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#res-rankonefloor}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos249-totient-reasoning-surface/section-12.md\\#res-rankonefloor-comparator}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% 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One set of upper Banach density zero supports the corrections, which satisfy any prescribed bound tending to infinity and eventually preserve each fixed coefficient and partial-sum congruence. For prime gaps with logarithmic corrections, block distributions remain close for lengths o(log log X), and cumulative positions remain asymptotic to n log n. Those positions need not be prime; the original irrationality question remains open.","owns":"the sparse correction construction and its prime-gap application; the long paper develops tail recurrences, local certificates and the limitations of other irrationality methods","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #251, which remains open","first_pass":["sec:problem","sec:construction","sec:prime-application"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-251-prime-gap-dyadic-series.pdf","hosted_pdf_path":"papers/erdos-251-prime-gap-dyadic-series.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/251/erdos-251-prime-gap-dyadic-series.tex","site_html":"maths/papers/erdos-251-prime-gap-dyadic-series.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/251/erdos-251-prime-gap-dyadic-series.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/251/erdos-251-prime-gap-dyadic-series.tex","body_pointer":"papers.records.erdos-251-prime-gap-dyadic-series.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% One of the Erdős Problem Notes: a short standalone treatment of a single\n% problem in the ErdosProblems expansion library.  Build with\n% `tectonic erdos-251-prime-gap-dyadic-series.tex` or `make`.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n\\usepackage{tikz}\n\\input{evidence/erdos-251-prime-gap-dyadic-series}\n% Problem-specific source pin: #251 advances independently of the corpus\n% default, while the #249/#257 notes retain their own reviewed snapshot.\n\\renewcommand{\\commit}{99f4bf47422abbd8757cbb22b50ba079d764d3a7}\n\\renewcommand{\\commitshort}{99f4bf47422a}\n\\hypersetup{\n  pdftitle={Sparse congruence-preserving perturbations of dyadic series (Erdős Problem 251)},\n  pdfsubject={Erdős Problem Notes: Problem 251},\n  pdfkeywords={irrationality, prime gaps, dyadic series, summation by parts, Lean 4},\n}\n\n\\runninghead{Erd\\H{o}s \\#251: sparse perturbations and exact criteria}\n\\title{Sparse Congruence-Preserving Perturbations\\\\of Dyadic Series}\n\\subtitle{What gap statistics can tell us about Erd\\H{o}s Problem~\\#251}\n\\author{Will Cook}\n\\date{September 2026}\n\n\\begin{document}\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\nWe perturb convergent dyadic series with nonnegative integer coefficients\nto attain both rational and irrational sums while preserving eventual\ncongruences and asymptotic block statistics.  Nonnegative corrections on\none set of upper Banach density zero attain a nondegenerate interval above\nthe original sum.  They eventually respect any prescribed bound tending\nto infinity and preserve coefficient and partial-sum residues for every\nfixed modulus, with target-independent cutoffs.  We vary adjacent correction pairs\nwith fixed ordinary totals; growing digit ranges let later weighted\nchoices fill the gaps between present choices.  Applied to prime gaps with\ncorrection bound $\\log(n+3)$, this preserves cumulative growth $n\\log n$\nand gives vanishing total variation error for blocks of length\n$o(\\log\\log X)$ sampled on $[X,2X)$.  Cumulative positions need not be\nprime; irrationality of the actual prime series remains open.\n\\end{abstract}\n\n\\section{Introduction}\\label{sec:problem}\nLet $p_0=2,p_1=3,\\ldots$ be the primes and put $g_n=p_{n+1}-p_n$.\nErd\\H{o}s asked whether\n\\[\n \\Pi=\\sum_{n\\ge0}\\frac{p_n}{2^{n+1}}\n\\]\nis irrational \\cite[p.~94]{erdos1958}\\cite[p.~62]{erdosgraham1980}\n\\cite[p.~103]{erdos1988}.  Summation by parts gives\n$\\Pi=2+\\sum_{n\\ge0}g_n2^{-(n+1)}$, with convergence justified in\nSection~\\ref{sec:parts}.  We study which properties of the gaps survive\nwhen the latter sum is changed to a prescribed value.\n\nThe support and every modulus cutoff are fixed before the target is\nchosen.  The interval contains rational and irrational sums with the\nsame eventual coefficient and partial-sum residues.  A block changes\nonly when it meets the support, whereas every correction enters the\ndyadic sum.  With polylogarithmic corrections, empirical block laws\nagree asymptotically at lengths $o(\\log\\log X)$; for prime gaps,\ncumulative growth remains $n\\log n$.  The comparison positions need\nnot be consecutive primes.\n\nWe use a standard interval-covering argument for series with finite choices.\nFridy's generalised-base lemma \\cite[p.~194]{fridy1966} treats prescribed\ndigit bounds and nonincreasing weights.  Crmari\\'{c} and Kova\\v{c}\n\\cite[Lemma~4]{crmarickovac2025} give the finite-choice form used here,\nand Kova\\v{c} and Tao \\cite[Lemma~5.1]{kovactao2024} use analogous\nintervals of reciprocal choices.  We must arrange these choices on a sparse set while preserving cumulative\ncongruences.  To do so, we redistribute a fixed correction between two adjacent\nindices.  The simplest useful calculation is\n\\[\n \\begin{array}{c|rrrr}\n (e_n,e_{n+1}) &(0,6)&(2,4)&(4,2)&(6,0)\\\\\n e_n+e_{n+1} &6&6&6&6\\\\\n 2^{n+2}\\bigl(e_n2^{-n-1}+e_{n+1}2^{-n-2}\\bigr)&6&8&10&12.\n \\end{array}\n\\]\nMoving $2$ from the later index to the earlier one changes the weighted\nsum while leaving the ordinary sum fixed.  If the cumulative correction\nbefore the pair is odd, an additional $1$ at $n-1$ clears its residue\nmodulo $2$.  Since the four pairs have the same sum, the choice of pair leaves every\nlater cumulative residue unchanged.  We repeat this operation with\nnested moduli and increasingly separated pairs.  The larger separations\nmake the weighted contributions smaller, so we enlarge the range of each\npair until the later contributions fill the gaps between consecutive\npresent choices.\n\nThroughout, $\\N=\\{0,1,\\ldots\\}$, intervals of indices contain integers,\nand $\\log$ is natural unless a base is displayed.  A set $S\\subseteq\\N$\nhas \\emph{upper Banach density zero} if\n\\[\n \\lim_{H\\to\\infty}\\sup_{u\\in\\N}\\frac{|S\\cap[u,u+H)|}{H}=0.\n\\]\nFor integers $X,m\\ge1$, let $\\mu_{a,X,m}$ denote the distribution of\n$(a_n,\\ldots,a_{n+m-1})$ when $n$ is uniform on $[X,2X)$.  The entries\nare not rescaled: we sample starting indices, not distinct block values.  We use\n$d_{\\rm TV}(\\mu,\\nu)=\\sup_B|\\mu(B)-\\nu(B)|$, with the supremum over all sets of blocks.\n\n\\begin{proposition}[sparse changes preserving congruences]\\label{res:sparserationalisation}\nLet $a:\\N\\to\\N$ satisfy $A=\\sum_{n\\ge0}a_n2^{-(n+1)}<\\infty$,\nlet $K\\in\\N$, and let $f:\\N\\to\\R$ tend to $+\\infty$.\nThere exist a set $S\\subseteq[K,\\infty)$ of upper Banach density zero\nand a nondegenerate interval $I\\subset(A,\\infty)$ such that, for every\n$r\\in I$, there is an integer correction $e:\\N\\to\\N$ satisfying\n\\[\n \\operatorname{supp}e\\subseteq S,\\qquad e_n\\le f(n)\\ \\text{eventually},\n \\qquad \\sum_{n\\ge0}(a_n+e_n)2^{-(n+1)}=r.\n\\]\nFor each integer $q\\ge1$ there is a cutoff $N_q$, independent of $r$,\nsuch that $q\\mid e_n$ and $q\\mid\\sum_{i<n}e_i$ for all $n\\ge N_q$.\nFor every $\\varepsilon>0$, the choice $f(n)=(\\log(n+3))^\\varepsilon$\ncan be made with\n\\[\n |S\\cap[X,2X)|=O_\\varepsilon\\!\\left(\\frac{X}{\\log\\log X}\\right),\n \\qquad\n d_{\\rm TV}(\\mu_{a,X,m},\\mu_{a+e,X,m})\n \\le\\frac{m|S\\cap[X,2X+m)|}{X}.\n\\]\nIn particular the distance tends to zero for every integer-valued\n$m=m(X)\\ge1$ with $m(X)=o(\\log\\log X)$, uniformly over target values.\n\\end{proposition}\n\nThe cutoff may depend on $q$, but $S$ and all cutoffs are fixed before\n$r$.  Only $e$ varies with the target, and its support may occupy just\npart of $S$.  Infinitely many changes are essential: finitely many\ninteger dyadic corrections add a rational number.  The allowance may\ngrow as slowly as $\\log\\log(n+3)$, but it cannot be bounded.  A modulus\nexceeding the bound would force $e$ eventually to vanish; its constant\ncumulative sum would then be divisible by every positive integer.\nNonnegativity forces $e=0$.  Polynomially growing coefficients satisfy\nthe convergence hypothesis; $a_n=2^n$ does not.\n\nSection~\\ref{sec:construction} proves the construction and\nSection~\\ref{sec:prime-application} transfers the stated prime-gap properties.\nFor the actual primes, integral tail shifts characterise rationality.\nTwo consecutive small differences with unequal gaps give a sufficient\ncondition for irrationality.  Finite enclosures certify individual\nwitnesses without supplying arbitrarily late ones.  The\n\\href{erdos251-prime-gap-reasoning-surface.pdf\\#context}{companion's literature discussion}\ncontains the further comparisons, including finite subsums and other\ndyadic coefficient sequences.\n\n\\section{Proof of Proposition~\\ref{res:sparserationalisation}}\\label{sec:construction}\n\\begin{proof}\n\\emph{Fixed totals.}\nUse $n_j-1$ to cancel a cumulative residue and $n_j,n_j+1$ for a\npair with fixed ordinary sum.  Varying the pair changes its weighted\ncontribution without altering any later residue to be cancelled.\nChoose an initial index $n_{-1}\\ge K$ and centres with\n$s_j=n_j-n_{j-1}\\ge4$, so the triples do not overlap.  Let\n$M_j$ be positive integers with $M_j\\mid M_{j+1}$, and let $D_j$\nbound the digit at stage $j$.  The spacing, moduli and digit bounds\nwill be chosen below.\n\nLet $C_j$ record the ordinary correction before the $j$th triple.\nStarting from $C_0=0$, prescribe\n\\[\n c_j=(-C_j)\\bmod M_j\\quad(0\\le c_j<M_j),\\qquad\n C_{j+1}=C_j+c_j+M_jD_j.\n\\]\nFor integers $0\\le d_j\\le D_j$, set\n\\begin{equation}\\label{eq:correction-triple}\n e_{n_j-1}=c_j,\\qquad e_{n_j}=M_jd_j,\\qquad\n e_{n_j+1}=M_j(D_j-d_j),\n\\end{equation}\nand put $e_n=0$ elsewhere.  The two sums of a triple are\n\\[\n \\begin{aligned}\n \\sum_{r=n_j-1}^{n_j+1}e_r\n    &=c_j+M_jD_j,\\\\\n \\sum_{r=n_j-1}^{n_j+1}e_r2^{-r-1}\n    &=c_j2^{-n_j}+M_jD_j2^{-n_j-2}+d_jM_j2^{-n_j-2}.\n \\end{aligned}\n\\]\nThe ordinary total contains no $d_j$, so the recurrence fixes every\n$c_j$ in advance.  In the weighted total only the last term varies:\ntransferring $M_j$ from $n_j+1$ to $n_j$ adds\n\\[\n w_j=M_j2^{-n_j-2}.\n\\]\n\nAfter $c_j$ has been inserted, the cumulative correction is divisible\nby $M_j$; the two entries of the pair preserve this divisibility.\nFix $q\\mid M_J$.  At index $n_J$ both the coefficient correction and\nthe cumulative correction are divisible by $q$.  At every later\ntriple, $q$ divides $C_j$ and $M_j$, so it also divides $c_j$ and\nboth pair entries.  The intervening corrections are zero.  Hence\n\\[\n q\\mid e_n,\\qquad q\\mid\\sum_{i<n}e_i\\qquad(n\\ge n_J).\n\\]\nThe entry $c_J$ need not be divisible by $q$, which is why it is placed at\n$n_J-1$, before the cutoff.  The index $n_J$ depends on $q$ and the chosen\nmoduli, but not on the digits or the value to be attained.\n\n\\par\\medskip\\noindent\\emph{Spacing and size.}\nThe choices at stage $j$ are $0,w_j,\\ldots,D_jw_j$.\nThe covering argument will work if the later ranges total at least\n$w_j$.  Taking $D_i=2^{s_i}-1$ compensates for separation by making\neach later range a scaled dyadic difference:\n\\[\n D_iw_i=(2^{n_i-n_{i-1}}-1)M_i2^{-n_i-2}\n       =M_i\\bigl(2^{-n_{i-1}-2}-2^{-n_i-2}\\bigr).\n\\]\nBecause $M_i\\ge M_j$ for $i>j$, replacing every later modulus by $M_j$\ngives the finite estimate\n\\begin{equation}\\label{eq:finite-overlap}\n \\begin{aligned}\n \\sum_{i=j+1}^{J}D_iw_i\n &\\ge M_j\\sum_{i=j+1}^{J}\n       \\bigl(2^{-n_{i-1}-2}-2^{-n_i-2}\\bigr)\\\\\n &=w_j-M_j2^{-n_J-2}\\qquad(J>j).\n \\end{aligned}\n\\end{equation}\nNo monotonicity of the weights is needed.  Each entry of\n\\eqref{eq:correction-triple} is at most $M_j2^{s_j}$, so the size cost of\nseparating the stages is exponential.  We must fit this product below\nthe allowance at all three indices, while $s_j\\to\\infty$ and every\nfixed positive integer eventually divides $M_j$.\n\nTo allow a nonmonotone bound $f$, put\n\\[\n h(n)=\\inf_{m\\ge n}\\min(f(m),m).\n\\]\nThis nondecreasing lower envelope tends to infinity even when $f$\noscillates.  A bound at $n_{j-1}$ therefore controls all three later\ncoordinates; the cap by $m$ ensures $e_n\\le n$ and hence summability\nfor every digit choice.\nEnlarge $n_{-1}\\ge K$ until $h(n_{-1})\\ge32$, and set\n\\[\n k_j=\\max\\{k\\ge2:k!2^{k+2}\\le h(n_{j-1})\\},\\qquad\n M_j=k_j!,\\qquad s_j=k_j+2,\\qquad n_j=n_{j-1}+s_j.\n\\]\nThe maximum exists: $k=2$ is admissible because $2!2^4=32$, and\n$k!2^{k+2}\\to\\infty$.  Since $n_j\\to\\infty$ and $h$ is nondecreasing\nand divergent, $k_j$ is nondecreasing and tends to infinity.  Thus\n$M_j\\mid M_{j+1}$, every fixed $q$ eventually divides $M_j=k_j!$,\nand $s_j=k_j+2\\to\\infty$.\nAt every coordinate $n$ of the $j$th triple, this schedule also gives\n\\[\n 0\\le e_n\\le M_j2^{s_j}\\le h(n_{j-1})\\le\\min(f(n),n).\n\\]\nLet $S=\\bigcup_{j\\ge0}\\{n_j-1,n_j,n_j+1\\}$.  For $R\\ge4$, all\nsufficiently late centres are at least $R$ apart.  Every interval of\nlength $H$ contains at most $3(H/R+2)$ points from those triples;\nthe earlier triples add a fixed number, independent of the interval's\nposition.  Divide by $H$, take the supremum over positions, then let\n$H\\to\\infty$.  The limiting upper bound is $3/R$; letting $R\\to\\infty$\nproves upper Banach density zero.\n\n\\par\\medskip\\noindent\\emph{Attaining the interval.}\nThe support and congruence requirements are now settled independently\nof the target.  Only the digits remain to be chosen.  Define\n\\[\n \\beta=\\sum_{j\\ge0}\\bigl(c_j2^{-n_j}+D_jw_j\\bigr),\\qquad\n F_j=\\sum_{i\\ge j}D_iw_i.\n\\]\nEven zero digits contribute $\\beta>0$; $F_j$ is the maximum variable\ncontribution from stage $j$ onwards.  The size bound makes both finite,\nwith $F_0>0$ and $F_j\\to0$.  The total correction is\n\\[\n \\sum_ne_n2^{-n-1}=\\beta+\\sum_jd_jw_j.\n\\]\nTaking $J\\to\\infty$ in \\eqref{eq:finite-overlap} now gives\n\\begin{equation}\\label{eq:overlap}\n F_{j+1}\\ge M_j2^{-n_j-2}=w_j.\n\\end{equation}\nThese bounds do not yet prove that every point of $[0,F_j]$ is\nattained.  A digit $d$ leaves a remainder in $[0,F_{j+1}]$ precisely\nfor current remainders in\n\\[\n [dw_j,dw_j+F_{j+1}],\\qquad d=0,\\ldots,D_j.\n\\]\nThese intervals meet or overlap by \\eqref{eq:overlap}, and their union\nis $[0,F_j]$ because $F_j=D_jw_j+F_{j+1}$.  Figure~\\ref{fig:correction-and-overlap}\nshows the two calculations.\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}[x=1cm,y=1cm,font=\\small]\n \\node[anchor=west] at (-.2,2.65) {\\textbf{A fixed ordinary total}};\n \\draw (-.15,1.72)--(9.5,1.72);\n \\foreach \\x in {1.05,4.4,7.85} {\\draw (\\x,1.64)--(\\x,1.80);}\n \\node at (1.05,2.0) {$n_j-1$};\n \\node at (4.4,2.0) {$n_j$};\n \\node at (7.85,2.0) {$n_j+1$};\n \\node at (1.05,1.37) {$c_j$};\n \\node at (4.4,1.37) {$M_jd_j$};\n \\node at (7.85,1.37) {$M_j(D_j-d_j)$};\n \\draw[->] (7.05,.88)--(5.10,.88);\n \\node[anchor=east] at (4.93,.88) {transfer $M_j$};\n \\node[anchor=west] at (-.2,.36)\n   {$\\text{ordinary total}=c_j+M_jD_j,\\qquad\n     \\text{weighted increment}=w_j=M_j2^{-n_j-2}$};\n \\node[anchor=west] at (-.2,-.30) {\\textbf{Consecutive target intervals}};\n \\draw[line width=.8pt] (0,-1.30)--(5.35,-1.30);\n \\draw (0,-.94)--(0,-1.38) (5.35,-.94)--(5.35,-1.38);\n \\draw[line width=.8pt] (3.3,-2.22)--(8.65,-2.22);\n \\draw (3.3,-2.14)--(3.3,-2.30) (8.65,-2.14)--(8.65,-2.30);\n \\draw[densely dashed] (3.3,-.96)--(3.3,-2.26);\n \\draw[densely dashed] (5.35,-.96)--(5.35,-2.26);\n \\node[anchor=south] at (0,-.93) {$dw_j$};\n \\node[anchor=south] at (5.35,-.93) {$dw_j+F_{j+1}$};\n \\node[anchor=north] at (3.3,-2.35) {$(d+1)w_j$};\n \\node[anchor=north] at (8.65,-2.35) {$(d+1)w_j+F_{j+1}$};\n \\node at (4.325,-1.76) {$F_{j+1}\\ge w_j$};\n\\end{tikzpicture}\n\\caption{Redistributing $M_j$ between adjacent entries keeps their ordinary\nsum fixed and increases the dyadic sum by $w_j$.  The candidate target\nintervals in the lower panel cover without gaps when $F_{j+1}\\ge w_j$.\nThe sketch shows strict overlap; the proof also allows equality and\nestablishes attainment by keeping the remainder in $[0,F_j]$ as $F_j\\to0$.}\n\\label{fig:correction-and-overlap}\n\\end{figure}\nGiven $x\\in[0,F_0]$, start with $\\rho_0=x$.  Having chosen the first\n$j$ digits, choose $d_j$ so that\n\\[\n \\rho_{j+1}=\\rho_j-d_jw_j\\in[0,F_{j+1}].\n\\]\nThe covering just proved makes this possible at every stage.  Since $0\\le\\rho_j\\le F_j$ and $F_j\\to0$, we obtain\n\\[\n x-\\sum_{i=0}^{j}d_iw_i=\\rho_{j+1}\\longrightarrow0.\n\\]\nThus every $x\\in[0,F_0]$ is attained, and we may take\n$I=(A+\\beta,A+\\beta+F_0)$.  The target $r\\in I$ determines only the\ndigits, through $x=r-A-\\beta$; it changes neither $S$, $I$ nor the\ncutoff $N_q=n_J$ chosen using $q\\mid M_J$.\nThis last selection is the standard finite-choice covering argument\nof \\cite[Lemma~4]{crmarickovac2025}.  The construction above supplies\nits summability and overlap hypotheses while imposing the congruences\nand requiring the support to lie in $S$.\n\n\\par\\medskip\\noindent\\emph{The logarithmic scale.}\nFor $f(n)=(\\log(n+3))^\\varepsilon$, the size cost $2^{s_j}$ leads to\nthe iterated logarithm.  We choose separations of order $\\log\\log n_j$\nand reserve part of the allowance for growing moduli.  Keep the same\ntriples and $D_j=2^{s_j}-1$, and set\n\\[\n \\begin{aligned}\n s_j&=\\left\\lfloor\\frac{\\varepsilon}{2}\n               \\log_2\\log(n_{j-1}+3)\\right\\rfloor,\\\\\n k_j&=\\max\\{k\\ge2:k!\\le(\\log(n_{j-1}+3))^{\\varepsilon/4}\\},\n \\end{aligned}\n\\]\nwith $M_j=k_j!$ and $n_j=n_{j-1}+s_j$.  Start far enough out that\n$s_j\\ge4$ and all choices exist.  The exponents $\\varepsilon/4$ and\n$\\varepsilon/2$ add to $3\\varepsilon/4<\\varepsilon$, so the corrections\nsatisfy the prescribed bound:\n\\[\n M_j2^{s_j}\\le(\\log(n_{j-1}+3))^{3\\varepsilon/4}\n             \\le(\\log(n+3))^\\varepsilon\n\\]\nat each coordinate $n$ of the triple.  After enlarging the initial\nindex, this is also at most $n$.  The moduli remain nested, and every fixed positive integer divides all\nsufficiently late moduli.  Summability, the congruence induction and\n\\eqref{eq:finite-overlap} therefore apply without change.  Since\n$s_j\\asymp_\\varepsilon\\log\\log n_j$, the relevant centres near $[X,2X)$\nare separated by $\\gg_\\varepsilon\\log\\log X$.  Each contributes at most\nthree support points, giving\n\\[\n |S\\cap[X,2X)|=O_\\varepsilon(X/\\log\\log X).\n\\]\n\nChoose the same uniform starting index $n\\in[X,2X)$ for both\nblocks.  They agree when the block avoids $S$.  A changed coordinate\n$t\\in S\\cap[X,2X+m)$ affects only starts in $[t-m+1,t]$, at most\n$m$ choices.  The union bound gives\n\\[\n \\begin{aligned}\n d_{\\rm TV}(\\mu_{a,X,m},\\mu_{a+e,X,m})\n &\\le\\mathbb P\\bigl(e_{n+i}\\ne0\\text{ for some }0\\le i<m\\bigr)\\\\\n &\\le\\frac{m|S\\cap[X,2X+m)|}{X}.\n \\end{aligned}\n\\]\nFor $m\\le X$, two dyadic bands cover $[X,2X+m)$, giving\n$O_\\varepsilon(m/\\log\\log X)$.  A block of length $o(\\log\\log X)$\ntherefore avoids $S$ with probability tending to one.  The convergence\nis uniform over targets because the same $S$ contains every correction\nsupport.\n\n\\end{proof}\n\nNo limiting block law is assumed.  A block-only test bounded in\nabsolute value by $B_0$ changes its mean by at most $2B_0d_{\\rm TV}$.\nFor such a test also depending on the starting index, the common-index\ncoupling gives $2B_0m|S\\cap[X,2X+m)|/X$ instead.\nVanishing absolute error can still erase a rare event; it supplies\nneither relative-frequency control nor a comparison of complete tails.  The scale $o(\\log\\log X)$ is sufficient, not proved\noptimal.  The companion's\n\\href{erdos251-prime-gap-reasoning-surface.pdf\\#sparse-construction}{Section~2}\nalso distinguishes upper Banach density from ordinary density.\n\n\\section{Application to prime gaps}\\label{sec:prime-application}\nThe elementary bound $p_n\\le1250(n+1)^4$ suffices for convergence of\nthe prime and gap series.  We use the central-binomial argument of\nErd\\H{o}s \\cite[pp.~194--196]{erdos1932};\n\\href{erdos251-prime-gap-reasoning-surface.pdf\\#prime-bound}{Appendix~A of the companion}\nderives the prime-power estimate and the bound.  We may therefore apply\nProposition~\\ref{res:sparserationalisation} to the gaps.\n\n\\begin{corollary}[a rational sum with the stated prime-gap statistics]\\label{res:jointcountermodel}\nLet $p_0=2,p_1=3,\\ldots$ be the primes and $g_n=p_{n+1}-p_n$.\nGiven $K\\in\\N$ and $0<\\varepsilon\\le1$, there is $b:\\N\\to\\N$ with\nrational dyadic sum such that $b_n=g_n$ for $n<K$, $b_n\\ge g_n$, and\n$b_n-g_n\\le(\\log(n+3))^\\varepsilon$ eventually.  For every fixed positive\nmodulus, both the coefficients and the cumulative positions eventually\nretain their corresponding residues.  The empirical distributions of\nunnormalised blocks have total variation distance tending to zero\nfor lengths $o(\\log\\log X)$, and for every fixed nonzero\n$F\\in\\Z[x_0,\\ldots,x_k]$,\n\\[\n \\bigl|\\{n<N:F(b_n,\\ldots,b_{n+k})=0\\}\\bigr|=o(N).\n\\]\nMoreover, with $P_n=2+\\sum_{i<n}b_i$,\n\\[\n 0\\le P_n-p_n=O_\\varepsilon\\!\\left(\n       \\frac{n(\\log(n+3))^\\varepsilon}{\\log\\log n}\\right),\n \\qquad P_n\\sim n\\log n.\n\\]\nThese positions are not asserted to be prime.\n\\end{corollary}\n\\evidenceremark{Lean proves this for every $\\varepsilon>0$, assuming\nSchlage-Puchta's Lemma~4 \\cite{schlagepuchta2011} only for the\nnonconcentration bound and the prime number theorem only for\n$P_n\\sim n\\log n$.}\n\n\\begin{proof}\nApply Proposition~\\ref{res:sparserationalisation} to $a=g$, choose a\nrational $r\\in I$, and fix the resulting sequence $b=g+e$.  The correction,\ncongruence and growing-block assertions follow at once.  It remains to\ntransfer polynomial nonconcentration and cumulative growth.\nFor fixed $k$ and nonzero $F\\in\\Z[x_0,\\ldots,x_k]$, a new zero requires\na block meeting $S$, so\n\\[\n \\begin{aligned}\n &\\bigl|\\{n<N:F(b_n,\\ldots,b_{n+k})=0\\}\\bigr|\\\\\n &\\quad\\le\\bigl|\\{n<N:F(g_n,\\ldots,g_{n+k})=0\\}\\bigr|\n       +\\sum_{i=0}^k|S\\cap[i,N+i)|.\n \\end{aligned}\n\\]\nThe first count is $o(N)$ by Schlage-Puchta's Lemma~4\n\\cite{schlagepuchta2011}; the finitely many support counts are $o(N)$\nbecause $S$ has density zero.  We count the event $F=0$, not averages\nof $F$, so no bound on its values is needed.  Since $b$ was chosen\nbefore $F,k$, the same sequence satisfies every fixed-polynomial condition.\n\nTo estimate $P_n-p_n=\\sum_{i<n}e_i$, first observe that\n$|S\\cap[0,n)|=O_\\varepsilon(n/\\log\\log n)$.  Indeed, the indices below\n$\\sqrt n$ contribute at most $\\sqrt n$ points, and on $[\\sqrt n,n)$ we\nsum the dyadic support bounds, whose lengths add to $O(n)$ and whose\n$\\log\\log X$ are comparable to $\\log\\log n$.  Multiplying by the\npointwise correction bound gives the stated error.  For\n$0<\\varepsilon\\le1$ this is $o(n\\log n)$, so the prime number theorem\n\\cite[Chapter~6]{mv2007} gives $P_n\\sim n\\log n$.  Finally, summing\nnonnegative terms in the opposite order, we obtain\n\\[\n \\sum_{n\\ge0}\\frac{P_n}{2^{n+1}}\n =2+\\sum_{i\\ge0}b_i\\sum_{n>i}2^{-n-1}\n =2+\\sum_{i\\ge0}\\frac{b_i}{2^{i+1}}=2+r\\in\\Q.\n\\]\n\\end{proof}\n\nFor $\\varepsilon>1$ we may use exponent $1$, which gives smaller\ncorrections and the error $O(n\\log(n+3)/\\log\\log n)$.  Thus the conclusions\nextend to every $\\varepsilon>0$.  For each fixed $y\\ge2$, eventual\ncongruence modulo $y!$ also gives $\\gcd(P_n,y!)=1$ once $p_n>y$.\nThe cutoff depends on $y$; primality would require excluding every\nprime up to $\\sqrt{P_n}$ at that same index.  Even prime-valued positions\ncould omit intervening primes.\n\nLand's draft \\cite[Theorem~2]{land2026} proves conditional irrationality\nunder a uniform Hardy--Littlewood hypothesis of the kind formulated by\nKuperberg \\cite[Conjecture~1.3]{kuperberg2023}.  Ringer's draft\n\\cite{ringer2026} obtains conditional normality.  Their quantitative\nprime-pattern hypotheses are stronger than the absolute block comparison\nabove and are not assumed here.  The\n\\href{erdos251-prime-gap-reasoning-surface.pdf\\#context}{companion's Section~3}\ncompares the hypotheses and sampling conventions.\n\n\\section{The prime series and its actual tails}\\label{sec:parts}\nWe now study complete tails of the actual prime gaps.  Put\n$G=\\sum_{n\\ge0}g_n2^{-(n+1)}$.  The polynomial bound from\nSection~\\ref{sec:prime-application} proves convergence and makes the\nendpoint in finite summation by parts vanish.\n\n\\begin{theorem}[prime-to-gap identity]\\label{res:infinite}\nThe actual prime and gap series satisfy $\\Pi=2+G$.  Their complete tails\n\\[\n T_N=\\sum_{j\\ge1}g_{N+j}2^{-j}\n\\]\nsatisfy $T_0=2G-1$ and $T_{N+1}=2T_N-g_{N+1}$.\nThus $\\Pi$, $G$ and $T_0$ have the same rationality status.\n\\end{theorem}\n\\begin{proof}\nFinite summation by parts gives\n\\[\n \\sum_{i=0}^{n}\\frac{p_i}{2^{i+1}}\n =2+\\sum_{i=0}^{n-1}\\frac{g_i}{2^{i+1}}-\\frac{p_n}{2^{n+1}}.\n\\]\nSince $p_n=O((n+1)^4)$, the endpoint tends to zero and we obtain\n$\\Pi=2+G$.  We obtain the tail recurrence by splitting off the first\nterm of $T_N$; removing $g_0=1$ from $G$ gives $T_0=2G-1$.\n\\end{proof}\nTao noted this reduction in the problem's forum discussion\n\\cite[7 October 2025]{erdosproblems251thread}.\n\n\\begin{corollary}[exact irrationality reformulation]\\label{res:irr-equivalence}\nThe prime-value dyadic series is irrational if and only if the prime-gap\ndyadic series is. Both series converge by the polynomial bound proved above;\nneither side is proved irrational.\n\\end{corollary}\n\\begin{proof}\nAdding the rational number $2$ preserves rationality and irrationality.\n\\end{proof}\n\nThe index $N$ is the last omitted coefficient, so $T_N$ starts with\n$g_{N+1}/2$.  Rescaling distinguishes these tails from the remainders\nof $G$.  The recurrence also admits an extra term $C2^N$; the\nvanishing boundary condition below excludes it.\n\\begin{lemma}[the boundary condition for the tail recurrence]\\label{res:true-tail}\nLet $\\sum_{j\\ge1}|a_j|2^{-j}<\\infty$ and $U_{N+1}=2U_N-a_{N+1}$.\nThen $U_N=\\sum_{j\\ge1}a_{N+j}2^{-j}$ for every $N$ if and only if\n$2^{-N}U_N\\to0$.\n\\end{lemma}\n\\begin{proof}\nIteration gives $2^{-N}U_N=U_0-\\sum_{j=1}^Na_j2^{-j}$.\nThe limit is zero exactly when $U_0$ is the full series.  Subtracting its\nfirst $N$ terms and multiplying by $2^N$ then gives the tail formula.\nConversely the formula implies\n$2^{-N}U_N=\\sum_{k>N}a_k2^{-k}\\to0$ by absolute convergence.\n\\end{proof}\n\n\\subsection{Integral shifts}\\label{sec:tail}\nFor $U_{N+1}=2U_N-a_{N+1}$ with $a_n\\in\\Z$, write\n$D_h(N)=U_{N+h}-U_N$.  Modulo integers, each step doubles, so an\nintegral shift is a return to the same fractional part.\nFor rational $U_0$, the power of two in its reduced denominator\ndetermines when returns of positive length can begin; the odd part\ndetermines their lengths.\n\n\\begin{theorem}[exact rationality classification]\\label{res:escape-irrational}\nFor a real integer-coefficient recurrence, the following are equivalent:\n$U_0\\in\\Q$; $D_h(N)\\in\\Z$ for some $h\\ge1,N\\ge0$; and, for some fixed\n$h\\ge1$, $D_h(N)\\in\\Z$ at every sufficiently large $N$.\nMore precisely, if $U_0=u/(2^sd)$ is in lowest terms, with $d$ odd, then\n\\[\n \\operatorname{den}(U_N)=2^{\\max(s-N,0)}d,\\qquad\n D_h(N)\\in\\Z\\ \\Longleftrightarrow\\ N\\ge s\\ \\text{and}\\ d\\mid2^h-1\n \\quad(h\\ge1).\n\\]\nConsequently $U_0$ is irrational exactly when every positive shift is\nnonintegral at every index, equivalently when for each fixed $h\\ge1$\nthere are arbitrarily late nonintegral shifts.\n\\end{theorem}\n\\begin{proof}\nIteration gives\n\\[\n U_N-2^NU_0\\in\\Z,\\qquad\n D_h(N)-2^N(2^h-1)U_0\\in\\Z.\n\\]\nAn integral shift with $h\\ge1$ makes the nonzero integer multiple\n$2^N(2^h-1)U_0$ integral, and therefore forces $U_0\\in\\Q$.\nConversely, write $U_0=u/(2^sd)$ in lowest terms.  Multiplication by\n$2^N$ cancels exactly $\\min(N,s)$ powers of two from the denominator.\nThe remaining factor $2^h-1$ is odd, so the second relation is integral\nexactly when $N\\ge s$ and $d\\mid2^h-1$.  For $d>1$, Euler's congruence\nprovides such a shift $h=\\varphi(d)$, and for $d=1$ we take $h=1$.\nThis shift works for every $N\\ge s$.  Negating the pointwise and eventual\ncriteria gives the two assertions about irrationality.\n\\end{proof}\n\nThe classification applies to any integer-coefficient recurrence;\nLemma~\\ref{res:true-tail} is needed only to identify complete tails.\nThe periodicity is of fractional parts, not necessarily tail values.\nFor the positive even coefficients $2,4,2,4,\\ldots$ starting at $n=1$,\nthe tails alternate between $8/3$ and $10/3$.  Length-$1$ shifts are\nnonintegral, but length-$2$ shifts vanish: one shift length cannot suffice.\n\n\\section{Small tail differences and finite tests}\\label{sec:local-certificate}\nFix $h\\ge1$ and put $D_N=T_{N+h}-T_N$ and\n$\\delta_N=g_{N+h+1}-g_{N+1}$.  Then\n$D_{N+1}=2D_N-\\delta_N$.  Both gap indices are positive, so neither\ninvolves the exceptional odd gap $g_0=1$; hence $\\delta_N$ is even.\nTwo integral differences of absolute value less than one must both\nvanish, forcing $\\delta_N=0$.  Evenness yields the sharper signed test below.\n\n\\begin{proposition}[two consecutive differences of absolute value less than one]\\label{res:signedwindow}\nLet $D,D'\\in\\R$, $\\delta\\in2\\Z$ and $D'=2D-\\delta$.\nThe conditions $|D|<1$, $|D'|<1$ and $\\delta\\ne0$ hold exactly when,\nfor some $s\\in\\{-1,1\\}$,\n\\[\n \\delta=2s,\\qquad \\tfrac12<sD<1.\n\\]\nIn that case $sD'\\in(-1,0)$ and both $D$ and $D'$ are nonintegral.\n\\end{proposition}\n\\begin{proof}\nThe bounds on $D,D'$ imply $-3<\\delta<3$, so a nonzero even $\\delta$\nequals $2s$ with $s\\in\\{-1,1\\}$.  Substituting in $D'=2D-2s$ gives\n$1/2<sD<1$.  Conversely this interval gives $sD'=2sD-2\\in(-1,0)$.\n\\end{proof}\nThe endpoints cannot be included: when $\\delta=2$, $D=1/2$ gives\n$D'=-1$, and $D=1$ already violates $|D|<1$.  The zero triple\n$D=D'=\\delta=0$ shows why the mismatch is also required.\n\n\\begin{problem}[two small tail differences at arbitrarily large indices]\\label{prob:smallpair}\nFor every integer $h\\ge1$ and every cutoff $N_0$, exhibit $N\\ge N_0$ with\n\\begin{equation}\\label{eq:smallpair}\n |T_{N+h}-T_N|<1,\\quad |T_{N+h+1}-T_{N+1}|<1,\\quad\n g_{N+h+1}\\ne g_{N+1}.\n\\end{equation}\n\\end{problem}\n\nTheorem~\\ref{res:escape-irrational} requires nonintegrality, not\nsmallness; \\eqref{eq:smallpair} is only a sufficient condition.\nAll three requirements must hold at the same arbitrarily late\nindices.  Separate infinite sets need not intersect.\n\nBy Proposition~\\ref{res:signedwindow}, such an index must have\n$\\delta_N=\\pm2$.  Schlage-Puchta's lemma\n\\cite[Lemma~4]{schlagepuchta2011}, applied to $x_h-x_0\\pm2$, shows that\nthese eligible indices already have density zero.  Density zero is\ncompatible with arbitrarily late witnesses; it supplies no lower bound\nfor their occurrence.  The absolute block comparison in\nProposition~\\ref{res:sparserationalisation} likewise gives no\nrelative-frequency estimate for this rare event.\n\n\\subsection{Finite separation from the integers}\nAn enclosure certifies nonintegrality only when it misses every integer.\nChoose $M(n)\\ge g_n$ with\n$\\sum_nM(n)2^{-n}<\\infty$, and write\n\\[\n S_{h,N,L}=\\sum_{j=1}^L(g_{N+h+j}-g_{N+j})2^{-j},\\qquad\n R_{h,N,L}(M)=\\sum_{j>L}(M(N+h+j)+M(N+j))2^{-j}.\n\\]\nSplitting the complete tails after $L$ terms gives\n\\[\n D_N=S_{h,N,L}+2^{-L}D_{N+L},\\qquad\n |D_N-S_{h,N,L}|\\le R_{h,N,L}(M).\n\\]\nThe uncomputed term is a rescaled later complete tail difference with\nthe same shift $h$.  An arbitrary real majorant gives an enclosure,\nbut need not give a computable radius.\n\nFor the actual prime gaps, one computable choice is $M(n)=1250(n+2)^4$.\nFor $P(x)=x^4+8x^3+36x^2+104x+150$, the identity\n$2P(x)=(x+1)^4+P(x+1)$ telescopes to\n$\\sum_{j=1}^{J}(x+j)^4 2^{-j}=P(x)-2^{-J}P(x+J)$.\nThe endpoint vanishes as $J\\to\\infty$, giving\n\\[\n R_{h,N,L}(M)=1250\\,2^{-L}\n             \\bigl(P(N+h+L+2)+P(N+L+2)\\bigr).\n\\]\n\n\\begin{proposition}[finite separation criterion]\\label{res:truncation}\nIf for every $h\\ge1$ and every cutoff $N_0$ there are $N\\ge N_0,L\\ge1$ with\n\\begin{equation}\\label{eq:truncation}\n \\operatorname{dist}(S_{h,N,L},\\Z)>R_{h,N,L}(M),\n\\end{equation}\nthen $\\Pi$ is irrational.\n\\end{proposition}\n\\begin{proof}\nThe enclosure gives\n$\\operatorname{dist}(D_N,\\Z)\\ge\n\\operatorname{dist}(S_{h,N,L},\\Z)-R_{h,N,L}(M)>0$.\nThus each fixed positive shift is nonintegral at arbitrarily large\nindices, and Theorem~\\ref{res:escape-irrational} applies.\n\\end{proof}\n\n\\begin{remark}[what arbitrary truncation does not gain]\\label{res:complete-truncation}\nWhenever $|D-S_L|\\le R_L\\to0$, nonintegrality of $D$ is equivalent to\n$\\operatorname{dist}(S_L,\\Z)>R_L$ for some $L$.  Indeed, for\n$\\delta=\\operatorname{dist}(D,\\Z)>0$, take $R_L<\\delta/2$ and use the\n$1$-Lipschitz property of distance.  Unrestricted depth therefore gives\nan exact reformulation, not an additional analytic estimate.  At a\nprescribed logarithmic depth one must also control the margin from the\nintegers as $N$ varies; a shrinking remainder alone does not do so.\nThe companion, Appendix~D.2, gives an explicit example distinguishing\nrefinement at a fixed value from enclosures of changing values.\n\\end{remark}\n\nA nonintegral value need not lie in either signed interval of\nProposition~\\ref{res:signedwindow}.  To certify \\eqref{eq:smallpair},\nsuppose $\\delta_N=2s$, with $s\\in\\{-1,1\\}$, and enclose $D_N$ using integers\n$A,Q,B$ with $Q>0$, $B\\ge0$ and $|QD_N-A|\\le B$.\nThen $sD_N\\in[(sA-B)/Q,(sA+B)/Q]$.  Placing its lower endpoint above\n$1/2$ and its upper endpoint below $1$ gives, respectively,\n\\[\n 2sA-Q>2B,\\qquad Q-sA>B.\n\\]\nWe retain the signed numerator: reduction modulo $Q$ preserves\ndistance to $Q\\Z$ but loses the enclosure's location.  The gap\ndifference fixes $s$, so it cannot be chosen to fit the enclosure.\nAt $h=1,N=2,L=40$, one has $\\delta_2=g_4-g_3=-2$ and\n\\[\n Q=2^{40},\\qquad A=-662838684750,\\qquad B=11764181250,\\qquad s=-1.\n\\]\nHere $A/Q=S_{1,2,40}$ and $B/Q=R_{1,2,40}(M)$, and the two strict\nmargins are $202637379224$ and $424908761776$.\nThe \\href{erdos251-prime-gap-reasoning-surface.pdf\\#finite-certificates}{companion's integer certificates}\ngive the prime-index calculation and remainder derivation.  This finite\npair leaves the quantifiers in Problem~\\ref{prob:smallpair} unresolved.\n\n\\section{Further consequences and questions}\\label{sec:open}\nWe can place the interval in Proposition~\\ref{res:sparserationalisation}\ninside $(A,A+\\eta)$ for any $\\eta>0$.  Choose $K'\\ge K$ such that\n$(K'+1)2^{-K'}<\\eta$, and start the construction after $K'$.\nThe bound $e_n\\le n$ from the proof gives\n\\[\n 0<\\sum_ne_n2^{-(n+1)}\\le(K'+1)2^{-K'}<\\eta.\n\\]\nThus rational and irrational targets remain available arbitrarily close\nto $A$, with the prescribed prefix unchanged.  The estimate controls the\ndyadic change, not the ordinary total.\n\n\\subsection{Nonperiodic coefficients}\\label{sec:carry}\nRationality forces eventual periodicity of the binary digits obtained\nafter carrying, not of the original integer coefficients.  For example,\n\\[\n \\sum_{i=0}^{n-1}\\frac{i-1}{2^{i+1}}=-\\frac{n}{2^n}\\longrightarrow0.\n\\]\nIts coefficients are unbounded and its first coefficient is $-1$.\nWe obtain a positive even example by taking $c_n=2(n^2+4n+2)$ and\n$U_n=2(n+4)^2$.  Direct substitution gives\n$U_{n+1}=2U_n-c_{n+1}$, while the polynomial growth gives\n$2^{-n}U_n\\to0$.  Lemma~\\ref{res:true-tail} therefore identifies $U$\nas the complete tail and yields\n\\[\n \\sum_{j\\ge1}c_j2^{-j}=32,\\qquad\n \\sum_{n\\ge0}c_n2^{-(n+1)}=18.\n\\]\nThe coefficients increase strictly and $c_{n+1}-c_n=4n+10$ is never\n$\\pm2$.  The \\href{erdos251-prime-gap-reasoning-surface.pdf\\#counterexamples}{companion's Section~8}\ngives further examples, including a bounded nonperiodic sequence in\nAppendix~D.  None of these examples constructs consecutive primes.\n\n\\begin{proposition}[prime gaps do not become periodic]\\label{res:gap-nonperiodic}\nFor every positive $h$, the actual consecutive-prime-gap sequence is not\neventually periodic with period $h$.\n\\end{proposition}\n\\begin{proof}\nIf the gaps were eventually periodic, they would be bounded.  For each\n$m\\ge2$, however, the $m-1$ integers $m!+2,\\ldots,m!+m$ are composite.\nThe consecutive primes on either side therefore have a gap of at least\n$m$, contradicting boundedness.\n\\end{proof}\n\nThe results on bounded prime gaps \\cite{zhang2014}, clusters of each\nfixed size \\cite{maynard2015,polymath2014}, and extreme large gaps\n\\cite{fgkmt2018} do not give the simultaneous inequalities in \\eqref{eq:smallpair}.\nThis remains so with Polymath's bound $\\liminf_n g_n\\le246$\n\\cite[Theorem~1.4(i)]{polymath2014}.  For fixed $h$ and $\\varepsilon>0$,\ntaking $L=\\lceil(4+\\varepsilon)\\log_2(N+2)\\rceil$ in the polynomial\nremainder gives an error $O_{h,\\varepsilon}(N^{-\\varepsilon})$.\nThe finite sum must still clear the relevant boundary by more than this\nerror to supply a witness for Problem~\\ref{prob:smallpair}.  The perturbation\ntheorem rules out no argument using additional primality information or\nquantitative rare-event estimates.\n\n\\section{Verification and reproducibility}\\label{sec:pinned-lean-sources}\n\\evidenceparagraph{\\evidencerecordurl}\nThe linked Lean declarations use Lean~4 \\cite{lean4} and mathlib\n\\cite{mathlib}.  The formal proof of\nProposition~\\ref{res:sparserationalisation} changes one coordinate at each\nstage, whereas the proof here uses triples.  Both choose the set containing the supports,\nthe interval and the congruence cutoffs before the value of the sum.\nFor Corollary~\\ref{res:jointcountermodel}, the formal statement covers\nevery $\\varepsilon>0$ and assumes Schlage-Puchta's lemma only for\nnonconcentration and the prime number theorem only for the asymptotic\n$P_n\\sim n\\log n$.  The source links record formal support for the linked\nstatements, rather than verification of the different printed proofs.\nThe \\href{erdos251-prime-gap-reasoning-surface.pdf\\#short-source-index}{companion's Appendix~G}\nretains the complete source index at this note's original pin, including\nthe finite telescoping identities and their nonperiodicity consequences.\nIts \\href{erdos251-prime-gap-reasoning-surface.pdf\\#context}{Section~3}\nretains the broader subsum and automatic-sequence comparisons.\n\n\\subsection*{Acknowledgements}\n\nThe author received no external funding and declares no competing interests.\nThe numbering follows Bloom's catalogue \\cite{erdosproblems}.\n\n\\begin{thebibliography}{99}\n\\small\\raggedright\\setlength{\\itemsep}{0pt}\n\\bibitem{erdos1958} Paul Erd\\H{o}s, \\href{https://users.renyi.hu/~p_erdos/1958-19.pdf}{\\emph{Sur certaines s\\'eries \\`a valeur irrationnelle}}.\n  L'Enseignement Mathématique \\textbf{4} (1958), 93--100, doi:\\href{https://doi.org/10.5169/seals-34629}{10.5169/seals-34629}.\n\\bibitem{erdosgraham1980} Paul Erd\\H{o}s and Ronald L. Graham, \\href{https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf}{\\emph{Old and New Problems and Results in Combinatorial Number Theory}}.\n  Monographies de L'Enseignement Mathématique, L'Enseignement Mathématique, 1980.\n\\bibitem{erdos1988} Paul Erd\\H{o}s, \\href{https://doi.org/10.1017/CBO9780511897184.009}{\\emph{On the irrationality of certain series: problems and results}}.\n  New Advances in Transcendence Theory, Cambridge University Press, 1988, pp.~102--109, doi:\\href{https://doi.org/10.1017/CBO9780511897184.009}{10.1017/CBO9780511897184.009}.\n\\bibitem{fgkmt2018} Kevin Ford, Ben Green, Sergei Konyagin, James Maynard and Terence Tao, \\href{https://arxiv.org/abs/1412.5029v3}{\\emph{Long gaps between primes}}.\n  Journal of the American Mathematical Society \\textbf{31} (2018), 65--105, doi:\\href{https://doi.org/10.1090/jams/876}{10.1090/jams/876}; arXiv:\\href{https://arxiv.org/abs/1412.5029v3}{1412.5029v3}.\n\\bibitem{fridy1966} John A. Fridy, \\href{https://www.fq.math.ca/Scanned/4-3/fridy.pdf}{\\emph{Generalized bases for the real numbers}}.\n  The Fibonacci Quarterly \\textbf{4} (1966), no.~3, 193--201.\n\\bibitem{kovactao2024} Vjekoslav Kova\\v{c} and Terence Tao, \\href{https://arxiv.org/abs/2406.17593v4}{\\emph{On several irrationality problems for Ahmes series}}.\n  Acta Mathematica Hungarica \\textbf{175} (2025), 572--608, doi:\\href{https://doi.org/10.1007/s10474-025-01528-0}{10.1007/s10474-025-01528-0}; arXiv:\\href{https://arxiv.org/abs/2406.17593v4}{2406.17593v4}. Statement and section numbers refer to arXiv version 4.\n\\bibitem{kuperberg2023} Vivian Kuperberg, \\href{https://arxiv.org/abs/2210.09775v2}{\\emph{Sums of singular series with large sets and the tail of the distribution of primes}}.\n  The Quarterly Journal of Mathematics \\textbf{74} (2023), no.~4, 1457--1479, doi:\\href{https://doi.org/10.1093/qmath/haad030}{10.1093/qmath/haad030}; arXiv:\\href{https://arxiv.org/abs/2210.09775v2}{2210.09775v2}. Statement numbers refer to arXiv version 2, 15 June 2023.\n\\bibitem{land2026} Johan Land, \\href{https://github.com/beetree/math_erdos_251}{\\emph{A conditional proof of the irrationality of $\\sum_{n\\ge1}p_n2^{-n}$ under a uniform Hardy--Littlewood prime-tuples conjecture}}.\n  Research draft, 5 September 2026. Formalisation material accompanies the draft.\n\\bibitem{lean4} Leonardo de Moura and Sebastian Ullrich, \\href{https://doi.org/10.1007/978-3-030-79876-5_37}{\\emph{The Lean 4 theorem prover and programming language}}.\n  Automated Deduction -- CADE 28, Lecture Notes in Computer Science, Springer, 2021, pp.~625--635, doi:\\href{https://doi.org/10.1007/978-3-030-79876-5_37}{10.1007/978-3-030-79876-5\\_37}.\n\\bibitem{mathlib} {The mathlib Community}, \\href{https://doi.org/10.1145/3372885.3373824}{\\emph{The Lean mathematical library}}.\n  Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, ACM, 2020, pp.~367--381, doi:\\href{https://doi.org/10.1145/3372885.3373824}{10.1145/3372885.3373824}. Describes a Lean 3-era snapshot; the repository lock identifies the library used by the present Lean 4 sources.\n\\bibitem{maynard2015} James Maynard, \\href{https://doi.org/10.4007/annals.2015.181.1.7}{\\emph{Small gaps between primes}}.\n  Annals of Mathematics \\textbf{181} (2015), 383--413, doi:\\href{https://doi.org/10.4007/annals.2015.181.1.7}{10.4007/annals.2015.181.1.7}.\n\\bibitem{mv2007} Hugh L. Montgomery and Robert C. Vaughan, \\href{https://doi.org/10.1017/CBO9780511618314}{\\emph{Multiplicative Number Theory I: Classical Theory}}.\n  Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2007, doi:\\href{https://doi.org/10.1017/CBO9780511618314}{10.1017/CBO9780511618314}.\n\\bibitem{zhang2014} Yitang Zhang, \\href{https://doi.org/10.4007/annals.2014.179.3.7}{\\emph{Bounded gaps between primes}}.\n  Annals of Mathematics \\textbf{179} (2014), 1121--1174, doi:\\href{https://doi.org/10.4007/annals.2014.179.3.7}{10.4007/annals.2014.179.3.7}.\n\\bibitem{erdosproblems} Thomas F. Bloom, \\href{https://www.erdosproblems.com/251}{\\emph{Erd\\H{o}s Problem \\#251}}.\n  2026. Accessed 6 September 2026.\n\\bibitem{erdosproblems251thread} {Erdős Problems contributors}, \\href{https://www.erdosproblems.com/forum/thread/251}{\\emph{Erd\\H{o}s Problem \\#251 discussion thread}}.\n  2026. Accessed 6 September 2026: Tao comment of 7 October 2025 and Land comments of 6 September 2026.\n\\bibitem{ringer2026} Stefan Ringer, \\href{https://github.com/StefanRinger/erdos-251}{\\emph{Local gap statistics, telescoping, and normality: a local-pattern approach to Erd\\H{o}s problem 251}}.\n  Preprint, 11 September 2026. Mutable main-branch TeX consulted 16 September 2026; the cited conditional statements were rechecked 18 September 2026. Formalisation material accompanies the draft.\n\\bibitem{crmarickovac2025} Ton\\'{c}i Crmari\\'{c} and Vjekoslav Kova\\v{c}, \\href{https://arxiv.org/abs/2504.18712v1}{\\emph{On the irrationality of certain super-polynomially decaying series}}.\n  Colloquium Mathematicum \\textbf{179} (2025), 55--68, doi:\\href{https://doi.org/10.4064/cm9628-5-2025}{10.4064/cm9628-5-2025}; arXiv:\\href{https://arxiv.org/abs/2504.18712v1}{2504.18712v1}. Lemma 4 is cited using arXiv version 1.\n\\bibitem{schlagepuchta2011} Jan-Christoph Schlage-Puchta, \\href{https://arxiv.org/abs/1105.1451v1}{\\emph{The irrationality of some number theoretical series}}.\n  Acta Arithmetica \\textbf{126} (2007), no.~4, 295--303, doi:\\href{https://doi.org/10.4064/aa126-4-1}{10.4064/aa126-4-1}; arXiv:\\href{https://arxiv.org/abs/1105.1451v1}{1105.1451v1}. Published in 2007; arXiv upload is from 2011. Lemma and preprint page locators refer to arXiv version 1.\n\\bibitem{erdos1932} Paul Erd\\H{o}s, \\href{https://users.renyi.hu/~p_erdos/1932-01.pdf}{\\emph{Beweis eines Satzes von Tschebyschef}}.\n  Acta Litterarum ac Scientiarum Szeged \\textbf{5} (1932), 194--198.\n\\bibitem{polymath2014} {D. H. J. Polymath}, \\href{https://arxiv.org/abs/1407.4897v4}{\\emph{Variants of the Selberg sieve, and bounded intervals containing many primes}}.\n  Research in the Mathematical Sciences \\textbf{1} (2014), article 12, doi:\\href{https://doi.org/10.1186/s40687-014-0012-7}{10.1186/s40687-014-0012-7}; arXiv:\\href{https://arxiv.org/abs/1407.4897v4}{1407.4897v4}. Theorem 1.4(i) refers to arXiv version 4 (22 December 2014); the journal numbers it Theorem 4(i). An erratum is recorded at doi:10.1186/s40687-015-0033-x.\n\\end{thebibliography}\n\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":43017,"sha256":"sha256:fa4220506e9c2822a70b5d554d7626dcc026497d3bb571ae9f07cc273a3d3957","cite_as":"Plectis reader corpus, paper:erdos-251-prime-gap-dyadic-series","source_repository":"plectis-erdos","source_path":"paper/251/erdos-251-prime-gap-dyadic-series.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:088d36d7ba3d3ee0cee2df99e8915927999698f19889f8ea0b8d55b3721ec778","byte_count":6923,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% This is the only copy of the shared preamble.  The eight short notes, the\n% long records assembled from paper/reasoning-parts other than #249 and #257\n% (which carry their own preambles), and the synthesis paper include it with\n% \\input{problem-note-preamble}; paper/Makefile builds from this directory\n% with the directory on the tectonic search path.\n%\n% Shared preamble for the Erdős Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erdős problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{8e58c395e9807ef3374d7365c62cf04d20ee6331}\n\\newcommand{\\commitshort}{9e231ce4371f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-erdos/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-erdos/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\usepackage{longtable}  % the notes' declaration tables; loaded here so the assembled records inherit it\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n% ---------------------------------------------------------------------------\n% Evidence marks.  paper-evidence.tex places \"Lean\" and \"Comparator\" links in\n% the margin beside each result from the paper's generated file\n% paper/evidence/<paper>.tex, which each paper inputs after this preamble.\n% ---------------------------------------------------------------------------\n\\input{paper-evidence}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Production provenance.  The wording is fixed by the corpus provenance\n% contract and reproduced verbatim in every manuscript in this repository, so\n% the papers cannot drift into different accounts of how they were made.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erdős problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every Lean mark in the margin points to propositions that the\n  pinned Lean kernel accepts, with no \\texttt{sorry}, no added axiom, and no\n  unchecked evaluation.  That claim concerns the formal statements only; it\n  says nothing about their mathematical interest, their novelty, or the\n  original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n\n% Compact opening card for the separately stated Comparator theorem selected\n% from each problem note.  The prose supplied by each note owns the\n% contribution and relevance judgements; the shared final sentence states the\n% mechanical assurance boundary once and points to the public evidence owner.\n\\newcommand{\\externalreviewobject}[3]{%\n  \\begin{quote}\\small\n  \\textbf{Contribution.} #1\\par\n  \\textbf{Relation to the open problem.} #2\\par\n  \\textbf{Executable review object.} #3  The repository's\n  external-verification job compares these exact Lean propositions with\n  separately declared challenge statements and an axiom budget, then asks\n  Lean's kernel to check the submitted proofs.  The\n  \\href{https://github.com/wcook04/plectis-erdos/blob/ca0e13f8acf5ccf48506e4bdb870953d3a0856fa/formalization.yaml}%\n  {formalisation manifest} and the commit-bound CI receipt record that check;\n  they do not assess novelty, significance, or whether the original problem is\n  solved.\n  \\end{quote}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  Under\n% Tectonic (XeTeX) the text and mathematics are set in TeX Gyre Pagella from\n% the bundled OpenType fonts, with a Heros sans range for \\mathsf and\n% Inconsolata for monospace; the pdflatex fallback keeps the same Palatino\n% design through newpx.  The A4 text block uses a classical measure with\n% quiet running heads and restrained ink.  Link colours: steel blue for\n% external (Lean-source and web) targets, deep burgundy for internal\n% cross-references and citations.\n\\NeedsTeXFormat{LaTeX2e}\n\\ProvidesPackage{paper-house-style}[2026/07/19 Pagella house style for the papers]\n\n\\RequirePackage{iftex}\n\\RequirePackage{amsmath}\n\\RequirePackage{amsthm}\n\\ifpdftex\n  \\RequirePackage[T1]{fontenc}\n  \\RequirePackage[utf8]{inputenc}\n  \\RequirePackage{amssymb}\n  \\RequirePackage{newpxtext}\n  \\RequirePackage{newpxmath}\n\\else\n  \\RequirePackage{fontspec}\n  \\RequirePackage{unicode-math}\n  \\setmainfont{texgyrepagella}[Extension=.otf,UprightFont=*-regular,\n    BoldFont=*-bold,ItalicFont=*-italic,BoldItalicFont=*-bolditalic]\n  \\setmathfont{texgyrepagella-math.otf}\n  \\setmathfont{texgyreheros-regular.otf}[range=sfup,Scale=MatchLowercase]\n  \\setmonofont{Inconsolatazi4}[Extension=.otf,UprightFont=*-Regular,\n    BoldFont=*-Bold,Scale=MatchLowercase]\n\\fi\n\n\\RequirePackage[a4paper,textwidth=418pt,textheight=636pt,hcentering,\n  top=68pt,headheight=15pt,headsep=20pt,footskip=34pt]{geometry}\n\\RequirePackage{microtype}\n\\linespread{1.045}\n\\clubpenalty=10000 \\widowpenalty=10000 \\tolerance=1600\n\n\\RequirePackage[dvipsnames,table]{xcolor}\n\\definecolor{linkexternal}{HTML}{275D8C}\n\\definecolor{linkinternal}{HTML}{6E2B2B}\n\\definecolor{papertableheader}{HTML}{E3EAF0}\n\\definecolor{papertablerow}{HTML}{F2F5F7}\n\\definecolor{papertablerule}{HTML}{7D8991}\n\\definecolor{paperresultrule}{HTML}{315B72}\n\n% Stable cross-document links between sibling PDFs. A named destination is\n% more durable than a page number: sections may move while their semantic role\n% remains the same. 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It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"},{"record_type":"paper_source_dependency","tex_reference":"evidence/erdos-251-prime-gap-dyadic-series","source_path":"paper/evidence/erdos-251-prime-gap-dyadic-series.tex","media_type":"text/x-tex","sha256":"sha256:1503e2c6bd58455fe1bd1c53e5288b1af4d8bd76c16108a050589063d20700ab","byte_count":6060,"licence":"CC-BY-4.0","body_format":"latex","body":"% Generated by scripts/paper_evidence.py from docs/paper_lean_coverage.json and the\n% Comparator replay receipts under evidence/comparator/.  Do not edit by hand.\n\\newcommand{\\evidencerecordurl}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md}\n\\DeclareResultEvidenceName{res:sparserationalisation}{Proposition 1.1}\n\\DeclareResultEvidenceRecord{res:sparserationalisation}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-sparserationalisation}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:sparserationalisation}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-sparserationalisation}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-sparserationalisation-comparator}\n\\DeclareResultEvidenceName{res:jointcountermodel}{Corollary 3.1}\n\\DeclareResultEvidenceRecord{res:jointcountermodel}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-jointcountermodel}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{res:jointcountermodel}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-jointcountermodel}{}\n\\DeclareResultEvidenceName{res:infinite}{Theorem 4.1}\n\\DeclareResultEvidenceRecord{res:infinite}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-infinite}{exact}{compared}\n\\DeclareResultEvidence{res:infinite}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-infinite}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-infinite-comparator}\n\\DeclareResultEvidenceName{res:irr-equivalence}{Corollary 4.2}\n\\DeclareResultEvidenceRecord{res:irr-equivalence}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-irr-equivalence}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:irr-equivalence}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-irr-equivalence}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-irr-equivalence-comparator}\n\\DeclareResultEvidenceName{res:true-tail}{Lemma 4.3}\n\\DeclareResultEvidenceRecord{res:true-tail}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-true-tail}{exact}{compared}\n\\DeclareResultEvidence{res:true-tail}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCompleteR20/TrueTail.lean\\#L57}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-true-tail-comparator}\n\\DeclareResultEvidenceName{res:escape-irrational}{Theorem 4.4}\n\\DeclareResultEvidenceRecord{res:escape-irrational}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-escape-irrational}{exact}{compared}\n\\DeclareResultEvidence{res:escape-irrational}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-escape-irrational}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-escape-irrational-comparator}\n\\DeclareResultEvidenceName{res:signedwindow}{Proposition 5.1}\n\\DeclareResultEvidenceRecord{res:signedwindow}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-signedwindow}{exact}{compared}\n\\DeclareResultEvidence{res:signedwindow}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-signedwindow}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-signedwindow-comparator}\n\\DeclareResultEvidenceName{res:truncation}{Proposition 5.3}\n\\DeclareResultEvidenceRecord{res:truncation}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-truncation}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:truncation}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperTailBoundsR7.lean\\#L273}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-truncation-comparator}\n\\DeclareResultEvidenceName{res:gap-nonperiodic}{Proposition 6.1}\n\\DeclareResultEvidenceRecord{res:gap-nonperiodic}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-gap-nonperiodic}{exact}{compared}\n\\DeclareResultEvidence{res:gap-nonperiodic}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCoreR7.lean\\#L175}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-251-prime-gap-dyadic-series.md\\#res-gap-nonperiodic-comparator}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/251/erdos-251-prime-gap-dyadic-series.tex","section_outline":["Introduction","Application to prime gaps","The prime series and its actual tails","Integral shifts","Small tail differences and finite tests","Finite separation from the integers","Further consequences and questions","Nonperiodic coefficients","Verification and reproducibility","Acknowledgements"]},"erdos251-prime-gap-reasoning-surface":{"paper_id":"erdos251-prime-gap-reasoning-surface","title":"Prime-Gap Dyadic Series: Perturbations, Exact Criteria and Certificates","form":"Reasoning surface","length_class":"long","pages":47,"subject":"Erdős #251","subject_kind":"problem","problem_id":"erdos_251","landing_role":"mathematics","question":"What exact tail criteria, sparse countermodels, and finite denominator bounds are proved for the prime-gap dyadic series, and why is irrationality still open?","precis":"A convergent dyadic series with nonnegative integer coefficients can be made rational by sparse nonnegative corrections of upper Banach density zero, eventually bounded by any f(n)→∞, while preserving every fixed modulus and, for a logarithmic allowance, short-block empirical distributions. For prime gaps the cumulative positions remain asymptotic to n log n; no primality conclusion is asserted. Summation by parts reduces the prime series to the gap series, and the tails T_N=∑_{j≥1} g_{N+j} 2^{-j} satisfy T_{N+1}=2T_N−g_{N+1}. That recurrence characterises rationality and yields finite tests with explicit remainder bounds. An exact finite certificate shows that a rational value of the prime series would have denominator at least 2^{39998}>10^{12040}. The bound does not establish irrationality.","owns":"the complete problem-specific reasoning surface for Erdős #251, including all registered result families and their boundaries","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #251, which remains open","first_pass":["long251:sec:tail","long251:sec:obstructions","long251:sec:open","sec:erdos-251-complete-family-map"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos251-prime-gap-reasoning-surface.pdf","hosted_pdf_path":"papers/erdos251-prime-gap-reasoning-surface.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/251/erdos251-prime-gap-reasoning-surface.tex","site_html":"maths/papers/erdos251-prime-gap-reasoning-surface.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/251/erdos251-prime-gap-reasoning-surface.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/251/erdos251-prime-gap-reasoning-surface.tex","body_pointer":"papers.records.erdos251-prime-gap-reasoning-surface.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Long-form reasoning record seeded from the complete problem note.\n% The short note and this record are now independent publication units.\n% This flat manuscript is generated from reviewable parts below.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n\\input{evidence/erdos251-prime-gap-reasoning-surface}\n% Compatibility labels for statements moved from the short paper.\n% amsmath permits only one ordinary label per equation. Its preserved kernel\n% label writer lets the aliases share the same number and PDF anchor.\n\\makeatletter\n\\newcommand{\\compatlabel}[1]{\\ltx@label{#1}}\n\\makeatother\n\\usepackage[htt]{hyphenat}\n\\usepackage{newunicodechar}\n\\newunicodechar{ℕ}{\\ensuremath{\\N}}\n\\newunicodechar{ℚ}{\\ensuremath{\\Q}}\n\\newunicodechar{ℤ}{\\ensuremath{\\Z}}\n\\newunicodechar{₀}{\\ensuremath{{}_0}}\n% Problem-specific source pin: #251 advances independently of the corpus\n% default, while the #249/#257 notes retain their own separately pinned snapshot.\n\\renewcommand{\\commit}{3d6d938d696fed0fb71dd55115a18a73738ff223}\n\\renewcommand{\\commitshort}{3d6d938d696f}\n\\renewcommand{\\PX}{lean/ErdosProblems}\n\\hypersetup{\n  pdftitle={Prime-gap dyadic series: perturbations, exact criteria and certificates},\n  pdfsubject={Erdős Problem Notes: Problem 251},\n  pdfkeywords={irrationality, prime gaps, dyadic series, summation by parts, Lean 4},\n}\n\n\\runninghead{Erd\\H{o}s \\#251: perturbations, criteria and certificates}\n\\title{Prime-Gap Dyadic Series:\\\\Perturbations, Exact Criteria and Certificates}\n\\subtitle{Proofs, counterexamples and computations for Erd\\H{o}s Problem~\\#251}\n\\author{Will Cook}\n\\date{September 2026}\n\n\\setlength{\\emergencystretch}{2em}\n\\def\\UrlBreaks{\\do\\.\\do\\_\\do\\/}\n\\begin{document}\n% ---- part core ----\n% Part: part core\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\nA sparse perturbation theorem shows that specified coefficient congruences\nand asymptotic block statistics are compatible with both rational and irrational\ndyadic sums.  For any convergent series with nonnegative integer\ncoefficients, one set of upper Banach density zero supports nonnegative\ncorrections attaining an interval of sums.  Any prescribed bound tending\nto infinity is eventually respected; coefficient and partial-sum residues\nare eventually preserved, with cutoffs independent of the target.\nFixed ordinary pair totals protect the congruences; growing digit ranges\nkeep the later weighted choices large enough to fill the interval.\nApplied to prime gaps with correction bound $\\log(n+3)$, the theorem\npreserves cumulative growth $n\\log n$ and gives vanishing total variation\nerror for blocks of length $o(\\log\\log X)$ sampled on $[X,2X)$, without\nensuring primality.\nFor the actual series, integral shifts of the complete tails characterise\nrationality.  We give sufficient conditions, counterexamples to weaker\ncoefficient conditions, and integer certificates excluding denominators\nbelow $2^{40062}>10^{12059}$.  Irrationality remains unresolved.\n\\end{abstract}\n\n\\section{Introduction}\\label{long251:sec:problem}\nThe primes $p_0=2,p_1=3,\\ldots$ define the dyadic series\n\\[\n \\Pi=\\sum_{n\\ge0}\\frac{p_n}{2^{n+1}}\n     =2/2+3/4+5/8+7/16+\\cdots.\n\\]\nIts irrationality is Erd\\H{o}s Problem~\\#251 \\cite{erdosproblems},\nrecorded in \\cite[p.~94]{erdos1958}, \\cite[p.~62]{erdosgraham1980} and\n\\cite[p.~103]{erdos1988}.  Writing $g_n=p_{n+1}-p_n$, summation by\nparts gives $\\Pi=2+\\sum_ng_n2^{-(n+1)}$.  We use this identity to\ncompare what can be deduced from properties of the coefficients with\nwhat would be needed from the actual prime-gap tails.\n\nTheorem~\\ref{long251:res:sparse-rationalisation} shows that a substantial\ncollection of coefficient properties does not determine rationality among\ncomparison sequences.  Starting with any convergent nonnegative integer\ndyadic series, it attains an interval of sums using corrections on one\nsparse set.  That set and every eventual coefficient and partial-sum\ncongruence cutoff are fixed before the target.  The family therefore\ncontains both rational and irrational sums, not merely a separately\nchosen example for each preserved property.  The corrections can\nrespect any prescribed bound tending to infinity.\n\nAt each stage a fixed ordinary correction total protects the congruence\ncalculations at later stages.  Redistributing that total between adjacent\nindices changes its dyadic contribution.  Growing digit ranges compensate for the dyadic decay\nacross the increasingly separated stages.\nA telescoping estimate then ensures that the remaining range spans every\ngap between adjacent present choices.\nWith a polylogarithmic correction bound, the empirical block laws remain\nasymptotically equal for lengths $o(\\log\\log X)$ sampled in $[X,2X)$.\nFor prime gaps, polynomial nonconcentration and cumulative growth\n$n\\log n$ also survive.  The missing requirement is that the cumulative\npositions be consecutive primes; neither residues nor these statistics\nsupply it.\n\nThe actual-prime arguments, beginning in Section~\\ref{long251:sec:parts},\nare independent of the construction.  Summation by parts and a vanishing\nboundary condition identify the complete tails.  Integral shifts then\ngive exact rationality criteria; two consecutive small differences with\nunequal gaps give a sufficient condition for irrationality.  Section~\\ref{long251:sec:local-certificate}\ncertifies individual differences, while\nSection~\\ref{long251:sec:denominator-floor} excludes finite ranges of\nrational denominators.  Neither finite calculation supplies arbitrarily\nlate witnesses.  Section~\\ref{long251:sec:obstructions} tests weaker\ncoefficient conditions by counterexamples; the appendices retain the\ninteger calculations, further criteria and source concordance.\n\nFor reference, the first primes and gaps are\n\\[\n \\begin{array}{c|cccccccccc}\n  i   & 0&1&2&3&4 &5 &6 &7 &8 &9\\\\\\hline\n  p_i & 2&3&5&7&11&13&17&19&23&29\\\\\n  g_i & 1&2&2&4&2 &4 &2&4&6&2\n \\end{array}\n\\]\nThus $g_0=1$ and all later gaps are even.  This parity restriction will be\nused to characterise two consecutive tail differences of absolute value\nless than one.  The alternative normalisation with denominator $2^i$ satisfies\n\\[\n \\sum_{i=0}^{n-1}\\frac{p_i}{2^i}\n =2\\sum_{i=0}^{n-1}\\frac{p_i}{2^{i+1}},\n\\]\nso it has the same rationality status.  The\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L111}{formal factor-of-two identity}\nuses these same finite sums.\n\n\\paragraph{Notation.}\nWe write $\\N=\\{0,1,\\ldots\\}$, use $\\ph$ for Euler's totient function,\nand let $\\operatorname{den}q$ be the positive denominator of $q\\in\\Q$\nin lowest terms.  A real number is \\emph{integral} if it belongs to $\\Z$,\nand $\\operatorname{dist}(x,\\Z)$ is its distance to the nearest integer.\nA sequence is eventually periodic with period $h\\ge1$ if\n$a_{n+h}=a_n$ for every sufficiently large $n$.  A set of indices has\ndensity zero if its counting function is $o(x)$, and a property holds\nfor almost all indices if its exceptional set has density zero.\nA property may hold at arbitrarily large indices even when those indices\nform a set of density zero.  An empty sum is zero.\n\n\\section{Sparse congruence-preserving perturbations}\n\\label{long251:sec:paired-corrections}\n\\papersectiontarget{sparse-construction}\n\nAt indices $n,n+1$, the pairs $(0,6),(2,4),(4,2),(6,0)$ all have\nordinary sum $6$, while their dyadic contributions are $6,8,10,12$\ndivided by $2^{n+2}$.  If the cumulative correction before the pair is\nodd, we first add $1$ at $n-1$.  Choosing the pair then leaves the\ncumulative residue after the triple unchanged.  We repeat this operation\nwith nested moduli eventually divisible by every fixed integer.\n\nFinite changes alone preserve rationality, so we use infinitely many\ntriples.  Making their separations tend to infinity gives upper Banach\ndensity zero, but also makes each weighted contribution smaller.  We\ncompensate by increasing the range of choices inside a pair.  The estimate\n\\eqref{long251:eq:sparse-overlap} shows that all later ranges together\ncover the gap between adjacent choices at the current stage.\nThe set containing the supports and the interval of sums are chosen before the value to be attained.\n\n\\paragraph{Density and block conventions.}\nAll intervals of indices contain integers, and $\\log$ is natural unless\na base is displayed.  Upper Banach density zero means\n\\[\n \\lim_{H\\to\\infty}\\sup_{u\\in\\N}\\frac{|S\\cap[u,u+H)|}{H}=0.\n\\]\nFor integers $X\\ge1$ and $m\\ge1$, let $\\mu_{a,X,m}$ be the empirical\nprobability measure obtained by choosing an integer $n\\in[X,2X)$ uniformly\nand observing $(a_n,\\ldots,a_{n+m-1})$.  The entries are not rescaled,\nand equal blocks are counted once for each starting index.\nWe use $d_{\\rm TV}(\\mu,\\nu)=\\sup_B|\\mu(B)-\\nu(B)|$, with the supremum over all sets of blocks.  For the same function\nof the block, bounded in absolute value by $B_0$, its means under the two\nmeasures differ by at most $2B_0d_{\\rm TV}(\\mu,\\nu)$.\nNo rescaling of the coefficients is implicit.\n\n\\begin{theorem}[sparse changes preserving congruences]\n\\label{long251:res:sparse-rationalisation}\nLet $a_n\\in\\N$ and $A=\\sum_{n\\ge0}a_n2^{-n-1}<\\infty$.  For every cutoff\n$K$ and every $f:\\N\\to\\R$ tending to infinity, there are a set\n$S\\subseteq[K,\\infty)$ of upper Banach density zero and a nondegenerate\ninterval $I\\subset(A,\\infty)$ with the following property.  Every $r\\in I$\nhas the form\n\\[\n r=\\sum_{n\\ge0}(a_n+e_n)2^{-n-1},\n\\]\nwhere $e_n\\in\\N$ is supported on $S$, $e_n\\le f(n)$ eventually, and, for\nevery integer $q\\ge1$, there is an $N_q$, chosen independently of $r$, such that\n\\[\n q\\mid e_n\\quad\\hbox{and}\\quad q\\mid\\sum_{i<n}e_i\n \\qquad(n\\ge N_q).\n\\]\nFor every $\\varepsilon>0$, the construction can instead be chosen with\n$e_n\\le(\\log(n+3))^\\varepsilon$ eventually and\n$|S\\cap[X,2X)|=O_\\varepsilon(X/\\log\\log X)$.  In that case the empirical\ndistributions of original and corrected unnormalised blocks of length\n$m(X)=o(\\log\\log X)$, sampled at the same integer starts in $[X,2X)$,\nhave total variation distance tending to zero, uniformly over $r$.\nPrecisely the coupling error is at most $m|S\\cap[X,2X+m)|/X$; with\n$\\|\\Phi\\|_\\infty\\le B_0$ the bounded-test error is at most twice this\nquantity times $B_0$, also for tests depending on the starting index.\n\\end{theorem}\n\nThe hypotheses allow any nonnegative integer sequence of polynomial\ngrowth, but not $a_n=2^n$, whose dyadic series diverges.  The bound on the\ncorrections can grow as slowly as $\\log\\log(n+3)$.  A fixed bound is not\npossible here: eventual boundedness together with eventual divisibility\nby every fixed modulus would make the corrections eventually zero; their\nfixed total would then have to be divisible by every modulus and hence be\nzero.  Nonnegative corrections could not change the sum.\nThe cutoff may depend on the modulus, but the common set $S$ and all\ncutoffs are fixed before the target.  The support of a correction may\noccupy only part of $S$.\nUpper Banach density zero is stronger than ordinary density zero.  For example, the union of the integer intervals\n$[k!,k!+k)$, $k\\ge3$, has ordinary density zero but upper Banach density\none.  If $k!\\le x<(k+1)!$, at most $k(k+1)/2$ indices have occurred,\nso their proportion is at most $k(k+1)/(2k!)\\to0$.  Nevertheless each\ninterval is filled and their lengths are unbounded.  Upper Banach density zero rules out this concentration in long intervals,\nhowever far apart those intervals lie.\n\n\\begin{proof}\n\\emph{Fixed totals.}\nUse $n_j-1$ to cancel a cumulative residue and $n_j,n_j+1$ for a\npair with fixed ordinary sum.  Varying the pair changes its weighted\ncontribution without altering any later residue to be cancelled.\nChoose an initial index $n_{-1}\\ge K$ and centres with\n$s_j=n_j-n_{j-1}\\ge4$, so the triples do not overlap.  Let\n$M_j$ be positive integers with $M_j\\mid M_{j+1}$, and let $D_j$\nbound the digit at stage $j$.  The spacing, moduli and digit bounds\nwill be chosen below.\n\nLet $C_j$ record the ordinary correction before the $j$th triple.\nStarting from $C_0=0$, prescribe\n\\[\n c_j=(-C_j)\\bmod M_j\\quad(0\\le c_j<M_j),\\qquad\n C_{j+1}=C_j+c_j+M_jD_j.\n\\]\nFor integers $0\\le d_j\\le D_j$, set\n\\begin{equation}\\label{long251:eq:sparse-triple}\n e_{n_j-1}=c_j,\\qquad e_{n_j}=M_jd_j,\\qquad\n e_{n_j+1}=M_j(D_j-d_j),\n\\end{equation}\nand put $e_n=0$ elsewhere.  The two sums of a triple are\n\\[\n \\begin{aligned}\n \\sum_{r=n_j-1}^{n_j+1}e_r\n    &=c_j+M_jD_j,\\\\\n \\sum_{r=n_j-1}^{n_j+1}e_r2^{-r-1}\n    &=c_j2^{-n_j}+M_jD_j2^{-n_j-2}+d_jM_j2^{-n_j-2}.\n \\end{aligned}\n\\]\nThe ordinary total contains no $d_j$, so the recurrence fixes every\n$c_j$ in advance.  In the weighted total only the last term varies:\ntransferring $M_j$ from $n_j+1$ to $n_j$ adds\n\\[\n w_j=M_j2^{-n_j-2}.\n\\]\n\nAfter $c_j$ has been inserted, the cumulative correction is divisible\nby $M_j$; the two entries of the pair preserve this divisibility.\nFix $q\\mid M_J$.  At index $n_J$ both the coefficient correction and\nthe cumulative correction are divisible by $q$.  At every later\ntriple, $q$ divides $C_j$ and $M_j$, so it also divides $c_j$ and\nboth pair entries.  The intervening corrections are zero.  Hence\n\\[\n q\\mid e_n,\\qquad q\\mid\\sum_{i<n}e_i\\qquad(n\\ge n_J).\n\\]\nThe entry $c_J$ need not be divisible by $q$, which is why it is placed at\n$n_J-1$, before the cutoff.  The index $n_J$ depends on $q$ and the chosen\nmoduli, but not on the digits or the value to be attained.\n\n\\par\\medskip\\noindent\\emph{Spacing and size.}\nThe choices at stage $j$ are $0,w_j,\\ldots,D_jw_j$.\nThe covering argument will work if the later ranges total at least\n$w_j$.  Taking $D_i=2^{s_i}-1$ compensates for separation by making\neach later range a scaled dyadic difference:\n\\[\n D_iw_i=(2^{n_i-n_{i-1}}-1)M_i2^{-n_i-2}\n       =M_i\\bigl(2^{-n_{i-1}-2}-2^{-n_i-2}\\bigr).\n\\]\nBecause $M_i\\ge M_j$ for $i>j$, replacing every later modulus by $M_j$\ngives the finite estimate\n\\begin{equation}\\label{long251:eq:finite-overlap}\n \\begin{aligned}\n \\sum_{i=j+1}^{J}D_iw_i\n &\\ge M_j\\sum_{i=j+1}^{J}\n       \\bigl(2^{-n_{i-1}-2}-2^{-n_i-2}\\bigr)\\\\\n &=w_j-M_j2^{-n_J-2}\\qquad(J>j).\n \\end{aligned}\n\\end{equation}\nNo monotonicity of the weights is needed.  Each entry of\n\\eqref{long251:eq:sparse-triple} is at most $M_j2^{s_j}$, so the size cost of\nseparating the stages is exponential.  We must fit this product below\nthe allowance at all three indices, while $s_j\\to\\infty$ and every\nfixed positive integer eventually divides $M_j$.\n\nTo allow a nonmonotone bound $f$, put\n\\[\n h(n)=\\inf_{m\\ge n}\\min(f(m),m).\n\\]\nThis nondecreasing lower envelope tends to infinity even when $f$\noscillates.  A bound at $n_{j-1}$ therefore controls all three later\ncoordinates; the cap by $m$ ensures $e_n\\le n$ and hence summability\nfor every digit choice.\nEnlarge $n_{-1}\\ge K$ until $h(n_{-1})\\ge32$, and set\n\\begin{equation}\\label{long251:eq:sparse-general-schedule}\n \\begin{split}\n k_j&=\\max\\{k\\ge2:k!2^{k+2}\\le h(n_{j-1})\\},\\\\\n M_j&=k_j!,\\qquad s_j=k_j+2,\\qquad n_j=n_{j-1}+s_j.\n \\end{split}\n\\end{equation}\nThe maximum exists: $k=2$ is admissible because $2!2^4=32$, and\n$k!2^{k+2}\\to\\infty$.  Since $n_j\\to\\infty$ and $h$ is nondecreasing\nand divergent, $k_j$ is nondecreasing and tends to infinity.  Thus\n$M_j\\mid M_{j+1}$, every fixed $q$ eventually divides $M_j=k_j!$,\nand $s_j=k_j+2\\to\\infty$.\nAt every coordinate $n$ of the $j$th triple, this schedule also gives\n\\[\n 0\\le e_n\\le M_j2^{s_j}\\le h(n_{j-1})\\le\\min(f(n),n).\n\\]\nLet $S=\\bigcup_{j\\ge0}\\{n_j-1,n_j,n_j+1\\}$.  For $R\\ge4$, all\nsufficiently late centres are at least $R$ apart.  Every interval of\nlength $H$ contains at most $3(H/R+2)$ points from those triples;\nthe earlier triples add a fixed number, independent of the interval's\nposition.  Divide by $H$, take the supremum over positions, then let\n$H\\to\\infty$.  The limiting upper bound is $3/R$; letting $R\\to\\infty$\nproves upper Banach density zero.\n\n\\par\\medskip\\noindent\\emph{Attaining the interval.}\nThe support and congruence requirements are now settled independently\nof the target.  Only the digits remain to be chosen.  Define\n\\[\n \\beta=\\sum_{j\\ge0}\\bigl(c_j2^{-n_j}+D_jw_j\\bigr),\\qquad\n F_j=\\sum_{i\\ge j}D_iw_i.\n\\]\nEven zero digits contribute $\\beta>0$; $F_j$ is the maximum variable\ncontribution from stage $j$ onwards.  The size bound makes both finite,\nwith $F_0>0$ and $F_j\\to0$.  The total correction is\n\\begin{equation}\\label{long251:eq:sparse-baseline}\n \\sum_ne_n2^{-n-1}=\\beta+\\sum_jd_jw_j.\n\\end{equation}\nTaking $J\\to\\infty$ in \\eqref{long251:eq:finite-overlap} now gives\n\\begin{equation}\\label{long251:eq:sparse-overlap}\n F_{j+1}\\ge M_j2^{-n_j-2}=w_j.\n\\end{equation}\nThese bounds do not yet prove that every point of $[0,F_j]$ is\nattained.  A digit $d$ leaves a remainder in $[0,F_{j+1}]$ precisely\nfor current remainders in\n\\[\n [dw_j,dw_j+F_{j+1}],\\qquad d=0,\\ldots,D_j.\n\\]\nThese intervals meet or overlap by \\eqref{long251:eq:sparse-overlap}, and their union\nis $[0,F_j]$ because $F_j=D_jw_j+F_{j+1}$.\n\nGiven $x\\in[0,F_0]$, start with $\\rho_0=x$.  Having chosen the first\n$j$ digits, choose $d_j$ so that\n\\[\n \\rho_{j+1}=\\rho_j-d_jw_j\\in[0,F_{j+1}].\n\\]\nThe covering just proved makes this possible at every stage.  Since $0\\le\\rho_j\\le F_j$ and $F_j\\to0$, we obtain\n\\[\n x-\\sum_{i=0}^{j}d_iw_i=\\rho_{j+1}\\longrightarrow0.\n\\]\nThus every $x\\in[0,F_0]$ is attained, and we may take\n$I=(A+\\beta,A+\\beta+F_0)$.  The target $r\\in I$ determines only the\ndigits, through $x=r-A-\\beta$; it changes neither $S$, $I$ nor the\ncutoff $N_q=n_J$ chosen using $q\\mid M_J$.\nThis is the finite-choice covering argument of\n\\cite[Lemma~4]{crmarickovac2025}: the $j$th choice set has largest gap\n$w_j$ and diameter $D_jw_j$, its maxima are summable, and the sum of\nsubsequent diameters is at least $w_j$.  Equality is allowed.\nFridy's generalised-base lemma\n\\cite[p.~194]{fridy1966} and the reciprocal-choice argument in\n\\cite[Lemma~5.1]{kovactao2024} are antecedents.  The finite-choice argument\nuses \\eqref{long251:eq:sparse-overlap}; the required comparison of moduli\nwas checked in \\eqref{long251:eq:finite-overlap}.\n\n\\par\\medskip\\noindent\\emph{The logarithmic scale.}\nFor $f(n)=(\\log(n+3))^\\varepsilon$, the size cost $2^{s_j}$ leads to\nthe iterated logarithm.  We choose separations of order $\\log\\log n_j$\nand reserve part of the allowance for growing moduli.  Keep the same\ntriples and $D_j=2^{s_j}-1$, and set\n\\begin{equation}\\label{long251:eq:sparse-polylog-schedule}\n \\begin{aligned}\n s_j&=\\left\\lfloor\\frac{\\varepsilon}{2}\n               \\log_2\\log(n_{j-1}+3)\\right\\rfloor,\\\\\n k_j&=\\max\\{k\\ge2:k!\\le(\\log(n_{j-1}+3))^{\\varepsilon/4}\\},\n \\end{aligned}\n\\end{equation}\nwith $M_j=k_j!$ and $n_j=n_{j-1}+s_j$.  Start far enough out that\n$s_j\\ge4$ and all choices exist.  The exponents $\\varepsilon/4$ and\n$\\varepsilon/2$ add to $3\\varepsilon/4<\\varepsilon$, so the corrections\nsatisfy the prescribed bound:\n\\[\n M_j2^{s_j}\\le(\\log(n_{j-1}+3))^{3\\varepsilon/4}\n             \\le(\\log(n+3))^\\varepsilon\n\\]\nat each coordinate $n$ of the triple.  After enlarging the initial\nindex, this is also at most $n$.  The moduli remain nested, and every fixed positive integer divides all\nsufficiently late moduli.  Summability, the congruence induction and\n\\eqref{long251:eq:finite-overlap} therefore apply without change.  Since\n$s_j\\asymp_\\varepsilon\\log\\log n_j$, the relevant centres near $[X,2X)$\nare separated by $\\gg_\\varepsilon\\log\\log X$.  Each contributes at most\nthree support points, giving\n\\[\n |S\\cap[X,2X)|=O_\\varepsilon(X/\\log\\log X).\n\\]\n\nChoose the same uniform starting index $n\\in[X,2X)$ for both\nblocks.  They agree when the block avoids $S$.  A changed coordinate\n$t\\in S\\cap[X,2X+m)$ affects only starts in $[t-m+1,t]$, at most\n$m$ choices.  The union bound gives\n\\[\n \\begin{aligned}\n d_{\\rm TV}(\\mu_{a,X,m},\\mu_{a+e,X,m})\n &\\le\\mathbb P\\bigl(e_{n+i}\\ne0\\text{ for some }0\\le i<m\\bigr)\\\\\n &\\le\\frac{m|S\\cap[X,2X+m)|}{X}.\n \\end{aligned}\n\\]\nFor $m\\le X$, two dyadic bands cover $[X,2X+m)$, giving\n$O_\\varepsilon(m/\\log\\log X)$.  A block of length $o(\\log\\log X)$\ntherefore avoids $S$ with probability tending to one.  The convergence\nis uniform over targets because the same $S$ contains every correction\nsupport.\nFor a test bounded in absolute value by $B_0$ that also depends on the\nstarting index, the coupled values agree at unchanged blocks and differ\nby at most $2B_0$ elsewhere.  Their mean difference is consequently at\nmost $2B_0m|S\\cap[X,2X+m)|/X$.  This uses the common starting index,\nnot merely the marginal distributions.  No limiting distribution,\nrandomness or independence of the original coefficients is assumed.\n\\end{proof}\nIdentical marginal distributions alone would not control an\nindex-dependent test.  For example, let $a_n=n\\bmod2$ and $b_n=1-a_n$.\nFor every even $X$, their one-coordinate empirical distributions on\n$[X,2X)$ are equal, but the test $\\mathbf{1}_{\\{x=n\\bmod2\\}}$ has mean\n$1$ for $(n,a_n)$ and $0$ for $(n,b_n)$.  This example is not a sparse\nperturbation; it isolates why the proof couples blocks at the same index.\n\nThe estimate gives a sufficient scale, not an optimal threshold.  It\ntends to zero for fixed block lengths and for\n$m(X)=\\lfloor\\sqrt{\\log\\log X}\\rfloor$, but gives no vanishing error for\n$m$ comparable to $\\log\\log X$.  Its error is also absolute, not relative:\nprobabilities $1/\\log X$ and $0$ differ by $o(1)$ despite the disappearance\nof the event.  Relative preservation would require an error small compared\nwith the event's probability.  Finally, agreement on a finite block does\nnot identify its complete weighted tail; later coefficients still contribute.\n\n\n\n\\begin{corollary}[target intervals arbitrarily close to the original sum]\\label{long251:res:local-targets}\nFor the sparse theorem for an arbitrary sequence, the target interval can additionally\nbe required to lie in $(A,A+\\eta)$ for any prescribed $\\eta>0$.\n\\end{corollary}\n\\begin{proof}\nIn the construction for a general bound $f$, every nonzero correction satisfies $e_n\\le n$.\nChoose an integer $K'\\ge K$ sufficiently large that\n$\\sum_{n\\ge K'}n2^{-(n+1)}=(K'+1)2^{-K'}<\\eta$, and start the construction\nbeyond $K'$.  This still leaves the originally prescribed prefix unchanged.\nThe complete interval of corrections has positive lower endpoint and\nupper endpoint at most this sum.  Here $\\eta$ is chosen before the\nconstruction, so the common support and interval may depend on $\\eta$.\nWithin that interval only the digits depend on the target.  This does\nnot make every target above $A$ attainable under one fixed size bound.\n\\end{proof}\n\n\\section{Context and mathematical dependencies}\\label{long251:sec:context}\n\\papersectiontarget{context}\nTao's forum comment of 7 October 2025 gives the prime-to-gap reduction\nand proposes sufficiently uniform quantitative control of about\n$\\log\\log n$ consecutive gaps as an input to irrationality\n\\cite[comment of 7 October 2025]{erdosproblems251thread}.\nTheorem~\\ref{long251:res:infinite} records the reduction with its endpoint\nand convergence proof.  An elementary prime bound suffices for it.\nWe use the prime number theorem separately, to deduce\n$P_n\\sim n\\log n$ in Corollary~\\ref{long251:res:jointcountermodel}.\n\n\\paragraph{Conditional prime-pattern results.}\nLand's draft gives conditional irrationality\n\\cite[Conjecture~1, Theorem~2]{land2026}, and Ringer's draft gives\nconditional normality \\cite{ringer2026}.  For each fixed integer base\n$b\\ge2$, Ringer's number is $\\sum_{n\\ge0}p_nb^{-(n+1)}$ and its\nnormality is to base $b$.  Varying $b$ varies the number, so the result\nmakes no assertion of absolute normality for one constant.\nThe authors supply formalisation material, which we have not rebuilt.\n\nThe uniform Hardy--Littlewood hypothesis in\n\\cite[Conjecture~1.3]{kuperberg2023} counts prime translates of each\nnonempty admissible tuple of $k\\le(\\log\\log x)^3$ distinct integer\nshifts in $[0,(\\log x)^2]$.  Admissibility means that the shifts omit a\nresidue class modulo each prime.  The main term is the singular series\nof the tuple multiplied by $\\int_2^x(\\log t)^{-k}\\,dt$, with absolute\nerror at most $Cx^{1-\\delta}$.  The constants $C,\\delta>0$ and the\nstarting threshold precede the choice of $x$ and of the tuple.\nReplacing the integral by its leading asymptotic can exceed this error,\nand qualitative asymptotics for individual fixed tuples do not give\nthe stated uniformity.\n\nIn the version rechecked on 18 September 2026, Ringer also states weaker\nhypotheses for the joint positions of the first $L$ primes after a prime\nbasepoint, with $L$ of order $\\log\\log X$.  The basepoints are uniform\namong primes in $(X,2X]$, where $X$ measures prime values.\nOne hypothesis controls a weighted sum of adverse tuple-count errors.\nThe adverse sign alternates with the order of inclusion--exclusion,\nso both undercounts and overcounts can matter.  Another hypothesis\nallows bounded domination by a comparison law in place of total variation\napproximation.  Neither is a single-gap marginal estimate or a consequence\nof an upper sieve bound alone.  We assume none of these hypotheses here.\n\nOur index band $[X,2X)$ corresponds to $[p_X,p_{2X})$.  The prime number\ntheorem gives $\\pi(2p_X)=2X+o(X)$, so its symmetric difference with the\nprime basepoints in $(p_X,2p_X]$ has size $o(X)$.  Each set has\n$X+o(X)$ elements.  The two uniform measures have common mass equal to\nthe intersection size divided by the larger cardinality, and hence\nhave total variation distance $o(1)$.  This compares the same uniformly\nbounded observable with the same normalisation at common basepoints.\nIt supplies no relative or quantitative control of the growing patterns\nrequired by the preceding conditional results.\n\n\\paragraph{Related irrationality problems.}\nErd\\H{o}s's 1958 article states that\n$\\sum_{n\\ge1}p_{n-1}^k/n!$ is irrational for each $k\\ge1$, and prints\na proof only for $k=1$, explaining that the higher-power proof is more\ncomplicated \\cite[pp.~93--95]{erdos1958}.\nSchlage-Puchta later proved the stronger linear independence over $\\Q$\nof $1,S_0,S_1,\\ldots$, where $S_k=\\sum_{n\\ge1}p_{n-1}^k/n!$\n\\cite[Theorem~3]{schlagepuchta2011}.\nThe fixed dyadic denominator question is stated separately in\n\\cite[p.~103]{erdos1988}, including the difficulty of the case $k=1$.\n\nThe same page proposes irrationality of\n$\\sum_{n\\ge1}p_{n-1}/(d_1\\cdots d_n)$ whenever $d_n\\ge2$ and\n$d_n=o(p_{n-1})$.  Kova\\v{c}'s note of 15 April 2026 disproves this\nvariable-denominator assertion with a sum equal to $1$\n\\cite[Theorem~1 and proof, pp.~1--2]{kovac2026}.\nIts printed attribution is ChatGPT~5.4 Pro, orchestrated by Vjeko Kova\\v{c}.\nSection~3 of the 1958 article also concerns variable product denominators\n\\cite[pp.~96--97]{erdos1958}.  We make no quantified reinterpretation\nof its nonstandard asymptotic notation in condition~(5), which is unused\nhere.  The endpoint example can be read directly: with\n$G_0=1$ and $G_n=\\prod_{j=1}^n(p_{j-1}+1)$,\n\\[\n \\frac{p_{n-1}}{G_n}=\\frac1{G_{n-1}}-\\frac1{G_n},\\qquad\n \\sum_{n\\ge1}\\frac{p_{n-1}}{G_n}=1.\n\\]\nThese examples leave fixed dyadic denominators unchanged as a separate\nquestion.\n\nA different dyadic series with bounded coefficients is due to Erd\\H{o}s\nand Pomerance.  If $P(m)$ is the largest prime factor of $m$, they proved\nirrationality of\n\\[\n \\sum_{m\\ge2}\\mathbf1_{\\{P(m)>P(m+1)\\}}2^{-m}\n \\quad\\text{\\cite[\\S7, p.~320]{erdospomerance1978}}.\n\\]\nErd\\H{o}s and Graham record the complementary indicator\n\\cite[p.~62]{erdosgraham1980}.  Since consecutive integers cannot have\nthe same largest prime factor, its series is $1/2$ minus the displayed\none and is also irrational.  These binary coefficients differ from the\nunbounded numerators of $\\Pi$.\nSchlage-Puchta's concatenation theorem\n\\cite[Theorem~2, p.~1]{schlagepuchta2011} concerns a base $b\\ge2$ that\nis not a proper power, with its stated monotonicity and growth assumptions.\nConcatenation places blocks according to their lengths, whereas our\ncoefficients occupy fixed dyadic positions.  We do not use that theorem.\nThe analytic input used below is his Lemma~4, proved by Selberg's sieve:\nfor every fixed nonzero $F\\in\\Z[x_0,\\ldots,x_k]$,\n$F(g_n,\\ldots,g_{n+k})\\ne0$ for almost all $n$\n\\cite[Lemma~4, pp.~5--6]{schlagepuchta2011}.\n\n\\paragraph{Interval covering and divisibility.}\nThe bounded perturbation in Theorem~\\ref{long251:res:boundedperturbation}\nis a binary expansion.  The sparse construction uses the finite-choice\nlemma of Crmari\\'{c} and Kova\\v{c} \\cite[Lemma~4]{crmarickovac2025}.\nFor finite nonnegative choice sets with at least two elements and\nsummable maxima, their lemma fills a single interval if each largest\nsuccessive gap is at most the sum of the subsequent diameters.\nHere the sets are $\\{0,w_j,\\ldots,D_jw_j\\}$, and the condition is\n$w_j\\le\\sum_{i>j}D_iw_i$.  The proof checks this inequality at every\nstage.  Fridy's lemma \\cite[Lemma, p.~194]{fridy1966} assumes nonincreasing\nweights, a restriction avoided by the finite-choice form.\nKova\\v{c} and Tao use reciprocal choices\n\\cite[Lemma~5.1, Theorem~2.5]{kovactao2024}.\n\nVan Doorn and Kova\\v{c} combine finite subsum covering with a distinguished\nsubsequence of denominators \\cite[Lemma~7, Proposition~8]{vandoornkovac2025}.\nEach distinguished denominator is divisible by all earlier denominators,\nand every positive integer divides some denominator.  Thus every fixed\ninteger divides all sufficiently late distinguished denominators, as it\ndoes our factorial moduli.  Their condition is imposed on the distinguished\nsubsequence, not on every denominator.  At a distinguished cutoff the\nfinite sums lie in the reciprocal grid of the last denominator.\nAdding the covering inequalities between successive cutoffs gives\nLemma~7 for the full finite prefix, leaving no gaps in that grid.\nOnce the denominator of a rational target divides the last denominator\nand the target lies in the available range, it has a finite representation.\nTheir proposition fills a half-open interval of rational targets by finite\nsubsums.  Our construction fills a fixed interval of real targets by\ninfinite choices.\n\nVan Doorn's exchange\n$\\{21d,28d\\}\\leftrightarrow\\{20d,30d\\}$ preserves the reciprocal sum\n$1/(12d)$ while changing the ordinary sum from $49d$ to $50d$\n\\cite[proof of Theorem~3]{vandoorn2025}.\nThe pair in \\eqref{long251:eq:sparse-triple} preserves the ordinary sum\nand changes the weighted one.  The analogy concerns the conserved\nquantity, and does not invoke his partition theorem.  The present\nargument adds no new interval-covering principle.\n\n\\paragraph{Binary subsums.}\nThe \\emph{achievement set} of a positive summable sequence consists of\nsums obtained by retaining or omitting each term.  The sequence\n$(2^{-n})_{n\\ge1}$ fills $[0,1]$.  The sequence $(3^{-n})_{n\\ge1}$\nomits $(1/6,1/3)$ because its terms after the first sum to $1/6$.\nNevertheless a failure of this simple overlap test at some indices does\nnot exclude interior elsewhere in the achievement set.\nBartoszewicz, Filipczak and Szymonik obtain interior from a central run\nof integer block sums in a multigeometric family\n\\cite[Theorem~2]{bartoszewicz2014}.\nPrus-Wi\\'sniowski and Ptak construct Cantorvals with density-zero overlap\nindices $\\{n:a_n\\le\\sum_{i>n}a_i\\}$ \\cite[Theorem]{prusptak2024}.\nThese compact sets are the closures of their interiors, and the endpoints\nof every nondegenerate interval component are accumulation points of\none-point components.\n\nThe surrounding classification is discussed in \\cite[\\S1]{glabprus2025}\nand in Nitecki's exposition \\cite[Theorem~14]{nitecki2015}, which credits\nGuthrie and Nymann.  The cited preprint's title and theorem numbers differ\nfrom those in the published Monthly article.\nEven prescribing the exact inequality pattern leaves the presence of\ninterior undetermined.  Marchwicki and Miska's construction\n\\cite[Theorem~2.1]{marchwickimiska2021}, with the proof repair in\n\\cite{miskaprusptak2023}, realises any infinite strict term-dominating\nindex set with a Cantor achievement set.  The repair preserves uniqueness,\nand the latter paper gives a simpler proof of a weaker conclusion without\nuniqueness.  Nowakowski's sufficient condition\n\\cite[Definition~2, Theorem~3.1]{nowakowski2025} instead requires every\nstage of the Star Procedure's positivity recursion to succeed.\nKnowing that neither term--tail inequality holds eventually therefore\ncannot replace the overlap calculation in our variable-digit proof.\n\n\\paragraph{Uniformity and finite-state comparisons.}\nAdamczewski, Drmota and M\\\"ullner compute logarithmic densities of each\nfixed automatic sequence along primes \\cite[Theorems~1.2, 1.4]{adm2022}.\nTheir Theorem~1.2 gives a sufficient condition for natural densities to\nexist and be rational, while Theorem~1.4 reduces logarithmic-density\ncalculation to integers coprime to a fixed modulus.  The general\nlogarithmic densities need not be rational.  Applying these results to\nour tails would require a specified finite-alphabet automaton computing\nthe observable, together with uniform error bounds if its state space\ngrows.  The theorems themselves concern neither unbounded consecutive\ngaps nor a growing family of carry automata.\n\nKuperberg's singular-series estimate permits a specified growing\ncardinality \\cite[Theorem~1.1]{kuperberg2023}.\nHer arithmetic-progression and smooth-weight estimates\n\\cite{kuperbergweighted2025} provide a fixed-parameter comparison, without\ngiving a law for ordered consecutive-gap blocks.\nJha's revised preprint \\cite{jha2026} studies growing short-interval\nprime counts under a strong Hardy--Littlewood hypothesis.  Such counts\nalone do not control the weighted consecutive-gap differences in our\ncriteria.  A fixed-dimensional limit theorem likewise leaves the required\ngrowing-block errors to be proved.\nAt the cited revision, the \\emph{Formal Conjectures} repository leaves\nProblem~\\#251 unproved \\cite{formalconjectures251}.\nIts zero-based sum starts with the zeroth prime divided by $2^0$,\nand is twice our normalisation.\n\n\\paragraph{The tail argument.}\nAn ordinary binary expansion is rational exactly when its digits are\neventually periodic.  Applying carries to integer coefficients changes\nthe sequence to which this criterion refers.  Even bounded nonperiodic\ncoefficients can give a rational sum, as\nProposition~\\ref{long251:xr:boundedpolignac} shows.\nA classical alternative is the growth criterion for Ahmes series\n$\\sum_k1/a_k$ with strictly increasing positive integer denominators:\n$a_k^{1/2^k}\\to\\infty$ suffices for irrationality\n\\cite{erdosstraus1963}\\cite[\\S1]{kovactao2024}.\nReplacing divergence by any sufficiently large fixed threshold fails,\nsince shifted Sylvester sequences grow like $C^{2^k}$ for arbitrarily\nlarge $C$ and have rational reciprocal sum.  The criterion is sufficient\nand is not necessary.  Splitting $p_i/2^{i+1}$ into $p_i$ equal unit\nfractions introduces repetitions, and even the distinct denominators\n$2^n$ satisfy $(2^n)^{1/2^n}\\to1$.\n\nWe instead track reduced tail denominators.  Rationality forces an\neventually integral shift, not repeating coefficients: the telescope\nwith $K_n=n$ in Proposition~\\ref{long251:res:telescope} has coefficients\n$n-1$ and sum zero.  Parity then gives a sufficient test using two small\ndifferences.  Its occurrence arbitrarily late for every positive shift\nremains unproved, as does its necessity.\n\n\\section{Summation by parts, with the endpoint retained}\\label{long251:sec:parts}\n\nWe return to the actual prime gaps\n$g_n=p_{n+1}-p_n$, with $p_0=2$ and $g_0=1$.  We write\n\\[\n \\Pi=\\sum_{n\\ge0}p_n2^{-n-1},\\qquad\n S=\\sum_{n\\ge0}g_n2^{-n-1}.\n\\]\nHere $S$ denotes the gap sum.  The set of indices used in\nSection~\\ref{long251:sec:paired-corrections} will not be needed below.\nThe short paper calls this same sum $G$.  The two calculations below\nhave different jobs: summation by parts gives $\\Pi=2+S$, and deleting\nthe first gap gives the initial tail $T_0=2S-1$.\n\nWe begin with a finite identity, so no convergence hypothesis is needed.\nIts endpoint identifies the estimate required before passing to the\ninfinite series.  The linked definitions are the\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L101}{dyadic partial sum}\nand the\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L121}{weighted sum of consecutive differences}.\nThe formula below writes both sums explicitly.\n\n\\begin{proposition}[finite summation by parts]\\label{long251:res:abel}\\compatlabel{res:abel}\nFor every rational sequence $P$ and every $n\\ge0$,\n\\[\n \\sum_{i=0}^{n}\\frac{P(i)}{2^{\\,i+1}}\n =P(0)+\\sum_{i=0}^{n-1}\\frac{P(i+1)-P(i)}{2^{\\,i+1}}\n       -\\frac{P(n)}{2^{\\,n+1}} .\n\\]\n\\end{proposition}\n\n\\begin{proof}\nAt $n=0$ both sides equal $P(0)/2$.  When $n$ is replaced by $n+1$,\nthe change on the right is\n\\[\n \\frac{P(n+1)-P(n)}{2^{n+1}}\n -\\frac{P(n+1)}{2^{n+2}}+\\frac{P(n)}{2^{n+1}}\n =\\frac{P(n+1)}{2^{n+2}}.\n\\]\nThis is the next term on the left, which proves the identity by induction.\n\\end{proof}\n\nSpecialising to $P(i)=p_i$, whose first value is $p_0=2$, and writing\n$g_i=p_{i+1}-p_i$ for the zero-based gaps, gives the reformulation.\n\n\\begin{theorem}[prime-gap reformulation]\\label{long251:res:parts}\\compatlabel{res:parts}\nLet $p_0=2,p_1=3,\\ldots$ be the primes in increasing order and\n$g_i=p_{i+1}-p_i$.  For every $n\\ge0$,\n\\[\n \\sum_{i=0}^{n}\\frac{p_i}{2^{\\,i+1}}\n =2+\\sum_{i=0}^{n-1}\\frac{g_i}{2^{\\,i+1}}-\\frac{p_n}{2^{\\,n+1}} .\n\\]\n\\end{theorem}\n\nThe leading $2$ is the first prime, not a normalising constant.\nAt $n=2$, for instance, the left side is $2/2+3/4+5/8=19/8$ and the right side\nis $2+(1/2+2/4)-5/8=19/8$.\n\n\\subsection{The infinite identity and the irrationality equivalence}\\label{long251:sec:infinite}\\compatlabel{sec:infinite}\n\nTo pass to the infinite series, let\n\\[\n u_n=\\frac{p_n}{2^{n+1}},\\qquad v_n=\\frac{g_n}{2^{n+1}}.\n\\]\nThus $\\sum_nu_n=\\Pi$.  The termwise identity $v_n=2u_{n+1}-u_n$ is the\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L202}{dyadic\ndiscrete derivative}.  It expresses each gap term as an integer combination of\ntwo consecutive prime terms, so once $(u_n)$ is summable the gap series can be\nsummed after proving absolute convergence, which justifies shifting\nand subtracting the prime series.\n\n\\begin{theorem}[infinite prime-gap identity]\\label{long251:res:infinite}\nBoth series converge and\n\\[\n \\sum_{n\\ge0}\\frac{p_n}{2^{\\,n+1}}\n \\;=\\;2+\\sum_{n\\ge0}\\frac{g_n}{2^{\\,n+1}} .\n\\]\n\\end{theorem}\n\n\\begin{proof}\nThe bound $p_n\\le1250(n+1)^4$ of Appendix~\\ref{long251:app:prime-bound} makes $(u_n)$\nsummable, and $0<g_n\\le p_{n+1}$ then makes $(v_n)$ summable.  The shifted\nsequence $(u_{n+1})$ is summable, so summing $v_n=2u_{n+1}-u_n$ and using\n$u_0=1$ gives\n\\[\n \\sum_{n\\ge0}v_n\n =2\\left(\\sum_{n\\ge0}u_n-u_0\\right)-\\sum_{n\\ge0}u_n\n =\\sum_{n\\ge0}u_n-2 . \\qedhere\n\\]\n\\end{proof}\n\nThe prime number theorem is needed neither for this convergence nor for\nthe identity.  Its asymptotic $p_n\\sim n\\log n$ is used in\nCorollary~\\ref{long251:res:nonconc-primes} and in the argument counting\nrepeated tail values~\\cite[Chapter~6]{mv2007}.\n\n\\begin{corollary}[exact irrationality reformulation]\\label{long251:res:irr-equivalence}\\label{res:irr-equivalence}\n$\\Pi$ is irrational if and only if $S=\\sum_{n\\ge0}g_n2^{-(n+1)}$ is irrational.\nThe corresponding zero-based series with denominator $2^n$ equals $4+2S$ and\nhas the same irrationality status.\n\\end{corollary}\n\\begin{proof}\nThe identity just proved gives $\\Pi=2+S$.  Adding a rational constant or\nmultiplying by a nonzero rational number preserves rationality and\nirrationality.\n\\end{proof}\n\nConcretely, $\\Pi=3.674643966\\ldots$ is irrational\nif and only if $S=1.674643966\\ldots$ is.  Neither irrationality statement is established here.\n\n\\section{The tail recurrence and the exact criteria}\\label{long251:sec:tail}\n\nWe rescale each tail so that its first denominator is again $2$.\nFor a convergent series $\\sum_{i\\ge0}a_i2^{-i-1}$ with $a_i\\in\\Z$,\nput\n\\[\n T_N=2^{N+1}\\sum_{i>N}a_i2^{-i-1}\n    =\\frac{a_{N+1}}2+\\frac{a_{N+2}}4+\\cdots.\n\\]\nThus $T_N$ starts with coefficient $a_{N+1}$, not $a_N$.  Doubling\nand removing that first coefficient gives\n$T_{N+1}=2T_N-a_{N+1}$.  Modulo integers, a step is simply doubling;\nthe detailed values of the integer coefficients disappear.  This is\nwhy the recurrence gives a rationality criterion for arbitrary integer\ncoefficients.\n\nAt $N=0$ we have\n$T_0=2\\sum_{i\\ge0}a_i2^{-i-1}-a_0$.  In particular, for the prime\ngaps $T_0=2S-1=2\\Pi-5$, so $T_0$, $S$ and $\\Pi$ have the same\nrationality status.  We first analyse the recurrence abstractly.\nWe then impose the vanishing boundary condition that singles out\ncomplete tails and apply the criterion to these prime-gap tails.\n\n\\begin{definition}\\label{long251:def:rec}\\compatlabel{def:rec}\nLet $g:\\N\\to\\Z$ and $T:\\N\\to\\Q$ or $T:\\N\\to\\R$.  Say $T$ satisfies the\n\\emph{dyadic tail recurrence} with \\emph{coefficients} $g$ when\n$T_{N+1}=2T_N-g_{N+1}$ for every $N$.  Write $\\sigma_h(N)=T_{N+h}-T_N$ for the\n\\emph{shift} of length $h$ at $N$, and call a real number \\emph{integral}\nwhen it belongs to $\\Z$.\n\\end{definition}\n\nFor a complete-tail example, take $g_n=2$ for odd $n\\ge1$ and $g_n=4$\nfor even $n\\ge1$.  Its complete tail at index $0$ is\n$T_0=(2/2+4/4)/(1-1/4)=8/3$, and the tails alternate between\n$8/3$ and $10/3$.  Thus $\\sigma_1(N)$ is always $2/3$ or $-2/3$,\nwhereas $\\sigma_2(N)=0$ for every $N$.  Nonintegrality at one prescribed\nshift length does not imply irrationality, even for positive even\ncoefficients and genuine tails.\n\nAn integral shift returns to the same fractional part.  To track the\ninteger accumulated in $h$ steps, define $B_{0,N}=0$ and\n$B_{h+1,N}=2B_{h,N}+g_{N+h+1}$,\nso that $B_{h,N}=g_{N+1}2^{\\,h-1}+\\cdots+g_{N+h}$ is the\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L647}{weighted integer sum}\naccumulated in those $h$ steps.\n\n\\begin{theorem}[block identity]\\label{long251:res:block}\\compatlabel{res:block}\nFor every $N$ and $h$,\n\\[\n T_{N+h}=2^{h}T_N-B_{h,N},\n \\qquad\\text{hence}\\qquad\n \\sigma_h(N)=(2^{h}-1)\\,T_N-B_{h,N} .\n\\]\n\\end{theorem}\n\n\\begin{proof}\nThe case $h=0$ follows from $B_{0,N}=0$.  If the first identity holds at\n$h$, then\n\\[\n T_{N+h+1}=2(2^hT_N-B_{h,N})-g_{N+h+1}\n          =2^{h+1}T_N-B_{h+1,N}.\n\\]\nSubtracting $T_N$ gives the second identity.\n\\end{proof}\n\nSubtracting the tail recurrences at $N+h$ and $N$ gives\n\\begin{equation}\\label{long251:eq:shift-step}\n \\sigma_h(N+1)=2\\sigma_h(N)-\\bigl(g_{N+h+1}-g_{N+1}\\bigr),\n\\end{equation}\nwhich is the\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L563}{recurrence for the tail differences}.\nSince $B_{h,N}$ is an integer, Theorem~\\ref{long251:res:block} also gives the\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L802}{integrality criterion}:\n$\\sigma_h(N)$ is integral if and only if $(2^h-1)T_N$ is integral.\nFor fixed $h$ in this general recurrence, $T_N$ therefore determines whether\nthe shift is integral, regardless of the later integer coefficients.\nFor a complete tail, however, changing those coefficients also changes\n$T_N$; the observation does not permit arbitrary changes to prime gaps.\n\nFor rational $T_N=u/v$ in lowest terms, the recurrence gives\n$T_{N+1}=(2u-g_{N+1}v)/v$.  The numerator has greatest common divisor\n$\\gcd(2,v)$ with $v$, because $\\gcd(u,v)=1$.  Thus\n\\begin{equation}\n \\operatorname{den}(T_{N+1})\n =\\frac{\\operatorname{den}(T_N)}{\\gcd(2,\\operatorname{den}(T_N))},\n \\qquad\n \\sigma_h(N)\\in\\Z\n \\iff\n \\operatorname{den}(T_N)\\mid 2^h-1 .\n\\label{long251:eq:den-law}\\compatlabel{eq:shift-denominator}\\compatlabel{res:shiftiff}\n\\end{equation}\nThe power of two in the denominator determines how many initial steps\nmust pass before a return is possible; the remaining odd factor $d$\ndetermines the possible return lengths.  Once that factor is reached,\nEuler's congruence supplies $d\\mid2^{\\ph(d)}-1$.  The least positive\nreturn length is the multiplicative order of $2$ modulo $d$, taken to\nbe $1$ when $d=1$.  Before the power of two is removed there is no return:\nfor example, $T_N=1/2$ gives no integral shift at $N$, since $2^h-1$ is\nalways odd.  This periodicity concerns fractional parts, not full tail\nvalues or coefficients.\n\n\\begin{theorem}[exact rationality classification]\\label{long251:res:escape-irrational}\\compatlabel{res:collapse}\nLet $T:\\N\\to\\R$ satisfy $T_{N+1}=2T_N-g_{N+1}$ with integer coefficients $g$.  The\nfollowing are equivalent:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item $T_0$ is rational;\n\\item $\\sigma_h(N)$ is an integer for some $h\\ge1$ and some $N$;\n\\item for some fixed $h\\ge1$, $\\sigma_h(N)$ is an integer at every\n  sufficiently large $N$.\n\\end{enumerate}\nConsequently $T_0$ is irrational if and only if every positive-length shift is\nnonintegral at every index, equivalently if and only if for every $h\\ge1$ and\nevery cutoff some later $N$ has $\\sigma_h(N)\\notin\\Z$.\n\\end{theorem}\n\n\\begin{proof}\nFor \\textup{(i)}$\\Rightarrow$\\textup{(iii)}, the block identity makes every\n$T_N$ rational.  By \\eqref{long251:eq:den-law}, its denominator is a fixed\nodd integer $d$ for all sufficiently large $N$.  Choose $h=\\ph(d)$;\nEuler's congruence and the same formula show that every subsequent\n$\\sigma_h(N)$ is integral.  The implication\n\\textup{(iii)}$\\Rightarrow$\\textup{(ii)} is immediate.  For\n\\textup{(ii)}$\\Rightarrow$\\textup{(i)}, the block identity gives\n$\\sigma_h(N)=(2^h-1)T_N-B_{h,N}$ with $B_{h,N}$ and $\\sigma_h(N)$ integers and\n$2^h-1\\ne0$, so $T_N$ is rational, and $T_N=2^NT_0-B_{N,0}$ then makes $T_0$\nrational.  Negating the three conditions gives the two irrationality\nformulations.\n\\end{proof}\n\n\\begin{lemma}[the boundary condition for the tail recurrence]\\label{long251:res:true-tail}\nLet $a_1,a_2,\\ldots$ be real numbers with\n$\\sum_{j\\ge1}|a_j|2^{-j}<\\infty$, and let $U_{N+1}=2U_N-a_{N+1}$.\nThen\n\\[\n U_N=\\sum_{j\\ge1}a_{N+j}2^{-j}\\quad\\hbox{for every }N\n \\quad\\Longleftrightarrow\\quad 2^{-N}U_N\\longrightarrow0.\n\\]\n\\end{lemma}\n\\begin{proof}\nIteration gives\n$2^{-N}U_N=U_0-\\sum_{j=1}^{N}a_j2^{-j}$.\nThe limit is zero exactly when $U_0$ equals the full series; subtracting\nits first $N$ terms then gives the claimed tail formula.  Without the boundary condition,\none may add $C2^N$ to every $U_N$ without changing the recurrence.\n\\end{proof}\n\nFor instance, zero coefficients and initial value $U_0=1/12$ give\n$\\tfrac1{12},\\tfrac16,\\tfrac13,\\tfrac23,\\tfrac43,\\ldots$, with\n$U_{N+h}-U_N=(2^h-1)2^N/12$.  Shifts of length $2$ become integral\nat index $2$; shifts of length $1$ never do.  These are not complete\ntails of the zero sequence, since $2^{-N}U_N=1/12$ does not tend to\nzero.  The denominator calculation remains valid, but the boundary\ncondition excludes this extra homogeneous term.\n\n\\paragraph{The actual prime-gap tails.}\nPut\n\\[\n T_N=\\sum_{j\\ge1}\\frac{g_{N+j}}{2^{\\,j}},\n\\]\nwhich converges by Theorem~\\ref{long251:res:infinite}.  Reindexing identifies\nthis sum with the rescaled complete tail of $S$, as in the\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/RealPrimeGapTail.lean\\#L31}{shifted-gap\nseries identity}.  Splitting off the first term gives\n$T_{N+1}=2T_N-g_{N+1}$ without a rationality hypothesis, as formalised in the\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/RealPrimeGapTail.lean\\#L48}{real\ntail recurrence}.  Since $T_0=2S-1=2.349287932\\ldots$, the numbers $\\Pi$, $S$\nand $T_0$ have the same rationality status, and\nTheorem~\\ref{long251:res:escape-irrational} specialises to the actual series:\nirrationality of $\\Pi$ is equivalent to nonintegrality arbitrarily far out\nfor each positive shift of $T$.  The formal statement is the\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/RealPrimeGapTail.lean\\#L73}{irrationality equivalence for the actual prime tails}.  Thus the passage from the series to its complete tails\nis proved.  What remains is to establish the required nonintegrality for\nthese prime-gap tails.\n\n\\subsection{Pairs of congruent indices and least-common-multiple shifts}\\label{long251:sec:freepair}\\compatlabel{sec:diagonal-collapse}\n\nInstead of prescribing the offset $M-N$, we may prescribe a positive\nmodulus $t$ and allow any pair with $M\\equiv N\\pmod t$.  This gives\nmore freedom in finding a witness, but the test must still work for\nevery $t$ and beyond every cutoff.\n\n\\begin{theorem}[pairs of congruent indices]\\label{long251:res:freepair}\n$S$ is irrational if and only if for every $t\\ge1$ and every $N_0$ there are\n$N,M\\ge N_0$ with $M\\equiv N\\pmod t$ and $T_M-T_N\\notin\\Z$.\n\\end{theorem}\n\n\\begin{proof}\nIf $S$ is irrational, take $M=N+t$ for the $N$ supplied by\nTheorem~\\ref{long251:res:escape-irrational} at shift length $t$.  Conversely, suppose\n$S$ is rational.  By \\eqref{long251:eq:den-law} the orbit reaches an index $N_0$ beyond\nwhich the reduced denominator is a fixed odd $d$; let $t$ be the\nmultiplicative order of $2$ modulo $d$, taking $t=1$ when $d=1$.  For $N,M\\ge N_0$ with $M\\equiv N$\nmodulo $t$, the difference $T_M-T_N$ is then an integer, contradicting the\nstated property at this $t$ and this cutoff.\n\\end{proof}\n\nThe obstruction under rationality is therefore one particular modulus:\nthe order of $2$ modulo the eventual odd denominator.  A successful\npair for some other modulus does not exclude it.\n\nEqual tail values do not help this criterion, since their difference is\nzero.  Section~\\ref{long251:sec:xr-compression} shows that rationality and\nthe prime number theorem would in fact force many equal tail values.\nIt also explains why, beyond the rational denominator cutoff, an interval\ncondition excluding integers already contradicts rationality at indices\ncongruent modulo the corresponding multiplicative order.  These are\nordinary deductions, not additional formalised statements.\n\nAlternatively, one can fix the pairs in advance.  Let $L_0=1$ and\n$L_j=\\lcm(1,\\ldots,j)$.  These numbers eventually exceed any starting\ncutoff and are eventually divisible by any proposed period.\n\n\\begin{theorem}[\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/OrderLatticeDiagonal.lean\\#L153}{criterion using least common multiples}]\\label{long251:res:lcmdiagonal}\\compatlabel{res:lcmdiagonal}\nLet $g:\\N\\to\\Z$ and $T:\\N\\to\\R$ satisfy $T_{N+1}=2T_N-g_{N+1}$.  Then\n$T_0$ is irrational if and only if\n$T_{2L_j}-T_{L_j}\\notin\\Z$ for every $j\\ge0$.\n\\end{theorem}\n\n\\begin{proof}\nTheorem~\\ref{long251:res:escape-irrational} gives the forward implication.  Conversely,\nif $T_0$ is rational then some positive shift length $h$ is integral at every\nbasepoint beyond an index $N_0$.  Choose $j$ so large that $N_0\\le L_j$ and\n$h\\mid L_j$.  The telescoping identity\n$\\sigma_{a+b}(N)=\\sigma_a(N)+\\sigma_b(N+a)$ shows by induction that every\npositive multiple of $h$ is integral at every such basepoint, so\n$T_{2L_j}-T_{L_j}$ is an integer.\n\\end{proof}\n\nOne could use factorials instead of least common multiples, since they\nhave the same eventual divisibility property.  The least common multiples\nare no larger at each index.  Theorems~\\ref{long251:res:freepair}\nand~\\ref{long251:res:lcmdiagonal} change which tail differences one must\ntest, but remain exact equivalences for integer-coefficient recurrences.\nNeither proves the needed nonintegrality for the actual prime gaps.\n\n\\section{A local certificate, and one actual pair}\\label{long251:sec:local-certificate}\n\nTo rule out an integral shift, we can bound two consecutive complete\ndifferences in $(-1,1)$ and check that the corresponding gaps are unequal.\nIndeed, simultaneous integrality would force both differences to vanish,\nand their recurrence would then give $g_{N+h+1}=g_{N+1}$.\nFor even gaps we can state these conditions as a single signed interval\nfor the first difference.\n\n\\begin{theorem}[two consecutive small tail differences]\\label{long251:res:smallpair}\\compatlabel{res:smallpair}\nLet $T$ satisfy the dyadic tail recurrence with integer coefficients $g$, and fix\n$h$ and $N$.  If\n\\[\n -1<\\sigma_h(N)<1,\\qquad -1<\\sigma_h(N+1)<1,\n \\qquad g_{N+h+1}\\ne g_{N+1},\n\\]\nthen $\\sigma_h(N)$ and $\\sigma_h(N+1)$ cannot both be integers.  Consequently,\nif such a pair occurs beyond every threshold, the $h$-shift is not eventually\nintegral.\n\\end{theorem}\n\n\\begin{proof}\nAn integer strictly between $-1$ and $1$ is zero.  If both shifts were\nintegers, both would vanish, and \\eqref{long251:eq:shift-step} would give\n$g_{N+h+1}=g_{N+1}$.  The final assertion chooses one such adjacent pair\nafter the alleged onset of integrality.\n\\end{proof}\n\n\\begin{corollary}[real form and the sufficient condition]\\label{long251:res:smallpair-real}\\compatlabel{res:smallpair-real}\nThe same statement holds for a real orbit, with the same proof.  If for every\n$h\\ge1$ and every cutoff some later $N$ satisfies the three displayed\nconditions for the actual prime gaps, then $\\Pi$ is irrational.\n\\end{corollary}\n\nThe unequal-gap condition alone is available: at $h=1$, $N=2$ it reads\n$g_4=2\\ne4=g_3$, and Proposition~\\ref{long251:res:gap-nonperiodic}\ngives such inequalities at arbitrarily large indices for each fixed $h$.\nThe missing assertion is that both small-tail inequalities and the\nunequal-gap condition hold at the same arbitrarily late indices.\nSeparate infinite sets of witnesses for the three requirements need not\nintersect.\n\n\\begin{proposition}[prime gaps do not become periodic]\\label{long251:res:gap-nonperiodic}\\label{res:gap-nonperiodic}\nFor every positive $h$, the actual consecutive-prime-gap sequence is not\neventually periodic with period $h$.\n\\end{proposition}\n\n\\begin{proof}\nFor $m\\ge2$ the integers $(m+1)!+2,\\ldots,(m+1)!+m+1$ are composite, so the\ngaps are unbounded.  An eventually periodic sequence of natural numbers has\nfinite range after its preperiod and a bounded initial segment, hence is\nbounded.\n\\end{proof}\n\n\\noindent Far stronger lower bounds for large gaps are\nknown~\\cite[Theorem~1]{fgkmt2018}; the\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L57}{unboundedness\nof the gaps} is all that is needed.\n\nWith even gap differences, the three conditions in\nTheorem~\\ref{long251:res:smallpair} reduce to two possible signed intervals,\neach accompanied by a prescribed gap difference.  For fixed $h$ write\n$D_N=\\sigma_h(N)$ and $\\delta_N=g_{N+h+1}-g_{N+1}$, so that\n$D_{N+1}=2D_N-\\delta_N$.\n\n\\begin{theorem}[\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/AffineShiftEscape.lean\\#L113}{an equivalent signed interval test}]\\label{long251:res:signedwindow}\nAssume $\\delta_N$ is even.  The conjunction\n$-1<D_N<1$, $-1<D_{N+1}<1$, $\\delta_N\\ne0$ is equivalent to\n\\[\n \\bigl(\\delta_N=2\\ \\hbox{ and }\\tfrac12<D_N<1\\bigr)\n \\quad\\hbox{or}\\quad\n \\bigl(\\delta_N=-2\\ \\hbox{ and }-1<D_N<-\\tfrac12\\bigr).\n\\]\n\\end{theorem}\n\n\\begin{proof}\nSince $\\delta_N=2D_N-D_{N+1}$, the bounds $|D_N|<1$ and\n$|D_{N+1}|<1$ give $-3<\\delta_N<3$.  A nonzero even integer in this\ninterval is $2$ or $-2$.  If $\\delta_N=2$, the inequality\n$-1<2D_N-2<1$, together with $|D_N|<1$, is equivalent to\n$1/2<D_N<1$.  Negating both differences gives the case $\\delta_N=-2$.\nEach substitution is reversible.\n\\end{proof}\n\nFor the actual gaps, both indices $N+1$ and $N+h+1$ are positive, so\n$\\delta_N$ is even for every $N\\ge0$.\nTheorem~\\ref{long251:res:signedwindow} therefore applies to the actual\ntails.  Its proof uses order, the recurrence and the evenness of\n$\\delta_N$; it does not require rational tail values.  This is the real\nversion used in Theorem~\\ref{long251:res:sparse}.\n\n\\subsection{An explicit remainder and a certified actual pair}\n\nHere $N$ locates the two tails, $h$ separates them, and $L$ is the\nnumber of terms retained from each.  A finite block supplies the centre\nof an enclosure, not the full difference.  The polynomial prime bound\nsupplies its radius; clearing denominators then makes each test an\ninteger comparison.\n\n\\begin{proposition}[explicit remainder]\\label{long251:res:explicit-remainder}\nFor integers $h,N\\ge0$ and $L\\ge1$ put\n\\[\n F_{h,N,L}=\\sum_{j=1}^{L}\\frac{g_{N+h+j}-g_{N+j}}{2^{\\,j}},\\qquad\n P(x)=x^4+8x^3+36x^2+104x+150,\n\\]\n\\[\n E_{h,N,L}=\\frac{1250}{2^{L}}\\bigl(P(N+h+L+2)+P(N+L+2)\\bigr).\n\\]\nThen $\\bigl|\\sigma_h(N)-F_{h,N,L}\\bigr|\\le E_{h,N,L}$.  In particular\n$|F_{h,N,L}|+E_{h,N,L}<1$ certifies $|\\sigma_h(N)|<1$, and\n$\\operatorname{dist}(F_{h,N,L},\\Z)>E_{h,N,L}$ certifies\n$\\sigma_h(N)\\notin\\Z$.\n\\end{proposition}\n\n\\begin{proof}\nSplitting the complete tails after $L$ terms gives\n\\[\n \\sigma_h(N)=F_{h,N,L}+2^{-L}\\sigma_h(N+L).\n\\]\nThe last term is the part not determined by the finite gap blocks.\nThe checked bound $p_n\\le1250(n+1)^4$ gives $0\\le g_m\\le p_{m+1}\\le1250(m+2)^4$,\nso its absolute value is at most\n\\[\n 1250\\sum_{j>L}\\frac{(N+h+j+2)^4+(N+j+2)^4}{2^{\\,j}} .\n\\]\nThe polynomial identity $2P(x)=(x+1)^4+P(x+1)$ telescopes to\n$\\sum_{k=1}^{J}(m+k)^42^{-k}=P(m)-P(m+J)2^{-J}$, whose last term tends to zero.\nApply it with $m=N+h+L+2$ and $m=N+L+2$ after writing $j=L+k$.  The two tests\nfollow from the triangle inequality and the definition of distance to $\\Z$.\n\\end{proof}\n\n\\begin{proposition}[a certified adjacent pair]\\label{long251:res:finite-smallpair}\nFor the actual prime gaps, $h=1$ and $N=2$ satisfy the three hypotheses of\nTheorem~\\ref{long251:res:smallpair}: both $\\sigma_1(2)$ and $\\sigma_1(3)$ lie in\n$(-1,1)$ and are nonintegral, and $g_4=2\\ne4=g_3$.\n\\end{proposition}\n\n\\begin{proof}\nTake $L=40$ and $Q=2^{40}=1099511627776$.  Exact integer arithmetic over the\nfirst $46$ primes gives\n\\[\n\\begin{array}{c|r|r}\n N&Q\\,F_{1,N,40}&Q\\,E_{1,N,40}\\\\\\hline\n 2&-662838684750&11764181250\\\\\n 3& 873345886050&12805761250\n\\end{array}\n\\]\nIn each row $|QF|+QE<Q$, which puts the whole certified interval inside\n$(-1,1)$, and $QE$ is smaller than the distance from $QF$ to the nearest\nmultiple of $Q$, which excludes every integer.  The margins are $424908761776$\nand $213359980476$ respectively.  These are strict integer comparisons rather\nthan inferences from rounded numerical tails.\nAppendix~\\ref{long251:app:certificates} replays them.\n\\end{proof}\n\nThis certifies one actual adjacent pair, not a recurrence theorem for\nsuch pairs.  Irrationality still requires a witness for every positive\n$h$ beyond every cutoff; the successful indices may depend on $h$.\n\n\\begin{proposition}[certifying two small differences from one enclosure]\\label{long251:res:one-tail-certificate}\nLet $D_{N+1}=2D_N-\\delta_N$ with real $D_N$, and suppose\n$\\delta_N=2s$ for $s\\in\\{-1,1\\}$.  If integers $A,B,Q$ satisfy\n$Q>0$, $B\\ge0$, $|QD_N-A|\\le B$, and\n\\[\n 2sA-Q>2B,\\qquad Q-sA>B,\n\\]\nthen $1/2<sD_N<1$, $|D_{N+1}|<1$, and both $D_N,D_{N+1}$ are\nnonintegral.  In particular, the two small-difference conditions are certified from\njust one tail enclosure.\n\\end{proposition}\n\\begin{proof}\nSince $|s|=1$, the enclosure gives\n\\[\n \\frac{sA-B}{Q}\\le sD_N\\le\\frac{sA+B}{Q}.\n\\]\nThe first strict inequality says that the lower endpoint exceeds $1/2$;\nthe second says that the upper endpoint is less than $1$.\nThus $sD_N\\in(1/2,1)$.  The recurrence and $\\delta_N=2s$ now give\n$sD_{N+1}=2sD_N-2\\in(-1,0)$.  Both intervals exclude integers, and\nthe second also gives $|D_{N+1}|<1$.\n\\end{proof}\nFor $h=1,N=2$, the exact data already used above give\n$Q=2^{40}$, $A=-662838684750$, $B=11764181250$ and $s=-1$.\nThe two positive integer margins are respectively\n$202637379224$ and $424908761776$.\nProposition~\\ref{long251:res:finite-smallpair} encloses each difference\nseparately.  Here the first enclosure and the exact gap difference give\nthe second interval through the recurrence.  The second enclosure is\ntherefore an independent cross-check, not a necessary input.\n\n\\section{Two lower bounds on a possible rational denominator}\\label{long251:sec:denominator-floor}\n\nHere the enclosure is for $\\Pi$ itself, rather than a tail difference.\nA finite prime sum gives the lower endpoint; the polynomial prime bound\ngives the upper endpoint.  Two fractions of determinant one bracket this\nentire interval, so every rational in it has denominator at least the sum\nof their denominators.  This last step is an integer identity, independent\nof how the fractions were found.  Appendix~\\ref{long251:app:certificates}\nprints the smaller certificate; the linked Lean development checks the\ninteger comparisons for both.  The bounds exclude finite denominator\nranges, not rationality.\n\n\\begin{theorem}[a denominator bound checked by exact integer comparisons]\\label{long251:res:denominatorfloor}\nLet $a\\in\\Z$ and let $b$ be a positive integer.  If $\\Pi=a/b$ then $b\\ge2^{589}>10^{177}$, and\nthe same floor holds for every rational equal to $S$.\n\\end{theorem}\n\n\\begin{proof}\nPut $c=1229$, the number of primes below $10^4$, and\n$A=\\sum_{i=0}^{c-1}p_i2^{\\,c-i-1}$, so that $A/2^{c}$ is the $c$th partial sum\nof $\\Pi$.  All terms are positive, so $A/2^{c}\\le\\Pi$.  For the upper bound,\n$p_{c+j}\\le1250(c+j+1)^4$.  Since $c\\ge9$, each successive ratio of\n$(c+1+j)^4$ is at most $(11/10)^4<3/2$.  Hence\n$(c+1+j)^4\\le(c+1)^4(3/2)^j$ for all $j\\ge0$.\nThe omitted dyadic terms are therefore bounded by a geometric series\nwith ratio $3/4$, whose sum gives\n\\[\n \\frac{A}{2^{c}}\\;\\le\\;\\Pi\\;\\le\\;\\frac{2A+5000(c+1)^4}{2^{\\,c+1}} .\n\\]\nThe four positive integers $u,v,u',v'$ printed in\nAppendix~\\ref{long251:app:certificates} satisfy\n\\[\n u'v-uv'=1,\\qquad\n \\frac uv<\\frac{A}{2^{c}},\\qquad\n \\frac{2A+5000(c+1)^4}{2^{\\,c+1}}<\\frac{u'}{v'},\\qquad\n v+v'\\ge2^{589},\n\\]\nall four being integer comparisons after clearing the positive denominators.\nIf $\\Pi=a/b$, the strict lower and upper bounds make $av-bu$ and\n$bu'-av'$ positive integers.  Each is therefore at least $1$.\nThe determinant identity expresses $b$ as their positive integer\ncombination of $v'$ and $v$:\n\\[\n b=b(u'v-uv')=(av-bu)v'+(bu'-av')v\\ge v+v' \\ge 2^{589}.\n\\]\nThe only properties needed of the bracketing fractions are their order,\npositive denominators and determinant one.  Their decimal expansions\nand the way they were found play no part in this denominator bound.\nSince $589\\log_{10}2>177$, the floor exceeds $10^{177}$.  If $S=a/b$, apply the\nsame argument to $\\Pi=(a+2b)/b$.\n\\end{proof}\n\n\\noindent The Lean kernel evaluates the four inequalities itself, on the\nsame integers, after recomputing the primes below $10^4$ by trial\ndivision; no compiled code is trusted.  Its analytic input is the same\nenclosure, derived from the\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L360}{polynomial\nbound} $p_n\\le1250(n+1)^4$.  The four integers $u,v,u',v'$ have $177$\nand $178$ digits.\n\n\\begin{theorem}[certified continued-fraction exclusion]\\label{long251:res:cfexclusion}\nEvery rational equal to $\\Pi$, and hence every rational equal to $S$, has\nreduced denominator $q\\ge2^{39997}$, and therefore $q>10^{12040}$.\n\\end{theorem}\n\nThe same determinant argument gives the stronger floor $2^{40062}$\nbelow.  The smaller constants in the statement and the different\ncontinued-fraction pairs are accounted for after the proof.\n\n\\begin{proof}\nPut $c=80200$, the number of primes below $1\\,023\\,068$, and\n$A=\\sum_{i=0}^{c-1}p_i2^{\\,c-i-1}$.  The estimate in the proof of\nTheorem~\\ref{long251:res:denominatorfloor} holds for every $c\\ge9$ and gives\n\\[\n \\frac{A}{2^{c}}\\;\\le\\;\\Pi\\;\\le\\;\\frac{2A+5000(c+1)^4}{2^{\\,c+1}} .\n\\]\nFour positive integers $u,v,u',v'$, described after this proof, satisfy\n\\[\n u'v-uv'=1,\\qquad\n \\frac uv<\\frac{A}{2^{c}},\\qquad\n \\frac{2A+5000(c+1)^4}{2^{\\,c+1}}<\\frac{u'}{v'},\\qquad\n v+v'\\ge2^{40062},\n\\]\nall four being integer comparisons after clearing the positive denominators.\nIf $\\Pi=a/b$ with $a\\in\\Z$ and $b$ a positive integer, the determinant\ncalculation in the proof of Theorem~\\ref{long251:res:denominatorfloor} gives\n\\[\n b\\;\\ge\\;v+v'\\;\\ge\\;2^{40062}\\;>\\;10^{12059}.\n\\]\nThe last inequality holds because $10^{12059}=(10^{59})^{204}\\,10^{23}$ and\n$2^{40062}=(2^{196})^{204}\\,2^{78}$, with $10^{59}<2^{196}$ and\n$10^{23}<2^{78}$.  The reduced denominator $q$ is one such $b$, so\n$q\\ge2^{40062}>2^{39997}$ and $q>10^{12059}>10^{12040}$.  If $S=a/b$, apply\nthe same argument to $\\Pi=(a+2b)/b$.\n\\end{proof}\n\n\\noindent The integers $u,v,u',v'$ come from the continued fractions of the\ntwo endpoints $A/2^{c}$ and $(2A+5000(c+1)^4)/2^{\\,c+1}$, which agree in their\nfirst $23407$ partial quotients.  The fraction $u/v$ is the convergent formed by\nthose partial quotients, and $u'/v'$ is the mediant of $u/v$ and the convergent\nbefore it.  Only the four comparisons enter the proof.  The numerators $u,u'$\nhave $12061$ decimal digits and the denominators $v,v'$ have $12060$.\n\nThe Lean kernel checks every step of this proof and evaluates every\ncomputation itself; no compiled code is trusted.  It recomputes $c$ and\n$A$ from the definition of primality, scanning the integers below\n$1\\,023\\,068$ in $250$ consecutive blocks and checking the prime count and\npartial numerator at the end of each block.  It decides primality by trial\ndivision below $1012$.  On $[1012,1013^2)$, which contains the rest of the\nscan because $1013^2=1\\,026\\,169$, it uses the\n\\href{https://github.com/wcook04/plectis-erdos/blob/e6c2d8f77ac24753c5216a49f4daf7f7388b309f/verification/Erdos251LargeCertificate/ErdosProblems/Erdos251/GcdPrimality.lean\\#L72}{fact}\nthat such an integer is prime exactly when it is coprime to the product of the\nprimes below $1012$, and it recomputes that product by trial division.  Lean\nproves that the two primality tests agree at every natural number.  The kernel\nthen decides the four comparisons, together with $10^{59}<2^{196}$ and\n$10^{23}<2^{78}$; the enclosure and the determinant step are proved in\ngeneral form.  Lean proves the\n\\href{https://github.com/wcook04/plectis-erdos/blob/e6c2d8f77ac24753c5216a49f4daf7f7388b309f/verification/Erdos251LargeCertificate/ErdosProblems/Erdos251/PaperLargeStreamingV5.lean\\#L76}{bound $b\\ge2^{40062}$ and $b>10^{12059}$ for both series}\nand derives the theorem as stated from it, using only Lean's three\nstandard axioms.\n\nAn independent\n\\href{https://github.com/wcook04/plectis-erdos/blob/beaf0866976c908539fba0a96075374dd658f44a/research/experiments/erdos251/exact_replay.py}{program}\noutside Lean recomputes $A$ by a prime sieve and uses exact integer\narithmetic.  Its \\href{https://github.com/wcook04/plectis-erdos/blob/beaf0866976c908539fba0a96075374dd658f44a/research/experiments/erdos251/receipts/251-exact-replay.json}{recorded output}\nhas the same $23407$ common partial quotients.  It brackets the interval\nwith the last two consecutive convergents, whose denominator sum has\nbit length $40062$, and checks the advertised bound $2^{39997}>10^{12040}$.\nThis pair differs from the convergent--mediant pair in the Lean certificate\nabove, whose denominator sum has bit length $40063$ and is at least\n$2^{40062}$.  The two bit lengths refer to different valid pairs.\n\nThe constants $2^{39997}$ and $10^{12040}$ in the statement come from a\nseparate computation with a bracket of width $2^{-80000}$, which used the best-approximation property of\nconvergents \\cite[\\S6, Theorems~16 and~17]{khinchin1964}; Short gives an\naccessible statement and proof~\\cite[Theorem~1.1]{short2009}.  Its\n\\href{https://github.com/wcook04/plectis-erdos/blob/dd6cc708f650f061de06594eba6ba43878f4de31/research/experiments/erdos251/receipts/certified-cf.json}{recorded output}\nlists the generating program and its digest and a last convergent whose\ndenominator has bit length $39998$, hence is at least $2^{39997}$; the\nprogram also checked its arithmetic against the known expansions of $e$\nand $\\pi$.  That bit length describes one convergent only; the bound for\nevery rational in the enclosure comes from the certified sum $v+v'$\nabove, and $2^{39997}>10^{12040}$ is a separate integer comparison.\n\nThese computations exclude finite denominator ranges.  Irrationality\nwould require unbounded exclusions, not just a larger finite bound.\nEven extending the prefix need not improve the bound at every step.\nNo irrationality-exponent estimate is asserted.\n\n\\section{Coefficient conditions and counterexamples}\\label{long251:sec:obstructions}\n\\papersectiontarget{counterexamples}\n\nThe examples test different proposed substitutes for information about\nthe actual tails.  Bounded changes preserve a fixed modulus and transfer\npolynomial nonconcentration through finitely many translates.  Sparse\nchanges preserve the larger collection of properties in the main theorem\nbecause most sampled blocks are untouched.  Later examples prescribe\ninteger tails first and derive coefficients with recurring values or\ngrowth conditions.  These mechanisms have different conclusions; none\nconstructs consecutive primes.\n\nWe also show that the actual small-pair witnesses, if they occur\narbitrarily late, must have density zero.  This does not rule them out:\ninfinitely many witnesses suffice, and positive density is not required.\n\n\\subsection{Bounded residue-preserving perturbations}\n\nFor one fixed positive modulus $M$, we may add either $0$ or $M$\nafter a prescribed prefix.  The weighted choices form a binary expansion,\nso we can choose a rational sum while preserving every coefficient\nmodulo $M$.  This construction permits changes at every index after the\nprefix; it imposes neither density-zero support nor eventual preservation\nof all moduli.\n\n\\begin{theorem}[bounded perturbations preserving congruences]\\label{long251:res:boundedperturbation}\nLet $a_n$ be natural numbers with $\\sum_{n\\ge0}a_n2^{-(n+1)}$ convergent.  For\nevery integer $M\\ge1$ and every cutoff $K$ there are digits\n$\\varepsilon_n\\in\\{0,1\\}$, zero for $n<K$, such that\n$\\sum_{n\\ge0}(a_n+M\\varepsilon_n)2^{-(n+1)}$ is rational.\n\\end{theorem}\n\n\\begin{proof}\nWrite $A=\\sum_{n\\ge0}a_n2^{-(n+1)}$ and choose a rational\n$r\\in\\bigl(A,A+M2^{-K}\\bigr)$.  A binary expansion of $(r-A)/M$ has the form\n$\\sum_{n\\ge K}\\varepsilon_n2^{-(n+1)}$ with $\\varepsilon_n\\in\\{0,1\\}$; set\n$\\varepsilon_n=0$ for $n<K$.  The perturbed series converges and equals $r$.\n\\end{proof}\n\n\\noindent Hence a property shared by every allowed perturbation cannot by\nitself force irrationality, since one such perturbation has a rational sum;\nthis consequence is not stated separately in Lean.  Lean also checks the construction at the actual consecutive prime gaps,\nfor every $M\\ge1$ and every $K$: the\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/BoundedPerturbationCountermodel.lean\\#L208}{perturbed\nprime-gap series is rational}, and each perturbed gap lies in\n$[g_n,g_n+M]$ and is\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/BoundedPerturbationCountermodel.lean\\#L234}{congruent\nto $g_n$ modulo $M$}.  For each fixed $k\\ge1$, the sum of $k$ consecutive gaps\nincreases by at most $kM$.  Thus infinitely many bounded gaps and bounded\nclusters of each fixed size survive, but their original numerical bounds\nneed not.  Nonnegative corrections also preserve lower bounds for large\ngaps.  Taking $M=2q$ preserves parity\nand the cumulative positions modulo a prescribed $q$, hence any limiting\nresidue frequencies of those positions.  These conclusions concern the\nsize and residue information in the results discussed in\nSection~\\ref{long251:sec:open}; they establish neither primality of the\nnew positions nor preservation of every quantitative assertion of those\ntheorems.\n\n\\subsection{Algebraic nonconcentration survives the rationalising perturbation}\n\nSchlage-Puchta's lemma states that\n$F(g_n,\\ldots,g_{n+k})\\ne0$ for almost all $n$, for every fixed nonzero\n$F\\in\\Z[x_0,\\ldots,x_k]$ \\cite[Lemma~4, pp.~5--6]{schlagepuchta2011}.\nWe say that an integer sequence has \\emph{fixed-block nonconcentration}\nif it satisfies this conclusion for every $k\\ge0$ and every such $F$.\nA bounded integer perturbation preserves the property: there are only\nfinitely many translated polynomials to consider for each fixed block.\nNo randomness or independence assumption is required.\n\n\nThis nonconcentration condition excludes every bounded integer sequence:\nif its values lie in a finite set $E$, the nonzero polynomial\n$\\prod_{c\\in E}(x_0-c)$ vanishes at every index.  It also excludes a\nsequence satisfying a fixed nonzero polynomial relation on every short\nblock, such as $a_n=n$, for which $x_1-x_0-1$ vanishes.  The density-zero exceptional set depends on the fixed polynomial.\nThere cannot be one such set for all polynomials: $x_0-a_n$ vanishes on\nthe block beginning at $n$, so a common exceptional set would contain\nevery index.  No uniform estimate for growing families of polynomials or\nblock lengths is asserted.\n\nIn the next proposition, $N,H$ are nonnegative integers, $h$ is an\ninteger, and all counts are over integer indices and integer triples.\n\n\\begin{proposition}[finite counting for shifted gap differences]\\label{long251:res:shiftedcount}\nWrite $p_n$ for the primes indexed from $p_0=2$ and $g_n=p_{n+1}-p_n$.\nFor $h\\ge2$ and $r\\in\\Z$, let $M_{h,r}(N)$ count $n<N$ with $g_{n+h}-g_n=r$.\nLet $Q_{N,H,r}$ count triples $(x,d,s)$ with $x<p_N$,\n$0<d<s\\le H$, $d+r>0$, and all four integers\n$x,x+d,x+s,x+s+d+r$ prime. Then, for every $N,H\\ge0$,\n\\[\n (H+1)M_{h,r}(N)\\le\n (h+1)p_{N+h+1}+(H+1)Q_{N,H,r}.\n\\]\n\\end{proposition}\n\n\\begin{proof}\nFor a counted index $n$, put $x=p_n$, $d=g_n$ and\n$s=p_{n+h}-p_n$.  If the total span $p_{n+h+1}-p_n$ is at most $H$,\nthen $0<d<s\\le H$, $d+r=g_{n+h}>0$, and the four primes are precisely\n$x,x+d,x+s,x+s+d+r$.  The count $Q_{N,H,r}$ need not require these\nprimes to be consecutive: it is an upper bound, and $x=p_n$ determines\n$n$, making the map injective.  For the remaining indices the integer\nspan is at least $H+1$.\nEach gap appears in at most $h+1$ of the spans, so their total is at most\n$(h+1)\\sum_{i<N+h}g_i<(h+1)p_{N+h+1}$.  Multiply the resulting bound\non the exceptional count by $H+1$ and add the small-span contribution.\n\\end{proof}\n\nIncreasing $H$ reduces the bound for long spans but enlarges the\nprime-pattern count.  To deduce zero density from this finite inequality,\nfor each $\\varepsilon>0$ and all large $N$ one needs a single choice of\n$H$ making the entire right side less than $\\varepsilon N(H+1)$.  For $h=1$, the same definitions give $s=d$, so $x+d=x+s$.\nThe strict condition $d<s$ in the four-prime count fails; that case\nrequires a three-prime count instead.\nZero density itself is already known for every fixed $h\\ge1$ and\n$r\\in\\Z$: apply Schlage-Puchta's lemma above to the nonzero polynomial\n$x_h-x_0-r$.  Thus the finite reduction should not be read as leaving\nthat upper bound open.  Neither upper bound supplies the late occurrences\nor weighted-tail separation required for irrationality.\n\nFor the perturbation claim, first suppose that $a$ has fixed-block\nnonconcentration and every correction is either $0$ or a fixed integer\n$M\\ge1$.  For a fixed $r\\in\\Z$, an equality $b_{n+1}-b_n=r$ then requires\n$a_{n+1}-a_n\\in\\{r-M,r,r+M\\}$.  Each of these three equalities holds on\na set of density zero.  The general argument uses the same finite-union\nstep for translated polynomials.\n\n\\begin{theorem}[nonconcentration is perturbation-stable]\\label{long251:res:nonconcentration}\nLet $a:\\N\\to\\Z$ have fixed-block nonconcentration, let $E\\subset\\Z$ be finite,\nand let $b_n=a_n+e_n$ with $e_n\\in E$ for every $n$.  Then $b$ has fixed-block\nnonconcentration.\n\\end{theorem}\n\n\\begin{proof}\nFix $k\\ge0$ and a nonzero $F\\in\\Z[x_0,\\ldots,x_k]$.  For a tuple\n$\\mathbf e=(e^{(0)},\\ldots,e^{(k)})\\in E^{k+1}$ put\n$F_{\\mathbf e}(x_0,\\ldots,x_k)=F(x_0+e^{(0)},\\ldots,x_k+e^{(k)})$.  The\nsubstitution $x_i\\mapsto x_i+e^{(i)}$ is a ring automorphism of\n$\\Z[x_0,\\ldots,x_k]$ with inverse $x_i\\mapsto x_i-e^{(i)}$, so\n$F_{\\mathbf e}\\ne0$.  If $F(b_n,\\ldots,b_{n+k})=0$ then\n$F_{\\mathbf e}(a_n,\\ldots,a_{n+k})=0$ for the tuple\n$\\mathbf e=(e_n,\\ldots,e_{n+k})$, whence\n\\[\n \\{\\,n:F(b_n,\\ldots,b_{n+k})=0\\,\\}\n \\;\\subseteq\\!\\!\n \\bigcup_{\\mathbf e\\in E^{k+1}}\\!\\!\n \\{\\,n:F_{\\mathbf e}(a_n,\\ldots,a_{n+k})=0\\,\\}.\n\\]\nEach set on the right has density zero by hypothesis, and there are\nonly finitely many such sets because $E^{k+1}$ is finite.  Their union\ntherefore has density zero.\n\\end{proof}\n\nFiniteness of the correction set is essential to this argument; no\nindependence between the corrections and the sequence is needed.\nWithout a restriction on the corrections, the choice $e_n=-a_n$ would\ngive $b_n=0$, which fails nonconcentration even for the polynomial $x_0$.\n\n\\begin{corollary}[nonconcentration does not force irrationality]\\label{long251:res:nonconc-primes}\nFix $M\\ge1$ and $K\\ge0$, and let $b$ be the perturbed sequence supplied by\nTheorem~\\ref{long251:res:boundedperturbation} at the actual prime gaps.  Then\n$\\sum_{n\\ge0}b_n2^{-(n+1)}$ is rational, $b_n=g_n$ for $n<K$,\n$b_n-g_n\\in\\{0,M\\}$ and $b_n\\equiv g_n\\pmod M$ for every $n$, $b$ has\nfixed-block nonconcentration, and the cumulative sequence\n$P_n=2+\\sum_{i<n}b_i$ satisfies $p_n\\le P_n\\le p_n+Mn$ and hence\n$P_n\\sim n\\log n$.\n\\end{corollary}\n\\evidenceremark{The Lean proof assumes Schlage-Puchta's Lemma~4\n\\cite{schlagepuchta2011} and the prime number theorem.}\n\n\n\\begin{proof}\nTheorem~\\ref{long251:res:boundedperturbation} gives the rational sum,\nprefix and congruence assertions.  Summing $0\\le b_i-g_i\\le M$ and using\n$\\sum_{i<n}g_i=p_n-2$ gives $p_n\\le P_n\\le p_n+Mn$.\nNow $p_n\\sim n\\log n$~\\cite[Chapter~6]{mv2007} implies $P_n\\sim n\\log n$.\nThe perturbation takes values in\nthe fixed two-element set $E=\\{0,M\\}$, so\nTheorem~\\ref{long251:res:nonconcentration} applies to the actual gaps, which\nhave fixed-block nonconcentration by Schlage-Puchta's\nlemma~\\cite[Lemma~4]{schlagepuchta2011}.\n\\end{proof}\n\nAll the corollary's properties hold for one rational-sum sequence; the\npositions $P_n$ need not be prime.  Taking $M$ even also preserves eventual\nevenness.  In particular $M=2q$ preserves parity and modulus $q$, with\n$b_n\\le g_n+2q$.  Nonconcentration remains a fixed-polynomial, fixed-block\nassertion, not a uniform estimate for growing families.\n\n\\begin{proposition}[nonconcentration under sparse changes]\\label{long251:res:sparse-nonconcentration}\nLet $a,b:\\N\\to\\Z$ agree off a set $S$ of ordinary density zero.\nIf $a$ has fixed-block nonconcentration, then so does $b$.\nNo boundedness assumption on $a-b$ is needed.\n\\end{proposition}\n\\begin{proof}\nFix $k$ and a nonzero $F\\in\\Z[x_0,\\ldots,x_k]$.  The two blocks agree\nunless $n+i\\in S$ for some $0\\le i\\le k$.  Consequently\n\\[\n \\begin{aligned}\n &\\bigl|\\{n<N:F(b_n,\\ldots,b_{n+k})=0\\}\\bigr|\\\\\n &\\quad\\le\\bigl|\\{n<N:F(a_n,\\ldots,a_{n+k})=0\\}\\bigr|\n       +\\sum_{i=0}^k|S\\cap[i,N+i)|.\n \\end{aligned}\n\\]\nThe first count is $o(N)$ by nonconcentration for $a$; the remaining\nfinitely many counts are $o(N)$ because $S$ has density zero.\nThe count concerns the event $F=0$, not the sizes of $F$ or the\ncorrections.  The sequence $b$ is already fixed, so this holds for\nevery fixed $F,k$, without a uniform estimate for growing polynomial\nfamilies or block lengths.\n\\end{proof}\n\n\\begin{samepage}\n\\begin{corollary}[a rational sum with the stated prime-gap statistics]\\label{long251:res:jointcountermodel}\nFor every prescribed finite prime-gap prefix and $0<\\varepsilon\\le1$,\nthere is an altered sequence $b=g+e$, with $P_n=2+\\sum_{i<n}b_i$,\nfor which one can simultaneously impose a rational dyadic value, nonnegative\ninteger corrections eventually at most $(\\log(n+3))^\\varepsilon$,\nall fixed eventual coefficient and cumulative congruences,\nfixed-block polynomial nonconcentration, and vanishing total variation distance between unnormalised block distributions for lengths $o(\\log\\log X)$.  The cumulative positions satisfy\n\\[\n 0\\le P_n-p_n=O_\\varepsilon\\left(\n    \\frac{n(\\log(n+3))^\\varepsilon}{\\log\\log n}\\right),\n \\qquad P_n\\sim n\\log n.\n\\]\n\\end{corollary}\n\\evidenceremark{Lean proves this for every $\\varepsilon>0$, assuming\nSchlage-Puchta's Lemma~4 \\cite{schlagepuchta2011} only for the\nnonconcentration clause and the prime number theorem only for\n$P_n\\sim n\\log n$.}\n\n\\end{samepage}\n\\begin{proof}\nApply Theorem~\\ref{long251:res:sparse-rationalisation} to the actual gaps,\nchoose a rational target in its interval, and fix the resulting $b=g+e$.\nProposition~\\ref{long251:res:sparse-nonconcentration} then transfers\nSchlage-Puchta's nonconcentration \\cite[Lemma~4]{schlagepuchta2011} to this\none sequence, for every fixed polynomial and block length.\nTo obtain the cumulative support bound, split $[\\sqrt n,n)$ into dyadic\nintervals, use $\\log\\log x\\asymp\\log\\log n$ there, and bound the first\n$\\sqrt n$ indices trivially.  Thus $|S\\cap[0,n)|=O(n/\\log\\log n)$.\nThe pointwise correction bound gives the displayed estimate, including\na fixed constant for the exceptional prefix.  Since\n$(\\log n)^{\\varepsilon-1}/\\log\\log n\\to0$ for $0<\\varepsilon\\le1$,\nthe prime number theorem \\cite[Chapter~6]{mv2007} gives the last conclusion.\nThe position series is rational as well: reversing the order of summation\nof nonnegative terms gives\n\\[\n \\sum_{n\\ge0}\\frac{P_n}{2^{n+1}}\n =2+\\sum_{i\\ge0}b_i\\sum_{n>i}2^{-n-1}\n =2+\\sum_{i\\ge0}\\frac{b_i}{2^{i+1}}\\in\\Q.\n\\]\nThe finite value on the right also proves convergence on the left.\n\\end{proof}\nThe case $\\varepsilon>1$ needs no new construction: use exponent $1$.\nIts corrections satisfy the larger allowance, while the stronger error\n$O(n\\log(n+3)/\\log\\log n)=o(n\\log n)$ still preserves cumulative growth.\nThus every positive exponent is available.  The construction supplies\nthe support and correction estimates; Schlage-Puchta's lemma supplies\nnonconcentration of the original gaps, and the prime number theorem\nsupplies their cumulative asymptotic.\n\nFor each fixed integer $y\\ge2$, take modulus $y!$.\nOnce the congruence holds and $p_n>y$, $\\gcd(P_n,y!)=1$ excludes every\nprime divisor at most $y$.  This is stronger than eventual oddness.\nThe cutoff depends on $y$; primality would require this exclusion up to\n$\\sqrt{P_n}$ at the same index.  Even prime-valued positions could omit\nintervening primes.\n\nThe bounded construction transfers nonconcentration by finitely many\npolynomial translates, not by preserving block frequencies.  The sparse\nconstruction transfers block laws by leaving most blocks unchanged, with\nabsolute error tending to zero for $m=o(\\log\\log X)$.  Neither argument\npreserves relative rare-event frequencies or the quantitative prime-pattern\nhypotheses in Section~\\ref{long251:sec:context}.\n\n\\subsection{Density of indices with two small tail differences}\n\nWhen the two tail differences have absolute value less than one and the\ngaps are unequal, the calculation above restricts the gap difference to\n$2$ or $-2$.  For each fixed shift, Schlage-Puchta's lemma gives density zero\nfor both events.  We use this to bound the set of possible witnesses,\nwithout excluding witnesses at arbitrarily large indices.\n\n\\begin{theorem}[density zero for the small-difference condition]\\label{long251:res:sparse}\nFix $h\\ge1$.  The set of $N\\ge1$ at which the three hypotheses of\nTheorem~\\ref{long251:res:smallpair} hold for the actual prime gaps has density zero.\nFor the same $h$, the set of $N$ with $g_{N+h+1}=g_{N+1}$ also has density\nzero.\n\\end{theorem}\n\\evidenceremark{The Lean proof assumes Schlage-Puchta's Lemma~4\n\\cite{schlagepuchta2011}.}\n\n\n\\begin{proof}\nFor $N\\ge1$ every gap involved is even, so $\\delta_N=g_{N+h+1}-g_{N+1}$ is\neven.  If $|\\sigma_h(N)|<1$, $|\\sigma_h(N+1)|<1$ and $\\delta_N\\ne0$, then\n$\\delta_N=2\\sigma_h(N)-\\sigma_h(N+1)$ lies in $(-3,3)$, so $\\delta_N=\\pm2$.\nApply Schlage-Puchta's lemma~\\cite[Lemma~4]{schlagepuchta2011} at index\n$n=N+1$ with $k=h$ to the two\npolynomials $x_h-x_0-2$ and $x_h-x_0+2$, neither of which vanishes\nidentically: each of $\\{N:\\delta_N=2\\}$ and $\\{N:\\delta_N=-2\\}$ has density\nzero, and the event is contained in their union.  The second assertion is the\nsame lemma applied to $x_h-x_0$.\n\\end{proof}\n\nUnequal gaps occur at almost every index, a stronger conclusion than\nProposition~\\ref{long251:res:gap-nonperiodic}.  But combining unequal gaps\nwith the two small tail differences forces their difference to be exactly\n$2$ or $-2$, which occurs only on a density-zero set.  Thus a proof of\nProblem~\\ref{long251:prob:smallpair} must find arbitrarily large successful\nindices in that sparse set.  A positive-density lower bound for the same\nevent is impossible.  An averaging argument that detects a sparse set is\nnot ruled out.\n\nThe proof of Schlage-Puchta's lemma gives a quantitative upper bound as\nwell.  For fixed $h$, it bounds the number of eligible indices in\n$[X,2X)$ by $O_h(X/\\log\\log X)$.  To see the two contributions, discard\nstarts whose block of $h+1$ gaps contains a gap exceeding\n$\\log X\\log\\log X$.  The sum of these block sums is $O_h(X\\log X)$\nby the prime number theorem, so there are at most\n$O_h(X/\\log\\log X)$ discarded starts.  For the remaining starts, the\nsieve calculation in the cited proof bounds the zeros of each of\n$x_h-x_0-2$ and $x_h-x_0+2$ by\n\\[\n O_h\\!\\left(\\frac{X(\\log\\log X)^{2h+2}}{\\log X}\\right)\n =o_h\\!\\left(\\frac{X}{\\log\\log X}\\right).\n\\]\nThe shift from $n=N+1$ changes only the band endpoints.  Thus the\nsame upper bound applies to the indices satisfying the three conditions\nof Theorem~\\ref{long251:res:smallpair}.  This is an ordinary\nconsequence of the cited sieve argument, not a new lower bound or an\nasymptotic formula for the number of successful indices.\n\nThe observed proportions in Section~\\ref{long251:sec:measurements} decrease\nacross the sampled bands.  These finite observations establish neither\nan asymptotic rate nor the required existence beyond every cutoff.\nCounting by prime size and counting by gap index also give different\ncutoffs and must not be compared without converting between them.\n\n\\subsection{Recurring gap values differing by two do not suffice}\n\nTheorem~\\ref{long251:res:signedwindow} needs values differing by $2$\nat a prescribed distance $h$ and a complete tail difference in a signed\ninterval.  Recurrence of two values in a residue class supplies neither\nrequirement.  We demonstrate this with an integer sequence whose dyadic\nsum is rational and whose complete tail differences are integral,\nwhile retaining the recurrence and growth properties stated below.\nThis is not a sequence of actual prime gaps.\n\nWe reverse the usual order: choose integer $U_n$ first, then set\n$a_n=2U_{n-1}-U_n$.  The construction must still make the coefficients\npositive and ensure $2^{-n}U_n\\to0$, so that these are actual tails.\nConstant tails $U_n=4$ give\nconstant coefficients $a_n=4$.  Raising one isolated tail to $6$\nchanges the two affected coefficients from $4,4$ to $2,8$, with the\nsame combined dyadic contribution.  A slowly growing even background\nwill supply cumulative growth $n\\log n$; factorial indices will impose\nthe recurrence in residue classes while keeping these changes sparse.\n\n\\begin{theorem}[recurring values are not enough]\\label{long251:res:polignacfail}\nThere is a sequence $(a_n)_{n\\ge1}$ of positive even integers with the\nfollowing properties.  The values $2$ and $4$ each occur infinitely often at\nindices divisible by every fixed $t\\ge1$.  The sequence is unbounded, not\neventually periodic, and satisfies $a_n=O(\\log n)$.  The series\n$\\sum_{n\\ge1}a_n2^{-n}$ equals $6$, and every scaled tail\n$\\sum_{j\\ge1}a_{N+j}2^{-j}$ is an integer, so every tail shift is integral.\nThe increasing odd sequence $P_n=3+\\sum_{j\\le n}a_j$ satisfies\n$P_n\\sim n\\log n$.\n\\end{theorem}\n\n\\begin{proof}\nPut $b_n=2\\lceil\\tfrac12\\log(n+64)\\rceil$, so\nthat $b_n$ is even, $b_n\\ge6$, and $b_n=\\log(n+64)+O(1)$.  Since\n$\\tfrac12\\log(n+65)-\\tfrac12\\log(n+63)<1$ for every $n\\ge1$, consecutive\nincrements of $b$ lie in $\\{0,2\\}$ and $b_{n+1}-b_{n-1}\\le2$.  Call an index\n$n$ \\emph{special} when $n=k!$ or $n=2\\,k!$ for some $k\\ge5$, and define\n\\[\n U_n=\\begin{cases}\n  2b_{n-1}-2,& n=k!,\\ k\\ge5,\\\\\n  2b_{n-1}-4,& n=2\\,k!,\\ k\\ge5,\\\\\n  b_n,&\\text{otherwise},\n \\end{cases}\n \\qquad a_n=2U_{n-1}-U_n\\quad(n\\ge1).\n\\]\nFor $k\\ge5$ one has $k!<2\\,k!<(k+1)!$, so all special indices are distinct.\nThey are even and therefore never adjacent.  Each $U_n$ is an even integer by construction, so\neach $a_n$ is an even integer.\n\nAt an ordinary index whose predecessor is also ordinary,\n$a_n=2b_{n-1}-b_n=b_{n-1}-(b_n-b_{n-1})\\ge6-2=4$.  At a special index the\npredecessor is ordinary and $a_n=2b_{n-1}-(2b_{n-1}-r)=r$, which is $2$ at\n$n=k!$ and $4$ at $n=2\\,k!$.  Immediately after a special index carrying $r$,\n\\[\n a_{n+1}=2(2b_{n-1}-r)-b_{n+1}\\ge4b_{n-1}-2r-(b_{n-1}+2)\n =3b_{n-1}-2r-2\\ge8 .\n\\]\nEvery $a_n$ is therefore a positive even integer, and $U_n=O(\\log n)$ gives\n$a_n=O(\\log n)$.  For $k\\ge5$, the index $n=k!+2$ and its predecessor are ordinary:\nboth lie strictly between $k!$ and $2\\,k!$, where there are no special\nindices.  Along these indices $a_n\\ge b_{n-1}-2\\to\\infty$.\nThus $(a_n)$ is unbounded and hence not eventually periodic.\n\nThe definition $a_n=2U_{n-1}-U_n$ is exactly $a_n2^{-n}=U_{n-1}2^{-(n-1)}-U_n2^{-n}$,\nso for every $N\\ge0$ and $m\\ge1$\n\\[\n \\sum_{j=1}^{m}\\frac{a_{N+j}}{2^{\\,j}}=U_N-\\frac{U_{N+m}}{2^{\\,m}} .\n\\]\nSince $U_n=O(\\log n)$ the endpoint tends to zero, so the tail at $N$ equals the\ninteger $U_N$; at $N=0$ the series equals $U_0=b_0=6$.  Every difference\n$U_{N+h}-U_N$ is an integer, so every tail shift is integral.\n\nFor each $t$ and every sufficiently large $k$ we have $t\\mid k!$, hence\n$t\\mid2\\,k!$, and $a_{k!}=2$ while $a_{2k!}=4$: both values recur infinitely\noften at indices congruent to $0$ modulo $t$.\n\nFinally, summing $a_j=2U_{j-1}-U_j$ gives\n$\\sum_{j=1}^{n}a_j=2U_0+\\sum_{j=1}^{n-1}U_j-U_n$.  The baseline\n$\\sum_{j<n}b_j=n\\log n+O(n)$.  The special indices up to $n$ number\n$O(\\log n/\\log\\log n)$ and each changes the summand by $O(\\log n)$, so their\ntotal contribution is $O((\\log n)^2)=o(n)$.  Hence\n$P_n=3+\\sum_{j\\le n}a_j\\sim n\\log n$, and $P_n$ is odd and increasing.\n\\end{proof}\n\nThe factorial indices used above give recurrence in the zero residue class.\nThe following separate construction strengthens this to every residue class.\n\n\n\\subsection{A rational series with recurring values in every residue class}\n\\label{long251:res:all-residue-log-countermodel}\nThere is a synthetic integer sequence $(a_n)_{n\\ge1}$ with all of the\nfollowing properties.  Each $a_n$ is positive and divisible by $2$, and\n\\[\n a_n\\le 4\\log(n+1)+24.\n\\]\nFor every $t\\ge1$, every $0\\le r<t$, and every cutoff $N$, there are\n$i,j\\ge N$ such that\n\\[\n i\\equiv j\\equiv r\\pmod t,\\qquad a_i=2,\\quad a_j=4.\n\\]\nFor every integer $B$ and cutoff $N$, some $n\\ge N$ satisfies $a_n>B$.\nFor every $h\\ge1$ there is no cutoff after which $a_{n+h}=a_n$ for all $n$.\nNevertheless, all the complete tails\n\\[\n U_N=\\sum_{j\\ge1}\\frac{a_{N+j}}{2^j}\\quad(N\\ge0)\n\\]\nare integers, $U_0=6$, and $U_{N+h}-U_N\\in\\mathbb Z$ for every $N,h\\ge0$.\nThe sequence\n\\[\n P_N=3+\\sum_{j=1}^{N}a_j\n\\]\nis strictly increasing and odd, with $P_N/(N\\log N)\\to1$.\nThese are synthetic positions; no $P_N$ is asserted to be prime.\n\nEnumerate the triples $(t,r,m)$ with $t\\ge1$, $0\\le r<t$ and $m\\ge0$,\neach once.  The extra index $m$ repeats each residue-class requirement\ninfinitely often; its parity will select the value $2$ or $4$.\nAt the $k$th step choose a centre $c_k\\equiv r\\pmod t$, subject to\n$c_0\\ge100$, $c_{k+1}\\ge c_k+3$ and $c_k\\ge2^{k^2}$ for $k\\ge1$.\nEach arithmetic progression is unbounded, so these lower bounds can be\nmet recursively.  Set $v_k=2$ for even $m$ and $v_k=4$ for odd $m$.\nThus each value is assigned beyond every cutoff in each residue class.\nUse the even baseline\n\\[\n b_n=6+2\\left\\lfloor\\frac{\\log(n+1)}2\\right\\rfloor,\n \\qquad\n U_n=\\begin{cases}2b_{n-1}-v_k,&n=c_k,\\\\b_n,&n\\notin\\{c_k:k\\ge0\\},\\end{cases}\n\\]\nand define $a_n=2U_{n-1}-U_n$.  The separation of the centres will ensure\nthat $a_{c_k}=v_k$.\n\nThe baseline is even and at least $6$.  For $n\\ge1$, the increase of\n$\\log(n+1)$ over two consecutive steps is less than $2$, so the floor\nin the definition gives $b_n-b_{n-1}\\in\\{0,2\\}$ and\n$b_{n+1}-b_{n-1}\\le2$.  Away from a centre and its successor,\n$a_n=2b_{n-1}-b_n\\ge4$.  At a centre, $a_{c_k}=v_k$.\nImmediately afterwards,\n\\[\n a_{c_k+1}=4b_{c_k-1}-2v_k-b_{c_k+1}\n \\ge3b_{c_k-1}-2v_k-2\\ge8.\n\\]\nSince the centres are at least three apart, these cases exhaust all\nindices.  Also $0<U_n\\le2b_n$, so\n$a_n\\le4b_{n-1}\\le4\\log(n+1)+24$.\nThe indices $n=c_k+2$ lie outside the centres and their successors,\nbecause consecutive centres are at least three apart.  Along these\nindices $a_n\\ge b_{n-1}-2\\to\\infty$.  Thus the sequence is unbounded\nand cannot be eventually periodic.\n\nFinite telescoping gives\n\\[\n \\sum_{j=1}^{m}a_{N+j}2^{-j}=U_N-U_{N+m}2^{-m}.\n\\]\nThe bound $U_n=O(\\log(n+1))$ makes the last term tend to zero.  Hence the\ncomplete tail equals the integer $U_N$, and its initial value is $6$.\nFor the cumulative growth, $b_n=\\log(n+1)+O(1)$ and there are at most\n$1+\\sqrt{\\log_2 N}$ centres up to $N$.  Replacing $b_n$ by $U_n$ changes\nits partial sum by $O((\\log N)^{3/2})$.  Therefore\n\\[\n \\sum_{j=1}^{N}a_j=2U_0+\\sum_{j=1}^{N-1}U_j-U_N\n                =N\\log N+O(N),\n\\]\nwhich gives the stated asymptotic for $P_N$.\n\nThe full statement is \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/AllResidueLogarithmicR9.lean\\#L503}{kernel-checked in Lean}.\n\n\nEven with all the displayed growth and recurrence properties, every\ntail difference is integral.  Requiring the values $2$ and $4$ in every\nresidue class therefore does not supply the signed tail interval.\nThe construction is not a prime-gap sequence: its logarithmic bound\nexcludes the extreme large gaps of the primes, so hypotheses using\nthose gaps are not tested by this example.\n\n\\subsection{Polynomial coefficients and telescoping sums}\n\nWe can construct further rational sums by starting with a polynomial\n$U_n$ and defining $c_{n+1}=2U_n-U_{n+1}$.  Polynomial growth gives the\nvanishing dyadic boundary condition, so the recurrence identifies $U_n$\nas the complete tail.  A quadratic choice already makes the coefficients\npositive and increasing.\n\n\\begin{proposition}[a quadratic sequence with integral tails]\\label{long251:res:polynomialcountermodel}\\compatlabel{res:polynomialcountermodel}\nPut $c_n=2(n^2+4n+2)$ and $U_n=2(n+4)^2$.  Then $c_n$ is positive, even and\nstrictly increasing, $U_{n+1}=2U_n-c_{n+1}$, every shift $U_{N+h}-U_N$ is\nintegral, $c_{n+1}-c_n=4n+10$ is never $\\pm2$, and\n\\[\n \\sum_{j\\ge1}\\frac{c_j}{2^{\\,j}}=32 .\n\\]\n\\end{proposition}\n\n\\begin{proof}\nDirect expansion gives the recurrence, and the finite telescope is\n$\\sum_{j=1}^{n}c_j2^{-j}=32-2(n+4)^22^{-n}$, whose last term tends to zero.\nThe remaining assertions follow from the integer values of $U_n$ and from\n$c_{n+1}-c_n=4n+10\\ge10$.\n\\end{proof}\n\n\\noindent The series starts at $j=1$ and equals the initial rescaled tail\n$U_0=32$.  Under the opening convention its sum is\n$\\sum_{n\\ge0}c_n2^{-(n+1)}=(c_0+32)/2=18$.\nPositivity, parity, strict growth, unboundedness and nonperiodicity are\ntherefore jointly compatible with rationality, and the adjacent-gap condition of\nTheorem~\\ref{long251:res:signedwindow} fails at every index.  The telescoping mechanism\nis the one used in the note of ChatGPT~5.4 Pro (orchestrated by Vjeko\nKova\\v{c}) against the variable-denominator expectation of\nErd\\H{o}s~\\cite[Theorem~1 and proof, pp.~1--2]{kovac2026}; the sequence is not a\nresult about Problem~\\#251.\n\nRationality also fails to force the coefficients to repeat.  Let $K:\\N\\to\\Q$ be\narbitrary and put $\\kappa_n=2K_n-K_{n+1}$: the\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1129}{coefficient of the telescoping series}.\n\n\\begin{proposition}[exact telescoping]\\label{long251:res:telescope}\\compatlabel{res:telescope}\\compatlabel{sec:carry}\nFor every $n\\ge0$,\n$\\sum_{i=0}^{n-1}\\kappa_i2^{-(i+1)}=K_0-K_n2^{-n}$.\n\\end{proposition}\n\n\\begin{proof}\nThe sum is empty when $n=0$.  Passing from $n$ to $n+1$ adds\n$(2K_n-K_{n+1})2^{-(n+1)}=K_n2^{-n}-K_{n+1}2^{-(n+1)}$,\nwhich replaces the terminal term by the required one.\n\\end{proof}\n\n\\noindent The identity $\\kappa_n=2K_n-K_{n+1}$ has the algebraic form used in\nthe rationality criterion of Erd\\H{o}s and Straus when $a_n=2$.  That criterion\nhas additional hypotheses: for integers $b_n$ and\npositive integers $a_n$ with $a_n>1$ for all large $n$ and\n$|b_n|/(a_{n-1}a_n)\\to0$, the series $\\sum_nb_n/(a_1\\cdots a_n)$ is rational\nexactly when some positive integer $B$ and integers $c_n$ satisfy\n$Bb_n=c_na_n-c_{n+1}$ and $|c_{n+1}|<a_n/2$ for all large\n$n$~\\cite[Theorem~2.1, pp.~85--86]{erdosstraus1974}.  With $a_n=2$,\nthe smallness hypothesis becomes $|b_n|/4\\to0$.  Since the $b_n$ are\nintegers, this forces $b_n=0$ eventually.  Thus the criterion in this\nspecialisation does not apply even to a bounded integer sequence that\nis nonzero infinitely often, let alone to the positive prime gaps.\nProposition~\\ref{long251:res:telescope}, by contrast, imposes no growth\ncondition on $K$.  The infinite series converges precisely when\n$2^{-n}K_n$ has a finite limit, and then its sum is\n$K_0-\\lim_{n\\to\\infty}2^{-n}K_n$.  The sum equals $K_0$ precisely when\nthat limit is zero; polynomial growth suffices.  For $K_n=2^n$, every\n$\\kappa_n$ is zero, so the series converges to $0$, not to $K_0=1$.\n\n\\noindent Taking $K_0=\\tfrac52$ and $K_n=2n+2$ for $n\\ge1$ gives\n$\\kappa_0=1$ and $\\kappa_n=2n$, so the coefficients $1,2,4,6,8,\\ldots$ are\npositive, even after the first term, unbounded and not eventually periodic,\nwhile their dyadic sum is $\\tfrac52$.  Rationality alone therefore cannot imply\neventual periodicity even for a positive, parity-correct integer coefficient\nsequence, and rationality cannot be contradicted merely by\nProposition~\\ref{long251:res:gap-nonperiodic}.\n\nThe arithmetic-progression formulations of nonintegrality do not create new\nprime-distribution information.  Their full definitions and exact\nhypotheses are preserved in Appendix~\\ref{long251:app:affine-detail}.\n\n\\section{Finite numerical searches}\\label{long251:sec:measurements}\n\nAn enclosure must control both the omitted tail and arithmetic error.\nThe scans supply no bound for accumulated double-precision rounding, so\ntheir counts remain observations even when the omitted tail is rigorously\nbounded.  By contrast, Proposition~\\ref{long251:res:finite-smallpair} and\nSection~\\ref{long251:sec:denominator-floor} use exact integer comparisons\nwith linked Lean checks.\n\nOver the $6\\,841\\,648$ primes below $1.2\\times10^{8}$, the scan tests\n$h=1,\\ldots,16$ separately.  For each offset it records indices satisfying\nthe signed inequalities in Theorem~\\ref{long251:res:signedwindow}, with\nobserved proportions between $0.00418$ and $0.008248$ and nearly balanced\nsigns.  The reported counts do not require one index to work for all offsets.  At $h=1$ it records $56\\,427$ hits among $6\\,841\\,564$ candidate\nindices, the last at the prime $119\\,995\\,753$.  Multiplying the observed proportion in each band\nby that band's mean of $\\log p$ gives $0.17671$ in the first band and\n$0.13148$ in the last.\nThis finite observation establishes no asymptotic counting law and no\noccurrence beyond every cutoff.  The\n\\href{https://github.com/wcook04/plectis-erdos/blob/dd6cc708f650f061de06594eba6ba43878f4de31/research/experiments/erdos251/receipts/adjacent-mismatch.json}{search output}\nis public.\n\nA second scan uses the $1\\,270\\,607$ primes below $2\\times10^{7}$ and\npairs of indices in the half-open range\n$1\\,143\\,545\\le n,m<1\\,270\\,540$.  It searches each residue class modulo\neach $1\\le t\\le20$ for the condition of\nTheorem~\\ref{long251:res:freepair}, selecting pairs with a next-gap difference of $2$ and a tail difference\nin the specified interval.  At $t=20$, every residue\nclass has at least $90\\,150$ counted pairs; pairs may share indices.\nFor each class, the program also records the larger index of a selected\nlate witness.  The minimum of those recorded indices is $1\\,270\\,520$;\nit need not come from the class with the fewest pairs.  For the saved sample pairs, floating-point arithmetic gives both\nsmall-difference inequalities.  The scan supplies neither arbitrary\nmoduli nor witnesses beyond every prescribed cutoff.  Its\n\\href{https://github.com/wcook04/plectis-erdos/blob/dd6cc708f650f061de06594eba6ba43878f4de31/research/experiments/erdos251/receipts/free-pair.json}{search output}\nis public.\nThe \\href{https://github.com/wcook04/plectis-erdos/blob/dd6cc708f650f061de06594eba6ba43878f4de31/research/experiments/erdos251/README.md}{computation guide}\ngives the programs, dependencies and commands to rerun both searches and\nthe $80000$-bit continued-fraction calculation of\nSection~\\ref{long251:sec:denominator-floor}.\nAppendix~\\ref{long251:sec:xr-corrections} corrects the stated\njustification of the tail bound in the free-pair program.\n\n\\section{Remaining prime-gap estimates}\\label{long251:sec:open}\n\nFor the actual tails, Theorem~\\ref{long251:res:escape-irrational} makes\nthe first question equivalent to irrationality.  The second requires\ntwo small differences and a gap mismatch at the same index.  A finite\ncertificate verifies one instance; both questions require witnesses\narbitrarily late for every positive shift.\n\n\\begin{problem}[nonintegral tail differences of each positive length]\\label{long251:prob:escape}\\compatlabel{prob:escape}\nFor every $h\\ge1$ and every $N_0$, prove that some $N\\ge N_0$ satisfies\n\\begin{equation}\n \\sum_{j\\ge1}\\frac{g_{N+h+j}-g_{N+j}}{2^{\\,j}}\\notin\\Z .\n\\label{long251:eq:shift-escape}\\compatlabel{eq:shift-escape}\n\\end{equation}\n\\end{problem}\n\n\\begin{problem}[two small tail differences at arbitrarily large indices]\\label{long251:prob:smallpair}\nFor every $h\\ge1$ and every $N_0$, prove that some $N\\ge N_0$ satisfies\n\\begin{equation}\n \\bigl|\\sigma_h(N)\\bigr|<1,\\qquad\n \\bigl|\\sigma_h(N+1)\\bigr|<1,\\qquad\n g_{N+h+1}\\ne g_{N+1} .\n\\label{long251:eq:smallpair}\n\\end{equation}\n\\end{problem}\n\nProblem~\\ref{long251:prob:escape}, like the congruent-index criterion of\nTheorem~\\ref{long251:res:freepair}, is exact;\nProblem~\\ref{long251:prob:smallpair} is only sufficient.\nThe counterexamples in Section~\\ref{long251:sec:obstructions} show why\nthe retained coefficient data do not supply the missing tail information.\nThey are not counterexamples about actual primes.\n\nThe successful indices have density zero by\nTheorem~\\ref{long251:res:sparse}, so the required existence theorem\nmust detect a sparse set.  A finite prefix is compatible with a rational\ncomparison series.  Supplemented by a proved remainder bound, however,\nit can certify one actual pair, as in\nProposition~\\ref{long251:res:finite-smallpair}; it does not establish\noccurrence beyond every cutoff.\n\nProposition~\\ref{long251:res:explicit-remainder} gives sufficient integer\ninequalities ensuring $|\\sigma_h(N)|<1$ at a chosen truncation length $L$.\nFailure of such a test need not mean that the complete tail lies outside\nthe required interval: its error interval may still cross an endpoint.\nFor fixed $h$ and $\\varepsilon>0$, the choice\n$L=\\lceil(4+\\varepsilon)\\log_2(N+2)\\rceil$ gives\n$E_{h,N,L}=O_{h,\\varepsilon}(N^{-\\varepsilon})$.\nIndeed, $L=O_\\varepsilon(\\log N)$, both arguments of the quartic $P$ are\n$O_{h,\\varepsilon}(N)$, and $2^{-L}\\le(N+2)^{-4-\\varepsilon}$.\nRequiring a fixed positive margin from both endpoints is stronger than\nthe strict inequalities in \\eqref{long251:eq:smallpair}.  At this depth,\nthe test bounds the signed integer $2^LF_{h,N,L}$, not just its residue\nmodulo $2^L$, and also requires $g_{N+h+1}-g_{N+1}=\\pm2$.\nFor fixed $h$ and $\\varepsilon$, checking one certificate uses\n$O_\\varepsilon(\\log N)$ gap values and the explicit remainder bound.\nThis counts data, not bit operations.  The unproved input is the\nexistence of certified witnesses beyond every cutoff for each fixed $h$;\nconstants uniform in $h$ are not required.\n\nThe following published results control gap sizes and clusters, not the\nsigned weighted sums required by this criterion.  Zhang's bounded-gap theorem~\\cite[Theorem~1, p.~1122]{zhang2014}\nproduces infinitely many bounded consecutive-prime gaps.  Maynard bounds\n$\\liminf_n(p_{n+m}-p_n)$ for every fixed $m$, giving bounded clusters of every\nfixed size~\\cite[Theorem~1.1, p.~384]{maynard2015}, with the explicit\nunconditional bound\n$\\liminf_n(p_{n+1}-p_n)\\le600$~\\cite[Theorem~1.3, p.~385]{maynard2015}.\nPolymath subsequently improved this to $246$\n\\cite[Theorem~1.4(i) of arXiv v4]{polymath2014}; neither numerical bound\ncontrols the signed weighted tails required here.  The large-gap\ntheorem of Ford, Green, Konyagin, Maynard and Tao gives an effective lower\nbound for the largest single consecutive-prime gap below\n$X$~\\cite[Theorem~1]{fgkmt2018}.  A theorem giving bounded clusters at\neach fixed cluster size does not by itself supply a weighted condition on\nblocks whose length grows with the basepoint, and none of these results is\nused as a proof input here.\n\nConditionally the picture is different.  Tao's comment names uniform\nquantitative prime-tuples control of about $\\log\\log n$ consecutive gaps as a\npossible approach~\\cite[comment of 7 October 2025]{erdosproblems251thread}, and\nLand's draft carries that out under Kuperberg's uniform prime-tuples\nconjecture~\\cite{land2026}.  The counterexamples in\nSection~\\ref{long251:sec:obstructions} show that the listed size,\ncongruence and block-statistical properties alone do not force\nirrationality.  These constructions are not shown to enumerate the\nconsecutive primes, so they do not rule out arguments that combine those\nproperties with further information about primes.\n\n\\paragraph{Funding and competing interests.} This work received no external\nfunding.  The author declares no competing interests.\n\n\\paragraph{Acknowledgements.}\n\nThe problem numbering follows the Erd\\H{o}s Problems catalogue maintained\nby Thomas Bloom~\\cite{erdosproblems}.  The logarithmic-scale construction\nand the transfer of nonconcentration come from earlier working materials\nof this project, which were not an independent mathematical review.\n\n\\appendix\n\\section{An elementary polynomial bound for the primes}\\label{long251:app:prime-bound}\n\\papersectiontarget{prime-bound}\n\nWe derive the bound $p_n\\le1250(n+1)^4$ by comparing the size and prime\nfactorisation of a central binomial coefficient, following\nErd\\H{o}s's proof of Chebyshev's theorem\n\\cite[\\S1, pp.~194--196]{erdos1932}.  With $\\pi(x)$ denoting the number\nof primes at most $x$, we first prove, for $m\\ge4$,\n\\begin{equation}\\label{long251:eq:binomial-bound}\n 4^m<m\\binom{2m}{m}\\le m(2m)^{\\pi(2m)} .\n\\end{equation}\nFor the first inequality, $m\\binom{2m}{m}/4^{m}$ equals $70/64$ at $m=4$ and\nits ratio at successive indices is $(2m+1)/(2m)>1$.  For the second, the\nexponent of a prime $\\ell$ in $\\binom{2m}{m}$ is\n$\\sum_{k\\ge1}\\bigl(\\lfloor2m/\\ell^{k}\\rfloor-2\\lfloor m/\\ell^{k}\\rfloor\\bigr)$,\neach summand is $0$ or $1$, and every summand with $\\ell^{k}>2m$ vanishes;\nthe exponent is therefore at most $\\lfloor\\log_\\ell(2m)\\rfloor$.\nThus the full power of each prime appearing in $\\binom{2m}{m}$ is at most\n$2m$.  Multiplying over at most $\\pi(2m)$ distinct primes gives the second\ninequality.\n\nNow fix $n\\ge0$, put $x=n+5$ and $m=x^4\\ge625$, and suppose $\\pi(2m)\\le n$.\nSince $x\\le2^{x}$ and $n+4(n+1)x=4x^2-15x-5\\le2x^4$,\n\\[\n m(2m)^{n}=2^{n}x^{4(n+1)}\n \\le2^{\\,n+4(n+1)x}\\le2^{\\,2x^4}=4^{m},\n\\]\ncontradicting \\eqref{long251:eq:binomial-bound}.  Hence $\\pi(2m)>n$, so at least $n+1$\nprimes lie below $2m$ and therefore\n\\[\n p_n\\le2(n+5)^4\\le1250(n+1)^4 ,\n\\]\nthe last step because $(n+5)\\le5(n+1)$ for $n\\ge0$.  The coarser bound\n$p_n\\le1250(n+1)^4$ is checked by the kernel at\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L360}{line~360}.\n\n\\section{Integer certificates}\\label{long251:app:certificates}\n\\papersectiontarget{finite-certificates}\n\nThis calculation checks two different finite conclusions from freshly\ncomputed primes below $10^4$: the two small-pair rows of\nProposition~\\ref{long251:res:finite-smallpair}, and the determinant and\nendpoint comparisons in Theorem~\\ref{long251:res:denominatorfloor}.\nAll arithmetic is integral; the four bracketing integers come from the\nlinked Lean proof.  The program checks those integers, not the analytic\nprime bound used to turn them into enclosures.  Adjacent quoted strings\nare concatenated by Python to keep the printed integers legible.\n{\\footnotesize\n\\begin{verbatim}\nfrom math import isqrt\n\nprimes = [n for n in range(2, 10000)\n          if all(n % d for d in range(2, isqrt(n) + 1))]\ngaps = [q - p for p, q in zip(primes, primes[1:])]\nP = lambda x: x**4 + 8*x**3 + 36*x*x + 104*x + 150\nQ = 2**40\nexpected = [(-662838684750, 11764181250),\n            (873345886050, 12805761250)]\nfor N, target in zip((2, 3), expected):\n    D = sum((gaps[N+1+j] - gaps[N+j])*2**(40-j)\n            for j in range(1, 41))\n    B = 1250*(P(N+43) + P(N+42))\n    distance = min(D % Q, Q - D % Q)\n    if (D, B) != target or not (abs(D)+B < Q and B < distance):\n        raise ArithmeticError(\"tail certificate failed\")\nif gaps[4] == gaps[3]:\n    raise ArithmeticError(\"gap mismatch failed\")\n\nu = int(\n    \"8065641857152652932176019632186898003271162829171466334827\"\n    \"3083607794415278717445033509407855988903369988525550746159\"\n    \"7355889792250084202344821020139160956663658789718168152662\"\n    \"0217\"\n)\n\nv = int(\n    \"2194945124413663232143970924541263312422069524635615360518\"\n    \"4247077351958221816830720189289904831662955084392690248683\"\n    \"1216291723988537733235173040607254496838530213867781442335\"\n    \"1745\"\n)\n\nup = int(\n    \"6539437101498162626882411892470905222108260008568555445975\"\n    \"3026163315521784668689909712759876562484620059098138417469\"\n    \"5232839888185316374272333277389611483117334000493867584923\"\n    \"912\"\n)\n\nvp = int(\n    \"1779611075789866551198427241621709630120136323281598176637\"\n    \"8490605249229735501968764904036152970729491656650873938580\"\n    \"6203025466313389810291243786005499164537499383612081805160\"\n    \"767\"\n)\n\nc = len(primes)\nA = sum(p*2**(c-1-i) for i, p in enumerate(primes))\nchecks = (c == 1229,\n          u > 0 and v > 0 and up > 0 and vp > 0,\n          up*v - u*vp == 1,\n          u*2**c < A*v,\n          (2*A + 5000*(c+1)**4)*vp < up*2**(c+1),\n          v+vp >= 2**589,\n          2**589 > 10**177)\nif not all(checks):\n    raise ArithmeticError(\"denominator certificate failed\")\nprint(\"Both tail rows and the denominator floor verified.\")\n\\end{verbatim}\n}\n\n\\section{Verification and reproducibility}\\label{long251:app:index}\\compatlabel{app:index}\n\\pdfbookmark[1]{Statements and declarations}{statements-declarations}\n\\evidenceparagraph{\\evidencerecordurl}\nThe source links identify definitions and proof steps at fixed revisions;\nthe short paper's source index retains its different revision in\nAppendix~\\ref{long251:app:short-source-index}. The formal sparse\nconstruction changes one coordinate per stage, whereas the printed\nproof uses triples. Both fix a set containing all correction supports,\nthe interval of sums and the modulus cutoffs before choosing a value. The formal proofs use the\npublished inputs stated below; the linked declarations do not check\nthe different printed arguments step by step.\n\nLean~4~\\cite{lean4} and mathlib~\\cite{mathlib} supply the proof-checking\nenvironment. The \\href{\\evidencerecordurl}{evidence record} lists the\ndeclarations and checks. The programs for the searches in\nSection~\\ref{long251:sec:measurements} are in the\n\\href{https://github.com/wcook04/plectis-erdos/blob/dd6cc708f650f061de06594eba6ba43878f4de31/research/experiments/erdos251/README.md}{computation guide}.\nThe program in Appendix~\\ref{long251:app:certificates} checks the\nadjacent pair and the bound $2^{589}$; the independent program of\nSection~\\ref{long251:sec:denominator-floor} checks the certificate of\nTheorem~\\ref{long251:res:cfexclusion}. Formal checking does not\nestablish novelty or attribution.\n\nThe summation-by-parts identity and the polynomial prime bound concern the\nactual primes.  The recurrence results apply to arbitrary integer\ncoefficients and, as stated, to rational or real sequences; substituting\nthe actual prime-gap tail requires its convergence and boundary condition.\nThe finite certificates use the same polynomial bound to control the\nomitted terms.  The counterexamples concern other coefficient sequences\nand construct no new primes.\n\nThe Lean proofs of three results assume published inputs that the Lean\ndevelopment does not prove.  The Lean proof of\nTheorem~\\ref{long251:res:sparse} assumes Schlage-Puchta's\nLemma~4 \\cite[Lemma~4, pp.~5--6]{schlagepuchta2011}; for a fixed nonzero\ninteger polynomial in a fixed block of consecutive prime gaps, it states\nthat the set of indices where the polynomial vanishes has density zero.\nThose of Corollaries~\\ref{long251:res:nonconc-primes}\nand~\\ref{long251:res:jointcountermodel} assume that lemma and the prime\nnumber theorem in the form $p_n\\sim n\\log n$~\\cite[Chapter~6]{mv2007}.\nWhat each statement adds to these inputs is checked, including the\nestimate that the corrected sequence of\nCorollary~\\ref{long251:res:jointcountermodel} differs from the prime gaps\nat $O_\\varepsilon(n/\\log\\log n)$ indices below $n$, which uses no\ninformation about the distribution of primes.\n\n% The exhaustive source manifest is retained in the authored TeX for automated\n% validation, and kept outside the printed note.\n\\iffalse\n\n\\begin{tabular}{@{}ll@{}}\ninformal statement & linked Lean source\\\\\nzero-based primes & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L40}{\\leanlabel{prime0}}\\\\\nzero-based gaps & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L44}{\\leanlabel{primeGap0}}\\\\\nfirst gap value & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L47}{\\leanlabel{primeGap0_zero}}\\\\\nsecond gap value & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L51}{\\leanlabel{primeGap0_one}}\\\\\nunbounded actual prime gaps & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L57}{\\leanlabel{exists_primeGap0_gt}}\\\\\ndyadic partial sum & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L101}{\\leanlabel{dyadicPartialSumQ}}\\\\\ndisplayed normalisation & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L106}{\\leanlabel{prime0DisplayedPartialSumQ}}\\\\\nfactor-of-two identity & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L111}{\\leanlabel{prime0DisplayedPartialSumQ_eq_two_mul}}\\\\\ndyadic difference sum & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L121}{\\leanlabel{dyadicDifferencePartialSumQ}}\\\\\npartial-sum step & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L124}{\\leanlabel{dyadicPartialSumQ_succ}}\\\\\ndifference-sum step & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L129}{\\leanlabel{dyadicDifferencePartialSumQ_succ}}\\\\\nsummation by parts & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L138}{\\leanlabel{dyadicPartialSumQ_eq_start_add_differences}}\\\\\nmonotonicity step & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L152}{\\leanlabel{prime0_mono_step}}\\\\\ngap cast identity & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L156}{\\leanlabel{primeGap0_cast}}\\\\\ngap partial sum & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L161}{\\leanlabel{primeGapPartialSumQ}}\\\\\nprime-gap summation by parts & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L172}{\\leanlabel{prime0_dyadic_summation_by_parts}}\\\\\nprime dyadic term & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L183}{\\leanlabel{primeDyadicTerm}}\\\\\ndisplayed prime dyadic term & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L188}{\\leanlabel{primeDisplayedDyadicTerm}}\\\\\nprime-gap dyadic term & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L192}{\\leanlabel{primeGapDyadicTerm}}\\\\\ninfinite factor-of-two identity & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L196}{\\leanlabel{primeDisplayedDyadicTerm_eq_two_mul}}\\\\\ngap discrete derivative & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L202}{\\leanlabel{primeGapDyadicTerm_eq}}\\\\\npolynomial prime bound & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L360}{\\leanlabel{prime0_le_polynomial}}\\\\\nprime-series summability & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L379}{\\leanlabel{summable_primeDyadicTerm}}\\\\\ngap summability transfer & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L385}{\\leanlabel{summable_primeGapDyadicTerm_of_summable_primeDyadicTerm}}\\\\\ninfinite gap identity & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L404}{\\leanlabel{tsum_primeDyadicTerm_eq_two_add_primeGap}}\\\\\nunconditional gap identity & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L427}{\\leanlabel{tsum_primeDyadicTerm_eq_two_add_primeGap_unconditional}}\\\\\nnormalised irrationality equivalence & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L435}{\\leanlabel{irrational_tsum_primeDyadicTerm_iff_primeGap}}\\\\\ndisplayed infinite identity & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L444}{\\leanlabel{tsum_primeDisplayedDyadicTerm_eq_four_add_two_primeGap}}\\\\\ndisplayed irrationality equivalence & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L459}{\\leanlabel{irrational_tsum_primeDisplayedDyadicTerm_iff_primeGap}}\\\\\ntail recurrence & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L482}{\\leanlabel{DyadicTailRecurrence}}\\\\\nrational actual-tail state & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L489}{\\leanlabel{rationalPrimeGapTailState}}\\\\\nhypothetical-rational tail representation & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L510}{\\leanlabel{exists_rationalPrimeGapTailState_representation_of_not_irrational}}\\\\\nrational actual-tail recurrence & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L527}{\\leanlabel{rationalPrimeGapTailState_recurrence}}\\\\\nshift & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L544}{\\leanlabel{tailShift}}\\\\\nshift step identity & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L563}{\\leanlabel{tailShift_succ}}\\\\\nintegrality & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L575}{\\leanlabel{RatIntegral}}\\\\\ntail block & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L647}{\\leanlabel{dyadicTailBlock}}\\\\\niterated block identity & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L653}{\\leanlabel{tail_iterate_eq_pow_mul_sub_block}}\\\\\nscaled shift identity & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L667}{\\leanlabel{tailShift_eq_scaled_sub_block}}\\\\\ninteger-shift invariance & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L677}{\\leanlabel{ratIntegral_sub_int_iff}}\\\\\nEuler multiplier & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L695}{\\leanlabel{ratIntegral_totientMultiplier_of_odd_den}}\\\\\npower-of-two/odd denominator scaling & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L723}{\\leanlabel{ratIntegral_scaled_of_den_eq_pow_two_mul_odd}}\\\\\nmultiplier at the two-adic exponent & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L756}{\\leanlabel{ratIntegral_multiplier_tail_at_twoExponent}}\\\\\nintegral-shift criterion & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L802}{\\leanlabel{tailShift_integral_iff_scaledTail}}\\\\\nfixed-denominator eventual shift & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L830}{\\leanlabel{rationalPrimeGapTailShift_eventuallyIntegral_of_den_eq}}\\\\\neventual integral shift & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L853}{\\leanlabel{rationalPrimeGapTailShift_eventuallyIntegral}}\\\\\ntotient shift & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L877}{\\leanlabel{tailShift_integral_totient_of_odd_den}}\\\\\none-step propagation & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L888}{\\leanlabel{tailShift_integral_succ}}\\\\\npropagation to every later index & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L900}{\\leanlabel{tailShift_integral_add}}\\\\\nadjacent small-shift obstruction & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L979}{\\leanlabel{tailShift_not_both_integral_of_small_pair_of_digit_ne}}\\\\\ncofinal small-mismatch theorem & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1006}{\\leanlabel{not_eventuallyIntegralTailShift_of_cofinal_small_mismatch}}\\\\\nprime gaps not eventually periodic & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1023}{\\leanlabel{primeGap0_not_eventually_periodic}}\\\\\nshift not eventually small & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1097}{\\leanlabel{rationalPrimeGapTail_has_positive_shift_not_eventually_small}}\\\\\nactual-prime-gap specialisation & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1112}{\\leanlabel{primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch}}\\\\\ncarry coefficient & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1129}{\\leanlabel{carryCoeff}}\\\\\ncarry partial sum & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1133}{\\leanlabel{carryPartialSum}}\\\\\ncarry telescoping identity & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1137}{\\leanlabel{carryPartialSum_eq}}\\\\\nnatural carry coefficient & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1177}{\\leanlabel{natCarryCoeff}}\\\\\nnatural-carry cast & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1180}{\\leanlabel{natCarryCoeff_cast}}\\\\\nexact denominator recurrence & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1217}{\\leanlabel{tail_den_succ}}\\\\\nodd-denominator persistence & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1226}{\\leanlabel{tail_den_succ_eq_of_odd}}\\\\\neven-denominator halving & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1236}{\\leanlabel{tail_den_succ_eq_half_of_even}}\\\\\nintegral shifts by denominator & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1279}{\\leanlabel{tailShift_integral_iff_den_dvd_mersenne}}\\\\\npower-of-two congruence & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1291}{\\leanlabel{tailShift_integral_iff_two_pow_modEq_one}}\\\\\none-shift rationality classifier & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1479}{\\leanlabel{not_irrational_initial_iff_exists_integral_positive_tailShift}}\\\\\neventual-shift rationality classifier & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1525}{\\leanlabel{not_irrational_initial_iff_exists_eventually_integral_positive_tailShift}}\\\\\npointwise irrationality classifier & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1551}{\\leanlabel{irrational_initial_iff_all_positive_tailShifts_nonintegral}}\\\\\ncofinal irrationality classifier & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1572}{\\leanlabel{irrational_initial_iff_cofinalNonintegralTailShifts}}\\\\\npolynomial countermodel recurrence & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1596}{\\leanlabel{polynomialTailOrbit_recurrence}}\\\\\npolynomial countermodel strict growth & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1632}{\\leanlabel{polynomialGapWord_strictMono}}\\\\\npolynomial countermodel excludes the adjacent $\\pm2$ trigger & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1644}{\\leanlabel{polynomialGapWord_no_adjacent_two_digit}}\\\\\nall polynomial-countermodel shifts are integral & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1654}{\\leanlabel{polynomialTailOrbit_shift_integral}}\\\\\npolynomial countermodel dyadic value & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PolynomialGapSeriesValue.lean\\#L92}{\\leanlabel{tsum_polynomialGapDyadicTerm_eq}}\\\\\npolynomial countermodel rationality & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/PolynomialGapSeriesValue.lean\\#L109}{\\leanlabel{not_irrational_tsum_polynomialGapDyadicTerm}}\\\\\nreal tail shifted-gap identity & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/RealPrimeGapTail.lean\\#L31}{\\leanlabel{realPrimeGapTail_eq_tsum_shifted_gaps}}\\\\\nreal tail recurrence & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/RealPrimeGapTail.lean\\#L48}{\\leanlabel{realPrimeGapTail_recurrence}}\\\\\nactual escape equivalence & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/RealPrimeGapTail.lean\\#L73}{\\leanlabel{irrational_primeSeries_iff_realPrimeGapTail_escape}}\\\\\nreal adjacent-pair consumer & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/RealPrimeGapTail.lean\\#L99}{\\leanlabel{realTailShift_not_both_integral_of_small_pair_of_digit_ne}}\\\\\nreal small-mismatch criterion & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/RealPrimeGapTail.lean\\#L120}{\\leanlabel{irrational_primeSeries_of_realPrimeGapTail_small_mismatch}}\\\\\nfree-pair lattice & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/FreePairReduction.lean\\#L92}{\\leanlabel{free_pair_integral_iff_modEq}}\\\\\nfree-pair lattice corollary & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/FreePairReduction.lean\\#L105}{\\leanlabel{exists_free_pair_lattice}}\\\\\nactual real tail recurrence & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/FreePairReduction.lean\\#L186}{\\leanlabel{primeGapRealTail_recurrence}}\\\\\nfree-pair equivalence for the actual series & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/FreePairReduction.lean\\#L236}{\\leanlabel{irrational_primeGap_tsum_iff_cofinalFreePairNonintegral}}\\\\\nkernel-decided certificate & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/KernelDenominatorFloor.lean\\#L310}{\\leanlabel{cert_10000}}\\\\\ndenominator floor, prime series & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/KernelDenominatorFloor.lean\\#L316}{\\leanlabel{kernel_denominator_floor}}\\\\\ndenominator floor, gap series & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/KernelDenominatorFloor.lean\\#L324}{\\leanlabel{kernel_denominator_floor_primeGap}}\\\\\nbounded-perturbation construction & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/BoundedPerturbationCountermodel.lean\\#L193}{\\leanlabel{exists_bounded_perturbation}}\\\\\nbinary-digit encoding & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/BoundedPerturbationCountermodel.lean\\#L134}{\\leanlabel{exists_bounded_perturbation_digits}}\\\\\nprime-anchored rational perturbation & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/BoundedPerturbationCountermodel.lean\\#L208}{\\leanlabel{exists_rational_bounded_perturbation_primeGap}}\\\\\nperturbed prime-gap series is rational & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/BoundedPerturbationCountermodel.lean\\#L224}{\\leanlabel{not_irrational_bounded_perturbation_primeGap}}\\\\\nperturbed gap bounds & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/BoundedPerturbationCountermodel.lean\\#L234}{\\leanlabel{perturbed_gap_bounds}}\\\\\nsigned two-window normal form & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/AffineShiftEscape.lean\\#L113}{\\leanlabel{adjacent_small_mismatch_iff_signed_two_window}}\\\\\npointwise affine identity & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/AffineCylinderCollapse.lean\\#L91}{\\leanlabel{ratAffinePowTwo_iff_evenIntegral}}\\\\\ncofinal affine circularity & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/AffineCylinderCollapse.lean\\#L162}{\\leanlabel{cofinal_affinePowTwo_escape_iff_not_eventuallyIntegral}}\\\\\nfixed-lattice circularity & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/AffineCylinderCollapse.lean\\#L308}{\\leanlabel{cofinal_blockResidue_escape_iff_not_eventuallyIntegral}}\\\\\nlcm-diagonal criterion & \\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos251/OrderLatticeDiagonal.lean\\#L153}{\\leanlabel{irrational_initial_iff_all_lcmDiagonal_nonintegral}}\\\\\n\\end{tabular}\n\\fi\n% Part: part extended_record\n\\section{Further criteria, examples and computational details}\n\\label{long251:sec:erdos-251-extended-record}\n\nThese further results concern the recurrence and actual-tail tests, not\nthe sparse construction.  We separate exact reformulations from stronger\nsufficient conditions, explain why refining one fixed enclosure differs\nfrom controlling enclosures at changing indices, and compare repeated\ntail values with the nonintegral differences the criteria require.\nA bounded counterexample, the complete numerical table and two corrections\nare retained below.\n\n\\subsection{Totient-length shifts and persistence of integrality}\n\\label{long251:sec:xr-totient}\n\nOnce the denominator is odd, Euler's congruence specifies an integral\nshift of totient length.  We record this choice and then prove that\nan integral shift persists under each subsequent recurrence step.\n\n\\begin{proposition}[a shift of totient length]\\label{long251:xr:totient}\\compatlabel{res:totient}\nLet $T:\\N\\to\\Q$ satisfy the dyadic tail recurrence with integer coefficients.  If the\nreduced denominator $d$ of $T_N$ is odd, then $\\sigma_{\\ph(d)}(N)$ is an\ninteger.\n\\end{proposition}\n\n\\begin{proof}\nSince $d$ is odd, $2$ and $d$ are coprime, so Euler's congruence gives\n$d\\mid2^{\\ph(d)}-1$.  Writing $2^{\\ph(d)}-1=dk$ and $T_N=u/d$ in lowest terms,\n$(2^{\\ph(d)}-1)T_N=ku$ is an integer, and the integral-shift criterion\ntransfers this to the shift.\n\\end{proof}\n\n\\begin{proposition}[propagation]\\label{long251:xr:propagate}\\compatlabel{res:propagate}\nLet $T:\\N\\to\\R$ satisfy $T_{n+1}=2T_n-a_{n+1}$ with integer coefficients,\nand define $\\sigma_h(N)=T_{N+h}-T_N$.  For fixed $h,N\\ge0$,\nif $\\sigma_h(N)$ is an integer, then $\\sigma_h(N+k)$ is an integer\nfor every $k\\ge0$.\n\\end{proposition}\n\n\\begin{proof}\nThe shift step identity gives\n$\\sigma_h(N+1)=2\\sigma_h(N)-(a_{N+h+1}-a_{N+1})$, an integer combination of an\ninteger and two coefficients.  Induct on $k$.\n\\end{proof}\n\n\\noindent These are the totient shift and the propagation theorems cited in the\nnote.  For example, if $\\operatorname{den}T_N=3$ then $\\ph(3)=2$ and\n$3T_N$ is an integer, so $\\sigma_2(N)$ is integral while $\\sigma_1(N)$ is not;\nif $\\operatorname{den}T_N=5$ then $\\sigma_4(N)$ is integral.  In the orbit with\nevery coefficient zero and $T_0=1/12$, the denominator is $12=2^2\\cdot3$, the orbit\nreaches $T_2=1/3$ with odd denominator at $s=2$, and $h=\\ph(3)=2$ is exactly the\nshift length seen to be integral from index $2$ onwards.  The totient supplies one length, while the exact criterion is\n$\\operatorname{den}(T_N)\\mid2^{h}-1$.  At an index with odd denominator\n$d>1$, the least positive length is the multiplicative order of $2$\nmodulo $d$.  When $d=1$, every positive length works; before the\ndenominator becomes odd, no positive length works.\n\n\\subsection{The finite truncation criterion}\\label{long251:sec:xr-truncation}\n\\papersectiontarget{truncation-details}\n\nFor the actual gaps, we enclose each complete weighted difference by a\nfinite sum and a bound on the omitted terms.  A nonnegative summable\nmajorant gives such an enclosure, but need not provide a computable\nremainder.  Proposition~\\ref{long251:res:explicit-remainder} supplies\nthe rational bound used in the finite tests.\n\n\\begin{samepage}\n\\begin{proposition}[finite truncation]\\label{long251:xr:truncation}\nLet $M:\\N\\to\\R$ satisfy $M(n)\\ge g_n$ for every $n$ and\n$\\sum_{n\\ge0}M(n)2^{-n}<\\infty$, and put\n\\[\n S_{h,N,L}=\\sum_{j=1}^{L}\\frac{g_{N+h+j}-g_{N+j}}{2^{\\,j}},\\qquad\n R_{h,N,L}(M)=\\sum_{j>L}\\frac{M(N+h+j)+M(N+j)}{2^{\\,j}} .\n\\]\nIf for every $h\\ge1$ and every $N_0$ there are $N\\ge N_0$ and $L\\ge1$ with\n$\\operatorname{dist}(S_{h,N,L},\\Z)>R_{h,N,L}(M)$, then the nonintegrality\ncondition in Problem~\\ref{long251:prob:escape} holds.\n\\end{proposition}\n\\end{samepage}\n\n\\begin{proof}\nThe part of $\\sum_{j\\ge1}(g_{N+h+j}-g_{N+j})2^{-j}$ omitted from $S_{h,N,L}$\nhas absolute value at most $R_{h,N,L}(M)$, so under the displayed inequality\nthe full sum lies at positive distance from every integer.\n\\end{proof}\n\nWith $M(n)=1250(n+2)^4$, the quantities $S_{h,N,L}$ and\n$R_{h,N,L}(M)$ are exactly $F_{h,N,L}$ and $E_{h,N,L}$ from\nProposition~\\ref{long251:res:explicit-remainder}.  The same enclosure\ntherefore tests separation from the integers, without requiring either\nof the two consecutive differences to be small.  To compute the distance\nfrom the truncated sum to $\\Z$, write the weighted block sum as\n\\[\n D_{h,N,L}=\\sum_{j=1}^{L}2^{\\,L-j}(g_{N+h+j}-g_{N+j}),\n \\qquad S_{h,N,L}=\\frac{D_{h,N,L}}{2^{L}}.\n\\]\nThen\n\\[\n \\operatorname{dist}(S_{h,N,L},\\Z)\n =2^{-L}\\min\\bigl\\{D_{h,N,L}\\bmod2^{L},\\;\n 2^{L}-(D_{h,N,L}\\bmod2^{L})\\bigr\\},\n\\]\nwhere $D_{h,N,L}\\bmod2^{L}$ is the least nonnegative residue, including when\n$D_{h,N,L}<0$.  The criterion requires this residue to be more than\n$2^LR_{h,N,L}(M)$ from both $0$ and $2^L$.  One successful comparison certifies nonintegrality at that index;\na failed comparison is inconclusive.  Irrationality requires such\ncertificates arbitrarily late for every positive $h$.  At prescribed\nlogarithmic depth, the needed residue must be farther from both endpoints\nthan the error bound.  With the polynomial majorant in\nProposition~\\ref{long251:res:explicit-remainder}, the remainder bound is\nrational and explicit, so the whole test reduces to integer comparisons.\nThe modular rewriting is an ordinary deduction, not a separately checked\nLean declaration.\n\n\\begin{proposition}[completeness of finite separation]\\label{long251:res:complete-truncation}\nSuppose $D\\in\\R$, $S_L\\in\\R$ and $R_L\\ge0$ satisfy\n$|D-S_L|\\le R_L$ and $R_L\\to0$.  Then\n\\[\n D\\notin\\Z\\quad\\Longleftrightarrow\\quad\n \\text{there exists }L\\text{ with }\\operatorname{dist}(S_L,\\Z)>R_L.\n\\]\n\\end{proposition}\n\\begin{proof}\nThe reverse implication follows from the triangle inequality.\nFor the forward implication put $\\delta=\\operatorname{dist}(D,\\Z)>0$.\nFor all sufficiently large $L$, one has $R_L<\\delta/2$, and the\n$1$-Lipschitz property of distance gives\n$\\operatorname{dist}(S_L,\\Z)\\ge\\delta-R_L>R_L$.\nNo monotonicity of the errors is required.\n\\end{proof}\nFor fixed $h,N$, increasing the depth eventually certifies every\nnonintegral $\\sigma_h(N)$.  This is not a decision procedure for\nintegrality: finitely many failed tests prove nothing, although a\nproof of failure at every depth implies integrality.  It gives neither\na prescribed depth as $N$ varies nor arbitrarily late successful indices.\nFor example, $[1-2^{-N},1+2^{-N}]$ contains both the noninteger\n$1+2^{-2N}$ and the integer $1$ for every $N\\ge1$.  The radii shrink,\nbut the values also change; no enclosure certifies nonintegrality.\nCompleteness keeps the value fixed while refining its enclosure.\n\n\\subsection{Nonintegral differences at multiples of each shift}\\label{long251:sec:xr-divisorhit}\n\nWe may ask for a nonintegral multiple of each prescribed shift length.\nThe following formulation still has the full strength of irrationality,\nas the argument after the question shows.\n\n\\begin{problem}[multiples of each shift]\\label{long251:xr:divisor-hit}\\compatlabel{prob:divisor-hit}\nFor every $r\\ge1$, is there an $m\\ge1$ such that the tail differences\nof length $mr$ are nonintegral at arbitrarily large indices, as in\n\\eqref{long251:eq:shift-escape}?\n\\end{problem}\n\nThis is an exact reformulation, not a weaker mathematical target.\nIf $T_0$ is irrational, every positive shift is nonintegral by the block\nidentity, so the stated condition follows with $m=1$.  Conversely,\nrationality gives a shift $h\\ge1$ that is integral at every sufficiently\nlate index.  For each positive integer $m$, the identity\n\\[\n T_{N+mh}-T_N=\\sum_{j=0}^{m-1}(T_{N+(j+1)h}-T_{N+jh})\n\\]\nmakes every multiple $mh$ eventually integral.  Taking $r=h$ contradicts\nthe stated condition.  This ordinary deduction uses the checked\nrationality classification and a finite telescope; rearranging the\nquantifiers supplies no new estimate for the actual prime gaps.\n\n\\subsection{Repeated tail values and congruent indices}\n\\label{long251:sec:xr-compression}\n\nFor the complete actual tails $T_N=\\sum_{j\\ge1}g_{N+j}2^{-j}$, the prime\nnumber theorem controls the mean size without assuming rationality.\nRationality would then restrict these tails to a fixed rational lattice,\nforcing many repetitions by pigeonhole.  Repetition gives difference zero,\nnot the nonintegral difference required at congruent indices.  We make\nboth steps explicit.  These are ordinary deductions; the linked formal\nresults establish the recurrence and the congruent-index criterion.\n\n\\paragraph{Repeated tail values.}\nFor an integer $X\\ge0$, summing $g_{N+1}=2T_N-T_{N+1}$ over\n$0\\le N<X$ gives the exact identity\n\\[\n \\sum_{N=0}^{X}T_N=p_{X+1}-p_1-T_0+2T_X\n \\le(p_{X+1}-p_0)+2T_X.\n\\]\nThe inequality uses $T_0\\ge0$ and $p_1\\ge p_0$.  The prime number theorem\n\\cite[Ch.~6]{mv2007} also gives\n\\begin{equation}\\label{long251:eq:tail-small-relative-prime}\n \\frac{T_X}{p_X}\n =\\sum_{j\\ge1}\\frac{g_{X+j}}{2^jp_X}\\longrightarrow0.\n\\end{equation}\nIndeed, for each fixed $j$, both $p_{X+j+1}/p_X$ and $p_{X+j}/p_X$\ntend to $1$, so their difference $g_{X+j}/p_X$ tends to zero.\nFor the uniform bound needed to pass the limit through the sum, the\ntwo-sided estimates $p_n\\asymp n\\log n$ give one constant $K$ such that,\nfor $X\\ge2$ and $j\\ge1$,\n\\[\n \\begin{aligned}\n \\frac{g_{X+j}}{p_X}\n &\\le K\\frac{(X+j+1)\\log(X+j+1)}{X\\log X}\\\\\n &\\le K(1+j)\\left(1+\\frac{\\log(1+j)}{\\log X}\\right)\n \\le K(1+j)^2.\n \\end{aligned}\n\\]\nThe second line uses $X+j+1\\le X(1+j)$ and\n$\\log(1+j)\\le j\\log2\\le j\\log X$.\nSince $\\sum_{j\\ge1}(1+j)^2 2^{-j}<\\infty$, dominated convergence applies.\nSubtracting the exact summation identities at $2X$ and $X$ now gives\n\\[\n \\sum_{X<N\\le2X}T_N\n =p_{2X+1}-p_{X+1}+2(T_{2X}-T_X)\n \\sim X\\log X.\n\\]\nIndeed, the prime difference is asymptotic to $X\\log X$, and\n\\eqref{long251:eq:tail-small-relative-prime} makes both tail terms\n$o(X\\log X)$.  Thus the mean complete tail in this band is asymptotic\nto $\\log X$, without a rationality assumption.  This is a mean, not a\npointwise estimate.  For the counting argument we only need the upper\nbound $\\sum_{X<N\\le2X}T_N\\le K_0X\\log X$, valid for any fixed $K_0>1$\nand all sufficiently large $X$.  For each fixed $C>1$, Markov's inequality\nthen leaves at least $(1-1/C)X$ indices with $T_N\\le K_0C\\log X$.\n\nUnder rationality, take $X$ beyond the point where the reduced denominator\nis the fixed odd integer $d$.  Among the indices just selected by Markov's\ninequality, the values lie in $d^{-1}\\Z_{\\ge0}\\cap[0,K_0C\\log X]$.\nThere are at most $dK_0C\\log X+1$ such values.  Pigeonhole therefore gives\none value occurring at least\n\\[\n \\frac{(1-1/C)X}{dK_0C\\log X+1}\n\\]\ntimes.  Taking, for example, $K_0=2$ gives a lower bound of order\n$\\gg_{C,d}X/\\log X$.  The value may depend on $X$; no single value\nrecurring in every band is established.\n\nThe growth input is essential: the rational quadratic example in\nProposition~\\ref{long251:res:polynomialcountermodel} has strictly increasing\ntails $T_N=2(N+4)^2$.  For the primes, however, repeated values give\nzero difference, not the required nonintegral one.\n\nThe terminal term in the exact identity cannot simply be omitted.  The factorial-gap argument in the proof of\nProposition~\\ref{long251:res:gap-nonperiodic} shows that\n$T_X\\ge g_{X+1}/2$ is unbounded, whereas $p_1+T_0$ is fixed.\nThe exact identity therefore gives\n$\\sum_{N=0}^{X}T_N>p_{X+1}$ for arbitrarily large $X$.\nIt does not establish that this strict inequality holds eventually.\n\n\\paragraph{The interval condition at congruent indices.}\nSuppose the sum is rational, let $d$ be the eventual odd reduced\ndenominator, and let $t$ be the order of $2$ modulo $d$, with $t=1$ when\n$d=1$.  Beyond the denominator cutoff, $M\\equiv N\\pmod t$ forces\n$T_M-T_N\\in\\Z$.  Neither this difference nor its negative can then lie in\n$(\\tfrac12,1)$.  Thus a witness in that interval, at two such congruent\nindices, would contradict rationality without a further condition on\nthe intervening gaps.  This uses only the interval occurring in\nTheorem~\\ref{long251:res:signedwindow}; it does not assert that its other\nhypotheses hold for arbitrary congruent pairs.\n\n\\subsection{A bounded sequence with recurring values and a rational sum}\n\\label{long251:sec:xr-boundedpolignac}\n\nThe following bounded example isolates the recurring-values argument\nwithout a growth estimate.  Logarithmic growth was needed in\nTheorem~\\ref{long251:res:polignacfail} to obtain cumulative size\n$n\\log n$, but is unnecessary for the recurrence itself.\n\n\\begin{proposition}[recurring values with integral tails]\\label{long251:xr:boundedpolignac}\nPut $U_0=4$ and, for $n\\ge1$, $U_n=6$ when $n=k!$ for some $k\\ge3$ and $U_n=4$\notherwise, and set $a_n=2U_{n-1}-U_n$ for $n\\ge1$.  Then $a_n\\in\\{2,4,8\\}$.\nFor every $k\\ge3$, the value $2$ occurs at index $k!$ and the value $4$\nat index $2\\,k!$, so both recur infinitely often at indices divisible by any fixed\n$t\\ge1$.  The series $\\sum_{n\\ge1}a_n2^{-n}$ equals $4$ and every tail\n$\\sum_{j\\ge1}a_{N+j}2^{-j}$ equals the integer $U_N$.\n\\end{proposition}\n\n\\begin{proof}\nFor $k\\ge3$ the index $k!$ is even and at least $6$, and $k!-1$ is odd, so the\npreceding tail is $4$; hence $a_{k!}=8-6=2$,\n$a_{k!+1}=12-4=8$, and $a_n=8-4=4$ elsewhere.  For $k\\ge2$ one has\n$k!<2\\,k!<(k+1)!$, so $2\\,k!$ is not a factorial.  Its odd predecessor\n$2\\,k!-1\\ge3$ is not a factorial either; therefore $a_{2k!}=4$.  Since $t\\mid k!$ for every\nlarge $k$, both values recur in the residue class $0$ modulo $t$.  The identity\n$a_n2^{-n}=U_{n-1}2^{-(n-1)}-U_n2^{-n}$ telescopes to\n$\\sum_{j=1}^{m}a_{N+j}2^{-j}=U_N-U_{N+m}2^{-m}$, and $U$ is bounded, so the\nendpoint vanishes.\n\\end{proof}\n\nThe indices with coefficient $2$ are precisely the factorials $k!$,\n$k\\ge3$.  They occur infinitely often with unbounded gaps, which is\nimpossible for a value recurring in an eventually periodic sequence.\nThus even this bounded coefficient sequence is nonperiodic, despite its\nrational dyadic sum.  Its cumulative sum has linear growth.\nThe logarithmic construction in\nTheorem~\\ref{long251:res:polignacfail} adds unboundedness and cumulative\ngrowth $N\\log N$ without changing the telescoping mechanism.  Neither\nconstruction guarantees prime cumulative positions.\n\n\\subsection{Numerical counts and proportions}\\label{long251:sec:xr-densities}\n\nThe scan in Section~\\ref{long251:sec:measurements} used the\n$6\\,841\\,648$ primes below $1.2\\times10^8$ and computed double-precision\ntails for offsets $h\\le16$.  We reproduce its reported counts for\n$h=1,\\ldots,4$.  The denominator of each proportion is the number of\ntested basepoints, $6\\,841\\,565-h$, rather than the prime count.\n\n\\begin{center}\n\\small\n\\begin{tabular}{@{}rrrrrr@{}}\n\\toprule\n$h$ & events & proportion & $\\delta=+2$ & $\\delta=-2$ & last event at prime\\\\\n\\midrule\n1 & 56\\,427 & 0.008248 & 28\\,022 & 28\\,405 & 119\\,995\\,753\\\\\n2 & 31\\,979 & 0.004674 & 15\\,742 & 16\\,237 & 119\\,993\\,807\\\\\n3 & 30\\,233 & 0.004419 & 15\\,093 & 15\\,140 & 119\\,998\\,321\\\\\n4 & 29\\,262 & 0.004277 & 14\\,606 & 14\\,656 & 119\\,993\\,473\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\nThese are separate counts for each offset $1\\le h\\le16$, not counts\nof indices successful for all offsets at once.  The proportions range\nfrom $0.00418$ to $0.008248$, with nearly balanced signs at each offset.\nFor $h=1$, the first row below gives the proportions in the eight bands;\nthe second multiplies each by its band's mean of $\\log p$:\n\\[\n\\begin{array}{l}\n 0.011285,\\;0.008951,\\;0.008303,\\;0.007936,\\;0.007651,\\;0.007507,\\;0.007253,\\;0.007094;\\\\[2pt]\n 0.17671,\\;0.15058,\\;0.14420,\\;0.14067,\\;0.13766,\\;0.13667,\\;0.13333,\\;0.13148 .\n\\end{array}\n\\]\nThe ratio of the last rescaled proportion to the first is $0.744$ at\n$h=1$ and lies between $0.744$ and $0.8121$ across the measured offsets.\nTheorem~\\ref{long251:res:sparse} gives limiting density zero for each\nfixed offset.  The bound $O_h(X/\\log\\log X)$ derived after its proof is\nonly an upper bound; it predicts neither a monotone decline nor an\nasymptotic proportion.  The finite-band ratios therefore do not\nestablish an asymptotic rate.\n\n\\subsection{Two corrections}\\label{long251:sec:xr-corrections}\n\nWe distinguish a bound for the actual tail from the polynomial series\nused to compute it, and then correct an insufficient offset condition.\n\n\\paragraph{The tail bound in the free-pair program.}\nThe $4096$-bit free-pair program linked through\nSection~\\ref{long251:sec:measurements} bounds $T_s$ by summing the first\n$400$ terms of $1250(s+j+2)^4 2^{-j}$ after rounding each down, then\nadding $1$.  This does not necessarily bound that polynomial series:\nat the recorded $s=1\\,270\\,306$, the returned value is $40$ below the\nexact sum $1250P(s+2)$, with $P$ as in\nProposition~\\ref{long251:res:explicit-remainder}.\n\nThe returned bound nevertheless has a valid justification.  The stronger\nestimate $p_n\\le2(n+5)^4$ proved in\nAppendix~\\ref{long251:app:prime-bound} gives\n\\[\n T_s\\le2P(s+6)<625(s+3)^4\\qquad(s\\ge0).\n\\]\nThe last expression is already the first summand of the program, and\nall its remaining summands are nonnegative.  The strict inequality\nfollows from\n\\[\n 625(s+3)^4-2P(s+6)\n =623s^4+7436s^3+32958s^2+62972s+40437>0.\n\\]\nThus the returned bound exceeds the actual tail, even though its stated\ncomparison with the larger polynomial series fails.  This validates the\nanalytic bound only; it neither controls floating-point errors in other\nscans nor reruns the denominator-exclusion search.\n\n\\paragraph{An insufficient condition.}  Theorem~\\ref{long251:res:freepair} allows the offset to vary.  For each\nmodulus $t$, one must still find pairs of congruent indices beyond every\nprescribed cutoff whose tail difference is nonintegral.\nTwo even coefficient values differing by $2$, each occurring infinitely\noften in the same residue class of indices, do not give the required\ntail difference for an integer-coefficient recurrence: the sequence of\nTheorem~\\ref{long251:res:polignacfail} has both properties and only\nintegral tail shifts.\n\n% Part: part family_catalogue\n\\section{Arithmetic-progression reformulations of integrality}\n\\label{long251:app:affine-detail}\nA longer observed block can appear to provide a stronger arithmetic test.\nHere it does not: the recurrence inserts the same block sum into the\nquantity tested and into the comparison progression, so that sum cancels.\nThe moving progression tests even integrality of the original difference.\nUnder the stated growth bound, separation from the fixed progression\n$2^r\\Z$ tests nonintegrality.  We record the two reductions separately.\n\nFix $h$ and abbreviate $D_N=\\sigma_h(N)$ and\n$\\delta_N=g_{N+h+1}-g_{N+1}$.  Iterating over a longer block gives\n\\[\n B_{h,N,r}=\\sum_{i=0}^{r-1}2^{r-1-i}\\delta_{N+i};\n \\qquad D_{N+r}=2^rD_N-B_{h,N,r}.\n\\]\nBoth the progression $-B_{h,N,r}+2^{r+1}\\Z$ and the value $D_{N+r}$\ncontain the same integer offset.  Substituting the recurrence cancels\nit, leaving a condition on $D_N$ alone.  Thus increasing the block\nlength in this particular test adds no arithmetic information.\n\n\\begin{theorem}[equivalent arithmetic-progression tests]\\label{long251:res:affinecollapse}\\compatlabel{res:affinecollapse}\nFor every rational dyadic tail recurrence and all $h,N,r\\ge0$,\n\\begin{equation}\n D_{N+r}\\in-B_{h,N,r}+2^{r+1}\\Z\\quad\\Longleftrightarrow\\quad D_N\\in2\\Z.\n\\label{eq:affinecollapse}\\compatlabel{long251:eq:affinecollapse}\n\\end{equation}\nConsequently, if every $\\delta_N$ is even, then\n\\begin{equation}\n \\begin{split}\n &\\bigl(\\forall N_0\\ \\exists N,r:\\ N_0<N\\text{ and }\n D_{N+r}\\notin-B_{h,N,r}+2^{r+1}\\Z\\bigr)\\\\\n &\\hspace{12mm}\\Longleftrightarrow\\quad\n D_N\\notin\\Z\\text{ for arbitrarily large }N.\n \\end{split}\n\\label{eq:affinecofinal}\\compatlabel{long251:eq:affinecofinal}\n\\end{equation}\n\nThere is a second equivalence.  Let $b:\\N\\to\\Q$ satisfy\n$|D_N|\\le b(N)$ for every $N$, and suppose that for every $N$ and every\npositive integer $q$ there is an $r$ with\n\\begin{equation}\n 2b(N+r)q<2^r.\n\\label{eq:dyadicscale}\\compatlabel{long251:eq:dyadicscale}\n\\end{equation}\nThen\n\\begin{equation}\n \\begin{split}\n &\\forall N_0\\ \\exists N,r:\\ N_0<N\\text{ and }\n   \\forall z\\in\\Z,\\quad\n   b(N+r)<|B_{h,N,r}-2^rz|\\\\\n &\\hspace{35mm}\\Longleftrightarrow\\quad\n D_N\\notin\\Z\\text{ for arbitrarily large }N.\n \\end{split}\n\\label{eq:fixedcollapse}\\compatlabel{long251:eq:fixedcollapse}\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nSubstituting $D_{N+r}=2^rD_N-B_{h,N,r}$ into the first membership condition cancels the\nobserved block from both sides and leaves $D_N=2z$.  This proves\n\\eqref{eq:affinecollapse}.  When every $\\delta_N$ is even, the recurrence\nalso gives $D_{N+1}\\in2\\Z$ if and only if $D_N\\in\\Z$.\nThus eventual integrality would force eventual even integrality and\ncontradict the left side of~\\eqref{eq:affinecofinal}.  This proves the\nforward implication.  Conversely, given a cutoff, choose a nonintegral\n$D_N$ beyond it and take $r=0$ in~\\eqref{eq:affinecollapse}.\n\nFor~\\eqref{eq:fixedcollapse}, eventual integrality and the block identity give\nsome $z\\in\\Z$ with\n\\[\n B_{h,N,r}-2^rz=-D_{N+r},\n\\]\nso the displayed strict separation contradicts $|D_{N+r}|\\le b(N+r)$.\nConversely, if $D_N$ is nonintegral, choose a positive integer $q$ with\n$1/q\\le\\operatorname{dist}(D_N,\\Z)$.  Such a $q$ exists because $\\Z$ is\nclosed and $D_N\\notin\\Z$.  Choose $r$ from \\eqref{eq:dyadicscale}.  The block identity and the triangle inequality\ngive, for every $z\\in\\Z$,\n\\[\n |B_{h,N,r}-2^rz|\n \\ge 2^r|D_N-z|-|D_{N+r}|\n \\ge \\frac{2^r}{q}-b(N+r)>b(N+r).\n\\]\nThe same $r$ works for all integers $z$, as required.\n\\end{proof}\n\nEvenness is essential in \\eqref{eq:affinecofinal}.  Without it, the\nrecurrence $T_N=N+1$, $g_n=n-1$ for $n\\ge1$, has\n$D_N=\\sigma_1(N)=1$ and $\\delta_N=1$ at every index.  Thus $D_N$ is never\nan even integer but is always an integer.  By\n\\eqref{eq:affinecollapse}, the left side of\n\\eqref{eq:affinecofinal} holds while the right side fails.\n\nThe growth hypothesis \\eqref{eq:dyadicscale} is equivalent to\n\\[\n \\liminf_{n\\to\\infty}2^{-n}b(n)=0.\n\\]\nIndeed, writing $n=N+r$, its inequality becomes\n$2^{-n}b(n)<1/(2^{N+1}q)$.  Since $b(n)\\ge0$, finding such an $n\\ge N$\nfor every $N$ and every positive integer $q$ is exactly the stated\nliminf condition.  In particular it holds for polynomial bounds and for\n$Cc^n$ with fixed $C>0$ and $1<c<2$, but fails for $b(n)=2^n$.\nA limit of zero is not necessary: the bound $b(n)=1$ at odd indices and\n$b(n)=2^n$ at even indices also satisfies the hypothesis.  This example\nconcerns the growth condition alone, not a bound for the prime tails.\nOnly one suitable scale is needed to magnify the positive distance to\n$\\Z$ beyond the error; no comparison at every sufficiently large scale\nis required.\n\nThe proof in fact uses no rationality of $D_N$: the integer $q$ is chosen\nfrom its positive distance to $\\Z$, not from a reduced denominator.\nThus \\eqref{eq:fixedcollapse} holds for real integer-coefficient\nrecurrences under the same bound and growth hypothesis.  The same is\ntrue of \\eqref{eq:affinecollapse}, and of \\eqref{eq:affinecofinal} when the\ncoefficient differences are even.  These extensions follow from the\nprinted argument; the Lean proof covers the rational case stated in the\ntheorem.\n\nThe first test is exactly even integrality of $D_N$, regardless of $r$.\nThe fixed-progression test compares $2^r\\operatorname{dist}(D_N,\\Z)$\nwith the bound on the remaining tail.  When $D_N$ is nonintegral, the\ngrowth hypothesis makes the former larger than twice the latter at a\nsuitable scale.  Rewriting the\ntest does not provide an estimate about the distribution of consecutive primes.\n\n\\subsection{A signed comparison with more analytic data}\n\\label{long251:sec:signed-dirichlet-comparison}\nA signed variant appears in the totient companion,\n\\href{erdos249-totient-reasoning-surface.pdf}{\\emph{The Binary Totient Series}},\nunder ``Signed interpolation with entire Dirichlet defects''.\nWe recall its different hypotheses to delimit the comparison. Around the totient sequence,\nsigned integer changes bounded by any positive $f(n)\\to\\infty$ fill an\ninterval at every reciprocal integer base. They can preserve the odd\ncoefficients, every fixed eventual congruence, all arithmetic-progression\nDirichlet singularities, and prescribed finite root-of-unity jets. The\nproof takes finite differences of increasing order on separated finite\nsets of indices.  The support of the changes has upper Banach density\nzero.  The union of the intervals spanned by those sets has ordinary\ndensity zero and upper Banach density one.\n\nThis theorem allows the bound on the corrections to grow arbitrarily slowly and retains more\nanalytic observations. It uses both signs, however, and does not subsume\nthe one-sided prime-gap perturbation theorem proved here. In particular,\nit constructs comparison integer sequences, not consecutive prime gaps,\nand neither theorem supplies an irrationality proof for the actual prime\nseries. The companion contains a full ordinary proof and the precise\nblock estimate in terms of the prescribed bound; no infinite formal theorem is claimed.\n\n\\section{Conclusions}\\label{sec:erdos-251-complete-family-map}\nThe construction's fixed ordinary totals protect cumulative residues;\nits variable weighted totals attain the target.  The resulting family\ncontains rational and irrational sums with the stated congruences and\nasymptotic block statistics, but does not preserve consecutive primality.\n\nFor the actual primes, the complete tails are identified and their integral\nshifts give an exact criterion.  The remaining task is to establish the\nrequired nonintegrality, not another recurrence identity.  The small-pair\nroute requires simultaneous witnesses arbitrarily late for every positive\nshift.  Finite separations and denominator exclusions do not establish\nthat infinite family of witnesses.\n\n% BEGIN pin-faithful-source-inventory\n\\section{Source index for the short paper}\\label{long251:app:short-source-index}\n\\papersectiontarget{short-source-index}\n\\begingroup\n\\renewcommand{\\commit}{99f4bf47422abbd8757cbb22b50ba079d764d3a7}\n\\renewcommand{\\commitshort}{99f4bf47422a}\nThe links below identify further formal definitions and statements used\nin the short paper, at its original revision\n\\texttt{99f4bf47422a} and line numbers.  This local source pin applies only\nto the index below; Appendix~\\ref{long251:app:index} retains the long\nrecord's separate pin.  Each link is\nlabelled by the mathematics it states.\n\\par\\medskip\n\\begingroup\n\\small\\raggedright\n\\noindent\\textbf{Finite series.} \\lword{Erdos251/PrimeGapDyadicTail.lean}{47}{primeGap0_zero}{The first prime gap is one};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{51}{primeGap0_one}{The second prime gap is two};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{101}{dyadicPartialSumQ}{Finite dyadic sums};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{121}{dyadicDifferencePartialSumQ}{Finite dyadic sums of consecutive differences};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{138}{dyadicPartialSumQ_eq_start_add_differences}{Summation by parts for a rational sequence};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{111}{prime0DisplayedPartialSumQ_eq_two_mul}{The two prime-series normalisations differ by a factor of two};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{172}{prime0_dyadic_summation_by_parts}{Summation by parts for the primes};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{183}{primeDyadicTerm}{Terms of the prime series};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{192}{primeGapDyadicTerm}{Terms of the prime-gap series};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{202}{primeGapDyadicTerm_eq}{The identity relating prime and gap terms}.\\par\n\n\\noindent\\textbf{Convergent series.} \\lword{Erdos251/PrimeGapDyadicTail.lean}{385}{summable_primeGapDyadicTerm_of_summable_primeDyadicTerm}{Convergence of the gap series from convergence of the prime series};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{404}{tsum_primeDyadicTerm_eq_two_add_primeGap}{The infinite prime-to-gap identity under summability};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{188}{primeDisplayedDyadicTerm}{Prime terms with denominator powers starting at zero};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{196}{primeDisplayedDyadicTerm_eq_two_mul}{The factor-of-two identity for individual terms};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{435}{irrational_tsum_primeDyadicTerm_iff_primeGap}{Equivalent irrationality of the prime and gap series};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{444}{tsum_primeDisplayedDyadicTerm_eq_four_add_two_primeGap}{The infinite identity in the second normalisation};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{459}{irrational_tsum_primeDisplayedDyadicTerm_iff_primeGap}{Equivalent irrationality in the second normalisation};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{360}{prime0_le_polynomial}{An elementary polynomial upper bound for the primes};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{379}{summable_primeDyadicTerm}{Convergence of the prime series};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{427}{tsum_primeDyadicTerm_eq_two_add_primeGap_unconditional}{The prime-to-gap identity with convergence proved}.\\par\n\n\\noindent\\textbf{Recurrences.} \\lword{Erdos251/PrimeGapDyadicTail.lean}{482}{DyadicTailRecurrence}{The recurrence satisfied by rescaled dyadic tails};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{544}{tailShift}{The difference between two terms of a recurrence};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{647}{dyadicTailBlock}{The weighted integer sum accumulated by iteration};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{653}{tail_iterate_eq_pow_mul_sub_block}{Iteration of the recurrence};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{667}{tailShift_eq_scaled_sub_block}{A shift as a multiple of a tail minus an integer};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{563}{tailShift_succ}{The recurrence for consecutive tail differences};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{677}{ratIntegral_sub_int_iff}{Subtracting an integer preserves integrality};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{802}{tailShift_integral_iff_scaledTail}{The criterion for a shift to be integral}.\\par\n\n\\noindent\\textbf{Denominators and integral shifts.} \\lword{Erdos251/PrimeGapDyadicTail.lean}{1279}{tailShift_integral_iff_den_dvd_mersenne}{The denominator divisibility criterion for an integral shift};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{695}{ratIntegral_totientMultiplier_of_odd_den}{Euler's congruence for an odd denominator};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{877}{tailShift_integral_totient_of_odd_den}{An integral shift of totient length};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{888}{tailShift_integral_succ}{An integral shift remains integral at the next index};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{900}{tailShift_integral_add}{An integral shift remains integral at every later index};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1217}{tail_den_succ}{The denominator after one recurrence step};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1226}{tail_den_succ_eq_of_odd}{An odd denominator is unchanged by a step};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1236}{tail_den_succ_eq_half_of_even}{An even denominator is halved by a step};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{723}{ratIntegral_scaled_of_den_eq_pow_two_mul_odd}{Removing the power of two from a rational denominator};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{756}{ratIntegral_multiplier_tail_at_twoExponent}{An integral multiple after the power of two is removed};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{830}{rationalPrimeGapTailShift_eventuallyIntegral_of_den_eq}{Eventual integral shifts from the denominator factorisation};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{853}{rationalPrimeGapTailShift_eventuallyIntegral}{Eventual integral shifts under rationality}.\\par\n\n\\noindent\\textbf{Rationality criteria.} \\lword{Erdos251/PrimeGapDyadicTail.lean}{489}{rationalPrimeGapTailState}{Rational representation of a rescaled prime-gap tail};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{527}{rationalPrimeGapTailState_recurrence}{The recurrence for that rational representation};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{510}{exists_rationalPrimeGapTailState_representation_of_not_irrational}{Existence of the rational representation under rationality};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1479}{not_irrational_initial_iff_exists_integral_positive_tailShift}{Rationality is equivalent to one integral positive-length shift};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1525}{not_irrational_initial_iff_exists_eventually_integral_positive_tailShift}{Rationality is equivalent to eventual integrality of a fixed shift};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1551}{irrational_initial_iff_all_positive_tailShifts_nonintegral}{Irrationality is equivalent to nonintegrality of every positive-length shift};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1572}{irrational_initial_iff_cofinalNonintegralTailShifts}{Irrationality is equivalent to arbitrarily late nonintegral shifts of each length}.\\par\n\n\\noindent\\textbf{Small differences and counterexamples.} \\lword{Erdos251/PrimeGapDyadicTail.lean}{979}{tailShift_not_both_integral_of_small_pair_of_digit_ne}{Two small differences with unequal coefficients cannot both be integral};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1006}{not_eventuallyIntegralTailShift_of_cofinal_small_mismatch}{Arbitrarily late small unequal pairs exclude eventual integrality};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1112}{primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch}{The small-pair implication for prime gaps};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{57}{exists_primeGap0_gt}{Prime gaps exceed every prescribed bound};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1023}{primeGap0_not_eventually_periodic}{Prime gaps are not eventually periodic};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1129}{carryCoeff}{Coefficients obtained by telescoping};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1137}{carryPartialSum_eq}{The finite telescoping identity};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1157}{carryPartialSum_natCast_eq}{The telescoping example with coefficients equal to the index minus one};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1163}{carryCoeff_natCast_not_eventually_periodic}{Nonperiodicity of the telescoping example};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1097}{rationalPrimeGapTail_has_positive_shift_not_eventually_small}{Rationality forces some fixed shift to fail eventual smallness}.\\par\n\n\\noindent\\textbf{Further finite identities.} \\lword{Erdos251/PrimeGapDyadicTail.lean}{106}{prime0DisplayedPartialSumQ}{Finite prime sums in the second normalisation};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{124}{dyadicPartialSumQ_succ}{Adding one term to a dyadic partial sum};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{129}{dyadicDifferencePartialSumQ_succ}{Adding one term to a difference partial sum};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{152}{prime0_mono_step}{Consecutive primes are in increasing order};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{156}{primeGap0_cast}{The natural-number gap agrees with subtraction in the rationals};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{161}{primeGapPartialSumQ}{Finite sums of weighted prime gaps};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1133}{carryPartialSum}{Finite sums for the telescoping coefficients};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1177}{natCarryCoeff}{A nonnegative version of the telescoping coefficients};\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{1180}{natCarryCoeff_cast}{Agreement of the nonnegative and integer formulas}.\\par\n\\endgroup\n\\endgroup\n% END pin-faithful-source-inventory\n\n% Part: part back\n\\begin{thebibliography}{99}\n\\small\\raggedright\\setlength{\\itemsep}{0pt}\n\\bibitem{erdos1958} Paul Erd\\H{o}s, \\href{https://users.renyi.hu/~p_erdos/1958-19.pdf}{\\emph{Sur certaines s\\'eries \\`a valeur irrationnelle}}.\n  L'Enseignement Mathématique \\textbf{4} (1958), 93--100, doi:\\href{https://doi.org/10.5169/seals-34629}{10.5169/seals-34629}.\n\\bibitem{erdosgraham1980} Paul Erd\\H{o}s and Ronald L. Graham, \\href{https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf}{\\emph{Old and New Problems and Results in Combinatorial Number Theory}}.\n  Monographies de L'Enseignement Mathématique, L'Enseignement Mathématique, 1980.\n\\bibitem{erdospomerance1978} Paul Erd\\H{o}s and Carl Pomerance, \\href{https://doi.org/10.1007/BF01818569}{\\emph{On the largest prime factors of $n$ and $n+1$}}.\n  Aequationes Mathematicae \\textbf{17} (1978), 311--321, doi:\\href{https://doi.org/10.1007/BF01818569}{10.1007/BF01818569}.\n\\bibitem{erdos1988} Paul Erd\\H{o}s, \\href{https://doi.org/10.1017/CBO9780511897184.009}{\\emph{On the irrationality of certain series: problems and results}}.\n  New Advances in Transcendence Theory, Cambridge University Press, 1988, pp.~102--109, doi:\\href{https://doi.org/10.1017/CBO9780511897184.009}{10.1017/CBO9780511897184.009}.\n\\bibitem{erdos1932} Paul Erd\\H{o}s, \\href{https://users.renyi.hu/~p_erdos/1932-01.pdf}{\\emph{Beweis eines Satzes von Tschebyschef}}.\n  Acta Litterarum ac Scientiarum Szeged \\textbf{5} (1932), 194--198.\n\\bibitem{erdosstraus1963} Paul Erd\\H{o}s and Ernst G. Straus, \\emph{On the irrationality of certain Ahmes series}.\n  Journal of the Indian Mathematical Society (N.S.) \\textbf{27} (1964), 129--133. MR0175848.\n\\bibitem{erdosstraus1974} Paul Erd\\H{o}s and Ernst G. Straus, \\href{https://doi.org/10.2140/pjm.1974.55.85}{\\emph{On the irrationality of certain series}}.\n  Pacific Journal of Mathematics \\textbf{55} (1974), no.~1, 85--92, doi:\\href{https://doi.org/10.2140/pjm.1974.55.85}{10.2140/pjm.1974.55.85}.\n\\bibitem{fgkmt2018} Kevin Ford, Ben Green, Sergei Konyagin, James Maynard and Terence Tao, \\href{https://arxiv.org/abs/1412.5029v3}{\\emph{Long gaps between primes}}.\n  Journal of the American Mathematical Society \\textbf{31} (2018), 65--105, doi:\\href{https://doi.org/10.1090/jams/876}{10.1090/jams/876}; arXiv:\\href{https://arxiv.org/abs/1412.5029v3}{1412.5029v3}.\n\\bibitem{fridy1966} John A. Fridy, \\href{https://www.fq.math.ca/Scanned/4-3/fridy.pdf}{\\emph{Generalized bases for the real numbers}}.\n  The Fibonacci Quarterly \\textbf{4} (1966), no.~3, 193--201.\n\\bibitem{khinchin1964} A. Ya. Khinchin, \\emph{Continued Fractions}.\n  University of Chicago Press, 1964.\n\\bibitem{kovactao2024} Vjekoslav Kova\\v{c} and Terence Tao, \\href{https://arxiv.org/abs/2406.17593v4}{\\emph{On several irrationality problems for Ahmes series}}.\n  Acta Mathematica Hungarica \\textbf{175} (2025), 572--608, doi:\\href{https://doi.org/10.1007/s10474-025-01528-0}{10.1007/s10474-025-01528-0}; arXiv:\\href{https://arxiv.org/abs/2406.17593v4}{2406.17593v4}. Statement and section numbers refer to arXiv version 4.\n\\bibitem{kuperberg2023} Vivian Kuperberg, \\href{https://arxiv.org/abs/2210.09775v2}{\\emph{Sums of singular series with large sets and the tail of the distribution of primes}}.\n  The Quarterly Journal of Mathematics \\textbf{74} (2023), no.~4, 1457--1479, doi:\\href{https://doi.org/10.1093/qmath/haad030}{10.1093/qmath/haad030}; arXiv:\\href{https://arxiv.org/abs/2210.09775v2}{2210.09775v2}. Statement numbers refer to arXiv version 2, 15 June 2023.\n\\bibitem{land2026} Johan Land, \\href{https://github.com/beetree/math_erdos_251}{\\emph{A conditional proof of the irrationality of $\\sum_{n\\ge1}p_n2^{-n}$ under a uniform Hardy--Littlewood prime-tuples conjecture}}.\n  Research draft, 5 September 2026. Formalisation material accompanies the draft.\n\\bibitem{lean4} Leonardo de Moura and Sebastian Ullrich, \\href{https://doi.org/10.1007/978-3-030-79876-5_37}{\\emph{The Lean 4 theorem prover and programming language}}.\n  Automated Deduction -- CADE 28, Lecture Notes in Computer Science, Springer, 2021, pp.~625--635, doi:\\href{https://doi.org/10.1007/978-3-030-79876-5_37}{10.1007/978-3-030-79876-5\\_37}.\n\\bibitem{mathlib} {The mathlib Community}, \\href{https://doi.org/10.1145/3372885.3373824}{\\emph{The Lean mathematical library}}.\n  Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, ACM, 2020, pp.~367--381, doi:\\href{https://doi.org/10.1145/3372885.3373824}{10.1145/3372885.3373824}. Describes a Lean 3-era snapshot; the repository lock identifies the library used by the present Lean 4 sources.\n\\bibitem{maynard2015} James Maynard, \\href{https://doi.org/10.4007/annals.2015.181.1.7}{\\emph{Small gaps between primes}}.\n  Annals of Mathematics \\textbf{181} (2015), 383--413, doi:\\href{https://doi.org/10.4007/annals.2015.181.1.7}{10.4007/annals.2015.181.1.7}.\n\\bibitem{mv2007} Hugh L. Montgomery and Robert C. Vaughan, \\href{https://doi.org/10.1017/CBO9780511618314}{\\emph{Multiplicative Number Theory I: Classical Theory}}.\n  Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2007, doi:\\href{https://doi.org/10.1017/CBO9780511618314}{10.1017/CBO9780511618314}.\n\\bibitem{schlagepuchta2011} Jan-Christoph Schlage-Puchta, \\href{https://arxiv.org/abs/1105.1451v1}{\\emph{The irrationality of some number theoretical series}}.\n  Acta Arithmetica \\textbf{126} (2007), no.~4, 295--303, doi:\\href{https://doi.org/10.4064/aa126-4-1}{10.4064/aa126-4-1}; arXiv:\\href{https://arxiv.org/abs/1105.1451v1}{1105.1451v1}. Published in 2007; arXiv upload is from 2011. Lemma and preprint page locators refer to arXiv version 1.\n\\bibitem{short2009} Ian Short, \\href{https://arxiv.org/abs/0912.1997v1}{\\emph{Ford Circles, Continued Fractions, and Rational Approximation}}.\n  The American Mathematical Monthly \\textbf{118} (2011), no.~2, 130--135, doi:\\href{https://doi.org/10.4169/amer.math.monthly.118.02.130}{10.4169/amer.math.monthly.118.02.130}; arXiv:\\href{https://arxiv.org/abs/0912.1997v1}{0912.1997v1}. Preprint title: Ford circles, continued fractions, and best approximation of the second kind. Preprint theorem locators refer to arXiv version 1.\n\\bibitem{vandoornkovac2025} Wouter van Doorn and Vjekoslav Kova\\v{c}, \\href{https://arxiv.org/abs/2509.24971v3}{\\emph{Lacunary sequences whose reciprocal sums represent all rational numbers in an interval}}.\n  Acta Arithmetica \\textbf{223} (2026), 275--295, doi:\\href{https://doi.org/10.4064/aa251001-13-1}{10.4064/aa251001-13-1}; arXiv:\\href{https://arxiv.org/abs/2509.24971v3}{2509.24971v3}. Statement numbers refer to arXiv version 3.\n\\bibitem{zhang2014} Yitang Zhang, \\href{https://doi.org/10.4007/annals.2014.179.3.7}{\\emph{Bounded gaps between primes}}.\n  Annals of Mathematics \\textbf{179} (2014), 1121--1174, doi:\\href{https://doi.org/10.4007/annals.2014.179.3.7}{10.4007/annals.2014.179.3.7}.\n\\bibitem{erdosproblems} Thomas F. Bloom, \\href{https://www.erdosproblems.com/251}{\\emph{Erd\\H{o}s Problem \\#251}}.\n  2026. Accessed 6 September 2026.\n\\bibitem{erdosproblems251thread} {Erdős Problems contributors}, \\href{https://www.erdosproblems.com/forum/thread/251}{\\emph{Erd\\H{o}s Problem \\#251 discussion thread}}.\n  2026. Accessed 6 September 2026: Tao comment of 7 October 2025 and Land comments of 6 September 2026.\n\\bibitem{kovac2026} {ChatGPT 5.4 Pro (orchestrated by Vjeko Kova\\v{c})}, \\href{https://web.math.pmf.unizg.hr/~vjekovac/files/Erdos_problem_251.pdf}{\\emph{On the Erd\\H{o}s problem \\#251}}.\n  Unpublished note, Department of Mathematics, University of Zagreb, 2026. Accessed 6 September 2026.\n\\bibitem{formalconjectures251} {The Formal Conjectures Authors}, \\href{https://github.com/google-deepmind/formal-conjectures/blob/f776d2f2039351b00737ffcafb9d7d7666e1d9af/FormalConjectures/ErdosProblems/251.lean}{\\emph{FormalConjectures.ErdosProblems.251}}.\n  2025. Pinned statement source, not a proof of Problem 251; accessed 28 July 2026.\n\\bibitem{ringer2026} Stefan Ringer, \\href{https://github.com/StefanRinger/erdos-251}{\\emph{Local gap statistics, telescoping, and normality: a local-pattern approach to Erd\\H{o}s problem 251}}.\n  Preprint, 11 September 2026. Mutable main-branch TeX consulted 16 September 2026; the cited conditional statements were rechecked 18 September 2026. Formalisation material accompanies the draft.\n\\bibitem{crmarickovac2025} Ton\\'{c}i Crmari\\'{c} and Vjekoslav Kova\\v{c}, \\href{https://arxiv.org/abs/2504.18712v1}{\\emph{On the irrationality of certain super-polynomially decaying series}}.\n  Colloquium Mathematicum \\textbf{179} (2025), 55--68, doi:\\href{https://doi.org/10.4064/cm9628-5-2025}{10.4064/cm9628-5-2025}; arXiv:\\href{https://arxiv.org/abs/2504.18712v1}{2504.18712v1}. Lemma 4 is cited using arXiv version 1.\n\\bibitem{adm2022} Boris Adamczewski, Michael Drmota and Clemens M\\\"ullner, \\href{https://arxiv.org/abs/2009.14773v2}{\\emph{(Logarithmic) densities for automatic sequences along primes and squares}}.\n  Transactions of the American Mathematical Society \\textbf{375} (2022), no.~1, 455--499, doi:\\href{https://doi.org/10.1090/tran/8476}{10.1090/tran/8476}; arXiv:\\href{https://arxiv.org/abs/2009.14773v2}{2009.14773v2}. Theorems 1.2 and 1.4 refer to arXiv version 2, 13 April 2021; journal publication is 2022.\n\\bibitem{vandoorn2025} Wouter van Doorn, \\href{https://arxiv.org/abs/2502.02200v2}{\\emph{Partitions with prescribed sum of reciprocals: asymptotic bounds}}.\n  2025; arXiv:\\href{https://arxiv.org/abs/2502.02200v2}{2502.02200v2}. Version 2, 23 July 2025. \n\\bibitem{glabprus2025} Szymon G{\\l}\\k{a}b and Franciszek Prus-Wi\\'sniowski, \\href{https://arxiv.org/abs/2512.17285v1}{\\emph{Achievement sets -- current results and open problems}}.\n  Real Analysis Exchange (2026), doi:\\href{https://doi.org/10.14321/realanalexch.1766383782}{10.14321/realanalexch.1766383782}; arXiv:\\href{https://arxiv.org/abs/2512.17285v1}{2512.17285v1}. Advance publication, first available in Project Euclid 8 June 2026. Consulted text remains arXiv:2512.17285v1 (19 December 2025).\n\\bibitem{prusptak2024} Franciszek Prus-Wi\\'sniowski and Jolanta Ptak, \\href{https://arxiv.org/abs/2412.08768v1}{\\emph{Achievable Cantorvals almost without reversed Kakeya conditions}}.\n  2024; arXiv:\\href{https://arxiv.org/abs/2412.08768v1}{2412.08768v1}. Version 1 submitted 11 December 2024. The sparse indices satisfy the overlap inequality, not its strict term-dominating reverse.\n\\bibitem{bartoszewicz2014} Artur Bartoszewicz, Ma{\\l}gorzata Filipczak and Emilia Szymonik, \\href{https://arxiv.org/abs/1304.4218v2}{\\emph{Multigeometric sequences and Cantorvals}}.\n  Central European Journal of Mathematics \\textbf{12} (2014), no.~7, 1000--1007, doi:\\href{https://doi.org/10.2478/s11533-013-0396-4}{10.2478/s11533-013-0396-4}; arXiv:\\href{https://arxiv.org/abs/1304.4218v2}{1304.4218v2}.\n\\bibitem{kuperbergweighted2025} Vivian Kuperberg, \\href{https://arxiv.org/abs/2301.06095v1}{\\emph{Sums of singular series along arithmetic progressions and with smooth weights}}.\n  International Journal of Number Theory \\textbf{21} (2025), no.~1, 53--74, doi:\\href{https://doi.org/10.1142/S1793042125500046}{10.1142/S1793042125500046}; arXiv:\\href{https://arxiv.org/abs/2301.06095v1}{2301.06095v1}. Preprint uploaded 15 January 2023; journal publication is 2025.\n\\bibitem{jha2026} Abhishek Jha, \\href{https://arxiv.org/abs/2605.23014v2}{\\emph{The Poisson Tail Conjecture for primes in short intervals}}.\n  2026; arXiv:\\href{https://arxiv.org/abs/2605.23014v2}{2605.23014v2}. Substantially revised version 2, 12 September 2026, 31 pages. Conditional statements must retain their strong Hardy--Littlewood hypotheses.\n\\bibitem{polymath2014} {D. H. J. Polymath}, \\href{https://arxiv.org/abs/1407.4897v4}{\\emph{Variants of the Selberg sieve, and bounded intervals containing many primes}}.\n  Research in the Mathematical Sciences \\textbf{1} (2014), article 12, doi:\\href{https://doi.org/10.1186/s40687-014-0012-7}{10.1186/s40687-014-0012-7}; arXiv:\\href{https://arxiv.org/abs/1407.4897v4}{1407.4897v4}. Theorem 1.4(i) refers to arXiv version 4 (22 December 2014); the journal numbers it Theorem 4(i). An erratum is recorded at doi:10.1186/s40687-015-0033-x.\n\\bibitem{nitecki2015} Zbigniew Nitecki, \\href{https://arxiv.org/abs/1106.3779v2}{\\emph{Cantorvals and Subsum Sets of Null Sequences}}.\n  The American Mathematical Monthly \\textbf{122} (2015), no.~9, 862--870, doi:\\href{https://doi.org/10.4169/amer.math.monthly.122.9.862}{10.4169/amer.math.monthly.122.9.862}; arXiv:\\href{https://arxiv.org/abs/1106.3779v2}{1106.3779v2}. Consulted preprint: Subsum Sets: Intervals, Cantor Sets, and Cantorvals, version 2 (8 July 2013). Theorem 14 is attributed there to Guthrie--Nymann; its locator is not journal pagination.\n\\bibitem{miskaprusptak2023} Piotr Miska, Franciszek Prus-Wi\\'sniowski and Jolanta Ptak, \\href{https://ruj.uj.edu.pl/server/api/core/bitstreams/d6630f7b-e6ee-4de8-8a1b-81c7b4c59d2e/content}{\\emph{More on Kakeya Conditions for Achievement Sets}}.\n  Results in Mathematics \\textbf{78} (2023), article 113, doi:\\href{https://doi.org/10.1007/s00025-023-01890-x}{10.1007/s00025-023-01890-x}. Repairs an estimate in the 2021 proof, preserving its uniqueness conclusion, and gives a simpler proof of a weaker theorem without that conclusion.\n\\bibitem{marchwickimiska2021} Jacek Marchwicki and Piotr Miska, \\href{https://link.springer.com/article/10.1007/s00025-021-01479-2}{\\emph{On Kakeya Conditions for Achievement Sets}}.\n  Results in Mathematics \\textbf{76} (2021), article 181, doi:\\href{https://doi.org/10.1007/s00025-021-01479-2}{10.1007/s00025-021-01479-2}. Theorem 2.1 is to be read with the proof repair in Miska--Prus-Wi\\'sniowski--Ptak (2023).\n\\bibitem{nowakowski2025} Piotr Nowakowski, \\href{https://arxiv.org/abs/2512.17761v1}{\\emph{On a new condition implying that an achievement set is a Cantorval and its applications}}.\n  2025; arXiv:\\href{https://arxiv.org/abs/2512.17761v1}{2512.17761v1}. Version 1, 19 December 2025. Theorem 3.1 requires the Star Procedure of Definition 2 never to break; no application to the present factorial weights is asserted.\n\\end{thebibliography}\n\n% ---- part extended_record ----\n% ---- part family_catalogue ----\n% ---- part back ----\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":185936,"sha256":"sha256:db2a5b20038e07a9770ef3d11493bbe706ee883da69d3add57267bc2534c84bd","cite_as":"Plectis reader corpus, paper:erdos251-prime-gap-reasoning-surface","source_repository":"plectis-erdos","source_path":"paper/251/erdos251-prime-gap-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:088d36d7ba3d3ee0cee2df99e8915927999698f19889f8ea0b8d55b3721ec778","byte_count":6923,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% This is the only copy of the shared preamble.  The eight short notes, the\n% long records assembled from paper/reasoning-parts other than #249 and #257\n% (which carry their own preambles), and the synthesis paper include it with\n% \\input{problem-note-preamble}; paper/Makefile builds from this directory\n% with the directory on the tectonic search path.\n%\n% Shared preamble for the Erdős Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erdős problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{8e58c395e9807ef3374d7365c62cf04d20ee6331}\n\\newcommand{\\commitshort}{9e231ce4371f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-erdos/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-erdos/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\usepackage{longtable}  % the notes' declaration tables; loaded here so the assembled records inherit it\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n% ---------------------------------------------------------------------------\n% Evidence marks.  paper-evidence.tex places \"Lean\" and \"Comparator\" links in\n% the margin beside each result from the paper's generated file\n% paper/evidence/<paper>.tex, which each paper inputs after this preamble.\n% ---------------------------------------------------------------------------\n\\input{paper-evidence}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Production provenance.  The wording is fixed by the corpus provenance\n% contract and reproduced verbatim in every manuscript in this repository, so\n% the papers cannot drift into different accounts of how they were made.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erdős problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every Lean mark in the margin points to propositions that the\n  pinned Lean kernel accepts, with no \\texttt{sorry}, no added axiom, and no\n  unchecked evaluation.  That claim concerns the formal statements only; it\n  says nothing about their mathematical interest, their novelty, or the\n  original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n\n% Compact opening card for the separately stated Comparator theorem selected\n% from each problem note.  The prose supplied by each note owns the\n% contribution and relevance judgements; the shared final sentence states the\n% mechanical assurance boundary once and points to the public evidence owner.\n\\newcommand{\\externalreviewobject}[3]{%\n  \\begin{quote}\\small\n  \\textbf{Contribution.} #1\\par\n  \\textbf{Relation to the open problem.} #2\\par\n  \\textbf{Executable review object.} #3  The repository's\n  external-verification job compares these exact Lean propositions with\n  separately declared challenge statements and an axiom budget, then asks\n  Lean's kernel to check the submitted proofs.  The\n  \\href{https://github.com/wcook04/plectis-erdos/blob/ca0e13f8acf5ccf48506e4bdb870953d3a0856fa/formalization.yaml}%\n  {formalisation manifest} and the commit-bound CI receipt record that check;\n  they do not assess novelty, significance, or whether the original problem is\n  solved.\n  \\end{quote}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  Under\n% Tectonic (XeTeX) the text and mathematics are set in TeX Gyre Pagella from\n% the bundled OpenType fonts, with a Heros sans range for \\mathsf and\n% Inconsolata for monospace; the pdflatex fallback keeps the same Palatino\n% design through newpx.  The A4 text block uses a classical measure with\n% quiet running heads and restrained ink.  Link colours: steel blue for\n% external (Lean-source and web) targets, deep burgundy for internal\n% cross-references and citations.\n\\NeedsTeXFormat{LaTeX2e}\n\\ProvidesPackage{paper-house-style}[2026/07/19 Pagella house style for the papers]\n\n\\RequirePackage{iftex}\n\\RequirePackage{amsmath}\n\\RequirePackage{amsthm}\n\\ifpdftex\n  \\RequirePackage[T1]{fontenc}\n  \\RequirePackage[utf8]{inputenc}\n  \\RequirePackage{amssymb}\n  \\RequirePackage{newpxtext}\n  \\RequirePackage{newpxmath}\n\\else\n  \\RequirePackage{fontspec}\n  \\RequirePackage{unicode-math}\n  \\setmainfont{texgyrepagella}[Extension=.otf,UprightFont=*-regular,\n    BoldFont=*-bold,ItalicFont=*-italic,BoldItalicFont=*-bolditalic]\n  \\setmathfont{texgyrepagella-math.otf}\n  \\setmathfont{texgyreheros-regular.otf}[range=sfup,Scale=MatchLowercase]\n  \\setmonofont{Inconsolatazi4}[Extension=.otf,UprightFont=*-Regular,\n    BoldFont=*-Bold,Scale=MatchLowercase]\n\\fi\n\n\\RequirePackage[a4paper,textwidth=418pt,textheight=636pt,hcentering,\n  top=68pt,headheight=15pt,headsep=20pt,footskip=34pt]{geometry}\n\\RequirePackage{microtype}\n\\linespread{1.045}\n\\clubpenalty=10000 \\widowpenalty=10000 \\tolerance=1600\n\n\\RequirePackage[dvipsnames,table]{xcolor}\n\\definecolor{linkexternal}{HTML}{275D8C}\n\\definecolor{linkinternal}{HTML}{6E2B2B}\n\\definecolor{papertableheader}{HTML}{E3EAF0}\n\\definecolor{papertablerow}{HTML}{F2F5F7}\n\\definecolor{papertablerule}{HTML}{7D8991}\n\\definecolor{paperresultrule}{HTML}{315B72}\n\n% Stable cross-document links between sibling PDFs. A named destination is\n% more durable than a page number: sections may move while their semantic role\n% remains the same. All public PDFs are copied to one directory at release.\n\\newcommand{\\papersectiontarget}[1]{\\hypertarget{#1}{}}\n\\newcommand{\\papersectionlink}[3]{\\href{#1\\##2}{#3}}\n\n\\RequirePackage{titlesec}\n\\titleformat{\\section}{\\Large\\bfseries}{\\thesection}{0.85em}{}\n\\titlespacing*{\\section}{0pt}{2.6ex plus .8ex minus .3ex}{1.5ex plus .3ex}\n\\titleformat{\\subsection}{\\large\\bfseries}{\\thesubsection}{0.8em}{}\n\\titlespacing*{\\subsection}{0pt}{2.1ex plus .6ex minus .2ex}{1.1ex plus .2ex}\n\\titlespacing*{\\paragraph}{0pt}{1.6ex plus .5ex minus .2ex}{0.85em}\n\n\\RequirePackage{fancyhdr}\n\\newcommand{\\paperrunninghead}{}\n\\newcommand{\\runninghead}[1]{\\renewcommand{\\paperrunninghead}{#1}}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\fancyhead[L]{\\small\\itshape\\paperrunninghead}\n\\fancyhead[R]{\\small\\thepage}\n\\renewcommand{\\headrulewidth}{0pt}\n\\fancypagestyle{plain}{\\fancyhf{}\\renewcommand{\\headrulewidth}{0pt}}\n\n\\RequirePackage{caption}\n\\captionsetup{font=small,labelfont=bf,skip=9pt,width=.92\\linewidth}\n\\RequirePackage{etoolbox}\n\\RequirePackage{pdflscape}\n\\renewcommand{\\arraystretch}{1.20}\n\\newcommand{\\papertablehead}{\\rowcolor{papertableheader}}\n\\newcommand{\\papertablebody}{\\rowcolors{2}{white}{papertablerow}}\n\\AtBeginEnvironment{tabular}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\\AtBeginEnvironment{longtable}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\n% A compact front-matter panel for the result a scanning reader should retain.\n% It is deliberately short and unbreakable: manuscripts should state one\n% result, its evidence, and its limit here, then supply the argument below.\n\\newenvironment{paperresult}[1]{%\n  \\par\\addvspace{0.7em}\n  \\begin{center}\\begin{minipage}{0.94\\linewidth}\n    {\\color{paperresultrule}\\rule{\\linewidth}{0.8pt}}\\par\\vspace{4pt}\n    {\\small\\sffamily\\bfseries\\color{paperresultrule}#1}\\par\\smallskip\n    \\small\\color{black}\n}{%\n    \\par\\vspace{4pt}{\\color{paperresultrule}\\rule{\\linewidth}{0.45pt}}\n    \\end{minipage}\\end{center}\\par\\addvspace{0.5em}\n}\n\n% Shared diagram grammar.  Reader-facing figures use the same restrained\n% hierarchy as the surrounding page: a short eyebrow, a substantial title,\n% one compact explanatory line, and an optional evidence badge.  The styles\n% are installed only when a manuscript loads TikZ.\n\\AtBeginDocument{%\n  \\@ifpackageloaded{tikz}{%\n    \\tikzset{\n      paper box/.style={draw=gray!48, line width=.45pt, rounded corners=3pt,\n        fill=white, align=left, inner xsep=7pt, inner ysep=6pt,\n        font=\\small},\n      paper card/.style={paper box, minimum height=13mm},\n      paper stage/.style={paper box, align=center, minimum height=14mm},\n      paper arrow/.style={-{Stealth[length=2.15mm,width=1.45mm]},\n        line width=.7pt, draw=gray!72!black},\n      paper feedback/.style={paper arrow, densely dashed,\n        draw=Violet!72!black},\n      paper eyebrow/.style={font=\\scriptsize\\sffamily\\bfseries,\n        text=gray!70!black},\n      paper label/.style={font=\\scriptsize\\sffamily,\n        text=gray!72!black, fill=white, inner sep=1.5pt},\n      paper badge/.style={rounded corners=2pt, inner xsep=4pt, inner ysep=2pt,\n        font=\\scriptsize\\sffamily\\bfseries},\n      paper boundary/.style={densely dashed, line width=.55pt,\n        draw=gray!62}\n    }%\n  }{}%\n}\n\n% Front matter: title block with an optional italic subtitle, small-caps\n% author, and a hairline rule; the abstract is set small with a small-caps\n% head and symmetric indents.\n\\newcommand{\\subtitle}[1]{\\gdef\\@papersubtitle{#1}}\n\\renewenvironment{abstract}{%\n  \\vspace{0.1em}\\small\n  \\begin{center}{\\normalsize\\scshape Abstract}\\end{center}\\vspace{-0.5em}\n  \\list{}{\\leftmargin 1.3em\\rightmargin\\leftmargin\\listparindent 0pt\n    \\parsep 0.32em\\itemindent 0pt}\\item\\relax}\n  {\\endlist\\vspace{0.5em}}\n\\def\\@maketitle{%\n  \\newpage\\null\n  \\vspace*{-2.6em}%\n  \\begin{center}%\n    {\\LARGE\\bfseries\\@title\\par}%\n    \\vskip 0.55em\n    {\\large\\itshape\\@papersubtitle\\par}%\n    \\vskip 1.1em\n    {\\scshape\\@author\\textsuperscript{*}\\par}%\n    \\vskip 0.25em\n    {\\small\\@date\\par}%\n    \\vskip 0.8em\n    \\rule{6.5em}{0.4pt}%\n  \\end{center}%\n  \\vskip 0.9em}\n\n% Production provenance, set at the foot of the first page.\n%\n% A reader who meets a single personal byline ordinarily infers that the named\n% author drafted the sentences.  For these papers that inference is false, and\n% a declaration reached only in the back matter would leave it standing for the\n% length of the paper.  The note therefore appears on page one, small and\n% visually subordinate to the title and abstract: enough to prevent the false\n% inference, not enough to make the production method the subject.\n%\n% It carries no mark, because it qualifies the whole manuscript rather than any\n% one passage.  \\footnotetext does not advance the footnote counter, and the\n% emptied \\thefootnote is confined to a group, so numbered footnotes elsewhere\n% in the paper are unaffected.\n%\n% The wording is not authored here.  It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"},{"record_type":"paper_source_dependency","tex_reference":"evidence/erdos251-prime-gap-reasoning-surface","source_path":"paper/evidence/erdos251-prime-gap-reasoning-surface.tex","media_type":"text/x-tex","sha256":"sha256:1d722846605b1bd9faaedd3a9034a8f33d36984f1d6826bccb4c0e722ee6284d","byte_count":24529,"licence":"CC-BY-4.0","body_format":"latex","body":"% Generated by scripts/paper_evidence.py from docs/paper_lean_coverage.json and the\n% Comparator replay receipts under evidence/comparator/.  Do not edit by hand.\n\\newcommand{\\evidencerecordurl}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md}\n\\DeclareResultEvidenceName{long251:res:sparse-rationalisation}{Theorem 2.1}\n\\DeclareResultEvidenceRecord{long251:res:sparse-rationalisation}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-sparse-rationalisation}{exact}{compared}\n\\DeclareResultEvidence{long251:res:sparse-rationalisation}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-sparse-rationalisation}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-sparse-rationalisation-comparator}\n\\DeclareResultEvidenceName{long251:res:local-targets}{Corollary 2.2}\n\\DeclareResultEvidenceRecord{long251:res:local-targets}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-local-targets}{exact}{compared}\n\\DeclareResultEvidence{long251:res:local-targets}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCompleteR20/LocalTargetInterval.lean\\#L38}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-local-targets-comparator}\n\\DeclareResultEvidenceName{long251:res:abel}{Proposition 4.1}\n\\DeclareResultEvidenceRecord{long251:res:abel}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-abel}{exact}{compared}\n\\DeclareResultEvidence{long251:res:abel}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L138}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-abel-comparator}\n\\DeclareResultEvidenceName{long251:res:parts}{Theorem 4.2}\n\\DeclareResultEvidenceRecord{long251:res:parts}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-parts}{exact}{compared}\n\\DeclareResultEvidence{long251:res:parts}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L172}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-parts-comparator}\n\\DeclareResultEvidenceName{long251:res:infinite}{Theorem 4.3}\n\\DeclareResultEvidenceRecord{long251:res:infinite}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-infinite}{exact}{compared}\n\\DeclareResultEvidence{long251:res:infinite}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCoreR7.lean\\#L47}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-infinite-comparator}\n\\DeclareResultEvidenceName{long251:res:irr-equivalence}{Corollary 4.4}\n\\DeclareResultEvidenceRecord{long251:res:irr-equivalence}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-irr-equivalence}{exact}{compared}\n\\DeclareResultEvidence{long251:res:irr-equivalence}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCoreR7.lean\\#L56}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-irr-equivalence-comparator}\n\\DeclareResultEvidenceName{long251:res:block}{Theorem 5.2}\n\\DeclareResultEvidenceRecord{long251:res:block}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-block}{exact}{compared}\n\\DeclareResultEvidence{long251:res:block}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCoreR7.lean\\#L76}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-block-comparator}\n\\DeclareResultEvidenceName{long251:res:escape-irrational}{Theorem 5.3}\n\\DeclareResultEvidenceRecord{long251:res:escape-irrational}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-escape-irrational}{exact}{compared}\n\\DeclareResultEvidence{long251:res:escape-irrational}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCoreR7.lean\\#L94}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-escape-irrational-comparator}\n\\DeclareResultEvidenceName{long251:res:true-tail}{Lemma 5.4}\n\\DeclareResultEvidenceRecord{long251:res:true-tail}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-true-tail}{exact}{compared}\n\\DeclareResultEvidence{long251:res:true-tail}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCompleteR20/TrueTail.lean\\#L57}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-true-tail-comparator}\n\\DeclareResultEvidenceName{long251:res:freepair}{Theorem 5.5}\n\\DeclareResultEvidenceRecord{long251:res:freepair}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-freepair}{exact}{compared}\n\\DeclareResultEvidence{long251:res:freepair}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCoreR7.lean\\#L189}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-freepair-comparator}\n\\DeclareResultEvidenceName{long251:res:lcmdiagonal}{Theorem 5.6}\n\\DeclareResultEvidenceRecord{long251:res:lcmdiagonal}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-lcmdiagonal}{exact}{compared}\n\\DeclareResultEvidence{long251:res:lcmdiagonal}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/OrderLatticeDiagonal.lean\\#L153}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-lcmdiagonal-comparator}\n\\DeclareResultEvidenceName{long251:res:smallpair}{Theorem 6.1}\n\\DeclareResultEvidenceRecord{long251:res:smallpair}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-smallpair}{exact}{compared}\n\\DeclareResultEvidence{long251:res:smallpair}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCoreR7.lean\\#L267}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-smallpair-comparator}\n\\DeclareResultEvidenceName{long251:res:smallpair-real}{Corollary 6.2}\n\\DeclareResultEvidenceRecord{long251:res:smallpair-real}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-smallpair-real}{exact}{compared}\n\\DeclareResultEvidence{long251:res:smallpair-real}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCoreR7.lean\\#L298}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-smallpair-real-comparator}\n\\DeclareResultEvidenceName{long251:res:gap-nonperiodic}{Proposition 6.3}\n\\DeclareResultEvidenceRecord{long251:res:gap-nonperiodic}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-gap-nonperiodic}{exact}{compared}\n\\DeclareResultEvidence{long251:res:gap-nonperiodic}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCoreR7.lean\\#L175}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-gap-nonperiodic-comparator}\n\\DeclareResultEvidenceName{long251:res:signedwindow}{Theorem 6.4}\n\\DeclareResultEvidenceRecord{long251:res:signedwindow}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-signedwindow}{exact}{compared}\n\\DeclareResultEvidence{long251:res:signedwindow}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCoreR7.lean\\#L135}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-signedwindow-comparator}\n\\DeclareResultEvidenceName{long251:res:explicit-remainder}{Proposition 6.5}\n\\DeclareResultEvidenceRecord{long251:res:explicit-remainder}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-explicit-remainder}{exact}{compared}\n\\DeclareResultEvidence{long251:res:explicit-remainder}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperTailBoundsR7.lean\\#L211}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-explicit-remainder-comparator}\n\\DeclareResultEvidenceName{long251:res:finite-smallpair}{Proposition 6.6}\n\\DeclareResultEvidenceRecord{long251:res:finite-smallpair}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-finite-smallpair}{exact}{compared}\n\\DeclareResultEvidence{long251:res:finite-smallpair}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperFiniteCertificatesR7.lean\\#L111}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-finite-smallpair-comparator}\n\\DeclareResultEvidenceName{long251:res:one-tail-certificate}{Proposition 6.7}\n\\DeclareResultEvidenceRecord{long251:res:one-tail-certificate}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-one-tail-certificate}{exact}{compared}\n\\DeclareResultEvidence{long251:res:one-tail-certificate}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCompleteR20/FiniteSeparation.lean\\#L6}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-one-tail-certificate-comparator}\n\\DeclareResultEvidenceName{long251:res:denominatorfloor}{Theorem 7.1}\n\\DeclareResultEvidenceRecord{long251:res:denominatorfloor}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-denominatorfloor}{exact}{compared}\n\\DeclareResultEvidence{long251:res:denominatorfloor}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-denominatorfloor}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-denominatorfloor-comparator}\n\\DeclareResultEvidenceName{long251:res:cfexclusion}{Theorem 7.2}\n\\DeclareResultEvidenceRecord{long251:res:cfexclusion}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-cfexclusion}{exact}{pending}\n\\DeclareResultEvidence{long251:res:cfexclusion}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/verification/Erdos251LargeCertificate/ErdosProblems/Erdos251/PaperLargeCertificateR7.lean\\#L29}{}\n\\DeclareResultEvidenceName{long251:res:boundedperturbation}{Theorem 8.1}\n\\DeclareResultEvidenceRecord{long251:res:boundedperturbation}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-boundedperturbation}{exact}{compared}\n\\DeclareResultEvidence{long251:res:boundedperturbation}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCoreR7.lean\\#L200}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-boundedperturbation-comparator}\n\\DeclareResultEvidenceName{long251:res:shiftedcount}{Proposition 8.2}\n\\DeclareResultEvidenceRecord{long251:res:shiftedcount}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-shiftedcount}{exact}{compared}\n\\DeclareResultEvidence{long251:res:shiftedcount}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/ShiftedGapCountingR9.lean\\#L152}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-shiftedcount-comparator}\n\\DeclareResultEvidenceName{long251:res:nonconcentration}{Theorem 8.3}\n\\DeclareResultEvidenceRecord{long251:res:nonconcentration}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-nonconcentration}{exact}{compared}\n\\DeclareResultEvidence{long251:res:nonconcentration}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperNonconcentrationR7.lean\\#L100}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-nonconcentration-comparator}\n\\DeclareResultEvidenceName{long251:res:nonconc-primes}{Corollary 8.4}\n\\DeclareResultEvidenceRecord{long251:res:nonconc-primes}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-nonconc-primes}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{long251:res:nonconc-primes}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-nonconc-primes}{}\n\\DeclareResultEvidenceName{long251:res:sparse-nonconcentration}{Proposition 8.5}\n\\DeclareResultEvidenceRecord{long251:res:sparse-nonconcentration}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-sparse-nonconcentration}{exact}{compared}\n\\DeclareResultEvidence{long251:res:sparse-nonconcentration}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCompleteR20/SparseNonconcentration.lean\\#L80}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-sparse-nonconcentration-comparator}\n\\DeclareResultEvidenceName{long251:res:jointcountermodel}{Corollary 8.6}\n\\DeclareResultEvidenceRecord{long251:res:jointcountermodel}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-jointcountermodel}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{long251:res:jointcountermodel}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-jointcountermodel}{}\n\\DeclareResultEvidenceName{long251:res:sparse}{Theorem 8.7}\n\\DeclareResultEvidenceRecord{long251:res:sparse}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-sparse}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{long251:res:sparse}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-sparse}{}\n\\DeclareResultEvidenceName{long251:res:polignacfail}{Theorem 8.8}\n\\DeclareResultEvidenceRecord{long251:res:polignacfail}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-polignacfail}{exact}{compared}\n\\DeclareResultEvidence{long251:res:polignacfail}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/LogarithmicCarryAsymptoticsR8.lean\\#L368}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-polignacfail-comparator}\n\\DeclareResultEvidenceName{long251:res:polynomialcountermodel}{Proposition 8.9}\n\\DeclareResultEvidenceRecord{long251:res:polynomialcountermodel}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-polynomialcountermodel}{exact}{compared}\n\\DeclareResultEvidence{long251:res:polynomialcountermodel}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCoreR7.lean\\#L27}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-polynomialcountermodel-comparator}\n\\DeclareResultEvidenceName{long251:res:telescope}{Proposition 8.10}\n\\DeclareResultEvidenceRecord{long251:res:telescope}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-telescope}{exact}{compared}\n\\DeclareResultEvidence{long251:res:telescope}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L1137}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-telescope-comparator}\n\\DeclareResultEvidenceName{long251:xr:totient}{Proposition D.1}\n\\DeclareResultEvidenceRecord{long251:xr:totient}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-xr-totient}{exact}{compared}\n\\DeclareResultEvidence{long251:xr:totient}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\\#L877}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-xr-totient-comparator}\n\\DeclareResultEvidenceName{long251:xr:propagate}{Proposition D.2}\n\\DeclareResultEvidenceRecord{long251:xr:propagate}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-xr-propagate}{exact}{compared}\n\\DeclareResultEvidence{long251:xr:propagate}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCompleteR20/RealPropagation.lean\\#L27}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-xr-propagate-comparator}\n\\DeclareResultEvidenceName{long251:xr:truncation}{Proposition D.3}\n\\DeclareResultEvidenceRecord{long251:xr:truncation}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-xr-truncation}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long251:xr:truncation}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperTailBoundsR7.lean\\#L261}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-xr-truncation-comparator}\n\\DeclareResultEvidenceName{long251:res:complete-truncation}{Proposition D.4}\n\\DeclareResultEvidenceRecord{long251:res:complete-truncation}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-complete-truncation}{exact}{compared}\n\\DeclareResultEvidence{long251:res:complete-truncation}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCompleteR20/FiniteSeparation.lean\\#L25}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-complete-truncation-comparator}\n\\DeclareResultEvidenceName{long251:xr:boundedpolignac}{Proposition D.6}\n\\DeclareResultEvidenceRecord{long251:xr:boundedpolignac}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-xr-boundedpolignac}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long251:xr:boundedpolignac}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperBoundedCarryR7.lean\\#L179}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-xr-boundedpolignac-comparator}\n\\DeclareResultEvidenceName{long251:res:affinecollapse}{Theorem E.1}\n\\DeclareResultEvidenceRecord{long251:res:affinecollapse}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-affinecollapse}{exact}{compared}\n\\DeclareResultEvidence{long251:res:affinecollapse}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos251/PaperCoreR7.lean\\#L241}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos251-prime-gap-reasoning-surface.md\\#long251-res-affinecollapse-comparator}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/251/erdos251-prime-gap-reasoning-surface.tex","section_outline":["Introduction","Sparse congruence-preserving perturbations","Context and mathematical dependencies","Summation by parts, with the endpoint retained","The infinite identity and the irrationality equivalence","The tail recurrence and the exact criteria","Pairs of congruent indices and least-common-multiple shifts","A local certificate, and one actual pair","An explicit remainder and a certified actual pair","Two lower bounds on a possible rational denominator","Coefficient conditions and counterexamples","Bounded residue-preserving perturbations","Algebraic nonconcentration survives the rationalising perturbation","Density of indices with two small tail differences","Recurring gap values differing by two do not suffice","A rational series with recurring values in every residue class","Polynomial coefficients and telescoping sums","Finite numerical searches","Remaining prime-gap estimates","An elementary polynomial bound for the primes","Integer certificates","Verification and reproducibility","Further criteria, examples and computational details","Totient-length shifts and persistence of integrality","The finite truncation criterion","Nonintegral differences at multiples of each shift","Repeated tail values and congruent indices","A bounded sequence with recurring values and a rational sum","Numerical counts and proportions","Two corrections","Arithmetic-progression reformulations of integrality","A signed comparison with more analytic data","Conclusions","Source index for the short paper"]},"erdos-257-mersenne-support-subseries":{"paper_id":"erdos-257-mersenne-support-subseries","title":"Irrationality criteria for Lambert subseries","form":"Problem note","length_class":"short","pages":26,"subject":"Erdős #257","subject_kind":"problem","problem_id":"erdos_257","landing_role":"mathematics","question":"Which supports are proved to give irrational Lambert subseries, and what still blocks the arbitrary infinite support that Erdős #257 asks about?","precis":"The series ∑_{a∈A}(b^a−1)^{-1} is irrational for every integer b≥2 and infinite A contained in E∪V when E satisfies a finite-prime weighted summability condition and V has a summable positive divisor cover. The two classes are incomparable, and their union reaches supports in neither class. A common average over dyadic lengths produces one index with both displacements small. Arbitrary infinite supports at base two remain unresolved.","owns":"the weighted, divisor-cover and mixed support criteria with complete proofs and method limits; the long paper treats finite denominators, the achievement set and rational membership","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #257, which remains open","first_pass":["sec:problem","sec:reciprocal-support","sec:eight-return-extensions","sec:map"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-257-mersenne-support-subseries.pdf","hosted_pdf_path":"papers/erdos-257-mersenne-support-subseries.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex","site_html":"maths/papers/erdos-257-mersenne-support-subseries.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/257/erdos-257-mersenne-support-subseries.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/257/erdos-257-mersenne-support-subseries.tex","body_pointer":"papers.records.erdos-257-mersenne-support-subseries.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Standalone Erdős Problem Note for #257.  Context is deliberately repeated\n% from the gateway so that this file remains independently readable.\n%\n% Round-5 editorial ordering: theorem, displacement mechanism, sufficient\n% extensions and their mathematical boundaries, then the actual target.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n% The selected math font supplies the standard small set-difference glyph.\n\\AtBeginDocument{\\let\\setminus\\smallsetminus}\n\\input{evidence/erdos-257-mersenne-support-subseries}\n\\renewcommand{\\commit}{99f4bf47422abbd8757cbb22b50ba079d764d3a7}\n\\renewcommand{\\commitshort}{99f4bf47422a}\n\\hypersetup{\n  pdftitle={Irrationality criteria for Lambert subseries (Erdős Problem 257)},\n  pdfsubject={Erdős Problem Notes: Problem 257},\n  pdfkeywords={irrationality, Mersenne numbers, Lambert series, divisor incidence, scaled tails, achievement sets, squarefree, Lean 4},\n}\n\n\\newcommand{\\Ach}{\\mathcal A}\n\n\\runninghead{Irrationality criteria for Lambert subseries}\n\\title{Irrationality criteria for Lambert subseries}\n\\subtitle{The series $\\sum_{a\\in A}(b^a-1)^{-1}$ and Erd\\H{o}s Problem~\\#257}\n\\author{Will Cook}\n\\date{30 September 2026}\n\n\\begin{document}\n\\maketitle\n\\noteprovenance\n\\begin{abstract}\nWe give two hereditary irrationality criteria for Lambert subseries.\nIf $E$ satisfies a finite-prime weighted summability condition at base two\nand $V$ admits a summable positive divisor cover, then\n$\\sum_{a\\in A}(b^a-1)^{-1}$ is irrational for every infinite\n$A\\subseteq E\\cup V$ and every integer $b\\ge2$. The two support\nclasses are incomparable, both reach beyond reciprocal summability,\nand their unions can lie in neither class. The proof averages first\nover multiples of a modulus and then over dyadic lengths. The second\naverage restores a reciprocal factor lost at incomplete residue periods;\nusing the same average for both criteria gives one index at which both\nerrors are small. Arbitrary infinite support at base two remains open.\n\\end{abstract}\n\n\\section{Introduction}\\label{sec:problem}\nFor a set $A$ of positive integers and a real number $b>1$, write\n\\[\n X_A(b)=\\sum_{a\\in A}\\frac1{b^a-1}.\n\\]\nThis series converges by comparison with a geometric series, since\n$(b^a-1)^{-1}\\le b^{1-a}/(b-1)$. Erd\\H{o}s proved irrationality\nfor full support at integer bases\n\\cite{erdos1948}, and later proved the same conclusion for pairwise\ncoprime supports of finite reciprocal sum \\cite[p.~222]{erdos1968}.\nOn that page he stated that the coprimality assumption could be removed,\nwithout giving the details. The question for arbitrary infinite support\nin base two is Problem~\\#257:\n\n\\begin{problem}[Erd\\H{o}s \\#257]\\label{res:problem}\nIs $X_A(2)$ irrational for every infinite $A\\subseteq\\Npos$?\n\\end{problem}\n\nWe seek conditions inherited by every infinite subset of the support.\nThe first replaces reciprocal summability by a divisibility-sensitive\nweight. For $a=2^km$ with $m$ odd, it uses\n$1/[m(b^{2^k}-1)]$ in place of $1/(2^km)$: a large power of $2$ can\ncompensate for many odd cofactors. More generally, fix a finite set $P$\nof primes and write $h(a)=\\prod_{p\\in P}p^{v_p(a)}$ for the\n\\emph{$P$-part} of $a$, where $v_p(a)$ is the exponent of $p$ in $a$.\nThe prime powers may grow with $a$, but the prime set is fixed.\n\n\\begin{theorem}[a weighted condition on the support]\n\\label{res:weighted-support}\nLet $b\\ge2$ be an integer, let $A\\subseteq\\Npos$ be infinite, and let $P$\nbe a finite nonempty set of primes.  Set\n$h(a)=\\prod_{p\\in P}p^{v_p(a)}$.  If\n\\begin{equation}\\label{eq:weighted-fixed-base}\n W_{b,P}(A):=\\sum_{a\\in A}\n \\frac{h(a)}{a(b^{h(a)}-1)}<\\infty,\n\\end{equation}\nthen $X_A(b)$ is irrational.  In particular, the base-two condition\n\\begin{equation*}\\label{eq:weighted-return}\n \\sum_{a\\in A}\\frac{h(a)}{a(2^{h(a)}-1)}<\\infty\n \\tag{W}\n\\end{equation*}\nimplies that $X_A(b)$ is irrational for every integer $b\\ge2$.  Both\nconclusions are hereditary under passage to infinite subsets.\n\\end{theorem}\n\nConsider the support\n\\[\n A_\\star=\\{2^km:k\\ge1,\\ m\\text{ odd},\\ m\\le2^{2^k}\\}.\n\\]\nFor a fixed $k$, the sum of $1/m$ over the odd cofactors is of order\n$2^k$. Thus the corresponding terms contribute at least $1/4$ to\n$\\sum_{a\\in A_\\star}1/a$, but at most $2^{1-k}$\nto~\\eqref{eq:weighted-return}.\nThe calculation in Section~\\ref{sec:eight-return-extensions} proves both\nbounds. Every infinite subset of $A_\\star$ consequently gives an\nirrational value at every integer base. Since $h/(2^h-1)\\le1$, the same\ntheorem also includes reciprocal-summable supports. Section~\\ref{sec:reciprocal-support}\ngives a direct proof of Erd\\H{o}s's stated extension.\n\nThe second condition uses joint divisibility: it bounds fractional\npowers of divisor counts by positive divisor sums\n(Theorem~\\ref{thm:variable-fractional-cover}). Neither condition implies\nthe other, but they combine: every infinite subset of $E\\cup V$ has\nirrational sum when $E$ satisfies the base-two weighted condition and\n$V$ has a summable divisor cover (Theorem~\\ref{res:mixed-supports}).\nTwo irrational subseries could sum to a rational number, so we must\ncontrol their errors at the same index. The separating constructions\nin Proposition~\\ref{res:weighted-cover-incomparability} give such a union\noutside both individual classes (Corollary~\\ref{res:strict-mixed-supports}).\nThese are sufficient conditions, not a characterisation of the supports\nin Problem~\\ref{res:problem}.\n\nThe proofs compare a multiple of $X_A(b)$ with an integer. For a\npositive integer $N$, put\n\\begin{equation}\n \\Delta_{b,A}(N)=\\sum_{a\\in A}\\frac{b^{N\\bmod a}-1}{b^a-1}\n =(b^N-1)X_A(b)-J_{b,A}(N),\\qquad\n J_{b,A}(N)=\\sum_{\\substack{a\\in A\\\\a\\le N}}\n             \\sum_{j=1}^{\\lfloor N/a\\rfloor}b^{N-ja}.\n \\label{eq:intro-displacement}\\tag{D}\n\\end{equation}\nDivision of $N$ by each $a$ gives the identity. At an integer base,\n$J_{b,A}(N)$ is an integer; for infinite $A$, the displacement is\npositive because some $a\\in A$ exceeds $N$. Thus, if $X_A(b)=p/q$\nwith $p\\in\\mathbb Z$ and $q\\ge1$ an integer, then\n$q\\Delta_{b,A}(N)$ is a positive integer and\n$\\Delta_{b,A}(N)\\ge1/q$. We seek displacements tending to zero;\n$\\Delta_{b,A}(N)$ itself is not a fractional part and may exceed $1$.\nErd\\H{o}s also proposed using fractional parts of powers times the series\nvalue \\cite[p.~226]{erdos1968}.\n\nDivisibility cancels finitely many exponents when $N$ is a multiple of\ntheir least common multiple. The remaining terms have small\ncomplete-period means when their gcd with the modulus is large.\nIncomplete-period bounds lack a factor $1/a$. Averaging over dyadic\nlengths restores it: an exponent enters the truncated error only when\nthe sampled range reaches it, and the reciprocals of subsequent lengths\nform a geometric tail bounded by a multiple of $1/a$. Exponents beyond\nthe sampled range are estimated separately. Explicit modulus dependence\nlets both criteria share this average.\n\nSections~\\ref{sec:eight-return-extensions}--\\ref{sec:mixed} prove the\nthree criteria; Sections~\\ref{sec:comparison}--\\ref{sec:map} separate\ntheir support classes and explain the method's limits. The appendices\ngive the direct reciprocal-summable proof, finite-denominator results\nand rational-membership tests. The last of these decide neither $1/2$\nnor $1/21$.\n\n\\section{Finite averages and the weighted criterion}\n\\label{sec:eight-return-extensions}\nFix the integer base $b$ and the finite prime set $P$. We first estimate\naverages over multiples of a fixed modulus $Q$, then choose $Q$ to\nexploit the weighted hypothesis. The role of the gcd is already visible\nat $b=2$, $Q=4$ and $a=6$: the residues are $4,2,0$, and\n\\[\n \\frac13\\left(\\frac{2^4-1}{63}+\\frac{2^2-1}{63}+0\\right)\n =\\frac2{21}\\le\\frac{2}{6(2^2-1)}=\\frac19.\n\\]\nThe final expression is the mean of $2^{4m\\bmod6}/63$ over the same\nperiod. If $6\\mid Q$, the term with exponent $6$ vanishes at every\nmultiple of $Q$. In the proof, divisibility removes finitely many\nexponents, and the gcd estimate bounds the remaining terms.\n\n\\begin{proof}[Proof of Theorem~\\ref{res:weighted-support}]\n\\emph{The contribution of one exponent.}\nPut $d_a(m)=(b^{m\\bmod a}-1)/(b^a-1)$ and $g=(Q,a)$.\nFor positive integers $Q,a,T$, the orbit has length $a/g$. Splitting\n$1,\\ldots,T$ into full orbits and a remainder gives\n\\begin{equation}\\label{eq:weighted-finite-orbit}\n \\frac1T\\sum_{t=1}^T d_a(tQ)\n \\le \\frac{g}{a(b^g-1)}+\\frac1{T(b^g-1)}.\n\\end{equation}\nDropping the $-1$ in the numerator gives the orbit sum\n$\\sum_{t=1}^{a/g}b^{tQ\\bmod a}/(b^a-1)=1/(b^g-1)$.\nComplete orbits give the first term in~\\eqref{eq:weighted-finite-orbit};\nnonnegativity bounds the remaining indices by one more orbit.\nThe theorem's weighted summand bounds the complete-period term whenever\n$g\\ge h(a)$. The modulus below ensures this for $h(a)\\le H$;\nlarger prime parts will instead give a uniform lower bound on $g$. Also, for $Y=QT$,\n\\begin{equation}\\label{eq:weighted-outer-short}\n \\frac1T\\sum_{t=1}^T\\sum_{\\substack{a\\in A\\\\a>Y}}d_a(tQ)\n \\le\\frac4T,\n\\end{equation}\nbecause $d_a(tQ)\\le2\\,2^{tQ-a}$ when $a>QT$ and\n$\\sum_{t=1}^T2^{tQ-QT}\\le2$.\n\n\\emph{Recovering the reciprocal factor.}\nThe error in~\\eqref{eq:weighted-finite-orbit} tends to zero for each\nfixed $a$, but without a factor $1/a$ this does not control its sum\nover the growing range $a\\le QT$. Take $T=2^j$ and sum first over\n$j$. The truncated error includes a fixed $a$ only when $2^j\\ge a/Q$;\nthe reciprocals of those lengths form a geometric tail of sum at most\n$2Q/a$.\nThus, for integers $Q,M\\ge1$ and $\\alpha_a\\ge0$ with\n$\\sum_a\\alpha_a/a<\\infty$,\n\\begin{equation}\\label{eq:weighted-dyadic-short}\n \\sum_{j=M}^{2M-1}\\frac1{2^j}\\sum_{a\\le Q2^j}\\alpha_a\n \\le2Q\\sum_{a\\ge1}\\frac{\\alpha_a}{a};\n\\end{equation}\nthe estimate is unchanged if the available dyadic lengths start later.\nDividing by $M$ will turn the cost $2Q$ into $2Q/M$.\n\n\\emph{Choosing the modulus.}\nFix $\\varepsilon>0$. Choose finite nonempty $F\\subseteq A$ so that the sum in\n\\eqref{eq:weighted-fixed-base} over $A\\setminus F$ is below $\\varepsilon$, and\nlet $L$ be any fixed positive common multiple of $F$. Besides cancelling\n$F$, the modulus should contain every $P$-prime power up to a growing\nthreshold $H$. Truncating each prime power separately keeps its size\npolynomial in $H$. With $p_*=\\max P$ and $H\\ge2p_*$, set\n\\[\n Q=L\\prod_{p\\in P}p^{\\lfloor\\log_pH\\rfloor},\\qquad\n G=\\left\\lfloor\\frac H{p_*}\\right\\rfloor.\n\\]\nFor $a\\in F$ the displacement term vanishes, since $a\\mid Q$.\nFor $a\\notin F$ with $h(a)\\le H$, we have $h(a)\\mid Q$ and hence\n$(Q,a)\\ge h(a)$.  The complete-period term\nin~\\eqref{eq:weighted-finite-orbit} is therefore bounded by\n$h(a)/[a(b^{h(a)}-1)]$, using the monotonicity of $n/(b^n-1)$.\nThe complete-period contributions therefore sum to less than\n$\\varepsilon$. The dyadic estimate will control their incomplete periods.\n\nFor $h(a)>H$, we instead have $(Q,a)\\ge G$. Indeed, if every\n$P$-prime-power component is at most $H$, then $h(a)\\mid Q$ and\n$(Q,a)\\ge h(a)>H$; otherwise some $p^{v_p(a)}>H$ contributes\n$p^{\\lfloor\\log_pH\\rfloor}>H/p\\ge H/p_*\\ge G$ to the gcd.\nSum the complete-period bounds over $a\\le QT$, using\n$\\sum_{a\\le QT}1/a\\le1+\\log(QT)$. There are at most $QT$\nincomplete-period terms, each at most $1/[T(b^G-1)]$, so the total\ncontribution from these exponents is at most\n\\[\n \\frac{G(1+\\log(QT))+Q}{b^G-1}.\n\\]\n\\emph{Balancing the two errors.}\nIncreasing $M$ trades a smaller incomplete-period cost $Q/M$ for a\nlarger large-gcd cost $GM/b^G$, since $T=2^j$ with $j<2M$.\nWith $F$ and $L$ fixed, $Q\\le LH^{|P|}$ is polynomial in $H$, whereas\n$G=H/p_*+O(1)$ is linear. Thus $M=\\lfloor b^{G/2}\\rfloor$ makes\nboth costs tend to zero. Here $M$ counts the scales $M\\le j<2M$;\ntheir observation lengths are $2^j$.\nApply~\\eqref{eq:weighted-dyadic-short} with\n$\\alpha_a={\\bf1}_A(a)/(b^{h(a)}-1)$; since $h(a)\\ge1$,\n$\\sum_a\\alpha_a/a\\le W_{b,P}(A)$.\nTogether with~\\eqref{eq:weighted-outer-short}, the preceding estimates give\n\\begin{equation}\\label{eq:weighted-main-bound}\n \\frac1M\\sum_{j=M}^{2M-1}\\frac1{2^j}\n \\sum_{t=1}^{2^j}\\Delta_{b,A}(tQ)\n \\le \\varepsilon+\\frac{2QW_{b,P}(A)}M\n +\\frac{G(1+\\log Q+2M\\log2)+Q}{b^G-1}+4\\,2^{-M}.\n\\end{equation}\nThe first two terms control complete and incomplete periods with\n$h(a)\\le H$; the last two control $h(a)>H$ within $a\\le QT$ and the\ntail $a>QT$. With $F$ and $L$ fixed, every term after\n$\\varepsilon$ tends to zero as $H\\to\\infty$. The left-hand side is a probability-weighted average: choose one of\nthe $M$ scales uniformly, then one of its $2^j$ indices uniformly.\nFor large $H$, at least one sampled $N=Qt$ therefore satisfies\n$\\Delta_{b,A}(N)<2\\varepsilon$. Since $\\varepsilon$ is arbitrary,\nthis contradicts the rational lower bound $1/q$. The argument gives\nno rate of decay in $N$.  Finally\n$W_{b,P}(A)\\le W_{2,P}(A)$ for $b\\ge2$, and the weighted sum decreases on taking\nsubsets.\n\\end{proof}\n\nDuverney--Tachiya~\\cite[Section~2, (2.3)--(2.9)]{duverneytachiya}\nalso bound an average on an arithmetic progression and select an index\ncontrolling a combined quantity. Their progression is built by the\nChinese remainder theorem. Here the additional dyadic average controls\nthe incomplete-period error as the modulus grows.\n\n\n\\subsection{An example beyond reciprocal summability}\nFor the support in the introduction, the two sums can be compared\nlayer by layer. Recall that\n\\[\n A_\\star=\\{2^k m:k\\ge1,\\ m\\text{ odd},\\ m\\le2^{2^k}\\}.\n\\]\nThe layers are disjoint, since their elements have different $2$-adic\nvaluations.  For $r\\ge2$, put\n\\[\n S_r=\\sum_{\\substack{m\\le2^r\\\\m\\text{ odd}}}\\frac1m.\n\\]\nEach block $2^j\\le m<2^{j+1}$, $1\\le j<r$, contains $2^{j-1}$ odd\nintegers, with reciprocal sum between $1/4$ and $1/2$.  Including\n$m=1$ gives $r/4\\le S_r\\le r$.  The sum of reciprocals in the $k$th layer\nis $2^{-k}S_{2^k}\\ge1/4$, so $\\sum_{a\\in A_\\star}1/a$ diverges.  For\n$P=\\{2\\}$, its contribution to the weighted sum satisfies\n\\[\n \\frac{S_{2^k}}{2^{2^k}-1}\n \\le\\frac{2^k}{2^{2^k}-1}\n \\le2^{1-k}.\n\\]\nThe last bound uses $2^{2^k}-1\\ge2^{2^k-1}$ and $2^k\\ge2k$.\nSummation proves~\\eqref{eq:weighted-return}, despite the divergent\nreciprocal sum.\n\n\nThe sets of primes for which the weighted sum converges can also be\nprescribed. Let $E$ be a finite set of primes and let $\\mathcal U$ be an\nupward-closed family of subsets of $E$ containing $E$ but not\n$\\varnothing$. A finite union of the constructions just described has\ndivergent reciprocal sum, and, for every $b\\ge2$ and finite prime set $P$,\nits weighted sum $W_{b,P}$ is finite exactly when $P\\cap E\\in\\mathcal U$.\nTaking $P=E$ shows that every infinite subset has an irrational sum at\nevery integer base.  The construction is given in\n\\papersectionlink{erdos257-mersenne-reasoning-surface.pdf}{record257:witness-rules}\n{Section~1.3 of the companion paper}.\n\n\\section{Positive divisor majorants}\\label{sec:common-kernel}\nThe weighted criterion treats exponents separately. A divisor cover\ninstead uses their joint divisibility. To describe it, write the series\nin terms of its divisor counts:\n\\begin{equation}\n c_A(n)=\\#\\{a\\in A:a\\mid n\\},\\qquad\n X_A(b)=\\sum_{n\\ge1}c_A(n)b^{-n}.\n \\label{eq:incidence}\n\\end{equation}\nExpand $(b^a-1)^{-1}=\\sum_{j\\ge1}b^{-aj}$ and interchange the\nnonnegative sums. The selector $\\mathbf1_A$ is zero or one, whereas\n$0\\le c_A(n)\\le\\tau(n)$, with $\\tau(n)$ the number of positive\ndivisors of $n$. These are power-series coefficients before any carrying;\nthey need not be base-$b$ digits.\n\nKaneko--Suzuki--Tachiya\n\\cite[Theorems~1 and~3]{kanekosuzukitachiya} require sparsity of the\nnonzero power-series coefficients. Sparsity of the selected Lambert\ndenominators does not suffice: for any nonempty $A$, $c_A(n)$ is positive\non every multiple of $\\min A$. Its nonzero positions therefore have\npositive lower density, violating those support-counting hypotheses.\nThe criteria do not apply directly to~\\eqref{eq:incidence}.\nThe density calculation and the distinction between their remote-tail\naverage and our displacement appear in\n\\papersectionlink{erdos257-mersenne-reasoning-surface.pdf}{record257:weighted-proof}\n{Section~1.2 of the companion paper}.\n\n\n\\subsection{A finite divisor set and the cover condition}\nFor finite $F\\subseteq\\Npos$, write $f_F(n)=\\#\\{a\\in F:a\\mid n\\}$.\nConsider the four exponents $F=\\{2,6,10,30\\}$. If $2\\nmid n$, the\ncount is zero. Otherwise each of the primes $3$ and $5$ that divides\n$n$ doubles the count. Thus\n\\[\n f_F(n)=\\mathbf1_{2\\mid n}(1+\\mathbf1_{3\\mid n})(1+\\mathbf1_{5\\mid n}).\n\\]\nFor $0<\\alpha\\le1$, put $z=2^\\alpha-1$. Taking the fractional power\nand expanding gives a positive divisor sum:\n\\[\n f_F(n)^\\alpha\n =\\mathbf1_{2\\mid n}+z\\mathbf1_{6\\mid n}\n  +z\\mathbf1_{10\\mid n}+z^2\\mathbf1_{30\\mid n},\\qquad\n C=\\frac12\\left(1+\\frac z3\\right)\\left(1+\\frac z5\\right).\n\\]\nSince $\\mathbf1_{d\\mid n}$ has mean $1/d$, $C$ is the mean of this\nexpansion. The same factorisation applies to\n$\\{qd:d\\mid\\prod_{p\\in P}p\\}$ when no prime in $P$ divides $q$.\nSmall $\\alpha$ reduces the coefficients, but the tail estimate costs\n$(2^\\alpha-1)^{-1}=1/z$. The condition below balances these effects\nover a sequence of finite sets.\n\n\\begin{theorem}[a summable divisor-cover criterion]\\label{thm:variable-fractional-cover}\nFor each $j\\ge1$, let $F_j\\subseteq\\Npos$ be finite, let\n$0<\\alpha_j\\le1$, and let $c_{j,d}\\ge0$ satisfy\n\\[\n f_{F_j}(n)^{\\alpha_j}\\le\\sum_{d\\mid n}c_{j,d}\\quad(n\\ge1).\n\\]\nSet $C_j=\\sum_{d\\ge1}c_{j,d}/d$.  If\n\\begin{equation*}\n \\sum_{j\\ge1}\\frac{C_j2^{j\\alpha_j}}{2^{\\alpha_j}-1}<\\infty,\n \\tag{V}\\label{eq:strengthened-cover}\n\\end{equation*}\nthen $X_A(b)$ is irrational for every infinite\n$A\\subseteq\\bigcup_jF_j$ and every integer $b\\ge2$.\n\\end{theorem}\n\nFor example, take $F_j=\\{4^j\\}$, $\\alpha_j=1$ and $c_{j,4^j}=1$,\nwith all other coefficients zero. Then $C_j=4^{-j}$, and the cost\nin~\\eqref{eq:strengthened-cover} is $\\sum_j2^{-j}$. Every finite set\nhas the majorant $\\alpha_j=1$, $c_{j,d}=\\mathbf1_{F_j}(d)$;\nthe restriction is summability of the full cost. Covering sets may\noverlap, and their majorants may use divisors outside them.\nSection~\\ref{sec:comparison} gives a necessary condition independent\nof these choices.\n\n\\subsection{A finite mean for shifted divisor tails}\nA divisor majorant becomes a sum of geometric tails, one for each\ndivisor. Indeed, the positive offsets $r$ with $d\\mid N+r$ are\n$d-(N\\bmod d),2d-(N\\bmod d),\\ldots$, so\n\\[\n \\sum_{\\substack{r\\ge1\\\\d\\mid N+r}}B^{-r}\n =\\frac{B^{N\\bmod d}}{B^d-1}.\n\\]\nFractional powers introduce $B=2^\\alpha$, possibly close to $1$.\nThe estimate must remain uniform after multiplication by $B-1$ and\nallow the modulus to grow: the mixed proof fixes the first covering\nsets before enlarging that modulus.\n\nFor $1<B\\le2$, positive integers $L,d,M$, and an integer $R\\ge0$, put\n\\[\n w_{B,d}(n)=\\frac{B^{n\\bmod d}}{B^d-1},\\qquad\n \\mathscr D_{L;R,M}F\n =\\frac1M\\sum_{k=R}^{R+M-1}\\frac1{2^k}\n      \\sum_{m=1}^{2^k}F(Lm).\n\\]\nChoose $k$ uniformly from $R,\\ldots,R+M-1$, then $m$ uniformly\nfrom $1,\\ldots,2^k$ and observe $N=Lm$. Scales, not individual\nobservations, have equal weight; repeated observations of $N$ add\ntheir weights.\n\nThe finite estimate\n\\begin{equation*}\n \\mathscr D_{L;R,M}w_{B,d}\n \\le\\frac{1+4L/M}{d(B-1)}\n \\tag{S}\\label{eq:mixed-finite-kernel}\n\\end{equation*}\nis uniform as $B\\downarrow1$ after multiplication by $B-1$:\nthe normalised bound is $(1+4L/M)/d$. The unnormalised factor\n$(B-1)^{-1}$ becomes the cost $(2^\\alpha-1)^{-1}$ in\n\\eqref{eq:strengthened-cover}. The bound is independent of the starting\nscale $R$ and tracks the ratio $L/M$, which tends to zero on the\nweighted proof's scales.\n\nFix $T=2^k$ and compare $d$ with the largest sampled multiple $LT$.\nWhen $d\\le LT$, the complete orbit has length $d/g$, with $g=(L,d)$,\nand total weight $1/(B^g-1)$. Split the sample into complete cycles\n(possibly none) and one remaining piece to obtain\n\\[\n \\frac1T\\sum_{m=1}^T w_{B,d}(Lm)\n \\le\\frac{g}{d(B^g-1)}+\\frac1{T(B^g-1)}\n \\le\\frac1{d(B-1)}+\\frac1{T(B-1)}.\n\\]\nAcross dyadic lengths satisfying $d\\le L2^k$, the reciprocal-length\nerrors sum to at most $2L/[d(B-1)]$.\nWhen $d>2LT$, we have $Lm\\bmod d=Lm$ and $2Lm\\le d-1$; hence\n$\\sum_{i=0}^{d-1}B^i\\ge dB^{(d-1)/2}\\ge dB^{Lm}$.\nEach term $w_{B,d}(Lm)$ is then at most $1/[d(B-1)]$.\nFor fixed $d$, the remaining transition range $LT<d\\le2LT$ occurs\nat most once as $T$ doubles. At that scale, geometric summation gives\n\\[\n \\frac1T\\sum_{m=1}^T w_{B,d}(Lm)\n =\\frac{B^L}{T(B^L-1)}\\frac{B^{LT}-1}{B^d-1}\n \\le\\frac{LB^L}{d(B^L-1)}\n \\le\\frac{2L}{d(B-1)}.\n\\]\nThe first inequality uses the convexity of $x\\mapsto B^x-1$ and its\nvalue $0$ at the origin to bound the ratio by $LT/d$. For the last\ninequality we used\n$(B^L-1)/(B-1)=\\sum_{i=0}^{L-1}B^i\\ge B^{L-1}$ and $B\\le2$.\nThe incomplete periods and the possible transition scale together\ncost at most $4L/[d(B-1)]$ before division by $M$. Adding the mean\nterm $1/[d(B-1)]$ proves~\\eqref{eq:mixed-finite-kernel}.\nThe corresponding \\href{https://github.com/wcook04/plectis-erdos/blob/c91562bd574a387cde904481e609c7b4cacebb14/lean/ErdosProblems/Erdos257/PaperCompleteR8/DyadicKernel.lean\\#L206}{source estimate} uses the same finite average.\nNonnegative interchange permits summation against any coefficients $c_d$\nwith $\\sum_dc_d/d<\\infty$.\n\n\\subsection{Proof of the cover criterion}\n\\begin{proof}[Proof of Theorem~\\ref{thm:variable-fractional-cover}]\nWrite $B_j=2^{\\alpha_j}$ and\n\\[\n U_j(N)=\\sum_{r\\ge1}2^{-r}f_{F_j}(N+r),\\qquad\n V_j(N)=\\sum_{d\\ge1}c_{j,d}w_{B_j,d}(N).\n\\]\nThe $U_j$ are the tails to be made small; the $V_j$ have means controlled\nby~\\eqref{eq:mixed-finite-kernel}. Subadditivity of the power $\\alpha_j$\nand the divisor majorant connect them:\n\\[\n U_j(N)^{\\alpha_j}\n \\le\\sum_{r\\ge1}B_j^{-r}f_{F_j}(N+r)^{\\alpha_j}\n \\le\\sum_{d\\ge1}c_{j,d}\n       \\sum_{\\substack{r\\ge1\\\\d\\mid N+r}}B_j^{-r}\n =V_j(N).\n\\]\nFor the displacement itself, the geometric-series identity gives\n\\[\n \\Delta_{2,F_j}(N)=U_j(N)-X_{F_j}(2)\\le U_j(N).\n\\]\nFor $j\\le J$, divisibility will cancel the displacement, so\n$U_j(N)=X_{F_j}(2)$. Only the tails with $j>J$ must be made small.\n\nFix $\\varepsilon>0$ and allot $t_j=\\varepsilon2^{-j}$ to the $j$th\ntail. These allowances sum to $\\varepsilon$. The sufficient condition\n$t_j^{-\\alpha_j}V_j<1$ for $U_j<t_j$ explains the factor\n$2^{j\\alpha_j}$ in the cover cost. Choose $J$ with\n\\[\n K_J:=\\sum_{j>J}\\frac{C_jt_j^{-\\alpha_j}}{B_j-1}<\\frac14.\n\\]\nThis is possible since $\\varepsilon^{-\\alpha_j}\\le\\max(1,\\varepsilon^{-1})$.\nChoose $L$ divisible by every member of the first $J$ finite sets.\nFor every $R\\ge0$ and $M\\ge4L$,~\\eqref{eq:mixed-finite-kernel}\nbounds the $\\mathscr D_{L;R,M}$-mean of\n$S_J(N):=\\sum_{j>J}t_j^{-\\alpha_j}V_j(N)$ by\n$(1+4L/M)K_J<1/2$. Choose a sample with $S_J(N)<1$.\nThis controls all countably many tails at once: each nonnegative\nsummand is below $1$, hence $U_j(N)<t_j$ for every $j>J$.\nThe first $J$ sets contribute no displacement; later sets cover the\nrest of $A$, with overlaps only enlarging the upper bound. Thus,\n\\[\n 0<\\Delta_{2,A}(N)\\le\\sum_{j>J}U_j(N)\\le\\varepsilon.\n\\]\nEquation~\\eqref{eq:intro-displacement} now excludes rationality at base two.\nFor $0\\le r<d$, the function $(b^r-1)/(b^d-1)$ is nonincreasing\non $b>1$.  For $r>0$, cancel $b-1$ and write it as\n$A(b)/(A(b)+C(b))$, where $A(b)=\\sum_{i<r}b^i$ and\n$C(b)=\\sum_{r\\le j<d}b^j$.  Its derivative has the sign of\n$A'C-AC'=\\sum_{i<r\\le j<d}(i-j)b^{i+j-1}<0$.\nThus $\\Delta_{b,A}(N)\\le\\Delta_{2,A}(N)$ for every real $b\\ge2$;\nwhen $b$ is an integer, this contradicts the lower bound obtained from\n\\eqref{eq:intro-displacement} under rationality. Any prescribed positive\ninteger can be included as a divisor of $L$, so the chosen indices\ncan also be required to be arbitrarily large.\n\\end{proof}\n\nThe same proof permits any positive weights $\\eta_j$ with $\\sum_j\\eta_j=1$:\nreplace $2^{j\\alpha_j}$ in~\\eqref{eq:strengthened-cover} by\n$\\eta_j^{-\\alpha_j}$ and take $t_j=\\varepsilon\\eta_j$.\nThese weights allocate error allowances; they do not reweight the\nLambert subseries.\nThe stronger condition\n$\\sum_jC_j2^{j\\alpha_j}2^{\\alpha_j}/(2^{\\alpha_j}-1)^2<\\infty$\nimplies~\\eqref{eq:strengthened-cover}, since each of its summands is the\ncorresponding summand of~\\eqref{eq:strengthened-cover} multiplied by\n$2^{\\alpha_j}/(2^{\\alpha_j}-1)>1$.\nThis proves inclusion of the hypotheses for a given cover; after\noptimisation over all covers, the inclusion of support classes is\nstrict by a \\href{https://github.com/wcook04/plectis-erdos/blob/3973d3b10bce72017b8f60093f0d6c3f4f592d80/lean/ErdosProblems/Erdos257/PaperCompleteR8/AnalyticIncomparability.lean\\#L41}{corresponding source theorem}.\n\n\\section{A common index for the two criteria}\\label{sec:mixed}\nSmall displacements at unrelated indices do not control the union.\nPut both finite cancellation requirements in one modulus, then bound\nboth remaining errors on the same finite distribution. The weighted\nproof enlarges the modulus; (S) keeps the cover estimate valid.\nWe work at base two and compare displacements for larger integer bases.\n\\begin{samepage}\n\\begin{theorem}[mixed weighted and cover supports]\\label{res:mixed-supports}\nLet $E,V\\subseteq\\Npos$.  Suppose $E$ satisfies\n\\eqref{eq:weighted-return} for a finite nonempty prime set $P$, and\n$V\\subseteq\\bigcup_jF_j$ for finite sets and nonnegative majorants\nsatisfying the hypotheses of Theorem~\\ref{thm:variable-fractional-cover},\nwith either~\\eqref{eq:strengthened-cover} or its positive-weight variant.\nThen $X_A(b)$ is irrational for every\ninfinite $A\\subseteq E\\cup V$ and every integer $b\\ge2$.\n\\end{theorem}\n\\end{samepage}\n\\begin{proof}\n\\emph{Fix the finite cancellation requirements.}\nFix $\\varepsilon>0$ and an integer $N_0\\ge1$, and put $\\rho=\\varepsilon/3$.\nUse the cover notation $B_j,U_j,V_j$ above, with $\\eta_j=2^{-j}$ in\nthe case~\\eqref{eq:strengthened-cover}.  Set $t_j=\\rho\\eta_j$ and choose\n$J$ so that\n\\[\n K_J=\\sum_{j>J}\\frac{C_jt_j^{-\\alpha_j}}{B_j-1}<\\frac1{16}.\n\\]\nChoose a finite $F\\subseteq E$ so that the weighted sum over $E\\setminus F$ is\n$\\kappa<\\rho/16$.  Let $L$ be a positive common multiple of $N_0$, all\nmembers of $F$, and all members of the first $J$ finite sets. Fix\n$J$, $F$ and $L$ before choosing scales; later moduli must be multiples\nof $L$.\n\n\\emph{Estimate both errors on one distribution.}\nWith this finite data fixed, use the base-two choices\n\\[\n Q=L\\prod_{p\\in P}p^{\\lfloor\\log_pH\\rfloor},\\qquad\n G=\\lfloor H/\\max P\\rfloor,\\qquad M=\\lfloor2^{G/2}\\rfloor.\n\\]\nBoth estimates use $\\mathscr D_{Q;M,M}$: the scale index $k$ runs\nfrom $M$ to $2M-1$, and each observation is $N=Qm$.\nThe estimate~\\eqref{eq:weighted-main-bound} allows any common multiple\nof $F$ as $L$, so it accommodates the cover's extra divisors. It also\nholds for finite or empty $E$: infinitude was needed only for the final\npositive displacement. Consequently\n\\[\n \\mathscr D_{Q;M,M}(\\Delta_{2,E}/\\rho)<\\frac18\n\\]\nfor sufficiently large $H$.  For the same finite distribution, (S) and\nnonnegative interchange give\n\\[\n \\mathscr D_{Q;M,M}S_J\\le(1+4Q/M)K_J<\\frac18,\n \\qquad S_J=\\sum_{j>J}t_j^{-\\alpha_j}V_j,\n\\]\nbecause $Q/M\\to0$.\n\\emph{Choose the index only after adding the errors.}\nOn this distribution,\n\\[\n \\mathscr D_{Q;M,M}\\bigl(\\Delta_{2,E}/\\rho+S_J\\bigr)<\\frac14.\n\\]\nLinearity of the mean suffices; no independence assumption is needed.\nChoose a sampled $N=Qm$ where the nonnegative sum is less than $1$.\nThen $\\Delta_{2,E}(N)<\\rho$, and every summand of $S_J(N)$ is less\nthan $1$. Since $U_j^{\\alpha_j}\\le V_j$, the same choice gives\n$U_j(N)<t_j$ for every $j>J$.\nThe displacement terms from the first $J$ finite sets vanish because\ntheir exponents divide $L$, and hence divide $N$. Therefore\n\\[\n \\Delta_{2,A}(N)\\le\\Delta_{2,E}(N)+\\Delta_{2,V}(N)<2\\rho<\\varepsilon,\n \\qquad N\\ge Q\\ge L\\ge N_0.\n\\]\nThis inequality remains valid when $E$ and $V$ overlap. Since $A$ is\ninfinite, $\\Delta_{2,A}(N)>0$. The comparison\n$0<\\Delta_{b,A}(N)\\le\\Delta_{2,A}(N)$ proved in the cover argument\ntherefore gives arbitrarily small positive values at every integer\nbase. Equation~\\eqref{eq:intro-displacement} rules out rationality.\n\\end{proof}\n\\section{Comparison of the support classes}\\label{sec:comparison}\nLet $\\mathcal W_b$ consist of supports satisfying the weighted condition\nat base $b$ for some finite nonempty prime set, and let $\\mathcal C$\nconsist of supports with a cover satisfying~\\eqref{eq:strengthened-cover}.\nMembership requires one finite prime set or one summable cover for the\nwhole support. The constructions below exclude every such choice.\n\n\\begin{proposition}[incomparable support criteria]\n\\label{res:weighted-cover-incomparability}\nThere are infinite positive supports $E$ and $V$ such that\n\\[\n E\\in\\mathcal W_2,\\quad E\\notin\\mathcal C,\\qquad\n V\\in\\mathcal C,\\quad V\\notin\\mathcal W_b\\ (b\\ge2),\\qquad\n \\sum_{a\\in V}\\frac1a=\\infty.\n\\]\n\\end{proposition}\n\nBoth constructions use $\\{qd:d\\mid\\prod_{p\\in P}p\\}$, with opposite\ndivisibility patterns. For $E$, growing powers $q=2^k$ make the weighted\nsum converge, while simultaneous divisors obstruct every cover.\nFor $V$, disjoint prime sets force every finite-prime weighted sum to\ndiverge, while fractional powers of the divisor counts make the cover\ncost summable.\n\n\\subsection{A lower bound independent of the cover}\nTo exclude every cover, we bound its cost below using only finite\nsubsets of the support. For finite $F$, let $\\mathbb E_F$ denote\nthe uniform mean modulo $\\operatorname{lcm}(F)$, with\n$\\operatorname{lcm}(\\varnothing)=1$ and $f_{\\varnothing}=0$.\nAllow arbitrary weights $\\eta_j>0$ with $\\sum_j\\eta_j=1$, so that the\nlower bound also survives reweighting the cover. Set\n\\[\n K=\\sum_j\\frac{C_j\\eta_j^{-\\alpha_j}}{2^{\\alpha_j}-1},\\qquad\n \\Psi(0)=0,\\quad\n \\Psi(t)=\\inf_{0<\\alpha\\le1}\\frac{t^\\alpha}{2^\\alpha-1}\\quad(t\\ge1).\n\\]\nHere $K$ depends on the cover, majorants, exponents and weights;\n$\\Psi$ minimises only over the scalar exponent $\\alpha$.\nWrite $\\log^+t=\\log\\max\\{1,t\\}$. For every finite\n$F\\subseteq\\bigcup_jF_j$,\n\\begin{equation}\n K\\ge\\mathbb E_F\\Psi(f_F)\\ge e\\,\\mathbb E_F\\log^+f_F.\n \\label{eq:cover-log-obstruction}\n\\end{equation}\nIndeed, if $f_F(n)=t>0$, coverage gives\n$\\sum_jf_{F_j}(n)\\ge t$, so some $j$ satisfies\n$f_{F_j}(n)\\ge\\eta_jt$. Multiply its divisor majorant by\n$\\eta_j^{-\\alpha_j}/(2^{\\alpha_j}-1)$ to obtain at least\n$t^{\\alpha_j}/(2^{\\alpha_j}-1)\\ge\\Psi(t)$.\nThe index may depend on $n$, but the sum of all these weighted majorants\nbounds $\\Psi(f_F(n))$ at every $n$.\nAverage over $1\\le n\\le X$. Each divisor indicator has mean\n$\\lfloor X/d\\rfloor/X\\le1/d$, so the weighted majorants have total\nmean at most $K$. As $X\\to\\infty$, the mean of $\\Psi(f_F)$ tends to\n$\\mathbb E_F\\Psi(f_F)$, proving the first inequality. Only this\nfinite-support function needs a period; the covering moduli need not\ndivide $\\operatorname{lcm}(F)$. The second is immediate for $t=0,1$;\nfor $t>1$, use $2^\\alpha-1\\le\\alpha$ and $e^u/u\\ge e$ with\n$u=\\alpha\\log t$.  The scalar inequality\n$t^\\alpha/(2^\\alpha-1)\\ge e\\log t$ also has a\n\\href{https://github.com/wcook04/plectis-erdos/blob/c91562bd574a387cde904481e609c7b4cacebb14/lean/ErdosProblems/Erdos257/PaperCompleteR7/CoverKernel.lean\\#L170}{corresponding source theorem}.\n\nFor $\\{qd:d\\mid\\prod_{p\\in P}p\\}$, Appendix~\\ref{app:elementary-details}\nobtains an asymptotically sharp cost in~\\eqref{eq:optimal-cube-cost}.\nThe separation below needs only~\\eqref{eq:cover-log-obstruction}.\n\n\\subsection{Two separating constructions}\n\\begin{proof}[Proof of Proposition~\\ref{res:weighted-cover-incomparability}]\nThe reciprocal sum over odd primes diverges, even after finitely many\nare removed.  Otherwise $\\prod_{p\\text{ odd}}(1+1/p)$ would bound the sum of reciprocals of\nodd squarefree integers; the decomposition $n=ds^2$ would\nthen make the odd harmonic series converge.\n\n\\emph{A weighted support with no divisor cover.}\nThe $k$th group of exponents will carry a factor $2^k$. The event $v_2(n)=k$ has density $2^{-(k+1)}$, so a prime\nreciprocal sum of order $2^k$ will give a fixed contribution to the\nmean logarithmic count. Taking its leading constant below $\\log2$\nleaves exponential decay in the weighted sum.\nChoose pairwise disjoint finite sets $P_k$ of odd primes such that\n$2^k/4\\le S_k:=\\sum_{p\\in P_k}1/p<2^k/4+1$.  Put\n$M_k=\\prod_{p\\in P_k}p$ and\n$E=\\bigcup_{k\\ge1}\\{2^kd:d\\mid M_k\\}$.\nFor $P=\\{2\\}$ the weighted sum is\n\\[\n \\sum_{k\\ge1}\\frac{\\prod_{p\\in P_k}(1+1/p)}{2^{2^k}-1}\n \\le 2e\\sum_{k\\ge1}\n       \\exp\\bigl(-(\\log2-1/4)2^k\\bigr)<\\infty.\n\\]\nFor $F_m=\\bigcup_{k=1}^m\\{2^kd:d\\mid M_k\\}$, average\n$\\log^+f_{F_m}$ over a period, using the disjoint events\n$v_2(n)=k<m$, of density $2^{-(k+1)}$.\nOn the $k$th event use only the $k$th group, giving\n$f_{F_m}(n)\\ge2^{Z_k(n)}$, where\n$Z_k(n)=\\sum_{p\\in P_k}\\mathbf1_{p\\mid n}$. Odd-prime divisibility\nis independent of that event, so\n\\[\n \\mathbb E_{F_m}\\log^+f_{F_m}\n \\ge(\\log2)\\sum_{k=1}^{m-1}2^{-(k+1)}S_k\n \\ge\\frac{m-1}{8}\\log2.\n\\]\nA cover satisfying~\\eqref{eq:strengthened-cover} would bound all these\nmeans by its fixed cost, contradicting their growth in $m$.\nThus $E\\notin\\mathcal C$.\n\n\\emph{A divisor cover with no finite-prime weighted condition.}\nChoose unused odd primes $q_j\\ge4^j$ and\ndisjoint finite sets $P_j$ of unused odd primes.  Stop each $P_j$\nat the first prefix with $R_j:=\\prod_{p\\in P_j}(1+1/p)\\ge q_j$.\nPrime reciprocal divergence permits this, and minimality gives\n$q_j\\le R_j<4q_j/3$.\nPut $S_j=\\sum_{p\\in P_j}1/p$, $M_j=\\prod_{p\\in P_j}p$, and\n$V=\\bigcup_{j\\ge1}F_j$, where $F_j=\\{q_jd:d\\mid M_j\\}$.\nNo prime divides exponents in two different sets $F_j$, and\n\\[\n \\sum_{a\\in F_j}\\frac1a=\\frac{R_j}{q_j}\\in[1,4/3).\n\\]\nThus $\\sum_{a\\in V}1/a$ diverges. Given a finite set $P$ of primes,\nall exponents in all but finitely many $F_j$ are coprime to\n$\\prod_{p\\in P}p$. On each such $F_j$ we have $h_P(a)=1$, so its\ncontribution to the base-$b$ weighted sum is\n$R_j/[q_j(b-1)]\\ge1/(b-1)$. Hence $V\\notin\\mathcal W_b$ for every\ninteger $b\\ge2$.\n\nIt remains to make the entire cover cost summable. The inequalities\n$\\log(1+x)\\ge x-x^2/2$ and $\\sum_{p\\in P_j}p^{-2}\\le1$ give\n$S_j\\le\\log R_j+1/2<\\log q_j+1$. Writing\n$z_j=2^{\\alpha_j}-1$, the index weight and divisor product contribute\n$(1+z_j)^j\\prod_{p\\in P_j}(1+z_j/p)\\le\\exp(z_j(j+S_j))$.\nWe choose $z_j(j+S_j)=1$ to keep this factor bounded while paying\nthe remaining cost $1/(q_jz_j)$.\nSet\n\\[\n z_j=(j+S_j)^{-1},\\qquad\n \\alpha_j=\\log_2(1+z_j),\\qquad\n c_{j,q_jd}=z_j^{\\omega(d)}\\quad(d\\mid M_j),\n\\]\nwith other coefficients zero and $\\omega(d)$ the number of prime\nfactors of $d$.  If $q_j\\mid n$ and $Z$ primes in $P_j$ divide $n$,\nthe divisor majorant equals $(1+z_j)^Z=f_{F_j}(n)^{\\alpha_j}$;\notherwise both sides vanish. The sum defining $C_j$ is\n$C_j=q_j^{-1}\\prod_{p\\in P_j}(1+z_j/p)$, whence\n\\[\n \\frac{C_j2^{j\\alpha_j}}{2^{\\alpha_j}-1}\n =\\frac{(1+z_j)^j}{q_jz_j}\n    \\prod_{p\\in P_j}(1+z_j/p)\n \\le\\frac{e(j+S_j)}{q_j}\n \\le e\\bigl((1+\\log4)j+1\\bigr)4^{-j}.\n\\]\nThe last bound uses that $(j+\\log x+1)/x$ decreases for $x\\ge4^j$.\nThe bound is summable in $j$ and includes the factor $2^{j\\alpha_j}$\nrequired by~\\eqref{eq:strengthened-cover}; hence $V\\in\\mathcal C$.\n\\end{proof}\n\n\n\\subsection{A strict enlargement and finite unions}\nThe mixed theorem now applies to a support outside both individual\nclasses.\n\\begin{corollary}[a support requiring the mixed criterion]\n\\label{res:strict-mixed-supports}\nThere is an infinite positive support $U$ with\n$U\\notin\\mathcal C$ and $U\\notin\\mathcal W_b$ for every integer $b\\ge2$,\nsuch that $X_A(b)$ is irrational for every infinite $A\\subseteq U$\nand every integer $b\\ge2$.\n\\end{corollary}\n\\begin{proof}\nLet $U=E\\cup V$, with $E,V$ as in\nProposition~\\ref{res:weighted-cover-incomparability}. A cover of $U$\nis also a cover of $E$, while convergence of the weighted sum over $U$\nwould imply convergence of the sum over $V$ for the same prime set,\nsince all summands are nonnegative. Thus $U$\nbelongs to neither individual class. Theorem~\\ref{res:mixed-supports}\ngives irrationality for every infinite subset of $U$ at every integer\nbase $b\\ge2$.\n\\end{proof}\nThe fixed-dyadic cover class in (V) is closed under finite unions.\nWhen interleaving two covers, we must control the index-dependent\nfactor $2^{j\\alpha_j}$. Place their $j$th finite sets at positions\n$2j-1,2j$ and halve each exponent; this factor then does not increase.\nThe old majorant coefficients still work, since\n$f^{\\alpha/2}\\le f^\\alpha$ for every nonnegative integer $f$.\nIt remains to compare the denominators in the cost. Writing\n$T_j=C_j2^{j\\alpha_j}/(2^{\\alpha_j}-1)$ for the original $j$th cost,\nits new cost is at most\n\\[\n \\frac{C_j2^{j\\alpha_j}}{2^{\\alpha_j/2}-1}\n =(1+2^{\\alpha_j/2})T_j\\le(1+\\sqrt2)T_j.\n\\]\nFor $r$ covers, interleave at positions at most $rj$ and divide each\nexponent by $r$; the cost ratio is\n$(2^\\alpha-1)/(2^{\\alpha/r}-1)<2r$.  Weighted supports are also closed under finite unions: take the union\nof their prime sets: $h_P(a)$ can only increase, and $h/(b^h-1)$\ndecreases with $h$.\nBoth criteria are hereditary and admit finite supports. Their mixed\nclass is therefore closed under finite unions and finite changes,\nalready with the fixed dyadic weights in (V).\nCountable unions need not remain in the mixed class: every prime\nsingleton is admitted, but full prime support fails the small-displacement\ncondition, as shown in Section~\\ref{sec:map}.\nThe reciprocal-summable class is contained in the weighted class, since\n$h/(2^h-1)\\le1$. A direct proof appears in\nAppendix~\\ref{sec:reciprocal-support}.\n\n\\section{Limits of the small-displacement criterion}\\label{sec:map}\nSmall displacement is a sufficient condition for irrationality, not a\nnecessary one. Full support already shows the distinction:\n\\[\n \\Delta_{2,\\Npos}(N)\n >(2^N-1)\\sum_{a>N}2^{-a}=1-2^{-N}\\ge\\frac12\n \\qquad(N\\ge1).\n\\]\nThe full-support value is nevertheless\n\\mword{Erdos249257/CertificateKernel.lean}{8328}{irrational_erdosSum_full_support}{irrational}~\\cite{erdos1948}.\nThus a proof for arbitrary support cannot rely on making these\ndisplacements tend to zero in every case. Other uses of the displacement,\ninteger linear forms and averaging arguments are not excluded.\n\nFull prime support gives a second obstruction. Although every prime\nsingleton is admitted by both support criteria, for the set of all\nprimes $\\mathcal P$ one has\n$\\Delta_{2,\\mathcal P}(N)>1/3$ for every $N\\ge1$: indeed\n$\\sum_{r\\ge1}2^{-r}\\omega(N+r)\\ge1$, whereas\n$X_{\\mathcal P}(2)\\le\\sum_{a\\ge2}(2^a-1)^{-1}<2/3$.\nHere $\\omega(n)$ is the number of distinct prime divisors of $n$.\n\n\nYet Tao--Ter\\\"av\\\"ainen prove the full-prime case at base~$2$\n\\cite[Theorem~1.3, p.~4]{taoteravainen2025}.  The paragraph following it\nstates extensions to prime support at every integer base and to full\nprime-power support at base~$2$, leaving the modifications to the reader.\nIt does not treat arbitrary infinite thinnings.  The proposed thinning\nextension is not a premise of any theorem here.\n\n\nThe logarithmic lower bound is not a converse: replacing a fractional\nmajorant by a logarithmic one need not control averages along the\nmultiples of a prescribed modulus.  The counterexample and the bound that remains valid on ordinary initial\nintervals are proved in\n\\papersectionlink{erdos257-mersenne-reasoning-surface.pdf}{record257:logarithmic-sampling}\n{Section~13 of the companion paper}; those proofs were first worked out in\nan AI-assisted note of 17 September 2026. These are statements about finite\naverages; they do not rule out an irrationality criterion based on other\nuses of logarithmic divisor counts.\n\n\nFor squarefree support, Duverney and Tachiya's theorem already gives\nirrationality.  It holds at every base $2^j$, $j\\ge1$, by Corollary~1.2 and Example~1.1\nof Duverney and Tachiya~\\cite[p.~4]{duverneytachiya}.  Their corollary\ncovers the $s$-free products of any pairwise coprime sequence of polynomial\ngrowth, and they present it as support for the conjecture of Erd\\H{o}s and\nGraham stated here as Problem~\\ref{res:problem}.\nFor the squarefree support with $1$ removed, the divisor count is\n$2^{\\omega(n)}-1$, which is odd for $n\\ge2$. This excludes a\nfirst-block-divisibility condition at even bases. Adjoining $1$ removes\nthat parity obstruction while changing the sum by a rational number.\nThe argument concerns that normalisation of the certificate, and gives\nno obstruction to irrationality of the sum. Details and the other known\nsupport families appear in\n\\papersectionlink{erdos257-mersenne-reasoning-surface.pdf}{record257:known-supports}\n{the comparison with known support theorems in the companion paper}.\n\nA growth-only criterion gives a different class: if\n$A=\\{c_1<c_2<\\cdots\\}$ and $\\limsup_n c_n/2^n=\\infty$, then\n$X_A(b)$ is irrational for every integer $b\\ge2$\n\\cite[Theorem~1]{erdos1975}.  The same section of the companion paper proves the translation from\ndenominator growth and gives examples\nshowing that this class and reciprocal summability are incomparable.\nThe interval-filling constructions in~\\cite{bkkkz2026} and the\nfreely chosen lacunary denominators of~\\cite{vandoornkovac} are not\nconstructions of subseries of these fixed Mersenne weights.\n\nPrime-incidence arguments require additional care as well: multiplying\nall prime exponents by $2$ creates positive correlations between their\ndivisibility indicators.  The identity and covariance calculation appear at the end of\n\\papersectionlink{erdos257-mersenne-reasoning-surface.pdf}{record257:mixed-proof}\n{the companion paper's discussion of mixed supports}.  Size conditions alone do not\nsupply that independence.\n\n\\subsection{Algebraic bases on separated supports}\n\\label{sec:separated-support-transfer}\nThe companion\n\\href{optimal-sparse-perturbations.pdf}{\\emph{Reading eight Erd\\H{o}s\nproblems together}}, subsection ``Divisibility cuts at every algebraic base'',\nstates a transcendence extension based on the number-field Subspace\nTheorem. Its hypothesis is an infinite set $H$ with indices $L_j<M_j$,\nwhere $L_j\\to\\infty$ and $M_j-L_j\\to\\infty$, such that every $n\\in H$\nwith $n\\le L_j$ divides $L_j$, and every $n\\in H$ with $n>L_j$ is a\nmultiple of $M_j$. The claimed conclusion is transcendence of\n$\\sum_{n\\in B}w_n/(t^n-1)$ for every infinite $B\\subseteq H$, bounded\npositive integer weights $w_n$ and real algebraic $t>1$.\n\nEvery divisibility chain has such cuts, as does\n$H_* = \\bigcup_j N_j\\{1,\\ldots,2^{N_j}\\}$ with $N_0=1$ and\n$N_{j+1}=2N_j\\operatorname{lcm}(1,\\ldots,2^{N_j})$.\nEach block has a reciprocal sum of at least $1/2$ and contains\nantichains of unbounded size. The weighted sum for one prime is at\nmost $2t^2/(t-1)^3$ at every real $t>1$. This connects the example\nwith the weighted criterion, although the separation condition is\nnot asserted for every support satisfying that criterion.\n\nThe companion proof is outside this paper and is not a premise of the\nweighted, cover or mixed theorem. It has two AI proof reviews, no Lean\nproof of transcendence and no independent human review. No historical\npriority is asserted for the extension. The case of arbitrary infinite\nsupport at base two remains unresolved.\n\n\\appendix\n\\section{Reciprocal-summable supports}\\label{sec:reciprocal-support}\nErd\\H{o}s stated the following extension of his pairwise-coprime theorem\n\\cite[p.~222]{erdos1968}. It follows from\nTheorem~\\ref{res:weighted-support}, since $h/(2^h-1)\\le1$, but the\ndirect argument explains why reciprocal summability makes the averaging\nsimpler.\n\n\\begin{theorem}[reciprocal-summable supports]\\label{res:reciprocal-support}\nLet $A\\subseteq\\Npos$ be infinite.  If\n\\[\n  \\sum_{a\\in A}\\frac1a<\\infty,\n\\]\nthen \\(X_A(b)\\) is irrational for every integer \\(b\\ge2\\).\n\\end{theorem}\n\nErd\\H{o}s also discussed weaker conditions \\cite[pp.~222, 226]{erdos1968}.\nWe do not identify our weighted hypothesis with the conditions he suggested.\n\nReciprocal summability permits two successive limits: first the\nobservation length grows with the modulus fixed, then the modulus\ngrows until each fixed exponent divides it. The two interchanges use\ndifferent summable bounds; neither requires a period for the infinite\nsupport.\n\nFix the integer base $b\\ge2$ and put\n\\[\n w_{b,d}(N)=\\frac{b^{N\\bmod d}}{b^d-1},\\qquad\n T_N^{(b)}=\\sum_{d\\in A}w_{b,d}(N).\n\\]\nThen $T_N^{(b)}-T_0^{(b)}=\\Delta_{b,A}(N)$.\nFor a fixed positive integer $Q$, the orbit of $Q$ modulo $d$ consists of\n$d/g$ residues, where $g=(Q,d)$.  Summing the geometric progression gives\n\\begin{equation}\n M_{Q,b}(d):=\\lim_{X\\to\\infty}\\frac1X\\sum_{m=1}^Xw_{b,d}(Qm)\n =\\frac{g}{d(b^g-1)}\\le\\frac1d.\n \\label{eq:gcd-orbit-mean}\n\\end{equation}\nThe example $b=2$, $Q=4$, $d=6$ in\nSection~\\ref{sec:eight-return-extensions} has mean $1/9$ for $w_{2,6}$,\nbut $2/21$ for its displacement. When $d\\mid Q$,\n$w_{b,d}(Qm)=w_{b,d}(0)$ and the displacement vanishes.\n\nTo interchange the sum over exponents and this limit, we use a bound\nsummable in $d$ for the fixed modulus $Q$.  Counting positive multiples of $d$ gives\n\\begin{align*}\n \\frac1X\\sum_{m=1}^Xw_{b,d}(Qm)\n &=\\sum_{r\\ge1}b^{-r}\n       \\frac{\\#\\{1\\le m\\le X:d\\mid Qm+r\\}}X\\\\\n &\\le\\sum_{r\\ge1}b^{-r}\\frac{Q+r}{d}\n =\\frac{Q/(b-1)+b/(b-1)^2}{d}\\le\\frac{Q+2}{d}.\n\\end{align*}\nIndeed, the counted multiples are distinct and at most $QX+r$.\nFor this fixed $Q$, the majorant $(Q+2)/d$ is summable over $A$.\nDominated convergence therefore takes the observation-length limit inside\nthe sum over $d$, giving the Ces\\`aro mean\n$\\sum_{d\\in A}M_{Q,b}(d)$.  This majorant is not uniform in growing $Q$.\n\nNext let $Q_t=\\operatorname{lcm}(1,\\ldots,t)$.\nFor each fixed $d$, eventually $d\\mid Q_t$ and\n$M_{Q_t,b}(d)=(b^d-1)^{-1}$.  A second dominated-convergence argument,\nusing $M_{Q_t,b}(d)\\le1/d$, yields\n\\begin{equation}\n \\sum_{d\\in A}M_{Q_t,b}(d)\\longrightarrow X_A(b).\n \\label{eq:lcm-prefix-orbit-limit}\n\\end{equation}\nThe limiting displacement means therefore tend to zero. Choose $t$,\nthen a sufficiently long finite average, then a term no larger than that\naverage. The resulting arbitrarily small positive displacements contradict\nthe rationality bound $1/q$ in~\\eqref{eq:intro-displacement}. This proves\nTheorem~\\ref{res:reciprocal-support} at every integer base.\n\nPowerful integers (all prime exponents at least two) have the form\n$u^2v^3$, so their reciprocal sum is at most $\\zeta(2)\\zeta(3)$.\nThe theorem therefore applies to every infinite subset. For full\nperfect-power supports, compare Duverney--Tachiya\n\\cite[Corollary~1.2, p.~4]{duverneytachiya} and Luca--Tachiya's earlier\nlinear-independence results \\cite{lucatachiya2014independence}.\n\n\\section{Finite-support denominator periods}\\label{sec:period}\n\nThe least common multiple of the selected exponents survives cancellation\nas the multiplicative order of the base modulo the reduced denominator.\n\n\\begin{theorem}[the exact denominator period]\\label{res:period}\nLet $F\\subseteq\\Npos$ be finite and nonempty, let $b\\ge2$ be an integer, and\nlet $D_F>0$ be the denominator of $X_F(b)$ in lowest terms.  Then $D_F$ is\ncoprime to $b$, and\n\\[\n  \\operatorname{ord}_{D_F}(b)=\\lcm\\{n:n\\in F\\}.\n\\]\nIf moreover $\\lcm(F)\\ge2$, then $\\lcm(F)<D_F$.  We use\n$\\operatorname{ord}_1(b)=1$, so the statement includes $F=\\{1\\}$\nat $b=2$.\n\\end{theorem}\n\nPut $L=\\operatorname{lcm}(F)$.  Clearing denominators gives\n$D_F\\mid b^L-1$, so the upper divisibility for the order is immediate.\nFor the reverse, fix a divisibility-maximal $n\\in F$ with $n\\ge2$ and\nchoose a prime $\\ell\\mid\\Phi_n(b)$. With $e=v_\\ell(b^n-1)$, the\ncyclotomic fact in Section~\\ref{app:elementary-details} gives\n$\\operatorname{ord}_{\\ell^e}(b)=n$ for the full prime power $\\ell^e$.\nEvery other selected exponent $m$ has $n\\nmid m$, hence\n$v_\\ell(b^m-1)<e$.  The $n$th summand has uniquely smallest $\\ell$-adic\nvaluation and cannot cancel.  Thus $\\ell^e\\mid D_F$ and\n$n\\mid\\operatorname{ord}_{D_F}(b)$.  Taking all maximal selected exponents\nproves the order statement; the size bound follows from\n$\\operatorname{ord}_{D_F}(b)\\mid\\varphi(D_F)<D_F$ when $L\\ge2$.\nThe case $F=\\{1\\}$ has order one directly.\n\nThe unique-valuation argument also permits signs $\\pm1$ on the finite\nsummands. This signed extension is not included in the formal statement.\nThe distinction between primes and prime powers is visible in\n\\[\n X_{\\{2,3\\}}(2)=\\frac{10}{21},\\qquad\n X_{\\{2,6\\}}(2)=\\frac{22}{63}.\n\\]\nIn both examples $2$ has multiplicative order six modulo the denominator.\nThe first combines orders two and three; the second retains the\norder-six prime power $9$, although no prime divisor of $63$ has order six.\nThe boundary $F=\\{1\\}$ at base two has $D_F=L=1$.\nThese lower denominator bounds give no upper height control for infinite\npartial sums and do not decide an infinite-support value.  Van Assche instead constructs approximants for the full Lambert series and proves\nboth nonvanishing and decay of the associated integer linear forms\n\\cite[Lemma~1 and (34)--(35)]{vanassche2001}.  That approximation mechanism\ndoes not follow from denominator survival alone.\n\n\nIn particular, $1/21$ is not a finite subseries sum. If it were, the order\nof $2$ modulo $21$ would force every selected exponent to divide $6$.\nThe exponent $1$ is excluded since its weight is $1$. Any sum containing\n$2$ or $3$ exceeds $1/21$, whereas the only nonzero remaining sum is\n$1/63$, from the exponent $6$. This will distinguish finite and infinite\nrepresentations in the next appendix.\n\n\\section{Rational membership}\\label{sec:rational-membership}\nThe preceding results start from a support and prove irrationality.\nWe now fix a target and ask whether any support represents it.\nThe first recurrence is a necessary condition on a rational value;\nthe greedy recurrence later gives an exact membership criterion.\nNeither supplies the missing arithmetic information for $1/2$ or $1/21$.\n\n\\subsection{Rational values and scaled tails}\\label{sec:forced}\nSuppose $X_A(2)=p/v$, where $p\\in\\Z$ and $v\\ge1$ is an integer.\nMultiplying the coefficient-series identity by $v2^N$ gives\n\\[\n z_N:=v\\sum_{r\\ge1}c_A(N+r)2^{-r}\n     =2^Np-v\\sum_{n=1}^{N}c_A(n)2^{N-n}\\in\\Z.\n\\]\nSeparating the first term of the tail then gives\n$z_{N+1}=2z_N-vc_A(N+1)$.\nAll the coefficients $c_A(n)$ are divisor counts of the same set $A$.\n\nFor nonempty $A$, every $a_0=\\min A$ consecutive integers contain\na multiple of $a_0$, so zero runs of $c_A$ have length at most $a_0-1$.\nFor infinite $A$, choose $k$ exponents and a common multiple $L$.\nThen $c_A(L)\\ge k$, and the scaled tail satisfies\n$\\sum_{r\\ge1}c_A(L-1+r)2^{-r}\\ge k/2$. Thus scaled tails are\nunbounded.\n\nThe compatibility condition on the coefficients is expressed by\nDirichlet convolution:\n$\\mu*c_A=\\mathbf1_A$, where $\\mu$ is the M\\\"obius function.\nA putative recurrence with integral forcing must therefore satisfy\n$(\\mu*c_A)(n)\\in\\{0,1\\}$ for all $n$.\nThe telescoping proof of the integer-recurrence criterion and the\nM\\\"obius inversion argument are given under\n\\papersectionlink{erdos257-mersenne-reasoning-surface.pdf}{record257:coefficient-recurrence}\n{Bounds for a general coefficient sequence} in the companion paper.  The example $A=\\{2\\}$ there has value $1/3$.\nThe correspondence permits finite supports; infinitude remains a separate\nrequirement for a counterexample to Problem~\\ref{res:problem}.\n\n\\subsection{The achievement set}\\label{sec:geometry}\nLet $w_n=(2^n-1)^{-1}$ and\n\\[\n \\Ach=\\left\\{\\sum_{n\\ge1}\\varepsilon_nw_n:\n                   \\varepsilon_n\\in\\{0,1\\}\\right\\},\\qquad\n R_N=\\sum_{n>N}w_n.\n\\]\nThe estimate\n\\[\n 2^{-N}<R_N\\le2^{-N}+\\frac23\\,4^{-N}<w_N\\qquad(N\\ge1)\n\\]\nfollows by writing $w_n=2^{-n}+4^{-n}/(1-2^{-n})$ and using\n$n\\ge N+1\\ge2$ for the middle bound. If two selectors first differ\nat $n$, their sums differ by at least $w_n-R_n>0$, proving uniqueness.\nSince every weight exceeds its tail, Hornich's theorem~\\cite{hornich1941}, as proved by\nNitecki~\\cite[Theorem~4(1), p.~9]{nitecki2013}, shows that the coding image is\na Cantor set of measure $\\lim_N2^NR_N$; the estimate above gives\n$2^NR_N\\to1$, so $\\lambda(\\Ach)=1$.\nConcretely, fixing the first $N$ digits gives $2^N$ disjoint closed\nintervals of length $R_N$.  These interval unions decrease to $\\Ach$,\nand continuity of Lebesgue measure from above gives the same limit.\nKova\\v{c}--Tao record this strict-tail inequality and the Cantor conclusion in the fixed-base setting~\\cite[Remark~4.1]{kovactao}.\nThe\n\\papersectionlink{erdos257-mersenne-reasoning-surface.pdf}{record257:geometry}\n{geometry section of the companion paper} also treats restricted supports.\nFor any allowed set $J$, removing exponents can only decrease the tail\nof each retained weight, so the same inequality gives injectivity. If $J$ omits exactly the\nfinite set $F$, its achievement set has measure $2^{-|F|}$; if $J$\nomits infinitely many exponents, that measure is zero. The\n\\href{https://github.com/wcook04/plectis-erdos/blob/c91562bd574a387cde904481e609c7b4cacebb14/lean/ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean\\#L30}{tail comparison},\n\\href{https://github.com/wcook04/plectis-erdos/blob/c91562bd574a387cde904481e609c7b4cacebb14/lean/ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean\\#L54}{injectivity} and\n\\href{https://github.com/wcook04/plectis-erdos/blob/c91562bd574a387cde904481e609c7b4cacebb14/lean/ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean\\#L397}{measure dichotomy} have separate source statements.\n\n\\subsection{The greedy recurrence}\\label{sec:actual-repairs}\nFor $x\\ge0$, the greedy rule starts with $r_0=x$ and, at rank $n\\ge1$,\nselects $n$ when $r_{n-1}\\ge(2^n-1)^{-1}$, subtracting that weight if\nselected.  Let $A_x$ be the resulting support and\n$c_x(n)=\\#\\{a\\in A_x:a\\mid n\\}$.  Set\n\\[\n P_0=0,\\qquad P_{N+1}=2P_N+c_x(N+1),\\qquad\n Q_N=\\lfloor2^Nx\\rfloor-P_N.\n\\]\nHere $P_N=\\sum_{j=1}^N2^{N-j}c_x(j)$ truncates the coefficient\nseries~\\eqref{eq:incidence}, not the selected Lambert weights.\nThus $Q_N$ measures coefficient truncation, whereas $r_N$ is the\nreal remainder after $N$ greedy decisions. The bound\n$P_N\\le2^NX_{A_x}(2)\\le2^Nx$ gives $Q_N\\ge0$, and\n\\begin{equation}\n Q_{N+1}=2Q_N+\\beta_N-c_x(N+1),\\qquad\n \\beta_N=\\lfloor2^{N+1}x\\rfloor-2\\lfloor2^Nx\\rfloor\\in\\{0,1\\}.\n \\label{eq:actual-repair-recurrence}\n\\end{equation}\n\nThe recurrence is integral for every real target. Write\n$\\delta=x-X_{A_x}(2)\\ge0$. The coefficient expansion gives\n\\[\n Q_N=\\left\\lfloor 2^N\\delta+\n             \\sum_{r\\ge1}c_x(N+r)2^{-r}\\right\\rfloor.\n\\]\nFor a represented target, $\\delta=0$ and $Q_N=O(\\sqrt N)$, leaving\ntoo little room for unit increases throughout a long window.\nFor an unrepresented target, $2^N\\delta$ eventually dominates the\ndivisor count in the recurrence and forces strict increase.\n\n\n\\begin{theorem}[membership and nonincreasing integer remainders]\\label{res:general-repair}\nFor every real $x\\ge0$, the following are equivalent:\n\\[\n \\begin{gathered}\n x\\in\\Ach;\\\\\n \\forall K\\ge0\\ \\exists N\\ge K:\\quad Q_{N+1}\\le Q_N;\\\\\n \\forall K\\ge0\\ \\exists N\\in[K,K+2\\lfloor\\sqrt K\\rfloor+12):\n \\quad Q_{N+1}\\le Q_N.\n \\end{gathered}\n\\]\n\\end{theorem}\n\n\\begin{proof}\nFor a represented target, the remaining sum is at least $w_n$ exactly\nwhen $n$ is selected, since $R_n<w_n$. Hence the greedy rule recovers\nthe support, and\n\\[\n 0\\le Q_N\\le\\sum_{r\\ge1}c_x(N+r)2^{-r}\n \\le2\\sqrt N+4,\n\\]\nsince $c_x(n)\\le\\tau(n)\\le2\\sqrt n$ and\n$\\sqrt{N+r}\\le\\sqrt N+\\sqrt r\\le\\sqrt N+r$.\nPut $s=\\lfloor\\sqrt K\\rfloor$ and $T=2s+12$.\nStrict increase at all $T$ steps would imply $Q_{K+T}\\ge T$.\nBut $K+T<(s+4)^2$ gives $Q_{K+T}<2s+12=T$.\nThis proves the window condition and hence cofinal nonincreases.\n\nConversely, if $x\\notin\\Ach$, then $\\delta>0$ and\n$Q_N\\ge2^N\\delta-1$, which eventually exceeds\n$c_x(N+1)\\le N+1$.\nEquation~\\eqref{eq:actual-repair-recurrence} then makes $Q_N$ strictly\nincreasing, contradicting cofinal nonincreases.\n\\end{proof}\n\nThe nonincreases must occur along the target's own greedy sequence.\nFor $x=0$, no exponent is selected and $Q_N=0$ at every rank.  A finite subseries sum is likewise represented and satisfies the\ncriterion.  The target $x=3/4$ lies strictly between $R_1<2/3$ and\n$w_1=1$, so it is not represented and its integer remainders eventually\nincrease strictly.  For $1/2$ and $1/21$, the theorem does not establish\nwhich alternative occurs.\n\nAn ordinary extension gives a uniform subpower window: for each\n$0<\\varepsilon<1$, its length is\n$\\lceil C_\\varepsilon(K+1)^\\varepsilon\\rceil$ for a suitable constant\n$C_\\varepsilon$. The linked formal proof establishes the square-root window.\nIndeed, $\\tau(n)\\le A_\\varepsilon n^\\varepsilon$ gives\n$Q_N\\le D_\\varepsilon(N+1)^\\varepsilon$ for represented targets,\nwhere $D_\\varepsilon=A_\\varepsilon\\sum_{r\\ge1}r^\\varepsilon2^{-r}$.\nChoose $C_\\varepsilon>D_\\varepsilon(C_\\varepsilon+3)^\\varepsilon$,\nwhich is possible because $\\varepsilon<1$.\nFor $T=\\lceil C_\\varepsilon(K+1)^\\varepsilon\\rceil$, strict increase\nat all $T$ steps would give\n\\[\n T\\le Q_{K+T}\\le D_\\varepsilon(K+T+1)^\\varepsilon\n \\le D_\\varepsilon(C_\\varepsilon+3)^\\varepsilon(K+1)^\\varepsilon<T,\n\\]\na contradiction. The converse still follows from exponential growth\nwhen $x$ is not represented. The shorter window does not decide\nmembership of a specified target.\n\nAt $x=1/2$, the digits $\\beta_N$ vanish for $N\\ge1$; at $x=1/21$ they\nare six-periodic.  Thus the unresolved arithmetic input is\n\\begin{equation}\n \\forall K\\ \\exists N\\ge K:\\qquad\n c_x(N+1)\\ge Q_N+\\beta_N,\n \\qquad x\\in\\{1/2,1/21\\}.\n \\label{eq:actual-selector-obligation}\n\\end{equation}\nBoth $Q_N$ and $c_x$ must arise from the same greedy support $A_x$.\nCounterexamples to fixed-multiplier schedules, with their finite phase\nmasks, are retained in the companion paper. They concern those schedules;\nthey do not decide either target.\n\n\\section{Integer quotients, approximation and further questions}\\label{sec:open}\nFor $1/21$ we compare integer greedy remainders with the real greedy\nsequence; for $1/2$ we allow arbitrary finite approximating supports.\nBoth tests require suitable data at unbounded depths.\n\n\\paragraph{Integer quotients for $1/21$.}\nFor $M,d\\ge1$, set\n\\[\n q_M(d)=\\left\\lfloor\\frac{2^M}{2^d-1}\\right\\rfloor,\n \\qquad T_M=\\left\\lfloor\\frac{2^M}{21}\\right\\rfloor.\n\\]\nFor $d\\ge2$, the geometric expansion gives\n$q_M(d)=\\sum_{j=1}^{\\lfloor M/d\\rfloor}2^{M-jd}$: the omitted fraction\n$2^{M\\bmod d}/(2^d-1)$ is less than $1$. Thus $q_M(d)$ is the\ncoefficient prefix contributed by exponent $d$.\nStarting with remainder $T_{2R}$, consider $d=2,\\ldots,R$ in order and\nsubtract $q_{2R}(d)$ whenever it does not exceed the current remainder.  Let $D_R$ be the set of selected exponents and $s_R$ the final\nremainder.  Thus\n\\[\n s_R=T_{2R}-\\sum_{d\\in D_R}q_{2R}(d)\\ge0.\n\\]\nThe precision is $4^R$, but $D_R$ records decisions only through\nrank $R$. The agreement condition below also uses a second run at\nthat same precision, continued through rank $2R$.\nFor example, $R=6$ gives $T_{12}=195$, $D_6=\\{5\\}$ and\n$s_6=195-\\lfloor4096/31\\rfloor=63\\le64$.  This is one row satisfying\nthe bound below, not evidence that such rows occur at unbounded depths.\n\n\nWrite $r_n=r_n(1/21)$ for the real greedy remainder and $A_{1/21}$ for its\nsupport.  The condition $\\mathcal F_{21}$ means that there exist integers\n$n,R_0,K_0\\ge0$ with all of the following properties:\n\\begin{enumerate}\n\\item $r_n>R_n$, the set of positive exponents outside $A_{1/21}$ is\nfinite, and every exponent greater than $n$ belongs to $A_{1/21}$;\n\\item for every $R\\ge R_0$, the integer rule with target $T_{2R}$ and\nweights $q_{2R}(2),\\ldots,q_{2R}(2R)$ selects exactly the same exponents\nas the real greedy rule on $\\{2,\\ldots,2R\\}$;\n\\item for every $K\\ge K_0$, the interval $(K,2K]$ contains an exponent\nof $A_{1/21}$.\n\\end{enumerate}\nA positive limiting remainder gives a tail violation and eventual\nselection of every exponent, hence the first and last clauses.\nThe middle clause additionally requires agreement with the integer rules.\n\n\\begin{theorem}[integer-quotient tests for $1/21$]\n\\label{res:one-over-twenty-one-frontier}\nThe following statements hold.\n\\begin{enumerate}\n\\item $1/21\\in\\Ach$ if and only if $\\mathcal F_{21}$ does not hold.\n\\item If there is an unbounded sequence of ranks $R$ with $s_R\\le 2^R$,\n  then $1/21\\in\\Ach$.\n\\item On $\\mathcal F_{21}$, eventually $s_R>2^R$, the boundary rank $R+1$\n  belongs to $D_{R+1}$, and, for all sufficiently large $R$,\n\\[\n \\begin{aligned}\n D_{R+1}&=D_R\\cup\\{R+1\\},\\\\\n s_{R+1}&=4s_R+\n \\left\\lfloor\\frac{4(2^{2R}\\bmod21)}{21}\\right\\rfloor\n -2c_{D_R}(2R+1)-c_{D_R}(2R+2)\\\\\n &\\hspace{3em}{}-(2^{R+1}+1).\n \\end{aligned}\n\\]\nHere $c_{D_R}(m)=\\#\\{d\\in D_R:d\\mid m\\}$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nWrite $x=1/21$, $G=A_{1/21}$ and $\\delta=x-X_G(2)\\ge0$.\nIf $x\\notin\\Ach$, then $\\delta>0$. The real remainder tends to\n$\\delta$, so every sufficiently small weight is selected and eventually\n$r_n>R_n$. For the integer comparisons, first choose $d_0$ with $w_d<\\delta/2$\nfor $d>d_0$. Comparisons through $d_0$ are strict,\nsince equality would give a finite representation of $x$.\nAfter division by $4^R$, each quotient weight differs from $w_d$ by\nless than $4^{-R}$, and the remainders differ by at most $d/4^R$\nwhenever earlier decisions agree. For large $R$, the first $d_0$\ndecisions therefore agree. At later ranks $d\\le2R$, the normalised\nremainder exceeds $\\delta/2>w_d$, since $2R/4^R<\\delta/2$;\nboth rules select every remaining rank through $2R$. This proves $\\mathcal F_{21}$. A represented target has\n$r_n\\le R_n$ at every rank, giving the converse in (1).\n\nFor every $R$, the rounding errors satisfy\n\\begin{equation}\n \\left|x-X_{D_R}(2)-\\frac{s_R}{4^R}\\right|\n \\le\\frac{R+1}{4^R}.\n \\label{eq:twenty-one-rounding}\n\\end{equation}\nThus $s_R\\le2^R$ at unbounded ranks gives finite subseries sums\ntending to $x$. Closedness of $\\Ach$ proves (2), without compatibility\nbetween the sets $D_R$; Section~\\ref{sec:period} excludes a finite\nrepresentation.\n\nOn $\\mathcal F_{21}$, the eventual agreement gives\n$D_R=G\\cap\\{2,\\ldots,R\\}$ and $D_{R+1}=D_R\\cup\\{R+1\\}$.\nThe rounding bound implies $s_R/4^R\\to\\delta>0$.\nFinally, for $d\\in D_R$ (so $d\\ge2$), use\n$q_{M+2}(d)=4q_M(d)+2\\mathbf1_{d\\mid M+1}+\\mathbf1_{d\\mid M+2}$,\ntogether with the analogous target identity and\n$q_{2R+2}(R+1)=2^{R+1}+1$, to obtain (3).\nThe full comparison induction and both quotient identities are given in\n\\papersectionlink{erdos257-mersenne-reasoning-surface.pdf}{record257:twenty-one-quotients}\n{the companion paper's integer-quotient proof for $1/21$}.\n\\end{proof}\n\n\n\\noindent In the list $q_{2R}(2),\\ldots,q_{2R}(R)$, each weight exceeds\nthe sum of its successors by at least $2^R+1$. Distinct subset sums\nare therefore separated by more than $2^R$, and the greedy sum is the\nlargest not exceeding the target. Thus, if\n$D\\subseteq\\{2,\\ldots,R\\}$ and an integer $s$ satisfy\n$\\sum_{d\\in D}q_{2R}(d)+s=T_{2R}$ with $0\\le s\\le2^R$, then\n$D=D_R$ and $s=s_R$\n(\\mword{Erdos249257/TwentyOneQuotientGreedy.lean}{231}{twentyOneClosedRow_forces_quotientGreedy}{uniqueness of a finite representation with the stated remainder bound}).\nThe cited finite crossing lemmas give additional consequences under their\nalignment hypotheses: an earlier finite prefix cannot occur, and a real\ngreedy exponent must be skipped\n(\\mword{Erdos249257/TwentyOneQuotientGreedy.lean}{5179}{twentyOneAlignedSaturatedCrossing_forces_canonical_ancestor_hole}{the missing-prefix consequence of an aligned crossing},\n\\mword{Erdos249257/TwentyOneQuotientGreedy.lean}{5223}{twentyOneAlignedSaturatedCrossing_forces_scaled_greedy_skip}{the real greedy skip forced by that crossing}).\n\n\\paragraph{Approximation to $1/2$ by finite supports.}\nFor $A\\subseteq\\{2,\\ldots,M\\}$ and $1\\le m\\le M$, define the integer\n\\[\n K_A(m)=2^{m-1}-\\sum_{j=2}^{m}2^{m-j}c_A(j).\n\\]\nIt measures the error in the truncated divisor-coefficient sum, after\nmultiplication by $2^m$.  In particular,\n$K_A(1)=1$ and $K_A(m+1)=2K_A(m)-c_A(m+1)$.\n\n\\begin{theorem}[finite approximations with vanishing scaled error]\n\\label{res:terminalhalf}\nSuppose there are integers $M_j\\ge1$ tending to infinity and sets\n$A_j\\subseteq\\{2,\\ldots,M_j\\}$ such that\n\\[\n \\frac{|K_{A_j}(M_j)|}{2^{M_j}}\\longrightarrow0.\n\\]\nThen $X_A(2)=1/2$ for some infinite set $A\\subseteq\\Npos$.\n\\end{theorem}\n\nOnly the terminal coefficient error needs control: the omitted tail\nhas the uniform bound below. Neither agreement between the supports\n$A_j$ nor bounds on earlier values of $K_{A_j}$ are required:\n\\[\n \\left|X_{A_j}(2)-\\frac12\\right|\n \\le \\frac{|K_{A_j}(M_j)|+2\\sqrt{M_j}+4}{2^{M_j}}.\n\\]\nIndeed, $c_{A_j}(n)\\le\\tau(n)\\le2\\sqrt n$ and\n$\\sqrt{M+r}\\le\\sqrt M+r$ bound the normalised tail by\n$\\sum_{r\\ge1}2^{-r}(2\\sqrt M+2r)=2\\sqrt M+4$.\nThus $X_{A_j}(2)\\to1/2$. Closedness of $\\Ach$ gives a representing\nsupport; the odd reduced denominators of finite sums make it infinite.\nAn $O(\\sqrt{M_j})$ terminal error suffices, but any $o(2^{M_j})$\nerror is allowed.  For a finite $D\\subseteq\\{2,\\ldots,M\\}$, the\nfloor-quotient identity gives\n\\[\n K_D(M)=2^{M-1}-\\sum_{d\\in D}\n                \\left\\lfloor\\frac{2^M}{2^d-1}\\right\\rfloor.\n\\]\nThe exact quotient rows considered in the companion paper satisfy\n$\\sum_{d\\in D}q_M(d)=2^{M-1}-1$, so they have $K_D(M)=1$.\nThe terminal criterion allows a larger error of either sign. At a fixed depth these are\ndifferent conditions, although cofinal existence of either kind of\napproximation characterises $1/2\\in\\Ach$.\n\nConversely, if $X_A(2)=1/2$, then $1\\notin A$, and the prefixes\n$A_M=A\\cap\\{2,\\ldots,M\\}$ satisfy\n\\[\n K_{A_M}(M)=\\sum_{r\\ge1}c_A(M+r)2^{-r}\n       \\le 2\\sqrt M+4.\n\\]\nThe right-hand side uses the full support $A$: truncation leaves its\ncoefficients through $M$ unchanged. Thus the existence hypothesis of\nTheorem~\\ref{res:terminalhalf} is equivalent to $1/2\\in\\Ach$.\nExistence at unbounded depths remains unproved; establishing it would\nrefute Problem~\\ref{res:problem}.\n\n\\paragraph{Finite families with a shared prefix.}\nPut $B(m)=2\\lfloor\\sqrt m\\rfloor+4$.  For $0\\le K\\le M$, consider a\nfamily $A_1,\\ldots,A_{B(M)}\\subseteq\\{2,\\ldots,M\\}$ with\n\\[\n 1\\le K_{A_k}(m)\\le B(m)\\quad(1\\le m\\le M),\n \\qquad K_{A_k}(M)=k.\n\\]\nRequire that all the $A_k$ agree on $\\{1,\\ldots,K\\}$ and that some\ninteger $E\\ge B(M)$ satisfies\n\\[\n \\sum_{j=K+1}^{M}1_{A_k}(j)\\,2^{M-j}+k=E\n \\qquad(1\\le k\\le B(M)).\n\\]\nThus the binary suffixes have consecutive values $E-k$. The hypothesis\nasks for a whole family at each available depth, not merely one\napproximant.\n\n\\begin{theorem}[unbounded shared-prefix families represent one half]\n\\label{res:cylinderhalf}\nSuppose that for every $N$ there are $M,K$ with\n$\\max\\{N,1\\}\\le M$, $0\\le K\\le M$, and a family satisfying all the\nconditions in the preceding paragraph.  Then $X_A(2)=1/2$ for some\ninfinite set $A\\subseteq\\Npos$.\n\\end{theorem}\n\nAt each available depth $M$, choose the member with terminal carry\n$1$. Its scaled terminal error is $2^{-M}$, so\nTheorem~\\ref{res:terminalhalf} applies along unbounded depths. The other\nshared-prefix conditions are not used in this compactness step.\nNo construction at unbounded depths is established here.\n\nThe tests above leave three kinds of arithmetic information to be\nsupplied: returns of the actual greedy sequence, finite approximants at\nunbounded depths, or an exclusion of a possible final skip. The following\nquestions specify those missing inputs.\n\n\\begin{problem}[membership of 1/21 in the Mersenne achievement set]\n\\label{prob:one-over-twenty-one-membership}\nFor the greedy remainder of $1/21$, prove that arbitrarily large $N$\nsatisfy\n\\[\n 2^{N+1}r_N(1/21)<\\frac{2^{N+1}}{2^{N+1}-1},\n \\quad\\text{equivalently}\\quad\n r_N(1/21)<\\frac1{2^{N+1}-1}.\n\\]\nEquivalently, exclude the condition $\\mathcal F_{21}$ defined above.\nOne sufficient route is to rule out the eventual recurrence in\nTheorem~\\ref{res:one-over-twenty-one-frontier}(3) together with $s_R>2^R$;\nanother is to prove $s_R\\le2^R$ at arbitrarily large ranks.  Neither\nsufficient route is claimed to be necessary by itself.\n\\end{problem}\n\n\\begin{problem}[bounded returns of the scaled remainder]\\label{prob:scaled-return}\nFor each of the targets $x=1/2$ and $x=1/21$, does the greedy remainder\n$r_N(x)$ return to one bounded interval after scaling by $2^N$?\n\\[\n \\exists B<\\infty\\ \\forall K\\ \\exists N\\ge K:\n \\qquad 2^N r_N(x)\\le B.\n\\]\n\\end{problem}\n\n\\begin{problem}[arithmetic tests for the greedy sequence]\\label{prob:actual-invariant}\nCan a finite-memory, $2$-adic or discrepancy argument prove $s_R\\le2^R$\nat arbitrarily large ranks, or rule out the eventual recurrence in\nTheorem~\\ref{res:one-over-twenty-one-frontier}(3)?  Such an argument must use\nthe divisor counts of the actual greedy support.  Can a bounded window of\n$R\\bmod6$, residues of $s_R$, endpoint divisor counts and the finite set of\neventual skips force a decrease or a contradiction?  Alternatively, can one\nshow that these bounded-memory data cannot distinguish the actual sequence\nfrom sequences that remain above $2^R$ but need not come from a support?\n\\end{problem}\n\n\\begin{problem}[final-skip Diophantine exclusion]\\label{prob:fatal-interval}\nLet $E=\\sum_{n\\ge1}(2^n-1)^{-1}$.  If $1/21\\notin\\Ach$, let $M$ be the last\nskipped exponent, $S_M$ its finite skipped prefix, and\n\\[\n a_M=\\frac1{21}+\\sum_{d\\in S_M}\\frac1{2^d-1}.\n\\]\nWrite $\\operatorname{gap}_M=(2^M-1)^{-1}-R_M>0$.\nCan the arithmetic restrictions on this last skipped exponent be used to\nprove $|E-a_M|\\ge\\operatorname{gap}_M$, contradicting\n\\[\n 0<a_M-E<\\operatorname{gap}_M?\n\\]\n\\end{problem}\n\n\n\\section{Elementary details used in the proofs}\\label{app:elementary-details}\n\\paragraph{Cyclotomic prime-power fact.}\nLet $b\\ge2$, $n\\ge2$, $\\ell\\mid\\Phi_n(b)$ be prime and\n$e=v_\\ell(b^n-1)$.  Then $\\operatorname{ord}_{\\ell^e}(b)=n$.\nWe include the $2$-adic case.\nFor odd $\\ell$, put $d=\\operatorname{ord}_{\\ell}(b)$ and\n$s=v_\\ell(b^d-1)$.  The elementary lifting identity\n$v_\\ell(b^{dt}-1)=s+v_\\ell(t)$ follows by factoring a geometric sum\nwhen $\\ell\\nmid t$ and by a binomial expansion for a factor $\\ell$.\nIn the factorisation $b^n-1=\\prod_{r\\mid n}\\Phi_r(b)$, subtracting\nthese valuations over proper divisors gives\n\\[\n v_\\ell(\\Phi_n(b))=\n \\begin{cases}s&n=d,\\\\1&n=d\\ell^j,\\ j\\ge1,\\\\0&\\text{otherwise.}\\end{cases}\n\\]\nIf $n=d$, no smaller positive exponent gives $b^m\\equiv1\\pmod\\ell$.\nIf $n=d\\ell^j$, the full valuation is $e=s+j$; the same lifting\nidentity shows that the least exponent giving valuation $e$ is\n$d\\ell^j=n$.\nFor $\\ell=2$, $b$ is odd.  Factoring $b^{2t}-1$ gives\n\\[\n v_2(b^{2^j}-1)=v_2(b-1)+v_2(b+1)+j-1\\qquad(j\\ge1).\n\\]\nThe only possible indices $n\\ge2$ are $n=2^j$:\n$v_2(\\Phi_2(b))=v_2(b+1)$ and\n$v_2(\\Phi_{2^j}(b))=1$ for $j\\ge2$; all other indices have\nvaluation zero.  The displayed identity again makes $2^j$ the least\nexponent with the full valuation.  This proves the fact.\nFinally $\\Phi_n(b)>1$: each primitive $n$th root $\\zeta$ satisfies\n$|b-\\zeta|>1$, and their product is $\\Phi_n(b)$.\n\n\\paragraph{Exact scalar cover cost.}\nFor $t\\ge1$, put $z=\\log_2t$.  The substitution $y=2^\\alpha$\nreduces the infimum defining $\\Psi(t)$ to $y^z/(y-1)$ on $1<y\\le2$.\nIts derivative has the sign of $(z-1)y-z$.  Hence\n\\[\n \\Psi(t)=\n \\begin{cases}t,&1\\le t\\le4,\\\\\n z^z/(z-1)^{z-1},&t>4.\n \\end{cases}\n\\]\nThe minimum is at $\\alpha=1$ for $t\\le4$ and in the interior for $t>4$;\nthe quotient diverges as $\\alpha\\downarrow0$. This scalar identity does\nnot supply a converse to the positive-cover criterion.\n\n\\subsection*{The cost for a finite set of divisors}\nFor the following finite divisor sets, the lower bound is asymptotically\nattained. Here $K_*(F)$ is the infimum of the cost $K$ from\nSection~\\ref{sec:comparison} over finite or countable covers of $F$, their majorants and exponents,\nand positive weights of total one. A one-set cover therefore has weight\n$1$, without the dyadic index factor in (V).\nFor $F(q,P)=\\{qd:d\\mid\\prod_{p\\in P}p\\}$, where $q\\ge2$ and no $p\\in P$\ndivides $q$, put $S=\\sum_{p\\in P}1/p$.  If $S\\ge1$, then\n\\begin{equation}\n \\frac{e(S-1)}q\\le K_*\\bigl(F(q,P)\\bigr)\\le\\frac{eS}q.\n \\label{eq:optimal-cube-cost}\n\\end{equation}\nFor the lower bound, we condition on $q\\mid n$ and write\n$f_F(n)=2^Z$.  The Chinese remainder theorem gives\n$\\mathbb EZ=S$.  For $z\\ge0$ and $v=\\alpha\\log2>0$,\n\\[\n \\frac{2^{\\alpha z}}{2^\\alpha-1}\n \\ge\\frac{e^{(z-1)v}}v\\ge e(z-1)\\quad(z>1);\n\\]\nfor $0\\le z\\le1$ the claimed lower bound is nonpositive.\nThus $\\Psi(2^z)\\ge e(z-1)$; averaging and multiplying by the\ndensity $1/q$ gives the lower bound. For the upper bound, use the\none-set cover with $z=1/S$ and\n$\\alpha=\\log_2(1+z)$; its exact positive expansion has cost\n\\[\n \\frac1{qz}\\prod_{p\\in P}(1+z/p)\\le\\frac{eS}q.\n\\]\nIn particular,\n$1-1/S\\le qK_*(F(q,P))/(eS)\\le1$.\nThus $K_*(F(q,P))\\sim eS/q$ as $S\\to\\infty$, uniformly over the\npermitted choices of $q$ and $P$.\n\n\\section{Sources, verification and reproducibility}\\label{app:sources}\n\n\\evidenceparagraph{\\evidencerecordurl}\n\nFor Erd\\H{o}s's reciprocal-summable extension and fractional-part\nargument, see pp.~222 and 226 of \\cite{erdos1968}. The periodic theorem\n\\cite{lucatachiya2014periodic} is also stated in Luca and Tachiya's account~\\cite[Theorem~A and Example~2, pp.~139--140]{lucatachiya2017},\nand Hornich's strict-tail theorem is proved in Nitecki's exposition\n\\cite[Theorem~4(1)]{nitecki2013}. These expositions are the sources used\nhere for those two results; the original articles were not independently\nretrieved.  The \\emph{Formal Conjectures}\nfile~\\cite{formalconjectures257} is statement-level prior art, not a\nproof dependency.\n\nFormal bindings concern statements, not every printed argument. The mixed\nimplication has a formal proof; a separate formal construction of a\nseparating support $V$ uses finite sets of squarefree divisors. The $A_\\star$ calculation, printed fresh-prime\nconstruction, Corollary~\\ref{res:strict-mixed-supports}, signed\nfinite-denominator extension and Theorem~\\ref{res:one-over-twenty-one-frontier}\nhave ordinary arguments here, without independent human review. The\nstrictness corollary has no formal binding in the record. The companion paper locates\nthe separate comparisons for its three logarithmic-sampling results.\nNo historical priority is asserted for the support comparison.\n\n\\paragraph{Reproducing the weighted theorem.}\nThe two declarations for Theorem~\\ref{res:weighted-support} are in\nthe public source at\n\\href{https://github.com/wcook04/plectis-erdos/tree/91ca3405b795a520825ac5ca04dcd591b9ddf3e3}{commit \\texttt{91ca3405}}.\nThat snapshot pins Lean~4.29.1 in \\texttt{lean-toolchain} and its Mathlib\nrevision in \\texttt{lake-manifest.json}.  From a complete checkout, with\n\\texttt{elan} installed, build the two modules with:\n\\begin{verbatim}\nlake exe cache get\npython3 scripts/lean_fast_build.py --jobs 2 \\\n  ErdosProblems.Erdos257.PaperCompleteR8.WeightedReturn \\\n  ErdosProblems.Erdos257.PaperCompleteR8.WeightedHereditaryClaim\n\\end{verbatim}\nThe cache download is optional.  The modules contain\n\\texttt{divisibilityWeightedClaim} and\n\\texttt{finitePrimeWeighted\\_fixedBase\\_hereditary}, respectively; their precise statements and associated checks are linked above.\n\nThe supplementary declarations are collected in\n\\papersectionlink{erdos257-mersenne-reasoning-surface.pdf}{record257:supplementary-sources}\n{the companion paper's final source section}, grouped by support criteria,\nfinite denominators, integer recurrences and achievement sets.\n\n\\begingroup\\small\n\\begin{thebibliography}{99}\n\\bibitem{duverneytachiya} D.~Duverney and Y.~Tachiya,\n  \\href{https://danielduverney.fr/documents/theorie-des-nombres/DuverneyTachiya190522.pdf}\n  {\\emph{Refinement of the Chowla--Erd\\H{o}s method and linear independence\n  of certain Lambert series}},\n  Forum Math.\\ 31 (2019), no.~6, 1557--1566,\n  \\href{https://doi.org/10.1515/forum-2018-0299}{DOI}.  Page numbers refer to\n  the linked author preprint.\n\\bibitem{kanekosuzukitachiya} H.~Kaneko, Y.~Suzuki, and Y.~Tachiya,\n  \\href{https://arxiv.org/abs/2601.20743v1}{\\emph{Refinements of Erd\\H{o}s's\n  irrationality criterion for certain sparse infinite series}},\n  arXiv:2601.20743v1 (2026).\n\\bibitem{taoteravainen2025} T.~Tao and J.~Ter\\\"av\\\"ainen,\n  \\href{https://arxiv.org/abs/2512.01739v2}{\\emph{Quantitative correlations and\n  some problems on prime factors of consecutive integers}}, arXiv:2512.01739v2\n  (submitted December 2025, revised April 2026).\n\\bibitem{kovactao} V.~Kova\\v{c} and T.~Tao, \\emph{On several irrationality\n  problems for Ahmes series}, Acta Math.\\ Hungar.\\ 175 (2025), no.~2, 572--608,\n  \\href{https://doi.org/10.1007/s10474-025-01528-0}{DOI}.  Page numbers refer\n  to arXiv:2406.17593v4.\n\\bibitem{erdos1948} P.~Erd\\H{o}s, \\emph{On arithmetical properties of Lambert\n  series}, J.~Indian Math.\\ Soc.\\ 12 (1948), 63--66.\n\\bibitem{formalconjectures257} The Formal Conjectures Authors,\n  \\href{https://github.com/google-deepmind/formal-conjectures/blob/f776d2f2039351b00737ffcafb9d7d7666e1d9af/FormalConjectures/ErdosProblems/257.lean}\n  {\\emph{FormalConjectures.ErdosProblems.\\texttt{257}}},\n  Lean source at commit \\texttt{f776d2f}, 2025, accessed 13 September 2026.\n\n\\bibitem{erdos1968} P.~Erd\\H{o}s,\n  \\href{https://users.renyi.hu/~p_erdos/1969-09.pdf}{\\emph{On the\n  irrationality of certain series}}, Math.\\ Student 36 (1968), 222--226\n  (issued 1969); \\href{https://www.renyi.hu/~p_erdos/1969-09.pdf}{five-page scan}.\n\\bibitem{erdos1975} P.~Erd\\H{o}s, \\emph{Some problems and results on the\n  irrationality of the sum of infinite series}, J.~Math.\\ Sci.\\ 10 (1975),\n  1--7.\n\\bibitem{bkkkz2026} K.~Barreto, J.~Kang, S.-H.~Kim, V.~Kova\\v{c}, and\n  S.~Zhang, \\href{https://arxiv.org/abs/2601.21442v3}{\\emph{Irrationality of\n  rapidly converging series: a problem of Erd\\H{o}s and Graham}},\n  arXiv:2601.21442v3 (2026), to appear in Bull.\\ London Math.\\ Soc.\n\\bibitem{nitecki2013} Z.~Nitecki,\n  \\href{https://arxiv.org/abs/1106.3779v2}{\\emph{Subsum sets: intervals, Cantor\n  sets, and Cantorvals}}, arXiv:1106.3779v2 (2013).\n\\bibitem{lucatachiya2017} F.~Luca and Y.~Tachiya,\n  \\href{https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2014-14.pdf}\n  {\\emph{Linear independence results for the values of divisor functions\n  series}}, RIMS K\\^oky\\^uroku No.~2014 (2017), 138--150.\n  Theorem~A restates their periodic-sequence theorem\n  (\\href{https://doi.org/10.1142/S1793042113501121}{DOI}).\n\\bibitem{lucatachiya2014periodic}\nF.~Luca and Y.~Tachiya,\n\\emph{Irrationality of Lambert series associated with a periodic sequence},\nInternational Journal of Number Theory \\textbf{10} (2014), no.~3, 623--636.\n\\href{https://doi.org/10.1142/S1793042113501121}{doi:10.1142/S1793042113501121}.\n\n\\bibitem{lucatachiya2014independence}\nF.~Luca and Y.~Tachiya,\n\\emph{Linear independence of certain Lambert series},\nProceedings of the American Mathematical Society \\textbf{142} (2014), no.~10, 3411--3419.\n\\href{https://doi.org/10.1090/S0002-9939-2014-12102-2}{doi:10.1090/S0002-9939-2014-12102-2}.\n\n\\bibitem{vanassche2001}\nW.~Van Assche,\n\\emph{Little $q$-Legendre polynomials and irrationality of certain Lambert series},\nThe Ramanujan Journal \\textbf{5} (2001), 295--310.\n\\href{https://doi.org/10.1023/A:1012930828917}{doi:10.1023/A:1012930828917};\n\\href{https://arxiv.org/abs/math/0101187v1}{arXiv:math/0101187v1}.\n\n\\bibitem{hornich1941}\nH.~Hornich,\n\\emph{\\\"Uber beliebige Teilsummen absolut konvergenter Reihen},\nMonatshefte f\\\"ur Mathematik und Physik \\textbf{49} (1941), 316--320.\n\\href{https://doi.org/10.1007/BF01707309}{doi:10.1007/BF01707309}.\n\n\n\\bibitem{vandoornkovac} W.~van Doorn and V.~Kova\\v{c},\n  \\emph{Lacunary sequences whose reciprocal sums represent all rational\n  numbers in an interval}, Acta Arith.\\ 223 (2026), 275--295.\n  \\href{https://doi.org/10.4064/aa251001-13-1}{DOI}.\n  Page references use \\url{https://arxiv.org/abs/2509.24971v3}.\n\n\\end{thebibliography}\n\\endgroup\n\n\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":79499,"sha256":"sha256:2676b3e9f6a5ba6f3ea8fceeed9684ff6ced0fa3bef9c3c0b6a3a878288af5d3","cite_as":"Plectis reader corpus, paper:erdos-257-mersenne-support-subseries","source_repository":"plectis-erdos","source_path":"paper/257/erdos-257-mersenne-support-subseries.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:088d36d7ba3d3ee0cee2df99e8915927999698f19889f8ea0b8d55b3721ec778","byte_count":6923,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% This is the only copy of the shared preamble.  The eight short notes, the\n% long records assembled from paper/reasoning-parts other than #249 and #257\n% (which carry their own preambles), and the synthesis paper include it with\n% \\input{problem-note-preamble}; paper/Makefile builds from this directory\n% with the directory on the tectonic search path.\n%\n% Shared preamble for the Erdős Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erdős problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{8e58c395e9807ef3374d7365c62cf04d20ee6331}\n\\newcommand{\\commitshort}{9e231ce4371f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-erdos/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-erdos/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\usepackage{longtable}  % the notes' declaration tables; loaded here so the assembled records inherit it\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n% ---------------------------------------------------------------------------\n% Evidence marks.  paper-evidence.tex places \"Lean\" and \"Comparator\" links in\n% the margin beside each result from the paper's generated file\n% paper/evidence/<paper>.tex, which each paper inputs after this preamble.\n% ---------------------------------------------------------------------------\n\\input{paper-evidence}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Production provenance.  The wording is fixed by the corpus provenance\n% contract and reproduced verbatim in every manuscript in this repository, so\n% the papers cannot drift into different accounts of how they were made.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erdős problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every Lean mark in the margin points to propositions that the\n  pinned Lean kernel accepts, with no \\texttt{sorry}, no added axiom, and no\n  unchecked evaluation.  That claim concerns the formal statements only; it\n  says nothing about their mathematical interest, their novelty, or the\n  original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n\n% Compact opening card for the separately stated Comparator theorem selected\n% from each problem note.  The prose supplied by each note owns the\n% contribution and relevance judgements; the shared final sentence states the\n% mechanical assurance boundary once and points to the public evidence owner.\n\\newcommand{\\externalreviewobject}[3]{%\n  \\begin{quote}\\small\n  \\textbf{Contribution.} #1\\par\n  \\textbf{Relation to the open problem.} #2\\par\n  \\textbf{Executable review object.} #3  The repository's\n  external-verification job compares these exact Lean propositions with\n  separately declared challenge statements and an axiom budget, then asks\n  Lean's kernel to check the submitted proofs.  The\n  \\href{https://github.com/wcook04/plectis-erdos/blob/ca0e13f8acf5ccf48506e4bdb870953d3a0856fa/formalization.yaml}%\n  {formalisation manifest} and the commit-bound CI receipt record that check;\n  they do not assess novelty, significance, or whether the original problem is\n  solved.\n  \\end{quote}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  Under\n% Tectonic (XeTeX) the text and mathematics are set in TeX Gyre Pagella from\n% the bundled OpenType fonts, with a Heros sans range for \\mathsf and\n% Inconsolata for monospace; the pdflatex fallback keeps the same Palatino\n% design through newpx.  The A4 text block uses a classical measure with\n% quiet running heads and restrained ink.  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It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"},{"record_type":"paper_source_dependency","tex_reference":"evidence/erdos-257-mersenne-support-subseries","source_path":"paper/evidence/erdos-257-mersenne-support-subseries.tex","media_type":"text/x-tex","sha256":"sha256:722ce3fc0640d2a265e5dd747e1c1d9a5d87a65aa5975d51a664acfed4135fef","byte_count":7186,"licence":"CC-BY-4.0","body_format":"latex","body":"% Generated by scripts/paper_evidence.py from docs/paper_lean_coverage.json and the\n% Comparator replay receipts under evidence/comparator/.  Do not edit by hand.\n\\newcommand{\\evidencerecordurl}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md}\n\\DeclareResultEvidenceName{res:weighted-support}{Theorem 1.2}\n\\DeclareResultEvidenceRecord{res:weighted-support}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-weighted-support}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:weighted-support}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-weighted-support}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-weighted-support-comparator}\n\\DeclareResultEvidenceName{thm:variable-fractional-cover}{Theorem 3.1}\n\\DeclareResultEvidenceRecord{thm:variable-fractional-cover}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#thm-variable-fractional-cover}{exact}{compared}\n\\DeclareResultEvidence{thm:variable-fractional-cover}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR8/PositiveCoverReturn.lean\\#L241}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#thm-variable-fractional-cover-comparator}\n\\DeclareResultEvidenceName{res:mixed-supports}{Theorem 4.1}\n\\DeclareResultEvidenceRecord{res:mixed-supports}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-mixed-supports}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:mixed-supports}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-mixed-supports}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-mixed-supports-comparator}\n\\DeclareResultEvidenceName{res:weighted-cover-incomparability}{Proposition 5.1}\n\\DeclareResultEvidenceRecord{res:weighted-cover-incomparability}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-weighted-cover-incomparability}{exact_or_stronger}{pending}\n\\DeclareResultEvidence{res:weighted-cover-incomparability}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-weighted-cover-incomparability}{}\n\\DeclareResultEvidenceName{res:strict-mixed-supports}{Corollary 5.2}\n\\DeclareResultEvidenceRecord{res:strict-mixed-supports}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-strict-mixed-supports}{none}{not_applicable}\n\\DeclareResultEvidenceName{res:reciprocal-support}{Theorem A.1}\n\\DeclareResultEvidenceRecord{res:reciprocal-support}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-reciprocal-support}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:reciprocal-support}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/AllBaseReciprocalSupportIrrationality.lean\\#L395}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-reciprocal-support-comparator}\n\\DeclareResultEvidenceName{res:period}{Theorem B.1}\n\\DeclareResultEvidenceRecord{res:period}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-period}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:period}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-period}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-period-comparator}\n\\DeclareResultEvidenceName{res:general-repair}{Theorem C.1}\n\\DeclareResultEvidenceRecord{res:general-repair}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-general-repair}{exact}{compared}\n\\DeclareResultEvidence{res:general-repair}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR20/GeneralRepairCorrespondence.lean\\#L15}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-general-repair-comparator}\n\\DeclareResultEvidenceName{res:one-over-twenty-one-frontier}{Theorem D.1}\n\\DeclareResultEvidenceRecord{res:one-over-twenty-one-frontier}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-one-over-twenty-one-frontier}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:one-over-twenty-one-frontier}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-one-over-twenty-one-frontier}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-one-over-twenty-one-frontier-comparator}\n\\DeclareResultEvidenceName{res:terminalhalf}{Theorem D.2}\n\\DeclareResultEvidenceRecord{res:terminalhalf}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-terminalhalf}{exact_or_stronger}{pending}\n\\DeclareResultEvidence{res:terminalhalf}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR20/TerminalSetCorrespondence.lean\\#L185}{}\n\\DeclareResultEvidenceName{res:cylinderhalf}{Theorem D.3}\n\\DeclareResultEvidenceRecord{res:cylinderhalf}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-cylinderhalf}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:cylinderhalf}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean\\#L287}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-257-mersenne-support-subseries.md\\#res-cylinderhalf-comparator}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/257/erdos-257-mersenne-support-subseries.tex","section_outline":["Introduction","Finite averages and the weighted criterion","An example beyond reciprocal summability","Positive divisor majorants","A finite divisor set and the cover condition","A finite mean for shifted divisor tails","Proof of the cover criterion","A common index for the two criteria","Comparison of the support classes","A lower bound independent of the cover","Two separating constructions","A strict enlargement and finite unions","Limits of the small-displacement criterion","Algebraic bases on separated supports","Reciprocal-summable supports","Finite-support denominator periods","Rational membership","Rational values and scaled tails","The achievement set","The greedy recurrence","Integer quotients, approximation and further questions","Elementary details used in the proofs","The cost for a finite set of divisors","Sources, verification and reproducibility"]},"erdos257-mersenne-reasoning-surface":{"paper_id":"erdos257-mersenne-reasoning-surface","title":"Reciprocal Mersenne Subseries","form":"Reasoning surface","length_class":"long","pages":152,"subject":"Erdős #257","subject_kind":"problem","problem_id":"erdos_257","landing_role":"mathematics","question":"What weighted and cover criteria prove hereditary Mersenne irrationality for Erdős #257, and which targets remain open?","precis":"A weighted summability criterion proves irrationality of reciprocal Mersenne subseries: if h is the part of a supported on a finite nonempty prime set P, every infinite A with ∑ h(a)/(a(b^{h(a)}−1)) finite has irrational ∑ (b^a−1)^{-1}. The base-two condition gives the conclusion at every integer base for every infinite subset, and permits an example with divergent reciprocal sum, while excluding full support and all odd exponents. The proof averages over a finite part of the support and over dyadic observation lengths. Erdős's weaker reciprocal-summable theorem is proved in full, and a common window extends both arguments to unions of weighted and positive-cover supports. Finite-denominator and achievement-set calculations formulate tests for the targets 1/2 and 1/21; those tests do not decide either target or the universal question. Prime support at base 2 and squarefree support at power-of-two bases are attributed as existing work.","owns":"a claim-bounded problem-specific reasoning surface for Erdős #257: selected checked premises, finite computations, closed routes, surviving obligations, and their exact evidence bands","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #257, whose universal and half-value questions remain open","first_pass":["sec:257-problem","sec:257-wall","sec:257-survivors","sec:257-howto"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos257-mersenne-reasoning-surface.pdf","hosted_pdf_path":"papers/erdos257-mersenne-reasoning-surface.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/257/erdos257-mersenne-reasoning-surface.tex","site_html":"maths/papers/erdos257-mersenne-reasoning-surface.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/257/erdos257-mersenne-reasoning-surface.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/257/erdos257-mersenne-reasoning-surface.tex","body_pointer":"papers.records.erdos257-mersenne-reasoning-surface.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Originally assembled from paper/reasoning-parts/erdos257. This cumulative\n% editorial return revises the assembled source directly. The complete text\n% is inlined; it is not a new output of the original assembly script.\n\\documentclass[11pt]{article}\n% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared preamble for the two per-problem reasoning-surface papers.\n% These documents are written to be read in full as working context: every\n% result appears with its exact hypotheses, its evidence band, the scale at\n% which it is proved, and the Lean declaration that carries it.\n\\usepackage{paper-house-style}\n\\usepackage{array,seqsplit}\n\\usepackage{adjustbox}\n\\usepackage{booktabs}\n\\usepackage{longtable}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage{xurl}\n\\usepackage{xspace}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdftitle={Reciprocal Mersenne Subseries: proofs, attribution and conditional membership tests for Erdos Problem 257},\n  pdfauthor={Will Cook},pdflang={en-GB}}\n% Lean declaration names are full of underscores and are typeset in text mode.\n% Category codes are fixed when TeX READS a line, not when a macro expands, so\n% no amount of \\catcode juggling inside \\lean can rescue an underscore that has\n% already been tokenised as a subscript. This package makes _ print as itself in\n% text while leaving it as a subscript in maths, which is exactly the split we\n% need. It also protects any stray underscore in authored prose.\n\\usepackage{underscore}\n\n% ===========================================================================\n% Theorem environments.\n% ===========================================================================\n\\theoremstyle{plain}\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{lem}[thm]{Lemma}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{cor}[thm]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{defn}[thm]{Definition}\n\\theoremstyle{remark}\n\\newtheorem{rem}[thm]{Remark}\n\\newtheorem{obs}[thm]{Observation}\n\n% ===========================================================================\n% Citation of a Lean declaration. The declaration name is the authority; the\n% file:line is where it sat at the pinned formal-source checkpoint. 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Defining it is\n% cheaper and safer than rewriting the prose around every occurrence.\n\\providecommand{\\m}[1]{\\ensuremath{#1}}\n\n% ===========================================================================\n% Evidence band. These are never blurred: a reader must always be able to see\n% what a kernel checked, what an exact computation checked, what was proved in\n% ordinary mathematics, what is quoted from the literature, and what is open.\n% ===========================================================================\n% [Lean] records a supplied source/receipt, not a fresh run in this revision.\n\\newcommand{\\ev}[1]{{\\small\\textsf{[#1]}}\\xspace}\n\n% Scale at which a statement is proved. This is the load-bearing axis of both\n% documents: the corpus proves at fixed or bounded scale, and both open\n% producers need cofinal scale.\n\\newcommand{\\scale}[1]{{\\small\\textsf{(#1)}}\\xspace}\n\n% The representation a statement is expressed in. 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This sibling\n% has no \\coord in a heading today; the guard keeps it that way by design.\n\\newcommand{\\coord}[1]{\\texorpdfstring{{\\small\\textit{#1}}}{#1}\\xspace}\n\\newcommand{\\codeid}[1]{\\nolinkurl{#1}}\n\n% ===========================================================================\n% Mathematical shorthand.\n% ===========================================================================\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\C}{\\mathbb{C}}\n\\newcommand{\\ph}{\\varphi}\n\\newcommand{\\EB}{E}\n\n\\allowdisplaybreaks\n\\emergencystretch=2em\n\n% These working records contain a small number of intentionally exhaustive\n% formulas. Keep ordinary displays at their natural size, but bound any display\n% wider than the classical text measure rather than letting it disappear past\n% the page edge.\n\\long\\def\\[#1\\]{%\n  \\leavevmode\\par\\vspace{\\abovedisplayskip}%\n  \\noindent\\makebox[\\linewidth][c]{%\n    \\begin{adjustbox}{max width=\\linewidth}%\n      \\(\\displaystyle #1\\)%\n    \\end{adjustbox}}%\n  \\par\\vspace{\\belowdisplayskip}%\n}\n\n\\runninghead{Erd\\H{o}s \\#257: reciprocal Mersenne subseries}\n\\title{Reciprocal Mersenne Subseries}\n\\subtitle{Support criteria, finite arithmetic and conditional orbit problems}\n\\author{Will Cook}\n\\date{30 September 2026}\n\n\\begin{document}\n% ---- part a257_front ----\n\\maketitle\n\n\\paperprovenance\n\n\\begin{abstract}\nPart~I proves that $\\sum_{a\\in A}(b^a-1)^{-1}$ is irrational for every\ninteger $b\\ge2$ and every infinite $A\\subseteq E\\cup V$ under two\nhypotheses. The set $E$ satisfies a finite-prime weighted summability\ncondition at base two, while $V$ admits a summable positive divisor cover.\nThe estimates for the two sets hold on one finite distribution, so the\nunion criterion reaches supports lying in neither individual class.\nThe record also develops finite-denominator arithmetic, achievement-set\ngeometry and integer-quotient tests for $1/2$ and $1/21$. Conditional\narguments retain their hypotheses, computations their finite ranges, and\ncounterexamples the estimates they refute. Both rational targets and the\nunrestricted base-two problem remain open.\n\\end{abstract}\n\n\\section*{Introduction}\nFor an integer $b\\ge2$, the series $X_A(b)=\\sum_{a\\in A}(b^a-1)^{-1}$\nis irrational for several classes of infinite supports $A$. Erd\\H{o}s proved the full-support case\nand later stated a criterion assuming only convergence of\n$\\sum_{a\\in A}1/a$; the periodic-support results belong to Luca and\nTachiya. We begin with these antecedents and prove two hereditary\ncriteria which also admit supports of divergent reciprocal sum.\nThe proofs subtract an explicitly chosen integer from $(b^N-1)X_A(b)$,\nleaving a positive difference $\\Delta_{b,A}(N)$. If $X_A(b)=p/q$ with\n$q>0$, then $q\\Delta_{b,A}(N)\\in\\mathbb Z$, so\n$\\Delta_{b,A}(N)\\ge1/q$. We rule out rationality by making this\ndifference arbitrarily small. In a finite average, the incomplete residue periods produce an error\nwithout the factor $1/a$ needed for summation over exponents. Averaging\nagain over dyadic lengths restores this factor. The decisive point for the\nunion theorem is stronger than proving the two criteria separately: both\nerrors are controlled on the same finite distribution, and one index is\nchosen only after they have been added. Keeping the modulus dependence\nexplicit makes that common distribution available to the cover estimate.\n\nPart~I gives the support theorems and the proofs followed by the short\npaper, \\emph{Irrationality criteria for Lambert subseries}.\nThe subsequent parts concern a different question: whether a prescribed\nrational number belongs to the Mersenne achievement set. The finite\nidentities in Sections~5--8 describe the integer quotients and their\nrelation to the real greedy sequence. Sections~9--12 state the additional\nuniform or unbounded-depth assertions needed to apply them. Notation\nspecific to a recurrence is introduced with that recurrence.\n\nSection~13 returns to the support criteria. A logarithmic divisor cost\ncannot uniformly replace the fractional-cover cost when sampling on\nmultiples of a prescribed modulus, even over complete periods. We prove\nthe counterexample and an initial-interval estimate that survives it.\n\nNone of these arguments decides the universal problem, half-membership\nor $1/21$-membership. The conditional results specify sufficient\nhypotheses or equivalent membership conditions, and the computations\ncover only the stated finite ranges. Each counterexample concerns the\nparticular estimate under discussion.\n\n\\subsection*{Notation used throughout}\nFor $A\\subseteq\\mathbb N_{>0}$ and an integer $b\\ge2$, put\n\\[\n X_A(b)=\\sum_{a\\in A}\\frac1{b^a-1},\\qquad\n c_A(n)=\\#\\{a\\in A:a\\mid n\\}\\quad(n\\ge1).\n\\]\nAt base two write $w_n=(2^n-1)^{-1}$, $R_n=\\sum_{j>n}w_j$ and\n$\\mathcal A=\\{X_A(2):A\\subseteq\\mathbb N_{>0}\\}$.  For $x\\ge0$ the\nreal greedy remainder is defined by $r_0(x)=x$ and\n\\[\n r_n(x)=r_{n-1}(x)-\\begin{cases}\n w_n,&w_n\\le r_{n-1}(x),\\\\\n 0,&w_n>r_{n-1}(x).\n \\end{cases}\n\\]\nThe greedy support $G_x$ consists of the selected indices; $G=G_{1/2}$.\nFor the half target index $1$ is skipped.  Thus a recurrence begun at index\n$2$ with remainder $1/2$ is the same process, with its index shifted.\n\nThe finite quotient arguments use\n\\[\n q(M,d)=\\left\\lfloor\\frac{2^M}{2^d-1}\\right\\rfloor,\\qquad\n Q(D,M)=\\sum_{d\\in D}q(M,d).\n\\]\nFor $M\\ge k$ their nonnegative binary remainder and signed next-step\nexpression are\n\\[\n \\begin{aligned}\n S(D,k,M)&=\\max\\{2^{M-k}-Q(D,M)-1,0\\},\\\\\n H(D,k,n)&=2S(D,k,n-1)+1-c_D(n)\\quad(n>k).\n \\end{aligned}\n\\]\nThe maximum matters: subtraction in the corresponding natural-number\nsource definition is truncated at zero.  Statements that remove the\nmaximum establish nonnegativity first.\n\nThe integer carry for a support omitting $1$ is\n\\[\n \\operatorname{ihc}(A,N)=2^N-\n \\sum_{j=2}^{N+1}2^{N+1-j}c_A(j),\\qquad\n C_A(N)=\\operatorname{ihc}(A,N)-1.\n\\]\nSome finite-depth statements use $K_A(m)=\\operatorname{ihc}(A,m-1)$.\nThe recurring bound $B(m)=2\\lfloor\\sqrt m\\rfloor+4$ is integral; bounds\nwritten with $\\sqrt m$ rather than its floor are real-valued bounds.\nQuantities specific to one argument are defined in that argument.\nSome formal sources allow a support to be a subset of $\\N$, including zero.\nIn all reciprocal sums in this paper only positive exponents contribute:\nfor such a support, $X_A(b)$ means $X_{A\\cap\\Npos}(b)$.  The expression\n$1/(b^0-1)$ is never evaluated.\n\n\\clearpage\n\\tableofcontents\n\\clearpage\n% ; - part a257_front ; -\n% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Front matter for the Erdos #257 reasoning-surface paper: the problem, the\n% barrier analysis, the surviving routes, and the reading contract. Everything\n% before the premise catalogue.\n\n\\clearpage\n\\part*{I. Support theorems and their proofs}\n\\addcontentsline{toc}{part}{I. Support theorems and their proofs}\n\\section{Support criteria and their proofs}\n\\label{sec:257-problem}\nReciprocal summability lets us average with a fixed modulus and then\nenlarge that modulus. We begin with this proof of the extension stated\nby Erd\\H{o}s. For the weighted and cover criteria the estimates must\ninstead remain controlled on a finite window as the modulus grows.\nThat common-window estimate is what permits the two conditions to be\nused together.\n\n\n\\subsection{Reciprocal-summable supports}\n\\papersectiontarget{record257:reciprocal-proof}\n\nFor every infinite set $A$ of positive integers with\n$\\sum_{a\\in A}1/a<\\infty$, the series\n\\[\n X_A(b)=\\sum_{a\\in A}\\frac1{b^a-1}\n\\]\nis irrational at every integer base $b\\ge2$. This is the\nreciprocal-summable theorem also stated in the short paper's\nAppendix~A. Erd\\H{o}s's\n\\href{https://users.renyi.hu/~p_erdos/1969-09.pdf}{1968 paper, p.~222}\nproves the pairwise-coprime case at every integer base and states the removal\nof coprimality without printing the details. The proof below gives the stated extension by averaging; no comparison\nwith the omitted proof is possible.  The pairwise-coprime case,\nTheorem~\\ref{thm:pairwise-coprime}, retains that extra hypothesis. The same\npaper says that $\\sum1/n_i<\\infty$ could be replaced by a weaker but more\ncomplicated condition (p.~222), and suggests that for pairwise coprime\nsupports Brun's method could probably replace it by\n$\\sum_{n_i<x}1/n_i=o(\\log\\log x)$ (p.~226).\nTheorem~\\ref{thm:257-weighted} below proves irrationality under one explicit\nweaker condition without coprimality; no identification with his suggestions\nis asserted.\n\nWe show that rationality would impose a positive lower bound and then\nconstruct smaller values. For a positive integer $N$, write\n$N=q_a a+r_a$, $0\\le r_a<a$. The identity\n\\[\n \\frac{b^N-1}{b^a-1}\n =b^{r_a}\\sum_{j=0}^{q_a-1}b^{ja}\n  +\\frac{b^{r_a}-1}{b^a-1}\n\\]\nshows that\n\\[\n \\Delta_{b,A}(N):=\\sum_{a\\in A}\\frac{b^{N\\bmod a}-1}{b^a-1}\n =(b^N-1)X_A(b)-J_N,\\qquad J_N\\in\\mathbb Z.\n\\]\nThe integer sum is finite, since $q_a=0$ for $a>N$. Every summand of\n$\\Delta_{b,A}(N)$ is nonnegative and an exponent $a>N$ makes it positive.\nThus $X_A(b)=p/q$ would imply $\\Delta_{b,A}(N)\\ge1/q$ for every $N>0$.\n\nPut $w_{b,d}(N)=b^{N\\bmod d}/(b^d-1)$. For a fixed positive modulus $Q$\nand $g=(Q,d)$, the orbit of $Qm\\bmod d$ has length $d/g$. Its geometric sum\ngives\n\\[\n \\lim_{X\\to\\infty}\\frac1X\\sum_{m=1}^Xw_{b,d}(Qm)\n =M_{Q,b}(d):=\\frac{g}{d(b^g-1)}\\le\\frac1d.\n\\]\nTo interchange this limit with the sum over the support, expand\n$w_{b,d}(Qm)$ as $\\sum_{r\\ge1}b^{-r}{\\bf1}_{d\\mid Qm+r}$.\nCounting positive multiples of $d$ then gives\n\\[\n \\frac1X\\sum_{m=1}^Xw_{b,d}(Qm)\n \\le\\sum_{r\\ge1}b^{-r}\\frac{Q+r}{d}\n \\le\\frac{Q+2}{d}.\n\\]\nWith $Q$ fixed, dominated convergence identifies the limiting mean of $\\sum_{d\\in A}w_{b,d}(Qm)$ with $\\sum_{d\\in A}M_{Q,b}(d)$.\n\nNow choose $Q_t=\\operatorname{lcm}(1,\\ldots,t)$. For each fixed $d$,\neventually $d\\mid Q_t$, so $M_{Q_t,b}(d)=(b^d-1)^{-1}$. The bound\n$M_{Q_t,b}(d)\\le1/d$ permits a second dominated-convergence argument:\n\\[\n \\sum_{d\\in A}M_{Q_t,b}(d)\\longrightarrow X_A(b).\n\\]\nConsequently the limiting mean of $\\Delta_{b,A}(Q_tm)$ tends to zero.\nGiven $\\varepsilon>0$, first choose $t$ and then a sufficiently long finite\naverage with mean below $\\varepsilon$; one term of that average is below\n$\\varepsilon$. Under rationality, choosing $\\varepsilon<1/q$\ncontradicts the lower bound above.\nThis proves the assertion.  The exact all-base statement, with hypotheses\n$b\\ge2$, infinitude of $A$, and summability of $a\\mapsto\\mathbf1_A(a)/a$,\nhas a\n\\href{https://github.com/wcook04/plectis-erdos/blob/c91562bd574a387cde904481e609c7b4cacebb14/lean/Erdos249257/AllBaseReciprocalSupportIrrationality.lean#L395}{kernel-checked Lean proof}.\n\nWe increased the observation length with $Q_t$ fixed, and only then\nincreased $t$; no independence of divisibility events was used. At base two and\nfull support, the contribution of exponents $a>N$ alone gives\n$\\Delta_{2,\\mathbb N_{>0}}(N)>1-2^{-N}\\ge1/2$. Full-support irrationality\ntherefore uses a different argument; see Theorem~\\ref{thm:full-support}.\n\n\\subsection{A weighted condition on the support}\n\\papersectiontarget{record257:weighted-proof}\nA prime-power factor can compensate for a large sum of reciprocals of\ncofactors. If $a=2^km$ with $m$ odd, the weighted term below is\n$1/[m(b^{2^k}-1)]$. Thus we can allow increasingly many odd cofactors\nas $k$ grows while keeping the weighted sum finite. The theorem extends\nthis observation to a finite set of primes.\n\n\n\\begin{thm}[A weighted summability criterion]\n\\label{thm:257-weighted}\nLet $b\\ge2$ be an integer, let $\\mathcal P$ be a finite nonempty set of\nprimes, and let $A\\subseteq\\mathbb N_{>0}$ be infinite.  Write\n\\[\n h(a)=\\prod_{p\\in\\mathcal P}p^{v_p(a)},\n\\]\nwhere $v_p(a)$ is the exponent of $p$ in $a$; thus $h(a)$ is the\n$\\mathcal P$-part of $a$.  If\n\\begin{equation}\\label{eq:257-weighted-mass}\n W_{b,\\mathcal P}(A):=\n \\sum_{a\\in A}\\frac{h(a)}{a\\bigl(b^{h(a)}-1\\bigr)}<\\infty,\n\\end{equation}\nthen\n\\[\n X_A(b)=\\sum_{a\\in A}\\frac1{b^a-1}\n\\]\nis irrational.  More precisely, for\n\\[\n \\Delta_{b,A}(m)=\n \\sum_{a\\in A}\\frac{b^{m\\bmod a}-1}{b^a-1},\n\\]\nfor every $\\varepsilon>0$ and every $N$ there is an $m\\ge N$ with\n$0<\\Delta_{b,A}(m)<\\varepsilon$.\n\\end{thm}\n\n\\paragraph{Examples and limits of the hypothesis.}\nSquares, powers of two and factorials satisfy reciprocal summability and\ntherefore satisfy the base-two weighted condition.  Full support does not:\nfor any fixed finite prime set $P$, its subset coprime to $\\prod_{p\\in P}p$\nhas weight $1/[a(b-1)]$ and divergent reciprocal sum.  The set of all odd\nexponents also fails, by restricting to integers coprime to\n$2\\prod_{p\\in P}p$.  Inclusion--exclusion in the harmonic sum proves both\ndivergences.  This is a limitation of the sufficient condition, not an\nargument for rationality; full support is already known to be irrational.\n\nWriting $a=hm$ with $h$ its $\\mathcal P$-part gives the exact decomposition\n\\[\n W_{b,\\mathcal P}(A)=\\sum_{h\\in\\mathcal H_{\\mathcal P}}\\frac1{b^h-1}\n       \\sum_{\\substack{m\\ge1,\\ hm\\in A\\\\(m,\\prod_{p\\in\\mathcal P}p)=1}}\\frac1m,\n\\]\nwhere $\\mathcal H_{\\mathcal P}$ consists of $1$ and the positive\nintegers with no prime factor outside $\\mathcal P$.  Each inner\nreciprocal sum must be finite; the displayed outer sum imposes the\nadditional summability across different $\\mathcal P$-parts.\nFor example, at base two with $\\mathcal P=\\{2\\}$, choose in each layer\n$2^k$, $k\\ge1$, a finite set of odd factors whose reciprocal\nsum is at least $2^{2^k}$.  Every inner sum is finite, but the\nouter weighted sum diverges.  Such finite sets exist by divergence\nof the harmonic sum over odd integers.  Prime support also fails\nthis criterion for every finite $\\mathcal P$: its contribution\nfrom primes outside $\\mathcal P$ is a constant multiple of their\ndivergent reciprocal sum.  These failures concern the sufficient\nhypothesis, not rationality of the resulting values.\nThe following construction shows, nonetheless, that this condition\nis strictly weaker than reciprocal summability.\nAt base two take\n\\[\n A_\\star=\\{2^k m:k\\ge1,\\ m\\text{ odd},\\ m\\le2^{2^k}\\}.\n\\]\nWrite $H_N=\\sum_{m=1}^N1/m$ and $N_k=2^{2^k}$. The sum of reciprocals\nof the exponents with $v_2(a)=k$ is\n\\[\n \\rho_k=2^{-k}\\!\\sum_{\\substack{m\\le N_k\\\\m\\text{ odd}}}\\frac1m\n =2^{-k}\\bigl(H_{N_k}-\\tfrac12H_{N_k/2}\\bigr)\n \\longrightarrow\\frac{\\log2}{2}.\n\\]\nThus $\\sum_{a\\in A_\\star}1/a$ diverges.  For $\\mathcal P=\\{2\\}$,\nthe weighted contribution of the same layer is\n\\[\n \\frac{2^k}{2^{2^k}-1}\\rho_k.\n\\]\nSince $H_N\\le1+\\log N$, these terms are at most\n$(1+2^k\\log2)2^{1-2^k}$, which is summable.  The theorem therefore proves\nirrationality at every integer base for every infinite subset of this set,\nalthough its reciprocal sum diverges.\n\n\n\\begin{proof}\nWe first derive the lower bound implied by rationality and then\nestimate a finite average of the same quantities. Division of $m$ by $a$, say $m=qa+r$ with\n$0\\le r<a$, gives\n\\[\n \\frac{b^m-1}{b^a-1}-\\frac{b^r-1}{b^a-1}\n   =\\sum_{j=1}^{q}b^{m-ja}.\n\\]\nConsequently\n\\begin{equation}\\label{eq:257-weighted-lattice}\n \\Delta_{b,A}(m)=(b^m-1)X_A(b)-J_{b,A}(m),\n \\qquad\n J_{b,A}(m)=\n \\sum_{\\substack{a\\in A\\\\a\\le m}}\n \\sum_{j=1}^{\\lfloor m/a\\rfloor}b^{m-ja}\\in\\mathbb Z.\n\\end{equation}\nThe integer sum is finite.  For $m\\ge1$, moreover,\n$\\Delta_{b,A}(m)>0$: since $A$ is infinite, some $a\\in A$ exceeds $m$,\nand its summand is positive.  At $m=0$ the displacement is zero.  Thus, if\n$X_A(b)=u/v$ with $v>0$, equation~\\eqref{eq:257-weighted-lattice} gives\n$v\\Delta_{b,A}(m)\\in\\mathbb Z$, whence\n\\begin{equation}\\label{eq:257-weighted-gap}\n \\Delta_{b,A}(m)\\ge \\frac1v\\qquad(m\\ge1).\n\\end{equation}\nWe therefore seek positive indices at which the displacement is\narbitrarily small.\n\nPut\n\\[\n \\phi_a(m)=\\frac{b^{m\\bmod a}}{b^a-1},\n \\qquad\n d_a(m)=\\phi_a(m)-\\frac1{b^a-1}.\n\\]\nThen $d_a(m)\\ge0$ and $\\Delta_{b,A}(m)=\\sum_{a\\in A}d_a(m)$.\nFor $Q,a,T\\ge1$, put $g=(Q,a)$.  The residues $tQ\\bmod a$ have period\n$a/g$ and visit $0,g,\\ldots,a-g$ once in a period, so\n\\[\n \\sum_{t=1}^{a/g}\\phi_a(tQ)\n =\\frac{1+b^g+\\cdots+b^{a-g}}{b^a-1}\n =\\frac1{b^g-1}.\n\\]\nComplete periods and at most one incomplete period therefore give\n\\begin{equation}\\label{eq:257-weighted-gcd-average}\n \\frac1T\\sum_{t=1}^{T}d_a(tQ)\n \\le \\frac{g}{a(b^g-1)}+\\frac1{T(b^g-1)}.\n\\end{equation}\nThe complete-period term explains the weight in\n\\eqref{eq:257-weighted-mass}: replacing $g$ by the $\\mathcal P$-part\n$h(a)$ gives its summand. The modulus below will permit this replacement\nfor $h(a)\\le H$; larger prime parts require a separate estimate.\nFor this comparison we use that $n/(b^n-1)$ decreases for positive\nintegers $n$; indeed\n\\[\n n(b^{n+1}-1)-(n+1)(b^n-1)\n =b^n\\bigl(n(b-1)-1\\bigr)+1>0.\n\\]\n\nThe part of the support beyond the observation range has a uniform bound.  If\n$Y=QT$ and $a>Y\\ge tQ$, then $tQ\\bmod a=tQ$ and\n$d_a(tQ)\\le2\\,2^{tQ-a}$.  Hence\n\\[\n \\sum_{a>Y}d_a(tQ)\\le2\\,2^{tQ-Y},\n \\qquad\n \\sum_{t=1}^{T}2^{tQ-Y}\n =\\sum_{j=0}^{T-1}2^{-jQ}\\le2,\n\\]\nand therefore\n\\begin{equation}\\label{eq:257-weighted-outer}\n \\frac1T\\sum_{t=1}^{T}\n \\sum_{\\substack{a\\in A\\\\a>QT}}d_a(tQ)\\le\\frac4T.\n\\end{equation}\n\nThe incomplete-period term in~\\eqref{eq:257-weighted-gcd-average}\nhas no factor $1/a$. Although it tends to zero for a fixed exponent,\nthere are more observed exponents as the window grows. To recover a\nsummable bound, average over lengths and sum in the other order.\nAn exponent $a$ first enters when the window reaches $a/Q$; choosing\ngeometrically spaced lengths makes the sum of their later reciprocals\nat most $2Q/a$. Let $\\alpha_a\\ge0$ with\n$\\sum_a\\alpha_a/a<\\infty$. For $Q,M\\ge1$, finite rearrangement gives\n\\begin{equation}\\label{eq:257-weighted-dyadic-average}\n \\sum_{j=M}^{2M-1}\\frac1{2^j}\n       \\sum_{a\\le Q2^j}\\alpha_a\n \\le 2Q\\sum_{a\\ge1}\\frac{\\alpha_a}{a}.\n\\end{equation}\nIndeed, for each fixed $a$ the admissible indices satisfy $2^j\\ge a/Q$ and\n\\[\n \\sum_{\\substack{M\\le j<2M\\\\a\\le Q2^j}}2^{-j}\\le\\frac{2Q}{a}.\n\\]\nOnly finitely many terms occur on the left, so no infinite interchange is\nhidden here.\n\nWe now choose the modulus to be divisible by a fixed finite subset\nof the support and by all powers of primes in $\\mathcal P$ up to a\ngrowing threshold. Fix $\\varepsilon>0$ and a lower bound $N$.  Select a finite $F\\subseteq A$ such that\n\\begin{equation}\\label{eq:257-weighted-tail-choice}\n \\sum_{a\\in A\\mathbin{\\backslash} F}\n \\frac{h(a)}{a(b^{h(a)}-1)}<\\varepsilon,\n\\end{equation}\nand put $L=\\operatorname{lcm}(F)$, with $L=1$ if $F$ is empty.  Write\n$p_*=\\max\\mathcal P$ and $r=|\\mathcal P|$.  For a large integer $H\\ge2p_*$,\nset\n\\[\n Q_0(H)=\\prod_{p\\in\\mathcal P}p^{\\lfloor\\log_pH\\rfloor},\n \\qquad Q=LQ_0(H),\n \\qquad G=\\left\\lfloor\\frac H{p_*}\\right\\rfloor.\n\\]\nThen $Q\\le LH^r$, $G\\ge1$, and every $a\\in F$ divides $Q$, so its\n$d_a(tQ)$ contribution vanishes.\n\nConsider first $a\\in A\\mathbin{\\backslash} F$ with $a\\le QT$ and $h(a)\\le H$.\nEvery prime-power factor of $h(a)$ occurs in $Q_0(H)$, hence\n$h(a)\\mid Q_0(H)$ and $(Q,a)\\ge h(a)$.  Applying\n\\eqref{eq:257-weighted-gcd-average} and the monotonicity above, the total\ncontribution of these $a$ is at most\n\\begin{equation}\\label{eq:257-weighted-small-part}\n \\varepsilon+\n \\frac1T\\sum_{\\substack{a\\in A, a\\le QT\\\\h(a)\\le H}}\n \\frac1{b^{h(a)}-1}.\n\\end{equation}\nThe first term is bounded using~\\eqref{eq:257-weighted-tail-choice}.\n\nFor $h(a)>H$, one has $(Q,a)\\ge G$.  If every $\\mathcal P$-prime-power\ncomponent of $h(a)$ is at most $H$, then $h(a)\\mid Q_0(H)$ and the gcd is\nlarger than $H$.  Otherwise some $p^{v_p(a)}>H$, and the gcd contains\n$p^{\\lfloor\\log_pH\\rfloor}>H/p\\ge H/p_*$.  Thus\n\\eqref{eq:257-weighted-gcd-average}, enlarged from the support to all\n$a\\le QT$, bounds the large-$h(a)$ part by\n\\begin{equation}\\label{eq:257-weighted-large-part}\n \\frac{G(1+\\log(QT))+Q}{b^G-1};\n\\end{equation}\nwe used $\\sum_{a\\le QT}a^{-1}\\le1+\\log(QT)$ and\n$\\#\\{a\\le QT\\}/T=Q$.\n\nLet\n\\[\n \\mathcal M(Q,T)=\\frac1T\\sum_{t=1}^{T}\\Delta_{b,A}(tQ).\n\\]\nCombining \\eqref{eq:257-weighted-outer},\n\\eqref{eq:257-weighted-small-part}, and\n\\eqref{eq:257-weighted-large-part} gives\n\\begin{equation}\\label{eq:257-weighted-master-average}\n \\mathcal M(Q,T)\\le\n \\varepsilon+\n \\frac1T\\sum_{\\substack{a\\in A, a\\le QT\\\\h(a)\\le H}}\n       \\frac1{b^{h(a)}-1}\n +\\frac{G(1+\\log(QT))+Q}{b^G-1}+\\frac4T.\n\\end{equation}\nIncreasing the number $M$ of dyadic lengths reduces the incomplete-period\nerror to order $Q/M$. It also makes $\\log T$ of order $M$ on the largest\nwindow, increasing the large-gcd contribution to order $GM/b^G$.\nThus the two quantities must be made small together. With $L$ fixed,\n$Q$ grows polynomially in $H$ and $G$ grows linearly, so an intermediate\nexponential choice of $M$ has both properties.  Choose\n\\[\n M=\\left\\lfloor b^{G/2}\\right\\rfloor\n\\]\nand average~\\eqref{eq:257-weighted-master-average} over\n$T=2^j$, $M\\le j<2M$.  Apply\n\\eqref{eq:257-weighted-dyadic-average} with\n$\\alpha_a={\\bf1}_A(a)/(b^{h(a)}-1)$.  Its hypothesis holds because\n\\[\n \\sum_a\\frac{\\alpha_a}{a}\n =\\sum_{a\\in A}\\frac1{a(b^{h(a)}-1)}\n \\le W_{b,\\mathcal P}(A).\n\\]\nThe resulting finite average satisfies\n\\begin{equation}\\label{eq:257-weighted-final-diagonal}\n \\frac1M\\sum_{j=M}^{2M-1}\\mathcal M(Q,2^j)\n \\le \\varepsilon+\\frac{2QW_{b,\\mathcal P}(A)}M\n +\\frac{G(1+\\log Q+2M\\log2)+Q}{b^G-1}\n +4\\,2^{-M}.\n\\end{equation}\n\nAs $H\\to\\infty$,\n\\[\n Q\\le LH^r,\\qquad G=H/p_*+O(1),\\qquad M\\asymp b^{G/2}.\n\\]\nConsequently $Q/M\\to0$, $GM/b^G\\to0$,\n$(G\\log Q+Q)/b^G\\to0$, and $2^{-M}\\to0$.  For sufficiently large $H$,\nwe may also arrange $Q\\ge N$ and make the right side of\n\\eqref{eq:257-weighted-final-diagonal} smaller than $2\\varepsilon$.\nThe left side is a finite average of positive numbers\n$\\Delta_{b,A}(tQ)$.  One of them therefore satisfies\n\\[\n 0<\\Delta_{b,A}(tQ)<2\\varepsilon,\n \\qquad tQ\\ge Q\\ge N.\n\\]\nThe construction is valid for every positive $\\varepsilon$; applying it with\n$\\varepsilon/2$ proves the stated cofinal bound with $\\Delta_{b,A}(m)<\\varepsilon$.\nThis contradicts~\\eqref{eq:257-weighted-gap} if $X_A(b)$ is rational.\n\\end{proof}\n\nSince $b^{h(a)}-1\\ge2^{h(a)}-1$ for every $b\\ge2$, one has\n$W_{b,\\mathcal P}(A)\\le W_{2,\\mathcal P}(A)$.  Thus the base-two weighted\nhypothesis gives the conclusion simultaneously at every integer base.  The\nhypothesis is also hereditary under passage to infinite subsets.  The\ntheorem does not cover all infinite supports and gives no quantitative\nirrationality measure; no priority is claimed for it.\n\n\\paragraph{Averaging antecedents.}\nBoth averaging proofs in this section end by choosing a term no larger than a\nfinite mean.\n\\href{https://danielduverney.fr/documents/theorie-des-nombres/DuverneyTachiya190522.pdf}{Duverney\nand Tachiya} choose their index in the same way in their refinement of the\nChowla--Erd\\H{o}s method: they average a local coefficient mass over an\narithmetic progression built by the Chinese remainder theorem, and a term\nattaining the minimum supplies the bounds that control the tail beyond it\n(Section~2, (2.3)--(2.9), pp.~5--6 of the author preprint).\n\\href{https://arxiv.org/abs/2601.20743}{Kaneko, Suzuki and Tachiya} measure\nthe average decay of scaled tails by $R_c(q,x,z)$, their~(1.7) on p.~3 of\narXiv v1.  Under a counting condition on the support and a bound on $R_c$,\nmost scaled tails in a range are small, and arbitrarily small nonzero scaled\ntails give irrationality (Lemmas~1 and~2, pp.~6--8).\nHere sparsity concerns the positions of nonzero power-series coefficients,\nnot the set of selected Lambert denominators. The divisor-incidence\ncoefficients $\\#\\{a\\in A:a\\mid n\\}$ are positive on every multiple of\n$\\min A$.  For any splitting into two sequences the two coefficient supports\ntherefore cover a set of positive lower density, which is incompatible with the\nsupport-counting conditions\n\\cite[Theorem~1(iii), p.~3, and Theorem~3(iv), p.~5]{kanekosuzukitachiya}\nrequiring both counts to be $o(x_n/z_n)$ with $z_n\\ge1$.  Their remote-tail\nquantity $R_c(q,x,z)$ sums only offsets $j\\ge z$, whereas the full scaled tail\n\\[\n \\sum_{r\\ge1}\\frac{\\#\\{a\\in A:a\\mid m+r\\}}{b^r}\n =\\sum_{a\\in A}\\phi_a(m)=X_A(b)+\\Delta_{b,A}(m)\n\\]\nis at least $X_A(b)$, which gives no lower bound for $R_c$.  The proofs above\naverage the displacement instead,\nalong multiples of a modulus that freezes a finite part of $A$; the weighted\nproof adds the dyadic average~\\eqref{eq:257-weighted-dyadic-average}.\n\n\\subsection{Dependence on the chosen primes}\\label{sec:257-finite-witness-rules}\n\\papersectiontarget{record257:witness-rules}\nThe weighted condition need not single out one required prime.\nWe construct a support for which convergence depends on a prescribed\nupward-closed condition on the chosen primes. This description holds\nat every integer base. We also allow the empty prime set, with\n$h(a)=1$; it gives a divergent weighted sum because the constructed\nsupport has divergent reciprocal sum.\n\n\\begin{prop}[realising finite monotone witness rules]\n\\label{prop:257-finite-witness-rule}\nLet $E$ be a finite set of primes, and let $\\mathcal U$ be an\nupward-closed family of subsets of $E$ with $E\\in\\mathcal U$ and\n$\\varnothing\\notin\\mathcal U$.  There is a set\n$A_{\\mathcal U}\\subseteq\\mathbb N_{>0}$ with\n$\\sum_{a\\in A_{\\mathcal U}}1/a=\\infty$ such that, for every integer\n$b\\ge2$ and every finite set $\\mathcal P$ of primes,\n\\begin{equation}\\label{eq:257-finite-witness-rule}\n W_{b,\\mathcal P}(A_{\\mathcal U})<\\infty\n \\quad\\Longleftrightarrow\\quad \\mathcal P\\cap E\\in\\mathcal U.\n\\end{equation}\nMoreover $X_B(b)$ is irrational for every infinite\n$B\\subseteq A_{\\mathcal U}$ and every integer $b\\ge2$.\n\\end{prop}\n\n\\begin{proof}\nFor each nonempty $C\\subseteq E$ put $D_C=\\prod_{p\\in C}p$ and\n\\[\n B_C=\\{D_C^km:k\\ge1,\\ 1\\le m\\le2^{D_C^k},\\ \\gcd(m,D_C)=1\\},\n\\]\nand let $g=\\prod_{p\\in C\\cap\\mathcal P}p$. The contribution to the\nweighted sum from the exponents with this value of $k$ lies between\npositive multiples of $g^k/(b^{g^k}-1)$, with constants depending only\non $C\\cup\\mathcal P$. For the upper bound, an element $D_C^km$ has\n$\\mathcal P$-part $g^ku$, and raising the $\\mathcal P$-part from $g^k$ to\n$g^ku$ multiplies the weight by\n$u/(1+b^{g^k}+\\cdots+b^{(u-1)g^k})\\le1$, while\n$\\sum_{m\\le2^r}1/m\\le2r$. For the lower bound, the cofactors $m\\equiv1\\pmod Q$, with\n$Q=\\prod_{p\\in C\\cup\\mathcal P}p$, have $\\mathcal P$-part exactly $g^k$,\nand grouping $1,\\dots,2^r$ into blocks of at most $Q$ integers gives\nreciprocal sum at least $H_{2^r}/Q\\ge r/(2Q)$, where $r=D_C^k$ and\n$H_n=\\sum_{j\\le n}1/j$.  The comparison terms equal $1/(b-1)$ when $g=1$\nand are summable when $g\\ge2$, since consecutive ratios are at most\n$g/b^{g^k}\\to0$.  Hence the weighted sum over $B_C$ converges exactly when\n$\\mathcal P$ meets $C$, and the case $Q=D_C$ shows that each layer adds at\nleast $1/(2D_C)$ to the ordinary reciprocal sum.\n\nLet $A_{\\mathcal U}$ be the union of the blocks $B_{E\\smallsetminus M}$ over\nthe forbidden sets $M\\subseteq E$, $M\\notin\\mathcal U$; each clause\n$E\\smallsetminus M$ is nonempty because $E\\in\\mathcal U$.  A subset of $E$ lies\nin $\\mathcal U$ exactly when it meets every clause: an allowed set cannot\nlie inside a forbidden one, and a forbidden set misses its own clause.\nThis finite correspondence is standard~\\cite[Section~2]{sedaghatstephenchindelevitch}.\nNonnegative weights make summability on a finite union equivalent to\nsummability on each block, which gives~\\eqref{eq:257-finite-witness-rule}\nand reciprocal divergence.  Since $E\\in\\mathcal U$,\n\\eqref{eq:257-finite-witness-rule} with $\\mathcal P=E$ gives\n$W_{b,E}(B)\\le W_{b,E}(A_{\\mathcal U})<\\infty$ for every infinite\n$B\\subseteq A_{\\mathcal U}$ and every $b\\ge2$, so\nTheorem~\\ref{thm:257-weighted} makes $X_B(b)$ irrational.\n\\end{proof}\n\nFor example, require the chosen prime set to contain at least two\nof $2,3,5$. No individual prime is then mandatory, and one prime\nalone does not suffice. The maximal\nforbidden sets give the three clauses $\\{2,3\\}$, $\\{2,5\\}$ and\n$\\{3,5\\}$. Using only these clauses gives the same witness classification\nas the formal statement, which uses every forbidden set. Historical\nnovelty of this arithmetic realisation has not been assessed.\n\n\\subsection{A finite estimate on common observation scales}\\label{sec:257-common-kernel}\n\\papersectiontarget{record257:common-kernel}\nA divisor majorant becomes a sum of geometric tails, one for each\ndivisor. The positive offsets $r$ with $d\\mid N+r$ are\n$d-(N\\bmod d),2d-(N\\bmod d),\\ldots$, and hence\n\\[\n \\sum_{\\substack{r\\ge1\\\\d\\mid N+r}}B^{-r}\n =\\frac{B^{N\\bmod d}}{B^d-1}.\n\\]\nThe cover argument raises a base-two tail to a fractional power $\\alpha$,\nreplacing its geometric factor by $B=2^\\alpha$. The estimate must\ntherefore allow $1<B\\le2$. It must also keep the modulus, first dyadic\nscale and number of scales separate. In the mixed theorem, the weighted\nconstruction chooses these parameters after the finite divisibility\nrequirements have been fixed.\n\nFor $1<B\\le2$, positive integers $L,d,M$, and an integer $R\\ge0$, put\n\\[\n w_{B,d}(n)=\\frac{B^{n\\bmod d}}{B^d-1},\\qquad\n \\mathscr D_{L;R,M}F\n =\\frac1M\\sum_{j=R}^{R+M-1}\\frac1{2^j}\n      \\sum_{m=1}^{2^j}F(Lm).\n\\]\nTo read this as sampling, first choose $j$ uniformly from\n$R,\\ldots,R+M-1$. Conditional on that scale, choose $m$ uniformly from\n$1,\\ldots,2^j$ and observe $N=Lm$. The scales have equal total weight;\nthe observations within a longer scale have smaller individual weights.\nRepeated occurrences of an integer $N$ retain their contributions to the\nsum. This is the distribution used for both parts of the mixed proof.\n\nThe finite estimate\n\\begin{equation}\n \\mathscr D_{L;R,M}w_{B,d}\n \\le\\frac{1+4L/M}{d(B-1)}\n \\label{eq:257-mixed-finite-kernel}\n\\end{equation}\nequivalently bounds $(B-1)\\mathscr D_{L;R,M}w_{B,d}$ by\n$(1+4L/M)/d$.  This normalised bound is uniform as $B\\downarrow1$.\nThe unnormalised right side grows like $(B-1)^{-1}$. In the cover\nargument $B=2^\\alpha$ can approach $1$, so the factor\n$(2^\\alpha-1)^{-1}$ must remain in the summability condition.\nThe ratio $L/M$ measures the cost of incomplete modular periods.\n\nFix $T=2^j$. We compare $d$ with the largest sampled multiple $LT$.\nFor $d\\le LT$ we use a complete-orbit bound with a remainder term;\nfor $d>2LT$ no residue is reduced modulo $d$. For fixed $d$, doubling\n$T$ moves from $d>2LT$ to $d\\le LT$ after at most one intermediate\nscale; hence $LT<d\\le2LT$ occurs for at most one dyadic length.\nWhen $d\\le LT$, the complete orbit has length $d/g$, with $g=(L,d)$,\nand total weight $1/(B^g-1)$. Splitting the sample into complete\ncycles---possibly none---and one remaining piece gives\n\\[\n \\frac1T\\sum_{m=1}^T w_{B,d}(Lm)\n \\le\\frac{g}{d(B^g-1)}+\\frac1{T(B^g-1)}\n \\le\\frac1{d(B-1)}+\\frac1{T(B-1)}.\n\\]\nAcross dyadic lengths satisfying $d\\le L2^j$, the reciprocal-length\nerrors sum to at most $2L/[d(B-1)]$.\n\\par\n\\begin{samepage}\n\\noindent\nWhen $d>2LT$, we have $Lm\\bmod d=Lm$ and $2Lm\\le d-1$; hence\n$\\sum_{i=0}^{d-1}B^i\\ge dB^{(d-1)/2}\\ge dB^{Lm}$.\nEach term $w_{B,d}(Lm)$ is then at most $1/[d(B-1)]$.\n\\par\n\\end{samepage}\nFinally, at most one dyadic length satisfies $LT<d\\le2LT$.\nFor that length the geometric sum and convexity give\n\\[\n \\frac1T\\sum_{m=1}^T w_{B,d}(Lm)\n =\\frac{B^L}{T(B^L-1)}\\frac{B^{LT}-1}{B^d-1}\n \\le\\frac{LB^L}{d(B^L-1)}\n \\le\\frac{2L}{d(B-1)}.\n\\]\nThe first inequality follows from convexity of $x\\mapsto B^x-1$,\nwhich vanishes at $0$: the ratio is at most $LT/d$. The last uses\n$(B^L-1)/(B-1)=\\sum_{i=0}^{L-1}B^i\\ge B^{L-1}$ and $B\\le2$.\nSumming the main terms and the two error bounds proves~\\eqref{eq:257-mixed-finite-kernel}.\nThe argument displayed here is an ordinary proof; the estimate is also\n\\href{https://github.com/wcook04/plectis-erdos/blob/c91562bd574a387cde904481e609c7b4cacebb14/lean/ErdosProblems/Erdos257/PaperCompleteR8/DyadicKernel.lean\\#L206}{proved in Lean}.\nNonnegative interchange permits summation against any coefficients $c_d$\nwith $\\sum_dc_d/d<\\infty$.\n\n\\subsection{Positive divisor majorants}\nFor a finite set $F\\subseteq\\Npos$, put\n$f_F(n)=\\#\\{a\\in F:a\\mid n\\}$. Consider the finite set\n$F=\\{2,6,10,30\\}$, obtained by multiplying each divisor of $15$ by $2$.\nFor $0<\\alpha\\le1$\nand $z=2^\\alpha-1$, its count has the exact expansion\n\\[\n f_F(n)^\\alpha\n =\\mathbf1_{2\\mid n}+z\\mathbf1_{6\\mid n}\n  +z\\mathbf1_{10\\mid n}+z^2\\mathbf1_{30\\mid n}.\n\\]\nIndeed, conditional on $2\\mid n$, each of $3$ and $5$ dividing $n$\ndoubles $f_F(n)$ and multiplies its fractional power by $1+z$.\nThe coefficient sum weighted by reciprocal divisors is\n$(1+z/3)(1+z/5)/2$. This illustrates the general positive expansion\nused in the separating constructions. The extra factor\n$(2^\\alpha-1)^{-1}$ in the averaging bound remains part of the\ncover cost, so small $\\alpha$ alone does not make that cost small.\nThe theorem permits a different fractional power for each finite set.\n\n\\begin{thm}[A summable family of divisor majorants]\\label{thm:257-variable-fractional-cover}\nFor each $j\\ge1$, let $F_j\\subseteq\\Npos$ be finite, let\n$0<\\alpha_j\\le1$, and let $c_{j,d}\\ge0$ satisfy\n\\[\n f_{F_j}(n)^{\\alpha_j}\\le\\sum_{d\\mid n}c_{j,d}\\quad(n\\ge1).\n\\]\nSet $C_j=\\sum_{d\\ge1}c_{j,d}/d$.  If\n\\begin{equation}\n \\sum_{j\\ge1}\\frac{C_j2^{j\\alpha_j}}{2^{\\alpha_j}-1}<\\infty,\n \\label{eq:257-strengthened-cover}\n\\end{equation}\nthen $X_A(b)$ is irrational for every infinite\n$A\\subseteq\\bigcup_jF_j$ and every integer $b\\ge2$.\n\\end{thm}\n\nEvery finite set admits such a majorant: take $\\alpha_j=1$ and\n$c_{j,d}=\\mathbf1_{F_j}(d)$.  The restrictive part is summability over the\nwhole family, not the existence of a majorant for an individual set.\nFor example, $F_j=\\{4^j\\}$ with these choices contributes $2^{-j}$\nto~\\eqref{eq:257-strengthened-cover}, so the theorem applies.  Full support\ncannot admit a cover satisfying the theorem, because the proof gives\narbitrarily small positive displacements and the full-support displacement\nis at least $1/2$.\n\n\\begin{proof}\nWrite $B_j=2^{\\alpha_j}$ and\n\\[\n U_j(N)=\\sum_{r\\ge1}2^{-r}f_{F_j}(N+r),\\qquad\n V_j(N)=\\sum_{d\\ge1}c_{j,d}w_{B_j,d}(N).\n\\]\nThe choice $B_j=2^{\\alpha_j}$ matches the decay after taking a\nfractional power.  Subadditivity and the divisor majorant give\n\\[\n U_j(N)^{\\alpha_j}\n \\le\\sum_{r\\ge1}B_j^{-r}f_{F_j}(N+r)^{\\alpha_j}\n \\le\\sum_{d\\ge1}c_{j,d}\n       \\sum_{\\substack{r\\ge1\\\\d\\mid N+r}}B_j^{-r}\n =V_j(N).\n\\]\n\\par\\noindent\n\\begin{minipage}{\\linewidth}\nThe last identity sums a geometric progression in each residue class.\nThe geometric-series identity also gives\n\\[\n \\Delta_{2,F_j}(N)=U_j(N)-X_{F_j}(2)\\le U_j(N).\n\\]\n\\end{minipage}\n\\par\nFor the first $J$ covering sets we will impose divisibility, making\n$\\Delta_{2,F_j}(N)$ vanish. At those indices the identity above gives\n$U_j(N)=X_{F_j}(2)$, which need not be zero. We bound only the remaining\ndisplacements by the corresponding tails $U_j(N)$.\n\nFix $\\varepsilon>0$ and allot $t_j=\\varepsilon2^{-j}$ to the $j$th tail;\nthese allowances sum to $\\varepsilon$. Since $U_j^{\\alpha_j}\\le V_j$,\nit suffices to make $t_j^{-\\alpha_j}V_j<1$. This normalisation explains\nthe factor $2^{j\\alpha_j}$ in the cover condition. Choose $J$ with\n\\[\n K_J:=\\sum_{j>J}\\frac{C_jt_j^{-\\alpha_j}}{B_j-1}<\\frac14.\n\\]\nThis is possible since $\\varepsilon^{-\\alpha_j}\\le\\max(1,\\varepsilon^{-1})$.\nChoose $L$ divisible by every member of the first $J$ covering sets.  Equation~\\eqref{eq:257-mixed-finite-kernel} bounds the finite mean of\n$S_J(N):=\\sum_{j>J}t_j^{-\\alpha_j}V_j(N)$ by\n$(1+4L/M)K_J<1/2$ whenever $M\\ge4L$.\nOne sample therefore has $S_J(N)<1$, forcing $U_j(N)<t_j$ for every $j>J$.\nEvery exponent in the first $J$ finite sets divides $L$, so those terms\nhave zero displacement.  Consequently,\n\\[\n 0<\\Delta_{2,A}(N)\\le\\sum_{j>J}U_j(N)\\le\\varepsilon.\n\\]\nThe positive lower bound under rationality excludes a rational value\nat base two. For $0\\le r<d$, the summand $(b^r-1)/(b^d-1)$ is nonincreasing\nin $b>1$.  For $r>0$, cancellation of $b-1$ writes it as\n$A(b)/(A(b)+C(b))$, with $A(b)=\\sum_{i<r}b^i$ and\n$C(b)=\\sum_{r\\le j<d}b^j$.  Its derivative has the sign of\n$A'C-AC'=\\sum_{i<r\\le j<d}(i-j)b^{i+j-1}<0$.\nThus $\\Delta_{b,A}(N)\\le\\Delta_{2,A}(N)$ for real $b\\ge2$;\nfor integer $b$ the same positive-integer argument gives irrationality.\nThis ordinary argument is sharper than the factor-two bound in the Lean\nsource's\n\\href{https://github.com/wcook04/plectis-erdos-lean/blob/52f29ad173b04e3bac941b3663f2b9aebe5de0bb/ErdosProblems/Erdos257/PaperCompleteR7/CoverKernel.lean#L232}{scalar comparison between bases}.\nAny prescribed positive integer can be included as a divisor of $L$,\nso the selected indices can also be required to be arbitrarily large.\n\\end{proof}\n\nThe same proof permits any positive weights $\\eta_j$ with $\\sum_j\\eta_j=1$:\nreplace $2^{j\\alpha_j}$ in~\\eqref{eq:257-strengthened-cover} by\n$\\eta_j^{-\\alpha_j}$ and take $t_j=\\varepsilon\\eta_j$.\n\\subsection{Combining the two support criteria}\n\\papersectiontarget{record257:mixed-proof}\nSeparate sequences of small weighted and covered displacements would not\ngive a small sum: their small values could occur on disjoint sets of\nindices. Instead, the weighted proof chooses one finite distribution and\nthe explicit modulus dependence in~\\eqref{eq:257-mixed-finite-kernel}\nlets the cover estimate use that same distribution. We add the two\nnonnegative errors there and choose an index only afterwards.\n\\begin{thm}[mixed weighted and cover supports]\\label{thm:257-mixed-supports}\nLet $E,V\\subseteq\\Npos$.  Suppose $E$ has finite weighted mass\n\\eqref{eq:257-weighted-mass} at $b=2$ for a finite nonempty prime set\n$\\mathcal P$.  Suppose also that $V\\subseteq\\bigcup_jF_j$ for sets and\nmajorants satisfying Theorem~\\ref{thm:257-variable-fractional-cover},\nwith~\\eqref{eq:257-strengthened-cover} or its positive-weight variant.\nThen $X_A(b)$ is irrational for every\ninfinite $A\\subseteq E\\cup V$ and every integer $b\\ge2$.\n\\end{thm}\n\\begin{proof}\nFix $\\varepsilon>0$ and an integer $N_0\\ge1$, and put $\\rho=\\varepsilon/3$.\nUse the cover notation $B_j,U_j,V_j$ above, with $\\eta_j=2^{-j}$ in\nthe case~\\eqref{eq:257-strengthened-cover}.  Set $t_j=\\rho\\eta_j$ and choose\n$J$ so that\n\\[\n K_J=\\sum_{j>J}\\frac{C_jt_j^{-\\alpha_j}}{B_j-1}<\\frac1{16}.\n\\]\nChoose a finite $F\\subseteq E$ whose complementary weighted mass is\n$\\kappa<\\rho/16$.  Let $L$ be a positive common multiple of $N_0$, all\nmembers of $F$, and all members of the first $J$ covering sets. The order\nis deliberate: first the finite pieces determine $L$; then the weighted\nargument enlarges $L$ to $Q$ and chooses the dyadic scales; only after both\nestimates are on that distribution is a sample selected. With the finite\nsets and $L$ fixed, choose the base-two weighted parameters\n\\[\n Q=L\\prod_{p\\in\\mathcal P}p^{\\lfloor\\log_pH\\rfloor},\\qquad\n G=\\lfloor H/\\max\\mathcal P\\rfloor,\\qquad M=\\lfloor2^{G/2}\\rfloor.\n\\]\nWe use $\\mathscr D_{Q;M,M}$ throughout the following estimates. Its\nscales run from $M$ to $2M-1$, and both quantities are evaluated at the\nsame sampled multiple $N=Qm$.\nThe finite estimate~\\eqref{eq:257-weighted-final-diagonal}, with complementary\nweighted mass $\\kappa$, remains valid for this $L$: its proof requires only\nthat every member of $F$ divide $L$.  It also applies to finite or empty $E$, since positivity was\nused only after the averaging estimate.  Thus\n\\[\n \\mathscr D_{Q;M,M}(\\Delta_{2,E}/\\rho)<\\frac18\n\\]\nfor sufficiently large $H$.  For the same finite distribution, \\eqref{eq:257-mixed-finite-kernel} and\nnonnegative interchange give\n\\[\n \\mathscr D_{Q;M,M}S_J\\le(1+4Q/M)K_J<\\frac18,\n \\qquad S_J=\\sum_{j>J}t_j^{-\\alpha_j}V_j,\n\\]\nbecause $Q/M\\to0$. Here the cover estimate is applied with modulus\n$Q$, not merely with the fixed divisor $L$; this is why its explicit\nmodulus dependence was needed. On this one distribution,\n\\[\n \\mathscr D_{Q;M,M}(\\Delta_{2,E}/\\rho+S_J)<\\frac14.\n\\]\nChoose a sampled $N=Qm$ at which the nonnegative sum is less than $1$.\nNonnegativity of both quantities and of every summand of $S_J$ gives\n$\\Delta_{2,E}(N)<\\rho$ and $U_j(N)<t_j$ for every $j>J$.\nFor $j\\le J$, it is the displacement $\\Delta_{2,F_j}(N)$ that vanishes,\nsince every exponent in $F_j$ divides $L$ and hence divides $N$. Therefore\n\\[\n \\Delta_{2,A}(N)\\le\\Delta_{2,E}(N)+\\Delta_{2,V}(N)<2\\rho<\\varepsilon,\n \\qquad N\\ge Q\\ge L\\ge N_0.\n\\]\nThe inequality also holds when $E$ and $V$ overlap. Since $A$ is\ninfinite, $\\Delta_{b,A}(N)>0$. The comparison\n$\\Delta_{b,A}(N)\\le\\Delta_{2,A}(N)$ from the cover proof makes these\nvalues arbitrarily small at every integer base, contradicting the\npositive lower bound imposed by rationality.\n\\end{proof}\n\n\\paragraph{Why the mixed criterion is strict.}\\label{record257:strict-mixed-proof}\nTake an infinite support $E$ satisfying the weighted condition but\nadmitting no strengthened positive cover, and an infinite support $V$\nadmitting such a cover but failing the weighted condition for every\nfinite prime set at every integer base.  The two constructions in the short paper's section on comparison of\nsupport classes supply these sets.  Put $U=E\\cup V$.  A strengthened cover of $U$ restricts to\none of $E$, since each divisor count decreases on a subset. Convergence\nof the weighted sum over $U$ would imply convergence over $V$ with\nthe same prime set, since every summand is nonnegative.  Hence $U$ belongs to neither individual class.\nFor every infinite $A\\subseteq U$, Theorem~\\ref{thm:257-mixed-supports}\napplies with these same sets $E,V$ and gives irrationality at every\ninteger base. Irrationality of the two summands separately would not\nsuffice. This proves the short paper's strictness corollary from the two\nconstructions, with the ordinary-proof qualification stated below.\n% END record257:strict-mixed-proof\n\nThe positive-weight variant of\nTheorem~\\ref{thm:257-variable-fractional-cover}, with its hereditary\nall-base conclusion, also has a\n\\href{https://github.com/wcook04/plectis-erdos/blob/c91562bd574a387cde904481e609c7b4cacebb14/lean/ErdosProblems/Erdos257/PaperCompleteR8/ArbitraryWeightMixedClaim.lean#L19}{kernel-checked Lean proof}.\nTheorem~\\ref{thm:257-mixed-supports} alone does not exhibit a support\noutside both individual classes; the preceding strictness argument also\nuses the two separating constructions. Existence of two sets with those\nproperties has a\n\\href{https://github.com/wcook04/plectis-erdos/blob/3973d3b10bce72017b8f60093f0d6c3f4f592d80/lean/ErdosProblems/Erdos257/PaperCompleteR8/AnalyticIncomparability.lean#L53}{Lean proof}\nwhose second set is formed from squarefree divisor sets with a different\nchoice of parameters.  The printed fresh-prime construction and the strictness corollary\nhave ordinary proofs, and none of these constructions has been independently\nreviewed.  The second set in the Lean proof also admits no cover with\n$\\sum_jC_j2^{j\\alpha_j}2^{\\alpha_j}/(2^{\\alpha_j}-1)^2<\\infty$, so this\nstronger cost defines a\n\\href{https://github.com/wcook04/plectis-erdos/blob/3973d3b10bce72017b8f60093f0d6c3f4f592d80/lean/ErdosProblems/Erdos257/PaperCompleteR8/AnalyticIncomparability.lean#L41}{strictly smaller support class}.\nThe mixed implication itself is kernel-checked.\nUniversal irrationality and the quantitative and prime-power thinning claims\nremain open.\n\nSupports satisfying the mixed criterion are closed under finite unions\nand finite changes, even with the fixed dyadic cover cost in\n\\eqref{eq:257-strengthened-cover}. For two covers, interleave their $j$th\nfinite sets at positions $2j-1,2j$ and halve each exponent. Halving\ncompensates for the increased index in $2^{j\\alpha_j}$.\nThe same coefficients still majorise the smaller powers, since integer\ndivisor counts satisfy $f^{\\alpha/2}\\le f^\\alpha$.\nThe denominator $2^\\alpha-1$ is the only remaining cost: replacing\n$\\alpha$ by $\\alpha/2$ increases the full cost of each finite set by at\nmost $1+\\sqrt2$.  For $r$ covers, positions at most $rj$ and\nexponents divided by $r$ give a cost ratio below $2r$.  For weighted\nsupports, take the union of the finite prime sets: the prime part can\nonly increase, while $h/(2^h-1)$ decreases. Each condition is inherited\nby subsets, and the weighted sum is finite on a finite support.\n\\href{https://github.com/wcook04/plectis-erdos/blob/3973d3b10bce72017b8f60093f0d6c3f4f592d80/lean/ErdosProblems/Erdos257/PaperCompleteR8/StrengthenedCoverFiniteUnion.lean#L167}{Lean proves}\nthese closures with the constant $3$ in place of $1+\\sqrt2$.\nFor a countable union the cover costs must still be summable.  Every prime singleton is admitted,\nbut for the full prime support $\\mathcal P$ one has\n$\\Delta_{2,\\mathcal P}(N)>1/3$ for every $N\\ge1$: indeed\n$\\sum_{r\\ge1}2^{-r}\\omega(N+r)\\ge1$, whereas\n$X_{\\mathcal P}(2)\\le\\sum_{a\\ge2}(2^a-1)^{-1}<2/3$.\nHere $\\omega(n)$ is the number of distinct prime divisors of $n$.\n\n\\paragraph{The elementary independence boundary.}\nFor the divisor counts $c_A$ and a fixed $a\\in A$,\nthe identity $c_A(an)-c_A(n)=1-\\mathbf1_{a\\mid n}$ holds for every\n$n\\ge1$ exactly when $a$ is coprime to every other member of $A$.\nEvery summand $\\mathbf1_{t\\mid an}-\\mathbf1_{t\\mid n}$ is nonnegative;\n$t=a$ supplies the right side, and coprimality makes all other terms zero.\nIf $g=(a,t)>1$ for some $t\\ne a$, take $n=t/g$ to obtain an additional\npositive term.  For distinct odd primes $p,q$, the covariance of\n$\\mathbf1_{2p\\mid n}$ and $\\mathbf1_{2q\\mid n}$ over a common period is\n$1/(2pq)-1/(4pq)=1/(4pq)$.  Thus dilating a prime support already destroys\nthe independence used in the prime-incidence argument.  This brief ordinary\ncalculation marks a boundary of that argument, not an irrationality criterion.\n\n\\subsection{The universal problem and the half-value question}\n\nFor an integer $n \\ge 1$ set $x_n := 1/(2^n-1)$, and for $A \\subseteq \\{1,2,3,\\dots\\}$ write\n$x_A := \\sum_{n \\in A} x_n$; the sum always converges, since $x_n = O(2^{-n})$. Erd\\H{o}s's\nProblem \\#257 asks a single universally quantified question about this family.\n\n\\begin{defn}[Universal \\#257, written (U)]\n\\label{defn:U}\nIs $x_A = \\sum_{n \\in A} 1/(2^n-1)$ irrational for \\emph{every} infinite $A \\subseteq \\mathbb N_{>0}$?\n\\textbf{OPEN.}\n\\end{defn}\n\n\\begin{defn}[The half-value question, written (H)]\n\\label{defn:H}\nLet the \\emph{Mersenne achievement set} be\n$\\mathcal{A} := \\{\\, y \\in \\R : \\exists A \\subseteq \\N,\\ 0 \\notin A,\\ y = \\sum_{n \\in A} x_n \\,\\}$\n(\\lean{mersenneAchievementSet}{GreedyAchievementSet.lean:571}). Is $1/2 \\in \\mathcal{A}$?\n\\textbf{OPEN.}\n\\end{defn}\n\nA representation of $1/2$ would disprove the universal assertion. Every $2^n-1$ is odd,\nso every \\emph{finite} subset sum of $\\{x_n\\}$ has odd reduced denominator and cannot equal $1/2$\n(\\lean{positiveMersenneSupportValue\\_coe\\_finset\\_ne\\_half}{HalfCutLocator.lean:243}). Hence a\nwitness to $1/2 \\in \\mathcal{A}$ is necessarily an infinite $A$ with $x_A = 1/2$ rational, which\nrefutes (U). Conversely, (U) implies $1/2 \\notin \\mathcal{A}$ as one instance among\nuncountably many, and a proof of $1/2 \\notin \\mathcal{A}$ leaves (U) entirely open. Thus (H) asks for one particular possible counterexample to (U).\nSection~\\ref{bar:asym} explains the corresponding difference between a\nfinite certificate of nonmembership and an unbounded survival condition.\n\nThe reciprocal-summable and finite-prime weighted theorems, together with the\nchecked support theorems below, leave (U) and (H) open. The later countermodels\nidentify the particular arithmetic information lost by individual approaches;\nthey do not prove that all such failures have one common cause.\n\n\\subsection{Known support theorems}\n\\papersectiontarget{record257:known-supports}\n\nFull-support irrationality is Erd\\H{o}s's theorem~\\cite{erdos1948}.  The\npairwise-coprime case is Erd\\H{o}s's theorem of 1968, where he also stated that\nthe coprimality condition can be removed~\\cite[p.~222]{erdos1968}.  Luca and\nTachiya proved irrationality for every purely periodic integer weight that is\nnot identically zero~\\cite[Theorem~A, p.~139]{lucatachiya2017}; the purely\nperiodic, eventually periodic, residue-class, odd-support and periodic-weight\nrows below follow from their theorem, and odd support is their\nExample~2~\\cite[p.~140]{lucatachiya2017}.  The lcm-gap supports, including factorials and powers of two, have\nsummable reciprocals, as the calculation following the table shows.\nThey are consequently included in the reciprocal-summable theorem\nstated by Erd\\H{o}s. The final column locates the corresponding\ndeclarations.\n\nThe two kinds of argument in the table use different data: divisibility\nof blocks of $c_A(n)$, or rational approximations obtained from a growing\nlcm gap. The hypotheses are stated separately for each row. The notation\n$\\operatorname{sc}_A(n)$ in later source excerpts means the same\nfunction as $c_A(n)$.  All bases in the table are integers $b\\ge2$.\nFor the lcm-gap row write\n$L_k=\\operatorname{lcm}(a_0,\\ldots,a_{k-1})$, with $L_0=1$.\n\n\\begin{center}\n\\begin{longtable}{@{}p{0.30\\textwidth}p{0.32\\textwidth}p{0.30\\textwidth}@{}}\n\\hline\n\\textbf{Support class} & \\textbf{Exact hypotheses} & \\textbf{Formal proof}  \\\\\n\\hline\n\\endhead\nFull support $A = \\Npos$ & $b \\ge 2$ & \\lean{irrational\\_erdosSum\\_full\\_support}{CertificateKernel.lean:8328} \\\\\nMultiples $A = d\\Npos$ & $b \\ge 2$, $d \\ge 1$ & \\lean{irrational\\_erdosSupportSeries\\_multiples}{CertificateKernel.lean:9103} \\\\\nPurely periodic & $m\\ge1$, $m$-periodic, contains a positive element & \\lean{irrational\\_erdosSupportSeries\\_periodic}{CertificateKernel.lean:11590} \\\\\nEventually periodic & $m\\ge1$, $m$-periodic from $N_0$, $A$ infinite & \\lean{irrational\\_erdosSupportSeries\\_eventuallyPeriodic}{CertificateKernel.lean:11604} \\\\\nResidue class & any $m \\ge 1$, any $c$ & \\lean{irrational\\_erdosSupportSeries\\_residueClass}{CertificateKernel.lean:11672} \\\\\nOdd support (density $1/2$) & $b\\ge2$ & \\lean{irrational\\_erdosSupportSeries\\_odd}{CertificateKernel.lean:11686} \\\\\nGrowing lcm gap & $a_k$ strictly increasing and positive; $a_k-L_k\\to\\infty$ & \\lean{irrational\\_erdosSum\\_of\\_lcm\\_gap}{CertificateKernel.lean:5883} \\\\\nFactorials; powers of two & instances of the lcm-gap theorem & \\lean{irrational\\_erdosSum\\_factorial\\_support}{CertificateKernel.lean:6035}, \\lean{irrational\\_erdosSum\\_two\\_pow\\_support}{CertificateKernel.lean:6059} \\\\\nPairwise coprime (Erd\\H{o}s 1968) & $A$ infinite, pairwise coprime, \\emph{and} $\\sum_{a \\in A} 1/a < \\infty$ & \\lean{irrational\\_erdosSupportSeries\\_pairwise\\_coprime}{CertificateKernel.lean:10776} \\\\\nSigned periodic weights & periodic integer weights; their divisor-transformed coefficients are nonnegative and frequently nonzero & \\lean{irrational\\_intWeightedErdosSeries\\_periodic\\_of\\_coeff\\_nonneg\\_of\\_frequently\\_ne\\_zero}{CertificateKernel.lean:14583} \\\\\n\\hline\n\\end{longtable}\n\\end{center}\n\nThe lcm-gap hypothesis does imply reciprocal summability.  Eventually\n$a_{k+2}>\\operatorname{lcm}(a_k,a_{k+1})$.  If\n$a_{k+1}\\ge2a_k$, monotonicity gives $a_{k+2}>2a_k$.  Otherwise\n$a_k\\nmid a_{k+1}$, so\n$\\operatorname{lcm}(a_k,a_{k+1})\\ge2a_{k+1}>2a_k$.\nThus each of the even- and odd-indexed subsequences eventually grows\ngeometrically, proving $\\sum_k1/a_k<\\infty$.  This explains the\ninclusion claimed above, rather than using ``lacunary'' as a substitute\nfor a growth hypothesis.  For eventually periodic weights, the\nperiodic tail must not be identically zero: an eventually zero weight\nsequence gives only a finite rational sum.\n\nFor $b=2$, the first row is the irrationality of the Erd\\H{o}s--Borwein\nconstant $E=\\sum_{n\\ge1}(2^n-1)^{-1}$. The all-base scope belongs to\nErd\\H{o}s's 1948 theorem; this record contributes its formalisation.\nThe other rows retain their displayed restrictions and leave arbitrary\ninfinite support open. A block-certificate argument may additionally\nrequire independence or a tail estimate for its particular support. Barrier~\\ref{bar:box}\ndiscusses those construction requirements; the averaging proof above does\nnot assume pairwise coprimality or independence of divisor events.\n\nA separate conditional result allows common factors confined to divisors\nof one fixed integer $Q$, while the remaining factors must be pairwise\ncoprime and have summable reciprocals\n(\\lean{irrational\\_erdosSupportSeries\\_of\\_orthogonalPetalBouquet}{SupportSunflowerDichotomy.lean:540}).\nIt also assumes that for every $K\\ge1$ there is an $N\\ge0$ such that\n\\[\n 2^K\\mid\\sum_{r=1}^K c_A(N+r)2^{K-r},\\qquad\n \\sum_{r\\ge1}c_A(N+K+r)2^{-r}\\le16.\n\\]\nThus relaxing coprimality does not remove the simultaneous divisibility\nand tail-bound requirement.  This additional hypothesis is not proved\nhere, and the source file does not establish a dichotomy for arbitrary\nsupports.\n\nThe record also gives a logarithmic bound on runs of zero divisor counts\nunder rationality: for each $\\varepsilon>0$, a suitable constant $B$ bounds\nthe length by $\\varepsilon\\log_2N+B$ beyond the stated threshold\n(\\lean{supportCoeffZeroWindow\\_length\\_le\\_eps\\_logb}{SublogDivisorCoverage.lean:435}).\nFor these supports there is already a simpler unconditional bound.  If\n$a_0=\\min A$, every $a_0$ consecutive positive integers contain a multiple\nof $a_0$, where $c_A$ is positive.  Hence every zero run has length at most\n$a_0-1$, whether or not $X_A(2)$ is rational.  In particular the logarithmic\nbound alone does not distinguish rational-valued supports.  The separate\nunbounded-tail result\n\\lean{shifted\\_state\\_unbounded\\_of\\_infinite\\_support}{RationalSupportCarrySkeleton.lean:2327}\nalso requires interpretation: an unbounded sequence can still satisfy\n$u(N)=o(2^N)$.\n\n\\begin{rem}[What is and is not covered in the universal direction]\nThe Lean sources contain no theorem for $A=$ the primes, but this is a\nformalisation boundary, not a mathematical open case: Tao and\nTer\\\"av\\\"ainen prove the base-$2$ prime-support value irrational by first\nidentifying it with $\\sum_{n\\ge1}\\omega(n)2^{-n}$.  Duverney and Tachiya\nlikewise settle squarefree support at every power-of-two base, and the\nperfect $i$th powers at every integer base (Corollary~1.2 and Example~1.1,\np.~4 of the author preprint).  Neither\nresult is formalised here.  The statement of Problem~\\#257 has been formalised\nbefore, as a conjecture with an unfilled proof, in the \\emph{Formal\nConjectures}\n\\href{https://github.com/google-deepmind/formal-conjectures/blob/f776d2f2039351b00737ffcafb9d7d7666e1d9af/FormalConjectures/ErdosProblems/257.lean}{collection};\nthat file also proves the Lambert identity for the full-support value and\nstates its irrationality, the 1948 theorem of Erd\\H{o}s, with proof\n\\texttt{sorry}.  It is statement-level prior art, and no theorem recorded here\nis derived from it.  This record has no general theorem for an\narbitrary positive-density nonperiodic or divisor-dense support, and the\nuniversal statement remains open.\n\\end{rem}\n\n\\paragraph{Periodic weights: the two hypothesis sets.}\nThe periodic sequence theorem belongs to Luca and Tachiya\n\\cite{lucatachiya2014periodic}; their later author-written account states\nit as Theorem~A~\\cite[p.~139]{lucatachiya2017}.  In the displayed Lean\nrow above, nonnegativity and frequent nonvanishing are assumptions on\n$c_w(n)=\\sum_{d\\mid n}w(d)$, not on $w(n)$ itself.  The formal row must\ntherefore be read as a separately specified instance, not as an exact\ntranscription of every clause of the published theorem.\n\n\\paragraph{A separate denominator-growth criterion.}\nErd\\H{o}s's growth theorem~\\cite{erdos1975}, as restated and recovered\nby Barreto, Kang, Kim, Kova\\v{c} and Zhang\n\\cite[Remark~4(4), p.~5]{bkkkz2026}, gives irrationality of\n$\\sum_n1/a_n$ for an increasing integer sequence with\n$a_n\\ge n^{1+\\tau}$ for some $\\tau>0$ and\n$\\limsup_n a_n^{1/2^n}=\\infty$.  For $a_n=b^{c_n}-1$, where\n$c_n$ are increasing positive exponents and $b\\ge2$ is an integer,\nthe growth hypothesis becomes\n\\[\n \\limsup_{n\\to\\infty}\\frac{c_n}{2^n}=\\infty.\n\\]\nIndeed $c_n\\ge n$, so the polynomial lower bound holds eventually,\nand $\\log_b(b^{c_n}-1)=c_n+O(1)$.  An eventual lower bound suffices:\ndeleting finitely many terms subtracts a rational number and changes\n$2^n$ only by a fixed factor after reindexing.\n\nThe two hypotheses can be separated explicitly. The squares $c_n=n^2$ have\nconvergent reciprocal sum but $c_n/2^n\\to0$.  In the other direction,\nset $n_1=1$ and $n_{k+1}=n_k+4^{n_k}$, and let block $k$ consist of\nall integers from $4^{n_k}$ to $2\\cdot4^{n_k}-1$.  The blocks are\nsuccessive disjoint increasing blocks, and their union has increasing\nenumeration satisfying $c_{n_k}=4^{n_k}$.  Thus\n$c_{n_k}/2^{n_k}=2^{n_k}\\to\\infty$, whereas each block has reciprocal\nsum at least $1/2$.  Its reciprocal sum therefore diverges.\nThese examples compare denominator growth with reciprocal summability\nonly; they leave the other support criteria out of the comparison.\n\n\\subsection{Geometry and rational-target membership}\n\\papersectiontarget{record257:geometry}\nAt the first differing digit, $w_n>R_n$ prevents all later digits from\ncancelling the difference. This proves uniqueness of the support coding.\nUniform convergence on the product of the two-point digit spaces gives\ncompactness. These facts have distinct uses below: uniqueness recovers\na represented target by the greedy rule, while closedness turns a\nconvergent sequence of finite subseries into a represented value.\n\n\n\\begin{thm}[Achievement-set geometry]\n\\label{thm:geometry}\n$\\mathcal{A}$ is compact, closed, perfect, totally disconnected and nowhere dense, and\n$\\operatorname{volume}(\\mathcal{A}) = 1$.  Thus its measure is positive\nalthough it contains no interval.  Its convex hull is $[0,E]$, where\n$E=\\sum_{n\\ge1}w_n$.  The positive-index digit coding onto\n$\\mathcal{A}$ is injective: each achievable real has \\emph{exactly one} support.\n\\end{thm}\n\n\\begin{thm}[Support-restricted refinement]\n\\label{thm:supported-dichotomy}\nUse zero-based indices in this statement: coordinate $j\\in\\N$ carries\nweight $w_{j+1}$.  For $J\\subseteq\\N$, consider the sums that use only\ncoordinates in $J$.  If $\\N\\smallsetminus J$ is finite, this achievement set\nhas measure $2^{-|\\N\\smallsetminus J|}$; if infinitely many coordinates are\nomitted, its measure is zero.  Injectivity survives every restriction;\nperfectness is proved when $J$ is infinite.  No perfectness claim is made for\nfinite $J$, whose coding range is finite.\n\\end{thm}\n\n\\paragraph{Hausdorff dimension of a restricted set.}\nFor example, retaining the positive exponents $2,4,6,\\ldots$ gives\na set of measure zero but Hausdorff dimension $1/2$.  More generally,\nlet $J\\subseteq\\{1,2,\\ldots\\}$ and put\n\\[\n S_J=\\left\\{\\sum_{n\\in J}\\varepsilon_nw_n:\n                  \\varepsilon_n\\in\\{0,1\\}\\right\\},\\qquad\n s_N=\\#(J\\cap\\{1,\\ldots,N\\}),\\qquad\n d=\\liminf_{N\\to\\infty}s_N/N.\n\\]\nThen $\\dim_H S_J=d$.  Here is an ordinary proof, separate from the\nformal statements above.  If $J$ is finite, both sides are zero,\nso suppose $J$ is infinite.\n\nWrite $w_n=2^{-n}+\\delta_n$, where\n$\\delta_n=\\sum_{r\\ge2}2^{-rn}$.  A geometric sum gives\n\\[\n \\sum_{k>n}\\delta_k\n =\\sum_{r\\ge2}\\frac{2^{-rn}}{2^r-1}\\le\\delta_n/3<\\delta_n.\n\\]\nThe unique Mersenne digit coding therefore defines a map from $S_J$\nonto the binary digit-restricted set\n$B_J=\\{\\sum_{n\\in J}\\varepsilon_n2^{-n}\\}$ by keeping the same digits.\nThis map is $1$-Lipschitz.  In fact, at the first differing digit $n$,\nthe binary difference is nonnegative after orienting the larger digit\nfirst, and the Mersenne difference exceeds it by at least\n$\\delta_n-\\sum_{k>n}\\delta_k>0$.\n\nThe $2^{s_N}$ Mersenne prefix intervals have length at most\n$R_N\\le2^{1-N}$.  For any $t>d$, a subsequence with\n$s_N/N<(d+t)/2$ makes their total $t$-power length tend to zero.\nHence $\\dim_H S_J\\le d$.\nFor the reverse inequality, give the allowed binary digits independent\nprobabilities $1/2$.  Each dyadic interval of length $2^{-N}$ has\nmeasure at most $2^{-s_N}$; endpoints have measure zero because $J$\nis infinite.  An interval $I$ with\n$2^{-N-1}<|I|\\le2^{-N}$ meets at most three such dyadic intervals.\nFor $0<s<d$ and all sufficiently large $N$, its measure is therefore\nat most $3\\cdot2^{-s_N}\\le3\\cdot2^s|I|^s$.\nSumming this inequality over any sufficiently fine cover of $B_J$\ngives a positive lower bound for the sum of its $s$-power lengths.\nThus $\\dim_H B_J\\ge s$; letting $s\\uparrow d$ proves\n$\\dim_H B_J\\ge d$.  A Lipschitz map cannot increase Hausdorff\ndimension, since it sends a cover to one of no larger diameters.\nThe map above now yields $\\dim_H S_J\\ge d$, as required.\nFor $d=0$ the lower bound is automatic.\n\n\\begin{rem}[Achievement-set terminology and sources]\nWe use the modern term \\emph{achievement set} for the subsum set that\n\\href{https://doi.org/10.11429/ptmps1907.7.14_250}{Kakeya's 1914 note} describes\nas a set of partial sums.  For an infinite summable sequence of positive\nterms, Kakeya proves perfectness, and nowhere density under an all-index strict-tail\ncondition; \\href{https://arxiv.org/abs/2406.17593v4}{Kova\\v{c}--Tao,\nRemark~4.1} verifies that strict-tail condition for the fixed-base Mersenne\nweights.  Nitecki proves that a subsum set whose terms all exceed their tails\nis a Cantor set of measure $\\lim_n2^nR_n$, and credits this to\nHornich~\\cite[Theorem~4(1), p.~9]{nitecki2013}; this is the classical source for\nthe topology and measure used here.  The\nvolume dichotomy, injectivity, and greedy-membership statements above and\nbelow are proved in Lean.  The sparse power-series criteria\nof \\href{https://arxiv.org/abs/2601.20743}{Kaneko--Suzuki--Tachiya}\n(Theorems~1--3, pp.~3--5 of arXiv v1) assume sparse coefficient supports;\ntheir relation to the divisor-incidence coefficients of these subseries is\nrecorded after Theorem~\\ref{thm:257-weighted}.\n\\end{rem}\n\n\\begin{thm}[Membership equals greedy survival; the fatal-gap dichotomy]\n\\label{thm:greedy-survival}\nFor a real target $x\\ge0$, let $r_n(x)$ be the remainder after the\ngreedy rule has processed weights $w_1,\\ldots,w_n$, and let\n$R_n=\\sum_{j>n}w_j$, with $r_0(x)=x$ and $R_0=E$.  Then\n\\[\n x\\in\\mathcal A\\quad\\Longleftrightarrow\\quad\n x\\ge0\\ \\text{ and }\\ r_n(x)\\le R_n\\ \\text{for every }n\\ge0.\n\\]\nFor every $x\\in[0,E]$, nonmembership is equivalent to a finite\nstrict gap between the two next-prefix intervals; see\nObservation~\\ref{obs:general-target-gap}.\n\\end{thm}\n\nTwo boundary degeneracies are excluded at every rank, unconditionally: $r_{k-1}(1/2)\\ne w_k$ by the\nodd-denominator argument above, and $r_{k-1}(1/2)\\ne R_k$ because the tail differs from $E$ by a\nrational while $E$ is irrational. Both greedy decisions are therefore strict at every rank.\n\nThe following finite checks use exact integer arithmetic.  Each has a\n\\href{https://github.com/wcook04/plectis-erdos/tree/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257}{published program and saved result},\nincluding the scan to row $200{,}000$ and the grouped computation to\ndepth $13{,}548{,}057$.  Decimal margins summarise those integer data and\ngive no asymptotic estimate.\nReset margins are measured by\n$\\log_2|\\mathrm{rem}(r+1)-2^{r+1}|-(r+5)/2$.\nThe separate real-greedy scan uses\n$\\log_2(4^k(r_{k-1}(1/2)-w_k))$ at a take and\n$\\log_2(4^k(R_k-r_{k-1}(1/2)))$ at a safe skip.\nThese are different normalisations, not one common margin statistic.\nTruncation rungs $J = 3,\\dots,22$ all \\emph{proved} to survive (not merely computed). Seam orbit\ncertified to row $200{,}000$: branch counts $\\mathrm{R{:}M{:}U} = 100197{:}49899{:}49898$, the\ntarget inequality failing at exactly one reset row $6\\le r\\le200{,}000$, namely $r = 7$; maximal pure-R run $19$, following\nthe reset at row $158{,}096$.\n\nIndependently, rows $6 \\le n \\le 2500$: $1209$ resets, zero classification\nanomalies, minimum in-scope margin $+1.1119$ bits at row $14$ growing to $\\approx 1246$ bits at row\n$2499$ (mean $627$), never re-approaching the threshold.\n\nSharp tail margin, ranks $2$ to $3000$: no\nfatal skip, take fraction $0.4992$, tightest take-margin $+0.0340$ bits at rank $7$ (the near-tie\n$1/126>1/127$).  In the stated units this is\n$\\log_2(4^7(1/126-1/127))=0.034035\\ldots$.\nIt is the smallest take-margin reported in this finite scan. The finite greedy prefix through $m=200{,}000$, certified rank by rank\nwith integer interval bounds\n(\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/greedy-prefix-200000.json}{saved run}),\nsatisfies\n$0\\le 1/2-\\sum_{d\\in G\\cap[1,m]}w_d<2^{-199999}$.\nThe sum here is finite; this statement does not evaluate the infinite greedy sum.\n\nA further exact finite computation, not used in the arguments here,\nregenerates the forced half-prefix support to depth $13{,}548{,}057$\n(\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/half_prefix_coefficient_macro_verify.cpp}{program},\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/coefficient-macro-13548057.json}{saved run}).\nThe reset rows $13$--$30$ are also\nverified in Lean by direct evaluation\n(\\lean{seamUpperResetDyadicBandEscape\\_through\\_thirty}{HalfCylinderUpperResetBandCertificates.lean:78}).\n\n\\begin{rem}[Literature, exactly]\n$E$ is irrational~\\cite{erdos1948}. Zudilin proves the irrationality measure bound\n$\\mu(E)\\le2.46497868\\ldots$~\\cite[Theorem~1, p.~154]{zudilin2004}, correcting an earlier\ncomputation (see the remark at the end of his Section~3, p.~159). Transcendence of $E$ is\nunknown~\\cite[p.~2]{duverneytachiya}, and the Erd\\H{o}s--Graham conjecture that every such subset\nsum is irrational, i.e.\\ (U), is itself open. For the binary digits of $E$, Erd\\H{o}s's proof\nproduces arbitrarily long runs of zeros. Crandall asked whether the block \\texttt{11} occurs\ninfinitely often in the binary expansion of $E$, and Campbell proves that it\ndoes~\\cite[Theorem~1, p.~12]{campbell2026}.\n\\end{rem}\n\nThe digit results concern particular patterns. An irrationality-measure\nbound controls lengths: a run of $L$ equal digits after position $N$\ngives a dyadic approximation to $E$ with denominator at most $2^N$, hence\n$L<(\\mu(E)-1+\\varepsilon)N$ for all sufficiently large $N$.\nThe reset argument needs a run bound for its inhomogeneous recurrence,\nnot for the binary digits of $E$; no implication from the cited digit\nresults to that bound is proved here.\nSuch a bound is an input to this particular approach to (H), not a\nnecessary ingredient of every possible proof of membership.\n\n\\clearpage\n\\part*{II. Limitations and extensions of recorded methods}\n\\addcontentsline{toc}{part}{II. Limitations and extensions of recorded methods}\n\\section{Limitations of the recorded methods}\n\\label{sec:257-wall}\nA bound for the actual greedy sequence and a bound uniform over a class\nof supports are different assertions. The following examples separate\nthose assertions, starting with uniqueness at a fixed target. They\nexclude the specified estimates or their proposed uniformity, without\ngiving an obstruction to every possible method.\n\n\n\\subsection{What the obstructions do and do not show}\n\nWe begin with uniqueness at a fixed target and depth, then compare the\nquantifiers required by the different estimates. The counterexamples\nidentify a failed hypothesis or a missing uniformity, while the\nequivalences retain the full difficulty of their target statements.\n\n\\begin{obs}[Uniqueness within the fixed-target coding]\n\\label{obs:no-ensemble}\nEach Mersenne weight exceeds the sum of all subsequent weights.\nConsequently the digit coding is injective (Theorem~\\ref{thm:geometry}):\na represented target has only one support, recovered by the greedy rule.\nFor the integer weights, both the scale and the cutoff matter.  If\n$2\\le k\\le M$ and\n$q_{M,d}=\\lfloor2^M/(2^d-1)\\rfloor$ for $2\\le d\\le k$, then\n\\[\n q_{M,d}-\\sum_{e=d+1}^{k}q_{M,e}\\ge2^{M-k}.\n\\]\nIndeed, with $w_d=(2^d-1)^{-1}$ and $R_k=\\sum_{e>k}w_e$,\n\\[\n w_d-\\sum_{e=d+1}^{k}w_e=w_d-R_d+R_k>R_k>2^{-k}.\n\\]\nRounding the head down loses less than one, while rounding the later\nterms down can only increase the difference.  The resulting integer\nis greater than $2^{M-k}-1$, proving the bound.  Comparing two words at\ntheir first differing rank gives the same separation for their sums.\nThe rows with $M=2s$, $k=s-1$ and $s\\ge3$ therefore have gap at least\n$2^{s+1}$; the full rows with $k=M$ have gap at least $1$.\nThe first bound must not be applied after enlarging the cutoff:\nat $M=6$, adjoining rank $3$ introduces the subset sum\n$q_{6,3}=9<16=2^{3+1}$.\n\nWhen a finite construction is proved to encode a fixed greedy target,\nits choices are constrained by that support.  Injectivity alone does\nnot identify an arbitrary finite approximation with a real greedy\nprefix.  Nor does it exclude averaging over indices, moduli, nearby\ntargets or auxiliary constructions.\n\\\\\n\\textit{Sources:} \\lean{weightedBoolSum\\_separated}{HalfCylinderIntegerGreedy.lean:447},\n\\lean{three\\_mul\\_tailWeight\\_add\\_exactLateGap\\_eq\\_three\\_mul\\_headWeight}{HalfCylinderLargestSkipGap.lean:230},\n\\lean{positiveMersenneDigitValue\\_injective}{GreedyAchievementSet.lean:1582}.\n\\end{obs}\n\nThe six limitations below require separate arguments.  Fixed-target\nuniqueness explains canonicalisation (\\ref{bar:canon}); the recorded\nidentities determine which sufficient conditions are equivalent to their\ngoal (\\ref{bar:collapse}).  Measure and category (\\ref{bar:measure}),\nobservation scales (\\ref{bar:scale}), finite versus infinite membership\n(\\ref{bar:asym}), and the hypotheses of the support certificate\n(\\ref{bar:box}) do not follow from injectivity alone.\n\n\\subsection{Uniqueness of the specified greedy words}\n\\label{bar:canon}\n\n\\begin{prop}[Uniqueness under the stated finite conditions]\n\\label{prop:canon}\nPut $G=\\mathrm{greedyMersenneSupport}(1/2)$,\n$w_n=(2^n-1)^{-1}$ and $R_d=\\sum_{n>d}w_n$.\nThe following are three separate uniqueness statements.\n\\begin{enumerate}[nosep,label=(\\roman*)]\n\\item If $D\\subseteq\\{1,\\ldots,d\\}$ and\n$X_D(2)\\le1/2\\le X_D(2)+R_d$, then $D=G\\cap\\{1,\\ldots,d\\}$.\n\\item If $c\\ge4$, $D\\subseteq\\{2,\\ldots,c-1\\}$ and\n$0<1/2-X_D(2)<w_c$, then $D=G\\cap\\{1,\\ldots,c-1\\}$.\n\\item Let positive integer weights $v_1,\\ldots,v_k$ satisfy\n$v_i\\ge g+\\sum_{j>i}v_j$ for a positive integer $g$.\nFor a nonnegative integer target $T$, let $y^*$ be the word obtained by\nvisiting these weights in order and taking each when it fits.\nAn admissible Boolean word $y$ has\n$0\\le T-\\sum_i y_iv_i<g$ if and only if $y=y^*$ and\n$0\\le T-\\sum_i y_i^*v_i<g$.\n\\end{enumerate}\nPart (iii) gives uniqueness under the small-remainder condition,\nnot existence.  Theorem~\\ref{record:257bm-i11a} gives a\none-weight counterexample to dropping that condition.\n\\end{prop}\n\\begin{proof}\nFor (i), truncate $D$ at each successive rank.  The target remains\nin the corresponding prefix interval because all discarded terms\nare bounded by the complete tail.  The two child intervals are\ndisjoint: the lower one forces a greedy skip and the upper one\nforces a greedy take.  Induction identifies every digit.\nPart (ii) implies (i) at depth $c-1$, since $w_c<R_{c-1}$.\nFor (iii), consider the first disagreement with the integer greedy\nword.  Taking a rejected weight would exceed $T$; rejecting an accepted\nweight leaves a deficit at least $g$, even after taking every later\nweight.  The converse is immediate.\n\\end{proof}\n\n\\textbf{Excludes at fixed depth.} Under the three canonicalisation\nhypotheses just stated, a proof cannot gain freedom by choosing a different\nadmissible word for the same target and depth.  This says nothing about\nfamilies satisfying different approximation conditions, or about averaging\nover depths or auxiliary data.\n\n\\textbf{What the finite tests show.} Three different calculations are\nrecorded here.  For unrestricted prefix residues, the proposed\n$2^{t/2}$ exclusion radius is reported to fail in $74\\%$ of the tested\nranks $t\\in[8,26]$.  Here the finite test uses the truncated,\nscaled fractional parts\n\\[\n b_d(t)=\\left\\lfloor 2^t\\left\\{\\frac{2^t}{2^d-1}\\right\\}\\right\\rfloor,\n \\qquad \\{y\\}=y-\\lfloor y\\rfloor.\n\\]\nAt $t=15$, the values for $d=2,3,5,6,10$ are respectively\n$21845,4681,1057,4161,1025$.  Their sum is $32769=2^{15}+1$,\nso the sum has circular distance one from $0$ modulo $2^{15}$.\nThis is a statement about these integer truncations, not the exact\nfractional-part sum.  A heuristic count of near-hits has\nsize $2^{t/6}$; that estimate is not a theorem about the greedy support.\n\nLong prescribed digit strings also occur for unrestricted finite words:\nthe recorded examples at $t=20$ and $t=22$ have respectively eighteen\nleading ones and twenty leading zeros.  A separate construction supplies\nabout $t/12$ independently adjustable bits, with the reported finite check\nextending to $t=601$.  These examples disprove the proposed static\nexclusions, not an assertion about the actual greedy word.\n\nThe integer-window search reported no counterexamples for $t\\le22$ or at\n$t=40,42,45$ over the tested cut positions.  That absence is a finite\nobservation.  Proposition~\\ref{prop:canon} explains why certain\n\\emph{specified} approximation conditions leave only the greedy word; it\ndoes not identify all three searches with one another or rule out averaging\nover other data.\n\n\\begin{rem}[A sufficient condition that is not necessary]\nThe natural sufficient condition via fractional split-mass is \\emph{not} necessary on real crossing\ncores: the example $D = \\{2,3\\}$, $c = 5$ is an explicit counterexample\n(\\lean{splitFractionMass\\_one\\_bound\\_not\\_necessary\\_fixture}{BooleanMobiusSkippedCoreCriticalCapacity.lean:183}).\nThe example refutes necessity of that sufficient condition.  It does not\nrefute using the condition on a class of cores for which it can be proved.\n\\end{rem}\n\n\\subsection{Equivalent formulations of half-membership}\n\\label{bar:collapse}\n\nThe six conditions below are equivalent to half-membership. Their common\nreason is the carry identity: a fixed nonzero difference from $1/2$\nproduces a term of exponentially growing magnitude, whereas the remaining\ncoefficient tail is only of order $\\sqrt N$. An absolute carry bound of\nthis order along an unbounded sequence therefore forces the difference to\nvanish. The proofs below distinguish a fixed support from finite approximants\nwhose supports may vary; their quantifier patterns alone would not justify\nthese equivalences. No claim is made that every conceivable sufficient\ncondition is equivalent to the goal.\n\n\\begin{lem}[Carry and residual value]\n\\label{lem:collapse-mech}\nLet $A \\subseteq \\N$ with $1 \\notin A$, put $\\delta := 1/2 - x_A$, and let\n$\\operatorname{ihc}(A,N)$ denote the integer half-carry at level $N$. Then\n\\[\n  \\operatorname{ihc}(A,N) \\;=\\; 2^{N+1}\\,\\delta \\;+\\; \\mathrm{T}(N+1),\n  \\qquad 0 \\;\\le\\; \\mathrm{T}(N+1) \\;\\le\\; 2\\sqrt{N+1} + 4 ,\n\\]\nwhere $\\mathrm T(m)=\\sum_{r\\ge1}c_A(m+r)2^{-r}$.\n\\end{lem}\n\n\\begin{prop}[A cofinal carry bound]\n\\label{prop:collapse}\nIf $|\\operatorname{ihc}(A,N)|\\le C\\sqrt{N+1}+C'$ for arbitrarily\nlarge $N$, with fixed constants $C,C'$, then $\\delta=0$.\nFor the particular greedy support $A=G$ of $1/2$, a one-sided upper bound\n$\\operatorname{ihc}(G,N)\\le C\\sqrt{N+1}+C'$ already suffices.\n\\end{prop}\n\n\\begin{proof}\nLemma~\\ref{lem:collapse-mech} gives\n\\[\n 2^{N+1}|\\delta|\n \\le |\\operatorname{ihc}(A,N)|+\\mathrm T(N+1).\n\\]\nThe right side is $O(\\sqrt{N+1})$ along the stated sequence, so division by\n$2^{N+1}$ yields $\\delta=0$.  For $A=G$, the greedy partial sums never\nexceed $1/2$, hence $\\delta\\ge0$\n(\\lean{erdosSupportSeries\\_greedyMersenneSupport\\_le}{HalfCarryReachability.lean:888}).\nSince $\\mathrm T(N+1)\\ge0$, the one-sided upper bound gives the same\nconclusion.  Conversely, $\\delta=0$ makes the carry equal to\n$\\mathrm T(N+1)$, so the bound holds with $C=2$ and $C'=4$.\nThis converse does not assert the bound for arbitrary prescribed constants.\nFor a prescribed $A$, the conclusion is $x_A=1/2$; it is not merely\nmembership witnessed by some other support.  For $A=G$, this equality is\nequivalent to $1/2\\in\\mathcal A$\nby \\lean{mem\\_mersenneAchievementSet\\_iff\\_greedy\\_survival}{GreedyAchievementSet.lean:1458}.\n\\end{proof}\n\n\\begin{lem}[The terminal bound at square depths]\n\\label{lem:sqrt-witness}\nSuppose $x_A = 1/2$ and $1 \\notin A$. Then for every $k \\ge 1$,\n\\[\n  \\operatorname{ihc}(A, k^2 - 1) \\;=\\; \\mathrm{T}(k^2) \\;\\le\\; 2k + 4 \\;=\\; \\ensuremath{B}(k^2).\n\\]\n\\end{lem}\n\n\\begin{proof}\nBy Lemma~\\ref{lem:collapse-mech} with $\\delta = 0$, $\\operatorname{ihc}(A,k^2-1) = \\mathrm{T}(k^2)\n\\le 2\\sqrt{k^2} + 4$. At a perfect square the real and truncated square roots agree exactly,\n$\\sqrt{k^2} = k = \\lfloor \\sqrt{k^2} \\rfloor$, so the real envelope coincides with the discrete\nstrip $\\ensuremath{B}(n) = 2\\lfloor\\sqrt n\\rfloor + 4$\n(\\lean{halfStripBound}{HalfCarryReachability.lean:32}) with no slack required.\n\\end{proof}\n\nAt the depths $M=k^2$, the real square root and the integer square root\nagree.  Thus the reverse implication for the terminal bound uses the\noriginal constant $4$, not a relaxed constant $6$.  Truncating an achieving\nsupport at $M=k^2$ leaves the carry at that depth unchanged, by\n\\lean{integerHalfCarry\\_inter\\_Iic\\_eq\\_of\\_succ\\_le}{HalfCarryReachability.lean:49}.\nAn achieving half-support excludes rank one because $w_1=1>1/2$.\n\n\\begin{prop}[Six equivalent membership conditions]\n\\label{prop:collapsed-list}\nEach of the following is logically \\emph{equivalent} to $1/2 \\in \\mathcal{A}$, not strictly weaker:\n\\begin{enumerate}[nosep,label=(\\alph*)]\n\\item $C_G(N)\\le2\\sqrt N+4$ for every $N\\ge0$;\n\\item there are arbitrarily large positive exponents outside $G$;\n\\item at arbitrarily large depths $n$, some $D\\subseteq\\{2,\\ldots,n\\}$\n  satisfies $Q(D,n)=2^{n-1}-1$;\n  no agreement between the sets $D$ at different depths is required;\n\\item at arbitrarily large $M$,\n  $\\operatorname{ihc}(G,M)\\le B(M+1)$;\n\\item at arbitrarily large positive depths $M$, some\n  $D\\subseteq\\{2,\\ldots,M\\}$ satisfies\n  $|\\operatorname{ihc}(D,M-1)|\\le B(M)$;\n\\item the half-target greedy remainder never exceeds the full remaining\n  tail, so no finite fatal gap exists; the seven equivalent\n  forms in the later classification express this same condition.\n\\end{enumerate}\nFor (d) and (e), the reverse directions use the square depths in\nLemma~\\ref{lem:sqrt-witness}.\nFor (a), membership gives\n$C_G(N)\\le2\\sqrt{N+1}+3\\le2\\sqrt N+4$ when $N\\ge1$, and $C_G(0)=0$.\nThe forward implication still needs the greedy inequality $x_G\\le1/2$.\n\\end{prop}\n\n\\textbf{Scope of the equivalences.} These conditions are equivalent to membership, not strictly\nweaker substitutes. Equivalence does not make a formulation useless as an invariant, but it does\nmean that proving any of them is the same problem as (H). The equivalent conditions remain\nunproved. In particular, mutually \\emph{incompatible} witnesses at different endpoints do not buy\na weaker sufficient condition, and the $\\Pi^0_2$ shape of these hypotheses is not evidence of extra\nroom.\n\n\\textbf{Scope limit.} Item (a) does \\emph{not} extend to every $A$ with\n$1 \\notin A$. Dropping the greedy-specific fact $x_G \\le 1/2$ yields only the one-sided statement\nthat the bound in (a), with $A$ in place of $G$, is equivalent to $1/2 - x_A \\le 0$. Thus the conclusion for a general support is a one-sided inequality,\nnot equality with $1/2$.\n\n\\subsection{The size of the required error bounds}\n\\label{bar:scale}\n\nOne sufficient half-membership argument requires the reset inequality\n\\[\n |\\mathrm{rem}(r+1)-2^{r+1}|>2^{(r+5)/2}\n \\qquad(r\\ge10\\text{ an upper or middle reset}).\n\\]\nFor the normalised deviation\n$\\lambda_{r+1}=(\\mathrm{rem}(r+1)-2^{r+1})/2^{r+1}$, this is\n$|\\lambda_{r+1}|>2^{(3-r)/2}$.  A near-integer estimate for a\nweighted divisor sum is a proposed way to obtain information about\nthis inequality, not an equivalent condition established here.\nSection~\\ref{sec:invent-R1} displays the exact recurrence and the\nadditional, changing skip-set term.  Each obstruction below has its\nown hypotheses; none excludes every method at the stated scale.\n\n\\textbf{Class (a): fixed precision or bounded state.} The following\nthree statements concern abstract recurrence models.  Their\nconclusions apply to the Mersenne problem only within the specified\nmodels; they do not use its full divisor-count constraints.\n\n\\begin{prop}[Limits of the specified local summaries]\n\\label{prop:local-void}\n(i) After $L$ common steps the endpoint residue mod $2^L$ of an affine binary orbit is\n\\emph{independent} of the initial carry: $u(L) - v(L) = 2^L(u_0 - v_0)$.\n(ii) Fix $m\\ge2$ and put $h=\\lfloor(m+1)/2\\rfloor$.  Consider the\ncoefficient sequences\n\\[\n c_r(m)=h-r,\\qquad c_r(m+1)=2r,\\qquad\n c_r(j)=0\\ (j\\ne m,m+1),\\qquad 0\\le r\\le h.\n\\]\nTheir binary sums are all $h2^{-m}$, and their scaled tails before\nposition $m$ agree, but their scaled tails immediately after $m$ are\n$r$.  Thus any exact label-and-decoder system recovering those latter\ntails must have at least $h+1=\\lfloor(m+1)/2\\rfloor+1$ labels.  A state\ndetermined only by the common preceding history, with no new input,\ncannot distinguish them. (iii) For any\nstarting carry and any finite list of prescribed valuations and\nodd unit residues modulo $2^u$, with $u\\ge1$, there are integer input\ncoefficients with those data whose successor carries lie in their\nprescribed centred intervals.  The unrestricted higher bits of the\ninput coefficients may be chosen separately at each step, as in\nProposition~\\ref{prop:2adic-nogo}, so these fixed-precision symbols alone do not exclude a completion in\nthat specified family.  This says nothing about extra constraints imposed\nby divisor counts from one common support.\n\\end{prop}\n\nFor an integer recurrence $J(M+1)=2J(M)-\\delta(M+1)$,\nreduction modulo $2$ gives $J(M+1)\\equiv\\delta(M+1)\\pmod2$.\nFor the particular recurrence $K(M+1)=2K(M)-(\\tau(M+1)-1)$,\nwith $K(2)=1$, this gives, for $M\\ge2$, that $K(M)$ is even\nexactly when $M$ is a square: divisors pair off except at a square.\nThis is a genuine restriction at a specified index, but it is already\nencoded in the known forcing term.  It supplies no additional\nrun-length estimate by itself.\n\n\\textbf{Class (b): global averages and Diophantine data.}\n\n\\begin{prop}[A bound for the reduced denominator]\n\\label{prop:exponent-gap}\nLet $n\\ge2$, let $\\mathrm{Skip}_n\\subseteq\\{2,\\ldots,n-1\\}$,\nand let $D_n$ be the reduced denominator of the associated finite sum.\nSince $D_n$ divides the product of its Mersenne denominators,\n\\[\n \\log_2D_n\\le\\sum_{d\\in\\mathrm{Skip}_n}\\log_2(2^d-1)\n \\le\\sum_{d\\in\\mathrm{Skip}_n}d\\le\\frac{n(n-1)}2-1.\n\\]\nThe middle inequality is strict when the skip set is nonempty; for an\nempty skip set both sums are zero.  This is an upper bound, not a\nquadratic asymptotic for $D_n$.  Combined with the cited fixed\nirrationality-measure exponent, it does not supply the required\n$2^{-3n/2}$-scale estimate.  Even an asymptotic for the exponent sum would\nneed additional information about the skip distribution.\n\\end{prop}\n\n\\textbf{Scope of these obstructions.} The common pre-index history in\nProposition~\\ref{prop:local-void}(ii) does not determine the post-index\ntail in its countermodel family.  This does not exclude a state using\nadditional arithmetic input from the actual support.  The displayed\ndenominator bound combined with the cited fixed irrationality-measure\nexponent does not yield the required run bound. These calculations do not\nexclude averaging over shifts, residues, or observation scales, nor a sharper analysis of the\nactual skip distribution.\n\n\\textbf{Prime steps and bounded windows.} In the universal recurrence,\na prime step subtracts $1$, so it sends $K(M)$ to $2K(M)-1$; it is\nnot exact doubling.  Observation~\\ref{obs:prime-doubling} gives an\nexplicit frozen-row drift with values between $2$ and $6$ at the eleven\nindices $607,\\ldots,617$, including the three prime indices $607$, $613$,\nand $617$.  This finite example shows that prime steps can occur within\na small-value window.  It neither proves statistical independence nor\nrules out every argument using prime-counting information.\n\n\\begin{rem}[Status of the exhaustion claim]\nThe preceding counterexamples concern the specified finite summaries and\nestimates.  They do not classify all available tools, even within the\nbroader categories of local information and global averages.  No exhaustion\nclaim is used in a subsequent deduction.\n\\end{rem}\n\n\\subsection{What finite certificates decide}\n\\label{bar:asym}\n\n\\begin{thm}[One-sidedness]\n\\label{thm:one-sided}\nNon-membership of $1/2$ in $\\mathcal A$ has an effectively checkable\nfinite-certificate formulation, hence a $\\Sigma^0_1$ formulation.\nMembership has the complementary $\\Pi^0_1$ formulation. Survival through a tested finite\ndepth alone does not establish membership; a uniform theorem or inductive invariant could.\nThe arithmetical-hierarchy form by itself proves neither undecidability nor the absence of finite\nproofs.\n\\end{thm}\n\n\\begin{proof}\nThe dichotomy $1/2 \\in \\mathcal{A}$ or $\\exists$ a fatal gap is unconditional\n(\\lean{half\\_mem\\_mersenneAchievementSet\\_or\\_exists\\_fatal\\_gap}{HalfCutLocator.lean:623}), and the\ntwo alternatives are exactly complementary\n(\\lean{existsFatalHalfGap\\_iff\\_half\\_not\\_mem\\_mersenneAchievementSet}{HalfCutLocator.lean:643}),\nwith \\lean{ExistsFatalHalfGap}{HalfCutLocator.lean:526} a finite witness type. In the other\ndirection, certified death implies non-membership\n(\\lean{certifiedGreedyMersenneDeath\\_not\\_mem}{GreedyAchievementSet.lean:1762}, over\n\\lean{CertifiedGreedyMersenneDeath}{GreedyAchievementSet.lean:1756}; $3/4$ is certified dead at\nlevel $1$, lookahead $0$, by\n\\lean{three\\_fourths\\_certifiedGreedyMersenneDeath}{GreedyAchievementSet.lean:1778}), whereas the\nabsence of a certificate through depth $D$ alone gives no conclusion\nabout later depths.  To make the search effective, note that a greedy\nremainder at a finite rank is rational.  The tail $R_n$ has rational\nupper bounds obtained by summing through a cutoff $M$ and bounding the\nremaining sum by $2^{-M}+(2/3)4^{-M}$.  These upper bounds converge to\n$R_n$.  Thus the strict inequality $r_n(1/2)>R_n$, when true, is witnessed\nby some finite cutoff and a comparison of rational numbers.  Enumerating\nranks and cutoffs gives the asserted finite-certificate search.\nMembership requires the survival inequality at every rank\n(Theorem~\\ref{thm:greedy-survival}).  Checking a finite initial segment\nalone does not supply the later inequalities; a separate uniform argument\nor induction would be needed.\n\\end{proof}\n\n\\textbf{Finite search versus infinite survival.}\nA discovered fatal-gap certificate proves non-membership.  Further\nsurviving prefixes give larger verified ranges, but do not establish\nmembership without an argument covering all ranks.  This distinction\nis unchanged whether one increases the geometric truncation depth,\nthe number of quotient rows, or the depth of a grouped computation.\nThe different finite parameters should not be reported as a single\nverification depth.\n\n\\subsection{Why measure and category do not decide a rational target}\n\\label{bar:measure}\n\n\\begin{rem}[Achievement-set geometry does not decide rational-target membership]\n\\label{prop:soft}\nThe achievement set has positive measure and is nowhere dense\n(Theorem~\\ref{thm:geometry}). Neither fact alone decides membership of a specified rational.\nInjectivity fixes the support of an achievable value, but it does not exclude averaging over\nindices, shifts, residues, observation scales or auxiliary constructions. The recorded\ncalculations obstruct the particular models and estimates analysed here; no impossibility theorem\nfor all probabilistic, measure-theoretic or averaging methods is claimed.\n\\end{rem}\n\nThe geometry itself is Lean-checked in Theorem~\\ref{thm:geometry}: $\\mathcal{A}$ has Lebesgue\nmeasure exactly $1$ inside an ambient span of $E \\approx 1.6067$, a relative density of about\n$62\\%$, and is simultaneously nowhere dense, perfect and totally disconnected. Those facts may hold\ntogether. They do not yield competing theorems about specified rationals. Showing that\n$\\mathcal{A}$ is ``small'' cannot decide (U) by measure: it is $62\\%$ of its ambient interval.\n\n\\subsection{Hypotheses of the CRT block-certificate argument}\n\\label{bar:box}\n\n\\begin{obs}[Two inputs to the CRT certificate argument]\n\\label{obs:box}\nThe CRT block-certificate arguments considered here require both\ncompatible divisibility prescriptions and a tail estimate strong enough\nfor the height inequality $q(C+N+L+2)<b^L$.  Pairwise coprimality is one\nway to obtain the first input; reciprocal summability is one way to obtain\nthe second.  This observation concerns those certificate arguments only.\nIt does not describe all theorems in Section~\\ref{sec:257-problem}:\nthe reciprocal-summable and weighted averaging proofs do not require\npairwise coprimality.\n\\end{obs}\n\n\\textbf{What remains to check.} One cannot apply a conditional CRT\ncertificate theorem to arbitrary infinite $A$ without establishing its\nhypotheses.  Shared factors impose compatibility conditions on the\nprescribed residues; they do not make every CRT construction impossible.\nThe certificate proof of Theorem~\\ref{thm:pairwise-coprime} uses the tail\nbudget in one step.  Deleting that assumption\ndoes not complete the proof: one needs a replacement estimate for that step.\nThis leaves open different constructions and different averaging arguments.\n\n\\textbf{Explains only a method boundary.} The primes are infinite and pairwise coprime, so they satisfy the\nfirst input.  The identity\n\\lean{compositeDilationDefect\\_eq\\_zero\\_of\\_prime\\_support}{CompositeDilationDefect.lean:103} makes\nthe dilation algebra exact there; yet this certificate theorem does not\napply because $\\sum_p1/p$ diverges, so the stated reciprocal-sum\nhypothesis fails.  This explains\nthe absence of a \\emph{Lean theorem from this certificate argument}; it does not make the\nbase-$2$ value open, since Tao--Ter\\\"av\\\"ainen settle it by a different\nanalytic method.  The sunflower generalisation inherits both axes and\ntherefore does not recover that external theorem.\n\n\\subsection{The distinct estimates still required}\n\nThe half-target analysis involves a geometrically weighted divisor sum\n$\\sum_{i\\le L}(\\tau(M+i)-1)2^{-i}$ and possible near-integer estimates.\nOne universal-support certificate instead asks for an opening divisible\nblock together with a controlled middle window.  These inputs are related\nthrough incidence identities, but the displayed arguments do not prove\nthem equivalent for all supports.  They should therefore be retained as\nseparate obligations.  Digit-pattern occurrence results do not by\nthemselves supply either required uniform estimate.\n\n\\section{Unproved inputs for further arguments}\n\\label{sec:257-survivors}\nThe preceding counterexamples leave estimates that could still be used\nin the conditional arguments. Some would imply membership; others are\nequivalent to it. The prime-support question asks for a formal recovery\nof a known theorem. For each proposed estimate we compare its\nhypotheses with the relevant counterexample. The list is not exhaustive.\n\n\\subsection*{Two further analytic tests}\nSmall displacements at a real base do not by themselves give the fixed\nrational lattice used in the integer-base proof.  For\n$\\beta=(3+\\sqrt5)/2$, let $u=\\beta^{-1}$ and\n$H=\\{2^km:k\\ge2,\\ m\\text{ odd},\\ m\\le2^{2^k}\\}$.\nThe finite prefix identity and an algebraic norm show that the condition\n\\[\n \\Delta_{\\beta,H}(N)\n \\left(1+\\sum_{\\substack{a\\in H\\\\a\\le N}}\n                 \\sum_{j=1}^{\\lfloor N/a\\rfloor}u^{N-ja}\\right)\n \\longrightarrow0\n\\]\nalong some sequence of positive integers $N\\to\\infty$ would exclude $X_A(\\beta)$ from $\\Q(\\sqrt5)$ for every\ninfinite $A\\subseteq H$.  The second factor controls the algebraic\nconjugate of the finite prefix; smallness of the displacement alone does\nnot control this product.  The product estimate remains unproved.\n\nTo justify the conditional implication, suppose instead that\n$X_A(\\beta)\\in\\Q(\\sqrt5)$, and choose an integer $q\\ge1$ with\n$qX_A(\\beta)\\in\\Z[\\beta]$.  The finite-prefix identity gives, for\n$N\\ge1$,\n\\[\n Y_N:=q\\Delta_{\\beta,A}(N)\n =q(\\beta^N-1)X_A(\\beta)\n  -q\\sum_{\\substack{a\\in A\\\\a\\le N}}\n           \\sum_{j=1}^{\\lfloor N/a\\rfloor}\\beta^{N-ja}\n \\in\\Z[\\beta].\n\\]\nThe infinite support makes $Y_N>0$.  Since\n$\\beta^2-3\\beta+1=0$, conjugation $\\sigma$ sends $\\beta$ to\n$u=3-\\beta\\in(0,1)$.  For $m,n\\in\\Z$ the norm of $m+n\\beta$\nis $m^2+3mn+n^2$, so the nonzero $Y_N$ has nonzero integer norm.\nMoreover,\n\\[\n |\\sigma(Y_N)|\\le q\\left(|\\sigma(X_A(\\beta))|\n       +\\sum_{\\substack{a\\in H\\\\a\\le N}}\n             \\sum_{j=1}^{\\lfloor N/a\\rfloor}u^{N-ja}\\right).\n\\]\nUsing $0<\\Delta_{\\beta,A}(N)\\le\\Delta_{\\beta,H}(N)$, we obtain\n\\[\n 1\\le |Y_N\\sigma(Y_N)|\n \\le q^2\\max\\{1,|\\sigma(X_A(\\beta))|\\}\\,\n       \\Delta_{\\beta,H}(N)\n       \\left(1+\\sum_{\\substack{a\\in H\\\\a\\le N}}\n                  \\sum_{j=1}^{\\lfloor N/a\\rfloor}u^{N-ja}\\right).\n\\]\nThe asserted limit would contradict this inequality.  The multiplier\n$q$ and the conjugate of the putative value are fixed, independently\nof $N$.  This proves only the implication from the displayed product\nestimate, not that estimate.\n\nA different extension requires squarefree remote-tail control and a\nweighted displacement to be small at the same indices.  CRT establishes\ncompatibility of the finite opening divisibility conditions, but not this\nsimultaneous smallness.  Neither argument is claimed to establish its\nrequired analytic hypothesis.\n\n\n\\subsection{A lower bound at reset ranks}\n\\label{ssec:route-1}\n\n\\textbf{What it needs.} Either of the first two sufficient conditions\nbelow. Their different scales do not establish an implication between\nthem. The sign condition (iii) is a separate question; no implication\nfrom it to half-membership is proved here.\n\\begin{enumerate}[nosep,label=(\\roman*)]\n\\item \\emph{Reset-deviation bound.} For every upper or middle reset\n  row $r\\ge10$, require\n  $|\\mathrm{rem}(r+1)-2^{r+1}|>2^{(r+5)/2}$.\n  Section~\\ref{sec:o4} states the exact conditional implication to\n  membership.  A run-length estimate is a possible way to approach\n  this bound, not an established exact reformulation of it.\n\\item \\emph{Band avoidance at resets, already proved to imply membership.}\n  For every upper reset $d \\ge 13$ (branch $\\mathrm U$ of\n  Theorem~\\ref{thm:dynamics}) and every $j \\le d$, the reset charge\n  $4\\cdot\\text{overshoot} + \\text{abovePulse}$ avoids the linear-width band\n  $(2^{d-j+1} - 2(d+j),\\, 2^{d-j+1}]$.\n\\item \\emph{Sign law alone.} At every reset, M-resets have $\\mathrm{dev} > 0$ and U-resets have\n  $\\mathrm{dev} < 0$. This specifies the sign but not a quantitative distance from zero.\n  Neither an implication to half-membership nor a strict comparison with\n  the first two conditions is established here.\n\\end{enumerate}\n\\textit{Sources.} \\lean{SeamUpperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:4566}\nwith conditional implication\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_upperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:4790};\nquantifier collapse \\lean{dyadicBandEscape\\_iff\\_exists\\_critical}{HalfUpperResetCriticalBand.lean:108},\nwhich reduces the $\\forall j \\in [0,d]$ band check to one nearest-boundary index per row; finite\nverification \\lean{seamUpperResetDyadicBandEscape\\_through\\_thirty}{HalfCylinderUpperResetBandCertificates.lean:78};\nalternative implication \\lean{LargestSkipLateStepSocket}{HalfCylinderLargestSkipInduction.lean:34} with\nconditional implication\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_largestSkipLateStepSocket}{HalfCylinderLargestSkipInduction.lean:167}\nand base case\n\\lean{largestSkipLateAt\\_fourteen}{HalfCylinderLargestSkipInduction.lean:73}.\n\n\\textbf{Logical strength.} The reset bound is a sufficient condition for\nhalf-membership in the cited argument.  No converse is established here.\nThe fact that it is quantitative, while membership is qualitative, does\nnot prove that the condition is strictly stronger: that would require a\nseparate implication or counterexample.  The carry identity of\nProposition~\\ref{prop:collapse} does not itself furnish this lower bound\non the reset deviation.\n\n\\textbf{What the finite checks establish.} The reported computation\nchecks the reset inequality over its stated range, with margins of\n$1.1119$ bits at row $14$ and about $1246$ bits at row $2500$; the reported\nmean is $627$ bits.  These are finite observations, not a proof of an\nasymptotic lower bound or of the inequality at every later reset.\n\nUniqueness in Section~\\ref{bar:canon} does not prevent this argument:\nthe proposed bound concerns the actual greedy sequence.  It does not\nrequire alternative supports for the same target.\n\n\\textbf{Run lengths.} The record reports a maximum right-branch run of\nlength $19$ after the reset at row $158{,}096$, far below the proposed threshold\n$(r-3)/2=79{,}046.5$ there.  Observing larger runs at larger ranks does not\nrefute the existence of some uniform finite bound.  Neither a uniform\nbound nor the weaker asymptotic assertion $L_r=o(r)$ is proved by these\ndata.  The stated implication needs the required inequality beyond the\nverified initial ranks.\n\nThe missing step is the quantitative lower bound itself.  The finite\nobservations do not supply the growing-precision estimate discussed in\nSection~\\ref{bar:scale}; it remains unproved.\n\n\\subsection{Increasing the depths of exact finite sums}\n\\label{ssec:route-2}\n\n\\textbf{What it needs.} Rule out the recycle branch returning to a bounded endpoint forever. Any\none of:\n\\begin{enumerate}[nosep,label=(\\alph*)]\n\\item \\emph{critical quotient supply}: for every below-half core $D$ bounded by $[2,c)$\n  with genuine crossing deficit, $2^{(2c-2)-1} \\le \\ensuremath{Q}\\,(\\mathrm{insert}\\\n  c\\ D)\\,(2c-2)$;\n\\item a direct proof that the crossing rank $c$ produced by the recycle branch cannot repeat the\n  value $4$, or any bounded value, infinitely often;\n\\item a growth law for $c$ as a function of the incoming endpoint $n$.\n\\end{enumerate}\nBy Proposition~\\ref{prop:canon}(ii), form (a) need only be checked along the \\emph{one} canonical\norbit, the half-target greedy prefix, not searched over $D$; and by the recorded sharp-capacity\nbiconditional it is exactly the $c-2$ bit capacity test\n$\\ensuremath{S}\\,D\\,1\\,(2c-2) < 2^{c-2}$. The example $D=\\{2,3\\}$, $c=5$ shows that the fractional-mass\nsufficient condition is not necessary.  It does not rule out proving that\ncondition for a suitably restricted family.\n\n\\textit{Sources.} \\lean{exactLocalMersenneHalfRow\\_double\\_or\\_recycle}{BooleanMobiusExactRowDichotomy.lean:26};\nthe formal counterexample\n\\lean{exists\\_seeded\\_bounded\\_double\\_or\\_recycle\\_model}{BooleanMobiusExactRowDichotomy.lean:91}\n(the model $n \\mapsto (n = 6)$ satisfies the seed and the exact transition shape yet is not\ncofinal, so the schema plus an endpoint-six seed provably does \\emph{not} give cofinality);\nconditional strict progress\n\\lean{skippedCoreSharpCapacity\\_below\\_or\\_strictly\\_laterExactRow}{BooleanMobiusExactRowCrossing.lean:233};\nunconditional recycling\n\\lean{exists\\_skippedCoreExactRow\\_of\\_value\\_above}{BooleanMobiusExactRowCrossing.lean:155};\nhypothesis \\lean{SkippedCoreCriticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:30};\ncapacity equivalence\n\\lean{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_criticalCapacity\\_iff}{BooleanMobiusSkippedCoreCriticalCapacity.lean:22};\nseed example $\\{2,3,6\\}$ at\n\\lean{exactLocalMersenneHalfRow\\_six}{BooleanMobiusExactRowSeed.lean:34}.\n\n\\textbf{The finite-row test and its unique candidate.}\nThe real-prefix argument in Proposition~\\ref{prop:canon}(ii) does not\nby itself identify an exact-row witness.  The integer weights have a\nseparate, decisive property: for fixed $n\\ge2$,\n\\[\n q_{n,d}=\\left\\lfloor\\frac{2^n}{2^d-1}\\right\\rfloor\n \\quad(2\\le d\\le n)\n\\]\nform a decreasing sequence in which each weight exceeds the sum of all\nlater weights by at least $1$.  Apply the integer greedy rule with\ncapacity $2^{n-1}-1$: visit $d=2,\\ldots,n$ and subtract $q_{n,d}$ when\nit fits.  By Theorem~\\ref{record:257bm-i11a} and Propositions~\\ref{record:257bm-i11b}--\\ref{record:257bm-i11c},\nany exact-row witness has zero remainder and must be this greedy word.\nThus there is at most one witness at a fixed depth, and its existence is\ntested by checking whether the final greedy remainder is zero.\nAn exhaustive search over $2^{n-1}$ subsets is unnecessary.\n\nThis integer greedy word should not be silently identified with the\nreal greedy prefix $G\\cap[2,n]$: the two rules use different weights.\nThe separation theorem proves uniqueness within the integer problem,\nnot that identification.\n\n\\textbf{Finite decidability and the cofinal requirement.}\nThe test decides the exact-row predicate at any specified depth.  The\nclaim that successful depths are arbitrarily large is still equivalent\nto half-membership, by Proposition~\\ref{prop:collapsed-list}(c).\nNeither uniqueness nor finite decidability supplies that cofinal\nassertion.  The double-or-recycle argument must prove genuine progress\nin the depths, not rely on a purported freedom to choose many witnesses\nat one depth.\n\n\\subsection{The known prime-support theorem}\n\\label{ssec:route-3}\n\nThe base-$2$ prime-support value is not an open target:\n\\[\n  \\sum_{p\\ {\\rm prime}}\\frac1{2^p-1}\n  =\\sum_{n\\ge1}\\frac{\\omega(n)}{2^n}\n\\]\nby expanding each reciprocal Mersenne term geometrically and collecting\ndivisors.  Tao and Ter\\\"av\\\"ainen prove the right-hand series irrational in\nTheorem~1.3 of\n\\href{https://arxiv.org/abs/2512.01739}{\\emph{Quantitative correlations and\nsome problems on prime factors of consecutive integers}}.  Their proof is\nanalytic and is cited here; it is not formalised in Lean.\n\nThe local pairwise-coprime certificate theorem still does not cover the\nprimes because it assumes summability of $\\sum_p1/p$.  That is a boundary of\nthis certificate theorem, not a boundary of the mathematics.  A\nwindowed-tail repair of that theorem can test whether the Lean\ncertificate theorem reproduces the known theorem; reproducing a cited\nresult is not new progress on \\#257.  After Theorem~1.3 the authors state the\nprime-support extension to every integer base $b\\ge2$ and, separately,\nthe full prime-power result at base $2$, leaving those modifications to\nthe reader.  These statements should not be combined into an attributed\nall-base theorem for arbitrary prime-power subsets.\n\n\\subsection{Equivalent membership conditions}\n\\label{ssec:route-4}\n\nProposition~\\ref{prop:collapsed-list} lists the equivalent conditions:\na pointwise upper bound on $C_G$, infinitely many skipped ranks, exact\nquotient sums at unbounded depths, and the two cofinal carry bounds.\nFor the carry conditions, the residual identity and its tail estimates are\nthe relevant inputs:\n\\lean{integerHalfCarry\\_eq\\_scaled\\_residual\\_add\\_tail}{HalfCarryReachability.lean:871},\n\\lean{binaryCoeffTail\\_nonneg}{GenericTailOrbitRigidity.lean:78},\n\\lean{binaryCoeffTail\\_supportCoeff\\_le\\_two\\_sqrt\\_add\\_four}{BooleanMobiusCarry.lean:290}, and, for the terminal-strip conditions, evaluation of the square root at\na perfect square.  The one-sided scope limitation in\nProposition~\\ref{prop:collapsed-list} still applies to general supports.\n\n\\textbf{Purpose of the equivalences.} These results identify the exact\nlogical content of the proposed hypotheses.  They do not settle\nhalf-membership, nor do they make an equivalent formulation useless: a\ndifferent expression of the same condition may expose an invariant or a\nmore convenient induction.  The formalisation task is separate from the\nunproved arithmetic assertion on either side.\n\n\\subsection{Why zero windows do not split the support class}\n\\label{ssec:route-5}\n\nOne proposed split asks whether every infinite support satisfies\n\\[\n \\begin{gathered}\n \\text{$c_A$ has arbitrarily late super-logarithmic zero windows}\\\\\n \\text{or a weighted block certificate exists at every precision.}\n \\end{gathered}\n\\]\nFor the present divisor counts, the first alternative never occurs.\nWriting $a_0=\\min A$, every run of zeros has length at most $a_0-1$,\nsince each $a_0$ consecutive integers contain a multiple of $a_0$.\nThe logarithmic estimate\n\\lean{supportCoeffZeroWindow\\_length\\_le\\_eps\\_logb}{SublogDivisorCoverage.lean:435}\ntherefore does not split the support class into two useful cases.\nThe lcm-gap theorem\n\\lean{irrational\\_erdosSum\\_of\\_lcm\\_gap}{CertificateKernel.lean:5883}\nremains an independent irrationality result, not an instance of this\nzero-window alternative.  The proposed assertion consequently requires\nblock certificates for every infinite support: for each required precision,\none must construct an opening block with\n$2^K\\mid\\sum_{r=1}^Kc_A(N+r)2^{K-r}$, a bounded middle contribution $C$,\nand $q(C+N+L+2)<2^L$.\n\nThis reformulation does not remove the CRT compatibility and tail-estimate\nrequirements of Section~\\ref{bar:box}.  It asks for a new construction that\nmeets them for general supports.  Nor does it identify that construction\nwith the reset estimate used for the half-target: the two arguments use\nrelated divisor-count identities, but no equivalence of their missing\nhypotheses is proved here.\n\n\\subsection{Comparison of the remaining inputs}\n\nRoute~4 packages recorded equivalences; it does not establish their\nunproved existence hypotheses.  Route~3 must be compared with the external\nprime-support theorem of Tao and Ter\\\"av\\\"ainen\n\\cite[Theorem~1.3]{taoteravainen2025}; reproducing it in this framework\nrequires its own analytic input.  Route~2 offers a different finite\nsearch space under the conditions stated there.  Routes~1 and~5 share\nsome incidence identities, but this is not a proof that every missing\nhypothesis is equivalent.  The appropriate comparison is the input each\nroute needs, its quantifiers, and what a successful estimate would imply.\n\n\\section{Reading the detailed record}\n\\label{sec:257-howto}\nPart~III groups the finite arithmetic by object: divisor certificates,\ngreedy coding, quotient identities, carries and support restrictions.\nThe implication tables show how to pass between these descriptions and\nwhere an additional hypothesis is needed. The preceding two sections\nsupply the corresponding comparisons with failed estimates and remaining\nanalytic questions.\n\nThe reported calculations concern $J\\le22$, quotient rows at most\n$200{,}000$, ranks at most $3000$, and, in a separate calculation, depth\nat most $13{,}548{,}057$. Their respective predicates and ranges are\nretained below. Theorem~\\ref{thm:one-sided} explains why a larger finite\nrange does not establish an assertion at unbounded depths. An\nobstruction for one recurrence also needs a proved map before it can\nbe applied to another formulation.\n% ; - part a257_p0 ; -\n\\clearpage\n\\part*{III. Detailed statement and arithmetic catalogue}\n\\addcontentsline{toc}{part}{III. Detailed statement and arithmetic catalogue}\n\\section{The series, the greedy rule and finite identities}\nExpanding a Mersenne weight as a binary geometric series converts a\nsupport into its divisor counts. We use that identity to compare real\ngreedy remainders with finite integer quotients. Rounding introduces an\nerror depending on both the scale and the cutoff; keeping those\nparameters separate is necessary for the later orbit estimates.\n\n\n\\subsection{The object}\n\nIn this part, write $x_n=w_n=(2^n-1)^{-1}$ for every positive\ninteger $n$, and $x_A=X_A(2)=\\sum_{n\\in A}x_n$ for\n$A\\subseteq\\mathbb N_{>0}$.  These series converge by comparison\nwith $\\sum_n2^{-n}$.  Exponent $1$ contributes the rational number\n$x_1=1$, so deleting or adjoining it does not affect irrationality.\nIt cannot occur in a representation of $1/2$; those calculations\ntherefore start at exponent $2$.\n\n\\begin{defn}[Universal Problem \\#257]\n\\textbf{Is $x_A = \\sum_{n \\in A} 1/(2^n-1)$ irrational for \\emph{every} infinite\n$A \\subseteq \\mathbb{N}_{>0}$?}\nThe quantifier ranges over all infinite sets of positive exponents. It is\n\\textbf{OPEN}.\n\\end{defn}\n\nThe support theorems prove irrationality for specified infinite\nfamilies.  The following question instead asks whether one particular\nrational number is represented.  A positive answer would disprove\nthe universal assertion; a negative answer would settle only that\ntarget.  The later analysis also treats $1/21$.\n\n\\begin{defn}[The distinguished half-value question]\n\\label{defn:half-question}\nDefine the \\emph{Mersenne achievement set}\n$\\mathcal{A} := \\{\\,y \\in \\mathbb{R} : \\exists A \\subseteq \\mathbb{N},\\ 0 \\notin A,\\ y =\n\\sum_{n \\in A} x_n\\,\\}$, the set of reals realizable as a subset sum of the positive-index Mersenne\nweights $\\{x_n\\}_{n \\ge 1}$. \\textbf{Is $1/2 \\in \\mathcal{A}$?} This is \\textbf{OPEN}.\n\\end{defn}\n\n\\begin{obs}[Why the half-value question controls universal \\#257]\n\\label{obs:half-controls-universal}\nAny finite subset sum of $\\{x_n\\}$ has odd reduced denominator, since every $2^n - 1$ is odd; hence\nno finite $A$ can sum to $1/2$ (\\lean{positiveMersenneSupportValue_coe_finset_ne_half}{HalfCutLocator.lean:243}).\nConsequently, \\emph{if} $1/2 \\in \\mathcal{A}$, the witnessing support $A$ is necessarily infinite, and\n$x_A = 1/2$ is rational. Such an $A$ is then a single infinite set for which the universal-\\#257\nsum is rational -- an outright refutation of universal \\#257. Conversely, universal \\#257 being true\nforces $1/2 \\notin \\mathcal{A}$ as one instance among uncountably many. The half-value question is therefore one concrete test of the negation,\nnot a restatement of the universal problem.  Excluding $1/2$ would\nnot exclude every other rational value, even though the support\ntheorems already settle many infinite families.  Representing $1/2$\nwould give an infinite-support counterexample to the universal\nassertion.\n\\end{obs}\n\n\\subsection{What is proved}\n\n\\subsubsection{Full-support irrationality, for every base}\n\n\\begin{thm}[Full-support irrationality, unconditional, every base $b \\ge 2$]\n\\label{thm:full-support}\nFor every integer $b \\ge 2$,\n\\[\n\\sum_{k=0}^{\\infty} \\frac{1}{b^{k+1}-1} \\quad \\text{is irrational.}\n\\]\n\\textbf{Hypotheses:} $b \\in \\mathbb{N}$, $b \\ge 2$. \\textbf{Conclusion:} unconditional irrationality\nof the full-support Erd\\H{o}s--Borwein-type series at every base, not merely base $2$.\nThe base-$2$ instance recovering the classical Erd\\H{o}s--Borwein constant's irrationality\nis a one-line corollary.\n\\end{thm}\n\nThis formalises the positive-integer-base scope of\n\\href{https://www.renyi.hu/~p_erdos/1948-04.pdf}{Erd\\H{o}s's classical 1948 full-support theorem},\nwhich already covered integer bases of absolute value greater than one. The base-two\nErd\\H{o}s--Borwein constant is a particular case; the Lean theorem is recorded with no open\nhypotheses.\n\n\\subsubsection{Structured infinite supports}\n\nBeyond full support ($A = \\{1,2,3,\\dots\\}$, i.e.\\ $m=1$ below), the following named infinite\nfamilies are proved irrational for every integer base $b\\ge2$.  The purely\nperiodic, eventually periodic, residue-class and odd supports follow from the periodic theorem of\nLuca and Tachiya~\\cite[Theorem~A, p.~139]{lucatachiya2017}; these cases also have Lean proofs\n(margin):\n\n\\begin{thm}[Purely periodic support]\\label{thm:purely-periodic-support}\nFor integers $b\\ge2$ and $m\\ge1$, and every $m$-periodic $A\\subseteq\\mathbb N$ (i.e.\\\n$n+m \\in A \\Leftrightarrow n \\in A$ for all $n$) containing a positive element, the support series\n$\\sum_{a\\in A,\\ a\\ge1}(b^a-1)^{-1}$ is irrational. \\textbf{Hypotheses:} $b \\ge 2$, $m \\ge 1$, $A$\n$m$-periodic, $\\exists a > 0,\\ a \\in A$.\n\\end{thm}\n\n\\begin{thm}[Eventually periodic support]\\label{thm:eventually-periodic-support}\nFor integers $b\\ge2$ and $m\\ge1$, if $A\\subseteq\\mathbb N$ is infinite and its membership is $m$-periodic from\nsome threshold $N_0$ on (i.e.\\ $n+m \\in A \\Leftrightarrow n \\in A$ for all $n \\ge N_0$), then\n$\\sum_{a\\in A,\\ a\\ge1}(b^a-1)^{-1}$ is irrational, by transferring irrationality across the finite\nsymmetric difference from the shifted purely periodic set $A_{\\mathrm{pure}} := \\{n : n + m N_0 \\in\nA\\}$. \\textbf{Hypotheses:} $b \\ge 2$, $m \\ge 1$, $N_0 \\in \\mathbb{N}$, eventual $m$-periodicity from\n$N_0$, $A$ infinite.\n\\end{thm}\n\n\\begin{thm}[Residue-class support]\\label{thm:residue-class-support}\nFor integers $b\\ge2$, $m\\ge1$ and $c$, the series\n$\\sum_{n\\ge1,\\ n\\equiv c\\pmod m}(b^n-1)^{-1}$ is irrational.\n\\end{thm}\n\n\\begin{thm}[Odd support]\\label{thm:odd-support}\nFor every integer $b\\ge2$, the series\n$\\sum_{n\\ge1,\\ n\\text{ odd}}(b^n-1)^{-1}$ is irrational. This density-$1/2$ support is\nthe case treated explicitly by Luca and Tachiya~\\cite[Example~2, p.~140]{lucatachiya2017}.\n\\end{thm}\n\nA support representing $1/2$ cannot be eventually periodic, by these\nirrationality theorems.  This is a restriction on a hypothetical\nrepresentation, not a proof that the actual greedy support is nonperiodic:\nif the greedy construction fails, it eventually selects every exponent.\nThe periodic-support results therefore decide neither alternative for\n$1/2\\in\\mathcal A$.\n\n\\subsubsection{Topology and measure of the achievement set}\n\n\\begin{thm}[Achievement-set topology and measure]\n\\label{thm:topology}\n$\\mathcal{A}$ (Definition~\\ref{defn:half-question}) is compact, closed, perfect, totally\ndisconnected, and nowhere dense; its Lebesgue measure is exactly $1$:\n$\\operatorname{volume}(\\mathcal{A}) = 1$. \\textbf{Hypotheses:} none. \\textbf{Conclusion:}\n$\\mathcal A$ is a Cantor set of positive measure, often called a\nfat Cantor set.  Its measure is $1$, not the length of its ambient\ninterval $[0,E]$, where $E\\approx1.6067$.\nStrict separation and summability give the compactness, unique coding, and Cantor topology, and\nHornich's strict-tail theorem, as proved by Nitecki~\\cite[Theorem~4(1), p.~9]{nitecki2013}, gives\nthe measure as $\\lim_N2^NR_N$ with $R_N=\\sum_{n>N}x_n$: each level-$N$ cylinder has length $R_N$,\nand the $2^N$ disjoint cylinders have total length $2^NR_N$. The Mersenne-specific input is\n$2^NR_N\\to1$. Other weight sequences require their own tail asymptotic.\\\\\n\\textit{Use:} Theorem~\\ref{thm:greedy-survival-record} below (compactness is exactly what powers\nevery ``limit of achieved points is achieved'' argument used downstream, including the seam-limit\nroute of Part~2).\n\\end{thm}\n\n\\begin{thm}[Membership equals greedy survival at every level]\n\\label{thm:greedy-survival-record}\nFor a real $x$, let $r_n(x)$ be its greedy remainder after rank $n$,\nand let $R_n=\\sum_{j>n}w_j$, as in the initial notation.  Then\n\\[\n x\\in\\mathcal A\\quad\\Longleftrightarrow\\quad\n x\\ge0\\ \\text{and}\\ r_n(x)\\le R_n\\text{ for every }n\\ge0.\n\\]\nThe inequality says that the unselected tail has enough total mass\nat every rank.  A failure at any one rank excludes membership;\nthe inequalities at all ranks give a representation.\n\\end{thm}\n\n\\subsection{The greedy rule for one half}\n\\label{ssec:greedy-orbit}\n\nFix the target $1/2$ and the weights $x_n = 1/(2^n-1)$. Here $x_1=1$, so exponent one cannot occur\nin a representation of $1/2$ by positive summands; the half-target discussion starts at exponent\ntwo. (The reciprocal formula is undefined only at exponent zero.) The greedy\nrule is deterministic.  Strict tail domination makes it recover the\nsupport of any represented target\n(Theorem~\\ref{thm:greedy-survival-record}):\n\n\\begin{defn}[The half-value greedy orbit]\n\\label{defn:greedy-orbit}\nLet $\\rho_k=r_{k-1}(1/2)$ be the residual entering rank $k$, so\n$\\rho_2=1/2$.  In this subsection $x_k=w_k$ and\n$T_{k+1}=R_k=\\sum_{j>k}w_j$.  At rank $k\\ge2$:\n\\begin{itemize}[nosep]\n\\item \\textbf{Take} iff $\\rho_k \\ge x_k$; if taken, $\\rho_{k+1} := \\rho_k - x_k$.\n\\item \\textbf{Skip} otherwise; $\\rho_{k+1} := \\rho_k$.\n\\end{itemize}\nA skip at rank $k$ is:\n\\begin{itemize}[nosep]\n\\item \\textbf{safe} iff $\\rho_k \\le T_{k+1} := \\sum_{j>k} x_j$\n(the total remaining mass does not yet rule out a representation; this\ninequality alone does not guarantee one);\n\\item \\textbf{fatal} iff $\\rho_k \\in (T_{k+1}, x_k)$ (the residual is too large for the remaining\ntail to ever reach, yet too small to have been taken at $k$ -- an impossible gap).\n\\end{itemize}\n\\end{defn}\n\nFor example, ranks $2$ and $3$ are selected, leaving\n$1/2-1/3-1/7=1/42$.  Ranks $4$ and $5$ are skipped, while rank $6$\nis selected and leaves $1/126$.  These decisions illustrate the rule;\nthey do not establish its infinite survival.\n\nBy Theorem~\\ref{thm:greedy-survival-record}, $1/2 \\in \\mathcal{A}$ iff no rank of this orbit is ever fatal.\nTwo boundary degeneracies are excluded unconditionally and independently of any open hypothesis:\n\n\\begin{rem}[No-ties: both greedy-orbit boundaries are strict]\n\\label{rem:no-ties}\n(a) $\\rho_k \\ne x_k$ at any rank: a finite Mersenne sum has odd reduced denominator (each $2^n-1$ is\nodd), so the residual $1/2$ minus a finite prefix cannot equal the odd-denominator\nnumber $x_k$, sharpening the general\ndenominator-parity fact used in Observation~\\ref{obs:half-controls-universal}.\n(b) $\\rho_k \\ne T_{k+1}$ at any rank: the tail $T_{k+1} = E - 1 - \\sum_{2 \\le j \\le k} x_j$, where\n$E := \\sum_{n\\ge1} x_n$ is the Erd\\H{o}s--Borwein constant, is irrational by Erd\\H{o}s's own 1948\ntheorem (the base-$2$ case of Theorem~\\ref{thm:full-support}), while $\\rho_k$ is always rational;\nequality is impossible. \\textbf{Conclusion:} both greedy-orbit boundaries are provably strict at\nevery rank, so the orbit's decisions stabilize after finitely many strict inequalities at any fixed\ntruncation depth.\n\\end{rem}\n\n\\subsection{Integer quotients and their remainder identity}\n\\label{ssec:seam-model}\n\nThe quotient rows are finite integer approximations to the real\ngreedy rule, not identical finite stages of that rule.  Their\nnormalised weights tend to the Mersenne weights and their targets\ntend to $1/2$.  Proposition~\\ref{prop:one-orbit} proves agreement\nat each fixed prefix once the row index is sufficiently large;\nit does not assert simultaneous agreement at all ranks.\nFix a row $n\\ge6$. The integer quotient weights at row $n$ are\n\\[\nw(n,d) := \\left\\lfloor \\frac{4^n}{2^d-1} \\right\\rfloor, \\qquad d = 2,\\dots,n-1,\n\\]\nand the integer target is $T_n := 2^{2n-1} - 2^n$. The integer greedy process takes ranks $d = 2,\\dots,n-1$\nin ascending order against $T_n$ exactly as in Definition~\\ref{defn:greedy-orbit}, producing a take\nset $D_n \\subseteq \\{2,\\dots,n-1\\}$ and skip set\n$\\mathrm{Skip}_n := \\{2,\\dots,n-1\\} \\mathbin{\\backslash} D_n$.\nDefine the \\emph{integer remainder} $\\mathrm{rem}(n) := T_n - \\sum_{d \\in D_n} w(n,d)$ and the\n\\emph{scaled remainder deviation} $\\Delta_n := \\mathrm{rem}(n) - 2^n$.\n\n\\begin{defn}[The full-support quotient recurrence]\nFor $M \\ge 2$, define\n\\[\nK(M) := 2^{M-1} - \\sum_{d=2}^{M} \\left\\lfloor \\frac{2^M}{2^d-1} \\right\\rfloor.\n\\]\nThis depends only on $M$, not on the selected support in a quotient\nrow.  The floor-quotient recurrence gives\n\\[\n K(2)=1,\\qquad K(M+1)=2K(M)-(\\tau(M+1)-1).\n\\]\nHere $\\tau(n)$ counts the positive divisors of $n$.  To see the correction term, double the quotients for $2\\le d\\le M$\nand add one for each such $d$ dividing $M+1$.  The new endpoint\n$d=M+1$ contributes one more quotient.  Together these are\nexactly the $\\tau(M+1)-1$ divisors at least two.\nThis is an identity for every $M\\ge2$, not merely a checked\nfinite recurrence.\n\\end{defn}\n\n\\begin{thm}[Decomposition of the integer remainder]\n\\label{thm:master-identity}\nFor every quotient row $n \\ge 6$,\n\\begin{equation*}\n\\Delta_n \\;=\\; K(2n) \\;+\\; \\sum_{d \\in \\mathrm{Skip}_n} \\left\\lfloor \\frac{4^n}{2^d-1} \\right\\rfloor.\n\\tag{I}\n\\end{equation*}\nHere the take and skip sets, capacity and remainder are those defined\nat the start of this subsection.  The identity separates the contribution\nof the full exponent range from that of the omitted ranks.  It makes no\nclaim that there are few omitted ranks; controlling their sum is a\nseparate part of any application.\\\\\n\\textbf{Proof.} For $d=n$ the quotient is $2^n+1$, and for\n$n<d\\le2n$ it is $2^{2n-d}$.  Hence\n\\[\n \\sum_{d=n}^{2n}\\left\\lfloor\\frac{4^n}{2^d-1}\\right\\rfloor\n =2^n+1+\\sum_{j=0}^{n-1}2^j=2^{n+1}.\n\\]\nSubstitute this in the definition of $K(2n)$, split\n$\\{2,\\ldots,n-1\\}=D_n\\sqcup\\mathrm{Skip}_n$, and use\n$\\Delta_n=2^{2n-1}-2^{n+1}-\\sum_{d\\in D_n}w(n,d)$.\nThis gives (I).  \\textbf{Certification:} rows $6\\le n\\le200$\nrechecked with zero mismatches by an exact\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/check_master_identity.py}{program}\n(\\S\\ref{ssec:anchors}).\\\\\n\\textit{Use:} (I) permits substitution of information about $K(2n)$.\nA resulting estimate for $\\Delta_n$ must also retain or bound the\nskip-set sum; information about $K$ alone need not control their\ncancellation.\n\\end{thm}\n\n\\subsection{A real-valued form of the quotient identity}\n\n\\begin{thm}[Real (non-integer) form of the quotient identity]\n\\label{thm:real-form}\nLet $x_d := 1/(2^d-1)$ and define the constant\n\\[\nC \\;:=\\; \\sum_{d \\ge 2} x_d \\;-\\; \\tfrac12 \\;=\\; E - \\tfrac32 \\;=\\; 0.1066951524152917\\ldots,\n\\]\nwhere $E$ is the Erd\\H{o}s--Borwein constant. Then, for every quotient row $n \\ge 6$,\n\\begin{equation*}\n\\Delta_n \\;=\\; 4^n \\Big( \\sum_{d \\in \\mathrm{Skip}_n} x_d \\;-\\; C \\Big) \\;+\\; \\eta_n, \\qquad\n|\\eta_n| < 2n+2.\n\\tag{II}\n\\end{equation*}\n\\textbf{Proof.} Splitting the full series at $n$ gives\n\\[\n \\eta_n=4^nR_{n-1}-2^{n+1}\n       +\\sum_{d\\in D_n}\\left\\{\\frac{4^n}{2^d-1}\\right\\}.\n\\]\nThe tail bounds $2^{-(n-1)}<R_{n-1}\\le\n2^{-(n-1)}+\\tfrac23 4^{-(n-1)}$ and $|D_n|\\le n-2$ show\n$0<\\eta_n<n+2/3<2n+2$.\n\n\\textbf{Use of the error bound.} For $H>2n+2$, the condition\n$|\\sum_{d\\in\\mathrm{Skip}_n}x_d-C|>(H+2n+2)/4^n$ is sufficient\nfor $|\\Delta_n|>H$.  Conversely, $|\\Delta_n|>H$ implies\n$|\\sum_{d\\in\\mathrm{Skip}_n}x_d-C|>(H-2n-2)/4^n$.\nThus a square-root-exponential deviation corresponds to a separation\nof order $2^{-3n/2}$, with the displayed additive error retained.\nThis is not an exact equivalence after simply discarding that error.\\\\\n\\textbf{Note:} $C = E - 3/2$ is Mersenne-specific, but the shape ``scaled deviation $=$\n$4^n \\cdot$(finite skip-sum $-$ target constant) $+ O(n)$'' is a template, not yet matched to any\nanalogous constant on the \\#249 side.\n\\end{thm}\n\n\\subsection{Dynamics}\n\n\\begin{thm}[Exact recurrences for the three branches]\n\\label{thm:dynamics}\nFix $n\\ge5$.  Among the quotient sums over subsets of\n$\\{2,\\ldots,n-1\\}$, let $D_n$ give the largest sum at most $T_n$\nand let $B_n$ give the smallest sum strictly greater than $T_n$.\nThus $D_n$ is the greedy support already defined.  Put\n\\[\n r=\\mathrm{rem}(n),\\qquad o=Q(B_n,2n)-T_n,\n\\]\nand define the two correction terms by\n\\[\n p^- =\\sum_{d\\in D_n}\\bigl(2\\mathbf1_{d\\mid2n+1}\n                              +\\mathbf1_{d\\mid2n+2}\\bigr),\\qquad\n p^+ =\\sum_{d\\in B_n}\\bigl(2\\mathbf1_{d\\mid2n+1}\n                              +\\mathbf1_{d\\mid2n+2}\\bigr).\n\\]\nBoth lie in $[0,2(n-2)]$, since $d\\ge2$ cannot divide both\nconsecutive integers.  The next remainder is\n\\[\n \\mathrm{rem}(n+1)=\n \\begin{cases}\n 2^{n+1}-4o-p^+,&\\text{if }4o+p^+\\le2^{n+1}\\quad(\\mathrm U),\\\\\n 4r+2^{n+1}-p^-,&\\text{if not }\\mathrm U\\text{ and }\n                   4r+2^{n+1}-p^-<2^{n+2}+4\\quad(\\mathrm M),\\\\\n 4r-2^{n+1}-p^--4,&\\text{otherwise}\\quad(\\mathrm R).\n \\end{cases}\n\\]\nConsequently, for $\\lambda_n=(\\mathrm{rem}(n)-2^n)/2^n$,\n\\begin{equation*}\n \\lambda_{n+1}=\n \\begin{cases}\n -(4o+p^+)/2^{n+1},&\\mathrm U,\\\\\n 2\\lambda_n+2-p^-/2^{n+1},&\\mathrm M,\\\\\n 2\\lambda_n-(p^-+4)/2^{n+1},&\\mathrm R.\n \\end{cases}\n \\tag{III}\n\\end{equation*}\n\\end{thm}\n\n\\begin{proof}\nApplying the one-step quotient identity twice gives\n\\[\n q(2n+2,d)=4q(2n,d)+2\\mathbf1_{d\\mid2n+1}\n                         +\\mathbf1_{d\\mid2n+2}.\n\\]\nThe new capacity is $T_{n+1}=4T_n+2^{n+1}$, and the new terminal\nweight, at exponent $n$, is $q(2n+2,n)=2^{n+2}+4$.\nWrite $g=2^{n+1}$.  The old quotient sums are separated by at\nleast $g$, so the corrected sums remain separated by at least\n$4g-2(n-2)>2g+4$.  This exceeds the new terminal weight.\nThe upper word exists: already the terms at $d=2,3,4$ have\nsum greater than $T_n$ for $n\\ge5$, since\n$1/3+1/7+1/15=19/35>1/2$ and their total floor error is less than $3$.\nConsequently, adding the terminal weight to a lower old word cannot\npass the next corrected old word.  Moreover, the increment\n$T_{n+1}-4T_n=g$ is smaller than this separation, so no old word\nbeyond the adjacent upper one can fit.  It is therefore enough\nto compare the two adjacent old words and the terminal weight.  The upper word fits precisely under\nthe condition $\\mathrm U$; its remainder is then too small to take\nthe terminal weight.  Otherwise the lower word is used, and\ncomparison with the terminal weight gives $\\mathrm M$ or\n$\\mathrm R$.  Substitution yields the displayed formulas.\n\\end{proof}\n\nOnly the $\\mathrm M$ and $\\mathrm R$ formulas have the asserted\nsmall correction to a fixed affine map.  To rewrite $\\mathrm U$\nin that form, let\n$\\gamma_n=r+o-2^{n+1}\\ge0$ be the excess of the actual adjacent\nspacing over its lower bound.  Then\n\\[\n \\lambda_{n+1}=2\\lambda_n-2-\n                   \\frac{4\\gamma_n+p^+}{2^{n+1}}\\qquad(\\mathrm U).\n\\]\nThere is no $O(n)$ bound on $\\gamma_n$ supplied by the separation\ninequality.  At $n=13$, exact calculation gives\n\\[\n r=15147,\\quad o=3997,\\quad p^+=4,\\quad\n \\gamma_{13}=2760,\\quad \\mathrm{rem}(14)=392.\n\\]\nThus $\\lambda_{14}-2\\lambda_{13}+2=-2761/4096$.\nIn particular, the proposed uniform error bound\n$|\\delta_n|\\le(2n+6)2^{-n}$ in a three-branch signed-digit formula\nis false already at this row.  The adjacent-spacing term cannot\nbe absorbed into the divisor correction.\n\nIterating the $\\mathrm R$ formula is still legitimate along a run\nof right branches.  Its accumulated correction and endpoint indices\nmust be retained in a reset estimate.  These exact recurrences do\nnot identify the proposed ceiling $L_r<(r-3)/2$ with a converse to\nthe conditional reset square-root estimate.\n\n\\subsection{Reported finite checks}\n\\label{ssec:anchors}\n\nThe following values and finite ranges come from exact integer\ncomputations for the quotient-greedy sequence of Section~\\ref{ssec:seam-model}.\nThey provide test cases for the formulas above. The low-row remainders\nat $n=14,15,32$ are exact integer values. Each larger certified range has a\n\\href{https://github.com/wcook04/plectis-erdos/tree/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257}{published program and saved run}.\n\n\\begin{obs}[Exact low-row values]\n\\label{obs:anchors-low}\nAt row $n=14$: $\\mathrm{rem}(14) = 392$, with take set $D_{14} = \\{2,3,6,7\\}$. At row $n=15$:\n$\\mathrm{rem}(15) = 34333$. At row $n=32$: $\\mathrm{rem}(32) = 3865046005$ and $\\Delta_{32} =\n-429921291$.\\\\\n\\textit{Source:} the values at rows $14$--$31$ are proved in Lean by direct\nevaluation (Theorem~\\ref{thm:upper-reset-band}), which uses them to verify the\nband-escape condition for reset indices $13\\le d\\le30$\n\\lean{seamUpperResetDyadicBandEscape\\_through\\_thirty}{HalfCylinderUpperResetBandCertificates.lean:78}.\nThe two ranges differ by one: the remainder used for reset $d$ is at\nrow $d+1$. The take set $D_{14}$, $\\mathrm{rem}(32)$ and $\\Delta_{32}$ are\nexact integer computations with the greedy rule of Section~\\ref{ssec:seam-model}.\n\\end{obs}\n\n\\begin{obs}[Depths of the reported computations]\n\\label{obs:anchors-depth}\nThe quotient identity (Theorem~\\ref{thm:master-identity}) is verified at rows $6 \\le n \\le 200$ with zero\nmismatches\n(\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/quotient-identity-200.json}{saved run}),\nand the quotient-greedy orbit itself is certified row by row to row $2500$\n(\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/seam-reset-crossing-2500.json}{saved run})\nand through the exact branch recursion to row $200{,}000$\n(\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/seam-orbit-200000.json}{saved run}; branch counts $\\mathrm{R{:}M{:}U} =\n100197{:}49899{:}49898$, the target inequality failing at exactly one reset row $6\\le r\\le 200{,}000$,\nnamely $r=7$).\n\\end{obs}\n\n\\begin{obs}[Four computations side by side]\n\\label{obs:anchors-coords}\nFour related computations are reported, with different targets or\ninteger representations, at the following ranges: truncation rung $J = 3,\\dots,22$ (all $20$ rungs proved to\nsurvive, not merely computed); quotient row $6 \\le n \\le 2500$\n(Theorem~\\ref{thm:master-identity} and the branch data underlying\nTheorem~\\ref{thm:dynamics}); integer margin scan to $m_c = 5\\times10^5$\n(an exploratory scan, which is not certified); sharp tail margin, ranks $2$ through\n$3000$ (no fatal skip at any rank; take fraction $0.4992$; tightest certified take-margin\n$0.0340$ bits at rank $7$, the near-tie $1/126 \\ge 1/127$).\\\\\n\\textit{Use:} truncated weights define a different greedy problem,\nand the quotient-row construction uses integer weights rather than the\noriginal real weights.  These are not automatically four readings of\nthe same decision stream.  Their relevance to full-weight membership\ncomes from the approximation, limit, or identification theorems cited\nfor each construction.  A certificate at one range does not transfer\nto another without such a theorem, and these reports do not decide the\nuniversal problem.\n\\end{obs}\n\n% ; - part a257_p1a ; -\n\\section{Detailed results and their hypotheses}\nThe catalogue begins with conditional membership tests, then gives the\nidentities and invariants used in their proofs. Its final groups record\nconsequences and counterexamples. Each result retains its hypotheses,\nand the margin identifies its associated formal proposition. The\nsupport results and these conditional tests leave the universal\nirrationality assertion unresolved.\n\n\\subsection{Conditional membership tests}\n\nThe finite tests below use $g_n=w_n-R_n$.  When written as functions,\n$R(n)=R_n$, $g(n)=g_n$, $r(x,n)=r_n(x)$ and $C(A,N)=C_A(N)$.\nFor a finite set $D$, $V(D)=X_D(2)$.  For $n\\ge2$, the integer remainder\n$s(n)=\\mathrm{rem}(n)$ uses weights $q(2n,d)$, $2\\le d<n$, and\ntarget $T_n=2^{2n-1}-2^n$; it is not the real remainder $r_n(1/2)$.\nIn the branch conditions below, $D_s,B_s,o_s,p_s^-,p_s^+$ denote the\nlower and upper supports, overshoot and correction terms of\nTheorem~\\ref{thm:dynamics}, with the row added as a subscript.\nThe labels $\\mathrm U,\\mathrm M,\\mathrm R$ mean precisely its three\narithmetic cases.  Finally, $F(k,J)=F_k(J)$ denotes the expression in\nTheorem~\\ref{thm:frozen-margin}.\n\n\n\n\\begin{thm}[Infinitely many greedy skips force membership]\n\\label{thm:fatal-absorbing}\nFix $x\\ge0$.  If $r_n(x)>R_n$ at some rank $n$, every later\nrank is selected and $r_{n+k}(x)>R_{n+k}$ for every $k\\ge0$.\nConsequently, infinitely many omitted positive ranks imply\n$x\\in\\mathcal A$.\n\\end{thm}\n\\begin{proof}\nThe identity $R_n=w_{n+1}+R_{n+1}$ gives\n$r_n(x)>w_{n+1}$, so the rule takes rank $n+1$ and leaves\n$r_{n+1}(x)=r_n(x)-w_{n+1}>R_{n+1}$.  Induction proves persistence.\nInfinitely many skips therefore force $0\\le r_n(x)\\le R_n$ at\nevery rank; since $R_n\\to0$, the greedy sum equals $x$.\n\\end{proof}\n\n\\begin{thm}[The quotient remainders converge to the greedy deficit]\n\\label{thm:seam-limit}\nLet $G$ be the real greedy support for $1/2$, and let $D_s$ be the\ninteger-greedy support at row $s$.  Then\n\\[\n X_{D_s}(2)\\longrightarrow X_G(2),\\qquad\n \\frac{\\mathrm{rem}(s)}{4^s}\\longrightarrow\n \\delta:=\\frac12-X_G(2)\\ge0.\n\\]\nIn particular, the following conditions are equivalent: $1/2\\in\\mathcal A$;\nthe full sequence $\\mathrm{rem}(s)/4^s$ tends to zero; and there\nexist $s_j\\to\\infty$ with $\\mathrm{rem}(s_j)/4^{s_j}\\to0$.\n\\end{thm}\n\\begin{proof}\nFor each fixed $d$, the scaled weight\n$4^{-s}q(2s,d)$ tends to $w_d$, and the scaled target tends to\n$1/2$.  The finite-prefix stability argument of\nProposition~\\ref{prop:one-orbit} therefore gives\n$D_s\\cap\\{2,\\ldots,K\\}=G\\cap\\{2,\\ldots,K\\}$ for all sufficiently\nlarge $s$, for each fixed $K$.  At such rows,\n\\[\n \\bigl|X_{D_s}(2)-X_G(2)\\bigr|\\le R_K.\n\\]\nFirst letting $s\\to\\infty$ and then $K\\to\\infty$ proves convergence\nof the values.  This uses summability, not an unproved uniform\nrate of prefix agreement.\n\nFor $s\\ge5$, put\n$\\Phi_s=\\sum_{d\\in D_s}\\{4^s/(2^d-1)\\}$, so\n$0\\le\\Phi_s<s-2$.  The quotient identity gives exactly\n\\[\n X_{D_s}(2)=\\frac12-2^{-s}\n            -\\frac{\\mathrm{rem}(s)}{4^s}+\\frac{\\Phi_s}{4^s}.\n\\]\nThe two error terms tend to zero, proving the asserted limit.\nThe real greedy sums never exceed $1/2$, so $\\delta\\ge0$.\nBy strict tail separation, a support representing $1/2$ must agree\nwith the greedy rule, as in Theorem~\\ref{thm:greedy-survival-catalogue}.\nThus membership is equivalent to $\\delta=0$, which is also equivalent\nto a zero limit along any one sequence $s_j\\to\\infty$.\n\\end{proof}\nConvergence itself is automatic; proving that its limit is zero is\nexactly the unresolved membership question.\n\n\\begin{thm}[Two-channel and dyadic cap sufficiency]\n\\label{thm:two-channel-cap}\nSuppose that for every $n\\ge0$ at which the next weight is skipped,\n\\[\n w_{n+1}>r_n(1/2)\\quad\\Longrightarrow\\quad\n r_n(1/2)\\le 2^{-(n+1)}+\\frac13\\,4^{-(n+1)}.\n\\]\nThen $1/2\\in\\mathcal A$.  The stronger bound\n$r_n(1/2)\\le2^{-(n+1)}$ at the same skipped ranks also suffices.\nThe first bound retains two positive geometric terms of the full tail;\nit is larger, and hence less restrictive, than the second.  Both are\nsufficient tests on the specified greedy orbit.  The result does not\nassert that either bound holds at all its skipped ranks.\n\\end{thm}\n\n\\begin{thm}[Second-channel phase separation sufficiency]\n\\label{thm:second-channel}\nDefine $P_n=4^n\\bigl(2r_n(1/2)-2^{-n}\\bigr)$.  If\n\\[\n \\frac16+\\frac{37}{56}\\,2^{-n}\\le\n \\left|P_n-\\frac13\\right|\\qquad\\text{for every }n\\ge1,\n\\]\nthen $1/2\\in\\mathcal A$.  These are exact rational inequalities because\nthe target and every finite greedy remainder are rational.  The cited\nfinite calculation establishes them for $1\\le n\\le6$; it therefore\nsuffices to prove them for every $n\\ge7$.\n\nUnlike Theorem~\\ref{thm:two-channel-cap}, this hypothesis concerns every\npositive rank, not only skipped ranks.  It excludes an explicit interval\naround $1/3$.  For example, $P_n\\notin(0,1)$ implies the displayed\ninequality when $n\\ge2$, but no such avoidance theorem for the whole\norbit is proved here.\n\\end{thm}\n\n\\begin{thm}[Straddle-prefix closed-set criterion]\n\\label{thm:straddle-closed-set}\nSuppose that for every $d\\ge0$ there is a finite support\n$D_d\\subseteq\\{1,\\ldots,d\\}$ such that\n\\[\n X_{D_d}(2)\\le t\\le X_{D_d}(2)+R_d.\n\\]\nThen $t\\in\\mathcal A$.  The supports need not agree at different depths:\ncompactness, together with $R_d\\to0$, supplies a limiting support.\nConversely, truncating a support representing $t$ supplies such finite\nsupports.  Thus this condition is an exact finite-approximation\nformulation of membership, not a criterion already known to hold for an\nunresolved target.\n\\end{thm}\n\n\\begin{thm}[Keeping the largest skipped rank beyond two thirds]\n\\label{thm:largest-skip-late}\nLet $D_s\\subseteq\\{2,\\ldots,s-1\\}$ be the support of the integer-greedy\nrow defined in Section~\\ref{ssec:seam-model}.  Assume the following implication\nfor every $s\\ge14$ and every largest omitted rank\n$d=\\max(\\{2,\\ldots,s-1\\}\\smallsetminus D_s)$:\n\\[\n 2s<3d\\quad\\Longrightarrow\\quad\n 2(s+1)<3d\\quad\\text{or}\\quad s\\notin D_{s+1}.\n\\]\nThen $1/2\\in\\mathcal A$.  The hypothesis says that either the same\nomitted rank remains beyond two thirds of the next row, or the next\nrow omits its terminal rank.  The latter alternative is exactly an\nupper or middle transition in the preceding classification.\n\nThe cited finite calculation establishes the initial inequality at\nrow $14$.  The displayed implication then preserves the inequality\nby induction and supplies omitted ranks tending to infinity.\nIt is an unproved condition on every later row; the verified base\ncase alone does not establish it.\n\\end{thm}\n\n\\begin{thm}[Two sufficient bounds at middle transitions]\n\\label{thm:middle-producer-escape}\nSuppose that at every middle transition with row $s\\ge13$,\n\\[\n |D_s|+p_s^-+5<4\\,\\mathrm{rem}(s).\n\\]\nThen $1/2\\in\\mathcal A$.  The stronger condition\n$\\mathrm{rem}(s)\\ge s$ at all such rows also suffices.\n\\end{thm}\nIndeed, $|D_s|\\le s-2$ and $p_s^-\\le2(s-2)$ give\n$|D_s|+p_s^-+5\\le3s-1<4s$.  This explains the implication\nbetween the two hypotheses.  The implication from the first bound to\nmembership is proved in Lean; no ordinary proof is printed here.  Neither all-row hypothesis\nis established by the finite computations.\n\n\\begin{thm}[The tail inequality at a final middle transition]\n\\label{thm:middle-allright-defect}\nSuppose row $D\\ge13$ is middle and all rows after it are right.\nLet $F_D=D_D\\cup\\{D\\}$ and put\n\\[\n C_D=4\\,\\mathrm{rem}(D)-p_D^--4,\\qquad\n \\Theta_D=\\sum_{j\\ge1}c_{F_D}(2D+2+j)2^{-j}.\n\\]\nThen $C_D<\\Theta_D$.  Since $0\\le\\Theta_D\\le|F_D|$,\nan estimate in the opposite direction would exclude this scenario.\nNo such reverse estimate is assumed or proved here.\nRemark~\\ref{rem:tail-dominance-open} states the corresponding sufficient\nhypothesis with the value $-3$ excepted.\n\\end{thm}\n\n\\begin{thm}[A conditional two-sided dyadic bound]\n\\label{thm:two-sided-dyadic}\nAssume the following two conditions for every row $s\\ge5$:\n\\[\n \\begin{aligned}\n \\mathrm M\\text{ at }s&\\quad\\Longrightarrow\\quad\n 4\\,\\mathrm{rem}(s)-p_s^--4\\notin\\{-3,-2,-1\\},\\\\\n \\mathrm R\\text{ at }s\\text{ and }o_s\\le2^s\n &\\quad\\Longrightarrow\\quad 4o_s+p_s^+\\le2^{s+2}.\n \\end{aligned}\n\\]\nThen at every row $s\\ge5$,\n\\[\n \\mathrm{rem}(s)\\le2^s\\quad\\text{or}\\quad o_s\\le2^s.\n\\]\nThe first hypothesis concerns only middle transitions.  The second\nconcerns only right transitions at which the old overshoot is at most\n$2^s$.  Neither condition is asserted here for the whole sequence.\n\\end{thm}\n\\begin{proof}\nAt row $5$, $D_5=\\{2,4\\}$ and $\\mathrm{rem}(5)=71$, whereas\n$B_5=\\{2,3\\}$ and $o_5=7\\le32$.  For the induction step use\nTheorem~\\ref{thm:dynamics}.  The upper branch gives\n$\\mathrm{rem}(s+1)\\le2^{s+1}$.  On a middle branch put\n$c=4\\,\\mathrm{rem}(s)-p_s^--4$.  If $c\\le-4$, the new remainder\nis $2^{s+1}+c+4\\le2^{s+1}$.  Otherwise the hypothesis forces\n$c\\ge0$.  Adding the terminal weight to the corrected lower word\nthen exceeds the target by $2^{s+1}-c>0$, so the new minimal\novershoot is at most $2^{s+1}$.\n\nOn a right branch, an old remainder at most $2^s$ gives a new\nremainder at most $2^{s+1}$ by the recurrence.  In the other case\nthe inductive hypothesis gives $o_s\\le2^s$; the corrected upper\nword exceeds the new target by $4o_s+p_s^+-2^{s+1}$, which is\npositive and at most $2^{s+1}$ by the second assumption.\n\\end{proof}\nThis conclusion is a disjunction.  It does not by itself give an\nupper bound for the remainder on rows where only the overshoot is\nsmall, and therefore is not by itself a membership criterion.\n\n\\begin{thm}[Upper-reset dyadic-band escape, checked for $13\\le d\\le30$]\n\\label{thm:upper-reset-band}\nThe upper-reset dyadic-band escape condition requires the following at every\nactual upper reset $d\\ge13$: for every $0\\le j\\le d$, the\nreset charge avoids a linear-width band immediately below the dyadic power\n$2^{d-j+1}$ ($2^{d-j+1}<\\mathrm{resetCharge}$ or\n$\\mathrm{resetCharge}+2(d+j)\\le 2^{d-j+1}$). Granted this, $1/2\\in\n\\ensuremath{\\mathcal A}$. The linked proof verifies this condition for\n$13\\le d\\le30$, using exact successor remainders at rows $14$--$31$\n(for example, $\\mathrm{rem}(14)=392$ and\n$\\mathrm{rem}(31)=4187487147$). This finite verification does not\nsupply the hypothesis for every $d\\ge13$.\n\\end{thm}\n\n\\begin{thm}[M\\\"obius-centred carry nonnegativity below $1/2$]\n\\label{thm:mobius-centred-nonneg}\nIf $1\\notin A$ and $X_A(2)<1/2$, then $C_A(N)\\ge0$ for every $N$.\nIndeed,\n\\[\n \\operatorname{ihc}(A,N)=2^{N+1}(1/2-X_A(2))+\n \\sum_{j>N+1}c_A(j)2^{N+1-j}>0.\n\\]\nThe carry is an integer, so it is at least $1$; subtracting $1$ gives\nthe asserted nonnegativity of the centred carry.  The strict inequality\nand integrality are both needed in this argument.  The statement is\nabout the divisor counts $c_A$, not a new irrationality criterion.\n\\end{thm}\n\n\\begin{thm}[A square-root bound on the greedy carry]\n\\label{thm:sqrt-bound-route}\nFor the greedy support $G=G_{1/2}$, the centred carry is nonnegative.\nIf in addition\n\\[\n C_G(N)\\le2\\sqrt N+4\\qquad\\text{for every }N\\ge0,\n\\]\nthen $G$ has infinitely many skipped indices and $X_G(2)=1/2$.\nThe unresolved part is the upper bound along this particular orbit.\nThe carry identity explains its strength: a positive gap\n$1/2-X_G(2)$ would contribute a term of order $2^N$, which cannot satisfy\na square-root bound.  This is distinct from the reset-deviation\nhypothesis of Theorem~\\ref{thm:upper-reset-band}; no equivalence between\nthe two hypotheses is asserted.\n\\end{thm}\n\n\\begin{thm}[Equivalent sign and vanishing conditions]\n\\label{thm:frozen-margin}\nFor the prefix $D=G\\cap\\{1,\\ldots,k\\}$ define\n\\[\n F_k(J)=\\sum_{i=1}^{J}c_D(k+1+i)2^{J-i}-2^J C_D(k).\n\\]\nThis is a finite integer expression: it compares the next $J$ divisor\ncounts of the fixed prefix with its centred carry.  Let $s_n$ denote the\ninteger greedy remainder at depth $2n$, with weights $q(2n,d)$ for\n$2\\le d<n$ and target $2^{2n-1}-2^n$.\nThe following conditions each imply $1/2\\in\\mathcal A$:\n\\begin{enumerate}[label=(\\roman*)]\n\\item For every skipped rank $n\\ge3$, $F_{n-1}(n)\\ge0$.\n\\item For every skipped rank $n\\ge3$ at which the real and integer\n      greedy words agree on $\\{2,\\ldots,n-1\\}$, $s_n=0$.\n\\item For every skipped rank $n\\ge3$, $B(2n)<s_n$.\n\\end{enumerate}\nConditions (i) and (ii) are equivalent.  Condition (iii) implies them;\nno converse or strict separation is established here.  These hypotheses\nmust hold at every indicated skipped rank, not merely through a finite\ncomputed range.\n\\end{thm}\n\n\\begin{thm}[Erd\\H{o}s--Borwein full-support irrationality, unconditional]\n\\label{thm:full-support-catalogue}\nFor every integer $b\\ge2$, the sum $\\sum_{n\\ge1}(b^n-1)^{-1}$\nis irrational.  This is Erd\\H{o}s's theorem~\\cite{erdos1948}; at base\n$2$ the sum is the Erd\\H{o}s--Borwein constant.  The result concerns\nfull support, not all its infinite subsets.\n\\end{thm}\n\n\\begin{thm}[Pairwise-coprime support irrationality, Erd\\H{o}s 1968]\n\\label{thm:pairwise-coprime}\nFor every integer $b\\ge2$ and every infinite pairwise-coprime support\n$A\\subseteq\\Npos$ with summable reciprocals, $\\sum_{a\\in A} 1/(b^a-1)$ is irrational.\n\\end{thm}\n\n\\begin{thm}[Irrationality from divisible coefficient blocks]\n\\label{thm:weighted-coeff-engine}\nLet $b\\ge2$ be an integer and let $c:\\mathbb N\\to\\mathbb N$ satisfy\n$c(n)\\le n$ for every $n$.  Suppose that for every integer $q\\ge1$\nthere are nonnegative integers $N,K,L,C$, with $K\\le L$, such that\n\\[\n \\begin{aligned}\n b^r&\\mid c(N+r) &&(1\\le r\\le K),\\\\\n \\sum_{r=K+1}^{L}c(N+r)b^{L-r}&\\le C,\\\\\n c(N+L+1+t)&>0 &&\\text{for some integer }t\\ge0,\\\\\n q(C+N+L+2)&<b^L.\n \\end{aligned}\n\\]\nThen $\\sum_{n\\ge1}c(n)b^{-n}$ is irrational.\n\\end{thm}\n\\begin{proof}\nWrite $x=\\sum_{n\\ge1}c(n)b^{-n}$.  Multiplication by $b^N$\nmakes the terms through $N$ integral.  The first condition also\nmakes the terms at $N+1,\\ldots,N+K$ integral.  Subtracting their\nsum leaves, for some integer $z$,\n\\[\n 0<b^Nx-z=\\sum_{r>K}\\frac{c(N+r)}{b^r}\n \\le\\frac{C+N+L+2}{b^L}<\\frac1q.\n\\]\nPositivity follows from the specified nonzero coefficient beyond $L$.\nThe middle sum contributes at most $C/b^L$, while $c(N+r)\\le N+r$\ngives\n\\[\n \\sum_{r>L}\\frac{c(N+r)}{b^r}\n \\le b^{-L}\\left(\\frac{N+L}{b-1}+\\frac{b}{(b-1)^2}\\right)\n \\le\\frac{N+L+2}{b^L}.\n\\]\nIf $x=p/q$ were rational, the positive difference $b^Nx-z$\ncould not be smaller than $1/q$.\n\\end{proof}\n\nA certificate at one precision is not enough: the same coefficient\nsequence must satisfy these conditions for every $q$.  The positive-tail\ncondition is essential; without it the zero sequence would satisfy the\nother conditions at every precision.  The full-support theorem supplies\nsuch certificates for $c=\\tau$.  Their existence for $c=\\varphi$ is not\nproved here.  This is a limitation of this particular sufficient\ncriterion, not a comparison of every approach to Problem~249.\n\n\n\\begin{thm}[Irrationality from a gap beyond the preceding least common multiple]\n\\label{thm:lcm-gap-engine}\nFor an integer base $b\\ge2$ and strictly increasing support $a:\\mathbb N\\to\\mathbb N$ with $a(0)\\ge\n1$: if $a(k)-\\mathrm{lcm}(a(0),\\dots,a(k-1))\\to\\infty$, then $\\sum'_k 1/(b^{a(k)}-1)$ is\nirrational. The base may vary over the integers, but the denominators remain\n$b^{a(k)}-1$.  Applying the argument to another denominator sequence\nwould require its own divisibility and tail estimates.\n\\end{thm}\n\n\\begin{thm}[Factorial-support and $2^k$-support instances]\n\\label{thm:factorial-twopow-support}\nFor every integer $b\\ge2$, both\n\\[\n \\sum_{k\\ge0}\\frac1{b^{(k+1)!}-1}\n \\quad\\text{and}\\quad\n \\sum_{k\\ge0}\\frac1{b^{2^k}-1}\n\\]\nare irrational.  These are instances of the preceding theorem.  For the\nfactorial support the preceding least common multiple is $k!$ when\n$k\\ge1$, so the gap is $k\\,k!$.  For the powers of two it is\n$2^{k-1}$, so the gap is $2^{k-1}$.  Both tend to infinity.\nThe conclusion concerns these two supports, not every infinite support.\n\\end{thm}\n\n\\begin{thm}[Multiples-support irrationality via dilation]\n\\label{thm:multiples-support}\nFor integers $b\\ge2$ and $d\\ge1$,\n\\[\n X_{d\\Npos}(b)=\\sum_{k\\ge1}\\frac1{(b^d)^k-1}\n\\]\nis irrational.  This is the full-support result\n(Theorem~\\ref{thm:full-support-catalogue}) at the integer base $b^d$;\nno new support argument is required.\n\\end{thm}\n\nLuca and Tachiya proved irrationality for every purely periodic integer weight that is not\nidentically zero~\\cite[Theorem~A, p.~139]{lucatachiya2017}.  The periodic, eventually periodic,\nresidue-class and odd-support statements below follow from their theorem; their Lean proofs are\nseparate and use the certificate method described here.\n\n\\begin{thm}[Periodic-support irrationality]\n\\label{thm:periodic-support}\nLet $b\\ge2$ and $m\\ge1$ be integers.  Suppose\n$A\\subseteq\\Npos$ is nonempty and satisfies\n$n\\in A$ if and only if $n+m\\in A$ for every $n\\ge1$.\nThen $X_A(b)$ is irrational.\n\nThe indicator of $A$ is a nonzero purely periodic integer weight,\nso this follows from Luca and Tachiya's theorem stated above.\nThe linked declaration supplies a separate formal proof.\nAt $m=1$ the only nonempty periodic support is full support;\nresidue classes and unions of residue classes give the other\nimmediate examples.\n\\end{thm}\n\n\\begin{thm}[Eventually-periodic support irrationality]\n\\label{thm:eventually-periodic}\nLet $b\\ge2$ and $m\\ge1$ be integers.  An infinite support whose\nmembership is $m$-periodic from some threshold $N_0$ onward has\nirrational $X_A(b)$.\n\\end{thm}\n\n\\begin{thm}[Residue-class and odd-support irrationality]\n\\label{thm:residue-odd}\nFor integers $b\\ge2$, $m\\ge1$ and a residue $c$,\n$\\sum_{\\substack{n\\ge1\\\\n\\equiv c\\pmod m}}(b^n-1)^{-1}$ is\nirrational. Specializing $m=2,c=1$: $\\sum_{n\\text{ odd}} 1/(b^n-1)$ is irrational for\nevery $b\\ge 2$, the case treated explicitly in~\\cite[Example~2, p.~140]{lucatachiya2017}.\n\\end{thm}\n\n\\paragraph{Comparison with the known theorem.} Luca and Tachiya proved irrationality for every\nnonzero purely periodic integer weight, of any signs~\\cite[Theorem~A, p.~139]{lucatachiya2017}, so\nthe terminating branch below occurs only for the zero weight.  The theorem records the dichotomy\nreached by the certificate method, together with its one-sided closures.\n\n\\begin{thm}[A formal dichotomy for signed periodic weights]\n\\label{thm:signed-periodic}\nLet $b\\ge2$ and $m\\ge1$ be integers, and let\n$w:\\Npos\\to\\Z$ be $m$-periodic.  Put\n\\[\n x=\\sum_{a\\ge1}\\frac{w(a)}{b^a-1},\\qquad\n c_w(n)=\\sum_{d\\mid n}w(d).\n\\]\nThe cited formal argument gives the dichotomy that $x$ is\nirrational or $b^kx\\in\\Z$ for some integer $k\\ge0$.\nIt excludes the latter alternative if $c_w$ has one sign\nthroughout and is nonzero at arbitrarily large indices.\nThese sign assumptions concern $c_w$, not $w$.\nAs noted above, Luca and Tachiya's theorem already excludes\nthe terminating alternative for every nonzero periodic $w$,\nwithout either sign restriction.  The zero weight gives $x=0$.\n\\end{thm}\n\nFor the next two results write $M_r=2^r-1$ and, for a positive\nsquarefree integer $r$, set\n\\[\n A_r=\\sum_{d\\mid r}\\mu(d)\\frac rd\\frac{M_r}{M_d}\\in\\Z,\n \\qquad B(r)=\\frac{A_r}{M_r}\n           =\\sum_{d\\mid r}\\frac{\\mu(d)(r/d)}{2^d-1}.\n\\]\nEvery quotient in $A_r$ is integral because $d\\mid r$.\nThe Lean source calls $A_r$ the M\\\"obius numerator; $B(r)$ is a finite\nrational sum, not the full-support Lambert series.\n\n\\begin{thm}[Denominators of finite Mersenne sums]\n\\label{thm:mersenne-channel-survival}\nLet $t\\ge1$ and $h\\ge1$ be integers, and let $r\\ge1$ be\nsquarefree with every prime factor at most $t$.  Let $P$ be a set of\nprime divisors of $r$ such that $t<2p$ for each $p\\in P$, and put\n$C=\\prod_{p\\in P}(2^p-1)$.  Then\n\\[\n \\frac{C}{\\gcd(C,h)}\\ \\bigm|\\ \\operatorname{den}\\bigl(hB(r)\\bigr).\n\\]\nIn particular, $C$ divides this reduced denominator when\n$\\gcd(C,h)=1$.  The factors in $C$ are pairwise coprime, since\n$\\gcd(2^p-1,2^q-1)=2^{\\gcd(p,q)}-1=1$ for distinct primes.\nThe assertion concerns this finite signed sum; it does not replace\nthe approximation hypotheses in Theorem~\\ref{thm:full-support-catalogue}.\n\\end{thm}\n\n\\begin{thm}[Exponential growth of the reduced denominators]\n\\label{thm:mersenne-channel-growth}\nFor an integer $t\\ge5$, put\n\\[\n H_t=\\operatorname{lcm}(1,\\ldots,t),\\quad\n r_t=\\prod_{p\\le t}p,\\quad h_t=H_t/r_t,\\quad\n D_t=\\operatorname{den}\\bigl(h_tB(r_t)\\bigr),\n\\]\nwhere the product is over primes.  Then $D_t\\ge2^{t/2}$.\nIts exact denominator is\n\\[\n D_t=\\frac{2^{r_t}-1}{\\gcd(2^{r_t}-1,h_tA_{r_t})}.\n\\]\n\\end{thm}\n\\begin{proof}\nApply Bertrand's postulate~\\cite[\\S\\S4--5, pp.~197--198]{erdos1932bertrand} to $m=\\lceil t/2\\rceil\\ge3$.\nThere is a prime $m<p<2m$, hence $p\\le t$ and $p-1\\ge t/2$.\nEvery prime divisor $\\ell$ of $2^p-1$ has multiplicative order\n$p$ for $2$ modulo $\\ell$.  Since $p$ is odd, $2p\\mid\\ell-1$;\nthus $\\ell>t$.  All prime factors of $h_t$ are at most $t$, so\nTheorem~\\ref{thm:mersenne-channel-survival} gives\n\\[\n 2^{t/2}\\le2^{p-1}<2^p-1\\le D_t.\n\\]\nThe exact expression for $D_t$ is the usual reduction of the\nfraction $h_tA_{r_t}/(2^{r_t}-1)$.\n\\end{proof}\nThe Lean proof of the bound uses natural-number division $t/2$,\nthat is, $\\lfloor t/2\\rfloor$.  The ceiling choice in the ordinary\nargument supplies the displayed real-exponent bound also when $t$\nis odd.  The Lean statement of the exact denominator uses an\nequivalent formula in its own numerator notation.  Neither denominator growth nor\nthat identity supplies an approximation-error estimate.\n\n\\subsection{Exact identities and reductions}\n\n\\begin{obs}[A finite half-value approximation and the first correction tail]\n\\label{obs:half-regression}\nThe finite prefix has value\n\\[\n \\frac12-X_{\\{2,3,6,7\\}}(2)=\\frac1{16002}.\n\\]\nThe excess of the full tail after rank $1$ over its dyadic value is\n\\[\n R_1-\\frac12\n =\\sum_{n\\ge2}\\left(\\frac1{2^n-1}-2^{-n}\\right)\n \\le\\frac1{12}+\\frac1{28}=\\frac5{42}.\n\\]\nIndeed, expand each summand as $4^{-n}+\\sum_{j\\ge3}2^{-jn}$;\nfor $n\\ge2$, the second sum is at most $2\\,8^{-n}$.  Thus\n$\\tfrac12-(R_1-\\tfrac12)\\ge\\tfrac8{21}>\\tfrac38$.\nThe first identity is an exact rational check, whereas the two tail\ninequalities use a convergent-series estimate.\n\\par\n\\lean{half\\_sub\\_four\\_term\\_prefix\\_eq}{HalfCutLocator.lean:664}\n\\lean{mersenneCorrectionTail\\_one\\_le}{HalfCutLocator.lean:670}\n\\end{obs}\n\n\\begin{obs}[Period-4 sign weight vanishes on a residue class]\n\\label{obs:period-four-obstruction}\nLet $w(n)$ have successive values $1,0,-1,0$, repeated\nwith period $4$.  For every $n\\equiv3\\pmod4$,\n\\[\n c_w(n)=\\sum_{d\\mid n}w(d)=0.\n\\]\nIndeed, $n$ is odd, and paired divisors $d,n/d$ have opposite\nvalues of $w$ because their product is $3$ modulo $4$.\nThere are no fixed pairs: such an $n$ is not a square.\nThus an argument requiring a nonzero coefficient at every\nchosen residue must check that requirement; periodicity alone\ndoes not provide it.  This cancellation does not conflict with\nirrationality of the full weighted series, which follows from\nLuca and Tachiya's theorem~\\cite[Theorem~A, p.~139]{lucatachiya2017}.\n\\par\n\\lean{intWeightedCoeff\\_periodFourSignWeight\\_eq\\_zero\\_of\\_mod\\_four\\_eq\\_three}{CertificateKernel.lean:14226}\n\\end{obs}\n\n\\begin{thm}[The Lambert-series identity for the M\\\"obius function]\n\\label{thm:mobius-lambert-identity}\nThe absolutely convergent signed series satisfies\n\\[\n \\sum_{d\\ge1}\\frac{\\mu(d)}{2^d-1}=\\frac12.\n\\]\nThis classical identity is also recorded in\n\\cite[Example~1.1, p.~4]{duverneytachiya}.\nExpand each denominator geometrically and interchange the\nabsolutely convergent sums.  The coefficient of $2^{-n}$ is\n$\\sum_{d\\mid n}\\mu(d)$, equal to $1$ for $n=1$ and $0$\notherwise.  This proves the identity.\nIts coefficients are $-1,0,1$, not indicators of a support;\nthe identity therefore does not represent $1/2$ as a subseries\nwith all coefficients in $\\{0,1\\}$.\n\\end{thm}\n\n\\begin{cor}[Negative-M\\\"obius Boolean support overshoots $1/2$]\n\\label{cor:negative-mobius-overshoot}\nLet $N=\\{d\\ge2:\\mu(d)=-1\\}$.  Isolating the $d=1$\nterm in Theorem~\\ref{thm:mobius-lambert-identity} gives\n\\[\n X_N(2)=\\frac12+\n       \\sum_{\\substack{d\\ge2\\\\\\mu(d)=1}}\\frac1{2^d-1}\n       \\ge\\frac12+\\frac1{63}>\\frac12,\n\\]\nsince $\\mu(6)=1$.  This rules out the particular candidate $N$.\nIt does not rule out other infinite supports representing $1/2$.\n\\end{cor}\n\n\\begin{obs}[Dyadic safety need not survive a take]\n\\label{obs:skip-block-non-invariant}\nFor a residual $p/(2D)$, taking rank $b$ subtracts $w_b$ and gives\n\\[\n \\frac{p}{2D}-\\frac1{2^b-1}\n =\\frac{p(2^b-1)-2D}{2D(2^b-1)}.\n\\]\nThe residual $7/34$ is at most $2^{-2}$.  Taking rank $3$ leaves\n$15/238$, which is strictly between $2^{-4}$ and $w_4=1/15$.\nThus rank $4$ is skipped but fails the dyadic safety test.  The displayed\nnumerator has increased from $7$ to $15$; that increase does not preserve\nsafety.\n\nThe same failure occurs with an initial denominator containing all\nMersenne denominators through rank $7$.  With\n$D=\\operatorname{lcm}(3,7,15,31,63,127)=1240155$, one has\n\\[\n \\frac{12149}{2D}-\\frac1{255}\n =\\frac{41179}{2\\cdot21082635}.\n\\]\nThe initial residual is at most $2^{-7}$.  After this rank-$8$ take,\nranks $9$ and $10$ are skipped, but the new residual exceeds $2^{-10}$.\nThe exact numerator of that dyadic excess is\n$2^9\\cdot41179-21082635=1013>0$.\nThese are counterexamples to the stated local safety implication,\nnot assertions that either state occurs on the real half-greedy orbit.\n\\par\n\\lean{safeBracket\\_and\\_numeratorMonotone\\_not\\_inductive\\_fixture}{DyadicPrefixCompression.lean:622}\n\\lean{lcmSaturated\\_safeBracket\\_not\\_inductive\\_fixture}{DyadicPrefixCompression.lean:647}\n\\end{obs}\n\n\\begin{thm}[The half-skip dichotomy via Erd\\H{o}s--Borwein irrationality]\n\\label{thm:half-skip-dichotomy}\nThe half target satisfies\n\\[\n \\frac12\\in\\mathcal A\\quad\\Longleftrightarrow\\quad\n \\mathbb N_{>0}\\smallsetminus G\\text{ is infinite},\n\\]\nwhere $G$ is the greedy support for $1/2$.  In the forward direction,\nfinitely many skipped exponents would express the full Mersenne sum as\n$1/2$ plus a finite rational sum, contradicting Erd\\H{o}s's full-support\nirrationality theorem.  The reverse implication uses the greedy\ntail criterion: once a remainder exceeds the whole available tail,\nevery later exponent is selected.  Infinitely many skips exclude that\nfailure.  The equivalence does not establish that infinitely many skips\nactually occur.\n\\end{thm}\n\n\\begin{thm}[Equivalent descriptions of half-membership]\n\\label{thm:nine-way-hub}\nEach of the following is equivalent to $1/2\\in\\mathcal A$:\n\\begin{enumerate}[label=(\\arabic*),nosep]\n\\item The integer-greedy sequence is not eventually always on branch $\\mathrm R$.\n\\item There are arbitrarily large $s\\ge5$ with $s\\notin D_{s+1}$.\n\\item Upper or middle transitions occur at arbitrarily large rows.\n\\item Some sequence $s_j\\to\\infty$ satisfies $s_j\\notin D_{s_j+1}$ for every $j$.\n\\item There are rows $s_j\\to\\infty$ and omitted ranks\n$d_j\\in\\{2,\\ldots,s_j-1\\}\\smallsetminus D_{s_j}$ with $d_j\\to\\infty$.\n\\item The positive ranks omitted by the real half-greedy rule form an infinite set.\n\\item The real half-greedy rule has no last omitted positive rank.\n\\end{enumerate}\n\\end{thm}\nItems (1)--(4) differ only by the branch classification and the choice\nof a sequence of rows.  Item (5) requires the omitted ranks themselves\nto tend to infinity, not merely their containing rows.  The cited\nidentification with the real greedy support supplies (6)--(7).\nThese equivalences do not prove that any of the conditions holds.\n\n\\begin{lem}[The values on either side of an eventual right continuation]\n\\label{lem:eventually-right-impossible}\nIf the seam eventually always extends ``true'' (right branch) from some row $S$ on with a\nfixed lower prefix $u$, the resulting cofinite-support value stays strictly below $1/2$; the matching\nalternative ``upper competitor'' word gives a strict excess \\emph{above} $1/2$. Neither of these two cofinite continuations represents $1/2$.\nThis does not rule out an eventually-right integer orbit; it describes\nthe two values in that case.  The inequalities are used in\nTheorem~\\ref{thm:final-middle-cell}.\n\\end{lem}\n\n\\begin{lem}[Each Mersenne weight exceeds its remaining tail]\n\\label{lem:mersenne-tail-weight}\nFor $n\\ge1$, let $w_n=(2^n-1)^{-1}$ and\n$R_n=\\sum_{j>n}w_j$.  Then\n\\[\n R_n=w_{n+1}+R_{n+1},\\qquad\n 2^{-n}<R_n\\le2w_{n+1}<w_n.\n\\]\nFor the upper bound, compare each tail term with\n$2^{1-j}w_{n+1}$ at index $n+j$, $j\\ge1$, and sum the\ngeometric series.  The last inequality follows by comparing\n$2/(2^{n+1}-1)$ with $1/(2^n-1)$.\nThis strict term-versus-tail inequality is the separation hypothesis\nused in the greedy arguments.\n\nIt is not available for the totient weights $\\varphi(n)/2^n$:\nthe terms at indices $4$ and $5$ already sum to\n$2/16+4/32=1/4=\\varphi(3)/2^3$, and the remaining tail is\npositive.  A general strict-tail argument therefore cannot be\ntransferred to that sequence without a different hypothesis or\nproof.\n\\end{lem}\n\n\\begin{thm}[Greedy survival and membership]\n\\label{thm:greedy-survival-catalogue}\nWith the greedy remainder $r_n(x)$ and complete tail $R_n$,\n\\[\n x\\in\\mathcal A\\quad\\Longleftrightarrow\\quad\n x\\ge0\\ \\text{and}\\ r_n(x)\\le R_n\\text{ for every }n\\ge0.\n\\]\nIf all inequalities hold, the nonnegative remainders tend to zero\nbecause $R_n\\to0$, so the greedy partial sums converge to $x$.\nConversely, strict tail domination $w_n>R_n$ forces the greedy\nchoices in any representation and therefore gives every inequality.\nThe argument applies to positive summable weights satisfying that\nseparation hypothesis; it is not a consequence of summability alone.\n\\end{thm}\n\n\\begin{lem}[The two next-prefix intervals and their gap]\n\\label{lem:rank-step-trichotomy}\nLet $d\\ge0$ be an integer, let $u\\subseteq\\{1,\\ldots,d\\}$, and suppose the target $t$ lies\nin the interval\n$[X_u(2),X_u(2)+R_d]$.\nAt depth $d+1$, either $t$ lies in the lower interval\n\\[\n [X_u(2),X_u(2)+R_{d+1}],\n\\]\nor in the upper interval\n\\[\n [X_u(2)+w_{d+1},X_u(2)+w_{d+1}+R_{d+1}],\n\\]\nor in the open gap between them.  These alternatives are disjoint\nbecause $R_{d+1}<w_{d+1}$, and exhaustive because\n$R_d=w_{d+1}+R_{d+1}$.\nFor $t=1/2$, Lemma~\\ref{lem:half-endpoint-kills} also excludes\nthe interval endpoints, so all relevant comparisons are strict.\nEndpoint exclusion is not what makes the two child intervals\ndisjoint; the strict-tail inequality does that.\n\\end{lem}\n\n\\begin{lem}[A greedy gap excludes every representation]\n\\label{lem:fatal-gap-exclusion}\nLet $d\\ge0$ be an integer and $u\\subseteq\\{1,\\ldots,d\\}$ a finite prefix.  If\n\\[\n X_u(2)+R_{d+1}<t<X_u(2)+w_{d+1},\n\\]\nthen no support agreeing with $u$ through rank $d$ represents $t$:\nomitting $d+1$ leaves value at most the lower endpoint, while\nincluding it gives value at least the upper endpoint.\nOther length-$d$ prefixes have disjoint containing intervals by\nthe first-difference argument using $w_n>R_n$.  Since the displayed\ngap lies inside the interval for $u$, none of those prefixes can\nrepresent $t$ either.  Thus a certified strict gap excludes every\nrepresentation, not just one proposed continuation.\n\\end{lem}\n\n\\begin{lem}[No finite Mersenne sum equals one half]\n\\label{lem:half-endpoint-kills}\nFor every finite $u\\subseteq\\mathbb N_{>0}$,\n$X_u(2)\\ne1/2$: its reduced denominator is odd.\nAlso $X_u(2)+R_d\\ne1/2$ for every $d\\ge0$, since $R_d$\nis the irrational full Mersenne sum minus a finite rational sum.\nThese observations exclude equality at the finite-prefix and\ncomplete-tail endpoints of Lemma~\\ref{lem:rank-step-trichotomy}.\n\\end{lem}\n\n\\begin{lem}[Straddle words are canonical: they agree with the greedy prefix]\n\\label{lem:straddle-agrees-greedy}\nLet $u\\subseteq\\{1,\\ldots,d\\}$ satisfy\n$X_u(2)\\le1/2\\le X_u(2)+R_d$.  Then\n\\[\n u=G\\cap\\{1,\\ldots,d\\},\n\\]\nwhere $G$ is the real greedy support for $1/2$.\nAt the first disagreement, if $u$ takes rank $k$ and the greedy rule\nskips it, then $X_u(2)>1/2$.  In the opposite case, the greedy\nremainder before $k$ is at least $w_k$, whereas the remaining\ncontribution allowed by $u$, including $R_d$, is at most\n$R_k<w_k$.  Both cases contradict the displayed interval.  Thus there is at most one such word\nat each depth; the lemma does not assert its existence at every depth.\n\\end{lem}\n\n\\begin{thm}[Last-skip iff local fatality: a pointwise criterion]\n\\label{thm:last-skip-iff-fatal}\nKeep $G$ for the selected half-greedy support, $r_M(1/2)$ for its\nremainder and $R_M$ for the complete tail after rank $M$.\nA positive rank $M$ is the last omitted rank if and only if\n$M\\notin G$ and $r_M(1/2)>R_M$.  Consequently,\n\\[\n \\frac12\\in\\mathcal A\n \\quad\\Longleftrightarrow\\quad\n r_M(1/2)\\le R_M\\quad\\text{for every positive }M\\notin G.\n\\]\nThus it suffices to check the tail inequality at the ranks actually\nomitted by the greedy rule.  This is an equivalence on that fixed orbit;\nit does not establish the inequality at its unboundedly many possible\nomitted ranks.\n\\end{thm}\n\n\\begin{lem}[Seam upper-or-middle classification]\n\\label{lem:seam-upper-or-middle}\nAt a row $s\\ge5$, the terminal rank is omitted in the next row,\n$s\\notin D_{s+1}$, if and only if either\n\\[\n 4o_s+p_s^+\\le2^{s+1},\n\\]\nor this inequality fails and\n\\[\n 4\\,\\mathrm{rem}(s)+2^{s+1}-p_s^-<2^{s+2}+4.\n\\]\nThese are exactly branches $\\mathrm U$ and $\\mathrm M$ of\nTheorem~\\ref{thm:dynamics}.  On either branch the new terminal weight\ndoes not fit; on $\\mathrm R$ it does.  This is an exact\nclassification, not a conjecture inferred from observed branch labels.\n\\end{lem}\n\n\\begin{lem}[The exact gap at the largest omitted rank]\n\\label{lem:largest-false-rank-algebra}\nLet $2\\le d<s$ and $2s<3d$.  Write\n$W_s(E)=\\sum_{e\\in E}\\lfloor4^s/(2^e-1)\\rfloor$ for a finite\nset $E\\subseteq\\{2,\\ldots,s-1\\}$.  For\n$u\\subseteq\\{2,\\ldots,d-1\\}$, put\n$E_-=u\\cup\\{d+1,\\ldots,s-1\\}$ and $E_+=u\\cup\\{d\\}$.\nThen\n\\[\n 3W_s(E_-)+3\\cdot2^{s+1}+2\\cdot4^{s-d}+4=3W_s(E_+).\n\\]\nThe correction is independent of the common prefix $u$.\nFor the integer-greedy rows $s\\ge5$, a right transition preserves the\nlargest omitted rank, whereas an upper or middle transition makes\n$s$ the largest omitted rank in the next row.\n\\end{lem}\n\\begin{proof}\nFor $d\\le e<s$, the inequality $2s<3d$ gives\n\\[\n \\left\\lfloor\\frac{4^s}{2^e-1}\\right\\rfloor\n =2^{2s-e}+4^{s-e}.\n\\]\nSubtract the sum over $d<e<s$ from the term at $d$.  The powers of\n$2$ leave $2^{s+1}$; the powers of $4$ leave\n$(2\\cdot4^{s-d}+4)/3$.  The common prefix $u$ cancels.\nThe branch statements follow from the support updates in\nTheorem~\\ref{thm:dynamics}: the right branch appends $s$ to the lower\nword, and the other two branches omit $s$.\n\\end{proof}\nFor example, $s=5$, $d=4$ and $u=\\{2\\}$ give\n$W_s(E_-)=341$ and $W_s(E_+)=409$; the displayed identity is\n$3\\cdot341+204=3\\cdot409$.  Both the factor $3$ on the right and\nthe restriction $2s<3d$ are essential to this formula.\n\n\n\\begin{thm}[Reduction to the nearest dyadic boundary]\n\\label{thm:critical-dyadic-band}\nLet $d,E$ be nonnegative integers with $E\\le2^{d+1}$, and let $j_*$\nbe the largest $j\\in\\{0,\\ldots,d\\}$ for which $E\\le2^{d-j+1}$.\nThe set is nonempty, so $j_*$ is well defined, including when $E=0$.\nThen\n\\[\n \\begin{aligned}\n &\\forall j\\in\\{0,\\ldots,d\\},\\quad\n  2^{d-j+1}<E\\ \\text{or}\\ E+2(d+j)\\le2^{d-j+1}\\\\\n &\\hspace{35mm}\\Longleftrightarrow\\quad\n E+2(d+j_*)\\le2^{d-j_*+1}.\n \\end{aligned}\n\\]\nFor $j>j_*$ the first alternative holds by maximality.  For $j\\le j_*$,\nthe left side of the required inequality increases with $j$ and its\nright side decreases, so the condition at $j_*$ implies all the others.\nThis is an elementary reduction from $d+1$ inequalities to one. Specialized to the seam reset charge\n(where the actual upper-reset condition gives $E\\le2^{d+1}$,\nas explained in Proposition~\\ref{record:257bm-c12}), the\nreduced hypothesis at the single index $j_*$ is proved logically equivalent to\nTheorem~\\ref{thm:upper-reset-band}'s band-avoidance hypothesis.\n\\end{thm}\n\n\\par\n\\noindent\\begin{minipage}{\\linewidth}\n\\begin{thm}[Excluding minus three at a final middle transition]\n\\label{thm:final-middle-cell}\nSuppose that row $D\\ge13$ is a middle transition in\nTheorem~\\ref{thm:dynamics} and every transition at a row $s\\ge D+1$\nis right.  Then\n\\[\n C_D:=4\\,\\mathrm{rem}(D)-p_D^--4\\ne-3.\n\\]\nThis excludes one particular value under the stated tail assumption;\nit neither excludes every final middle transition nor proves the\nall-middle-row hypothesis of Theorem~\\ref{thm:two-sided-dyadic}.\n\\end{thm}\n\\end{minipage}\\par\nTo see the contradiction, complete $D_D$ by all ranks greater than $D$.\nThe resulting support $B=D_D\\cup\\{D+1,D+2,\\ldots\\}$ has\n$X_B(2)<1/2$ under the all-right assumption.  The exact carry identity\nof the Lean proof (margin) gives $C_B(2D+1)=C_D+3$.  If $C_D=-3$, this is zero.\nThe recurrence\n$C_B(N+1)=2C_B(N)+1-c_B(N+2)$ and nonnegativity from\nTheorem~\\ref{thm:mobius-centred-nonneg} first force $C_B(2D+2)=0$:\nindeed $2D+3\\in B$, so $c_B(2D+3)\\ge1$.\nNext $D+2,2D+4\\in B$ imply $c_B(2D+4)\\ge2$, giving\n$C_B(2D+3)<0$, a contradiction.  The stronger conclusion\n$C_D\\ge-2$ is proved in Theorem~\\ref{thm:cd-neg3-impossible}.\n\n\n\\begin{lem}[The signed position of a skipped greedy prefix]\n\\label{lem:skipped-endpoint-trichotomy}\nLet $s\\ge5$ be omitted by the real greedy support $G$, and put\n\\[\n H_s=G\\cap\\{2,\\ldots,s-1\\},\\qquad\n f_s=\\sum_{d\\in H_s}\\left\\lfloor\\frac{4^s}{2^d-1}\\right\\rfloor-T_s.\n\\]\nExactly one of the following holds:\n\\[\n \\begin{array}{lll}\n f_s<0:& H_s=D_s,& f_s=-\\mathrm{rem}(s),\\quad \\mathrm{rem}(s)\\ge1;\\\\\n f_s=0:& H_s=D_s,& \\mathrm{rem}(s)=0;\\\\\n f_s>0:& H_s=B_s,& f_s=o_s.\n \\end{array}\n\\]\nThus the real greedy prefix is one of the two adjacent integer words.\nThis classifies its signed distance $f_s$ from $T_s$; it does not\nexclude the negative case.  Nor does it identify $f_s$ with the\ndifferent coordinate $C_s=4\\mathrm{rem}(s)-p_s^- -4$ used in the\nfinal-middle argument.\n\\end{lem}\n\n\\begin{lem}[Spacing of reverse-carry words]\n\\label{lem:reverse-carry-word}\nFor $i=1,2$, let integer sequences $a_i,b_i,u_i$ satisfy\n\\[\n b_i(m)+2u_i(m)=a_i(m)+u_i(m+1).\n\\]\nLet $k,L\\ge0$ be integers.  Suppose that at index $k$ the coefficients agree and\n$b_1(k)-b_2(k)=1$.  Suppose also that both coefficients and bits\nagree at indices $k+1,\\ldots,k+L$.  Then\n\\[\n u_1(k+L+1)-u_2(k+L+1)\n   =2^L\\bigl(2(u_1(k)-u_2(k))+1\\bigr).\n\\]\nThe factor in parentheses is an odd integer.  If the two terminal\ncarries have absolute values at most $B_1,B_2$, respectively, then\n$2^L\\le B_1+B_2$.  For a common bound $B$ the conclusion is\n$2^L\\le2B$, not $2^L\\le B$.\nThe formula follows by subtracting the two recurrences: the\ninitial bit difference gives the odd factor, and each subsequent\nagreement doubles it.  An application must supply the stated\nagreements and terminal bounds.\n\\end{lem}\n\n\\begin{lem}[Linear functionals factoring through one value]\n\\label{lem:linear-channel-nogo}\nLet $V$ be a vector space over $\\mathbb Q$, let\n$\\mathrm{ev}:V\\to\\mathbb Q$ be linear, and choose $e\\in V$ with\n$\\mathrm{ev}(e)=1$.  Suppose that each linear functional\n$\\ell_j:V\\to\\mathbb Q$ vanishes on $\\ker(\\mathrm{ev})$.\nThen $v-\\mathrm{ev}(v)e\\in\\ker(\\mathrm{ev})$ gives\n$\\ell_j(v)=\\ell_j(e)\\mathrm{ev}(v)$.\nConsequently any finite evaluation matrix $(\\ell_j(v_i))_{i,j}$ is an\nouter product and has rank at most one.  Every square minor of order\nat least two therefore vanishes.  This excludes determinant arguments\nformed from these particular functionals, not determinant methods\nwith additional independent information.\n\\end{lem}\n\n\\begin{thm}[Two-thirds band: exact localisation of post-take skip-unsafety]\n\\label{thm:two-thirds-band}\nWrite a positive residual as $1/R$.  A skipped rank $k$ passes\nthe sufficient dyadic test precisely when $R\\ge2^k$.\nSuppose a weight at rank $b$ is taken without exhausting the\nresidual: with $q=2^b-1$, assume $0<R<q$.\nThe new reciprocal residual is $Rq/(q-R)$.\nIf the next take is at rank $c\\ge b+2$, put $m=2^{c-1}$.\nThe last skipped rank is dyadically unsafe exactly when\n\\[\n m-1<\\frac{Rq}{q-R}<m,\n \\quad\\text{equivalently}\\quad\n \\frac{q(m-1)}{q+m-1}<R<\\frac{qm}{q+m}.\n\\]\nClearing the positive denominators proves the equivalence.\nThe interval has width $q^2/((q+m)(q+m-1))$.\n\nFor a single skipped rank, $c=b+2$ and $m=2q+2$.\nThe width is then $q^2/((3q+1)(3q+2))<1/9$, and the\ninterval lies inside $2q<3R<2q+2/3$.\nIn this single-skip case, the cited corollaries exclude integral\n$R$ and show that a reduced pre-take residual $p/(2D)$ with\n$p,D,q$ odd can be unsafe only if $p\\ge7$.\nThe latter conclusion uses the divisibility by $4$ of\n$6D-2pq$ under the unsafe-band inequalities.\n\nThese are local transition statements, not bounds on the actual\nhalf-greedy orbit.  Dyadic safety is only sufficient for the\ncurrent tail test.  The odd-coprime data $(p,D,b)=(17,41,3)$\ngive an unsafe single-skip example; no conclusion that the\nhalf-greedy orbit avoids such data follows from the band calculation.\n\\end{thm}\n\n\\begin{thm}[A weaker sufficient test against the remaining tail]\n\\label{thm:sharp-fatal-gap}\nLet $k,u,L$ be positive integers, and put\n$a=2L-(2^k-1)u$.  Suppose $a>0$, equivalently that\n$\\rho=u/(2L)<w_k$ and the greedy rule skips weight $w_k$.\nThe dyadic sufficient test $\\rho\\le2^{-k}$ is equivalent to $u\\le a$.\nThe weaker sufficient condition\n\\[\n 2u\\le3a\n\\]\nensures $\\rho<R_k$, by comparison with\n$2^{-k}+(3\\cdot4^k)^{-1}+(7\\cdot8^k)^{-1}<R_k$.\nThis conclusion excludes a fatal tail-mass deficit at the current step;\nit does not assert that $\\rho$ is representable by the remaining\nweights, or that all future greedy steps survive.\n\nThe containment of these sufficient conditions is strict even for\nvalid rational data: $(k,u,L,a)=(2,7,13,5)$ gives\n$1/4<7/26<R_2$.  In contrast, the scalar pair $(u,a)=(3,2)$ in the\nlinked inequality lemma does not arise from integral $L$ under the\npresent parity relation.  Unit numerators are nonfatal at such a\nskipped step, since then $a\\ge1$.  Conversely, a fatal step requires\n$3a<2u$, hence $u\\ge2$, or $u\\ge3$ when $u$ is odd.\n\\end{thm}\n\n\\begin{lem}[Summability of one gap length per level]\n\\label{lem:gap-mass-summability}\nFor $n\\ge1$, put $g_n=w_n-R_n>0$.  The sum of one gap\nlength per level satisfies, for $N\\ge0$,\n\\[\n \\sum_{n>N}g_n\\le\\frac29\\,4^{-N}+\\frac37\\,8^{-N}.\n\\]\nThis follows by summing the per-level upper bounds geometrically.\nIt is not the measure of the union of all gaps: level $n$ has\n$2^{n-1}$ translated gaps of length $g_n$.  The corresponding weighted\nsum and its geometric interpretation are given after\nTheorem~\\ref{record:257hg-i2}.  Neither estimate decides membership\nof a specified point such as $1/2$.\n\\end{lem}\n\n\\begin{lem}[The effect of adding one divisor]\n\\label{lem:half-divisor-unit-drop}\nLet $N\\ge0$ and let finite supports $D_0,D_1$ differ only by\n$D_1=D_0\\cup\\{N+1\\}$, with $N+1\\notin D_0$.\nFor every positive integer $m$,\n\\[\n c_{D_1}(m)-c_{D_0}(m)=\\mathbf1_{N+1\\mid m}.\n\\]\nIn particular, the difference is $1$ at $m=2(N+1)$.\nThis follows directly from the definition of a divisor count.\nIts use in another coefficient sequence requires proving that\nsequence has the same support-incidence description.\n\\end{lem}\n\n\\begin{thm}[Uniqueness of an integer recurrence with a vanishing scaled limit]\n\\label{thm:tempered-orbit-rigidity}\nFor any nonnegative-integer coefficient sequence $c:\\mathbb N\\to\\mathbb N$ with $c(n)\\le\nn$: the binary coefficient series $X_c = \\sum'_{n\\ge 1} c(n)/2^n$ is rational iff there\nexists a positive integer multiplier $v$ and an integer orbit $u:\\mathbb N\\to\\mathbb Z$\nsatisfying the exact carry recurrence $u(N{+}1)=2u(N)-v\\cdot c(N{+}1)$ together with the\ncondition $u(N)/2^N\\to0$.  Every integer sequence satisfying both conditions obeys $u(N) = v\\cdot T_c(N)$ exactly, where $T_c(N)=\\sum_{j\\ge 1} c(N{+}j)/2^j$ is\nthe scaled tail.  For each fixed $v$ there is at most one such sequence. Positivity of the orbit alone\nis deliberately \\emph{not} used as an equivalent criterion: a homogeneous $2^N$-scaled\nperturbation can be added to any orbit without breaking the recurrence, so the limit condition cannot be dropped.  The choices $c=c_A$ for\nProblem~257 and $c=\\varphi$ for Problem~249 both satisfy $0\\le c(n)\\le n$.\nThe telescoping argument is given again, with an example, in\nTheorem~\\ref{record:257bm-i-t7}.\n\\end{thm}\n\n\\begin{lem}[Changing finitely many support elements]\n\\label{lem:tail-transfer}\nLet $b\\ge2$ be an integer, and let $A,B\\subseteq\\mathbb N_{\\ge1}$.\nIf $X_A(b)$ is irrational and $A,B$ have finite symmetric difference, then $X_B(b)$ is irrational too: the two sums differ by a finite sum of\nrational numbers.  This observation supplies the finite modifications in\nTheorem~\\ref{thm:eventually-periodic}.\n\\end{lem}\n\n\\begin{lem}[An integer test for the interval between the dyadic and Mersenne weights]\n\\label{lem:dyadic-excess-reformulation}\nFor positive integers $p,L$ and $n\\ge0$, write the residual as $p/(2L)$\nand put $E=2^np-L$.  Clearing positive denominators gives\n\\[\n \\frac{p}{2L}\\le\\frac1{2^{n+1}}\\quad\\Longleftrightarrow\\quad E\\le0,\n \\qquad\n \\frac1{2^{n+1}}<\\frac{p}{2L}<\\frac1{2^{n+1}-1}\n \\quad\\Longleftrightarrow\\quad 0<E<\\frac p2.\n\\]\nThe first test concerns the dyadic bound, not the greedy selection\nthreshold itself.  The second identifies precisely the interval in\nwhich a Mersenne weight is skipped but the dyadic safety test fails.\n\\end{lem}\n\n\\begin{lem}[The odd denominator survives dyadic subtraction]\n\\label{lem:denominator-sandwich}\nLet $r/D$ be a reduced fraction with $D>0$ odd, let $p\\in\\mathbb Z$,\nand let $c\\ge0$ be an integer.  The reduced denominator of\n$p/2^c-r/D$ is divisible by $D$ and divides $2^cD$.\nIndeed, its unreduced numerator $pD-2^cr$ is coprime to $D$,\nsince $\\gcd(r,D)=\\gcd(2^c,D)=1$.  Reduction can therefore remove\nonly powers of $2$ from the displayed denominator.  No property\nof Mersenne weights is used.\n\\end{lem}\n\n\\begin{lem}[A guaranteed divisor of a reduced denominator]\n\\label{lem:denominator-survival}\nLet $a\\in\\mathbb Z$, $D\\in\\mathbb N_{>0}$ and $m,C,h\\in\\mathbb N$.\nWrite $\\operatorname{den}(x)$ for the positive reduced denominator of\na rational number $x$.  If $m\\mid D$ and $\\gcd(m,|a|)=1$, then\n$m\\mid\\operatorname{den}(a/D)$.  If $C\\mid D$ and $\\gcd(C,|a|)=1$, then\n\\[\n \\frac{C}{\\gcd(C,h)}\\mid\\operatorname{den}(ha/D).\n\\]\nThus scaling may remove part of the guaranteed divisor.\nIndeed, reduction divides $D$ by $\\gcd(D,|a|)$, so no prime power in $m$\ncan be lost. After scaling, a prime $p\\mid C$ can lose at most $v_p(h)$\npowers from the guaranteed divisor. This is a lower bound on the reduced\ndenominator, not an equality: for $D=12$, $C=6$, $a=1$, $h=2$, it guarantees\nthe divisor $3$, while the reduced denominator is $6$.\nThe lemma feeds Theorem~\\ref{thm:mersenne-channel-survival} directly.\n\\end{lem}\n\n\\begin{lem}[Commuting prime-power differences]\n\\label{lem:mixed-prime-power-layer}\nFor a function $g:\\mathbb N_{>0}\\to\\mathbb Z$, a positive integer $p$\nand an integer $e\\ge1$, define\n\\[\n (\\Delta_{p,e}g)(n)=g(p^en)-g(p^{e-1}n).\n\\]\nThese operators commute.  If $p\\ne q$ are primes, $e,f\\ge1$,\n$A\\subseteq\\mathbb N_{>0}$, $n\\ge1$ and $\\gcd(n,pq)=1$, then\n\\[\n (\\Delta_{q,f}\\Delta_{p,e}c_A)(n)\n   =\\sum_{d\\mid n}\\mathbf1_A(p^eq^fd).\n\\]\nThus the mixed difference counts precisely the support elements\nwhose $p$- and $q$-adic exponents are $e$ and $f$ in this divisor sum.\n\\end{lem}\n\\begin{proof}\nExpanding either order gives the same four terms:\n\\[\n \\begin{aligned}\n &g(p^eq^fn)-g(p^{e-1}q^fn)\\\\\n &\\qquad{}-g(p^eq^{f-1}n)+g(p^{e-1}q^{f-1}n).\n \\end{aligned}\n\\]\nThis commutation needs no primality or coprimality.  For $g=c_A$,\nwrite each divisor of $p^eq^fn$ uniquely as $p^iq^jd$, with\n$d\\mid n$, $0\\le i\\le e$ and $0\\le j\\le f$.  The two differences\ncancel all terms except $i=e$, $j=f$.\n\\end{proof}\nFor $A=\\{12\\}$, $p=2$, $e=2$, $q=3$, $f=1$, $n=1$, the\nfour counts are $1,0,0,0$, so the mixed difference is $1$.\nThis coefficient identity alone does not transfer rationality through\nthe subsequence operation $g(n)\\mapsto g(pn)$.\n\n\n\\begin{defn}[The Mersenne achievement set and the support-series object]\n\\label{defn:achievement-set}\nFor a support $A\\subseteq\\mathbb N_{\\ge1}$, its Mersenne subsum is\n\\[\n x_A=\\sum_{n\\ge1}\\frac{\\mathbf1_A(n)}{2^n-1},\n \\qquad\n \\mathcal A=\\{x_A:A\\subseteq\\mathbb N_{\\ge1}\\}.\n\\]\nFinite, empty, and infinite supports are all allowed in $\\mathcal A$.\nMore generally, for an integer base $b\\ge2$,\n$X_A(b)=\\sum_{n\\ge1}\\mathbf1_A(n)/(b^n-1)$ and\n$c_A(n)=\\sum_{d\\mid n}\\mathbf1_A(d)=(\\mathbf1_A*1)(n)$ for $n\\ge1$,\nwhere $*$ denotes Dirichlet convolution.  Full support gives the series\nin Theorem~\\ref{thm:full-support-catalogue}.  The linked definitions\nimplement these positive-index sums.\n\\par\n\\lean{mersenneAchievementSet}{GreedyAchievementSet.lean:571}\n\\lean{erdosSupportSeries}{CertificateKernel.lean:8897}\n\\lean{supportCoeff}{CertificateKernel.lean:8856}\n\\end{defn}\n\n\\begin{prop}[Achievement-set topology: compact, closed, perfect, measure exactly one]\n\\label{prop:achievement-set-topology}\n$\\ensuremath{\\mathcal A}$ is compact (continuous image of the binary-sequence\nCantor space $\\mathbb N\\to\\mathrm{Fin}\\,2$ under the product topology, via\nthe positive-index digit map), hence closed; it is also perfect, totally\ndisconnected, and nowhere dense, with Lebesgue measure exactly $1$. The\ncompactness/closedness argument (binary coding $\\to$ Cantor space $\\to$ continuous image)\nis a fully generic technique for characterizing the achievement set of \\emph{any}\nabsolutely convergent digit-weighted series, not specific to Mersenne denominators ;\nreusable for a $\\varphi(n)/2^n$ subsum set after checking summability.\nNo separation of successive weights is needed for compactness or\nclosedness; the stronger topological conclusions require their own\nhypotheses. Closedness alone is what powers every ``limit of a sequence of\nachieved points is achieved'' argument in this catalogue (e.g.\\\nTheorem~\\ref{thm:seam-limit}, Theorem~\\ref{thm:straddle-closed-set}).\n\\end{prop}\n\n\\begin{thm}[Nonmembership of one half and a finite fatal gap]\n\\label{thm:master-dichotomy}\nThe value $1/2$ is not in $\\mathcal A$ if and only if there are\nan integer $d\\ge0$ and a finite set $u\\subseteq\\{1,\\ldots,d\\}$\nsuch that\n\\[\n X_u(2)+R_{d+1}<\\frac12<X_u(2)+w_{d+1}.\n\\]\nOne implication is Lemma~\\ref{lem:fatal-gap-exclusion}.  For the\nother, follow the greedy prefixes: nonmembership forces a first\nfailure of the tail inequality, hence one of these gaps.\nThe endpoint equalities are excluded by\nLemma~\\ref{lem:half-endpoint-kills}.\n\nA finite prefix and strict gap provide a nonmembership witness.\nThe tail inequality can be certified by rational truncation bounds\nas in Theorem~\\ref{thm:one-sided}.  Membership, in contrast, asserts\nthat no such witness exists; failure to find one in a finite search\nis not a proof.  The reduction uses $w_n>R_n$, not compactness\nalone.  The totient weights fail this hypothesis, as the example\nafter Lemma~\\ref{lem:mersenne-tail-weight} shows.\n\\end{thm}\n\n\\begin{defn}[Separated integer values and bounded perturbations]\n\\label{defn:perturbed-family}\nLet $\\alpha$ be a set, let $s:\\alpha\\to\\N$ be injective, and let\n$p:\\alpha\\to\\N$.  Suppose integers $g\\ge1$ and $B\\ge0$ satisfy\n\\[\n s(x)<s(y)\\ \\Longrightarrow\\ s(x)+g\\le s(y),\n \\qquad 0\\le p(x)\\le B<3g.\n\\]\nThe updated values are $t(x)=4s(x)+p(x)$.\nFor a capacity $C\\ge0$, an adjacent pair consists of $x_-,x_+\\in\\alpha$\nwith $s(x_-)$ the largest value at most $C$ and $s(x_+)$ the\nsmallest value greater than $C$.  Existence of both values is part\nof the assumption; it is not guaranteed for an arbitrary family.\nSet\n\\[\n r=C-s(x_-),\\qquad o=s(x_+)-C,\\qquad\n p_-=p(x_-),\\quad p_+=p(x_+).\n\\]\nThe inequality $B<3g$ preserves the order of the updated values.\nThe next result uses the stronger bound $B<g$ to ensure that the\nselected updated value is admissible at capacity $4C+g$.\nThe linked structures encode these data.\n\\par\n\\lean{PerturbedFamily}{HalfCylinderIntegerGreedy.lean:1286}\n\\lean{AdjacentCut}{HalfCylinderIntegerGreedy.lean:1320}\n\\end{defn}\n\n\\begin{thm}[Perturbed-family maximality and the three-branch recurrence]\n\\label{thm:perturbed-family-maximality}\nUse the family and adjacent pair of\nDefinition~\\ref{defn:perturbed-family}, and assume in addition that\n$B<g$.  At capacity $C'=4C+g$, the largest admissible updated value\nis $t(x_+)$ if $4o+p_+\\le g$, and $t(x_-)$ otherwise.\nAfter this choice, apply the take-if-possible rule to an additional\nweight $W=2g+4$.  The resulting remainder is\n\\[\n \\begin{cases}\n g-4o-p_+,&4o+p_+\\le g,\\\\\n 4r+g-p_-,&4o+p_+>g\\ \\text{and}\\ 4r+g-p_-<W,\\\\\n 4r-g-p_--4,&4o+p_+>g\\ \\text{and}\\ 4r+g-p_-\\ge W.\n \\end{cases}\n\\]\n\\end{thm}\n\n\\begin{proof}\nThe lower candidate satisfies $t(x_-)\\le4C+B<C'$, and\n$t(x_+)\\le C'$ is exactly the test $4o+p_+\\le g$.\nOrder preservation excludes earlier values from improving the\nselected one.  Any value after $x_+$ has\n$s(x)\\ge s(x_+)+g>C+g$, and hence $t(x)>C'$.\nThus no other candidate is missed.\n\nOn the first branch the remainder is at most $g<W$, so the extra\nweight is not taken.  The other two branches compare the lower\ncandidate's remainder $4r+g-p_-$ with $W$.\nAll displayed differences are nonnegative on their respective branches.\n\\end{proof}\n\nThe theorem describes a two-stage rule, not automatically the maximum\nof $\\{t(x)+\\varepsilon W:x\\in\\alpha,\\ \\varepsilon\\in\\{0,1\\}\\}$\nbelow $C'$.  For example, take old values $4,5,7$, $g=1$, $B=0$,\n$p=0$ and $C=6$.  Then $C'=25$ and $W=6$.  The rule chooses\n$20$ and cannot add $6$, but $16+6=22$ is a larger admissible\nextended value.  A sufficient extra hypothesis for global maximality\nis $4g-B\\ge W$: every earlier updated value plus $W$ is then at\nmost the selected updated value.  In the Mersenne application\n$g=2^{s+1}$, $B=2(s-2)$ and $s\\ge5$, so this inequality holds.\nThus the concrete recurrence is unaffected, but an application to\nanother weight system must check this separation as well.\n\nThe hypothesis $B<g$ is additional; it does not follow from\n$B<3g$ in Definition~\\ref{defn:perturbed-family}.\nThe weaker bound alone does not ensure admissibility of the chosen value.\nTake $g=2$, $B=3$, old values $0,2,4$, perturbations $0,3,0$, and $C=2$.\nThe rule selects the old value $2$, whose update is $11>C'=10$,\nalthough the updated value $0$ is admissible.\nThis example explains why the stronger hypothesis must accompany\nthe maximality statement.\n% ; - part a257_p1b ; -\n\\subsection{Consequence theorems}\n\nThis subsection concerns finite sets $D$ whose floor-quotient sums nearly\nrepresent $1/2$.  Its main implication is elementary to state: if such\nfinite sums approximate $1/2$ at unbounded depths with an error tending to\nzero, compactness supplies a representing support.  The difficult input\nis the existence of the finite sets, not the final compactness argument.\nDefinitions and conditional implications are separated below.  An\nunproved hypothesis is not established merely because the\nimplication using it has a formal proof.\n\n\\subsubsection*{Finite quotients and their remainders}\n\n\\begin{defn}[An exact finite quotient sum]\\label{record:257bm-d1}\nFor an integer $n\\ge1$, an exact row at depth $n$ is a set\n$D\\subseteq\\{2,\\ldots,n\\}$ satisfying\n\\[\n Q(D,n)=\\sum_{d\\in D}\\left\\lfloor\\frac{2^n}{2^d-1}\\right\\rfloor\n       =2^{n-1}-1.\n\\]\nThe word ``exact'' refers to this integer equality, not to\n$X_D(2)=1/2$, which is impossible for finite $D$.\nThe formal definition also allows $n=0$ using truncated\nnatural-number subtraction.  That degenerate case is immaterial\nto existence at unbounded depths.\n\\par\\noindent   \\lean{ExactLocalMersenneHalfRow}{BooleanMobiusCofinalExactRows.lean:31}\n\\end{defn}\n\n\\begin{defn}[Floor quotients and the signed endpoint expression]\\label{record:257bm-d2}\n\\[\n\\ensuremath{q}(M,d) := \\left\\lfloor \\frac{2^M}{2^d-1} \\right\\rfloor,\\qquad\n\\ensuremath{Q}(D,M) := \\sum_{d\\in D} \\ensuremath{q}(M,d),\n\\]\nThe signed endpoint expression has three arguments:\n\\[\n \\ensuremath{H}(D,k,n)\n =2\\,\\ensuremath{S}(D,k,n-1)+1-c_D(n),\n \\qquad c_D(n)=\\#\\{d\\in D:d\\mid n\\}.\n\\]\nThe suffix in this formula is defined next.  The quotients are integers;\nthis final expression is allowed to be negative.\n\\par\\noindent   \\lean{localPrefixQuotient}{BooleanMobiusLocalRepair.lean:78} \\lean{localMersenneQuotient}{BooleanMobiusLocalRepair.lean:20} \\lean{localRepairInteger}{BooleanMobiusLocalRepair.lean:123}\n\\end{defn}\n\n\\begin{defn}[The remaining binary suffix]\\label{record:257bm-d3}\n\\[\n\\ensuremath{S}(D,k,M)\n :=\\max\\{2^{M-k}-\\ensuremath{Q}(D,M)-1,0\\}\n \\qquad(M\\ge k),\n\\]\nthe nonnegative remainder after subtracting the quotient contributions\nfrom the nonterminating binary expansion of $2^{-k}$ through place $M$.\nThe formal definition uses truncated natural-number subtraction; in the\napplications where the displayed difference is nonnegative, the maximum\nmay be omitted.  The formal convention for $M<k$ also truncates $M-k$\nto zero, a case not used in these capacity estimates.\n\\par\\noindent   \\lean{localBinarySuffix}{BooleanMobiusLocalRepair.lean:102}\n\\end{defn}\n\n\\begin{defn}[Exact quotient sums at unbounded depths]\\label{record:257bm-d4}\nThe cofinal exact-row condition says that for every integer $N$\nthere are $n\\ge\\max\\{N,1\\}$ and\n$D\\subseteq\\{2,\\ldots,n\\}$ with\n\\[\n Q(D,n)=2^{n-1}-1.\n\\]\nNo agreement is required between supports chosen at different depths.\nThis removes a compatibility requirement from the data to be supplied,\nnot from the underlying membership question:\nProposition~\\ref{prop:collapsed-list} identifies this cofinal\nexistence statement with half-membership.  A failed construction\nof compatible rows therefore does not refute the condition,\nand the condition itself is not proved here.\n\\par\\noindent   \\lean{CofinalExactLocalMersenneHalfRows}{BooleanMobiusCofinalExactRows.lean:38}\n\\end{defn}\n\n\\subsubsection*{Two implications yielding half-membership}\n\n\\begin{thm}[Compactness from exact finite sums]\\label{record:257bm-c1}\nSuppose the cofinal exact-row condition in\nDefinition~\\ref{record:257bm-d4} holds.  Then $1/2\\in\\mathcal A$.\nFor each $N$, choose an exact row $D_N$ at depth\n$n_N\\ge\\max\\{N,1\\}$.  Proposition~\\ref{record:257bm-i9} gives\n\\[\n \\left|X_{D_N}(2)-\\frac12\\right|\n       \\le\\frac{n_N+1}{2^{n_N}}\\longrightarrow0.\n\\]\nEvery $X_{D_N}(2)$ belongs to the closed set $\\mathcal A$, so\nits limit does too.  No finite support represents $1/2$;\nthe representing support is therefore infinite.\nThe proof does not require the chosen rows to be nested.\nThe unproved input is their existence at unbounded depths.\n\\end{thm}\n\n\\begin{thm}[Infinitely many greedy skips suffice]\\label{record:257bm-c2}\nSuppose that the greedy remainder for $1/2$ satisfies\n\\[\n \\forall N\\ \\exists c\\ge\\max\\{N,4\\}:\\qquad\n 0<r_{c-1}(1/2)<w_c.\n\\]\nThen exact quotient rows exist at unbounded depths, and $1/2\\in\\mathcal A$.\nA single such skip at $c$ gives an exact row at depth $2c-2$ without\nan additional capacity hypothesis.  Taking arbitrarily large $c$\ntherefore supplies the rows needed for the compactness implication\nin Theorem~\\ref{record:257bm-c1}.\nThe existence of arbitrarily late positive skips is the hypothesis,\nnot an established property of the greedy sequence.  It is equivalent\nto half-membership and remains unproved here.\n\\end{thm}\n\n\\subsubsection*{Compatible finite approximations}\n\n\\begin{thm}[A compatible family of finite supports]\\label{record:257bm-c3}\nLet $b_{n,d}\\in\\{0,1\\}$ satisfy $b_{n+1,d}=b_{n,d}$ whenever\n$2d\\le n$.  Define\n\\[\n E_n=\\{d:2\\le d\\le n,\\ b_{n,d}=1\\},\\qquad\n D_n=E_n\\cap\\{2,\\ldots,\\lfloor n/2\\rfloor\\}.\n\\]\nThe four conditions on the finite rows, required for every $n\\ge2$, are\n\\[\n \\begin{aligned}\n 2^{\\max\\{c_{D_n}(n)-1,0\\}}-1&\\le S(D_n,1,n-1),\\\\\n \\sum_{\\substack{d\\in E_n\\\\d>\\lfloor n/2\\rfloor}}2^{n-d}\n     &=H(D_n,1,n),\\\\\n H(D_n,1,n)&<2^{n-\\lfloor n/2\\rfloor},\\\\\n Q(E_n,n)&=2^{n-1}-1.\n \\end{aligned}\n\\]\nThe first condition implies that the integer in the next two lines is\nnonnegative.  The second specifies the binary value of the upper half of\nthe row, the third bounds that value by its available number of bits, and\nthe fourth is the exact quotient equation.  The maximum in the first line\nrecords natural-number subtraction at the case $c_{D_n}(n)=0$.\n\nThese are the finite hypotheses of the compatible-limit construction.\nThe bit condition fixes coordinate $d$ from row $2d$ onward.  Unlike the\nindependent finite supports in Definition~\\ref{record:257bm-d4}, the rows\ntherefore have a prescribed common limit.  This extra structure is not,\nby itself, a proof of strictness between the corresponding existence\nstatements.\n\\end{thm}\nThe existence of rows satisfying these four conditions for every $n\\ge2$\nis not proved.\n\n\\subsubsection*{A quotient bound at a crossing}\n\n\\begin{thm}[A sufficient quotient bound at a crossing]\\label{record:257bm-c4}\nConsider the following condition on finite sets $D$ and integers $c\\ge4$:\n\\[\n \\begin{gathered}\n D\\subseteq\\{2,\\ldots,c-1\\},\\qquad\n X_D(2)<\\frac12<X_D(2)+w_c\\\\[2pt]\n \\Longrightarrow\\qquad Q(D\\cup\\{c\\},2c-2)\\ge2^{2c-3}.\n \\end{gathered}\n\\]\nIt requires the quotient inequality at every strict crossing of one half\nby an added weight.  Theorem~\\ref{record:257bm-i5} identifies this\ninequality with the binary bound needed in the finite construction.\nIf the condition holds for all such $D,c$, the induction below produces\nexact sums at unbounded depths and hence an infinite support of value\n$1/2$.  The universal crossing condition itself is unproved.\n\\end{thm}\n\n\\begin{thm}[Induction from the depth-six example]\\label{record:257bm-c5}\nAn already-formalised induction from the endpoint-six seed (Definition~\\ref{record:257bm-i-seed}) consumes Theorem~\\ref{record:257bm-c4} at\nevery step: each protected exact row either doubles below half\n(unconditional) or recycles at its first crossing rank $e > $ cutoff, giving endpoint\n$2e-2 > $ previous endpoint; protection (endpoint $< 2\\cdot$cutoff, new ranks $>$\ncutoff) is exactly what converts the non-growing recycle endpoint of the bare dichotomy\n(Theorem~\\ref{record:257bm-k1}) into strict progress. The below-half branch never fires twice from the seed\narithmetic, so the supply is needed at essentially every step. Its conclusion is\nthe cofinal exact-row condition of Definition~\\ref{record:257bm-d4}.\n\\end{thm}\n\n\\begin{thm}[The quotient condition on greedy prefixes]\\label{record:257bm-c6}\nLet $D_c=G\\cap\\{2,\\ldots,c-1\\}$.  The condition on the actual greedy\nprefixes is\n\\[\n \\begin{gathered}\n c\\ge4,\\quad c\\notin G\\\\\n \\Longrightarrow\\qquad Q(D_c\\cup\\{c\\},2c-2)\\ge2^{2c-3}.\n \\end{gathered}\n\\]\nIt asks for the quotient bound at every skipped rank of the specified\nhalf-greedy orbit.  This restricts the crossing test of\nTheorem~\\ref{record:257bm-c4} to its actual greedy prefixes.  The finite\nuniqueness results justify that restriction where the cited induction\nuses it; they do not prove the displayed inequality.\n\\end{thm}\n\n\\begin{thm}[Two consecutive skips]\\label{record:257bm-c6a}\nIf $c\\ge6$ and the real half-greedy rule skips both $c$ and $c+1$,\nput $D=G\\cap\\{2,\\ldots,c-1\\}$.  Then\n\\[\n S(D,1,2c-3)<2^{c-3}.\n\\]\nThe one-step quotient recurrence gives\n$S(D,1,2c-2)<2^{c-2}$, so\nTheorem~\\ref{record:257bm-c7} supplies an exact row at depth $2c-2$\nwhose new ranks are all greater than $c$.\nThe hypothesis concerns this pair of skipped ranks; no unbounded\nsequence of such pairs is proved here.  For the stated precritical-suffix\ncondition at every skipped rank, the remaining tests are the\nskip-then-take cases, with ranks $4$ and $5$ handled separately in\nthe cited proof.\n\\end{thm}\n\\begin{proof}\nThe two skips imply $0<\\rho=1/2-X_D(2)<w_{c+1}$.\nPut $M=2c-3$.  The fractional-part error in $Q(D,M)$ is less\nthan $c-2$, because $D\\subseteq\\{2,\\ldots,c-1\\}$.  Therefore\n\\[\n S(D,1,M)\n <\\frac{2^{2c-3}}{2^{c+1}-1}+c-3\n <2^{c-4}+c-2\\le2^{c-3},\n\\]\nwhere the last inequality holds for $c\\ge6$.\nThe suffix is nonnegative because $X_D(2)<1/2$, so its next-step\nidentity is $S(D,1,M+1)=2S(D,1,M)+1-c_D(M+1)$.\nIntegrality gives the required strict bound at $M+1$.\n\\end{proof}\n\n\n\\begin{thm}[A later skip after a selected block]\\label{record:257bm-c6b}\nSuppose the real half-greedy rule skips rank $c$, takes\n$c+1,\\ldots,c+t-1$, and skips $c+t$.  If\n\\[\n 0<t\\le c-3,\\qquad c-2\\le2^{c-t-3},\n\\]\nthen the precritical suffix bound at $c$ holds.  It yields sharp\ncapacity and hence an exact row at depth $2c-2$.\nThe case $t=1$ includes Theorem~\\ref{record:257bm-c6a}; the proof\nalso permits longer selected blocks.  The arithmetic condition is\nprecisely $t\\le c-3-\\lceil\\log_2(c-2)\\rceil$.\nIts role in the proof is to make the dyadic allowance $2^{c-t-3}$\ncover the bound $|D|\\le c-2$ for the earlier selected support.\n\nTo obtain cofinal exact rows by this result, the stated gap condition\nmust hold at cofinally many skipped ranks $c$, not just at one rank\nor throughout a finite sample.  A finite empirical skip frequency\nalone gives no such pointwise gap bound.\n\\end{thm}\n\n\\subsubsection*{Filling the remaining binary positions}\n\n\\begin{thm}[Filling the remaining binary positions]\\label{record:257bm-c7}\nFor any $c\\ge4$ and any finite $D\\subseteq[2,c)$ with $\\mathrm{value}(D)<1/2$ and the sharp\ncapacity\n\\[\n\\ensuremath{S}(D,1,2c-2) < 2^{c-2},\n\\]\nthere is an exact row $E$ at endpoint $2c-2$ with $D\\subseteq E$ and every new rank strictly\nabove $c$. The implication requires the displayed capacity bound,\nbut neither a crossing condition nor a specified real greedy prefix.\nThe crossing hypothesis used in Theorem~\\ref{record:257bm-c4} is\none way to seek that input, not an additional premise of this filling\nresult.  No strict comparison between the corresponding cofinal\nexistence statements is asserted.\n\\end{thm}\n\n\\begin{thm}[An exact sum at depth $2c-2$]\\label{record:257bm-c8}\nUnder the hypotheses of Theorem~\\ref{record:257bm-c7}, there is a\nset $E\\subseteq\\{2,\\ldots,2c-2\\}$ with\n\\[\n Q(E,2c-2)=2^{2c-3}-1.\n\\]\nThis conclusion forgets the additional support-extension information\nin that theorem.  It is the form used when only the existence of an\nexact row is needed.\n\\end{thm}\n\n\\begin{prop}[Quotients in the upper half of the index range]\\label{record:257bm-c9}\nFor integers $d\\ge2$ and $d\\le M<2d$,\n\\[\n q(M,d)=\\left\\lfloor\\frac{2^M}{2^d-1}\\right\\rfloor=2^{M-d}.\n\\]\nIndeed, the quotient is $2^{M-d}+2^{M-d}/(2^d-1)$, whose second\nterm is strictly between zero and one.  In\nTheorem~\\ref{record:257bm-c7}, the available ranks\n$d=c+1,\\ldots,2c-2$ therefore supply the binary weights\n$2^{c-3},\\ldots,1$.  They represent every integer from $0$ to\n$2^{c-2}-1$, which explains both the capacity bound and the\nchoice of terminal depth.\n\\end{prop}\n\n\\begin{thm}[A finite sum from a skipped prefix]\\label{record:257bm-c10}\nLet $c\\ge4$ and $D\\subseteq\\{2,\\ldots,c-1\\}$ satisfy\n\\[\n 0<\\frac12-X_D(2)<\\frac1{2^c-1}.\n\\]\nThen there is $E$ with $D\\subseteq E\\subseteq\\{2,\\ldots,2c-2\\}$ and\n$Q(E,2c-2)=2^{2c-3}-1$.  No separate sharp-capacity assumption is\nrequired.  The binary completion in the linked proof uses ranks\n$c,\\ldots,2c-2$; it may therefore insert $c$.\nTheorem~\\ref{record:257bm-c7} instead assumes the sharper\n$(c-2)$-bit bound so that every added rank is strictly greater than $c$.\n\\end{thm}\n\nFor a finite $F\\subseteq\\{2,\\ldots,n\\}$ with $X_F(2)>1/2$, take\n$c$ to be its first crossing rank and $D=F\\cap\\{2,\\ldots,c-1\\}$.\nThen $c\\ge4$ because $1/3+1/7<1/2$, and the displayed hypotheses\nhold: equality at the lower endpoint is excluded by odd denominators.\nThe Lean proofs use the theorem in exactly this first-crossing form.\n\n\\subsubsection*{Reducing the dyadic-boundary checks}\n\n\\begin{thm}[Testing one nearest dyadic boundary]\\label{record:257bm-c11}\nLet $d,E$ be nonnegative integers with $E\\le2^{d+1}$, and let\n\\[\n j_*:=\\max\\{0\\le j\\le d:E\\le2^{d-j+1}\\}.\n\\]\nThe boundary $2^{d-j_*+1}$ is the smallest power at least $E$ among\nthe finite list $2,4,\\ldots,2^{d+1}$; in particular $j_*=d$ when $E\\le2$.\nThe following conditions are equivalent:\n\\[\n \\begin{gathered}\n \\text{for every }0\\le j\\le d,\\quad\n 2^{d-j+1}<E\\ \\text{or}\\ E+2(d+j)\\le2^{d-j+1};\\\\\n E+2(d+j_*)\\le2^{d-j_*+1}.\n \\end{gathered}\n\\]\n\\end{thm}\n\\begin{proof}\nFor $j>j_*$ the boundary is below $E$; for $j<j_*$ the boundary\nis larger and the required width $2(d+j)$ is smaller.  This proves\nsufficiency of the single inequality; necessity is its instance\nat $j_*$.\n\\end{proof}\n\nFor example, $d=5$, $E=20$ gives $j_*=1$ and\n$20+12=32$, so equality at the allowed edge satisfies every band condition.\n\nAt an actual upper reset, Proposition~\\ref{record:257bm-c12}\nsupplies $E\\le2^{d+1}$.  The equivalence replaces all its local\nband tests by one test.  The arithmetic inequality at every required\nreset remains an unproved hypothesis of the membership implication.\n\n\\begin{prop}[An upper bound for the reset expression]\\label{record:257bm-c12}\nLet $d\\ge5$ be an actual upper-reset row.  Write $E_d$ for its\nreset charge, the sum of four times the adjacent upper overshoot and\nthe corresponding nonnegative correction term.  The upper-reset identity is\n\\[\n \\mathrm{rem}(d+1)+E_d=2^{d+1}.\n\\]\nThe nonnegativity of the successor remainder therefore gives\n$E_d\\le2^{d+1}$, the side condition needed for\nTheorem~\\ref{record:257bm-c11}.  This argument uses the upper-reset\nbranch assumption.  It does not require the separate, conditional\ntwo-sided bound of Theorem~\\ref{thm:two-sided-dyadic}, and does not\nassert the charge bound at arbitrary rows.\n\\end{prop}\n\n\\subsubsection*{Nonnegativity of the omitted-tail margin}\n\nThe next three statements compare the conditions of\nTheorem~\\ref{thm:frozen-margin}.  The first two are equivalent; the third\nis sufficient for them.\n\n\\begin{defn}[A nonnegativity condition at skipped ranks]\\label{record:257bm-c13}\n\\[\n\\forall\\ \\text{skipped rank } n\\ge3,\\quad 0 \\le \\ensuremath{F}(n-1,n).\n\\]\nThis condition is not established here.  Its usefulness is the following implication:\nassuming it, a Lean theorem proves\n$1/2\\in\\ensuremath{\\mathcal A}$.\n\\par\\noindent   \\lean{HalfGreedySkippedFullShellNonnegative}{HalfCylinderFullShellSeamBridge.lean:604} \\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_skippedFullShellNonnegative}{HalfCylinderFullShellSeamBridge.lean:633}\n\\end{defn}\n\n\\begin{thm}[An equivalent vanishing condition]\\label{record:257bm-c14}\n\\[\n\\begin{gathered}\n \\forall\\ \\text{skipped rank }n\\ge3\\text{ whose actual word equals the seam-greedy word},\\\\\n \\text{the integer remainder at }n\\text{ is zero}.\n\\end{gathered}\n\\]\nThis condition is equivalent to Definition~\\ref{record:257bm-c13}.\nAt an actual skipped rank, the source proves that\n$F(n-1,n)<0$ holds exactly when the real prefix agrees with the\ninteger greedy word and its integer remainder is positive.\nOn agreement, $F(n-1,n)$ is the negative of that nonnegative\nremainder.  Thus nonnegativity of the margin at every skip is\nequivalent to vanishing of the remainder at every aligned skip.\nThe equivalence is proved; neither condition is established for all\nrequired ranks.\n\\end{thm}\n\n\\begin{thm}[A sufficient lower bound for the integer remainder]\\label{record:257bm-c15}\n\\[\n\\forall\\ \\text{skipped rank }n\\ge3,\\quad B(2n) < \\mathrm{rem}(n),\n\\]\na sufficient condition for the nonnegativity condition above.\nIndeed, a negative margin at an actual skip would force\n$1\\le\\mathrm{rem}(n)\\le B(2n)$, contradicting this bound.\nThis proves the implication, not a converse or strict separation of the conditions.\n\\end{thm}\nThe displayed lower bound itself is not established.\n\n\\subsubsection*{Square-root bounds for half-carries}\n\n\\begin{thm}[A square-root carry bound implies half-membership]\\label{record:257rig-c16}\nWrite $G$ for the greedy support and $C_G(N)$ for its centred carry. If\n\\[\n\\forall N,\\qquad C_G(N) \\le 2\\sqrt N + 4,\n\\]\nthen the greedy selected support $G$ is infinite and $X_G(2)=1/2$.\nNonnegativity of $C_G$ is unconditional (see\nTheorem~\\ref{thm:mobius-centred-nonneg} and the greedy case above); only the upper\n$2\\sqrt N+4$ bound remains open. This is a different hypothesis from the dyadic-band\ncondition and the largest-skip condition; logical independence is not\nasserted.\n\\end{thm}\n\n\\begin{thm}[Terminal bounds at unbounded depths]\\label{record:257rig-c17}\nSuppose that for every $N\\ge0$ there are $M\\ge\\max\\{N,1\\}$ and\n$D\\subseteq\\{2,\\ldots,M\\}$ such that\n\\[\n |\\operatorname{ihc}(D,M-1)|\\le B(M)\n       =2\\lfloor\\sqrt M\\rfloor+4.\n\\]\nThen some infinite $A\\subseteq\\Npos$ satisfies $X_A(2)=1/2$.\nNo compatibility between different finite supports or bound on their\nearlier carries is required.\n\nThese are not the exact-row conditions of\nDefinition~\\ref{record:257bm-d4}.  For such a finite support,\n\\[\n \\operatorname{ihc}(D,M-1)=2^{M-1}-Q(D,M),\n\\]\nso an exact row has carry $1$, whereas the displayed bound allows\nseveral positive and negative values.  At $M=6$, for example,\n$D=\\{2,3\\}$ has $Q(D,6)=30$ and carry $2$: it satisfies the\nterminal bound but is not an exact row.\nBoth cofinal existence statements nevertheless imply, and are\nimplied by, half-membership, as explained in\nProposition~\\ref{prop:collapsed-list}.  The difference between their\nfinite data must not be confused with a strict logical weakening\nof the membership problem.\n\\end{thm}\nThe hypothesis is not established.\n\n\\begin{thm}[Cofinal returns of the greedy carry]\\label{record:257rig-c18}\nSuppose that for every $N\\ge0$ there is $M\\ge N$ such that\n\\[\n \\operatorname{ihc}(G,M)\\le B(M+1)\n       =2\\lfloor\\sqrt{M+1}\\rfloor+4.\n\\]\nThen $G$ is infinite and $X_G(2)=1/2$.  Unlike\nTheorem~\\ref{record:257rig-c16}, this hypothesis bounds the actual greedy\ncarry only at unboundedly many indices.  It still concerns the same fixed\nsupport $G$, not independently chosen finite supports.\nComparing this carry with the carry obtained by fixing a finite prefix introduces a contribution from later omitted ranks. No equivalence with the finite-support hypothesis\nof Theorem~\\ref{record:257bm-c7} follows without controlling that\nadditional contribution. The hypothesis is not established.\n\\end{thm}\n\n\\subsubsection*{A margin at an unspecified later horizon}\n\n\\begin{thm}[An eventual nonnegative margin suffices]\\label{record:257bm-c19}\nFix a positive depth $k$ and put $D=G\\cap\\{2,\\ldots,k\\}$.\nThe following conditions are equivalent:\n\\[\n r_k(1/2)<2^{-(k+1)}\n \\quad\\Longleftrightarrow\\quad\n F_k(J)\\ge0\\ \\text{for some }J\\ge0.\n\\]\nHere $F_k(J)$ is the integer expression defined in\nTheorem~\\ref{thm:frozen-margin}.  Its normalised value is\n\\[\n 2^{-J}F_k(J)=\\sum_{i=1}^{J}c_D(k+1+i)2^{-i}-C_D(k).\n\\]\nThis normalised expression, not necessarily $F_k(J)$ itself, is\nnondecreasing in $J$.  Its limit is\n$1-2^{k+1}r_k(1/2)$, by the finite-support Lambert identity.\nThis is positive exactly under the strict dyadic inequality on the left.  At positive $k$, oddness of the\nreduced excess numerator excludes equality at zero.\n\nThe result supplies some finite horizon, not the horizon $J=c-3$\nrequired in the crossing application.  More quantitatively, write\n$\\eta=1-2^{k+1}r_k(1/2)>0$.  Since\n$\\sum_{r\\ge1}c_D(m+r)2^{-r}\\le m+2$, the omitted tail shows that\n$(k+J+3)2^{-J}<\\eta$ suffices for $F_k(J)>0$.\nThe value $\\eta$ is rational for the finite support $D$, so this is an\neffective sufficient test.  Proving it at the prescribed horizon is a\nseparate arithmetic obligation.\n\\end{thm}\n\n\\subsubsection*{Equivalent conditions for infinitely many greedy skips}\n\n\\begin{thm}[Half-membership and infinitely many skips]\\label{record:257bm-c20}\n\\[\n\\tfrac12\\in\\ensuremath{\\mathcal A} \\;\\Longleftrightarrow\\; \\text{the set of ranks skipped by the greedy rule for }\\tfrac12\\text{ is infinite}.\n\\]\nOdd-denominator parity supplies the positivity clause in Theorem~\\ref{record:257bm-c2} automatically. Thus its\ncofinal-positive-skip hypothesis is exactly the assertion that the greedy skipped support is\ninfinite, which the displayed theorem identifies with half-membership. See Observation~\\ref{record:257bm-k5} for the\nreason this restatement does not itself prove membership.\n\\end{thm}\n\n\\subsection{Invariants}\n\nThis subsection collects identities and bounds used in the preceding\ncriteria.  Each statement retains its own hypotheses; a bound for one\nbranch is not an invariant of the entire recurrence.\n\n\\subsubsection*{One-step quotient identities}\n\n\\begin{thm}[The next floor quotient]\\label{record:257bm-i1a}\nFor $d\\ge2$ and $M\\ge0$, the floor quotient satisfies\n\\[\n q(M+1,d)=2q(M,d)+\\mathbf1_{d\\mid M+1}.\n\\]\nIndeed, $q(M,d)=\\sum_{j=1}^{\\lfloor M/d\\rfloor}2^{M-jd}$.\nIncreasing $M$ doubles each existing term and adds $1$ exactly when\n$M+1$ is divisible by $d$.  Thus a quotient doubles or doubles plus\none; it does not remain unchanged except when both values are zero.\nSumming over a fixed finite support gives the corresponding update\nwith added term $c_D(M+1)$.\n\\end{thm}\n\n\\begin{thm}[The next quotient sum]\\label{record:257bm-i1b}\nFor a fixed finite set $D\\subseteq\\{2,3,\\ldots\\}$ and $M\\ge0$,\n\\[\n Q(D,M+1)=2Q(D,M)+c_D(M+1).\n\\]\nThis is the sum of Theorem~\\ref{record:257bm-i1a} over $d\\in D$.\nKeeping $D$ fixed is essential: changing the support between steps\nadds a separate difference of quotient sums.  The exclusion of\n$d=1$ is also essential for the displayed correction term, since\n$q(M,1)=2^M$ doubles without an added unit.\n\\end{thm}\n\n\\begin{thm}[The signed endpoint recurrence]\\label{record:257bm-i1c}\nThe signed next-step expression is\n$H(D,k,n)=2S(D,k,n-1)+1-c_D(n)$.  The last term counts the selected\nexponents dividing the new endpoint. This is the same recurrence shape as the\ngeneric tempered-orbit recurrence $u(N+1)=2u(N)-v\\cdot c(N+1)$ from the\ntempered-orbit criterion (Theorem~\\ref{record:257bm-i-t7} below), specialised to Mersenne\nlocal repair. A matching recurrence alone does not imply rationality:\nTheorem~\\ref{record:257bm-i-t7} also requires one fixed coefficient\nsequence, integer states and a vanishing scaled limit.  Those additional\nconditions must be checked before that criterion can be applied to these\nfinite-row quantities.\n\\end{thm}\n\n\\subsubsection*{Bounds for the remaining binary positions}\n\n\\begin{defn}[The sharper binary bound]\\label{record:257bm-i-cap}\nLet $c\\ge4$, $D\\subseteq\\{2,\\ldots,c-1\\}$, and $X_D(2)<1/2$.\nThe sharper bound is the condition\n\\[\n S(D,1,2c-2)<2^{c-2}.\n\\]\nIt permits a binary completion using ranks $c+1,\\ldots,2c-2$,\nwith rank $c$ omitted, as explained below.\n\\par\\noindent   \\lean{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_criticalCapacity\\_iff}{BooleanMobiusSkippedCoreCriticalCapacity.lean:22}\n\\end{defn}\n\n\\begin{thm}[Equivalent forms of the sharper bound]\\label{record:257bm-i5}\nUnder the hypotheses of Definition~\\ref{record:257bm-i-cap}:\n\\[\n\\ensuremath{S}(D,1,2c-2) < 2^{c-2} \\;\\Longleftrightarrow\\; 2^{(2c-2)-1} \\le \\ensuremath{Q}(D\\cup\\{c\\},\\ 2c-2).\n\\]\nThe threshold on the right is one greater than the exact-row target\n$2^{2c-3}-1$.  Thus this is a test for omitting rank $c$, not for\nincluding it in the completed row.\n\\end{thm}\n\n\\begin{proof}\nPut $M=2c-2$.  Since $X_D(2)<1/2$, one has\n$Q(D,M)<2^{M-1}$, and the integer remainder is\n$S(D,1,M)=2^{M-1}-Q(D,M)-1\\ge0$ without truncation.\nThe geometric quotient identity gives\n$q(M,d)=2^{M-d}$ for $c\\le d\\le M$, since $2d>M$.\nIn particular $q(M,c)=2^{c-2}$, so\n\\[\n S(D,1,M)<2^{c-2}\n \\quad\\Longleftrightarrow\\quad\n Q(D\\cup\\{c\\},M)\\ge2^{M-1}.\n\\]\nFor the completion, use the ranks $c+1,\\ldots,M$ instead.\nTheir quotient weights are $2^{c-3},\\ldots,1$ and represent every\ninteger from $0$ to $2^{c-2}-1$.  A subset of them therefore has\nquotient sum $S(D,1,M)$ and completes $D$ to the exact target\n$2^{M-1}-1$ while omitting $c$.  These finite identities prove\nthe equivalence and explain the sharper bound.  They do not assert\nthat a completed row stays below one half in real value.\n\\end{proof}\n\nFor example, $D=\\{2,3\\}$ and $c=5$ give $M=8$, $Q(D,8)=121$\nand remainder $6<8$.  Inserting rank $5$ adds $8$, giving $129$\nrather than the exact target $127$.  The correct completion adds\nranks $6$ and $7$, whose quotients are $4$ and $2$.\n\n\\begin{thm}[An unconditional bound with one extra bit]\\label{record:257bm-i6}\nUnconditionally, for every $c\\ge4$ and every below-half core $D\\subseteq[2,c)$ with deficit\n$<\\ensuremath{w}(c)$:\n\\[\n\\ensuremath{S}(D,1,2c-2) < 2^{c-1}.\n\\]\nThe bound holds uniformly in $c$, but is twice the threshold needed\nin Theorem~\\ref{record:257bm-i5} to apply Theorem~\\ref{record:257bm-c7}.\nBefore using $|D|\\le c-2\\le2^{c-2}$, the proof gives the sharper\nadditive estimate\n$\\ensuremath{S}(D,1,2c-2)<2^{c-2}+|D|$.\nThus the critical-capacity inequality would follow by excluding the\ninteger band $[2^{c-2},\\,2^{c-2}+c-3]$, which contains $c-2$ integers. This is the same band\nshape as Theorem~\\ref{record:257bm-c11}'s dyadic-band condition.\nIts width is linear in $c$, whereas the square-root reset condition\ndiscussed in Section~\\ref{sec:o4} has width $2^{(r+5)/2}$ in a\ndifferent parameter $r$.  Comparing these widths alone proves no implication\nbetween the two hypotheses.\n\\end{thm}\n\n\\subsubsection*{Doubling the endpoint}\n\n\\begin{thm}[The binary bound after doubling]\\label{record:257bm-i7}\nLet $n\\ge6$ and $D\\subseteq\\{2,\\ldots,n\\}$ satisfy\n$2\\in D$, $Q(D,n)=2^{n-1}-1$, and $X_D(2)<1/2$.  Then\n\\[\n S(D,1,2n-1)<2^{n-1}.\n\\]\nThe missing quotient can therefore be filled using only the ranks\n$n+1,\\ldots,2n-1$, without changing $D$.\n\\end{thm}\n\n\\begin{proof}\nWrite $F_M=\\sum_{d\\in D}\\{2^M/(2^d-1)\\}$ and $P=2^{n-1}$.\nThe exact row at $n$ gives\n$2^n(1/2-X_D(2))=1-F_n$.\nAt $M=2n-1$ this becomes\n\\[\n S(D,1,2n-1)=P(1-F_n)+F_{2n-1}-1.\n\\]\nThe below-half assumption makes this an ordinary integer difference,\nso no truncation is needed.  Since $2\\in D$, one has $F_n\\ge1/3$;\nalso $F_{2n-1}\\le|D|\\le n-1$.  Therefore\n\\[\n S(D,1,2n-1)\\le\\frac23P+n-2<P,\n\\]\nusing $3(n-2)<2^{n-1}$ for $n\\ge6$.\nThe ranks $n+1,\\ldots,2n-1$ supply exactly $n-1$ binary digits,\nwith total capacity $2^{n-1}-1$.  The binary expansion of the\nnonnegative missing quotient gives the required extension.\n\\end{proof}\n\nFor example, $n=6$ and $D=\\{2,3,6\\}$ give $X_D(2)=31/63$,\n$Q(D,11)=1006$ and missing quotient $1023-1006=17$.\nThe ranks $7$ and $11$ supply $16+1$.\nThe resulting exact row $E=\\{2,3,6,7,11\\}$ has\n$X_E(2)-1/2=13955/32756094>0$.\nThus the extension is exact at its integer target, but need not\nremain below half in real value.\n\n\\begin{prop}[Every exact sum contains exponent two]\\label{record:257bm-i-rank2}\nLet $n\\ge3$ and let $D\\subseteq\\{2,\\ldots,n\\}$ satisfy\n$Q(D,n)=2^{n-1}-1$.  Then $2\\in D$.\nThus the rank-two hypothesis in Theorem~\\ref{record:257bm-i7}\nis automatic for its exact rows.  The interval restriction on\n$D$ is part of the assertion, not an assumption about arbitrary\nfinite quotient sums.\n\\end{prop}\n\n\\subsubsection*{The depth-six example}\n\n\\begin{defn}[An exact sum at depth six]\\label{record:257bm-i-seed}\nThe set $\\{2,3,6\\}$ satisfies\n\\[\n\\ensuremath{Q}(\\{2,3,6\\},6) = 2^5-1 = 31,\\qquad \\ensuremath{V}(\\{2,3,6\\}) < 1/2,\n\\]\nhence it is an exact row at depth $6$ (Definition~\\ref{record:257bm-d1}).\nBoth facts are direct numerical checks.\nThis is the chosen seed for the $n\\ge6$ transition theorem; the induction\nstarts at this depth.  It is not the first\nexact quotient row without that restriction: at $n=2$, $D=\\{2\\}$ already\ngives $Q(D,2)=1=2^{2-1}-1$.\n\\par\\noindent   \\lean{exactRowSixSupport}{BooleanMobiusExactRowSeed.lean:14} \\lean{exactLocalMersenneHalfRow\\_six}{BooleanMobiusExactRowSeed.lean:34}\n\\end{defn}\n\n\\subsubsection*{Approximation error and finite denominators}\n\n\\begin{prop}[Error in the finite subseries value]\\label{record:257bm-i9}\nAn exact row at endpoint $n$ has real value within $O(n/2^n)$ of $1/2$: the quantitative bound\n$|y_n-1/2|\\le(n+1)/2^n$ consumed by Theorem~\\ref{record:257bm-c1} to turn a sequence of exact rows tending to depth\ninfinity into a sequence of values tending to $1/2$.\n\\end{prop}\n\n\\begin{prop}[No finite support has value one half]\\label{record:257bm-i10}\nA finite sum of reciprocals of odd integers has odd denominator in\nlowest terms, so it cannot equal $1/2$. Applied to the denominators\n$2^a-1$, this proves that a support representing $1/2$ must be infinite,\nas used in Theorem~\\ref{record:257bm-c1}.\n\\end{prop}\n\n\\subsubsection*{Uniqueness for gap-dominated finite weights}\n\n\\begin{thm}[Small remainder forces the greedy word]\\label{record:257bm-i11a}\nLet $g\\ge1$ and let $w_1,\\ldots,w_m$ be nonnegative integer weights\nsatisfying\n\\[\n w_i\\ge g+\\sum_{j>i}w_j\\qquad(1\\le i\\le m).\n\\]\nFix an integer capacity $C\\ge0$. The greedy word $\\gamma$ is obtained\nby taking $w_i$ exactly when it does not exceed the remaining capacity.\nFor an admissible Boolean word $\\varepsilon$, meaning\n$\\sum_i\\varepsilon_iw_i\\le C$, one has\n\\[\n C-\\sum_i\\varepsilon_iw_i<g\n \\quad\\Longleftrightarrow\\quad\n \\varepsilon=\\gamma\\ \\text{and}\\\n C-\\sum_i\\gamma_iw_i<g.\n\\]\nIndeed, two different words have sums separated by at least $g$, as\nseen at their first differing index; the greedy sum is the largest\nadmissible sum. Thus there is at most one admissible word with remainder\nbelow $g$, and any such word is greedy. The greedy word alone need not\nhave remainder below $g$: for weights $(5)$, gap $g=2$ and capacity\n$C=3$, its remainder is $3$.\n\\end{thm}\n\n\\begin{prop}[Gap domination of the integer weights]\\label{record:257bm-i11b}\nFor integers $1\\le d\\le R\\le M$, the quotient weights satisfy\n\\[\n \\left\\lfloor\\frac{2^M}{2^d-1}\\right\\rfloor\n \\ge 2^{M-R}+\n \\sum_{j=d+1}^{R}\\left\\lfloor\\frac{2^M}{2^j-1}\\right\\rfloor.\n\\]\nThus the finite weight list at ranks $2,\\ldots,R$ satisfies the\nhypothesis of Theorem~\\ref{record:257bm-i11a} with $g=2^{M-R}$.\nAt $M=2R-1$ this gap is $2^{R-1}$, and at $M=2R$ it is $2^R$.\nThe inequality is unconditional; it does not assert that the greedy\nremainder is smaller than the gap.\n\\end{prop}\n\n\\begin{prop}[Uniqueness of the finite greedy representation]\\label{record:257bm-i11c}\nLet $1\\le M$, $0\\le R\\le M$, and $0\\le a<2^{M-R}$. If\n\\[\n \\sum_{d=2}^{R}\\varepsilon_d\n       \\left\\lfloor\\frac{2^M}{2^d-1}\\right\\rfloor+a\n       =2^{M-1}-1,\\qquad \\varepsilon_d\\in\\{0,1\\},\n\\]\nthen $\\varepsilon$ is the greedy word for capacity $2^{M-1}-1$\nand $a$ is its remainder. This follows by applying\nTheorem~\\ref{record:257bm-i11a} with the preceding gap inequality.\nThe statement identifies any such representation; it does not prove\nthat a representation with $a<2^{M-R}$ exists. The degenerate formal\ncase $M=0$ has empty word and zero target, using truncated natural-number\nsubtraction.\n\\end{prop}\n\n\\subsubsection*{A geometric form of the quotient condition}\n\n\\begin{thm}[A division-free form of the binary bound]\\label{record:257bm-i12}\nFor $M\\ge0$ and $d\\ge2$,\n\\[\n \\left\\lfloor\\frac{2^M}{2^d-1}\\right\\rfloor\n   =\\sum_{j=1}^{\\lfloor M/d\\rfloor}2^{M-jd},\n\\]\nwhere an empty sum is zero. To see this, write $M=qd+r$ with\n$0\\le r<d$ and expand the finite geometric sum. The remaining fraction\nis $2^r/(2^d-1)$, which lies strictly between $0$ and $1$.\nSubstitution in Theorem~\\ref{record:257bm-i5} expresses its quotient\ncondition as a finite sum of powers of $2$. This is an exact rewriting,\nnot a weaker hypothesis or a new existence result.\n\\end{thm}\n\n\\subsubsection*{Bounds for a general coefficient sequence}\n\\papersectiontarget{record257:coefficient-recurrence}\nRationality of a binary coefficient series gives an integer recurrence.\nThe necessary decay condition is on the scaled recurrence, and allows\nunbounded integer values. We first bound the coefficient tail, then\nrecover the series value by telescoping. For support series, M\\\"obius\ninversion imposes an additional condition on the coefficients.\n\n\n\\begin{thm}[An upper bound for a binary coefficient tail]\\label{record:257bm-i2}\nLet $c:\\N\\to\\N$ satisfy $c(n)\\le n$ for every $n$.  Then\n\\[\n \\sum_{r\\ge1}c(N+r)2^{-r}\\le N+2\\qquad(N\\ge0).\n\\]\nIndeed, $c(N+r)\\le N+r$, while $\\sum_{r\\ge1}2^{-r}=1$ and\n$\\sum_{r\\ge1}r2^{-r}=2$.  Thus the scaled tail is $O(N)$, and in\nparticular $o(2^N)$, as required in Theorem~\\ref{record:257bm-i-t7}.\nThe coefficient bound also holds for Euler's totient function.\n\\end{thm}\n\n\\begin{thm}[Rationality through an integer recurrence]\\label{record:257bm-i-t7}\nLet $c:\\N\\to\\N$ satisfy $c(n)\\le n$ for every $n$.  Then\n$\\sum_{n\\ge1}c(n)2^{-n}$ is rational if and only if there exist a positive\ninteger $v$ and a sequence $u:\\N\\to\\Z$ such that\n\\[\n u(N+1)=2u(N)-v c(N+1)\\quad(N\\ge0),\\qquad\n \\frac{u(N)}{2^N}\\longrightarrow0.\n\\]\nThe limit condition is $u(N)=o(2^N)$; it does not assume that $u$ is bounded.\nNo divisor-count hypothesis is imposed on $c$.  Thus the ordinary statement\nalso applies to $c=\\varphi$; the identification with the separate formal\n\\#249 development, including its indexing conventions, is not asserted here.\n\\end{thm}\n\nTo prove the forward implication from a recurrence, we telescope:\n\\[\n \\frac{u(N)}{2^N}=u(0)-v\\sum_{n=1}^N\\frac{c(n)}{2^n}.\n\\]\nThe limit condition gives the rational value $u(0)/v$.  Conversely, if the\nseries equals $p/v$, define\n$u(N)=2^Np-v\\sum_{n=1}^N2^{N-n}c(n)$.  This is an integer and equals\n$v\\sum_{r\\ge1}c(N+r)2^{-r}$, so the preceding tail bound proves the limit.\nThis argument also explains why an arbitrary integer solution of the\nrecurrence is insufficient: its homogeneous part can grow like $2^N$.\n\nFor example, the finite support $A=\\{2\\}$ has\n$c_A(n)=\\mathbf1_{2\\mid n}$ and $X_A(2)=1/3$.  With $v=3$, the sequence\n$u(N)=1$ for even $N$ and $u(N)=2$ for odd $N$ satisfies the recurrence.\nThe criterion therefore includes finite supports; it does not provide the\ninfinite support needed to contradict Problem~257.\n\n\\begin{thm}[M\\\"obius inversion of the divisor counts]\\label{record:257bm-i-mob}\nFor a set $A\\subseteq\\N$ and each positive integer $n$,\n\\[\n \\sum_{d\\mid n}\\mu(d)c_A(n/d)=\\mathbf1_A(n),\n\\]\nwhere $\\mu$ is the M\\\"obius function and $c_A(n)$ counts the positive\nelements of $A$ dividing $n$.  Thus the divisor counts determine the\npositive support exactly.  In particular their M\\\"obius transform takes\nonly the values $0$ and $1$.  The identity does not depend on a base.\n\\end{thm}\n\n\\begin{thm}[The support series as a coefficient series]\\label{record:257bm-i-bridge}\nFor $A\\subseteq\\Npos$,\n\\[\n X_A(2)=\\sum_{a\\in A}\\frac1{2^a-1}\n       =\\sum_{n\\ge1}\\frac{c_A(n)}{2^n}.\n\\]\nExpanding each denominator as a geometric series and interchanging\nnonnegative sums gives the identity.  Since $c_A(n)\\le\\tau(n)\\le n$,\nTheorem~\\ref{record:257bm-i-t7} applies.  Together with\nTheorem~\\ref{record:257bm-i-mob}, it expresses rationality through an\ninteger recurrence whose coefficients all come from the same support.\n\\end{thm}\n\n\\subsubsection*{Restrictions on a support with rational value}\n\nThe next results impose separate restrictions on rational-valued supports.\nThe related strict-gap result is Proposition~\\ref{record:257bm-i10}.\n\n\\begin{thm}[A restriction on dyadic rational values]\\label{record:257rig-i2}\nIf an infinite support $A\\subseteq\\Npos$ has $X_A(2)=p/2^c$ for\nintegers $p$ and $c\\ge0$, then $\\sum_{a\\in A}1/a$ either diverges or\nconverges to a value greater than $1$. The cited proof averages the\nshifted integer recurrence and uses a common multiple of two distinct\nsupport elements. Thus a convergent reciprocal sum of at most $1$ is\nexcluded. This is a separate necessary condition: the reciprocal-summable\ncriterion proved earlier already excludes every support with a convergent\nreciprocal sum, not just those whose sum is at most $1$.\n\\end{thm}\n\n\\begin{thm}[Unboundedness of a positive shifted recurrence]\\label{record:257rig-i3}\nLet $A\\subseteq\\Npos$ be infinite, fix a shift $c\\ge0$ and a positive\ninteger $v$, and let $u$ be a positive integer sequence satisfying\n\\[\n u(n+1)+v c_A(c+n+1)=2u(n)\\qquad(n\\ge0).\n\\]\nThen $u$ is unbounded. Indeed, under a proposed bound $u(n)\\le B$,\nchoose $2B+1$ distinct support elements and a common multiple larger\nthan $c$. At the corresponding index the divisor count is at least\n$2B+1$, contradicting the recurrence. This does not conflict with\nTheorem~\\ref{record:257bm-i-t7}, whose condition is $u(n)/2^n\\to0$,\nnot boundedness. An unbounded sequence can still satisfy that limit.\n\\end{thm}\n\n\\begin{thm}[Bounds for intervals with zero divisor counts]\\label{record:257rig-i4a}\nSuppose that $X_A(2)=p/(2^cv)$ for an infinite positive support $A$\nand an odd positive integer $v$. For every $\\varepsilon>0$ there is\n$B$ such that, for $N\\ge1$, a run of $h$ zero divisor counts starting\nafter $c+N$ satisfies $h\\le\\varepsilon\\log_2N+B$.\nIn fact this conclusion holds without rationality: fix any $a\\in A$.\nEvery $a$ consecutive positive integers include a multiple of $a$, where\n$c_A$ is positive. Hence $h\\le a-1$, so one can take $B=a-1$.\nThe zero-run conclusion therefore imposes no additional restriction on\na fixed nonempty support, regardless of the tail estimates used in the\nlinked proof.\n\\end{thm}\n\n\\begin{prop}[A subpower bound for divisor counts]\\label{record:257rig-i4b}\nFor positive integers $n,k$,\n$\\tau(n)^k \\le (k^{2^k})^k\\cdot n$, or equivalently\n$\\tau(n)\\le k^{2^k}n^{1/k}$. The constant absorbs the finitely many primes\nbelow $2^k$; for larger primes, $(\\nu+1)^k\\le p^\\nu$ controls each factor\nof the divisor product. This estimate is independent of the support\nproblem. The elementary zero-run bound above already follows from a\nsingle positive support element, without this estimate or a recurrence.\n\\end{prop}\n\n\\begin{prop}[Two distinct prime-power differences commute]\\label{record:257rig-i5}\nLemma~\\ref{lem:mixed-prime-power-layer} gives the four-term expansion\nand its divisor-count interpretation.  Commutation is elementary\nfor arbitrary positive multipliers.  The extraction formula uses\ndistinct primes, \\emph{positive} exponents $e,f$ and $\\gcd(n,pq)=1$.\nThe example $A=\\{12\\}$ shows the extracted coefficient explicitly.\nNo rationality statement about multiplicative subsequences follows\nfrom commutation alone.\n\\end{prop}\n\n\\subsubsection*{The first crossing of the half-value}\n\n\\begin{defn}[The first crossing rank]\\label{record:257bm-i-cross}\nFor a finite set $E$, its crossing ranks are\n\\[\n\\bigl\\{c\\in E:\\ \\tfrac12 < \\ensuremath{V}(\\{e\\in E:e\\le c\\})\\bigr\\},\n\\]\nthe ranks at which $E$'s inclusive ordered prefix is already above one half.\n\\par\\noindent   \\lean{localMersenneCrossingRanks}{BooleanMobiusExactRowCrossing.lean:24}\n\\end{defn}\n\n\\begin{thm}[Existence of a first crossing]\\label{record:257bm-i-cross2}\nLet $E\\subseteq\\{2,3,\\ldots\\}$ be finite and suppose $X_E(2)>1/2$.\nThere is a least $c\\in E$ such that\n\\[\n X_{E\\cap[2,c)}(2)<\\frac12\n       <X_{E\\cap[2,c]}(2),\\qquad c\\ge4.\n\\]\nPositivity of the summands makes the partial sums increasing.  Their\nodd reduced denominators exclude equality with $1/2$, so the first\ncrossing is strict on both sides.  Finally\n$w_2+w_3=10/21<1/2$, which excludes a crossing before rank $4$.\nThis is the finite crossing data used in the later exact-row\nconstructions; the lower bound on $c$ and strictness are specific\nto these weights.\n\\end{thm}\n\n\\subsubsection*{Centred carries at skipped ranks}\n\n\\begin{thm}[Nonnegative centred carries below the half-value]\\label{record:257hg-i6}\nIf $1\\notin A$ and $X_A(2)<1/2$, then\n\\[\n C_A(N)=\\operatorname{ihc}(A,N)-1\\ge0\\qquad(N\\ge0).\n\\]\nIndeed, Lemma~\\ref{lem:collapse-mech} gives\n\\[\n \\operatorname{ihc}(A,N)\n =2^{N+1}\\bigl(1/2-X_A(2)\\bigr)\n     +\\sum_{r\\ge1}c_A(N+1+r)2^{-r}>0.\n\\]\nThe first term is strictly positive and the tail is nonnegative.\nSince the half-carry is an integer, it is at least $1$; subtracting\n$1$ proves the centred bound.  Nonnegativity alone, without this\nstrictness and integrality step, would only give $C_A(N)\\ge-1$.\nThe argument supplies the lower bound used with the conditional\nupper bound in Theorem~\\ref{record:257rig-c16}.\n\\end{thm}\n\n\\begin{thm}[Three possibilities at a skipped endpoint]\\label{record:257hg-i7}\nFor a rank $s\\ge5$ omitted by the real greedy support, let $H_s$\nand $f_s$ be as in Lemma~\\ref{lem:skipped-endpoint-trichotomy}.\nThat lemma identifies the actual prefix as $D_s$ when $f_s\\le0$\nand as $B_s$ when $f_s>0$, with the exact remainder or overshoot\nin each case.  To obtain the nonnegative-margin condition in\nDefinition~\\ref{record:257bm-c13}, the negative case would still have\nto be excluded at every required skipped rank.  The trichotomy\nitself does not exclude it.  In particular, the value $-3$\nexcluded for $C_D$ under the all-right-tail hypothesis of\nTheorem~\\ref{thm:final-middle-cell} is not an exclusion for $f_s$:\nthe coordinates and the hypotheses are different.\n\\end{thm}\n\n\\subsubsection*{Gap lengths and the measure of their union}\n\n\\begin{defn}[The gap after a Mersenne weight]\\label{record:257hg-i1}\nFor $n\\ge1$, let $R_n=\\sum_{j>n}w_j$ and $g_n=w_n-R_n>0$.\nThe quantity $g_n$ is the length of each gap created at level $n$, not\nthe sum of all gap lengths created at that level.\n\\par\\noindent   \\lean{mersenneGap}{GreedyAchievementSet.lean:2337} \\lean{mersenneGap\\_pos}{GreedyAchievementSet.lean:2346}\n\\end{defn}\n\n\\begin{thm}[The unweighted sum of remaining gap lengths]\\label{record:257hg-i2}\nFor every $N\\ge0$, the unweighted sum of gap lengths satisfies\n\\[\n \\sum_{n>N}g_n\\le\\frac29\\,4^{-N}+\\frac37\\,8^{-N},\n\\]\nwhich tends to zero.  The estimate follows by summing the per-level\nbound; it counts one length per level.  It does not, by itself, bound\nthe measure of the union of all gaps, because there are $2^{n-1}$\ndisjoint translated gaps at level $n$.\n\\end{thm}\n\nThe measure of the union of gaps created after level $N$ is instead\n\\[\n \\sum_{n>N}2^{n-1}g_n=2^N R_N-1\n \\le\\frac23\\,2^{-N}\\qquad(N\\ge1).\n\\]\nIndeed, the finite sum from $N+1$ to $M$ telescopes to\n$2^N R_N-2^M R_M$, and $2^M R_M\\to1$.  The final bound uses\n$R_N\\le2^{-N}+(2/3)4^{-N}$.  Disjointness follows from the strict-tail\nconstruction of the achievement set.  This is an ordinary geometric\ndeduction, not the statement of the preceding linked gap-sum lemma.\nA small union measure still gives no answer about a prescribed point.\n\n\\subsubsection*{A sufficient test for a safe skip}\n\n\\begin{defn}[Three geometric terms in the tail lower bound]\\label{record:257hg-i3}\nFor $k\\ge1$ put $\\ell_3(k)=1/2^k + 1/(3\\cdot4^k) + 1/(7\\cdot8^k)$.  Then\n$\\ell_3(k)<\\ensuremath{R}(k)$: this is a strict rational lower bound obtained by truncating\nthe Lambert-type series at three terms.\n\\par\\noindent   \\lean{mersenneTailLB3}{HalfGreedyFatalGap.lean:56} \\lean{three\\_div\\_lt\\_mersenneTailLB3}{HalfGreedyFatalGap.lean:70} \\lean{mersenneTailLB3\\_lt\\_mersenneTail}{HalfGreedyFatalGap.lean:216}\n\\end{defn}\n\n\\begin{thm}[A sufficient inequality for a safe skip]\\label{record:257hg-i4}\nLet $k,u,L\\ge1$ be integers, set $a=2L-(2^k-1)u$, and\nsuppose $a>0$.  For the skipped rational remainder $\\rho=u/(2L)$,\n$\\rho\\le2^{-k}$ is equivalent to $u\\le a$, whereas\n\\[\n 2u\\le3a\\quad\\Longrightarrow\\quad\\rho<R_k.\n\\]\nThe latter implication uses the three-term lower bound above.  More\nexplicitly, with $t=2^k\\ge2$,\n\\[\n \\left(\\frac1t+\\frac1{3t^2}+\\frac1{7t^3}\\right)\n       -\\frac3{3t-1}\n =\\frac{2t-3}{21t^3(3t-1)}>0.\n\\]\nThe inequality $2u\\le3a$ gives $\\rho\\le3/(3t-1)$, proving the\nclaim.  The first two terms alone would not prove this comparison.\nThis is a sufficient exclusion of the current tail-mass deficit, not\nan exact test for membership in the remaining achievement set.\n\nThe sufficient condition is weaker than $u\\le a$.  A realizable example\nis $(k,u,L,a)=(2,7,13,5)$; it passes $2u\\le3a$ but not $u\\le a$.\nThe linked scalar example $(u,a)=(3,2)$ proves strict containment of the\ninequalities, but is not integral rational data for this substitution.\nFor comparison, the exact mass threshold is\n$a/u\\ge R_k^{-1}-(2^k-1)$; its right-hand side lies strictly between\n$0$ and $2/3$.  None of these comparisons asserts that the half-greedy\norbit satisfies the sufficient inequality at every skipped rank.\n\\end{thm}\n\n\\begin{prop}[Two unconditional safety cases]\\label{record:257hg-i5}\nUnder the positive-integer and skipped-step hypotheses of\nTheorem~\\ref{record:257hg-i4}, $u=1$ implies $a\\ge1$, so the current\nremainder is less than $R_k$.  Conversely, a fatal tail-mass deficit\nforces $3a<2u$, hence $u\\ge2$, or $u\\ge3$ when $u$ is odd.\nThese are statements about the current step, not infinite survival.\n\\end{prop}\n\n\\subsubsection*{Two-sided bounds near dyadic endpoints}\n\n\\begin{thm}[Two-sided dyadic bounds]\\label{record:257bm-i13}\nUnder the local hypothesis excluding the specified three middle cells and imposing the right-pulse\nbound, induction (base case at row 5 verified by direct computation) propagates it to the universal\ntwo-sided bound\n\\[\n\\forall s\\ge5,\\quad \\min(\\mathrm{rem}(s),\\ \\mathrm{overshoot}(s)) \\;\\le\\; 2^s.\n\\]\nThis conditional bound says that at least one of the remainder and\nthe adjacent overshoot is at most $2^s$.  It does not assert that both\nare at most $2^s$; that stronger assertion would replace the minimum\nby a maximum.  It is not needed for the charge bound at an actual upper reset:\nProposition~\\ref{record:257bm-c12} obtains that bound directly from the\nupper-reset identity.  Applying the inductive argument to another\nrecurrence would require its own transition and separation estimates.\n\\end{thm}\n\n\\begin{prop}[A finite band check for $13\\le d\\le30$]\\label{record:257bm-i14}\nFor every actual upper-reset index $13\\le d\\le30$ and every\n$0\\le j\\le d$, the linked certificate verifies\n\\[\n2^{d-j+1}<\\mathrm{resetCharge}\n\\quad\\text{or}\\quad\n\\mathrm{resetCharge}+2(d+j)\\le2^{d-j+1}.\n\\]\nIt computes the successor remainder at each row $d+1$, from\n$\\mathrm{rem}(14)=392$ through $\\mathrm{rem}(31)=4187487147$.\nThis proves the required condition on a finite\nrange, not the universal condition for all $d\\ge13$. A proof for\nunbounded reset indices must supply a further argument; repeatedly\nextending the table alone does not establish that statement.\n\\end{prop}\n\n\\subsubsection*{General identities for perturbed greedy recurrences}\n\n\\begin{thm}[A general perturbed greedy recurrence]\\label{record:257bm-i15}\nLet the old integer values be separated by at least $g\\ge1$,\nand let the update be $t(x)=4s(x)+p(x)$ with\n$0\\le p(x)\\le B<g$.  Suppose $x_-$ and $x_+$ are the adjacent\nold values on either side of capacity $C$.\nTheorem~\\ref{thm:perturbed-family-maximality} gives the largest\nadmissible updated value at capacity $4C+g$ and the exact three-branch\nremainder after testing the extra weight $2g+4$.\n\nThe underlying structure in\nDefinition~\\ref{defn:perturbed-family} assumes only $B<3g$;\nthat suffices for order preservation, but the cited maximality\napplication has the additional hypothesis $B<g$.\nThe counterexample after Theorem~\\ref{thm:perturbed-family-maximality}\nshows what can fail without it.  For a maximum over both choices of the extra weight, rather than\nthe stated two-stage rule, one also needs a separation argument;\n$4g-B\\ge2g+4$ suffices, as shown after that theorem.\n\\end{thm}\n\n\\begin{thm}[Spacing of compatible reverse-carry words]\\label{record:257bm-i16}\nConsider two integer carry recurrences\n$b_i(m)+2u_i(m)=a_i(m)+u_i(m+1)$, $i=1,2$.\nIf $a_1(k)=a_2(k)$ and the output bits at $k$ are $1$ and $0$,\nthen the carry difference at $k+1$ is odd.  If the coefficients\nand bits agree for the following $L$ positions, the difference\nat $k+L+1$ is $2^L$ times that odd integer.\nConsequently absolute terminal bounds $B_1,B_2$ give\n$2^L\\le B_1+B_2$.\nLemma~\\ref{lem:reverse-carry-word} states the exact identity and\nproves it by subtraction.  The linked formal statements encode\nthese agreements; they do not supply terminal bounds for an\nunrelated digit system.\n\\end{thm}\n\n\\begin{prop}[Changing one support bit at a doubled rank]\\label{record:257bm-i17}\nLet two supports agree except that the second includes $N+1$\nwhile the first does not.  At the argument $2(N+1)$ their divisor\ncounts differ by exactly $1$, because $N+1$ divides $2(N+1)$.\nMore generally the difference at a positive argument $m$ is\n$\\mathbf1_{N+1\\mid m}$.  The linked statements apply this identity\nto the finite supports in their hypotheses.  No conclusion about\na different coefficient sequence follows without identifying its\nown support change.\n\\end{prop}\n\n\\subsection{Obstructions and countermodels}\n\nThe results below test particular proposed implications.  A countermodel\nto an abstract transition rule shows that the rule needs an additional\nhypothesis; it does not disprove a more structured arithmetic\nargument.  Each entry states the information retained by its model\nand the conclusion that this information fails to imply.\n\n\\begin{thm}[A bounded model of the doubling-or-return alternative]\\label{record:257bm-k1}\nThe dichotomy $\\mathrm{ExactLocalMersenneHalfRow}(2n{-}1) \\vee \\exists c,\\,4\\le c\\le n\\wedge\n\\mathrm{ExactLocalMersenneHalfRow}(2c{-}2)$ (proved for $n\\ge6$ at Theorem~\\ref{record:257bm-k-dich} below) is not by\nitself enough for cofinality. Countermodel: $\\mathrm{boundedDoubleOrRecycleModel}(n) := (n=6)$\nsatisfies exactly the same two-branch transition shape (seed at 6, and the\nmodel reproduces the $\\vee$ shape by always taking\nthe recycle branch with $c=4$, conclusion back at $2\\cdot4-2=6$), yet\n\\[\n\\neg\\big(\\forall N,\\ \\exists n\\ge N,\\ \\mathrm{boundedDoubleOrRecycleModel}(n)\\big) \\qquad \\text{(not cofinal ;  only ever true at } n=6\\text{)}.\n\\]\nThe\ndouble-or-recycle transition shape, even together with an endpoint-six seed, is logically\ninsufficient to conclude cofinal exact rows. A genuinely new progress input (strict endpoint\ngrowth, or an independent cofinality argument) is required. The countermodel retains only the seed and this transition rule.  It\ntherefore tests those premises, not additional arithmetic information\nthat a more structured induction may use.\n\\end{thm}\n\n\\begin{thm}[Doubling or returning to an earlier depth]\\label{record:257bm-k-dich}\nFor $n\\ge6$, $\\mathrm{ExactLocalMersenneHalfRow}(n)$:\n\\[\n\\mathrm{ExactLocalMersenneHalfRow}(2n-1) \\;\\vee\\; \\exists c,\\ 4\\le c\\le n \\wedge \\mathrm{ExactLocalMersenneHalfRow}(2c-2).\n\\]\nEvery exact row of depth at least $6$ has at least one of the two\nasserted continuations.  The proof separates the below-half case from\nthe above-half case, but the two existential conclusions need not be\nexclusive.  The returned depth $2c-2$ is not claimed to exceed $n$. The finite Mersenne value cannot equal $1/2$, by\nObservation~\\ref{record:257bm-k11}; this says nothing about equality\nof the returned and original depths.  For example, at $n=6$, the\nsupport $\\{2,3,6\\}$ is an exact row, and so is\n$\\{2,3,6,7,11\\}$ at depth $11$.  The second conclusion also holds\nwith $c=4$, since $2c-2=6$.  Thus both conclusions hold in this example,\nand the second allows no depth increase.  What is\nmissing, per Theorem~\\ref{record:257bm-k1}, is not another dichotomy but strict endpoint progress in the recycle\nbranch, or a proof the below-half branch recurs ;  re-deriving this disjunction without an additional progress argument\nwould not establish cofinality.\n\\end{thm}\n\n\\begin{prop}[A sufficient fractional-mass bound need not hold]\\label{record:257bm-k2}\nTake $D=\\{2,3\\}$ and $c=5$.  Direct calculation gives\n\\[\n X_D(2)=\\frac{10}{21}<\\frac12\n <\\frac{331}{651}=X_{D\\cup\\{5\\}}(2),\\qquad S(D,1,8)=6<8.\n\\]\nHowever, the combined fractional mass is\n\\[\n \\sum_{d\\in D\\cup\\{5\\}}\\frac{2^8\\bmod(2^d-1)}{2^d-1}\n =\\frac{757}{651}>1.\n\\]\nThe linked Lean proof checks this example by direct computation.  Thus the\ncombined fractional-mass bound by $1$, which suffices for the sharp\ncapacity estimate, is not necessary for that estimate.\nThus the fractional-mass bound cannot hold at every real crossing core.\nA proof using it would need to handle the exceptional cores separately;\nthe example does not rule out all uses of fractional-mass estimates.\nTheorem~\\ref{record:257rig-k6} identifies the support at a critical\ncrossing with a real greedy prefix, but does not prove the missing bound\non those prefixes.\n\\end{prop}\n\n\\begin{obs}[A doubled finite sum can exceed one half]\\label{record:257bm-k3}\nTheorem~\\ref{record:257bm-i7}'s doubling extension is not preserved iteratively: below-halfness is not preserved\nunder one application of doubling, and demonstrably fails at the very first step. From the\nidentity $\\mathrm{value}(E)-\\tfrac12 = \\big(\\mathrm{localFractionMass}(E,2n{-}1)-1\\big)/2^{2n-1}$,\nat the seed $D=\\{2,3,6\\}$ (Definition~\\ref{record:257bm-i-seed}) with $M=11$ the core alone contributes\n\\[\n\\frac{2^{11}\\bmod3}{3} + \\frac{2^{11}\\bmod7}{7} + \\frac{2^{11}\\bmod63}{63} = \\frac23+\\frac47+\\frac{32}{63} \\approx 1.746 > 1,\n\\]\nso the doubled row is strictly above half. Iterating the cheap (doubling) arm is therefore\nimpossible from the only recorded seed; the chain is forced into the capacity-gated recycle arm\nat step one. What would close the doubling route instead: cofinally many below-half exact rows,\n$\\forall N\\ \\exists n\\ge N\\ \\exists D$ exact at $n$ with $\\mathrm{localFractionMass}(D,n)<1$; a\nsingle such row at each of cofinally many $n$ would make doubling unnecessary; a\nself-reproducing one would close everything.\n\\par\\noindent   \\lean{exists\\_exactRowStrictUpperExtension\\_two\\_mul\\_sub\\_one\\_of\\_exact\\_below}{BooleanMobiusExactRowDoubling.lean:185}\n\\end{obs}\n\n\\begin{thm}[The returning endpoint need not be larger]\\label{record:257bm-k4}\nTheorem~\\ref{record:257bm-c10} is unconditional, but its\nwitness is $\\exists c\\le n$, not $\\exists c$ large: $2c-2$ may be $\\le n$, so the endpoint need\nnot grow. The countermodel of Theorem~\\ref{record:257bm-k1} is the\nexplicit falsifier of the naive hope that growth comes for free:\n$\\mathrm{boundedDoubleOrRecycleModel}(n):=(n=6)$ satisfies the same transition schema plus a\nseed and is not cofinal. Growth is recovered only inside\n$\\mathrm{ProtectedExactLocalMersenneRow}$ (Theorem~\\ref{record:257bm-c5}), whose invariants\n$\\mathrm{endpoint}<2\\cdot\\mathrm{cutoff}$ and $\\mathrm{new\\_above\\_cutoff}$ force $c>\\mathrm{cutoff}$\nhence $2c-2>\\mathrm{endpoint}$ ;  and maintaining those invariants is precisely what needs the\nstrict-upper (sharp capacity) fill of Theorem~\\ref{record:257bm-c7} rather than this general recycling theorem.\n\\end{thm}\n\n\\begin{obs}[The skip condition is equivalent to membership]\\label{record:257bm-k5}\nThe positivity conjunct of Theorem~\\ref{record:257bm-c2}'s hypothesis,\n$0<\\ensuremath{r}(1/2,c{-}1)$, holds unconditionally: the greedy remainder is $1/2$ minus a\nfinite Mersenne sum, which is nonzero by the odd-denominator parity fact of\nObservation~\\ref{record:257bm-k11}. Hence the hypothesis of Theorem~\\ref{record:257bm-c2}\nholds if and only if the set of ranks skipped by the greedy rule for $1/2$ is infinite,\nwhich Theorem~\\ref{record:257bm-c20} proves equivalent to $1/2\\in\\ensuremath{\\mathcal A}$. Proving Theorem~\\ref{record:257bm-c2}'s hypothesis would therefore settle\nthe half-target question.  This equivalence does not by itself make the\nhypothesis easier, but it also does not preclude a useful argument in\ngreedy coordinates.  Theorem~\\ref{record:257bm-c7} offers a different\nsufficient condition whose finite supports need not be greedy prefixes.\n\\par\\noindent   \\lean{localMersennePrefixValue\\_halfGreedy\\_lt\\_half}{BooleanMobiusCriticalCapacityCofinal.lean:138}\n\\end{obs}\n\n\\begin{thm}[Uniqueness at a critical crossing]\\label{record:257rig-k6}\nFor $c\\ge4$, $D$ bounded $[2,c)$, below-half, with genuine crossing deficit\n$\\tfrac12-\\mathrm{value}(D) < \\ensuremath{w}(c)$:\n\\[\nD \\;=\\; \\mathrm{halfGreedyPrefixSupport}(c-1).\n\\]\nThus $D=G\\cap\\{1,\\ldots,c-1\\}$: at a critical crossing the support is\nfixed by $c$.  The proof uses the strict inequality between each Mersenne\nweight and the sum of all later weights, as in\nTheorem~\\ref{record:257bm-i11a}.  Consequently, the support quantifier in\nTheorem~\\ref{record:257bm-c4} does not allow arbitrary choices of $D$.\nThis reduces that hypothesis to a statement about the actual greedy\nsequence.  It does not rule out proving a new bound for that sequence.\n\\end{thm}\n\n\\begin{obs}[Different requirements on finite witnesses]\\label{record:257rig-k7}\nTheorem~\\ref{record:257rig-c17} (the cofinal terminal-strip condition) tolerates any support inside the depth-$M$\nwindow but demands the carry inside a $\\sim2\\sqrt M$ strip; Theorem~\\ref{record:257bm-c7} tolerates a carry up to\n$2^{c-2}$ (exponentially looser) but demands the support be confined to ranks $<c$, because the\nexact fill encodes the residue in the pure upper window where the Mersenne quotient is exactly\n$2^{M-d}$ (Proposition~\\ref{record:257bm-c9}). The two conditions therefore differ in the permitted support as well as\nthe size of the carry.  If the square-root witnesses could also be chosen\nwith support in $[2,c)$ at depth $2c-3$, then\n$2\\sqrt{2c}+4<2^{c-2}$ for $c\\ge8$ would give the numerical bound needed\nfor the fill.  The required support restriction is not proved here.\nComparing the displayed bounds alone does not order the two existence\nhypotheses.\n\\par\\noindent   \\lean{HalfCarryCofinalTerminalOnlyStrip}{TerminalOnlyCofinal.lean:34}\n\\end{obs}\n\n\\begin{obs}[An unsafe example in the two-thirds interval]\\label{record:257hg-k8}\nWrite a greedy residual as $\\mathrm{rem}=1/R$; a skip at rank $k$ is dyadically safe if and\nonly if $R\\ge2^k$.  Two unconditional corollaries hold: an integral reciprocal $R$ is never\nunsafe, and an unsafe run with odd $p,D,q$ forces $p\\ge7$.  Dyadic safety is sufficient but\nnot necessary for the true greedy process to survive: the true test uses the actual tail\n$\\sum_{j>k}w_j$, which exceeds $2^{-k}$.  Nothing here asserts that the actual half-greedy\norbit avoids the band: $(p,D,b)=(17,41,3)$ is an explicit odd-coprime example that is\ndyadically unsafe.  Theorem~\\ref{record:257hg-i4}'s criterion $2u\\le3a$ narrows exactly this\ngap in the single-skip case; whether the fatal region meets a band reachable by the actual\ngreedy trajectory is not settled.\n\\par\\noindent   \\lean{TwoThirdsBand}{HalfGreedyTwoThirdsBand.lean:120} \\lean{not\\_twoThirdsBand\\_of\\_int}{HalfGreedyTwoThirdsBand.lean:185} \\lean{seven\\_le\\_of\\_intBand\\_odd}{HalfGreedyTwoThirdsBand.lean:231}\n\\end{obs}\n\n\\begin{thm}[Vanishing of the specified linear-channel determinant]\\label{record:257bm-k9}\nLet $V$ be a vector space over $\\Q$, let $e:V\\to\\Q$ be linear, and\nlet $(\\ell_j)_{j\\in\\iota}$ be a finite family of linear functionals\nvanishing on $\\ker e$.  For any vectors $(v_i)_{i\\in\\iota}$, the matrix\n$(\\ell_j(v_i))_{i,j\\in\\iota}$ has rank at most one, so every square\nminor of size at least two vanishes.\nIndeed, the functionals descend to $V/\\ker e$, which has dimension at\nmost one.  This elementary linear-algebra argument applies at every\nmatrix size.  It does not cover additional functionals that fail to\nvanish on $\\ker e$.\n\\end{thm}\n\n\\begin{obs}[The additive error in the binary bound]\\label{record:257bm-k10}\nTheorem~\\ref{record:257bm-i6}'s $2^{c-1}$ bound and Theorem~\\ref{record:257bm-i5}'s sharp $2^{c-2}$ requirement differ by one bit, and the\nproof of the loose bound shows exactly where: it derives $A<2^{c-2}+|D|$ and discards\n$|D|\\le c-2\\le2^{c-2}$. Under this additive upper bound, the remaining assertion is that the\ninteger suffix avoids $[2^{c-2},\\,2^{c-2}+c-3]$.  This identifies the\nunresolved interval; it does not prescribe how to exclude it.  A sharper\nestimate, an arithmetic restriction on the suffix, or another argument\ncould in principle establish the required inequality.\n\\par\\noindent   \\lean{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_upperHalfCapacity}{BooleanMobiusSkippedCoreExactRow.lean:123}\n\\end{obs}\n\n\\begin{obs}[Odd denominators of finite sums]\\label{record:257bm-k11}\nFor finite $D\\subseteq\\Npos$, each denominator $2^d-1$ is odd.\nThe sum $V(D)$ can therefore be written over an odd common denominator;\nits reduced denominator, being a divisor of that denominator, is odd.\nIn particular, $V(D)\\ne1/2$.  This elementary argument supplies the\nparity fact used in Proposition~\\ref{record:257bm-k2} and\nProposition~\\ref{record:257bm-i10}.\n\\par\\noindent   \\lean{finiteErdosSum\\_den\\_odd}{DyadicPrefixCompression.lean:1513}\n\\end{obs}\n\n\\begin{thm}[Excluding the cell with value minus three]\\label{record:257hg-k12}\nAt a middle row $D\\ge13$ followed only by right transitions,\n$4\\,\\mathrm{rem}(D)-p_D^--4\\ne-3$, by\nTheorem~\\ref{thm:final-middle-cell}.  Its proof uses the nonnegative\ncentred carry for the completed support, not a finite search.\nThe values $-2,-1$ remain among the three exceptional negative cells.\nExcluding them under this extra tail assumption would still not\nexclude nonnegative values of the coordinate, or establish the\nall-middle-row and right-branch hypotheses in\nTheorem~\\ref{thm:two-sided-dyadic}.  The complete remaining tail\ninequality is stated in Remark~\\ref{rem:tail-dominance-open}.\n\\end{thm}\n% ; - part a257_newdecls ; -\n\\subsection{Further finite and conditional results}\n\nThis subsection records further finite identities and conditional\nmembership results.  They concern the\nmiddle transition, finite families with a common prefix, and rational\ntargets other than $1/2$.  Their hypotheses are stated separately from\nthe existence assertions still needed to apply them.\n\n\\subsubsection{The integer update in a middle interval}\n\nThe following identities explain the two-sided bound and the reset-band\ncondition used above.  They also distinguish the signed middle coordinate\n$C_s=4\\,\\mathrm{rem}(s)-p_s^- -4$ from the nonnegative label in the\nnext estimate.\n\n\\begin{prop}[The unsafe middle range is exactly three integers]\\label{prop:unsafe-middle-range-exactly-three}\nFor $s\\ge5$, write $R=\\mathrm{rem}(s)$ and $P=p_s^-$.  Since\n$4R-P-4$ is an integer,\n\\[\n\\neg(4R-P-4\\le-4\\ \\vee\\ 0\\le 4R-P-4)\\ \\Longleftrightarrow\\ 4R-P-4\\in\\{-3,-2,-1\\}\n\\]\n\\end{prop}\nThe complement of the two safe integer ranges is precisely\n$\\{-3,-2,-1\\}$.  The conclusion is only a classification; excluding\nany of these values requires an additional argument.\n\nFor a row $s\\ge5$, set\n\\[\n E_s=\\sum_{d\\in D_s}\\left\\{\\frac{4^s}{2^d-1}\\right\\},\\qquad\n Z_s=\\mathrm{rem}(s)-E_s,\\qquad\n c=4\\,\\mathrm{rem}(s)-p_s^-=C_s+4.\n\\]\nHere braces denote fractional part.  Assume $c\\ge0$, as in the linked\nstatements below.  In particular, the exceptional signed values\n$C_s=-3,-2,-1$ correspond to $c=1,2,3$, not to negative labels.\nThe floor-quotient recurrence gives $p_s^-\\le4E_s$, and hence\n\\[\n 4Z_s\\le c.\n\\]\nThis inequality has a Lean proof\n(\\lean{seamMiddleCell\\_four\\_mul\\_floorZ\\_le}{HalfCylinderMiddleCarryLowerBound.lean:4142}).\nOn a middle transition $D_{s+1}=D_s$.  Writing\n$\\rho=1/2-X_{D_s}(2)$, the exact quotient identity therefore gives\n\\[\n 4^{s+1}\\rho=2^{s+2}+4Z_s\\le2^{s+2}+c,\n\\]\nalso proved in Lean\n(\\lean{seamMiddleCell\\_scaledRationalRemainder\\_le}{HalfCylinderMiddleCarryLowerBound.lean:4167}).\n\nFor $C_s=-3$, the correct label is $c=1$.  This estimate does have a\nlocal consequence: since\n$4^{s+1}R_s>2^{s+2}+4/3$, it gives $\\rho<R_s$.\nThus the completed support $D_s\\cup\\{s+1,s+2,\\ldots\\}$ has value\ngreater than $1/2$.  It does not prove that every later greedy step\nsurvives.  For $s\\ge13$, under the additional all-right-tail hypothesis, the same\ncompleted support would have value less than $1/2$, giving the\ncontradiction in\n\\lean{finalMiddleCell\\_neg\\_three\\_not\\_last}{HalfCylinderFinalMiddleCellEscape.lean:587}.\nThat tail hypothesis must not be dropped.  For $c=2,3$, the displayed\nupper bound alone no longer gives $\\rho<R_s$.\n\nA parallel equivalence holds for the right branch: an overshoot witness $\\le2^s$ produces a\npulse leak above $2^{s+2}$ exactly when the overshoot lies within an explicit distance $k$ of\n$2^s$ with $4k<\\mathrm{abovePulse}$\n(\\lean{seamRightPulseLeak\\_iff\\_distanceCell}{HalfCylinderMiddleCarryLowerBound.lean:4202}).\nThe remainder left by the integer greedy rule with the row-$s$ quotient weights and target\n$2^{2s-1}$ (\\lean{seamPureHalfPrefixRemainder}{HalfCylinderMiddleCarryLowerBound.lean:4232})\nhas an exact case-split formula\n(\\lean{seamPureHalfPrefixRemainder\\_eq\\_ite}{HalfCylinderMiddleCarryLowerBound.lean:4238}),\nand a bound on it is proved equivalent to the two-sided bound\n(\\lean{seamPureHalfPrefixRemainder\\_le\\_iff\\_twoSided}{HalfCylinderMiddleCarryLowerBound.lean:4356}).\n\n\\begin{obs}[The reset-band hypothesis has a verified base case]\nThe upper-reset dyadic-band escape condition (discussed after Section~\\ref{sec:o4}) is\nreduced by three lemmas to a quarter-scale overshoot exclusion: an equivalence\n(\\lean{seamUpperReset\\_band\\_iff\\_successorRemainder\\_avoids}{HalfCylinderMiddleCarryLowerBound.lean:4469})\nrewrites the reset-charge band as a successor-remainder avoidance; an identity\n(\\lean{upperResetCharge\\_dyadicDistance\\_add\\_abovePulse}{HalfCylinderMiddleCarryLowerBound.lean:4490})\nrescales it by a factor of $4$; and a third lemma\n(\\lean{seamUpperReset\\_band\\_of\\_overshoot\\_band}{HalfCylinderMiddleCarryLowerBound.lean:4514})\nderives the band from a quarter-scale overshoot exclusion plus the pulse bound $\\mathrm{abovePulse}\\le\n2(d-2)$. Row $d=13$, the first row of the late regime, is then proved unconditionally to escape\n\\emph{every} band simultaneously: its successor remainder is exactly $392$, by exact\ncomputation\n(\\lean{seamUpperResetDyadicBandEscape\\_at\\_thirteen}{HalfCylinderMiddleCarryLowerBound.lean:4547}).\nThis is one verified row of a hypothesis required for every $d\\ge13$; it shows only that the\nbase case is not itself an obstruction.\n\\end{obs}\n\n\\subsubsection{Finite approximations without compatibility}\n\\papersectiontarget{record257:terminal-approximation}\nFor a finite support $D\\subseteq\\{2,\\ldots,M\\}$, the integer\n\\[\n K_D(M)=2^{M-1}-\\sum_{j=2}^{M}2^{M-j}\\#\\{d\\in D:d\\mid j\\}\n\\]\nis the half-carry at the terminal rank.  It is\n$\\operatorname{ihc}(D,M-1)$ in the notation used earlier.  The condition\nbelow asks for $M_j\\to\\infty$ and finite sets $D_j$ with\n$|K_{D_j}(M_j)|/2^{M_j}\\to0$.  It does not require the sets to agree or\nbound the carries at earlier ranks.\nThe estimate\n\\[\n \\left|\\sum_{d\\in D_j}\\frac1{2^d-1}-\\frac12\\right|\n \\le\\frac{|K_{D_j}(M_j)|+2\\sqrt{M_j}+4}{2^{M_j}}\n\\]\nfollows by expanding each Mersenne weight. Indeed,\n\\[\n X_D(2)-\\frac12=-2^{-M}K_D(M)\n       +\\sum_{n>M}c_D(n)2^{-n}.\n\\]\nHere $c_D(n)\\le\\tau(n)\\le2\\sqrt n$, and\n$\\sqrt{M+r}\\le\\sqrt M+r$ gives\n\\[\n \\sum_{r\\ge1}2^{-r}c_D(M+r)\\le2\\sqrt M+4.\n\\]\nEach finite value belongs to the closed achievement set; its limit\ntherefore belongs too, even when the finite supports disagree.\nNo compatibility between them is needed for this implication.\nThe finite approximants themselves remain an unproved input.\nA square-root terminal bound is sufficient and stronger as a bound on\nthese particular witnesses; it does not establish a strict logical\nseparation between the corresponding existence statements.\n\n\nHere is the finite family used in the cited construction.  Put\n$B(m)=2\\lfloor\\sqrt m\\rfloor+4$.  A depth-$M$ family with a shared\nprefix through $K\\le M$ consists of sets\n$A_1,\\ldots,A_{B(M)}\\subseteq\\{2,\\ldots,M\\}$ such that\n\\[\n 1\\le K_{A_k}(m)\\le B(m)\\quad(1\\le m\\le M),\n \\qquad K_{A_k}(M)=k,\n\\]\nall $A_k$ agree through $K$, and one integer $E\\ge B(M)$ satisfies\n\\[\n \\sum_{j=K+1}^{M}1_{A_k}(j)2^{M-j}+k=E\n \\quad(1\\le k\\le B(M)).\n\\]\nThe suffixes therefore have consecutive binary values $E-k$.\nThe following statements explain how these particular families supply\nfinite approximations; they do not construct them at arbitrary depths.\n\n\\begin{prop}\\label{prop:finite-approximations-without-compatibility}\nEvery shared-prefix family just defined at depth $N$ contains a finite\nsupport $A$ with $|K_A(N)|\\le B(N)$, and at a feedback row this survives \\emph{both} outputs of\nthe corpus's total feedback theorem; full-cylinder advance or a localized one-hole seam, which\ncan delete at most one of carries $3,4$.\n\\end{prop}\nFour concrete instances, at depths $51$--$54$, are verified by exact computation of the\nunderlying stages\n(\\lean{fullCylinderStage51\\_halfTerminalOnlyStripWitness}{SuffixCylinderTerminalOnlyBridge.lean:231},\n\\lean{fullCylinderStage54\\_halfTerminalOnlyStripWitness}{SuffixCylinderTerminalOnlyBridge.lean:257}).\nThis is a finite chain of witnesses; it does not reach unbounded depths.\n\n\\begin{obs}[Shared-prefix families at unbounded depths]\nThe conditional theorem\n\\[\n\\left(\\begin{gathered}\\text{such shared-prefix families exist}\\\\\\text{at arbitrarily large depths }M\\end{gathered}\\right)\n\\ \\Longrightarrow\\ \\exists A\\text{ infinite with }X_{A}(2)=1/2\n\\]\nis proved unconditionally\n(\\lean{exists\\_infinite\\_support\\_half\\_of\\_cofinalCylinderStages}{SuffixCylinderTerminalOnlyBridge.lean:277}),\nwith a positive-support refinement using the same odd-denominator lemma\n(\\lean{finite\\_boolSupport\\_ne\\_half}{HalfCarryReachability.lean:589};\n\\lean{exists\\_infinite\\_positive\\_support\\_half\\_of\\_cofinalCylinderStages}{SuffixCylinderTerminalOnlyBridge.lean:287}).\nThe hypothesis requires these finite families at arbitrarily large\ndepths and is unproved.  The terminal bound then implies half-membership.\nThis is a sufficient condition; no converse or strict comparison with\nother sufficient conditions is established here.\n\\end{obs}\n\nFor the conditional implication, choose the member with terminal carry\n$1$ from each shared-prefix family. Its error in approximating $1/2$\nis at most $(1+2\\sqrt M+4)2^{-M}$ by the preceding estimate.\nAt unbounded depths these errors tend to zero, so compactness gives a\nrepresentation. Every finite sum has odd reduced denominator; hence\nthat representation uses an infinite support. This deduction needs no\ncompatibility between the chosen members of different families. It\nstill assumes that the stated families exist at unbounded depths.\n\n\\subsubsection{Removing one case from a sufficient tail condition}\n\nFive Lean declarations reduce the cases that remain to be checked in the\nsquare-root lower bound on the producer carry at late middle rows (discussed in\nPart~IV), using the proved exclusion of the $-3$ cell.  No strict separation of the\nexistence conditions is inferred from dropping that case.\n\n\\begin{defn}\nThe \\emph{middle tail condition} is the hypothesis that at every genuine middle\ntransition ($s\\ge13$, no successor carry, below the terminal-weight threshold) the terminal-augmented\nbinary coefficient tail is strictly below the signed carry\n(\\lean{SeamMiddleProducerTailEscape}{HalfCylinderLastProducerContradiction.lean:34}).\nThe \\emph{reduced middle tail condition} is the same hypothesis restricted to cells\n$\\ne-3$ (\\lean{SeamMiddleProducerTailEscapeExceptNegThree}{HalfCylinderLastProducerContradiction.lean:54}).\n\\end{defn}\nThe middle tail condition trivially implies the reduced one\n(\\lean{SeamMiddleProducerTailEscape.toExceptNegThree}{HalfCylinderLastProducerContradiction.lean:76});\nthe substance is in the converse direction. The reduced condition, with cell\n$-3$ discharged by the already proved exclusion\n(\\lean{middleProducer\\_neg\\_three\\_not\\_last}{HalfCylinderLastProducerContradiction.lean:364}),\nalready suffices for half-membership:\n\\[\n\\text{reduced middle tail condition}\\ \\Longrightarrow\\ \\tfrac12\\in\\mathcal{A}\n\\]\n(\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_middleProducerTailEscapeExceptNegThree}{HalfCylinderLastProducerContradiction.lean:436}).\nThe unreduced condition and the square-root bound both imply it\n(\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_middleProducerTailEscape}{HalfCylinderLastProducerContradiction.lean:451}).\nSo the square-root bound implies the middle tail condition, which implies the reduced\ncondition, which implies $1/2\\in\\mathcal A$: a chain of sufficient implications.  The\narrows do not prove that any implication is strict; all three antecedents remain unproved.\n\n\\subsubsection{An eventual nonnegative margin}\n\nThe finite-horizon condition can be stated directly.  Recall the integer\nmargin $F_k(J)$ from Theorem~\\ref{thm:frozen-margin} and its equivalence with a\nnegative excess\n(\\lean{exists\\_greedyHalfFrozenMargin\\_nonneg\\_iff\\_excess\\_neg}{HalfCylinderFiniteShadow.lean:1229}).\n\n\\begin{defn}[A nonnegative margin by the next depth]\nAt every skipped step $k+1$, require\n\\[\n \\exists J\\in\\{0,\\ldots,k+1\\},\\qquad F_k(J)\\ge0.\n\\]\n\\par\\noindent\n\\lean{HalfGreedyGovernedFrozenMarginProducer}{HalfCylinderFiniteShadow.lean:1267}\n\\end{defn}\nIt implies $1/2\\in\\mathcal A$\n(\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_governedFrozenMarginProducer}{HalfCylinderFiniteShadow.lean:1275},\nthrough\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_skipped\\_dyadicCap}{GreedyAchievementSet.lean:1444}).\nIndeed, the finite coefficient window is at most its\ninfinite tail, so $F_k(J)\\ge0$ implies\n$r_k(1/2)\\le2^{-(k+1)}$ at that skip.  This is below $R_{k+1}$.\nAt $k=0$ the remainder is exactly $1/2$ and the same inequality\nholds; the strict version in Theorem~\\ref{record:257bm-c19}\nrequires positive $k$.\n\nThe bound $J\\le k+1$ is \\emph{not used} in this implication.\nAllowing any finite $J$ at each skip already suffices; requiring\nfirst passage by $k+1$ is a stronger finite-horizon demand, not a\nnecessary ingredient of the membership proof.  Neither all-skip\nexistence condition is established here, and no equivalence between\nthe bounded and unbounded horizons is asserted.\n\n\\subsubsection{Rewriting the signed carry recurrence}\n\nNumeral-adjacent boundary words have more structure than the previously used\ncoefficient bound.  If the lower suffix numeral is one larger than the upper\nsuffix numeral, there is a unique carry pivot $d$: rank $d$ enters the lower\nsupport, every rank $e$ with $d<e\\le N$ flips from lower-unselected to\nupper-selected, and all ranks below $d$ agree\n(\\lean{exists\\_carry\\_pivot\\_of\\_wordSuffixNumeral\\_succ}{SuffixCylinderCarryPivot.lean:141}).\nHence for\nevery positive coefficient row $m\\ge1$ the boundary discrepancy is the exact divisor-incidence\nidentity\n\\[\n c_{\\rm low}(m)+\\#\\{e:d<e\\le N,\\ e\\mid m\\}\n =c_{\\rm up}(m)+\\mathbf 1_{d\\mid m},\n\\]\nnot merely an inequality\n(\\lean{supportCoeff\\_add\\_deepDivisorCount\\_eq\\_of\\_carry\\_pivot}{SuffixCylinderCarryPivot.lean:163}).\n\nThe stage-level theorem consumes adjacent profiled prefixes directly\n(\\lean{profiledGapStage\\_carry\\_pivot\\_of\\_hasAdjacentPrefixes}{SuffixCylinderCarryPivot.lean:228}).\nThis is a local normal form for the two-sheet boundary.  It does\nnot produce adjacent stages at unbounded depths and therefore decides neither\nhalf-membership nor universal \\#257.\n\n\\subsubsection{Further restrictions on represented values}\n\nThese results concern achievement-set geometry and do not settle the universal\nirrationality problem.  For an arbitrary support restriction $J\\subseteq\\mathbb{N}$, let\n$\\mathcal A_J$ be the set of Mersenne digit values supported on $J$\n(\\lean{supportedMersenneAchievementSet}{ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean:76}). It is compact\n(\\lean{isCompact\\_supportedMersenneAchievementSet}{ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean:90}),\nnowhere dense for every $J$ (\\lean{isNowhereDense\\_supportedMersenneAchievementSet}{ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean:112}),\nand perfect, that is compact with no isolated points, whenever $J$ is infinite\n(\\lean{perfect\\_supportedMersenneAchievementSet}{ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean:167}), via a\nCantor-space no-isolated-point argument on the coding space itself\n(\\lean{preperfect\\_supportedMersenneDigitSet}{ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean:120}).\n\n\\begin{prop}[Exact Lebesgue-measure dichotomy]\\label{prop:exact-lebesgue-measure-dichotomy}\nEither $J=F^c$ for a finite $F$, and $\\mathrm{vol}(\\mathcal A_J)=2^{-|F|}$\nexactly, or $J^c$ is infinite and the volume is exactly $0$.\n\\end{prop}\nThe finite case is an exact doubling induction over inserted\ncoordinates (\\lean{volume\\_supportedMersenneAchievementSet\\_insert}{ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean:286}),\nusing disjointness of the two coordinate-split faces\n(\\lean{disjoint\\_supportedMersenneAchievementSet\\_translate}{ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean:262});\nthis is where unique binary coding is used. The dichotomy is the case of retained coordinates of\nHornich's strict-tail measure formula~\\cite[Theorem~4(1), p.~9]{nitecki2013}: the retained Mersenne\nweights still exceed their tails, and the Mersenne tail estimate supplies the values $2^{-|F|}$ and\n$0$.  None of this concerns irrationality of any subseries value.\n\n\\subsubsection{Conditional counterexamples at rational targets}\n\nFinite supports whose scaled terminal carries tend to zero give an infinite $A$ with\n$X_{A}(2)=1/2$\n(\\lean{exists\\_rational\\_half\\_counterexample\\_of\\_terminalScaledVanishing}{ErdosProblems/Erdos257/HalfCounterexampleFrontier.lean:31}),\nand hence refute the universal statement\n(\\lean{not\\_universal\\_of\\_terminalScaledVanishing}{ErdosProblems/Erdos257/HalfCounterexampleFrontier.lean:39};\nthe universal statement itself has its own\n\\lean{UniversalMersenneSubseriesIrrationality}{ErdosProblems/Erdos257/HalfCounterexampleFrontier.lean:26}).\nThe required finite supports have only been checked to a large finite cutoff.  This restates\nthe open half-membership question; it is not a new result, since any representation of $1/2$\nalready refutes the universal statement.\n\nA Lean theorem also proves an unconditional finite-support obstruction at\nthe second rational target $1/21$:\n\\[\n  F\\subseteq\\{n:n\\ge2\\}\\quad\\Longrightarrow\\quad\n  \\sum_{n\\in F}\\frac1{2^n-1}\\ne\\frac1{21}\n\\]\nfor every finite $F$\n(\\lean{finiteErdosSum\\_ne\\_one\\_div\\_twenty\\_one}{ErdosProblems/Erdos257/HalfCounterexampleFrontier.lean:61}).\nThe denominator-order identity forces\n$\\operatorname{lcm}(F)=6$, after which the eight subsets of $\\{2,3,6\\}$ are exact arithmetic.\nThis removes finite termination for that target; it does not establish achievement-set membership.\n\n\\subsubsection{Multiplicity of the stated two--three representations}\n\nConsider the primitive representations $2p+3q=n$ with $p,q>0$ and $\\gcd(p,q)=1$\n(\\lean{IsPrimitive23Solution}{Primitive23Multiplicity.lean:20}).  Every $n\\ge11$\nhas such a representation\n(\\lean{exists\\_primitive23\\_solution\\_of\\_eleven\\_le}{Primitive23Multiplicity.lean:52}),\nwhile $n=10$ has none\n(\\lean{no\\_primitive23\\_solution\\_ten}{Primitive23Multiplicity.lean:26}).\nThe value $n=11$ already has two distinct representations\n(\\lean{exists\\_two\\_primitive23\\_solutions\\_eleven}{Primitive23Multiplicity.lean:38}),\nand so does every $n=10k$ with $k\\ge2$\n(\\lean{exists\\_two\\_primitive23\\_solutions\\_mul\\_ten}{Primitive23Multiplicity.lean:86}).\nThese facts explain an approach and give no counterexample: nothing here proves\nthat the associated integer-multiplicity expansion of $1/21$ can be made Boolean.  The recurring multiplicities are precisely\nthe unresolved collision data, not zero--one support digits.\n\n\\subsubsection{Integer-quotient tests for one twenty-first}\n\\papersectiontarget{record257:twenty-one-quotients}\n\nFor $x=1/21$, nonmembership forces the real greedy sequence eventually\nto select every exponent. The integer quotient rows then agree with\nthat sequence at growing cutoffs. Let\n$\\mathcal F_{21}$ denote the conjunction consisting\nof a fatal greedy gap, finite skipped support (hence cofinite selection),\neventual alignment between the quotient rows and the rational greedy prefix,\nand eventual visits to every doubling block.  Then\n\\[\n \\frac1{21}\\in\\ensuremath{\\mathcal A}\n \\quad\\Longleftrightarrow\\quad\n \\text{$\\mathcal F_{21}$ does not hold}\n\\]\n(\\lean{one\\_div\\_twenty\\_one\\_mem\\_iff\\_not\\_fatalAlignedBranch}{TwentyOneQuotientGreedy.lean:3507}).\n\nOn the fatal branch the canonical quotient remainder $s_R$ is eventually\nstrictly above the closed capacity $2^R$\n(\\lean{twentyOneFatalAlignedBranch\\_eventually\\_strict\\_supercapacity}{TwentyOneQuotientGreedy.lean:5625}),\nand thereafter the support appends $R+1$ while $s_R$ follows one explicit\naffine recurrence\n(\\lean{twentyOneFatalAlignedBranch\\_eventually\\_affine\\_supercapacity}{TwentyOneQuotientGreedy.lean:5658}).\nConversely, closed rows $s_R\\le2^R$ along any unbounded sequence decay after\nnormalisation and put $1/21$ in the achievement set\n(\\lean{twentyOneCofinalEvenQuotientGreedyDecay\\_of\\_closedRows}{TwentyOneQuotientGreedy.lean:5554};\n\\lean{one\\_div\\_twenty\\_one\\_mem\\_mersenneAchievementSet\\_of\\_cofinalGreedyDecay}{TwentyOneQuotientGreedy.lean:5458}).\n\nWe give the definitions and the complete ordinary proof, including the\nrounding estimate and the agreement through rank $2R$ needed in the\nshort paper. Put $T_M=\\lfloor2^M/21\\rfloor$ and\n$q_M(d)=\\lfloor2^M/(2^d-1)\\rfloor$. Starting with $T_{2R}$, apply the\ninteger greedy rule to the weights $q_{2R}(d)$, in increasing order of\n$d$, selecting a weight exactly when it does not exceed the current\nremainder. Let $D_R$ be the selected exponents in $\\{2,\\ldots,R\\}$,\nand let\n\\[\n s_R=T_{2R}-\\sum_{d\\in D_R}q_{2R}(d).\n\\]\nThe alignment in $\\mathcal F_{21}$ refers to the full integer word\nthrough $2R$ agreeing with the real greedy word through $2R$, for all\nsufficiently large $R$. Its other clauses are a finite fatal gap\n$r_n(1/21)>R_n$ with all later exponents selected, and a selected\nexponent in every sufficiently late block $(K,2K]$.\nOn this conjunction, the eventual recurrence is\n\\begin{equation}\n \\begin{split}\n s_{R+1}={}&4s_R+\n \\left\\lfloor\\frac{4(4^R\\bmod21)}{21}\\right\\rfloor\n -2c_{D_R}(2R+1)-c_{D_R}(2R+2)\\\\\n &{}-(2^{R+1}+1),\\qquad\n D_{R+1}=D_R\\cup\\{R+1\\},\\quad s_R>2^R.\n \\end{split}\n \\label{eq:record257-twenty-one-recurrence}\n\\end{equation}\n\n\\begin{proof}\nPut $x=1/21$, let $G=G_{1/21}$, and write $\\delta=x-X_G(2)\\ge0$.\nIf $x\\notin\\mathcal A$, then $\\delta>0$ and $r_n\\downarrow\\delta$.\nSince $w_n\\to0$, every sufficiently large exponent is selected. Choose\n$n$ beyond that point with $R_n<\\delta$. This proves the first clause of\n$\\mathcal F_{21}$, and cofinite support gives its third clause.\n\nWe verify the integer comparisons as well. Set $B=2^{2R}$. When the\ninteger and real rules agree through $d-1$, their remainders after\nnormalisation by $B$ differ by at most $d/B$, because each floor changes\nits argument by less than one. The normalised integer weight differs\nfrom $w_d$ by less than $1/B$. Each fixed real comparison is strict:\nequality would give a finite representation of $x$, and hence\n$x\\in\\mathcal A$. Choose $d_0$ such that $w_d<\\delta/2$ for $d>d_0$.\nFor large $R$ the finitely many comparisons through $d_0$ agree, and\n$2R/B<\\delta/2$. At every subsequent step $d\\le2R$, the normalised\ninteger remainder is at least $\\delta-d/B>\\delta/2>w_d$, so that both\nrules select $d$. Induction proves the second clause. Conversely, a\nrepresented target is recovered by the real greedy rule, and its\nremainder always satisfies $r_n\\le R_n$. Thus $\\mathcal F_{21}$ is\nimpossible for a represented target, proving the membership equivalence.\n\nFor any $R$, rounding the target and the selected weights gives\n\\begin{equation}\n \\left|x-X_{D_R}(2)-\\frac{s_R}{4^R}\\right|\n \\le\\frac{R+1}{4^R}.\n \\label{eq:record257-twenty-one-rounding}\n\\end{equation}\nIf $s_R\\le2^R$ on an unbounded sequence, the finite subseries values\n$X_{D_R}(2)$ tend to $x$. The achievement set is closed, so $x\\in\\mathcal A$.\nThis proves the closed-return implication. A finite representation is\nexcluded by the finite-denominator argument already given.\n\nOn $\\mathcal F_{21}$, the integer comparison and cofinite support give\n$D_R=G\\cap\\{2,\\ldots,R\\}$ and $D_{R+1}=D_R\\cup\\{R+1\\}$ for all large\n$R$. Equation~\\eqref{eq:record257-twenty-one-rounding} yields\n$s_R/4^R\\to\\delta>0$, and hence $s_R>2^R$ eventually. For $d\\ge2$,\nthe finite geometric expansion of $q_M(d)$ gives\n\\[\n q_{M+2}(d)=4q_M(d)+2\\mathbf1_{d\\mid M+1}+\\mathbf1_{d\\mid M+2}.\n\\]\nAlso\n\\[\n T_{2R+2}=4T_{2R}+\n \\left\\lfloor\\frac{4(2^{2R}\\bmod21)}{21}\\right\\rfloor,\n \\qquad q_{2R+2}(R+1)=2^{R+1}+1.\n\\]\nSubtract the contributions of $D_R$ and of the new exponent $R+1$ to\nobtain~\\eqref{eq:record257-twenty-one-recurrence}.\n\\end{proof}\n\nThe remaining question is whether this eventual recurrence can occur\nfor the actual greedy word, or whether closed returns occur at\nunbounded depths. Neither assertion is established here. Since no\nfinite support represents $1/21$, membership would give an\ninfinite-support rational counterexample to the universal problem.\n\n% ; - part a257_p2 ; -\n\\clearpage\n\\part*{IV. Scales, conditional interfaces and unresolved inputs}\n\\addcontentsline{toc}{part}{IV. Scales, conditional interfaces and unresolved inputs}\n\\section{Finite ranges and unbounded parameters}\n\\label{sec:scale-ladder}\nHalf-membership follows from exact rows at unbounded depths or from\ninfinitely many greedy skips. The reset criterion has a different form:\nit assumes a bound at every reset beyond a threshold. We compare these\nquantifiers before using the shared transition identities. A finite\nwitness cannot replace either unbounded existence or a uniform bound.\n\nFor example, $\\forall n\\ge N_0,\\,P(n)$ yields cofinal witnesses to\n$P$, by taking $n=\\max\\{N,N_0\\}$. An implication\n$H(n)\\Rightarrow C(n)$ yields no such witness before $H(n)$ is proved,\nand a finite verified band says nothing about later indices.\nCompatibility between cofinal witnesses is a further condition only\nwhere it is imposed explicitly.\n\nTable~\\ref{tab:scale-ladder} records both the exact parameter ranges and\nthe reasons for the assertions: ordinary or formal proofs, cited\ntheorems, finite calculations and unproved hypotheses. The lower rows\ninclude proved conditional implications as well as open antecedents.\nNeither their truth nor their evidence is determined by the quantifier\nprefix alone.\n\n\\begingroup\n\\small\n\\setlength{\\tabcolsep}{3pt}\n\\begin{longtable}{@{}p{0.60\\linewidth}p{0.37\\linewidth}@{}}\n\\toprule\n\\papertablehead\n\\textbf{Result} & \\textbf{Quantifier prefix} \\\\\n\\midrule\n\\endfirsthead\n\\toprule\n\\papertablehead\n\\textbf{Result} & \\textbf{Quantifier prefix} \\\\\n\\midrule\n\\endhead\n\\bottomrule\n\\endfoot\n\\papertablehead\\multicolumn{2}{l}{\\emph{One concrete witness or value}} \\\\\nSeed for continuation from $n\\ge6$: support $\\{2,3,6\\}$ satisfies\n$\\ensuremath{Q}=2^5-1$ (exact computation). \\lean{exactLocalMersenneHalfRow\\_six}{BooleanMobiusExactRowSeed.lean:34}\n& $n=6$ \\\\\nOne-point countermodel: the double-or-recycle transition shape alone, even seeded, does not\ngive exact rows at unbounded depths. \\lean{boundedDoubleOrRecycleModel\\_not\\_cofinal}{BooleanMobiusExactRowDichotomy.lean:79}\n& $n=6$ (free-standing model) \\\\\nThe combined-residue-mass $\\le 1$ sufficient test for sharp capacity is not necessary:\nexample $D=\\{2,3\\}$, $c=5$ (exact computation). \\lean{splitFractionMass\\_one\\_bound\\_not\\_necessary\\_fixture}{BooleanMobiusSkippedCoreCriticalCapacity.lean:183}\n& $D=\\{2,3\\},\\ c=5$ \\\\\nExact residual: $\\tfrac12-W(\\{2,3,6,7\\})=\\tfrac1{16002}$ (exact computation).\n\\lean{half\\_sub\\_four\\_term\\_prefix\\_eq}{HalfCutLocator.lean:664}\n& none (numeric instance) \\\\\nClosed-form M\\\"obius--Lambert value $\\sum_{d\\ge1}\\mu(d)/(2^d-1)=1/2$ (classical identity, cited).\n\\lean{tsum\\_moebius\\_div\\_two\\_pow\\_sub\\_one\\_eq\\_half}{MersenneLambertLadder.lean:587}\n& none (closed-form value) \\\\\nThe one natural negative-M\\\"obius candidate support overshoots: $1/2 <\n\\sum_{d\\in N}1/(2^d-1)$ for $N=\\{d\\ge2:\\mu(d)=-1\\}$.\n\\lean{half\\_lt\\_tsum\\_negativeMobius}{MobiusSignSupportNoGo.lean:164}\n& one fixed candidate support $N$ \\\\\n\\papertablehead\\multicolumn{2}{l}{\\emph{Finite ranges and witnesses bounded by the input}} \\\\\nFrom an exact row, either repeated doubling is available or an exact row is obtained at some $2c-2$ with $c\\le n$; these alternatives need not be disjoint.\n\\lean{exactLocalMersenneHalfRow\\_double\\_or\\_recycle}{BooleanMobiusExactRowDichotomy.lean:26}\n& $\\forall n\\ge6$, given a row at $n$: $(\\ldots)\\lor\\exists c\\le n(\\ldots)$ \\\\\nEvery above-half bounded support recycles into a genuine exact row at some $2c-2$ with $c\\le n$.\n\\lean{exists\\_skippedCoreExactRow\\_of\\_value\\_above}{BooleanMobiusExactRowCrossing.lean:155}\n& $E\\subseteq[2,n]$, $X_E(2)>1/2$: $\\exists c\\le n$ \\\\\n(A cross-check for Problem~\\#249.) $\\sum\\varphi(n)/2^n$ matches no rational whose denominator divides\n$2^{14}(2^h-1)$, for some $1\\le h\\le16$ (exact computation).\n\\lean{totient\\_series\\_ne\\_rat\\_of\\_den\\_dvd\\_upto\\_sixteen}{CarrySurvivorExtinction.lean:587}\n& $1\\le h\\le16$ (explicit bound) \\\\\nTruncation-rung ladder: $\\mathrm{HalfRung}(J)$ proved for every $3\\le J\\le22$ via the finite\ndecision procedure (\\S\\ref{sec:four-coordinates}, Theorem~\\ref{thm:tr-finite-decision});\nordinary argument with exact computation\n(\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/notes/truncation-rung-ladder-2026-07-24.md}{working note},\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/check_truncation_rung_ladder.py}{program}).\n& $3\\le J\\le22$ (finite table, exhaustive) \\\\\n\\papertablehead\\multicolumn{2}{l}{\\emph{Uniform assertions and conditional implications; premises retained}} \\\\\nMembership equals greedy survival.\n\\lean{mem\\_mersenneAchievementSet\\_iff\\_greedy\\_survival}{GreedyAchievementSet.lean:1458}\n& $\\forall x\\ge0$, membership iff survival at every rank \\\\\nMembership iff every actually skipped rank survives.\n\\lean{no\\_lastHalfGreedySkip\\_iff\\_every\\_skip\\_survives}{HalfCylinderFixedTailSocket.lean:57}\n& $\\forall M\\in$ skipped support \\\\\nEvery crossing core is forced to be the canonical rational half-greedy prefix.\n\\lean{eq\\_halfGreedyPrefixSupport\\_of\\_critical\\_crossing}{BooleanMobiusCriticalCapacityCofinal.lean:50}\n& $c\\ge4$, $D\\subseteq[2,c)$, $0<1/2-X_D(2)<w_c$ \\\\\nLargest-skip conditional implication used in the reset argument.\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_largestSkipLateStepSocket}{HalfCylinderLargestSkipInduction.lean:167}\n& $\\forall s\\ge14$, given the hypothesis \\\\\nBand avoidance verified by exact computation at reset indices $13\\le r\\le30$, using\nsuccessor remainders at indices $14,\\ldots,31$.  This is a finite range\nof the conditional hypothesis. \\lean{seamUpperResetDyadicBandEscape\\_through\\_thirty}{HalfCylinderUpperResetBandCertificates.lean:78}\n& $13\\le r\\le30$, exhaustive \\\\\nCritical-band-index reduction: a $(d{+}1)$-way check collapses to one nearest-boundary check.\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_upperResetCriticalBandEscape}{HalfUpperResetCriticalBand.lean:896}\n& $\\forall d,E$ with $E\\le2^{d+1}$ \\\\\nParity obstruction: no finite support has value $1/2$. \\lean{finite\\_boolSupport\\_ne\\_half}{HalfCarryReachability.lean:589}\n& $\\forall A$ finite, $0\\notin A$ \\\\\nFull-support irrationality at every integer base.\n\\lean{irrational\\_erdosSum\\_full\\_support}{CertificateKernel.lean:8328}; base two:\n\\lean{irrational\\_erdosBorwein\\_series}{CertificateKernel.lean:8335}\n& $\\forall b\\ge2$ \\\\\nErd\\H{o}s 1968: any infinite, pairwise-coprime, reciprocal-summable index set gives an irrational\nsupport series. \\lean{irrational\\_erdosSupportSeries\\_pairwise\\_coprime}{CertificateKernel.lean:10776}\n& $\\forall b\\ge2\\ \\forall A$ (infinite, coprime, summable) \\\\\nBlock-certificate criterion, independent of the problem: a weighted-coefficient series is\nirrational given a divisibility and window certificate at every precision.\n\\lean{irrational\\_coeff\\_series\\_of\\_weighted\\_coeff\\_block\\_certificates}{CertificateKernel.lean:8665}\n& $\\forall b\\ge2\\ \\forall c\\,(\\forall m,c(m)\\le m)$, given $\\forall q\\,\\exists N,K,L,C$ \\\\\nFinite greedy-prefix denominators are odd (source of the parity boundary exclusion,\n\\S\\ref{sec:four-coordinates}). \\lean{halfGreedyPrefixDenominator\\_odd}{DyadicPrefixCompression.lean:1521}\n& $\\forall n$ \\\\\n\\papertablehead\\multicolumn{2}{c}{\\textbf{Unproved hypotheses and the proved implications that use them}} \\\\\n\\papertablehead\\multicolumn{2}{l}{\\emph{Unbounded-witness and all-reset conditions have different quantifiers}} \\\\\nThe exact-row existence condition: exact rows recur at arbitrarily large endpoints, with no coherence\nrequired between witnesses (unproved). \\lean{CofinalExactLocalMersenneHalfRows}{BooleanMobiusCofinalExactRows.lean:38}\n& $\\forall N\\ \\exists n\\ge N$: an exact row at depth $n$ \\\\\nThe preceding condition implies $1/2\\in\\mathcal A$ by compactness (theorem proved;\nhypothesis open). \\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_cofinalExactLocalRows}{BooleanMobiusCofinalExactRows.lean:71}\n& given $\\forall N\\,\\exists n\\ge N(\\ldots)$ \\\\\nAn equivalent membership condition: the canonical rational half-greedy orbit itself has infinitely many\npositive skips (unproved). \\lean{CofinalPositiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:22}\n& $\\forall N\\ \\exists c\\ge\\max(N,4),\\ 0<\\mathrm{rem}(c{-}1)<w_c$ \\\\\nInfinitely many positive greedy skips imply $1/2\\in\\mathcal A$ (theorem proved; hypothesis open).\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_positiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:97}\n& given $\\forall N\\,\\exists c\\ge N(\\ldots)$ \\\\\nMembership iff the greedy skip set is infinite (equivalence proved; both sides assert\nunboundedly many indices).\n\\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_greedySkippedSupport\\_infinite}{GreedyAchievementSet.lean:2583}\n& the skipped support is infinite \\\\\nSeven equivalent forms of membership (Theorem~\\ref{thm:nine-way-hub}): the integer-greedy\nsequence is not eventually always on branch $\\mathrm R$, the terminal rank is omitted at\nunboundedly many rows, there are cofinally many skipped ranks, \\emph{etc.}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_not\\_seamGreedyEventuallyRight}{HalfCylinderHalfMembershipClassification.lean:112}\n& every right-hand side asserts unboundedly many indices \\\\\nLadder dichotomy: either infinitely many rungs survive (\\#257 false, explicit limit support) or an\neventual bad-rank obstruction closes only this route (ordinary argument,\nCorollary~8 of the\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/notes/truncation-rung-ladder-2026-07-24.md}{truncation-rung note}).\n& open branch (a): $\\exists^\\infty J,\\ \\mathrm{HalfRung}(J)$ \\\\\nReset square-root escape, a sufficient all-reset hypothesis (unproved).\nTheorem~\\ref{thm:lower-bound-every-reset} gives the conditional\nimplication; the run-length\ncomparison is discussed separately in Section~\\ref{sec:o4}.\n& $\\forall r\\ge10$ that are resets,\n$|\\mathrm{rem}(r+1)-2^{r+1}|>2^{(r+5)/2}$ \\\\\n\\caption{Quantifiers for the stated results.  The groups\nseparate finite tests, uniform implications, unbounded witness\nconditions and all-reset hypotheses.  A theorem conditional on an\nunproved hypothesis is not a proof of that hypothesis.}\n\\label{tab:scale-ladder}\n\\end{longtable}\n\\endgroup\n\n\\begin{rem}[Finite checks and quantified conclusions]\n\\label{rem:promotion-audit}\nAn earlier review examined eighteen possible implications from recorded\nresults to the required unbounded-witness conditions.  It found no\nsuch implication.  This is a report about those attempts, not a theorem\nexcluding other arguments.\n\nFor example, \\lean{seamUpperResetDyadicBandEscape\\_through\\_thirty}{HalfCylinderUpperResetBandCertificates.lean:78}\nchecks reset indices $13$ through $30$; it has no conclusion beyond that\nrange.  Likewise, a bound depending on the depth need not produce a\nwitness at a larger depth.  Conversely, a proved eventual assertion of\nthe \\emph{same} witness predicate immediately gives cofinal witnesses.\nWhat decides the matter is the precise premise and conclusion, whatever\ntactic a finite proof uses.\n\\end{rem}\n\n\\section{Approximations to the greedy orbit}\n\\label{sec:four-coordinates}\nFinite quotient procedures approximate the real greedy sequence for\n$1/2$. Agreement on a fixed prefix does not give agreement at a cutoff\nthat grows with the scale. We track the cutoff in each of the four\ncalculations below and state the uniform estimate that would be needed\nto pass to unbounded depths.\n\n\n\\subsection*{Four reported calculations}\n\nFix the weights $x_n:=1/(2^n-1)$, $n\\ge2$, and the target $1/2$. The single object under study\nthroughout this programme is the greedy orbit for this target: a residual $\\rho$ starts at $1/2$,\nand at rank $k$ the orbit \\emph{takes} iff $\\rho\\ge x_k$. A skip at rank $k$ is \\emph{safe} iff\n$\\rho\\le T_{k+1}:=\\sum_{j>k}x_j$ and \\emph{fatal} iff $\\rho\\in(T_{k+1},x_k)$; the fatal interval is\nexactly the gap no later tail can repair. The four calculations below concern this target, but their finite\nparameters and certification records differ.  Agreement with the real\ngreedy prefix requires a convergence argument for the approximants used\nin each calculation; agreement over a tested range is not that argument.\n\n\\begin{itemize}\n\\item \\textbf{Truncation rung $J$.} Replace the full weight $x_n$ by the $J$-term truncation\n$w_n^{(J)}:=\\sum_{q=1}^{J}2^{-qn}$ and ask whether some Boolean support hits $1/2$ exactly under\nthe truncated weights. Certified for $3\\le J\\le22$ (Table~\\ref{tab:scale-ladder}, finite ranges;\nTheorem~\\ref{thm:tr-witness-exclusion}--\\ref{thm:tr-finite-decision} below).\n\\item \\textbf{Seam row $s$.} The base-4 seam recursion tracking the deviation $\\Delta_s:=\\mathrm{rem}(s)-2^s$\nof the integer greedy orbit from its dyadic target. Certified rows $6$ through $2500$ for the\nreset-crossing classification, and the underlying orbit separately certified to row\n$200{,}000$, with exactly one violation of the target inequality at reset rows $6\\le r\\le200{,}000$, at $r=7$\n(saved runs to\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/seam-reset-crossing-2500.json}{row $2500$}\nand to\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/seam-orbit-200000.json}{row $200{,}000$}).\n\\item \\textbf{Integer margin $m_c$.} An exploratory, unverified scan of the integer-greedy\nmargin out to $5\\times10^5$, tracking how far the certified region sits from the danger\nthreshold.\n\\item \\textbf{Sharp tail margin.} The rank-by-rank take/skip margin against the exact tail, checked\nfor ranks $2$ through $3000$: $1497$ takes against $1502$ skips, zero fatal skips anywhere, minimum\nsafe-skip margin $+2.9922$ bits at rank $5$, and the smallest\nreported take-margin $+0.0340$ bits at rank $7$, where the remainder\n$1/126$ exceeds the weight $1/127$.  Here the margins are the base-two\nlogarithms of the corresponding positive slack multiplied by $4^k$\n(\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/tail-margins-3000.json}{saved run};\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/notes/reset-crossing-unification-2026-07-24.md}{working note},\nsection~8).\n\\end{itemize}\n\n\\begin{prop}[Stability of each fixed greedy prefix]\n\\label{prop:one-orbit}\nLet $t_j\\to1/2$ and, for each fixed $n\\ge2$, let\n$v_n^{(j)}\\to x_n=(2^n-1)^{-1}$, with $v_n^{(j)}>0$.\nApply the greedy rule with target $t_j$ and weights $v_n^{(j)}$ in\nincreasing order of $n$, through depths $m_j\\to\\infty$.\nFor every fixed depth $K$, the decisions at ranks $2,\\ldots,K$\neventually agree with those of the real half-greedy rule.\nThis assertion concerns finite prefixes, not survival at all ranks.\n\\end{prop}\n\\begin{proof}\nAt rank $2$, the target and the weight converge to their real counterparts.\nTheir comparison is strict by Lemma~\\ref{lem:no-ties}, so its sign is\neventually unchanged.  Inductively, once all earlier decisions agree,\nthe approximate remainder is the target minus a fixed finite sum of\nconvergent weights.  It therefore converges to the real remainder.\nThe next comparison is again strict, so the next decision eventually\nagrees.  A finite induction through $K$ proves the claim.\n\\end{proof}\n\\begin{lem}[No-ties lemma]\n\\label{lem:no-ties}\nAt every rank $k$ of the full greedy orbit for target $1/2$, both defining comparisons are strict:\n$\\rho\\ne x_k$ and $\\rho\\ne T_{k+1}$.\n\\end{lem}\n\\begin{proof}\nThe two strict comparisons have different reasons: parity excludes\nequality with a weight, and irrationality excludes equality with the\ncomplete tail.\n\n\\emph{Take boundary excluded (parity).} Write a finite Mersenne sum as $p/q=\\sum_{d\\in F}x_d$ in lowest\nterms.  Its denominator $q$ is odd, because every $2^d-1$ is odd\n(equivalently, \\lean{halfGreedyPrefixDenominator\\_odd}{DyadicPrefixCompression.lean:1521}, itself\ntraced to \\lean{finiteErdosSum\\_den\\_odd}{DyadicPrefixCompression.lean:1513}). Hence the residual\nafter any finite prefix is $\\rho=\\tfrac12-\\tfrac pq=\\tfrac{q-2p}{2q}$: its reduced denominator\nis \\emph{even}. Every weight $x_k=1/(2^k-1)$ has \\emph{odd} denominator. An even-denominator\nrational can never equal an odd-denominator rational, so $\\rho=x_k$ is impossible at any rank:\nthe take comparison is never a tie.\n\n\\emph{Skip-safety boundary excluded (irrationality).} The tail past rank $k$ is\n$T_{k+1}=E-1-\\sum_{2\\le j\\le k}x_j$, where $E=\\sum_{n\\ge1}1/(2^n-1)$ is the Erdős--Borwein\nconstant: a rational shift of $E$. Erdős's own 1948 theorem gives $E$ irrational\n(\\lean{irrational\\_erdosBorwein\\_series}{CertificateKernel.lean:8335}, instantiated at $b=2$), so\n$T_{k+1}$ is irrational for every $k$. The residual $\\rho$ arising from any finite Boolean prefix\nis rational. An irrational number never equals a rational one, so $\\rho=T_{k+1}$ is impossible at\nany rank; the skip-safety comparison is never a tie either.\n\nBoth boundaries of the fatal interval $(T_{k+1},x_k)$ are therefore approached only strictly, at\nevery rank, unconditionally.\n\\end{proof}\n\\begin{rem}\nThe hypotheses of Proposition~\\ref{prop:one-orbit} can be checked for\ntwo concrete approximations.  The truncated geometric weights\n$\\sum_{q=1}^{J}2^{-qn}$ increase to $x_n$.  For the integer quotient\nrows, division by $4^s$ gives weights\n$4^{-s}\\lfloor4^s/(2^n-1)\\rfloor\\to x_n$ and target\n$4^{-s}(2^{2s-1}-2^s)=1/2-2^{-s}\\to1/2$.\nHence both greedy rules agree with each fixed real prefix once their\nparameters are large enough.  Convergent upper and lower bounds for\n$R_k$ likewise determine the strict tail comparison at a fixed rank.\nNo conclusion about the exploratory margin scan follows without specifying\nits approximants and proving their convergence.  Fixed-prefix agreement is not uniform agreement through a growing\ndepth.  For the quotient rows, however, summability bounds the unmatched\ntail by $R_K$ and does give convergence of the coded values, as proved\nin Theorem~\\ref{thm:seam-limit}.  It identifies their limit as\n$X_G(2)$; it does not establish that this value is $1/2$.\n\\end{rem}\n\n\\subsection*{Successive tail truncations}\n\nFinite truncations of the geometric expansion give rational weights\nand permit a finite decision procedure for each fixed truncation\norder $J$.  The bounds below explain why only finitely many ranks\nneed to be tested.  Fix $J\\ge2$ and set\n\\[\n\\begin{aligned}\nw_n^{(J)}&:=\\sum_{q=1}^{J}2^{-qn}, &\nT_{n+1}^{(J)}&:=\\sum_{q=1}^{J}\\frac{2^{-qn}}{2^q-1},\\\\\n\\mathrm{HalfRung}(J)&:\\Longleftrightarrow\n\\exists A\\subseteq\\{2,3,\\ldots\\},&\n\\sum_{n\\in A}w_n^{(J)}&=\\tfrac12.\n\\end{aligned}\n\\]\nThe arguments below are ordinary mathematics.  The accompanying exact\ninteger computation rechecks the finite instances $3\\le J\\le22$; neither\nthe arguments nor that computation are presented as a Lean verification.\n\n\\begin{lem}[Forced greedy]\n\\label{lem:tr-forced-greedy}\nFor every $J\\ge2$: $w_n^{(J)}>T_{n+1}^{(J)}$ for all $n$ (the $q=1$ terms agree exactly, and every\n$q\\ge2$ tail term is strictly smaller than the corresponding weight term). Consequently the greedy support is the unique candidate support, and\n$\\mathrm{HalfRung}(J)$ holds iff the greedy orbit for $1/2$ under weights $w_n^{(J)}$ never lands in\na fatal interval $(T_{n+1}^{(J)},w_n^{(J)})$. Rank $1$ is always a safe skip; ranks $2$ and $3$ are\nalways takes.\n\\end{lem}\n\n\\begin{lem}[Parity forces infinite support]\n\\label{lem:tr-parity}\nFor every $J\\ge2$, no finite $A\\subseteq\\{2,3,\\ldots\\}$ attains\n$\\mathrm{HalfRung}(J)$.\n\\end{lem}\n\\begin{proof}\nThe empty set has value zero.  For a nonempty finite set, put $m=\\max A$.\nAfter multiplication by $2^{Jm}$, the $m$th summand is odd and every earlier\nsummand is even, whereas the target $2^{Jm-1}$ is even.  This parity\ncontradiction excludes finite support.\n\\end{proof}\n\n\\begin{thm}[Witness exclusion]\n\\label{thm:tr-witness-exclusion}\nDefine the misalignment mass\n$$\\mu_J(M):=\\sum_{q=2}^{J}\\frac{2^{M\\bmod q}}{2^q-1}.$$\nLet $J\\ge3$, $n\\ge4$, and suppose some $M\\in[n,2n-2]$ has $\\mu_J(M)\\le\\tfrac{11}{15}$. Then no\nBoolean prefix $D\\subseteq\\{2,\\ldots,n-1\\}$ satisfies\n$$T_{n+1}^{(J)}<\\tfrac12-\\sum_{d\\in D}w_d^{(J)}<w_n^{(J)};$$\nThis excludes the specified gap for every Boolean prefix, not only the\ngreedy prefix.\n\\end{thm}\n\\begin{proof}\nSuppose such a prefix exists and write\n$\\rho=1/2-\\sum_{d\\in D}w_d^{(J)}$.  Since\n$T_{n+1}^{(J)}\\ge2^{-n}$ and $n\\le M\\le2n-2$,\n\\[\n 0<2^M\\rho-2^{M-n}\n   <\\sum_{q=2}^{J}2^{M-qn}\n   \\le\\frac{2^{M-2n}}{1-2^{-n}}\\le\\frac4{15}.\n\\]\nHere $n\\ge4$ gives the last inequality.  Split each scaled summand\n$2^{M-qd}$ according to whether $qd\\le M$.  The terms with\n$qd\\le M$ are integers.  The other terms have total\n\\[\n F=\\sum_{d\\in D}\\sum_{\\substack{1\\le q\\le J\\\\qd>M}}2^{M-qd}\n \\le\\sum_{q=2}^{J}\\sum_{d>\\lfloor M/q\\rfloor}2^{M-qd}\n =\\mu_J(M)\\le\\frac{11}{15}.\n\\]\nThere is no $q=1$ contribution to $F$, since $d<n\\le M$.\nThus $2^M\\rho+F-2^{M-n}$ is an integer strictly between zero and one,\na contradiction.  The same calculation also applies when $J=2$.\n\\end{proof}\n\n\\begin{cor}[Half-LCM horizon]\n\\label{cor:tr-half-lcm}\nLet $J\\ge2$ and $L_J:=\\mathrm{lcm}(2,3,\\ldots,J)$.  For every\n$n\\ge\\max\\{4,L_J/2+1\\}$ there is an $M\\in[n,2n-2]$ with\n$\\mu_J(M)<11/15$.  Thus witness exclusion leaves only the finite\nwindow $[4,L_J/2]$ to check.\n\\end{cor}\n\\begin{proof}\nLet $M$ be the least multiple of $L_J$ with $M\\ge n$.\nIf $M=L_J$, then $M\\le2n-2$ is the assumed inequality.\nIf $M=kL_J$ with $k\\ge2$, then $n\\ge(k-1)L_J+1$ gives\n$2n-2\\ge2(k-1)L_J\\ge M$.  At this witness,\n\\[\n \\mu_J(M)=\\sum_{q=2}^{J}\\frac1{2^q-1}\n <\\sum_{q=2}^{\\infty}\\frac{4}{3\\cdot2^q}\n =\\frac23<\\frac{11}{15}.\n\\]\nThe bound uses $2^q-1\\ge3\\cdot2^{q-2}$, with strict inequality for\n$q>2$, so no numerical approximation to the full series is needed.\nApply Theorem~\\ref{thm:tr-witness-exclusion}; its proof also covers\n$J=2$.  Ranks $2$ and $3$ are takes, as already noted.\n\\end{proof}\n\n\\begin{lem}[Mod-12 filter, $J\\ge7$]\n\\label{lem:tr-mod12}\nFor $J\\ge7$, the inequality $\\mu_J(M)\\le11/15$ implies $12\\mid M$.\nThus only multiples of $12$ need be tested as witnesses in each interval\n$[n,2n-2]$; no potentially uncovered rank is discarded.\n\\end{lem}\n\\begin{proof}\nIf $M$ is odd, the $q=2,3$ terms give at least $2/3+1/7>11/15$.\nIf $M\\equiv2\\pmod4$, the $q=2,3,4$ terms give at least\n$1/3+1/7+4/15=26/35>11/15$.  Finally, if $4\\mid M$ but $3\\nmid M$,\nthe $q=2,3,4,6$ terms give at least\n$1/3+2/7+1/15+4/63=236/315>11/15$.\nEach case contradicts the assumed upper bound.\n\\end{proof}\n\n\\begin{thm}[Finite decision procedure]\n\\label{thm:tr-finite-decision}\nCall $n\\in[4,L_J/2]$ \\textbf{bad} if no $M\\in[n,2n-2]$ has $\\mu_J(M)\\le\\tfrac{11}{15}$, and set\n$B(J):=\\max(\\mathrm{bad}\\cup\\{3\\})$. Then $\\mathrm{HalfRung}(J)$ holds iff the greedy orbit for\n$1/2$ under weights $w_n^{(J)}$ survives every rank from $2$ through\n$B(J)$.  This is a finite exact decision procedure.\n\\end{thm}\n\\begin{proof}\nSurvival at all ranks plainly implies survival through $B(J)$.\nConversely, every later rank at most $L_J/2$ has a witness by the\ndefinition of $B(J)$, and every larger rank has one by\nCorollary~\\ref{cor:tr-half-lcm}.  Witness exclusion rules out a fatal gap\nat all these ranks.  The strict-tail criterion then gives membership.\nFor $J=2$, the same proof works because $\\mu_2(M)\\le2/3$ for every $M$;\nthe window is empty and $B(2)=3$.\n\\end{proof}\n\n\\begin{rem}[Verified truncations and the unbounded-index question]\nThe finite decision procedure verifies $\\mathrm{HalfRung}(J)$ for every\n$3\\le J\\le22$.  The checking program uses exact integers: it verifies\nwitness intervals covering the ranks beyond $B(J)$, proves for $J\\le16$\nthat the listed bad ranks have no witness, and tests the greedy remainders\nthrough $B(J)$.  In particular, it reproduces $B(19)=1008$ and\n$B(20)=B(21)=B(22)=1530$.\nThe\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/check_truncation_rung_ladder.py}{checking program}\nand its saved outputs for\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/truncation-ladder-3-16.json}{$3\\le J\\le16$}\nand\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/truncation-ladder-17-22.json}{$17\\le J\\le22$}\nare published.  For $J\\ge17$ a floating-point filter proposes witnesses\nand each accepted witness is rechecked with integers, so a missed witness\ncould only lengthen the exact greedy check.  These are finite\ncomputer-assisted proofs using the ordinary lemmas above, not Lean\nproofs and not a test of the full weights through all ranks.\n\nIf $A$ represents $1/2$ with the $J$-truncated weights, then\n\\[\n 0\\le X_A(2)-\\tfrac12\n \\le\\sum_{q>J}\\frac{2^{-2q}}{1-2^{-q}}\n \\le2\\sum_{q>J}4^{-q}=\\frac23\\,4^{-J}.\n\\]\nConsequently, such representations for an unbounded sequence of $J$\nwould put $1/2$ in the closed set $\\mathcal A$.  Its representing support\nwould be infinite, since finite Mersenne sums have odd denominator\n(Lemma~\\ref{lem:no-ties}); Lemma~\\ref{lem:tr-parity} separately excludes\nfinite support for each truncation.  The unbounded-index assertion is\nstill unproved.  Twenty verified values of $J$ do not establish it.\n\\end{rem}\n% ; - part a257_p3 ; -\n\\section{Which hypotheses remain unproved}\nThree quantified conditions underlie the membership arguments below.\nWe compare the available estimates with each condition, distinguishing\na stronger assumption from an equivalent reformulation and a finite\nrange from an unbounded assertion. The missing estimates remain open;\nthese comparisons decide neither \\#249 nor \\#257.\n\n\\subsection{The three \\#257 obligations, stated exactly}\n\n\\begin{defn}[257-reset]\nFor every upper or middle reset row $r\\ge10$, require\n$\\Delta_{r+1}^2>2^{r+5}$, where\n$\\Delta_{r+1}=\\mathrm{rem}(r+1)-2^{r+1}$.\nThe finite range $10\\le r<31$ must be covered as well as the tail.\nSection~\\ref{sec:o4} gives the exact conditional implication from this\nhypothesis.  A proposed bound on right-run lengths is a different\nstatement until the pulse corrections and endpoint conventions are\nmatched.  Finite observations establish neither failure of every\nconstant run bound nor logarithmic asymptotic growth.\n\\end{defn}\n\n\\begin{defn}[257-cofinal-rows]\nFor every $N$ there is $n\\ge N$ with an exact row at depth $n$\n(Definition~\\ref{record:257bm-d1}); that is, there are finite sets at unboundedly many depths\nwhose quotient sums land exactly on the half-value target.  This condition implies\n$1/2\\in\\mathcal A$ unconditionally\n(\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_cofinalExactLocalRows}{BooleanMobiusCofinalExactRows.lean:71}).\nNo coherence between the witnessing rows is demanded and none should be assumed: the rows may\nbe mutually incompatible; only their depths need tend to infinity. Requiring compatibility would add a condition on the witnesses.\nWhether that condition can be obtained from this existence statement\nneeds a separate extraction argument; it is not a strict logical\ncomparison merely because the witnesses are described differently.\n\\end{defn}\n\n\\begin{defn}[257-universal]\nThe original question is the base-$2$ statement:\n$\\sum_{n\\in A}(2^n-1)^{-1}$ is irrational for every infinite\n$A\\subseteq\\mathbb N_{\\ge1}$.  Asking the same for every integer\nbase $b\\ge2$ is a stronger all-base extension, not the definition of\nProblem~\\#257 itself.\n\\end{defn}\n\n\\subsection{Reading a row}\n\nEach entry identifies the available result, the assumptions needed to\napply it, and the further estimate still required.  A source link supports\nthe stated declaration, not every proposed consequence in the discussion.\nOrdinary deductions and finite computations are identified separately.\n\n\\subsection{Reset estimates and their missing hypotheses}\n\n\\begin{obs}[Remainders along a right-branch run]\n\\lean{upperOrMiddle\\_within\\_of\\_two\\_pow\\_deficit}{HalfCylinderResetDeficitEscape.lean:261}.\nFor integers $s\\ge5$ and $k\\ge0$, the cited theorem assumes\n$\\mathrm{rem}(s)\\le2^s$ and\n$2^k(2^s-\\mathrm{rem}(s))\\ge2^s$ and gives an upper or middle\ntransition at some $t\\in[s,s+k]$.  Its literal contrapositive concerns\n\\emph{all} $k+1$ transition indices in that closed interval.\n\nFor a run of exactly $k\\ge1$ right transitions, at indices\n$s,\\ldots,s+k-1$, a direct recurrence gives the sharper conventionally\nindexed statement.  Put $D_j=2^{s+j}-\\mathrm{rem}(s+j)$ and\n$p_j=p^-_{s+j}\\ge0$.  The right-branch identity yields\n\\[\n D_{j+1}=4D_j+p_j+4,\\qquad\n 4^kD_0<D_k\\le2^{s+k}.\n\\]\nThe strict inequality uses $k\\ge1$, and the last bound uses\n$\\mathrm{rem}(s+k)\\ge0$.  Hence $D_0<2^{s-k}$.\nIn particular, $D_0\\ge1$ excludes $k\\ge s$ consecutive right\ntransitions.  A stronger lower bound on $D_0$ shortens that run, but\nmust be established at the starting row of the run.  Applying this\nargument after a reset requires the post-reset index and the\n\\emph{deficit-side} condition; it does not prove the absolute reset\nbound on both signs.\n\\end{obs}\n\n\\begin{obs}[The affine excess recurrence]\n\\label{obs:affine-excess-recurrence}\n\\lean{affineRightExcess\\_exactIterate}{HalfCylinderMiddleCarryLowerBound.lean:3320},\n\\lean{affineRightExcess\\_charge\\_lt\\_scaledInitial}{HalfCylinderMiddleCarryLowerBound.lean:3350},\n\\lean{eventualRightTail\\_excess\\_exactIterate}{HalfCylinderMiddleCarryLowerBound.lean:3361}\n(recurrence \\lean{rightBranch\\_excess\\_succ\\_eq}{HalfCylinderMiddleCarryLowerBound.lean:2497}).\nFor $k$ right transitions from row $S\\ge5$, let\n$X_j=\\mathrm{rem}(S+j)-2^{S+j}$ and $p_j=p^-_{S+j}$.\nThe cited identity is\n\\[\n X_0=\\frac{X_k}{4^k}+\\sum_{j=0}^{k-1}\\frac{p_j+4}{4^{j+1}}.\n\\]\nThe source also gives $0\\le p_j\\le2(S+j-2)$.  Thus the following\nconsequences are ordinary deductions from the identity:\n\\[\n \\sum_{j=0}^{k-1}\\frac{p_j+4}{4^{j+1}}\n \\le\\sum_{j\\ge0}\\frac{2(S+j)}{4^{j+1}}\n =\\frac{2S}{3}+\\frac29.\n\\]\nIf the terminal row satisfies $\\mathrm{rem}(S+k)<2^{S+k+1}$, then\n\\[\n X_0<2^{S-k}+\\frac{2S}{3}+\\frac29.\n\\]\nOnly this terminal bound is needed; no bound at every intermediate row\nis required.  If instead the largest omitted rank $d$ at the terminal\nrow satisfies $2(S+k)<3d$, the next entry gives the different bound\n\\[\n X_0<2^{S-k}+\\frac{2\\,4^{S+k-d}+4}{3\\,4^k}\n                +\\frac{2S}{3}+\\frac29.\n\\]\nThe extra term must be retained.  Lateness of an omitted rank is not\nknown to be equivalent to $\\mathrm{rem}(s)<2^{s+1}$; the cited estimate\nalone does not imply that sharper ceiling.  Neither upper bound supplies\nthe missing lower bound on an actual reset deviation.\n\\end{obs}\n\n\\begin{obs}[A bound at later rows]\n\\lean{three\\_mul\\_remainder\\_lt\\_exactLateGap}{HalfCylinderResetDeficitEscape.lean:117}.\nFor $s\\ge5$, if $d$ is the largest omitted rank and $2s<3d$,\nthe cited theorem gives\n\\[\n \\mathrm{rem}(s)<2^{s+1}+\\frac{2\\,4^{s-d}+4}{3}.\n\\]\nSince $s-d<s/3$, its additive error is at most\n$(2\\cdot2^{2s/3}+4)/3$, hence $O(2^{2s/3})$.\nThe undivided error $2\\,4^{s-d}+4$ need not be less than\n$2^{2s/3}$: already $s=7,d=5$ gives $36>2^{14/3}$.\nThese numbers test the numerical estimate, not an asserted pair on the\ngreedy orbit.  The theorem supplies an upper bound only under the\nlargest-omitted-rank and lateness assumptions.  When it is used in the\npreceding recurrence, its additive error remains part of the result.\n\\end{obs}\n\n\\begin{obs}[Exclusion of equality at a branch boundary]\n\\lean{seamGreedyFloorZ\\_ne\\_takeThreshold}{HalfCylinderMiddleCarryLowerBound.lean:1665}, with\n\\lean{halfGreedy\\_skipped\\_endpoint\\_trichotomy}{HalfCylinderSkippedEndpointClassifier.lean:246}.\nThe first cited theorem excludes equality between the corrected\nreal-valued quotient expression and the terminal \\emph{take threshold}:\nsuch equality would give a finite Mersenne sum equal to $1/2$, contrary\nto denominator parity.  The second classifies the signed integer\nposition of a real greedy prefix at a skipped rank; its statement\nexplicitly permits the zero-position case.\n\nNeither statement identifies its boundary with\n$\\mathrm{rem}(r)=2^r$.  They therefore do not, by themselves, prove\n$|\\mathrm{rem}(r)-2^r|\\ge1$.  Integrality gives that inequality only\nafter nonvanishing of this particular deviation has been proved.\nAn estimate of the form $2^{r/2+o(r)}$ would not settle the required\nconstant either: $2^{r/2}/r$ has that form but is smaller than\n$2^{(r+5)/2}$.  The reset hypothesis requires its stated constant,\nstrict inequality and post-reset index, not merely an exponent\napproaching $1/2$.\n\\end{obs}\n\n\\begin{obs}[A sharper test for a fatal greedy gap]\n\\lean{skipSafe\\_of\\_two\\_mul\\_le\\_three\\_mul}{Erdos249257/HalfGreedyFatalGap.lean:107},\n\\lean{three\\_mul\\_lt\\_two\\_mul\\_of\\_fatal}{Erdos249257/HalfGreedyFatalGap.lean:146},\n\\lean{two\\_le\\_of\\_fatal}{Erdos249257/HalfGreedyFatalGap.lean:161},\n\\lean{three\\_le\\_of\\_fatal\\_of\\_odd}{Erdos249257/HalfGreedyFatalGap.lean:173}.\nLet $k\\ge1$ be a skipped rank and write its positive rational\nremainder in lowest terms as $\\rho=u/(2L)<w_k$, with $u,L$ odd and\npositive.  Set $a=2L-(2^k-1)u>0$.  The cited sufficient condition\n$2u\\le3a$ guarantees $\\rho<R_k$ and improves the simpler condition\n$u\\le a$.  Its contrapositive gives $3a<2u$ at a fatal skip,\nand hence $u\\ge3$ when $u$ is odd.\n\nThis is not the exact threshold.  With\n$\\theta_k=R_k^{-1}-(2^k-1)$, direct rearrangement gives\n\\[\n \\rho<R_k\\quad\\Longleftrightarrow\\quad u\\theta_k<a.\n\\]\nThe tail estimate gives $\\theta_k<2/3$ and $\\theta_k\\to2/3$;\n$2u\\le3a$ is therefore sufficient, not necessary.\nFor example, at $k=2$ the local rational state $\\rho=77/282$ has\n$u=77$, $L=141$, $a=51$, so $2u>3a$, yet\n$\\sum_{j=3}^{12}(2^j-1)^{-1}>77/282$ proves $\\rho<R_2$.\nThis is a local state, not an asserted point of the half-greedy orbit.\nImproving a bounded tail ratio alone does not provide a lower bound\ngrowing exponentially in $k$ on an arithmetic numerator; such a bound\nwould require additional information about the actual prefix.\n\\end{obs}\n\n\\begin{obs}[Exclusion of a linear-width band at rows 13--30]\n\\lean{seamUpperResetDyadicBandEscape\\_through\\_thirty}{HalfCylinderUpperResetBandCertificates.lean:78}.\n\\textbf{Conclusion.} At every upper reset row $d$, $13 \\le d \\le 30$, the reset charge $E_d =\n4\\cdot\\mathrm{overshoot}_d + \\mathrm{abovePulse}_d$ avoids the linear-width band\n$(2^{d-j+1}-2(d+j),\\ 2^{d-j+1}]$ for every $j \\le d$; the finite segment of\n\\lean{SeamUpperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:4566}, whose full\nform would give half-membership by\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_upperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:4790},\nand so refute the universal statement.\n\\textbf{Parameter range.} Proved at 18 explicit rows by exact computation checked in the Lean\nkernel; the obligation needs all $d \\ge 13$. The excluded band has width $O(d)$, whereas the reset condition uses\na width of order $2^{(r+5)/2}$.  This is a comparison of numerical scales,\nnot a proof that one hypothesis on the orbit implies the other.\n\\textbf{Missing step.} A structural lower bound on the distance from $E_d$ to the nearest dyadic\npower at or above it, of width $2(d+j_0)$ at the single critical index $j_0$; already collapsed\nfrom $\\forall j$ to one inequality by\n\\lean{HalfUpperResetCriticalBand.dyadicBandEscape\\_iff\\_exists\\_critical}{HalfUpperResetCriticalBand.lean:108}.\nOnly $O(d)$ separation is required here; separation at the square root of the dyadic scale is not needed.\n\\end{obs}\n\n\\begin{obs}[Middle predecessors and right-branch runs]\n\\lean{seamMiddleBranch\\_nextRemainder\\_ge\\_row}{HalfCylinderMiddleCarryLowerBound.lean:2655},\n\\lean{seamMiddleBranch\\_nextRemainder\\_ge\\_expBarrier}{HalfCylinderMiddleCarryLowerBound.lean:2680},\n\\lean{SeamUpperResetDyadicBandEscape.remainder\\_ge\\_row}{HalfCylinderMiddleCarryLowerBound.lean:4641}.\n\\textbf{Conclusion.} Unconditional, uniform in $s\\ge5$: a middle (M) branch forces $s+1 \\le\n\\mathrm{rem}(s+1)$, using no lower bound on the source remainder; with the row-scale source\nbound this upgrades to $2^{s+1}+(s+1) \\le \\mathrm{rem}(s+1)$. A minimal-counterexample argument\nin the same Lean source then shows that any row whose remainder is smaller than its index must have an upper-reset (U) ancestor, since M-ancestors\nand R-runs are both excluded.\n\\textbf{Scope of the hypothesis.} Two of the three ancestor channels (M, R) are\nclosed unconditionally at all scales; only the U channel remains, and it needs only the\n$O(d)$-wide band escape of the preceding observation.  The 257-reset statement demands escape\nat the square root of the dyadic scale, whereas linear-width escape at U-resets alone suffices\nfor this argument.\n\\textbf{Missing step.} The single remaining channel: for every $d \\ge 13$ with successor\ncarries, the reset charge avoids the width-$2(d+j_0)$ band at its critical dyadic index.  Then\nthe remainder is at least the row index at every late middle row, which is the hypothesis of\nthe next observation, and hence $1/2 \\in \\ensuremath{\\mathcal A}$; each of these implications is\nalready proved in Lean.\n\\end{obs}\n\n\\begin{obs}[A linear lower bound at middle transitions]\n\\lean{SeamMiddleProducerRowEscape}{HalfCylinderMiddleCarryLowerBound.lean:1769},\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_middleProducerRowEscape}{HalfCylinderMiddleCarryLowerBound.lean:2371}\n(via its cardinality and tail forms, \\lean{}{HalfCylinderMiddleCarryLowerBound.lean:1510}).\n\\textbf{Conclusion.} If at every late middle row ($s \\ge 13$, no carry, middle branch)\n$s \\le \\mathrm{rem}(s)$, then $1/2 \\in \\ensuremath{\\mathcal A}$.  The hypothesis is much\nweaker than an exponential bound, but stronger than the square-root-scale carry bound.\n\\textbf{Scope of the hypothesis.} This hypothesis asks for a \\emph{linear-in-row}\nlower bound on the remainder itself; the obligation supplies an \\emph{exponential} lower bound\non the deviation from $2^s$ at reset rows. For unrestricted numerical values, a large absolute deviation from\n$2^s$ permits $\\mathrm{rem}(s)=0$, which violates the linear lower bound.\nThat observation is not a counterexample within the actual recurrence,\nso incomparability of the two orbit hypotheses is not established.\n\\textbf{Missing step.} $\\forall s \\ge 13$ at a middle transition, $s \\le \\mathrm{rem}(s)$.\nThe upper-reset band condition described above is one sufficient input\nto this linear bound; no converse is asserted.\n\\end{obs}\n\n\\begin{obs}[A square-root lower bound in the row index]\n\\lean{SeamMiddleProducerSqrtEscape}{HalfCylinderLastProducerContradiction.lean:84},\nits implication to the exact tail bound,\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_middleProducerSqrtEscape}{HalfCylinderLastProducerContradiction.lean:458}.\n\\textbf{Conclusion.} If at every late middle row $s$ the producer carry exceeds $2\\sqrt{2s+2}+4$, then\n$1/2 \\in \\ensuremath{\\mathcal A}$; the producer carry equals the middle coordinate\n$4\\,\\mathrm{rem}-\\mathrm{belowPulse}-4$\n(\\lean{}{HalfCylinderProducerLowerBound.lean:150}), so the requirement is essentially\n$\\mathrm{rem}(s) \\gtrsim (\\sqrt{s}+\\mathrm{belowPulse})/4$.\n\\textbf{Comparison.} This is a square root of the \\emph{row index}\n$s$ (a tail majorant for the divisor counts), not the square root of the \\emph{dyadic scale}\n$2^{(r+5)/2}$.  The two square roots in this record are exponentially far apart and easy to\nconflate; this hypothesis is a bound in the row index.  The cited conditional theorem uses it to rule out a\nlast non-right transition.  An implication between the different orbit\nhypotheses would require a further argument.\n\\textbf{Missing step.} $\\forall s \\ge 13$ at a middle transition, $4\\,\\mathrm{rem}(s) -\n\\mathrm{belowPulse}(s) - 4 > 2\\sqrt{2s+2}+4$; a row-scale, not dyadic-scale, lower bound on the\nmiddle coordinate.\n\\end{obs}\n\n\\begin{obs}[The remaining interval of possible integer values]\n\\lean{upperProducer\\_not\\_last}{HalfCylinderLastProducerContradiction.lean:299},\n\\lean{middleProducer\\_neg\\_three\\_not\\_last}{HalfCylinderLastProducerContradiction.lean:364},\n\\lean{finalMiddleCell\\_neg\\_three\\_not\\_last}{HalfCylinderFinalMiddleCellEscape.lean:587},\n\\lean{middleProducer\\_allRight\\_landingExcess\\_window}{HalfCylinderMiddleCarryLowerBound.lean:2547},\n\\lean{middleThenAllRight\\_landingExcess\\_two\\_le}{HalfCylinderMiddleCarryLowerBound.lean:2319}.\n\\textbf{Conclusion.} Under the hypothesis of a last non-right transition\nat row $D\\ge13$, the upper case is excluded.  In the middle case put\n\\[\n X_{D+1}:=\\mathrm{rem}(D+1)-2^{D+1}\n          =4\\,\\mathrm{rem}(D)-p_D^-=C_D+4.\n\\]\nThe cited bounds give $2\\le X_{D+1}<2\\sqrt{2D+2}+8$.\nThis permits nonnegative $C_D$ as well as the negative exceptional\nvalues $-2,-1$; the latter do not exhaust the remaining interval.\n\\textbf{Scope of the hypothesis.} the residual bad set is an explicit\n$O(\\sqrt D)$-wide integer window at each candidate last non-right transition row, not an interval at dyadic\nscale. The reset hypothesis would exclude this window.  Its width is much\nsmaller than $2^{(r+5)/2}$, but width alone does not measure the difficulty\nof proving that the actual sequence avoids it.\n\\textbf{Missing step.} Exclude every integer\n\\[\n 2\\le X_{D+1}\\le\\bigl\\lceil2\\sqrt{2D+2}\\bigr\\rceil+7\n\\]\nat a hypothetical last middle transition $D\\ge13$.\nThe ceiling is necessary when converting the strict real upper bound\nto an integer interval.  Excluding only $C_D=-2,-1$ does not suffice.\n\\end{obs}\n\n\\begin{obs}[Excluding three integer values]\n\\lean{SeamTwoSidedDyadicCellEscape}{HalfCylinderMiddleCarryLowerBound.lean:4374},\n\\lean{SeamTwoSidedDyadicCellEscape.step}{HalfCylinderMiddleCarryLowerBound.lean:4398},\n\\lean{SeamTwoSidedDyadicCellEscape.twoSided}{HalfCylinderMiddleCarryLowerBound.lean:4448}.\nTheorem~\\ref{thm:two-sided-dyadic} states the two needed hypotheses\nand their row ranges explicitly.  Excluding $-3,-2,-1$ is required\nonly on middle branches; the overshoot inequality is required only\non right branches where $o_s\\le2^s$.  The resulting invariant bounds\nat least one of the two distances from the target, not necessarily\nthe lower remainder.  The exclusion of $-3$ in the special\nlast-middle-then-all-right configuration does not establish the\nunrestricted middle-branch hypothesis.  Both universal assumptions\nremain to be proved for the actual sequence.\n\\end{obs}\n\n\\begin{obs}[An upper bound on the remainder]\n\\lean{SeamGreedyRemainderGapBound}{HalfCylinderSeamLimit.lean:192},\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_seamRemainderGapBound}{HalfCylinderSeamLimit.lean:303}\nand the cofinal implication \\lean{}{HalfCylinderSeamLimit.lean:181}.\nThe estimate $\\mathrm{rem}(s)<2^{s+1}$ for every $s\\ge6$ would\ngive $\\mathrm{rem}(s)/4^s\\to0$ and hence $1/2\\in\\mathcal A$.\nTheorem~\\ref{thm:seam-limit} identifies the normalized limit\nunconditionally; requiring its value to be zero, even only along\none unbounded sequence, is equivalent to membership.  No strict\nseparation of these conditions on the actual sequence is proved.\nThe upper estimate is also the missing ceiling in the excess-side\nright-run argument.  The existing bound on rows satisfying $2s<3d$,\nwhere $d$ is the largest omitted rank, still requires that row\ncondition wherever it is applied.  The finite checks do not supply\nit at every later row.\n\\end{obs}\n\n\\begin{obs}[Reported computation through row 200{,}000]\n\\textbf{Conclusion.} Exact integer arithmetic to row 200{,}000\n(\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/seam-orbit-200000.json}{saved run}); branch counts $R\\!:\\!M\\!:\\!U =\n100197\\!:\\!49899\\!:\\!49898$; within the tested range $6\\le r\\le200{,}000$ the obligation fails only at $r=7$; maximum R-run $19$ after the reset at row\n$158{,}096$ against the required threshold $\\approx (r-3)/2 \\approx 79{,}000$ (margin\n$\\sim\\!4000\\times$). Independently: 1209 resets checked to row 2500\n(\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/seam-reset-crossing-2500.json}{saved run}), zero classification\nanomalies, exactly two crossing cells in rows $6\\le s\\le2500$ ($s{=}7,d{=}5$ and $s{=}10,d{=}7$), both below hypothesis\nscope.\n\\textbf{Parameter range.} fixed-scale exhaustive verification versus a statement\nat every reset $r \\ge 31$.  This is the difference between a finite and an unbounded\nparameter range: a finite verified list supplies no witnesses at unbounded depths, however\nlarge the margin.\n\\textbf{Missing step.} The computation records the tested range and its\nexception at $r=7$; it does not determine a valid threshold for the infinite\norbit. A theorem must still establish the proposed bound at every reset\nbeyond its stated threshold.\n\\end{obs}\n\n\\begin{obs}[Structure of resets after long runs]\n\\textbf{Conclusion.} At a dangerous reset $r$ ($|\\mathrm{rem}(r{+}1)-2^{r+1}| \\le 2^{(r+5)/2}$) whose\npreceding reset $r_0$ has pure R-run length $L' = r-r_0-1 > (r+5)/4$, the deviation\n$w_{r_0+1}$ is pinned to at most two adjacent integers determined by the divisor-pulse stream\nalone; a chain of $n$ dangerous resets forces $n{-}1$ nested exact integer-coincidence towers.\n\\textbf{Scope of the hypothesis.} applies only to dangerous resets preceded by a\nlong pure R-run ($L' > (r+5)/4$); an isolated dangerous reset after a short run is untouched.\nPinning to two integers is not yet a contradiction: no theorem excludes those two values.  This is\nan ordinary argument; it is not formalised.\n\\textbf{Missing step.} (i) a congruence or pulse-stream obstruction showing the two pinned integer\nvalues are unattainable, and (ii) removal of the long-preceding-run hypothesis, or a proof that\ndangerous resets must enter chains to which the exclusion applies.\nExcluding the two pinned values addresses only resets satisfying the\nlong-run hypothesis; covering all remaining resets is a separate\nrequirement.  Neither step alone proves a global exclusion.\n\\end{obs}\n\n\\begin{obs}[The sign condition and growth of the margin]\n\\textbf{Conclusion.} At all 1209 observed resets, M $\\Rightarrow$ deviation $>0$ and U $\\Rightarrow$\ndeviation $<0$, with zero exceptions; the margin over the $2^{(r+5)/2}$ threshold grows from\n1.11 bits at $r=14$ to $\\sim$1246 bits at $r=2499$. The record isolates this as a separate,\npossibly more tractable target than the full square-root escape.\n\\textbf{Different restrictions.} The upper sign follows from the\npositive reset charge: $w_{r+1}=-(4o_r+p_r^+)<0$.\nThe middle sign is equivalent to $4\\mathrm{rem}(r)>p_r$ and needs\nfurther information about the orbit.  Neither sign implies a\nsquare-root lower bound on the absolute deviation.\n\\textbf{Missing step.} Prove that middle inequality and then a\nquantitative margin on each branch.  The exact branch calculation\nis given in Section~\\ref{sec:invent-R1}; it is not a substitute for\nthe magnitude estimate.\n\\end{obs}\n\n\\begin{obs}[Comparison of finite truncation lengths]\nThe exact finite procedure in\nTheorem~\\ref{thm:tr-finite-decision} verifies the twenty truncations\n$3\\le J\\le22$.  The proof combines finite integer certificates with a\nwitness bound valid beyond the finite search horizon; it does not infer\nan infinite statement from a long finite run.  Saved runs for\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/truncation-ladder-3-16.json}{$3\\le J\\le16$} and\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/truncation-ladder-17-22.json}{$17\\le J\\le22$} reproduce it.\nRepresentations for an unbounded sequence of $J$ would yield a\nrepresentation of $1/2$ for the full weights by the compactness estimate\nfollowing that theorem.  That unbounded-index premise remains unproved.\nThe size of the least-common-multiple horizon affects this finite\nalgorithm, but is not an impossibility theorem for other arguments.\n\\end{obs}\n\n\\subsection{Finite quotient sums and unbounded depths}\n\n\\begin{obs}[Exact rows from a sharp capacity bound]\n\\lean{exists\\_exactRowStrictUpperFill\\_of\\_skippedCoreSharpCapacity}{BooleanMobiusSkipRow.lean:238},\n\\lean{exactLocalMersenneHalfRow\\_two\\_mul\\_sub\\_two\\_of\\_skippedCoreSharpCapacity}{BooleanMobiusSkipRow.lean:332}.\n\\textbf{Conclusion.} For any $c \\ge 4$ and any finite $D \\subseteq [2,c)$ with\n$\\ensuremath{V}\\,D < 1/2$ and $\\ensuremath{S}\\,D\\,1\\,(2c{-}2) <\n2^{c-2}$: an exact row $E$ at endpoint $2c{-}2$ with $D \\subseteq E$, every new rank strictly\nabove $c$.  No additional crossing condition or identification with a greedy prefix is needed.\n\\textbf{Scope of the hypothesis.} needs a per-$c$ input\n$\\ensuremath{S}\\,D\\,1\\,(2c{-}2) < 2^{c-2}$, and this record supplies that\ninput only through the crossing-deficit route, which\n\\lean{eq\\_halfGreedyPrefixSupport\\_of\\_critical\\_crossing}{BooleanMobiusCriticalCapacityCofinal.lean:50}\nthen identifies the core with\n$D=\\mathrm{halfGreedyPrefixSupport}(c{-}1)$, under its critical-crossing\npremises.  The fill implication does not require that identification;\nthis difference in its formulation is not a proof of strictness\nbetween the associated cofinal existence statements.\n\\textbf{Closes it ($\\star$):} $\\forall N\\ \\exists c \\ge N,\\ \\exists$ finite $D \\subseteq [2,c)$\nwith $\\ensuremath{V}\\,D<1/2$ and $\\ensuremath{S}\\,D\\,1\\,(2c{-}2) <\n2^{c-2}$; equivalently $\\ensuremath{C}\\,{\\uparrow}D\\,(2c{-}3) < 2^{c-2}$. No\ncoherence between the $D$'s, no requirement that $D$ be a greedy prefix.\n\\end{obs}\n\n\\begin{obs}[A conditional induction for unbounded depths]\n\\lean{SkippedCoreCriticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:30} $\\to$\n\\lean{exists\\_laterProtectedExactLocalMersenneRow}{BooleanMobiusCriticalCapacityCofinal.lean:1139}\n$\\to$\n\\lean{cofinalExactLocalMersenneHalfRows\\_of\\_criticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:1309}\n(\\lean{ProtectedExactLocalMersenneRow}{BooleanMobiusCriticalCapacityCofinal.lean:1087}; strict\nprogress \\lean{}{BooleanMobiusExactRowCrossing.lean:233}).\n\\textbf{Conclusion.} Under the sharp-capacity supply hypothesis below,\nthe induction starts from the depth-six seed and constructs exact rows\nat unbounded depths.  A protected row is either doubled while its real\nvalue stays below $1/2$, or extended using its first crossing rank $e$.\nIn the latter case protection gives $2e-2$ strictly larger than the old\nendpoint.\n\\textbf{Scope of the hypothesis.} The recycle branch uses the critical quotient supply\nhypothesis whenever recycling is required.\nProtection (endpoint $<2\\cdot$cutoff and new ranks $>$cutoff) makes that\nextension strictly increase the endpoint.  The induction establishes the\nconditional implication, not the supply hypothesis.\n\\textbf{Missing step.} The sharp inequality at one crossing rank per step, canonical form\n\\lean{HalfGreedySkippedCriticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:178}:\n$\\forall c \\ge 4$ skipped by the rational half-greedy orbit, $2^{(2c-2)-1} \\le\n\\ensuremath{Q}(\\mathrm{insert}\\,c\\,(\\mathrm{halfGreedyPrefixSupport}(c{-}1)))\\,(2c{-}2)$.\n\\end{obs}\n\n\\begin{obs}[Unconditional capacity, off by one bit]\n\\lean{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_upperHalfCapacity}{BooleanMobiusSkippedCoreExactRow.lean:123}.\n\\textbf{Conclusion.} Unconditionally, $\\forall c \\ge 4$ and every below-half core $D \\subseteq [2,c)$\nwith deficit $< \\ensuremath{w}\\,c$: $\\ensuremath{S}\\,D\\,1\\,(2c{-}2) <\n2^{c-1}$. Uniform in $c$, no table, no case split.\n\\textbf{Parameter range.} One bit short. The strict-upper fill needs\n$<2^{c-2}$, this gives $<2^{c-1}$. The proof derives $A < 2^{c-2}+|D|$ then discards\n$|D| \\le c-2 \\le 2^{c-2}$; the loss is purely additive. The true unproved statement is only that\n$A$ avoids the linear-width band $[2^{c-2},\\,2^{c-2}+c-3]$, width $c{-}2$ inside a range of size\n$2^{c-2}$.\n\\textbf{Missing step.} A linear-width dyadic-band anti-concentration: at cofinally many crossing\nranks $c$, $\\ensuremath{S}\\,D\\,1\\,(2c{-}2) \\notin [2^{c-2},\\,2^{c-2}+c-3]$. Same band\nshape as the earlier linear-band conditions.  The reset condition\nuses a different coordinate and parameter, with width $2^{(r+5)/2}$;\nno implication between the hypotheses follows merely from this\ncomparison of widths.\n\\end{obs}\n\n\\begin{obs}[Doubling extension, fails at the only recorded seed]\n\\lean{exists\\_exactRowStrictUpperExtension\\_two\\_mul\\_sub\\_one\\_of\\_exact\\_below}{BooleanMobiusExactRowDoubling.lean:185};\nrank two is automatic\n(\\lean{two\\_mem\\_of\\_exact\\_localMersenneQuotient}{BooleanMobiusExactRowRankTwo.lean:23}).\n\\textbf{Conclusion.} $\\forall n \\ge 6$: a below-half exact row at $n$ produces an exact row at\n$2n{-}1$, support-extending, all new ranks $>n$.  This is the only unconditional\nendpoint-doubling step in this record.\n\\textbf{Scope of the hypothesis.} Below-halfness is not preserved and\ndemonstrably fails at the first step. For the seed $D=\\{2,3,6\\}$ at $M=11$ the core alone\ncontributes $2/3+4/7+32/63 \\approx 1.746 > 1$, so the doubled row is strictly \\emph{above} half.\nIterating the cheap arm is therefore impossible from the only recorded seed; the chain is forced\ninto the capacity-gated recycle arm at step one.\n\\textbf{Missing step.} Cofinally many below-half exact rows; $\\forall N\\ \\exists n \\ge N\\ \\exists D$\nexact at $n$ with $\\sum_{d\\in D}2^{n\\bmod d}/(2^d-1) < 1$. A single such row at each of cofinally\nmany $n$ makes doubling unnecessary; a self-reproducing one closes everything.\n\\end{obs}\n\n\\begin{obs}[Consecutive-skip precritical bound]\n\\lean{halfGreedy\\_precriticalSuffix\\_lt\\_of\\_next\\_skip}{BooleanMobiusCriticalCapacityCofinal.lean:682}\n(used in \\lean{halfGreedySkippedCriticalQuotientSupply\\_of\\_precriticalSuffix}{BooleanMobiusCriticalCapacityCofinal.lean:1001}).\n\\textbf{Conclusion.} Unconditional: if $c \\ge 6$ and both rank $c$ and rank $c{+}1$ are skipped by\nthe rational half-greedy orbit, then $\\ensuremath{S}(\\mathrm{halfGreedyPrefixSupport}\n(c{-}1))\\,1\\,(2c{-}3) < 2^{c-3}$, doubling into sharp capacity at $c$, hence an exact row at\n$2c{-}2$, with no supply hypothesis.\n\\textbf{Scope of the hypothesis.} It needs two \\emph{consecutive} skipped ranks.\nThe residual open set is exactly the skip-then-take rows:\n\\lean{halfGreedySkippedPrecriticalSuffixSupply\\_iff\\_preTake}{BooleanMobiusCriticalCapacityCofinal.lean:838}\nproves the whole hypothesis equivalent to\n\\lean{HalfGreedyPreTakePrecriticalSuffixSupply}{BooleanMobiusCriticalCapacityCofinal.lean:200},\nwith $c=4,5$ already settled by direct computation.\n\\textbf{Missing step.} Cofinally many $c \\ge 6$ at which two consecutive ranks $c,c{+}1$ are both\nskipped by the rational half-greedy orbit; or dually, the pre-take residual at cofinally many\nskipped-then-taken ranks. Either one closes the obligation with no further input.\n\\end{obs}\n\n\\begin{obs}[Bounded-gap precritical bound]\n\\lean{halfGreedy\\_precriticalSuffix\\_lt\\_of\\_future\\_skip\\_after\\_takenBlock}{BooleanMobiusCriticalCapacityCofinal.lean:603}\n(with \\lean{precriticalCrossingTax\\_of\\_futureThreshold}{BooleanMobiusCriticalCapacityCofinal.lean:472},\n\\lean{sub\\_two\\_le\\_two\\_pow\\_sub\\_four}{BooleanMobiusCriticalCapacityCofinal.lean:432}).\n\\textbf{Conclusion.} Unconditional and uniform in \\emph{both} parameters: skip at $c$, takes at\n$c{+}1,\\dots,c{+}t{-}1$, skip at $c{+}t$, with $0<t\\le c-3$ and $c-2\\le 2^{c-t-3}$\n$\\Rightarrow$ precritical suffix bound at $c$ $\\Rightarrow$ sharp capacity $\\Rightarrow$ exact\nrow at $2c{-}2$. Handles any skip gap $t \\lesssim c-3-\\log_2(c-2)$, not just $t=1$.\n\\textbf{Scope of the hypothesis.} The only missing input is an orbit-level\nskip-gap bound: that the next skipped rank after $c$ occurs at $c{+}t$ with $c-2 \\le\n2^{c-t-3}$. The proof is fully uniform in $(c,t)$, every constant explicit, no table.\n\\textbf{Missing step.} For cofinally many skipped ranks $c$ of the rational half-greedy orbit, the\nnext skipped rank is at most $c+(c-3-\\lceil\\log_2(c-2)\\rceil)$.\nThe finite skip-density observation does not imply this pointwise gap\nbound at unbounded ranks.  Establishing the latter would supply the\nhypothesis of the stated conditional argument.\n\\end{obs}\n\n\\begin{obs}[A computable horizon, not the required horizon]\n\\lean{exists\\_greedyHalfFrozenMargin\\_nonneg\\_of\\_excess\\_neg}{HalfCylinderFiniteShadow.lean:1198},\n\\lean{exists\\_greedyHalfFrozenMargin\\_nonneg\\_iff\\_excess\\_neg}{HalfCylinderFiniteShadow.lean:1229}.\n\\textbf{Conclusion.} At every dyadically safe rank $k>0$, $\\ensuremath{F}\\,k\\,J \\ge\n0$ for \\emph{some} horizon $J$; the finite coefficient window eventually covers the centred\ncarry. Nonnegativity is monotone up in $J$.\n\\textbf{Parameter range.} For a fixed prefix with positive safety margin,\nthe displayed tail estimate gives a computable sufficient horizon: increase\n$J$ until $(k+J+3)2^{-J}$ is smaller than that margin. What is not proved is\nsuccess by the prescribed horizon needed by the exact-row construction.\nThe exact-row route needs the crossing by the specific horizon $J=c-3$\n(\\lean{halfGreedy\\_precriticalSuffix\\_lt\\_iff\\_frozenMargin\\_nonneg}{BooleanMobiusCriticalCapacityCofinal.lean:866},\n$k=c-1$). The quantitative ingredients are already available:\n\\lean{binaryCoeffTail\\_eq\\_finiteCoeffWindow\\_add\\_shiftedTail}{HalfCylinderFiniteShadow.lean:203} gives\n$\\mathrm{gap}=\\mathrm{tail}(k{+}1{+}J)/2^J$, and\n\\lean{binaryCoeffTail\\_le}{GenericTailOrbitRigidity.lean:85} gives $\\mathrm{tail}(m)\\le m+2$.\n\\textbf{Available bound.} Margin$(k,J)\\ge0$ as soon as $(k{+}J{+}3)/2^J <\n|\\mathrm{halfGreedyNextDyadicExcessNumerator}\\,k|/\\mathrm{halfGreedyPrefixDenominator}\\,k$.\n\\textbf{Missing step.} At the prescribed horizon $J=k-2$, the displayed bound\nwould suffice if one proved\n$|E_k|/D_k > (2k{+}1)/2^{k-2}$ at every skipped rank.\nThe tail estimate uses only lemmas already proved; this stronger margin\nbound is not established.\n\\end{obs}\n\n\\begin{obs}[A nonnegative margin at an unspecified horizon]\n\\lean{HalfGreedySkippedFullShellNonnegative}{HalfCylinderFullShellSeamBridge.lean:604},\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_skippedFullShellNonnegative}{HalfCylinderFullShellSeamBridge.lean:633},\n\\lean{HalfGreedySkippedSeamEscape}{HalfCylinderFullShellSeamBridge.lean:687}.\n\\textbf{Open property and proved conditional implication.} The open property asks\nthat every skipped rank $n\\ge3$ satisfy\n$0\\le\\ensuremath{F}(n{-}1)\\,n$. Assuming this property, a Lean theorem proves\n$1/2\\in\\ensuremath{\\mathcal A}$.\n\\textbf{Order of quantifiers.} The horizon is exactly three doubling steps short.\nThe exact-row hypothesis needs margin$(k,k{-}2)\\ge0$; this gives margin$(k,k{+}1)\\ge0$. By the\nsucc-recurrence, margin$(k,k{+}1)=8\\cdot$margin$(k,k{-}2)+4\\,\\mathrm{sc}(2k)+2\\,\\mathrm{sc}(2k{+}1)\n+\\mathrm{sc}(2k{+}2)$, where $\\mathrm{sc}$ is the divisor count, bounded by $\\tau\\le 2\\sqrt{2k+2}$. Since\nnonnegativity is monotone \\emph{up} in $J$, the cited statement is the weaker one and cannot\nsupply the obligation as it stands.\n\\textbf{Missing step.} A horizon-tightening; at every skipped rank, margin$(k,k{+}1) \\ge\n4\\,\\mathrm{sc}(2k)+2\\,\\mathrm{sc}(2k{+}1)+\\mathrm{sc}(2k{+}2)$: the full-shell margin must exceed\nan explicit $O(\\sqrt k)$ divisor-count quantity, on the same square-root scale as the carry\nbounds above.\n\\end{obs}\n\n\\begin{obs}[Cofinal positive skips \\emph{is} the obligation]\n\\lean{CofinalPositiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:22},\n\\lean{cofinalExactLocalMersenneHalfRows\\_of\\_positiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:84},\n\\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_greedySkippedSupport\\_infinite}{GreedyAchievementSet.lean:2583},\n\\lean{}{GreedyAchievementSet.lean:1515}; positivity\n\\lean{}{BooleanMobiusCriticalCapacityCofinal.lean:138}.\n\\textbf{Conclusion.} Cofinally many positive rational-greedy skips $\\Rightarrow$\nthe existence of exact quotient sums at unbounded depths $\\Rightarrow 1/2 \\in\n\\ensuremath{\\mathcal A}$.\n\\textbf{Scope of the hypothesis.} It is equivalent to the problem itself. The positivity\nconjunct $0 < \\ensuremath{r}(1/2)(c{-}1)$ holds unconditionally by the odd-denominator parity\nfact, so the hypothesis says exactly that the greedy rule for $1/2$ skips infinitely many\nranks, which Theorem~\\ref{record:257bm-c20} proves equivalent to\n$1/2\\in\\mathcal A$.  Thus proving this hypothesis would settle the half-target\nquestion itself.\n\\textbf{What remains.} One must prove that this fixed greedy sequence has\ninfinitely many skips.  Equivalence neither supplies that proof nor rules\nout finding a useful invariant for the sequence.  The condition in\nTheorem~\\ref{record:257bm-c7} instead permits finite supports that are\nnot greedy prefixes.\n\\end{obs}\n\n\\begin{obs}[Terminal bounds and restrictions on finite support]\n\\lean{HalfCarryCofinalTerminalOnlyStrip}{TerminalOnlyCofinal.lean:34},\n\\lean{HalfTerminalOnlyStripWitness}{TerminalOnlyCofinal.lean:24},\n\\lean{exists\\_infinite\\_support\\_half\\_of\\_cofinalTerminalOnlyStrip}{TerminalOnlyCofinal.lean:192}.\n\\textbf{Conclusion.} Cofinally many depths $M$ carrying a finite normalized word $a$ with\n$|\\ensuremath{\\operatorname{ihc}}(\\operatorname{supp}a)(M{-}1)| \\le \\ensuremath{B}\\,M\n\\Rightarrow \\exists A$ infinite with $X_{A}(2)=1/2$. Same architecture\nas the obligation: cofinal, no coherence, mutually incompatible witnesses allowed.\n\n\\par\\medskip\n\n\\textbf{Different restrictions.} The terminal condition permits \\emph{any} support inside the depth-$M$ window but\ndemands the carry inside a $\\sim 2\\sqrt M$ strip; the obligation form ($\\star$) tolerates a carry\nup to $2^{M/2-1}$, exponentially looser, but demands the support be confined to ranks $<\nM/2+1$, since the exact fill encodes the residue in the pure upper window where the Mersenne\nquotient is exactly $2^{M-d}$.  The essential difference is where the support may sit.\n\n\\par\\medskip\n\n\\textbf{Missing step.} Confinement of the strip witnesses to the lower half: $\\forall N\\\n\\exists c\\ge N$ with a strip-scale witness whose support lies in $[2,c)$ at depth $2c{-}3$. Then\n$2\\sqrt{2c}+4<2^{c-2}$ for $c\\ge8$ gives the required numerical\ncapacity bound.  The confinement of the support remains an additional,\nunproved condition; the comparison does not establish a simpler\nexistence theorem.\n\\end{obs}\n\n\\begin{obs}[The live greedy carry and the carry of a fixed prefix]\n\\lean{GreedyHalfCarryCofinalStripReturn}{CofinalStripReturn.lean:76},\n\\lean{greedy\\_half\\_infinite\\_of\\_cofinalStripReturn}{CofinalStripReturn.lean:130},\n\\lean{infinite\\_support\\_half\\_of\\_mobiusCenteredHalfCarry\\_sqrtBound}{HalfCarryReachability.lean:817}\n(nonnegativity from\n\\lean{}{HalfCylinderFinalMiddleCellEscape.lean:94}).\n\\textbf{Conclusion.} Cofinal returns of the actual greedy carry to the square-root strip\n(Theorem~\\ref{record:257rig-c18}), or a pointwise $2\\sqrt N+4$ bound on the M\\\"obius-centred\ncarry (Theorem~\\ref{record:257rig-c16}), each give an infinite support with value exactly $1/2$.\n\\textbf{Comparison.} These bound the carry of the \\emph{infinite}\ngreedy support $G$ of $1/2$; sharp capacity is about the \\emph{frozen\nfinite prefix} $\\mathrm{halfGreedyPrefixSupport}(c{-}1)$. The two carries agree only at index $k$\n(inside \\lean{greedyHalfFrozenMargin\\_eq\\_actual\\_skip\\_margin}{HalfCylinderFiniteShadow.lean:1120}),\nnot at depth $2c{-}3$ where the fill needs it. The bridge is already recorded:\n\\lean{halfGreedy\\_precriticalSuffix\\_lt\\_iff\\_futureSkipCoverage}{BooleanMobiusCriticalCapacityCofinal.lean:949}:\nsharp capacity $\\Leftrightarrow \\ensuremath{C}\\,G\\,(2c{-}4) \\le \\Phi_c$, where\n$\\Phi_c=\\sum_{i=1}^{c-3}2^{c-3-i}\\mathbf 1[c+i\\notin G]$ is the future-skip capacity;\ntransport costs exactly this term.\n\\textbf{Missing step.} Either a strip bound stated for the frozen prefix rather than the live orbit,\nor $\\Phi_c \\ge \\ensuremath{B}(2c{-}3)$; holds as\nsoon as one skip occurs early in $[c{+}1,\\,2c{-}4]$, since the capacity is dyadically weighted.\nSame input as the bounded-gap precritical bound above.\n\\end{obs}\n\n\\begin{obs}[Above-support recycling, no growth guarantee]\n\\lean{exists\\_skippedCoreExactRow\\_of\\_value\\_above}{BooleanMobiusExactRowCrossing.lean:155},\n\\lean{exactLocalMersenneHalfRow\\_of\\_first\\_localMersenne\\_crossing}{BooleanMobiusExactRowCrossing.lean:191},\n\\lean{exactLocalMersenneHalfRow\\_two\\_mul\\_sub\\_two\\_of\\_skippedCore}{BooleanMobiusSkipRow.lean:149}.\n\\textbf{Conclusion.} Unconditional: every finite above-half support bounded by $n$ recycles through\nits first crossing rank $c$ into a genuine exact row at endpoint $2c{-}2$, with $4 \\le c \\le n$.\nNo capacity hypothesis at all.\n\\textbf{Order of quantifiers.} The witness is $\\exists c \\le n$, not $\\exists c$\nlarge; $2c{-}2$ may be $\\le n$, so the endpoint need not grow.\nThe countermodel of Theorem~\\ref{record:257bm-k1}\n(\\lean{exists\\_seeded\\_bounded\\_double\\_or\\_recycle\\_model}{BooleanMobiusExactRowDichotomy.lean:91})\nsatisfies the same transition schema plus a seed and does \\emph{not} give rows at unbounded\ndepths.  Growth is recovered only for protected exact rows, whose invariants force\n$c>$cutoff, hence $2c{-}2>$ endpoint; and maintaining those invariants is precisely what needs the sharp-capacity fill\nrather than this general recycling.\n\\textbf{Missing step.} A crossing-location lower bound; for the supports actually produced, the\nfirst crossing rank satisfies $2c{-}2>n$. Equivalently, keep protection alive without sharp\ncapacity, e.g.\\ a fill placing all new ranks above $c$ using only the recorded $c{-}1$-bit\ncapacity.\n\\end{obs}\n\n\\begin{obs}[Double-or-recycle dichotomy, proved insufficient]\n\\lean{exactLocalMersenneHalfRow\\_double\\_or\\_recycle}{BooleanMobiusExactRowDichotomy.lean:26},\n\\lean{boundedDoubleOrRecycleModel\\_not\\_cofinal}{BooleanMobiusExactRowDichotomy.lean:79},\n\\lean{exists\\_seeded\\_bounded\\_double\\_or\\_recycle\\_model}{BooleanMobiusExactRowDichotomy.lean:91}.\n\\textbf{Conclusion.} $\\forall n \\ge 6$, an exact row at $n$ gives an exact row at $2n{-}1$ or at\n$2c{-}2$ for some $4 \\le c \\le n$; plus a formal proof that this shape, even seeded at endpoint\nsix, is logically insufficient for cofinality.\n\\textbf{Order of quantifiers.} The dichotomy is proved by splitting on the sign\nof the witness value (odd denominators exclude equality); the countermodel pins the\ndeficit precisely: what is missing is not another dichotomy but strict endpoint progress in the\nrecycle branch, or a proof that the below-half branch recurs. Re-deriving this dichotomy without controlling repeated returns to\nbounded depths would leave the same gap.\n\\textbf{Missing step.} Either (a) the recycle branch upgraded to $2c{-}2>n$, or (b) cofinally many\nbelow-half exact rows so the doubling branch alone is cofinal. These are two sufficient ways to obtain progress; the countermodel\ndoes not rule out other assumptions that also force unbounded depths.\n\\end{obs}\n\n\\subsection{Universal support criteria and their limits}\n\n\\begin{obs}[A sufficient block-certificate condition]\n\\lean{irrational\\_erdosSupportSeries\\_of\\_weighted\\_coeff\\_certificates}{CertificateKernel.lean:9031}\n(carry variant \\lean{}{CertificateKernel.lean:9053}).\n\\textbf{Conclusion.} $\\forall b\\ge2,\\ \\forall A\\subseteq\\mathbb N$ (no infinitude, no structure\nhypothesis at all): if $\\forall q>0\\ \\exists N,K,L,C$ with $K\\le L$, $b^r \\mid\nc_{A}(N{+}r)$ for $r\\in[1,K]$, $\\sum_{r=K+1}^L c_{A}(N{+}r)\nb^{L-r}\\le C$, $\\exists t,\\ 0<c_{A}(N{+}L{+}1{+}t)$, and $q(C{+}N{+}L{+}2)<b^L$,\nthen $X_{A}(b)$ is irrational. The cited full-support, multiples, periodic, residue-class,\npairwise-coprime and fixed-core constructions use this criterion.  No\nsuch factorisation is asserted for the weighted support theorem in\nSection~\\ref{sec:257-problem}.\n\\textbf{Scope of the hypothesis.} The criterion is already universally quantified over $A$;\nthe universal quantifier is \\emph{not} the gap.  The certificate hypothesis must hold at every\nprecision, and it is supplied only for named classes; certificate existence for an arbitrary\ninfinite $A$ remains open.\n\\textbf{Limit of this sufficient condition.} Supplying these\ncertificates for every infinite support would imply the universal\nstatement, but Proposition~\\ref{prop:squarefree} rules out that supply\nfor the two specified certificate schemes.  The carry variant replaces\ndigit-wise divisibility by aggregate divisibility\n$b^K\\mid\\sum_r c_A(N+r)b^{K-r}$; the squarefree obstruction applies to\nthat variant too.  Universal certificate supply is therefore not being\nposed as an unresolved equivalent formulation of Problem~257.\n\\end{obs}\n\n\\begin{obs}[Rational targets and infinitely many greedy skips]\n\\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_greedySkippedSupport\\_infinite}{GreedyAchievementSet.lean:2583}\n(forward \\lean{}{GreedyAchievementSet.lean:2482}; reverse\n\\lean{mem\\_mersenneAchievementSet\\_of\\_greedySkippedSupport\\_infinite}{GreedyAchievementSet.lean:1528}).\nFor every rational $x\\ge0$,\n\\[\n x\\in\\mathcal A\\quad\\Longleftrightarrow\\quad\n \\text{the greedy algorithm for $x$ skips infinitely many ranks}.\n\\]\nThe reverse implication holds for every real $x\\ge0$:\na fatal remainder $r_n(x)>R_n$ forces every subsequent weight to\nbe taken, because $R_n=w_{n+1}+R_{n+1}$.\nHence infinitely many skips exclude a fatal remainder, and\nTheorem~\\ref{thm:greedy-survival} gives membership.\nFor the forward implication, membership identifies the greedy\nsum with $x$.  Finitely many skips would make that sum $E$ minus\na finite rational sum, which is irrational by\nErd\\H{o}s's theorem~\\cite{erdos1948}.  This contradicts rationality\nof $x$ and proves the stated extension as an ordinary argument.\n\nInfinitely many skips do not mean infinitely many selected ranks:\n$x=1/3$ has the single-element support $\\{2\\}$ and skips every\nother rank.  Thus the equivalence does not decide which rational\ntargets are represented, or provide an infinite support with rational\nvalue.  The Lean declarations above concern the half target; the\nextension to every rational target is an ordinary argument and is not\nformalised.\n\\end{obs}\n\n\\begin{obs}[Finite sums exclude every rational with even denominator]\n\\lean{finite\\_boolSupport\\_ne\\_half}{HalfCarryReachability.lean:589}; the underlying parity\nlemma has its own \\lean{finiteErdosSum\\_den\\_odd}{DyadicPrefixCompression.lean:1513}.\n\\textbf{Conclusion.} $\\forall A$ finite with $0\\notin A$: $X_{A}(2) \\ne\n1/2$; any support achieving exactly $1/2$ must be infinite.\n\\textbf{Parameter range.} The statement is specific to the target $1/2$, while the underlying\nparity lemma (the reduced denominator of a finite sum is odd) holds for every finite support.\n\\textbf{The same argument in general.} For every rational $t$ with even reduced denominator\nand every finite $A$ with $0\\notin A$, $X_{A}(2) \\ne t$; the proof is the same. Thus a represented rational with even reduced denominator would refute universal\n\\#257 at $b=2$. The converse is not established. If $\\mathcal F$ denotes the set of finite-support\nvalues, the exact unrestricted condition is\n$(\\mathbb Q\\cap\\ensuremath{\\mathcal A})\\smallsetminus\\mathcal F\\ne\\varnothing$.\n\\end{obs}\n\n\\begin{obs}[The denominator in the reciprocal-mass estimate]\n\\lean{dyadic\\_support\\_fraction\\_reciprocalMass\\_diverges\\_or\\_gt\\_one}{RationalSupportCarrySkeleton.lean:2210};\nthe version for general $v$ has its own\n\\lean{shiftedBooleanCollision\\_wrap\\_bound\\_of\\_common\\_multiple}{RationalSupportCarrySkeleton.lean:1968}.\n\\textbf{Conclusion.} For $A$ infinite with $X_{A}(2) = p/2^c$ (dyadic\nrational): $\\sum_{a\\in A}1/a$ diverges or exceeds $1$.\n\\textbf{Comparison.} The result is restricted to denominators $2^c$ ($v{=}1$), while\nthe obligation needs every rational $p/(2^c v)$.  The restriction is one of application, not\nof proof: every supporting lemma allows an arbitrary positive $v$, and the general-$v$\nconclusion is already recorded for arbitrary $(v,h,L,F)$.\n\\textbf{Missing step.} Instantiate the recorded generic theorem at $F=\\{a,b\\}$, $L=\\mathrm{lcm}\\,a\\,b$,\n$h=$ multiplicative order of $2$ mod $v$, giving $\\rho(A) \\ge\n\\mathrm{doublingWrapCount}(p,v,h)/h + 1/\\mathrm{lcm}(a,b)$ for every rational value. The one\ngenuinely new step: for $v{>}1$ the excess can be $0$, so strictness needs $1 \\le\n\\mathrm{doublingWrapCount}(p,v,h)/h + \\lfloor|F|/2\\rfloor/L$ (equivalently: the\nmean least residue of $p\\cdot2^n \\bmod v$ is $\\ge v(1-|F|/2L)$).\n\\end{obs}\n\n\\begin{obs}[Zero windows of a fixed support]\n\\lean{supportCoeffZeroWindow\\_length\\_le\\_eps\\_logb}{SublogDivisorCoverage.lean:435}.\n\\textbf{Conclusion.} For any $A$ with a positive element and\n$X_{A}(2)=p/(2^c v)$, $v{>}0$ (any rational, not just dyadic):\nfor every $\\varepsilon>0$ there is $B\\ge0$ such that, for all $N\\ge1$, every run of $h$ zero\ndivisor counts starting after $c+N$ has $h \\le \\varepsilon\\log_2 N + B$. The constant $B$ may depend on the fixed support and the other parameters.\nMore basically, if $a_0$ is any positive element of $A$, then each block\nof $a_0$ consecutive integers contains a multiple of $a_0$.  Every zero\nwindow of $c_A$ therefore has length at most $a_0-1$, whether or not\n$X_A(2)$ is rational.  For a fixed nonempty support, the displayed\nlogarithmic bound thus gives no additional restriction.\nA proposed alternative between super-logarithmic zero windows and general\nblock certificates cannot split this class: the first branch is already\nimpossible.  It would require the certificate branch for every support,\nwhich the obstruction in Proposition~\\ref{prop:squarefree} rules out for\nthe specified schemes.\n\\end{obs}\n\n\\begin{obs}[Prime support and the reciprocal-sum hypothesis]\n\\lean{irrational\\_erdosSupportSeries\\_pairwise\\_coprime}{CertificateKernel.lean:10776}\n(its certificates:\n\\lean{exists\\_weighted\\_coeff\\_certificates\\_supportCoeff\\_pairwise\\_coprime}{CertificateKernel.lean:9902}).\n\\textbf{Conclusion.} For every $b\\ge2$, an infinite pairwise-coprime support with\nsummable reciprocal mass has irrational support series.\n\\textbf{Method boundary:} the primes satisfy the other hypotheses but\n$\\sum_p1/p$ diverges, so this Lean theorem does not apply.  It is false,\nhowever, to infer that the base-$2$ prime-support value is open.\nTao--Ter\\\"av\\\"ainen prove\n\\[\n \\sum_{p\\ {\\rm prime}}\\frac1{2^p-1}\n =\\sum_{n\\ge1}\\frac{\\omega(n)}{2^n}\n\\]\nirrational in Theorem~1.3 of\n\\href{https://arxiv.org/abs/2512.01739}{their 2025 preprint}.  That result is\ncited, not formalised here.\n\\textbf{What remains is a test of the method:} replacing the global tail budget by a\nwindowed one would ask whether this particular certificate theorem can recover\nthe known theorem.  It is no longer an open mathematical target or a claimed\nnew route for \\#257.\n\\end{obs}\n\n\\begin{obs}[The two hypotheses of the sunflower criterion]\n\\lean{irrational\\_erdosSupportSeries\\_of\\_orthogonalPetalBouquet}{SupportSunflowerDichotomy.lean:540}.\n\\textbf{Conclusion.} For $A=$ (finite exceptional set dividing $Q$) $\\cup \\{\\mathrm{core}_i \\cdot\n\\mathrm{petal}_i\\}$ with cores $\\mid Q$, petals pairwise coprime and coprime to $Q$, $\\sum\n1/\\mathrm{petal}_i<\\infty$: if $A$ satisfies the tail-selection hypothesis below, then its\nseries is irrational.\n\n\\par\\medskip\n\n\\textbf{Different restrictions.} Two gaps. (i) On the support: the bouquet class strictly\ngeneralises pairwise-coprime supports but still demands globally pairwise-coprime, summable\npetals; it inherits the exclusion of the primes noted in the preceding observation and adds\nnothing in the divisor-dense regime. (ii) On the selector: the tail-selection hypothesis must\nhold at every precision and is unproved.\n\n\\par\\medskip\n\n\\textbf{Missing step.} For the selector, $\\forall K{>}0\\ \\exists N$, $2^K \\mid$ the carried first\nblock $\\wedge\\ \\sum_{j\\ge1}c_{A}(N{+}K{+}j)2^{-j}\\le16$, from the\nbouquet's persistent-reduced-modulus averaging. For the class: the assertion that every infinite $A$ admits a\nbouquet structure is \\emph{false} (take $A=$ all multiples of 6: there is no coprime\npetal decomposition).  A theorem for every infinite support would\ntherefore require a different structural hypothesis, not merely a\nproof of that false existence assertion.\n\\end{obs}\n\n\\begin{obs}[What eventual periodicity supplies]\n\\lean{irrational\\_erdosSupportSeries\\_eventuallyPeriodic}{CertificateKernel.lean:11604}.\n\\textbf{Conclusion.} $\\forall b\\ge2\\ \\forall m{>}0\\ \\forall N_0$: $A$ infinite and $m$-periodic above\n$N_0$ $\\Rightarrow$ irrational. Plus: the class of supports with irrational series is closed\nunder finite symmetric difference in both directions. The statement follows from the periodic\ntheorem of Luca and Tachiya~\\cite[Theorem~A, p.~139]{lucatachiya2017}.\n\n\\par\\medskip\n\n\\textbf{Scope of the hypothesis.} The theorem allows every positive\nperiod and every finite starting threshold.  It still requires an\neventually periodic support, so it does not include all aperiodic\nsupports covered by the other criteria in this record.  Finite\nsymmetric difference preserves eventual periodicity.  Thus the\nfinite-perturbation argument alone cannot extend this theorem to an\naperiodic tail; a further hypothesis or argument is needed.\n\n\\par\\medskip\n\n\\textbf{Missing step.} A ``locally periodic at cofinally many scales'' version; $\\forall q{>}0\\\n\\exists N\\ \\exists m\\le f(N)$ such that $A\\cap[N,N{+}L]$ is $m$-periodic for $L$ large enough\nthat the periodic sieve manufactures one block certificate. The periodic argument already\nneeds periodicity only across the certificate window, so this is the closest genuine\nwidening; but proving that periodic windows exist at unboundedly many scales for arbitrary $A$\nis a new statement.\n\\end{obs}\n\n\\begin{obs}[The reduced denominator in a gap criterion]\n\\lean{irrational\\_erdosSum\\_of\\_lcm\\_gap}{CertificateKernel.lean:5883} (instances\n\\lean{irrational\\_erdosSum\\_factorial\\_support}{CertificateKernel.lean:6035},\n\\lean{irrational\\_erdosSum\\_two\\_pow\\_support}{CertificateKernel.lean:6059}).\nLet $A=\\{a_0<a_1<\\cdots\\}\\subseteq\\mathbb N_{>0}$, let $b\\ge2$ be an integer, and\nwrite the first $k$ terms in lowest terms as\n\\[\n S_k=\\sum_{j<k}\\frac1{b^{a_j}-1}=\\frac{P_k}{Q_k},\\qquad Q_k\\ge1.\n\\]\nA sufficient irrationality condition is\n\\[\n Q_k b^{-a_k}\\longrightarrow0,\n \\quad\\text{equivalently}\\quad a_k-\\log_b Q_k\\longrightarrow+\\infty.\n\\]\nThe next omitted exponent is $a_k$, not the last exponent included\nin $S_k$.  Since $a_{k+j}\\ge a_k+j$, the positive omitted tail obeys\n\\[\n 0<X_A(b)-S_k\n \\le\\left(\\frac{b}{b-1}\\right)^2b^{-a_k}.\n\\]\nIf $X_A(b)=p/q$ were rational, positivity would give\n$qQ_k(X_A(b)-S_k)\\ge1$.  The displayed tail estimate contradicts\nthis under the proposed condition.  This is the denominator-times-error\nargument in \\lean{irrational\\_of\\_den\\_mul\\_abs\\_sub\\_tendsto\\_zero}{CertificateKernel.lean:5371},\nwith all indices specified.\n\nFor $L_k=\\operatorname{lcm}(a_0,\\ldots,a_{k-1})$ one has\n$Q_k\\mid b^{L_k}-1$, hence $\\log_b Q_k<L_k$.\nThus the earlier condition $a_k-L_k\\to\\infty$ suffices.\nUsing actual reduced denominators can improve this estimate, but\nrequires an \\emph{upper} bound for $Q_k$ relative to $b^{a_k}$.\nA surviving-divisor \\emph{lower} bound does not provide that upper\nbound.  In particular, the divisibility statement in\n\\lean{}{RationalDenominatorSurvival.lean:17} and the lower bound in\n\\lean{upperHalfMersenneProduct\\_lower\\_bound}{MersenneShadowDenominatorGrowth.lean:60}\ndo not alone establish the proposed condition for an arbitrary\nsupport.  No such condition is proved here for the remaining dense\nsupports.\n\\end{obs}\n\n\\begin{obs}[Rationality expressed by the integer carry]\n\\lean{erdosSupportSeries\\_rational\\_iff\\_exists\\_temperedCarry}{BooleanMobiusCarry.lean:384};\nthe general criterion of Theorem~\\ref{record:257bm-i-t7} has its own\n\\lean{binaryCoeffSeries\\_rational\\_iff\\_exists\\_temperedBinaryOrbit}{GenericTailOrbitRigidity.lean:426}.\n\\textbf{Conclusion.} For every $A\\subseteq\\mathbb N$, with no hypothesis:\n$X_{A}(2)$ is rational if and only if there are $q>0$ and $U:\\mathbb N\\to\\mathbb Z$ with\n$U(N+1)=2U(N)-q\\,c_A(N+1)$ and $U(N)/2^N\\to0$.\n\\textbf{Comparison.} This is already universal over $A$; it is the exact form in which the\nobligation should be stated, and the argument literally shared with \\#249 (take\n$c=\\varphi$ in Theorem~\\ref{record:257bm-i-t7}). It does not refute the orbit:\n\\lean{temperedBinaryOrbit\\_eq\\_scaledTail}{GenericTailOrbitRigidity.lean:339} pins the orbit\nuniquely to $u(N)=qT_c(N)$, and $T_c(N)\\le N{+}2$ confirms it is never ruled out on growth\ngrounds for any $c$ with $c(n)\\le n$.\n\\textbf{Missing step.} A support-specific non-existence theorem: for every infinite $A$ and\nevery $q>0$, no such $U$ exists.  Because the criterion is an equivalence and the orbit is\nunique, this is logically equivalent to the obligation; its value is to name the single object\n(the tempered carry orbit of $c_{A}$) whose non-existence is the entire content, for every $A$\nat once.\n\\end{obs}\n\n\\begin{obs}[Why unbounded carry alone is insufficient]\n\\lean{shifted\\_state\\_unbounded\\_of\\_infinite\\_support}{RationalSupportCarrySkeleton.lean:2327},\n\\lean{exists\\_unbounded\\_shifted\\_odd\\_tail\\_nat\\_state\\_of\\_support\\_fraction}{RationalSupportCarrySkeleton.lean:2383},\n\\lean{one\\_add\\_mul\\_card\\_le\\_two\\_mul\\_shifted\\_state}{RationalSupportCarrySkeleton.lean:2237}.\nLet $A\\subseteq\\Npos$ be infinite, let $c\\ge0$ and $v\\ge1$ be integers,\nand let $u(n)$ be positive integers satisfying\n\\[\n u(n+1)+v c_A(c+n+1)=2u(n)\\qquad(n\\ge0).\n\\]\nFor a finite $F\\subseteq A$ and a common multiple $L$ of its\nelements with \\emph{$L>c$}, put $n=L-c-1\\ge0$.  Directly from\nthe recurrence,\n\\[\n 2u(L-c-1)=u(L-c)+v c_A(L)\\ge1+v|F|.\n\\]\nThis argument uses positivity of the next state, not an assumed\ntail representation or a limit condition.  A general positive\nsolution may also have an exponentially growing homogeneous part.\nThe condition $L>c$ ensures that the coefficient row really is\n$L$; truncated natural-number subtraction cannot replace it.\n\nFinite subsets $F$ of arbitrarily large size, followed by positive\ncommon multiples exceeding $c$, prove that $u$ is unbounded.\nIn the tempered case the separate identity\n$u(n)=v\\sum_{r\\ge1}c_A(c+n+r)2^{-r}$ gives\n$u(n)\\le v(c+n+2)$.  Unboundedness is consistent with this linear\nupper bound and with $u(n)/2^n\\to0$.  A contradiction would require\nan incompatible upper estimate at the same chosen indices.\n\\end{obs}\n\n\\begin{obs}[A finite-gap criterion for every target in the convex hull]\n\\lean{half\\_mem\\_mersenneAchievementSet\\_or\\_exists\\_fatal\\_gap}{HalfCutLocator.lean:623},\n\\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_no\\_existsFatalHalfGap}{HalfCutLocator.lean:654};\nthe exclusion for a general support has its own\n\\lean{positiveMersenneSupportValue\\_ne\\_of\\_rank\\_gap}{HalfCutLocator.lean:380}.\n\\label{obs:general-target-gap}\nFor every real $x\\in[0,E]$, not just rational targets, one has\n\\[\n x\\notin\\mathcal A\\quad\\Longleftrightarrow\\quad\n \\exists m\\ge1\\ \\exists D\\subseteq\\{1,\\ldots,m-1\\},\\qquad\n X_D(2)+R_m<x<X_D(2)+w_m.\n\\]\nHere $R_m=\\sum_{j>m}w_j$.  The implication to the right has a\nshort greedy proof.  If $x\\notin\\mathcal A$, choose the first $m$\nwith $r_m(x)>R_m$ using Theorem~\\ref{thm:greedy-survival}.\nSince $r_0(x)=x\\le E$, we have $m\\ge1$ and\n$r_{m-1}(x)\\le R_{m-1}=w_m+R_m$.  A take at $m$ would leave a\nremainder at most $R_m$, so this step must be a skip.  Thus\n$R_m<r_{m-1}(x)<w_m$, giving the displayed gap with the greedy\nprefix $D$.\n\nConversely, the displayed gap contains no achievable value.\nA support with prefix $D$ either skips $m$, giving value at most\n$X_D(2)+R_m$, or takes $m$, giving value at least $X_D(2)+w_m$.\nA different prefix first differs at some $j<m$; the inequality\n$w_j>R_j$ orders its entire continuation interval strictly on one\nside of the interval with prefix $D$.\nBoth gap endpoints themselves are achievable, by the supports\n$D\\cup\\{m+1,m+2,\\ldots\\}$ and $D\\cup\\{m\\}$.\nConsequently no parity argument or irrationality assertion about\n$E$ is needed for this ordinary general-target dichotomy.\nThe restriction to $[0,E]$ is essential to this formulation:\ntargets outside the convex hull are already excluded and need not\nlie in an internal gap.  The general-target equivalence and both endpoint\nconstructions are Lean-checked in\n\\lean{paper\\_general\\_target\\_gap}{lean/ErdosProblems/Erdos257/PaperCompleteR20/GeneralTargetGap.lean:173}.\n\\end{obs}\n\n\\begin{obs}[Why the signed periodic theorem still requires periodicity]\n\\lean{irrational\\_or\\_bpow\\_mul\\_eq\\_intCast\\_intWeightedErdosSeries\\_periodic}{CertificateKernel.lean:14175},\n\\lean{irrational\\_intWeightedErdosSeries\\_periodic\\_of\\_coeff\\_nonneg\\_of\\_frequently\\_ne\\_zero}{CertificateKernel.lean:14583}.\n\\textbf{Conclusion.} $\\forall b\\ge2\\ \\forall m{>}0\\ \\forall$ $\\mathbb Z$-valued $m$-periodic weight\n$w$: irrational$(\\sum w(a)/(b^a-1))$ $\\vee$ $\\exists k,z,\\ b^k x=z$; the terminating branch closes\nunconditionally as soon as $c_w$ is one-signed and frequently nonzero.\nLuca and Tachiya's theorem already gives irrationality for every nonzero periodic\nweight~\\cite[Theorem~A, p.~139]{lucatachiya2017}.\n\n\\par\\medskip\n\n\\textbf{Different restrictions.} Two mismatches. (i) Periodicity of the weight is still\nrequired; this is the signed version of the periodic-support theorem and keeps its periodicity hypothesis.\n(ii) The one-sidedness clause is automatic for \\#257 (a $0/1$ support indicator gives\n$c_{A}\\ge0$), so the sign conditions add nothing here; they were designed for the signed\n\\#249 argument.\nThe genuinely transferable content is the dichotomy shape itself: irrational-or-$b$-adically-\nterminating, terminating branch killable by a positive far tail.\n\n\\par\\medskip\n\n\\textbf{Missing step.} An aperiodic version of the dichotomy; for arbitrary $A$, either\n$X_{A}(2)$ is irrational or $2^k\\cdot$it is an integer. For arbitrary infinite $A$, exclusion of the terminating branch is itself an additional unproved\nassertion; finite-support odd-denominator parity does not pass automatically to an infinite limit.\nAn aperiodic dichotomy together with a valid terminating-branch exclusion would close the\nobligation. Periodicity is used only to manufacture the full-block certificates,\nso removing it again requires the missing supply of certificates for arbitrary supports.\n\\end{obs}\n\n\\subsection{Three combined conditional arguments}\n\nEntry A has a Lean proof, and entry B is an ordinary argument about what\nits hypotheses would imply.  The facts behind entry C, the survival equivalence of\nTheorem~\\ref{thm:greedy-survival-record} and the one-sided certificates,\nare proved in Lean.  None of the three establishes an unproved orbit\nhypothesis or decides~\\#257.\n\n\\begin{thm}[A: a sufficient lower bound at every reset]\\label{thm:lower-bound-every-reset}\nAssume that\n$|\\mathrm{rem}(r+1)-2^{r+1}|>2^{(r+5)/2}$ for every upper or\nmiddle reset $r\\ge10$.  Then $1/2\\in\\mathcal A$.\n\nHere is the role of the constants.  Put\n\\[\n B(r)=2^{\\lfloor(r+4)/2\\rfloor}+2r+3.\n\\]\nFor $r\\ge10$, one has $B(r)^2\\le2^{r+5}$.\nA right branch at the first crossing of the largest omitted rank $r$\nwould force $|\\Delta_{r+1}|\\le B(r)$, by tracing the exact affine\nrecurrence back to that reset.  The assumed strict lower bound\nexcludes this crossing.  The largest-skip induction then gives\nhalf-membership.  The hypothesis concerns every actual reset beyond\nthe threshold; neither a long finite check nor an asserted\nrun-length asymptotic supplies it.\n\\end{thm}\n\n\\begin{rem}[B: rigidity of dangerous resets]\n\\label{rem:dangerous-reset-chain-scope}\nFor a dangerous reset preceded by a sufficiently long pure right run,\nObservation~\\ref{obs:dangerous-reset-rigidity} confines the deviation at\nthe preceding reset to at most two adjacent integers determined by the\ndivisor-pulse stream. Applying that observation along a chain of $n$\ndangerous resets gives $n-1$ nested integer conditions, provided each\nintervening run meets its length threshold. The argument leaves a\ndangerous reset after a short run untouched. In particular, failure of\nsquare-root escape alone has not been shown to produce such a chain.\nNo probability model is supplied and the pinned values are not excluded.\nThis is an ordinary argument under the stated long-run hypotheses.\n\\end{rem}\n\n\\begin{thm}[C: the one-sided finite decision boundary]\\label{thm:one-sided-finite-decision-boundary}\nThe equivalence of Theorem~\\ref{thm:greedy-survival-record} separates two logically different kinds of evidence. A fatal greedy gap found at a finite rank is\na finite certificate that $1/2\\notin\\mathcal A$. By contrast, membership requires survival at\nevery rank; this paper supplies no finite certificate that the orbit survives forever and no\ncompletion theorem turning a long surviving prefix into membership.\n\nThe reset-crossing hypotheses, truncation-rung ladder, and sharp-capacity inequalities feed\nsufficient conditions for membership. They are not proved equivalent to one another,\nnor is failure to find one at a given depth evidence of survival. Consequently these searches\ngive at most a semi-decision procedure for \\emph{nonmembership of the specific value $1/2$}.\nHalting does not prove the universal statement in \\#257, and non-halting does not prove that\n$1/2$ is represented. In particular, neither truth nor falsity of the universal problem is\nshown semi-decidable here. Deeper negative search alone establishes nothing beyond the tested\nfinite ranks; this is the quantifier boundary of Theorem~\\ref{thm:one-sided}.\n\\end{thm}\n\n\\subsection{Four limitations of the stated implications}\n\nThe following conditions are equivalent to their target statements\nwithin the specified scope.  An equivalence may still be useful by\nexposing a quantity that is easier to estimate.  What it does not\nsupply is a proof of that estimate, or a strict weakening of the\nlogical obligation.\n\n\\begin{rem}[The centred-carry bound is equivalent to the conclusion, for the greedy support only]\n\\lean{greedy\\_half\\_infinite\\_of\\_mobiusCenteredHalfCarry\\_sqrtBound}{HalfCarryReachability.lean:834}\nand its packaged form\n\\lean{greedy\\_half\\_infinite\\_of\\_mobiusCenteredHalfCarry\\_upperBound}{HalfCarryReachability.lean:953}\nassume the bound $C_A(N)\\le2\\sqrt N+4$ for every $N$. For the greedy support $A=G$ of $1/2$\nspecifically, this bound is logically \\emph{equivalent} to the theorem's own conclusion\n$X_{A}(2)=1/2$, not a strictly weaker sufficient condition. This follows\nfrom the exact residual identity of Lemma~\\ref{lem:collapse-mech}, the unconditional\nsquare-root tail bound\n(\\lean{binaryCoeffTail\\_supportCoeff\\_le\\_two\\_sqrt\\_add\\_four}{BooleanMobiusCarry.lean:290}), the\nunconditional tail-nonnegativity lemma\n(\\lean{binaryCoeffTail\\_nonneg}{GenericTailOrbitRigidity.lean:78}), and the greedy-specific fact\n$X_{A}(2)\\le1/2$. \\textbf{The scope limit is exact:} this equivalence does \\emph{not} extend to\nevery $A$ with $1\\notin A$; dropping the greedy-specific fact yields only that the bound is\nequivalent to $\\delta\\le0$ (a one-sided condition), not to $\\delta=0$. For the related hypothesis\n\\lean{GreedyHalfCarryCofinalStripReturn}{CofinalStripReturn.lean:76}, the former gap between\nthe real and the integer square root is removed by restricting cofinal witness depths\nto perfect squares. Any remaining gap lies in the hypothesis itself, not in rounding.\n\\end{rem}\n\n\\begin{rem}[What the terminal carry condition proves]\n\\lean{HalfCarryCofinalTerminalOnlyStrip}{TerminalOnlyCofinal.lean:34} at the original strip\nconstant is logically equivalent to $1/2\\in\\ensuremath{\\mathcal A}$ after restricting\ncofinal witness depths to perfect squares and using the recorded tail bound. The square-root\nconversion raises no further issue; the open content is the hypothesis itself.\n\\end{rem}\n\n\\begin{rem}[Positive greedy skips and half-membership]\nAs noted above, the positivity conjunct of\n\\lean{CofinalPositiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:22} holds\nunconditionally, by parity and greedy nonnegativity\n(\\lean{}{BooleanMobiusCriticalCapacityCofinal.lean:138}).  The hypothesis therefore says\nexactly that the greedy rule for $1/2$ skips infinitely many ranks, which is\n\\emph{equivalent} to $1/2\\in\\ensuremath{\\mathcal A}$\n(\\lean{}{GreedyAchievementSet.lean:2514}).  A proof or disproof of this statement settles\nthe half-target question, and conversely; it is not an intermediate step toward it.\n\\end{rem}\n\n\\begin{rem}[Scope discipline for the equivalence class]\nEach equivalence uses its stated domain.  The centred-carry upper\nbound concerns the actual greedy support $G$, not every support omitting\n$1$.  The cofinal terminal condition already works with the original\nconstant $4$, because perfect-square depths avoid a rounding loss.\nThe positive-skip equivalence is stated here at the target $1/2$;\nextending it to another target requires checking the target-dependent\narguments.  None of these qualifications supplies the still-missing\nunbounded sequence of witnesses.\n\\end{rem}\n% ; - part a257_p4 ; -\n\\section{Hypothesis-specific obstructions}\nAn abstract recurrence can satisfy some identities of the greedy orbit\nand still violate the proposed reset bound. Such a counterexample applies\nto the actual orbit only after the correspondence of states has been\nproved. We keep these examples separate from calculations on genuine\nfinite orbits. Additional arithmetic hypotheses may exclude the abstract\ncounterexamples.\n\nThroughout, $\\mathrm{rem}(s)$ is the quotient-row remainder, $\\lambda_n := \\Delta_n/2^n$ the normalised seam\ndeviation of Theorem~\\ref{thm:dynamics}, and $\\mathcal A$ the Mersenne achievement set. Reset square-root escape\nfrom row 10 is the statement $|\\mathrm{rem}(r+1)-2^{r+1}| > 2^{(r+5)/2}$ for every reset row $r\\ge 10$.\nIt is a sufficient input for the cited\nhalf-membership argument, not a necessary condition for every\npossible proof.\n\n\\subsection{Four tested approaches to the reset estimate}\n\n\\begin{obs}[Prime steps can occur within a bounded drift window]\n\\label{obs:prime-doubling}\n\\textbf{Route as conceived.} The universal recursion $K(M+1) = 2K(M) - (\\tau(M+1)-1)$ doubles at every step and subtracts the divisor-excess $\\tau(M+1)-1$. At a prime $M+1=p$,\n$\\tau(p)=2$ so the subtracted term is $1$; the naive hope is that primality of an index forces the\nstep to be \"almost pure doubling,\" and that prime density (Chebyshev/PNT-scale) then bounds how\nlong a run of near-pure-doubling steps can last, which would bound the danger run-length $L_r$ in\nthe right-branch recurrence in Theorem~\\ref{thm:dynamics}.\nThe upper branch has a different, adjacent-spacing correction, so\na run-length argument cannot use the discarded uniform-error\nformula across resets.\n\n\\textbf{A finite test.} Use the skip set of the quotient row $n=607$\nfrom Section~\\ref{ssec:seam-model}, and define the frozen-row drift\n\\[\n J_{607}(M)=K(M)+\\sum_{d\\in\\mathrm{Skip}_{607}}\n                   \\left\\lfloor\\frac{2^M}{2^d-1}\\right\\rfloor.\n\\]\nExact integer arithmetic gives, in order at $M=607,\\ldots,617$,\n\\[\n 5,\\ 5,\\ 4,\\ 4,\\ 6,\\ 2,\\ 3,\\ 4,\\ 5,\\ 2,\\ 3.\n\\]\nThere are eleven values and ten transitions.  The indices $607$, $613$,\nand $617$ are prime.  For $M+1\\ge607$ the recurrence is\n\\[\n J_{607}(M+1)=2J_{607}(M)\n -\\bigl(\\tau(M+1)-1-c_{\\mathrm{Skip}_{607}}(M+1)\\bigr).\n\\]\nAt a prime index the last coefficient is $1$.  The composite-index\ncoefficients can compensate for those prime steps, as the displayed\nvalues illustrate.  This table is a new finite arithmetic check; the\nlonger earlier run search was not repeated.\n\n\\textbf{Scope.} Primality at some indices does not force this drift to\nleave a small interval immediately.  The example supplies no universal\nrun-length bound and no statistical conclusion about prime indices.\nIt also does not exclude a proof that combines prime counts with\nquantitative information about the intervening divisor counts.\n\\end{obs}\n\n\\begin{obs}[Parity and perfect squares in the recurrence]\n\\label{obs:parity-squares}\n\\textbf{Route as conceived.} $K(M)$ is even exactly when $M$ is a perfect square (a true,\nprovable fact). The hope was to use this parity law to forbid certain residues or run patterns in\nthe digit stream of $K$, since parity constraints are a classical first attack on digit-run\nquestions.\n\n\\textbf{What the parity calculation says.} The recurrence implies\n$J(M+1)\\equiv\\delta(M+1)\\pmod2$ for every integer forcing sequence.\nFor $K$, the additional arithmetic fact that $\\tau(M)$ is odd exactly\nat squares gives the stated parity pattern.  One should not call that\npattern a property of every forcing sequence, or say that it excludes\nno residue: at each fixed index it does specify the parity.\n\n\\textbf{Scope.} Restating a parity consequence of a known recurrence\ndoes not add a new independent constraint.  To turn it into a\nrun-length bound would require further information about the sizes or\narrangement of the forcing terms.  No argument here rules out using\nparity together with such information.\n\n\\textbf{Use of the observation.} The parity law does constrain\n$K(M)$ at each specified index and is a useful check on any proposed\nrun pattern.  The missing implication is from this residue information\nto a quantitative bound on run lengths.  No such implication is\nproved by the parity calculation alone.\n\\end{obs}\n\n\\begin{obs}[What the denominator estimate does not give]\n\\label{obs:irrationality-measure}\nOne proposed separation argument would prevent\n$\\sum_{d\\in\\mathrm{Skip}_n}x_d$ from approximating\n$C:=E-\\tfrac32=0.1066951524152917\\ldots$ at scale $2^{-3n/2}$.\nAn irrationality-measure bound is relevant only after relating the\nreduced denominator of that finite sum to $n$.\n\nFor $n\\ge2$ and $\\mathrm{Skip}_n\\subseteq\\{2,\\ldots,n-1\\}$, its\nreduced denominator $D_n$ satisfies\n\\[\n \\log_2 D_n\\le\\sum_{d\\in\\mathrm{Skip}_n}\\log_2(2^d-1)\n \\le\\sum_{d\\in\\mathrm{Skip}_n}d\n \\le\\frac{n(n-1)}2-1.\n\\]\nThe middle inequality is strict when the skip set is nonempty.\nThis is an upper bound; it is not an asymptotic formula for $D_n$.\nIn particular, cardinality alone does not justify an asymptotic of\norder $n^2/4$.\n\nZudilin proves $\\mu(E)\\le2.46497868\\ldots$~\\cite[Theorem~1, p.~154]{zudilin2004}.\nRational translation preserves the irrationality exponent, so for\nany fixed $\\nu>2.46497868\\ldots$ there is a constant $c_\\nu>0$\nsuch that every reduced rational $p/q$, with $q\\ge1$, satisfies\n$|C-p/q|\\ge c_\\nu q^{-\\nu}$.\nUsing only the displayed denominator estimate therefore gives\n\\[\n \\left|C-\\sum_{d\\in\\mathrm{Skip}_n}x_d\\right|\n \\ge c_\\nu\\,2^{-\\nu(n(n-1)/2-1)}.\n\\]\nThis guaranteed lower bound is eventually smaller than $2^{-3n/2}$,\nso it does not prove the desired separation.  A sharper estimate\nfor the actual reduced denominators, or an approximation result\nusing additional structure of these numerators, could change the\ncomparison.  Neither is excluded by this calculation.\n\\end{obs}\n\nRetaining only the parity of an input coefficient leaves considerable\nfreedom.  In $e'=2e+c$, every incoming integer $e$ admits the odd\nchoice $c=-1-2e$, which gives $e'=-1$.  The next proposition uses\nthe same centred-remainder construction when more unit bits and\na valuation are prescribed.\n\n\\begin{prop}[Centred completion of fixed-precision 2-adic data]\n\\label{prop:2adic-nogo}\nFix $u\\ge1$ and a finite list $(v_i,a_i)_{0\\le i<m}$, where\n$v_i\\ge0$ are integers and the integers $a_i$ are odd.  For every\ninitial integer $e_0$, there are integers $z_i,e_{i+1}$ such that\n\\[\n e_{i+1}=2e_i+2^{v_i}(a_i+2^u z_i),\\qquad\n |e_{i+1}|\\le2^{v_i+u-1}\\qquad(0\\le i<m).\n\\]\nThus one may prescribe the valuation and $u$ bits of the odd unit\nof each input coefficient while keeping every \\emph{successor}\nstate in its stated centred interval.  The initial state is arbitrary.\n\\end{prop}\n\n\\begin{proof}\nFor a single step, put $M=2^{v_i+u}$ and $R=M/2$, and choose\n\\[\n e_{i+1}=\\bigl((2e_i+2^{v_i}a_i+R)\\bmod M\\bigr)-R.\n\\]\nThe remainder convention $0\\le x\\bmod M<M$ gives\n$-R\\le e_{i+1}<R$.  Also\n$e_{i+1}-2e_i\\equiv2^{v_i}a_i\\pmod M$, so this difference has\nthe required form $2^{v_i}(a_i+2^u z_i)$.  Repeating the\nconstruction proves the finite-list statement.  It uses no\nrelation between the freely chosen quotients $z_i$ at different\nsteps.\n\\end{proof}\n\nThe formal-source statements are\n\\lean{fixedPrecisionTropicalNoGo}{TropicalCurvatureCarry.lean:137},\n\\lean{vu\\_step\\_has\\_centred\\_completion}{TropicalCurvatureCarry.lean:69}\nand\n\\lean{vu\\_word\\_has\\_prefix\\_locked\\_completion}{TropicalCurvatureCarry.lean:115}.\nThey concern this unrestricted congruence model, not coefficients\nalready fixed by a divisor-count sequence.  Fixed-precision data\nalone cannot exclude every completion in this model.  The proposition\ndoes not show that the completion comes from a Mersenne support,\nsatisfies additional arithmetic relations, or belongs to the\nparticular greedy orbit.  In particular, it is not a general\nobstruction to every fixed-window argument for Problem~249 or~257.\n\nThese observations have separate scopes: a finite prime-window\nexample, a parity consequence of a known recurrence, an insufficient\ndenominator estimate, and a completion theorem for specified\nfixed-precision symbols.  They do not exhaust the possible approaches\nto the reset estimate or force every proof to use digit-carry\nmachinery.  The next section studies that machinery because it gives\nexplicit conditional implications, not because other methods have\nbeen ruled out.\n\n\\subsection{Excluding particular final dyadic intervals}\n\\label{sec:branch-exclusions}\n\nThe two-sided bound\n$\\min(\\mathrm{rem}(s),\\mathrm{overshoot}(s))\\le2^s$ for all $s\\ge5$\nis conditional.  It propagates by induction from a local hypothesis which at every row\nexcludes three specified integer values on the middle branch and imposes\none further inequality on the right branch:\n\\[\n \\begin{gathered}\n s\\ge5,\\quad\\text{no carry and a middle transition}\n \\quad\\Longrightarrow\\quad C_s\\notin\\{-3,-2,-1\\};\\\\\n s\\ge5,\\quad\\text{no carry and a right transition},\\quad\n \\mathrm{overshoot}(s)\\le2^s\\\\\n \\Longrightarrow\\quad\\mathrm{charge}(s)\\le2^{s+2}.\n \\end{gathered}\n\\]\nHere $C_s=4\\,\\mathrm{rem}(s)-p_s^- -4$.  On a middle transition,\n$\\mathrm{rem}(s+1)-2^{s+1}=C_s+4$; on a right transition,\n$\\mathrm{rem}(s+1)=C_s-2^{s+1}$.  Thus $C_s$ must not be identified\nwith the next row's centred remainder.\n\\lean{Erdos249257.SeamTwoSidedDyadicCellEscape}{HalfCylinderMiddleCarryLowerBound.lean:4374}\n\n\\begin{prop}[The upper (carries) successor needs no exceptional-cell exclusion]\n\\label{prop:upper-unconditional}\nLet $s\\ge5$ and suppose the upper branch occurs at row $s$, that is,\nthe transition at row $s$ is on branch $\\mathrm U$ of Theorem~\\ref{thm:dynamics}.\nThen\n\\[\n  \\mathrm{rem}(s+1)\\le2^{s+1}.\n\\]\nIndeed, the reset identity expresses $2^{s+1}$ as\n$\\mathrm{rem}(s+1)$ plus a nonnegative reset charge.\nThus the upper branch needs no exceptional-value exclusion for the\nnext-row two-sided invariant.  This conclusion concerns that induction\nstep, not the absence of upper transitions on the orbit.\n\\end{prop}\n\n\\begin{thm}[$C_D=-3$ is impossible at the final middle transition, $D\\ge 13$]\n\\label{thm:cd-neg3-impossible}\nConsider a hypothetical \\emph{final middle transition}: a middle transition at row $D\\ge 13$\n($\\lnot\\mathrm{carries}$, and the middle-branch inequality\n$4\\mathrm{rem}(D)+\\mathrm{gap}-\\mathrm{belowPulse} < \\mathrm{terminalWeight}$ holds at $D$), followed\nby an all-right tail forever after ($\\forall s\\ge D{+}1$, $\\mathrm{seamGreedyWord}(s+1) =\n\\mathrm{seamGreedyWord}(s).\\mathrm{extend}\\ \\mathrm{true}$). Under these assumptions,\n\\[\n  \\mathrm{belowPulse}(D) + 2 \\;\\le\\; 4\\cdot\\mathrm{rem}(D),\n\\]\ni.e.\\ $C_D \\ge -2$ ;  the cell $C_D=-3$ (and every more negative value) is excluded.\n\n\\textbf{Proof of the bound.} The all-right-tail assumption puts the\nterminal-augmented finite prefix strictly above $1/2$.\nConsequently the producer carry is below its complete incidence tail,\nor equivalently\n$Z_D<1+1/(2^D-1)$, where\n$Z_D=\\mathrm{rem}(D)-\\mathrm{seamWordFloorError}(D)$.\nThe lower bound uses the other endpoint: under the same tail assumption,\nthe finite value at $D$ plus the complete Mersenne tail remains below\n$1/2$.  The exact scaled remainder identity therefore gives\n\\[\n Z_D>4^D\\sum_{e>D}\\frac1{2^e-1}-2^D>\\frac13.\n\\]\nThe last inequality follows by retaining the first two geometric\nchannels of the complete Mersenne tail.  Put\n$E_D=\\mathrm{seamWordFloorError}(D)$.  Pulse absorption gives\n$\\mathrm{belowPulse}(D)\\le4E_D$, so\n\\[\n 4\\,\\mathrm{rem}(D)-\\mathrm{belowPulse}(D)\n \\ge4(\\mathrm{rem}(D)-E_D)=4Z_D>\\frac43.\n\\]\nThe left side is an integer, hence is at least $2$, as required.\nThe upper threshold inequality alone would not give this lower bound.\n\n\\textbf{Scope.} This excludes $C_D=-3$ at a middle row $D\\ge13$\nunder the additional all-right-tail assumption.  The general induction\nrequires exclusion of all three cells at every middle row, not just\nunder this extra hypothesis, together with its right-branch bound. The two statements share the same coordinate $C_D$\nbut have different hypotheses.  Theorem~\\ref{thm:cd-neg3-impossible}\nis universal over rows satisfying the additional all-right-tail\nassumption; it does not assert that such a row exists.  The induction\nhypothesis concerns every middle row, without that tail assumption.\nExcluding $C_D=-3$ in the former statement therefore does not supply\nthe exclusion required in the latter.\n\\textbf{Object or representation:} about the \\emph{object}\nunder the stated hypothesis ;  a genuine arithmetic consequence of the all-right-tail assumption\nvia the fatal-gap orbit, not a coordinate artifact.\n\\end{thm}\n\n\\begin{cor}[Only $C_D\\in\\{-2,-1\\}$ remain among the exceptional dyadic cells]\n\\label{cor:cd-remaining}\nAt a final middle row $D\\ge13$, Theorem~\\ref{thm:cd-neg3-impossible}\nremoves $-3$ from the exceptional set $\\{-3,-2,-1\\}$, leaving\n$-2,-1$ \\emph{within that set}.  It does not assert\n$C_D\\in\\{-2,-1\\}$: nonnegative values have not been excluded.\nProposition~\\ref{prop:upper-unconditional} handles upper transitions\nin the two-sided induction; it supplies no additional restriction on\na middle coordinate.  For the all-middle-row induction, the exclusion\nof all three values still needs proof without an all-right-tail\nassumption, together with the separate right-branch inequality.\n\\end{cor}\n\n\\begin{rem}[A sufficient tail inequality and its source]\n\\label{rem:tail-dominance-open}\nLet $U_D$ be the finite set of exponents selected by the below\nword at row $D$, and put $F_D=U_D\\cup\\{D\\}$.  With\n$c_{F_D}(n)=\\#\\{d\\in F_D:d\\mid n\\}$, define\n\\[\n \\Theta_D:=\\sum_{j\\ge1}c_{F_D}(2D+2+j)2^{-j}.\n\\]\nThis is a tail sum, not a ratio.  Since\n$0\\le c_{F_D}(n)\\le |F_D|$, it satisfies\n$0\\le\\Theta_D\\le |F_D|$.  The integer\n$C_D=4\\,\\mathrm{rem}(D)-\\mathrm{belowPulse}(D)-4$\nis the producer carry in the cited source.  Under the final-middle\nand all-right-tail assumptions, that source proves $C_D<\\Theta_D$:\n\\lean{Erdos249257.middleProducer\\_allRight\\_forces\\_carry\\_lt\\_tail}{HalfCylinderMiddleCarryLowerBound.lean:1814}.\n\nAn inequality in the opposite direction would exclude such a final\nmiddle transition.  The following sufficient condition retains the\nexception already handled by Theorem~\\ref{thm:cd-neg3-impossible}:\n\\[\n \\textbf{(TD)}\\qquad\n \\forall D\\ge13,\\quad\n \\bigl(D\\text{ is a middle row and }C_D\\ne-3\\bigr)\n \\ \\Longrightarrow\\ \\Theta_D<C_D.\n\\]\nBecause $\\Theta_D\\ge0$, this condition in particular excludes\n$C_D=-2$ and $C_D=-1$.  It requires an inequality at the other indicated\nmiddle rows as well, so it is not merely a reformulation of excluding\ntwo integer values.  Nor does it supply the separate right-branch\nhypothesis used in the two-sided induction.\n\nThis is the reduced middle tail condition of Part~III, and half-membership follows from it\n(\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_middleProducerTailEscapeExceptNegThree}{HalfCylinderLastProducerContradiction.lean:436}).\nThe inequality itself is not established here.  Its role is a\nconditional implication, not a newly identified necessary condition.\n\nThe two more concrete sufficient bounds in\nTheorem~\\ref{thm:middle-producer-escape} imply the tail inequality even\nwithout the exception.  Their Lean forms are\n\\lean{Erdos249257.SeamMiddleProducerCardEscape}{HalfCylinderMiddleCarryLowerBound.lean:279}\nand\n\\lean{Erdos249257.SeamMiddleProducerRowEscape}{HalfCylinderMiddleCarryLowerBound.lean:1769}.\nThe cardinality condition is\n\\[\n |U_D|+\\mathrm{belowPulse}(D)+5<4\\,\\mathrm{rem}(D).\n\\]\nIt gives $|F_D|=|U_D|+1<C_D$, hence $\\Theta_D<C_D$.\nThe simpler row condition $D\\le\\mathrm{rem}(D)$ is stronger:\n$\\mathrm{belowPulse}(D)\\le2|U_D|$ and $|U_D|\\le D-2$ give\n$|U_D|+\\mathrm{belowPulse}(D)+5\\le3D-1<4D$.\nThe corresponding half-membership implication is\n\\lean{Erdos249257.half\\_mem\\_mersenneAchievementSet\\_of\\_middleProducerCardEscape}{HalfCylinderMiddleCarryLowerBound.lean:2364}.\nNo converse between these sufficient conditions is claimed.\n\\end{rem}\n\n\\subsection{Related counterexamples and restrictions}\n\nThe next results give counterexamples to particular summaries and\nestimates for integer carries.  Their hypotheses matter: a countermodel\nfor one summary is not a prohibition on every proof that uses carries\nor on every other input.\n\n\\begin{prop}[A finite-state restriction for the stated pulse family]\n\\label{prop:finite-state-nogo}\nFor a ``balanced pulse'' family at location $m\\ge2$ (radius $\\rho=\\lfloor(m{+}1)/2\\rfloor$, parameters\n$0\\le r\\le\\rho$, moving mass between positions $m$ and $m{+}1$ with the weighted total\n$2c(m)+c(m{+}1)=2\\rho$ of the two coefficients fixed), if\na predecessor state is constant across the whole family, then no function\n$\\mathrm{decode}:\\mathrm{State}\\to\\mathbb N$ can recover the parameter $r$ from $\\mathrm{state}(r)$\nfor every $r$. The family has exactly $\\rho+1=\\lfloor(m{+}1)/2\\rfloor+1$ members, so the fan-out is\nunbounded in $m$. More strongly, any finite set of states carrying an exact decoder for the family\nhas at least $\\rho+1$ elements, unbounded in $m$.\n\n\\textbf{Scope.} Excludes bounded-state encodings of pre-$m$ history that must distinguish every\nmember of the displayed balanced-pulse family, whether the coefficients are $\\varphi$ or a\nMöbius-support indicator. Applying it to \\#249 or \\#257 requires showing that the relevant orbit\nrealises that family, and it does not rule out every finite-state proof strategy.\n\\textbf{Object or representation:} about a \\emph{representation}\nclass, the encodings of history that separate the balanced-pulse family; it says nothing about $C$\nor $\\mathcal A$ directly.\n\\end{prop}\n\n\\begin{prop}[Möbius-support countermodel: the natural negative-sign candidate overshoots]\n\\label{prop:mobius-nogo}\nThe signed Lambert identity $\\sum_{d\\ge1}\\mu(d)/(2^d-1) = 1/2$ is exact. Writing\n$N:=\\{d:\\mu(d)=-1\\}$: $\\sum_{d\\in N}1/(2^d-1) = 1/2 + \\sum_{d\\in P}1/(2^d-1)$ where $P:=\\{d\\ge2:\n\\mu(d)=1\\}$, and quantitatively $1/2 + 1/63 \\le \\sum_{d\\in N}1/(2^d-1)$ (using the first positive\ntail term $d=6$, $\\mu(6)=1$) ;  the negative-Möbius support strictly \\emph{overshoots} $1/2$ by at\nleast $1/63$.\n\n\\textbf{Scope.} Rules out exactly one natural candidate infinite Boolean support (the negative-Möbius\nset) as a witness for $1/2\\in\\mathcal A$. A route-sufficiency no-go only ;  it says nothing about\nwhether \\emph{some other} infinite Boolean support sums to $1/2$.\n\\textbf{Object or representation:} about the \\emph{object} ;  a genuine\nvalue inequality for one specific candidate set, not a coordinate artifact. Margin $1/63$ is the\nconcrete number any repair attempt (adding/removing finitely many elements) must close or exceed.\n\\end{prop}\n\n\\begin{prop}[Finite computation and half-membership]\n\\label{prop:finite-boolSupport-and-onesided}\nNo finite positive-index Boolean support has value exactly $1/2$: the reduced denominator of any\nfinite Mersenne subset-sum is provably \\textbf{odd} (each $2^n-1$ is odd), while $1/2$ needs an even\ndenominator. Separately,\n$\\mathsf{CertifiedGreedyMersenneDeath}$ is a finite-depth certificate\nof $x\\notin\\mathcal A$, obtained by decidable tests on rational input.\nFor example, the supplied source excludes $3/4$ at level $1$ with\nlookahead $0$.\nFailure to find such a certificate at a given depth does not establish\nmembership.  It records survival of that finite test, not survival\nat every depth.\n\n\\textbf{Scope.} Any support representing $1/2$, if it exists, must\nbe infinite.  This follows from denominator parity, independently\nof the certificate search.  The search supplies a different fact:\na successful exclusion certificate proves nonmembership, whereas\nits absence in a bounded search does not prove membership or\nrationality.  The same distinction must be checked separately for\nany other certificate family.\n\\end{prop}\n\n\\begin{prop}[Period exclusion for the totient series]\n\\label{prop:carry-survivor-extinction}\nFor $\\sum_n\\varphi(n)/2^n$, the cited finite test proves that\n$\\mathrm{totientTail}(N+h)-\\mathrm{totientTail}(N)$ is not an integer\nby excluding every possible integer state in a bounded range within\nfinitely many steps.  If the series were rational, some positive\nperiod $h_0$ would make these tail differences integral for every\nsufficiently large $N$.  Telescoping would then give integrality\nalso for every positive multiple $mh_0$.\n\nConsequently, the required certificate supply is: for every $h_0\\ge1$\nand every lower bound $N_0$, there exist $m\\ge1$, $N\\ge N_0$ and a\nfinite test length $K$ excluding integrality for $(mh_0,N)$.  The\nunbounded choice of $N$ is essential; a counterexample before the\nunknown eventual threshold does not exclude an eventual period.\nIt is sufficient instead to obtain such certificates for\n$h=\\mathrm{lcm}(1,\\ldots,t)$ with both $t$ and $N$ exceeding arbitrary\nprescribed lower bounds.\nThe supplied finite theorem excludes the tail differences for\n$1\\le h\\le16$ at $(N,L)=(14,9)$, and hence excludes rational values\nwhose reduced denominator divides $2^{14}(2^h-1)$ for one of those\n$h$.  It does not establish the unbounded certificate supply.\n\n\\textbf{Scope.} The Lean declarations used in this comparison\nconcern the totient series of Problem~249.  Their carry recurrences\nmay suggest constructions for Problem~257, but applying them there\nrequires a new argument for the divisor transform of its chosen\nsupport.  The totient-specific theorems do not supply that argument.\n\\end{prop}\n\n\\subsection{The scalar-localisation height obstruction}\n\n\\begin{lem}[Denominator complement survives scaling]\n\\label{lem:scalar-localization}\nFor $x:\\mathbb Q$, $c:\\mathbb Z$, $H:\\mathbb N$: if $H\\mid x.\\mathrm{den}$ and\n$(c\\cdot x).\\mathrm{den}\\mid H$ ;  i.e.\\ multiplying by the integer $c$ shrinks the displayed\ndenominator down \\emph{into} $H$ ;  then the \\textbf{complementary} denominator factor\n$x.\\mathrm{den}/H$ divides $c$:\n\\[\n  H\\mid x.\\mathrm{den} \\ \\land\\ (c\\cdot x).\\mathrm{den}\\mid H \\implies x.\\mathrm{den}/H \\mid |c|.\n\\]\nEquivalently, writing $D=x.\\mathrm{den}/H$, we have $D\\mid c$,\nso $t=c/D$ is an integer and $Hcx=t\\,x.\\mathrm{num}$.\nHere $H>0$ follows from $H\\mid x.\\mathrm{den}$, since the\nreduced denominator is positive.\n\n\\textbf{Scope.} This is elementary rational arithmetic, independent\nof the problem.  If $c\\ne0$, divisibility gives the size bound\n$|c|\\ge x.\\mathrm{den}/H$.  The nonzero condition is essential:\n$c=0$ clears every denominator and satisfies the divisibility\nconclusion without any positive lower bound on $|c|$.  The lemma\ntherefore constrains denominator clearing by a nonzero bounded\nmultiplier, not arbitrary multiplication.\n\\textbf{Object or representation:} about a\n\\emph{representation} ;  pure denominator bookkeeping for $\\mathbb Q$, with zero problem-specific\ncontent; the upstream primitive\nis the domain-neutral extraction this lemma builds on.\n\\end{lem}\n\n\\begin{cor}[Mersenne specialisation]\n\\label{cor:mersenne-height}\nLet $x$ be a positive rational number and let $r\\ge0$, $n\\ge1$\nbe integers.  If $2^r\\mid x.\\mathrm{num}.\\mathrm{natAbs}$ and\n$x<2/(2^n-1)$, then\n$2^r\\cdot(2^n-1) < 2\\cdot x.\\mathrm{den}$ ;  a numerator $2$-power lower bound plus a Mersenne-scale\nupper bound on $x$ together force a denominator lower bound, \\emph{without} introducing a global\nprefix LCM.\nTo see the inequality directly, write $x=a/b$ in lowest terms\nwith $a,b>0$.  The hypotheses give $a\\ge2^r$ and\n$a(2^n-1)<2b$.  The conclusion follows by substitution.\nAn application of Lemma~\\ref{lem:scalar-localization} must in\naddition supply its denominator-divisibility assumptions.\n\\end{cor}\n\n\\subsection{The final-skip band formula, and what it does not show}\n\n\\begin{defn}[Upper-reset dyadic band escape]\n\\label{defn:band-escape}\nAt each upper-reset row $d\\ge13$, write\n$E_d=4\\,\\mathrm{overshoot}(d)+\\mathrm{abovePulse}(d)$ for the reset\nexpression in the preceding transition formulas.  The condition is\n\\[\n \\text{for every }0\\le j\\le d,\\qquad\n E_d>2^{d-j+1}\\quad\\text{or}\\quad\n E_d+2(d+j)\\le2^{d-j+1}.\n\\]\nThus $E_d$ must avoid the interval\n$(2^{d-j+1}-2(d+j),\\,2^{d-j+1}]$ at every indicated scale.\n\\lean{Erdos249257.SeamUpperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:4566}.\nIn words: at every upper-reset row $d\\ge13$, the reset charge must avoid a linear-width forbidden\nband immediately below every dyadic threshold $2^{d-j+1}$, for every $j\\le d$ simultaneously.\nGranted this hypothesis,\n\\lean{Erdos249257.half\\_mem\\_mersenneAchievementSet\\_of\\_upperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:4790}\ngives $1/2\\in\\mathcal A$ outright.\n\\end{defn}\n\n\\begin{prop}[Quantifier collapse: one critical index suffices, not $d{+}1$]\n\\label{prop:critical-band-index}\nLet $d,E\\in\\mathbb N$ and assume $E\\le2^{d+1}$. Then the purely combinatorial statement\n$\\mathsf{DyadicBandEscape}(d,E) \\iff \\exists j,\\ \\mathsf{CriticalDyadicBandIndex}(d,E,j)\\land\nE+2(d+j)\\le 2^{d-j+1}$ collapses the $\\forall j\\in[0,d]$ band-avoidance condition (formally $d{+}1$\nseparate inequalities) to checking exactly \\emph{one} nearest-boundary index $j$.\nThe range assumption ensures that a dyadic threshold lies at or above $E$.\nChoose the smallest such threshold: smaller thresholds are already below\n$E$, while larger thresholds have narrower forbidden bands. Without the\nrange assumption all bands escape automatically when $E>2^{d+1}$, but no\ncritical index exists; $(d,E)=(0,3)$ is the smallest example.\nSpecialised to the\nconcrete seam reset charge, $\\mathsf{SeamUpperResetCriticalBandEscape}$ is proved logically\n\\emph{equivalent} to Definition~\\ref{defn:band-escape}'s hypothesis.\nZero Mersenne/seam content in the core lemma ;  pure $(d,E,j)$ arithmetic over powers of 2.\n\\end{prop}\n\n\\begin{obs}[A verified finite range of reset indices]\n\\label{cert:band-through-31}\nThe linked certificate proves the condition in\nDefinition~\\ref{defn:band-escape} for each reset index $13\\le d\\le30$.\nIt uses successor remainders at rows $14$--$31$, including\n$\\mathrm{rem}(14)=392$ and $\\mathrm{rem}(31)=4{,}187{,}487{,}147$.\nThus the last remainder row is $31$, but the last reset index covered\nby this theorem is $30$.\n\\lean{Erdos249257.seamUpperResetDyadicBandEscape\\_through\\_thirty}{HalfCylinderUpperResetBandCertificates.lean:78}.\n\\end{obs}\n\n\\begin{rem}[What this does \\emph{not} show]\n\\label{rem:band-caveat}\nDefinition~\\ref{defn:band-escape} is a $\\forall d\\ge13$ statement; Observation~\\ref{cert:band-through-31}\nis a finite verification in the cited Lean source for $13\\le d\\le30$. \\textbf{The\nfinite certificate does not establish the universal hypothesis.} Nothing here proves the band\ncondition for all $d\\ge13$; it is the unproved hypothesis of\nTheorem~\\ref{thm:upper-reset-band}, stated here in its own form.\n\n\\par\\medskip\n\nThe finite theorem supplies no reset index beyond $30$. It therefore\ndoes not prove that the orbit avoids the band at unboundedly many reset\nindices, let alone at every reset. The universal band condition remains\nan unproved sufficient hypothesis.\n\n\\par\\medskip\n\nFor the branch convention of Theorem~\\ref{thm:dynamics}, direct\ninteger recomputation at reset-test indices $6\\le r\\le2500$\ngives $605$ middle and $604$ upper resets, hence $1209$ resets.\nThere are $1286$ right transitions.  The earlier reported count\n$1264$ does not agree with this convention.  The recomputation\nchecks the sign inequalities at these resets, the square-root\nbound at resets $r\\ge10$, and the displayed band condition at\nupper resets $r\\ge13$, with no failures.  This remains a finite\nordinary computation, distinct from the cited Lean theorem for\n$13\\le r\\le30$ and from an assertion at every later reset.\n\\end{rem}\n% ; - part a257_invent ; -\n\\section{Unproved inputs for the recorded implications}\n\\label{sec:invent-257}\nThe following estimates would supply inputs to particular conditional\narguments. We compare each with the bounds and finite calculations\nalready available, retaining its quantifiers and evidence basis. The\ncomparisons leave both rational targets and the universal problem open,\nand do not supply a single obstruction common to all approaches.\n\nFor example, the first two weighted divisor counts after $M=3$\ngive $(\\tau(4)-1)/2+(\\tau(5)-1)/4=5/4$.\nThe finite weighted divisor sum below records the same calculation\nfor any starting index and length.  Its role in particular universal-support\ncriteria is discussed separately; it is not a proved common necessary\nobstruction for the two problems.\n\n\\begin{defn}[The short-window divisor phase]\n\\label{defn:theta}\nFor $M\\ge 1$ and $L\\ge 1$ put\n\\[\n\\Theta_L(M)\\ :=\\ \\sum_{i=1}^{L}\\bigl(\\tau(M+i)-1\\bigr)\\,2^{-i}\\ \\in\\ \\Q_{\\ge 0},\n\\]\nwhere $\\tau$ is the number-of-divisors function, so that $\\tau(n)-1=\\#\\{d\\ge 2: d\\mid n\\}$ counts\nthe divisors of $n$ other than $1$.\n\\end{defn}\n\nIteration of the integer recurrence\n$K(M+1)=2K(M)-(\\tau(M+1)-1)$ gives the exact identity\n\\[\n \\Theta_L(M)=K(M)-2^{-L}K(M+L)\\qquad(M\\ge2,\\ L\\ge1).\n\\]\nConsequently $2^L\\Theta_L(M)$ is an integer, and\n\\[\n \\operatorname{dist}(\\Theta_L(M),\\mathbb Z)\n =2^{-L}\\operatorname{dist}(K(M+L),2^L\\mathbb Z).\n\\]\nA nonzero distance is at least $2^{-L}$, but nonvanishing itself\nrequires an argument.  For instance $\\Theta_1(3)=1$ has distance\nzero.  The identity concerns $K$; the quotient deviation also\ncontains the skip-set correction in\nTheorem~\\ref{thm:master-identity}.  No positive lower bound for\nthis phase is asserted to be necessary for the universal support\nproblem.\n\n\\subsection{Reset bounds and finite weighted divisor sums}\n\\label{sec:invent-R1}\n\n\\subsubsection*{The statement needed}\n\nThe first two conditions below each have a conditional implication to\nhalf-membership.  No implication between them is asserted.  The sign\ncondition is weaker than a quantitative margin in its conclusion,\nbut its relation to either membership hypothesis must still be\nproved.\n\n\\emph{Form (i), the reset-deviation bound.} Write\n$w_s=\\mathrm{rem}(s)-2^s$.  The actual hypothesis is\n\\[\n \\forall r\\ge10\\ \\text{that are upper or middle resets}:\\qquad\n |w_{r+1}|>2^{(r+5)/2}.\n\\]\nEquivalently, $|\\lambda_{r+1}|>2^{(3-r)/2}$ for\n$\\lambda_{r+1}=w_{r+1}/2^{r+1}$.  This normalisation concerns\ndistance of the deviation from zero, not distance of an unrelated\nphase from the nearest integer.\nThe run-length estimate $L_r<(r-3)/2$ for resets $r\\ge31$ is a\nproposed comparison target.  Its exact relation to the displayed\nbound requires the pulse terms and the initial range $10\\le r<31$;\nit is not stated as an equivalence here.  A sublinear estimate gives\nan eventual ceiling, not automatically the required finite initial\ncases.  An exact computation\n(\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/seam-orbit-200000.json}{saved run})\ngives a maximum run of length\n$19$ through row $200{,}000$, following the reset at row $158{,}096$.\nThat observation is finite evidence, not an asymptotic theorem.\n\n\\emph{Form (ii), band avoidance, already proved to imply membership.}  For every\nupper reset $d\\ge13$ and every $j\\le d$,\n\\[\n2^{\\,d-j+1} < E_d\\quad\\lor\\quad E_d + 2(d+j)\\ \\le\\ 2^{\\,d-j+1},\n\\qquad\nE_d := 4\\cdot\\mathrm{overshoot}_d+\\mathrm{abovePulse}_d\n\\]\n(\\lean{SeamUpperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:4566}), with the\nconditional implication to membership\n\\lean{half_mem_mersenneAchievementSet_of_upperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:4790}.\nThe $\\forall j\\le d$ band check collapses to a single nearest-boundary index per row by\n\\lean{dyadicBandEscape_iff_exists_critical}{HalfUpperResetCriticalBand.lean:108}, and rows\n$13\\le d\\le 30$ are verified in Lean by\n\\lean{seamUpperResetDyadicBandEscape_through_thirty}{HalfCylinderUpperResetBandCertificates.lean:78}.\n\n\\emph{Form (iii), the sign of the reset deviation.}\n\\[\n\\forall\\ \\text{reset rows } r:\\qquad\nr \\text{ is an } M\\text{-reset}\\Rightarrow w_{r+1}>0,\n\\qquad\nr \\text{ is a } U\\text{-reset}\\Rightarrow w_{r+1}<0 .\n\\]\nA finite recomputation (\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/seam-reset-crossing-2500.json}{saved run}) verifies these signs\nat all $1209$ resets in rows $6$--$2500$, with $605$ middle and $604$ upper resets.\nThe upper sign follows directly from the exact branch identity below.\nThe middle sign remains an unproved assertion about the orbit; neither\nsign alone gives the required lower bound on the magnitude.\n\n\\subsubsection*{Divisor residues and the changing skip set}\n\nExpanding each divisor count separates its residue conditions.\nThis is a finite counting identity.  It does not prescribe whether\nan eventual proof should be analytic, combinatorial, or use another\nmethod.\n\n\\begin{lem}[Divisor-residue form of the short-window phase]\n\\label{lem:odometer}\nFor all $M,L\\ge 1$,\n\\[\n\\Theta_L(M)\\ =\\ \\sum_{d\\ge 2}\\ \\sum_{\\substack{1\\le i\\le L\\\\ i\\,\\equiv\\,-M\\ (\\mathrm{mod}\\ d)}} 2^{-i}\n\\ =\\ \\sum_{d\\ge 2}\\ 2^{-i_d(M)}\\cdot\\frac{1-2^{-d\\,m_d}}{1-2^{-d}},\n\\]\nwhere $i_d(M)\\in[1,d]$ is the least $i\\ge1$ with $d\\mid M+i$ (so $i_d(M)$ depends only on\n$M \\bmod d$), $m_d := \\#\\{1\\le i\\le L: d\\mid M+i\\}$, and terms with $i_d(M)>L$ are empty. In\nparticular, for every integer $D\\ge M+L$,\n\\[\n\\Theta_L(M)\\ =\\ \\sum_{d=2}^{D}\\ \\sum_{i=1}^{L} 2^{-i}\\,\\mathbf 1_{d\\mid M+i},\n\\]\nand the right-hand side, with an arbitrary integer $x\\ge0$ in place of $M$, depends on $x$ only\nthrough its residues modulo the integers $d$ with $2\\le d\\le D$. The cutoff $D$ has to be fixed\nbefore the argument varies: at $(M,L)=(1,1)$ the cutoff is $M+L=2$, the integers $1$ and $3$ have\nthe same residue modulo every $d$ with $2\\le d\\le 2$, and yet $\\Theta_1(1)=1/2$ while\n$\\Theta_1(3)=1$.\n\\end{lem}\n\\begin{proof}\n$\\tau(n)-1=\\sum_{d\\ge2}[\\,d\\mid n\\,]$; exchange the order of summation and sum the geometric\nprogression $i_d, i_d+d, i_d+2d,\\dots$ inside $[1,L]$. No $d>M+L$ divides any $M+i$ with\n$1\\le i\\le L$, which gives the cutoff form, and each indicator $\\mathbf 1_{d\\mid x+i}$ depends\nonly on $x \\bmod d$.\n\\end{proof}\nThe identity was also checked by exact rational computation at\n$(M,L)\\in\\{1000,10001,123456\\}\\times\\{12,20\\}$.\n\nA finite divisor cutoff makes the averaging question precise.  For\nintegers $D\\ge2$ and $L\\ge1$, define\n\\[\n \\Psi_{L,D}(x)=\\sum_{d=2}^{D}\\sum_{i=1}^{L}2^{-i}\n                  \\mathbf1_{d\\mid x+i},\\qquad\n Q_D=\\operatorname{lcm}(1,\\ldots,D).\n\\]\nThis is the cutoff sum of Lemma~\\ref{lem:odometer}.  It is periodic in $x$\nwith period $Q_D$, and\n$\\Theta_L(M)=\\Psi_{L,D}(M)$ whenever $M\\ge1$ and $D\\ge M+L$.\nAveraging each indicator over a complete period gives\n\\[\n \\frac1{Q_D}\\sum_{x=0}^{Q_D-1}\\Psi_{L,D}(x)\n   =(1-2^{-L})\\sum_{d=2}^{D}\\frac1d.\n\\]\nThe mean diverges as $D\\to\\infty$, even for fixed $L$.\nThus this family does not converge to a finite integrable observable\nwhose limiting mean could simply be used in an equidistribution\nargument.\n\nThe same issue arises if one uses the profinite integers\n$\\widehat{\\mathbb Z}$, whose Haar probability measure is uniform\non each finite residue quotient.  The finite-cutoff functions are\ncontinuous there.  Their increasing limit is infinite almost\neverywhere: the events $p\\mid x+1$ are independent over primes by\nthe Chinese remainder theorem, have probabilities $1/p$, and occur\ninfinitely often almost surely.  Each contributes $1/2$ to the\nterm at offset $1$.  The positive integer inputs $M$, for which\n$\\Theta_L(M)$ is finite, do not justify treating this divergent\nlimit as a continuous or integrable function.  Only the finite\nperiod average above is used in the present argument.\n\nThere is a related residue expression in\nTheorem~\\ref{thm:real-form}, but its index and divisor set differ:\n\\[\n \\sum_{d\\in D_n}\\left\\{\\frac{4^n}{2^d-1}\\right\\}\n =\\sum_{d\\in D_n}\\frac{2^{-i_d(2n)}}{1-2^{-d}}.\n\\]\nIndeed $i_d(2n)=d-(2n\\bmod d)$, with value $d$ when the\nremainder is zero.  The argument is $2n$, not $n$, because the\nquotient scale is $4^n=2^{2n}$.  For example, at $n=7,d=3$ the\nfractional part is $4/7$; using $i_3(7)$ would instead give $2/7$.\nThis sum uses the selected finite set $D_n$, whereas $\\Theta_L$\nuses all divisors with offsets truncated at $L$.  Their common\ngeometric summand does not make the two sums equal.\n\nTo keep the changing support visible, put\n$S_n=\\sum_{d\\in\\mathrm{Skip}_n}\\lfloor4^n/(2^d-1)\\rfloor$.\nThe master identity $\\Delta_n=K(2n)+S_n$ and the recurrence for $K$\ngive, for $n\\ge6$ and $\\ell\\ge1$,\n\\[\n 2^{-2\\ell}\\Delta_{n+\\ell}\n =\\Delta_n-\\Theta_{2\\ell}(2n)\n       +2^{-2\\ell}S_{n+\\ell}-S_n.\n\\]\nA phase estimate would have to control this additional term and\nidentify the required reset rows before implying the reset bound.\nNo such implication is proved here.  The scales also differ from\na logarithmic-window assertion: $\\ell$ quotient transitions use\n$2\\ell$ binary steps starting at $M=2n$.  If $\\ell$ is proportional\nto $n$, the phase length is comparable to $M$, not to $\\log M$.\nA theorem for logarithmic windows would need a separate argument\nconnecting its range to the reset condition.\n\nFinite residue expansions, divisor-sum estimates and probabilistic\nlocal estimates are possible sources of further information.  None\nis an input already supplied here.  An application must state the\ndivisor cutoff, offset length, error at the required precision, and\ncontrol on the changing skip-set sum, rather than appeal to the\nname of a method or to equidistribution of a different statistic.\n\n\\subsubsection*{The specific obstacle}\n\n\\textbf{What the cited short-interval theorems control.}  Matom\\\"aki and\nRadziwi\\l{}\\l{} prove that a real-valued multiplicative function bounded by $1$ has, in almost all\nwindows $[x,x+h]$ with $h\\to\\infty$, essentially its long\naverage~\\cite[Theorem~1, pp.~1--2]{matomakiradziwill}.  Mangerel extends such bounds to a class of\ndivisor-bounded multiplicative\nfunctions~\\cite[Theorem~1.7, p.~7, and Corollary~1.8, pp.~7--8]{mangerel2021}, and Sun describes the\nalmost-all transition profile of the $k$-fold divisor function at its critical polylogarithmic\nwindow~\\cite[Theorem~1.1 and Corollary~1.2, p.~3]{sun2026}.  These results control short-interval averages.  They do not by\nthemselves supply a reset-deviation estimate or its unproved\nweighted-phase substitute, for two reasons.\n\nFirst, their conclusions hold for almost all $x$, whereas the proposed\napplication needs every reset row beyond its threshold.  An\nexceptional set of density zero may still contain infinitely many\nrequired indices.  No density assertion about the reset set, or\navoidance of the exceptional set by that orbit, is proved here.\n\nSecond, $\\tau$ is not bounded by $1$, and the relevant statistic is the distance to $\\Z$ of a\n\\emph{dyadically weighted} sum of $\\tau$-values at a precision shrinking exponentially in the window\nparameter.  The first term $(\\tau(M+1)-1)/2$ has fixed weight\n$1/2$, independent of the window length; an unweighted average\nover that window does not by itself control this term.  An application through the phase would need both an exceptional-set\nargument along the specified indices and control of the weighted\nsum.  It would still have to justify the passage from that sum to\nthe reset deviation, including the skip-set correction.\n\n\\textbf{The required statistic is different.} A central limit theorem\nfor prime-factor counts, or an almost-all theorem for unweighted\ninterval averages, is not by itself a bound for the distance to an\ninteger of this dyadically weighted divisor sum.  An application must\nstate its precision, its dependence on the growing window length,\nand its exceptional set.  The references cited above do not supply\nthat orbitwise estimate in the form used here.  This is a limitation\nof the proposed application, not a survey claim that every relevant\nlocal estimate is unavailable.\n\nThe countermodels in the record impose similarly specific\nlimitations.  They concern fixed-precision residues\n(\\lean{affineBinaryOrbit_mod_twoPow_eq}{GenericTailOrbitRigidity.lean:307}),\na decoder without fresh arithmetic input\n(\\lean{balancedPulse_no_autonomous_decoder}{GenericTailOrbitRigidity.lean:247}),\nand fixed-precision signatures\n(\\lean{fixedPrecisionTropicalNoGo}{TropicalCurvatureCarry.lean:137}).\nThey do not partition all possible proof methods.  Together with the\ndenominator estimate, they explain why the particular shortcuts\nexamined here are insufficient; they do not prove that a\nweighted-window method is necessary or impossible.\n\n\\subsubsection*{Possible estimates to prove}\n\n\\emph{A conditional bound along a right-branch run.}\n\nLet $r\\ge6$, $L\\ge1$, and suppose that the transitions at rows\n$r,\\ldots,r+L-1$ all take the right branch.  Put\n$w_s=\\mathrm{rem}(s)-2^s$ and let $p_s$ be the lower-prefix\npulse, so $0\\le p_s\\le2(s-2)$.  Iterating the exact recurrence\n$w_{s+1}=4w_s-4-p_s$ gives\n\\[\n w_{r+L}=4^Lw_r-\n       \\sum_{j=0}^{L-1}4^{L-1-j}(4+p_{r+j}).\n\\]\nThe reverse triangle inequality and the finite geometric sum give\n\\[\n |w_{r+L}|\n \\ge4^L\\left(|w_r|-\\frac{4+2(r+L)}3\\right).\n\\]\nThis estimate alone does not force a right-branch run to end.\nFor that conclusion one also needs an upper bound on its terminal\ndeviation.  More precisely, if $C>0$ and\n\\[\n |w_{r+L}|\\le C2^{r+L},\\qquad\n |w_r|>\\frac{4+2(r+L)}3,\n\\]\nthen division by $4^L$ and taking logarithms give\n\\[\n L\\le r+\\log_2 C-\n \\log_2\\left(|w_r|-\\frac{4+2(r+L)}3\\right).\n\\]\nEvery term and hypothesis is explicit here.  A separate window\nestimate must supply the displayed terminal upper bound; it does\nnot follow from the right-branch recurrence alone.  This explains\nthe possible connection between the initial deviation and the\nrun length without asserting a sharp threshold, a run-length\nequivalence, or an unproved bound for all later rows.  This conditional\ncalculation is an ordinary argument.\n\n\\emph{An exact finite phase experiment.}\n\nFor the $200{,}000$ integers $10^6\\le M<1.2\\times10^6$, an\ninteger divisor sieve through $1{,}200{,}039$ computes\n$2^{40}\\Theta_{40}(M)$ exactly.  Reducing that integer modulo\n$2^{40}$ gives the following counts and empirical proportions for\n$\\operatorname{dist}(\\Theta_{40}(M),\\mathbb Z)<t$.\n\\begin{center}\n\\begin{tabular}{lccccc}\n\\hline\n$t$ & $2^{-6}$ & $2^{-8}$ & $2^{-10}$ & $2^{-12}$ & $2^{-14}$\\\\\n\\hline\nCount & $7083$ & $1858$ & $491$ & $125$ & $37$\\\\\nProportion & $0.035415$ & $0.009290$ & $0.002455$ & $0.000625$ & $0.000185$\\\\\nProportion$/t$ & $2.267$ & $2.378$ & $2.514$ & $2.560$ & $3.031$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\nThe experiment samples consecutive integers, not reset rows.  Its median distance\nis $0.24470815549\\ldots$, compared with $1/4$ for a uniform\nfractional part.  A uniform fractional part has probability $2t$\nof distance less than $t$, for $0<t\\le1/2$, which explains the\ncomparison in the last row.  Agreement of a finite histogram with\nthat model is not a distribution theorem.\n\nTruncation matters at the smallest tested thresholds.  With the\nsame starting indices, the counts below $2^{-14}$ are $22,36,37,37$\nfor $L=16,24,32,40$, respectively.  Thus similar central quantiles\ndo not justify ignoring the truncation error in a rare-event test.\nMore generally, for $L'\\ge L\\ge1$ and $B\\ge0$, if\n$\\tau(M+i)-1\\le B$ over $L<i\\le L'$, then\n\\[\n 0\\le\\Theta_{L'}(M)-\\Theta_L(M)\n \\le B(2^{-L}-2^{-L'})\\le B2^{-L}.\n\\]\nDistance to the nearest integer is $1$-Lipschitz.  A threshold\ncomparison is therefore guaranteed to remain unchanged when\nthe distance at the shorter cutoff is farther than this error\nfrom the threshold.  This is a sufficient stability test, not a\nnecessary one.\nThe estimate uses a bound over the specified finite range, not an\nunproved uniform bound on the divisor function.\n\nA probabilistic application requires further assumptions.  For\nexample, if events $E_r$ in a specified probability space satisfied\n$\\mathbb P(E_r)\\le C2^{(3-r)/2}$ for every $r\\ge R$, the union\nbound would give\n\\[\n \\mathbb P\\left(\\bigcup_{r\\ge R}E_r\\right)\n \\le \\frac{C2^{(3-R)/2}}{1-2^{-1/2}}.\n\\]\nSummability would also imply only finitely many failures almost\nsurely, not zero failures at every index.  Neither the finite\nhistogram nor its fit to a uniform-phase model proves these\nprobability bounds, identifies their events with reset failures,\nor defines a probability for the actual deterministic orbit.\nIn particular, no membership conclusion is drawn from small\nexpected counts computed in such a model.  The counts are\nexact finite computations; the error and probability inequalities are\nordinary arguments; an application to resets is unproved.\n\n\\emph{The reset signs and the exact branch identities.}\nWrite $p_s$ for the nonnegative lower-prefix pulse and\n$w_s=\\mathrm{rem}(s)-2^s$.  On a middle transition, where the upper\ncarry does not occur, the update is\n\\[\n \\mathrm{rem}(s+1)=4\\mathrm{rem}(s)+2^{s+1}-p_s,\n \\qquad\n w_{s+1}=4w_s+2^{s+2}-p_s=4\\mathrm{rem}(s)-p_s.\n\\]\nThe natural-number subtraction can be read as signed subtraction\nbecause $p_s\\le2(s-2)<2^{s+1}$ for $s\\ge5$.\nThe middle-branch condition is\n$4\\mathrm{rem}(s)<2^{s+1}+4+p_s$, together with failure of the\nupper carry.  Thus a positive middle-reset deviation is exactly the\nadditional inequality $4\\mathrm{rem}(s)>p_s$.\nThe pulse bound and branch inequality alone do not prove it:\n$s=5$, $\\mathrm{rem}=1$, $p=5$ satisfy both numerical bounds but give\n$4\\mathrm{rem}-p=-1$.  This is a counterexample to those numerical\npremises, not a claim that the concrete orbit reaches that state.\n\nOn an upper transition, let $o_s>0$ be the overshoot and $p_s^+\\ge0$\nthe upper-prefix pulse.  The carry condition pays the subtraction,\nso\n\\[\n w_{s+1}=-(4o_s+p_s^+)<0.\n\\]\nThis sign already follows from the exact upper-branch identity;\nonly the middle sign needs further orbit information.  On a right\ntransition the corresponding formula is\n$w_{s+1}=4w_s-4-p_s$.\nThese identities separate the\nsign calculation from the unproved magnitude estimate needed for square-root escape.  At the\nreported middle step $s=14$, for example,\n$\\mathrm{rem}(14)=392$, $p_{14}=3$ and\n$\\mathrm{rem}(15)=34333$, giving $w_{15}=1565$.\n\n\\subsection{Progress to larger endpoint depths}\n\\label{sec:invent-R2}\n\n\\subsubsection*{The statement needed}\n\nThe exact-row transition\n\\lean{exactLocalMersenneHalfRow_double_or_recycle}{BooleanMobiusExactRowDichotomy.lean:26}\nsays every exact row $n\\ge6$ either doubles to $2n-1$ or recycles to $2c-2$ for some\n$4\\le c\\le n$.  Exact rows at unbounded depths, which give $1/2\\in\\mathcal A$, need the recycle\nbranch not to return to a bounded endpoint forever. Three sufficient forms:\n\\[\n\\text{(a)}\\quad\n\\text{critical quotient supply}:\\ \\forall D\\subseteq[2,c),\\ 4\\le c,\\\nv(D)<\\tfrac12\\ \\wedge\\ \\tfrac12-v(D)<\\ensuremath{w}\\,c\n\\]\n\\[\n\\Longrightarrow\\quad 2^{\\,(2c-2)-1}\\ \\le\\ \\ensuremath{Q}\\,(\\{c\\}\\cup D)\\,(2c-2),\n\\]\nwhere $v(D):=\\ensuremath{V}\\ D$\n(\\lean{SkippedCoreCriticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:30});\n\\[\n\\text{(b)}\\quad\n\\text{the crossing rank } c \\text{ produced by the recycle branch does not repeat a bounded value infinitely often};\n\\]\n\\[\n\\text{(c)}\\quad\nc\\ge g(n),\\qquad 2g(n)-2>n\n\\quad\\text{at every reachable recycle input }n.\n\\]\n\nUnboundedness of $g$ alone would not suffice in (c).  For example,\n$g(n)=\\lfloor n/2\\rfloor+1$ tends to infinity but allows the fixed\ntransition $n=6$, $c=4$, $2c-2=6$.  The displayed strict-progress\ninequality excludes such cycles; the doubling branch already\nincreases every endpoint $n\\ge6$.\n\n\\subsubsection*{What kind of object would furnish it}\n\nThe real and integer greedy rules must be distinguished here.\nA real straddling word agrees with the real greedy prefix\n(\\lean{IsStraddlePrefix.half_agrees_greedy}{HalfCutLocator.lean:442}).\nAn exact local row instead satisfies\n$\\ensuremath{Q}\\ D\\ n=2^{n-1}-1$.  Although floor division is not\ninjective in general, the quotient weights at fixed $n$ have a gap of\nat least $1$ between distinct subset sums.  A witness is therefore\nunique when it exists and is recovered by the integer greedy test\n(Theorem~\\ref{record:257bm-i11a} and\nPropositions~\\ref{record:257bm-i11b}--\\ref{record:257bm-i11c}).\nThere is no exponential family of witnesses to choose from at that\ndepth.  What remains to prove is that successful depths are unbounded.\n\nOne possible additional ingredient is an integer-valued function\n$\\Phi(n,D)\\ge0$ on reachable exact rows that increases by at least one\nunder each permitted transition and is bounded on rows of bounded\nendpoint.  Along an infinite trajectory, $\\Phi$ would then tend to\ninfinity, forcing the endpoints to be unbounded.  Strict increase only\non recycle steps would need a separate check of what doubling does to\n$\\Phi$; an unspecified real-valued increasing quantity need not tend\nto infinity.\nThis describes one possible proof mechanism, not a necessary class of\nall proofs. The record already has the sharp\ncapacity test in exact form: by\n\\lean{localBinarySuffix_two_mul_sub_two_lt_criticalCapacity_iff}{BooleanMobiusSkippedCoreCriticalCapacity.lean:22},\nform (a) is exactly the statement that the $c-2$-bit suffix capacity\n$\\ensuremath{S}\\,D\\,1\\,(2c-2)<2^{\\,c-2}$ is met.\n\n\\subsubsection*{The specific obstacle}\n\nThe formal counterexample concerns exactly the transition schema:\nthat schema \\emph{plus} a seed provably does not give cofinality. The theorem\n\\lean{exists_seeded_bounded_double_or_recycle_model}{BooleanMobiusExactRowDichotomy.lean:91}\nexhibits the one-point predicate $P(n):=(n=6)$, which satisfies the seed\n(\\lean{exactLocalMersenneHalfRow_six}{BooleanMobiusExactRowSeed.lean:34}, the explicit witness\n$\\{2,3,6\\}$) and the exact shape of the transition, and is not cofinal. So no argument that uses\nonly the dichotomy and the base case can work, however cleverly the induction is arranged. Any\nproof must use arithmetic content of exact rows that the abstract\nschema does not see.\n\nThere is a second, more specific trap, and it is also formalised. The natural sufficient\ncondition via fractional split mass is \\emph{not necessary}: the example at $D=\\{2,3\\}$, $c=5$\n(\\lean{splitFractionMass_one_bound_not_necessary_fixture}{BooleanMobiusSkippedCoreCriticalCapacity.lean:183})\nexhibits a real crossing core violating it. Thus the fractional-mass bound is not necessary for an individual\ncrossing core.  This counterexample does not rule out proving that\nsufficient bound on a more restricted family.\n\n\\subsubsection*{Possible estimates to prove}\n\n\\emph{The uniquely determined prefix at a crossing.} The identification\nbelow is proved; the required capacity bound remains open.\n\nForm (a) is universally quantified over cores $D$, which looks like an exponential obligation.\nIt is not. Its hypotheses are exactly those of\n\\lean{eq_halfGreedyPrefixSupport_of_critical_crossing}{BooleanMobiusCriticalCapacityCofinal.lean:50},\nwhich forces $D=G\\cap\\{1,\\ldots,c-1\\}$. So the $\\forall D$ collapses\nto a single orbit and form (a) is a statement to be checked along \\emph{one} sequence indexed by\n$c$, namely\n\\[\n\\forall c\\ge4\\ \\text{skipped by the real half-greedy orbit}:\\qquad\n\\ensuremath{S}\\bigl(G\\cap\\{1,\\ldots,c-1\\},\\,1,\\,2c-2\\bigr)\\ <\\ 2^{\\,c-2}.\n\\]\nThe crossing hypothesis has not disappeared: these are the ranks\nwhere the real greedy rule skips the next term.  Each resulting\nfinite inequality is decidable.  Computing its margin can identify\nexamples to explain a proposed estimate, but a finite table neither\nselects a proof method nor establishes a cofinal supply.\n\nThe progress-function criterion above remains unproved.  Endpoint\ndoubling is not a proved doubling law for the largest selected\nexponent, and an unbounded function of the incoming endpoint alone\ndoes not rule out a bounded cycle.\n\n\\subsection{Prime support and the limits of one formal argument}\n\\label{sec:invent-R3}\n\nThe base-$2$ statement\n\\[\n\\mathrm{Irrational}\\Bigl(\\sum_{p\\ {\\rm prime}}\\frac1{2^p-1}\\Bigr)\n\\]\nis known.  Expanding geometrically and collecting divisors gives\n\\[\n \\sum_{p\\ {\\rm prime}}\\frac1{2^p-1}\n =\\sum_{n\\ge1}\\frac{\\omega(n)}{2^n},\n\\]\nand Tao--Ter\\\"av\\\"ainen prove the latter series irrational in Theorem~1.3\nof\n\\href{https://arxiv.org/abs/2512.01739}{\\emph{Quantitative correlations and\nsome problems on prime factors of consecutive integers}}.  This is a cited\nanalytic result, not a theorem formalised in Lean.\n\nThe existing pairwise-coprime Lean theorem does not recover it because that\ntheorem assumes $\\sum_{a\\in A}1/a<\\infty$, whereas $\\sum_p1/p$ diverges.  The\nblock-certificate criterion behind it would also demand, in the prime case,\n\\[\n  2^r\\mid\\omega(N+r)\\qquad(1\\le r\\le K),\n\\]\na simultaneous exact divisibility pattern far stronger than the input used\nby Tao--Ter\\\"av\\\"ainen.  Small computational witnesses for $K=2,3$ diagnose\nthat the condition is not vacuous, but they neither prove the known theorem\nnor define a new open case.\n\nThus the contribution of this analysis is negative and specific to\nthe method: it identifies why the present formal certificate\ncriterion fails to reproduce a theorem already available by quantitative\ncorrelation estimates.  The earlier proposal to prove the prime-support\nirrationality as new mathematics is withdrawn.  Extensions to other bases\nor prime-power supports are mentioned by the cited authors with proof\nmodifications omitted; this paper promotes only the fully written base-$2$\ntheorem.\n\n\\subsection{Equivalent carry conditions at perfect-square depths}\n\\label{sec:invent-R4}\n\n\\subsubsection*{The statement needed}\n\nThe following five conditions are equivalent to $1/2\\in\\mathcal A$:\n\\begin{enumerate}[label=(\\arabic*)]\n\\item $C_G(N)\\le2\\sqrt N+4$ for every $N\\ge0$.\n\\item The set $\\mathbb N_{>0}\\smallsetminus G$ is infinite.\n\\item For arbitrarily large $n$ there is\n$D\\subseteq\\{2,\\ldots,n\\}$ with $Q(D,n)=2^{n-1}-1$.\n\\item For arbitrarily large $M$,\n$\\operatorname{ihc}(G,M)\\le B(M+1)$.\n\\item For arbitrarily large $M\\ge1$ there is\n$D\\subseteq\\{2,\\ldots,M\\}$ with\n$|\\operatorname{ihc}(D,M-1)|\\le B(M)$.\n\\end{enumerate}\nHere $G$ is the actual greedy support for $1/2$.\nThe forward directions of (2)--(5) are proved in Lean\n(\\lean{half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:97},\n\\lean{half_mem_mersenneAchievementSet_of_cofinalExactLocalRows}{BooleanMobiusCofinalExactRows.lean:71},\n\\lean{half_mem_mersenneAchievementSet_of_cofinalStripReturn}{CofinalStripReturn.lean:157},\n\\lean{half_mem_mersenneAchievementSet_of_cofinalTerminalOnlyStrip}{TerminalOnlyCofinal.lean:134}). The reverse implications are ordinary deductions in\nthis paper.  The perfect-square argument below supplies the terminal\nbounds without changing the constant in the original statements.\n\n\\subsubsection*{The carry bound at perfect-square depths}\n\n\\begin{lem}[The terminal bound at square depths]\n\\label{lem:sqwitness}\nLet $A\\subseteq\\N$ with $1\\notin A$ and $X_{A}(2)=1/2$. Then for every\n$k\\ge1$,\n\\[\n\\bigl(\\ensuremath{\\operatorname{ihc}}\\ A\\ (k^{2}-1)\\ :\\ \\R\\bigr)\n\\ =\\ \\mathtt{binaryCoeffTail}\\ (c_{A})\\ (k^{2})\n\\ \\le\\ 2k+4\\ =\\ \\ensuremath{B}\\ (k^{2}).\n\\]\n\\end{lem}\n\\begin{proof}\nBy \\lean{integerHalfCarry_eq_scaled_residual_add_tail}{HalfCarryReachability.lean:871},\n\\[\n\\bigl(\\ensuremath{\\operatorname{ihc}}\\ A\\ N:\\R\\bigr)\n= 2^{\\,N+1}\\Bigl(\\tfrac12-X_{A}(2)\\Bigr)\n+\\sum_{j\\ge1}c_{A}(N+1+j)2^{-j},\n\\]\nand the first term vanishes by hypothesis. Take $N=k^{2}-1$, so $N+1=k^{2}$. By\n\\lean{binaryCoeffTail_supportCoeff_le_two_sqrt_add_four}{BooleanMobiusCarry.lean:290}\n(unconditional) the tail at $k^2$ is at most $2\\sqrt{k^{2}}+4=2k+4$, and by\n\\lean{halfStripBound}{HalfCarryReachability.lean:32}, $\\ensuremath{B}\\ n\n= 2\\lfloor\\sqrt n\\rfloor+4$, which at $n=k^{2}$ equals $2k+4$ exactly. Non-negativity of the tail\n(\\lean{binaryCoeffTail_nonneg}{GenericTailOrbitRigidity.lean:78}) gives the lower side.\n\\end{proof}\n\nThe difference between the real and the integer square root disappears when cofinal witness\ndepths are restricted to perfect squares: at $n=k^{2}$ the two roots coincide\nexactly and no widening is needed. Applying Lemma~\\ref{lem:sqwitness} along $M=k^{2}-1$ gives (4)\ndirectly; taking $a$ to be the indicator of an achieving support truncated to\n$\\{0,\\ldots,M\\}$ at $M=k^{2}$, and using\n\\lean{integerHalfCarry_inter_Iic_eq_of_succ_le}{HalfCarryReachability.lean:49} together with\n$1\\notin A$ (forced because $\\ensuremath{w}\\,1=1>1/2$), gives the witness required by\n\\lean{HalfTerminalOnlyStripWitness}{TerminalOnlyCofinal.lean:24} and hence (5).\n\n\\subsubsection*{What the equivalences establish}\n\nEach biconditional is a theorem about the exact logical content of a\nproposed hypothesis.  It is also a reformulation of the membership\nquestion.  These are compatible descriptions, not competing evidence\nclasses. The equivalence identifies the missing content but does\nnot show that the reformulation cannot expose a useful invariant. In particular, (3) shows that the existence of finite witnesses at\nunbounded depths, even without compatibility between those witnesses,\nis equivalent to membership.  The absence of a compatibility requirement\nmay still be useful in constructing witnesses; equivalence alone says\nnothing about which formulation is easier to prove.\n\n\\textbf{Scope of the equivalences.} (1) does \\emph{not} extend to every\n$A$ with $1\\notin A$: without\n\\lean{erdosSupportSeries_greedyMersenneSupport_le}{HalfCarryReachability.lean:888} (which supplies\n$\\delta:=1/2-S_G\\ge0$ for the greedy support specifically) one obtains only the one-sided\nequivalence of the bound in (1), with $A$ in place of $G$, and $1/2-S_A\\le0$. The mechanism in all five cases is the same and\nshould be recorded once: $2^{\\,N+1}\\delta$ dominates any $\\sqrt{N}$ envelope, so a strip\nhypothesis forces $\\delta=0$; and $\\delta=0$ makes the unconditional tail bound supply the strip.\nMembership is then read off by\n\\lean{mem_mersenneAchievementSet_iff_greedy_survival}{GreedyAchievementSet.lean:1458}.\n\n\\subsection{Sparse and dense supports under the certificate criterion}\n\\label{sec:invent-R5}\n\n\\subsubsection*{The statement needed}\n\n\\[\n \\begin{gathered}\n A\\subseteq\\mathbb N_{>0},\\quad A\\text{ infinite}\\quad\\Longrightarrow\\\\\n \\text{$c_A$ has arbitrarily late super-logarithmic zero windows}\\\\\n \\text{or the specified block certificates exist at every precision.}\n \\end{gathered}\n\\]\nFor a fixed nonempty $A$, let $a_0=\\min A$.  Every interval of $a_0$\nconsecutive integers contains a multiple of $a_0$, so $c_A$ has no zero\nwindow longer than $a_0-1$.  The first alternative is therefore\nimpossible, independently of rationality.  The cited logarithmic result\n\\lean{supportCoeffZeroWindow_length_le_eps_logb}{SublogDivisorCoverage.lean:435}\ndoes not create a sparse subclass here.  The separate lcm-gap theorem\n\\lean{irrational_erdosSum_of_lcm_gap}{CertificateKernel.lean:5883}\nremains a valid sufficient criterion, but does not prove this proposed\nalternative.\nThe proposal would consequently require the following certificates for\nevery infinite support:\n\\[\n\\exists N,K,L,C,\\quad K\\le L,\\quad\n\\forall r\\in[1,K]:\\ 2^{r}\\mid c_{A}(N+r),\n\\]\n\\[\n\\sum_{r=K+1}^{L}c_{A}(N+r)\\,2^{\\,L-r}\\le C,\n\\qquad\nq\\,(C+N+L+2)<2^{L}.\n\\]\n\n\\subsubsection*{What kind of object would furnish it}\n\nThe identity $c_A=\\mathbf1_A*\\mathbf1$ is a Dirichlet convolution,\nand M\\\"obius inversion gives $\\mathbf1_A=\\mu*c_A$.  These identities\nhold for every support, but do not supply the displayed divisibility\nor tail estimates.  Shared factors restrict which residue prescriptions\ncan be realised; they do not make the Chinese remainder theorem\ninapplicable.  For positive moduli $m_i$, a finite system\n$N\\equiv a_i\\pmod{m_i}$ is solvable precisely when\n\\[\n a_i\\equiv a_j\\pmod{\\gcd(m_i,m_j)}\\qquad\\text{for every }i,j.\n\\]\nNecessity follows by subtraction.  For sufficiency, keep the strongest\nprime-power modulus for each prime; compatibility makes all weaker\nprescriptions agree with it, and the ordinary CRT combines the\nresulting coprime prime powers~\\cite[Theorem~3.1, pp.~2--3]{conradcrt}.\nFor example, $N\\equiv1\\pmod4$ and $N\\equiv1\\pmod6$ give\n$N\\equiv1\\pmod{12}$, whereas $4\\mid N$ and $2\\nmid N$ are\nincompatible.  A block-certificate construction must satisfy these\ncompatibility conditions and its separate tail budget.  Neither\nM\\\"obius inversion nor CRT alone provides both inputs.\n\n\\subsubsection*{The specific obstacle, proved}\n\nThe dense branch as stated is \\emph{false}: an explicit, natural, divisor-dense support refutes\nit.  This also corrects an earlier claim that universal \\#257 reduces with no loss to the\nexistence of certificates for every infinite support.\n\n\\begin{prop}[Squarefree support: a limit of the certificate method, not an open value]\n\\label{prop:squarefree}\nLet $A=\\{n\\ge2:n\\text{ is squarefree}\\}$.  Then\n\\[\n  c_{A}(n)=2^{\\omega(n)}-1,\n\\]\nwhich is odd for every $n\\ge2$.  Consequently neither the digitwise nor the\ncarry-aware divisibility-first block-certificate schema has an instance at\nany even base.\n\\end{prop}\n\nThis does not leave the squarefree value open.  Duverney--Tachiya's\nCorollary~1.2 and Example~1.1 prove that, for every $h\\ge1$,\n\\[\n  1,\\quad\n  \\sum_{n\\ {\\rm squarefree}}\\frac1{2^n-1},\\quad\\ldots,\\quad\n  \\sum_{n\\ {\\rm squarefree}}\\frac1{2^{hn}-1}\n\\]\nare linearly independent over $\\Q$.  After deleting the rational $n=1$\nterm, squarefree support is therefore irrational at every base $2^j$.\nAmong the integer bases, the cited theorem leaves the non-powers of two\nuncovered.  This is cited mathematics, not formalised here.  Proposition~\\ref{prop:squarefree}\nis a failure of two proof schemas on a value already known to be irrational.\n\nThe dependence on the choice of support is exact.  Adjoin $1$ to the support.  The series\nchanges by the rational number $(b-1)^{-1}$, so its irrationality is\nunchanged, while the incidence becomes $2^{\\omega(n)}$:\n\\lean{ErdosProblems.Erdos257.supportCoeff_squarefreeShiftedSupport}{ErdosProblems/Erdos257/SquarefreeSupportIncidence.lean:314}\nand\n\\lean{ErdosProblems.Erdos257.irrational_erdosSupportSeries_squarefreeSupport_iff_shifted}{ErdosProblems/Erdos257/SquarefreeSupportIncidence.lean:335}.\nAt base $2$, CRT then supplies arbitrarily long opening blocks with\n$\\omega(N+r)\\ge r$:\n\\lean{ErdosProblems.Erdos257.exists_omega_ge_block}{ErdosProblems/Erdos257/SquarefreeSupportIncidence.lean:441}\nand\n\\lean{ErdosProblems.Erdos257.exists_digitwise_block_squarefreeShiftedSupport}{ErdosProblems/Erdos257/SquarefreeSupportIncidence.lean:485}.\nThe middle-window and height conditions remain; no certificate or new\nirrationality proof follows.  The durable conclusion is methodological:\ntest a no-go statement under rational finite changes before attributing it\nto the underlying value.\n\n\\subsection{What the experiments and obstructions do not imply}\n\\label{sec:invent-shape}\n\nEverything in this subsection is inference from the recorded failures, not a theorem.  Where\nthe evidence does not determine the answer, the text says so.\n\n\\subsubsection*{Finite observations and their limits}\nThe recorded obstructions concern specified representations.  Fixed\nprecision, a bounded carry state and a prescribed suffix can lose\ninformation needed by the actual-selector condition.  They do not show\nthat the underlying deterministic sequence has no arithmetic structure.\nThe relevant exact statements remain\n\\lean{fixedPrecisionTropicalNoGo}{TropicalCurvatureCarry.lean:137},\n\\lean{balancedPulse_no_autonomous_decoder}{GenericTailOrbitRigidity.lean:247}\nand\n\\lean{affineBinaryOrbit_mod_twoPow_eq}{GenericTailOrbitRigidity.lean:307}.\nTheir hypotheses, rather than the fact that an attempted argument failed,\ndetermine their scope.\n\nThe inherited experiment log reports $200{,}000$ seam rows with counts\n$R:M:U=100197:49899:49898$, which a\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/seam-orbit-200000.json}{saved run}\nreproduces, a roughly geometric run-length histogram, and a\nlongest equal run of $15$ in $40{,}000$ binary digits.  It also reports\nagreement with OEIS A211706 and a near-integer phase constant of about\n$2.5$.  The other figures and the external digit comparison come from\nthat log alone.  They are descriptive reports, not certificates of\nnormality, independence, a random sampling model, or an infinite\nanti-concentration estimate.  A statement about ``sampling noise'' would\nrequire a specified probabilistic model that is not supplied here.\n\nThe exceptional row $r=7$ and the crossing $(s,d)=(10,7)$ are likewise\nstatements about the tested ranges only: reset rows $6\\le r\\le200{,}000$\nand rows $s\\le2500$ (\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos257/receipts/seam-reset-crossing-2500.json}{saved run}).  No claim that this is\nthe only exceptional row of the infinite orbit is made.\n\nThese representations all descend from the same greedy carry;\nthey are not independent tests of a common random hypothesis.  Strict\ntail domination gives an injective coding, but says neither that its\ndigits are random nor that averaging over indices is impossible.  The\nmeasure-one and nowhere-dense properties of the achievement set do not\ndecide membership of a specified rational.  The correct research question\nis whether a precise arithmetic estimate can be proved for the actual\nselector, not whether finite data resemble a random model.\n\n\\subsubsection*{Why the support and rational-target problems remain distinct}\n\nProposition~\\ref{prop:squarefree} is a reason to separate the two halves\nrather than assimilate them.  On the universal side the present\ncertificate method first meets a parity obstruction that depends on the choice of support.\nThe squarefree value is nevertheless already known to be irrational at\npower-of-two bases, and adjoining $1$ removes the opening-block obstruction\nwithout changing irrationality.  Thus the obstruction concerns this method; it\ndoes not concern the value or the universal problem.\n\nShort-window near-integer anti-concentration is one possible missing\ninput for the particular mechanisms analysed here.  No equivalence shows\nthat it is necessary for either problem, and no argument identifies it as\na common necessary obstruction to both.  The squarefree example shows why\nfinite rational changes of support must be checked before an opening-block\nfailure is interpreted as a property of the value.\n\n\\subsubsection*{What the evidence does and does not determine}\n\nIt does not determine whether a short proof exists: nothing in a catalogue of failed methods\nbounds the length of a successful one.\n\nIt records limitations of specified methods.  The fixed-precision\ncompletion theorem and the balanced-pulse countermodel concern the\nparticular summaries defined above, not all fixed-precision arithmetic\nor all finite-state methods with additional arithmetic input. An\nirrationality-measure bound on $E$ does not classify a specified rational in $\\mathcal{A}$.\nInstantiating the present block-certificate method at a general support is blocked, on the\nuniversal side, by Proposition~\\ref{prop:squarefree}. Survival through any tested finite depth does\nnot by itself establish membership; a fatal-gap certificate would establish non-membership. These\nfacts do not decide rational-target membership, and they do not yield an impossibility theorem for\nall probabilistic, measure-theoretic or averaging methods.\n% ; - part a257_p5 ; -\n\\clearpage\n\\part*{V. Outstanding obligations and provenance}\n\\addcontentsline{toc}{part}{V. Outstanding obligations and provenance}\n\\section{What is open, stated exactly}\nThe support theorems and finite arithmetic leave three target questions:\nthe universal base-two assertion, half-membership and $1/21$-membership.\nWe collect their quantified forms here. Some are equivalent formulations,\nsome are sufficient conditions, and some have only finite computed\ninstances. The table in \\S\\ref{sec:v-table} identifies the equivalences;\nnone of them supplies the missing membership argument.\n\n\\subsection{Notation fixed for this section}\n\nThe base-$2$ support series is\n$X_{A}(2) := \\sum_{n\\in A} 1/(2^n-1)$\n(\\lean{erdosSupportSeries}{CertificateKernel.lean:8897}). $c_{A}(n) :=\n|\\{d\\mid n : d\\in A\\}|$ (\\lean{supportCoeff}{CertificateKernel.lean:8856}) is the divisor-indicator\ncoefficient whose generating tail enters every certificate below.\n$\\mathcal{A}=\\{X_A(2):A\\subseteq\\mathbb N_{\\ge1}\\}$\n(\\lean{mersenneAchievementSet}{GreedyAchievementSet.lean:571}).\n$G$ is the greedy support for target $1/2$;\n$\\mathrm{rem}(s)$ is its integer seam\nremainder at row $s$, and a \\emph{reset} at row $r$ is a row where the greedy orbit returns to the\nnear-$2^r$ regime after a run of takes or skips.\n\n\\subsection{The universal statement}\n\n\\begin{defn}[Universal \\#257]\n\\[\n\\mathrm{U257} \\;:=\\; \\forall\\, A\\subseteq\\mathbb{N}_{\\ge 1},\\ A\\text{ infinite} \\;\\Longrightarrow\\;\n\\mathrm{Irrational}\\!\\left(X_{A}(2)\\right).\n\\]\n\\end{defn}\nThis is the literal statement of Erd\\H{o}s \\#257; it is open.  The sufficient condition used by\nthe certificate method of this record is\n\\[\n\\mathrm{Cert}(A)\\ :=\\ \\forall q>0,\\ \\exists N,K,L,C,\\ K\\le L,\\\n\\left(\\forall r\\in[1,K],\\ 2^r\\mid c_{A}(N+r)\\right)\\wedge{}\n\\]\n\\[\n\\sum_{r=K+1}^{L}c_{A}(N+r)\\cdot 2^{L-r}\\le C\\ \\wedge\\\n\\exists t,\\ 0<c_{A}(N+L+1+t)\\ \\wedge\\ q(C+N+L+2)<2^L,\n\\]\nwith $\\mathrm{Cert}(A)\\Rightarrow \\mathrm{Irrational}(X_{A}(2))$\nalready proved unconditionally for \\emph{every} $A$\n(\\lean{irrational_erdosSupportSeries_of_weighted_coeff_certificates}{CertificateKernel.lean:9031};\nthe theorem is universally quantified over $A$, and only its hypothesis is missing).  Therefore the stronger statement\n\\[\n\\mathrm{O1'}\\ :=\\ \\forall A,\\ A\\text{ infinite}\\Longrightarrow \\mathrm{Cert}(A).\n\\]\nwould imply $\\mathrm{U257}$, but the reduction is \\emph{not} lossless and\n$\\mathrm{O1'}$ is actually false.  For\n$A=\\{n\\ge2:n\\text{ squarefree}\\}$ the incidence is\n$2^{\\omega(n)}-1$, hence odd for every $n\\ge2$, and the digitwise\ncertificate schema has no instance at any even base\n(\\lean{ErdosProblems.Erdos257.not_exists_digitwise_certificates_squarefreeSupport}{ErdosProblems/Erdos257/SquarefreeSupportIncidence.lean:292}).\nThis is a limit of the method, not a counterexample to irrationality:\nDuverney--Tachiya prove the squarefree value irrational at every\npower-of-two base.  Likewise the pairwise-coprime theorem does not cover the\nprimes because $\\sum_p1/p$ diverges, even though Tao--Ter\\\"av\\\"ainen prove\nthe base-$2$ prime-support value irrational.  The universal target is\n$\\mathrm{U257}$ itself; $\\mathrm{O1'}$ is a rejected sufficient\nstrengthening, not an open equivalent normal form.\n\n\\subsection{Whether one half is represented}\n\n\\begin{defn}[Half membership]\n\\[\n\\mathrm{HALF}\\ :=\\ \\tfrac12\\in\\mathcal{A}.\n\\]\n\\end{defn}\n$\\mathrm{HALF}$ is a \\emph{single instance} of the negation of $\\mathrm{U257}$: any witness\n$A$ with $0\\notin A$, $X_{A}(2)=1/2$ is infinite\n(\\lean{finite_boolSupport_ne_half}{HalfCarryReachability.lean:589};\nfinite supports have odd-denominator value, so can never equal a dyadic rational), hence a\ngenuine counterexample to $\\mathrm{U257}$. Thus $\\mathrm{HALF}\\Rightarrow\\neg\\mathrm{U257}$.  The reverse\nimplication is not established: failure of half-membership would not decide $\\mathrm{U257}$ if false, since $\\mathrm{U257}$ could still fail at a\ndifferent target $t\\in\\mathbb{Q}$.  $\\mathrm{HALF}$ is open.  It is logically equivalent, by an\nunconditional Lean theorem, to a purely combinatorial statement about one fixed orbit:\n\\[\n\\mathrm{HALF}\\ \\Longleftrightarrow\\ \\text{the set of ranks skipped by the greedy rule for }\\tfrac12\\text{ is infinite}\n\\]\n(\\lean{half_mem_mersenneAchievementSet_iff_greedySkippedSupport_infinite}{GreedyAchievementSet.lean:2583};\nthe reverse direction holds for every target, and the forward direction uses only that the target\nis rational, so the same theorem holds verbatim for every rational $t\\ge 0$ in place of $1/2$).\n\n\\subsubsection{Conditions equivalent to half-membership}\n\nThe following three conditions are equivalent to half-membership.\nThey express the same unresolved assertion in different terms.  The\nproofs of equivalence identify the content still to be established;\nthey do not compare the difficulty of possible proofs.\n\n\\begin{prop}[Infinitely many positive skips are equivalent to half-membership]\n\\label{prop:cpgs-equiv}\nDefine\n\\[\n\\mathrm{CPGS}\\ :=\\ \\forall N,\\ \\exists c\\ge N,\\ c\\ \\text{skipped by the rational half-greedy\norbit}\\ \\wedge\\ 0<\\ensuremath{r}(1/2)(c-1)\n\\]\nThe positivity\nconjunct is unconditionally true for every skipped $c$\n(an odd-denominator parity fact), hence deletable without changing\nthe Prop's meaning. What survives, $\\forall N\\,\\exists c\\ge N$ skipped, is exactly\n$(\\mathtt{greedyMersenneSkippedSupport}(1/2)).\\mathrm{Infinite}$, so\n\\[\n\\mathrm{CPGS}\\ \\Longleftrightarrow\\ \\mathrm{HALF}.\n\\]\n\\end{prop}\nThe implication\n\\lean{cofinalExactLocalMersenneHalfRows_of_positiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:84}\ntherefore gives an equivalent formulation, not a proof of membership.\n\n\\begin{prop}[Terminal carry bounds are equivalent to half-membership]\n\\label{prop:strip-equiv}\nConsider finite sets $D\\subseteq\\{2,\\ldots,M\\}$ at arbitrarily large\ndepths $M$ with\n\\[\n |\\operatorname{ihc}(D,M-1)|\\le2\\lfloor\\sqrt M\\rfloor+4.\n\\]\nThis is the exact terminal-strip condition.\nIt is equivalent to $1/2\\in\\mathcal A$: the forward direction follows\nfrom the displayed finite-approximation estimate; for the converse,\ntruncate an achieving support at $M=k^2$ and apply\nLemma~\\ref{lem:sqrt-witness}.  Thus constant $4$ already suffices at\ncofinally many depths.  The relaxed constant $6$ gives a bound at every\ndepth, using $\\sqrt M\\le\\lfloor\\sqrt M\\rfloor+1$, but that relaxation is\nnot needed for the cofinal statement.\n\\end{prop}\nThe existence condition is therefore not logically weaker than half-membership.\nAn equivalent formulation may still be useful for finding a proof.\n\n\\begin{obs}\nPropositions~\\ref{prop:cpgs-equiv} and~\\ref{prop:strip-equiv} do not decide\nhalf-membership.  They show that the proposed existence conditions are\nequivalent to it, not that those formulations have no possible use.  The\nupper bound on the centred carry in\n\\lean{greedy_half_infinite_of_mobiusCenteredHalfCarry_sqrtBound}{HalfCarryReachability.lean:834}\nhas the same equivalence for the greedy support $A=G$.  For arbitrary\n$A$ with $1\\notin A$, an upper bound alone yields only\n$x_A\\ge1/2$, as the identity\n$\\operatorname{ihc}(A,N)=2^{N+1}(1/2-x_A)+\\mathrm T(N+1)$ shows.\nThe greedy inequality $x_G\\le1/2$ is what turns that conclusion into\nequality.\n\\end{obs}\n\n\\subsection{Exact finite sums at unbounded depths}\n\n\\begin{defn}\nFor every integer $N$, require some $n\\ge\\max\\{N,1\\}$ and\n$D\\subseteq\\{2,\\ldots,n\\}$ with\n\\[\n \\sum_{d\\in D}\\left\\lfloor\\frac{2^n}{2^d-1}\\right\\rfloor\n       =2^{n-1}-1.\n\\]\n(\\lean{CofinalExactLocalMersenneHalfRows}{BooleanMobiusCofinalExactRows.lean:38}).  This\ncondition is unproved.  Supports at different depths need not agree.\nThis is the condition in Definition~\\ref{record:257bm-d4};\nno strict weakening of half-membership is asserted.\n\\end{defn}\nThe implication from this condition to $\\mathrm{HALF}$ is proved unconditionally\nvia compactness of $\\mathcal{A}$ (a continuous image of the space of $0/1$ sequences under the\ndigit-coding map, hence closed)\n(\\lean{half_mem_mersenneAchievementSet_of_cofinalExactLocalRows}{BooleanMobiusCofinalExactRows.lean:71}).\nThe cited theorem already proves that a representing support exists,\nbecause that is what membership of $\\mathcal A$ means.  It need not\nchoose a support by a specified algorithm from the row witnesses.\nAn ordinary diagonal subsequence argument gives a compatible limiting\nsupport: successively fix each coordinate along an infinite\nsubsequence, then use the uniform tail bound to pass to its value.\nThis uses continuity of the digit map, also recorded in\n\\lean{continuous_positiveMersenneDigitValue}{GreedyAchievementSet.lean:628}.  The diagonal argument preserves $2\\in A$, since all sufficiently deep exact rows\ncontain exponent two\n(\\lean{two_mem_of_exact_localMersenneQuotient}{BooleanMobiusExactRowRankTwo.lean:23}).\nThis describes an ordinary existence proof; no additional\nformal extraction theorem or effective algorithm from purely\nexistential row data is claimed.\n\nProposition~\\ref{prop:collapsed-list}(c) already identifies\nthis condition as logically equivalent to $1/2\\in\\mathcal{A}$.\nEquivalence of a membership condition, constructive extraction of a coherent support, and a\nrejected sufficient strengthening are not interchangeable descriptions.  Absence of a Lean\nextraction theorem does not make this condition a strictly weaker substitute for $\\mathrm{HALF}$.\nThe remaining work here is to supply exact rows at unbounded depths, or any equivalent\nmembership condition.\n\n\\subsubsection{A capacity condition for exact sums at unbounded depths}\nThe finite construction uses a $(c-2)$-bit capacity test. Write\n\\[\n\\ensuremath{S}\\ D\\ k\\ M\\ :=\\ 2^{M-k}-\\ensuremath{Q}\\ D\\ M - 1\n\\]\n(\\lean{localBinarySuffix}{BooleanMobiusLocalRepair.lean:102}), with\nnatural-number subtraction understood as truncated at zero.  The loose\nbound below requires a critical crossing: $c\\ge4$,\n$D\\subseteq\\{2,\\ldots,c-1\\}$, and\n$0<1/2-V(D)<w_c$.  A merely below-half core is not enough:\n$D=\\varnothing$ gives $S(D,1,2c-2)=2^{2c-3}-1$.\nFor a critical-crossing core the two bounds are\n\\begin{align}\n&\\text{(loose, proved)}\\quad \\ensuremath{S}\\ D\\ 1\\ (2c-2)\\ <\\ 2^{c-1}\n\\tag{$\\ast$}\\\\\n&\\text{(sharp, needed)}\\quad \\ensuremath{S}\\ D\\ 1\\ (2c-2)\\ <\\ 2^{c-2}\n\\tag{$\\ast\\ast$}\n\\end{align}\nThe implication $(\\ast)$ from the critical-crossing hypotheses is\nproved uniformly in $c$\n(\\lean{localBinarySuffix_two_mul_sub_two_lt_upperHalfCapacity}{BooleanMobiusSkippedCoreExactRow.lean:123}).\n\n$(\\ast\\ast)$ is exactly one bit sharper and, via\n\\lean{localBinarySuffix_two_mul_sub_two_lt_criticalCapacity_iff}{BooleanMobiusSkippedCoreCriticalCapacity.lean:22}\n(under the stated core hypotheses), is equivalent to\n$Q(D\\cup\\{c\\},2c-2)\\ge2^{2c-3}$.\nHere $c$ is the added exponent; the quotient is evaluated at depth\n$2c-2$, not at depth $c$.\nThe proof of $(\\ast)$ gives the more informative additive bound\n$S<2^{c-2}+|D|$ before using $|D|\\le c-2$.\nOne possible cofinal supply is therefore the following exclusion\nof a specified integer interval:\n\\[\n\\begin{gathered}\n\\text{(capacity band)}\\qquad \\forall N\\ \\exists c\\ge\\max(N,4)\\ \\exists D\\subseteq\\{2,\\ldots,c-1\\},\\\\\n0<1/2-V(D)<w_c,\\qquad\nS(D,1,2c-2)\\notin[\\,2^{c-2},\\,2^{c-2}+c-3\\,].\n\\end{gathered}\n\\]\nThe excluded band contains $c-2$ integers.  The sharp capacity\nbound itself allows $2^{c-2}$ nonnegative integer values.\nThe numerical widths differ from those in Section~\\ref{sec:o4}, but the\nconditions concern different quantities.  Width alone does not prove a\nlogical implication between them.\n\nThe sharp-fill implication\n\\lean{exists_exactRowStrictUpperFill_of_skippedCoreSharpCapacity}{BooleanMobiusSkipRow.lean:238}\nassumes $(\\ast\\ast)$ for a below-half core\n$D\\subseteq\\{2,\\ldots,c-1\\}$, but does not require a critical\ncrossing or identify that core with a real greedy prefix.  This gives\nanother formulation in which to seek the required finite rows.\nIt does not establish a strict logical weakening of half-membership,\nnor does it supply cores satisfying the bound at unbounded ranks.\nMoreover, the integer gap separation fixes the resulting exact row\nwhenever it exists; the formulation does not create multiple exact\nrows at a fixed depth.\n\nThe complete induction from the endpoint-six seed to exact sums at unbounded depths is already\nformalised\n(\\lean{cofinalExactLocalMersenneHalfRows_of_criticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:1309});\nit uses the sharp capacity input, in its reduced canonical form\n\\[\n c\\ge4,\\ c\\notin G\\quad\\Longrightarrow\\quad\n Q\\bigl((G\\cap\\{2,\\ldots,c-1\\})\\cup\\{c\\},2c-2\\bigr)\n \\ge2^{2c-3}.\n\\]\n(\\lean{HalfGreedySkippedCriticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:178}),\nat essentially every step; this input is unproved.  This condition is required along the specified greedy prefixes.  That\nrestriction on the witnesses does not by itself prove a strict logical\nordering with the cofinal condition above.\n\n\\subsection{Reset bounds and runs of right transitions}\n\\label{sec:o4}\n\n\\begin{defn}[Reset $\\sqrt{\\text{-escape}}$ from row 10]\n\\[\n\\mathrm{SQRTESC}\\ :=\\ \\forall r\\ge10\\ \\text{that are resets},\\quad\n\\big|\\mathrm{rem}(r+1)-2^{r+1}\\big|>2^{(r+5)/2}\n\\qquad\\Big(\\text{equivalently } \\Delta_{r+1}^2 > 2^{r+5}\\Big).\n\\]\n\\end{defn}\nThe conditional implication uses the integer bound\n$B(r)=2^{\\lfloor(r+4)/2\\rfloor}+2r+3$ described in Theorem~\\ref{thm:lower-bound-every-reset}:\n$B(r)^2\\le2^{r+5}$ for $r\\ge10$, while a right branch at the\ncritical crossing would force $|\\Delta_{r+1}|\\le B(r)$.\nThus the actual implication chain is\n\\[\n \\mathrm{SQRTESC}\\ \\Longrightarrow\\\n \\text{the hypothesis of Theorem~\\ref{thm:largest-skip-late}}\\ \\Longrightarrow\\ \\mathrm{HALF}.\n\\]\nThe last implication is\n\\lean{half_mem_mersenneAchievementSet_of_largestSkipLateStepSocket}{HalfCylinderLargestSkipInduction.lean:167};\nthe first is proved in Theorem~\\ref{thm:lower-bound-every-reset}.\nThese are sufficient implications, not an equivalence with\nhalf-membership.\n\nFor comparison one may ask for\n\\[\n \\mathrm{RUNBOUND}:\\qquad\n \\forall\\,\\text{reset }r\\ge31,\\quad L_r<(r-3)/2,\n\\]\nwhere $L_r$ counts the consecutive right branches after reset $r$.\nAlong $L$ such steps the exact relation is\n\\[\n w_{r+1+L}=4^Lw_{r+1}\n   -\\sum_{j=0}^{L-1}4^{L-1-j}(4+p_{r+1+j}).\n\\]\nThe pulse sum and the threshold ranges cannot be dropped to claim\n$\\mathrm{RUNBOUND}\\Leftrightarrow\\mathrm{SQRTESC}$.\nThe pointwise run/deviation equivalence stated in earlier source\nnotes is not established by the bound used in the conditional proof.\nThe observation at $r=7$ (deviation $9$, square-root threshold $64$)\nis outside both proposed late ranges; that fact alone does not\njustify replacing the initial threshold $10$ by $31$.\n\n\\emph{Finite evidence.} Exact integer arithmetic to row $200{,}000$ shows that the\ntarget inequality holds at every reset row $6\\le r\\le200{,}000$ except $r=7$; the maximum observed R-run is $19$, after\nthe reset at row $158{,}096$, against a required threshold there of $\\approx79{,}046$, a margin of about\n$4000$ times.  This is a finite verified list; it supplies no witnesses at unbounded depths.\n\n\\begin{obs}[The sign of the reset deviation]\nAt all $1209$ observed resets, the sign of the deviation is perfectly predicted by branch type\n(M$\\Rightarrow$dev$>0$, U$\\Rightarrow$dev$<0$, zero exceptions), and the margin over the threshold\ngrows from $1.11$ bits at $r=14$ to $\\sim1246$ bits by row $2500$ (exact computation). Proving the sign\nlaw reduces to the middle inequality\n$4\\mathrm{rem}(r)>p_r$.  For an upper reset the sign already follows\nfrom $w_{r+1}=-(4o_r+p_r^+)<0$, with positive overshoot and\nnonnegative pulse.  The middle inequality remains unproved.\nEven after both signs are known, a separate lower bound for their\nmagnitudes is required for $\\mathrm{SQRTESC}$.\n\\end{obs}\n\n\\begin{obs}[Rigidity of dangerous resets]\n\\label{obs:dangerous-reset-rigidity}\nIf reset $r$ is \\emph{dangerous} ($|\\mathrm{rem}(r+1)-2^{r+1}|\\le2^{(r+5)/2}$) and the preceding\nreset $r_0$ has a pure R-run of length $L'=r-r_0-1>(r+5)/4$, then $w_{r_0+1}$ is confined to at\nmost two adjacent integers, determined by the divisor-pulse stream alone. A chain of $n$\nconsecutive dangerous resets with long runs forces $n-1$ nested exact congruence towers. This\nshows the failure mode is a tower of exact integer coincidences, not drift; it is an ordinary\nargument, and it does \\emph{not} by itself exclude the two pinned values; an isolated dangerous reset after a\nshort run is untouched by this theorem.\n\\end{obs}\n\n\\subsection{Three middle-cell exclusions and the right-branch bound}\n\n\\begin{defn}\nThe \\emph{two-sided cell condition} excludes three integer values of the middle coordinate,\n$4\\cdot\\mathrm{rem}(s)-\\mathrm{belowPulse}(s)-4\\notin\\{-3,-2,-1\\}$, plus one right-pulse-leak\nbound $4\\cdot\\mathrm{overshoot}(s)+\\mathrm{abovePulse}(s)\\le2^{s+2}$ under\n$\\mathrm{overshoot}(s)\\le2^s$ (\\lean{SeamTwoSidedDyadicCellEscape}{HalfCylinderMiddleCarryLowerBound.lean:4374}).\n\\end{defn}\nGranted this, induction from a base case at row $5$, checked by direct computation, propagates\n$\\forall s\\ge5,\\ \\mathrm{rem}(s)\\le2^s\\vee\\mathrm{overshoot}(s)\\le2^s$\n(\\lean{SeamTwoSidedDyadicCellEscape.twoSided}{HalfCylinderMiddleCarryLowerBound.lean:4448},\ngiven the hypothesis). The final-middle theorem excludes $-3$ at a\nmiddle row $D\\ge13$ followed by an all-right tail\n(\\lean{finalMiddleCell_neg_three_not_last}{HalfCylinderFinalMiddleCellEscape.lean:587}).\nIt does not supply the $-3$ exclusion on each eligible non-carry\nmiddle branch required by this induction. To complete this route, the\nthree exclusions must hold on those branches of the actual orbit,\nwithout an all-right-tail assumption, and the right-branch bound must\nhold under its own antecedents. Explicitly, at every row $s\\ge5$\nwhere the upper carry does not occur, that is,\n$4o_s+p_s^+>2^{s+1}$, the required implications are\n\\[\n \\begin{gathered}\n 4\\,\\mathrm{rem}(s)+2^{s+1}-p_s^-<2^{s+2}+4\n \\quad\\Longrightarrow\\quad C_s\\notin\\{-3,-2,-1\\};\\\\[3pt]\n 4\\,\\mathrm{rem}(s)+2^{s+1}-p_s^-\\ge2^{s+2}+4,\\quad o_s\\le2^s\n \\quad\\Longrightarrow\\quad 4o_s+p_s^+\\le2^{s+2}.\n \\end{gathered}\n\\]\nHere $C_s=4\\,\\mathrm{rem}(s)-p_s^- -4$, and $o_s,p_s^-,p_s^+$\nare the overshoot and correction terms of Theorem~\\ref{thm:dynamics}.\nThese conditions exclude specified integer values in one coordinate. The\nsquare-root reset condition instead bounds the magnitude of a\ndifferent quantity by a rank-dependent threshold.  Their numerical\nwidths do not order the two hypotheses, and no implication between\nthem is established here.\n\nA distinct sufficient condition concerns upper resets only.  For\nevery $d\\ge13$ at which the upper carry occurs, set\n$E_d=4\\mathrm{overshoot}(d)+\\mathrm{abovePulse}(d)$ and require,\nfor every $0\\le j\\le d$,\n\\[\n 2^{d-j+1}<E_d\\quad\\text{or}\\quad\n E_d+2(d+j)\\le2^{d-j+1}.\n\\]\nThis is the upper-reset band condition of Theorem~\\ref{thm:upper-reset-band}\n(\\lean{SeamUpperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:4566}).  It is\nunproved in general, and verified by exact finite computation for rows $13\\le d\\le30$\n(\\lean{seamUpperResetDyadicBandEscape_through_thirty}{HalfCylinderUpperResetBandCertificates.lean:78}).\nThis band has linear width $O(d)$, whereas the other bound uses a width\nof order $2^{d/2}$.  The quantities being bounded are different, so this\ncomparison of widths does not prove a logical implication or strict\nweakening.  The linear-width condition suffices for half-membership\n(\\lean{half_mem_mersenneAchievementSet_of_upperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:4790}).\nIt is proved logically equivalent to a single-critical-index check per row\n(\\lean{seamUpperResetCriticalBandEscape_iff}{HalfUpperResetCriticalBand.lean:883}), which is a genuine\nquantifier-complexity reduction, not a strength reduction.\n\n\\subsection{Whether one twenty-first is represented}\n\nThe integer-quotient proof at\n\\papersectionlink{erdos257-mersenne-reasoning-surface.pdf}{record257:twenty-one-quotients}\n{the end of the detailed finite results} gives the equivalence\n\\[\n \\frac1{21}\\in\\mathcal A\n \\quad\\Longleftrightarrow\\quad\n \\text{$\\mathcal F_{21}$ does not hold}\n\\]\n(\\lean{one\\_div\\_twenty\\_one\\_mem\\_iff\\_not\\_fatalAlignedBranch}{TwentyOneQuotientGreedy.lean:3507}).\nThe fatal branch $\\mathcal F_{21}$ means a finite fatal greedy gap followed\nby cofinite exact selection, eventual quotient/rational alignment, and\neventual visits to every doubling block.  If it occurs, the canonical\nquotient remainder is eventually strictly larger than $2^R$ and follows one\nexplicit affine recurrence:\n\\lean{twentyOneFatalAlignedBranch\\_eventually\\_strict\\_supercapacity}{TwentyOneQuotientGreedy.lean:5625}\nand\n\\lean{twentyOneFatalAlignedBranch\\_eventually\\_affine\\_supercapacity}{TwentyOneQuotientGreedy.lean:5658}.\nAny unbounded sequence of closed returns instead proves membership.\n\nNo finite support on ranks at least $2$ sums to $1/21$\n(\\lean{finiteErdosSum\\_ne\\_one\\_div\\_twenty\\_one}{ErdosProblems/Erdos257/HalfCounterexampleFrontier.lean:61}).\nThus membership would already be an infinite-support rational counterexample\nto $\\mathrm{U257}$.  The missing step is to exclude the fatal aligned\naffine-supercapacity branch (or force unbounded closed returns); neither is\nproved.\n\n\\subsection{Implication table}\n\\label{sec:v-table}\n\nRead left to right: $\\Rightarrow$ records a proved implication\nwith the hypotheses stated in the relevant section, and\n$\\Leftrightarrow$ records an equivalence.  Lean proofs and ordinary\narguments are distinguished as in the text.  ``No implication stated'' means that\nno comparison is established here.  Quantitative bounds are not ranked\nsolely by their widths.\n\n{\\small\n\\begin{longtable}{@{}p{0.25\\linewidth}p{0.43\\linewidth}p{0.23\\linewidth}@{}}\n\\toprule\n\\papertablehead\nTarget & Relation to its endpoint & Status \\\\\n\\midrule\nUniversal irrationality & Would exclude both rational targets; the stronger universal certificate condition is false & open \\\\\nHalf-membership & Would give a counterexample to universal irrationality & open \\\\\n$1/21$ membership & Equivalent to excluding the specified failure branch; would give a counterexample & open \\\\\nInfinitely many greedy skips (Proposition~\\ref{prop:cpgs-equiv}) & $\\Leftrightarrow$ half-membership & open, equivalent condition \\\\\nTerminal-only strip (Prop.~\\ref{prop:strip-equiv}) & $\\Leftrightarrow$ half-membership & open, equivalent condition \\\\\nThe centred square-root bound for $G$ & $\\Leftrightarrow$ half-membership & open, equivalent condition \\\\\nExact finite sums at unbounded depths & $\\Leftrightarrow$ half-membership (Prop.~\\ref{prop:collapsed-list}(c)); Lean proves $\\Rightarrow$ & open, equivalent condition \\\\\nLinear-width capacity band & $\\Rightarrow$ exact sums at unbounded depths (Lean, given a lemma) & open \\\\\nThe quotient condition on skipped greedy prefixes & $\\Rightarrow$ exact sums at unbounded depths (Lean) & open; no strict comparison asserted \\\\\nReset square-root bound & $\\Rightarrow$ half-membership (Lean); no run-length equivalence asserted & open \\\\\nThe reset sign condition & no implication to half-membership stated & open \\\\\nRigidity after long runs & partial rigidity, no exclusion & open \\\\\nThe remaining middle-cell exclusions and right-branch bound & $\\Rightarrow$ two-sided bound (Lean, given the hypothesis) & open \\\\\nThe upper-reset band condition & $\\Rightarrow$ half-membership (Lean); linear width & open; verified for $13\\le r\\le30$ \\\\\n\\bottomrule\n\\end{longtable}}\n\n\\subsection{Possible inputs and their scope}\n\nThe following comparisons identify missing hypotheses in the recorded\narguments.  They neither predict the length of a proof nor classify all\npossible methods.\n\n\\textbf{The universal statement.} The question is $\\mathrm{U257}$ itself. The stronger schema\n$\\mathrm{O1'}=\\forall A\\,\\mathrm{Cert}(A)$ is a rejected sufficient strengthening, not future work:\nProposition~\\ref{prop:squarefree} already exhibits a squarefree support on which neither the\ndigitwise nor the carry-aware block certificate exists, while Duverney--Tachiya still\nprove that squarefree value irrational. Supplying $\\mathrm{Cert}(A)$ for arbitrary infinite\nsupports is therefore not an open equivalent normal form of $\\mathrm{U257}$.  The remaining work\nis the universal irrationality statement, or a different method that covers supports the\ncertificate schemas cannot.\n\n\\textbf{Half-membership, exact sums and middle cells.} The sharp-capacity gap $(\\ast)\\to(\\ast\\ast)$ and the\n$-2,-1$ middle-cell exclusion are both, at bottom, requests for anti-concentration of an explicit\ndivisor-count quantity ($\\ensuremath{S}$, or $4\\cdot\\mathrm{rem}-\\mathrm{belowPulse}-4$)\naway from specified short integer windows.  A congruence argument,\na quantitative distribution estimate, or another method would have to\nsupply the stated inequalities.  The reformulations do not decide\nwhich kind of argument can do so.\n\n\\textbf{Reset separation.} The specified quotient-row argument\nneeds $|\\mathrm{rem}(r+1)-2^{r+1}|>2^{(r+5)/2}$ at its reset rows.\nA weighted divisor window is one proposed source of information.\nAfter division by $2^{r+1}$, the displayed condition becomes\n$|\\lambda_{r+1}|>2^{(3-r)/2}$.  A bound for\n$\\operatorname{dist}(\\Theta_L(M),\\mathbb Z)$ concerns a different\nquantity; Section~\\ref{sec:invent-R1} displays the additional\nskip-set term that a transfer argument would need to control.\nNo such implication or uniform estimate is supplied.\nPart~IV compares this with a totient-window hypothesis for a particular\nargument about Problem~249.  The analogy does not identify the two\nstatements, transfer a theorem between them, or show that either\nhypothesis is necessary for its original problem.\n\n\\subsection{Summary}\n\nThe results here do not decide the universal statement\n$\\mathrm{U257}$ or the two rational membership targets $1/2$ and\n$1/21$.  The conditions surveyed here have different logical roles.\nThose equivalent to half-membership, including exact sums at unbounded\ndepths by Proposition~\\ref{prop:collapsed-list}(c), do not establish\nmembership merely by reformulating it.  The sufficient conditions (the\nreset bound, the remaining middle-cell exclusions and the capacity band)\nhave not been established at unboundedly many indices, and\nno converse is proved for them.  The proposed universal certificate\nstrengthening $\\mathrm{O1'}$ is a separate, rejected statement: it is\nalready false on squarefree support.  Finite computational checks,\nhowever large their margins, are distinct from all these quantified\nassertions. The sufficient support theorems in the first part are proved for their\nstated classes.  The identities, finite checks and equivalences in this\nsection organise the remaining questions but do not resolve either\nrational target or the universal problem.\n\n\\clearpage\n\\section{Logarithmic cost under arithmetic sampling}\\label{sec:257-logarithmic-sampling}\n\\papersectiontarget{record257:logarithmic-sampling}\nA logarithmic divisor majorant is too weak to give the uniform\narithmetic-progression estimate suggested by the fractional-cover proof.\nWe construct a counterexample already visible on complete periods, so\nan improved incomplete-period error cannot remove it. We then prove an\ninitial-interval estimate. This distinguishes the available logarithmic\nbound from a converse to the short paper's divisor-cube estimates.\n\nThe ordinary proofs originate in an AI-assisted note of 17 September\n2026. The margin gives the three corresponding formal results and\ntheir separate comparison records. The ordinary proofs have not had\nindependent human review.\nFor the underlying Lambert subseries, Kova\\v{c}--Tao treat the same fixed-base\nobject in~\\cite[Section~2.1.2 and Remark~4.1]{kovactao}; their strict-tail\nargument gives distinct support coding and Cantor topology.\nVan Doorn--Kova\\v{c} treat a freely chosen denominator sequence.\nTheir convention is $n_{i+1}/n_i\\ge\\lambda$ at every index.\nFor each $1<\\lambda<2$, they construct such a sequence whose finite\nreciprocal sums include every rational in $[0,2]$; no sequence with\n$n_{i+1}/n_i\\ge2$ can do this on any nonempty open\ninterval~\\cite[Theorem~1, Lemma~4 and Corollary~5]{vandoornkovac}.\nThis is a statement about the rational points of an interval,\nnot an interval consisting entirely of finite sums; there are\nonly countably many finite subsets.\nThose results provide context, not an arithmetic logarithmic estimate or a\nsolution of fixed-base rational membership.  The proof dependencies below\nare positive divisor majorisation, elementary prime selection, the Chinese\nremainder theorem, and the original kernel estimate\n\\eqref{eq:257-mixed-finite-kernel}.  The short paper's logarithmic lower bound\nand its bound for $F(q,P)$ concern the same fractional-cover cost; their\nstructured upper bound is not a uniform logarithmic converse.\n\n\\subsection{The two finite costs}\nLet $F$ be a finite nonempty subset of $\\mathbb N_{>0}$, and write\n\\[\n Q=\\operatorname{lcm}(F),\\qquad f_F(n)=\\#\\{a\\in F:a\\mid n\\},\\qquad\n U_F(N)=\\sum_{r\\ge1}2^{-r}f_F(N+r)\n       =\\sum_{a\\in F}\\frac{2^{N\\bmod a}}{2^a-1}.\n\\]\nFor $0<t\\le1$ define, with natural logarithms,\n\\begin{equation}\n \\kappa_1(F;t)=\\min\\left\\{\\sum_{d\\mid Q}\\frac{c_d}{d}:\n c_d\\ge0,\\quad\n \\log(1+f_F(s)/t)\\le\\sum_{d\\mid s}c_d\\quad(s\\mid Q)\\right\\}.\n \\label{eq:257-logarithmic-cost}\n\\end{equation}\nThe constraints also hold at every positive integer $n$, by replacing $n$\nwith $\\gcd(n,Q)$.  The programme is finite and feasible; a bounded cost\nsublevel bounds each coefficient, so its minimum is attained.\nFor $L\\ge1$, let $\\mathbb P_L$ be uniform sampling of $N=Lm$ over a\nperiod of $U_F(Lm)$, for which $Q/\\gcd(Q,L)$ is a permissible length.\nThis notation does not assert that the displayed period is least.\n\nWe use $K_*(F)$ for the infimum of the existing fractional-cover cost\n\\begin{equation}\n K=\\sum_j\\frac{C_j\\eta_j^{-\\alpha_j}}{2^{\\alpha_j}-1},\\qquad\n C_j=\\sum_{d\\ge1}\\frac{c_{j,d}}d,\n \\label{eq:257-optimised-cover-cost}\n\\end{equation}\nover finite or countable families of finite sets $F_j$ of positive integers covering\n$F$, exponents $0<\\alpha_j\\le1$, weights $\\eta_j>0$ with\n$\\sum_j\\eta_j=1$, and coefficients $c_{j,d}\\ge0$ satisfying\n$f_{F_j}(n)^{\\alpha_j}\\le\\sum_{d\\mid n}c_{j,d}$ for every $n\\ge1$.\nThe covering sets need not be disjoint or contained in $F$.  Covers of infinite\ncost do not lower the infimum, and a finite-cost cover exists, for example\nthe single set $F$ with exponent one.  This is the fractional cost of\nTheorem~\\ref{thm:257-variable-fractional-cover},\nnot a redefinition of the logarithmic cost.\nThe two costs optimise different majorants. The quantity $\\kappa_1$\nuses one logarithmic majorant of $f_F$, whereas $K_*$ also permits\nsplitting the support, varying the fractional exponents and allocating\npositive weights. The comparison below allows all of those choices.\n\n\\begin{thm}[arithmetic logarithmic counterexample]\\label{thm:257-logarithmic-counterexample}\nFor every integer $H\\ge2$ and every real $A_0\\ge0$, there are a\nsquarefree positive integer $L$ and a finite nonempty set $F$ of distinct\nsquarefree positive integers such that\n\\[\n \\min F>\\max\\{L,A_0\\},\\qquad\n \\kappa_1(F;1)\\le\\frac{30\\log2}{H},\\qquad\n \\mathbb P_L(U_F>1)\\ge1-e^{-1}.\n\\]\nFor these $F,L$, some $R\\ge0$ satisfies\n$\\mathscr D_{L;R,L}\\mathbf1_{\\{U_F>1\\}}>1/2$.\nConsequently, no absolute constant $C$ satisfies\n\\begin{equation*}\n \\mathscr D_{L;R,M}\\mathbf1_{\\{U_F>t\\}}\n \\le C(1+L/M)\\kappa_1(F;t)\n \\tag{E}\n\\end{equation*}\nuniformly in finite $F$, positive integers $L,M$, integers $R\\ge0$,\nand $0<t\\le1$.\n\\end{thm}\n\n\\begin{proof}\nWe choose the modulus so that several small divisibility conditions hold\nautomatically. Fresh tag primes make the remaining conditions independent.\nTaking all least common multiples then makes the divisor count\nexponential in the number of tags. Its logarithm has small cost, but the\nshifted potential exceeds one at sufficiently many offsets to prevent\nthe proposed uniform bound.\n\nChoose a finite set of primes $P$ such that, with\n\\[\n L=\\prod_{p\\in P}p,\\qquad D=\\tau(L),\\qquad\n B=\\prod_{p\\in P}(1+1/p),\n\\]\nwe have $B\\ge H^2$.  Such a set exists because the sum of reciprocal primes\ndiverges and $\\log(1+1/p)\\ge1/(2p)$.\nHere $\\tau$ counts positive divisors and $\\sigma$ denotes their sum.\nComplementary divisors satisfy\n\\[\n \\frac{\\sigma(r)}r\\,\n \\frac{\\sigma(L/r)}{L/r}=B\\qquad(r\\mid L).\n\\]\nThus the set\n$G=\\{r\\mid L:\\sigma(r)/r\\ge H\\}$ has at least $D/2$ elements.\nAlso $D\\ge B\\ge H^2$.\n\nChoose a finite set $\\mathcal B$ of parent primes, all exceeding\n$\\max\\{L,H,D,A_0\\}$, with\n\\[\n \\frac4D\\le\\sum_{q\\in\\mathcal B}\\frac1q\\le\\frac5D.\n\\]\nDivergence of the prime reciprocal sum after any finite cutoff permits\nstopping at the first crossing of $4/D$; the overshoot is less than $1/D$.\nFor each $q\\in\\mathcal B$ and $d\\mid L$, choose a finite set $T_{q,d}$\nof tag primes, all greater than $H$, disjoint from $P$, from $\\mathcal B$,\nand from every other tag set, such that\n\\[\n \\frac{4d}{H}\\le S_{q,d}:=\\sum_{p\\in T_{q,d}}\\frac1p\\le\\frac{5d}{H}.\n\\]\nThe same stopping argument works successively: each new reciprocal is\nless than $1/H\\le d/H$, and only finitely many primes have been excluded.\nNo bound on the size of these prime sets is needed.\n\nLet $\\mathcal I_q=\\{(d,p):d\\mid L,\\ p\\in T_{q,d}\\}$, and define\n\\[\n F_q=\\left\\{q\\operatorname{lcm}\\{dp:(d,p)\\in I\\}:\n                         I\\subseteq\\mathcal I_q\\right\\},\n \\qquad F=\\bigcup_{q\\in\\mathcal B}F_q,\n\\]\nwhere the least common multiple of the empty set is one.\nEvery tag prime occurs in just one labelled pair.  Its presence in the\nleast common multiple therefore recovers whether that pair was selected.\nHence different subsets give distinct exponents.  All exponents are\nsquarefree; different parent primes give disjoint sets, since no parent\nprime occurs in any other set.  Their minimum is greater than\n$\\max\\{L,A_0\\}$.  The nonempty tag sets with $d=L$ also show $L\\mid Q$.\n\nFor an integer $n\\ge1$ put\n$Z_q(n)=\\sum_{(d,p)\\in\\mathcal I_q}\\mathbf1_{\\{dp\\mid n\\}}$.\nThe least-common-multiple encoding gives the exact identity\n\\[\n f_{F_q}(n)=\\mathbf1_{\\{q\\mid n\\}}2^{Z_q(n)}.\n\\]\nIndeed, when $q\\mid n$, a selected least common multiple divides $n$\nexactly when every selected pair does.  Consequently\n\\[\n \\log(1+f_{F_q}(n))\n \\le(\\log2)\\left(\\mathbf1_{\\{q\\mid n\\}}\n          +\\sum_{(d,p)\\in\\mathcal I_q}\\mathbf1_{\\{qdp\\mid n\\}}\\right).\n\\]\nAll moduli in this majorant divide $\\operatorname{lcm}(F_q)$.\nIts cost is at most\n\\begin{equation}\n \\frac{\\log2}{q}\\left(1+\\sum_{d\\mid L}\\frac{S_{q,d}}d\\right)\n \\le\\frac{\\log2}{q}(1+5D/H).\n \\label{eq:257-parent-log-cost}\n\\end{equation}\n\nNow sample $N$ uniformly modulo $Q$, subject to $L\\mid N$.\nFor $r\\in G$, the event $q\\mid N+r$ has probability $1/q$.\nFor a fixed parent these events are disjoint as $1\\le r\\le L<q$.\nConditional on one such event, the tag-prime events remain independent\nby the Chinese remainder theorem.  Since $d\\mid N+r$ is equivalent to\n$d\\mid r$, the mean of $Z_q(N+r)$ is\n\\[\n \\mu_r=\\sum_{d\\mid r}S_{q,d}\n \\ge\\frac4H\\sigma(r)\\ge4r,\n\\]\nand its variance is at most $\\mu_r$.\nChebyshev's inequality gives\n\\[\n \\mathbb P_L\\bigl(Z_q(N+r)<r\\mid q\\mid N+r\\bigr)\n \\le\\frac{\\mu_r}{(\\mu_r-r)^2}\n \\le\\frac4{9r}\\le\\frac49.\n\\]\nThus success has conditional probability at least $1/2$.\nOn success the $r$th term of $U_{F_q}(N)$ is at least\n$2^{-r}2^r=1$.  Later positive terms, for example those supplied by the\nexponent $q\\in F_q$, make $U_{F_q}(N)>1$ strictly.\nThe event $E_q$ of success for some $r\\in G$ consequently satisfies\n\\[\n \\mathbb P_L(E_q)\\ge\\frac{|G|}{2q}\\ge\\frac{D}{4q}.\n\\]\nDifferent parent groups depend on disjoint prime coordinates, so the\n$E_q$ are independent under this conditional uniform measure.  It follows\nthat\n\\[\n \\mathbb P_L(U_F>1)\\ge\n 1-\\prod_{q\\in\\mathcal B}(1-\\mathbb P_L(E_q))\n \\ge1-\\exp\\left(-\\frac D4\\sum_{q\\in\\mathcal B}\\frac1q\\right)\n \\ge1-e^{-1}.\n\\]\nSince $f_F=\\sum_qf_{F_q}$ and\n$\\log(1+\\sum_qx_q)\\le\\sum_q\\log(1+x_q)$ for $x_q\\ge0$,\nwe may sum the positive majorants in~\\eqref{eq:257-parent-log-cost}:\n\\[\n \\kappa_1(F;1)\\le\\frac{5\\log2}{D}(1+5D/H)\n \\le\\frac{30\\log2}{H}.\n\\]\n\nSince $L\\mid Q$, write $P_0=Q/L$ and $p=\\mathbb P_L(U_F>1)$.  The bounded periodic\nsequence $\\mathbf1_{\\{U_F(Lm)>1\\}}$ satisfies, for every integer $T\\ge1$,\n\\[\n \\left|\\frac1T\\sum_{m=1}^{T}\\mathbf1_{\\{U_F(Lm)>1\\}}-p\\right|\n \\le\\frac{P_0}{T}.\n\\]\nSumming over the dyadic window gives\n\\[\n \\left|\\mathscr D_{L;R,M}\\mathbf1_{\\{U_F>1\\}}-p\\right|\n \\le\\frac{2P_0}{M2^R}.\n\\]\nTake $M=L$ and $2^R\\ge16P_0$.  The error is at most $1/8$, whereas\n$p\\ge1-e^{-1}>5/8$, so the mean exceeds $1/2$.\nA putative bound (E) would instead give at most $60C\\log2/H$.\nChoosing an integer $H>120C\\log2$ contradicts it.\n\\end{proof}\n\n\\subsection{The arithmetic lower bound for every fractional cover}\n\\begin{cor}[finite-functional separation]\\label{cor:257-logarithmic-separation}\nFor every finite nonempty $F$, with $Q=\\operatorname{lcm}(F)$,\n\\begin{equation}\n K_*(F)\\ge\\max_{\\ell\\mid Q}\n   \\mathbb P\\bigl(U_F(N)>1\\mid\\ell\\mid N\\bigr),\n \\label{eq:257-arithmetic-cover-lower}\n\\end{equation}\nwhere $N$ is uniform modulo $Q$.\nThe supports of Theorem~\\ref{thm:257-logarithmic-counterexample} satisfy\n$K_*(F)\\ge1-e^{-1}$ and $\\kappa_1(F;1)\\le30\\log2/H$.\nIn particular, no absolute $C$ gives $K_*(F)\\le C\\kappa_1(F;1)$\nfor all finite nonempty $F$, even after all covering sets, exponents, coefficients\nand positive weights have been optimised.\n\\end{cor}\n\\begin{proof}\nFix one finite-cost cover in~\\eqref{eq:257-optimised-cover-cost}, and write\n$U_j=U_{F_j}$ and $B_j=2^{\\alpha_j}$.\nCoverage gives $U_F\\le\\sum_jU_j$.  If $U_F>1$, then $U_j>\\eta_j$\nfor at least one index.  Subadditivity and the divisor majorants therefore\ngive the pointwise estimate\n\\[\n \\mathbf1_{\\{U_F>1\\}}\n \\le\\sum_j\\eta_j^{-\\alpha_j}U_j^{\\alpha_j}\n \\le\\sum_{j,d}\\eta_j^{-\\alpha_j}c_{j,d}w_{B_j,d}.\n\\]\nNonnegative interchange and the original geometric-kernel estimate\n\\eqref{eq:257-mixed-finite-kernel}, with its factor $1+4L/M$, imply\n\\[\n \\mathscr D_{L;R,M}\\mathbf1_{\\{U_F>1\\}}\\le(1+4L/M)K.\n\\]\nKeep the cover and $L$ fixed. For each fixed $M$, let $R\\to\\infty$:\nperiodicity of $\\mathbf1_{\\{U_F>1\\}}$ gives\n$\\mathbb P_L(U_F>1)\\le(1+4L/M)K$.\nNow let $M\\to\\infty$ to remove the incomplete-period factor, giving\n$\\mathbb P_L(U_F>1)\\le K$.\nOnly the indicator for the finite set $F$ needs a period; no common\nperiod for the covering sets has been assumed.\nFor $L=\\ell\\mid Q$ the resulting probability is the one in the statement.\nSince this bound holds for each admissible cover and each such $\\ell$,\ntake the infimum over covers and the maximum over the finitely many\nchoices of $\\ell$.  Finally use the constructed\n$L\\mid Q$ and let $H$ tend to infinity.\n\\end{proof}\n\n\\subsection{What survives without arithmetic restriction}\n\\begin{prop}[ordinary initial intervals]\\label{prop:257-logarithmic-initial-interval}\nFor every finite nonempty $F$, integer $X\\ge1$, and $0<t\\le1$,\n\\[\n \\frac1X\\#\\{1\\le N\\le X:U_F(N)>t\\}\n \\le\\frac{2}{\\log(4/3)}\\kappa_1(F;t).\n\\]\n\\end{prop}\n\\begin{proof}\nChoose a positive logarithmic divisor majorant\n$g(n)=\\sum_{d\\mid n}c_d$ of cost $K=\\sum_{d\\mid Q}c_d/d$, putting\n$c_d=0$ when $d\\nmid Q$, and let $a=\\log(4/3)$.\nFor positive integers $n,Y$,\n\\[\n g(n)\\le Kn,\\qquad\n \\sum_{n=1}^{Y}g(n)=\\sum_{d\\mid Q}c_d\\lfloor Y/d\\rfloor\\le KY.\n\\]\nIf $U_F(N)>t$, there is an $r\\ge1$ with\n$\\log(1+f_F(N+r)/t)>ar$.\nOtherwise every $f_F(N+r)/t\\le(4/3)^r-1$, and\n\\[\n \\frac{U_F(N)}t\\le\\sum_{r\\ge1}2^{-r}\\bigl((4/3)^r-1\\bigr)=1,\n\\]\na contradiction.\nIf $K\\ge a/2$ the required counting bound is trivial.\nFor $K<a/2$, a witnessing integer $n=N+r$ satisfies\n$a(n-N)<g(n)\\le Kn$, whence $n<aN/(a-K)\\le2X$ when $N\\le X$.\nA fixed $n$ can witness at most $g(n)/a$ integer values of\n$N$, since $1\\le n-N<g(n)/a$.\nThe union bound thus gives\n\\[\n \\#\\{1\\le N\\le X:U_F(N)>t\\}\n \\le\\frac1a\\sum_{n=1}^{2X}g(n)\\le\\frac{2KX}{a}.\n\\]\nTake the infimum over the admissible majorants.\n\\end{proof}\n\nThe proposition also bounds every $L=1$ dyadic observation window, since\nsuch a window averages ordinary initial-interval means.  It does not bound\nconditional arithmetic sampling uniformly in $L$.  Freezing a finite\ninitial support in the irrationality argument requires exactly that\narithmetic restriction.  An irrationality criterion with logarithmic\ndivisor costs in place of fractional ones therefore does not follow from\nthis proof; it is not disproved either.\nThe corollary above separates finite functionals, not infinite-support\nirrationality classes.  The weighted, fractional-cover, mixed and\nfinite-period noncollapse results are not affected.  No conclusion here\ndecides universal \\#257 or membership of $1/2$ or $1/21$.\n\n\\clearpage\n\\section*{Two proof details and sources}\n\\papersectiontarget{record257:supplementary-sources}\nThe comparison of displacement atoms works for real bases:\n$\\Delta_{b,A}(N)\\le\\Delta_{2,A}(N)$ whenever $b\\ge2$.\nTo see this, cancel $b-1$ and differentiate\n$\\sum_{i<r}b^i/\\sum_{i<d}b^i$. The numerator of the derivative is\n$\\sum_{i<r\\le j<d}(i-j)b^{i+j-1}\\le0$.\nAn integer base is needed only when the displacement is placed on a\nrational lattice.\n\nFor the scalar positive-cover cost, put $z=\\log_2t$.  Direct differentiation\nof $y^z/(y-1)$ on $1<y\\le2$ gives\n\\[\n \\Psi(t)=t\\quad(1\\le t\\le4),\\qquad\n \\Psi(t)=\\frac{z^z}{(z-1)^{z-1}}\\quad(t>4).\n\\]\nThese two calculations supply the elementary details used in the\nshort paper.\n\nThe original periodic and linear-independence papers of Luca and Tachiya\nare now distinguished from their later overview\n\\cite{lucatachiya2014periodic,lucatachiya2014independence,lucatachiya2017}.\nTheir theorem formulations used here were read in the author-written\noverview; the two original publisher PDFs were not retrieved.  Van Assche's nonvanishing and\nsmall-linear-form argument for the full Lambert series\n\\cite[Lemma~1 and (34)--(35)]{vanassche2001} is a comparator for the missing\nupper-height control, not a result for arbitrary thinnings.\n\nFor achievement-set background, Hornich's original bibliographic record\n\\cite{hornich1941} is separated from the exposition actually used here\n\\cite{nitecki2013}.  G\\l\\k{a}b and Prus-Wi\\'sniowski's survey\n\\cite{glabprus2025} provides a modern map of geometry and cardinality\nquestions, not a rational-membership theorem for the present set.\n\n\\begingroup\n% Keep the full declaration names and original fixed-revision links from\n% the short paper's relocated supplementary source catalogue.\n\\newcommand{\\mword}[4]{\\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/#1\\#L#2}{#4}}\n\\newcommand{\\lword}[4]{\\href{https://github.com/wcook04/plectis-erdos/blob/99f4bf47422abbd8757cbb22b50ba079d764d3a7/ErdosProblems/#1\\#L#2}{#4}}\n\\paragraph{Library declarations.}\nThe following fixed-revision links locate the supplementary formal\nstatements used in this note and its companion.\nPairwise-coprime supports: \\mword{Erdos249257/CertificateKernel.lean}{6082}{erdos257_family_factorial_instance}{factorial-support instance}; \\mword{Erdos249257/CertificateKernel.lean}{6090}{erdos257_family_two_pow_instance}{powers-of-two-support instance}; \\mword{Erdos249257/CertificateKernel.lean}{10776}{irrational_erdosSupportSeries_pairwise_coprime}{pairwise-coprime theorem}.\nEventually periodic weights: \\mword{Erdos249257/CertificateKernel.lean}{12811}{irrational_ratWeightSeries_eventuallyPeriodic}{irrationality for the stated eventually periodic weights}.\nFixed-core supports: \\mword{Erdos249257/SupportSunflowerDichotomy.lean}{531}{sunflowerForcedCarrySupply_of_orthogonalPetalBouquet}{carry certificates for the stated fixed-core support}; \\mword{Erdos249257/SupportSunflowerDichotomy.lean}{540}{irrational_erdosSupportSeries_of_orthogonalPetalBouquet}{irrationality for the stated fixed-core support}.\nFinite supports and denominators: \\mword{Erdos249257/CertificateKernel.lean}{5246}{finite_period_noncollapse_rat_den}{noncollapse for the reduced rational denominator}.\nRational values and carries: \\mword{Erdos249257/BooleanMobiusCarry.lean}{949}{exists_normalized_support_fraction_iff_exists_booleanMobiusCarry}{rational support values and M\\\"obius-inverted carries}; \\mword{Erdos249257/GenericTailOrbitRigidity.lean}{426}{binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit}{rationality and integer recurrences with vanishing scaled error}; \\mword{Erdos249257/RationalSupportCarrySkeleton.lean}{1480}{one_div_oddOrder_le_reciprocalMass_of_support_fraction}{a lower bound for reciprocal mass}; \\mword{Erdos249257/RationalSupportCarrySkeleton.lean}{2383}{exists_unbounded_shifted_odd_tail_nat_state_of_support_fraction}{unboundedness of the shifted odd-tail recurrence}; \\mword{Erdos249257/SublogDivisorCoverage.lean}{392}{supportCoeffZeroWindow_length_le_eps_logb_add}{lengths of intervals with zero divisor count}.\nComposite dilation: \\mword{Erdos249257/CompositeDilationDefect.lean}{30}{supportCoeff_mul_eq_add_defect}{the divisor-count identity under dilation}; \\mword{Erdos249257/CompositeDilationDefect.lean}{151}{compositeDilationDefect_le_exceptional_add_petalCoeff}{the upper bound for the dilation correction}; \\mword{Erdos249257/CompositeDilationDefect.lean}{218}{compositeDilationDefect_two_six_fixture}{the example with exponents two and six}.\nThe achievement set: \\mword{Erdos249257/GreedyAchievementSet.lean}{996}{volume_mersenneAchievementSet}{Lebesgue measure one}; \\mword{Erdos249257/GreedyAchievementSet.lean}{1656}{perfect_mersenneAchievementSet}{perfect}; \\mword{Erdos249257/GreedyAchievementSet.lean}{1672}{isTotallyDisconnected_mersenneAchievementSet}{totally disconnected}; \\mword{Erdos249257/GreedyAchievementSet.lean}{1681}{isNowhereDense_mersenneAchievementSet}{nowhere dense}; \\mword{Erdos249257/GreedyAchievementSet.lean}{1458}{mem_mersenneAchievementSet_iff_greedy_survival}{membership and greedy tail bounds}; \\mword{Erdos249257/GreedyAchievementSet.lean}{1784}{three_fourths_not_mem_mersenneAchievementSet}{non-membership of $3/4$}; \\mword{Erdos249257/GreedyAchievementSet.lean}{2583}{half_mem_mersenneAchievementSet_iff_greedySkippedSupport_infinite}{the greedy form}; \\mword{Erdos249257/HalfCarryReachability.lean}{589}{finite_boolSupport_ne_half}{the finite-support exclusion}.\nRestricted achievement sets: \\lword{Erdos257/MersenneSubseriesRigidity.lean}{397}{volume_supportedMersenneAchievementSet_dichotomy}{the formal dichotomy}.\nGreedy skips and exact rows: \\mword{Erdos249257/BooleanMobiusSkipRowCofinal.lean}{22}{CofinalPositiveHalfGreedySkips}{the cofinal-skip hypothesis}; \\mword{Erdos249257/BooleanMobiusSkipRowCofinal.lean}{55}{exactLocalMersenneHalfRow_of_positiveHalfGreedySkip}{an exact finite sum from one greedy skip}; \\mword{Erdos249257/BooleanMobiusSkipRowCofinal.lean}{84}{cofinalExactLocalMersenneHalfRows_of_positiveHalfGreedySkips}{exact finite sums from arbitrarily late greedy skips}; \\mword{Erdos249257/BooleanMobiusSkipRowCofinal.lean}{97}{half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips}{membership from infinitely many greedy skips}; \\mword{Erdos249257/BooleanMobiusSkippedCoreExactRow.lean}{228}{exists_upperHalfBooleanWord_of_skippedCore}{the upper-half Boolean fill}.\nMembership of $1/2$: \\mword{Erdos249257/HalfCylinderHalfMembershipClassification.lean}{126}{half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse}{the terminal-bit form}; \\mword{Erdos249257/HalfCylinderHalfMembershipClassification.lean}{213}{half_mem_mersenneAchievementSet_iff_exists_unboundedSkippedRanksAlong}{the skipped-rank form}.\nThe two-thirds band: \\mword{Erdos249257/HalfGreedyTwoThirdsBand.lean}{88}{postTakeUnsafeAt_iff_band}{general band localization}; \\mword{Erdos249257/HalfGreedyTwoThirdsBand.lean}{127}{singletonUnsafe_iff_twoThirdsBand}{two-thirds band}; \\mword{Erdos249257/HalfGreedyTwoThirdsBand.lean}{185}{not_twoThirdsBand_of_int}{integral safety}; \\mword{Erdos249257/HalfGreedyTwoThirdsBand.lean}{231}{seven_le_of_intBand_odd}{odd numerator bound}.\n\n\n\\endgroup\n\n\\section*{Verification and reproducibility}\n\\evidenceparagraph{\\evidencerecordurl}\nLinks marked \\textsc{Lean source} identify the declaration associated\nwith a statement or proof step. The integer-quotient proof for $1/21$\nand other arguments without a formal link remain ordinary proofs; no\nnew formal or independent human verification is claimed here. The\nlinked records identify the propositions and source revisions checked.\n\n\\section*{Statements and declarations}\n\n\\paragraph{Artefact and data availability.} The Lean declarations, fixed\ntoolchain and library versions are given in the evidence record linked above.\n\n\\paragraph{Scope of the formal checks.} Kernel checking establishes that a proposition\nwas proved; it does not validate the exposition or the literature judgements, and it\ndoes not solve Problem 257.\n\n\\paragraph{Funding and competing interests.} This work received no external\nfunding. The author declares no competing interests.\n\n\\begin{thebibliography}{99}\n\\small\n\\raggedright\n\\setlength{\\itemsep}{0pt}\n\\bibitem{erdos1948} P.~Erd\\H{o}s,\n  \\href{https://www.renyi.hu/~p_erdos/1948-04.pdf}{\\emph{On arithmetical\n  properties of Lambert series}}, J.~Indian Math.\\ Soc.\\ 12 (1948), 63--66.\n\\bibitem{erdos1968} P.~Erd\\H{o}s,\n  \\href{https://users.renyi.hu/~p_erdos/1969-09.pdf}{\\emph{On the\n  irrationality of certain series}}, Math.\\ Student 36 (1968), 222--226\n  (issued 1969); \\href{https://www.renyi.hu/~p_erdos/1969-09.pdf}{five-page scan}.  The theorem on p.~222 treats pairwise-coprime support with\n  convergent reciprocal sum at every integer base $b\\ge2$; the claimed\n  removal of pairwise coprimality is stated without proof.  The same page\n  says the reciprocal-sum condition could be replaced by a weaker but more\n  complicated condition, and p.~226 suggests\n  $\\sum_{n_i<x}1/n_i=o(\\log\\log x)$ for pairwise coprime supports.\n\\bibitem{lucatachiya2017} F.~Luca and Y.~Tachiya,\n  \\href{https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2014-14.pdf}\n  {\\emph{Linear independence results for the values of divisor functions\n  series}}, RIMS K\\^oky\\^uroku No.~2014 (2017), 138--150.  Theorem~A (p.~139)\n  restates Theorem~1.1 of their paper \\emph{Irrationality of Lambert series\n  associated with a periodic sequence}, Int.\\ J.~Number Theory 10 (2014),\n  no.~3, 623--636,\n  \\href{https://doi.org/10.1142/S1793042113501121}{DOI}: for a purely periodic\n  integer sequence $a(n)$, not identically zero,\n  $\\sum_{n\\ge1}a(n)/(q^n-1)$ is irrational for every integer $q$ with $|q|>1$.\n  Example~2 (p.~140) treats the odd support.\n\\bibitem{nitecki2013} Z.~Nitecki,\n  \\href{https://arxiv.org/abs/1106.3779v2}{\\emph{Subsum sets: intervals, Cantor\n  sets, and Cantorvals}}, arXiv:1106.3779v2 (2013).  Theorem~4 (p.~9) shows\n  that when every term exceeds its tail the subsum set is a Cantor set of\n  Lebesgue measure $\\lim_n2^nX_n$, where $X_n$ is the $n$th tail, and credits\n  this to H.~Hornich (1941).\n\\bibitem{sedaghatstephenchindelevitch} N.~Sedaghat, T.~Stephen, and\n  L.~Chindelevitch, \\href{https://doi.org/10.4230/LIPIcs.SEA.2018.6}\n  {\\emph{Speeding up dualization in the Fredman--Khachiyan Algorithm B}},\n  17th International Symposium on Experimental Algorithms (SEA 2018),\n  LIPIcs 103 (2018), 6:1--6:13.\n\\bibitem{kanekosuzukitachiya} H.~Kaneko, Y.~Suzuki, and Y.~Tachiya,\n  \\href{https://arxiv.org/abs/2601.20743v1}{\\emph{Refinements of Erd\\H{o}s's\n  irrationality criterion for certain sparse infinite series}},\n  arXiv:2601.20743v1 (2026).  The averaged tail $R_c(q,x,z)$, a sum over\n  offsets $j\\ge z$, is~(1.7) and Theorem~1 is on p.~3; its rational-integer\n  form, Theorem~3, is on p.~5.  Lemma~1 (pp.~6--7) derives irrationality from\n  arbitrarily small nonzero scaled tails, and Lemma~2 (pp.~7--8) shows that\n  most scaled tails in a range are small under a counting condition on the\n  support.  Their criteria assume sparse coefficient supports.\n\\bibitem{duverneytachiya} D.~Duverney and Y.~Tachiya,\n  \\href{https://danielduverney.fr/documents/theorie-des-nombres/DuverneyTachiya190522.pdf}\n  {\\emph{Refinement of the Chowla--Erd\\H{o}s method and linear independence\n  of certain Lambert series}},\n  Forum Math.\\ 31 (2019), no.~6, 1557--1566,\n  \\href{https://doi.org/10.1515/forum-2018-0299}{DOI}.  Page numbers refer to\n  the linked author preprint.  Theorem~1.1 (pp.~2--3) is the refined\n  Chowla--Erd\\H{o}s criterion; its proof in Section~2 (pp.~5--7) selects an\n  index by minimising a local coefficient mass along an arithmetic\n  progression, (2.3)--(2.9).  Corollary~1.2 gives the general $F_s(E)$\n  theorem and its monomial images under $|q|\\operatorname{lcm}(1,\\dots,\\ell)\\le s$, and at\n  every integer base when $s=\\infty$; Example~1.1 gives the joint squarefree\n  family at all bases \\(2^j\\), \\(j\\ge1\\).  Both are on p.~4, which also\n  relates Corollary~1.2 to the conjecture of Erd\\H{o}s and Graham; the proof\n  of Corollary~1.2 is on pp.~10--11.\n\\bibitem{taoteravainen2025} T.~Tao and J.~Ter\\\"av\\\"ainen,\n  \\href{https://arxiv.org/abs/2512.01739v2}{\\emph{Quantitative correlations and\n  some problems on prime factors of consecutive integers}}, arXiv:2512.01739v2\n  (submitted December 2025, revised April 2026).  Theorem~1.3 (p.~4) proves\n  $\\sum_{n\\ge1}\\omega(n)/2^n=\\sum_p(2^p-1)^{-1}$ irrational, settling the\n  prime-support case of \\#257 at base~$2$; the paragraph after it states that\n  the prime-support result extends to every integer base $b\\ge2$, and\n  separately that full prime-power support is irrational at base $2$.\n  The modifications for those two assertions are left to the reader;\n  arbitrary subsets of the primes or prime powers are not treated there.  Theorem~3.1 is on p.~24, and the\n  proof of Theorem~1.3 is Section~5, pp.~44--56.\n\\bibitem{zudilin2004} W.~Zudilin, \\emph{Heine's basic transform and a\n  permutation group for $q$-harmonic series}, Acta Arith.\\ 111 (2004), no.~2,\n  153--164, \\href{https://doi.org/10.4064/aa111-2-4}{DOI}.  Theorem~1 (p.~154)\n  gives $\\mu(h_p(1))\\le2.46497868\\ldots$ for $h_p(1)=\\sum_{n\\ge1}1/(p^n-1)$,\n  uniformly in the integer $p$ with $|p|\\ge2$; the remark at the end of\n  Section~3 (p.~159) corrects the computation of the author's earlier paper.\n\\bibitem{campbell2026} J.~M.~Campbell,\n  \\href{https://arxiv.org/abs/2605.24160v1}{\\emph{On the binary digits of the\n  Erd\\H{o}s--Borwein constant}}, arXiv:2605.24160v1 (2026).  Theorem~1 (p.~12)\n  proves that the block \\texttt{11} occurs infinitely often in the binary\n  expansion of $\\sum_{n\\ge1}1/(2^n-1)$, answering a question of Crandall\n  recalled in the introduction.\n\\bibitem{matomakiradziwill} K.~Matom\\\"aki and M.~Radziwi\\l\\l,\n  \\emph{Multiplicative functions in short intervals}, Ann.\\ of Math.\\ (2) 183\n  (2016), no.~3, 1015--1056,\n  \\href{https://doi.org/10.4007/annals.2016.183.3.6}{DOI}.  Page numbers\n  refer to arXiv:1501.04585v4.\n\\bibitem{mangerel2021} A.~P.~Mangerel, \\emph{Divisor-bounded multiplicative\n  functions in short intervals}, Res.\\ Math.\\ Sci.\\ 10 (2023), no.~1, Paper\n  No.~12, \\href{https://doi.org/10.1007/s40687-023-00376-0}{DOI}.  Page numbers\n  refer to arXiv:2108.11401v2.\n\\bibitem{sun2026} Y.-C.~Sun,\n  \\href{https://arxiv.org/abs/2401.08432v3}{\\emph{The critical-window profile\n  for the $k$-fold divisor function in almost all short intervals}},\n  arXiv:2401.08432v3 (2026).\n\\bibitem{lucatachiya2014periodic}\nF.~Luca and Y.~Tachiya,\n\\emph{Irrationality of Lambert series associated with a periodic sequence},\nInternational Journal of Number Theory \\textbf{10} (2014), no.~3, 623--636.\n\\href{https://doi.org/10.1142/S1793042113501121}{doi:10.1142/S1793042113501121}.\n\n\\bibitem{lucatachiya2014independence}\nF.~Luca and Y.~Tachiya,\n\\emph{Linear independence of certain Lambert series},\nProceedings of the American Mathematical Society \\textbf{142} (2014), no.~10, 3411--3419.\n\\href{https://doi.org/10.1090/S0002-9939-2014-12102-2}{doi:10.1090/S0002-9939-2014-12102-2}.\n\n\\bibitem{vanassche2001}\nW.~Van Assche,\n\\emph{Little $q$-Legendre polynomials and irrationality of certain Lambert series},\nThe Ramanujan Journal \\textbf{5} (2001), 295--310.\n\\href{https://doi.org/10.1023/A:1012930828917}{doi:10.1023/A:1012930828917};\n\\href{https://arxiv.org/abs/math/0101187v1}{arXiv:math/0101187v1}.\n\n\\bibitem{hornich1941}\nH.~Hornich,\n\\emph{\\\"Uber beliebige Teilsummen absolut konvergenter Reihen},\nMonatshefte f\\\"ur Mathematik und Physik \\textbf{49} (1941), 316--320.\n\\href{https://doi.org/10.1007/BF01707309}{doi:10.1007/BF01707309}.\n\n\\bibitem{glabprus2025}\nS.~G\\l\\k{a}b and F.~Prus-Wi\\'sniowski,\n\\emph{Achievement sets -- current results and open problems},\narXiv:2512.17285v1 (2025), preprint.\n\\url{https://arxiv.org/abs/2512.17285v1}.\n\n\\bibitem{kovactao} V.~Kova\\v{c} and T.~Tao, \\emph{On several irrationality\n  problems for Ahmes series}, Acta Math.\\ Hungar.\\ 175 (2025), no.~2, 572--608,\n  \\href{https://doi.org/10.1007/s10474-025-01528-0}{DOI}.  Page numbers refer\n  to arXiv:2406.17593v4.  Remark~4.1 (p.~13) records the strict tail\n  inequality and the Cantor structure.  Theorem~2.3 (p.~5; proof pp.~13--14)\n  constructs rational merged sums from several bases under its mass\n  hypothesis; the construction merges several bases and leaves fixed-base \\#257 untouched.\n\\bibitem{vandoornkovac} W.~van Doorn and V.~Kova\\v{c},\n  \\emph{Lacunary sequences whose reciprocal sums represent all rational\n  numbers in an interval}, Acta Arith.\\ 223 (2026), 275--295,\n  \\href{https://doi.org/10.4064/aa251001-13-1}{DOI}.\n  The theorem and page references used here refer to\n  arXiv:2509.24971v3 (3 December 2025),\n  \\url{https://arxiv.org/abs/2509.24971v3}.\n  Theorem~1 is on p.~2; Lemma~4 and Corollary~5 are on pp.~5--7.\n\n\\bibitem{erdos1975} P.~Erd\\H{o}s, \\emph{Some problems and results on the\n  irrationality of the sum of infinite series}, J.~Math.\\ Sci.\\ 10 (1975),\n  1--7.\n\n\\bibitem{bkkkz2026} K.~Barreto, J.~Kang, S.-H.~Kim, V.~Kova\\v{c}, and\n  S.~Zhang, \\href{https://arxiv.org/abs/2601.21442v3}{\\emph{Irrationality of\n  rapidly converging series: a problem of Erd\\H{o}s and Graham}},\n  arXiv:2601.21442v3 (2026), to appear in Bull.\\ London Math.\\ Soc.\n\n\\bibitem{conradcrt} K.~Conrad,\n  \\href{https://kconrad.math.uconn.edu/blurbs/ugradnumthy/crt.pdf}\n  {\\emph{The Chinese remainder theorem}}, expository notes,\n  Theorem~3.1 (pp.~2--3).  The compatibility criterion for noncoprime\n  moduli used above is obtained by the displayed prime-power reduction.\n\n\\bibitem{erdos1932bertrand} P.~Erd\\H{o}s,\n  \\href{https://www.renyi.hu/~p_erdos/1932-01.pdf}\n  {Beweis eines Satzes von Tschebyschef}, 1932, pp.~194--198,\n  especially \\S\\S4--5.\n\n\\end{thebibliography}\n% ---- part a257_p0 ----\n% ---- part a257_p1a ----\n% ---- part a257_p1b ----\n% ---- part a257_newdecls ----\n% ---- part a257_p2 ----\n% ---- part a257_p3 ----\n% ---- part a257_p4 ----\n% ---- part a257_invent ----\n% ---- part a257_p5 ----\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":495052,"sha256":"sha256:47cf6a152738f17c6879bd5070adb73c34a1254775afd44dbf7b6ccd346aa100","cite_as":"Plectis reader corpus, paper:erdos257-mersenne-reasoning-surface","source_repository":"plectis-erdos","source_path":"paper/257/erdos257-mersenne-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"evidence/erdos257-mersenne-reasoning-surface","source_path":"paper/evidence/erdos257-mersenne-reasoning-surface.tex","media_type":"text/x-tex","sha256":"sha256:3ae0d45633dae1966b9dbcad7d3ef837698acfd238679766000f37b47d213f37","byte_count":121561,"licence":"CC-BY-4.0","body_format":"latex","body":"% Generated by scripts/paper_evidence.py from docs/paper_lean_coverage.json and the\n% Comparator replay receipts under evidence/comparator/.  Do not edit by hand.\n\\newcommand{\\evidencerecordurl}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface.md}\n\\DeclareResultEvidenceName{thm:257-weighted}{Theorem 1.1}\n\\DeclareResultEvidenceRecord{thm:257-weighted}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-1.md\\#thm-257-weighted}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:257-weighted}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-1.md\\#thm-257-weighted}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-1.md\\#thm-257-weighted-comparator}\n\\DeclareResultEvidenceName{prop:257-finite-witness-rule}{Proposition 1.2}\n\\DeclareResultEvidenceRecord{prop:257-finite-witness-rule}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-1.md\\#prop-257-finite-witness-rule}{exact_or_stronger}{pending}\n\\DeclareResultEvidence{prop:257-finite-witness-rule}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/WitnessLogicIrrational.lean\\#L39}{}\n\\DeclareResultEvidenceName{thm:257-variable-fractional-cover}{Theorem 1.3}\n\\DeclareResultEvidenceRecord{thm:257-variable-fractional-cover}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-1.md\\#thm-257-variable-fractional-cover}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:257-variable-fractional-cover}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR8/PositiveCoverReturn.lean\\#L241}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-1.md\\#thm-257-variable-fractional-cover-comparator}\n\\DeclareResultEvidenceName{thm:257-mixed-supports}{Theorem 1.4}\n\\DeclareResultEvidenceRecord{thm:257-mixed-supports}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-1.md\\#thm-257-mixed-supports}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:257-mixed-supports}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-1.md\\#thm-257-mixed-supports}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-1.md\\#thm-257-mixed-supports-comparator}\n\\DeclareResultEvidenceName{thm:geometry}{Theorem 1.8}\n\\DeclareResultEvidenceRecord{thm:geometry}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-1.md\\#thm-geometry}{exact}{compared}\n\\DeclareResultEvidence{thm:geometry}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR20/AchievementGeometry.lean\\#L40}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-1.md\\#thm-geometry-comparator}\n\\DeclareResultEvidenceName{thm:supported-dichotomy}{Theorem 1.9}\n\\DeclareResultEvidenceRecord{thm:supported-dichotomy}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-1.md\\#thm-supported-dichotomy}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:supported-dichotomy}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-1.md\\#thm-supported-dichotomy}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-1.md\\#thm-supported-dichotomy-comparator}\n\\DeclareResultEvidenceName{thm:greedy-survival}{Theorem 1.11}\n\\DeclareResultEvidenceRecord{thm:greedy-survival}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-1.md\\#thm-greedy-survival}{exact}{compared}\n\\DeclareResultEvidence{thm:greedy-survival}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR20/GeneralTargetGap.lean\\#L183}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-1.md\\#thm-greedy-survival-comparator}\n\\DeclareResultEvidenceName{prop:canon}{Proposition 2.2}\n\\DeclareResultEvidenceRecord{prop:canon}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#prop-canon}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{prop:canon}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#prop-canon}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#prop-canon-comparator}\n\\DeclareResultEvidenceName{lem:collapse-mech}{Lemma 2.4}\n\\DeclareResultEvidenceRecord{lem:collapse-mech}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#lem-collapse-mech}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{lem:collapse-mech}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#lem-collapse-mech}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#lem-collapse-mech-comparator}\n\\DeclareResultEvidenceName{prop:collapse}{Proposition 2.5}\n\\DeclareResultEvidenceRecord{prop:collapse}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#prop-collapse}{exact}{compared}\n\\DeclareResultEvidence{prop:collapse}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#prop-collapse}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#prop-collapse-comparator}\n\\DeclareResultEvidenceName{lem:sqrt-witness}{Lemma 2.6}\n\\DeclareResultEvidenceRecord{lem:sqrt-witness}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#lem-sqrt-witness}{exact}{compared}\n\\DeclareResultEvidence{lem:sqrt-witness}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR20/CarryCollapseCorrespondence.lean\\#L8}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#lem-sqrt-witness-comparator}\n\\DeclareResultEvidenceName{prop:collapsed-list}{Proposition 2.7}\n\\DeclareResultEvidenceRecord{prop:collapsed-list}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#prop-collapsed-list}{exact}{compared}\n\\DeclareResultEvidence{prop:collapsed-list}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR20/SixMembershipConditions.lean\\#L123}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#prop-collapsed-list-comparator}\n\\DeclareResultEvidenceName{prop:local-void}{Proposition 2.8}\n\\DeclareResultEvidenceRecord{prop:local-void}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#prop-local-void}{exact_or_stronger}{pending}\n\\DeclareResultEvidence{prop:local-void}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#prop-local-void}{}\n\\DeclareResultEvidenceName{prop:exponent-gap}{Proposition 2.9}\n\\DeclareResultEvidenceRecord{prop:exponent-gap}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#prop-exponent-gap}{exact}{compared}\n\\DeclareResultEvidence{prop:exponent-gap}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#prop-exponent-gap}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#prop-exponent-gap-comparator}\n\\DeclareResultEvidenceName{thm:one-sided}{Theorem 2.11}\n\\DeclareResultEvidenceRecord{thm:one-sided}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#thm-one-sided}{exact}{compared}\n\\DeclareResultEvidence{thm:one-sided}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#thm-one-sided}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-2.md\\#thm-one-sided-comparator}\n\\DeclareResultEvidenceName{thm:full-support}{Theorem 5.4}\n\\DeclareResultEvidenceRecord{thm:full-support}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-full-support}{exact}{compared}\n\\DeclareResultEvidence{thm:full-support}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/CertificateKernel.lean\\#L8328}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-full-support-comparator}\n\\DeclareResultEvidenceName{thm:purely-periodic-support}{Theorem 5.5}\n\\DeclareResultEvidenceRecord{thm:purely-periodic-support}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-purely-periodic-support}{exact}{compared}\n\\DeclareResultEvidence{thm:purely-periodic-support}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/CertificateKernel.lean\\#L11590}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-purely-periodic-support-comparator}\n\\DeclareResultEvidenceName{thm:eventually-periodic-support}{Theorem 5.6}\n\\DeclareResultEvidenceRecord{thm:eventually-periodic-support}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-eventually-periodic-support}{exact}{compared}\n\\DeclareResultEvidence{thm:eventually-periodic-support}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/CertificateKernel.lean\\#L11604}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-eventually-periodic-support-comparator}\n\\DeclareResultEvidenceName{thm:residue-class-support}{Theorem 5.7}\n\\DeclareResultEvidenceRecord{thm:residue-class-support}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-residue-class-support}{exact}{compared}\n\\DeclareResultEvidence{thm:residue-class-support}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/ResidueClassSupport.lean\\#L21}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-residue-class-support-comparator}\n\\DeclareResultEvidenceName{thm:odd-support}{Theorem 5.8}\n\\DeclareResultEvidenceRecord{thm:odd-support}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-odd-support}{exact}{compared}\n\\DeclareResultEvidence{thm:odd-support}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/CertificateKernel.lean\\#L11686}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-odd-support-comparator}\n\\DeclareResultEvidenceName{thm:topology}{Theorem 5.9}\n\\DeclareResultEvidenceRecord{thm:topology}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-topology}{exact}{compared}\n\\DeclareResultEvidence{thm:topology}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-topology}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-topology-comparator}\n\\DeclareResultEvidenceName{thm:greedy-survival-record}{Theorem 5.10}\n\\DeclareResultEvidenceRecord{thm:greedy-survival-record}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-greedy-survival-record}{exact}{compared}\n\\DeclareResultEvidence{thm:greedy-survival-record}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-greedy-survival-record}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-greedy-survival-record-comparator}\n\\DeclareResultEvidenceName{thm:master-identity}{Theorem 5.14}\n\\DeclareResultEvidenceRecord{thm:master-identity}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-master-identity}{exact}{compared}\n\\DeclareResultEvidence{thm:master-identity}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR20/QuotientRowIdentity.lean\\#L89}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-master-identity-comparator}\n\\DeclareResultEvidenceName{thm:real-form}{Theorem 5.15}\n\\DeclareResultEvidenceRecord{thm:real-form}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-real-form}{exact}{compared}\n\\DeclareResultEvidence{thm:real-form}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-real-form}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-real-form-comparator}\n\\DeclareResultEvidenceName{thm:dynamics}{Theorem 5.16}\n\\DeclareResultEvidenceRecord{thm:dynamics}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-dynamics}{exact}{compared}\n\\DeclareResultEvidence{thm:dynamics}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-dynamics}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-5.md\\#thm-dynamics-comparator}\n\\DeclareResultEvidenceName{thm:fatal-absorbing}{Theorem 6.1}\n\\DeclareResultEvidenceRecord{thm:fatal-absorbing}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-fatal-absorbing}{exact}{compared}\n\\DeclareResultEvidence{thm:fatal-absorbing}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/GreedyGapCriteria.lean\\#L27}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-fatal-absorbing-comparator}\n\\DeclareResultEvidenceName{thm:seam-limit}{Theorem 6.2}\n\\DeclareResultEvidenceRecord{thm:seam-limit}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-seam-limit}{exact}{compared}\n\\DeclareResultEvidence{thm:seam-limit}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-seam-limit}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-seam-limit-comparator}\n\\DeclareResultEvidenceName{thm:two-channel-cap}{Theorem 6.3}\n\\DeclareResultEvidenceRecord{thm:two-channel-cap}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-two-channel-cap}{exact}{compared}\n\\DeclareResultEvidence{thm:two-channel-cap}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-two-channel-cap}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-two-channel-cap-comparator}\n\\DeclareResultEvidenceName{thm:second-channel}{Theorem 6.4}\n\\DeclareResultEvidenceRecord{thm:second-channel}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-second-channel}{exact}{compared}\n\\DeclareResultEvidence{thm:second-channel}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-second-channel}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-second-channel-comparator}\n\\DeclareResultEvidenceName{thm:straddle-closed-set}{Theorem 6.5}\n\\DeclareResultEvidenceRecord{thm:straddle-closed-set}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-straddle-closed-set}{exact}{compared}\n\\DeclareResultEvidence{thm:straddle-closed-set}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-straddle-closed-set}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-straddle-closed-set-comparator}\n\\DeclareResultEvidenceName{thm:largest-skip-late}{Theorem 6.6}\n\\DeclareResultEvidenceRecord{thm:largest-skip-late}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-largest-skip-late}{exact}{compared}\n\\DeclareResultEvidence{thm:largest-skip-late}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-largest-skip-late}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-largest-skip-late-comparator}\n\\DeclareResultEvidenceName{thm:middle-producer-escape}{Theorem 6.7}\n\\DeclareResultEvidenceRecord{thm:middle-producer-escape}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-middle-producer-escape}{exact}{compared}\n\\DeclareResultEvidence{thm:middle-producer-escape}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-middle-producer-escape}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-middle-producer-escape-comparator}\n\\DeclareResultEvidenceName{thm:middle-allright-defect}{Theorem 6.8}\n\\DeclareResultEvidenceRecord{thm:middle-allright-defect}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-middle-allright-defect}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:middle-allright-defect}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-middle-allright-defect}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-middle-allright-defect-comparator}\n\\DeclareResultEvidenceName{thm:two-sided-dyadic}{Theorem 6.9}\n\\DeclareResultEvidenceRecord{thm:two-sided-dyadic}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-two-sided-dyadic}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:two-sided-dyadic}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/HalfCylinderMiddleCarryLowerBound.lean\\#L4448}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-two-sided-dyadic-comparator}\n\\DeclareResultEvidenceName{thm:upper-reset-band}{Theorem 6.10}\n\\DeclareResultEvidenceRecord{thm:upper-reset-band}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-upper-reset-band}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:upper-reset-band}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-upper-reset-band}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-upper-reset-band-comparator}\n\\DeclareResultEvidenceName{thm:mobius-centred-nonneg}{Theorem 6.11}\n\\DeclareResultEvidenceRecord{thm:mobius-centred-nonneg}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-mobius-centred-nonneg}{exact}{compared}\n\\DeclareResultEvidence{thm:mobius-centred-nonneg}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-mobius-centred-nonneg}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-mobius-centred-nonneg-comparator}\n\\DeclareResultEvidenceName{thm:sqrt-bound-route}{Theorem 6.12}\n\\DeclareResultEvidenceRecord{thm:sqrt-bound-route}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-sqrt-bound-route}{exact}{compared}\n\\DeclareResultEvidence{thm:sqrt-bound-route}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-sqrt-bound-route}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-sqrt-bound-route-comparator}\n\\DeclareResultEvidenceName{thm:frozen-margin}{Theorem 6.13}\n\\DeclareResultEvidenceRecord{thm:frozen-margin}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-frozen-margin}{exact}{compared}\n\\DeclareResultEvidence{thm:frozen-margin}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-frozen-margin}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-frozen-margin-comparator}\n\\DeclareResultEvidenceName{thm:full-support-catalogue}{Theorem 6.14}\n\\DeclareResultEvidenceRecord{thm:full-support-catalogue}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-full-support-catalogue}{exact}{compared}\n\\DeclareResultEvidence{thm:full-support-catalogue}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/CertificateKernel.lean\\#L8328}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-full-support-catalogue-comparator}\n\\DeclareResultEvidenceName{thm:pairwise-coprime}{Theorem 6.15}\n\\DeclareResultEvidenceRecord{thm:pairwise-coprime}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-pairwise-coprime}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:pairwise-coprime}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/CertificateKernel.lean\\#L10776}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-pairwise-coprime-comparator}\n\\DeclareResultEvidenceName{thm:weighted-coeff-engine}{Theorem 6.16}\n\\DeclareResultEvidenceRecord{thm:weighted-coeff-engine}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-weighted-coeff-engine}{exact}{compared}\n\\DeclareResultEvidence{thm:weighted-coeff-engine}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/CertificateKernel.lean\\#L8665}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-weighted-coeff-engine-comparator}\n\\DeclareResultEvidenceName{thm:lcm-gap-engine}{Theorem 6.17}\n\\DeclareResultEvidenceRecord{thm:lcm-gap-engine}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-lcm-gap-engine}{exact}{compared}\n\\DeclareResultEvidence{thm:lcm-gap-engine}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/CertificateKernel.lean\\#L5883}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-lcm-gap-engine-comparator}\n\\DeclareResultEvidenceName{thm:factorial-twopow-support}{Theorem 6.18}\n\\DeclareResultEvidenceRecord{thm:factorial-twopow-support}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-factorial-twopow-support}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:factorial-twopow-support}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-factorial-twopow-support}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-factorial-twopow-support-comparator}\n\\DeclareResultEvidenceName{thm:multiples-support}{Theorem 6.19}\n\\DeclareResultEvidenceRecord{thm:multiples-support}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-multiples-support}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:multiples-support}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-multiples-support}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-multiples-support-comparator}\n\\DeclareResultEvidenceName{thm:periodic-support}{Theorem 6.20}\n\\DeclareResultEvidenceRecord{thm:periodic-support}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-periodic-support}{exact_or_stronger}{pending}\n\\DeclareResultEvidence{thm:periodic-support}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR20/PositivePeriodicSupport.lean\\#L49}{}\n\\DeclareResultEvidenceName{thm:eventually-periodic}{Theorem 6.21}\n\\DeclareResultEvidenceRecord{thm:eventually-periodic}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-eventually-periodic}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:eventually-periodic}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/CertificateKernel.lean\\#L11604}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-eventually-periodic-comparator}\n\\DeclareResultEvidenceName{thm:residue-odd}{Theorem 6.22}\n\\DeclareResultEvidenceRecord{thm:residue-odd}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-residue-odd}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:residue-odd}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-residue-odd}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-residue-odd-comparator}\n\\DeclareResultEvidenceName{thm:signed-periodic}{Theorem 6.23}\n\\DeclareResultEvidenceRecord{thm:signed-periodic}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-signed-periodic}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:signed-periodic}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-signed-periodic}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-signed-periodic-comparator}\n\\DeclareResultEvidenceName{thm:mersenne-channel-survival}{Theorem 6.24}\n\\DeclareResultEvidenceRecord{thm:mersenne-channel-survival}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-mersenne-channel-survival}{exact}{compared}\n\\DeclareResultEvidence{thm:mersenne-channel-survival}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-mersenne-channel-survival}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-mersenne-channel-survival-comparator}\n\\DeclareResultEvidenceName{thm:mersenne-channel-growth}{Theorem 6.25}\n\\DeclareResultEvidenceRecord{thm:mersenne-channel-growth}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-mersenne-channel-growth}{exact}{compared}\n\\DeclareResultEvidence{thm:mersenne-channel-growth}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-mersenne-channel-growth}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-mersenne-channel-growth-comparator}\n\\DeclareResultEvidenceName{thm:mobius-lambert-identity}{Theorem 6.28}\n\\DeclareResultEvidenceRecord{thm:mobius-lambert-identity}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-mobius-lambert-identity}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:mobius-lambert-identity}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/MersenneLambertLadder.lean\\#L587}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-mobius-lambert-identity-comparator}\n\\DeclareResultEvidenceName{cor:negative-mobius-overshoot}{Corollary 6.29}\n\\DeclareResultEvidenceRecord{cor:negative-mobius-overshoot}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#cor-negative-mobius-overshoot}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{cor:negative-mobius-overshoot}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#cor-negative-mobius-overshoot}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#cor-negative-mobius-overshoot-comparator}\n\\DeclareResultEvidenceName{thm:half-skip-dichotomy}{Theorem 6.31}\n\\DeclareResultEvidenceRecord{thm:half-skip-dichotomy}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-half-skip-dichotomy}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:half-skip-dichotomy}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-half-skip-dichotomy}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-half-skip-dichotomy-comparator}\n\\DeclareResultEvidenceName{thm:nine-way-hub}{Theorem 6.32}\n\\DeclareResultEvidenceRecord{thm:nine-way-hub}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-nine-way-hub}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:nine-way-hub}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-nine-way-hub}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#thm-nine-way-hub-comparator}\n\\DeclareResultEvidenceName{lem:eventually-right-impossible}{Lemma 6.33}\n\\DeclareResultEvidenceRecord{lem:eventually-right-impossible}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#lem-eventually-right-impossible}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{lem:eventually-right-impossible}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#lem-eventually-right-impossible}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#lem-eventually-right-impossible-comparator}\n\\DeclareResultEvidenceName{lem:mersenne-tail-weight}{Lemma 6.34}\n\\DeclareResultEvidenceRecord{lem:mersenne-tail-weight}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#lem-mersenne-tail-weight}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{lem:mersenne-tail-weight}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#lem-mersenne-tail-weight}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-1.md\\#lem-mersenne-tail-weight-comparator}\n\\DeclareResultEvidenceName{thm:greedy-survival-catalogue}{Theorem 6.35}\n\\DeclareResultEvidenceRecord{thm:greedy-survival-catalogue}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-greedy-survival-catalogue}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:greedy-survival-catalogue}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/GreedyAchievementSet.lean\\#L1458}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-greedy-survival-catalogue-comparator}\n\\DeclareResultEvidenceName{lem:rank-step-trichotomy}{Lemma 6.36}\n\\DeclareResultEvidenceRecord{lem:rank-step-trichotomy}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-rank-step-trichotomy}{exact_or_stronger}{pending}\n\\DeclareResultEvidence{lem:rank-step-trichotomy}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-rank-step-trichotomy}{}\n\\DeclareResultEvidenceName{lem:fatal-gap-exclusion}{Lemma 6.37}\n\\DeclareResultEvidenceRecord{lem:fatal-gap-exclusion}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-fatal-gap-exclusion}{exact}{compared}\n\\DeclareResultEvidence{lem:fatal-gap-exclusion}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-fatal-gap-exclusion}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-fatal-gap-exclusion-comparator}\n\\DeclareResultEvidenceName{lem:half-endpoint-kills}{Lemma 6.38}\n\\DeclareResultEvidenceRecord{lem:half-endpoint-kills}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-half-endpoint-kills}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{lem:half-endpoint-kills}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-half-endpoint-kills}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-half-endpoint-kills-comparator}\n\\DeclareResultEvidenceName{lem:straddle-agrees-greedy}{Lemma 6.39}\n\\DeclareResultEvidenceRecord{lem:straddle-agrees-greedy}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-straddle-agrees-greedy}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{lem:straddle-agrees-greedy}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/HalfCutLocator.lean\\#L442}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-straddle-agrees-greedy-comparator}\n\\DeclareResultEvidenceName{thm:last-skip-iff-fatal}{Theorem 6.40}\n\\DeclareResultEvidenceRecord{thm:last-skip-iff-fatal}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-last-skip-iff-fatal}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:last-skip-iff-fatal}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-last-skip-iff-fatal}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-last-skip-iff-fatal-comparator}\n\\DeclareResultEvidenceName{lem:seam-upper-or-middle}{Lemma 6.41}\n\\DeclareResultEvidenceRecord{lem:seam-upper-or-middle}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-seam-upper-or-middle}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{lem:seam-upper-or-middle}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean\\#L57}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-seam-upper-or-middle-comparator}\n\\DeclareResultEvidenceName{lem:largest-false-rank-algebra}{Lemma 6.42}\n\\DeclareResultEvidenceRecord{lem:largest-false-rank-algebra}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-largest-false-rank-algebra}{exact}{compared}\n\\DeclareResultEvidence{lem:largest-false-rank-algebra}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-largest-false-rank-algebra}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-largest-false-rank-algebra-comparator}\n\\DeclareResultEvidenceName{thm:critical-dyadic-band}{Theorem 6.43}\n\\DeclareResultEvidenceRecord{thm:critical-dyadic-band}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-critical-dyadic-band}{exact}{compared}\n\\DeclareResultEvidence{thm:critical-dyadic-band}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-critical-dyadic-band}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-critical-dyadic-band-comparator}\n\\DeclareResultEvidenceName{thm:final-middle-cell}{Theorem 6.44}\n\\DeclareResultEvidenceRecord{thm:final-middle-cell}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-final-middle-cell}{exact}{compared}\n\\DeclareResultEvidence{thm:final-middle-cell}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-final-middle-cell}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-final-middle-cell-comparator}\n\\DeclareResultEvidenceName{lem:skipped-endpoint-trichotomy}{Lemma 6.45}\n\\DeclareResultEvidenceRecord{lem:skipped-endpoint-trichotomy}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-skipped-endpoint-trichotomy}{exact}{compared}\n\\DeclareResultEvidence{lem:skipped-endpoint-trichotomy}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/HalfCylinderSkippedEndpointClassifier.lean\\#L246}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-skipped-endpoint-trichotomy-comparator}\n\\DeclareResultEvidenceName{lem:reverse-carry-word}{Lemma 6.46}\n\\DeclareResultEvidenceRecord{lem:reverse-carry-word}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-reverse-carry-word}{exact}{compared}\n\\DeclareResultEvidence{lem:reverse-carry-word}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-reverse-carry-word}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-reverse-carry-word-comparator}\n\\DeclareResultEvidenceName{lem:linear-channel-nogo}{Lemma 6.47}\n\\DeclareResultEvidenceRecord{lem:linear-channel-nogo}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-linear-channel-nogo}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{lem:linear-channel-nogo}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-linear-channel-nogo}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-linear-channel-nogo-comparator}\n\\DeclareResultEvidenceName{thm:two-thirds-band}{Theorem 6.48}\n\\DeclareResultEvidenceRecord{thm:two-thirds-band}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-two-thirds-band}{exact}{compared}\n\\DeclareResultEvidence{thm:two-thirds-band}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/PostTakeBandLocalisation.lean\\#L96}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-two-thirds-band-comparator}\n\\DeclareResultEvidenceName{thm:sharp-fatal-gap}{Theorem 6.49}\n\\DeclareResultEvidenceRecord{thm:sharp-fatal-gap}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-sharp-fatal-gap}{exact}{compared}\n\\DeclareResultEvidence{thm:sharp-fatal-gap}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/GreedyGapCriteria.lean\\#L299}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-sharp-fatal-gap-comparator}\n\\DeclareResultEvidenceName{lem:gap-mass-summability}{Lemma 6.50}\n\\DeclareResultEvidenceRecord{lem:gap-mass-summability}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-gap-mass-summability}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{lem:gap-mass-summability}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-gap-mass-summability}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-gap-mass-summability-comparator}\n\\DeclareResultEvidenceName{lem:half-divisor-unit-drop}{Lemma 6.51}\n\\DeclareResultEvidenceRecord{lem:half-divisor-unit-drop}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-half-divisor-unit-drop}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{lem:half-divisor-unit-drop}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-half-divisor-unit-drop}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-half-divisor-unit-drop-comparator}\n\\DeclareResultEvidenceName{thm:tempered-orbit-rigidity}{Theorem 6.52}\n\\DeclareResultEvidenceRecord{thm:tempered-orbit-rigidity}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-tempered-orbit-rigidity}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:tempered-orbit-rigidity}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-tempered-orbit-rigidity}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-tempered-orbit-rigidity-comparator}\n\\DeclareResultEvidenceName{lem:tail-transfer}{Lemma 6.53}\n\\DeclareResultEvidenceRecord{lem:tail-transfer}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-tail-transfer}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{lem:tail-transfer}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-tail-transfer}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-tail-transfer-comparator}\n\\DeclareResultEvidenceName{lem:dyadic-excess-reformulation}{Lemma 6.54}\n\\DeclareResultEvidenceRecord{lem:dyadic-excess-reformulation}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-dyadic-excess-reformulation}{exact_or_stronger}{pending}\n\\DeclareResultEvidence{lem:dyadic-excess-reformulation}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-dyadic-excess-reformulation}{}\n\\DeclareResultEvidenceName{lem:denominator-sandwich}{Lemma 6.55}\n\\DeclareResultEvidenceRecord{lem:denominator-sandwich}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-denominator-sandwich}{exact_or_stronger}{pending}\n\\DeclareResultEvidence{lem:denominator-sandwich}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-denominator-sandwich}{}\n\\DeclareResultEvidenceName{lem:denominator-survival}{Lemma 6.56}\n\\DeclareResultEvidenceRecord{lem:denominator-survival}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-denominator-survival}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{lem:denominator-survival}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-denominator-survival}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-denominator-survival-comparator}\n\\DeclareResultEvidenceName{lem:mixed-prime-power-layer}{Lemma 6.57}\n\\DeclareResultEvidenceRecord{lem:mixed-prime-power-layer}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-mixed-prime-power-layer}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{lem:mixed-prime-power-layer}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-mixed-prime-power-layer}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#lem-mixed-prime-power-layer-comparator}\n\\DeclareResultEvidenceName{prop:achievement-set-topology}{Proposition 6.59}\n\\DeclareResultEvidenceRecord{prop:achievement-set-topology}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#prop-achievement-set-topology}{exact}{compared}\n\\DeclareResultEvidence{prop:achievement-set-topology}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/AchievementSetTopologyAndFiniteHalf.lean\\#L27}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#prop-achievement-set-topology-comparator}\n\\DeclareResultEvidenceName{thm:master-dichotomy}{Theorem 6.60}\n\\DeclareResultEvidenceRecord{thm:master-dichotomy}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-master-dichotomy}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:master-dichotomy}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-master-dichotomy}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-master-dichotomy-comparator}\n\\DeclareResultEvidenceName{thm:perturbed-family-maximality}{Theorem 6.62}\n\\DeclareResultEvidenceRecord{thm:perturbed-family-maximality}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-perturbed-family-maximality}{exact_or_stronger}{pending}\n\\DeclareResultEvidence{thm:perturbed-family-maximality}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#thm-perturbed-family-maximality}{}\n\\DeclareResultEvidenceName{record:257bm-c1}{Theorem 6.67}\n\\DeclareResultEvidenceRecord{record:257bm-c1}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#record-257bm-c1}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-c1}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#record-257bm-c1}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#record-257bm-c1-comparator}\n\\DeclareResultEvidenceName{record:257bm-c2}{Theorem 6.68}\n\\DeclareResultEvidenceRecord{record:257bm-c2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#record-257bm-c2}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-c2}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#record-257bm-c2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#record-257bm-c2-comparator}\n\\DeclareResultEvidenceName{record:257bm-c3}{Theorem 6.69}\n\\DeclareResultEvidenceRecord{record:257bm-c3}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#record-257bm-c3}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-c3}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#record-257bm-c3}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#record-257bm-c3-comparator}\n\\DeclareResultEvidenceName{record:257bm-c4}{Theorem 6.70}\n\\DeclareResultEvidenceRecord{record:257bm-c4}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#record-257bm-c4}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-c4}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#record-257bm-c4}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#record-257bm-c4-comparator}\n\\DeclareResultEvidenceName{record:257bm-c5}{Theorem 6.71}\n\\DeclareResultEvidenceRecord{record:257bm-c5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#record-257bm-c5}{exact}{pending}\n\\DeclareResultEvidence{record:257bm-c5}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-2.md\\#record-257bm-c5}{}\n\\DeclareResultEvidenceName{record:257bm-c6}{Theorem 6.72}\n\\DeclareResultEvidenceRecord{record:257bm-c6}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c6}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-c6}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/BooleanMobiusCriticalCapacityCofinal.lean\\#L1068}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c6-comparator}\n\\DeclareResultEvidenceName{record:257bm-c6a}{Theorem 6.73}\n\\DeclareResultEvidenceRecord{record:257bm-c6a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c6a}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-c6a}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c6a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c6a-comparator}\n\\DeclareResultEvidenceName{record:257bm-c6b}{Theorem 6.74}\n\\DeclareResultEvidenceRecord{record:257bm-c6b}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c6b}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-c6b}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c6b}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c6b-comparator}\n\\DeclareResultEvidenceName{record:257bm-c7}{Theorem 6.75}\n\\DeclareResultEvidenceRecord{record:257bm-c7}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c7}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-c7}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/BooleanMobiusSkipRow.lean\\#L238}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c7-comparator}\n\\DeclareResultEvidenceName{record:257bm-c8}{Theorem 6.76}\n\\DeclareResultEvidenceRecord{record:257bm-c8}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c8}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-c8}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/BooleanMobiusSkipRow.lean\\#L332}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c8-comparator}\n\\DeclareResultEvidenceName{record:257bm-c9}{Proposition 6.77}\n\\DeclareResultEvidenceRecord{record:257bm-c9}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c9}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-c9}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c9}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c9-comparator}\n\\DeclareResultEvidenceName{record:257bm-c10}{Theorem 6.78}\n\\DeclareResultEvidenceRecord{record:257bm-c10}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c10}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-c10}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/MersenneQuotientRowRecurrences.lean\\#L264}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c10-comparator}\n\\DeclareResultEvidenceName{record:257bm-c11}{Theorem 6.79}\n\\DeclareResultEvidenceRecord{record:257bm-c11}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c11}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-c11}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c11}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c11-comparator}\n\\DeclareResultEvidenceName{record:257bm-c12}{Proposition 6.80}\n\\DeclareResultEvidenceRecord{record:257bm-c12}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c12}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-c12}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c12}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c12-comparator}\n\\DeclareResultEvidenceName{record:257bm-c14}{Theorem 6.82}\n\\DeclareResultEvidenceRecord{record:257bm-c14}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c14}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-c14}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c14}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c14-comparator}\n\\DeclareResultEvidenceName{record:257bm-c15}{Theorem 6.83}\n\\DeclareResultEvidenceRecord{record:257bm-c15}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c15}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-c15}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c15}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c15-comparator}\n\\DeclareResultEvidenceName{record:257rig-c16}{Theorem 6.84}\n\\DeclareResultEvidenceRecord{record:257rig-c16}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257rig-c16}{exact}{compared}\n\\DeclareResultEvidence{record:257rig-c16}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257rig-c16}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257rig-c16-comparator}\n\\DeclareResultEvidenceName{record:257rig-c17}{Theorem 6.85}\n\\DeclareResultEvidenceRecord{record:257rig-c17}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257rig-c17}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{record:257rig-c17}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257rig-c17}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257rig-c17-comparator}\n\\DeclareResultEvidenceName{record:257rig-c18}{Theorem 6.86}\n\\DeclareResultEvidenceRecord{record:257rig-c18}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257rig-c18}{exact}{compared}\n\\DeclareResultEvidence{record:257rig-c18}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257rig-c18}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257rig-c18-comparator}\n\\DeclareResultEvidenceName{record:257bm-c19}{Theorem 6.87}\n\\DeclareResultEvidenceRecord{record:257bm-c19}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c19}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{record:257bm-c19}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c19}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c19-comparator}\n\\DeclareResultEvidenceName{record:257bm-c20}{Theorem 6.88}\n\\DeclareResultEvidenceRecord{record:257bm-c20}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c20}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-c20}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c20}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-c20-comparator}\n\\DeclareResultEvidenceName{record:257bm-i1a}{Theorem 6.89}\n\\DeclareResultEvidenceRecord{record:257bm-i1a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i1a}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i1a}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i1a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i1a-comparator}\n\\DeclareResultEvidenceName{record:257bm-i1b}{Theorem 6.90}\n\\DeclareResultEvidenceRecord{record:257bm-i1b}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i1b}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i1b}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/BooleanMobiusExactTransition.lean\\#L131}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i1b-comparator}\n\\DeclareResultEvidenceName{record:257bm-i1c}{Theorem 6.91}\n\\DeclareResultEvidenceRecord{record:257bm-i1c}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i1c}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i1c}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i1c}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i1c-comparator}\n\\DeclareResultEvidenceName{record:257bm-i5}{Theorem 6.93}\n\\DeclareResultEvidenceRecord{record:257bm-i5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i5}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i5}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/BooleanMobiusSkippedCoreCriticalCapacity.lean\\#L22}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i5-comparator}\n\\DeclareResultEvidenceName{record:257bm-i6}{Theorem 6.94}\n\\DeclareResultEvidenceRecord{record:257bm-i6}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i6}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i6}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i6}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i6-comparator}\n\\DeclareResultEvidenceName{record:257bm-i7}{Theorem 6.95}\n\\DeclareResultEvidenceRecord{record:257bm-i7}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i7}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i7}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i7}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i7-comparator}\n\\DeclareResultEvidenceName{record:257bm-i-rank2}{Proposition 6.96}\n\\DeclareResultEvidenceRecord{record:257bm-i-rank2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i-rank2}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i-rank2}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/BooleanMobiusExactRowRankTwo.lean\\#L23}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i-rank2-comparator}\n\\DeclareResultEvidenceName{record:257bm-i9}{Proposition 6.98}\n\\DeclareResultEvidenceRecord{record:257bm-i9}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i9}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i9}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/BooleanMobiusGlobalRepair.lean\\#L243}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i9-comparator}\n\\DeclareResultEvidenceName{record:257bm-i10}{Proposition 6.99}\n\\DeclareResultEvidenceRecord{record:257bm-i10}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i10}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i10}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i10}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i10-comparator}\n\\DeclareResultEvidenceName{record:257bm-i11a}{Theorem 6.100}\n\\DeclareResultEvidenceRecord{record:257bm-i11a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i11a}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i11a}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/BooleanMobiusGreedyReduction.lean\\#L918}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i11a-comparator}\n\\DeclareResultEvidenceName{record:257bm-i11b}{Proposition 6.101}\n\\DeclareResultEvidenceRecord{record:257bm-i11b}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i11b}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i11b}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i11b}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i11b-comparator}\n\\DeclareResultEvidenceName{record:257bm-i11c}{Proposition 6.102}\n\\DeclareResultEvidenceRecord{record:257bm-i11c}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i11c}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i11c}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/BooleanMobiusGreedyReduction.lean\\#L997}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i11c-comparator}\n\\DeclareResultEvidenceName{record:257bm-i12}{Theorem 6.103}\n\\DeclareResultEvidenceRecord{record:257bm-i12}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i12}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i12}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i12}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i12-comparator}\n\\DeclareResultEvidenceName{record:257bm-i2}{Theorem 6.104}\n\\DeclareResultEvidenceRecord{record:257bm-i2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i2}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{record:257bm-i2}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i2-comparator}\n\\DeclareResultEvidenceName{record:257bm-i-t7}{Theorem 6.105}\n\\DeclareResultEvidenceRecord{record:257bm-i-t7}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i-t7}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{record:257bm-i-t7}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/GenericTailOrbitRigidity.lean\\#L435}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-3.md\\#record-257bm-i-t7-comparator}\n\\DeclareResultEvidenceName{record:257bm-i-mob}{Theorem 6.106}\n\\DeclareResultEvidenceRecord{record:257bm-i-mob}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i-mob}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{record:257bm-i-mob}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i-mob}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i-mob-comparator}\n\\DeclareResultEvidenceName{record:257bm-i-bridge}{Theorem 6.107}\n\\DeclareResultEvidenceRecord{record:257bm-i-bridge}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i-bridge}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{record:257bm-i-bridge}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i-bridge}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i-bridge-comparator}\n\\DeclareResultEvidenceName{record:257rig-i2}{Theorem 6.108}\n\\DeclareResultEvidenceRecord{record:257rig-i2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-i2}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{record:257rig-i2}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-i2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-i2-comparator}\n\\DeclareResultEvidenceName{record:257rig-i3}{Theorem 6.109}\n\\DeclareResultEvidenceRecord{record:257rig-i3}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-i3}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{record:257rig-i3}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-i3}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-i3-comparator}\n\\DeclareResultEvidenceName{record:257rig-i4a}{Theorem 6.110}\n\\DeclareResultEvidenceRecord{record:257rig-i4a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-i4a}{exact}{compared}\n\\DeclareResultEvidence{record:257rig-i4a}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-i4a}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-i4a-comparator}\n\\DeclareResultEvidenceName{record:257rig-i4b}{Proposition 6.111}\n\\DeclareResultEvidenceRecord{record:257rig-i4b}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-i4b}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{record:257rig-i4b}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-i4b}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-i4b-comparator}\n\\DeclareResultEvidenceName{record:257rig-i5}{Proposition 6.112}\n\\DeclareResultEvidenceRecord{record:257rig-i5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-i5}{exact}{compared}\n\\DeclareResultEvidence{record:257rig-i5}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-i5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-i5-comparator}\n\\DeclareResultEvidenceName{record:257bm-i-cross2}{Theorem 6.114}\n\\DeclareResultEvidenceRecord{record:257bm-i-cross2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i-cross2}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i-cross2}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/BooleanMobiusExactRowCrossing.lean\\#L31}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i-cross2-comparator}\n\\DeclareResultEvidenceName{record:257hg-i6}{Theorem 6.115}\n\\DeclareResultEvidenceRecord{record:257hg-i6}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-i6}{exact}{compared}\n\\DeclareResultEvidence{record:257hg-i6}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-i6}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-i6-comparator}\n\\DeclareResultEvidenceName{record:257hg-i7}{Theorem 6.116}\n\\DeclareResultEvidenceRecord{record:257hg-i7}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-i7}{exact}{compared}\n\\DeclareResultEvidence{record:257hg-i7}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/HalfCylinderSkippedEndpointClassifier.lean\\#L246}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-i7-comparator}\n\\DeclareResultEvidenceName{record:257hg-i2}{Theorem 6.118}\n\\DeclareResultEvidenceRecord{record:257hg-i2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-i2}{exact}{compared}\n\\DeclareResultEvidence{record:257hg-i2}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-i2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-i2-comparator}\n\\DeclareResultEvidenceName{record:257hg-i4}{Theorem 6.120}\n\\DeclareResultEvidenceRecord{record:257hg-i4}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-i4}{exact}{compared}\n\\DeclareResultEvidence{record:257hg-i4}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-i4}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-i4-comparator}\n\\DeclareResultEvidenceName{record:257hg-i5}{Proposition 6.121}\n\\DeclareResultEvidenceRecord{record:257hg-i5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-i5}{exact}{compared}\n\\DeclareResultEvidence{record:257hg-i5}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-i5}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-i5-comparator}\n\\DeclareResultEvidenceName{record:257bm-i13}{Theorem 6.122}\n\\DeclareResultEvidenceRecord{record:257bm-i13}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i13}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i13}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/DyadicBandAndTwoSidedBounds.lean\\#L105}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i13-comparator}\n\\DeclareResultEvidenceName{record:257bm-i14}{Proposition 6.123}\n\\DeclareResultEvidenceRecord{record:257bm-i14}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i14}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i14}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i14}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i14-comparator}\n\\DeclareResultEvidenceName{record:257bm-i15}{Theorem 6.124}\n\\DeclareResultEvidenceRecord{record:257bm-i15}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i15}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i15}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i15}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i15-comparator}\n\\DeclareResultEvidenceName{record:257bm-i16}{Theorem 6.125}\n\\DeclareResultEvidenceRecord{record:257bm-i16}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i16}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-i16}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i16}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i16-comparator}\n\\DeclareResultEvidenceName{record:257bm-i17}{Proposition 6.126}\n\\DeclareResultEvidenceRecord{record:257bm-i17}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i17}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{record:257bm-i17}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i17}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-i17-comparator}\n\\DeclareResultEvidenceName{record:257bm-k1}{Theorem 6.127}\n\\DeclareResultEvidenceRecord{record:257bm-k1}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-k1}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-k1}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-k1}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-k1-comparator}\n\\DeclareResultEvidenceName{record:257bm-k-dich}{Theorem 6.128}\n\\DeclareResultEvidenceRecord{record:257bm-k-dich}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-k-dich}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-k-dich}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-k-dich}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-k-dich-comparator}\n\\DeclareResultEvidenceName{record:257bm-k2}{Proposition 6.129}\n\\DeclareResultEvidenceRecord{record:257bm-k2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-k2}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-k2}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-k2}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-k2-comparator}\n\\DeclareResultEvidenceName{record:257bm-k4}{Theorem 6.131}\n\\DeclareResultEvidenceRecord{record:257bm-k4}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-k4}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-k4}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-k4}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-k4-comparator}\n\\DeclareResultEvidenceName{record:257rig-k6}{Theorem 6.133}\n\\DeclareResultEvidenceRecord{record:257rig-k6}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-k6}{exact}{compared}\n\\DeclareResultEvidence{record:257rig-k6}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/ExactRowDichotomyCountermodels.lean\\#L231}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257rig-k6-comparator}\n\\DeclareResultEvidenceName{record:257bm-k9}{Theorem 6.136}\n\\DeclareResultEvidenceRecord{record:257bm-k9}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-k9}{exact}{compared}\n\\DeclareResultEvidence{record:257bm-k9}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-k9}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257bm-k9-comparator}\n\\DeclareResultEvidenceName{record:257hg-k12}{Theorem 6.139}\n\\DeclareResultEvidenceRecord{record:257hg-k12}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-k12}{exact}{compared}\n\\DeclareResultEvidence{record:257hg-k12}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-k12}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#record-257hg-k12-comparator}\n\\DeclareResultEvidenceName{prop:unsafe-middle-range-exactly-three}{Proposition 6.140}\n\\DeclareResultEvidenceRecord{prop:unsafe-middle-range-exactly-three}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#prop-unsafe-middle-range-exactly-three}{exact}{compared}\n\\DeclareResultEvidence{prop:unsafe-middle-range-exactly-three}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/LinearChannelAndMiddleCellExclusion.lean\\#L83}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#prop-unsafe-middle-range-exactly-three-comparator}\n\\DeclareResultEvidenceName{prop:finite-approximations-without-compatibility}{Proposition 6.142}\n\\DeclareResultEvidenceRecord{prop:finite-approximations-without-compatibility}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#prop-finite-approximations-without-compatibility}{exact}{compared}\n\\DeclareResultEvidence{prop:finite-approximations-without-compatibility}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#prop-finite-approximations-without-compatibility}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#prop-finite-approximations-without-compatibility-comparator}\n\\DeclareResultEvidenceName{prop:exact-lebesgue-measure-dichotomy}{Proposition 6.146}\n\\DeclareResultEvidenceRecord{prop:exact-lebesgue-measure-dichotomy}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#prop-exact-lebesgue-measure-dichotomy}{exact}{compared}\n\\DeclareResultEvidence{prop:exact-lebesgue-measure-dichotomy}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/SharedPrefixFamiliesAndMeasureDichotomy.lean\\#L68}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-6-4.md\\#prop-exact-lebesgue-measure-dichotomy-comparator}\n\\DeclareResultEvidenceName{prop:one-orbit}{Proposition 8.1}\n\\DeclareResultEvidenceRecord{prop:one-orbit}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#prop-one-orbit}{exact}{compared}\n\\DeclareResultEvidence{prop:one-orbit}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#prop-one-orbit}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#prop-one-orbit-comparator}\n\\DeclareResultEvidenceName{lem:no-ties}{Lemma 8.2}\n\\DeclareResultEvidenceRecord{lem:no-ties}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#lem-no-ties}{exact}{compared}\n\\DeclareResultEvidence{lem:no-ties}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#lem-no-ties}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#lem-no-ties-comparator}\n\\DeclareResultEvidenceName{lem:tr-forced-greedy}{Lemma 8.4}\n\\DeclareResultEvidenceRecord{lem:tr-forced-greedy}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#lem-tr-forced-greedy}{exact_or_stronger}{pending}\n\\DeclareResultEvidence{lem:tr-forced-greedy}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#lem-tr-forced-greedy}{}\n\\DeclareResultEvidenceName{lem:tr-parity}{Lemma 8.5}\n\\DeclareResultEvidenceRecord{lem:tr-parity}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#lem-tr-parity}{exact}{compared}\n\\DeclareResultEvidence{lem:tr-parity}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/TruncatedRungWitnessHorizon.lean\\#L176}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#lem-tr-parity-comparator}\n\\DeclareResultEvidenceName{thm:tr-witness-exclusion}{Theorem 8.6}\n\\DeclareResultEvidenceRecord{thm:tr-witness-exclusion}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#thm-tr-witness-exclusion}{exact}{compared}\n\\DeclareResultEvidence{thm:tr-witness-exclusion}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/TruncatedRungWitnessHorizon.lean\\#L250}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#thm-tr-witness-exclusion-comparator}\n\\DeclareResultEvidenceName{cor:tr-half-lcm}{Corollary 8.7}\n\\DeclareResultEvidenceRecord{cor:tr-half-lcm}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#cor-tr-half-lcm}{exact}{compared}\n\\DeclareResultEvidence{cor:tr-half-lcm}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/TruncatedRungWitnessHorizon.lean\\#L467}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#cor-tr-half-lcm-comparator}\n\\DeclareResultEvidenceName{lem:tr-mod12}{Lemma 8.8}\n\\DeclareResultEvidenceRecord{lem:tr-mod12}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#lem-tr-mod12}{exact}{compared}\n\\DeclareResultEvidence{lem:tr-mod12}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/TruncatedRungWitnessHorizon.lean\\#L544}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#lem-tr-mod12-comparator}\n\\DeclareResultEvidenceName{thm:tr-finite-decision}{Theorem 8.9}\n\\DeclareResultEvidenceRecord{thm:tr-finite-decision}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#thm-tr-finite-decision}{exact}{compared}\n\\DeclareResultEvidence{thm:tr-finite-decision}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/TruncatedRungGreedyDecision.lean\\#L896}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-8.md\\#thm-tr-finite-decision-comparator}\n\\DeclareResultEvidenceName{thm:lower-bound-every-reset}{Theorem 9.46}\n\\DeclareResultEvidenceRecord{thm:lower-bound-every-reset}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-9.md\\#thm-lower-bound-every-reset}{exact}{compared}\n\\DeclareResultEvidence{thm:lower-bound-every-reset}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-9.md\\#thm-lower-bound-every-reset}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-9.md\\#thm-lower-bound-every-reset-comparator}\n\\DeclareResultEvidenceName{thm:one-sided-finite-decision-boundary}{Theorem 9.48}\n\\DeclareResultEvidenceRecord{thm:one-sided-finite-decision-boundary}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-9.md\\#thm-one-sided-finite-decision-boundary}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{thm:one-sided-finite-decision-boundary}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/CentredCompletionAndDecisionBoundary.lean\\#L77}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-9.md\\#thm-one-sided-finite-decision-boundary-comparator}\n\\DeclareResultEvidenceName{prop:2adic-nogo}{Proposition 10.4}\n\\DeclareResultEvidenceRecord{prop:2adic-nogo}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-2adic-nogo}{exact}{compared}\n\\DeclareResultEvidence{prop:2adic-nogo}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/CentredCompletionAndDecisionBoundary.lean\\#L34}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-2adic-nogo-comparator}\n\\DeclareResultEvidenceName{prop:upper-unconditional}{Proposition 10.5}\n\\DeclareResultEvidenceRecord{prop:upper-unconditional}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-upper-unconditional}{exact}{compared}\n\\DeclareResultEvidence{prop:upper-unconditional}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/BranchCellHorizonExclusions.lean\\#L31}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-upper-unconditional-comparator}\n\\DeclareResultEvidenceName{thm:cd-neg3-impossible}{Theorem 10.6}\n\\DeclareResultEvidenceRecord{thm:cd-neg3-impossible}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#thm-cd-neg3-impossible}{exact}{compared}\n\\DeclareResultEvidence{thm:cd-neg3-impossible}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/BranchCellHorizonExclusions.lean\\#L50}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#thm-cd-neg3-impossible-comparator}\n\\DeclareResultEvidenceName{cor:cd-remaining}{Corollary 10.7}\n\\DeclareResultEvidenceRecord{cor:cd-remaining}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#cor-cd-remaining}{exact}{compared}\n\\DeclareResultEvidence{cor:cd-remaining}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/BranchCellHorizonExclusions.lean\\#L81}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#cor-cd-remaining-comparator}\n\\DeclareResultEvidenceName{prop:finite-state-nogo}{Proposition 10.9}\n\\DeclareResultEvidenceRecord{prop:finite-state-nogo}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-finite-state-nogo}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{prop:finite-state-nogo}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-finite-state-nogo}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-finite-state-nogo-comparator}\n\\DeclareResultEvidenceName{prop:mobius-nogo}{Proposition 10.10}\n\\DeclareResultEvidenceRecord{prop:mobius-nogo}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-mobius-nogo}{exact}{compared}\n\\DeclareResultEvidence{prop:mobius-nogo}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-mobius-nogo}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-mobius-nogo-comparator}\n\\DeclareResultEvidenceName{prop:finite-boolSupport-and-onesided}{Proposition 10.11}\n\\DeclareResultEvidenceRecord{prop:finite-boolSupport-and-onesided}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-finite-boolsupport-and-onesided}{exact}{compared}\n\\DeclareResultEvidence{prop:finite-boolSupport-and-onesided}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/MobiusSignAndFiniteCertificates.lean\\#L101}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-finite-boolsupport-and-onesided-comparator}\n\\DeclareResultEvidenceName{prop:carry-survivor-extinction}{Proposition 10.12}\n\\DeclareResultEvidenceRecord{prop:carry-survivor-extinction}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-carry-survivor-extinction}{exact}{compared}\n\\DeclareResultEvidence{prop:carry-survivor-extinction}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-carry-survivor-extinction}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-carry-survivor-extinction-comparator}\n\\DeclareResultEvidenceName{lem:scalar-localization}{Lemma 10.13}\n\\DeclareResultEvidenceRecord{lem:scalar-localization}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#lem-scalar-localization}{exact}{compared}\n\\DeclareResultEvidence{lem:scalar-localization}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#lem-scalar-localization}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#lem-scalar-localization-comparator}\n\\DeclareResultEvidenceName{cor:mersenne-height}{Corollary 10.14}\n\\DeclareResultEvidenceRecord{cor:mersenne-height}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#cor-mersenne-height}{exact}{compared}\n\\DeclareResultEvidence{cor:mersenne-height}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/ScalarLocalisationHeightObstruction.lean\\#L77}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#cor-mersenne-height-comparator}\n\\DeclareResultEvidenceName{prop:critical-band-index}{Proposition 10.16}\n\\DeclareResultEvidenceRecord{prop:critical-band-index}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-critical-band-index}{exact}{compared}\n\\DeclareResultEvidence{prop:critical-band-index}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/CriticalDyadicBandCollapse.lean\\#L20}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-10.md\\#prop-critical-band-index-comparator}\n\\DeclareResultEvidenceName{lem:odometer}{Lemma 11.2}\n\\DeclareResultEvidenceRecord{lem:odometer}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-11.md\\#lem-odometer}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{lem:odometer}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-11.md\\#lem-odometer}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-11.md\\#lem-odometer-comparator}\n\\DeclareResultEvidenceName{lem:sqwitness}{Lemma 11.3}\n\\DeclareResultEvidenceRecord{lem:sqwitness}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-11.md\\#lem-sqwitness}{exact}{compared}\n\\DeclareResultEvidence{lem:sqwitness}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-11.md\\#lem-sqwitness}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-11.md\\#lem-sqwitness-comparator}\n\\DeclareResultEvidenceName{prop:squarefree}{Proposition 11.4}\n\\DeclareResultEvidenceRecord{prop:squarefree}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-11.md\\#prop-squarefree}{exact}{compared}\n\\DeclareResultEvidence{prop:squarefree}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/SquarefreeSupportEngineCeiling.lean\\#L20}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-11.md\\#prop-squarefree-comparator}\n\\DeclareResultEvidenceName{prop:cpgs-equiv}{Proposition 12.3}\n\\DeclareResultEvidenceRecord{prop:cpgs-equiv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-12.md\\#prop-cpgs-equiv}{exact}{compared}\n\\DeclareResultEvidence{prop:cpgs-equiv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/SquareDepthAndHalfMembershipEquivalences.lean\\#L44}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-12.md\\#prop-cpgs-equiv-comparator}\n\\DeclareResultEvidenceName{prop:strip-equiv}{Proposition 12.4}\n\\DeclareResultEvidenceRecord{prop:strip-equiv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-12.md\\#prop-strip-equiv}{exact}{compared}\n\\DeclareResultEvidence{prop:strip-equiv}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-12.md\\#prop-strip-equiv}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-12.md\\#prop-strip-equiv-comparator}\n\\DeclareResultEvidenceName{thm:257-logarithmic-counterexample}{Theorem 13.1}\n\\DeclareResultEvidenceRecord{thm:257-logarithmic-counterexample}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-13.md\\#thm-257-logarithmic-counterexample}{exact}{compared}\n\\DeclareResultEvidence{thm:257-logarithmic-counterexample}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-13.md\\#thm-257-logarithmic-counterexample}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-13.md\\#thm-257-logarithmic-counterexample-comparator}\n\\DeclareResultEvidenceName{cor:257-logarithmic-separation}{Corollary 13.2}\n\\DeclareResultEvidenceRecord{cor:257-logarithmic-separation}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-13.md\\#cor-257-logarithmic-separation}{exact}{compared}\n\\DeclareResultEvidence{cor:257-logarithmic-separation}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-13.md\\#cor-257-logarithmic-separation}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-13.md\\#cor-257-logarithmic-separation-comparator}\n\\DeclareResultEvidenceName{prop:257-logarithmic-initial-interval}{Proposition 13.3}\n\\DeclareResultEvidenceRecord{prop:257-logarithmic-initial-interval}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-13.md\\#prop-257-logarithmic-initial-interval}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{prop:257-logarithmic-initial-interval}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos257/PaperCompleteR21/LogarithmicInitialInterval.lean\\#L491}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos257-mersenne-reasoning-surface/section-13.md\\#prop-257-logarithmic-initial-interval-comparator}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% 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It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/257/erdos257-mersenne-reasoning-surface.tex","section_outline":["Introduction","Notation used throughout","Support criteria and their proofs","Reciprocal-summable supports","A weighted condition on the support","Dependence on the chosen primes","A finite estimate on common observation scales","Positive divisor majorants","Combining the two support criteria","The universal problem and the half-value question","Known support theorems","Geometry and rational-target membership","Limitations of the recorded methods","What the obstructions do and do not show","Uniqueness of the specified greedy words","Equivalent formulations of half-membership","The size of the required error bounds","What finite certificates decide","Why measure and category do not decide a rational target","Hypotheses of the CRT block-certificate argument","The distinct estimates still required","Unproved inputs for further arguments","Two further analytic tests","A lower bound at reset ranks","Increasing the depths of exact finite sums","The known prime-support theorem","Equivalent membership conditions","Why zero windows do not split the support class","Comparison of the remaining inputs","Reading the detailed record","The series, the greedy rule and finite identities","The object","What is proved","Full-support irrationality, for every base","Structured infinite supports","Topology and measure of the achievement set","The greedy rule for one half","Integer quotients and their remainder identity","A real-valued form of the quotient identity","Dynamics"]},"erdos257-expository-sums-never-fractions":{"paper_id":"erdos257-expository-sums-never-fractions","title":"Why certain sums of 1/(2^n-1) are irrational","form":"Expository paper","length_class":"short","pages":16,"subject":"Erdős #257","subject_kind":"problem","problem_id":"erdos_257","landing_role":"mathematics","question":"Why do the fractions 1/(2^n−1) over an infinite set of exponents n add up to an irrational number whenever 1/n, added up over the same exponents, is finite?","precis":"Erdős asked whether infinitely many of the fractions 1, 1/3, 1/7, 1/15, …, each of the form 1/(2^n−1) and each used at most once, can ever add up to a fraction. The paper proves that the chosen fractions add up to an irrational number whenever 1/n, added up over the chosen exponents n, is finite; for the exponents 2, 4, 8, 16, … this makes 1/3+1/15+1/255+1/65535+… irrational. Sections 1 to 8 explain the proof with examples; Appendix A supplies supporting details and formal-proof correspondence. The finite reconstruction here is an ordinary proof; the linked Lean and Comparator receipts establish the formal theorem and its own proof. For an arbitrary infinite choice the question is still open.","owns":"an account of Theorem A.1 of the #257 short paper at base 2 for a reader new to the problem: a guided proof through worked examples, with supporting details, exercises and the formal-proof correspondence in appendices","not_authority_for":"the validity of the Lean theorem it quotes, which belongs to the cited kernel-checked source, or a solution to Erdős #257, which remains open","first_pass":["sec:intro","sec:test","sec:doubling","sec:proof"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos257-expository-sums-never-fractions.pdf","hosted_pdf_path":"papers/erdos257-expository-sums-never-fractions.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/257/erdos257-expository-sums-never-fractions.tex","site_html":"maths/papers/erdos257-expository-sums-never-fractions.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/257/erdos257-expository-sums-never-fractions.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/257/erdos257-expository-sums-never-fractions.tex","body_pointer":"papers.records.erdos257-expository-sums-never-fractions.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Expository paper for Erdős Problem #257: the theorem of Appendix A of the\n% short paper, reconstructed as a guided proof in Sections 1 to 8.\n% Appendix A supplies supporting details and formal-proof correspondence.\n\\documentclass[10pt]{article}\n\\usepackage{paper-house-style}\n\\usepackage{xurl}\n\\usepackage{tikz}\n\\usetikzlibrary{arrows.meta,decorations.pathreplacing}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,citecolor=linkinternal]{hyperref}\n\\hypersetup{pdftitle={Why certain reciprocal Mersenne sums are irrational},pdfauthor={Will Cook},pdflang={en-GB},bookmarksnumbered=true}\n\\runninghead{Irrationality of reciprocal Mersenne subseries}\n\\title{Why certain sums of $1/(2^n-1)$ are irrational}\n\\subtitle{An irrationality proof through examples\\\\ (Erd\\H{o}s Problem \\#257)}\n\\author{Will Cook}\n\\date{12 October 2026}\n\\emergencystretch=1.5em\n\\renewcommand{\\bottomfraction}{0.4}\n\n\\theoremstyle{plain}\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}{Problem}\n\\newtheorem{exercise}{Exercise}\n\\newtheorem{readercheck}{Question}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\lft}{\\ell}\n\\newcommand{\\rmod}{\\mskip5mu\\mathrm{mod}\\mskip5mu}\n\n\\begin{document}\n\\maketitle\n\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the\nclaims when he could. AI agents did most of the research and drafting.\nCook did not independently verify every claim.\nAppendix~\\ref{app:lean} identifies the linked formal proof and the\nlimits of its coverage of this exposition.}\n\n\\section{The question: which infinite sums are irrational?}\\label{sec:intro}\n\nEvery finite sum of fractions is a fraction. An infinite sum can be one too:\n\\[\n \\frac12+\\frac14+\\frac18+\\dots=1.\n\\]\nNow consider\n\\[\n S=\\frac13+\\frac1{15}+\\frac1{255}+\\frac1{65535}+\\dots,\n\\]\nwhere the denominators are $2^2-1,2^4-1,2^8-1,2^{16}-1,\\dots$.\nThe exponents double at each step. We will prove that this sum cannot equal\nany fraction.\n\nMultiply by $255$, the third denominator:\n\\[\n\\begin{aligned}\n 255S&=85+17+1+\\frac{255}{65535}+\\dots\\\\\n     &=103+\\frac1{257}+\\dots.\n\\end{aligned}\n\\]\nThe first three terms have become the integer $103$, leaving a positive\nremainder. If $S=p/q$, with integers $p,q$ and $q>0$, multiplying by an\ninteger and subtracting an integer leaves an integer multiple of $1/q$.\nA positive remainder must therefore be at least $1/q$. We will repeat the\nconstruction to make the \\emph{whole} remainder smaller than this fixed gap,\nwhatever denominator $q$ is proposed.\n\nThis example belongs to a larger family. Its fractions are obtained by\ntaking a power of $2$,\nsubtracting $1$, and taking the reciprocal:\n\\begin{center}\n\\begin{tabular}{l|rrrrrrc}\n\\papertablehead\nexponent $n$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $\\dots$\\\\\n$2^n$ & $2$ & $4$ & $8$ & $16$ & $32$ & $64$ & $\\dots$\\\\\ndenominator $2^n-1$ & $1$ & $3$ & $7$ & $15$ & $31$ & $63$ & $\\dots$\\\\\nterm $1/(2^n-1)$ & $1$ & $\\tfrac13$ & $\\tfrac17$ & $\\tfrac1{15}$ & $\\tfrac1{31}$ & $\\tfrac1{63}$ & $\\dots$\n\\end{tabular}\n\\end{center}\nChoose infinitely many of these terms, each at most once, and add them.\nChoosing the terms is equivalent to choosing a set $A$ of positive integer\nexponents. The opening example uses $A=\\{2,4,8,16,\\dots\\}$; choosing all\neven exponents instead gives\n\\[\n A=\\{2,4,6,8,\\dots\\}:\\qquad\n \\frac13+\\frac1{15}+\\frac1{63}+\\frac1{255}+\\dots.\n\\]\nThe exponent $0$ is excluded, since $2^0-1=0$. We write\n\\[\n S_A=\\sum_{n\\in A}\\frac1{2^n-1}.\n\\]\nThe symbol $\\sum_{n\\in A}$ means that we add the terms whose exponents\nbelong to $A$, taking those exponents in increasing order. A \\emph{partial\nsum} stops after finitely many selected terms.\nHere \\emph{irrational} means that $S_A$ cannot equal $p/q$ for integers\n$p,q$ with $q>0$.\n\n\\begin{samepage}\nFor the geometric series at the start, the partial sums are\n\\begin{center}\n\\begin{tabular}{l|cccc}\n\\papertablehead\nnumber of terms $N$ & $1$ & $2$ & $3$ & $4$\\\\\n$\\tfrac12+\\dots+\\tfrac1{2^N}$ & $\\tfrac12$ & $\\tfrac34$ & $\\tfrac78$ & $\\tfrac{15}{16}$\\\\\ngap to $1$ & $\\tfrac12$ & $\\tfrac14$ & $\\tfrac18$ & $\\tfrac1{16}$\n\\end{tabular}\n\\end{center}\n\\end{samepage}\nIn general the partial sum is $1-2^{-N}$. Its gap to $1$ tends to zero,\nwhich is precisely the assertion\n$\\tfrac12+\\tfrac14+\\tfrac18+\\dots=1$.\n\nEvery series $S_A$ converges: for $n\\ge1$,\n\\[\n 0<\\frac1{2^n-1}\\le\\frac1{2^{n-1}},\\qquad\n \\sum_{n=1}^{\\infty}\\frac1{2^{n-1}}=2.\n\\]\nThe inequality follows from $2^n-1\\ge2^{n-1}$. Consequently the partial sums\nof $S_A$ increase and are bounded by $2$. A basic property of real numbers\nsays that an increasing, bounded sequence has a finite limit\n(Appendix~\\ref{app:sums}). The issue is whether this limit is rational.\n\n\\section{Erd\\H{o}s's question and the theorem}\\label{sec:problem}\n\nIn a 1988 account, Erd\\H{o}s wrote \\cite[p.~105]{erdos1988} (excerpt):\n\\begin{quote}\n``let $n_1<n_2<\\dots$. Is it then true that\n$\\sum \\frac{1}{2^{n_k}-1}$ is irrational, \\ldots''\n\\end{quote}\nThe indices $k=1,2,\\dots$ count the chosen exponents $n_k$.\nIn these terms, we prove:\n\n\\begin{theorem}\\label{thm:main}\nLet $n_1<n_2<\\dots$ be an infinite sequence of positive integers.\nIf $\\sum_{k=1}^{\\infty}1/n_k<\\infty$, then\n\\[\n \\sum_{k=1}^{\\infty}\\frac1{2^{n_k}-1}\n\\]\nis irrational.\n\\end{theorem}\n\nTaking $A=\\{n_1,n_2,\\dots\\}$, this series is exactly $S_A$, and the\nadditional hypothesis is exactly $\\sum_{n\\in A}1/n<\\infty$.\nThe question for arbitrary infinite $A$ remains open \\cite{erdosproblems257}.\n\nThe condition concerns the reciprocals of the \\emph{exponents}. The\nconclusion concerns the fractions with denominators $2^n-1$. For the\ndoubling exponents, the distinction is visible in one table:\n\\begin{center}\n\\begin{tabular}{l|cccc c}\n\\papertablehead\nchosen exponent $n$ & $2$ & $4$ & $8$ & $16$ & $\\dots$\\\\\nterm in the hypothesis, $1/n$ & $\\tfrac12$ & $\\tfrac14$ & $\\tfrac18$ & $\\tfrac1{16}$ & $\\dots$\\\\\nterm in $S_A$, $1/(2^n-1)$ & $\\tfrac13$ & $\\tfrac1{15}$ & $\\tfrac1{255}$ & $\\tfrac1{65535}$ & $\\dots$\n\\end{tabular}\n\\end{center}\nThe second row sums to $1$. The theorem therefore says that the third row\nsums to an irrational number. The hypothesis is an additional restriction\non $A$; it is not needed merely to make $S_A$ converge.\n\n\\begin{samepage}\nWe will use the hypothesis in this form: beyond a sufficiently large exponent,\nthe remaining reciprocals $1/n$ together add up to less than any positive\ntarget we specify. This remaining sum is called a \\emph{tail}.\\par\n\\end{samepage}\n\n\\begin{samepage}\nThe square exponents $A=\\{1,4,9,16,\\dots\\}$ also satisfy the condition.\nIndeed, for $m\\ge2$,\n\\[\n \\frac1{m^2}<\\frac1{m(m-1)}=\\frac1{m-1}-\\frac1m,\n\\]\nso the comparison sum cancels:\n\\[\n (1-\\tfrac12)+(\\tfrac12-\\tfrac13)+(\\tfrac13-\\tfrac14)\n +\\dots+(\\tfrac1{N-1}-\\tfrac1N)=1-\\tfrac1N<1.\n\\]\nThus $\\sum_{m=2}^{N}1/m^2<1$. Hence\n$\\sum_{m=1}^{\\infty}1/m^2$ converges, and the theorem proves that\n$1+\\tfrac1{15}+\\tfrac1{511}+\\tfrac1{65535}+\\dots$ is irrational.\\par\n\\end{samepage}\n\nBy contrast, choosing every positive integer gives the divergent harmonic\nseries in the hypothesis. Group its terms as $1$; $1/2$; $1/3+1/4$;\n$1/5+1/6+1/7+1/8$; and so on. For example,\n\\[\n \\frac15+\\frac16+\\frac17+\\frac18\\ge4\\cdot\\frac18=\\frac12.\n\\]\nEvery subsequent block doubles the number of terms and halves their smallest\nvalue, so again contributes at least $1/2$. These repeated contributions\nmake the partial sums unbounded. Even exponents also fail the condition, since their reciprocal sum\nis half the harmonic series. The theorem makes no assertion for these\nchoices. Infinitude is essential too: $A=\\{2,4\\}$ has a finite reciprocal\nsum, but $S_A=\\tfrac13+\\tfrac1{15}=\\tfrac25$.\n(Question~\\ref{chk:hypotheses}.)\n\nErd\\H{o}s proved the theorem in 1968 under the extra assumption that no two\nchosen exponents share a divisor greater than $1$, and stated that this\nassumption could be removed without supplying the details \\cite[p.~222]{erdos1968}.\nWe explain a proof without that assumption.\n\nThe proof has three stages. First, rationality forces a fixed lower bound\non certain positive remainders. Next, the denominators $2^n-1$ let us\ncalculate those remainders explicitly. Finally, a small average guarantees\nat least one choice whose \\emph{total} remainder is below that lower bound. The doubling exponents show the\nconstruction at work before the averaging is needed.\nThe main proof ends in Section~\\ref{sec:allchoices}.\nAppendices~\\ref{app:sums}--\\ref{app:proof} justify the supporting infinite-sum\nand divisibility facts; they can be read as needed or after the main proof.\nThe other-base extension and formal-proof correspondence are further reading.\n\n\\section{A rational number has a fixed positive gap}\\label{sec:test}\n\nStart with $2/5$. Multiplying it by an integer and subtracting an integer\nalways gives an integer multiple of $1/5$:\n\\[\n 3\\cdot\\frac25-1=\\frac15,\\quad\n 4\\cdot\\frac25-1=\\frac35,\\quad\n 5\\cdot\\frac25-2=0,\\quad\n 1\\cdot\\frac25-1=-\\frac35.\n\\]\nThere is no such value strictly between $0$ and $1/5$.\nThe denominator fixes a smallest possible positive result.\n\nThe same reasoning applies to any fraction. From now on write $S=S_A$\nfor the series belonging to our chosen set $A$, and suppose $S=p/q$,\nwhere $p\\in\\mathbb Z$ and $q$ is a positive integer. For integers $u,j$,\n\\[\n uS-j=\\frac{up-jq}{q}.\n\\]\nIts numerator is an integer. If $uS-j>0$, then $up-jq\\ge1$, and therefore\n\\[\n uS-j\\ge\\frac1q.\n\\]\nPositivity is necessary: zero and negative values do not contradict\nrationality.\n\nTo prove our sum irrational, we will therefore suppose that $S=p/q$ and\nconstruct integers $u,j$, which may depend on $q$, satisfying\n\\begin{equation}\\label{eq:goal}\n 0<uS-j<\\frac1q.\n\\end{equation}\n\n\\begin{samepage}\n\\section{Multiplying the series and retaining fractional parts}\\label{sec:split}\n\nWe want a multiplier that makes some terms integers. Choose a positive integer\n$k$ and consider $2^k-1$. Here $k$ is the multiplier exponent; it no longer\nindexes the list $n_1,n_2,\\dots$.\nFor example, $15=2^4-1$ is divisible by $3=2^2-1$.\nLet us first see exactly what multiplying by $15$ does:\n\\begin{center}\n\\begin{tabular}{c|c|c|c|c}\n\\papertablehead\nexponent $n$ & original term & multiplied term & integer part & fractional part\\\\\n$2$ & $1/3$ & $15/3=5$ & $5$ & $0$\\\\\n$3$ & $1/7$ & $15/7=2+1/7$ & $2$ & $1/7$\\\\\n$4$ & $1/15$ & $15/15=1$ & $1$ & $0$\\\\\n$5$ & $1/31$ & $15/31$ & $0$ & $15/31$\n\\end{tabular}\n\\end{center}\n\\end{samepage}\nThese rows illustrate the operation; in the chosen series we retain only\nrows with $n\\in A$. For $x\\ge0$, its integer part $\\lfloor x\\rfloor$ is\nthe greatest integer at most $x$. Subtracting it leaves the fractional part\n$\\{x\\}=x-\\lfloor x\\rfloor\\in[0,1)$. For example, $2\\le15/7<3$,\nso we subtract $2$ and retain $1/7$. This choice keeps every contribution\nnonnegative.\n\n\\paragraph{The remainder can be read from the exponents.}\nKeep the denominator $2^n-1$ fixed and add $n$ to $k$. The identity\n\\[\n 2^{k+n}-1=2^k(2^n-1)+(2^k-1)\n\\]\ngives\n\\[\n \\frac{2^{k+n}-1}{2^n-1}=2^k+\\frac{2^k-1}{2^n-1}.\n\\]\nThe added term $2^k$ is an integer, so the fractional part is unchanged.\nWrite $k=an+r$ with $0\\le r<n$. Repeatedly subtracting $n$ from the exponent\nreduces the calculation to $r$. Since $0\\le2^r-1<2^n-1$,\n\\begin{equation}\\label{eq:fracpart-main}\n \\lft_n(k):=\\left\\{\\frac{2^k-1}{2^n-1}\\right\\}\n =\\frac{2^{k\\bmod n}-1}{2^n-1}.\n\\end{equation}\nHere $k\\bmod n$ denotes the integer remainder $r$ in division by $n$.\nIt lies between $0$ and $n-1$; $\\lft_n(k)$ is the fractional part calculated\nfrom it. We will sum these fractional parts to obtain $R(k)$.\nFor example, $8=2\\cdot3+2$, so $\\{255/7\\}=(2^2-1)/7=3/7$.\nThe fractional part is zero exactly when $n$ divides $k$. Thus the\nmultiplier $2^k-1$ turns a selected term into an integer exactly when its\nexponent divides $k$. This is why we chose this multiplier: we can make\nselected contributions vanish by choosing $k$ divisible by their exponents.\n\n\\pagebreak[0]\n\\begin{samepage}\n\\paragraph{What happens to the entire sum.}\nApply the same multiplier to every selected term,\nsubtract each integer part, and sum the fractional parts:\n\\[\n R(k)=\\sum_{n\\in A}\\lft_n(k),\\qquad\n J(k)=\\sum_{\\substack{n\\in A\\\\ n\\le k}}\n       \\left\\lfloor\\frac{2^k-1}{2^n-1}\\right\\rfloor.\n\\]\nOnly exponents $n\\le k$ contribute to $J(k)$: when $n>k$, the multiplied\nterm is already less than $1$. Thus $J(k)$ is a finite sum of integers.\nSplitting the terms gives\n\\begin{equation}\\label{eq:split-main}\n (2^k-1)S=J(k)+R(k).\n\\end{equation}\nThe series defining $R(k)$ converges by comparison with $(2^k-1)S$;\nAppendix~\\ref{app:split} justifies the identity using partial sums.\\par\n\\end{samepage}\n\nThe sum $R(k)$ need not itself be a fractional part. For the three exponents\n$2,3,6$ and $k=5$, the separate fractional parts sum to\n\\[\n \\left\\{\\frac{31}3\\right\\}+\\left\\{\\frac{31}7\\right\\}\n +\\left\\{\\frac{31}{63}\\right\\}\n =\\frac13+\\frac37+\\frac{31}{63}=\\frac{79}{63}>1.\n\\]\nThe fractional part of the whole product is instead $16/63$.\nWe retain the separate contributions because each has the explicit formula\n\\eqref{eq:fracpart-main}; their sum still has the required form $uS-j$.\n\nFinally, if $A$ is infinite, there is always a selected exponent $n>k$.\nIts multiplied term is positive and less than $1$, so it contributes\npositively to $R(k)$; all other contributions are nonnegative. Therefore\n\\[\n S=p/q\\quad\\Longrightarrow\\quad\n R(k)=(2^k-1)S-J(k)\\ge\\frac1q\\quad\\hbox{for every }k\\ge1.\n\\]\nOur task is now to find \\emph{one} $k$ with $R(k)<1/q$.\n(Question~\\ref{chk:positive}.)\n\n\\section{A complete example: the doubling exponents}\\label{sec:doubling}\n\nReturn to the opening example, $A=\\{2,4,8,16,\\dots\\}$.\nWe now know why $255$ cleared its first three terms: it is $2^8-1$,\nand the exponents $2,4,8$ all divide $8$. In our new notation that calculation is\n\\[\n 255S=103+R(8),\\qquad R(8)=\\frac1{257}+\\frac1{16843009}+\\dots>0.\n\\]\nOne small remainder rules out only some denominators. To rule out every\n$q$, we need remainders as small as we please. For any doubling exponent\n$k$, all selected $n\\le k$ divide $k$ and contribute zero. Every later\nselected exponent is at least $2k$, and\n\\[\n 0<\\lft_n(k)=\\frac{2^k-1}{2^n-1}\n <\\frac{2^k}{2^{n-1}}=\\frac2{2^{n-k}}.\n\\]\nFor $k=8$, the gaps $n-k$ are $8,24,56,\\dots$. In general they are\ndistinct integers at least $k$. Insert the missing positive terms so that\nthe gaps run through \\emph{all} integers $k,k+1,k+2,\\dots$; this can only\nincrease the upper bound:\n\\[\n 0<R(k)<\\sum_{e=k}^{\\infty}\\frac2{2^e}=\\frac4{2^k}.\n\\]\nThe last equality is the geometric-series formula: the first term is\n$2/2^k$ and each successive term is half its predecessor.\n\n\\begin{samepage}\nFor $k=8$ this\ngives $R(8)<1/64$, ruling out every denominator $q\\le64$: such a\ndenominator would force $R(8)\\ge1/q\\ge1/64$.\nFor $k=16$ the bound is $R(16)<1/16384$.\nGiven any proposed denominator $q$, choose a doubling exponent $k$ with\n$2^k>4q$. Then $0<R(k)<1/q$, contradicting rationality. This completes\nthe proof for the doubling exponents.\\par\n\\end{samepage}\n\nThe special feature was that $k$ both eliminated all earlier terms and\nleft the next exponent at least $2k$. A general set $A$ need not have that\nspacing. The next section explains how averaging replaces it.\n\n\\begin{samepage}\n\\section{Why averaging works}\\label{sec:general}\n\n\\paragraph{Larger multipliers need not help.}\nFor the square exponents $A=\\{1,4,9,16,\\dots\\}$, both $k=4$ and $k=8$\nare divisible by the first two selected exponents $1,4$. All later selected\nexponents exceed $8$, so their integer parts vanish for both multipliers.\nConsequently\n\\[\n R(4)=15\\left(\\frac1{511}+\\frac1{65535}+\\dots\\right),\\qquad\n R(8)=255\\left(\\frac1{511}+\\frac1{65535}+\\dots\\right)=17R(4).\n\\]\nIn fact $R(4)\\approx0.0296$ and $R(8)\\approx0.503$. Increasing $k$ while\npreserving the same zero contributions has enlarged the remaining terms.\nEven eliminating longer initial groups need not suffice: in\nExercise~\\ref{ex:6}, the next selected exponent is $k+1$, leaving\n$(2^k-1)/(2^{k+1}-1)$, which approaches $1/2$. The choices of $k$ matter,\nnot just how many initial rows they make zero.\\par\n\\end{samepage}\n\n\\subsection{One table, many attempts}\n\nKeep $A$, and hence the original number $S$, fixed. For each new choice of\n$k$, repeat the whole calculation: multiply every selected term by $2^k-1$,\nsubtract its integer part, and add what remains to obtain $R(k)$.\n\\emph{Each column below is one entire attempt to obtain the contradiction.}\nEach row follows the contribution from one original term across the attempts.\nWe need one column whose total is less than $1/q$.\n\nChoose candidate exponents spaced $Q$ apart:\n$k=Q,2Q,\\dots,MQ$. Here $Q$ is the spacing, $M$ is the number of attempts,\nand $m$ labels the attempt $k=mQ$; the actual multiplier is $2^{mQ}-1$.\nNeither $Q$ nor $k$ has to belong to $A$.\nWe choose $Q$ divisible by an initial group of selected exponents so those\nrows stay zero in every column. For the square exponents, $Q=4$ gives:\n\\begin{center}\n\\begin{tabular}{c|cccc}\n\\papertablehead\nselected exponent $n$ & $k=4$ & $k=8$ & $k=12$ & $k=16$\\\\\n$1$ & $0$ & $0$ & $0$ & $0$\\\\\n$4$ & $0$ & $0$ & $0$ & $0$\\\\\n$9$ & $15/511$ & $255/511$ & $7/511$ & $127/511$\\\\\n$16$ & $15/65535$ & $255/65535$ & $4095/65535$ & $0$\\\\\n$\\vdots$ & $\\vdots$ & $\\vdots$ & $\\vdots$ & $\\vdots$\\\\\\hline\ncolumn total & $R(4)$ & $R(8)$ & $R(12)$ & $R(16)$\n\\end{tabular}\n\\end{center}\nEvery entry is a fractional part $\\lft_n(k)$. For example, the cell\n$n=9,k=12$ uses $12=1\\cdot9+3$, so\n$\\lft_9(12)=(2^3-1)/(2^9-1)=7/511$.\nEach column uses the same multiplier throughout, including in the rows\nnot displayed. (Question~\\ref{chk:columns}.)\n\n\\begin{samepage}\n\\paragraph{Average complete attempts.}\nDifferent rows may be small in different columns; choosing a separate\ncolumn for each row would not give one multiplier for the original series.\nInstead average \\emph{every row across the same columns}, then add those\nrow averages. This gives the average of the complete column totals:\nboth calculations add the same entries and divide by the same number $M$.\nIn symbols,\n\\begin{equation}\\label{eq:average-main}\n \\frac1M\\sum_{m=1}^{M}R(mQ)\n =\\sum_{n\\in A}\\frac1M\\sum_{m=1}^{M}\\lft_n(mQ).\n\\end{equation}\n\\end{samepage}\nFor infinitely many rows we first use finite portions and then let them\ngrow; convergence preserves the equality (Appendix~\\ref{app:proof}).\nIf this average is below $1/q$, at least one whole column has total below\n$1/q$. Otherwise every column total would be at least $1/q$, making their\naverage at least $1/q$ too.\n\n\\paragraph{What the two estimates will do.}\nWe will divide the rows into three groups. An initial finite group stays\nzero. A further finite group can complete its cycles if we choose $M$\nappropriately; we will bound each such row's average by $1/n$.\nFor the infinitely many remaining rows, we need a bound valid for every\n$M$: we will prove $2Q/n$. Once $Q$ is fixed, we can make the sum of these\nbounds as small as needed by moving far enough down the rows: the remaining\nreciprocals $1/n$ have arbitrarily small total. Section~\\ref{sec:proof} will choose the two\nrow cutoffs and then $M$ so that both nonzero groups have small total averages.\n\n\\subsection{Complete cycles give small averages}\\label{sec:cycles}\n\nTo derive a rule for any row, use $n=3$ and $n=6$, which are not\nexponents of the square series. Start with $n=3$ and consecutive $k$:\n\\begin{center}\n\\begin{tabular}{l|ccc|ccc}\n\\papertablehead\n$k$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$\\\\\n$k\\bmod3$ & $1$ & $2$ & $0$ & $1$ & $2$ & $0$\\\\\n$\\lft_3(k)$ & $1/7$ & $3/7$ & $0$ & $1/7$ & $3/7$ & $0$\n\\end{tabular}\n\\end{center}\nOne repeating block, or \\emph{cycle}, has sum $4/7<1$ and average\n$4/21<1/3$. This bounds the average, not every entry: $3/7>1/3$.\nFor general $n$, a consecutive cycle has remainders $0,1,\\dots,n-1$.\nThe powers $2^0,2^1,\\dots,2^{n-1}$ add up to $2^n-1$, exactly the\ndenominator. Each actual numerator is one less than the corresponding power,\nso the cycle's total is\n\\[\n \\frac{(2^0-1)+(2^1-1)+\\dots+(2^{n-1}-1)}{2^n-1}\n <\\frac{2^0+2^1+\\dots+2^{n-1}}{2^n-1}=1.\n\\]\nDividing by its $n$ entries gives a cycle average below $1/n$.\nThese reciprocals bound the row averages, which is why the hypothesis\ncontrols their sum.\n\n\\pagebreak[0]\n\\begin{samepage}\n\\paragraph{Keep the required spacing.}\nOur candidates use steps of $Q$, not necessarily $1$, because $Q$ preserves\nthe initial zero rows. With $n=6$ and $Q=4$:\n\\begin{center}\n\\begin{tabular}{l|ccc}\n\\papertablehead\nsample number $m$ & $1$ & $2$ & $3$\\\\\n$k=4m$ & $4$ & $8$ & $12$\\\\\n$k\\bmod6$ & $4$ & $2$ & $0$\\\\\n$\\lft_6(k)$ & $15/63$ & $3/63$ & $0$\n\\end{tabular}\n\\end{center}\nThe last two rows repeat every three samples. Only the even remainders\noccur, and the average is $(15+3+0)/(3\\cdot63)=2/21<1/6$.\\par\n\\end{samepage}\n\n\\newpage\n\\begin{samepage}\nLet $g=\\gcd(n,Q)$ be the \\emph{greatest common divisor}, the largest\npositive integer dividing both $n$ and $Q$, and put $c=n/g$.\nIn the example $n=6,Q=4$, these are $g=2,c=3$.\nWe will show that one cycle has exactly $c$ samples, visiting\n$0,g,2g,\\dots,n-g$ once each, in some order. Knowing which remainders\noccur will let us compare the sampled mean with the mean over all remainders.\nHere are the four steps.\n\n\\emph{Possible remainders.} Each remainder is a multiple of $g$: it is\n$mQ$ minus a multiple of $n$, and both are divisible by $g$.\nBetween $0$ and $n-1$ there are exactly $c$ possibilities,\n$0,g,2g,\\dots,n-g$.\n\n\\emph{A return after $c$ steps.} Since\n\\[\n cQ=\\frac ng Q=n\\frac Qg,\n\\]\ntaking another $c$ steps adds a multiple of $n$ and leaves the remainder\nunchanged. So $c$ is a period; we still have to exclude earlier repetition.\n\n\\emph{No earlier repetition.} Two integers have the same remainder on\ndivision by $n$ exactly when their difference is a multiple of $n$.\nFor samples $m_1<m_2$, that difference is\n$m_2Q-m_1Q=(m_2-m_1)Q$. Dividing out $g$, equal remainders therefore mean\nthat $n/g$ divides $(m_2-m_1)(Q/g)$.\nThe integers $n/g$ and $Q/g$ have no common divisor greater than $1$;\nthey are called \\emph{coprime}. This lets us cancel $Q/g$: $c=n/g$\nmust divide $m_2-m_1$. In the example $n=6,Q=4$, write $d=m_2-m_1$:\nif $3$ divides $2d$, it also divides $d=2(2d)-3d$, since both terms on\nthe right are multiples of $3$. Appendix~\\ref{app:cycle} proves the\ngeneral cancellation fact. Thus samples fewer than $c$ steps apart cannot\nhave the same remainder.\n\n\\emph{All possibilities are visited.} The first $c$ remainders are distinct\nand there are only $c$ possible values. Hence every possible remainder\noccurs exactly once per cycle.\\par\n\\end{samepage}\n\n\\paragraph{Compare means without calculating a new formula.}\nSort the sampled remainders $4,2,0$ as $0,2,4$, preserving the sampled mean.\nThese start the pairs $(0,1)$, $(2,3)$, $(4,5)$. Since\n\\mbox{$(2^r-1)/(2^n-1)$} increases with $r$, each pair's first remainder gives\nits smaller fractional part. Copy each sampled value twice to compare\nwith all six consecutive-cycle values.\n\\begin{center}\n\\captionsetup{hypcap=false}\n\\input{257/figures/cycle-mean.tex}\n\\captionof{figure}{For $n=6,Q=4$, two copies of each sample double\nboth sum and count, preserving the mean. Each filled value is at most its\noutlined counterpart, so the sampled mean is at most the consecutive-cycle\nmean, which is below $1/6$.}\n\\label{fig:cycle-mean}\n\\end{center}\n\\begin{samepage}\nIn general, each group has $g$ consecutive remainders, and the sample\nselects its first: $0,g,\\dots,n-g$. Repeating each selected value $g$ times\ngives a sum no larger than the sum over all $n$ remainders. Divide by $n$:\nthe copied list retains the mean of the $c$ samples, while the consecutive-cycle\nmean is below $1/n$. Thus, whenever $M$ is a positive multiple of $c$,\n\\begin{equation}\\label{eq:cycle-main}\n \\frac1M\\sum_{m=1}^{M}\\lft_n(mQ)<\\frac1n.\n\\end{equation}\nThe bound $1/n$ does not grow with $Q$: making the initial rows zero can\nrequire a large spacing without worsening this bound, provided $M$ completes\nthe cycles. If $n$ divides $Q$, the cycle has one entry, zero, which also\nsatisfies the bound.\\par\n\\end{samepage}\n\n\\pagebreak[0]\n\\begin{samepage}\n\\subsection{A bound that also covers unfinished cycles}\\label{sec:unfinished}\n\nAn unfinished cycle can miss its final zero. For $n=5$ and consecutive\ncandidates ($Q=1$), one cycle is\n\\[\n \\frac1{31},\\quad\\frac3{31},\\quad\\frac7{31},\\quad\\frac{15}{31},\\quad0.\n\\]\nThe first four entries average $26/(4\\cdot31)=13/62>1/5$;\nall five average $26/(5\\cdot31)=26/155<1/5$.\nThe increasing positive list and the full cycle therefore have different\naverage bounds. (Question~\\ref{chk:partial} gives a spaced example.)\\par\n\\end{samepage}\n\nNor can finitely many columns complete infinitely many rows: if $n>MQ$,\nthe last sample has not even reached the first positive multiple of $n$.\n\nWe need another estimate, valid for \\emph{every} positive $M$:\n\\begin{equation}\\label{eq:uniform-main}\n \\frac1M\\sum_{m=1}^{M}\\lft_n(mQ)\\le\\frac{2Q}{n}.\n\\end{equation}\nFor fixed $Q$, this bound holds for every $M$, so choosing the number of\ncolumns later cannot invalidate it. We prove it using the same cycles.\n\n\\paragraph{First bound all the consecutive entries.}\nThe $M$ sampled exponents $Q,2Q,\\dots,MQ$ lie among the first $MQ$\npositive integers. We first bound the sum over all $MQ$ entries.\nFor $n\\ge2$, consecutive entries beginning at $k=1$ repeat the block\n\\[\n \\underbrace{\\frac1{2^n-1},\\frac3{2^n-1},\\dots,\n \\frac{2^{n-1}-1}{2^n-1}}_{n-1\\text{ positive entries, increasing}},\\quad 0.\n\\]\n\\begin{samepage}\nFor $n=2$, the positive list consists only of $1/3$.\nThe positive entries sum to less than $1$, as the complete-cycle calculation\nshowed. Their average is therefore less than $1/(n-1)$.\nAny beginning portion of this \\emph{positive} list has no larger average:\neach omitted entry is at least every included entry, so appending the\nomitted entries cannot lower the average. Thus a nonempty beginning portion\nhas average at most $2/n$, since $1/(n-1)\\le2/n$ for $n\\ge2$.\nWe excluded the final zero for this comparison; the whole block including\nzero already has the stronger bound $1/n$.\\par\n\\end{samepage}\n\nThe first $MQ$ entries consist of complete blocks followed, possibly,\nby a beginning portion of the positive list. For example, the first seven\nentries when $n=5$ have numerators $(1,3,7,15,0)$ followed by $(1,3)$,\nall over $31$. Each piece has sum at most its number of entries times $2/n$.\nThe pieces contain $MQ$ entries altogether, so their total is at most $2MQ/n$.\nFor $n=1$, all entries are zero and the bound still holds.\n\n\\paragraph{Keep only the sampled entries.}\nRemoving nonnegative entries cannot increase the sum. Hence\n\\[\n \\sum_{m=1}^{M}\\lft_n(mQ)\n \\le\\sum_{k=1}^{MQ}\\lft_n(k)\\le\\frac{2MQ}{n}.\n\\]\nDivide by the number $M$ of actual candidates to obtain\n\\eqref{eq:uniform-main}. This explains the factor $Q$: we bounded $M$\nsampled entries by including all $MQ$ consecutive entries.\nWe have not changed our candidates or assumed that their sampled values\nare increasing.\n\n\\section{Combining the estimates}\\label{sec:proof}\n\nSuppose $S=p/q$. Every column total $R(k)$ is then at least $1/q$.\nWe will contradict this by making a finite average of column totals\nless than $1/q$.\n\n\\subsection{The construction with actual numbers}\\label{app:numbers}\n\nUse the doubling exponents again, this time to see the general method\nat work. We already have a shorter proof for them. Assume the proposed\ndenominator is $q=5$. It is enough to make the average of the column totals\nless than $1/5$; allow less than $1/10$ from each of the two nonzero groups.\n\nBeyond a doubling exponent $2^h$, the reciprocal tail is\n$1/2^{h+1}+1/2^{h+2}+\\dots=1/2^h$.\n\\begin{enumerate}\n\\item \\emph{Decide which early rows to make zero.}\n Set the first cutoff at exponent $t=16$. The reciprocal sum of all later\n rows is $1/16<1/10$, so any middle group will have enough room in its bound.\n The earlier cutoff $8$ would give $1/8>1/10$.\n\\item \\emph{Preserve those zeros in every attempt.}\n Choose $Q=16$, divisible by $2,4,8,16$. Every candidate $k=16m$ therefore\n gives zero in these four rows.\n\\item \\emph{Make the infinite far group small despite the factor $2Q$.}\n Applying $2Q/n$ to every row beyond $t=16$ would bound the sum of their\n averages by $32/16=2$. Increasing a doubling cutoff $t$ together with $Q=t$ still\n gives $2Q/t=2$. This shows a limitation of the estimate, not a lower\n bound on $R(k)$. Instead keep $Q=16$ and put the second cutoff at\n $T=512$. The remaining rows have exponents $1024,2048,\\dots$.\n Adding the bounds $32/n$ on their averages gives\n \\[\n  32\\left(\\frac1{1024}+\\frac1{2048}+\\dots\\right)\n  =\\frac{32}{512}<\\frac1{10}.\n \\]\n A cutoff of $256$ would leave $32/256=1/8$, above the reserved $1/10$.\n\\item \\emph{Complete all cycles between the cutoffs.}\n The middle exponents $32,64,128,256,512$ have cycle lengths\n $2,4,8,16,32$. Choose $M=64$, a common multiple of these lengths.\n Every middle row completes its cycles, and the far bound is still valid\n because it holds for every $M$.\n\\end{enumerate}\nHere $t$ and $T$ are cutoffs on \\emph{exponent values}, not counts of terms.\nThe last candidate is $MQ=1024$, whereas the second cutoff is $T=512$:\nthe number of columns and the division into groups of rows have different jobs.\n$M=32$ would also work; choosing $64$ shows that we may use more columns\nwithout changing either cutoff.\n\nThe initial rows contribute zero. The middle row averages together are\nless than\n\\[\n \\frac1{32}+\\frac1{64}+\\frac1{128}+\\frac1{256}+\\frac1{512}\n =\\frac{31}{512}.\n\\]\nThe far row averages add up to at most $2Q/T=32/512$. Hence\n\\[\n \\frac1{64}\\sum_{m=1}^{64}R(16m)\n <\\frac{31}{512}+\\frac{32}{512}=\\frac{63}{512}<\\frac15.\n\\]\nAt least one of the 64 complete attempts has $0<R(k)<1/5$, contradicting\n$S=p/5$. We have proved that a candidate works without identifying which\none. That existence is enough for the contradiction.\n\nThe first cutoff determines which rows $Q$ must keep zero; the second is\nchosen after $Q$ is fixed, to compensate for the factor $2Q$.\nOnly then do we choose how many columns to use.\n(For a change of $Q$, see Question~\\ref{chk:order}.)\n\n\\begin{samepage}\n\\subsection{The same choices for any admissible set}\\label{sec:allchoices}\n\nLet $A$ satisfy the theorem's hypotheses, and suppose $S_A=p/q$, where\n$q$ is a positive integer. Since only finitely many positive integers lie\nbelow any fixed bound, the infinite set $A$ has arbitrarily large elements.\nMoving a cutoff farther out can only decrease a reciprocal tail. We may\ntherefore choose cutoffs $t,T\\in A$, with $T>t$: the initial group then\ncontains $t$, and the middle group contains $T$.\nMake the choices in this order:\n\\[\n t\\ \\longrightarrow\\ Q\\ \\longrightarrow\\ T\\ \\longrightarrow\\ M.\n\\]\n\\begin{enumerate}\n\\item Choose $t\\in A$ so that\n $\\sum_{n\\in A,\\,n>t}1/n<1/(2q)$.\n This is possible because the reciprocal tails tend to zero.\n\\item Choose a positive common multiple $Q$ of the selected exponents\n $n\\le t$. Their product works.\n These exponents divide every candidate $mQ$, so their rows are zero.\n\\item With $Q$ fixed, choose $T\\in A$, with $T>t$, so that\n $2Q\\sum_{n\\in A,\\,n>T}1/n<1/(2q)$.\n The tails again tend to zero, now with a fixed coefficient $2Q$.\n\\item For the finitely many selected $t<n\\le T$, choose $M$ to be a\n positive common multiple of their cycle lengths $n/\\gcd(n,Q)$.\n Their product works.\n The far-tail bound remains valid for this final choice of $M$.\n\\end{enumerate}\n\\end{samepage}\nAdd the row averages over these same $M$ columns:\n\\begin{center}\n\\small\n\\begin{tabular}{l|p{4cm}|p{5.4cm}}\n\\papertablehead\nselected exponents & reason for the bound & total contribution\\\\\n$n\\le t$ & each $n$ divides $Q$ & $0$\\\\\n$t<n\\le T$ & complete cycles; row average below $1/n$\n & below $\\sum_{n\\in A,\\,n>t}1/n<1/(2q)$\\\\\n$n>T$ & bound $2Q/n$ for every $M$\n & at most $2Q\\sum_{n\\in A,\\,n>T}1/n<1/(2q)$\n\\end{tabular}\n\\end{center}\nBy \\eqref{eq:average-main}, the average of the column totals is therefore\n\\[\n \\frac1M\\sum_{m=1}^{M}R(mQ)<0+\\frac1{2q}+\\frac1{2q}=\\frac1q.\n\\]\nAt least one total is below $1/q$, contradicting the rationality lower bound\nfor every column. This proves Theorem~\\ref{thm:main}.\n\nInfinitude supplied positivity; rationality supplied the fixed gap;\nreciprocal convergence supplied both small tails. The order of choices\nensured that all bounds apply to the same finite collection of complete\nattempts.\n\n\\begin{samepage}\n\\section{Consequences and limits}\\label{sec:limits}\n\nFor every integer $b\\ge2$, the same method proves\n\\[\n A\\text{ infinite},\\quad\\sum_{n\\in A}\\frac1n<\\infty\n \\quad\\Longrightarrow\\quad\n \\sum_{n\\in A}\\frac1{b^n-1}\\text{ is irrational}.\n\\]\nReplace the multiplier by $b^k-1$. The fractional parts are\n$(b^{k\\bmod n}-1)/(b^n-1)$, and their cycle averages are still below $1/n$.\nAppendix~\\ref{app:bases} checks both estimates.\n\\end{samepage}\n\nThe theorem covers arbitrary selections from the square-exponent series\n$1+\\tfrac1{15}+\\tfrac1{511}+\\dots$.\nIf infinitely many remain, their sum is irrational: deleting terms cannot\nincrease the reciprocal-exponent sum\n$\\sum_{m\\ge1}1/m^2<\\infty$, so Theorem~\\ref{thm:main} still applies.\nIf only finitely many remain, their sum is rational.\nThus a selection from this series has rational sum exactly when it is finite;\nno pattern in the retained exponents is required.\nReplacing $2$ by any integer $b\\ge2$ gives the same conclusion for the\nsame retained exponents.\n\nThe reciprocal-sum condition is sufficient, not necessary. These particular\nremainders $R(k)$ need not become arbitrarily small for every infinite $A$.\nWhen every positive exponent is selected, the terms with $n>k$ alone give\n\\[\n R(k)\\ge\\sum_{n>k}\\frac{2^k-1}{2^n-1}\n >\\sum_{n>k}\\frac{2^k-1}{2^n}\n =1-2^{-k}\\ge\\frac12.\n\\]\nThus these remainders can never be arbitrarily small. Nevertheless,\nErd\\H{o}s proved that the full sum is irrational in 1948 \\cite{erdos1948}.\nFailure of our method does not imply rationality.\nTao and Ter\\\"av\\\"ainen proved irrationality for the prime exponents\n\\cite[Theorem~1.3]{taoteravainen2025}, another case with divergent reciprocal\nsum. The short paper \\cite{shortpaper} gives further sufficient conditions;\nthe long record \\cite{longrecord} develops the proofs and the obstructions\nto extending them. The question for every infinite set $A$ remains open.\n\nAppendix~\\ref{app:lean} identifies the Lean-checked theorem and explains how\nits proof differs from the finite-average proof given here.\n\n\\appendix\n\n\\section{Supporting details and formal-proof correspondence}\\label{app:full}\n\nThis appendix supplies convergence and limit details, the divisibility\nargument for sampled cycles, the other-base estimates, and an optional\naccount of formal verification.\n\n\\subsection{Convergence and tails}\\label{app:sums}\n\nFor nonnegative terms $a_n$, the partial sums\n$\\sum_{n=1}^{N}a_n$ increase. If they are bounded above, let $L$ be their\nleast upper bound: the smallest real number at least every partial sum.\nFor every $\\varepsilon>0$, some partial sum exceeds $L-\\varepsilon$;\notherwise $L-\\varepsilon$ would be a smaller upper bound. Every later\npartial sum then lies in $(L-\\varepsilon,L]$, so the series converges to $L$. Its tail after $N$ equals\n$L-\\sum_{n=1}^{N}a_n$ and tends to zero. This is the property used twice\nwhen choosing the cutoffs in Section~\\ref{sec:proof}.\n\nComparison follows by comparing partial sums. In particular,\n$0\\le a_n\\le b_n$ and $\\sum b_n<\\infty$ imply\n$\\sum a_n\\le\\sum b_n<\\infty$. If one term is strictly smaller, the sums\ndiffer by at least that positive difference. Restricting a nonnegative\nseries to a subset of its indices can only decrease its sum.\n\nFor $0<x<1$, multiplication and subtraction of the finite geometric sum\ngive\n\\[\n (1-x)(1+x+\\dots+x^{N-1})=1-x^N.\n\\]\nLetting $N\\to\\infty$ yields $\\sum_{j=0}^{\\infty}x^j=1/(1-x)$.\nAt $x=1/2$, this gives $\\sum_{j=1}^{\\infty}2^{-j}=1$ and, by scaling,\n$\\sum_{j=h}^{\\infty}2^{-j}=2^{1-h}$.\nConsequently $1/(2^n-1)\\le2^{1-n}$ proves convergence of $S_A$ for every\nset $A$, irrespective of whether $\\sum_{n\\in A}1/n$ converges.\n\n\\subsection{Passing to the full transformed sum}\\label{app:split}\n\nFix $k$. Each $\\lft_n(k)$ is nonnegative and at most\n$(2^k-1)/(2^n-1)$, so $R(k)$ converges by comparison with $(2^k-1)S$.\nOnly selected exponents $n\\le k$ have nonzero integer parts. For any\ninteger $L\\ge k$, splitting the finitely many terms up to $L$ gives\n\\[\n (2^k-1)\\sum_{\\substack{n\\in A\\\\n\\le L}}\\frac1{2^n-1}\n =J(k)+\\sum_{\\substack{n\\in A\\\\n\\le L}}\\lft_n(k).\n\\]\nAs $L$ grows, $J(k)$ stays fixed and the two convergent partial sums tend\nto $S$ and $R(k)$. This proves \\eqref{eq:split-main}.\n\n\\subsection{Why the sampled remainders form a cycle}\\label{app:cycle}\n\nThe elementary cancellation fact is: if positive integers $c,u$ are\ncoprime and $c$ divides $du$, then $c$ divides $d$.\nThe example in Section~\\ref{sec:cycles} used $c=3,u=2$.\nFor general coprime $c,u$, the same argument uses integers $a,b$ with\n$au+bc=1$. Here is why they exist. Among the positive values\n$au+bc$, allowing $a,b$ to be any integers, choose the smallest, $v$;\nthere is at least one since $u=1\\cdot u+0\\cdot c$ is such a combination.\nWrite $v=au+bc$ for coefficients attaining this minimum. Division gives\n$u=sv+r$ with $0\\le r<v$, so\n\\[\n r=u-sv=(1-sa)u-sbc.\n\\]\nThus $r$ is another integer combination of $u,c$. If it were positive,\nit would contradict the choice of $v$ as the smallest positive combination.\nHence $r=0$, so $v$ divides $u$. The same reasoning with $c$ shows that\n$v$ divides $c$ too. Since $u,c$ are coprime, $v=1$.\nNow $d=a(du)+b(dc)$ proves cancellation.\n\nReturn to $g=\\gcd(n,Q)$, $c=n/g$ and $u=Q/g$. The integers $c,u$ are\ncoprime: a larger common divisor would give a common divisor of $n,Q$\nlarger than $g$. Write $x\\mid y$ to mean that $x$ divides $y$.\nTwo sampled exponents $m_1Q$ and $m_2Q$ have equal remainders exactly when\ntheir difference is divisible by $n$, that is,\n\\[\n n\\mid(m_2-m_1)Q\n \\quad\\Longleftrightarrow\\quad\n c\\mid(m_2-m_1)u\n \\quad\\Longleftrightarrow\\quad\n c\\mid m_2-m_1.\n\\]\nThus the first $c$ sampled remainders are distinct and repeat after $c$\nsteps. Every remainder is divisible by $g$, since both $mQ$ and $n$ are.\nThere are exactly $c$ possibilities $0,g,\\dots,n-g$, so each occurs once.\nThis supplies the remainder rule used in Section~\\ref{sec:cycles}.\n\n\\subsection{Adding the row averages}\\label{app:proof}\n\nFor finitely many rows $n\\le H$, adding the row averages equals averaging\nthe column totals: both operations add the same entries and divide by $M$.\nKeep $Q$ and $M$ fixed. Given any positive target $\\varepsilon$, each of\nthe $M$ convergent column sums $R(mQ)$ has tail less than $\\varepsilon$\nbeyond some row cutoff. Take the largest of these finitely many cutoffs.\nAll column tails, and therefore their average, are less than $\\varepsilon$\nbeyond it. Consequently the finite sums of row averages tend to the\naverage of the full column totals. This proves \\eqref{eq:average-main};\nno interchange of two infinite sums is needed.\n\n\\subsection{Other bases}\\label{app:bases}\n\nLet $b\\ge2$ be an integer. For $b=3$, exponents $2,4,8,\\dots$ give\n$1/8+1/80+1/6560+\\dots$. Convergence and the fractional-part formula carry\nover because $b^n-1\\ge2^{n-1}$ and\n$b^{k+n}-1=b^k(b^n-1)+(b^k-1)$.\n\nFor a consecutive cycle, the sum of the fractional parts is\n\\[\n \\frac{(b^0-1)+\\dots+(b^{n-1}-1)}{b^n-1}\n <\\frac{b^0+b^1+\\dots+b^{n-1}}{b^n-1}\n =\\frac1{b-1}\\le1.\n\\]\nHere we used $1+b+\\dots+b^{n-1}=(b^n-1)/(b-1)$ and $b\\ge2$.\nThe fractional parts still increase with the remainder, so the equal-group\ncomparison bounds a complete sampled-cycle average by the consecutive-cycle\naverage, which is below $1/n$.\n\nFor $n\\ge2$ the $n-1$ positive entries are still increasing and have total\nless than $1$. The beginning-portion argument in\nSection~\\ref{sec:unfinished} therefore gives the same $2/n$ bound for\nconsecutive entries starting at $k=1$. Keeping only the sampled entries\nthen gives the bound $2Q/n$ for any sampled length $M$. For $n=1$ the\nfractional parts are all zero: $(b^k-1)/(b-1)$ is an integer.\nBoth estimates required for the three groups remain valid. With the same\nchoices of $t,Q,T,M$, the proof establishes irrationality of\n$\\sum_{n\\in A}1/(b^n-1)$ whenever $A$ is infinite and\n$\\sum_{n\\in A}1/n<\\infty$ \\cite[Theorem~A.1]{shortpaper}.\n\n\\subsection{Formal verification and its scope}\\label{app:lean}\n\n\\begin{center}\n\\begin{minipage}{0.9\\linewidth}\n\\small\\ttfamily\ntheorem irrational\\_erdosSupportSeries\\_of\\_summable\\_reciprocal\\\\\n\\hspace*{2em}(b : $\\mathbb{N}$) (A : Set $\\mathbb{N}$) (hb : 2 $\\le$ b) (hA : A.Infinite)\\\\\n\\hspace*{2em}(hsum : Summable (reciprocalSupportTerm A)) :\\\\\n\\hspace*{2em}Irrational (erdosSupportSeries b A)\n\\end{minipage}\n\\end{center}\nLine by line: $b$ is a nonnegative integer and \\texttt{hb} assumes $b\\ge2$; $A$ is a\nset of nonnegative integers and \\texttt{hA} assumes it is infinite;\n\\texttt{reciprocalSupportTerm A} is a sequence indexed by $n=0,1,\\dots$, with entry $1/n$ when\n$n$ is in $A$ and $0$ otherwise, and \\texttt{hsum} assumes its sum is finite;\nthe conclusion says that \\texttt{erdosSupportSeries b A}, the sum of\n$1/(b^n-1)$ over $A$, is irrational. Lean's nonnegative integers start at $0$, and in\nLean dividing by $0$ gives $0$, so the exponent $0$ counts as $0$ in both sums;\nremoving it changes neither sum and leaves an infinite set infinite. So for\n$b=2$ the statement says the same as Theorem~\\ref{thm:main}. It is the formal\nversion of \\cite[Theorem~A.1]{shortpaper}.\n\nThe \\href{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/Erdos249257/AllBaseReciprocalSupportIrrationality.lean#L395}{Lean source at a fixed commit}\ncontains this statement and its proof. The Lean kernel, a small program,\nhas checked every step of that proof.\nComparator checked that the proof establishes a separately written formal\nstatement. Its \\href{https://github.com/wcook04/plectis-erdos/blob/0064fc9b2a2693de97aebc8abec26090c3cd1095/evidence/comparator/replay-35935225572/receipt-E257_47.json}{verification receipt}\nrecords acceptance by Lean's kernel and by a second, independent kernel\nchecker. At base $2$ the Lean proof follows \\cite[Appendix~A]{shortpaper},\nwhich takes limits of averages; it reaches the other bases by comparing each\nbase-$b$ fractional part with twice the base-$2$ one, a different route from\nAppendix~\\ref{app:bases}.\n\nWith exponent $0$ omitted as above, the shifted quantity used in Lean is\n\\[\n \\sum_{n\\in A}\\frac{2^{k\\bmod n}}{2^n-1}=S_A+R(k).\n\\]\nIts value at $k=0$ is $S_A$. A value strictly between $S_A$ and\n$S_A+\\varepsilon$ therefore means exactly $0<R(k)<\\varepsilon$, the target of our\nproof. Subtracting $S_A$, we can describe the two limits in terms of our\nfractional parts.\n\nFirst fix $Q$ and let the averaging length tend to infinity.\nEach row's average tends to its cycle mean: complete cycles have that mean,\nand the unfinished last cycle has bounded length, so its weight in the\naverage tends to zero. Throughout this limit the row averages are bounded\nby $2(Q+1)/n$.\n\nTo pass from the row limits to the whole sum, choose a row cutoff so far\nout that the sum of these bounds beyond it is small. This is possible\nbecause $Q$ is fixed and $\\sum_{n\\in A}1/n$ converges.\nThe same bound holds for each row's limit, so it controls both the tails\nbefore taking the limit and the tail of the limiting row averages.\nOnly finitely many earlier rows remain, and their limits can be taken one\nat a time. Combining that finite part with the two small tails justifies\npassing the limit through the sum.\n\nNext consider $Q=\\operatorname{lcm}(1,\\dots,t)$. For every $n\\le t$, the\nlimiting row average is zero, because $n$ divides $Q$. All remaining\nlimiting row averages are bounded by $1/n$, independently of $t$.\nTheir total is therefore at most $\\sum_{n\\in A,\\,n>t}1/n$, which tends to\nzero. This is the second use of small reciprocal tails.\nThe averaging length tends to infinity with $Q$ fixed before $t$ varies.\nTo obtain one witness, the proof chooses a suitable $Q$ first and a\nsufficiently long average second.\n\nOur cutoffs $t,T$ and number $M$ of candidate columns make those choices explicit,\nusing the separately proved estimate $2Q/n$. Under the rationality assumption, both arguments make the integer\n$qR(k)$ strictly between $0$ and $1$.\nThe finite reconstruction and its estimates are ordinary proofs\nprovided here; the historical Lean and Comparator receipts establish the\nlinked formal theorem and its own proof.\n\n\\clearpage\n\\begin{samepage}\n\\section{Questions and exercises}\\label{sec:exercises}\n\n\\begin{readercheck}\\label{chk:hypotheses}\nWhich of $\\{2,4\\}$, $\\{2,4,8,16,\\dots\\}$ and $\\{2,4,6,8,\\dots\\}$ does\nTheorem~\\ref{thm:main} apply to?\n\\end{readercheck}\n\\end{samepage}\n\n\\begin{readercheck}\\label{chk:positive}\nSuppose $S=p/q$. Why would $0<R(k)<1/q$ be impossible, even if $k$\nwas chosen using $q$? Why do we need $R(k)>0$, and where does infinitude\nof $A$ supply it?\n\\end{readercheck}\n\n\\begin{readercheck}\\label{chk:columns}\nIn the square-exponent table, what stays fixed and what changes when we\nmove from $k=4$ to $k=8$? Why can we not choose the smallest entry in each\nrow and regard their sum as one successful attempt?\n\\end{readercheck}\n\n\\begin{readercheck}\\label{chk:partial}\nWhy is the mean over a complete consecutive cycle below $1/n$? Why\ndoes the same bound hold for a complete sampled cycle, which may have fewer\nthan $n$ entries? For $n=6$ and $Q=4$, the full cycle averages\n$\\tfrac2{21}<\\tfrac16$. Is the first value alone also below $\\tfrac16$?\n\\end{readercheck}\n\n\\begin{readercheck}\\label{chk:order}\nIn Section~\\ref{app:numbers}, replace $Q=16$ by $Q'=64$ and keep\n$T=512$. Does the far bound still give less than $\\tfrac1{10}$? What must\nbe chosen again?\n\\end{readercheck}\n\n\\begin{exercise}\\label{ex:2}\nFor $n=5$ and $Q=1$, calculate the mean of the first seven entries.\nSeparate them into a complete cycle and a beginning portion of the next.\nWhy is the mean of all seven not simply half the sum of the two piece-means?\n\\end{exercise}\n\n\\begin{exercise}\\label{ex:3}\nUse the fractional-part formula~\\eqref{eq:fracpart-main} to show that $2^{12}-1$ is divisible by $7$ and\nby $63$, and find its remainder on division by $2^5-1=31$.\n\\end{exercise}\n\n\\begin{exercise}\\label{ex:4}\nAdapt the proof in Section~\\ref{sec:doubling} to the exponents\n$3,9,27,81,\\dots$, using $k=3^j$.\n\\end{exercise}\n\n\\begin{exercise}\\label{ex:5}\nShow that $5\\sqrt2-7=(\\sqrt2-1)^3$, and that multiplying a number\n$a\\sqrt2-c$, where $a,c$ are nonnegative integers, by $3-2\\sqrt2$ gives\nanother number $a'\\sqrt2-c'$, where $a',c'$ are nonnegative integers. Use the test of\nSection~\\ref{sec:test} to show that $\\sqrt2$ is irrational.\n\\end{exercise}\n\n\\begin{exercise}\\label{ex:6}\nLet $a_1=2$ and let each later exponent be one more than the product of the\nearlier ones, giving $2,3,7,43,1807,\\dots$. Set $A=\\{a_1,a_2,\\dots\\}$\nand write $P_j=a_1a_2\\cdots a_j$.\n(a) Show that $\\tfrac1{a_1}+\\dots+\\tfrac1{a_j}=1-\\tfrac1{P_j}$. (b) Show that\nwith $k=P_j$ the first $j$ exponents leave $0$ but the exponent $P_j+1$ leaves at\nleast $\\tfrac37$, so $R(P_j)\\ge\\tfrac37$ for every $j$.\n\\end{exercise}\n\n\\section{Answers}\\label{app:answers}\n\\begingroup\n\\setlength{\\parskip}{0.7ex plus 0.2ex}\n\\setlength{\\parindent}{0pt}\n\n\\emph{Question~\\ref{chk:hypotheses}.} Only $\\{2,4,8,16,\\dots\\}$: $\\{2,4\\}$ is\nfinite (its sum $\\tfrac25$ is rational), and for $\\{2,4,6,\\dots\\}$ the sum\n$\\tfrac12+\\tfrac14+\\tfrac16+\\dots$ is half of $1+\\tfrac12+\\tfrac13+\\dots$,\nwhich diverges.\n\n\\emph{Question~\\ref{chk:positive}.} The identity\n$R(k)=((2^k-1)p-qJ(k))/q$ has an integer numerator. If positive, that\nnumerator is at least $1$, so $R(k)\\ge1/q$. Choosing $k$ using $q$ does not\nchange this argument: $2^k-1$ and $J(k)$ remain integers. Zero would give no\ncontradiction. Infinitude supplies a selected $n>k$, whose multiplied term\nis positive and less than $1$; all of it contributes to $R(k)$, and the\nother contributions are nonnegative.\n\n\\emph{Question~\\ref{chk:columns}.} The exponents $A$ and the original sum\n$S$ stay fixed. The multiplier changes from $15$ to $255$, and every\nselected term is transformed using that same multiplier within its column.\nSeparate row minima may come from different columns and need not describe\nany one multiplier applied to $S$.\n\n\\emph{Question~\\ref{chk:partial}.} A consecutive cycle has total below $1$\nand $n$ entries, hence mean below $1/n$. A sampled cycle selects the smallest\nremainder from each equal-sized group. Since the fractional parts increase\nwith the remainder, its mean is no larger, even when it has fewer entries.\nThe first value\nalone, however, is $\\tfrac{15}{63}=\\tfrac5{21}=\\tfrac{10}{42}$, more than\n$\\tfrac16=\\tfrac7{42}$. The complete-cycle bound need not hold for a\nbeginning portion; the separate bound~\\eqref{eq:uniform-main} covers these too.\n\n\\emph{Question~\\ref{chk:order}.} No:\n$2Q'/T=128/512=1/4>1/10$. Choose $T$ again: $T'=1024$ still fails,\nbut $T'=2048$ works, since $128/2048=1/16<1/10$.\nThe middle exponents $32,64,\\dots,2048$ now have cycle lengths\n$1,1,2,4,8,16,32$, so $M=64$ still works. Its validity had to be\nrechecked, even though its value did not change. This is why $Q$ is fixed\nbefore $T$.\n\n\\emph{Exercise~\\ref{ex:2}.} The first five entries sum to $26/31$;\nthe next two sum to $4/31$. All seven therefore average $30/217$.\nThe two piece-means, $26/155$ and $2/31$, must be weighted by their lengths:\n$\\bigl(5\\cdot\\tfrac{26}{155}+2\\cdot\\tfrac2{31}\\bigr)/7=30/217$.\nGiving the two pieces equal weight would give each entry in the shorter\npiece more weight than each entry in the longer one.\n\n\\emph{Exercise~\\ref{ex:3}.} $3$ and $6$ divide $12$, so $7$ and $63$ divide\n$2^{12}-1=4095$. The remainder of $12$ divided by $5$ is $2$, so the remainder\non division by $31$ is $2^2-1=3$: indeed $4095=31\\times132+3$.\n\n\\emph{Exercise~\\ref{ex:4}.} For $k=3^j$, exponents $3^i$ with $i\\le j$ divide\n$k$ and leave nothing. Later exponents $n=3^i$ leave less than\n$\\tfrac2{2^{n-k}}$, and the gaps $3^i-3^j$ are different and at least\n$2\\times3^j$, so $R(3^j)<\\sum_{e\\ge2\\times3^j}\\tfrac2{2^e}=\\tfrac4{2^{2\\times3^j}}$,\nwhich is below $\\tfrac1q$ for large $j$.\n\n\\emph{Exercise~\\ref{ex:5}.} $(\\sqrt2-1)^2=3-2\\sqrt2$ and\n$(\\sqrt2-1)(3-2\\sqrt2)=5\\sqrt2-7$. Also\n$(a\\sqrt2-c)(3-2\\sqrt2)=(3a+2c)\\sqrt2-(4a+3c)$. Start with $\\sqrt2-1=1\\sqrt2-1$. The displayed recurrence preserves\nnonnegative integer values of $a$ and $c$, so every odd power\n$(\\sqrt2-1)^{2i+1}$ is $m\\sqrt2-j$ for nonnegative integers $m$ and $j$, and it is\npositive. Since $1^2<2<(3/2)^2$, we have $0<\\sqrt2-1<\\tfrac12$. Hence\n$0<(\\sqrt2-1)^{2i+1}<\\tfrac1{2\\cdot4^i}$, which is below $\\tfrac1q$ once\n$2\\cdot4^i>q$. By the test, $\\sqrt2$ is irrational.\n\n\\begin{samepage}\n\\emph{Exercise~\\ref{ex:6}.} (a) With $P_0=1$, $a_j=P_{j-1}+1$ and\n$P_j=P_{j-1}a_j$, so\n$\\tfrac1{P_{j-1}}-\\tfrac1{P_j}=\\tfrac{a_j-1}{P_j}=\\tfrac1{a_j}$, and adding\nthese for $1,\\dots,j$ leaves $1-\\tfrac1{P_j}$. Since $P_j\\ge2^j$,\nthese partial sums tend to $1$. The set $A$ is infinite and satisfies\nthe theorem's reciprocal-sum hypothesis. (b) The first $j$ exponents\ndivide $P_j$. The exponent $P_j+1$ is bigger than $k=P_j$, so it leaves\n$\\tfrac{2^{P_j}-1}{2^{P_j+1}-1}$, which is at least $\\tfrac37$ because\n$7(2^{P_j}-1)-3(2^{P_j+1}-1)=2^{P_j}-4\\ge0$. Thus the particular choices\n$k=P_j$ fail to make the remainders small. Other multiples of $P_j$ preserve\nthe same initial zeros; averaging lets us search among those choices.\\par\n\\end{samepage}\n\n\\endgroup\n\n\\begin{thebibliography}{9}\n\\setlength{\\itemsep}{2pt}\n\\setlength{\\parsep}{0pt}\n\\setlength{\\parskip}{0pt}\n\\bibitem{erdos1948} P.~Erd\\H{o}s, \\emph{On arithmetical properties of Lambert\n series}, J.~Indian Math.\\ Soc.\\ 12 (1948), 63--66.\n\\bibitem{erdos1968} P.~Erd\\H{o}s,\n \\href{https://users.renyi.hu/~p_erdos/1969-09.pdf}{\\emph{On the irrationality of\n certain series}}, Math.\\ Student 36 (1968), 222--226 (issued 1969).\n\\bibitem{erdos1988} P.~Erd\\H{o}s,\n \\href{https://users.renyi.hu/~p_erdos/1988-22.pdf}{\\emph{On the irrationality of\n certain series: problems and results}}, in A.~Baker (ed.), \\emph{New Advances\n in Transcendence Theory}, Cambridge University Press, 1988, 102--109.\n\\bibitem{erdosproblems257} T.~F.~Bloom, \\emph{Erd\\H{o}s Problem \\#257},\n \\url{https://www.erdosproblems.com/257}; the statement is quoted in the Formal\n Conjectures file \\texttt{FormalConjectures/ErdosProblems/257.lean}.\n\\bibitem{taoteravainen2025} T.~Tao and J.~Ter\\\"av\\\"ainen,\n \\href{https://arxiv.org/abs/2512.01739v2}{\\emph{Quantitative correlations and\n some problems on prime factors of consecutive integers}}, arXiv:2512.01739v2\n (submitted December 2025, revised April 2026).\n\\bibitem{shortpaper} W.~Cook,\n \\href{https://github.com/wcook04/plectis-erdos/blob/0064fc9b2a2693de97aebc8abec26090c3cd1095/paper/257/erdos-257-mersenne-support-subseries.pdf}{\\emph{Irrationality\n criteria for Lambert subseries}}, 30 September 2026.\n\\bibitem{longrecord} W.~Cook,\n \\href{https://github.com/wcook04/plectis-erdos/blob/0064fc9b2a2693de97aebc8abec26090c3cd1095/paper/257/erdos257-mersenne-reasoning-surface.pdf}{\\emph{Reciprocal\n Mersenne subseries: support criteria, finite arithmetic and conditional orbit\n problems}}, 30 September 2026.\n\\end{thebibliography}\n\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":53432,"sha256":"sha256:5efdc16fe9917b634a2af0068bab9ce0e17127b58d5e82e6ab3e412c1e791352","cite_as":"Plectis reader corpus, paper:erdos257-expository-sums-never-fractions","source_repository":"plectis-erdos","source_path":"paper/257/erdos257-expository-sums-never-fractions.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"257/figures/cycle-mean.tex","source_path":"paper/257/figures/cycle-mean.tex","media_type":"text/x-tex","sha256":"sha256:6155b89fc5370c923a1b66e349ae7086bbb3e30c9d34bd3ca09a0583064893da","byte_count":2100,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-License-Identifier: CC-BY-4.0\n% The arrow repeats values; it does not represent chronological sampling.\n% Each tuple is remainder/original numerator/repeated numerator, over 63.\n% Common linear height scale and equal widths preserve the comparison.\n\\begin{tikzpicture}[x=1mm,y=1mm,font=\\small]\n  \\node[anchor=west,align=left,inner sep=0pt] at (0,23)\n    {3 sampled values\\\\(sorted)};\n  \\node[anchor=west,inner sep=0pt] (samples) at (0,15)\n    {$0,\\ \\dfrac{3}{63},\\ \\dfrac{15}{63}$};\n  % The arrow ends at the copied-value row, not at the original bar labels.\n  \\draw[-{Stealth[length=2mm]},line width=.6pt]\n    ([yshift=-2mm]samples.south) -- (samples.south |- 0,-10.5) -- (61,-10.5);\n  \\node[anchor=west,inner sep=0pt] at (19,4) {copy each twice};\n  % Adjacent keys distinguish the constructed copies from the comparison data.\n  \\filldraw[fill=paperresultrule!30,draw=black!75,line width=.55pt]\n    (33,20.75) rectangle (36.5,24.25);\n  \\node[anchor=west,align=left,inner sep=0pt] at (39,22.5)\n    {copied sample\\\\values};\n  \\draw[black!75,line width=.55pt] (33,12.75) rectangle (36.5,16.25);\n  \\node[anchor=west,align=left,inner sep=0pt] at (39,14.5)\n    {consecutive\\\\cycle values};\n  \\foreach \\r/\\v/\\w in {0/0/0,1/1/0,2/3/3,3/7/3,4/15/15,5/31/15}{\n    \\pgfmathsetmacro{\\xx}{62+10*\\r+3*floor(\\r/2)}\n    \\fill[paperresultrule!30] (\\xx,0) rectangle (\\xx+7,.55*\\w);\n    \\draw[black!75,line width=.55pt] (\\xx,0) rectangle (\\xx+7,.55*\\v);\n    \\node[anchor=south,inner sep=1pt] at (\\xx+3.5,.55*\\v+1.3)\n      {$\\dfrac{\\v}{63}$};\n    \\node[anchor=north,inner sep=1pt] at (\\xx+3.5,-1.3) {$\\r$};\n  }\n  \\node[anchor=east,inner sep=0pt] at (58,-3) {remainder $r$};\n  \\foreach \\a in {62,85,108}\n    \\draw[decorate,decoration={brace,mirror,amplitude=3pt},black!65]\n      (\\a,-6.5) -- (\\a+17,-6.5);\n  % List the zero copies explicitly rather than inventing positive bar heights.\n  \\node[anchor=north,inner sep=1pt] at (70.5,-8.5) {$0,\\ 0$};\n  \\node[anchor=north,inner sep=1pt] at (93.5,-8.5)\n    {$3/63,\\ 3/63$};\n  \\node[anchor=north,inner sep=1pt] at (116.5,-8.5)\n    {$15/63,\\ 15/63$};\n\\end{tikzpicture}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  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It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/257/erdos257-expository-sums-never-fractions.tex","section_outline":["The question: which infinite sums are irrational?","A rational number has a fixed positive gap","Multiplying the series and retaining fractional parts","A complete example: the doubling exponents","Why averaging works","One table, many attempts","Complete cycles give small averages","A bound that also covers unfinished cycles","Combining the estimates","The construction with actual numbers","The same choices for any admissible set","Consequences and limits","Supporting details and formal-proof correspondence","Convergence and tails","Passing to the full transformed sum","Why the sampled remainders form a cycle","Adding the row averages","Other bases","Formal verification and its scope","Questions and exercises","Answers"]},"erdos269-running-lcm-reasoning-surface":{"paper_id":"erdos269-running-lcm-reasoning-surface","title":"Running least common multiples: distinct values and multiplicities","form":"Reasoning surface","length_class":"long","pages":73,"subject":"Erdős #269","subject_kind":"problem","problem_id":"erdos_269","landing_role":"mathematics","question":"How do distinct running-LCM irrationality, infinite-rank kernels and repeated-sum tail arithmetic fit together, and which residue windows remain open?","precis":"For the prime set {2,3,5}, a five-map argument proves irrationality of the sum over distinct running LCMs. Two-prime sums reduce to a transcendental Hecke–Mahler value. For the repeated three-prime sum, an equivalent residue criterion leaves an arithmetic condition open. Complete ordinary arguments for general distinct-running-LCM sums and prime-power subseries are retained as remarks awaiting formalisation.","owns":"distinct-height irrationality, all-rank kernel structure and the repeated-sum tail criteria, preserving the multiplicity distinction and the remaining open residue-window obligation","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #269, which remains open","first_pass":["long269:sec:problem","long269:sec:rank","long269:sec:open","sec:erdos-269-complete-family-map"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos269-running-lcm-reasoning-surface.pdf","hosted_pdf_path":"papers/erdos269-running-lcm-reasoning-surface.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/269/erdos269-running-lcm-reasoning-surface.tex","site_html":"maths/papers/erdos269-running-lcm-reasoning-surface.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/269/erdos269-running-lcm-reasoning-surface.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/269/erdos269-running-lcm-reasoning-surface.tex","body_pointer":"papers.records.erdos269-running-lcm-reasoning-surface.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Extended proofs and source references accompanying the short note.\n% The immutable source-link revisions below are deliberately unchanged.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n\\input{evidence/erdos269-running-lcm-reasoning-surface}\n% Number the ordinary-proof remarks so their labels identify the assertion.\n% This local presentation change leaves the shared preamble untouched.\n\\theoremstyle{remark}\n\\newtheorem{recordremark}[theorem]{Remark}\n\\renewenvironment{remark}[1][]{\\begin{recordremark}[#1]}{\\end{recordremark}}\n\\theoremstyle{plain}\n\n\\usepackage{longtable}\n\\usepackage{colortbl}\n\\usepackage[htt]{hyphenat}\n\\usepackage{newunicodechar}\n\\newunicodechar{ℕ}{\\ensuremath{\\N}}\n\\newunicodechar{ℚ}{\\ensuremath{\\Q}}\n\\newunicodechar{ℤ}{\\ensuremath{\\Z}}\n\\newunicodechar{₀}{\\ensuremath{{}_0}}\n% Historical default for macros that require one source coordinate. Individual\n% source links below remain pinned to their own immutable revisions.\n\\renewcommand{\\commit}{ee650b32b8b2cb98b94e5500df5370d85f7403b8}\n\\renewcommand{\\commitshort}{ee650b32b8b2}\n% The distinct-height programs, at the commit that added them, and the Lean\n% proof of the {2,3,5} theorem, at the commit that added it.\n\\newcommand{\\distinctheighturl}{https://github.com/wcook04/plectis-erdos/tree/10178c4df40cb27d83b5ebb1ec337dd588b82998/research/experiments/erdos269/distinct_height}\n\\newcommand{\\distinctheightlean}{https://github.com/wcook04/plectis-erdos/blob/5154f4c46fa5d5ab2cff8df857a0b10f310cbeeb/lean/ErdosProblems/Erdos269/DistinctHeightIrrationality.lean}\n\\newcommand{\\distinctheightblocklean}{https://github.com/wcook04/plectis-erdos/blob/5154f4c46fa5d5ab2cff8df857a0b10f310cbeeb/lean/ErdosProblems/Erdos269/DistinctHeightBlockRadix.lean}\n\\hypersetup{\n  pdftitle={Running least common multiples: distinct values and multiplicities},\n  pdfsubject={Erdős Problem Notes: Problem 269},\n  pdfkeywords={irrationality, transcendence, Hecke-Mahler values, least common multiple, smooth numbers, Cantor series, lattice sums, Lean 4},\n}\n\n% The three pure-power coordinates and the running height.\n\\newcommand{\\hgt}{\\operatorname{H}}\n\\newcommand{\\lo}{\\lambda}\n\\newcommand{\\hi}{\\eta}\n\\newcommand{\\Lprefix}{\\operatorname{L}}\n\\newcommand{\\Kernel}{\\operatorname{K}}\n\\newcommand{\\lpr}{\\operatorname{lpr}}\n\\newcommand{\\Esc}{\\mathsf E}\n\n\\runninghead{Running least common multiples}\n\\title{Running least common multiples:\\\\distinct values and multiplicities}\n\\subtitle{Erd\\H{o}s Problem~\\#269}\n\\author{Will Cook}\n\\date{July 2026; revised 5 October 2026}\n\n\\setlength{\\emergencystretch}{2em}\n% Permit breaks inside long declaration names in the generated catalogue.\n\\usepackage{xurl}\n\\begin{document}\n% ---- part core ----\n% Part: core\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\nWe study the running least common multiples of integers whose prime\nfactors belong to a fixed finite set.\nFor $\\{2,3,5\\}$, a five-map argument proves irrationality of the sum over\ndistinct running LCMs. Two-prime sums reduce to a transcendental Hecke--Mahler\nvalue. For the repeated three-prime sum we give an equivalent residue\ncriterion, whose arithmetic condition remains open. Complete ordinary\narguments for general sums over distinct running LCMs and prime-power subseries are\nretained as remarks awaiting formalisation.\n\\end{abstract}\n\n\\section{Two sums from running least common multiples}\\label{long269:sec:problem}\nLet $P$ be a finite set of at least two primes, and let $\\mathcal S_P$\nconsist of $1$ and all positive integers whose prime factors belong to\n$P$. For $x\\ge1$, put\n\\[\n \\Lprefix(x)=\\lcm\\{u\\in\\mathcal S_P:u\\le x\\},\\qquad\n \\hgt_P(x)=\\prod_{p\\in P}p^{\\lfloor\\log_p x\\rfloor}.\n\\]\nEvery integer in this finite set divides $\\hgt_P(x)$, and the set\ncontains the largest power of each prime in $P$ below or equal to $x$.\nTheir product therefore equals the least common multiple:\n$\\Lprefix(x)=\\hgt_P(x)$. The value changes only at a positive prime\npower. Summing its distinct reciprocals gives\n\\[\n \\mathcal D_P=1+\\sum_{\\substack{t=p^n\\\\p\\in P,\\ n\\ge1}}\\frac1{\\hgt_P(t)}.\n\\]\nWe begin with an irrationality result for three primes.\n\n\\begin{theorem}[the sum over distinct running LCMs at $\\{2,3,5\\}$]\\label{long269:res:distinct-height-235}\nThe sum $\\mathcal D_{\\{2,3,5\\}}$ is irrational.\n\\end{theorem}\n\nThe proof in Section~\\ref{long269:sec:distinct-235} gives the primes three\ndifferent roles. Powers of $2$ delimit the blocks; powers of $5$ occur\noften enough to separate the five affine tail images; powers of $3$\nhave an irrational block frequency. Separation makes each tail determine\nthe next block and tail. Under rationality there are only finitely many\ntails, so repetition would force an impossible periodic block sequence.\n\nErd\\H{o}s asserted the general conclusion for every finite $P$ with\n$|P|\\ge2$ in his letter of 1 January 1973, without printing a\nproof~\\cite[p.~335]{erdos1974letter}. The theorem above is a special case\nof that assertion. The general argument in\nSection~\\ref{long269:sec:distinct-all} compares nearby tails and finds an\nisolated interchange of jumps on a torus. We retain it as an ordinary\nargument awaiting formalisation, with its intermediate claims stated as\nremarks. The specialised five-map proof is independent of that argument.\n\nThe question in Erd\\H{o}s Problem~\\#269 retains multiplicities. If\n$a_1<a_2<\\cdots$ enumerates $\\mathcal S_P$, it asks about\n\\[\n \\mathcal R_P=\\sum_{n\\ge1}\\frac1{[a_1,\\ldots,a_n]}\n             =\\sum_{u\\in\\mathcal S_P}\\frac1{\\Lprefix(u)}\n\\]\n\\cite[p.~65]{erdosgraham1980}\\cite[p.~106]{erdos1988}; the numbering is\nBloom's~\\cite{erdosproblems}. For $P=\\{2,3,5\\}$, the integers $5$ and $6$\nboth have running LCM $60$. They contribute $2/60$ to $\\mathcal R_P$ and\n$1/60$ to $\\mathcal D_P$. Thus Theorem~\\ref{long269:res:distinct-height-235}\nconcerns the sum over distinct values, whereas the problem concerns\nthe sum with multiplicity. The arguments in this paper leave\n$\\mathcal R_P$ unresolved when $P$ is finite and $|P|\\ge3$.\n\nFor two primes the conclusion is stronger.\n\n\\begin{theorem}[two-prime transcendence]\\label{long269:res:lead-two-prime}\nLet $p$ and $q$ be distinct primes.  Then $\\mathcal R_{\\{p,q\\}}$ and\n$\\mathcal D_{\\{p,q\\}}$ are transcendental.\n\\end{theorem}\n\\evidenceremark{The Lean proof assumes the transcendence theorem of Bugeaud\nand Laurent.}\n\nFan's post of 26 June 2026 gave the finite-prime running-LCM identity,\nthe factorisation of $\\mathcal R_{\\{p,q\\}}$, and the Hecke--Mahler reduction\nwith its transcendence conclusion~\\cite{fan2026comment}. We reproduce that\ncalculation in Section~\\ref{long269:sec:two-prime}. Both values are\nnonconstant polynomials over $\\mathbb Q$ in one boundary sum, whose\ntranscendence follows from Loxton and van der Poorten\n\\cite[Theorem~8, p.~40]{loxtonvdp1977}, in the formulation of Bugeaud and\nLaurent~\\cite[Theorem~1.1]{bugeaudlaurent2023}. No priority is claimed for\nthis deduction.\n\nRetaining multiplicities preserves the bases but changes the numerators.\nThe repeated tails satisfy $X_{a+1}=b_aX_a-m_a$, with\n$b_a\\in\\{2,6,10,30\\}$, while both $m_a$ and $X_a$ grow quadratically.\nThe preceding finite-tail repetition argument no longer applies. Rationality would\nstill make every sufficiently late $BX_a$ integral, where $B$ is the\ndenominator part coprime to $30$. A finite segment excludes such an\ninteger tail when its least positive residue exceeds the quadratic\nbound. Irrationality is equivalent to finding these segments for every\n$B$ beyond every prescribed onset. The finite computations do not\nestablish either unbounded quantifier.\n\nThe five-map proof comes first. Sections~\\ref{long269:sec:lcm}\nand~\\ref{long269:sec:two-prime} then supply the running LCM geometry and\ntwo-prime formulae. For general distinct sums,\nSection~\\ref{long269:sec:distinct-all} uses comparison of two tails rather\nthan recovery from one; the finite-state, subseries and linear-flow\nextensions follow as ordinary arguments pending formalisation.\nSection~\\ref{long269:sec:rank} separates exact kernel rank from approximation.\nFor the repeated target, Sections~\\ref{long269:sec:blocks}--\\ref{long269:sec:escape}\nderive the coefficients, bounds and residue criterion; the remaining\nsections distinguish finite certificates from missing arithmetic inputs.\nSection~\\ref{long269:sec:proof-sources} records formal coverage. The\n\\href{erdos-269-three-prime-running-lcm.pdf\\#r4-five-map-theorem}{short paper}\nconcentrates on the five-map theorem and the\n\\href{erdos-269-three-prime-running-lcm.pdf\\#r4-repeated-criterion}{repeated-tail criterion}.\n\nWe use $\\N=\\{0,1,2,\\ldots\\}$ for exponents and braces for fractional parts.\nIn the finite-geometry section and in the kernel calculation, $P=\\{p,q,r\\}$ consists\nof three pairwise distinct primes, $\\hgt=\\hgt_P$, and\n\\[\n \\Kernel(i,j,k)=\\hgt(p^iq^jr^k)^{-1}.\n\\]\nThe other sections specify their prime sets separately. For $\\{2,3,5\\}$,\n$\\mathcal S_P$ consists of the usual $5$-smooth integers. This fixed support\ndiffers from a varying smoothness bound in $\\Psi(x,y)$\n\\cite{hildebrand1986}. Tijdeman and Meijer~\\cite[Sections~1--4]{tijdemanmeijer1974}\nand the two-prime study of Languasco, Luca, Moree and Togb\\'e\n\\cite{languasco2025} give the relevant lattice-counting context; the\nestimates below use no smooth-number density asymptotic or gap theorem.\nKova\\v{c} and Tao~\\cite{kovactao2024} study other unit-fraction series by\nelementary methods, without providing an input to these proofs.\n\n\\medskip\n\\noindent\\textbf{Keywords.} irrationality; transcendence; least common\nmultiple; smooth numbers; Cantor series; separated rank.\n\\qquad\\textbf{MSC 2020.} 11J72 (primary); 11A05, 11N25, 68V20 (secondary).\n\n\\section{Recovering the prime-power jumps}\\label{long269:sec:distinct-235}\n\\papersectiontarget{r3-five-maps}\nAs in long division, periodicity requires both finitely many remainders\nand a deterministic rule for the next step. The varying bases make the\nsecond requirement nontrivial. We obtain it by separating the five\npossible block images, using a bound on the actual arithmetic tails.\n\n\\begin{proof}[Proof of Theorem~\\ref{long269:res:distinct-height-235}]\n\\emph{Finitely many tails under rationality.}\nWrite $\\hgt=\\hgt_{\\{2,3,5\\}}$ in this proof and put $P_a=\\hgt(2^a)$.\nEvery running LCM at or before $2^a$ divides $P_a$. We therefore normalise\nthe tail after $2^a$ by\n\\begin{equation}\\label{long269:eq:five-tail}\n Y_a=P_a\\sum_{\\substack{t>2^a\\\\t=2^n,3^n\\text{ or }5^n,\\ n\\ge1}}\n                   \\frac1{\\hgt(t)}.\n\\end{equation}\nEach subsequent jump multiplies the running LCM by at least $2$, and a jump\nby $3$ eventually occurs. Comparison with $\\sum_{j\\ge1}2^{-j}$ gives\n$0<Y_a<1$.\n\nSuppose that $\\mathcal D_{\\{2,3,5\\}}=N/K$, where $N$ is an integer and\n$K\\ge1$ is an integer. Clearing the finite prefix gives\n\\[\n KY_a=P_aN-K\\left(P_a+\n        \\sum_{\\substack{t\\le2^a\\\\t=2^n,3^n\\text{ or }5^n,\\ n\\ge1}}\n                 \\frac{P_a}{\\hgt(t)}\\right)\\in\\mathbb Z.\n\\]\nThus the $Y_a$ lie in the finite set $(K^{-1}\\mathbb Z)\\cap(0,1)$.\nIt remains to show why equality of two tails propagates.\n\n\\medskip\n\\emph{Recovering a block from its tail.}\nConsider the jumps between $16$ and $32$, including the last endpoint.\nThey occur at $25,27,32$, multiplying the running LCM successively by $5,3,2$.\nAfter normalisation at $16$, their contribution is\n$1/5+1/15+1/30$. The remaining tail is normalised at a running LCM $30$\ntimes as large, so\n\\[\n Y_4=\\frac15+\\frac1{15}+\\frac1{30}+\\frac{Y_5}{30}\n     =\\frac{9+Y_5}{30}.\n\\]\nIn the block $(64,128]$, the interior jumps occur in the opposite order,\nat $81,125$. Its constant term is $1/3+1/15+1/30=13/30$.\nThe multiplier $30$ records the primes, but not their order; the\nnumerators $9$ and $13$ retain the order. The remaining question is\nwhether the unknown following tail can conceal this difference.\n\nIn general, $(2^a,2^{a+1}]$ contains its terminal power of $2$, at most\none power of $3$ and at most one power of $5$. Let $\\tau_a$ denote the\nsequence of interior prime labels, omitting the terminal $2$. Splitting\noff the block gives\n\\begin{equation}\\label{long269:eq:five-maps}\n Y_a=G_{\\tau_a}(Y_{a+1}),\\qquad\n G_\\tau(y)=\\frac{\\mu_\\tau+y}{b_\\tau},\\qquad\n \\begin{array}{c|ccccc}\n \\tau&\\varnothing&3&5&35&53\\\\\\hline\n b_\\tau&2&6&10&30&30\\\\\n \\mu_\\tau&1&3&3&13&9\n \\end{array}\n\\end{equation}\nHere $b_\\tau$ is the factor by which the running LCM increases across the\nblock, and $\\mu_\\tau/b_\\tau$ is the block's normalised contribution.\n\nThe five formulae alone do not recover $\\tau_a$ from $Y_a$.\nThe coarse images $G_{53}([0,1])=[3/10,1/3]$ and\n$G_5([0,1])=[3/10,2/5]$ overlap. We need a smaller interval known to\ncontain the next \\emph{arithmetic} tail. The table gives the lower bound\n$Y_a\\ge3/10$. With that lower endpoint, separating these two images\nrequires an upper endpoint $U$ satisfying\n\\[\n G_{53}(U)<G_5(3/10),\\qquad\n \\frac{9+U}{30}<\\frac{33}{100},\\qquad U<\\frac9{10}.\n\\]\nThis specifies the improvement needed from the arithmetic sequence.\n\nA power $5^f$ belongs to block $\\lfloor f\\log_2 5\\rfloor$. These\nindices start at $2$ and their successive differences are $2$ or $3$.\nEvery three consecutive blocks therefore contain a power of $5$.\nGiven $a$, let $j\\in\\{a,a+1,a+2\\}$ be the first such block. The maps\nfor blocks containing $5$ give $Y_j\\le7/15$, since $Y_{j+1}<1$ and\n$G_{35}(1)=7/15$ bounds $G_5(1)$ and $G_{53}(1)$ as well.\nWe now work \\emph{backwards} from $j$ to $a$. Each intervening block\nhas map $G_\\varnothing$ or $G_3$, and\n$G_3(y)\\le G_\\varnothing(y)=(1+y)/2$ for $y\\ge0$.\nAt most two backward steps give the successive bounds\n$7/15$, $11/15$, $13/15$. Hence\n\\begin{equation}\\label{long269:eq:five-interval}\n Y_a\\in I=\\left[\\frac3{10},\\frac{13}{15}\\right],\\qquad\n G_\\varnothing\\bigl(G_\\varnothing(7/15)\\bigr)=\\frac{13}{15}<\\frac9{10}.\n\\end{equation}\nThe bound uses the occurrence of a power of $5$ within three blocks.\nIt does not assert invariance under arbitrary words in the five maps:\nindeed, $G_\\varnothing(13/15)=14/15$ lies outside $I$.\n\nThe following table gives the exact images of $I$. They\nare pairwise disjoint, with successive gaps\n$1/900,17/300,79/900,1/180$. Writing $L=3/10$ and $U=13/15$, the two\nnarrow gaps are\n\\[\n G_5(L)-G_{53}(U)=\\frac{3L-U}{30}=\\frac1{900},\\qquad\n G_\\varnothing(L)-G_3(U)=\\frac{3L-U}{6}=\\frac1{180}.\n\\]\nBoth are positive because $U<3L=9/10$.\n\n\\[\n \\begin{array}{c|c}\n \\tau&G_\\tau(I)\\\\\\hline\n 53&[31/100,74/225]\\\\\n 5&[33/100,29/75]\\\\\n 35&[133/300,104/225]\\\\\n 3&[11/20,29/45]\\\\\n \\varnothing&[13/20,14/15]\n \\end{array}\n\\]\n\nSince $Y_{a+1}\\in I$, the actual tail $Y_a$ belongs to exactly one of\nthese five image intervals, identifying $\\tau_a$. For instance,\n$Y_a\\in[31/100,74/225]$ forces $\\tau_a=53$ and gives $Y_{a+1}=30Y_a-9$.\nIn general we calculate $Y_{a+1}=b_{\\tau_a}Y_a-\\mu_{\\tau_a}$, identify its\nimage interval, and continue. Each recovered value is the next arithmetic\ntail, so the same bound $I$ remains available at every step.\n\n\\medskip\n\\emph{Repetition would give a rational frequency.}\nThe finitely many tail values obtained above include a repeated value\n$Y_u=Y_v$ with $u<v$. Since the image intervals are disjoint, this\nforces $\\tau_u=\\tau_v$. Applying the corresponding inverse map gives\n$Y_{u+1}=Y_{v+1}$. Induction yields\n$\\tau_{u+j}=\\tau_{v+j}$ for every $j\\ge0$. Thus the entire later block\nword has period $v-u$.\n\nTo exclude this possibility, let $\\alpha=\\log2/\\log3$. The number of powers\nof $3$ in block $a$ is\n$\\lfloor(a+1)\\alpha\\rfloor-\\lfloor a\\alpha\\rfloor$, and is determined by\n$\\tau_a$. If the block word has eventual period $\\ell\\ge1$, let $c$ be the\nnumber of $3$-powers in one period. For every sufficiently large fixed $a$\nand every $n\\ge1$ we then have\n\\[\n \\lfloor(a+n\\ell)\\alpha\\rfloor=\\lfloor a\\alpha\\rfloor+nc.\n\\]\nDividing by $n$ and taking the limit gives $\\ell\\alpha=c$, contrary to\nunique factorisation, since it would imply $2^\\ell=3^c$.\n\\end{proof}\nThe Lean proof follows this route.  It\n\\href{\\distinctheightlean\\#L60}{identifies $\\hgt(t)$ with the running least\ncommon multiple}, \\href{\\distinctheightlean\\#L698}{shows that the sum counts\neach distinct value once}, identifies the contribution of each dyadic block\nwith one of the five pairs $(b_\\tau,\\mu_\\tau)$, and proves that\n\\href{\\distinctheightlean\\#L437}{every three consecutive blocks contain a power\nof $5$} and that \\href{\\distinctheightlean\\#L494}{the blocks containing a power\nof $3$ are not eventually periodic}.  It then applies an\n\\href{\\distinctheightblocklean\\#L340}{irrationality criterion for block series\nwith these five pairs}, whose proof contains the bounds, the separation and\nthe return above.\n\nThe first twelve block types are\n$\\varnothing,3,5,3,53,\\varnothing,35,3,\\varnothing,53,\\varnothing,35$.\nThe saved numerical calculation over the first $3000$ blocks reports tails\nin $[0.3186,0.7799]$. These approximations are separate from the rational\nbounds used in the proof.\n\nGrouping the jumps into dyadic blocks matters.  One jump at a time, the\ntails $x_k=(q_1\\cdots q_k)\\sum_{j>k}(q_1\\cdots q_j)^{-1}$\nsatisfy $x_{k}=(1+x_{k+1})/q_{k+1}$ and $1/4\\le x_k\\le1$, and the images of\n$[1/4,1]$ under $x\\mapsto(1+x)/2$ and $x\\mapsto(1+x)/3$ overlap on\n$[5/8,2/3]$.  A tail before a jump by $2$ cannot be told from a tail before a\njump by $3$ from these bounds alone.\n\nListing the jumps one at a time writes $\\mathcal D_{\\{2,3,5\\}}$ as a\nCantor series with every digit equal to $1$ and every base in $\\{2,3,5\\}$.  The criteria of\nErd\\H{o}s--Straus \\cite[Theorem~2.1]{erdosstraus1974} and\nHan\\v{c}l--Tijdeman \\cite[Theorem~3.1]{hancltijdeman2004}, recalled in\nSection~\\ref{long269:sec:actual-orbit}, assume that each digit divided by\nthe product of the last two bases in its denominator tends to $0$.  Here that\nquotient is at least $1/25$, so neither criterion applies.\n\n\\section{Intervals of constancy of the running LCM}\\label{long269:sec:lcm}\n\nWe first describe which smooth integers contribute the same running value.\nThe LCM is determined by the largest exponent of each prime in the prefix.\nFor the unrestricted prefix this gives\n$\\operatorname{lcm}(1,\\ldots,N)=\\prod_{t\\le N}t^{\\lfloor\\log_t N\\rfloor}$,\nwith $t$ running over primes, and hence\n$\\log\\operatorname{lcm}(1,\\ldots,N)=\\psi(N)$; see Apostol~\\cite{apostol1976}\nand Montgomery and Vaughan~\\cite{montgomeryvaughan2007}.\nWe apply the same exponent rule to a fixed prime support, keeping track\nof the intervals on which the running LCM is constant.\n\nThe smooth numbers up to $x$ are indexed by the exponent triples $(i,j,k)$ with\n$i\\le\\lfloor\\log_p x\\rfloor$, $j\\le\\lfloor\\log_q x\\rfloor$,\n$k\\le\\lfloor\\log_r x\\rfloor$ and $p^{i}q^{j}r^{k}\\le x$.\nThe coordinate bounds alone need not describe the prefix: each prime-power\nfactor may be at most $x$ while their product exceeds $x$.  For instance,\nat $x=6$ the factors $4$, $3$ and $5$ satisfy their coordinate bounds,\nbut the corresponding smooth number is $60$.\n\n\\begin{theorem}[the running least common multiple]\\label{long269:res:lcm}\nLet $p,q,r$ be pairwise distinct primes and $x\\ge1$.  Then\n$\\Lprefix(x)=\\hgt(x)$.\n\\end{theorem}\n\n\\begin{proof}\nEvery smooth $n\\le x$ has exponents bounded by the corresponding integer\nlogarithms, so $n\\mid\\hgt(x)$ and hence $\\Lprefix(x)\\mid\\hgt(x)$.  Conversely\nthe three pure powers $p^{\\lfloor\\log_p x\\rfloor}$,\n$q^{\\lfloor\\log_q x\\rfloor}$ and $r^{\\lfloor\\log_r x\\rfloor}$ are themselves\nsmooth numbers not exceeding $x$, so each divides $\\Lprefix(x)$, and distinct\nprimes have coprime powers, so their product divides $\\Lprefix(x)$ as well.\n\\end{proof}\n\n\nAt $(p,q,r)=(2,3,5)$ the first ten values are\n\\[\n\\begin{array}{c|cccccccccc}\nx&1&2&3&4&5&6&7&8&9&10\\\\ \\hline\n\\Lprefix(x)&1&2&6&12&60&60&60&120&360&360\n\\end{array}\n\\]\nSo $\\Lprefix(6)=4\\cdot3\\cdot5=60$, which exceeds $6$: the running value at a\nsmooth cutoff already contains powers of the other two primes that the cutoff\nitself does not.  The kernel must therefore account for all three\nmaximal powers, not just the factorisation of the cutoff.\nAlso $\\hgt(x)\\le x^3$, since each factor is at most $x$.\nThe exponent is the number of generating primes.\n\nSay that $x$ and $y$ lie in the same \\emph{logarithmic cell} when\n$\\lfloor\\log_b x\\rfloor=\\lfloor\\log_b y\\rfloor$ for each of $b=p,q,r$.  By Theorem~\\ref{long269:res:lcm} the running value depends on $x$ only\nthrough the three integer logarithms, so it is constant on cells and moves only\nwhere one logarithm moves.\n\n\\begin{proposition}[constancy and jump ratios]\\label{long269:res:cell}\nIf $x,y\\ge1$ lie in the same logarithmic cell then $\\Lprefix(x)=\\Lprefix(y)$,\nand the same holds for the kernel at two smooth points of one cell.  If\n$\\lfloor\\log_p y\\rfloor=\\lfloor\\log_p x\\rfloor+1$ while the other two\nlogarithms agree, then $\\Lprefix(y)=p\\,\\Lprefix(x)$, and similarly with $q$ or\n$r$ in place of $p$.\n\\end{proposition}\n\n\\begin{proof}\nBoth parts are immediate from Theorem~\\ref{long269:res:lcm}: the running LCM depends on $x$\nonly through the three integer logarithms, and advancing one of them multiplies\nexactly one factor by its base.\n\\end{proof}\n\n\\label{long269:res:jump}\n\n\\begin{proposition}[jump count]\\label{long269:res:count}\nLet $n\\ge0$.  The set of the first $n$ positive powers of $p$, of $q$ and of\n$r$ has exactly $3n$ elements, and adjoining the common origin $1$ gives\nexactly $3n+1$.\n\\end{proposition}\n\n\\begin{proof}\nFor a fixed prime $b$, the powers $b,b^{2},\\ldots,b^{n}$ are distinct.\nA common value for two different primes would contradict unique\nfactorisation.  Finally $1$ is not a\npositive power of any prime.\n\\end{proof}\n\n\nThe later tail estimates use two elementary counting facts.  Write\n$\\mathcal B(h_p,h_q,h_r)$ for the exponent triples with\n$i\\le h_p$, $j\\le h_q$ and $k\\le h_r$, and $F(H)$ for the points of this\nbox whose running LCM equals $H$.\n\n\\begin{proposition}[grouping equal running LCMs]\\label{long269:res:fibre}\nFor every box $\\mathcal B$,\n\\[\n \\sum_{(i,j,k)\\in\\mathcal B}\\Kernel(i,j,k)=\\sum_{H}\\#F(H)/H,\n\\]\nthe outer sum ranging over the running LCMs attained on $\\mathcal B$.\n\\end{proposition}\n\n\\begin{proof}\nPartition $\\mathcal B$ into the fibres of the running LCM map.  On $F(H)$ every\nsummand is $1/H$, so the fibre contributes $\\#F(H)/H$.\n\\end{proof}\n\n\nNow fix an interval $[\\lo,\\hi)$ with $0\\le\\lo<\\hi$ and write\n$\\mathcal S$ for the exponent triples of $\\mathcal B(h_p,h_q,h_r)$ whose\nvalue $p^iq^jr^k$ lies in it.  In the next lemma, the exponents $a,a'$ are\nnonnegative integers, and $w\\ge0$ is the product of the fixed factors.\n\n\\begin{lemma}[uniqueness in a short interval]\\label{long269:res:short}\nLet $b\\ge1$ and $\\hi\\le b\\,\\lo$.  If $b^{a}w$ and $b^{a'}w$ both lie in\n$[\\lo,\\hi)$ then $a=a'$.\n\\end{lemma}\n\n\\begin{proof}\nIf $a<a'$ then $\\hi\\le b\\,\\lo\\le b^{a+1}w\\le b^{a'}w<\\hi$, which is impossible;\nthe case $a>a'$ is symmetric.\n\\end{proof}\n\nThe short-interval condition says that multiplying by the omitted base\nmoves a point beyond the interval.  It holds for $[L,2L)$ when that base\nis at least two.  A wider interval need not have this property: $1$ and\n$2$ both lie in $[1,3)$ and have the same odd part.  Thus the ratio bound,\nnot just finiteness of the interval, permits the injective projection.\n\n\\begin{proposition}[counting a block by two coordinates]\\label{long269:res:drop}\n\\label{long269:res:shell}\nIf $\\hi\\le r\\,\\lo$ then $\\#\\mathcal S\\le(h_p+1)(h_q+1)$, and if\n$\\hi\\le p\\,\\lo$ then $\\#\\mathcal S\\le(h_q+1)(h_r+1)$.  If moreover\n$\\hi\\le r\\,\\lo$ and $h_p\\le h_q\\le h_r$ with $h_p+h_q+h_r=j$, then\n$9\\,\\#\\mathcal S\\le(j+3)^{2}$.\n\\end{proposition}\n\n\\begin{proof}\nSuppose $\\hi\\le r\\,\\lo$.  If two triples of $\\mathcal S$ agree in their first\ntwo coordinates, Lemma~\\ref{long269:res:short} with $b=r$ and $w=p^{i}q^{j}$ forces\ntheir third coordinates to agree, so the projection forgetting the third\ncoordinate is injective on $\\mathcal S$ and its image lies in a rectangle with\n$(h_p+1)(h_q+1)$ points.  The other case is the same with the first coordinate\nprojected away.  Under the sorting hypothesis the two surviving coordinates are\nthe two smallest, so it suffices that $a\\le b\\le c$ with $a+b+c=j$ gives\n$9(a+1)(b+1)\\le(j+3)^{2}$.  From $a\\le b\\le c$ we get $a+2b\\le j$, so it is\nenough that $9(a+1)(b+1)\\le(a+2b+3)^{2}$; writing $b=a+d$ with $d\\ge0$, the\ndifference of the two sides is $d(3a+4d+3)\\ge0$.\n\\end{proof}\n\n\n\\section{The two-prime Hecke--Mahler reduction}\n\\label{long269:sec:two-prime}\n\\papersectiontarget{r3-two-prime}\n\nLet $P=\\{p,q\\}$ with $p<q$, and put\n$L_{p,q}(t)=p^{\\lfloor\\log_p t\\rfloor}q^{\\lfloor\\log_q t\\rfloor}$.\nThe proof of Theorem~\\ref{long269:res:lcm}, with one coordinate omitted,\nidentifies this with the running LCM. The two sums are therefore\n\\[\n \\mathcal D_{\\{p,q\\}}\n =1+\\sum_{t\\in\\{p,p^2,\\ldots\\}\\cup\\{q,q^2,\\ldots\\}}\\frac1{L_{p,q}(t)},\n \\qquad\n \\mathcal R_{\\{p,q\\}}=\\sum_{i,j\\ge0}\\frac1{L_{p,q}(p^iq^j)}.\n\\]\nWe first express the contribution from powers of $q$ in terms of that\nfrom powers of $p$. The repeated kernel then factors, and this relation\nputs both factors in a single transcendental parameter. This is the\nfactorisation and Hecke--Mahler reduction credited to Fan in the introduction.\n\n\\begin{proof}[Proof of Theorem~\\ref{long269:res:lead-two-prime}]\nSet\n\\[\n \\theta=\\frac{\\log p}{\\log q},\\qquad x=\\frac1p,\\qquad y=\\frac1q,\\qquad\n m_n=\\lfloor n\\theta\\rfloor,\\qquad \\delta_n=m_{n+1}-m_n .\n\\]\nHere $0<\\theta<1$, and $\\theta$ is irrational, since a rational value would\ngive $p^{b}=q^{a}$ for positive integers $a,b$.  Consequently\n$\\delta_n\\in\\{0,1\\}$.  Put\n\\[\n A=\\sum_{n\\ge0}x^ny^{m_n},\n \\qquad\n B_\\ast=\\sum_{n\\ge0}\\delta_nx^ny^{m_n+1} .\n\\]\nThe initial value and the positive powers of $p$ contribute $A$, since\n$L_{p,q}(p^n)=p^nq^{m_n}$.  A power of $q$ lies strictly between $p^n$ and\n$p^{n+1}$ exactly when $\\delta_n=1$, it is then $q^{m_n+1}$, and its post-jump\nreciprocal is $x^ny^{m_n+1}$; so the positive powers of $q$ contribute $B_\\ast$ and\n$\\mathcal D_{\\{p,q\\}}=A+B_\\ast$.  All these series converge absolutely.\n\nSince $y^{m_{n+1}}-y^{m_n}=\\delta_ny^{m_n}(y-1)$, an index shift gives\n$A-1-xA=x(y-1)B_\\ast/y$, so $B_\\ast=\\bigl((p-1)A-p\\bigr)/(1-q)$ and\n\\begin{equation}\\label{long269:eq:two-prime-affine}\n \\mathcal D_{\\{p,q\\}}=\\frac{(q-p)A+p}{q-1} .\n\\end{equation}\nAt a smooth point, $\\log_p(p^iq^j)=i+j/\\theta$ and\n$\\log_q(p^iq^j)=j+i\\theta$, so\n$L_{p,q}(p^iq^j)=p^{\\,i+\\lfloor j/\\theta\\rfloor}q^{\\,j+m_i}$ and absolute\nconvergence permits the factorisation\n\\[\n \\mathcal R_{\\{p,q\\}}=A\\sum_{j\\ge0}y^jx^{\\lfloor j/\\theta\\rfloor}.\n\\]\nFor $j\\ge1$, the index $n=\\lfloor j/\\theta\\rfloor$ is precisely the one\nfor which $p^n<q^j<p^{n+1}$; the inequalities are strict because distinct\nprimes have no common positive power.  This interval contains at most one\npower of $q$, since $p<q$.  Thus $\\delta_n=1$ and $j=m_n+1$.\nConversely, every $n$ with $\\delta_n=1$ contains that unique power of $q$.\nThe second factor is therefore $1+B_\\ast$, giving\n\\begin{equation}\\label{long269:eq:two-prime-quadratic}\n \\mathcal R_{\\{p,q\\}}\n =\\frac{(p+q-1)A-(p-1)A^{2}}{q-1} .\n\\end{equation}\n\nIt remains to prove that $A$ is transcendental.  For the Hecke--Mahler series\n\\[\n F_\\theta(x,y)=\\sum_{n\\ge1}\\sum_{k=1}^{\\lfloor n\\theta\\rfloor}x^ny^k\n\\]\na finite geometric sum gives\n$\\bigl((1-y)/y\\bigr)F_\\theta(x,y)=x/(1-x)-(A-1)$, that is\n\\begin{equation}\\label{long269:eq:hecke-mahler-boundary}\n A=\\frac1{1-x}-\\frac{1-y}{y}F_\\theta(x,y).\n\\end{equation}\nBugeaud and Laurent's Theorem~1.1 states, in particular, that\n$F_\\theta(\\beta,\\alpha)$ is transcendental when $\\theta\\in(0,1)$ is irrational,\n$\\alpha$ and $\\beta$ are nonzero algebraic numbers, $|\\beta|<1$ and\n$|\\beta\\alpha^\\theta|<1$ \\cite[Theorem~1.1]{bugeaudlaurent2023}; the\n$\\rho=0$ case used here goes back to Loxton and van der Poorten\n\\cite[Theorem~8, p.~40]{loxtonvdp1977}.  Take $(\\beta,\\alpha)=(x,y)$: then\n$|\\beta|=1/p<1$ and\n\\[\n |xy^{\\theta}|=\\frac1p\\left(\\frac1q\\right)^{\\log p/\\log q}=\\frac1{p^{2}}<1 .\n\\]\nSo $F_\\theta(x,y)$ is transcendental, and \\eqref{long269:eq:hecke-mahler-boundary}\nmakes $A$ transcendental.  The coefficient of $A$\nin~\\eqref{long269:eq:two-prime-affine} is $(q-p)/(q-1)\\ne0$ and the coefficient of\n$A^{2}$ in~\\eqref{long269:eq:two-prime-quadratic} is $-(p-1)/(q-1)\\ne0$, both rational.\nIf either value were algebraic, its identity would exhibit $A$ as a root of a\nnonzero polynomial over the algebraic numbers.\n\\end{proof}\n\n\\label{long269:res:two-prime-transcendence}\\label{long269:res:two-prime-repeated-transcendence}\nA product of two transcendental numbers need not be transcendental.\nWhat proves the repeated sum transcendental is its nonconstant quadratic\nexpression in the single value $A$, not the factorisation by itself.\nThe value theorem adds no unverified hypothesis in this two-prime case:\ndistinct primes give an irrational slope and the displayed reciprocal\narguments satisfy its size conditions.  Primality is stronger than the\ncalculation needs.  For coprime integers $1<p<q$, enumerate the monoid\n$\\{p^iq^j:i,j\\ge0\\}$, not all integers supported on the prime factors of\n$pq$.  Unique exponent pairs, the running-LCM product and the irrationality\nof $\\log p/\\log q$ still hold, so both identities and transcendence\nconclusions remain valid.  For example, this applies to generators $4,9$.\nBy contrast, the monoid generated by $4,8$ has running LCM $8$ at the\ncutoff $8$, not $4^{\\lfloor\\log_4 8\\rfloor}8^{\\lfloor\\log_8 8\\rfloor}=32$;\nits slope is rational as well.  The coprime extension was already noted\nin the forum discussion (Section~\\ref{long269:long:history}).\n\nA third prime introduces an additional floor term that cannot be separated\nin this way; Section~\\ref{long269:sec:rank} makes that obstruction precise.\nFor the sums over distinct running LCMs, the next section proves irrationality for\nevery finite set of primes by an elementary argument that uses no value\ntheorem.\n\n\\section{The sums over distinct running LCMs}\n\\label{long269:sec:distinct}\n\nWe record the ordinary arguments for general finite prime supports.\nTheir assertions, including the block and finite-state criteria below,\nare presented as remarks because this part has not been formalised.\nThe complete proofs are retained. The five-map theorem in\nSection~\\ref{long269:sec:distinct-235} has a separate formal proof.\nNo argument here gives transcendence for more than two primes.\n\nLet $P$ be a finite set of at least two primes, with least element $p$, and\nput $\\hgt_P(t)=\\prod_{r\\in P}r^{\\lfloor\\log_rt\\rfloor}$.  The proofs of\nTheorem~\\ref{long269:res:lcm} and Propositions~\\ref{long269:res:cell}\nand~\\ref{long269:res:count} apply unchanged to any finite set of primes: the\nrunning least common multiple of the $P$-smooth integers up to $t$ is\n$\\hgt_P(t)$, it changes only at the prime powers $r^n$ ($r\\in P$, $n\\ge1$),\nwhere it is multiplied by $r$, and distinct prime powers are distinct\nnumbers.  List the prime powers as $t_1<t_2<\\cdots$ and let $q_k$ be the prime\nof $t_k$.  The distinct running values are then $1$ and\n$\\hgt_P(t_k)=q_1\\cdots q_k$, so\n\\begin{equation}\\label{long269:eq:distinct-sum}\n \\mathcal D_P=1+\\sum_{k\\ge1}\\frac1{\\hgt_P(t_k)}\n =\\sum_{k\\ge0}\\frac1{q_1\\cdots q_k}.\n\\end{equation}\nThis is the sum $\\Sigma'$ of Erd\\H{o}s's letter~\\cite{erdos1974letter},\nwhich counts each distinct value of $[a_1,\\ldots,a_n]$ once.  For\n$P=\\{2,3,5\\}$ the prime powers $2,3,4,5,8,9,16,25,27,32$ give the following terms,\nincluding the initial $1$\n\\[\n 1,\\frac12,\\frac16,\\frac1{12},\\frac1{60},\\frac1{120},\\frac1{360},\\frac1{720},\n \\frac1{3600},\\frac1{10800},\\frac1{21600},\\ldots .\n\\]\nNumerically, $\\mathcal D_{\\{2,3,5\\}}=1.77961698329474905045919064931698327\\ldots$.\nPut $Q_k=q_1\\cdots q_k=\\hgt_P(t_k)$.\n\n\\subsection{Proof for an arbitrary finite prime set}\\label{long269:sec:distinct-all}\n\\papersectiontarget{r3-distinct-proof}\n\\begin{remark}[the sums over distinct running LCMs]\\label{long269:res:distinct-height-all}\nFor every finite set $P$ of primes with $|P|\\ge2$, the sum $\\mathcal D_P$\nis irrational.\n\\end{remark}\n\nHere we compare two tails rather than recover a word from one tail.\nDiscreteness is used twice. Under rationality with denominator $K$, the\ntails lie in $K^{-1}\\mathbb Z$; a long common prefix makes two of them\ncloser than $1/K$ and hence equal. Next, corresponding blocks contain\nthe same prime labels in possibly different orders. Their denominator\nproducts agree. Clearing that product leaves an integer difference of\nblock numerators equal to a difference of tails in $(-1,1)$. Both must\nvanish, and this conclusion propagates through the blocks.\n\nWe therefore need one rearranged block with a changed numerator.\nFor two or three distinct primes, every change of order does this;\nfor four or more, some do not. The torus construction isolates a pair\nof neighbouring jumps and reverses it. Other blocks may change too,\nbut equality constrains every block separately, so one changed\ntwo-letter numerator suffices. We first establish the word identity,\nthen construct the two arithmetic tails.\n\n\\paragraph{Words and tails.}\\label{long269:sec:words}\nFor an infinite word $u=u_1u_2\\cdots$ over $P$, define\n$V(u)=\\sum_{j\\ge1}(u_1\\cdots u_j)^{-1}$. For a nonempty finite word\n$\\sigma=\\sigma_1\\cdots\\sigma_n$, put\n\\[\n \\Pi(\\sigma)=\\sigma_1\\cdots\\sigma_n,\\qquad\n f(\\sigma)=\\sum_{j=1}^{n}\\sigma_{j+1}\\cdots\\sigma_n.\n\\]\nThe final term in $f$ is $1$. Thus $f$ is the numerator of the block\ncontribution when its denominator is the product of the block letters.\nFor two distinct letters $a,b$, we have $f(ab)=b+1$ and $f(ba)=a+1$.\nReversing this pair changes $f$ by the nonzero integer $b-a$.\nRemoving a prefix gives\n\\begin{equation}\\label{long269:eq:word-shift}\n V(\\sigma u)=\\frac{f(\\sigma)+V(u)}{\\Pi(\\sigma)}.\n\\end{equation}\nWe apply this identity to\n$x_k=V(q_{k+1}q_{k+2}\\cdots)=\\sum_{j>k}Q_k/Q_j$.\nEvery suffix contains every prime of $P$, so\n$0<x_k<1/(\\min P-1)\\le1$.\n\n\\begin{remark}[integral tails]\\label{long269:res:dp-integral-tails}\nIf $\\mathcal D_P=N/K$ with integers $N$ and $K\\ge1$, then $Kx_k$ is an\ninteger for every $k\\ge1$.\n\\end{remark}\n\\begin{proof}\nEach $Q_i$ with $i\\le k$ divides $Q_k$. Multiplication of the defining\nseries by $Q_k$ therefore gives\n\\[\n Kx_k=Q_kN-K\\left(Q_k+\\sum_{i\\le k}\\frac{Q_k}{Q_i}\\right)\\in\\mathbb Z.\n\\]\n\\end{proof}\n\nTwo words with a long common prefix have close values. When the words\ndiffer only by rearrangements, equality of their values imposes a stronger\ncondition on every rearranged block.\n\n\\begin{remark}[rearranged blocks]\\label{long269:res:dp-blocks}\nLet $u=\\sigma_1\\sigma_2\\cdots$ and $u'=\\sigma'_1\\sigma'_2\\cdots$ be infinite\nwords, where each $\\sigma_i$ is a nonempty finite word and $\\sigma'_i$ is a\nrearrangement of it, and suppose that every suffix of $u$ or $u'$ starting\nat a block boundary has value in $(0,1)$.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item If $V(u)=V(u')$, then $f(\\sigma_i)=f(\\sigma'_i)$ for every $i$.\n\\item If $\\sigma_i=\\sigma'_i$ for every $i<i_0$, then\n$|V(u)-V(u')|<1/\\Pi(\\sigma_1\\cdots\\sigma_{i_0-1})$.\n\\end{enumerate}\n\\end{remark}\n\\begin{proof}\nLet $U_i$ and $U'_i$ be the values starting at block $i$. Since the two\nblocks have the same product, \\eqref{long269:eq:word-shift} yields\n\\[\n \\Pi(\\sigma_i)(U_i-U'_i)\n =f(\\sigma_i)-f(\\sigma'_i)+U_{i+1}-U'_{i+1}.\n\\]\nIf $U_i=U'_i$, the integer on the right equals a number in $(-1,1)$,\nand hence both differences vanish. Induction proves (i). If the preceding\nblocks agree, repeated removal of their common prefix gives\n\\[\n V(u)-V(u')=\\frac{U_{i_0}-U'_{i_0}}\n {\\Pi(\\sigma_1\\cdots\\sigma_{i_0-1})},\n\\]\nwhich proves (ii).\n\\end{proof}\n\n\\begin{remark}[short rearrangements]\\label{long269:res:dp-short}\nThe map $f$ is injective on the orderings of any set of at most three\ndistinct primes.\n\\end{remark}\n\\begin{proof}\nFor two letters, $f(ab)=1+b$. For three,\n$f(abc)=1+c(1+b)$. Suppose two orderings of $\\{a,b,c\\}$ have the same\nvalue. If their last letters agree, their middle letters agree too.\nOtherwise we may write\n$a(1+s)=b(1+t)$, with $s\\in\\{b,c\\}$ and $t\\in\\{a,c\\}$.\nThe choices $(s,t)=(b,a)$ and $(c,c)$ imply $a=b$; the other two imply\n$b\\mid a$ or $a\\mid b$. Each is impossible for distinct primes.\n\\end{proof}\n\nThere is a collision on four letters:\n$f(5,7,3,2)=f(7,2,3,5)=51$. Consequently, a change in order alone will not\nsuffice for larger $P$. We shall arrange an isolated interchange of two\nletters, for which $f(ab)-f(ba)=b-a\\ne0$.\n\n\\paragraph{The logarithmic flow.}\\label{long269:sec:flow}\nConsider\n\\[\n \\mathbb T=\\prod_{q\\in P}\\mathbb R/(\\log q)\\mathbb Z,\n \\qquad \\Theta(s)=(s\\bmod\\log q)_{q\\in P}.\n\\]\nFor $\\theta\\in\\mathbb T$, a $q$-crossing is a time $t$ at which\n$\\theta_q+t\\in(\\log q)\\mathbb Z$. At\n$\\theta_k=\\Theta(\\log t_k)$ the positive crossings occur at\n$\\log(q^j/t_k)$ with $q^j>t_k$, so their labels, read in time order, are\n$q_{k+1}q_{k+2}\\cdots$. Crossings at this phase never coincide.\n\nLet $T_0$ be the closure of $\\Theta(\\mathbb R)$. Every forward orbit in\n$T_0$ is dense there. Indeed, the pigeonhole principle gives arbitrarily\nlarge $d>0$ with $\\Theta(d)$ arbitrarily close to $0$, so negative times\nare limits of positive ones. Translation then proves the same assertion\nfor every starting point and every later starting time.\n\nIn the normalised coordinates $\\varphi_q=\\theta_q/\\log q$, Kronecker's\ntheorem identifies $T_0$ with the image of the smallest rational subspace\n$L\\subseteq\\mathbb R^P$ containing $\\omega=(1/\\log q)_{q\\in P}$.\nAny two coordinate forms $\\varphi_a,\\varphi_b$ are linearly independent on\n$L$. To see this, their dependence would put a nonzero vector supported\non $\\{a,b\\}$ in the rational annihilator of $L$. That intersection is\nrational, so it would contain a nonzero integer vector. Its pairing with\n$\\omega$ would force $\\log a/\\log b\\in\\mathbb Q$, contrary to unique\nfactorisation. In particular,\n$(\\log b)\\varphi_b-(\\log a)\\varphi_a$ is a nonzero form on $L$.\nWe write it as $\\theta_b-\\theta_a$ when using real representatives.\n\nTo interchange just two jumps, we seek a phase at which exactly those\ntwo crossings coincide. The next argument finds such a phase inside\n$T_0$; it does not require this orbit closure to be the whole torus\n$\\mathbb T$.\n\n\\begin{remark}[two walls]\\label{long269:res:dp-two-walls}\nIf $|P|\\ge4$, there is $x\\in T_0$ with $\\theta_q(x)=0$ for exactly two\nprimes $q\\in P$.\n\\end{remark}\n\\begin{proof}\nChoose a rational plane in $L$ on which every pair of coordinate forms\nremains independent. Such a plane exists: for each pair, the determinant\non $L\\times L$ is a nonzero polynomial, and finitely many of these\npolynomials cannot vanish on all rational pairs. An integer basis of the\nplane identifies its image with a two-dimensional subtorus of $T_0$.\nOn this subtorus each wall $\\theta_q=0$ has the equation\n$\\langle\\gamma_q,s\\rangle\\in\\mathbb Z$, where\n$\\gamma_q\\in\\mathbb Z^2\\smallsetminus\\{0\\}$, and the vectors $\\gamma_q$ are\npairwise nonparallel. A wall may have several components if $\\gamma_q$\nis not primitive.\n\nThe walls form a finite arrangement of closed geodesics. Let $k_v$ count\nthe wall families through a vertex $v$. Each such family contributes two\nincident half-edges, so $E=\\sum_v k_v$. Two families alone divide the torus\ninto open parallelograms: their two integer linear forms give a finite\ncovering of the standard torus. Adding the remaining straight lines\nsubdivides these parallelograms into convex polygons, each with at least\nthree sides. Thus $2E\\ge3F$, while Euler's formula gives $V-E+F=0$.\nTogether these imply\n\\[\n \\sum_v(k_v-3)=E-3V\\le0.\n\\]\nEvery vertex belongs to at least two wall families, and the last\ninequality says that the average number of families through a vertex is\nat most three. The origin belongs to all $|P|\\ge4$ families. Another\nvertex must therefore belong to exactly two, as required.\n\\end{proof}\n\n\\paragraph{Comparison of two tails.}\\label{long269:sec:tail-comparison}\n\\begin{proof}[Proof of Remark~\\ref{long269:res:distinct-height-all}]\nSuppose that $\\mathcal D_P=N/K$, with $K\\ge1$. Fix $p\\in P$ and a jump\n$t_k$ which is a power of $p$. We shall compare the tails after $t_k$ and\n$t_m=t_kp^n$. We first choose $n$ so that the phase returns close enough\nto its initial value to preserve a long initial crossing word. After\nfixing this return, we\nshall find a later group of crossings whose order it changes. The\ncommon initial word will force equality of the two tails; the later\nrearrangement will contradict that equality.\n\nFor the common word to force equality, its reciprocal product must be\nat most $1/K$. Choose $T>0$ so that the number $\\nu$ of crossings in\n$(0,T]$ satisfies $2^\\nu\\ge K$, and let $g>0$ be the minimum gap between\ndistinct crossings in $[0,T+1]$. Put\n$\\beta=\\min_{q\\ne p}\\operatorname{dist}(\\theta_{k,q},0)>0$.\nThe gap $g$ will keep these early crossings in order, while $\\beta$\nwill prevent a crossing from moving past the starting time $0$.\nWhen $|P|\\ge4$, also choose the point $x$ and primes $a,b$ from\nRemark~\\ref{long269:res:dp-two-walls}, and set\n\\[\n \\eta=\\min\\left(\\frac1{10},\n       \\min_{c\\ne a,b}\\operatorname{dist}(\\theta_c(x),0)\\right)>0.\n\\]\nIn this case $\\eta$ leaves room to isolate the $a$- and $b$-crossings\nfrom all the others. Take $\\varepsilon<\\min(\\beta,g/2,\\log2/(2|P|))$, and when\n$|P|\\ge4$ require also $\\varepsilon<\\eta/8$. Dirichlet's simultaneous\napproximation theorem gives arbitrarily large $n$ and integers $m_q\\ge1$\nfor $q\\ne p$ such that\n\\[\n \\delta_p=0,\\qquad \\delta_q=n\\log p-m_q\\log q,\\qquad\n \\max_q|\\delta_q|<\\varepsilon.\n\\]\nThe numbers $\\delta_q$ are pairwise distinct, since an equality would\nidentify positive powers of distinct primes.\n\nAt $\\theta_m=\\theta_k+\\Theta(n\\log p)$ a $q$-crossing moves by\n$-\\delta_q$. For $q\\ne p$ every crossing of $\\theta_k$ is at distance at\nleast $\\beta$ from $0$, so signs are preserved; the $p$-crossings do not\nmove. We therefore obtain a bijection between the positive crossings of\nthe two phases.\n\nCut the crossings of $\\theta_k$ whenever successive times are more than\n$2\\varepsilon$ apart, and call each resulting block a chain. Different\nchains keep their order after the shift. A chain contains at most $|P|$\ncrossings, all of different primes: otherwise its first $|P|+1$ crossings\nwould span less than $2\\varepsilon|P|<\\log2$, although some prime would\nhave to occur twice. The same argument excludes a repeated prime in a\nshorter chain. Thus the two tail words are concatenations of the same\nchains, with the letters possibly rearranged inside each chain.\n\nEvery chain meeting $(0,T]$ is a singleton, by the choice of $g$ and\n$\\varepsilon$. The first $\\nu$ letters consequently agree, and\nRemark~\\ref{long269:res:dp-blocks}(ii) gives\n$|x_k-x_m|<2^{-\\nu}\\le1/K$. The integral-tail lemma forces $x_k=x_m$.\nBy Remark~\\ref{long269:res:dp-blocks}(i), every chain $C$ and its rearrangement $C'$\nnow satisfy $f(C)=f(C')$.\n\nThe return, and hence every displacement $\\delta_q$, is now fixed.\nIt remains to find a chain that violates this last equality. For at\nmost three primes, any change of order will do; for larger sets we\nwill isolate a chain of length two.\n\nWhen $|P|\\le3$, choose $q\\ne p$. If $\\delta_q>0$, density of\n$j\\log p-i\\log q+\\log t_k$ for large positive $i,j$ supplies crossings\n$s_p,s_q>0$ with $s_q-\\delta_q<s_p<s_q$. Their order reverses. The case\n$\\delta_q<0$ is the same with the inequalities reversed. The two crossings\nbelong to one chain, which has at most three distinct primes and changes\nits order. This contradicts Remark~\\ref{long269:res:dp-short}.\n\nFor $|P|\\ge4$, put $D=\\delta_a-\\delta_b\\ne0$. Since the form\n$\\theta_b-\\theta_a$ is nonzero on $L$, choose a small $y\\in L$ with\n$|\\theta_q(y)|<\\eta/8$ for all $q$ and\n$\\theta_b(y)-\\theta_a(y)$ strictly between $0$ and $D$.\nAt the phase\n\\[\n z=x+y+\\Theta(-\\eta/2)\n\\]\nthere is one $a$-crossing and one $b$-crossing in\n$[-\\eta/4,5\\eta/4]$. Their times $\\tau_a,\\tau_b$ lie in\n$(\\eta/4,3\\eta/4)$, and $\\tau_a-\\tau_b$ is strictly between $0$ and $D$.\nThere are no other crossings in that interval: for $c\\ne a,b$ the nearest\none is at distance at least $7\\eta/8$ from $\\eta/2$.\n\nThe phase $z$ thus has an isolated pair which the fixed displacement\nreverses. To obtain such a pair in the arithmetic crossing word, keep\nall the strict margins above and choose a neighbourhood $U$ of $z$ in\n$T_0$ where they still hold. Density gives $t^*>T+2$ with\n$\\theta_k+\\Theta(t^*)\\in U$. The two corresponding crossings of\n$\\theta_k$ are less than $|D|<2\\varepsilon$ apart, whereas every other\ncrossing is more than $\\eta/2>4\\varepsilon$ away. They form a chain of\nlength two, beyond the initial word already used to equate the tails.\nAfter shifting, their time difference becomes\n$(\\tau_a-\\tau_b)-D$, which has the opposite sign. Thus $C=ab,C'=ba$\nor conversely, contradicting $f(ab)-f(ba)=b-a\\ne0$.\n\\end{proof}\n\n\\paragraph{Reversals near the origin.}  A shorter route would take, at a\nreturn of the orbit near the origin of $T_0$, the chain of crossings there.\nThat shortcut need not work when $\\dim T_0=2$, which happens exactly when the numbers\n$1/\\log q$ span a two-dimensional space over $\\Q$; the proof makes no assumption excluding this case. Linear-form estimates for logarithms do not supply the required independence of their reciprocals.  In that case every small vector of $L$, in the coordinates\n$\\theta_q$, has the form $\\alpha(1,\\ldots,1)+\\beta w$ for one fixed $w$, so\nnear the origin all the forms $\\theta_b-\\theta_a$ are proportional.  The\nchain at such a return contains every prime of $P$, and the only\nrearrangement it can undergo is the full reversal.  Full reversals can\ncollide:\n\\[\n f(43,13,3,53)=f(53,3,13,43)=2280,\n\\]\nand among the sets of four primes below $100$ there are five such\ncollisions.  Remark~\\ref{long269:res:dp-two-walls} avoids the question by\nproducing a point on exactly two walls for every $P$ with $|P|\\ge4$; a point\non three walls would also serve, by Remark~\\ref{long269:res:dp-short}.\nSwapping one adjacent pair always changes $f$, since\n$f(\\sigma)-f(\\sigma')=(\\sigma_{i+1}-\\sigma_i)\\sigma_{i+2}\\cdots\\sigma_n$\nwhen $\\sigma'$ swaps $\\sigma_i$ and $\\sigma_{i+1}$; a rearranged chain of\nfour or more primes can differ from its image by more than one swap.  Nor can the argument stop at\nthe first rearranged chain: for four primes that chain can collide, as\n$f(5,7,3,2)=f(7,2,3,5)=51$ shows, which is why\nRemark~\\ref{long269:res:dp-blocks}(i) is applied to every chain.\n\n\\paragraph{Exact illustrations.}  The program \\texttt{two\\_tail.py} of the\n\\href{\\distinctheighturl}{programs for sums over distinct values} checks the finite facts\nabove in exact arithmetic; the proof uses none of them.  Among the sets of\ntwo or three primes below $200$, $f$ has no collision; among the $2380$\nsets of four primes below $60$ it has $30$.  For $P=\\{2,3,5,7\\}$,\n$t_k=2^{60}$ and the return $n=283526$, with $\\varepsilon\\approx0.0117$,\nthe first $4000$ crossings correspond exactly, every chain stays in place,\nand of the $39$ rearranged chains $35$ are transpositions; none has equal\n$f$-values.  For the closer return $n=2547450$, with\n$\\varepsilon\\approx0.0034$, the difference $x_k-x_m$ computed exactly has\n$\\log_{10}|x_k-x_m|=-501.1234$, the value predicted by its first\nrearranged chain, $(5,2)$ against $(2,5)$ after $1005$ letters.  On $400$\nrandom arrangements of four to six line families on $\\R^2/\\Z^2$, some with\nnon-primitive normal vectors, $\\sum_v(k_v-3)\\le-4$; the square with both\ndiagonals attains $0$ and still has a vertex on exactly two lines.\n\n\\subsection{Blocks and affine maps}\\label{long269:sec:distinct-blocks}\nWe now extend the calculation of Section~\\ref{long269:sec:distinct-235}\nto an arbitrary finite prime set.\n\nTo obtain a second description of the sum over distinct running LCMs, group its jumps\ninto $(p^a,p^{a+1}]$, where $p$ is the least prime of $P$, and define\n\\[\n P_a=\\hgt_P(p^a),\\qquad\n Y_a=P_a\\sum_{t_k>p^a}\\frac1{\\hgt_P(t_k)}.\n\\]\nWrite $\\tau_a=(r_1,\\ldots,r_v)$ for the labels of the jumps strictly\ninside block $a$, in increasing order of their powers. This is the block's\n\\emph{type}; write $\\varnothing$ when there is no interior jump.\nThe final jump at $p^{a+1}$ always has label $p$.\n\n\\begin{remark}[blocks, tails and integrality]\\label{long269:res:distinct-height-blocks}\nLet $P$ be a finite set of at least two primes with least element $p$.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item Block $a$ ends with the jump at $p^{a+1}$ and contains at most one\npower of each prime $q\\in P\\smallsetminus\\{p\\}$, lying strictly inside it.  The\npower $q^f$ with $f\\ge1$ lies in block $\\lfloor f\\log_pq\\rfloor$.\n\\item If block $a$ has type $\\tau=(r_1,\\ldots,r_v)$, then\n$Y_a=G_\\tau(Y_{a+1})$, where\n\\[\n G_\\tau(y)=\\frac{\\mu_\\tau+y}{b_\\tau},\\qquad b_\\tau=pr_1\\cdots r_v,\\qquad\n \\mu_\\tau=1+b_\\tau\\sum_{j=1}^{v}\\frac1{r_1\\cdots r_j}.\n\\]\nEach $G_\\tau$ is increasing with slope $1/b_\\tau\\le1/p$ and maps\n$[0,1/(p-1)]$ into itself, and $0<Y_a\\le1/(p-1)$.\n\\item If $\\mathcal D_P=N/K$ with integers $N$ and $K\\ge1$, then $KY_a$ is\nan integer for every $a\\ge0$.\n\\end{enumerate}\n\\end{remark}\n\n\n\\begin{proof}\n(i) The endpoint $p^{a+1}$ is a jump.  Consecutive powers of $q$ have ratio\n$q>p$, so the block contains at most one of them, and no power of $q$ equals\na power of $p$.  The power $q^f$ lies in $(p^a,p^{a+1})$ exactly when\n$a<f\\log_pq<a+1$.\n\n(ii) The jumps of block $a$ are the interior powers, with running LCMs\n$P_ar_1$, $P_ar_1r_2$, \\ldots, $P_ar_1\\cdots r_v$, followed by the jump at\n$p^{a+1}$, whose running LCM is $P_{a+1}=b_\\tau P_a$.  Splitting off the block,\n\\[\n Y_a=\\sum_{j=1}^{v}\\frac1{r_1\\cdots r_j}+\\frac1{b_\\tau}\n    +\\frac{P_a}{P_{a+1}}Y_{a+1}=\\frac{\\mu_\\tau+Y_{a+1}}{b_\\tau}.\n\\]\nEach jump multiplies the running LCM by at least $p$, so\n$0<Y_a\\le\\sum_{k\\ge1}p^{-k}=1/(p-1)$.  Since every $r_j>p$,\n$G_\\tau(1/(p-1))\\le\\sum_{j=1}^{v}p^{-j}+p^{-v}/(p-1)=1/(p-1)$, and\n$G_\\tau(0)>0$.\n\n(iii) If $t\\le p^a$, each exponent in $\\hgt_P(t)$ is at most the\ncorresponding exponent in $P_a$, so $\\hgt_P(t)\\mid P_a$.  Hence\n$P_a\\mathcal D_P-Y_a=P_a\\bigl(1+\\sum_{t_k\\le p^a}\\hgt_P(t_k)^{-1}\\bigr)$ is\nan integer, and so is $KY_a=P_aN-K(P_a\\mathcal D_P-Y_a)$.\n\\end{proof}\n\n\\subsection{A decoding criterion for every finite set of primes}\n\\label{long269:sec:distinct-criterion}\nFor other prime sets, a useful tail interval may depend on the recent\noccurrence of each prime. Put $Q=P\\smallsetminus\\{p\\}$ and choose the\ninteger $g_q$ by $p^{g_q}<q<p^{g_q+1}$. Successive blocks containing a\n$q$-power have indices $\\lfloor f\\log_p q\\rfloor$ and\n$\\lfloor(f+1)\\log_p q\\rfloor$, by\nRemark~\\ref{long269:res:distinct-height-blocks}(i). Their difference is\n$g_q$ or $g_q+1$. The gaps give a finite memory of the last occurrences. We use it to\nenclose tails of every allowed path, a class containing all sufficiently\nlate arithmetic paths but possibly others. Separation on this larger\nclass suffices for recovery; overlap need not give equal arithmetic tails.\nThe memory sharpens the enclosures, but the decoder must ultimately work\nwithout being told the state.\n\nThese constraints define a finite automaton.  Its states are the tuples\n$s=(s_q)_{q\\in Q}$ with $1\\le s_q\\le g_q+1$; the entry $s_q$ counts the blocks\nsince the last block containing a power of $q$.  A \\emph{transition} from $s$\nis a pair $(\\tau,s')$ in which $\\tau$ is an ordering of a set $T\\subseteq Q$\nwith\n\\[\n \\{q:s_q=g_q+1\\}\\subseteq T\\subseteq\\{q:s_q\\ge g_q\\},\n\\]\nand $s'_q=1$ for $q\\in T$, $s'_q=s_q+1$ for $q\\notin T$.  A path from $s$\nis a sequence of transitions $(u_1,s_1),(u_2,s_2),\\ldots$ with $(u_1,s_1)$ a\ntransition from $s$ and each $(u_{i+1},s_{i+1})$ a transition from $s_i$.\nThe \\emph{value} of an infinite path is the limit of\n$G_{u_1}\\circ\\cdots\\circ G_{u_n}(y)$ as $n\\to\\infty$; by\nRemark~\\ref{long269:res:distinct-height-blocks}(ii) these numbers lie, for\n$y\\in[0,1/(p-1)]$, in nested intervals of length at most $p^{-n}/(p-1)$, so\nthe limit exists and does not depend on $y$.\n\nEvery prime $q\\in Q$ lies in block $g_q$.  For $a>\\max_qg_q$ let $s_a$ record,\nfor each $q$, the number $a-\\ell$, where $\\ell<a$ is the index of the last\nblock before $a$ containing a power of $q$.  The next such block has index\n$\\ell+g_q$ or $\\ell+g_q+1$, and it is at least $a$.  Hence $s_a$ is a state,\nblock $a$ contains a power of $q$ when $(s_a)_q=g_q+1$ and does not when\n$(s_a)_q<g_q$, and $(\\tau_a,s_{a+1})$ is a transition from $s_a$.  Since\n$Y_a=G_{\\tau_a}\\circ\\cdots\\circ G_{\\tau_{a+n-1}}(Y_{a+n})$ with\n$Y_{a+n}\\in[0,1/(p-1)]$, the tail $Y_a$ is the value of the actual path\n$(\\tau_a,s_{a+1}),(\\tau_{a+1},s_{a+2}),\\ldots$ from $s_a$.\n\n\\begin{remark}[a decoding criterion]\\label{long269:res:distinct-height-criterion}\nLet $P$ be a finite set of at least two primes, and for every state $s$ let\n$J(s)$ be a closed interval containing the values of all infinite paths from\n$s$.  For $k\\ge1$ and each type $\\tau$, let $U_\\tau$ be the union of the\nintervals $G_{u_1}\\circ\\cdots\\circ G_{u_k}\\bigl(J(s_k)\\bigr)$ over all states\n$s$ and all paths $(u_1,s_1),\\ldots,(u_k,s_k)$ from $s$ with $u_1=\\tau$.  If\nthe sets $U_\\tau$ are pairwise disjoint, then $\\mathcal D_P$ is irrational.\n\\end{remark}\n\n\nThe union defining $U_\\tau$ runs over \\emph{all} starting states.\nThus membership can identify the first block from $Y_a$ alone; separating\ntypes only within each known state would not give this conclusion.\n\n\\begin{proof}\nLet $a>\\max_qg_q$.  The first $k$ transitions of the actual path from $s_a$\nform a path of length $k$ with first type $\\tau_a$, and\n$Y_a=G_{\\tau_a}\\circ\\cdots\\circ G_{\\tau_{a+k-1}}(Y_{a+k})$, where\n$Y_{a+k}\\in J(s_{a+k})$ because it is the value of an infinite path from\n$s_{a+k}$.  Thus $Y_a\\in U_{\\tau_a}$, and by disjointness $\\tau_a$ is the\nonly type with this property.  So $Y_a$ determines $\\tau_a$, then\n$Y_{a+1}=b_{\\tau_a}Y_a-\\mu_{\\tau_a}$, and by induction the word\n$\\tau_a\\tau_{a+1}\\cdots$.\n\nSuppose $\\mathcal D_P=N/K$.  By\nRemark~\\ref{long269:res:distinct-height-blocks}(ii) and~(iii), the numbers\n$KY_a$ are integers in $[0,K/(p-1)]$, so $Y_a=Y_{a'}$ for some\n$\\max_qg_q<a<a'$, and the word $(\\tau_a)$ is eventually periodic.  Fix\n$q\\in Q$ and put $\\beta=\\log p/\\log q$, which lies in $(0,1)$ and is\nirrational because no power of $p$ equals a power of $q$.  By\nRemark~\\ref{long269:res:distinct-height-blocks}(i), block $a$ contains\n$\\lfloor(a+1)\\beta\\rfloor-\\lfloor a\\beta\\rfloor$ powers of $q$, a number\nrecorded by $\\tau_a$.  As in the proof of\nTheorem~\\ref{long269:res:distinct-height-235}, eventual periodicity with\nperiod $\\ell$ forces $\\ell\\beta\\in\\Z$, a contradiction.\n\\end{proof}\n\nIntervals $J(s)$ with the required property are easy to produce.  The\nconstant interval $[0,1/(p-1)]$ qualifies by\nRemark~\\ref{long269:res:distinct-height-blocks}(ii).  If a family $J$\nqualifies, so does any family $J'$ with\n\\[\n J'(s)\\supseteq G_\\tau\\bigl(J(s')\\bigr)\\quad\\text{for every transition }(\\tau,s')\\text{ from }s,\n\\]\nbecause an infinite path from $s$ is a transition $(\\tau,s')$ followed by an\ninfinite path from $s'$.  Starting from the constant family, take the interval hull of the images\nand round its endpoints outwards. Every round preserves containment, though\na hull may fill genuine gaps. Keep the separate image intervals when\nforming the final unions $U_\\tau$. The smallest enclosing interval at state\n$s$ is $J(s)=[L(s),U(s)]$, where $L(s)$ and $U(s)$ are the least and\ngreatest infinite-path values.  They satisfy\n$L(s)=\\min G_\\tau(L(s'))$ and $U(s)=\\max G_\\tau(U(s'))$ over the transitions\nfrom $s$, and these equations have unique solutions because each $G_\\tau$\ncontracts by at least the factor $1/p$.\n\nFor $\\{2,3,5\\}$ the constant interval $I=[3/10,13/15]$ has the required\nproperty: every path takes a transition with $5\\in T$ within any three\nsteps, which is the only fact the bounds used.  With $k=1$ the criterion is\nthen the proof given above.\n\n\\subsection{Certificates for three and four primes}\n\\label{long269:sec:distinct-certificates}\n\nThe saved computations apply the interval criterion in exact rational\narithmetic to the following finite family. They accompany the ordinary\nproof of Remark~\\ref{long269:res:distinct-height-criterion} and have not\nbeen formalised. The exclusions in part (ii) concern these interval\ncalculations; the ordinary torus argument also treats those sets.\n\n\n\\begin{remark}[three and four primes up to $31$]\\label{long269:res:distinct-height-sets}\nLet $P$ be a set of primes, each at most $31$.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item If $|P|=3$, then $\\mathcal D_P$ is irrational.\n\\item If $|P|=4$ and $P$ is not one of the $38$ sets listed below, then\n$\\mathcal D_P$ is irrational.\n\\end{enumerate}\n\\end{remark}\n\n\n\\begin{proof}\nFor each set, the program \\texttt{certificate.py} of the\n\\href{\\distinctheighturl}{programs for sums over distinct values} builds the automaton,\ncomputes intervals $J(s)$ by $80$ rounds of the operation above from\n$[0,1/(p-1)]$, with endpoints rounded outwards to multiples of $2^{-96}$, and\ntests the disjointness of the sets $U_\\tau$ for $k=1,2,\\ldots$.  Every\ncomparison is made in exact rational arithmetic, and each $U_\\tau$ is\ncomputed as a union of intervals merged where they meet.  The test passes for\neach of the $165$ sets of three primes with some $k\\le4$, and for each set of\nfour primes outside the list with some $k\\le5$.\nRemark~\\ref{long269:res:distinct-height-criterion} then gives\nirrationality.\n\\end{proof}\n\nFor three primes the least passing depth is $k=1$ for $141$ sets and $k=2$\nfor the $17$ sets\n$\\{2,3,11\\}$, $\\{2,3,13\\}$, $\\{2,3,17\\}$, $\\{2,3,19\\}$, $\\{2,3,23\\}$,\n$\\{2,3,29\\}$, $\\{2,3,31\\}$, $\\{2,5,7\\}$, $\\{2,11,17\\}$, $\\{2,13,19\\}$,\n$\\{2,17,19\\}$, $\\{2,19,29\\}$, $\\{2,19,31\\}$, $\\{3,13,17\\}$, $\\{3,17,23\\}$,\n$\\{3,23,31\\}$ and $\\{3,29,31\\}$.  It is $k=3$ for $\\{2,3,7\\}$,\n$\\{2,11,13\\}$, $\\{2,17,29\\}$ and $\\{2,19,23\\}$, and $k=4$ for\n$\\{2,17,31\\}$, $\\{2,23,29\\}$ and $\\{2,29,31\\}$.  The smallest gap between\nsets $U_\\tau$ of different types, among all passing triples, is about\n$9.4\\cdot10^{-6}$, for $\\{2,17,31\\}$; for $\\{2,3,5\\}$ it is\n$27983/963915\\approx0.029$ at depth $1$, much larger than the gap $1/900$ of\nthe single interval used above.  For four primes the least passing depth is\n$1,2,3,4,5$ for $184$, $34$, $28$, $42$ and $4$ sets respectively, and the\nsmallest gap is about $1.8\\cdot10^{-8}$, for $\\{3,7,17,19\\}$.  All these gaps\nare far above the rounding scale $2^{-96}$.  The $38$ four-prime sets with no\npassing depth $k\\le5$ are $\\{2,3,5,7\\}$, $\\{2,3,5,11\\}$, $\\{2,3,5,13\\}$,\n$\\{2,3,5,17\\}$, $\\{2,3,5,19\\}$, $\\{2,3,5,23\\}$, $\\{2,3,7,11\\}$,\n$\\{2,3,7,13\\}$, $\\{2,3,7,17\\}$, $\\{2,3,7,23\\}$, $\\{2,3,11,13\\}$,\n$\\{2,3,11,17\\}$, $\\{2,3,11,19\\}$, $\\{2,3,11,29\\}$, $\\{2,3,11,31\\}$,\n$\\{2,3,13,19\\}$, $\\{2,3,13,29\\}$, $\\{2,3,17,19\\}$, $\\{2,3,17,29\\}$,\n$\\{2,3,29,31\\}$, $\\{2,5,7,11\\}$, $\\{2,5,17,29\\}$, $\\{2,5,23,29\\}$,\n$\\{2,5,23,31\\}$, $\\{2,11,13,19\\}$, $\\{2,11,17,23\\}$, $\\{2,11,17,29\\}$,\n$\\{2,11,23,31\\}$, $\\{2,13,19,31\\}$, $\\{2,19,23,31\\}$, $\\{3,5,13,17\\}$,\n$\\{3,5,17,19\\}$, $\\{3,5,23,31\\}$, $\\{3,7,11,13\\}$, $\\{3,19,23,29\\}$,\n$\\{3,19,23,31\\}$, $\\{3,19,29,31\\}$ and $\\{7,13,29,31\\}$.  Twenty of them\ncontain both $2$ and $3$.\n\nA second computation replaces the rounded intervals by the exact extremes\n$[L(s),U(s)]$, found by policy iteration and checked by substitution into\ntheir defining equations.  It returns the same least depth for all $495$\nsets.  A third check compares the automaton with the first $400$ actual\nblocks of every set: every actual word is a path, and every exact enclosure\nof an actual tail lies in the interval of its state.  This comparison is a\nconsistency check only.  The whole triple computation takes about three\nseconds, and the four-prime computation about seventy seconds, on one\nlaptop; the saved logs record every depth and gap.\n\nFor $\\{2,3,5,7\\}$ the automaton has $18$ states and $16$ types, and the test\nfails through depth $7$.  Its tail intervals range from about $0.188$ to\n$0.785$, so the largest exceeds three times the smallest, and the images of\nthe empty block and of the block with one power of $3$ overlap.  The largest\noverlap between two first types is always of this pair.  For each depth $k$\nthe table gives the number of merged intervals in the sets $U_\\tau$ and the\nlength of that largest overlap.\n\\begin{center}\\small\n\\begin{tabular}{@{}rrr@{}}\n\\hline\n$k$ & intervals & largest overlap\\\\\\hline\n$1$ & $16$ & $2.4\\cdot10^{-2}$\\\\\n$2$ & $58$ & $1.8\\cdot10^{-3}$\\\\\n$3$ & $386$ & $1.0\\cdot10^{-4}$\\\\\n$4$ & $1413$ & $4.6\\cdot10^{-5}$\\\\\n$5$ & $8395$ & $7.6\\cdot10^{-7}$\\\\\n$6$ & $38494$ & $2.4\\cdot10^{-7}$\\\\\n$7$ & $193905$ & $1.8\\cdot10^{-8}$\\\\\\hline\n\\end{tabular}\n\\end{center}\nAt depth $7$ two further pairs of types still overlap, $(5,7,3)$ with $(7)$\nby about $1.5\\cdot10^{-9}$ and $(3,5,7)$ with $(3,7)$ by about\n$5.8\\cdot10^{-10}$.\n\nThe persistent overlap is not a rounding error or merely slow convergence.\nAn exact collision explains why no depth succeeds. The two-block words\n$(5,7,3),\\varnothing$ and $(7),(3,5)$ have the same map,\n\\[\n G_{(5,7,3)}\\circ G_\\varnothing(y)=G_{(7)}\\circ G_{(3,5)}(y)=\\frac{103+y}{420},\n\\]\nwhich is the collision $f(5,7,3,2)=f(7,2,3,5)=51$ of\nSection~\\ref{long269:sec:distinct-all} read in blocks: the letters are\n$5,7,3,2,2$ and $7,2,3,5,2$, and $f=103$ for both.  From the states\n$(1,2,2)$ and $(1,2,3)$, written as $(s_3,s_5,s_7)$, both words continue by\nthe blocks $(3),(5,7)$ and end in the state $(2,1,1)$, so the four-block\nwords have the same map and the same end state.  Hence the sets $U_{(5,7,3)}$\nand $U_{(7)}$ of Remark~\\ref{long269:res:distinct-height-criterion} meet at\nevery depth, for any valid choice of intervals, and the criterion cannot\ndecide $\\{2,3,5,7\\}$ with this automaton.  Both four-block words occur in\nthe actual block sequence, $12$ and $40$ times among the first $20000$\nblocks.  Within this automaton, a tail value therefore need not determine its first block. The identity does not exhibit equal tails at two positions of the actual prime-power sequence: those tails have different later phases. It proves failure of this interval criterion. The proof of Remark~\\ref{long269:res:distinct-height-all} uses the isolated reversal of two actual jumps instead.\n\n\\subsection{Repeated tails and prime-power subseries}\n\\label{long269:sec:distinct-stop}\n% Ordinary subseries arguments: no formal proof is claimed.\n\nWe compare the two normalisations before considering the sums over pure\nprime powers. The subseries assertions in this subsection have ordinary\nproofs and remain unformalised. For $P=\\{2,3,5\\}$ the two tail recurrences\nhave the same bases,\n$b_{\\tau_a}=P_{a+1}/P_a=b_a$, but their coefficients behave differently:\n\\[\n Y_{a+1}=b_aY_a-\\mu_{\\tau_a},\\qquad\n X_{a+1}=b_aX_a-m_a.\n\\]\nThe first identity is Remark~\\ref{long269:res:distinct-height-blocks}(ii);\nthe second is Theorem~\\ref{long269:res:actual-orbit}, with the notation of\nSection~\\ref{long269:sec:blocks}. The five types give only the coefficients $1,3,9,13$, whereas the\nweighted block counts satisfy\n$m_a=\\Theta((a+1)^2)$ by Lemma~\\ref{long269:res:literal-triangle}.\nConsequently $X_a>m_a/30$ is unbounded. Under rationality the integral\nvalues $BX_a$ need not repeat, and the maps $y\\mapsto(m_a+y)/b_a$ form\nan infinite family depending on $a$. Even equality of two tails would\nnot identify their later coefficients. The residue criterion in\nSection~\\ref{long269:sec:escape} compares an endpoint residue with a\nquadratic bound instead. The example after\nProposition~\\ref{long269:long:windowconsumer} shows the limit of an argument\nusing just bases and growth: the same bases admit quadratic numerators\n$\\widehat m_a$ with $\\sum_a\\widehat m_a/P_{a+1}=9$.\n\nThe two-tail proof of Remark~\\ref{long269:res:distinct-height-all} stops\nat the same place.  Over the prime powers the repeated sum is\n$\\mathcal R_P=\\sum_{k\\ge0}c_k/Q_k$, where $c_k$ counts the smooth integers\nin $[t_k,t_{k+1})$, with $t_0=1$.  The two tails after $t_k$ and after\n$t_kp^n$ still have words that differ by rearranging chains, and a rational\nvalue still makes the scaled tails integers of one denominator.  But the\nscaled tails are unbounded, as the $X_a$ are, and\nRemark~\\ref{long269:res:dp-blocks} needs every tail in $(0,1)$: the integer\n$f(\\sigma_i)-f(\\sigma'_i)$ is shown to vanish because it equals a\ndifference of two tails.  With unbounded tails that step fails, the digits\n$c_k$ move with the rearrangement as well, and equality of two tails no\nlonger forces agreement on each rearranged run.  The irrationality of\n$\\mathcal R_P$ for $|P|\\ge3$ remains open.\n\n\\paragraph{Prime-power subseries.}\nThe comparison of two tails applies to each prime-power subseries.\nFor $p\\in P$ put\n\\[\n E_p=\\sum_{\\alpha\\ge0}\\frac1{\\Lprefix(p^\\alpha)}=\\sum_{\\alpha\\ge0}\\frac1{\\hgt_P(p^\\alpha)},\n\\]\nthe sub-sum of $\\mathcal R_P$ over the terms at which the smooth integer is\na power of $p$.  For $P=\\{p,q\\}$ with $p<q$ these are the numbers $A=E_p$\nand $1+B_\\ast=E_q$ of the proof of Theorem~\\ref{long269:res:lead-two-prime}.\n\n\\begin{remark}[sub-sums and the catalogue sum]\\label{long269:res:single-prime-identities}\nLet $P$ be a finite set of primes with $|P|\\ge2$.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item $\\mathcal D_P=\\sum_{q\\in P}E_q-(|P|-1)$.\n\\item If $P=\\{p,q\\}$, then $\\hgt_P(p^iq^j)=\\hgt_P(p^i)\\hgt_P(q^j)$ for all\n$i,j\\ge0$, and $\\mathcal R_P=E_pE_q$.\n\\item If $P=\\{p,q,r\\}$, then\n$\\hgt_P(p^iq^jr^k)=\\hgt_P(p^i)\\hgt_P(q^j)\\hgt_P(r^k)\\,\np^{\\kappa_p(j,k)}q^{\\kappa_q(i,k)}r^{\\kappa_r(i,j)}$, where\n$\\kappa_p(j,k)=\\lfloor j\\log_pq+k\\log_pr\\rfloor-\\lfloor j\\log_pq\\rfloor-\\lfloor\nk\\log_pr\\rfloor\\in\\{0,1\\}$ and $\\kappa_q,\\kappa_r$ are defined in the same\nway.  For a general finite $P$ the corresponding exponents lie in\n$\\{0,\\ldots,|P|-2\\}$.\n\\end{enumerate}\n\\end{remark}\n\n\n\\begin{proof}\n(i) Each term $1/Q_k$ of $\\mathcal D_P$ with $k\\ge1$ is a term of $E_{q_k}$\nand of no other $E_q$, and each $E_q$ also contains the term $1$.\n(ii) and (iii) The exponent of $p$ in $\\hgt_P(\\prod_{q\\in P}q^{e_q})$ is\n$e_p+\\lfloor\\sum_{q\\ne p}e_q\\log_pq\\rfloor$, and its exponent in\n$\\prod_{q\\in P}\\hgt_P(q^{e_q})$ is $e_p+\\sum_{q\\ne p}\\lfloor e_q\\log_pq\\rfloor$.\nFor $m$ nonnegative reals the floor of the sum exceeds the sum of the floors\nby an integer in $\\{0,\\ldots,m-1\\}$.  With two primes $m=1$, so the running LCMs\nfactor, and summing $\\hgt_P(p^i)^{-1}\\hgt_P(q^j)^{-1}$ over $i,j\\ge0$ gives\n$E_pE_q$.\n\\end{proof}\n\nWith three primes $\\mathcal R_P$ is therefore the product of the three\nsingle-prime sums twisted by the carries $\\kappa$, and the plain product\n$E_pE_qE_r$ differs from it.\n\n\\papersectiontarget{r3-prime-subseries}\n\\begin{remark}[the single-prime sub-sums]\\label{long269:res:single-prime-subsums}\nLet $P$ be a finite set of primes with $|P|\\ge2$ and let $p\\in P$.  Then\n$E_p$ is irrational.\n\\end{remark}\n\n\nFor $P=\\{p,q\\}$, $E_p$ is $A$ or $1+B_\\ast$ in the proof of\nTheorem~\\ref{long269:res:lead-two-prime}, an affine function of the\nHecke--Mahler value $F_\\theta(1/p,1/q)$ with nonzero slope; for $p<q$,\n$E_p=p/(p-1)-(q-1)F_\\theta(1/p,1/q)$ by~\\eqref{long269:eq:hecke-mahler-boundary}.\nIt is transcendental by the theorem of Bugeaud and Laurent\n\\cite[Theorem~1.1]{bugeaudlaurent2023}, in the case due to Loxton and van der\nPoorten \\cite[Theorem~8, p.~40]{loxtonvdp1977}.  We cite these sources for\nthis case and have not re-verified its earlier attribution history.\n\n\\begin{proof}\nWe modify the word argument so that it counts only occurrences of $p$.\nFor an infinite word $u$ and a finite word $\\sigma=\\sigma_1\\cdots\\sigma_n$,\nput\n\\[\n V_p(u)=\\sum_{i:u_i=p}\\frac1{u_1\\cdots u_i},\\qquad\n f_p(\\sigma)=\\sum_{j:\\sigma_j=p}\\sigma_{j+1}\\cdots\\sigma_n.\n\\]\nThen $V_p(\\sigma u)=(f_p(\\sigma)+V_p(u))/\\Pi(\\sigma)$. Writing\n$y_k=V_p(q_{k+1}q_{k+2}\\cdots)$, we have\n\\[\n E_p=1+\\sum_{k:q_k=p}\\frac1{Q_k},\\qquad\n Q_kE_p=Q_k+\\sum_{j\\le k,\\ q_j=p}\\frac{Q_k}{Q_j}+y_k.\n\\]\nIn an actual suffix, the prefix product at the $j$th occurrence of $p$\nis at least $p^j$, with strict inequality after another prime has\noccurred. Hence $0<V_p(u)<1/(p-1)\\le1$, and rationality $E_p=N/K$\nwould imply $Ky_k\\in\\mathbb Z$. The proof of\nRemark~\\ref{long269:res:dp-blocks} therefore applies to $V_p,f_p$ as well.\nOn a chain of distinct primes containing $p$, only one term remains in\n$f_p$: the product of the primes following $p$. Unique factorisation\ndetects a change in that set, without recovering its order. This weaker\nrequirement avoids the collisions of the full block numerator $f$.\n\nFor each $q\\ne p$, the form $\\theta_p-\\theta_q$ is nonzero on the\nrational subspace $L$ used in Section~\\ref{long269:sec:distinct-all}.\nWe may therefore choose $v\\in L$ with $|v_q|\\le1$ for every $q$ and\n$v_q\\ne v_p$ for $q\\ne p$, where $v_q=\\theta_q(v)$, and put\n$\\mu=\\min_{q\\ne p}|v_p-v_q|>0$. Suppose $E_p=N/K$. Choose a jump\n$t_k$ labelled $p$, and choose $T,\\nu,g,\\beta$ as in the general\nirrationality proof, with $2^\\nu\\ge K$. Set $\\eta=1/10$ and first fix\n\\[\n 0<\\varepsilon<\\min\\{\\beta,g/2,\\log2/(2|P|),\\eta/4,\\eta\\mu/4\\}.\n\\]\nDirichlet's theorem then supplies arbitrarily large $n$ and integers\n$m_q\\ge1$ such that $\\delta_p=0$ and\n$0<|\\delta_q|=|n\\log p-m_q\\log q|<\\varepsilon$ for $q\\ne p$.\nWe compare $t_k$ with $t_m=t_kp^n$ and cut crossings at gaps greater than\n$2\\varepsilon$. As before, each chain consists of distinct primes and\nis merely rearranged by the phase shift. The common prefix of $\\nu$\nletters and the integral-tail argument give $y_k=y_m$, and hence\n$f_p(C)=f_p(C')$ for every chain and its rearrangement.\n\nWe shall contradict this equality near the origin of $T_0$. Put\n\\[\n \\begin{aligned}\n b_0&=\\min_{q\\ne p}\\frac{|\\delta_q|}{|v_p-v_q|},\n &B&=\\{q\\ne p:\\delta_q<0\\},\\\\\n A_+&=\\{q\\ne p:v_q<v_p\\},\n &A_-&=\\{q\\ne p:v_q>v_p\\}.\n \\end{aligned}\n\\]\nSince $0<b_0\\le\\varepsilon/\\mu<\\eta/4$ and $A_+,A_-$ partition the\nnonempty set $P\\smallsetminus\\{p\\}$, one of these two sets, say $A_s$ with\n$s\\in\\{+1,-1\\}$, differs from $B$. Take $b=sb_0/2$ and consider\n$z=bv+\\Theta(-\\eta/2)$. For each $q$, its crossing nearest $\\eta/2$\noccurs at\n\\[\n \\tau_q=\\eta/2-bv_q\\in(3\\eta/8,5\\eta/8),\\qquad\n \\tau_q-\\tau_p=b(v_p-v_q).\n\\]\nEvery other $q$-crossing lies outside $(-\\eta/4,5\\eta/4)$, and for\n$q\\ne p$ the displayed difference is nonzero with absolute value at most\n$|\\delta_q|/2$. We choose a neighbourhood $U$ of $z$ in $T_0$ on which\nthere are no crossings in $[-\\eta/4,\\eta/4]\\cup[3\\eta/4,5\\eta/4]$,\neach prime has a crossing in $(\\eta/4,3\\eta/4)$, and every difference\n$\\tau_q-\\tau_p$ retains its sign and has absolute value less than\n$|\\delta_q|$. These conditions are open and hold strictly at $z$.\n\nBy density, $\\theta_k+\\Theta(t^*)\\in U$ for some $t^*>T+2$. The resulting\ncrossings near $t^*$ form one chain: each is within $\\varepsilon$ of\nthe $p$-crossing and every other crossing is more than\n$\\eta/2>2\\varepsilon$ away. In this chain the followers of $p$ form\n$A_s$. After the shift, $q$ follows $p$ precisely when\n$\\tau_q-\\tau_p>\\delta_q$, which, by the bound\n$|\\tau_q-\\tau_p|<|\\delta_q|$, is equivalent to $\\delta_q<0$.\nIts followers now form $B\\ne A_s$, contradicting $f_p(C)=f_p(C')$.\n\\end{proof}\n\nThis argument uses a change of followers and so needs no two-wall lemma.\nEither of $A_+,A_-$ may be empty, but they are distinct when $|P|\\ge2$.\nFor $P=\\{p\\}$ the sum is $p/(p-1)$, and this is precisely where the choice\nof two distinct follower sets fails. The strict conditions defining $U$\nare needed to carry the construction to an actual orbit.\n\nFor $P=\\{2,3,5\\}$, $p=2$,\n$t_k=2^{20}$ and $n=65$, the first rearranged chain is $(2,3)$ against\n$(3,2)$ after $131$ letters, with $f_2$ equal to $3$ and $1$, and the\ndifference of the two tails, computed exactly to $300$ letters, has\n$\\log_{10}|y_k-y_m|=-58.098$.\n\nFor the repeated sum $\\mathcal R_P$, changing the scale from $t_k$ to\n$t_kp^n$ brings in early terms from the rays $p^\\alpha w$, with $w$\nsupported on $P\\smallsetminus\\{p\\}$. The comparison above accounts only for\nthe ray $w=1$, so it gives no corresponding estimate for the difference\nof the full repeated tails. In the case $P=\\{2,3,5\\}$ those normalised\ntails are unbounded, as the lower bounds below show.\n\n\\section{Words obtained from a linear flow}\n\\label{long269:sec:clock-transfer}\n\nWe apply the interchange argument to the order in which finitely many\nlinear functions take integer values. Their slopes will be assumed\nlinearly independent over $\\Q$; this hypothesis is not assumed for\nreciprocal prime logarithms. Interchanging two late crossings then\ndetermines the rational relations among the corresponding subseries,\nwithout requiring recovery of every letter. The arguments in this\nsection remain ordinary proofs awaiting formalisation.\n\nLet $A$ be a finite alphabet.  Give $a\\in A$ an integer radix $b_a\\ge2$\nand a rational coefficient $c_a$.  For $w=w_1w_2\\cdots$, define\n\\[\n Q_0=1,\\qquad Q_n=\\prod_{j=1}^n b_{w_j},\\qquad\n V_c(w)=\\sum_{n\\ge1}\\frac{c_{w_n}}{Q_n},\\qquad\n T_a(x)=\\frac{c_a+x}{b_a}.\n\\]\nWrite $\\sigma w=w_2w_3\\cdots$ and\n$X_w=\\overline{\\{\\sigma^nw:n\\ge0\\}}$ in the one-sided product topology.\nA word belongs to $X_w$ precisely when each of its finite prefixes begins\nsome shifted word. It need not itself be a shift. The first letter acts\nby the outermost map: if $u=u_1\\cdots u_m$, then\n$V_c(uz)=T_{u_1}\\circ\\cdots\\circ T_{u_m}(V_c(z))$.  Put\n$B(u)=\\prod_{j\\le m}b_{u_j}$, with $B(\\varnothing)=1$.\n\n\\begin{remark}[interchanging two letters after a long common prefix]\n\\label{long269:lem:late-swap}\nSuppose $V_c(w)$ is rational.  If for distinct letters $a,b$ and arbitrarily\nlong words $u$ there is a common infinite suffix $z$ such that both\n$uabz$ and $ubaz$ belong to $X_w$, then\n\\[\n c_a(b_b-1)=c_b(b_a-1).\n\\]\nIf such interchanges join every pair of letters along a connected graph,\n$V_c(w)$ is rational precisely when there is $\\lambda\\in\\Q$ with\n$c_a=\\lambda(b_a-1)$ for all $a$; in that case $V_c(w)=\\lambda$.\n\\end{remark}\n\n\\begin{proof}\nFirst clear the common denominator of the coefficients, so that all $c_a$ are\nintegers.  The recurrence\n$V_c(\\sigma w)=b_{w_1}V_c(w)-c_{w_1}$ shows that, if\n$V_c(w)=p/K$, then every shifted value belongs to $K^{-1}\\Z$.\nFor $\\lambda_a=c_a/(b_a-1)$, the positive weights\n\\[\n \\frac{b_{w_n}-1}{Q_n}=\\frac1{Q_{n-1}}-\\frac1{Q_n}\n\\]\nsum to one.  Hence $V_c(v)$ lies between $\\min_a\\lambda_a$ and\n$\\max_a\\lambda_a$ for every word $v$.  Sharing $m$ initial letters\nchanges a value by at most the width of this interval divided by $B(u)$.\nThus $V_c$ is continuous. Since $K^{-1}\\Z$ is closed, the lattice\ncondition passes from the actual shifts to all of $X_w$.\n\nThe order of the maps matters.  A direct calculation gives, for every $x$,\n\\[\n T_a(T_b(x))-T_b(T_a(x))\n  =\\frac{c_a(b_b-1)-c_b(b_a-1)}{b_ab_b}.\n\\]\nThe values of $uabz$ and $ubaz$ therefore differ by this number divided\nby $B(u)$.  Their difference belongs to $K^{-1}\\Z$.  If its numerator\nwere nonzero, its absolute value would be positive but smaller than $1/K$\nfor a sufficiently long $u$, a contradiction.  On a connected interchange\ngraph all $\\lambda_a$ agree.  Conversely, for\n$c_a=\\lambda(b_a-1)$ the displayed positive weights telescope and give\n$V_c(v)=\\lambda$ for every $v$.  Dividing by the earlier common denominator restores the statement for rational coefficients.\n\\end{proof}\n\nThe common suffix is essential: distinct words can have the same affine\nmap.  For example, the blocks $(5,7,3,2)$ and $(7,2,3,5)$ have the same\nmap $(51+x)/210$.  Choosing the second block at block indices\n$1,2,4,8,\\ldots$ produces a non-eventually-periodic word with rational\nvalue $51/209$.  Thus aperiodicity and bounded tails do not suffice, even with integer\nbases; the interchange hypothesis is essential to the argument.  Nor does the lemma\nassert sufficiency when its graph is disconnected.\n\nNow let $r_1,\\ldots,r_d>0$, $d\\ge2$, be linearly independent over $\\Q$.\nRecord the letter $i$ at each positive time $t$ for which\n$r_it+\\theta_i\\in\\Z$, where $\\theta_i\\in[0,1)$. Assume that no two of\nthese functions take integer values simultaneously at positive time.\nIn chronological order, the crossings give an infinite word $w$ on\n$\\{1,\\ldots,d\\}$.\n\n\\begin{remark}[isolated interchanges in the crossing sequence]\n\\label{long269:lem:clock-swaps}\nEvery pair of distinct labels satisfies the late-interchange\nhypothesis of Remark~\\ref{long269:lem:late-swap} for $X_w$.\n\\end{remark}\n\n\\begin{proof}\nFix a label $a$.  Immediately after its sufficiently late crossings,\nthe remaining coordinates of the flow are\n\\[\n \\theta_i-\\frac{r_i}{r_a}\\theta_a\n       +k\\frac{r_i}{r_a}\\pmod1\\qquad(i\\ne a),\n\\]\nwhere $k$ runs through all sufficiently large integers.  These points\nare dense in the $(d-1)$-torus.  Indeed, $1$ and the $r_i/r_a$ for $i\\ne a$\nare rationally independent.  For every nonzero integral character of\nthe torus, its average along this orbit is a geometric sum divided by\nits length and tends to zero.  Trigonometric approximation of continuous\nfunctions then shows that every nonempty open set is visited infinitely\noften.  This also proves density after any finite initial segment is\ndiscarded.\n\nFix $b\\ne a$ and an arbitrarily large integer $L$, and work on the section\njust after an $a$-crossing.  Set $T=L/r_a$ and choose the phase of $b$ so\nthat the $b$-coordinate crosses at $T$.  This phase is nonzero, since\n$Lr_b/r_a\\notin\\Z$.  Choose the remaining phases outside the countable\nsets that cause a crossing at zero, a third crossing at $T$, or any other\npositive simultaneous crossing.  The resulting singular configuration has\nexactly the one double crossing of $a,b$ at $T$: a second meeting of these\ntwo coordinates would make $r_b/r_a$ rational.\n\nPerturb the $b$ phase on opposite sides. On each bounded time interval,\na sufficiently small perturbation changes only this pair's order.\nTaking one-sided limits gives $uabz$ and $ubaz$ with the \\emph{same}\ninfinite suffix $z$. No fixed nonzero perturbation is required to preserve\nall later crossings: agreement on each finite interval suffices for these\nlimiting words.  Their common prefix has at least $L-1$ letters,\nfrom the earlier $a$-crossings.  For each finite prefix of either limit,\na small open set of nonsingular section phases realises that prefix.\nThe dense actual $a$-section orbit meets that set, so the prefix occurs\nat the beginning of an actual shifted word.  Consequently both limiting\nwords belong to $X_w$.  Letting $L$ grow proves the assertion.\n\\end{proof}\n\n\\begin{remark}[rational relations among the subseries]\n\\label{long269:thm:clock-relations}\nUnder the preceding hypotheses on the linear flow, choose integers $b_i\\ge2$, put\n$Q_n=\\prod_{k\\le n}b_{w_k}$ and\n\\[\n S_i=\\sum_{n:\\,w_n=i}\\frac1{Q_n}\\qquad(1\\le i\\le d).\n\\]\nFor every $c_i\\in\\Q$,\n\\[\n \\sum_{i=1}^dc_iS_i\\in\\Q\n \\quad\\Longleftrightarrow\\quad\n c_i=\\lambda(b_i-1)\\quad\\hbox{for all $i$ and some $\\lambda\\in\\Q$}.\n\\]\nThe rational value is $\\lambda$.  Equivalently, all rational relations\n$a_0+\\sum_i a_iS_i=0$ are multiples of\n$(-1,b_1-1,\\ldots,b_d-1)$.  Thus the span of\n$1,S_1,\\ldots,S_d$ has dimension $d$; each $S_i$ and each nonempty\nproper unweighted sub-sum is irrational.  The full unweighted sum is\nirrational exactly when the radices are not all equal.\n\\end{remark}\n\nThe following argument proves the assertion of the remark in ordinary\nmathematics; formalisation and external peer review remain outstanding.\n\n\\begin{proof}\nAbsolute convergence gives $\\sum_i c_iS_i=V_c(w)$.  The previous\ntwo remarks make the interchange graph complete, so they give both\ndirections and the stated value.  The relation space follows by moving\nthe rational value to the constant coordinate.  An indicator vector of\na nonempty proper set has both zero and one entries and cannot be a\nmultiple of the strictly positive vector $(b_i-1)_i$.  The all-ones\nvector is such a multiple precisely when all the $b_i$ agree.\n\\end{proof}\n\nFor example, take\n\\[\n (r_1,r_2,r_3,r_4)=(1,\\sqrt2,\\sqrt3,\\sqrt6),\\qquad\n (b_1,b_2,b_3,b_4)=(2,3,5,7),\\qquad \\theta_i=0.\n\\]\nThe four rates form the standard basis of the biquadratic field\n$\\Q(\\sqrt2,\\sqrt3)$ over $\\Q$; the ratio of any two distinct rates is\nirrational, so positive crossings never coincide.  Write\n$(d_1,d_2,d_3,d_4)=(1,2,3,6)$.  Counting crossings up to the $n$th\ncrossing of the $i$-th coordinate gives the explicit series\n\\begin{equation}\\label{long269:eq:four-clocks}\n S_i=\\sum_{n\\ge1}\\left(\n       \\prod_{j=1}^4 b_j^{\\lfloor n\\sqrt{d_j/d_i}\\rfloor}\n                        \\right)^{-1}.\n\\end{equation}\nThe only rational relation among $1,S_1,S_2,S_3,S_4$ is generated by\n$S_1+2S_2+4S_3+6S_4=1$.  In particular their rational span has\ndimension four and $\\sum_iS_i$ is irrational.  This does not provide an\nirrationality measure or transcendence result.  In the related\ntwo-letter Hecke--Mahler setting, stronger transcendence results exist\n\\cite{bugeaudlaurent2023}; the example's four independent rates do not\nsatisfy that two-letter hypothesis.  The ordinary two-wall argument for the distinct running-LCM sum avoids\nthis assumption on reciprocal prime logarithms.  The repeated\nrunning-LCM question remains open.\n\n\n\\section{Separation of the three-prime kernel}\\label{long269:sec:rank}\n\\papersectiontarget{r3-rank}\n\nWe test an exact kernel identity: can finitely many products separate\none exponent from the other two? A single binary carry rules this out\nfor three primes. Diagonal rescaling leaves threshold columns; independent\nrow and column choices produce staircase minors of every order.\nThis says nothing yet about identities after summation or approximation,\nwhich we treat separately. The initial two-generator identity is valid\nfor real generators; distinct primes enter the later density argument.\n\n\\begin{proposition}[two generators separate]\\label{long269:res:two-prime-rank}\nFor real $p,q>1$, with $L_{p,q}(t)=p^{\\lfloor\\log_p t\\rfloor}q^{\\lfloor\\log_q t\\rfloor}$\nas above, and all integers $i,j\\ge0$, the two-prime kernel\n$\\Kernel_2(i,j)=1/L_{p,q}(p^iq^j)$ is the outer product\n\\[\n \\Kernel_2(i,j)\n =\\bigl(p^{i}q^{\\lfloor\\log_q p^{i}\\rfloor}\\bigr)^{-1}\n  \\bigl(p^{\\lfloor\\log_p q^{j}\\rfloor}q^{j}\\bigr)^{-1},\n\\]\nso every two-by-two minor of $\\Kernel_2$ vanishes.\n\\end{proposition}\n\n\\begin{proof}\nSince $i$ and $j$ are integers,\n$\\lfloor\\log_p(p^iq^j)\\rfloor=i+\\lfloor\\log_p q^{j}\\rfloor$ and\n$\\lfloor\\log_q(p^iq^j)\\rfloor=j+\\lfloor\\log_q p^{i}\\rfloor$.  Hence\n$L_{p,q}(p^iq^j)$ is the product of $p^{i}q^{\\lfloor\\log_q p^{i}\\rfloor}$,\nwhich depends on $i$ alone, and $p^{\\lfloor\\log_p q^{j}\\rfloor}q^{j}$, which\ndepends on $j$ alone.  A matrix whose entries are a product of a row function\nand a column function has vanishing two-by-two minors.  For distinct primes\n$p,q$ the product $L_{p,q}$ is the running least common multiple by the\nargument of Theorem~\\ref{long269:res:lcm}.\n\\end{proof}\n\n\nAt three primes the smallest rectangle already disproves a single\nproduct. A factorisation $f(i)g(j)h(k)$ would force\n$\\Kernel(0,0,0)\\Kernel(1,1,0)=\\Kernel(1,0,0)\\Kernel(0,1,0)$.\n\n\\begin{proposition}[non-separability at $\\{2,3,5\\}$]\\label{long269:res:rank}\nWith $(p,q,r)=(2,3,5)$,\n\\[\n \\det\\begin{pmatrix}\n \\Kernel(0,0,0)&\\Kernel(0,1,0)\\\\\n \\Kernel(1,0,0)&\\Kernel(1,1,0)\n \\end{pmatrix}\n =\\det\\begin{pmatrix}1&1/6\\\\1/2&1/60\\end{pmatrix}\n =-\\frac1{15}\\ne0 .\n\\]\n\\end{proposition}\n\n\\begin{proof}\nThe four values are computed from $\\hgt(1)=1$, $\\hgt(2)=2$,\n$\\hgt(3)=2\\cdot3=6$ and $\\hgt(6)=4\\cdot3\\cdot5=60$, so the determinant is\n$1/60-1/12=-1/15$.\n\\end{proof}\n\n\n\\begin{theorem}[no finite separation of the kernel]\\label{long269:res:infinite-rank}\n\\label{long269:res:lead-infinite-rank}\nLet $p,q,r$ be primes with $p\\ne q$, $p\\ne r$ and $q\\ne r$.  For every\n$n\\ge1$ there are injective maps $I,J:\\{0,\\ldots,n-1\\}\\to\\N$ such that, for\nevery $k\\ge0$,\n\\[\n \\det\\bigl(\\Kernel(I(a),J(b),k)\\bigr)_{0\\le a,b<n}\\ne0 .\n\\]\nConsequently, for no finite $d$ do there exist rational-valued functions\n$f_\\ell:\\N\\to\\Q$ and $G_\\ell:\\N^{2}\\to\\Q$, $0\\le\\ell<d$, satisfying\n$\\Kernel(i,j,k)=\\sum_{\\ell<d}f_\\ell(i)G_\\ell(j,k)$ for all $i,j,k$.\n\\end{theorem}\n\n\\begin{proof}\nFor fixed $k$, the $p$- and $q$-exponents still split into row and column\nterms. The $r$-exponent also includes the carry from adding two fractional\nparts. To isolate it, put $\\alpha=\\log_r p$, $\\beta=\\log_r q$,\n$x_i=\\{i\\alpha\\}$ and $y_j=\\{j\\beta\\}$. Write $v_p(N)$ for the exponent\nof the prime $p$ in the positive integer $N$. The three prime exponents are\n\\[\n \\begin{split}\n v_p(\\hgt(p^iq^jr^k))&=i+\\lfloor j\\log_pq+k\\log_pr\\rfloor,\\\\\n v_q(\\hgt(p^iq^jr^k))&=j+\\lfloor i\\log_qp+k\\log_qr\\rfloor,\\\\\n v_r(\\hgt(p^iq^jr^k))&=k+\\lfloor i\\alpha\\rfloor+\\lfloor j\\beta\\rfloor+\n   \\lfloor x_i+y_j\\rfloor .\n \\end{split}\n\\]\nEvery term except the last floor depends on $i$ and $k$ alone or on $j$ and $k$\nalone, so with positive rational $R_i(k)$ and $C_j(k)$,\n\\begin{equation}\\label{long269:eq:carry-factorisation}\n \\Kernel(i,j,k)=R_i(k)\\,C_j(k)\\,t^{\\lfloor x_i+y_j\\rfloor},\n \\qquad t=r^{-1} .\n\\end{equation}\nThe remaining matrix is independent of $k$. Since $x_i+y_j\\in[0,2)$,\nits entry is $1$ when there is no carry and $t$ when there is one.\n\nEach of $\\alpha$ and $\\beta$ is irrational, since a rational value would give\nan equality of positive powers of distinct primes, so each fractional-part\norbit is dense in $[0,1]$ and each is injective.  The row and column indices\nare chosen independently, so no density theorem for a single orbit of\npairs is required.  First choose indices with\n$0<x_{I(0)}<\\cdots<x_{I(n-1)}<1$.  We want the columns to cross their\nthresholds at successive selected rows, so that consecutive row differences\nwill leave a triangular matrix.  Write $s_b=1-y_{J(b)}$, and use density\nof the second orbit to choose\n\\[\n s_0\\in(0,x_{I(0)}),\\qquad\n s_b\\in(x_{I(b-1)},x_{I(b)})\\quad(1\\le b<n).\n\\]\nThe intervals are disjoint, so the $J(b)$ are distinct.  Their strict\nendpoints ensure that no selected entry lies on a threshold, and\n$x_{I(a)}+y_{J(b)}\\ge1$ holds exactly when $b\\le a$.  Dividing row $a$ by\n$R_{I(a)}(k)$ and column $b$ by $C_{J(b)}(k)$ therefore leaves\n\\[\n C_n(t)=\\begin{pmatrix}\n t&1&1&\\cdots&1\\\\\n t&t&1&\\cdots&1\\\\\n \\vdots&\\vdots&\\ddots&\\ddots&\\vdots\\\\\n t&t&\\cdots&t&1\\\\\n t&t&\\cdots&t&t\n \\end{pmatrix},\n \\qquad\n \\det C_n(t)=t(t-1)^{n-1}\\ne0 ,\n\\]\nthe determinant following by subtracting from each row its predecessor, working\nupwards from the last.  Before the division, the determinant equals\n\\[\n \\det C_n(t)\\prod_{a<n}R_{I(a)}(k)\\prod_{b<n}C_{J(b)}(k),\n\\]\nwhich is nonzero for every $k$; this also explains why the same indices work\nin every layer.\n\nFor the last assertion, suppose that a representation with $d$ summands\nexists and fix $k$.  On the selected rows $I(0),\\ldots,I(d)$ and columns\n$J(0),\\ldots,J(d)$ the resulting matrix factors as\n\\[\n (f_\\ell(I(a)))_{0\\le a\\le d,\\,\\ell<d}\n (G_\\ell(J(b),k))_{\\ell<d,\\,0\\le b\\le d}.\n\\]\nIt has rank at most $d$ and hence zero determinant, contradicting the\nnonzero minor of order $d+1$.\n\\end{proof}\n\n\nIndex selection is essential.  In the $k=0$ layer of the $\\{2,3,5\\}$\nkernel, the leading $4\\times4$ minor is singular because\n$\\Kernel(3,j,0)=\\Kernel(0,j,0)/120$ for $0\\le j\\le3$.\nThis is not a proportionality of the full rows: at the next column,\n\\[\n \\Kernel(3,4,0)-\\frac{\\Kernel(0,4,0)}{120}=-\\frac1{19440000}.\n\\]\nA singular leading minor therefore does not settle the rank.\n\nThe selected minors put thresholds at distinct cuts among the row phases. A general sample\nneed not do so: several columns can have the same cut, and two can be\nconstant. Counting the distinct cuts gives its rank.\n\n\\begin{proposition}[rank of threshold columns]\\label{long269:res:finite-cut-rank}\nLet $m\\ge1$ and let $c$ lie in a field with $c\\ne0,1$.  For $0\\le h\\le m$,\nlet $v_h$ be the length-$m$ column whose first $h$ entries are $1$ and whose\nremaining entries are $c$.  If the distinct columns of a matrix are the\n$v_h$ with $h$ in a nonempty set $E\\subseteq\\{0,\\ldots,m\\}$, then its rank is\n\\[\n |E|-\\mathbf 1_{\\{0,m\\}\\subseteq E}.\n\\]\n\\end{proposition}\n\n\\begin{proof}\nWrite $E=\\{h_1<\\cdots<h_s\\}$.  The $s-1$ consecutive differences are\n\\[\n v_{h_{a+1}}-v_{h_a}\n   =(1-c)\\mathbf 1_{\\{h_a,\\ldots,h_{a+1}-1\\}},\n\\]\nso they are linearly independent because their nonempty supports are\ndisjoint.  If $h_1>0$ or $h_s<m$, those supports miss a coordinate on which\n$v_{h_1}$ is nonzero, and adjoining $v_{h_1}$ gives rank $s$.  If\n$h_1=0$ and $h_s=m$, the differences sum to $(1-c)\\mathbf 1$ while\n$v_0=c\\mathbf 1$, so $v_0$ is already in their span and the rank is $s-1$.\n\\end{proof}\n\nThe restrictions $c\\ne0,1$ exclude the zero column at $c=0$ and the\ncollapse of all columns at $c=1$.  They hold automatically at $c=1/r$ over\n$\\mathbb Q$.\nTo apply the proposition, fix $k$ and sort the chosen row phases $x_i$.\nEach normalised column is a threshold column $v_h$, where $h$ is the\nnumber of sampled phases strictly below $1-y_j$.  Repeated threshold\npositions give proportional columns before column normalisation.\nThe row and column factors in~\\eqref{long269:eq:carry-factorisation} are\nnonzero, so the formula gives the exact rank of every nonempty rectangular\nsample within this layer, independently of $k$.  Repeated rows or columns\ndo not change rank.\n\nThis also gives an exact algorithm.  Order the sampled rows by the\nrational numbers $p^i/r^{\\lfloor\\log_r p^i\\rfloor}$.  For column $j$, count\nthose strictly below $r^{\\lfloor\\log_r q^j\\rfloor+1}/q^j$, then apply the\nendpoint correction to the set of resulting counts.  Indeed, raising\n$x_i<1-y_j$ to base $r$ gives exactly this rational comparison, and\n$x_i+y_j=1$ belongs to the $c$ side of the threshold.  The integer\nlogarithms are found by comparing powers, so the whole calculation uses\ninteger arithmetic rather than numerical logarithms.\n\nThe fixed-layer restriction is essential.  With $(p,q,r)=(2,3,5)$, rows\n$i=0,1$ and columns indexed by $(j,k)=(0,0),(0,1)$ give\n\\[\n \\begin{pmatrix}\n  \\Kernel(0,0,0)&\\Kernel(0,0,1)\\\\\n  \\Kernel(1,0,0)&\\Kernel(1,0,1)\n \\end{pmatrix}\n =\\begin{pmatrix}1&1/60\\\\1/2&1/360\\end{pmatrix},\n \\qquad \\det=-\\frac1{180}.\n\\]\nBoth columns have the same phase $y_0=0$, yet their rank is $2$, not $1$.\nWhen $k$ varies with the column, the row factor also varies with that\ncolumn and cannot be removed by one common diagonal rescaling.  The\nthreshold formula is not a rank formula for such mixed-layer samples.\n\n\\paragraph{Scope of the rank theorem.}\nFan's comment of 26 June 2026~\\cite{fan2026comment} notes that the\ntwo-prime argument does not seem to generalise immediately to $|P|\\ge3$.\nThe theorem above gives a precise obstruction: no finite exact separation\nof the stated form exists.  The same finite-dimensional proof works for\nreal or complex factors, without continuity or boundedness assumptions.\nThis does not exclude approximation or imply irrationality or\ntranscendence of the sum.\nThe threshold construction needs $\\log_r p$ and $\\log_r q$ irrational,\nbut not $p\\ne q$.  The latter condition is needed for the three-prime\nrunning-LCM interpretation, not for the two independent dense orbits.\n\n\\subsection{Smith factors of the selected staircase}\n\\label{long269:sec:selected-kernel-height}\nThe preceding argument only uses that a determinant is nonzero. An\nirrationality argument would also need control of integer coefficients\nafter denominators are cleared. We examine that separate question for the\nselected minors. For integers $N\\ge1,r\\ge2$, let $A_N(r)$ have entry $1$\non and below the diagonal and $r$ above it. This is the staircase used in\nTheorem~\\ref{long269:res:infinite-rank}. Its Smith factors determine the\nlattice index; they do not by themselves give a small integral linear form.\n\n\\begin{remark}[Smith factors of the selected staircase]\n\\label{long269:lem:staircase-smith}\nThe Smith factors of $A_N(r)$ are\n$1,r-1,\\ldots,r-1$, and\n$\\det A_N(r)=(1-r)^{N-1}$.\n\\end{remark}\n\nThe Smith calculation and its consequence for integral coefficients are\nordinary arguments here; the existing Lean result proves nonvanishing of\nthe selected minors, not these arithmetic statements.\n\n\\begin{proof}\nWorking from bottom to top, subtract row $i-1$ from row $i$ for every\n$i>1$.  Row $i$ then has just $1-r$ in column $i$.  The first row is\n$(1,r,\\ldots,r)$; subtract $r$ times its first column from each later\ncolumn.  The resulting diagonal is $(1,1-r,\\ldots,1-r)$.  These are\nunimodular integral operations, and $1$ divides $r-1$, so this is its\nSmith form up to signs.  Its determinant has the displayed sign.\n\\end{proof}\n\nFor the selected $\\{2,3,5\\}$ minor, the positive integers $U_i,V_j$ give\n\\[\n K=\\operatorname{diag}(U_i)^{-1}A_N(5)\\operatorname{diag}(5V_j)^{-1}.\n\\]\nConsequently\n\\[\n \\det K=\\frac{(-4)^{N-1}}{W},\\qquad\n W=5^N\\prod_iU_i\\prod_jV_j.\n\\]\nMore generally, an entrywise-integral diagonal clearing\n$\\operatorname{diag}(s)K\\operatorname{diag}(t)$ has diagonal entries\n$z_i=s_it_i/(5U_iV_i)\\in\\Z\\smallsetminus\\{0\\}$ and determinant\n$(-4)^{N-1}\\prod_i z_i$.  Its absolute determinant is therefore at\nleast $4^{N-1}$, attained by $s_i=5U_i$, $t_j=V_j$.\nThis optimality is restricted to diagonal entrywise clearings.\n\nFor the selected carry pattern, the injective nonnegative index families\n$I,J$ satisfy\n$v_2(W)\\ge\\sum_iI(i)+\\sum_jJ(j)\\ge N(N-1)\\ge2(N-1)$.\nThus $4^{N-1}$ divides $W$, and the \\emph{reduced} determinant is\n$\\pm1/D$, where $D=W/4^{N-1}\\in\\N_{>0}$.\nThus the factor $4^{N-1}$ in the integer determinant is absorbed entirely\nby its denominator. For a linear form in one target value, a small\nrational prefactor likewise disappears when the coefficients are reduced\nto coprime integers. Indeed, for $K\\in\\mathrm{GL}_N(\\Q)$ and rational\n$u,v$, the Schur complement gives\n\\[\n \\det\\begin{pmatrix}K&u\\\\v^T&X\\end{pmatrix}\n =\\det K\\bigl(X-v^TK^{-1}u\\bigr).\n\\]\nWrite $v^TK^{-1}u=s/t$ in lowest terms, with $t>0$. If a nonzero\nrational scalar multiple of $\\det K(X-s/t)$ has integer coefficients\n$aX+b$, then $b=-as/t$. Since $\\gcd(s,t)=1$, we have $t\\mid a$.\nDividing the two coefficients by their common factor therefore leaves\n$\\pm(tX-s)$. This is the primitive integral coefficient polynomial,\nregardless of the smallness of $\\det K$ or a compatible rational change\nof basis. The conclusion concerns this construction with one target and\nthe diagonal clearings above. It does not exclude new recurrences,\nmultiple target values or stronger approximation data, and it does not\nsettle the repeated running-LCM problem. Smith form is\nused here in its standard sense~\\cite{stanley2016}.\n\n\\subsection{Uniform approximation of the rescaled kernel}\\label{long269:sec:approximation}\n\\papersectiontarget{r3-approximation}\n\nExact rank does not decide approximation. For the rescaled kernel,\ninfinitely many columns remain a fixed distance apart, giving a uniform\nlower bound against every finite rank. For the original kernel, decay\ninstead permits summable truncation. We prove these statements for the\ntwo different matrices. Write\n\\begin{equation}\\label{long269:eq:carry-matrix}\n C(i,j)=t^{\\lfloor x_i+y_j\\rfloor},\\qquad\n x_i=\\{i\\log_r p\\},\\quad y_j=\\{j\\log_r q\\},\\quad t=r^{-1}.\n\\end{equation}\nThis is the remaining factor in\n\\eqref{long269:eq:carry-factorisation} after removing the row and column\nfactors. A matrix has \\emph{finite separated rank} if\n$A(i,j)=\\sum_{\\ell<d}f_\\ell(i)g_\\ell(j)$ for some finite $d$.\nEquivalently, its columns span a finite-dimensional space of real\nsequences. No continuity or boundedness of $f_\\ell,g_\\ell$ is assumed.\n\n\\begin{theorem}[distance from matrices of finite separated rank]\\label{long269:res:uniform-rank}\nLet $p,q,r$ be pairwise distinct primes and let $C$ be as\nin~\\eqref{long269:eq:carry-matrix}.  Then\n\\[\n \\inf_{A}\\ \\sup_{i,j\\ge0}\\ |C(i,j)-A(i,j)|=\\frac{1-t}{2}=\\frac{r-1}{2r},\n\\]\nthe infimum being over all matrices $A$ of finite separated rank, and it is\nattained by the constant matrix of value $(1+t)/2$.\n\\end{theorem}\n\n\\begin{proof}\nWe first show that any two distinct columns of $C$ have supremum distance\n$1-t$. Every entry belongs to $\\{1,t\\}$, with $C(i,j)=t$ precisely when\n$x_i+y_j\\ge1$. Given $j\\ne k$, irrationality of $\\log_r q$ makes\n$y_j\\ne y_k$, so suppose $y_j<y_k$. By density of $(x_i)$ we can choose\n$i$ with $1-y_k<x_i<1-y_j$; the two entries in this row are $t$ and $1$.\n\nSuppose a finite-rank matrix $A$ approximated $C$ with error\n$\\varepsilon=\\sup_{i,j}|C(i,j)-A(i,j)|<(1-t)/2$. Its columns $A_j$ would\nbe bounded sequences with $\\|A_j\\|_\\infty\\le1+\\varepsilon$, and their\nspan $V$ would be finite-dimensional. Although no boundedness was assumed\nfor the separated factors, the supremum norm is defined on this span\nbecause each column is bounded. For distinct columns the triangle\ninequality gives\n\\[\n \\|A_j-A_k\\|_\\infty\\ge\\|C_j-C_k\\|_\\infty-2\\varepsilon\n =1-t-2\\varepsilon>0.\n\\]\nThus a bounded subset of $V$ would contain infinitely many points at a\nfixed positive distance from one another, contrary to total boundedness\nin finite dimension. This proves the lower bound.\n\nFor equality, take the constant matrix $A(i,j)=(1+t)/2$. It has rank one,\nand its distance from every entry of $C$ is $(1-t)/2$.\n\\end{proof}\n\nThe lower bound concerns the entire normalised matrix, not each finite\nrestriction.  For example, at $t=1/5$ the matrices\n\\[\n T=\\begin{pmatrix}1/5&1\\\\1/5&1/5\\end{pmatrix},\\qquad\n A=\\begin{pmatrix}3/10&9/10\\\\1/10&3/10\\end{pmatrix}\n\\]\nsatisfy $\\det A=0$ and $\\|T-A\\|_{\\max}=1/10<2/5$.\nHere $\\|\\cdot\\|_{\\max}$ is the largest absolute entry.  This finite\nexample has no bearing on the infinite family of separated columns used\nin the proof.\n\nFor the original kernel, the decaying row and column factors instead\nallow approximation in the summation norm.  Let $K^{(N)}(i,j,k)=\\Kernel(i,j,k)$ for $i<N$ and\n$K^{(N)}(i,j,k)=0$ otherwise.  This is a sum of at most $N$ terms separated\nbetween $i$ and $(j,k)$.  Since\n$\\hgt(x)>x^3/(pqr)$ for $x\\ge1$, geometric summation gives\n\\begin{equation}\\label{long269:eq:l1-finite-rank-approximation}\n \\sum_{i,j,k\\ge0}|\\Kernel(i,j,k)-K^{(N)}(i,j,k)|\n \\le\n \\frac{pqr\\,p^{-3N}}\n {(1-p^{-3})(1-q^{-3})(1-r^{-3})}.\n\\end{equation}\nThe same bound controls the supremum of the entrywise errors, so these\nfinite separated-rank approximants converge both in $\\ell^1$ and uniformly\nto the original kernel.  The positive uniform lower bound in\nTheorem~\\ref{long269:res:uniform-rank} belongs to the rescaled matrix.\nDiagonal rescaling preserves exact rank, but the rescaling factors here\nare unbounded and do not preserve uniform error estimates.  Neither the\nrank obstruction nor these approximation bounds decide whether the scalar\nsum is rational.\n\n\\section{The recurrence for tails between powers of two}\\label{long269:sec:blocks}\n\nReturn to $S=\\mathcal R_{\\{2,3,5\\}}$ and $P=\\{2,3,5\\}$. In place of\nthe three jumps in $(16,32]$, we now count all seven smooth integers in\n$[16,32)$: $16,18,20,24,25,27,30$. Their running LCMs are\n$720,720,720,720,3600,10800,10800$. Two changes need keeping separate:\nwe retain multiplicities, and we include the left rather than the right\nendpoint. The first changes the coefficients; the second changes the\nnormalisation.\n\nWe group the terms of $S$ into $[2^a,2^{a+1})$, including the left\nendpoint. The prefix to be cleared is therefore strictly before $2^a$:\nit excludes the jump by $2$ at that boundary. Put $P_a=\\hgt(2^a)$ and\n$h_a=P_a/2$. Every running LCM in that prefix divides $h_a$.\nHere $h_0=1/2$, and $h_a$ is an integer for $a\\ge1$.\nThe powers of $3$ and $5$ selected at $2^a$ exceed $2^a/3$ and $2^a/5$,\nrespectively, while the power of $2$ is $2^a$ itself. Thus\n\\[\n \\frac{8^a}{15}<P_a\\le8^a\\qquad(a\\ge0).\n\\]\nWith this normalisation, define the block masses and tails by\n\\begin{equation}\\label{long269:eq:actual-tail}\n s_a=\\sum_{\\substack{i,j,k\\ge0\\\\2^a\\le2^i3^j5^k<2^{a+1}}}\n         \\frac1{\\hgt(2^i3^j5^k)},\\qquad\n T_a=\\sum_{j\\ge a}s_j,\\qquad X_a=h_aT_a.\n\\end{equation}\nFor $a\\ge1$, the prefix has exponent of $2$ at most $a-1$ and other\nexponents at most their boundary values. This proves the divisibility\nby $P_a/2$ needed in Lemma~\\ref{long269:res:all-scale-lattice}.\n\nThe jumps after $2^a$ and up to $2^{a+1}$ consist of any powers of $3$ or\n$5$ strictly inside that interval, followed by the factor $2$ at its right\nendpoint.  There is at most one power of each odd prime: successive\npowers have ratio greater than two.  Let $I_a$ list the pairs $(p,e)$ with\n$p\\in\\{3,5\\}$ and $2^a<p^e<2^{a+1}$, ordered by the value $p^e$.\n\n\\begin{proposition}[four possible bases]\\label{long269:res:dyadic-alphabet}\nFor every $a$,\n\\begin{equation}\\label{long269:eq:dyadic-alphabet}\n b_a=\\frac{P_{a+1}}{P_a}=2\\prod_{(p,e)\\in I_a}p\\in\\{2,6,10,30\\},\n \\qquad\\text{so}\\qquad 2\\le b_a\\le30 .\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nA power of $3$ may occur and a power of $5$ may occur, each at most once.\nThe resulting factor is $2$, $2\\cdot3$, $2\\cdot5$ or $2\\cdot3\\cdot5$.\n\\end{proof}\n\n\nThe coefficient subtracted at step $a$ is the block mass with its\ndenominators cleared:\n\\[\n m_a=h_{a+1}s_a\n =\\sum_{\\substack{x\\text{ smooth}\\\\2^a\\le x<2^{a+1}}}\n       \\frac{P_{a+1}}{2\\hgt(x)}.\n\\]\nEach summand is an integer, by the same strict-cutoff argument.\nFor example, $[2,4)$ contains just $2$ and $3$, so\n$m_1=6(1/2+1/6)=4$ and $X_2=6X_1-4$.\nThe first four pairs $(b_a,m_a)$, starting at $a=1$, are\n$(6,4)$, $(10,7)$, $(6,7)$ and $(30,65)$.\nIn particular $m_3>b_3$: these coefficients are not positional digits.\n\nIn the block $[16,32)$, the internal jumps are $25$ and $27$, in that\norder. Multiplication by $h_5=10800$ gives the weights for the seven\npoints listed above.  The four smooth numbers\n$16,18,20,24$ precede both jumps and have weight $15$; $25$ precedes only\nthe jump at $27$ and has weight $3$; and $27,30$ have weight $1$.  Thus\n\\[\n m_4=4\\cdot15+1\\cdot3+2\\cdot1\n     =7+(5-1)\\cdot4\\cdot3+(3-1)\\cdot5=65.\n\\]\nThe second expression starts with the seven points and adds the\ncorrections before $25$ and $27$.  It is this form that extends to every\nblock.\n\nTo compute $m_a$ in general, count the smooth numbers before each\nprime-power jump.\nLet $\\mathcal N(t)$ be the number of positive $\\{2,3,5\\}$-smooth integers\nstrictly below $t$.  First give every point in $[2^a,2^{a+1})$ weight one.\nAt an internal jump $p^e$, the points before that jump need an additional\nweight $p-1$, multiplied by the prime factors at all later internal jumps.\nThus, for $a\\ge1$,\n\\begin{equation}\\label{long269:eq:actual-digit}\n\\begin{aligned}\n m_a&=\\mathcal N(2^{a+1})-\\mathcal N(2^a)\\\\\n &\\quad+\\sum_{(p,e)\\in I_a}(p-1)\n       \\bigl(\\mathcal N(p^e)-\\mathcal N(2^a)\\bigr)\n       \\prod_{\\substack{(q,f)\\in I_a\\\\p^e<q^f}}q,\\\\\n m_0&=1.\n\\end{aligned}\n\\end{equation}\nThe first difference counts the whole block; each later difference counts\nits points strictly before the indicated jump.  The proof below justifies\nthese weights, establishes convergence and derives the tail identities.\n\n\\begin{theorem}[the tail recurrence]\\label{long269:res:actual-orbit}\nThe block masses are summable and $S=\\sum_{a\\ge0}s_a$.  For every $a\\ge0$,\n\\begin{equation}\\label{long269:eq:shell-digit-identity}\n m_a=h_{a+1}s_a\\in\\N_{>0},\n \\qquad\n X_{a+1}=b_aX_a-m_a,\n \\qquad\n X_a=\\sum_{j\\ge a}\\frac{m_j}{b_ab_{a+1}\\cdots b_j} .\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nEvery exponent of $3$ or $5$ in the $a$th block is at most $a$, and for each\nsuch pair Lemma~\\ref{long269:res:short} leaves at most one exponent of $2$ placing the\npoint in $[2^{a},2^{a+1})$.  Each running LCM is at least $2^{a}$ and the block\ncontains $2^{a}$, so $0<s_a\\le(a+1)^{2}2^{-a}$.  The majorant is summable,\nwhich justifies every tail splitting below, and unique prime factorisation\nidentifies $\\sum_a s_a$ with the original repeated series.\n\nWrite the internal jumps as $t_\\ell=p_\\ell^{e_\\ell}$, $1\\le\\ell\\le v$, and set\n$t_0=2^{a}$, $t_{v+1}=2^{a+1}$ and $n_\\ell=\\mathcal N(t_\\ell)$.\nOn $[t_\\ell,t_{\\ell+1})$ the\nrunning LCM is $P_a\\prod_{u\\le\\ell}p_u$, and the terminal jump has factor two, so\n\\[\n h_{a+1}s_a\n =\\sum_{\\ell=0}^{v}(n_{\\ell+1}-n_\\ell)\\prod_{u>\\ell}p_u\n =n_{v+1}-n_0+\\sum_{\\ell=1}^{v}(p_\\ell-1)\n        \\Bigl(\\prod_{u>\\ell}p_u\\Bigr)(n_\\ell-n_0).\n\\]\nThe second equality is finite summation by parts and is exactly\n\\eqref{long269:eq:actual-digit}.  The count itself is finite: for $p=2,3,5$,\nwith $q,r$ the other primes, counting by the exponent of $p$ gives\n\\[\n \\mathcal N(p^e)=\\sum_{u=1}^e\\#\\{(i,j)\\in\\N^2:q^ir^j<p^u\\}.\n\\]\nAt $a=0$ the block is the single point $1$, giving $m_0=1$.  Positivity follows from $s_a>0$.  Splitting $T_a=s_a+T_{a+1}$ and\nmultiplying by $h_a$ gives the recurrence, and\n$b_a\\cdots b_j=P_{j+1}/P_a$ with $m_j=h_{j+1}s_j$ gives\n$m_j/(b_a\\cdots b_j)=h_as_j$, whose summation is the last identity.\n\\end{proof}\n\nFor estimates, it is useful to count by the odd part $3^j5^k$ instead.\nEach such part below $2^{a+1}$ has exactly one power-of-two multiple in\n$[2^a,2^{a+1})$.  This turns the same numerator into a two-dimensional\ncount, with weights determined by the remaining odd-prime jumps.\n\n\\begin{lemma}[the block numerator as a weighted lattice count]\n\\label{long269:res:literal-triangle}\nPut $\\lambda_3=\\log_2 3$, $\\lambda_5=\\log_2 5$ and\n$\\theta_p=1/\\lambda_p$ for $p=3,5$.\nFor $j,k\\ge0$ write $w_{j,k}=j\\lambda_3+k\\lambda_5$ and\n$t_{j,k}=\\{w_{j,k}\\}$.  The block numerator is\n\\[\n m_a=\\sum_{\\substack{j,k\\ge0\\\\w_{j,k}<a+1}}\n 3^{\\lfloor(a+1)\\theta_3\\rfloor-\\lfloor(a+t_{j,k})\\theta_3\\rfloor}\n 5^{\\lfloor(a+1)\\theta_5\\rfloor-\\lfloor(a+t_{j,k})\\theta_5\\rfloor}.\n\\]\nEvery summand is in $\\{1,3,5,15\\}$.  In particular, for $a\\ge0$,\n\\[\n \\bigl(\\lfloor a/(2\\lambda_3)\\rfloor+1\\bigr)\n \\bigl(\\lfloor a/(2\\lambda_5)\\rfloor+1\\bigr)\n \\le m_a\\le15(a+1)^2.\n\\]\nThus $m_a=\\Theta((a+1)^2)$ and the numerator sequence is unbounded.\nThese are integer numerators, not positional digits restricted to\n$\\{0,\\ldots,b_a-1\\}$.\n\\end{lemma}\n\\begin{proof}\nFor each pair with $w_{j,k}<a+1$, exactly one exponent\n$i=a-\\lfloor w_{j,k}\\rfloor\\ge0$ puts\n$2^i3^j5^k=2^{a+t_{j,k}}$ in $[2^a,2^{a+1})$.\nThis pairs each point of the triangle with exactly one smooth integer\nin the block.  Substituting in\n$P_{a+1}/(2\\hgt(2^i3^j5^k))$ cancels the power of $2$ and gives the displayed\nweight.  Each remaining floor increment is zero or one since\n$0<\\theta_p<1$.  Finally $\\lambda_3,\\lambda_5>1$ implies\n$j,k\\le a$ for every summation pair, proving the upper bound.\nFor the lower bound, restrict to\n$0\\le j\\le\\lfloor a/(2\\lambda_3)\\rfloor$ and\n$0\\le k\\le\\lfloor a/(2\\lambda_5)\\rfloor$.\nThen $w_{j,k}\\le a$ and each weight is at least one.\n\\end{proof}\nThe quadratic order comes from the size of the triangle, not from a\npolynomial formula for $m_a$. As $a$ changes, new points enter and the\nweights of old points can change at floor crossings. A finite-difference\nargument must control both effects; polynomial growth alone cancels neither.\nQuadratic growth does not make this count a quadratic polynomial.\n\nThe recurrence also restricts how persistently a nonintegral tail can\napproach the integers.  The following alternative uses only the integer\ncoefficients and the bounds $2\\le b_a\\le30$.\n\n\\begin{proposition}[integer tails or repeated separation from the integers]\\label{long269:res:actual-dichotomy}\nFor every integer $B\\ge1$, either $BX_a\\in\\Z$ for some $a\\ge0$ and every\nlater $a$, or for every $a_0$ there is $a\\ge a_0$ with\n$|BX_a-z|\\ge1/31$ for every $z\\in\\Z$.\n\\end{proposition}\n\n\\begin{proof}\nIf the second alternative fails, there are $A$ and integers $z_a$ such\nthat $e_a=BX_a-z_a$ satisfies $|e_a|<1/31$ for all $a\\ge A$.\nThe recurrence gives\n$z_{a+1}-b_az_a+Bm_a=b_ae_a-e_{a+1}$.  Since $Bm_a$ is integral, the\nleft side is an integer; the right side has absolute value less than\n$(30+1)/31=1$.  Both sides therefore vanish, so $e_{a+1}=b_ae_a$.\nHence $|e_{A+k}|\\ge2^k|e_A|$ for every $k$, whereas\n$|e_{A+k}|<1/31$.  Thus $e_A=0$, and the integer recurrence propagates\nintegrality from $BX_A$ to every later $BX_a$.\n\\end{proof}\n\nThe proposition does not determine which alternative holds for the\ntails of $S$.\nFor comparison, Erd\\H{o}s--Taylor\n\\cite[Theorem~1, p.~600]{erdostaylor1957} prove a countability statement for\nincreasing integer sequences with bounded successive ratios; Fan\n\\cite[Lemma~3.1, arXiv v1]{fan2026strongly} allows an unbounded, not\nnecessarily increasing sequence, still with a uniform upper bound on\nsuccessive ratios.\nFor our divisibility chain the same error argument identifies the entire\nexceptional set, not just its cardinality.  For each fixed integer $B\\ge1$,\n\\[\n \\bigl\\{\\xi\\in\\R:\\operatorname{dist}(Bh_a\\xi,\\Z)\\longrightarrow0\\bigr\\}\n =\\frac1B\\Z[1/30],\n \\qquad\n \\Z[1/30]=\\{m/30^t:m\\in\\Z,\\ t\\in\\N\\}.\n\\]\nIndeed, apply the proof above to $Bh_a\\xi$, whose successive terms are\nrelated by multiplication by $b_a$.  Convergence of the distances forces\n$Bh_A\\xi\\in\\Z$ for some $A\\ge1$.  Since $h_A$ has no prime factors outside\n$2,3,5$, this gives membership in the right-hand side.  Conversely, every\nfixed product of powers of $2,3,5$ eventually divides $h_a$, so every\nmember of the right-hand side gives integral values for all large $a$.\n\nSeparation from the integers for one multiplier does not prove\nirrationality.  For example, $\\xi=1/7$ satisfies\n$\\operatorname{dist}(h_a\\xi,\\Z)\\ge1/7$ for every $a\\ge1$, because\n$7\\nmid h_a$; with $B=7$ the values $Bh_a\\xi$ are all integers.\nFor $S$, the denominator clearing in\nTheorem~\\ref{long269:res:actual-cancellation} below shows which multipliers\nmatter: the integral alternative for the actual tails must be excluded\nfor every $B\\ge1$ coprime to $30$, not merely for $B=1$.\n\n\\section{Bounding the tails and clearing a rational denominator}\n\\label{long269:sec:actual-orbit}\n\\papersectiontarget{r3-repeated-bounds}\n\nWe sharpen the quadratic estimate by counting points of equal running LCM.\nAs the cutoff increases, one prime exponent increases at each jump, so\nthe sum of the exponents indexes the nonempty cells. At $2^a$ it is\n\\[\n n_a=a+\\lfloor\\log_3(2^a)\\rfloor+\\lfloor\\log_5(2^a)\\rfloor.\n\\]\nCounting the points in each cell and summing their geometric weights\ngives the polynomial\n\\[\n Q(n)=\\frac{n^2+8n+18}{9}.\n\\]\n\n\\begin{theorem}[a quadratic upper bound]\\label{long269:res:actual-tail-bound}\nFor every $a\\ge0$, $0<X_a\\le Q(n_a)$.\n\\end{theorem}\n\n\\begin{proof}\nPartition the smooth integers $x\\ge2^{a}$ by the exponent vector of their running LCM\n\\[\n (A,B,C)=(\\lfloor\\log_2x\\rfloor,\\lfloor\\log_3x\\rfloor,\\lfloor\\log_5x\\rfloor).\n\\]\nThe cell of a vector is the interval $[\\lo,\\hi)$, where\n\\[\n \\lo=\\max(2^{A},3^{B},5^{C}),\\qquad\n \\hi=\\min(2^{A+1},3^{B+1},5^{C+1}).\n\\]\nThus $\\hi\\le2^{A+1}\\le2\\lo$.  By Lemma~\\ref{long269:res:short}, fixing the\nexponents of $3$ and $5$ leaves at most one exponent of $2$.  Since $A\\ge B\\ge C$, the cell\ncontains at most\n\\[\n (B+1)(C+1)\\le\\frac{(A+B+C+3)^{2}}{9}\n\\]\nsmooth points: writing $u=B+1\\ge v=C+1$ and using $A+1\\ge u$, the difference\n$(2u+v)^{2}-9uv=(u-v)(4u-v)$ is nonnegative.  Unique factorisation identifies\nthese exponent triples with distinct smooth integers.\n\nAs the cutoff increases all three prime exponents are nondecreasing, so two\nnonempty cells with the same exponent sum are the same cell, and there is at most one nonempty cell with each exponent sum.  Every cell above\n$2^a$ has exponent sum at least $n_a$, and a cell with sum $n_a+k$ has\nrunning LCM at least $P_a2^k$,\nsince each of the $k$ extra prime factors is at least $2$.  Nonnegative summation over exponent sums, allowing empty cells, gives\n\\[\n X_a\\le\\frac1{18}\\sum_{k\\ge0}\\frac{(n_a+k+3)^{2}}{2^{k}}\n      =\\frac{n_a^{2}+8n_a+18}{9},\n\\]\nthe evaluation using the geometric moments $\\sum2^{-k}=2$, $\\sum k2^{-k}=2$\nand $\\sum k^{2}2^{-k}=6$.  Positivity follows from the block at $2^{a}$.\n\\end{proof}\n\nEach extra prime factor costs at least two in the denominator; each\ncell contains at most quadratically many points. This is a geometric\nsum of elementary projection counts, not an asymptotic smooth-number\nestimate or a Hecke--Mahler argument.\n\nThe quadratic order is also necessary.  The positive next tail and the\nrecurrence give\n\\[\n X_a=\\frac{m_a+X_{a+1}}{b_a}>\\frac{m_a}{b_a}\\ge\\frac{m_a}{30}.\n\\]\nThe lower bound of Lemma~\\ref{long269:res:literal-triangle}, together\nwith $n_a\\le3a$, therefore gives $X_a=\\Theta((a+1)^2)$.  In particular,\nthe normalised tails are not uniformly bounded.  This order estimate\ndoes not assert an asymptotic constant or optimality of $Q$.\n\nThe bound is used at the endpoint of a segment, where the natural index is the\npositive prime-power jump count strictly below the cutoff.  Put\n\\begin{equation}\\label{long269:eq:endpoint-index}\n j_a=\\#\\{p^{e}<2^{a}:p\\in\\{2,3,5\\},\\ e\\ge1\\}=n_a-1\\qquad(a\\ge1),\n\\end{equation}\nthe equality holding for $a\\ge1$ because the powers of $2$ below $2^{a}$ number\n$a-1$ while the powers of $3$ and of $5$ below $2^{a}$ number\n$\\lfloor\\log_3 2^{a}\\rfloor$ and $\\lfloor\\log_5 2^{a}\\rfloor$; at $a=0$ the\ncount is $j_0=0$ and $n_0-1=-1$.  Substituting $n_a=j_a+1$ into $Q$ gives the\ninteger bound used throughout the rest of the note:\n\\begin{equation}\\label{long269:eq:actual-bound}\n K(B,a)=\\bigl\\lfloor B\\,Q(n_a)\\bigr\\rfloor,\n \\qquad\n K(B,a)=\\left\\lfloor\\frac{B(j_a^{2}+10j_a+27)}9\\right\\rfloor\\quad(a\\ge1).\n\\end{equation}\nThe strict cutoff in \\eqref{long269:eq:endpoint-index} is $2^a$.\nThere are two distinct uses of the bound: $BX_a\\le BQ(n_a)$ holds for the\nactual real tails, whereas $BX_a\\le K(B,a)$ follows only when $BX_a$\nis integral. For example, $Q(n_3)=83/9$ gives the integer bound\n$K(1,3)=9$, not an upper bound of $9$ for an arbitrary real quantity\nbelow $83/9$. The short paper instead uses\n$K_{\\rm short}(B,a)=90B(a+1)^2$, valid without integrality. Here $K$\nalways means \\eqref{long269:eq:actual-bound}; either bound may be used\nin the residue contradiction.\n\n\n\n\\begin{lemma}[finite denominator clearing]\\label{long269:res:all-scale-lattice}\nFor all integers $0\\le u\\le b$ the sum over the segment $h_b\\sum_{a=u}^{b-1}s_a$ is a\nnatural number.  If $S=N/D$ with $N\\in\\Z$ and $D\\in\\N_{>0}$, then $DX_a\\in\\Z$\nfor every $a\\ge1$, and there are indices $1\\le i<j\\le D+1$ for which\n$X_i-X_j\\in\\Z$.\n\\end{lemma}\n\n\\begin{proof}\nAn empty segment has mass zero.  Otherwise $b\\ge1$, and every integer $x<2^{b}$\nhas exponent of $2$ at most $b-1$ while its other prime exponents are at\nmost those at $2^{b}$, so $\\hgt(x)\\mid P_b/2=h_b$ and every term of the\nfinite segment clears at $h_b$.  Since $h_a\\sum_{u<a}s_u$ is an integer,\n\\[\n DX_a=h_aN-Dh_a\\sum_{u<a}s_u\\in\\Z.\n\\]\nAmong $D+1$ of these integers two share a residue\nmodulo $D$, and the corresponding tails differ by an integer.\n\\end{proof}\n\nThe strict upper endpoint is what permits division by two.  A cutoff\nincluding $2^b$ would not clear its term at the normalising factor $h_b=P_b/2$.\nThe pigeonhole argument identifies two tails modulo $\\Z$, not two equal\ntails. Since $X_a$ is unbounded, the lattice condition does not force\nthe exact repetition used in the distinct-value proof.\n\n\\begin{theorem}[rationality gives positive integer tails]\n\\label{long269:res:actual-cancellation}\\label{long269:res:lead-carry-bridge}\n\\label{long269:res:actual-carry-bound}\\label{long269:res:denominator-reduction}\nSuppose $S=N/D$ with $N\\in\\Z$, $D\\in\\N_{>0}$, and write\n\\[\n D=2^{u}3^{v}5^{w}B,\\qquad u,v,w\\in\\N,\\quad B\\in\\N_{>0},\\quad\\gcd(B,30)=1,\n \\qquad a_D=u+1+2v+3w .\n\\]\nThen for every $a\\ge a_D$ the number $d_a=BX_a$ is a positive integer and\n\\[\n d_{a+1}=b_ad_a-Bm_a,\\qquad 1\\le d_a\\le K(B,a)\\le90B(a+1)^{2} .\n\\]\n\\end{theorem}\n\n\\begin{proof}\nLet $M=2^{u}3^{v}5^{w}$.  For $a\\ge a_D$ we have $2^{a}\\ge2^{u+1}$,\n$2^{a}\\ge3^{v}$ and $2^{a}\\ge5^{w}$, using $3<2^{2}$ and $5<2^{3}$; hence\n$M\\mid h_a$.  By Lemma~\\ref{long269:res:all-scale-lattice}, $X_a$ differs from $h_aN/D$\nby an integer.  Since $M\\mid h_a$ and $D=MB$, multiplying by $B$ shows\nthat $BX_a$ is an integer.  Positivity and the\nrecurrence come from~\\eqref{long269:eq:shell-digit-identity}, and the upper bound is\nTheorem~\\ref{long269:res:actual-tail-bound} with the floor taken, since $d_a$ is an\ninteger at most $BQ(n_a)$.  Finally $n_a\\le3a$, so\n$Q(n_a)\\le a^{2}+\\tfrac83a+2\\le90(a+1)^{2}$.\n\\end{proof}\n\nThe factorisation restricts no possible denominator: remove its powers\nof $2$, $3$ and $5$, leaving $B$ coprime to $30$. These parts have\ndifferent roles. The removed part controls the onset of integrality;\n$B$ scales the integer tails and their bound. The sufficient $a_D$ need\nnot be the first clearing index. For the exact answer, write\n$\\operatorname{den}(x)$ for the positive reduced denominator of a rational $x$.\n\n\\papersectiontarget{r3-denominator}\n\\begin{proposition}[exact denominators and minimal clearing]\n\\label{long269:res:exact-denominator}\nSuppose $S=N/(MB)$ is in lowest terms, with $M=2^u3^v5^w$ and\n$\\gcd(B,30)=1$.  For every $a\\ge1$,\n\\[\n \\operatorname{den}(X_a)=\\frac{MB}{\\gcd(M,h_a)},\\qquad\n \\operatorname{den}(BX_a)=\\frac{M}{\\gcd(M,h_a)}.\n\\]\nHence the first integral reduced tail occurs at\n\\[\n a_* =\\min\\{a\\ge1:2^a\\ge\\max(2^{u+1},3^v,5^w)\\}\\le a_D,\n\\]\nand $BX_a$ is integral exactly for $a\\ge a_*$.  This onset is computable by\ninteger powers; it is not an estimate obtained by rounding logarithms.\n\\end{proposition}\n\\begin{proof}\nThe finite prefix clears at $h_a$, so Lemma~\\ref{long269:res:all-scale-lattice} gives $X_a-h_aN/(MB)\\in\\mathbb Z$. Since $\\gcd(N,MB)=1$ and every prime factor of $h_a$ is in $\\{2,3,5\\}$, cancellation removes exactly $\\gcd(M,h_a)$ from the denominator. Multiplication by $B$ gives the second formula.\nNow $M\\mid h_a$ means\n$a-1\\ge u$, $\\lfloor\\log_3 2^a\\rfloor\\ge v$ and\n$\\lfloor\\log_5 2^a\\rfloor\\ge w$, precisely the three integer inequalities.\nAll three persist when $a$ increases, giving the first and every later\nintegral reduced tail.  The earlier bounds $3<4$ and $5<8$ give $a_*\\le a_D$.\n\\end{proof}\nFor example, a hypothetical reduced denominator $2^3 3^2 5\\cdot7$ would\nrequire $2^a\\ge\\max(16,9,5)$, so $a_*=4$ rather than the sufficient\n$a_D=11$.  Lowest terms matter for this exact answer: unreduced factors\ncould cancel against the numerator sooner.\n\nThe recurrence preserves integrality forward.  Its homogeneous equation\nalso determines how fast two distinct solutions separate.\n\n\\begin{proposition}[propagation of integrality and uniqueness of a small solution]\\label{long269:res:pinning}\nFor every $a$, $X_a=(m_a+X_{a+1})/b_a>0$, and if $X_a\\in\\Z$ then $X_n\\in\\Z$ for\nevery $n\\ge a$.  Moreover, fix $A$, a positive width function $w$ with\n$w(A+k)/8^{k}\\to0$, and a real sequence $(y_n)_{n\\ge A}$ satisfying\n$y_{n+1}=b_ny_n-m_n$.  If $y_n$ and $X_n$ both lie in\n$(m_n/b_n,\\;m_n/b_n+w(n)]$ for every $n\\ge A$, then $y_A=X_A$.\n\\end{proposition}\n\n\\begin{proof}\nThe identity is the recurrence solved for $X_a$, and positivity holds because\nevery block contains its dyadic left endpoint.  Integer coefficients preserve integrality at every later step.  For the last assertion,\n$y_{A+k}-X_{A+k}=(P_{A+k}/P_A)(y_A-X_A)$.\nThe dyadic running LCM bounds from Section~\\ref{long269:sec:blocks} give\n$P_{A+k}/P_A>8^k/15$, whereas the common interval bounds the absolute\ndifference by $w(A+k)$.  Thus\n$|y_A-X_A|<15w(A+k)/8^k\\to0$.\n\\end{proof}\n\nThe width condition allows every positive polynomial width and widths\n$\\rho^n$ with $1<\\rho<8$, but not $8^n$ itself.  Both solutions must lie in\nthe stated intervals; the recurrence alone does not supply that condition.\nMore generally, fix an integer $B\\ge1$.  Every real solution of\n$y_{a+1}=b_ay_a-Bm_a$ starting at an index $A$ satisfies\n\\[\n y_a=BX_a+\\frac{P_a}{P_A}(y_A-BX_A)\\qquad(a\\ge A).\n\\]\nSubtracting the recurrence for $BX_a$ proves the identity.  Since\n$X_a=O((a+1)^2)$ and $P_a=\\Theta(8^a)$, the unique solution with\n$y_a=o(8^a)$ is $y_a=BX_a$; every other solution has\n$|y_a|=\\Theta(8^a)$.  This classifies growth; integrality of $BX_a$\nremains a separate question.\nThe same formula explains a numerical precaution: a starting error\n$\\varepsilon$ is multiplied by $P_a/P_A$ under forward iteration.\nA decimal approximation propagated forwards is therefore not a\ncertificate of the tail floors.  The digit calculation in\nSection~\\ref{long269:sec:open} uses rational intervals with an explicit\ninfinite-tail bound instead.\n\n\\subsection{A bound using the order of the jumps}\\label{long269:sec:smaller-bound}\nThe bound in Theorem~\\ref{long269:res:actual-tail-bound} uses only the fact\nthat every new prime factor is at least $2$. Successive powers of $2$\ncannot continue indefinitely without a power of $3$ between them.\nIncorporating that restriction improves the geometric majorant.\n\n\\begin{proposition}[a smaller quadratic bound]\n\\label{long269:res:jump-constrained-bound}\nFor every $a\\ge0$,\n\\[\n 0<X_a\\le\\widetilde Q(n_a)<Q(n_a),\\qquad\n \\widetilde Q(n)=\\frac{1210n^2+9130n+18847}{11979}.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor a point $x\\ge2^a$, let $d_p$ be the increase in the exponent of $p$ in its running LCM\nfrom the boundary $2^a$, and put $k=d_2+d_3+d_5$.\nThe definition of $d_3$ gives\n$x<3^{\\lfloor\\log_3 2^a\\rfloor+d_3+1}\\le2^a3^{d_3+1}\n<2^{a+2d_3+2}$, hence $d_2\\le2d_3+1$.\nIt follows that $k\\le3(d_3+d_5)+1$, so at least\n$j=\\lfloor(k+1)/3\\rfloor$ of the $k$ new prime factors are odd.\nEach of these contributes at least $3$, and each remaining factor at\nleast $2$.  Hence\n\\[\n \\frac{\\hgt(x)}{P_a}\\ge2^{k-j}3^j=:d(k).\n\\]\nThere is at most one interval of constancy with each exponent sum, with at most\n$(n_a+k+3)^2/9$ supported points by the earlier projection bound.\nConsequently\n\\[\n X_a\\le\\sum_{k\\ge0}\\frac{(n_a+k+3)^2}{18d(k)}.\n\\]\nNow $d(3m)=12^m$, $d(3m+1)=2\\cdot12^m$ and\n$d(3m+2)=6\\cdot12^m$.  Grouping in threes and summing the quadratic\ngeometric series gives $\\widetilde Q(n_a)$.\nFinally $11979\\bigl(Q(n)-\\widetilde Q(n)\\bigr)\n=121n^2+1518n+5111>0$ for $n\\ge0$.\n\\end{proof}\nThe estimate uses the original block multiplicities and the order of\nthe prime-power jumps.  It is not a bound for an arbitrary recurrence\nwith the same bases.  For an integral $BX_a$, the bound can be\nrounded down to $\\lfloor B\\widetilde Q(n_a)\\rfloor$.  This improves the\nbound available in a rationality contradiction.\nTheorem~\\ref{long269:res:actual-escape-endpoint} and the recorded finite\ntests use $K$, not this smaller bound.  No optimality claim is made for\neither quadratic bound.\n\nSection~\\ref{long269:sec:escape} explains how the smaller bound can also\nreplace $K$ in the residue argument.\n\n\\paragraph{Relation to Cantor-series criteria.}\nSince $h_0=1/2$, the value $S/2=X_0$ is the Cantor series\n$\\sum_{a\\ge0}m_a/P_{a+1}$, with $P_0=1$ and $P_{a+1}=b_aP_a$.\nIts normalised tails $X_a$ are the usual objects in the rationality\ncriteria for Cantor series.  Under a small-numerator hypothesis,\nErd\\H{o}s and Straus characterise rationality by the existence of a\npositive integer $B$ and integers $c_a$ satisfying, eventually,\n\\[\n Bm_a=b_ac_a-c_{a+1},\\qquad |c_{a+1}|<b_a/2\n\\]\n\\cite[Theorem~2.1 and (2.4), pp.~85--86]{erdosstraus1974}.\nThe second condition is part of the criterion, not a consequence of the\nrecurrence alone.  Their proof already chooses nearest integers.\nHan\\v{c}l and Tijdeman make that choice part of the criterion: they\nprescribe $c_a$ as a nearest integer to $Bm_a/b_a$ and require the recurrence\n\\cite[\\S\\S2--3 and Theorem~3.1, pp.~372--375]{hancltijdeman2004}.\nKoutsoukou-Argyraki and Li formalised the Erd\\H{o}s--Straus criteria in\nIsabelle/HOL~\\cite{afperdosstraus2020}.\n\nBoth cited criteria require $m_a/(b_{a-1}b_a)\\to0$ in our notation.\nHere the weighted-triangle count gives $m_a=\\Theta((a+1)^2)$, and\n\\[\n \\frac{m_a}{900}\\le\\frac{m_a}{b_{a-1}b_a}\\le\\frac{m_a}{4}\n \\qquad(a\\ge1).\n\\]\nThe ratio therefore has quadratic order and tends to infinity.\nThere is also a direct lower count that does not use the real-logarithm\ntriangle estimate.  For $a\\ge1$, each pair\n$0\\le j,k\\le\\lfloor a/5\\rfloor$ has $3^j5^k<2^a$, since $15<2^5$,\nand hence a unique power-of-two multiple in $[2^a,2^{a+1})$.\nEach such point contributes at least one to $m_a$.  Thus\n$m_a\\ge(\\lfloor a/5\\rfloor+1)^2$, an elementary lower bound sufficient\nto see directly that the small-numerator condition fails.\n\nThe actual scaled tails also satisfy\n\\[\n BX_a-\\frac{Bm_a}{b_a}=\\frac{BX_{a+1}}{b_a}\n =\\Theta((a+1)^2)\n\\]\nfor each fixed $B\\ge1$, by the recurrence, the quadratic tail bounds and\n$2\\le b_a\\le30$.  Thus any eventual integer tails supplied by a\nhypothetical rational value could not be the prescribed nearest integers.\nIn fact, those prescribed integers fail the recurrence at every\nsufficiently large index, regardless of how ties are resolved.  If $c_a$\nis a nearest integer to $Bm_a/b_a$, then\n\\[\n |b_ac_a-Bm_a|\\le15,\\qquad\n c_{a+1}\\ge\\frac{Bm_{a+1}}{30}-\\frac12\\longrightarrow\\infty.\n\\]\nConsequently $c_{a+1}=b_ac_a-Bm_a$ is impossible for all sufficiently\nlarge $a$.  This is a failure of the proposed construction of integer coefficients, not an irrationality\ncontradiction: the criterion's small-numerator hypothesis also fails.\n\nThe denominator clearing itself is standard; the calculations specific\nto this series retain its multiplicities, use the strict endpoint to\ndivide the normalising factor by two, and bound the resulting positive tails.\n\nFor polynomial Cantor data, Han\\v{c}l--Tijdeman\n\\cite[Theorems~2.2 and 3.1]{hancltijdeman2008} give a polynomial\ncancellation criterion and a division mechanism.  Their nonconstant\npolynomial radix is not our bounded $(b_a)$.  Their separate\nTheorem~4.2 uses a finite-product rearrangement.  We do not use an\nunrestricted infinite reindexing from that argument: for the integer\ndecompositions considered here, the terminal terms must be checked\nseparately.  In the first-order case\n$m_a=b_ac_a-c_{a+1}$, finite summation gives\n\\[\n \\sum_{a=0}^{N-1}\\frac{m_a}{P_{a+1}}\n =c_0-\\frac{c_N}{P_N}.\n\\]\nEvery integer $c_0$ generates an integer solution; integrality alone\nplaces no restriction on the initial value. The terminal condition is the\nrestriction: $c_N/P_N\\to c_0-S/2$, so decay to zero selects the actual\ninitial tail $c_0=S/2$. This is the same uniqueness mechanism as for the\nsmall solutions above, not a consequence of solving the recurrence.\n\nFor these actual numerators the terminal term cannot vanish for\n\\emph{any} integer $c_0$.  Indeed, $P_3=120$, $n_3=5$ and\n$(m_0,m_1,m_2)=(1,4,7)$, so the tail bound gives\n\\[\n 0<\\frac S2\n =\\frac12+\\frac4{12}+\\frac7{120}+\\frac{X_3}{120}\n \\le\\frac{107}{120}+\\frac{83}{1080}\n =\\frac{523}{540}<1.\n\\]\nThus $c_0-S/2\\ne0$ for every integer $c_0$.  Since\n$8^N/15<P_N\\le8^N$, every such unscaled integer solution satisfies\n$|c_N|=\\Theta(8^N)$.  This is a concrete reason that quadratic growth of\n$m_a$ does not imply polynomial growth of recursively defined carries.\nThe starting index is essential: this does not exclude the actual tails\nbecoming integral at a later index.  After replacing $m_a$ by $D m_a$ for an integer $D\\ge1$, however, a\nvanishing terminal term would require $DS/2=c_0$.  Excluding that\npossibility for every $D$ would require irrationality itself; the\nunscaled calculation does not do so.\n\nIn the cited rearrangement the transformed integer numerator must be\n$o(b_a)$, which for bounded radices means eventual zero.  Only after the\nterminal terms have been controlled does this cancellation determine the\ninfinite sum.  A polynomial bound on each individual decomposition\ncoefficient would suffice for a fixed number of shifts: the corresponding\nseries are absolutely convergent, since their denominators are products\nof at least $2$ at each step.  A polynomial bound on the combined\nnumerators is not a substitute, as the exponentially growing integer solutions above demonstrate.  Nor does quadratic growth force an integer solution to be a polynomial in the floor coordinates.\n\n\\section{A residue criterion and the bounds it allows}\\label{long269:sec:escape}\n\\papersectiontarget{r3-residue}\n\nTwo iterations already show how the recurrence determines an endpoint\nresidue:\n\\[\n X_2=6X_1-4,\\qquad X_3=10X_2-7=60X_1-47.\n\\]\nWhenever $BX_1$ is integral, $BX_3\\equiv-47B\\pmod{60}$.\nAn integral starting tail therefore puts the terminal tail in an\narithmetic progression. Positivity and the quadratic bound restrict its\npossible integer values. If the least positive representative exceeds\nthe bound, the starting tail cannot have been integral. For the zero\nclass, that representative is the modulus, not zero. To exclude every\nrational denominator, the test must succeed for every eligible $B$\nbeyond every prescribed clearing onset.\n\nA segment starts at $\\ell\\ge0$ and consists of $h\\ge0$ steps.  Its product\nof bases and accumulated numerator are integers, even when the tails are\nnot.  Set\n\\[\n W_{\\ell,0}=1,\\quad F_{\\ell,0}=0,\\qquad\n W_{\\ell,h+1}=b_{\\ell+h}W_{\\ell,h},\\quad\n F_{\\ell,h+1}=b_{\\ell+h}F_{\\ell,h}+m_{\\ell+h}.\n\\]\nInduction gives\n\\begin{equation}\\label{long269:eq:window-identity}\n X_{\\ell+h}=W_{\\ell,h}X_\\ell-F_{\\ell,h},\\qquad\n d_{\\ell+h}=W_{\\ell,h}d_\\ell-BF_{\\ell,h}\n\\end{equation}\nfor any sequence satisfying $d_{n+1}=b_nd_n-Bm_n$.\nSince $b_a=P_{a+1}/P_a$, the product telescopes to\n$W_{\\ell,h}=P_{\\ell+h}/P_\\ell$.  For the actual block masses,\n$m_a=h_{a+1}s_a$ and $h_a=P_a/2$ identify the two terms after division\nby $W_{\\ell,h}$:\n\\[\n \\frac{F_{\\ell,h}}{W_{\\ell,h}}\n =h_\\ell\\sum_{a=\\ell}^{\\ell+h-1}s_a,\n \\qquad\n \\frac{X_{\\ell+h}}{W_{\\ell,h}}=h_\\ell T_{\\ell+h}.\n\\]\nThus the accumulated numerator records a finite part of the original\nreciprocal sum, in the same normalisation as the starting tail. No\nassumption of rationality or integrality is used in these identities.\nThe previously proved bounds $8^a/15<P_a\\le8^a$ also give\n$W_{\\ell,h}>8^h/15$.\nFor a positive integer $C$ and an integer $N$, use\n$\\lpr_C(N)=1+((N-1)\\bmod C)\\in\\{1,\\ldots,C\\}$.\nIn particular $\\lpr_C(0)=C$.  Positivity of this representative is what\nallows comparison with a positive integer tail.\n\n\\begin{proposition}[least positive residues]\\label{long269:res:consumer}\nLet $C>0$ and let $c$ be an integer with $0<c$ and $|c|\\le K$.  If\n$c\\equiv N\\pmod C$ and $K<\\lpr_C(N)$, then the hypotheses are contradictory.\n\\end{proposition}\n\n\\begin{proof}\nEvery positive integer congruent to $N$ is at least $\\lpr_C(N)$.\nThus $\\lpr_C(N)\\le c=|c|\\le K$, contrary to the strict inequality.\n\\end{proof}\n\n\nFor a bound $G:\\N_{>0}\\times\\N\\to\\N$ let $\\Esc(G)$ be the statement\n\\begin{equation}\\label{long269:eq:actual-escape}\n \\begin{gathered}\n \\text{for every }B\\ge1\\text{ with }\\gcd(B,30)=1\\text{ and every }a_0\\ge1,\\\\\n \\text{there are }\\ell\\ge a_0\\text{ and }h\\ge1\\text{ with }\n \\lpr_{W_{\\ell,h}}(-BF_{\\ell,h})>G(B,\\ell+h) .\n \\end{gathered}\n\\end{equation}\nThe segment may depend on both $B$ and $a_0$.  All quantities in the\ninequality are finite integers.  The quantifiers over denominators and\nprescribed onsets are nevertheless unbounded; a search over a finite\nrectangle of $(B,\\ell)$ does not verify them.  Every $b_a>0$, so\n$W_{\\ell,h}>0$ is automatic.\n\nHere $K$ is the integer bound obtained above from the jump count.\nUnder the hypothesis that $BX_\\ell$ is integral, the recurrence makes\n$BX_{\\ell+h}$ integral as well, so $0<BX_{\\ell+h}\\le K(B,\\ell+h)$.\nEquivalently, the starting integer must lie in\n\\[\n \\left(\\frac{BF_{\\ell,h}}{W_{\\ell,h}},\n       \\frac{BF_{\\ell,h}+K(B,\\ell+h)}{W_{\\ell,h}}\\right].\n\\]\nA residue exceeding $K$ says exactly that this interval contains no\ninteger. The floor in the definition of $K$ is legitimate here because\nintegrality is being assumed. The short paper uses the coarser real-tail\nbound $90B(a+1)^2$ instead. The argument needs a valid bound on the\npossible integer tail and the stated growth condition, rather than a\nparticular numerical choice.\n\nThe two hypotheses on $G$ in the next theorem serve different purposes.\nDomination of $K$ lets a residue above $G$ exclude a positive integer tail.\nThe limit $G(B,a)/8^a\\to0$ lets every sufficiently long segment from a\nfixed nonintegral start overtake $G$.  It is a sufficient growth condition,\nnot a necessary one: if only existence of a segment is required, a\nsubsequence of small endpoint bounds can suffice, as we prove below.\nThe stated theorem already covers every polynomial upper bound that\ndominates $K$, and also $\\max\\{K(B,a),\\lceil B\\rho^a\\rceil\\}$ for\n$1<\\rho<8$.  The zero bound fails to control the possible integer tail.\n\n\\begin{theorem}[a residue criterion for every dominating bound of size $o(8^a)$]\\label{long269:res:actual-escape-endpoint}\n\\label{long269:res:lead-escape-equivalence}\\label{long269:res:windowconsumer}\nLet $G:\\N_{>0}\\times\\N\\to\\N$ satisfy $K(B,a)\\le G(B,a)$ for all $B$ and $a$,\nand $G(B,a)/8^{a}\\to0$ as $a\\to\\infty$ for each fixed $B$.  Then\n\\[\n \\Esc(G)\\quad\\Longleftrightarrow\\quad S\\notin\\Q .\n\\]\nBoth $K$ of~\\eqref{long269:eq:actual-bound} and $K_0(B,a)=90B(a+1)^{2}$ satisfy these hypotheses, so $\\Esc(K)$, $\\Esc(K_0)$ and irrationality of $S$ are mutually\nequivalent.  By contrast, $\\Esc(0)$ holds automatically, since every\nleast positive residue is at least $1$; its truth alone therefore\nprovides no contradiction to an integral tail.\n\\end{theorem}\n\n\\begin{proof}\nSuppose $\\Esc(G)$ and suppose $S=N/D$ were rational.\nTheorem~\\ref{long269:res:actual-cancellation} supplies $B\\ge1$ coprime to $30$, an\nonset $a_D$, and positive integers $d_a=BX_a\\le K(B,a)\\le G(B,a)$ for\n$a\\ge a_D$ satisfying the cleared recurrence.  Apply~\\eqref{long269:eq:actual-escape}\nwith $a_0=a_D$ to obtain a segment $(\\ell,h)$ with $\\ell\\ge a_D$.\nBy~\\eqref{long269:eq:window-identity}, $d_{\\ell+h}\\equiv-BF_{\\ell,h}$ modulo\n$W_{\\ell,h}$, while $0<d_{\\ell+h}\\le G(B,\\ell+h)<\\lpr_{W_{\\ell,h}}(-BF_{\\ell,h})$.\nProposition~\\ref{long269:res:consumer} is the contradiction, so $S$ is irrational.\n\nConversely suppose $S\\notin\\Q$, and fix $B\\ge1$ coprime to $30$ and\n$a_0\\ge1$.  Set $\\ell=a_0$ and\n$\\delta=\\lceil BX_\\ell\\rceil-BX_\\ell\\in(0,1)$; the prefix identity makes\n$BX_\\ell$ irrational.  For all sufficiently large $h$,\n$0<\\delta+BX_{\\ell+h}/W_{\\ell,h}<1$, because $X_{\\ell+h}=O((\\ell+h+1)^2)$ and\n$W_{\\ell,h}>8^h/15$ as shown above.\nThe integer $\\lceil BX_\\ell\\rceil W_{\\ell,h}-BF_{\\ell,h}$ is therefore\nexactly\n\\[\n \\lpr_{W_{\\ell,h}}(-BF_{\\ell,h})\n =\\delta W_{\\ell,h}+BX_{\\ell+h}.\n\\]\nIts first term eventually exceeds $G(B,\\ell+h)$, since\n$G(B,a)=o(8^a)$.  Thus every sufficiently long segment from this fixed\nstart satisfies the residue inequality, which is stronger than the required existence.\n\nFor the two stated bounds, $n_a\\le3a$ gives\n$K(B,a)\\le BQ(3a)\\le90B(a+1)^2=K_0(B,a)$.\nBoth are $O((a+1)^2)$ for fixed $B$, so both satisfy the required limit.\nFinally, $\\lpr_C(N)\\ge1>0$ always, independently of whether any tail is\nintegral; a residue greater than zero cannot exclude a positive integer.\n\\end{proof}\n\nThe domination assumption is sufficient, not asserted to be necessary\nfor equivalence.  In particular, Proposition~\\ref{long269:res:jump-constrained-bound}\nalready supplies the smaller valid bound\n$\\lfloor B\\widetilde Q(n_a)\\rfloor$.  Replacing $K$ by that bound in the\nforward implication leaves the argument unchanged; the converse uses only\n$G(B,a)=o(8^a)$.  The zero-bound example explains why a size comparison\nwith the possible integer tail is needed in the contradiction, not why\nthis particular $K$ is indispensable.\n\nAt a fixed start, the same identity distinguishes two behaviours. If\n$BX_\\ell$ is nonintegral, the least positive residue eventually occupies\na fixed positive fraction of the modulus. If $BX_\\ell$ is integral, the\nresidue eventually equals the normalised tail $BX_{\\ell+h}$ itself.\nThe next proposition gives both conclusions without assuming that $S$\nis irrational.\n\n\\begin{proposition}[the fixed-start residue limit]\n\\label{long269:res:residue-limit}\nFor fixed integers $B,\\ell\\ge1$, write\n\\[\n R_h=\\lpr_{W_{\\ell,h}}(-BF_{\\ell,h}).\n\\]\nFor all sufficiently large $h$,\n\\[\n R_h=(\\lceil BX_\\ell\\rceil-BX_\\ell)W_{\\ell,h}+BX_{\\ell+h},\n\\]\nand consequently\n\\[\n \\lim_{h\\to\\infty}\\frac{R_h}{W_{\\ell,h}}\n =\\lceil BX_\\ell\\rceil-BX_\\ell.\n\\]\nIn particular, if $BX_\\ell$ is integral, then $R_h=BX_{\\ell+h}$\nfor all sufficiently large $h$.\n\\end{proposition}\n\\begin{proof}\nThe least positive residue is the modulus times the gap to the next\nstrictly larger integer.  Division with remainder gives\n\\[\n \\frac{R_h}{W_{\\ell,h}}\n =\\left\\lfloor\\frac{BF_{\\ell,h}}{W_{\\ell,h}}\\right\\rfloor+1\n       -\\frac{BF_{\\ell,h}}{W_{\\ell,h}}.\n\\]\nBy the iterated recurrence,\n$BF_{\\ell,h}/W_{\\ell,h}=BX_\\ell-BX_{\\ell+h}/W_{\\ell,h}$.\nThese finite sums approach $BX_\\ell$ strictly from below, since\n$BX_{\\ell+h}>0$ and $BX_{\\ell+h}/W_{\\ell,h}\\to0$.\nTheir floors therefore eventually equal $\\lceil BX_\\ell\\rceil-1$,\nincluding when $BX_\\ell$ is an integer.  Substitution proves both claims.\nThe strict approach from below matters at an integer limit; the residue\nconvention alone would assign a zero congruence class the value\n$W_{\\ell,h}$, not zero.\n\\end{proof}\nThus a nonintegral $BX_\\ell$, even for a rational $S$, gives arbitrarily\nlong successful segments against any bound $G(B,a)=o(8^a)$ for that fixed\n$B$.  The rationality contradiction instead uses the denominator's\nspecific multiplier and a start beyond its clearing onset.  Success for\none pair $(B,\\ell)$ does not replace either quantifier.\n\n\\paragraph{The exact growth restriction on a dominating bound.}\nA bound need not be small at every late endpoint.  For example, set\n\\[\n G(B,a)=\n \\begin{cases}\n K(B,a),&a\\text{ even},\\\\\n K(B,a)+8^a,&a\\text{ odd}.\n \\end{cases}\n\\]\nAt an odd endpoint $a=\\ell+h$, no segment with $\\ell\\ge1$ can pass:\nits residue is at most $W_{\\ell,h}=P_a/P_\\ell\\le8^a/2<G(B,a)$.\nAt even endpoints the test is unchanged.  If $S$ is irrational, the\nproof above therefore supplies successful segments ending at every\nsufficiently large even index, from each fixed start.  Thus\n$G(B,a)/8^a\\to0$ is not necessary for the equivalence.\n\nMore precisely, for every integer-valued $G\\ge K$,\n\\[\n \\Esc(G)\\quad\\Longleftrightarrow\\quad\n \\left\\{\\begin{gathered}\n S\\notin\\Q,\\\\\n \\liminf_{a\\to\\infty}\\frac{G(B,a)}{8^a}=0\n \\quad\\text{for every }B\\ge1\\text{ with }\\gcd(B,30)=1.\n \\end{gathered}\\right.\n\\]\nSuppose first that $\\Esc(G)$\nholds.  The earlier rationality contradiction uses only $G\\ge K$.\nFor a successful segment starting at $\\ell\\ge a_0$, put $a=\\ell+h$.\nSince a least positive residue is at most its modulus,\n\\[\n 0\\le\\frac{G(B,a)}{8^a}\n <\\frac{P_a}{P_\\ell8^a}\\le\\frac1{P_\\ell}\n \\le\\frac1{P_{a_0}}.\n\\]\nSuch endpoints satisfy $a>a_0$, and $P_{a_0}\\to\\infty$; hence the\nlower limit is zero.  Conversely, suppose $S$ is irrational and these\nlower limits vanish.  Fix $B$ and $\\ell\\ge1$.  By the fixed-start\nresidue limit, with $\\delta=\\lceil BX_\\ell\\rceil-BX_\\ell>0$,\n\\[\n \\frac{\\lpr_{W_{\\ell,h}}(-BF_{\\ell,h})}{P_{\\ell+h}}\n \\longrightarrow\\frac{\\delta}{P_\\ell}>0.\n\\]\nThe bounds $8^a/15<P_a\\le8^a$ imply\n$\\liminf_a G(B,a)/P_a=0$.  Along a sufficiently late subsequence of\nendpoints the residue therefore exceeds $G$.  This gives arbitrarily\nlong successful segments from each fixed start, but need not give\nsuccess at every sufficiently large length, as the parity example shows.\n\nIn particular, $G(B,a)=K(B,a)+8^a$ allows no segment at all.  More\ngenerally, an eventual lower bound $G(B,a)\\ge c8^a$, for one eligible\n$B$ and some $c>0$, excludes every sufficiently late start: choose\n$P_\\ell\\ge1/c$ and compare the bound with $W_{\\ell,h}\\le8^{\\ell+h}/P_\\ell$.\nThese are restrictions on the test bound, not additional assumptions\nabout the series.  The original quadratic bounds satisfy the lower-limit\ncondition, so the remaining problem is still the existence of the\nsegments.  Equivalence alone says nothing about the relative difficulty\nof the formulations; a useful arithmetic proof must exploit further\ninformation about the numerators $m_a$.\n\nA fixed segment can have a lower bound greater than $1$.  Write $W$ and\n$F$ for its product and accumulated numerator.  Every prime factor of\n$W$ lies in $\\{2,3,5\\}$, so an eligible multiplier $B$ is a unit modulo\n$W$.  Congruence then gives\n\\[\n \\gcd\\bigl(\\lpr_W(-BF),W\\bigr)=\\gcd(F,W).\n\\]\nFor $(W,F)=(6,4)$, the possible residues are $2$ and $4$, attained at\n$B=1$ and $11$; thus $2$ is a lower bound for this segment, and the residues\nare not units.  For $(W,F)=(60,47)$, however,\n$\\lpr_{60}(-37\\cdot47)=1$, with $37$ coprime to $30$.\nThus no lower bound greater than $1$ works for all segments and eligible\nmultipliers, although a particular segment may have one.\n\n\\subsection{How fast the product of the bases grows}\n\nThe growth scale in the preceding criterion comes from the running\nLCM formula. The elementary estimate $W_{\\ell,h}\\ge2^h$ would suffice\nfor bounds $G(B,a)=o(2^a)$ with $B$ fixed; the exact formula gives\n$W_{\\ell,h}\\asymp8^h$ uniformly in $\\ell$ and allows $G(B,a)=o(8^a)$.\nFor an irrational $S$, the preceding lower-limit condition also covers\ndominating bounds outside these two classes. The same growth estimate\nwill give a necessary lower bound on successful segment lengths.\n\n\\begin{proposition}[growth of the product of consecutive bases]\\label{long269:res:window-growth}\nPut $\\theta_3=\\log_32$ and $\\theta_5=\\log_52$.  For all $\\ell\\ge0$ and\n$h\\ge1$,\n\\[\n W_{\\ell,h}=2^{h}\\,\n 3^{\\lfloor(\\ell+h)\\theta_3\\rfloor-\\lfloor\\ell\\theta_3\\rfloor}\\,\n 5^{\\lfloor(\\ell+h)\\theta_5\\rfloor-\\lfloor\\ell\\theta_5\\rfloor},\n \\qquad\n \\frac{8^{h}}{15}<W_{\\ell,h}<15\\cdot8^{h} .\n\\]\n\\end{proposition}\n\n\\begin{proof}\nTelescoping \\eqref{long269:eq:dyadic-alphabet} gives\n$W_{\\ell,h}=P_{\\ell+h}/P_\\ell$, and the displayed formula is that\nquotient written out.  Each floor difference differs from $h\\theta_p$ by less\nthan one, and $3^{\\theta_3}=5^{\\theta_5}=2$, so the $3$-factor lies strictly\nbetween $2^{h}/3$ and $3\\cdot2^{h}$ and the $5$-factor strictly between\n$2^{h}/5$ and $5\\cdot2^{h}$.  Multiplying the three ranges gives the bounds.\n\\end{proof}\n\n\\begin{corollary}[a fixed maximum length cannot cover arbitrarily late starts]\\label{long269:res:no-bounded-length}\nFix $B\\ge1$ coprime to $30$ and $H\\ge1$.  Only finitely many starts $\\ell$\nadmit a segment of length at most $H$ satisfying the residue inequality against $K$.\n\\end{corollary}\n\n\\begin{proof}\nA least positive residue never exceeds its modulus, so the residue inequality at $(\\ell,h)$\nrequires $K(B,\\ell+h)<W_{\\ell,h}<15\\cdot8^{h}\\le15\\cdot8^{H}$.  On the other\nhand $j_a\\ge a-1$, since the powers $2,\\ldots,2^{a-1}$ already lie below\n$2^{a}$, so $K(B,\\ell+h)\\ge\\lfloor B((\\ell-1)^{2}+10(\\ell-1)+27)/9\\rfloor$,\nwhich tends to infinity with $\\ell$.\n\\end{proof}\n\n\\noindent Corollary~\\ref{long269:res:no-bounded-length} is the exact reason a finite\nscan cannot approach the cofinal quantifier by widening its denominator range\nalone.  For $\\ell\\ge1$, the inequality\n$K(B,\\ell+h)>B\\ell^2/9$ shows that any segment satisfying the residue inequality must also satisfy\n\\[\n 3h>\\log_2K(B,\\ell+h)-\\log_215\n    >\\log_2B+2\\log_2\\ell-\\log_2135.\n\\]\nThis is only a necessary lower bound, not an asymptotic formula or an\nupper bound for the first successful length.\nAn explicit threshold beyond which the residue inequality holds can be obtained from a positive\nlower bound for $\\delta=\\lceil BX_\\ell\\rceil-BX_\\ell$ and for $1-\\delta$,\ntogether with the displayed tail and product estimates.  No such uniform information about the tails of $S$ is proved here.\n\n\\section{Finite computations and their limits}\\label{long269:sec:evidence}\n\nWe first calculate individual residue certificates, then enclose the\nsum using twelve blocks to exclude small rational denominators. The\nlarger scan and two larger exclusions are separate saved computations;\ntheir programs and outputs are in the\n\\href{https://github.com/wcook04/plectis-erdos/tree/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos269/large-computations}{large-computation directory}.\nEach certificate excludes its stated finite family. None establishes\nthe criterion for every denominator beyond every starting threshold.\n\n\\subsection{Checking individual segments}\n\nWe calculate each row using the integer-only\n\\href{\\sourceurl/scripts/check_erdos269_dyadic_windows.py}{integer-arithmetic checker}.\nFor block $a$, enumerate the pairs with $3^j5^k<2^{a+1}$, choose the unique\n$i\\ge0$ for which $2^i3^j5^k\\in[2^a,2^{a+1})$, and sum\n$P_{a+1}/(2\\hgt(2^i3^j5^k))$. The recursions for $W$ and $F$ then give\nthe residue in each row. Powers and comparisons are integral throughout.\nThe columns give the denominator $B$, the start\n$\\ell$, the length $h$, the endpoint jump index $j_{\\ell+h}$, the product\n$W$ of the bases, the accumulated numerator $F$, the residue\n$R=\\lpr_W(-BF)$ and the bound $K$ of~\\eqref{long269:eq:actual-bound}.\n\\[\n\\begin{array}{c|c|c|c|r|r|r|r}\nB&\\ell&h&j_{\\ell+h}&W&F&R&K\\\\ \\hline\n1&1&2&4&60&47&13&9\\\\\n7&1&3&6&360&289&137&95\\\\\n16&1&4&9&10800&8735&640&352\n\\end{array}\n\\]\nThe first row reads as follows.  The segment starts at $\\ell=1$ and has length\n$2$, so $W=b_1b_2=6\\cdot10=60$; the accumulated numerator is $F=47$; and\n$\\lpr_{60}(-47)=13$, since $-47+60=13$, which exceeds\n$K(1,3)=\\lfloor(16+40+27)/9\\rfloor=9$.  The third row lies outside the domain\nof~\\eqref{long269:eq:actual-escape}, since $\\gcd(16,30)=2$, and is displayed to\nillustrate the finite calculation at greater depth.\n\nThe larger scan covers $B\\le5000$ coprime to $30$ and\n$100\\le\\ell\\le3000$: all $3{,}869{,}934$ pairs satisfy the residue inequality, with first successful length\nat most $18$ in a search to length $24$\n(\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos269/large-computations/receipts/scan-5000-100-3000.json}{saved run}).\nIts histogram in Section~\\ref{long269:long:experiments} is a finite\nobservation that the product-growth bound does not predict.\nThe accompanying integer-only check reconstructs the blocks\nand tests all $2496$ pairs with $1\\le B\\le97$, $\\gcd(B,30)=1$ and\n$1\\le\\ell\\le96$.  Every pair satisfies the residue inequality; the first successful lengths range\nfrom $1$ to $10$, with a search limit of $24$.  This check uses the\nenumeration and finite recursions above, without floating-point\nlogarithms.  Neither computation proves the residue inequality for unbounded $B$ or\narbitrarily late starts; Corollary~\\ref{long269:res:no-bounded-length}\nrules out covering the latter quantifier with any fixed maximum length.\n\n\\subsection{Finite denominator bounds}\n\n\\paragraph{A denominator bound from twelve blocks.}\nWe obtain a denominator bound by enclosing $X_1=S-1$ between two rational\nnumbers. Here $h_1=1$, and summation through block $11$, followed by the\nbound $Q(n_{12})=Q(24)=262/3$ with $P_{12}=27993600000$, gives\n\\[\n X_1=2\\sum_{a=1}^{11}\\frac{m_a}{P_{a+1}}\n          +\\frac{2X_{12}}{P_{12}},\n\\]\n\\[\n \\frac{11327881097}{13996800000}<X_1\n \\le\\frac{33983643553}{41990400000}.\n\\]\nThe enclosure is strictly inside the interval\n$(11498/14207,\\,6409/7919)$, as integer cross-multiplication verifies.\nThe endpoint determinant is $14207\\cdot6409-11498\\cdot7919=1$.\nFor any rational $u/v$ strictly between them, with $v>0$, both integers\n$14207u-11498v$ and $6409v-7919u$ are positive.  Consequently\n\\[\n v=7919(14207u-11498v)+14207(6409v-7919u)\\ge22126.\n\\]\nThus, if $S$ is rational, its reduced denominator is at least $22126$.\nThis is the elementary denominator bound between two fractions with\ncross-determinant one, often expressed using Farey neighbours.  It is\nalso the best lower bound on a possible denominator obtainable from\nthis enclosure alone: the mediant\n\\[\n \\frac{11498+6409}{14207+7919}=\\frac{17907}{22126}\n\\]\nis in lowest terms and lies strictly inside the certified enclosure.\nThus the interval alone remains compatible with denominator $22126$;\nwe have neither identified $X_1$ with this mediant nor determined a\ndenominator for $S$.\n\nThe certificate uses the actual block coefficients and proved tail\nbound.  A narrower certified enclosure excludes further denominators;\nthe continued-fraction exclusion below uses one built from $15{,}500$\nblocks.\n\n\\paragraph{A denominator bound from $128$ consecutive blocks.}\n\\label{long269:res:lead-block-exclusion}\nA finite lattice computation, outside the theorem inputs, excludes the\nfollowing denominators.  Using segments of length $128$ and $64$ starting\nindices, the first $10005$, no rational value\nof the $\\{2,3,5\\}$ running-LCM series has reduced denominator $MB$ with $M$ a\ndivisor supported on $\\{2,3,5\\}$ of $2^{10005}3^{6312}5^{4308}$, $\\gcd(B,30)=1$ and\n$1<B\\le B_{\\max}$, where $B_{\\max}$ is the $106$-digit integer\n\\[\n\\begin{aligned}\n B_{\\max}={}&1134599670999687767349520845707093359257353022286558739363600235\\\\\n &016103207564063373270305324172145281971729 ,\n\\end{aligned}\n\\]\nso that $\\log_2B_{\\max}=348.9846\\ldots$.\n\nThe exponent triple $(10005,6312,4308)$ is that of $\\hgt(2^{10005})$.\nThe normalisation therefore uses the full running LCM at its first\nstart, whereas Lemma~\\ref{long269:res:all-scale-lattice} uses the half\nrunning LCM $h_{10005}$.  This family of smooth denominators is the\nlarger one.\n\nThe\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos269/large-computations/regenerated/block_certificate.json}{certificate}\nand its exact\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos269/large-computations/verify_recovered.py}{checker}\nare published.  Its starts $a_1<\\dots<a_{64}$ are the first $a\\ge10005$\nwith $W_{a,128}=W_{10005,128}=W=2^{128}3^{81}5^{56}$, so\n$\\log_2W=386.40993\\ldots$; put $m_0=W/2$, $F_j=F_{a_j,128}$ and\n$w_j=(a_j+128)^2+6(a_j+128)+11$.  The certificate lists an integer\nmatrix $C=(c_{ij})$ with $\\sum_jc_{ij}F_j\\equiv0\\pmod{m_0}$ and\n$\\sum_j|c_{ij}|w_j\\le184{,}560{,}461{,}211$ for every row $i$, an\ninteger matrix $E$ with $CE=EC=m_0I$, and\n$B_{\\max}=\\lfloor(W-1)/184{,}560{,}461{,}211\\rfloor$.  If\n$B\\le B_{\\max}$ and integers $f_j=Ws_j-2BF_j$ satisfy $0<f_j\\le Bw_j$,\neach $\\sum_jc_{ij}f_j$ is a multiple of $W$ of absolute value less than\n$W$, hence zero, and invertibility of $C$ forces every $f_j$ to vanish,\na contradiction.  The weights bound the tails: since $n_b\\le2.07b$,\nTheorem~\\ref{long269:res:actual-tail-bound} gives\n$2X_b\\le2Q(n_b)\\le b^2+6b+11$ for every $b\\ge0$.  The reduction from a\nrational value $N/(MB)$ to the $64$ residue inequalities, with\n$s_j=2BX_{a_j}$ and $f_j=2BX_{a_j+128}$, uses the iterated recurrence at full\nrunning LCM and is not machine-checked.\n\n\\paragraph{Continued-fraction exclusion.}\n\\label{long269:res:cf-exclusion}\nA finite computation, outside the theorem inputs, shows that neither\n$X_1$ nor $S$ is a fraction with reduced denominator at most\n$2^{22482}$, about $10^{6768}$.  Let $W$ and $F$ be the product of the bases and\naccumulated numerator of blocks $1$ to $15{,}500$.  The tail bound used\nfor the twelve-block calculation, at $Q(n_{15501})=Q(31956)=340480534/3$,\ngives $F/W<X_1\\le(F+Q)/W$.  Every block is built with integers, and\nblocks $1$ to $30$ are also checked against a direct enumeration of the\nsmooth numbers.  The two endpoints share their first $13{,}540$ partial\nquotients, and the numbers with a given prefix of partial quotients form\nan interval \\cite[Lemma~10.1.2, p.~106]{bosmaCF}.  The first $13{,}109$\ndigits, $a_0=0$ included, define a cylinder whose endpoints have\ncross-determinant $1$ and strictly contain the enclosure, so every\nfraction in the enclosure has denominator at least the sum of the two\nendpoint denominators, a $22{,}483$-bit integer.  The\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos269/large-computations/regenerated/tail_certificate.json}{enclosure},\nthe\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos269/large-computations/regenerated/cf_common_prefix.json}{common digits},\nthe\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos269/large-computations/regenerate_269.py}{regeneration program}\nand the exact\n\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos269/large-computations/verify_recovered.py}{checker}\nare published.  A separate continued-fraction probe\n(\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos269/large-computations/receipts/cf-statistics.json}{saved run})\ncertifies $13{,}109$ partial quotients after $a_0$ and records\na largest denominator of $22{,}483$ bits, a largest partial quotient of\n$129{,}114$, a mean partial quotient of $23.4133$, observed Gauss--Kuzmin\nfrequencies $0.4208$, $0.1665$, $0.0917$, $0.0575$, $0.0391$ against the\npredicted $0.4150$, $0.1699$, $0.0931$, $0.0589$, $0.0406$, and a L\\'evy\nconstant of $1.18869$ against $\\pi^{2}/(12\\log2)=1.18657$.  The predicted\nvalues are the almost-everywhere frequencies of Gauss's law and L\\'evy's\nalmost-everywhere constant \\cite[pp.~288--289 and footnote~5]{levy1936}.\nThese statistics describe a finite prefix and do not bear on whether $X_1$ is a\nLiouville number, algebraic, or rational with a larger denominator.\n\nThe two exclusions concern different finite families: one uses a\nlattice at a fixed starting index and fixed smooth part, the other a continued-fraction\nenclosure at $a=1$.  Both have published witnesses and exact checkers.\nEach excludes only its stated denominator range, and neither settles an\ninstance of the problem.\n\n\\section{The remaining arithmetic questions}\\label{long269:sec:open}\n\\papersectiontarget{r3-open}\n\nThe finite recursions are explicit; the missing step is to prove that\nsuccessful segments exist with the required unbounded quantifiers.\nWeighted shifts preserve the value of the series, but preservation does\nnot supply cancellation. We calculate the surviving boundary terms,\nthen identify the extra hypotheses needed by integer-recurrence and\nfunction-theoretic approaches. The irrationality of $S$ remains open here.\n\n\\subsection{An exact weighted-shift identity for the repeated series}\n\\label{long269:sec:weighted-shifts}\nSet $\\alpha=S/2$ and retain the boundary running LCMs $P_a$. To combine\nshifted tails, we first express them with the same denominators. After\nmultiplication by $P_t$, the term with index $a+t$ becomes\n\\[\n \\frac{P_t m_{a+t}}{P_{a+t+1}}\n =\\frac{P_tP_{a+1}}{P_{a+t+1}}\\,\n   \\frac{m_{a+t}}{P_{a+1}}.\n\\]\nThe first factor on the right is the weight $\\gamma_{a,t}$ below. It is\nforced by the change of denominator, rather than chosen to produce\ncancellation. The removed finite prefix supplies the rational correction.\nThe identity concerns the repeated sum $S$ only. In this subsection and\nthe next, $r$ is a positive integer shift; the prime-generator notation is\nno longer in use.\n\n\\begin{lemma}[weighted shifts preserve the actual value]\n\\label{long269:res:weighted-shift-identity}\nFix integers $c_0,\\ldots,c_\\sigma$, not all zero, independently of the\npositive integer shift $r$.  Put\n\\[\n \\gamma_{a,t}=\\frac{P_tP_{a+1}}{P_{a+t+1}},\\qquad\n D_{r,a}=15\\sum_{j=0}^{\\sigma}c_j\\gamma_{a,jr}m_{a+jr}.\n\\]\nThen $\\gamma_{a,t}\\in\\{1,1/3,1/5,1/15\\}$ and $D_{r,a}\\in\\Z$.\nFurthermore, with\n\\[\n A_r=15\\sum_{j=0}^{\\sigma}c_jP_{jr},\\qquad\n Z_r=15\\sum_{j=0}^{\\sigma}c_jP_{jr}\n       \\sum_{k<jr}\\frac{m_k}{P_{k+1}}\\in\\Z,\n\\]\nwe have the absolutely convergent identity\n\\begin{equation}\\label{long269:eq:weighted-shift-identity}\n A_r\\alpha-Z_r=\\sum_{a\\ge0}\\frac{D_{r,a}}{P_{a+1}}.\n\\end{equation}\nOne may take\n$C=225\\sum_{j=0}^{\\sigma}|c_j|\\max(1,j)^2$ in\n$|D_{r,a}|\\le C(a+r+1)^2$.  If $J$ is the largest index with $c_J\\ne0$,\nthen $A_r\\ne0$ whenever\n$\\sum_{j<J}|c_j|2^{-(J-j)r}<|c_J|$; an empty sum is zero.\n\\end{lemma}\n\\begin{proof}\nThe exponent of $2$ in $\\gamma_{a,t}$ is $t+(a+1)-(a+t+1)=0$.\nFor $p=3,5$, the exponent is\n$\\lfloor t\\theta_p\\rfloor+\\lfloor(a+1)\\theta_p\\rfloor\n-\\lfloor(a+t+1)\\theta_p\\rfloor$, which is $0$ or $-1$ by floor addition.\nThus the only possible denominator factors are one $3$ and one $5$;\nthe prefactor $15$ in $D_{r,a}$ clears both.\nFor $k<jr$, $P_{k+1}$ divides $P_{jr}$, so $Z_r$ is an integer.\nFor each $t$, absolute convergence gives\n\\[\n \\sum_{a\\ge0}\\frac{\\gamma_{a,t}m_{a+t}}{P_{a+1}}\n =P_t\\sum_{k\\ge t}\\frac{m_k}{P_{k+1}}\n =P_t\\left(\\alpha-\\sum_{k<t}\\frac{m_k}{P_{k+1}}\\right).\n\\]\nThe finite linear combination proves the identity.\nLemma~\\ref{long269:res:literal-triangle}, $0<\\gamma\\le1$ and\n$a+jr+1\\le\\max(1,j)(a+r+1)$ give the stated constant.\nFor $j<J$, $P_{jr}/P_{Jr}\\le2^{-(J-j)r}$.  Dividing $A_r$ by\n$15P_{Jr}$ and bounding the remaining terms proves the nonvanishing\ncondition, which holds for all sufficiently large $r$.\n\\end{proof}\nFor any fixed nonzero coefficient vector, including $(1,-3,3,-1)$,\nthe nonvanishing condition is automatic for sufficiently large $r$:\neach term on its left tends to zero.  No arithmetic assumption on $S$ is\nneeded.  The identity still represents $S/2$, but gives no sparsity or\ncancellation of $D_{r,a}$.\n\n\\subsection{Boundary terms in weighted finite differences}\n\\label{long269:sec:boundary-difference}\nFollow one lattice point through four successive triangles. If it lies\nin all four and its weight is unchanged, the coefficients $(1,-3,3,-1)$\nsum to zero. If it first enters the second triangle, its coefficient is\n$-3+3-1=-1$; entry in the third gives $3-1=2$, and entry in the fourth\ngives $-1$. Thus a finite difference depends both on which triangles\ncontain the point and on changes of its weight. We separate these two\neffects below. Avoiding floor crossings controls the weights, but does\nnot remove the newly added strips.\n\nRecall that\n$w_{j,k}=j\\log_2 3+k\\log_2 5$, $t_{j,k}=\\{w_{j,k}\\}$ and\n$\\theta_p=1/\\log_2 p$ for $p=3,5$.\nTo distinguish the shared points from the new ones, let\n$\\mathcal T_a=\\{(j,k)\\in\\mathbb N^2:w_{j,k}<a+1\\}$ and, for $0\\le t<1$, set\n\\[\n \\omega_a(t)=\\prod_{p\\in\\{3,5\\}}\n p^{\\lfloor(a+1)\\theta_p\\rfloor-\\lfloor(a+t)\\theta_p\\rfloor}.\n\\]\nThis weight is defined even for pairs outside $\\mathcal T_a$.  For $\\nu\\ge0$, put\n\\[\n \\begin{split}\n \\kappa_p(a,t,\\nu r)\n &=\\lfloor(a+t+\\nu r)\\theta_p\\rfloor\n   -\\lfloor(a+t)\\theta_p\\rfloor-\\lfloor\\nu r\\theta_p\\rfloor,\\\\\n \\chi_\\nu(a,t)&=3^{-\\kappa_3(a,t,\\nu r)}5^{-\\kappa_5(a,t,\\nu r)}.\n \\end{split}\n\\]\nEach $\\kappa_p$ is $0$ or $1$.  A value $\\kappa_p=1$ records a floor-addition carry.  This is different\nfrom a new lattice point entering $\\mathcal T_a$ as $a$ increases.\n\n\\begin{proposition}[an exact interior-and-strip decomposition]\n\\label{long269:res:strip-decomposition}\nFor a fixed operator $c_0,\\ldots,c_\\sigma$ and $r\\ge1$, let\n$E_0=\\mathcal T_a$ and $E_s=\\mathcal T_{a+sr}\\smallsetminus \\mathcal T_{a+(s-1)r}$ for $1\\le s\\le\\sigma$.\nThen\n\\[\n \\frac{D_{r,a}}{15}=\n \\sum_{s=0}^{\\sigma}\\ \\sum_{(j,k)\\in E_s}\n \\omega_a(t_{j,k})\\sum_{\\nu=s}^{\\sigma}c_\\nu\\chi_\\nu(a,t_{j,k}).\n\\]\nFor the cubic operator $(1,-3,3,-1)$, if all these crossing bits vanish,\nthen\n\\[\n \\frac{D_{r,a}}{15}=-M_0+2M_1-M_2,\\qquad\n M_s=\\sum_{(j,k)\\in \\mathcal T_{a+(s+1)r}\\smallsetminus \\mathcal T_{a+sr}}\\omega_a(t_{j,k}).\n\\]\nThus absence of floor crossings cancels the common interior, but not\nnecessarily the three boundary strips.\n\\end{proposition}\n\\begin{proof}\nFor each $p=3,5$, the exponent of $p$ on either side of\n\\[\n \\gamma_{a,\\nu r}\\omega_{a+\\nu r}(t)\n   =\\omega_a(t)\\chi_\\nu(a,t)\n\\]\nis\n\\[\n \\lfloor(a+1)\\theta_p\\rfloor+\\lfloor\\nu r\\theta_p\\rfloor\n -\\lfloor(a+t+\\nu r)\\theta_p\\rfloor.\n\\]\nOn the left the upper-boundary exponents cancel; on the right the\n$\\lfloor(a+t)\\theta_p\\rfloor$ terms cancel.  This proves the pointwise\nidentity.\nInsert the triangle formula for each $m_{a+\\nu r}$ and group by the first\ntriangle containing a pair.  A pair in $E_s$ occurs exactly in the terms\n$\\nu\\ge s$, proving the first formula.  Under the no-crossing hypothesis,\n$\\chi_\\nu=1$.  The full cubic sum is zero, while its three successive suffix\nsums are $-1,2,-1$, proving the second formula.\n\\end{proof}\n\n\\begin{example}[cubic cancellation can fail with no floor crossings]\n\\label{long269:res:no-crossing-counterexample}\nTake $a=0$, $r=35$ and $(c_0,c_1,c_2,c_3)=(1,-3,3,-1)$.  Exact integer\ncomparisons give\n\\[\n\\begin{array}{c|rrrr}\n u&0&35&70&105\\\\\\hline\n \\lfloor\\log_3 2^u\\rfloor=\\lfloor\\log_3 2^{u+1}\\rfloor&0&22&44&66\\\\\n \\lfloor\\log_5 2^u\\rfloor=\\lfloor\\log_5 2^{u+1}\\rfloor&0&15&30&45\\\\\n m_u&1&195&723&1582\n\\end{array}\n\\]\nThe first two rows imply $b_u=2$ and\n$\\lfloor(u+t)\\theta_p\\rfloor=\\lfloor u\\theta_p\\rfloor$ for every\n$0\\le t<1$, $p=3,5$.  Hence every relevant $\\kappa_p(0,t,u)$ vanishes,\n$\\gamma_{0,u}=1$, and all triangle weights at these four indices are $1$.\nThe last row is therefore the count of pairs satisfying\n$3^j5^k<2^{u+1}$, not a floating-point estimate.  The strips have sizes\n$194,528,859$, so\n\\[\n D_{35,0}=15(-194+2\\cdot528-859)=45\\ne0.\n\\]\nExact enumeration reproduces all four counts and\nchecks the endpoint power inequalities using integers.  This example disproves the claim that avoiding floor crossings alone\nforces the third difference to vanish.  It does not\nexclude a different fixed operator, different shifts, a boundary correction\nor a sparse-defect statement restricted to a suitable infinite subsequence.\n\\end{example}\n\n\n\\paragraph{What the Hecke--Mahler comparison does and does not supply.}\nIn the published Luca--Ouaknine--Worrell article\n\\cite[Definition~5 and Theorem~6]{lucaouaknineworrell2025}, the same fixed\ninteger coefficients are applied to shifts of one integer sequence.\nThe resulting nonzero terms must have expanding gaps and uniform\npolynomial variation; the original sequence must also have polynomial\ngrowth.  Theorem~6 then gives a fixed-base value criterion.  Theorem~8 verifies that\ncondition for $f(\\lfloor m\\vartheta+\\rho\\rfloor)$ when\n$f\\in\\mathbb Z[x]$ is nonconstant, $\\vartheta,\\rho\\in(0,1)$ and\n$\\vartheta$ is irrational.  These interval restrictions belong to that\nnormalised combinatorial statement.  Their main value theorem,\nTheorem~1, allows every real $\\rho$ and every irrational real $\\vartheta$,\nwith any algebraic base $\\beta$ satisfying $|\\beta|>1$.\nClaim~10 uses a finite difference of order $\\deg f+1$, which vanishes\nwhen no floor crossing occurs.  This is a polynomial evaluated at a\nfloor, not merely an integer sequence of polynomial growth.\nOur $m_a$ is instead a weighted lattice count.  Proposition~\\ref{long269:res:strip-decomposition}\nseparates floor crossings from its moving boundary, and\nExample~\\ref{long269:res:no-crossing-counterexample} shows that the latter\ncan survive a cubic difference even with no crossings.\n\nThere is a second distinction.  In our displayed shift identity the\ncoefficient of $m_{a+jr}$ is $15c_j\\gamma_{a,jr}$, which can depend on $a$.\nFor example, $P_1=2$, $P_2=12$ and $P_3=120$ give\n$\\gamma_{0,1}=1/3$ but $\\gamma_{1,1}=1/5$.  These factors are required by\nthe denominator chain.  A support estimate for $D_{r,a}$ is therefore not\nby itself a verification of Definition~5 for the sequence $(m_a)$.\nNo fixed-coefficient representation to which that theorem applies has\nbeen established here.\n\nFor a variable-denominator analogue, a useful hypothesis would be a fixed\nchoice of coefficients and shifts $r_n\\to\\infty$\nfor which $\\Delta_n=\\{a\\ge0:D_{r_n,a}\\ne0\\}$ is infinite and distinct\nmembers are at least $\\eta r_n$ apart, for a fixed $\\eta>0$.\nEven this would leave the uniform polynomial-variation requirement, for\nexample\n\\[\n |D_{r_n,a'}|\\le C\\bigl((a'-a)^d+|D_{r_n,a}|\\bigr)\n \\quad(a<a',\\ a,a'\\in\\Delta_n),\n\\]\nwith $C,d$ independent of $n$, $a$ and $a'$.\nThis is a condition on the gaps between nonzero terms, not merely on\ntheir density: even a set of density zero can contain adjacent pairs.\nIt also requires infinitely many nonzero terms for each selected shift.  A small\nbut nonzero term at every index would fail it, and an identically zero\ndifference would also fail the infinitude requirement.  The cited paper\nconstructs suitable shifts for the polynomial-floor sequences just\ndescribed; no such result is proved for our weighted counts.  The bound $O((a+r+1)^2)$ controls\nabsolute size: it depends on the location $a$ itself.  The variation\ncondition instead bounds a later nonzero term by the gap from an earlier\none and the size of that earlier term, with constants uniform in the\nshift.  The absolute-size bound alone gives no such comparison.\nThe denominator chain itself is not a fixed-base power sequence.  The\nnext proposition shows what is lost in two direct fixed-base recodings.\nA value criterion for the original chain would need a separate proof,\nincluding non-cancellation and the Diophantine height estimates required\nby a Subspace-Theorem argument. These heights measure the numerators and\ndenominators of the approximating algebraic quantities; they are unrelated\nto the running LCM $\\hgt$.\nThe finite set of prime divisors alone does not provide those estimates.\n\n\\paragraph{Why a finite alphabet of bases is not enough.}\nKebis, Luca, Ouaknine, Scoones and Worrell\n\\cite[Definition~3, Theorem~6 and Claim~7]{kebis2024echoing}\nwork with fixed-base series whose coefficients lie in a finite algebraic\nalphabet.  Their echoing condition requires long near-repetitions,\nseparated intervals containing the mismatches, and nonzero weighted\nmismatch sums on at least two of those intervals.  Claim~7 supplies the\nnon-cancellation step in the proof of the value theorem.  Merely having\nfinitely many letters does not give these properties.\nThe four-letter radix sequence is not the numerator sequence: the actual\n$m_a$ is unbounded.  A recoding would have to preserve the scalar value,\nidentify its coefficient alphabet and verify the echoing conditions.\nNo fixed-base recoding with verified echoing properties is proved here.\n\nTwo natural bases expose the tradeoff.  The least integer base whose\npowers clear every $P_n$ is $30$, while base $8$ matches the growth of\n$P_n=\\hgt(2^n)$.  The former keeps integrality; the latter keeps polynomial\nsize.\n\n\\begin{proposition}[what direct fixed-base recoding preserves]\n\\label{long269:res:fixed-base-recoding}\nThe following identities converge absolutely:\n\\[\n \\alpha=\\sum_{a\\ge0}\\frac{e_a}{30^{a+1}}\n        =\\sum_{a\\ge0}\\frac{v_a}{8^{a+1}},\\qquad\n e_a=\\frac{m_a30^{a+1}}{P_{a+1}},\\quad\n v_a=\\frac{m_a8^{a+1}}{P_{a+1}}.\n\\]\nHere $e_a\\in\\mathbb Z_{>0}$ and $e_a\\ge(15/4)^{a+1}$, whereas\n$v_a\\in\\mathbb Z[1/15]$ and $0<v_a<225(a+1)^2$.\nFor an integer $q\\ge2$, the termwise divisibility $P_n\\mid q^n$ for every\n$n\\ge1$ holds exactly when $30\\mid q$; that direct recoding then has\ncoefficients at least $(q/8)^{a+1}$.\n\\end{proposition}\n\\begin{proof}\nEach exponent in $P_n$ is at most $n$, so $P_n\\mid30^n$.  The estimate\n$8^n/15<P_n\\le8^n$ was proved in Section~\\ref{long269:sec:blocks}.\nThese facts, $m_a\\ge1$ and\n$m_a\\le15(a+1)^2$, give all the coefficient bounds and identities.\nNecessity of $30\\mid q$ follows because each of $2,3,5$ divides some $P_n$;\nsufficiency follows from $P_n\\mid30^n$.  The general lower bound follows\nagain from $P_n\\le8^n$.\n\\end{proof}\nThe base-$30$ coefficients are integral but grow exponentially, so the\ncited polynomial-growth criterion does not apply.  The base-$8$\ncoefficients have polynomial size but are not all integral: already\n$v_1=4\\cdot8^2/12=64/3$.  In fact, their reduced denominators grow\nexponentially even after cancellation.  Since $P_{a+1}/2^{a+1}$ is odd,\nthe expression\n\\[\n v_a=\\frac{m_a4^{a+1}}{P_{a+1}/2^{a+1}}\n\\]\ncan cancel an odd factor only through $m_a$.  Consequently,\n\\[\n \\operatorname{den}(v_a)\n \\ge \\frac{P_{a+1}}{2^{a+1}m_a}\n >\\frac{4^{a+1}}{225(a+1)^2}.\n\\]\nThe last inequality uses $P_{a+1}>8^{a+1}/15$ and\n$m_a\\le15(a+1)^2$.  Thus the bound on absolute value does not control\nDiophantine height even for these particular coefficients; fixed\ndenominator-prime support is not enough.\n\nMore generally, no fixed integer base gives this termwise recoding both\neventually integral coefficients and polynomial growth.  If $30\\mid q$,\nthe proposition gives exponential growth.  Otherwise choose\n$p\\in\\{2,3,5\\}$ not dividing $q$.  Cancellation by $m_a$ removes at most\na factor $m_a$, so\n\\[\n \\operatorname{den}\\!\\left(\\frac{m_aq^{a+1}}{P_{a+1}}\\right)\n \\ge\\frac{p^{\\lfloor(a+1)\\log_p2\\rfloor}}{m_a}\n >\\frac{2^{a+1}}{15p(a+1)^2}.\n\\]\nThis denominator tends to infinity.  Thus matching the denominator growth by base $8$ keeps the coefficients\nsmall but introduces large rational denominators; making the coefficients\nintegral by base $30$ makes them exponentially large. This obstruction\nconcerns termwise rescaling only, not regrouping or carrying.\n\nCarrying resolves a different problem: it bounds the digits but leaves\nthe radices variable. It also requires the integer parts of the actual\ntails, not just the known block numerators. Set\n\\[\n \\delta_a=m_a+\\lfloor X_{a+1}\\rfloor-b_a\\lfloor X_a\\rfloor\n          =\\lfloor b_a\\{X_a\\}\\rfloor.\n\\]\nThe equality follows from the tail recurrence and $m_a\\in\\Z$.  Thus\n$0\\le\\delta_a<b_a$ and $\\{X_a\\}=(\\delta_a+\\{X_{a+1}\\})/b_a$.\nIterating gives the finite identity\n\\[\n \\{X_0\\}=\\sum_{a=0}^{N-1}\\frac{\\delta_a}{P_{a+1}}\n              +\\frac{\\{X_N\\}}{P_N}.\n\\]\nThe last term is nonnegative, is less than $1/P_N\\le2^{-N}$, and tends\nto zero.  This is the greedy mixed-radix expansion of the fractional part\nof $X_0=S/2$, with digits in $\\{0,\\ldots,29\\}$.\nFor the actual series its first eight digits are\n\\[\n (\\delta_0,\\ldots,\\delta_7)=(1,4,8,3,10,1,8,5).\n\\]\nTo certify them, use the twelve-block enclosure from\nSection~\\ref{long269:sec:evidence}, either propagated by the recurrence\nor evaluated directly as\n\\[\n P_a\\sum_{j=a}^{11}\\frac{m_j}{P_{j+1}}<X_a\n \\le P_a\\sum_{j=a}^{11}\\frac{m_j}{P_{j+1}}\n       +\\frac{P_a}{P_{12}}Q(n_{12})\\qquad(0\\le a<12).\n\\]\nBoth descriptions give the same rational endpoints. At indices $0$\nthrough $8$, each interval lies strictly between consecutive integers,\ncertifying all nine floors needed for the eight digits. Width alone\nwould not suffice: an interval crossing an integer leaves its floor\nundetermined.  For example,\n$\\lfloor X_4\\rfloor=2$ and $\\lfloor X_5\\rfloor=5$, giving\n$\\delta_4=65+5-30\\cdot2=10$, not the uncarried numerator $m_4=65$.\nThis finite calculation makes no claim about the later digit complexity.\n\nA rational number need not have eventually periodic digits when the radices vary.  For example,\ntake radix $5$ at the indices $1,2,4,8,\\ldots$ and radix $3$ elsewhere.\nThe constant tail fraction $1/2$ gives digit $2$ at those indices and\n$1$ elsewhere.  These digits are not eventually periodic, but their\nmixed-radix expansion sums to $1/2$.  This example uses different radices\nfrom our four possible bases; it shows why fixed-base eventual\nperiodicity cannot be transferred to a variable-radix expansion.  Our digits also\ndepend on the actual tail fractions, not just on the four-letter radix\nsequence.  The construction supplies neither a fixed-base expansion\nwith verified echoing properties nor a bound on its digit complexity.\n\nAdamczewski--Bugeaud's complexity theorem\n\\cite[Theorem~1]{adamczewskibugeaud2007} concerns the actual digits of an\ninteger-base expansion: for an algebraic irrational their length-$n$ block\ncomplexity divided by $n$ tends to infinity.  Neither $(b_a)$ nor the\nuncarried $(e_a)$ is such a digit expansion of $\\alpha$.\n\n\n\\begin{problem}[the repeated three-prime target]\n\\label{long269:prob:producer}\\label{long269:prob:tails269}\nProve irrationality of $S=\\mathcal R_{\\{2,3,5\\}}$.  Equivalent forms are\n$\\Esc(K)$ and\n\\begin{equation}\\label{long269:eq:tail-nonintegrality}\n BX_a\\notin\\mathbb Z\n \\quad\\text{for every }B\\ge1\\text{ with }\\gcd(B,30)=1\\text{ and every }a\\ge1.\n\\end{equation}\n\\end{problem}\nThe residue formulation uses finite integer computations, but asks for a\nsuccess for every eligible denominator and beyond every prescribed start.\nThe tail formulation requires that no positive integer multiplier\ncoprime to $30$ make any $X_a$, $a\\ge1$, integral.  These are\nreformulations of the same irrationality assertion, not extra assumptions\non the series, and neither follows from the rank obstruction.  The next\nproposition proves the tail reformulation, complementing\nTheorem~\\ref{long269:res:actual-escape-endpoint}.\n\n\\begin{proposition}[irrationality is equivalent to nonintegrality of every reduced tail]\n\\label{long269:res:tails-equivalence}\nStatement~\\eqref{long269:eq:tail-nonintegrality}, quantified over every $B\\ge1$\ncoprime to $30$ and every $a\\ge1$, is equivalent to irrationality of $S$.\n\\end{proposition}\n\n\\begin{proof}\nIf $BX_a\\in\\Z$ for some such $B$ and $a$, then $X_a\\in\\Q$, and since\n$S=\\sum_{j<a}s_j+X_a/h_a$ with $h_a$ a positive rational and the prefix a\nfinite sum of rationals, $S\\in\\Q$.  Conversely if $S\\in\\Q$, then\nTheorem~\\ref{long269:res:actual-cancellation} produces $B$ coprime to $30$ with\n$BX_a\\in\\Z$ for every $a\\ge a_D$.\n\\end{proof}\n\nIntegrality at one index propagates forwards, but nonintegrality at one\nearly index does not propagate forever: a rational denominator can clear\nlater.  Proposition~\\ref{long269:res:exact-denominator} describes that onset\nexactly.  The dichotomy in Proposition~\\ref{long269:res:actual-dichotomy}\nalso permits integral tails, so it does not settle this question.\n\n\\subsection{Function-theoretic representations}\\label{long269:sec:function-representation}\n\nThe elementary proof of irrationality of $\\mathcal D_{\\{2,3,5\\}}$ gives\nno transcendence conclusion. Recall that\n\\[\n \\mathcal D_{\\{2,3,5\\}}=1+\n \\sum_{t\\in\\{2^n,3^n,5^n:n\\ge1\\}}\n \\frac1{2^{\\lfloor\\log_2t\\rfloor}3^{\\lfloor\\log_3t\\rfloor}\n             5^{\\lfloor\\log_5t\\rfloor}}.\n\\]\nEach running LCM is counted once at its first jump, unlike the repeated sum.\nThe bases in \\eqref{long269:eq:dyadic-alphabet} come from these same jumps.\nFor two primes a representation by one Hecke--Mahler value proves\ntranscendence (Section~\\ref{long269:sec:two-prime}). A corresponding\nthree-prime representation has not been constructed here.\n\n\\begin{problem}[an exact function representing the distinct-height sum]\n\\label{long269:prob:representation}\nTranscendence of $\\mathcal D_{\\{2,3,5\\}}$ is not proved here.  For a functional\napproach to it, first specify the function, its coefficient field and\nconvergence domain, then prove an exact identity for $\\mathcal D_{\\{2,3,5\\}}$,\nand only then apply a stated value theorem.  A conditional theorem must\ndisplay the extra nondegeneracy assumption; a no-representation theorem\nmust specify the class it excludes.\n\\end{problem}\n\nFor $(p,q,r)=(2,3,5)$, the two slopes in the rank proof are\n$\\log_5 2=\\theta_5$ and $\\log_5 3=\\theta_5/\\theta_3$, not $\\theta_3$ and\n$\\theta_5$.  Each is irrational: a rational value would equate a positive\npower of $5$ with a positive power of $2$ or $3$.  The proof chooses the\nrow and column indices independently.  It therefore needs no rational\nindependence of $1,\\theta_3,\\theta_5$ and no density assertion for the\nsingly indexed orbit $(\\{n\\theta_3\\},\\{n\\theta_5\\})$.  Any functional\napproach that needs such a stronger hypothesis must establish it separately.\n\nA finite-dimensional encoding is useful here only if its data determine\nthe represented function: equal encodings must give equal functions.\nThe Lean sources impose this through a factorisation of the linear map\nto functions through a finite-dimensional space. The resulting space of\nfunctions is then finite-dimensional\n(\\lword{Erdos269/WeightedPhaseCarry.lean}{109}{carry_eq_residueDigit_add_coboundary}{identity\nfor the carry},\n\\lword{Erdos269/WeightedPhaseCarry.lean}{293}{CartierRealisation}{map to\nfunctions},\n\\lword{Erdos269/WeightedPhaseCarry.lean}{334}{finite_realisedSpan_of_factorisation}{finite-dimensional\nconclusion}).  They also bound the carry residue and its digit in their\nstated intervals\n(\\lword{Erdos269/WeightedPhaseCarry.lean}{150}{carryResidue_mem_interval}{residue\nbound},\n\\lword{Erdos269/WeightedPhaseCarry.lean}{157}{residueDigit_mem_interval}{digit\nbound}).  These conditional linear-algebra statements neither construct a\nfunction representing $\\mathcal D_{\\{2,3,5\\}}$ nor show that a corresponding\nfunction space must be infinite-dimensional.\n\n\\paragraph{The cited value theorems and their hypotheses.}\nPellarin's rank-one theorem concerns quadratic irrational slopes and\nalgebraic evaluation points in the convergence domain\n\\cite[Theorem~1.1 and Remark~1.2, pp.~330--332]{pellarin2006}.\nThe prime-logarithm slopes used here are not quadratic: each $\\log_r p$\nis irrational by unique factorisation, and the Gelfond--Schneider theorem\nthen makes it transcendental, since $r^{\\log_r p}=p$ is algebraic.\\footnote{For this logarithmic form of the Gelfond--Schneider theorem, see Keith\nConrad, \\emph{Transcendence of $e$}, p.~1, footnote~1,\n\\url{https://kconrad.math.uconn.edu/blurbs/analysis/transcendence-e.pdf}\n(accessed 18 September 2026).}\nThis rules out direct substitution of those slopes into Pellarin's\nquadratic-slope theorem.  It does not rule out a different representation\nof the scalar sum, and it does not affect the two-prime argument above,\nwhich uses a theorem for general irrational slopes.\n\nIn Pellarin's reduced-slope normalisation, the rank-one condition says\nthat the evaluation points arise from a common point by the quadratic\norder's monomial action, up to\nmultiplication by torsion points.  For a single evaluation point it is\nautomatic: use that point itself and the identity action.  For several\npoints it is a condition to check, not a consequence of having two\nlattice coordinates.  Under it, algebraic independence of the values is\nequivalent to $\\Q$-linear independence of the specified auxiliary formal\nLaurent series; the rank-one condition alone does not imply independence.\nThis module rank is not the rank of our reciprocal kernel.  No exact\nrepresentation suitable for this theorem has been constructed here, but\ninfinite kernel rank alone does not preclude one.\n\nFor one variable, Adamczewski and Faverjon's proofs of Nishioka's theorem\nand the lifting theorem are regular-point results\n\\cite[Theorems~1 and 2]{adamczewskifaverjon2023}.  Regularity means that the\nsystem matrix stays defined and invertible along the evaluation orbit.\nTheir multivariate lifting theorem\n\\cite[Theorem~3.3]{adamczewskifaverjon2026} additionally assumes an admissible\ntransformation--point pair.  Its growth, decay and nonvanishing conditions\nare specified in Definition~5.1; none follows merely from writing a double\nsum.  Lifting transfers a relation among values to one among functions;\na transcendence application must exclude the latter.  A Mahler system\nalone cannot do this, as the constant function $1$ illustrates.\n\nRegularity of a preselected system is not, however, a universal prerequisite\nfor Mahler's method.  The same paper's Theorem~1.1 treats univariate Mahler\nfunctions at algebraic points in the punctured unit disc where their values\nare defined, without that regularity hypothesis.  The hypotheses must\ntherefore be matched to the particular theorem being invoked.  No exact\nMahler representation or suitable evaluation data for this sum over distinct running LCMs are constructed here.  The locators for this paper refer to its 68-page\nauthor manuscript, not the older 52-page arXiv version.\n\n\\subsection{Hypotheses for auxiliary integer recurrences}\\label{long269:sec:carry-hypotheses}\n\nFor ordered primes $p_1<\\cdots<p_s$, the interval\n$(p_1^a,p_1^{a+1})$ contains at most one power of each $p_i>p_1$.\nThe radix including the final $p_1$-jump consequently belongs to\n\\[\n \\left\\{p_1\\prod_{i=2}^s p_i^{\\varepsilon_i}:\n                  \\varepsilon_i\\in\\{0,1\\}\\right\\}.\n\\]\nThis finite alphabet determines neither the frequency of the radices nor\nthe error terms in the weighted block counts. Estimates for those\nquantities, or for specified shifted kernels, would be additional\narithmetic inputs. The conditional statements about integer recurrences below keep such\ninputs explicit.\n\nSome auxiliary results explain why bounded integer approximations alone\nare insufficient.  Here are their hypotheses in explicit form.\n\nLet $j(n)\\in\\{2,3,5\\}$ label a sequence of jumps, and suppose each label\noccurs.  A perturbation $\\varepsilon_n$ in an additive abelian group has\nsum zero on every\ninterval $[a,b)$ whose endpoints have the same label exactly when\n\\[\n \\varepsilon_n=u(j(n+1))-u(j(n))\n\\]\nfor three potential values $u(2),u(3),u(5)$.  One direction follows by\ntelescoping.  Conversely, the zero-sum assumption makes\n$\\sum_{k<n}\\varepsilon_k$ the same at any two indices with the same\nlabel.  Assign that common value to $u(j(n))$; every label occurs, so all\nthree values are defined.  Consecutive partial sums then give the\ndisplayed identity.  In the usual terminology, the perturbation is a\ncoboundary of the function $u$ on the labels: it is the difference of\nsuccessive potential values.  This is a telescoping condition, not a bound\non the size of the perturbation.  It does not by\nitself force the perturbation to vanish: unequal potential values give\nnonzero differences when the label changes.  If the perturbation also\nvanishes on a genuine $2\\to3$ transition and on a genuine $2\\to5$\ntransition, all three potential values agree, so every perturbation is\nzero.\n\nFor an integer sequence satisfying $c_{n+1}=b_nc_n-Dm_n$, compare it with\nan integer sequence $z_n$.  Define the error $e_n=Dz_n-c_n$ and the\nperturbation $\\varepsilon_n=b_nz_n-z_{n+1}-m_n$.  Then\n\\[\n e_{n+1}=b_ne_n-D\\varepsilon_n.\n\\]\nUnder the preceding zero-sum and transition hypotheses,\n$\\varepsilon_n=0$ and hence\n$e_N=(\\prod_{n<N}b_n)e_0$.  If every $b_n\\ge2$ and the integer $e_0$ is\nnonzero, then $|e_N|\\ge2^N$.  This contradicts even one bound\n$|e_N|<2^N$, and therefore also excludes a uniform bound for all $N$.\nThe nonzero initial error is essential: the identically zero error causes\nno contradiction.\n\nA calculation at four consecutive indices gives a related obstruction.  If\n$0<T_i<1$, $z_i\\in\\Z$ and $|z_i-T_i|<1$, then $z_i$ is $0$ or $1$.\nFor bases $2,3,2,5$, the equalities\n$2z_0-z_1-1=0$ and $2z_2-z_3-1=0$ force all four integers to be $1$.\nThe sum of the first two perturbations is then\n\\[\n (2z_0-z_1-1)+(3z_1-z_2-1)=1,\n\\]\nnot zero.  Thus unit accuracy, the two exact equalities and a zero sum on\nthat block cannot all hold.  Unit accuracy alone does not give either\nexact equality.\n\nNo integer approximation to the tails of $S$ is constructed here that\nsatisfies all these conditions.\nRationality supplies integer carries of at most quadratic growth, but\nnot the additional zero-sum and transition identities.  The carries obtained from $S$ satisfy a weighted block identity\ninstead.  Applying the auxiliary results\nwould require that identity to imply their hypotheses, or would require\na different construction of the integer approximations.\n\nThere is also a conditional criterion for a sum supported on the powers\nof $2$.  Its elementary ingredients can be stated without introducing\nanother formal vocabulary.  From $y'=by-1$ with $b>0$ and $0<y'<1$ one\ngets $1/b<y<2/b$.  For integers $a>0$ and $p\\ge3$, if $u>1/a$ and\n$v<2/(pa)$, then\n$u-v>1/(3a)$.  Finally, a real number $x$ is irrational if, for every\npositive integer $d$, there are integers $n,k$ with\n$0<|nx-k|<1/d$.\n\nTo use these observations for a series, the conditional result\nrequires exact identities $y_M=H_Mx-z_M$, with $H_M,z_M\\in\\Z$, and,\nfor every positive $d$, two indices with different $H_M$, different\n$y_M$, and $|y_M-y_{M'}|<1/d$.  Their difference gives the required\nnonzero linear form.  Equal tail values would give zero and would not suffice;\na merely bounded gap would not suffice either.  Indeed, for a rational\n$x$ of denominator $d$, every nonzero difference of this form has size at\nleast $1/d$.  No concrete series in this record is shown to satisfy all\nthese clearing and arbitrarily small nonzero-gap conditions.\n\n\\subsection{The unresolved repeated-sum question}\\label{long269:sec:remaining-question}\n\nThe distinct sum at $\\{2,3,5\\}$ is settled by\nTheorem~\\ref{long269:res:distinct-height-235}. The general and subseries\narguments remain ordinary proofs pending formalisation\n(Remarks~\\ref{long269:res:distinct-height-all}\nand~\\ref{long269:res:single-prime-subsums}). The finite-state calculations\nof Remark~\\ref{long269:res:distinct-height-sets} have their own scope;\nthe $\\{2,3,5,7\\}$ automaton collision limits that interval criterion only.\n\nFor $\\mathcal R_{\\{2,3,5\\}}$, the missing ingredient is not just another\nnormalisation or one sharper finite enclosure. The coefficients, recurrence,\nquadratic bounds and denominator-clearing formula are explicit. We must\nexclude eventual integral reduced tails for every possible denominator,\nor equivalently prove the residue inequalities beyond every prescribed\nonset. The fixed-start formula explains why an early success does not\nsuffice; the strip identity explains why avoiding floor crossings does\nnot supply the proposed cancellation.\n\nThe rank and uniform-norm results concern the kernel, not the arithmetic\nof its sum.  Finite tests, countability, radix recoding and denominator\nsupport do not close the remaining argument.  The settled two-prime case\nuses Fan's earlier reduction to the cited Hecke--Mahler value theorem.\n\n\\pdfbookmark[1]{Statements and declarations}{statements-declarations}\n\\subsection*{Verification and reproducibility}\\label{long269:sec:proof-sources}\n\\evidenceparagraph{\\evidencerecordurl}\n\n\\paragraph{Proof sources.}\nThe evidence record fixes the source versions and individual statement\nassociations. These are earlier checks; this revision makes no fresh\nformal-verification claim.\n\nThe general distinct-value assertion and its four word-and-torus\narguments are stated as remarks with complete ordinary proofs in\nSection~\\ref{long269:sec:distinct-all}; they have no Lean proofs. The source record reports that the general and\nprime-power subseries arguments were found on 26 September 2026 and checked\nby a second AI agent, whose corrections were incorporated. It records\nno human review; the present revision does not constitute a new independent\nreview. No separate literature search for the prime-power subseries theorem\nat $|P|\\ge3$ is recorded, and no priority claim is made for it. The\nsubseries identities also have ordinary proofs. The program\n\\texttt{two\\_tail.py} checks their finite instances in exact arithmetic.\n\nFor $\\{2,3,5\\}$ the separate theorem for the sum over distinct values has a\n\\href{\\distinctheightlean\\#L689}{Lean proof}; its source record says\nthat Comparator had not been run on it. The general affine-block calculation,\ndecoding criterion and finite-set application are also remarks with\nordinary proofs; the last additionally uses exact computation. Their\nformal coverage is restricted to the $\\{2,3,5\\}$ case with one block\nand the interval of Section~\\ref{long269:sec:distinct-235}.\nThe linear-flow and Smith-factor results likewise have the ordinary\nproofs printed here; no additional formal coverage is inferred for them.\n\nThe two-prime deduction is credited to Fan~\\cite{fan2026comment}.\nIts formal proof assumes the cited Hecke--Mahler value theorem, as specified\nin Section~\\ref{long269:sec:coverage}; its algebraic identities are\nunconditional. The weighted-triangle and shift identities, denominator\nformula, scaled dichotomy, uniqueness argument, residue limit, strip\ndecomposition and recoding also have written proofs here. Their individual\nformal associations are retained, without a blanket assertion that every\nresult in the record is formalised. Kernel acceptance uses the standard\naxioms allowed by the linked checks and is not an independent review of\nthe paper-to-statement correspondence.\n\nThe twelve-block enclosure, denominator bound and eight mixed-radix digits\nhave finite exact proofs. The two large exclusions have saved witnesses\nand exact checkers, and the larger scan has a saved run. None of these\nfinite computations proves the residue inequalities beyond every starting\nindex. No Comparator replay is claimed for the general or subseries\narguments.\n\n\\paragraph{Artefact and data availability.} The links in the\nsource inventory and to the integer-arithmetic checker open a\n\\href{\\sourceurl}{fixed revision} of the repository.  The\nwitnesses and exact checkers for the two large numerical exclusions and\nthe saved larger scan are in the\n\\href{https://github.com/wcook04/plectis-erdos/tree/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos269/large-computations}{large-computation directory}.\nThe \\href{\\distinctheighturl}{programs for sums over distinct values} recompute the\nconstants of Theorem~\\ref{long269:res:distinct-height-235} and the\ncertificates of Remark~\\ref{long269:res:distinct-height-sets} in exact\nrational arithmetic, with two independent computations of the tail\nintervals, and store their outputs; a fast test checks them against the\nstored outputs.  The same directory checks the finite illustrations of\nSection~\\ref{long269:sec:distinct-all} and the block identity for\n$\\{2,3,5,7\\}$.\n\n\\paragraph{Funding and competing interests.} This work received no external\nfunding.  The author declares no competing interests.\n\n\\paragraph{Authorship and review.} Will Cook directed the work.  AI agents\ndid most of the research and drafting.  Cook has not independently\nverified every mathematical claim, and these manuscripts have not had\nindependent human mathematical review.\n\n\\paragraph{Acknowledgements.}  Steve Fan's forum post of 26 June 2026 supplied the\ntwo-prime factorisation and its Hecke--Mahler reduction before this\nmanuscript; it also records the elementary running-LCM identity for finite\nprime sets~\\cite{fan2026comment}.  The\ntranscendence input is due to Yann Bugeaud and Michel Laurent and to the\nearlier work of Loxton and van der Poorten cited by them.  The\nproblem numbering and its status as of 28 July 2026 are taken from the\nErd\\H{o}s Problems catalogue maintained by Thomas Bloom~\\cite{erdosproblems}.\n% Part: extended_record\n\\section{Further examples, conditional lemmas and source references}\n\\label{long269:long:extended}\n\nWe collect the historical statements and then separate the lemmas on integer recurrences\nfrom their three-prime application. Their explicit divisibility and\nsegment hypotheses specify what another recurrence must supply. Finite\nexamples and the declaration inventory complete the record.\n\n\\subsection{The historical and catalogue record}\\label{long269:long:history}\n\nFor a singleton $P=\\{p\\}$, we have $a_n=p^{n-1}$ and\n$[a_1,\\ldots,a_n]=p^{n-1}$, so both sums equal $p/(p-1)$.\nErd\\H{o}s's 1988 account calls the infinite-prime case a simple exercise\nand presents irrationality for a finite number of primes greater than one\nas a probable extension~\\cite[p.~106]{erdos1988}.\nBloom's catalogue records an open problem at the access date of\n28 July 2026~\\cite{erdosproblems}. These are dated source statements,\nnot a fresh determination of the catalogue's present status.\n\nIn the letter written on 1 January 1973 and published in 1974, Erd\\H{o}s\ncalls the irrationality of $\\mathcal R_P$ a conjecture that he cannot prove,\nand says that he can show the sum over the distinct values of\n$[a_1,\\ldots,a_k]$ to be irrational \\cite[p.~335]{erdos1974letter}.  That\nsum is $\\mathcal D_P$.  The letter introduces its primes as $p_1,\\ldots,p_s$,\nlater writing $p_r$, without restricting their number, so the singleton\ncalculation forces the qualification $|P|\\ge2$.  It prints no argument.  The\ncatalogue has cited the letter for Problem~\\#269 since 28 December 2025,\nafter the letter was quoted in the site's discussion of missing\nproblems~\\cite{erdosproblemsmissing}, and its page records Erd\\H{o}s's\nstatement about the distinct values.\nThe catalogue page says that Erd\\H{o}s proved the statement; the letter\nasserts it and prints no argument.\nRemark~\\ref{long269:res:distinct-height-all} proves it for every finite\n$P$ with $|P|\\ge2$.  For $|P|=2$ it was already a consequence of the\nHecke--Mahler transcendence theorem of Loxton and van der Poorten\n\\cite[Theorem~8, p.~40]{loxtonvdp1977} and Bugeaud and Laurent\n\\cite[Theorem~1.1]{bugeaudlaurent2023}, through the affine identity of\nSection~\\ref{long269:sec:two-prime}, by the route Fan used for\n$\\mathcal R_{\\{p,q\\}}$.  We do not know what\nargument Erd\\H{o}s had in mind.\n\nThe literature search recorded here did not locate a proof of the\nassertion in print. Erd\\H{o}s and Graham state only the question for $\\mathcal R_P$: the sum is irrational\nfor infinite $P$, and they ask what happens for finite $P$ with more than\none element \\cite[p.~65]{erdosgraham1980}; the sum over distinct running LCMs does not\nappear there.  Erd\\H{o}s's survey of\n1975 on the irrationality of series \\cite{erdos1975} does not mention the\nrunning least common multiple of the $P$-smooth integers, his survey of\n1980 on combinatorial number theory \\cite{erdos1980survey} does not discuss\nirrationality, and his survey of\n1988 states only the question for $\\mathcal R_P$ \\cite[p.~106]{erdos1988}.\nThe Hecke--Mahler transcendence theorems cited here, of Loxton and\nvan der Poorten, Bugeaud and Laurent, and\nLuca--Ouaknine--Worrell~\\cite{lucaouaknineworrell2025}, have a single\nslope; the identifications in Section~\\ref{long269:sec:two-prime}\napply to $\\mathcal D_P$ when $|P|=2$.  The rationality\ncriterion of Diananda and Oppenheim~\\cite{dianandaoppenheim1955}, which\ncondenses a Cantor series into blocks of consecutive terms, has the form of\nthe dyadic argument of Section~\\ref{long269:sec:distinct-235}; it restates\nrationality and does not decide $\\mathcal D_P$.\nThe discussion thread of Problem~\\#269, as inspected for that search,\nsupplied no argument for $\\mathcal D_P$ with $|P|\\ge3$, and the\n\\emph{Formal Conjectures} statement\n\\cite{formalconjectures269} concerns $\\mathcal R_P$ only.  The Cantor-series\ncriteria of Erd\\H{o}s--Straus~\\cite[Theorem~2.1]{erdosstraus1974} and\nHan\\v{c}l--Tijdeman~\\cite[Theorem~3.1]{hancltijdeman2004} do not apply to\n$\\mathcal D_P$, as explained in Section~\\ref{long269:sec:distinct-235}.  The\nsearch did not reach the sections on irrational series in Guy's\n\\emph{Unsolved Problems in Number Theory}, or MathSciNet reviews.  The unresolved target of this record is the\nrepeated series $\\mathcal R_P$; a transcendence statement for $\\mathcal D_P$\nwith $|P|\\ge3$ is a separate question\n(Problem~\\ref{long269:prob:representation}).\n\nOn 26 June 2026 Steve Fan posted the two-prime factorisation, the Hecke--Mahler reduction and the transcendence\nconclusion in the discussion thread of the problem's page\n\\cite{fan2026comment}; the comment itself notes that the argument does not seem\nto generalise immediately to $|P|\\ge3$, and a reply there observes that it\napplies to arbitrary coprime pairs.  The first public version of this\nnote is dated 22 July 2026, after Fan's post.  We retain\nthe calculation as exposition and make no priority claim for it.\n\nThe statement of the problem has been formalised as a conjecture with an\nunfilled proof in the \\emph{Formal Conjectures} collection\n\\cite{formalconjectures269}.  In the cited source, the rational,\nirrational and infinite-prime assertions end in \\texttt{sorry}.  Its Nat-indexed series includes the empty-prefix\nleast-common-multiple term, so its value differs from the conventional one by a\nrational constant; transporting a theorem across that boundary needs an\nexplicit series-identification lemma, which is not supplied here.\n\n\\subsection{Integer recurrences in their general form}\n\\label{long269:long:general-carries}\n\nConsider an arbitrary integer recurrence with positive integer radices.\nDivisibility and the existence of segments satisfying the residue inequality must now be supplied\nas hypotheses. For the actual three-prime radices, the criterion below\nshows that each integer solution is eventually divisible by the smooth part of a\ndenominator. It still leaves the residue-inequality hypothesis to be proved.\nThe comparison with Cantor-series criteria is in\nSection~\\ref{long269:sec:actual-orbit}.\n\nLet $D$ and $B$ be positive integers with $D=D_{\\mathrm{sm}}B$, where\n$D_{\\mathrm{sm}}=2^u3^v5^w$, $u,v,w\\in\\N$ and $\\gcd(B,30)=1$.\nLet $(c_n)$ be an integer sequence satisfying $c_{n+1}=b_nc_n-Dm_n$\nfor integer sequences $(b_n)$ and $(m_n)$.  For these general sequences,\nform $W$ and $F$ by the recursions in Section~\\ref{long269:sec:escape};\na segment satisfying the residue inequality must have $W\\ne0$, and its modulus is $|W|$.\nIn the factorisation below the quotients $d_n$ are required to be integers;\nwriting $c_n=D_{\\mathrm{sm}}d_n$ asserts divisibility, not just an identity\nin $\\mathbb Q$.\n\n\\begin{proposition}[conditional denominator reduction]\n\\label{long269:long:denominator-reduction}\nIf $c_n=D_{\\mathrm{sm}}d_n$ for every $n$, with $D_{\\mathrm{sm}}>0$, then the\nrecurrence and iterated recurrence for $(d_n)$ have multiplier $B$ in place\nof $D$.  Moreover, for every $n$ and every real $t$,\n\\[\n 0<c_n\\le Dt\\quad\\Longleftrightarrow\\quad0<d_n\\le Bt.\n\\]\n\\end{proposition}\n\n\\begin{proof}\nSubstitute $c_n=D_{\\mathrm{sm}}d_n$ and $D=D_{\\mathrm{sm}}B$ in the\nrecurrence and divide by $D_{\\mathrm{sm}}$.  Dividing\n$0<c_n\\le D_{\\mathrm{sm}}Bt$ by this positive factor gives the stated\nbound equivalence.  Dividing the iterated recurrence\n$c_{\\ell+h}=W_{\\ell,h}c_\\ell-DF_{\\ell,h}$ likewise gives\n$d_{\\ell+h}=W_{\\ell,h}d_\\ell-BF_{\\ell,h}$.\n\\end{proof}\n\nFor a general bound $c_n\\le G(D,n)$, division and integrality give only\n\\[\n d_n\\le\\left\\lfloor\\frac{G(D,n)}{D_{\\mathrm{sm}}}\\right\\rfloor,\n\\]\nnot automatically $d_n\\le G(B,n)$.  The bounds used for the actual tails\nare linear in the multiplier before rounding.  For any real $t$,\n\\[\n \\left\\lfloor\\frac{\\lfloor Dt\\rfloor}{D_{\\mathrm{sm}}}\\right\\rfloor\n =\\lfloor Bt\\rfloor.\n\\]\nIndeed, an integer $z$ satisfies $D_{\\mathrm{sm}}z\\le\\lfloor Dt\\rfloor$\nexactly when $D_{\\mathrm{sm}}z\\le Dt$, or $z\\le Bt$.\nThus rounding does not change the reduced bound in this case.\n\nGrowth of the radix product is not enough: $b_n=3$, $m_n=1$, $D=2$\nand $c_n=1$ satisfy the recurrence, but $2$ never divides $c_n$.\nThe precise condition follows by reducing the recurrence modulo\n$D_{\\mathrm{sm}}$.  For any starting index $A$ and every $n\\ge A$,\n\\[\n \\begin{aligned}\n c_n&\\equiv\\left(\\prod_{j=A}^{n-1}b_j\\right)c_A\n       \\pmod{D_{\\mathrm{sm}}},\\\\\n D_{\\mathrm{sm}}\\mid c_n\n &\\quad\\Longleftrightarrow\\quad\n \\frac{D_{\\mathrm{sm}}}{\\gcd(D_{\\mathrm{sm}},c_A)}\n       \\mid\\prod_{j=A}^{n-1}b_j.\n \\end{aligned}\n\\]\nThe congruence is an induction, since $D_{\\mathrm{sm}}\\mid D$.\nFor the equivalence, cancel the positive greatest common divisor;\nthe remaining factor of $c_A$ is coprime to the remaining modulus.\nThus it is growth of the required prime valuations, not growth of the\nproduct as a real number, that ensures divisibility.\n\nFor the actual radices the product is $P_n/P_A$.  Its valuations at\n$2,3,5$ all tend to infinity, so every fixed $D_{\\mathrm{sm}}$ eventually\ndivides $c_n$, for any integral initial value $c_A$.  Under rationality\n$S=N/D$ in lowest terms, one may start at $A=1$:\n$c_1=DX_1=N-D$ is integral and coprime to $D_{\\mathrm{sm}}$.\nSince $P_1=2$, the criterion becomes\n$D_{\\mathrm{sm}}\\mid P_n/2=h_n$, exactly the onset already obtained\nfrom the finite-prefix identity in Proposition~\\ref{long269:res:exact-denominator}.\nThis is an alternative proof of the divisibility step, not an additional\narithmetic hypothesis left to verify for $S$.  The finite-prefix proof\nin Theorem~\\ref{long269:res:actual-cancellation} remains valid.\n\n\\begin{proposition}[segments satisfying the residue inequality exclude a positive bounded integer solution]\n\\label{long269:long:windowconsumer}\nLet $(b_n)$ and $(m_n)$ be sequences of nonnegative integers, let\n$G:\\N_{>0}\\times\\N\\to\\N$, and assume the residue condition\n\\eqref{long269:eq:actual-escape} for these sequences and $G$, using\n$|W_{\\ell,h}|>0$ as the modulus.\nFix $B>0$ coprime to $30$.  There is no integral sequence $(d_n)$ satisfying\nsimultaneously $d_{n+1}=b_nd_n-Bm_n$, $d_n>0$ and $|d_n|\\le G(B,n)$ for every\n$n\\ge0$.\n\\end{proposition}\n\n\\begin{proof}\nChoose one segment satisfying the residue inequality $(\\ell,h)$.  The iterated recurrence gives\n$d_{\\ell+h}\\equiv-BF_{\\ell,h}$ modulo $|W_{\\ell,h}|$.  The endpoint value is\npositive and at most $G(B,\\ell+h)$, whereas the least positive residue of\nthe right-hand side exceeds that bound, so Proposition~\\ref{long269:res:consumer}\napplies.\n\\end{proof}\n\nProduct growth gives room for a large residue; it does not put the\nactual residue there. For the constant data $b_n=2$, $m_n=1$, the positive\nsolution is $d_n=B$. Every segment has $W=2^h$, $F=2^h-1$ and\n$-BF\\equiv B\\pmod W$. Its least positive residue is therefore at most\n$B$, however large $W$ becomes, so no bound $G(B,n)\\ge B$ can pass the\ntest.  For the actual three-prime coefficients and $G=K$,\nTheorem~\\ref{long269:res:actual-escape-endpoint} makes the residue-inequality hypothesis\nequivalent to irrationality of $S$.  That equivalence does not hold for\nan arbitrary choice of $G$.\n\nCoprimality with $30$ selects the denominators covered by the hypothesis;\nonce a segment is fixed, the finite contradiction does not use it.  The conventions cover the edge cases: a zero product of the bases is excluded, a zero residue is\nrepresented by the full modulus, and positivity prevents the endpoint value from vanishing.  The version stated with a nonzero smooth factor assumes the exact\nfactorisation $c_n=D_{\\mathrm{sm}}d_n$, the corresponding recurrence for\n$(c_n)$ and the positive upper bound for $(d_n)$, and gives the same\ncontradiction.\n\nEven retaining the actual radix sequence and quadratic growth does not\nforce irrationality if the numerators are changed.  For the same $b_a$,\nprescribe the positive integer carries $(a+3)^2$ and set\n\\[\n \\widehat m_a=b_a(a+3)^2-(a+4)^2.\n\\]\nThe shift by three ensures positivity even at $a=0$: since $b_a\\ge2$,\n\\[\n \\widehat m_a\\ge2(a+3)^2-(a+4)^2=a^2+4a+2>0.\n\\]\nBoth these integer coefficients and the prescribed carries have quadratic\norder, because $2\\le b_a\\le30$.  Nevertheless, finite telescoping gives\n\\[\n \\sum_{a=0}^{N-1}\\frac{\\widehat m_a}{P_{a+1}}\n =9-\\frac{(N+3)^2}{P_N}\\longrightarrow9.\n\\]\nThis example keeps the exact radices, not the actual block multiplicities\nor their numerical upper bound $Q$.  It shows why the arithmetic of the\nspecific $m_a$, rather than the radix alphabet and growth orders alone,\nmust enter a proof for $S$.\n\n\\subsection{Worked finite examples}\\label{long269:long:finite-geometry}\n\nAt the first ten integer cutoffs, the running LCM changes by a factor $2$\nat $2,4,8$, by $3$ at $3,9$, and by $5$ at $5$.\nThe table in Section~\\ref{long269:sec:lcm} also shows constant values on\n$\\{5,6,7\\}$ and $\\{9,10\\}$, as in\nProposition~\\ref{long269:res:cell}. In particular,\n$\\Lprefix(10)=8\\cdot9\\cdot5=360$ is the LCM of\n$1,2,3,4,5,6,8,9,10$, the supported integers at most $10$.\n\nFor Proposition~\\ref{long269:res:fibre} take $(p,q,r)=(2,3,5)$ and the box\n$\\mathcal B(1,1,1)$, whose eight points carry the smooth values\n$1,2,3,5,6,10,15,30$ and the running LCMs\n\\[\n\\begin{array}{c|cccccccc}\np^{i}q^{j}r^{k}&1&2&3&5&6&10&15&30\\\\ \\hline\n\\hgt&1&2&6&60&60&360&360&10800\n\\end{array}\n\\]\nSix running LCMs occur, two of them twice: the points $5$ and $6$ share the running LCM\n$60$, and $10$ and $15$ share the running LCM $360$.  The identity reads\n\\[\n 1+\\tfrac12+\\tfrac16+\\tfrac1{60}+\\tfrac1{60}+\\tfrac1{360}+\\tfrac1{360}\n +\\tfrac1{10800}\n =1+\\tfrac12+\\tfrac16+\\tfrac2{60}+\\tfrac2{360}+\\tfrac1{10800}\n =\\tfrac{18421}{10800},\n\\]\nand the two coefficients $2$ carry the whole content of the regrouping on this\nbox.  Where the running LCMs are pairwise distinct the identity is a relabelling.\n\nMultiplying block radices along a run of blocks gives the product of the\nprime multipliers at the jumps in that run: for instance $b_1b_2=60$ is the product of the\nfour multipliers at $3,4,5,8$.  Block $4$ is the only one among the first six\ncontaining an internal power of each odd prime, whereas block $5$ contains neither, so\nits radix falls back to the terminal factor alone.\n\n\\subsection{The finite-scan histogram}\\label{long269:long:experiments}\n\nFor completeness we reproduce the first-success counts from the larger\nscan in Section~\\ref{long269:sec:evidence}. It tested $B\\le5000$ coprime\nto $30$ and $100\\le\\ell\\le3000$, stopping each search at depth $24$.\nThe saved output\n(\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos269/large-computations/receipts/scan-5000-100-3000.json}{scan-5000-100-3000.json})\ngives $3{,}869{,}934$ pairs, a segment satisfying the inequality in every case, and the following\nfirst-success histogram:\n\\[\n\\resizebox{\\linewidth}{!}{$\\begin{array}{c|rrrrrrrrrrrrrrr}\nh&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18\\\\ \\hline\n\\#&1&104&812&5437&51409&237423&735450&1431226&1132756&236752&34910&3076&521&49&8\n\\end{array}$}\n\\]\nThe first case attaining the maximal observed length $18$ has\n$B=917$ and $\\ell=2980$.  The smaller run\n(\\href{https://github.com/wcook04/plectis-erdos/blob/f757115e03bebe5c04431a147da559015c680a37/research/experiments/erdos269/large-computations/receipts/scan-1000-100-500.json}{scan-1000-100-500.json})\nover $B\\le1000$ coprime to $30$ and $100\\le\\ell\\le500$ gives $106{,}666$\npairs and maximal first successful length $14$, first attained at\n$B=359$ and $\\ell=291$.  The two cases have the following data.\n\\begin{center}\\small\n\\begin{tabular}{@{}lrr@{}}\n\\hline\n & Larger scan & Smaller scan\\\\\\hline\nDenominator $B$ & $917$ & $359$\\\\\nStart $\\ell$ & $2980$ & $291$\\\\\nLength $h$ & $18$ & $14$\\\\\nEndpoint jump index & $6179$ & $627$\\\\\nProduct of bases & $18139852800000000$ & $5038848000000$\\\\\nAccumulated numerator & $13196471407660025821045$ & $25864575212865807$\\\\\nLeast positive residue & $76322101735$ & $213175287$\\\\\nUpper bound & $3896420420$ & $15932659$\\\\\\hline\n\\end{tabular}\n\\end{center}\n\nThese histograms concern a bounded region.  The product-growth law gives\na necessary lower bound on a possible length of such a segment, not a distribution\nlaw for residues or first successful lengths.  No generic-residue model is\nused as evidence for the residue inequality at arbitrarily late starts.\n\n\\subsection{Source inventory}\\label{long269:long:sources}\n\\papersectiontarget{r3-source-inventory}\n\nThe table gives declaration locations at their recorded revisions.\nSection~\\ref{long269:sec:proof-sources} explains the assumptions attached\nto each kind of association. Before using the table, two conventions\nrequire attention: integer division at the initial index and removal of\nthe first block. The Lean version of\nthe normalising factor $h_a$ is a natural number computed by integer division:\nits value at $0$ is $0$, whereas $h_0=1/2$.  It agrees with $h_a$ for\n$a\\ge1$, as the identity for half the running LCM assumes.  The Lean version of the\ntail $X_a$ divides after casting to $\\mathbb R$, so agrees with\n$X_a=h_aT_a$ at every index.\n\nThe denominator-clearing lemmas and the lemmas on division of an integer recurrence assume $T_1$ rational.\nThe first block consists of $1$ alone, so $T_1=S-1$; thus $S=N/D$ supplies\n\\[\n T_1=\\frac{N-D}{D},\\qquad \\gcd(N-D,D)=\\gcd(N,D).\n\\]\nLean proves this translation, which preserves the denominator, its\nsmooth factor and the clearing onset.\n\n\\begin{longtable}{@{}p{.35\\linewidth}p{.59\\linewidth}@{}}\ninformal statement & linked Lean source\\\\\n\\endhead\ninteger represented by an exponent triple & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{32}{smooth3Val}\\\\\npure-power running LCM & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{37}{threePrimeHeight}\\\\\nlattice kernel & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{41}{threePrimeKernelQ}\\\\\nexponent triples below the cutoff & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{47}{smoothPrefixExponents}\\\\\nrunning least common multiple & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{54}{smoothPrefixLcm}\\\\\nprefix value divides the running LCM & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{60}{smooth3Val_dvd_threePrimeHeight_of_mem}\\\\\ndivisibility into the running LCM & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{78}{smoothPrefixLcm_dvd_threePrimeHeight}\\\\\nfirst pure-power membership & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{85}{pureFirst_mem_smoothPrefixExponents}\\\\\nsecond pure-power membership & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{98}{pureSecond_mem_smoothPrefixExponents}\\\\\nthird pure-power membership & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{111}{pureThird_mem_smoothPrefixExponents}\\\\\nrunning-lcm identity & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{124}{smoothPrefixLcm_eq_threePrimeHeight}\\\\\ncell relation & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{164}{SameThreePrimeLogCell}\\\\\ncell constancy of the running LCM & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{170}{threePrimeHeight_eq_of_sameLogCell}\\\\\ncell constancy of the running value & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{180}{smoothPrefixLcm_eq_of_sameLogCell}\\\\\ncell constancy of the kernel & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{192}{threePrimeKernelQ_eq_of_sameLogCell}\\\\\npositive power sets & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{205}{positivePrimePowers}\\\\\ncounting positive prime powers & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{209}{positivePrimePowers_card}\\\\\nexclusion of the origin & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{220}{one_not_mem_positivePrimePowers}\\\\\ndisjoint positive prime powers & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{230}{positivePrimePowers_disjoint}\\\\\nunion of positive prime powers & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{244}{threePrimePositiveJumpSet}\\\\\npositive jump count & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{250}{threePrimePositiveJumpSet_card}\\\\\njump set with the origin & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{274}{threePrimeJumpSetWithOrigin}\\\\\njump count with the origin & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{279}{threePrimeJumpSetWithOrigin_card}\\\\\nfirst running LCM step & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{295}{threePrimeHeight_firstLogStep}\\\\\nsecond running LCM step & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{306}{threePrimeHeight_secondLogStep}\\\\\nthird running LCM step & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{317}{threePrimeHeight_thirdLogStep}\\\\\nfirst coordinate step & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{327}{smoothPrefixLcm_firstLogStep}\\\\\nsecond coordinate step & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{340}{smoothPrefixLcm_secondLogStep}\\\\\nthird coordinate step & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{353}{smoothPrefixLcm_thirdLogStep}\\\\\nexponent box & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{369}{smoothExponentBox}\\\\\npoint running LCM & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{374}{smoothPointHeight}\\\\\nrunning LCM fibre & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{378}{smoothHeightFiber}\\\\\nfibre sum & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{385}{smoothHeightFiber_kernel_sum}\\\\\ngrouping equal denominators in a finite sum & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{407}{finiteSmoothKernelSum_groupedByHeight}\\\\\nupper bound by the cube of the cutoff & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{431}{threePrimeHeight_le_cube}\\\\\norigin value & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{444}{kernel_235_origin}\\\\\nvalue at two & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{451}{kernel_235_two}\\\\\nvalue at three & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{459}{kernel_235_three}\\\\\nvalue at six & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{468}{kernel_235_six}\\\\\nexample excluding a product of two factors & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{480}{kernel_235_not_rankOne}\\\\\nvariable-base tail step & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{488}{tailStateStep}\\\\\nexpanded tail step & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{491}{tailStateStep_eq}\\\\\nsmooth exponent block & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{501}{smoothExponentShell}\\\\\nshort-interval uniqueness & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{510}{exponent_unique_in_short_interval}\\\\\nfirst-coordinate projection & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{544}{smoothExponentShell_card_le_dropFirst}\\\\\nthird-coordinate projection & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{586}{smoothExponentShell_card_le_dropThird}\\\\\nsorted quadratic estimate & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{630}{sorted_pair_quadratic}\\\\\nquadratic block bound & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{647}{smoothExponentShell_card_quadratic}\\\\\ndyadic internal power & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{663}{DyadicInternalPower}\\\\\ninternal-power uniqueness & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{669}{dyadicInternalPower_exponent_unique}\\\\\ndyadic block base & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{751}{dyadicBlockBase235}\\\\\nexact dyadic radix alphabet & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{762}{dyadicBlockBase235_cases}\\\\\nbounded-radix consequence & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{774}{dyadicBlockBase235_mem_interval}\\\\\na nonzero $2\\times2$ determinant & \\lref{Erdos269/ThreePrimeRunningLcm.lean}{827}{kernel_235_minor_eq_neg_one_fifteen}\\\\\nno integer rotation orbit & \\lref{Erdos269/KernelCarryRank.lean}{154}{noIntegerOrbit_logb_of_prime}\\\\\nrunning LCM factorisation & \\lref{Erdos269/KernelCarryRank.lean}{218}{threePrimeHeight_factorisation}\\\\\nkernel factorisation & \\lref{Erdos269/KernelCarryRank.lean}{259}{threePrimeKernelQ_factorisation}\\\\\ntwo-prime outer product & \\lref{Erdos269/KernelCarryRank.lean}{278}{twoPrimeKernelQ_eq_outer_product}\\\\\ntwo-prime vanishing minors & \\lref{Erdos269/KernelCarryRank.lean}{298}{twoPrimeKernelQ_minor_two_eq_zero}\\\\\nuniform minors, general form & \\lref{Erdos269/KernelCarryRank.lean}{313}{exists_uniform_nonsingular_threePrimeKernel_minor}\\\\\nuniform minors for primes & \\lref{Erdos269/KernelCarryRank.lean}{376}{exists_uniform_nonsingular_threePrimeKernel_minor_of_prime}\\\\\nno finite separation & \\lref{Erdos269/KernelCarryRank.lean}{388}{not_finite_separable_threePrimeKernel}\\\\\nsecond-order minor & \\lref{Erdos269/KernelCarryRank.lean}{560}{kernel_235_minor2_eq_neg_one_fifteen}\\\\\nthird-order minor & \\lref{Erdos269/KernelCarryRank.lean}{570}{kernel_235_minor3_eq_one_over_81000}\\\\\nmisleading proportional row & \\lref{Erdos269/KernelCarryRank.lean}{583}{kernel_235_row_three_eq_smul_row_zero}\\\\\nfailure of that proportionality & \\lref{Erdos269/KernelCarryRank.lean}{592}{kernel_235_row_three_ne_smul_row_zero_at_four}\\\\\ninteger numerator for one dyadic block & \\lref{Erdos269/DyadicBlockThresholdPartition.lean}{150}{dyadicOrderedBlockDigit235}\\\\\nrecurrence for normalised tails & \\lref{Erdos269/DyadicOrderedTailRecurrence.lean}{110}{dyadicNormalizedTailStateR235_succ}\\\\\nblock summability & \\lref{Erdos269/DyadicShellSummability.lean}{142}{summable_dyadicShellMassR235}\\\\\ninfinite tail recurrence & \\lref{Erdos269/DyadicShellSummability.lean}{177}{dyadicNormalizedShellTsumTailR235_succ}\\\\\ninteger tails or separation from the integers & \\lref{Erdos269/DyadicShellSummability.lean}{188}{dyadicShellTsumTail_integer_or_cofinal_far}\\\\\nboundary divisibility & \\lref{Erdos269/RationalLatticeReduction.lean}{51}{smoothHeight_mul_prime_dvd_boundaryHeight}\\\\\nhalf the running LCM for $a\\ge1$ & \\lref{Erdos269/RationalLatticeReduction.lean}{101}{two_mul_heightNormalizer235}\\\\\nsegment clearing identity & \\lref{Erdos269/RationalLatticeReduction.lean}{158}{heightNormalizer235_mul_windowMass_eq_int}\\\\\nclearing a rational $T_1$ at $a\\ge1$ & \\lref{Erdos269/RationalLatticeReduction.lean}{196}{qsmul_normalizedTailState_eq_int_of_value_eq_rat}\\\\\ncongruence between normalised tails & \\lref{Erdos269/RationalLatticeReduction.lean}{296}{exists_normalizedTailState_collision_of_value_eq_rat}\\\\\nleast positive residue & \\lref{Erdos269/ResidueEscape.lean}{26}{leastPositiveResidue}\\\\\npositive representative range & \\lref{Erdos269/ResidueEscape.lean}{31}{leastPositiveResidue_pos_le}\\\\\nrepresentative congruence & \\lref{Erdos269/ResidueEscape.lean}{52}{leastPositiveResidue_modEq}\\\\\nresidue inequality & \\lref{Erdos269/ResidueEscape.lean}{71}{ResidueEscapesWindow}\\\\\nexcluding a bounded positive natural number & \\lref{Erdos269/ResidueEscape.lean}{76}{no_bounded_positive_state_of_residue_escape}\\\\\ncontrapositive form & \\lref{Erdos269/ResidueEscape.lean}{96}{residue_le_bound_of_bounded_positive_state}\\\\\nexcluding a bounded positive integer & \\lref{Erdos269/ResidueEscape.lean}{110}{no_bounded_positive_int_state_of_leastPositiveResidue}\\\\\nidentifying the least positive representative & \\lref{Erdos269/ResidueEscape.lean}{138}{leastPositiveResidue_eq_natAbs_of_pos_le_modEq}\\\\\nproduct of the bases & \\lref{Erdos269/RestrictedFloorSum.lean}{417}{windowBase}\\\\\naccumulated numerator in a segment & \\lref{Erdos269/RestrictedFloorSum.lean}{422}{windowForcing}\\\\\naffine iterated recurrence & \\lref{Erdos269/RestrictedFloorSum.lean}{455}{affineRecurrence_window}\\\\\nscaling the accumulated numerator & \\lref{Erdos269/RestrictedFloorSum.lean}{469}{windowForcing_const_mul}\\\\\niterating an integer recurrence & \\lref{Erdos269/RestrictedFloorSum.lean}{480}{integralCarry_window}\\\\\nendpoint residue identification & \\lref{Erdos269/RestrictedFloorSum.lean}{497}{leastPositiveResidue_windowForcing_eq_carry}\\\\\ndivisibility by the smooth part of the denominator & \\lref{Erdos269/RestrictedFloorSum.lean}{548}{smooth3Val_dvd_threePrimeHeight_of_le}\\\\\ncommon-factor cancellation & \\lref{Erdos269/RestrictedFloorSum.lean}{581}{integralCarry_cancel_commonFactor}\\\\\nbound after dividing out the common factor & \\lref{Erdos269/RestrictedFloorSum.lean}{601}{reducedCarry_pos_le_of_commonFactor}\\\\\niterated recurrence after division & \\lref{Erdos269/RestrictedFloorSum.lean}{612}{reducedIntegralCarry_window}\\\\\ncofinal segment hypothesis & \\lref{Erdos269/RestrictedFloorSum.lean}{629}{CofinalLocalWindowEscape}\\\\\nthe residue inequality excludes a positive integer solution of the reduced recurrence & \\lref{Erdos269/RestrictedFloorSum.lean}{645}{no_positive_reducedCarry_of_cofinalLocalWindowEscape}\\\\\nthe same contradiction with a common factor & \\lref{Erdos269/RestrictedFloorSum.lean}{689}{no_positive_absorbedCarry_of_cofinalLocalWindowEscape}\\\\\nreal iterated recurrence & \\lref{Erdos269/CofinalWindowEscapeEquivalence.lean}{68}{trueNormalizedState_window}\\\\\nirrationality implies the residue inequality, general bound & \\lref{Erdos269/CofinalWindowEscapeEquivalence.lean}{327}{cofinalLocalWindowEscape_of_irrational_of_beaten}\\\\\nirrationality implies the residue inequality, quadratic family & \\lref{Erdos269/CofinalWindowEscapeEquivalence.lean}{355}{cofinalLocalWindowEscape_of_irrational_of_quadratic}\\\\\nirrationality implies the residue inequality, actual bound & \\lref{Erdos269/CofinalWindowEscapeEquivalence.lean}{379}{cofinalLocalWindowEscape_of_irrational}\\\\\nresidue criterion equivalent to an irrational tail & \\lref{Erdos269/CofinalWindowEscapeEquivalence.lean}{392}{actualCofinalLocalWindowEscape_iff}\\\\\nresidue criterion equivalent to an irrational sum & \\lref{Erdos269/CofinalWindowEscapeEquivalence.lean}{398}{actualCofinalLocalWindowEscape_iff_irrational_value}\\\\\nrational $T_1$ gives eventual integer carries & \\lref{Erdos269/RationalityCarryBridge.lean}{324}{exists_reducedCarry_of_value_eq_rat}\\\\\nresidue condition for the actual series & \\lref{Erdos269/RationalityCarryBridge.lean}{397}{ActualCofinalLocalWindowEscape}\\\\\nirrationality from the residue condition & \\lref{Erdos269/RationalityCarryBridge.lean}{479}{irrational_value_of_cofinalLocalWindowEscape}\\\\\nbounded-radix alternative & \\lref{Erdos269/BoundedRadixTailEscape.lean}{89}{boundedRadix_zero_or_cofinal_far}\\\\\nrational value from an integral scaled tail & \\lref{Erdos269/BoundedRadixTailEscape.lean}{183}{rational_of_scaledTail_integer}\\\\\nzero block sums as differences of potentials & \\lref{Erdos269/ThreeChannelBlockRigidity.lean}{59}{channelBlockNull_iff_channelPotential}\\\\\na nonzero error exceeds the bound at one index & \\lref{Erdos269/CarryLiftExtinction.lean}{178}{no_carryLift_of_errorBound_below_twoPow}\\\\\na nonzero error cannot stay uniformly bounded & \\lref{Erdos269/CarryLiftExtinction.lean}{238}{no_bounded_carryLift_of_blockNull_twoAnchors}\\\\\nfirst-block sum & \\lref{Erdos269/CarryLiftExtinction.lean}{289}{binaryLift_twoAnchors_force_firstBlock_sum_one}\\\\\nfour consecutive values cannot meet all three conditions & \\lref{Erdos269/CarryLiftExtinction.lean}{308}{no_unitAccurateLift_with_twoAnchors_and_firstTwoBlockNull}\\\\\ncarry as a residue plus a potential difference & \\lref{Erdos269/WeightedPhaseCarry.lean}{109}{carry_eq_residueDigit_add_coboundary}\\\\\ncarry interval & \\lref{Erdos269/WeightedPhaseCarry.lean}{150}{carryResidue_mem_interval}\\\\\ndigit interval & \\lref{Erdos269/WeightedPhaseCarry.lean}{157}{residueDigit_mem_interval}\\\\\nmap from formal expressions to functions & \\lref{Erdos269/WeightedPhaseCarry.lean}{293}{CartierRealisation}\\\\\nfinite-dimensional span of the resulting functions & \\lref{Erdos269/WeightedPhaseCarry.lean}{334}{finite_realisedSpan_of_factorisation}\\\\\ninterval containing a Cantor-series tail & \\lref{Erdos269/PurePowerIrrationality.lean}{61}{state_mem_Ioo}\\\\\nlower bound for a difference of tails & \\lref{Erdos269/PurePowerIrrationality.lean}{74}{stateGap_lower_bound}\\\\\nsmall-form criterion & \\lref{Erdos269/PurePowerIrrationality.lean}{94}{irrational_of_small_nonzero_forms}\\\\\nirrationality from clearing and small nonzero gaps & \\lref{Erdos269/PurePowerIrrationality.lean}{156}{irrational_of_clearing_and_small_gaps}\\\\\nindices giving the threshold staircase & \\lword{Shared/IrrationalRotationStaircase.lean}{271}{exists_staircase_indices}{staircase indices}\\\\\nfinite threshold-column rank, including the extreme-column defect & \\href{https://github.com/wcook04/plectis-erdos/blob/d11bd0b16f2c0b68d722777762ebbd4f96a12db8/lean/ErdosProblems/Erdos269/PaperR7FiniteCutRank.lean#L182}{finite cut rank}\\\\\nuniform admissible modular minors, with indices chosen before modulus and layer & \\href{https://github.com/wcook04/plectis-erdos/blob/d11bd0b16f2c0b68d722777762ebbd4f96a12db8/lean/ErdosProblems/Erdos269/PaperR7ModularMinors.lean#L133}{uniform modular minors}\\\\\n\\end{longtable}\n% Part: family_catalogue\n\\clearpage\n\\section{Dependencies and remaining questions}\n\\label{sec:erdos-269-complete-family-map}\nThe table separates the established special cases from the ordinary\ngeneralisations awaiting formalisation and the unresolved repeated-sum\ncondition. Exact kernel rank concerns separation of a function; the\nresidue criterion concerns a scalar sum and still needs arithmetic\nexistence for its segments.\n\n\\begin{center}\\small\n\\begin{tabular}{p{0.26\\linewidth}p{0.64\\linewidth}}\n\\hline\nResult & Dependency and limit\\\\\\hline\nRunning LCMs and intervals of constancy & Prime-power divisibility and unique factorisation.\nTheir geometry alone gives no irrationality statement.\\\\\nTwo-prime values & Earlier factorisation plus the cited external\nHecke--Mahler value theorem.  The reduction to that theorem is checked in\nLean; the theorem itself is a cited input (Section~\\ref{long269:sec:coverage}).\\\\\nDistinct values, every $P$ & Integrality of the scaled tails,\nrearranged chains at a return of the logarithmic flow, Kronecker's theorem\nand an Euler characteristic count on a two-dimensional subtorus.  Ordinary\nproof checked by a second agent; no Lean proof.  It stops for the repeated sums because\ntheir scaled tails are unbounded.\\\\\nPrime-power subseries & The same two-tail comparison with the integer $f_p$\nand a chain chosen near the origin of the torus; no two-wall lemma.\nOrdinary proof checked by a second agent; no Lean proof.  Says nothing about the whole\nrepeated sum.\\\\\nDistinct values, blocks & Dyadic blocks, finitely many affine maps with\ndisjoint images, integrality of the scaled tails and the irrationality of\n$\\log p/\\log q$.  The theorem for $\\{2,3,5\\}$ is checked in Lean; the\ngeneral criterion has an ordinary proof, and\nRemark~\\ref{long269:res:distinct-height-sets} adds exact finite\ncomputations.  No statement about the repeated sums.\\\\\nArbitrary-order rank & Diagonal rescaling, separate one-dimensional\ndensities and a staircase determinant; checked in Lean.\\\\\nTail recurrence & Counting the original multiplicities, normalising by\nhalf the boundary running LCM, and convergence; an abstract recurrence would\nnot identify the sum.\\\\\nDenominator reduction & A finite prefix with cleared denominators and\nan upper bound for positive integer tails.  The first valid index is\ndetermined exactly (Proposition~\\ref{long269:res:exact-denominator}); checked\nin Lean.\\\\\nResidue criterion & The actual recurrence, a dominating bound and segment\nproducts comparable to $8^h$; the inequality at arbitrarily late starts remains\nunproved.\\\\\nWeighted differences & Exact reindexing and the contributions of the\nnew boundary strips; integer enumeration checks the displayed\ncounterexample to third-difference cancellation.\\\\\nFinite computations & Direct checks settle only their individual\nsegments.  The two large exclusions have published witnesses and exact\ncheckers, and the larger scan a published saved run; none is a theorem\ninput.\\\\\\hline\n\\end{tabular}\n\\end{center}\nNone of these results proves irrationality of the repeated sum\n$\\mathcal R_P$ for three or more primes.\n\n\\subsection*{The formal two-prime reduction}\n\\label{long269:sec:coverage}\n\\papersectiontarget{coverage}\n\nThe source note in Section~\\ref{long269:sec:proof-sources} distinguishes\nordinary proofs from individual formal associations. For\nTheorem~\\ref{long269:res:lead-two-prime}, the Lean development\nverifies the reduction of that theorem to the transcendence of the Hecke--Mahler boundary sum\nat an irrational $\\theta\\in(0,1)$ and nonzero algebraic $\\beta,\\alpha$ with\n$|\\beta|<1$ and $|\\beta||\\alpha|^\\theta<1$, which is Theorem~1.1 of Bugeaud\nand Laurent~\\cite{bugeaudlaurent2023} in the case $\\rho=0$, due to Loxton and\nvan der Poorten~\\cite[Theorem~8, p.~40]{loxtonvdp1977}.  Lean does not\nprove that input; the Lean theorem assumes it.  The formal development checks the following parts of the reduction:\nthe running least-common-multiple identity, the selection of the single power\nof $q$ between consecutive powers of $p$, the factorisation of the repeated\nsum, the identification of the distinct running values, the telescoping\nidentity, the boundary\nidentity~\\eqref{long269:eq:hecke-mahler-boundary} expressing $A$ through the\nHecke--Mahler sum, the irrationality of $\\log p/\\log q$, and each side\ncondition verified before the value theorem is applied.  The two identities\nexpressing $\\mathcal D_{\\{p,q\\}}$ and $\\mathcal R_{\\{p,q\\}}$ as polynomials in\n$A$ are unconditional and carry no additional input.\n\nA theorem whose own statement is conditional is formalised exactly as stated.\nTheorem~\\ref{long269:res:lead-two-prime} is asserted unconditionally in this\nrecord, and its Lean counterpart carries the additional named input described\nhere.\n% Part: back\n\n\\begin{thebibliography}{99}\n\\bibitem{erdosgraham1980} Paul Erd\\H{o}s and Ronald L. 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Meijer,\n  \\emph{On integers generated by a finite number of fixed primes}, Compositio Mathematica \\textbf{29}, no.~3 (1974), 273--286,\n  \\href{https://www.numdam.org/article/CM_1974__29_3_273_0.pdf}{source}.\n\\bibitem{languasco2025} Alessandro Languasco, Florian Luca, Pieter Moree and Alain Togb\\'{e},\n  \\emph{Sequences of integers generated by two fixed primes}, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg \\textbf{95} (2025), 123--148,\n  \\href{https://doi.org/10.1007/s12188-025-00293-9}{doi:\\nolinkurl{10.1007/s12188-025-00293-9}}; arXiv:\\href{https://arxiv.org/abs/2309.12806}{2309.12806}.\n\\bibitem{adamczewskibugeaud2007} Boris Adamczewski and Yann Bugeaud,\n  \\emph{On the complexity of algebraic numbers I. Expansions in integer bases},\n  Annals of Mathematics \\textbf{165}, no.~2 (2007), 547--565,\n  \\href{https://doi.org/10.4007/annals.2007.165.547}{doi:\\nolinkurl{10.4007/annals.2007.165.547}}.\n\\bibitem{pellarin2006} Federico Pellarin,\n  \\emph{On the arithmetic properties of complex values of Hecke--Mahler series I.\n  The rank one case}, Annali della Scuola Normale Superiore di Pisa,\n  Classe di Scienze (5) \\textbf{5}, no.~3 (2006), 329--374,\n  \\href{https://www.numdam.org/item/ASNSP_2006_5_5_3_329_0/}{published source}.\n\\bibitem{adamczewskifaverjon2023} Boris Adamczewski and Colin Faverjon,\n  \\emph{A new proof of Nishioka's theorem in Mahler's method},\n  Comptes Rendus. Math\\'ematique \\textbf{361} (2023), 1011--1028,\n  \\href{https://doi.org/10.5802/crmath.458}{doi:\\nolinkurl{10.5802/crmath.458}}.\n\\bibitem{adamczewskifaverjon2026} Boris Adamczewski and Colin Faverjon,\n  \\emph{Mahler's method in several variables and finite automata},\n  Annals of Mathematics \\textbf{204}, no.~2 (2026), 455--533,\n  \\href{https://doi.org/10.4007/annals.2026.204.2.1}{doi:\\nolinkurl{10.4007/annals.2026.204.2.1}}.\n  Online 13 September 2026; locators here refer to the\n  \\href{https://faverjon.perso.math.cnrs.fr/AdamczewskiFaverjon_MahlerFiniteAutomata.pdf}{68-page author manuscript}.\n\\end{thebibliography}\n% ---- part extended_record ----\n% ---- part family_catalogue ----\n% ---- part back ----\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":236046,"sha256":"sha256:27b68b4b0cf24ce2b4ca045ed96a303fc713164e33a2e646792c62fcddfa4f4c","cite_as":"Plectis reader corpus, paper:erdos269-running-lcm-reasoning-surface","source_repository":"plectis-erdos","source_path":"paper/269/erdos269-running-lcm-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:088d36d7ba3d3ee0cee2df99e8915927999698f19889f8ea0b8d55b3721ec778","byte_count":6923,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% This is the only copy of the shared preamble.  The eight short notes, the\n% long records assembled from paper/reasoning-parts other than #249 and #257\n% (which carry their own preambles), and the synthesis paper include it with\n% \\input{problem-note-preamble}; paper/Makefile builds from this directory\n% with the directory on the tectonic search path.\n%\n% Shared preamble for the Erdős Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erdős problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{8e58c395e9807ef3374d7365c62cf04d20ee6331}\n\\newcommand{\\commitshort}{9e231ce4371f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-erdos/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-erdos/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\usepackage{longtable}  % the notes' declaration tables; loaded here so the assembled records inherit it\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n% ---------------------------------------------------------------------------\n% Evidence marks.  paper-evidence.tex places \"Lean\" and \"Comparator\" links in\n% the margin beside each result from the paper's generated file\n% paper/evidence/<paper>.tex, which each paper inputs after this preamble.\n% ---------------------------------------------------------------------------\n\\input{paper-evidence}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Production provenance.  The wording is fixed by the corpus provenance\n% contract and reproduced verbatim in every manuscript in this repository, so\n% the papers cannot drift into different accounts of how they were made.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erdős problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every Lean mark in the margin points to propositions that the\n  pinned Lean kernel accepts, with no \\texttt{sorry}, no added axiom, and no\n  unchecked evaluation.  That claim concerns the formal statements only; it\n  says nothing about their mathematical interest, their novelty, or the\n  original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n\n% Compact opening card for the separately stated Comparator theorem selected\n% from each problem note.  The prose supplied by each note owns the\n% contribution and relevance judgements; the shared final sentence states the\n% mechanical assurance boundary once and points to the public evidence owner.\n\\newcommand{\\externalreviewobject}[3]{%\n  \\begin{quote}\\small\n  \\textbf{Contribution.} #1\\par\n  \\textbf{Relation to the open problem.} #2\\par\n  \\textbf{Executable review object.} #3  The repository's\n  external-verification job compares these exact Lean propositions with\n  separately declared challenge statements and an axiom budget, then asks\n  Lean's kernel to check the submitted proofs.  The\n  \\href{https://github.com/wcook04/plectis-erdos/blob/ca0e13f8acf5ccf48506e4bdb870953d3a0856fa/formalization.yaml}%\n  {formalisation manifest} and the commit-bound CI receipt record that check;\n  they do not assess novelty, significance, or whether the original problem is\n  solved.\n  \\end{quote}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  Under\n% Tectonic (XeTeX) the text and mathematics are set in TeX Gyre Pagella from\n% the bundled OpenType fonts, with a Heros sans range for \\mathsf and\n% Inconsolata for monospace; the pdflatex fallback keeps the same Palatino\n% design through newpx.  The A4 text block uses a classical measure with\n% quiet running heads and restrained ink.  Link colours: steel blue for\n% external (Lean-source and web) targets, deep burgundy for internal\n% cross-references and citations.\n\\NeedsTeXFormat{LaTeX2e}\n\\ProvidesPackage{paper-house-style}[2026/07/19 Pagella house style for the papers]\n\n\\RequirePackage{iftex}\n\\RequirePackage{amsmath}\n\\RequirePackage{amsthm}\n\\ifpdftex\n  \\RequirePackage[T1]{fontenc}\n  \\RequirePackage[utf8]{inputenc}\n  \\RequirePackage{amssymb}\n  \\RequirePackage{newpxtext}\n  \\RequirePackage{newpxmath}\n\\else\n  \\RequirePackage{fontspec}\n  \\RequirePackage{unicode-math}\n  \\setmainfont{texgyrepagella}[Extension=.otf,UprightFont=*-regular,\n    BoldFont=*-bold,ItalicFont=*-italic,BoldItalicFont=*-bolditalic]\n  \\setmathfont{texgyrepagella-math.otf}\n  \\setmathfont{texgyreheros-regular.otf}[range=sfup,Scale=MatchLowercase]\n  \\setmonofont{Inconsolatazi4}[Extension=.otf,UprightFont=*-Regular,\n    BoldFont=*-Bold,Scale=MatchLowercase]\n\\fi\n\n\\RequirePackage[a4paper,textwidth=418pt,textheight=636pt,hcentering,\n  top=68pt,headheight=15pt,headsep=20pt,footskip=34pt]{geometry}\n\\RequirePackage{microtype}\n\\linespread{1.045}\n\\clubpenalty=10000 \\widowpenalty=10000 \\tolerance=1600\n\n\\RequirePackage[dvipsnames,table]{xcolor}\n\\definecolor{linkexternal}{HTML}{275D8C}\n\\definecolor{linkinternal}{HTML}{6E2B2B}\n\\definecolor{papertableheader}{HTML}{E3EAF0}\n\\definecolor{papertablerow}{HTML}{F2F5F7}\n\\definecolor{papertablerule}{HTML}{7D8991}\n\\definecolor{paperresultrule}{HTML}{315B72}\n\n% Stable cross-document links between sibling PDFs. A named destination is\n% more durable than a page number: sections may move while their semantic role\n% remains the same. All public PDFs are copied to one directory at release.\n\\newcommand{\\papersectiontarget}[1]{\\hypertarget{#1}{}}\n\\newcommand{\\papersectionlink}[3]{\\href{#1\\##2}{#3}}\n\n\\RequirePackage{titlesec}\n\\titleformat{\\section}{\\Large\\bfseries}{\\thesection}{0.85em}{}\n\\titlespacing*{\\section}{0pt}{2.6ex plus .8ex minus .3ex}{1.5ex plus .3ex}\n\\titleformat{\\subsection}{\\large\\bfseries}{\\thesubsection}{0.8em}{}\n\\titlespacing*{\\subsection}{0pt}{2.1ex plus .6ex minus .2ex}{1.1ex plus .2ex}\n\\titlespacing*{\\paragraph}{0pt}{1.6ex plus .5ex minus .2ex}{0.85em}\n\n\\RequirePackage{fancyhdr}\n\\newcommand{\\paperrunninghead}{}\n\\newcommand{\\runninghead}[1]{\\renewcommand{\\paperrunninghead}{#1}}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\fancyhead[L]{\\small\\itshape\\paperrunninghead}\n\\fancyhead[R]{\\small\\thepage}\n\\renewcommand{\\headrulewidth}{0pt}\n\\fancypagestyle{plain}{\\fancyhf{}\\renewcommand{\\headrulewidth}{0pt}}\n\n\\RequirePackage{caption}\n\\captionsetup{font=small,labelfont=bf,skip=9pt,width=.92\\linewidth}\n\\RequirePackage{etoolbox}\n\\RequirePackage{pdflscape}\n\\renewcommand{\\arraystretch}{1.20}\n\\newcommand{\\papertablehead}{\\rowcolor{papertableheader}}\n\\newcommand{\\papertablebody}{\\rowcolors{2}{white}{papertablerow}}\n\\AtBeginEnvironment{tabular}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\\AtBeginEnvironment{longtable}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\n% A compact front-matter panel for the result a scanning reader should retain.\n% It is deliberately short and unbreakable: manuscripts should state one\n% result, its evidence, and its limit here, then supply the argument below.\n\\newenvironment{paperresult}[1]{%\n  \\par\\addvspace{0.7em}\n  \\begin{center}\\begin{minipage}{0.94\\linewidth}\n    {\\color{paperresultrule}\\rule{\\linewidth}{0.8pt}}\\par\\vspace{4pt}\n    {\\small\\sffamily\\bfseries\\color{paperresultrule}#1}\\par\\smallskip\n    \\small\\color{black}\n}{%\n    \\par\\vspace{4pt}{\\color{paperresultrule}\\rule{\\linewidth}{0.45pt}}\n    \\end{minipage}\\end{center}\\par\\addvspace{0.5em}\n}\n\n% Shared diagram grammar.  Reader-facing figures use the same restrained\n% hierarchy as the surrounding page: a short eyebrow, a substantial title,\n% one compact explanatory line, and an optional evidence badge.  The styles\n% are installed only when a manuscript loads TikZ.\n\\AtBeginDocument{%\n  \\@ifpackageloaded{tikz}{%\n    \\tikzset{\n      paper box/.style={draw=gray!48, line width=.45pt, rounded corners=3pt,\n        fill=white, align=left, inner xsep=7pt, inner ysep=6pt,\n        font=\\small},\n      paper card/.style={paper box, minimum height=13mm},\n      paper stage/.style={paper box, align=center, minimum height=14mm},\n      paper arrow/.style={-{Stealth[length=2.15mm,width=1.45mm]},\n        line width=.7pt, draw=gray!72!black},\n      paper feedback/.style={paper arrow, densely dashed,\n        draw=Violet!72!black},\n      paper eyebrow/.style={font=\\scriptsize\\sffamily\\bfseries,\n        text=gray!70!black},\n      paper label/.style={font=\\scriptsize\\sffamily,\n        text=gray!72!black, fill=white, inner sep=1.5pt},\n      paper badge/.style={rounded corners=2pt, inner xsep=4pt, inner ysep=2pt,\n        font=\\scriptsize\\sffamily\\bfseries},\n      paper boundary/.style={densely dashed, line width=.55pt,\n        draw=gray!62}\n    }%\n  }{}%\n}\n\n% Front matter: title block with an optional italic subtitle, small-caps\n% author, and a hairline rule; the abstract is set small with a small-caps\n% head and symmetric indents.\n\\newcommand{\\subtitle}[1]{\\gdef\\@papersubtitle{#1}}\n\\renewenvironment{abstract}{%\n  \\vspace{0.1em}\\small\n  \\begin{center}{\\normalsize\\scshape Abstract}\\end{center}\\vspace{-0.5em}\n  \\list{}{\\leftmargin 1.3em\\rightmargin\\leftmargin\\listparindent 0pt\n    \\parsep 0.32em\\itemindent 0pt}\\item\\relax}\n  {\\endlist\\vspace{0.5em}}\n\\def\\@maketitle{%\n  \\newpage\\null\n  \\vspace*{-2.6em}%\n  \\begin{center}%\n    {\\LARGE\\bfseries\\@title\\par}%\n    \\vskip 0.55em\n    {\\large\\itshape\\@papersubtitle\\par}%\n    \\vskip 1.1em\n    {\\scshape\\@author\\textsuperscript{*}\\par}%\n    \\vskip 0.25em\n    {\\small\\@date\\par}%\n    \\vskip 0.8em\n    \\rule{6.5em}{0.4pt}%\n  \\end{center}%\n  \\vskip 0.9em}\n\n% Production provenance, set at the foot of the first page.\n%\n% A reader who meets a single personal byline ordinarily infers that the named\n% author drafted the sentences.  For these papers that inference is false, and\n% a declaration reached only in the back matter would leave it standing for the\n% length of the paper.  The note therefore appears on page one, small and\n% visually subordinate to the title and abstract: enough to prevent the false\n% inference, not enough to make the production method the subject.\n%\n% It carries no mark, because it qualifies the whole manuscript rather than any\n% one passage.  \\footnotetext does not advance the footnote counter, and the\n% emptied \\thefootnote is confined to a group, so numbered footnotes elsewhere\n% in the paper are unaffected.\n%\n% The wording is not authored here.  It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"},{"record_type":"paper_source_dependency","tex_reference":"evidence/erdos269-running-lcm-reasoning-surface","source_path":"paper/evidence/erdos269-running-lcm-reasoning-surface.tex","media_type":"text/x-tex","sha256":"sha256:f62bede753be77e31c5015dc5546ed26b63a49bdfb0931c7c552418a39063de4","byte_count":28731,"licence":"CC-BY-4.0","body_format":"latex","body":"% Generated by scripts/paper_evidence.py from docs/paper_lean_coverage.json and the\n% Comparator replay receipts under evidence/comparator/.  Do not edit by hand.\n\\newcommand{\\evidencerecordurl}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md}\n\\DeclareResultEvidenceName{long269:res:distinct-height-235}{Theorem 1.1}\n\\DeclareResultEvidenceRecord{long269:res:distinct-height-235}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-distinct-height-235}{exact}{pending}\n\\DeclareResultEvidence{long269:res:distinct-height-235}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/DistinctHeightIrrationality.lean\\#L689}{}\n\\DeclareResultEvidenceName{long269:res:lead-two-prime}{Theorem 1.2}\n\\DeclareResultEvidenceRecord{long269:res:lead-two-prime}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-lead-two-prime}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{long269:res:lead-two-prime}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-lead-two-prime}{}\n\\DeclareResultEvidenceName{long269:res:lcm}{Theorem 3.1}\n\\DeclareResultEvidenceRecord{long269:res:lcm}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-lcm}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:lcm}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperCompleteR20/RealCutoffs.lean\\#L49}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-lcm-comparator}\n\\DeclareResultEvidenceName{long269:res:cell}{Proposition 3.2}\n\\DeclareResultEvidenceRecord{long269:res:cell}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-cell}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:cell}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-cell}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-cell-comparator}\n\\DeclareResultEvidenceName{long269:res:count}{Proposition 3.3}\n\\DeclareResultEvidenceRecord{long269:res:count}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-count}{exact}{compared}\n\\DeclareResultEvidence{long269:res:count}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR7BasicAssembly.lean\\#L69}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-count-comparator}\n\\DeclareResultEvidenceName{long269:res:fibre}{Proposition 3.4}\n\\DeclareResultEvidenceRecord{long269:res:fibre}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-fibre}{exact}{compared}\n\\DeclareResultEvidence{long269:res:fibre}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean\\#L407}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-fibre-comparator}\n\\DeclareResultEvidenceName{long269:res:short}{Lemma 3.5}\n\\DeclareResultEvidenceRecord{long269:res:short}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-short}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:short}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperCompleteR20/RealBaseShortInterval.lean\\#L10}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-short-comparator}\n\\DeclareResultEvidenceName{long269:res:drop}{Proposition 3.6}\n\\DeclareResultEvidenceRecord{long269:res:drop}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-drop}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:drop}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperCompleteR20/RealCutoffs.lean\\#L285}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-drop-comparator}\n\\DeclareResultEvidenceName{long269:res:distinct-height-all}{Remark 9, p. 10}\n\\DeclareResultEvidenceRecord{long269:res:distinct-height-all}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-distinct-height-all}{none}{not_applicable}\n\\DeclareResultEvidenceName{long269:res:dp-integral-tails}{Remark 10, p. 11}\n\\DeclareResultEvidenceRecord{long269:res:dp-integral-tails}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-dp-integral-tails}{none}{not_applicable}\n\\DeclareResultEvidenceName{long269:res:dp-blocks}{Remark 11, p. 11}\n\\DeclareResultEvidenceRecord{long269:res:dp-blocks}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-dp-blocks}{none}{not_applicable}\n\\DeclareResultEvidenceName{long269:res:dp-short}{Remark 12, p. 12}\n\\DeclareResultEvidenceRecord{long269:res:dp-short}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-dp-short}{none}{not_applicable}\n\\DeclareResultEvidenceName{long269:res:dp-two-walls}{Remark 13, p. 12}\n\\DeclareResultEvidenceRecord{long269:res:dp-two-walls}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-dp-two-walls}{none}{not_applicable}\n\\DeclareResultEvidenceName{long269:res:distinct-height-blocks}{Remark 14, p. 15}\n\\DeclareResultEvidenceRecord{long269:res:distinct-height-blocks}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-distinct-height-blocks}{none}{not_applicable}\n\\DeclareResultEvidenceName{long269:res:distinct-height-criterion}{Remark 15, p. 17}\n\\DeclareResultEvidenceRecord{long269:res:distinct-height-criterion}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-distinct-height-criterion}{none}{not_applicable}\n\\DeclareResultEvidenceName{long269:res:distinct-height-sets}{Remark 16, p. 17}\n\\DeclareResultEvidenceRecord{long269:res:distinct-height-sets}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-distinct-height-sets}{none}{not_applicable}\n\\DeclareResultEvidenceName{long269:res:single-prime-identities}{Remark 17, p. 20}\n\\DeclareResultEvidenceRecord{long269:res:single-prime-identities}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-single-prime-identities}{none}{not_applicable}\n\\DeclareResultEvidenceName{long269:res:single-prime-subsums}{Remark 18, p. 20}\n\\DeclareResultEvidenceRecord{long269:res:single-prime-subsums}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-single-prime-subsums}{none}{not_applicable}\n\\DeclareResultEvidenceName{long269:lem:late-swap}{Remark 19, p. 22}\n\\DeclareResultEvidenceRecord{long269:lem:late-swap}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-lem-late-swap}{none}{not_applicable}\n\\DeclareResultEvidenceName{long269:lem:clock-swaps}{Remark 20, p. 23}\n\\DeclareResultEvidenceRecord{long269:lem:clock-swaps}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-lem-clock-swaps}{none}{not_applicable}\n\\DeclareResultEvidenceName{long269:thm:clock-relations}{Remark 21, p. 24}\n\\DeclareResultEvidenceRecord{long269:thm:clock-relations}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-thm-clock-relations}{none}{not_applicable}\n\\DeclareResultEvidenceName{long269:res:two-prime-rank}{Proposition 7.1}\n\\DeclareResultEvidenceRecord{long269:res:two-prime-rank}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-two-prime-rank}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:two-prime-rank}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperCompleteR20/RealTwoPrimeKernel.lean\\#L63}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-two-prime-rank-comparator}\n\\DeclareResultEvidenceName{long269:res:rank}{Proposition 7.2}\n\\DeclareResultEvidenceRecord{long269:res:rank}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-rank}{exact}{compared}\n\\DeclareResultEvidence{long269:res:rank}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR7BasicAssembly.lean\\#L102}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-rank-comparator}\n\\DeclareResultEvidenceName{long269:res:infinite-rank}{Theorem 7.3}\n\\DeclareResultEvidenceRecord{long269:res:infinite-rank}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-infinite-rank}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:infinite-rank}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR7BasicAssembly.lean\\#L22}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-infinite-rank-comparator}\n\\DeclareResultEvidenceName{long269:res:finite-cut-rank}{Proposition 7.4}\n\\DeclareResultEvidenceRecord{long269:res:finite-cut-rank}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-finite-cut-rank}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:finite-cut-rank}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR7FiniteCutRank.lean\\#L182}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-finite-cut-rank-comparator}\n\\DeclareResultEvidenceName{long269:lem:staircase-smith}{Remark 26, p. 28}\n\\DeclareResultEvidenceRecord{long269:lem:staircase-smith}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-lem-staircase-smith}{none}{not_applicable}\n\\DeclareResultEvidenceName{long269:res:uniform-rank}{Theorem 7.6}\n\\DeclareResultEvidenceRecord{long269:res:uniform-rank}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-uniform-rank}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:uniform-rank}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR8UniformRank.lean\\#L320}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-uniform-rank-comparator}\n\\DeclareResultEvidenceName{long269:res:dyadic-alphabet}{Proposition 8.1}\n\\DeclareResultEvidenceRecord{long269:res:dyadic-alphabet}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-dyadic-alphabet}{exact}{compared}\n\\DeclareResultEvidence{long269:res:dyadic-alphabet}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-dyadic-alphabet}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-dyadic-alphabet-comparator}\n\\DeclareResultEvidenceName{long269:res:actual-orbit}{Theorem 8.2}\n\\DeclareResultEvidenceRecord{long269:res:actual-orbit}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-actual-orbit}{exact}{compared}\n\\DeclareResultEvidence{long269:res:actual-orbit}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR7SeriesIdentification.lean\\#L189}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-actual-orbit-comparator}\n\\DeclareResultEvidenceName{long269:res:literal-triangle}{Lemma 8.3}\n\\DeclareResultEvidenceRecord{long269:res:literal-triangle}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-literal-triangle}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:literal-triangle}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperCompleteR20/LiteralTriangleReal.lean\\#L193}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-literal-triangle-comparator}\n\\DeclareResultEvidenceName{long269:res:actual-dichotomy}{Proposition 8.4}\n\\DeclareResultEvidenceRecord{long269:res:actual-dichotomy}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-actual-dichotomy}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:actual-dichotomy}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR7ActualOrbit.lean\\#L132}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-actual-dichotomy-comparator}\n\\DeclareResultEvidenceName{long269:res:actual-tail-bound}{Theorem 9.1}\n\\DeclareResultEvidenceRecord{long269:res:actual-tail-bound}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-actual-tail-bound}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:actual-tail-bound}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR8RankMajorant.lean\\#L357}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-actual-tail-bound-comparator}\n\\DeclareResultEvidenceName{long269:res:all-scale-lattice}{Lemma 9.2}\n\\DeclareResultEvidenceRecord{long269:res:all-scale-lattice}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-all-scale-lattice}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:all-scale-lattice}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperCompleteR20/BoundedLatticeCollision.lean\\#L56}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-all-scale-lattice-comparator}\n\\DeclareResultEvidenceName{long269:res:actual-cancellation}{Theorem 9.3}\n\\DeclareResultEvidenceRecord{long269:res:actual-cancellation}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-actual-cancellation}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:actual-cancellation}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-actual-cancellation}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-actual-cancellation-comparator}\n\\DeclareResultEvidenceName{long269:res:exact-denominator}{Proposition 9.4}\n\\DeclareResultEvidenceRecord{long269:res:exact-denominator}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-exact-denominator}{exact}{compared}\n\\DeclareResultEvidence{long269:res:exact-denominator}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-exact-denominator}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-exact-denominator-comparator}\n\\DeclareResultEvidenceName{long269:res:pinning}{Proposition 9.5}\n\\DeclareResultEvidenceRecord{long269:res:pinning}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-pinning}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:pinning}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperCompleteR20/EightScaleRigidity.lean\\#L77}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-pinning-comparator}\n\\DeclareResultEvidenceName{long269:res:jump-constrained-bound}{Proposition 9.6}\n\\DeclareResultEvidenceRecord{long269:res:jump-constrained-bound}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-jump-constrained-bound}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:jump-constrained-bound}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/ActualSharpTailMajorantR10.lean\\#L339}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-jump-constrained-bound-comparator}\n\\DeclareResultEvidenceName{long269:res:consumer}{Proposition 10.1}\n\\DeclareResultEvidenceRecord{long269:res:consumer}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-consumer}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:consumer}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/ResidueEscape.lean\\#L110}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-consumer-comparator}\n\\DeclareResultEvidenceName{long269:res:actual-escape-endpoint}{Theorem 10.2}\n\\DeclareResultEvidenceRecord{long269:res:actual-escape-endpoint}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-actual-escape-endpoint}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:actual-escape-endpoint}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperCompleteR20/OcticEscapeWhole.lean\\#L50}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-actual-escape-endpoint-comparator}\n\\DeclareResultEvidenceName{long269:res:residue-limit}{Proposition 10.3}\n\\DeclareResultEvidenceRecord{long269:res:residue-limit}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-residue-limit}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:residue-limit}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-residue-limit}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-residue-limit-comparator}\n\\DeclareResultEvidenceName{long269:res:window-growth}{Proposition 10.4}\n\\DeclareResultEvidenceRecord{long269:res:window-growth}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-window-growth}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:window-growth}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR7WindowResults.lean\\#L196}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-window-growth-comparator}\n\\DeclareResultEvidenceName{long269:res:no-bounded-length}{Corollary 10.5}\n\\DeclareResultEvidenceRecord{long269:res:no-bounded-length}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-no-bounded-length}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:no-bounded-length}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR7WindowResults.lean\\#L267}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-no-bounded-length-comparator}\n\\DeclareResultEvidenceName{long269:res:weighted-shift-identity}{Lemma 12.1}\n\\DeclareResultEvidenceRecord{long269:res:weighted-shift-identity}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-weighted-shift-identity}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:weighted-shift-identity}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperCompleteR20/WeightedShiftValue.lean\\#L216}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-weighted-shift-identity-comparator}\n\\DeclareResultEvidenceName{long269:res:strip-decomposition}{Proposition 12.2}\n\\DeclareResultEvidenceRecord{long269:res:strip-decomposition}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-strip-decomposition}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:strip-decomposition}{Lean}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-strip-decomposition}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-strip-decomposition-comparator}\n\\DeclareResultEvidenceName{long269:res:fixed-base-recoding}{Proposition 12.4}\n\\DeclareResultEvidenceRecord{long269:res:fixed-base-recoding}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-fixed-base-recoding}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:fixed-base-recoding}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperCompleteR20/FixedBaseRecoding.lean\\#L154}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-fixed-base-recoding-comparator}\n\\DeclareResultEvidenceName{long269:res:tails-equivalence}{Proposition 12.6}\n\\DeclareResultEvidenceRecord{long269:res:tails-equivalence}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-tails-equivalence}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:res:tails-equivalence}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR7RationalBridge.lean\\#L214}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-res-tails-equivalence-comparator}\n\\DeclareResultEvidenceName{long269:long:denominator-reduction}{Proposition 13.1}\n\\DeclareResultEvidenceRecord{long269:long:denominator-reduction}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-long-denominator-reduction}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:long:denominator-reduction}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperCompleteR20/RealBoundDenominatorReduction.lean\\#L25}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-long-denominator-reduction-comparator}\n\\DeclareResultEvidenceName{long269:long:windowconsumer}{Proposition 13.2}\n\\DeclareResultEvidenceRecord{long269:long:windowconsumer}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-long-windowconsumer}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long269:long:windowconsumer}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/RestrictedFloorSum.lean\\#L645}{https://github.com/wcook04/plectis-erdos/blob/b4d9b60327e3c4d4cc50acb37e62268788cd77e0/evidence/erdos269-running-lcm-reasoning-surface.md\\#long269-long-windowconsumer-comparator}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/269/erdos269-running-lcm-reasoning-surface.tex","section_outline":["Two sums from running least common multiples","Recovering the prime-power jumps","Intervals of constancy of the running LCM","The two-prime Hecke--Mahler reduction","The sums over distinct running LCMs","Proof for an arbitrary finite prime set","Blocks and affine maps","A decoding criterion for every finite set of primes","Certificates for three and four primes","Repeated tails and prime-power subseries","Words obtained from a linear flow","Separation of the three-prime kernel","Smith factors of the selected staircase","Uniform approximation of the rescaled kernel","The recurrence for tails between powers of two","Bounding the tails and clearing a rational denominator","A bound using the order of the jumps","A residue criterion and the bounds it allows","How fast the product of the bases grows","Finite computations and their limits","Checking individual segments","Finite denominator bounds","The remaining arithmetic questions","An exact weighted-shift identity for the repeated series","Boundary terms in weighted finite differences","Function-theoretic representations","Hypotheses for auxiliary integer recurrences","The unresolved repeated-sum question","Verification and reproducibility","Further examples, conditional lemmas and source references","The historical and catalogue record","Integer recurrences in their general form","Worked finite examples","The finite-scan histogram","Source inventory","Dependencies and remaining questions","The formal two-prime reduction"]},"erdos-269-three-prime-running-lcm":{"paper_id":"erdos-269-three-prime-running-lcm","title":"Distinct running least common multiples","form":"Problem note","length_class":"short","pages":16,"subject":"Erdős #269","subject_kind":"problem","problem_id":"erdos_269","landing_role":"mathematics","question":"Why is the sum over distinct running LCMs of the 5-smooth integers irrational?","precis":"A five-map argument proves irrationality of the reciprocal sum over distinct running least common multiples of the 5-smooth integers. The spacing of powers of five separates the tail images, so rationality would force a periodic jump sequence. Counting running LCMs with multiplicity gives a different number, whose irrationality remains open here.","owns":"the five-map proof for distinct running LCMs over {2,3,5}; the long paper treats general prime sets, the sum with multiplicity, kernels and residue criteria","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #269, which remains open","first_pass":["sec:problem","sec:distinct"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-269-three-prime-running-lcm.pdf","hosted_pdf_path":"papers/erdos-269-three-prime-running-lcm.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/269/erdos-269-three-prime-running-lcm.tex","site_html":"maths/papers/erdos-269-three-prime-running-lcm.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/269/erdos-269-three-prime-running-lcm.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/269/erdos-269-three-prime-running-lcm.tex","body_pointer":"papers.records.erdos-269-three-prime-running-lcm.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% One of the Erdős Problem Notes: a short standalone treatment of a single\n% problem in the ErdosProblems expansion library.  Build with\n% `tectonic erdos-269-three-prime-running-lcm.tex` or `make`.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n\\input{evidence/erdos-269-three-prime-running-lcm}\n\\usepackage{longtable}\n\\usepackage{colortbl}\n\\usepackage{xurl}\n\\usepackage{tikz}\n% Preserve the original immutable source revision used by the links below.\n\\renewcommand{\\commit}{99f4bf47422abbd8757cbb22b50ba079d764d3a7}\n\\renewcommand{\\commitshort}{99f4bf47422a}\n% The distinct-height programs, at the commit that added them, and the Lean\n% proof of the {2,3,5} theorem, at the commit that added it.\n\\newcommand{\\distinctheighturl}{https://github.com/wcook04/plectis-erdos/tree/10178c4df40cb27d83b5ebb1ec337dd588b82998/research/experiments/erdos269/distinct_height}\n\\newcommand{\\distinctheightlean}{https://github.com/wcook04/plectis-erdos/blob/5154f4c46fa5d5ab2cff8df857a0b10f310cbeeb/lean/ErdosProblems/Erdos269/DistinctHeightIrrationality.lean}\n\\newcommand{\\distinctheightblocklean}{https://github.com/wcook04/plectis-erdos/blob/5154f4c46fa5d5ab2cff8df857a0b10f310cbeeb/lean/ErdosProblems/Erdos269/DistinctHeightBlockRadix.lean}\n\\hypersetup{\n  pdftitle={Distinct running least common multiples (Erdős Problem 269)},\n  pdfsubject={Erdős Problem Notes: Problem 269},\n  pdfkeywords={irrationality, transcendence, Hecke-Mahler values, least common multiple, smooth numbers, Cantor series, lattice sums, Lean 4},\n}\n\n% The three pure-power coordinates and the running height.\n\\newcommand{\\hgt}{\\operatorname{H}}\n\\newcommand{\\Lprefix}{\\operatorname{L}}\n\\newcommand{\\Kernel}{\\operatorname{K}}\n\n\\runninghead{Distinct running least common multiples}\n\\title{Distinct running least common multiples}\n\\subtitle{Affine tails and Erd\\H{o}s Problem~\\#269}\n\\author{Will Cook}\n\\date{July 2026; revised 6 October 2026}\n\n\n% A local PDF link uses its destination name directly. The shared web-style\n% helper would make hyperref emit the literal GoToR name \"nameddest=...\".\n\\newcommand{\\longrecord}[2]{\\href{erdos269-running-lcm-reasoning-surface.pdf\\##1}{#2}}\n\n\\begin{document}\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\nWe give a five-map proof that the reciprocals of the distinct running least\ncommon multiples of the $5$-smooth integers have an irrational sum.\nThe spacing of powers of $5$ separates the affine tail images; rationality\nwould then force periodicity, contrary to the frequency of powers of $3$.\nFor the sum counted with multiplicity, the tails instead give an integer\nrecurrence and an equivalent residue criterion for irrationality. Proving\nthat criterion for the actual coefficients remains open here.\n\\end{abstract}\n\n\\section{Two sums from running least common multiples}\\label{sec:problem}\nLet $P$ be a finite set of primes and $\\mathcal S_P$ the positive integers\nwith prime factors in $P$, including $1$. For $P=\\{2,3,5\\}$, the running\nleast common multiples of these $5$-smooth integers begin\n$1,2,6,12,60,60,120,360,360,\\ldots$. Put\n\\[\n \\Lprefix(x)=\\lcm\\{u\\in\\mathcal S_P:u\\le x\\},\\qquad\n \\hgt_P(x)=\\prod_{p\\in P}p^{\\lfloor\\log_p x\\rfloor}\\quad(x\\ge1).\n\\]\nEvery member of the prefix divides $\\hgt_P(x)$, while the prefix contains\nthe largest power of each $p\\in P$ at most $x$. Thus\n$\\Lprefix(x)=\\hgt_P(x)$, with a jump by $p$ exactly at each positive\npower of $p$. Counting distinct running LCMs or retaining multiplicities gives\n\\[\n \\mathcal D_P=1+\\sum_{\\substack{t=p^n\\\\p\\in P,\\ n\\ge1}}\\frac1{\\hgt_P(t)},\n \\qquad\n \\mathcal R_P=\\sum_{u\\in\\mathcal S_P}\\frac1{\\Lprefix(u)}.\n\\]\nFor $P=\\{2,3,5\\}$, $\\Lprefix(5)=\\Lprefix(6)=60$ contributes $1/60$ to\n$\\mathcal D_P$ and $2/60$ to $\\mathcal R_P$. Erd\\H{o}s Problem~\\#269 asks about\n$\\mathcal R_P$~\\cite[p.~65]{erdosgraham1980} and~\\cite[p.~106]{erdos1988}.\nPositive powers of distinct primes never coincide; geometric growth of\nsuccessive running LCMs gives convergence of $\\mathcal D_P$.\nWe use $\\N=\\{0,1,2,\\ldots\\}$.\n\n\\papersectiontarget{r4-five-map-theorem}\n\\begin{theorem}[the sum over distinct running LCMs for $\\{2,3,5\\}$]\\label{res:distinct-height-235}\nThe number $\\mathcal D_{\\{2,3,5\\}}$ is irrational.\n\\end{theorem}\n\nRationality would leave only finitely many normalised tails, but finiteness\nalone does not force the jumps to repeat. The decisive step is to recover\nthe next block and the following tail from the current tail. Powers of\n$2$ delimit the blocks; the spacing of powers of $5$ separates their five\naffine images. Equal tails then have identical futures, contradicting\nthe irrational frequency of powers of $3$.\n\nErd\\H{o}s asserted irrationality of $\\mathcal D_P$ for every finite\n$P$ with $|P|\\ge2$ in a letter dated 1 January 1973, without printing a\nproof~\\cite[p.~335]{erdos1974letter}. Thus the theorem above is a special\ncase of that earlier assertion. We claim no priority for its conclusion.\nThe \\longrecord{r3-distinct-proof}{companion's general argument} uses an\nisolated interchange of jumps on a torus. That argument is retained there\nas an ordinary proof awaiting formalisation. The proof below is independent\nof that argument.\n\nOur arguments leave the repeated sum $\\mathcal R_P$ unresolved for finite\n$P$ with $|P|\\ge3$. For $P=\\{p\\}$ both sums equal $p/(p-1)$.\n\nAfter the five-map proof in Section~\\ref{sec:distinct-235}, we retain the\nmultiplicities. The bases remain the same, but the block coefficients and\ntails grow quadratically. Sections~\\ref{sec:lcm} and~\\ref{sec:escape} replace\nfinite-state repetition by a test for excluding integer tails.\nFor two primes, Fan's factorisation~\\cite{fan2026comment} reduces both sums\nto a Hecke--Mahler value and yields transcendence\n(Section~\\ref{sec:two-prime}). The appendices supply the running LCM\nidentities and show why exact finite separation of the three-prime kernel\ncannot extend that factorisation.\n\n\\section{Recovering the prime-power jumps}\\label{sec:distinct}\\label{sec:distinct-235}\nLong division combines a finite set of remainders with a deterministic\nrule for the next digit and remainder. Here the bases vary with the\nprime-power jumps, so that rule must be proved. We first obtain finitely\nmany tails under rationality, then recover the jumps from each tail.\n\n\\begin{proof}[Proof of Theorem~\\ref{res:distinct-height-235}]\n\\emph{Finitely many tails under rationality.}\nWrite $\\hgt=\\hgt_{\\{2,3,5\\}}$ in this proof and put $P_a=\\hgt(2^a)$.\nEvery running LCM at or before $2^a$ divides $P_a$. We therefore normalise\nthe tail after $2^a$ by\n\\begin{equation}\\label{eq:five-tail}\n Y_a=P_a\\sum_{\\substack{t>2^a\\\\t=2^n,3^n\\text{ or }5^n,\\ n\\ge1}}\n                   \\frac1{\\hgt(t)}.\n\\end{equation}\nEach subsequent jump multiplies the running LCM by at least $2$, and a jump\nby $3$ eventually occurs. Comparison with $\\sum_{j\\ge1}2^{-j}$ gives\n$0<Y_a<1$.\n\nSuppose that $\\mathcal D_{\\{2,3,5\\}}=N/K$, where $N$ is an integer and\n$K\\ge1$ is an integer. Clearing the finite prefix gives\n\\[\n KY_a=P_aN-K\\left(P_a+\n        \\sum_{\\substack{t\\le2^a\\\\t=2^n,3^n\\text{ or }5^n,\\ n\\ge1}}\n                 \\frac{P_a}{\\hgt(t)}\\right)\\in\\mathbb Z.\n\\]\nThus the $Y_a$ lie in the finite set $(K^{-1}\\mathbb Z)\\cap(0,1)$.\nIt remains to show why equality of two tails propagates.\n\n\\medskip\n\\emph{Recovering a block from its tail.}\nConsider the jumps between $16$ and $32$, including the last endpoint.\nThey occur at $25,27,32$, multiplying the running LCM successively by $5,3,2$.\nAfter normalisation at $16$, their contribution is\n$1/5+1/15+1/30$. The remaining tail is normalised at a running LCM $30$\ntimes as large, so\n\\[\n Y_4=\\frac15+\\frac1{15}+\\frac1{30}+\\frac{Y_5}{30}\n     =\\frac{9+Y_5}{30}.\n\\]\nIn the block $(64,128]$, the interior jumps occur in the opposite order,\nat $81,125$. Its constant term is $1/3+1/15+1/30=13/30$.\nThe multiplier $30$ records which primes occur, but not their order.\nThe numerators $9$ and $13$ retain that order. To read it from a tail,\nwe must also control the unknown following tail.\n\nEvery block $(2^a,2^{a+1}]$ contains its terminal power of $2$, at most\none power of $3$ and at most one power of $5$. We record the interior\njumps in their order by a word $\\tau_a$ on the letters $3,5$, omitting\nthe terminal $2$. Splitting off the block gives\n\\begin{equation}\\label{eq:five-maps}\n Y_a=G_{\\tau_a}(Y_{a+1}),\\qquad\n G_\\tau(y)=\\frac{\\mu_\\tau+y}{b_\\tau},\\qquad\n \\begin{array}{c|ccccc}\n \\tau&\\varnothing&3&5&35&53\\\\\\hline\n b_\\tau&2&6&10&30&30\\\\\n \\mu_\\tau&1&3&3&13&9\n \\end{array}\n\\end{equation}\nHere $b_\\tau$ is the factor by which the running LCM increases across the\nblock, and $\\mu_\\tau/b_\\tau$ is the block's normalised contribution.\n\nThe bound $0<Y_{a+1}<1$ does not yet recover $\\tau_a$:\n$G_{53}([0,1])=[3/10,1/3]$ and $G_5([0,1])=[3/10,2/5]$ overlap.\nThe table improves the lower bound to $Y_a\\ge3/10$. To separate these\ntwo images, an upper bound $U$ must satisfy\n\\[\n G_{53}(U)<G_5(3/10),\\qquad\n \\frac{9+U}{30}<\\frac{33}{100},\\qquad U<\\frac9{10}.\n\\]\nThe spacing of powers of $5$ supplies such a bound.\nA power $5^f$ belongs to block $\\lfloor f\\log_2 5\\rfloor$. These\nindices start at $2$ and their successive differences are $2$ or $3$.\nEvery three consecutive blocks therefore contain a power of $5$.\nGiven $a$, let $j\\in\\{a,a+1,a+2\\}$ be the first such block. The maps\nfor blocks containing $5$ give $Y_j\\le7/15$, since $Y_{j+1}<1$ and\n$G_{35}(1)=7/15$ bounds $G_5(1)$ and $G_{53}(1)$ as well.\nWe now work \\emph{backwards} from $j$ to $a$. Each intervening block\nhas map $G_\\varnothing$ or $G_3$, and\n$G_3(y)\\le G_\\varnothing(y)=(1+y)/2$ for $y\\ge0$.\nAt most two backward steps give the successive bounds\n$7/15$, $11/15$, $13/15$. Hence\n\\begin{equation}\\label{eq:five-interval}\n Y_a\\in I=\\left[\\frac3{10},\\frac{13}{15}\\right],\\qquad\n G_\\varnothing\\bigl(G_\\varnothing(7/15)\\bigr)=\\frac{13}{15}<\\frac9{10}.\n\\end{equation}\nThis bound holds for the tails of the prime-power sequence because\nevery three consecutive blocks contain a power of $5$. The interval\n$I$ is not invariant under arbitrary compositions of the five maps:\nfor example, $G_\\varnothing(13/15)=14/15$ lies outside $I$.\n\nFigure~\\ref{fig:five-map-images} gives the exact images of $I$. They\nare pairwise disjoint, with successive gaps\n$1/900,17/300,79/900,1/180$. Writing $L=3/10$ and $U=13/15$, the two\nnarrow gaps are\n\\[\n G_5(L)-G_{53}(U)=\\frac{3L-U}{30}=\\frac1{900},\\qquad\n G_\\varnothing(L)-G_3(U)=\\frac{3L-U}{6}=\\frac1{180}.\n\\]\nThus the same inequality $U<3L=9/10$ separates both pairs.\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{tikzpicture}[x=1cm,y=1cm,font=\\small]\n  % All five rows use the same horizontal scale, from 3/10 to 19/20.\n  \\node[anchor=west] at (0,3.65) {$\\tau$};\n  \\node[anchor=west] at (.8,3.65) {$G_\\tau(I)$};\n  \\node[anchor=west] at (4.8,3.65) {Common scale};\n  \\foreach \\y/\\name/\\lo/\\hi in {\n    3.05/53/{31/100}/{74/225},\n    2.50/5/{33/100}/{29/75},\n    1.95/35/{133/300}/{104/225},\n    1.40/3/{11/20}/{29/45},\n    .85/\\varnothing/{13/20}/{14/15}} {\n      \\node[anchor=west] at (0,\\y) {$\\name$};\n      \\draw[black!15,line width=.3pt] (4.8,\\y)--(13.2,\\y);\n      \\pgfmathsetmacro{\\leftend}{4.8+8.4*((\\lo)-.3)/.65}\n      \\pgfmathsetmacro{\\rightend}{4.8+8.4*((\\hi)-.3)/.65}\n      \\draw[line width=1.4pt] (\\leftend,\\y)--(\\rightend,\\y);\n      \\draw[line width=.6pt] (\\leftend,\\y-.075)--(\\leftend,\\y+.075)\n                             (\\rightend,\\y-.075)--(\\rightend,\\y+.075);\n  }\n  \\node[anchor=west] at (.8,3.05) {$[31/100,\\,74/225]$};\n  \\node[anchor=west] at (.8,2.50) {$[33/100,\\,29/75]$};\n  \\node[anchor=west] at (.8,1.95) {$[133/300,\\,104/225]$};\n  \\node[anchor=west] at (.8,1.40) {$[11/20,\\,29/45]$};\n  \\node[anchor=west] at (.8,.85) {$[13/20,\\,14/15]$};\n  \\draw[line width=.4pt] (4.8,.40)--(13.2,.40);\n  \\foreach \\v/\\lab in {.3/{3/10},.4/{2/5},.5/{1/2},.6/{3/5},.7/{7/10},.8/{4/5},.9/{9/10}} {\n    \\pgfmathsetmacro{\\xx}{4.8+8.4*(\\v-.3)/.65}\n    \\draw (\\xx,.35)--(\\xx,.45);\n    \\node[below,font=\\footnotesize] at (\\xx,.33) {$\\lab$};\n  }\n  % Independent enlargements of the two smallest gaps. Endpoint values,\n  % not graphical resolution, establish that the intervals are disjoint.\n  \\node[anchor=west,font=\\footnotesize] at (0,-.40) {Two smallest gaps (independent enlargements)};\n  \\draw[line width=1.4pt] (.4,-1.05)--(2.25,-1.05);\n  \\draw[line width=1.4pt] (3.05,-1.05)--(5.25,-1.05);\n  \\draw (2.25,-1.13)--(2.25,-.97) (3.05,-1.13)--(3.05,-.97);\n  \\node[above] at (1.25,-.98) {$G_{53}(I)$};\n  \\node[above] at (4.3,-.98) {$G_5(I)$};\n  \\node[below left,font=\\footnotesize] at (2.25,-1.13) {$74/225$};\n  \\node[below right,font=\\footnotesize] at (3.05,-1.13) {$33/100$};\n  \\draw[<->,line width=.4pt] (2.25,-1.82)--(3.05,-1.82);\n  \\node[below,font=\\footnotesize] at (2.65,-1.82) {$1/900$};\n  \\draw[line width=1.4pt] (7.3,-1.05)--(9.2,-1.05);\n  \\draw[line width=1.4pt] (10.2,-1.05)--(12.8,-1.05);\n  \\draw (9.2,-1.13)--(9.2,-.97) (10.2,-1.13)--(10.2,-.97);\n  \\node[above] at (8.05,-.98) {$G_3(I)$};\n  \\node[above] at (11.7,-.98) {$G_\\varnothing(I)$};\n  \\node[below left,font=\\footnotesize] at (9.2,-1.13) {$29/45$};\n  \\node[below right,font=\\footnotesize] at (10.2,-1.13) {$13/20$};\n  \\draw[<->,line width=.4pt] (9.2,-1.82)--(10.2,-1.82);\n  \\node[below,font=\\footnotesize] at (9.7,-1.82) {$1/180$};\n\\end{tikzpicture}\n\\caption{The five images of the arithmetic tail interval\n$I=[3/10,13/15]$. Their separation determines the next block from an\nactual tail. The full image $G_\\varnothing(I)$ extends beyond $I$;\nthe figure does not assert invariance under arbitrary words. Only the\nadjacent endpoints are shown in the two lower enlargements.}\n\\label{fig:five-map-images}\n\\end{figure}\n\nEach actual $Y_a$ lies in exactly one image interval, identifying\n$\\tau_a$ and then $Y_{a+1}=b_{\\tau_a}Y_a-\\mu_{\\tau_a}$.\nFor instance, $Y_a\\in[31/100,74/225]$ forces $\\tau_a=53$ and\n$Y_{a+1}=30Y_a-9$. This is the next arithmetic tail, not an arbitrary\ninverse image; it lies in $I$, so recovery can be repeated.\n\n\\medskip\n\\emph{Repetition would give a rational frequency.}\nThe finitely many tail values obtained above include a repeated value\n$Y_u=Y_v$ with $u<v$. Since the image intervals are disjoint, this\nforces $\\tau_u=\\tau_v$. Applying the corresponding inverse map gives\n$Y_{u+1}=Y_{v+1}$. Induction yields\n$\\tau_{u+j}=\\tau_{v+j}$ for every $j\\ge0$. Thus the entire later block\nword has period $v-u$.\n\nTo exclude this possibility, let $\\alpha=\\log2/\\log3$. The number of powers\nof $3$ in block $a$ is\n$\\lfloor(a+1)\\alpha\\rfloor-\\lfloor a\\alpha\\rfloor$, and is determined by\n$\\tau_a$. If the block word has eventual period $\\ell\\ge1$, let $c$ be the\nnumber of $3$-powers in one period. For every sufficiently large fixed $a$\nand every $n\\ge1$ we then have\n\\[\n \\lfloor(a+n\\ell)\\alpha\\rfloor=\\lfloor a\\alpha\\rfloor+nc.\n\\]\nDividing by $n$ and taking the limit gives $\\ell\\alpha=c$, contrary to\nunique factorisation, since it would imply $2^\\ell=3^c$.\n\\end{proof}\n\n\\section{The repeated three-prime sum}\n\\label{sec:lcm}\\label{sec:shell}\\label{sec:actual-orbit}\nWe now retain every smooth integer, not just the jumps of its running\nLCM. Put $P=\\{2,3,5\\}$, $S=\\mathcal R_P$ and $\\hgt=\\hgt_P$ throughout\nthis section. The denominator changes at the same prime powers; the\nnumber of terms at each denominator is what changes the argument.\n\nFor example, $[16,32)$ contains the seven smooth integers\n$16,18,20,24,25,27,30$. Grouping them by their running LCM gives\n\\[\n \\begin{array}{c|c|c}\n \\text{smooth integers}&\\text{running LCM}&\\text{contribution}\\\\\\hline\n 16,18,20,24&720&4/720\\\\\n 25&3600&1/3600\\\\\n 27,30&10800&2/10800\n \\end{array}\n\\]\nThe distinct sum counts each displayed running LCM once; the repeated\nsum assigns the coefficients $4,1,2$. The jumps alone therefore no longer\ndetermine the block contribution. Appendix~\\ref{sec:height-identities}\ngives the general finite grouping identity.\n\nWith blocks $[2^a,2^{a+1})$, the tail now includes $2^a$. The prefix\nexcludes the jump by $2$ there, so half the running LCM clears it. Put\n\\[\n s_a=\\sum_{\\substack{x\\in\\mathcal S_P\\\\2^a\\le x<2^{a+1}}}\\frac1{\\hgt(x)},\n \\qquad T_a=\\sum_{j\\ge0}s_{a+j},\\qquad\n h_a=\\frac{\\hgt(2^a)}2,\\qquad X_a=h_aT_a.\n\\]\nFor $a\\ge1$, each prefix LCM has exponent of $2$ at most $a-1$ and\ndivides the integer $h_a$; at $a=0$, $h_0=1/2$. The factor $1/2$ reflects\nthe endpoint change from Section~\\ref{sec:distinct-235}, not multiplicity.\n\nSplitting $T_a=s_a+T_{a+1}$ and using $h_{a+1}=b_ah_a$ gives\n$X_{a+1}=b_aX_a-m_a$, where $m_a=h_{a+1}s_a$. Explicitly,\n\\begin{equation}\\label{eq:actual-digit}\n b_a=\\frac{\\hgt(2^{a+1})}{\\hgt(2^a)},\\qquad\n m_a=\\sum_{\\substack{x\\in\\mathcal S_P\\\\2^a\\le x<2^{a+1}}}\n       \\frac{\\hgt(2^{a+1})}{2\\hgt(x)}.\n\\end{equation}\nSince $h_0=1/2$, we also have $h_{a+1}=(b_0\\cdots b_a)/2$ and\n$m_a/(b_0\\cdots b_a)=s_a/2$.\nFor $[2,4)$, the running LCMs at $2,3$ are $2,6$ and $h_2=6$, giving\n$m_1=6(1/2+1/6)=4$ and $b_1=6$. For the block displayed above,\n$h_5=10800$ and\n\\[\n m_4=4\\cdot15+3+2\\cdot1=65,\\qquad b_4=30.\n\\]\n\\begin{lemma}[integer coefficients and four possible bases]\n\\label{res:dyadic-alphabet}\nFor every $a\\ge0$, $m_a$ is a positive integer and\n$b_a\\in\\{2,6,10,30\\}$.  The word ``numerator'' does not impose the\npositional-digit restriction $m_a<b_a$; that restriction need not hold.\n\\end{lemma}\n\\begin{proof}\nFor $x<2^{a+1}$ we have $2\\hgt(x)\\mid\\hgt(2^{a+1})$, so every summand\nin $m_a$ is integral. The interval contains $2^a$, giving positivity.\nBetween successive powers of $2$ there is at most one power of $3$ and\nat most one power of $5$. Along with the final factor $2$, these give\n$b_a\\in\\{2,6,10,30\\}$.\n\\end{proof}\n\n\\begin{proposition}[the tail recurrence and a quadratic bound]\n\\label{res:actual-orbit}\nThe series defining $S,T_a$ converge.  For every $a\\ge0$,\n\\[\n X_{a+1}=b_aX_a-m_a,\\qquad\n 0<X_a\\le\\frac{8640}{343}(a+1)^2<90(a+1)^2.\n\\]\nFor every integer $B\\ge1$, either $BX_a$ is integral at some index (and hence at every later index), or $\\operatorname{dist}(BX_a,\\mathbb Z)\\ge1/31$\nat arbitrarily large indices.\n\\end{proposition}\n\\begin{proof}\nWe count at most $(a+1)^2$ smooth integers in $[2^a,2^{a+1})$: each pair\nof exponents of $3,5$, both at most $a$, allows at most one exponent of $2$.\nBy \\eqref{res:cube},\n$s_a\\le30(a+1)^2/8^a$. As $h_a\\le8^a/2$ and\n$a+j+1\\le(a+1)(j+1)$, we have\n\\[\n X_a\\le15(a+1)^2\\sum_{j\\ge0}\\frac{(j+1)^2}{8^j}\n     =\\frac{8640}{343}(a+1)^2.\n\\]\nThis also proves convergence. Splitting off $s_a$ gives the recurrence,\nsince $h_{a+1}=b_ah_a$ and $m_a=h_{a+1}s_a$.\n\nFor the final assertion, suppose that every sufficiently late distance\nis strictly less than $1/31$, and write $BX_a=z_a+e_a$, where $z_a$ is\nintegral and $|e_a|<1/31$. The recurrence makes $e_{a+1}-b_ae_a$ an\ninteger of absolute value less than $(1+b_a)/31\\le1$. Thus\n$e_{a+1}=b_ae_a$. A nonzero error would therefore at least double at\neach step, contradicting $|e_a|<1/31$. All sufficiently late errors vanish;\nintegrality then persists by the recurrence.\n\\end{proof}\n\nThe loss of bounded tails is genuine. For $a\\ge1$, every odd part\n$3^j5^k<2^a$ has a unique multiple by a power of $2$ in\n$[2^a,2^{a+1})$. Taking $0\\le j,k\\le\\lfloor a/5\\rfloor$ ensures this\ninequality because $15<2^5$. Each such integer contributes at least $1$\nto $m_a$, so\n\\[\n m_a\\ge(\\lfloor a/5\\rfloor+1)^2,\\qquad X_a>m_a/30.\n\\]\nRationality would still put the tails in a fixed lattice, but not in a\nfinite set. Repetition alone also gives no recovery rule: the maps\n$y\\mapsto(m_a+y)/b_a$ now depend on $a$ and form an infinite family.\nThe proof in Section~\\ref{sec:distinct-235} needed both finiteness and\nunique recovery. We instead use the quadratic bound to limit the positive\ninteger tails required by rationality.\n\nThe Cantor-series form $S/2=\\sum_{a\\ge0}m_a/(b_0\\cdots b_a)$ also\nexplains why the criteria in \\cite[Theorem~2.1]{erdosstraus1974} and\n\\cite[Theorem~3.1]{hancltijdeman2004} do not apply: their small-numerator\nhypothesis $m_a/(b_{a-1}b_a)\\to0$ fails with bounded $b_a$ and these\nunbounded $m_a$. The \\longrecord{r3-repeated-bounds}{companion's tail estimates}\ngive sharper bounds.\n\n\\begin{theorem}[rationality gives positive integer tails]\n\\label{res:denominator-reduction}\nIf $S=A/D$ in lowest terms, where\n$D=2^u3^v5^wB$ and $\\gcd(B,30)=1$, then for every\n$a\\ge a_0=u+1+2v+3w$,\n\\[\n d_a=BX_a\\in\\mathbb Z_{>0},\\qquad\n d_{a+1}=b_ad_a-Bm_a,\\qquad d_a\\le90B(a+1)^2.\n\\]\n\\end{theorem}\n\\begin{proof}\nFor $a\\ge1$, clearing the finite prefix gives\n\\begin{equation}\\label{eq:prefix-lattice}\n X_a=h_aS-\\sum_{\\substack{x\\in\\mathcal S_P\\\\x<2^a}}\n                 \\frac{h_a}{\\hgt(x)},\\qquad\n \\sum_{\\substack{x\\in\\mathcal S_P\\\\x<2^a}}\\frac{h_a}{\\hgt(x)}\\in\\mathbb Z.\n\\end{equation}\nIf $a\\ge u+1+2v+3w$, the exponent $a-1$ of $2$ in $h_a$ is at least $u$,\nand $2^a\\ge3^v,5^w$. Therefore $2^u3^v5^w\\mid h_a$, and $BX_a$ differs\nfrom $h_aA/(2^u3^v5^w)$ by an integer. Positivity, recurrence and the bound\nfollow from Proposition~\\ref{res:actual-orbit}.\n\\end{proof}\n\n\\begin{corollary}[the first index at which the denominator clears]\\label{res:exact-onset}\nUnder the same lowest-terms hypothesis, put $M=2^u3^v5^w$ and let\n$\\operatorname{den}$ denote the positive reduced denominator.  For $a\\ge1$,\n\\[\n \\operatorname{den}(BX_a)=\\frac{M}{\\gcd(M,h_a)}.\n\\]\nConsequently $BX_a$ is integral exactly when\n$2^a\\ge\\max(2^{u+1},3^v,5^w)$.  The first such $a$ can be found by\ninteger comparisons, without logarithmic rounding.\n\\end{corollary}\n\\begin{proof}\nEquation \\eqref{eq:prefix-lattice} shows that $BX_a$ and $h_aA/M$ have\nthe same reduced denominator. Since $\\gcd(A,M)=1$, this denominator is\n$M/\\gcd(M,h_a)$. Divisibility by each of $2^u,3^v,5^w$ gives the three\nstated inequalities.\n\\end{proof}\nThe two denominator parts have different roles: $M$ determines when\nclearing occurs; $B$ is the multiplier left after clearing. For\n$D=2^3 3^2 5\\cdot7$, the first integral $7X_a$ would occur at $a=4$,\nnot the sufficient index $11$ in the theorem. Thus an early nonintegral\ntail need not stay nonintegral. The\n\\longrecord{r3-denominator}{companion's denominator calculation} also gives\nthe reduced denominator of $X_a$ itself.\n\n\\section{A residue criterion}\\label{sec:escape}\\label{sec:open}\nA finite segment bounds the starting tail. Its integrality would\nconstrain the endpoint tail to a residue class. The residue inequality\nexpresses the same obstruction in either form.\n\nFor two steps the recurrence reads\n$X_{\\ell+2}=b_{\\ell+1}b_\\ell X_\\ell-(b_{\\ell+1}m_\\ell+m_{\\ell+1})$.\nTo keep track of the multiplicative and additive terms over $h$ steps,\nwrite\n\\[\n W_{\\ell,0}=1,\\quad F_{\\ell,0}=0,\\qquad\n W_{\\ell,h+1}=b_{\\ell+h}W_{\\ell,h},\\quad\n F_{\\ell,h+1}=b_{\\ell+h}F_{\\ell,h}+m_{\\ell+h}.\n\\]\nInduction yields\n\\begin{equation}\\label{eq:actual-tail}\n X_{\\ell+h}=W_{\\ell,h}X_\\ell-F_{\\ell,h}.\n\\end{equation}\nSince $b_a=h_{a+1}/h_a$, its product telescopes to\n$W_{\\ell,h}=h_{\\ell+h}/h_\\ell$. Thus\n\\[\n \\frac{F_{\\ell,h}}{W_{\\ell,h}}\n =h_\\ell\\sum_{a=\\ell}^{\\ell+h-1}s_a,\n \\qquad\n \\frac{X_{\\ell+h}}{W_{\\ell,h}}=h_\\ell T_{\\ell+h}.\n\\]\nThese finite and remaining contributions sum to $X_\\ell$, both in the\nstarting normalisation $h_\\ell$, by \\eqref{eq:actual-tail}.\nThroughout this section let\n\\begin{equation}\\label{eq:actual-bound}\n K(B,a)=90B(a+1)^2.\n\\end{equation}\nThe bound $0<BX_{\\ell+h}\\le K(B,\\ell+h)$ and\n\\eqref{eq:actual-tail} enclose the starting tail in the interval\n\\[\n \\frac{BF_{\\ell,h}}{W_{\\ell,h}}<BX_\\ell\n \\le\\frac{BF_{\\ell,h}+K(B,\\ell+h)}{W_{\\ell,h}}.\n\\]\nWidth below $1$ gives at most one integer, not necessarily none. With\n$W=W_{\\ell,h}$ and $F=F_{\\ell,h}$, the interval contains no integer exactly\nwhen\n\\[\n BF+K(B,\\ell+h)<W\\bigl(\\lfloor BF/W\\rfloor+1\\bigr).\n\\]\nThe right side is $W$ times the first integer strictly above $BF/W$.\nIts excess over $BF$ is the least positive residue of $-BF$ modulo $W$:\nthe residue measures the gap to that integer in units of $1/W$.\n\nFor integers $W\\ge1,t$, define\n$\\operatorname{lpr}_W(t)=1+((t-1)\\bmod W)$, with remainder in\n$\\{0,\\ldots,W-1\\}$. Its values lie in $\\{1,\\ldots,W\\}$, so the zero class\nis represented by $W$. This matches the open left endpoint: if $W\\mid BF$,\nthe next possible integer is one full unit above $BF/W$, not $BF/W$ itself.\n\n\\begin{lemma}[least positive residues]\\label{res:consumer}\nIf $d$ is a positive integer with $d\\le K$ and $d\\equiv -BF\\pmod W$,\nwhere $W\\ge1$, then $\\operatorname{lpr}_W(-BF)\\le K$.\n\\end{lemma}\n\\begin{proof}\nThe least positive representative is at most every positive representative\nof its class, and in particular at most $d$.\n\\end{proof}\nIn~\\eqref{eq:actual-tail}, an integral $BX_\\ell$ makes $BX_{\\ell+h}$\nsuch a positive representative.\n\nFor $\\ell=1,h=6$, direct iteration gives\n$W_{1,6}=648000$ and $F_{1,6}=524431$. Since $K(1,7)=5760$, the interval is\n\\[\n \\frac{524431}{648000}<X_1\\le\\frac{530191}{648000}<1.\n\\]\nEquivalently, $\\operatorname{lpr}_{W_{1,6}}(-F_{1,6})=123569>5760$\ncertifies $X_1\\notin\\mathbb Z$, treating only $B=1$ at $\\ell=1$.\nFor denominators $MB$, varying $B$ covers the part coprime to $30$;\narbitrarily late starts allow each supported part $M$ to clear.\n\n\\papersectiontarget{r4-repeated-criterion}\n\\begin{theorem}[a residue criterion for irrationality]\\label{res:windowconsumer}\nThe number $S$ is irrational if and only if\n\\begin{equation}\\label{eq:escape}\n \\begin{gathered}\n \\text{for every }B\\ge1\\text{ with }\\gcd(B,30)=1\n \\text{ and every }a_0\\ge1,\\\\\n \\text{there are }\\ell\\ge a_0,\\ h\\ge1\\text{ such that}\n \\operatorname{lpr}_{W_{\\ell,h}}(-BF_{\\ell,h})>K(B,\\ell+h).\n \\end{gathered}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nSuppose that \\eqref{eq:escape} holds and $S$ is rational. By\nTheorem~\\ref{res:denominator-reduction}, $d_a=BX_a$ is a positive integer\nfor every sufficiently large $a$. Choose a segment beyond that onset.\nEquation \\eqref{eq:actual-tail} gives\n$d_{\\ell+h}\\equiv-BF_{\\ell,h}\\pmod{W_{\\ell,h}}$, contrary to\nLemma~\\ref{res:consumer} and \\eqref{eq:escape}.\n\nConversely, suppose that $S$ is irrational and fix $B,\\ell\\ge1$.\nEquation~\\eqref{eq:prefix-lattice} makes $BX_\\ell$ nonintegral.\nThe distance upwards to the next integer,\n$\\delta=\\lceil BX_\\ell\\rceil-BX_\\ell\\in(0,1)$, is now fixed as the\nsegment length $h$ grows. We have $W_{\\ell,h}\\ge2^h$, whereas\n$X_{\\ell+h}$ has only quadratic growth. Consequently\n$\\delta+BX_{\\ell+h}/W_{\\ell,h}$ lies in $(0,1)$ for sufficiently large\n$h$. This ensures that the following representative lies strictly\nbetween $0$ and the modulus, so it is the least positive one:\n\\[\n \\operatorname{lpr}_{W_{\\ell,h}}(-BF_{\\ell,h})\n =\\lceil BX_\\ell\\rceil W_{\\ell,h}-BF_{\\ell,h}\n =\\delta W_{\\ell,h}+BX_{\\ell+h}.\n\\]\nThe positive constant $\\delta$ multiplies an exponentially growing\n$W_{\\ell,h}$, so the first term eventually exceeds $K(B,\\ell+h)$.\nThis proves the inequality from every fixed start, and hence after\nevery prescribed onset $a_0$. No bound uniform in $B$ or $\\ell$ is\nneeded for this direction.\n\\end{proof}\n\nAt a fixed start, success rules out integrality of $BX_\\ell$.\nIf $BX_\\ell$ is integral, the recurrence, positivity and the bound prevent\nsuccess at any length; for all sufficiently large $h$,\n$\\operatorname{lpr}_{W_{\\ell,h}}(-BF_{\\ell,h})=BX_{\\ell+h}$.\nIf $BX_\\ell$ is nonintegral, every sufficiently long segment succeeds,\neven when $S$ is rational with a denominator not yet cleared at $\\ell$.\nIncreasing the length does not remove the need for starts beyond every\ncutoff. A finite failed search distinguishes neither case.\n\nThe converse obtains $\\delta>0$ from irrationality. Proving~\\eqref{eq:escape}\nwithout that assumption is the remaining task. The upper-bound property of $K$\nis essential: a zero bound would make every residue inequality automatic.\nThe \\longrecord{r3-residue}{companion's residue analysis} treats the fixed-start\nformula and smaller or more general bounds.\n\n\\noindent\\begin{minipage}{\\linewidth}\nThe repeated three-prime question can therefore be written in three\nequivalent forms.\n\\begin{problem}[the repeated three-prime target]\nProve that $S=\\mathcal R_{\\{2,3,5\\}}$ is irrational.\n\\end{problem}\n\n\\begin{problem}[no reduced tail is an integer]\\label{prob:tails269}\nProve that for every $a\\ge1$ and every integer $B\\ge1$ coprime to $30$,\n\\begin{equation}\\label{eq:tail-nonintegrality}\n BX_a\\notin\\mathbb Z.\n\\end{equation}\n\\end{problem}\n\\begin{problem}[residue inequalities after every starting index]\\label{prob:producer}\nProve~\\eqref{eq:escape}.\n\\end{problem}\n\\end{minipage}\n\nThe equivalence follows from \\eqref{eq:prefix-lattice}, denominator\nclearing and Theorem~\\ref{res:windowconsumer}; none of the three assertions\nis proved for $S$. The \\longrecord{r3-open}{companion's discussion of the\nrepeated sum} records finite calculations and the hypotheses missing\nfrom proposed finite-difference and function-theoretic arguments.\n\n\\section{Two-prime transcendence}\\label{sec:two-prime}\nFor two primes, the distinct sum is affine in one Hecke--Mahler value;\nthe repeated sum is quadratic in it.\nFan gave the factorisation of the repeated sum and its transcendence\nconsequence in his post of 26 June 2026~\\cite{fan2026comment}.\n\n\\begin{theorem}[both two-prime sums]\n\\label{res:two-prime-transcendence}\n\\label{res:two-prime-repeated-transcendence}\nLet $p<q$ be distinct primes.  Put $\\theta=\\log p/\\log q$ and\n$A=\\sum_{n\\ge0}p^{-n}q^{-\\lfloor n\\theta\\rfloor}$.  Let\n$\\mathcal R_{p,q}$ sum the reciprocal running LCM at every positive\n$\\{p,q\\}$-smooth integer, and let $\\mathcal D_{p,q}$ count each distinct\nrunning LCM once.  Then\n\\begin{equation}\\label{eq:two-prime-affine}\n \\mathcal D_{p,q}=\\frac{(q-p)A+p}{q-1},\\qquad\n \\mathcal R_{p,q}=\\frac{(p+q-1)A-(p-1)A^2}{q-1}.\n\\end{equation}\nBoth numbers are transcendental.\n\\end{theorem}\n\\evidenceremark{The Lean proof assumes the transcendence theorem of Bugeaud\nand Laurent; the two identities in~\\eqref{eq:two-prime-affine} are proved\nin Lean without it.}\n\\begin{proof}\nWrite $x=1/p$, $y=1/q$, $m_n=\\lfloor n\\theta\\rfloor$ and\n$\\delta_n=m_{n+1}-m_n$.  Unique factorisation makes $\\theta$ irrational,\nand $0<\\theta<1$ gives $\\delta_n\\in\\{0,1\\}$.  The initial value and the\npositive powers of $p$ contribute $A=\\sum_{n\\ge0}x^ny^{m_n}$.  There is\none $q$-power strictly between $p^n$ and $p^{n+1}$ exactly when\n$\\delta_n=1$, so the positive powers of $q$ contribute\n\\[\n B=\\sum_{n\\ge0}\\delta_nx^ny^{m_n+1},\\qquad \\mathcal D_{p,q}=A+B.\n\\]\nAbsolute convergence allows an index shift. Multiplying\n$y^{m_{n+1}}-y^{m_n}=\\delta_ny^{m_n}(y-1)$ by $x^{n+1}$ and summing gives\n$(A-1)-xA=x(y-1)B/y$. Therefore\n\\[\n B=\\frac{p-(p-1)A}{q-1}.\n\\]\nFor the repeated sum, the running LCM at $p^iq^j$ is\n$p^{i+\\lfloor j/\\theta\\rfloor}q^{j+m_i}$. Its reciprocal is the product\nof $x^iy^{m_i}$, which depends only on $i$, and\n$y^jx^{\\lfloor j/\\theta\\rfloor}$, which depends only on $j$. Absolute\nconvergence therefore gives\n\\[\n \\mathcal R_{p,q}\n =A\\sum_{j\\ge0}y^jx^{\\lfloor j/\\theta\\rfloor}=A(1+B).\n\\]\nFor the last equality, $n=\\lfloor j/\\theta\\rfloor$ locates $q^j$\nbetween $p^n$ and $p^{n+1}$; hence $m_n=j-1$ and $\\delta_n=1$.\nThis matches the positive-$j$ terms with those of $B$, while $j=0$\ncontributes $1$. Substitution proves~\\eqref{eq:two-prime-affine}.\n\nFactorisation alone would not prove transcendence of the product.\nThe two identities instead express both sums as nonconstant polynomials\nover $\\mathbb Q$ in the same $A$. It remains to prove $A$ transcendental.\nFor this we use the Hecke--Mahler series\n$F_\\theta(x,y)=\\sum_{n\\ge1}\\sum_{k=1}^{m_n}x^ny^k$.\nGeometric summation gives\n\\begin{equation}\\label{eq:hecke-mahler-boundary}\n A=\\frac1{1-x}-\\frac{1-y}{y}F_\\theta(x,y).\n\\end{equation}\nBugeaud and Laurent's theorem \\cite[Theorem~1.1]{bugeaudlaurent2023} applies\nwith intercept $\\rho=0$, $\\beta=x$ and $\\alpha=y$: the slope $\\theta$ is\nirrational and lies in $(0,1)$, $x$ and $y$ are nonzero algebraic numbers,\n$|x|<1$ and $|xy^\\theta|=p^{-2}<1$.  Thus $F_\\theta(x,y)$ is transcendental;\nthis case $\\rho=0$ is due to Loxton and van der Poorten\n\\cite[Theorem~8, p.~40]{loxtonvdp1977}.  Thus $A$ is transcendental.\nThe displayed affine and quadratic polynomials are nonconstant, so an\nalgebraic value of either would force $A$ to be algebraic.\n\\end{proof}\n\nThe \\longrecord{r3-two-prime}{companion's two-prime calculation} records\nan ordinary extension to the monoid generated by coprime integers\n$1<p<q$, such as $4,9$. That monoid need not contain all integers\nsupported on the prime divisors of $pq$; multiplicatively dependent\ngenerators such as $4,8$ require a different calculation.\n\nFor larger prime sets, the sums over pure powers\n$E_p=\\sum_{\\alpha\\ge0}\\Lprefix(p^\\alpha)^{-1}$ satisfy\n$\\mathcal D_P=\\sum_{p\\in P}E_p-(|P|-1)$. The\n\\longrecord{r3-prime-subseries}{companion's prime-power subseries argument}\ntreats their irrationality as a remark with an ordinary proof awaiting\nformalisation. The source record contains no separate priority search for\nthis claim. Irrationality of the individual subseries would not imply\nirrationality of their sum. The repeated series also contains mixed\nproducts of primes, whose multiplicities change the tail estimates.\n\n\\appendix\n\\section{Identities for the running LCM and multiplicities}\\label{sec:height-identities}\n\\label{sec:cells}\\label{sec:fibre}\nLet $p,q,r$ be pairwise distinct primes. Write\n$\\hgt=\\hgt_{\\{p,q,r\\}}$ and\n$\\Kernel(i,j,k)=\\hgt(p^iq^jr^k)^{-1}$ throughout this appendix and\nthe next. The following identities underlie the grouping of the\nrepeated series in Section~\\ref{sec:lcm}.\n\n\\begin{proposition}[the running least common multiple]\\label{res:lcm}\nLet $p,q,r$ be pairwise distinct primes and $x\\ge1$. Then the running\nleast common multiple is given by $\\Lprefix(x)=\\hgt(x)$.\n\\end{proposition}\n\\begin{proof}\nWe have $u\\mid\\hgt(x)$ for every supported $u\\le x$. For the reverse\ndivisibility, take the maximal powers of $p,q,r$ below $x$: they belong\nto the prefix and are pairwise coprime, so their product divides its LCM.\n\\end{proof}\nFor each prime, $x/p<p^{\\lfloor\\log_p x\\rfloor}\\le x$. Multiplying gives\n\\begin{equation}\\label{res:cube}\n x^3/(pqr)<\\hgt(x)\\le x^3.\n\\end{equation}\n\n\\begin{proposition}[cells and jumps]\\label{res:cell}\nThe running LCM is constant when the three integer logarithms are constant.\nA jump in exactly one logarithm multiplies it by the corresponding prime.\nThe first $n$ positive powers of each prime, together with $1$, form\n$3n+1$ distinct points.\n\\end{proposition}\n\\begin{proof}\nThe running LCM is a product of the three maximal pure powers, which proves\nthe constancy and jump assertions. Unique factorisation separates the\n$3n$ positive pure powers from one another and from $1$.\n\\end{proof}\n\n\\begin{proposition}[grouping terms with the same running LCM]\\label{res:fibre-prop}\nFor a finite exponent box $\\mathcal B$, set\n$F(H)=\\{(i,j,k)\\in\\mathcal B:\\hgt(p^iq^jr^k)=H\\}$.  Then\n\\begin{equation}\\label{res:fibre}\n \\sum_{(i,j,k)\\in\\mathcal B}\\Kernel(i,j,k)=\\sum_H\\frac{\\#F(H)}H.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nPartition the finite exponent box by the value of the running LCM. The terms\nin the part $F(H)$ all equal $1/H$.\n\\end{proof}\n\n\\section{The three-prime kernel}\\label{sec:rank}\nThe two-prime proof separates the summand by exponents. For three primes,\nthe carry in adding two fractional parts prevents any finite separation\nof $i$ from $(j,k)$. Diagonal rescaling leaves just two entry values,\n$1$ and $1/r$; choosing their threshold positions gives nonzero minors\nof every order. This concerns exact kernel identities, not identities\nafter summation.\n\n\n\\begin{example}\\label{res:rank}\nFor $(p,q,r)=(2,3,5)$, the leading two-by-two determinant is $-1/15$:\nits four entries are $1,1/6,1/2,1/60$, so it equals $1/60-1/12$.\n\\end{example}\n\nA sum of $d$ separated products has rank at most $d$ in each layer.\nThus the two-by-two example excludes only rank one; excluding every\nfinite $d$ requires nonzero minors of arbitrary order. Their indices\nmay depend on the order, but not on $k$.\n\n\\begin{theorem}[no finite separation of the kernel]\\label{res:infinite-rank}\nLet $p,q,r$ be primes with $p\\ne q$, $p\\ne r$ and $q\\ne r$.  For every\n$n\\ge0$ there are injective maps $I,J:\\{0,\\ldots,n-1\\}\\to\\N$ such that, for\nevery $k\\ge0$,\n\\[\n \\det\\bigl(\\Kernel(I(a),J(b),k)\\bigr)_{0\\le a,b<n}\\ne0.\n\\]\nConsequently, for no finite $d$ do there exist rational-valued functions\n$f_\\ell(i)$ and $G_\\ell(j,k)$, $0\\le\\ell<d$, satisfying\n\\[\n \\Kernel(i,j,k)=\\sum_{\\ell<d}f_\\ell(i)G_\\ell(j,k)\n \\qquad\\hbox{for all }i,j,k.\n\\]\n\\end{theorem}\n\n\\begin{proof}\nFix $k$. The powers of $p$ and $q$ separate into row and column factors,\nas before. Only the power of $r$ retains an interaction between $i$ and $j$. Put $c=r^{-1}$, $x_i=\\{i\\log_r p\\}$ and\n$y_j=\\{j\\log_r q\\}$, where braces denote fractional parts. Then\n\\[\n \\lfloor i\\log_r p+j\\log_r q\\rfloor\n =\\lfloor i\\log_r p\\rfloor+\\lfloor j\\log_r q\\rfloor\n   +\\mathbf1_{\\{x_i+y_j\\ge1\\}}.\n\\]\nThe last term is the carry when the two fractional parts are added.\nPutting all the other prime powers into row and column factors gives\n\\[\n \\Kernel(i,j,k)=U_i(k)^{-1}C_{ij}V_j(k)^{-1},\\qquad\n C_{ij}=c^{\\mathbf1_{\\{x_i+y_j\\ge1\\}}},\n\\]\nwhere\n\\[\n\\begin{aligned}\n U_i(k)&=p^i q^{\\lfloor\\log_q(p^ir^k)\\rfloor}\n            r^{k+\\lfloor\\log_r p^i\\rfloor},\\\\\n V_j(k)&=q^j p^{\\lfloor\\log_p(q^jr^k)\\rfloor}\n            r^{\\lfloor\\log_r q^j\\rfloor}.\n\\end{aligned}\n\\]\nThe positive factors preserve nonvanishing of minors and contain all\ndependence on $k$. In column $j$, the entry changes from $1$ to $c$\nat the threshold $1-y_j$. Both $\\log_r p$ and $\\log_r q$ are irrational,\nso their fractional-part orbits are dense separately; no joint orbit\nis needed. For $n\\ge1$, choose rows with\n$0<x_{I(0)}<\\cdots<x_{I(n-1)}<1$. Then choose each column threshold\nbetween consecutive row phases. Writing $s_b=1-y_{J(b)}$, take\n\n\\[\n s_0\\in(0,x_{I(0)}),\\qquad\n s_b\\in(x_{I(b-1)},x_{I(b)})\\quad(1\\le b<n).\n\\]\n\nThe intervals for the $s_b$ are disjoint, so the column indices are\ndistinct.  The strict inequalities also avoid the threshold itself, and give\n$x_{I(a)}+y_{J(b)}\\ge1$ exactly when $b\\le a$.  Thus multiplying row $a$\nby $U_{I(a)}(k)$ and column $b$ by $V_{J(b)}(k)$ reduces the selected kernel\nminor to\n\\[\n T_n(c)=\\begin{pmatrix}\n c&1&\\cdots&1\\\\\n c&c&\\cdots&1\\\\\n \\vdots&\\vdots&\\ddots&\\vdots\\\\\n c&c&\\cdots&c\n \\end{pmatrix},\\qquad\n \\det T_n(c)=c(c-1)^{n-1}.\n\\]\nIndeed, subtract each preceding row from the next, working upwards from\nthe last row. Each row difference has just one nonzero entry, $c-1$, in\nits own column. The resulting triangular matrix has diagonal entries\n$c,c-1,\\ldots,c-1$.\nThe original determinant is therefore\n\\[\n c(c-1)^{n-1}\n \\prod_{a<n}U_{I(a)}(k)^{-1}\\prod_{b<n}V_{J(b)}(k)^{-1}\\ne0.\n\\]\nThe indices were chosen from $C$, independently of $k$.  For $n=0$ the\nempty determinant equals $1$.\n\nFor the final assertion, suppose a separation with $d$ summands existed and\nfix any $k$.  On the rows $I(0),\\ldots,I(d)$ and columns\n$J(0),\\ldots,J(d)$ its matrix would be the product of the $(d+1)\\times d$\nmatrix $(f_\\ell(I(a)))_{a,\\ell}$ and the $d\\times(d+1)$ matrix\n$(G_\\ell(J(b),k))_{\\ell,b}$.  Its rank would be at most $d$, so its\ndeterminant would vanish, contrary to the minor just constructed.\n\\end{proof}\n\nNo continuity or boundedness is assumed for the separated factors.\n\n\\begin{corollary}[the same minors modulo integers coprime to $30$]\n\\label{res:admissible-modular-minors}\nFor $(p,q,r)=(2,3,5)$ and every $n\\ge1$, there are injective maps\n$I,J:\\{0,\\ldots,n-1\\}\\to\\N$, chosen independently of $B$ and $k$, such that\nfor every $B\\ge2$ coprime to $30$ and every $k\\ge0$, the selected $n\\times n$\nkernel matrix has unit determinant over $\\mathbb Z/B\\mathbb Z$, with each\nreciprocal prime power interpreted by its modular inverse.\n\\end{corollary}\n\\begin{proof}\nChoose the maps from Theorem~\\ref{res:infinite-rank}.  Every row and column\nfactor is a unit modulo $B$.  The normalised determinant is\n$5^{-1}(-4/5)^{n-1}$, also a unit.\n\\end{proof}\nThe modulus may be composite, for example $49$ or $77$.  Coprimality\nwith $30$ makes both the reciprocal entries and the determinant units:\nthe latter introduces only a power of $4$ in its numerator.\n\n\n\n\n\n\\paragraph{Leading minors.}\nThe leading $4\\times4$ block at $\\{2,3,5\\}$ is singular:\n\\[\n \\Kernel(3,j,0)=\\frac1{120}\\Kernel(0,j,0)\\qquad(0\\le j<4).\n\\]\nEvaluate the running LCM at $3^j$ and $8\\cdot3^j$ to obtain these equalities.\nAt $j=4$, however,\n$\\Kernel(3,4,0)-\\Kernel(0,4,0)/120=-1/19440000$.\nThe selection in the theorem avoids this problem by choosing column\nthresholds at distinct cuts among the row phases. An arbitrary\nsample may miss gaps; its rank counts distinct cuts, with one exception.\n\n\n\\begin{proposition}[rank of a matrix of threshold columns]\\label{res:finite-cut-rank}\nLet $m\\ge1$ and let $c$ lie in a field with $c\\ne 0,1$.\nFor $0\\le k\\le m$ write $v_k$ for the length-$m$ column with a $1$ in each of\nthe first $k$ coordinates and $c$ thereafter.\nA matrix whose distinct columns are $v_k$ for $k$ in a nonempty set $E$ has\nrank $|E|-\\mathbf 1_{\\{0,m\\}\\subseteq E}$.\n\\end{proposition}\n\nThe exception is the pair of constant columns, $v_0=c v_m$.\nAll other distinct cuts contribute independently.\n\n\\begin{proof}\nList $E=\\{k_1<\\cdots<k_t\\}$.  The $t-1$ differences\n$v_{k_{j+1}}-v_{k_j}=(1-c)\\mathbf 1_{\\{k_j,\\ldots,k_{j+1}-1\\}}$\nhave disjoint nonempty supports, hence are linearly independent.\nIf $k_1>0$ or $k_t<m$, their union misses a coordinate where $v_{k_1}$ is\nnonzero, so the rank is $t$.\nIf $k_1=0$ and $k_t=m$, then $v_0=c\\,\\mathbf 1$ lies in the span of the\ndifferences (their sum is $(1-c)\\mathbf 1$), so the rank is $t-1$.\n\\end{proof}\n\n\nOrder the sampled row phases and take $c=1/r$ over $\\mathbb Q$.\nThe formula then gives the rank of every nonempty rectangular sample,\nindependently of $k$: the layer changes only the nonzero diagonal\nfactors. The \\longrecord{r3-rank}{companion's kernel analysis} gives the\nexact rational comparisons.\n\n\n\\paragraph{Approximation.}\nThe diagonal rescaling preserves rank but removes the kernel's decay.\nRetaining rows $i<N$ in the original kernel gives $N$ separated terms;\n$\\hgt(x)>x^3/(pqr)$ makes the error tend to zero uniformly and in\n$\\ell^1(\\mathbb N^3)$. The rescaled matrix instead has uniform distance\n$(1-1/r)/2$ from finite-rank matrices. The\n\\longrecord{r3-approximation}{companion's approximation theorem} proves\nboth statements. Neither implies irrationality of the scalar sum.\n\n\n\\section{Verification, reproducibility and sources}\\label{app:sources}\n\\evidenceparagraph{\\evidencerecordurl}\nThe five-map proof is in \\href{\\distinctheightlean}{the distinct-value\ndevelopment} and \\href{\\distinctheightblocklean}{its block-series\ncriterion}. Its statement has no recorded Comparator comparison. The\nevidence record fixes the source versions; the named input for\ntwo-prime transcendence is stated beside that theorem.\n\nThe \\href{\\distinctheighturl}{programs for sums over distinct values} calculate the\ninterval endpoints and finite-state examples using exact arithmetic;\nthe proof in Section~\\ref{sec:distinct-235} derives its constants directly.\n\nThe \\longrecord{r3-five-maps}{companion} compares one-jump and dyadic bounds.\nIts finite-state extension uses earlier blocks to sharpen the intervals.\nSaved exact computations satisfy the criterion for every three-prime set\nand $292$ of the $330$ four-prime sets of primes at most $31$. The criterion\nand computations remain remarks awaiting formalisation.\nFor $\\{2,3,5,7\\}$, two admissible automaton words induce the same affine map,\ndefeating first-block recovery at every refinement. Equal maps alone do\nnot give equal arithmetic tails.\n\nThe \\longrecord{r3-source-inventory}{companion's source inventory} covers\nthe ordinary general and subseries arguments, the 26 September 2026 AI\ncheck and its corrections. No independent human mathematical review has\nbeen reported.\n\n\\subsection*{Further comparisons}\n\nThe companion compares our bounded-base series with the criteria of\nErd\\H{o}s--Straus~\\cite{erdosstraus1974},\nHan\\v{c}l--Tijdeman~\\cite{hancltijdeman2004} and\nDiananda--Oppenheim~\\cite{dianandaoppenheim1955}; the bounded-ratio argument\nwith Erd\\H{o}s--Taylor~\\cite[Theorem~1, p.~600]{erdostaylor1957} and\nFan~\\cite[Lemma~3.1, p.~7]{fan2026strongly}; and integer recurrences with\nHan\\v{c}l--Tijdeman's polynomial Cantor-series results\n\\cite[Theorems~2.2, 3.1 and 4.2]{hancltijdeman2008}.\nKoutsoukou-Argyraki and Li's Isabelle development~\\cite{afperdosstraus2020}\nformalises classical criteria, not this repeated series.\n\nThe transcendence comparisons concern finite differences\n\\cite{lucaouaknineworrell2025}, echoing \\cite{kebis2024echoing}, fixed-base\ndigit complexity \\cite{adamczewskibugeaud2007} and multivariate Mahler theory\n\\cite{adamczewskifaverjon2026}. The companion exhibits nonzero\nthird-difference boundary terms and distinguishes variable-base digits\nfrom fixed-base expansions: quadratic growth supplies neither the missing\ncancellation nor recurrence hypotheses. Fixed-prime semigroups\n\\cite{tijdemanmeijer1974,languasco2025} and other Ahmes-series problems\n\\cite{kovactao2024} supply context, not estimates used in our counting proof.\n\n\\pdfbookmark[1]{Statements and declarations}{statements-declarations}\n\\subsection*{Statements and declarations}\n\\paragraph{Authorship and availability.}\nAI agents carried out most of the research and drafting. The companion\ncontains the longer proofs and exact computations, together with the\nhypotheses and limitations of the unsuccessful approaches.\n\\paragraph{Funding and competing interests.}\nThis work received no external funding. The author declares no competing\ninterests.\n\\paragraph{Acknowledgements.}\nThe problem number and its status as of 28 July 2026 are taken from Thomas\nBloom's Erd\\H{o}s Problems catalogue~\\cite{erdosproblems}.\n\n\\begin{thebibliography}{99}\n\\setlength{\\itemsep}{3pt plus .5pt minus .5pt}\n\\setlength{\\parsep}{0pt}\\setlength{\\parskip}{0pt}\n\\bibitem{erdosgraham1980} Paul Erd\\H{o}s and Ronald L. Graham,\n  \\emph{Old and New Problems and Results in Combinatorial Number Theory}, Monographies de L'Enseignement Math\\'{e}matique \\textbf{28}, L'Enseignement Math\\'{e}matique (1980),\n  \\href{https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf}{source}.\n\\bibitem{erdos1988} Paul Erd\\H{o}s,\n  \\emph{On the irrationality of certain series: problems and results}, in \\emph{New Advances in Transcendence Theory}, Cambridge University Press (1988), 102--109,\n  \\href{https://doi.org/10.1017/CBO9780511897184.009}{doi:\\nolinkurl{10.1017/CBO9780511897184.009}}.\n\\bibitem{erdos1974letter} Paul Erd\\H{o}s,\n  \\emph{Letter to the Editor}, Fibonacci Quarterly \\textbf{12}, no.~4 (1974), 335,\n  \\href{https://www.fq.math.ca/Scanned/12-4/letter.pdf}{source}.\n\\bibitem{erdosproblems} Thomas F. Bloom,\n  \\emph{Erd\\H{o}s Problem \\#269} (2026),\n  \\href{https://www.erdosproblems.com/269}{source}. Accessed 28 July 2026.\n\\bibitem{bugeaudlaurent2023} Yann Bugeaud and Michel Laurent,\n  \\emph{Transcendence and continued fraction expansion of values of Hecke--Mahler series}, Acta Arithmetica \\textbf{209} (2023), 59--90,\n  \\href{https://doi.org/10.4064/aa220323-18-1}{doi:\\nolinkurl{10.4064/aa220323-18-1}}; arXiv:\\href{https://arxiv.org/abs/2203.12901}{2203.12901}.\n\\bibitem{loxtonvdp1977} John H. Loxton and Alfred J. van der Poorten,\n  \\emph{Arithmetic properties of certain functions in several variables III}, Bulletin of the Australian Mathematical Society \\textbf{16} (1977), 15--47,\n  \\href{https://doi.org/10.1017/S0004972700022978}{doi:\\nolinkurl{10.1017/S0004972700022978}}.\n\\bibitem{fan2026comment} Steve Fan,\n  \\emph{Comment on Erd\\H{o}s Problem \\#269, thread 269, post 7218} (2026),\n  \\href{https://www.erdosproblems.com/forum/thread/269#post-7218}{source}. Public forum post, 26 June 2026, thread 269, post 7218.\n\\bibitem{dianandaoppenheim1955} P. H. Diananda and A. Oppenheim,\n  \\emph{Criteria for irrationality of certain classes of numbers~II},\n  American Mathematical Monthly \\textbf{62}, no.~4 (1955), 222--225.\n  Cited in the form stated by Serbenyuk, arXiv:\\href{https://arxiv.org/abs/1706.03124}{1706.03124}, Theorem~1.\n\\bibitem{erdosstraus1974} Paul Erd\\H{o}s and Ernst G. Straus,\n  \\emph{On the irrationality of certain series}, Pacific Journal of Mathematics \\textbf{55}, no.~1 (1974), 85--92,\n  \\href{https://doi.org/10.2140/pjm.1974.55.85}{doi:\\nolinkurl{10.2140/pjm.1974.55.85}}.\n\\bibitem{hancltijdeman2004} Jaroslav Han\\v{c}l and Robert Tijdeman,\n  \\emph{On the irrationality of Cantor and Ahmes series}, Publicationes Mathematicae Debrecen \\textbf{65}, no.~3--4 (2004), 371--380,\n  \\href{https://doi.org/10.5486/PMD.2004.3254}{doi:\\nolinkurl{10.5486/PMD.2004.3254}}.\n\\bibitem{erdostaylor1957} Paul Erd\\H{o}s and S. James Taylor,\n  \\emph{On the set of points of convergence of a lacunary trigonometric series and the equidistribution properties of related sequences}, Proceedings of the London Mathematical Society \\textbf{s3-7}, no.~1 (1957), 598--615,\n  \\href{https://doi.org/10.1112/plms/s3-7.1.598}{doi:\\nolinkurl{10.1112/plms/s3-7.1.598}}.\n\\bibitem{fan2026strongly} Steve Fan,\n  \\emph{Strongly complete sets and a conjecture of Erd\\H{o}s} (2026),\n  \\href{https://arxiv.org/abs/2607.14071v1}{source}; arXiv:\\href{https://arxiv.org/abs/2607.14071}{2607.14071}. The cited Lemma~3.1 is in arXiv v1, 15 July 2026.\n\\bibitem{hancltijdeman2008} Jaroslav Han\\v{c}l and Robert Tijdeman,\n  \\emph{On the irrationality of polynomial Cantor series}, Acta Arithmetica \\textbf{133}, no.~1 (2008), 37--52,\n  \\href{https://doi.org/10.4064/aa133-1-3}{doi:\\nolinkurl{10.4064/aa133-1-3}}.\n\\bibitem{lucaouaknineworrell2025} Florian Luca, Jo\\\"{e}l Ouaknine and James Worrell,\n  \\emph{Transcendence of Hecke--Mahler Series}, Bulletin of the London Mathematical Society \\textbf{57}, no.~5 (2025), 1360--1368,\n  \\href{https://doi.org/10.1112/blms.70033}{doi:\\nolinkurl{10.1112/blms.70033}}; arXiv:\\href{https://arxiv.org/abs/2412.07908}{2412.07908}. Numbered references use the published article.\n\\bibitem{kebis2024echoing} Pavol Kebis, Florian Luca, Jo\\\"{e}l Ouaknine, Andrew Scoones and James Worrell,\n  \\emph{On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words}, in \\emph{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, Leibniz International Proceedings in Informatics \\textbf{297}, Schloss Dagstuhl -- Leibniz-Zentrum f\\\"{u}r Informatik (2024), 144:1--144:15,\n  \\href{https://doi.org/10.4230/LIPIcs.ICALP.2024.144}{doi:\\nolinkurl{10.4230/LIPIcs.ICALP.2024.144}}.\n\\bibitem{tijdemanmeijer1974} Robert Tijdeman and H. G. Meijer,\n  \\emph{On integers generated by a finite number of fixed primes}, Compositio Mathematica \\textbf{29}, no.~3 (1974), 273--286,\n  \\href{https://www.numdam.org/article/CM_1974__29_3_273_0.pdf}{source}.\n\\bibitem{languasco2025} Alessandro Languasco, Florian Luca, Pieter Moree and Alain Togb\\'{e},\n  \\emph{Sequences of integers generated by two fixed primes}, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg \\textbf{95} (2025), 123--148,\n  \\href{https://doi.org/10.1007/s12188-025-00293-9}{doi:\\nolinkurl{10.1007/s12188-025-00293-9}}; arXiv:\\href{https://arxiv.org/abs/2309.12806}{2309.12806}.\n\\bibitem{kovactao2024} Vjekoslav Kova\\v{c} and Terence Tao,\n  \\emph{On several irrationality problems for Ahmes series}, Acta Mathematica Hungarica \\textbf{175} (2025), 572--608,\n  \\href{https://doi.org/10.1007/s10474-025-01528-0}{doi:\\nolinkurl{10.1007/s10474-025-01528-0}}; arXiv:\\href{https://arxiv.org/abs/2406.17593}{2406.17593}.\n\\bibitem{afperdosstraus2020} Angeliki Koutsoukou-Argyraki and Wenda Li,\n  \\emph{Irrationality Criteria for Series by Erd\\H{o}s and Straus}, Archive of Formal Proofs (2020),\n  \\href{https://isa-afp.org/entries/Irrational_Series_Erdos_Straus.html}{source}. Entry dated 12 May 2020; proof-document version consulted: 6 February 2026.\n\\bibitem{adamczewskibugeaud2007} Boris Adamczewski and Yann Bugeaud,\n  \\emph{On the complexity of algebraic numbers I. Expansions in integer bases},\n  Annals of Mathematics \\textbf{165}, no.~2 (2007), 547--565,\n  \\href{https://doi.org/10.4007/annals.2007.165.547}{doi:\\nolinkurl{10.4007/annals.2007.165.547}}.\n\\bibitem{adamczewskifaverjon2026} Boris Adamczewski and Colin Faverjon,\n  \\emph{Mahler's method in several variables and finite automata},\n  Annals of Mathematics \\textbf{204}, no.~2 (2026), 455--533,\n  \\href{https://doi.org/10.4007/annals.2026.204.2.1}{doi:\\nolinkurl{10.4007/annals.2026.204.2.1}}.\n  Online 13 September 2026; locators here refer to the\n  \\href{https://faverjon.perso.math.cnrs.fr/AdamczewskiFaverjon_MahlerFiniteAutomata.pdf}{68-page author manuscript}.\n\\end{thebibliography}\n\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":52465,"sha256":"sha256:ac61a4c170358ab7006ac37c42a8201bd1c4dc8502d010204fcff66995fc1e2f","cite_as":"Plectis reader corpus, paper:erdos-269-three-prime-running-lcm","source_repository":"plectis-erdos","source_path":"paper/269/erdos-269-three-prime-running-lcm.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:088d36d7ba3d3ee0cee2df99e8915927999698f19889f8ea0b8d55b3721ec778","byte_count":6923,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% This is the only copy of the shared preamble.  The eight short notes, the\n% long records assembled from paper/reasoning-parts other than #249 and #257\n% (which carry their own preambles), and the synthesis paper include it with\n% \\input{problem-note-preamble}; paper/Makefile builds from this directory\n% with the directory on the tectonic search path.\n%\n% Shared preamble for the Erdős Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erdős problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{8e58c395e9807ef3374d7365c62cf04d20ee6331}\n\\newcommand{\\commitshort}{9e231ce4371f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-erdos/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-erdos/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\usepackage{longtable}  % the notes' declaration tables; loaded here so the assembled records inherit it\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n% ---------------------------------------------------------------------------\n% Evidence marks.  paper-evidence.tex places \"Lean\" and \"Comparator\" links in\n% the margin beside each result from the paper's generated file\n% paper/evidence/<paper>.tex, which each paper inputs after this preamble.\n% ---------------------------------------------------------------------------\n\\input{paper-evidence}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Production provenance.  The wording is fixed by the corpus provenance\n% contract and reproduced verbatim in every manuscript in this repository, so\n% the papers cannot drift into different accounts of how they were made.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erdős problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every Lean mark in the margin points to propositions that the\n  pinned Lean kernel accepts, with no \\texttt{sorry}, no added axiom, and no\n  unchecked evaluation.  That claim concerns the formal statements only; it\n  says nothing about their mathematical interest, their novelty, or the\n  original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n\n% Compact opening card for the separately stated Comparator theorem selected\n% from each problem note.  The prose supplied by each note owns the\n% contribution and relevance judgements; the shared final sentence states the\n% mechanical assurance boundary once and points to the public evidence owner.\n\\newcommand{\\externalreviewobject}[3]{%\n  \\begin{quote}\\small\n  \\textbf{Contribution.} #1\\par\n  \\textbf{Relation to the open problem.} #2\\par\n  \\textbf{Executable review object.} #3  The repository's\n  external-verification job compares these exact Lean propositions with\n  separately declared challenge statements and an axiom budget, then asks\n  Lean's kernel to check the submitted proofs.  The\n  \\href{https://github.com/wcook04/plectis-erdos/blob/ca0e13f8acf5ccf48506e4bdb870953d3a0856fa/formalization.yaml}%\n  {formalisation manifest} and the commit-bound CI receipt record that check;\n  they do not assess novelty, significance, or whether the original problem is\n  solved.\n  \\end{quote}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  Under\n% Tectonic (XeTeX) the text and mathematics are set in TeX Gyre Pagella from\n% the bundled OpenType fonts, with a Heros sans range for \\mathsf and\n% Inconsolata for monospace; the pdflatex fallback keeps the same Palatino\n% design through newpx.  The A4 text block uses a classical measure with\n% quiet running heads and restrained ink.  Link colours: steel blue for\n% external (Lean-source and web) targets, deep burgundy for internal\n% cross-references and citations.\n\\NeedsTeXFormat{LaTeX2e}\n\\ProvidesPackage{paper-house-style}[2026/07/19 Pagella house style for the papers]\n\n\\RequirePackage{iftex}\n\\RequirePackage{amsmath}\n\\RequirePackage{amsthm}\n\\ifpdftex\n  \\RequirePackage[T1]{fontenc}\n  \\RequirePackage[utf8]{inputenc}\n  \\RequirePackage{amssymb}\n  \\RequirePackage{newpxtext}\n  \\RequirePackage{newpxmath}\n\\else\n  \\RequirePackage{fontspec}\n  \\RequirePackage{unicode-math}\n  \\setmainfont{texgyrepagella}[Extension=.otf,UprightFont=*-regular,\n    BoldFont=*-bold,ItalicFont=*-italic,BoldItalicFont=*-bolditalic]\n  \\setmathfont{texgyrepagella-math.otf}\n  \\setmathfont{texgyreheros-regular.otf}[range=sfup,Scale=MatchLowercase]\n  \\setmonofont{Inconsolatazi4}[Extension=.otf,UprightFont=*-Regular,\n    BoldFont=*-Bold,Scale=MatchLowercase]\n\\fi\n\n\\RequirePackage[a4paper,textwidth=418pt,textheight=636pt,hcentering,\n  top=68pt,headheight=15pt,headsep=20pt,footskip=34pt]{geometry}\n\\RequirePackage{microtype}\n\\linespread{1.045}\n\\clubpenalty=10000 \\widowpenalty=10000 \\tolerance=1600\n\n\\RequirePackage[dvipsnames,table]{xcolor}\n\\definecolor{linkexternal}{HTML}{275D8C}\n\\definecolor{linkinternal}{HTML}{6E2B2B}\n\\definecolor{papertableheader}{HTML}{E3EAF0}\n\\definecolor{papertablerow}{HTML}{F2F5F7}\n\\definecolor{papertablerule}{HTML}{7D8991}\n\\definecolor{paperresultrule}{HTML}{315B72}\n\n% Stable cross-document links between sibling PDFs. A named destination is\n% more durable than a page number: sections may move while their semantic role\n% remains the same. All public PDFs are copied to one directory at release.\n\\newcommand{\\papersectiontarget}[1]{\\hypertarget{#1}{}}\n\\newcommand{\\papersectionlink}[3]{\\href{#1\\##2}{#3}}\n\n\\RequirePackage{titlesec}\n\\titleformat{\\section}{\\Large\\bfseries}{\\thesection}{0.85em}{}\n\\titlespacing*{\\section}{0pt}{2.6ex plus .8ex minus .3ex}{1.5ex plus .3ex}\n\\titleformat{\\subsection}{\\large\\bfseries}{\\thesubsection}{0.8em}{}\n\\titlespacing*{\\subsection}{0pt}{2.1ex plus .6ex minus .2ex}{1.1ex plus .2ex}\n\\titlespacing*{\\paragraph}{0pt}{1.6ex plus .5ex minus .2ex}{0.85em}\n\n\\RequirePackage{fancyhdr}\n\\newcommand{\\paperrunninghead}{}\n\\newcommand{\\runninghead}[1]{\\renewcommand{\\paperrunninghead}{#1}}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\fancyhead[L]{\\small\\itshape\\paperrunninghead}\n\\fancyhead[R]{\\small\\thepage}\n\\renewcommand{\\headrulewidth}{0pt}\n\\fancypagestyle{plain}{\\fancyhf{}\\renewcommand{\\headrulewidth}{0pt}}\n\n\\RequirePackage{caption}\n\\captionsetup{font=small,labelfont=bf,skip=9pt,width=.92\\linewidth}\n\\RequirePackage{etoolbox}\n\\RequirePackage{pdflscape}\n\\renewcommand{\\arraystretch}{1.20}\n\\newcommand{\\papertablehead}{\\rowcolor{papertableheader}}\n\\newcommand{\\papertablebody}{\\rowcolors{2}{white}{papertablerow}}\n\\AtBeginEnvironment{tabular}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\\AtBeginEnvironment{longtable}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\n% A compact front-matter panel for the result a scanning reader should retain.\n% It is deliberately short and unbreakable: manuscripts should state one\n% result, its evidence, and its limit here, then supply the argument below.\n\\newenvironment{paperresult}[1]{%\n  \\par\\addvspace{0.7em}\n  \\begin{center}\\begin{minipage}{0.94\\linewidth}\n    {\\color{paperresultrule}\\rule{\\linewidth}{0.8pt}}\\par\\vspace{4pt}\n    {\\small\\sffamily\\bfseries\\color{paperresultrule}#1}\\par\\smallskip\n    \\small\\color{black}\n}{%\n    \\par\\vspace{4pt}{\\color{paperresultrule}\\rule{\\linewidth}{0.45pt}}\n    \\end{minipage}\\end{center}\\par\\addvspace{0.5em}\n}\n\n% Shared diagram grammar.  Reader-facing figures use the same restrained\n% hierarchy as the surrounding page: a short eyebrow, a substantial title,\n% one compact explanatory line, and an optional evidence badge.  The styles\n% are installed only when a manuscript loads TikZ.\n\\AtBeginDocument{%\n  \\@ifpackageloaded{tikz}{%\n    \\tikzset{\n      paper box/.style={draw=gray!48, line width=.45pt, rounded corners=3pt,\n        fill=white, align=left, inner xsep=7pt, inner ysep=6pt,\n        font=\\small},\n      paper card/.style={paper box, minimum height=13mm},\n      paper stage/.style={paper box, align=center, minimum height=14mm},\n      paper arrow/.style={-{Stealth[length=2.15mm,width=1.45mm]},\n        line width=.7pt, draw=gray!72!black},\n      paper feedback/.style={paper arrow, densely dashed,\n        draw=Violet!72!black},\n      paper eyebrow/.style={font=\\scriptsize\\sffamily\\bfseries,\n        text=gray!70!black},\n      paper label/.style={font=\\scriptsize\\sffamily,\n        text=gray!72!black, fill=white, inner sep=1.5pt},\n      paper badge/.style={rounded corners=2pt, inner xsep=4pt, inner ysep=2pt,\n        font=\\scriptsize\\sffamily\\bfseries},\n      paper boundary/.style={densely dashed, line width=.55pt,\n        draw=gray!62}\n    }%\n  }{}%\n}\n\n% Front matter: title block with an optional italic subtitle, small-caps\n% author, and a hairline rule; the abstract is set small with a small-caps\n% head and symmetric indents.\n\\newcommand{\\subtitle}[1]{\\gdef\\@papersubtitle{#1}}\n\\renewenvironment{abstract}{%\n  \\vspace{0.1em}\\small\n  \\begin{center}{\\normalsize\\scshape Abstract}\\end{center}\\vspace{-0.5em}\n  \\list{}{\\leftmargin 1.3em\\rightmargin\\leftmargin\\listparindent 0pt\n    \\parsep 0.32em\\itemindent 0pt}\\item\\relax}\n  {\\endlist\\vspace{0.5em}}\n\\def\\@maketitle{%\n  \\newpage\\null\n  \\vspace*{-2.6em}%\n  \\begin{center}%\n    {\\LARGE\\bfseries\\@title\\par}%\n    \\vskip 0.55em\n    {\\large\\itshape\\@papersubtitle\\par}%\n    \\vskip 1.1em\n    {\\scshape\\@author\\textsuperscript{*}\\par}%\n    \\vskip 0.25em\n    {\\small\\@date\\par}%\n    \\vskip 0.8em\n    \\rule{6.5em}{0.4pt}%\n  \\end{center}%\n  \\vskip 0.9em}\n\n% Production provenance, set at the foot of the first page.\n%\n% A reader who meets a single personal byline ordinarily infers that the named\n% author drafted the sentences.  For these papers that inference is false, and\n% a declaration reached only in the back matter would leave it standing for the\n% length of the paper.  The note therefore appears on page one, small and\n% visually subordinate to the title and abstract: enough to prevent the false\n% inference, not enough to make the production method the subject.\n%\n% It carries no mark, because it qualifies the whole manuscript rather than any\n% one passage.  \\footnotetext does not advance the footnote counter, and the\n% emptied \\thefootnote is confined to a group, so numbered footnotes elsewhere\n% in the paper are unaffected.\n%\n% The wording is not authored here.  It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"},{"record_type":"paper_source_dependency","tex_reference":"evidence/erdos-269-three-prime-running-lcm","source_path":"paper/evidence/erdos-269-three-prime-running-lcm.tex","media_type":"text/x-tex","sha256":"sha256:3423b65aad1bfd226e7dde1b807177e87e05cc3d618bef4928f830acb38d80f2","byte_count":9190,"licence":"CC-BY-4.0","body_format":"latex","body":"% Generated by scripts/paper_evidence.py from docs/paper_lean_coverage.json and the\n% Comparator replay receipts under evidence/comparator/.  Do not edit by hand.\n\\newcommand{\\evidencerecordurl}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md}\n\\DeclareResultEvidenceName{res:distinct-height-235}{Theorem 1.1}\n\\DeclareResultEvidenceRecord{res:distinct-height-235}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-distinct-height-235}{exact}{pending}\n\\DeclareResultEvidence{res:distinct-height-235}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/DistinctHeightIrrationality.lean\\#L689}{}\n\\DeclareResultEvidenceName{res:dyadic-alphabet}{Lemma 3.1}\n\\DeclareResultEvidenceRecord{res:dyadic-alphabet}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-dyadic-alphabet}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:dyadic-alphabet}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperCompleteR20/DyadicAlphabetWhole.lean\\#L19}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-dyadic-alphabet-comparator}\n\\DeclareResultEvidenceName{res:actual-orbit}{Proposition 3.2}\n\\DeclareResultEvidenceRecord{res:actual-orbit}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-actual-orbit}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:actual-orbit}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR7SeriesIdentification.lean\\#L168}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-actual-orbit-comparator}\n\\DeclareResultEvidenceName{res:denominator-reduction}{Theorem 3.3}\n\\DeclareResultEvidenceRecord{res:denominator-reduction}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-denominator-reduction}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:denominator-reduction}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR7RationalBridge.lean\\#L80}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-denominator-reduction-comparator}\n\\DeclareResultEvidenceName{res:exact-onset}{Corollary 3.4}\n\\DeclareResultEvidenceRecord{res:exact-onset}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-exact-onset}{exact}{compared}\n\\DeclareResultEvidence{res:exact-onset}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-exact-onset}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-exact-onset-comparator}\n\\DeclareResultEvidenceName{res:consumer}{Lemma 4.1}\n\\DeclareResultEvidenceRecord{res:consumer}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-consumer}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:consumer}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR7BasicAssembly.lean\\#L138}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-consumer-comparator}\n\\DeclareResultEvidenceName{res:windowconsumer}{Theorem 4.2}\n\\DeclareResultEvidenceRecord{res:windowconsumer}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-windowconsumer}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:windowconsumer}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR7WindowResults.lean\\#L51}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-windowconsumer-comparator}\n\\DeclareResultEvidenceName{res:two-prime-transcendence}{Theorem 5.1}\n\\DeclareResultEvidenceRecord{res:two-prime-transcendence}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-two-prime-transcendence}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{res:two-prime-transcendence}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-two-prime-transcendence}{}\n\\DeclareResultEvidenceName{res:lcm}{Proposition A.1}\n\\DeclareResultEvidenceRecord{res:lcm}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-lcm}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:lcm}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperCompleteR20/RealCutoffs.lean\\#L49}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-lcm-comparator}\n\\DeclareResultEvidenceName{res:cell}{Proposition A.2}\n\\DeclareResultEvidenceRecord{res:cell}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-cell}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:cell}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-cell}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-cell-comparator}\n\\DeclareResultEvidenceName{res:fibre-prop}{Proposition A.3}\n\\DeclareResultEvidenceRecord{res:fibre-prop}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-fibre-prop}{exact}{compared}\n\\DeclareResultEvidence{res:fibre-prop}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean\\#L407}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-fibre-prop-comparator}\n\\DeclareResultEvidenceName{res:infinite-rank}{Theorem B.2}\n\\DeclareResultEvidenceRecord{res:infinite-rank}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-infinite-rank}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:infinite-rank}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR7BasicAssembly.lean\\#L22}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-infinite-rank-comparator}\n\\DeclareResultEvidenceName{res:admissible-modular-minors}{Corollary B.3}\n\\DeclareResultEvidenceRecord{res:admissible-modular-minors}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-admissible-modular-minors}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:admissible-modular-minors}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR7ModularMinors.lean\\#L133}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-admissible-modular-minors-comparator}\n\\DeclareResultEvidenceName{res:finite-cut-rank}{Proposition B.4}\n\\DeclareResultEvidenceRecord{res:finite-cut-rank}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-finite-cut-rank}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:finite-cut-rank}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos269/PaperR7FiniteCutRank.lean\\#L182}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-269-three-prime-running-lcm.md\\#res-finite-cut-rank-comparator}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/269/erdos-269-three-prime-running-lcm.tex","section_outline":["Two sums from running least common multiples","Recovering the prime-power jumps","The repeated three-prime sum","A residue criterion","Two-prime transcendence","Identities for the running LCM and multiplicities","The three-prime kernel","Verification, reproducibility and sources","Further comparisons","Statements and declarations"]},"erdos-1041-lemniscate-newton-flow":{"paper_id":"erdos-1041-lemniscate-newton-flow","title":"Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections","form":"Problem note","length_class":"short","pages":19,"subject":"Erdős #1041","subject_kind":"problem","problem_id":"erdos_1041","landing_role":"mathematics","question":"How does the degree-seven counterexample obstruct short lemniscate connections, and why do trinomials admit them?","precis":"The degree-seven polynomial constructed by ani has all its zeros in the open unit disc, but every connected subset of {|f|<1} containing two zeros has Hausdorff measure greater than two. The proof confines a slit preimage near a critical point and applies radial projection on its two inverse sheets. For monic trinomials with zeros in the open unit disc, the root equation instead keeps every root-to-origin segment inside the lemniscate.","owns":"ani's degree-seven counterexample and the trinomial connection theorem; the long paper develops critical-value estimates, other subclasses and subsidiary formal checks","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or the current status of Erdős #1041, including the later one-polynomial Formal Conjectures path-image Hausdorff-length check and its unreviewed historical curve-length correspondence; those distinctions belong to docs/claims.json in the mathematics repository","first_pass":["sec:problem","sec:counterexample","sec:trinomial"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-1041-lemniscate-newton-flow.pdf","hosted_pdf_path":"papers/erdos-1041-lemniscate-newton-flow.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex","site_html":"maths/papers/erdos-1041-lemniscate-newton-flow.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1041/erdos-1041-lemniscate-newton-flow.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1041/erdos-1041-lemniscate-newton-flow.tex","body_pointer":"papers.records.erdos-1041-lemniscate-newton-flow.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Standalone Erdős Problem Note for #1041.  Context is deliberately repeated\n% from the sibling notes so that this file remains independently readable.\n%\n% The public March manuscript is attributable only to the erdosproblems.com\n% handle `shtuka`; the PDF contains no real-name author metadata.  The\n% bibliography therefore cites that handle and public URL conservatively.\n%\n% Build from the public paper directory with `make`.\n\\documentclass[11pt]{article}\n\\usepackage{longtable}\n\\input{problem-note-preamble}\n\\usepackage{tikz}\n\\usetikzlibrary{arrows.meta}\n% Number remarks locally so their retained labels have distinct destinations.\n% The shared preamble defines an unnumbered remark environment.\n\\let\\remark\\relax\n\\let\\endremark\\relax\n\\newtheorem{remark}[theorem]{Remark}\n\n\\input{evidence/erdos-1041-lemniscate-newton-flow}\n\\renewcommand{\\commit}{7380b7871687b6bcc41ca0143c61f232e8af6500}\n\\renewcommand{\\commitshort}{7380b7871687}\n\\renewcommand{\\PX}{lean/ErdosProblems}\n\\hypersetup{\n  pdftitle={Paths in polynomial lemniscates: a degree-seven counterexample and radial connections},\n  pdfsubject={Erdős Problem Notes: Problem 1041},\n  pdfkeywords={lemniscate, counterexample, Hausdorff measure, critical values, trinomials, weighted Poisson inequality, Lean 4},\n}\n\n% Source targets retain their original pinned identifiers.\n\n\\runninghead{Paths in polynomial lemniscates}\n\\title{Paths in Polynomial Lemniscates:\\\\A Degree-Seven Counterexample\\\\and Radial Connections}\n\\subtitle{Erd\\H{o}s Problem~\\#1041}\n\\author{Will Cook}\n\\date{29 September 2026}\n\n% Hyperref remote PDF links take the destination name after the hash.\n\\newcommand{\\recordlink}[3]{\\href{#1\\##2}{#3}}\n\n\\begin{document}\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\nA monic degree-seven polynomial $f$ constructed by \\texttt{ani} answers\nErd\\H{o}s Problem~1041 negatively: all zeros lie in the open unit disc,\nbut every connected subset of $\\{|f|<1\\}$ containing two zeros has\none-dimensional Hausdorff measure greater than $2$. The obstruction lies in a single two-root\ncomponent. Its slit preimage is confined near the critical point, so\nradial projection on the two inverse sheets forces a length greater\nthan $2$. In contrast, for every monic trinomial with zeros in the open\nunit disc, the root equation keeps all root-to-origin segments inside\nthe lemniscate. Further analytic criteria use small or isolated critical\nvalues; their proofs are not fully formalised in Lean.\n\\end{abstract}\n\n\\section{Introduction}\\label{sec:problem}\n\nTwo points in the open unit disc are less than $2$ apart. The extra\ncondition in the following question is that the whole connecting curve\nstay where a given polynomial has modulus less than $1$.\nFor a monic polynomial $f$, write\n\\[\n \\Omega_f=\\{z:|f(z)|<1\\},\\qquad K_t(f)=\\{z:|f(z)|\\le t\\}.\n\\]\nErd\\H{o}s, Herzog and Piranian \\cite[Problem~5, p.~139]{ehp1958} asked\nwhether two zeros in the open unit disc could always be connected by a\nshort curve in $\\Omega_f$. Their question is listed as Problem~1041 in\nBloom's catalogue \\cite{bloom}.\n\n\\begin{problem}[Erd\\H{o}s \\#1041]\\label{res:problem}\nLet $f(z)=\\prod_{i=1}^{n}(z-z_i)$ be monic with $n\\ge2$ and all\n$z_i$ in the open unit disc.  Must two root occurrences be joined by a curve\nof length less than $2$ inside $\\{|f|<1\\}$?\n\\end{problem}\n\nA repeated root allows a constant curve between two occurrences, so we\nconsider distinct zeros. Pendyala\n\\cite[Theorem~1 and Lemma~1]{june2026} proved the degree-four case using\na close-pair chord or radial segments through the centre of a smallest\nenclosing disc. In degree seven the answer is negative. The example was\nconstructed by the erdosproblems.com contributor\n\\href{https://www.erdosproblems.com/forum/thread/1041\\#post-8861}{\\texttt{ani}}.\n\n\\papersectiontarget{counterexample-statement}\n\\begin{theorem}[\\texttt{ani}'s degree-seven example]\\label{res:ani-degree-seven-counterexample}\nThe monic polynomial $f$ in \\eqref{eq:ani-f} has seven distinct zeros in\nthe open unit disc. Every connected set $K\\subset\\Omega_f$ containing\ntwo of its zeros satisfies $\\mathcal H^1(K)>2$.\n\\end{theorem}\n\nHere $\\mathcal H^1$ is one-dimensional Hausdorff measure. A rectifiable\npath image has measure at most its total variation, so the theorem\nexcludes short paths as well as more general connected sets.\n\nFive components of $\\Omega_f$ contain one zero each; the sixth contains\ntwo and is the only possible place to join a pair. On this component,\n$f$ has degree two. Cut the value disc from its critical value to the\nunit circle. Inside the two-root component, the \\emph{whole} slit\npreimage lies in a small disc about the critical point. Connectedness\nforces a root-joining set to approach this disc on both inverse sheets.\nThe two radial projections then give a lower bound: the sum of the root\ndistances from the critical point, minus twice the small disc's radius.\nFor the chosen pair this bound exceeds $2$.\n\nSection~\\ref{sec:counterexample} contrasts the quadratic with the\ncounterexample; Section~\\ref{sec:trinomial} treats trinomials and the\nobstruction introduced by a fourth term. Section~\\ref{sec:critical-proximity}\nseparates root proximity from containment. Sections~\\ref{sec:constant-factor}\nand~\\ref{sec:separation} construct paths by component growth and continuation\naround an isolated critical value, respectively. These independent\nanalytic arguments remain unformalised. The\n\\href{erdos1041-lemniscate-reasoning-surface.pdf}{companion paper}\ngives their proofs, the counterexample's rational data, and the arguments\nbehind the later estimates and questions.\n\n\\section{A component with a narrow bottleneck}\\label{sec:counterexample}\n\n\\subsection{A quadratic model}\nConsider $p(z)=z^2-r^2$ with $0<r<1$. Its critical point is $0$,\nits critical value is $-r^2$, and the outward slit in the value disc is\n$J=(-1,-r^2]$. We can compute its preimage without choosing a branch:\n\\[\n p^{-1}(J)=\\{iy:|y|<\\sqrt{1-r^2}\\}.\n\\]\nThis vertical interval separates the two inverse sheets containing $r$\nand $-r$. It shrinks to the critical point as $r\\uparrow1$. Yet the\nhorizontal segment $[-r,r]$ stays in $\\{|p|<1\\}$ and has length $2r<2$.\nA narrow neck alone therefore does not give the desired counterexample.\nFor the radial-distance estimate below to exceed $2$, the two root\ndistances from the critical point must themselves have sum greater than $2$.\n\n\\subsection{A length estimate for a double cover}\nThe degree on a component counts preimages \\emph{in that component},\nwith multiplicity; it equals the number of zeros there, not necessarily\nthe degree of the polynomial \\cite[proof of Proposition~2.1]{eks2010}.\nNear a simple critical point $c$, write $p(c+z)-p(c)=z^2A(z)$.\nThe quadratic model suggests a spatial scale $\\sqrt{\\delta/M}$:\nthe value slit has length $\\delta=1-|p(c)|$, while\n$M=|A(0)|=|p''(c)|/2$ converts squared distance into value distance.\nThe next lemma controls this comparison on a disc of radius $h$.\nDegree two serves a different purpose. It ensures that the two local\npreimages near $c$ exhaust every fibre in the chosen component, so a\nconnection cannot pass through an unexamined part of the slit preimage.\n\n\\begin{lemma}\\label{lem:two-sheet-bottleneck}\nLet $p$ be a polynomial and $U$ a component of $\\{|p|<1\\}$ on which $p$\nhas degree two, with one simple critical point $c$ and $v=p(c)\\ne0$. Write\n$p(c+z)-v=z^2A(z)$ and put $M=|A(0)|$, $\\delta=1-|v|$.\nLet $h>0$. If $|A(z)/A(0)-1|\\le1/4$ for $|z|\\le h$ and\n$\\delta<Mh^2/4$, then every connected subset $K$ of $U$ containing its\ntwo roots $a,b$ satisfies\n\\[\n \\mathcal H^1(K)\\ge |a-c|+|b-c|-\\frac83\\sqrt{\\delta/M}.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nCut the value disc along $J=\\{tv:1\\le t<1/|v|\\}$. The slit avoids\n$0$ and removes the only critical value of $p|_U$. Over its simply\nconnected complement, the double cover splits into two inverse sheets\n$U_1,U_2$, containing $a,b$ respectively.\n\nThe sheet boundaries inside $U$ lie over the slit. To confine them,\nput $r_0=(4/3)\\sqrt{\\delta/M}<2h/3$. The relative-error hypothesis\nmakes $A$ nonvanishing on this disc, and on $|z|=r_0$ gives\n\\[\n |p(c+z)-v|\\ge\\tfrac34Mr_0^2=\\tfrac43\\delta>\\delta.\n\\]\nEvery value on $J$ is less than $\\delta$ from $v$. Rouch\\'e's theorem\ncompares its preimages with the double zero of $p-v$ and gives exactly\ntwo in $B(c,r_0)$, counted with multiplicity. As the value moves from\n$v$ along $J$, these preimages continue from $c$ without crossing the\ncircle. Their values stay in the unit disc, so they remain in $U$.\nThe degree-two hypothesis now identifies these local preimages with the\nentire fibre in $U$. Thus\n\\[\n p^{-1}(J)\\cap U\\subset B(c,r_0).\n\\]\n\nIt remains to count how much of $K$ lies on each sheet. For\n$r_0<t<|a-c|$, suppose $K\\cap U_1$ missed the circle $|z-c|=t$.\nThen $K\\cap U_1\\cap\\{|z-c|>t\\}$ would be both open and closed in $K$:\nthe boundary of $U_1$ inside $U$ lies in $B(c,r_0)$, and the remaining\nboundary is on the missing circle. This set contains $a$ and excludes\n$b$, contradicting connectedness. Hence the radial image of $K\\cap U_1$\ncontains $(r_0,|a-c|)$ whenever this interval is nonempty. The map\n$z\\mapsto|z-c|$ is $1$-Lipschitz, and therefore\n\\[\n \\mathcal H^1(K\\cap U_1)\\ge(|a-c|-r_0)_+.\n\\]\nHere $x_+=\\max(x,0)$. The same argument on $U_2$ gives the contribution\nfrom $b$. The radial intervals may overlap on the real line; it is the\ndisjointness of the \\emph{sheets in the plane} that permits addition. Hausdorff outer\nmeasure adds across disjoint open sets, without a measurability\nassumption on $K$, so\n\\begin{align*}\n \\mathcal H^1(K)\n &\\ge (|a-c|-r_0)_+ + (|b-c|-r_0)_+\\\\\n &\\ge |a-c|+|b-c|-2r_0.\n\\end{align*}\nThis is the required bound. No parametrisation of $K$ has been used.\n\\end{proof}\n\n\\begin{figure}[!htbp]\n\\centering\n\\papersectiontarget{bottleneck-geometry}\n\\begin{tikzpicture}[x=1cm,y=1cm,font=\\small,\n    >=Stealth, line cap=round, line join=round]\n  % Schematic only. No coordinates are data for the degree-seven polynomial.\n  \\node at (1.55,1.58) {Value plane};\n  \\draw[thin] (1.55,0) circle (1.08);\n  \\fill (1.55,0) circle (1.5pt) node[below left] {$0$};\n  \\fill (2.14,0) circle (1.5pt) node[below] {$v$};\n  \\draw[line width=1.2pt] (2.14,0) -- (2.63,0);\n  \\draw[fill=white] (2.63,0) circle (1.6pt);\n  \\node[above] at (2.39,.08) {$J$};\n  \\node at (1.55,-1.43) {$|w|<1$};\n  \\draw[->] (4.72,.18) -- (3.35,.18) node[midway,above] {$p$};\n  \\node[align=center] at (4.04,-.36) {degree two};\n\n  \\node at (8.7,1.58) {Component $U$};\n  % The two shaded regions represent inverse sheets, not a parametrised set K.\n  \\path[fill=black!6]\n    (8.7,.25) .. controls (7.95,.27) and (7.7,1.06) .. (6.8,1.06)\n    .. controls (5.15,1.06) and (5.15,-1.06) .. (6.8,-1.06)\n    .. controls (7.7,-1.06) and (7.95,-.27) .. (8.7,-.25) -- cycle;\n  \\draw[thin]\n    (8.7,.25) .. controls (7.95,.27) and (7.7,1.06) .. (6.8,1.06)\n    .. controls (5.15,1.06) and (5.15,-1.06) .. (6.8,-1.06)\n    .. controls (7.7,-1.06) and (7.95,-.27) .. (8.7,-.25)\n    .. controls (9.45,-.27) and (9.7,-1.06) .. (10.6,-1.06)\n    .. controls (12.25,-1.06) and (12.25,1.06) .. (10.6,1.06)\n    .. controls (9.7,1.06) and (9.45,.27) .. (8.7,.25);\n  \\draw[densely dashed] (8.7,0) circle (.63);\n  \\draw[line width=1.2pt] (8.7,-.25) -- (8.7,.25);\n  \\fill (8.7,0) circle (1.6pt) node[left] {$c$};\n  \\draw[thin] (8.7,0) -- (9.20,.38) node[midway,above left] {$r_0$};\n  \\node at (8.7,.99) {$B(c,r_0)$};\n  \\fill (6.12,-.06) circle (1.6pt) node[below] {$a$};\n  \\fill (11.25,.06) circle (1.6pt) node[below] {$b$};\n  \\node at (6.95,.38) {$U_1$};\n  \\node at (10.42,.38) {$U_2$};\n  \\node at (8.7,-1.43) {$p^{-1}(J)\\cap U\\subset B(c,r_0)$};\n\n  % These are one-dimensional radial images, not curves in U.\n  \\node[align=center] at (1.85,-2.18) {Radial images\\\\under $z\\mapsto|z-c|$};\n  \\node[above] at (6.9,-2.07) {$K\\cap U_1$};\n  \\draw[densely dotted,line width=.8pt] (5.77,-2.27) -- (8.02,-2.27);\n  \\draw[fill=white] (5.77,-2.27) circle (1.6pt);\n  \\fill (8.02,-2.27) circle (1.6pt);\n  \\node[below] at (5.77,-2.30) {$r_0$};\n  \\node[below] at (8.02,-2.30) {$|a-c|$};\n  \\node[above] at (10.52,-2.07) {$K\\cap U_2$};\n  \\draw[densely dotted,line width=.8pt] (9.40,-2.27) -- (11.65,-2.27);\n  \\draw[fill=white] (9.40,-2.27) circle (1.6pt);\n  \\fill (11.65,-2.27) circle (1.6pt);\n  \\node[below] at (9.40,-2.30) {$r_0$};\n  \\node[below] at (11.65,-2.30) {$|b-c|$};\n\\end{tikzpicture}\n\\caption{The slit and its preimage in Lemma~\\ref{lem:two-sheet-bottleneck}.\nThe component and radii are schematic, with both zeros outside the small\ndisc. Connectedness forces the radial image on each sheet to contain the\ndotted interval. These intervals are displayed separately, although they\nmay overlap as subsets of the real line. The two measure contributions\nadd because $U_1$ and $U_2$ are disjoint in the plane, giving\n$|a-c|+|b-c|-2r_0$. The dotted intervals are not paths in $U$.}\n\\label{fig:two-sheet-bottleneck}\n\\end{figure}\n\n\\subsection{The polynomial}\nStarting from $z^7-1$, we put the roots on $|z|=\\rho<1$ and\nsplit the multiple critical point at scale $\\varepsilon$. The linear,\nquadratic and cubic perturbations below all have order $\\varepsilon^7$\nat $z=\\varepsilon w$, so they alter the critical geometry without\ncomparable root displacement. We must identify the only component\ncontaining a pair and make its distance surplus exceed the loss in\nLemma~\\ref{lem:two-sheet-bottleneck}. The fixed coefficients are those\nof \\texttt{ani}'s construction.\nPut $s=10^{-6}$, $\\varepsilon=s^2=10^{-12}$ and $\\rho=1-s^{16}$, and let\n\\begin{gather*}\n A=-\\frac{329507}{1600},\\qquad B=\\frac{551827}{800},\\qquad\n C=-\\frac{23013813}{32000},\\\\\n a=A-is,\\qquad b=iB+\\frac95s,\\qquad c_0=-C-\\frac{162}{25}is.\n\\end{gather*}\nDefine\n\\begin{align}\n F(z)&=z^7-1+\\varepsilon^4(az^3-\\bar a z^4)\n      +\\varepsilon^5(bz^2-\\bar b z^5)\n      +\\varepsilon^6(c_0z-\\bar c_0z^6),\\label{eq:ani-F}\\\\\n f(z)&=\\rho^7F(z/\\rho).\\label{eq:ani-f}\n\\end{align}\n\\paragraph{Putting the roots on a circle.}\nBy $F(z)=-z^7\\overline{F(1/\\bar z)}$, the Cayley substitution\n$z=(1+ix)/(1-ix)$ gives a real polynomial after clearing the denominator\nand removing a constant phase. Symmetry alone does not locate the zeros:\nthe decisive step is seven sign changes on disjoint rational intervals.\nThe map parametrises the unit circle except $-1$, so these give seven\ndistinct circle zeros $\\zeta_j$, exhausting the degree. Thus $-1$ is\nnot a zero and $b_j=\\rho\\zeta_j$ satisfies $|b_j|=\\rho<1$.\nThe distance estimate below absorbs the contraction $1-\\rho=\\varepsilon^8$.\n\n\\paragraph{Resolving the critical points.}\nAt the smaller scale $z=\\rho\\varepsilon w$, write\n\\[\n F(\\varepsilon w)=-1+\\varepsilon^7Q(w),\\qquad\n Q(w)=c_0w+bw^2+aw^3-\\varepsilon\\bar a w^4\n                  -\\varepsilon^3\\bar b w^5-\\varepsilon^5\\bar c_0w^6+w^7.\n\\]\nRouch\\'e estimates on six disjoint rational discs locate the six zeros\nof $Q'$. Exactly one, $w_s$, corresponds to a critical value of modulus\nless than one. It satisfies\n\\[\n |w_s-(0.0000000039+0.8232474662i)|<10^{-6}.\n\\]\nLet $c_s=\\rho\\varepsilon w_s$, $v=f(c_s)$, $\\delta=1-|v|$, and write\n$f(c_s+z)-v=z^2A_s(z)$. Since $|\\zeta_j|=1$,\n\\[\n |b_j-c_s|=\\rho|\\zeta_j-\\varepsilon w_s|\n \\ge\\rho\\bigl(1-\\varepsilon\\operatorname{Re}(w_s\\bar\\zeta_j)\\bigr).\n\\]\nThe negative projection sum for $\\zeta_3,\\zeta_6$ gives a positive\ndistance surplus of order $\\varepsilon$; the contraction has only\norder $\\varepsilon^8$. The exact bounds at the fixed parameter are:\n\\begin{align}\n M:=|A_s(0)|&>180\\rho^5\\varepsilon^5,&\n \\delta&<\\frac{36}{5\\cdot10^6}\\rho^7\\varepsilon^7,\n       \\label{eq:ani-estimates}\\\\\n \\left|A_s(z)/A_s(0)-1\\right|&<\\frac14\n       \\quad\\left(|z|\\le\\frac{\\rho\\varepsilon}{10}\\right),&\n |b_3-c_s|+|b_6-c_s|&>\n       \\rho\\left(2+\\frac{143}{1000}\\varepsilon\\right).\n       \\label{eq:ani-distances}\n\\end{align}\nThe subscripts $3,6$ refer to Cayley roots in the intervals\n$(43812,43813)/10^4$ and $(-4816,-4815)/10^4$.\nThe Taylor bound gives the lemma's quadratic control, while the bounds\non $M$ and $\\delta$ give $\\sqrt{\\delta/M}<\\rho\\varepsilon/5000$.\nThus the slit loss is smaller than the displayed distance surplus.\nThis comparison matters only if the selected roots share a component;\nthe following paths establish that membership, not a short connection.\n\n\\paragraph{Identifying the two roots in the component.}\nIn the $w$-plane, polygonal arcs connect a small disc about $w_s$ to\npoints near $8\\zeta_3$ and $8\\zeta_6$. Short joining segments then\nreach those exact root directions, and radial segments continue to the\nroots. For the four inner polygonal segments we use\n\\[\n 1-|f(\\rho\\varepsilon w)|^2\\ge\n 2\\rho^{14}\\varepsilon^7\n \\left(7\\varepsilon+\\operatorname{Re}Q(w)\n                       -\\tfrac12\\varepsilon^7|Q(w)|^2\\right).\n\\]\nThis follows by expanding the square and using\n$1-\\rho^{14}\\ge14\\rho^{14}\\varepsilon^8$.\nOn each segment, the expression in parentheses is a polynomial of degree\nat most fourteen in the segment parameter. All fifteen coefficients in\nits degree-fourteen Bernstein basis are positive. The basis polynomials\nare nonnegative and sum to one, so the expression is positive along the\nwhole segment, which therefore stays inside $\\{|f|<1\\}$ after scaling.\nThe companion bounds $Q$ directly on the short joining segments.\nFor the radial portions we use the root identity\n\\[\n F(t\\zeta)=t^7-1+\n       \\sum_{k=1}^6\\varepsilon^{7-k}q_k\\zeta^k(t^k-t^7),\n \\qquad Q(w)=\\sum_{k=1}^7q_kw^k,\n\\]\nwhich gives $|F(t\\zeta)|<1$ for $8\\varepsilon\\le t\\le1$.\nAll rational intervals, vertices and coefficient tests are specified in\n\\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{counterexample-certificate}\n{the companion's finite verification}. At the regular level $|f|=1$,\nthe component-wise Riemann--Hurwitz calculation\n\\cite[proof of Proposition~2.1]{eks2010} gives $k-1$ critical points\nin a component with $k$ zeros, counted with multiplicity. Since $c_s$\nis the only critical point in $\\Omega_f$, its component contains\nexactly the two roots just reached; every other component contains one.\n\n\n\\begin{proof}[Proof of Theorem~\\ref{res:ani-degree-seven-counterexample}]\nApply Lemma~\\ref{lem:two-sheet-bottleneck} to the component containing\n$b_3,b_6$, with $h=\\rho\\varepsilon/10>0$. The finite estimates give\n$\\sqrt{\\delta/M}<\\rho\\varepsilon/5000$ and\n$\\delta<Mh^2/4$. Every connected set in that component containing the\ntwo zeros therefore has measure greater than\n\\[\n \\rho\\left(2+\\left(\\frac{143}{1000}-\\frac8{15000}\\right)\n                        \\varepsilon\\right)>2.\n\\]\nAll other components contain one zero. A connected set containing two\nzeros must lie in the component just considered, which proves the theorem.\n\\end{proof}\n\nThe choice $s=10^{-6}$ is fixed throughout. Extending these estimates to\nthe small-parameter family reported by \\texttt{ani} remains unproved\nhere. The argument estimates the measure of the connected set itself;\na bound for the total variation of a chosen parametrisation would not\nsuffice.\n\n\\section{Trinomials and radial segments}\\label{sec:trinomial}\n\nIf $f(z)=z^n+b$ has a zero $\\zeta$ in the open unit disc, then\n$|b|=|\\zeta|^n<1$. The identity $f(t\\zeta)=b(1-t^n)$, $0\\le t\\le1$,\nkeeps the whole segment $[0,\\zeta]$ in the lemniscate.\nThe same cancellation survives a middle monomial. Its coefficient may\nbe large, so we use $f(\\zeta)=0$ to eliminate that term \\emph{before}\ntaking absolute values. The remaining quantities are $b$ and\n$-\\zeta^n$, both of modulus less than one.\n\n\\begin{theorem}[all-degree monic trinomials]\n\\label{res:trinomial-all-degree}\nLet $1\\le m<n$ and\n\\[\n f(z)=z^n+az^m+b,\n\\]\nwith every zero in the open unit disc.  For any zero $\\zeta$, the entire\nsegment $[0,\\zeta]$ lies in $\\{|f|<1\\}$.  Distinct zeros $\\zeta_1,\\zeta_2$\nare therefore joined by the broken line $\\zeta_1\\to0\\to\\zeta_2$ of length\n$\\|\\zeta_1\\|+\\|\\zeta_2\\|<2$ inside the open unit lemniscate.\n\\end{theorem}\n\n\\begin{proof}\nFix a zero $\\zeta$. Eliminating $a\\zeta^m$ with the root equation gives\n\\[\n f(t\\zeta)=(1-t^m)b+(t^m-t^n)(-\\zeta^n).\n\\]\nVieta's formula gives $|b|<1$. For $0\\le t<1$, the two weights\nare nonnegative, the first is positive, and their sum is $1-t^n$. Hence\n\\[\n |f(t\\zeta)|\\le |b|(1-t^m)+|\\zeta|^n(t^m-t^n)<1-t^n\\le1.\n\\]\nAt $t=1$ the value is zero. We join any two distinct zeros by concatenating\ntheir segments through the origin.\n\\end{proof}\n\nAbel summation identifies exactly what this argument needs. For\n$f(z)=\\sum_{k=0}^n a_kz^k$ and a zero $\\zeta$, put\n$S_j=\\sum_{k=0}^j a_k\\zeta^k$. Since $S_n=0$, summation by parts gives\n\\[\n f(t\\zeta)=\\sum_{j=0}^{n-1}(t^j-t^{j+1})S_j.\n\\]\nFor $0\\le t<1$, the weights are nonnegative and sum to $1-t^n$.\nThus $f(t\\zeta)/(1-t^n)$ is a convex combination of the $S_j$.\nFor a trinomial, $S_j=b$ before the middle term and $S_j=-\\zeta^n$\nafter it. Thus every partial sum lies in the unit disc. A fourth term\nintroduces another partial sum for which root locations alone need not\ngive that bound, as the next example shows. The conclusion concerns\nsegments ending at zeros, not star-shapedness of the whole lemniscate.\n\n\\begin{proposition}[failure of a prescribed radial segment]\n\\label{res:sextic-spoke}\nThere exist $r\\in(0,1)$ for which every zero of\n\\[\n f_r(z)=z^6+\\tfrac15 r^2 z^4-\\tfrac15 r^4 z^2-r^6\n\\]\nlies in the open unit disc, yet the radial segment from the origin to the zero\n$r$ leaves $\\{|f_r|<1\\}$.\n\\end{proposition}\n\n\\begin{proof}\nWe substitute $z=rw$ and factor the polynomial as\n$r^6(w^2-1)(w^4+\\tfrac65w^2+1)$. The quadratic in $w^2$ has\nconjugate roots of product one, so all six original zeros have modulus\n$r$. At the midpoint of that segment, $f_r(r/2)=-(327/320)r^6$;\nchoosing $320/327<r^6<1$ proves the assertion.\n\\end{proof}\n\nThis failure concerns the prescribed segment only. In\nSection~\\ref{sec:solved-families} we use the partial-sum identity again,\nfirst for power substitutions in a cubic and then for a quintic whose\ntwo missing coefficients select a suitable pair of segments.\n\n\\section{A small least critical value}\\label{sec:constant-factor}\n\\papersectiontarget{low-critical-short}\n\nA small critical value leaves a long interval of levels before the unit\nlemniscate is reached. For a squarefree polynomial, put\n$\\mu=\\min_{f'(c)=0}|f(c)|>0$ and follow a component from its first\nmerger at level $\\mu$ to level $1$. If no two roots admit a connection\nshorter than $2$, conformal length estimates force them far apart in\nthe component's hyperbolic metric. Packing then bounds the root count\nfrom below; inverse-ray averaging turns that count into a lower bound\nfor area growth. When $\\log(1/\\mu)$ is large enough, this growth\ncontradicts P\\'olya's area bound.\n\nThe companion proves the analytic reduction and gives the rational\ncomparison. The resulting criterion needs no restriction on root\nlocations, but its analytic proof has not been formalised.\n\n\\begin{remark}[unformalised small-critical-value assertion]\n\\label{res:low-critical-thirteen-twentyfifths}\nLet $f$ be squarefree and monic of degree $n\\ge2$, and let $\\mu$ be its least\ncritical-value modulus.  If $\\mu\\le13/25$, then two distinct roots are joined\ninside $\\{|f|<1\\}$ by a rectifiable curve of length strictly less than $2$.\n\\end{remark}\n\\evidenceremark{This assertion has an ordinary analytic argument and an\nexact rational comparison, but no Lean proof. The argument is recorded\nat \\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{low-critical-proof}\n{the small-critical-value proof}.}\n\n\\begin{remark}[scaling of the preceding assertion]\n\\label{res:scaled-low-critical}\nEvery squarefree monic polynomial of degree $n\\ge2$ has two distinct roots\njoined in $\\{|f|<(25/13)\\mu\\}$ by a curve of length less than\n\\[\n 2\\bigl((25/13)\\mu\\bigr)^{1/n}.\n\\]\nIn every degree the length can be chosen less than $(5/2)\\mu^{1/n}$.\n\\end{remark}\n\\evidenceremark{Lean checks this scaling implication under the named\ninput \\texttt{LowCriticalThirteenTwentyFifths}, the unformalised\nassertion in Remark~\\ref{res:low-critical-thirteen-twentyfifths}.}\n\n\\begin{proof}[Scaling implication]\nFor $n\\ge3$, apply the preceding assertion to\n$g(z)=s^{-n}f(sz)$, where $s=((25/13)\\mu)^{1/n}$. Its least\ncritical-value modulus is $13/25$, and rescaling its connection gives\nthe first bound. The estimate $2(25/13)^{1/n}<5/2$ holds for $n\\ge3$.\nFor $n=2$, write $f(z)=(z-h)^2-d^2$ and use the root segment, of length\n$2|d|=2\\sqrt\\mu$, contained in $K_\\mu(f)$.\n\\end{proof}\n\n\\subsection{The analytic argument}\nAssume that no pair has a connection of length less than $2$ in\n$\\Omega_f$. At a regular level $t\\in(\\mu,1)$, let $C_t$ contain a\nchosen first merger and let $k\\ge2$ be its root count. Put\n$a=\\operatorname{Area}(C_t)/\\pi$ and $x=\\log(t/\\mu)$; P\\'olya's\ninequality gives $a\\le1$ \\cite[Theorems~1 and~6]{crane}.\nUniformise $C_t$ by the disc, with $d(0,s)=2\\operatorname{artanh}s$\nfor $0\\le s<1$. For roots at hyperbolic distance $d$, choose\n$\\varphi:\\mathbb D\\to C_t$ sending $\\pm s$ to them. Then\n$d=4\\operatorname{artanh}s$ and\n$\\int_{\\mathbb D}|\\varphi'|^2\\,dA=\\pi a$. Cauchy--Schwarz with the\nBergman kernel gives\n\\[\n \\operatorname{length}(\\varphi([-s,s]))^2\n \\le a\\int_{-s}^{s}\\int_{-s}^{s}\\frac{du\\,dv}{(1-uv)^2}\n =2a\\log\\cosh(d/2).\n\\]\nThus pairwise root distances are at least $D$, where\n$\\cosh(D/2)=e^{2/a}$.\n\nTo control the distances from the centre as well, choose a point $h$\non a connected set through the first pair in $K_\\mu(f)$ whose intrinsic\ndistance from every root is at least $1$. Here intrinsic distance means\nthe infimum of Euclidean curve lengths in $C_t$. Such an $h$ exists:\nthe intrinsic open unit balls about distinct roots are disjoint, since\nan intersection would give a path shorter than $2$, and disjoint open\nsets cannot cover a connected set containing two roots. Choose the\nuniformisation to send $0$ to $h$. If $d_j$ are the resulting\nhyperbolic root distances from $0$, then\n\\[\n \\lambda(d_j)\\le\\tfrac12\\delta(a),\\qquad\n \\sum_j\\lambda(d_j)\\ge x,\\qquad\n \\lambda(d)=-\\log\\tanh(d/2),\\quad\n \\delta(a)=-\\log(1-e^{-1/a}).\n\\]\nThe one-root Bergman estimate gives the first bound because every root\nhas intrinsic distance at least $1$ from $h$. For the second, the component-wise Blaschke representation\n\\cite[(2.2)]{eks2010} expresses $f/t$ in disc coordinates.\nSince $h$ is not a root and lies in $K_\\mu(f)$, evaluation at $0$ gives\n$\\sum_j\\lambda(d_j)=\\log(t/|f(h)|)\\ge x$.\nDividing the required sum $x$ by the maximum contribution\n$\\delta(a)/2$ gives $k\\ge2x/\\delta(a)$. This uses only distances\nfrom the centre. To exploit the pairwise separation too, we intersect\nthe disjoint hyperbolic balls about the roots with a common circle.\nThe passage through a first merger, including simultaneous mergers, is\npart of the analytic input still to be formalised.\n\n\\hypertarget{circle-packing-inputs}{}\nOn a fixed hyperbolic circle of radius $r>0$ centred at $0$, the balls\nof radius $D/2$ cut out disjoint arcs. Their angular half-widths satisfy\n\\[\n w(d,r)=\\arccos\\!\\left(\\operatorname{clamp}_{[-1,1]}\n \\frac{\\cosh d\\cosh r-\\cosh(D/2)}{\\sinh d\\sinh r}\\right),\n \\qquad \\sum_jw(d_j,r)\\le\\pi.\n\\]\nHere $\\operatorname{clamp}_{[-1,1]}$ truncates to $[-1,1]$.\nThis circle-slice inequality is formalised separately. To turn it into\na root count, majorise the contribution of one root by weighted slice\nwidths: choose $\\sigma_i\\ge0$, $r_i>0$ and $U>0$ such that\n$\\lambda(d)\\le U+\\sum_i\\sigma_iw(d,r_i)$ whenever $d>0$ and\n$\\lambda(d)\\le\\delta(a)/2$. Summing at the roots and using the angular\nbound at each $r_i$ gives\n\\begin{equation}\\label{eq:short-packing}\n k\\ge\\frac{x-\\pi\\sum_i\\sigma_i}{U}.\n\\end{equation}\nThe circle must be fixed before taking these intersections: angular\nprojections of disjoint balls at different distances can overlap.\n\nWe now convert the root count into area growth. Lift one common value\nradius from all $k$ roots to $\\partial C_t$ and join successive endpoints\nby boundary arcs. All pieces stay in $\\Omega_f$ because $t<1$.\nSplitting the lift integral at level $\\mu$ bounds the mean total lift\nlength by $\\sqrt{ka(x+2)/2}$. Each of the $k$ resulting root-to-root\nconnections has length at least $2$; their sum counts each lift twice\nand the boundary once. Hence\n$2k\\le\\sqrt{2ka(x+2)}+\\mathcal H^1(\\partial C_t)$.\nCombining this with\n$\\mathcal H^1(\\partial C_t)^2\\le2\\pi k t\\,(d/dt)\\operatorname{Area}(C_t)$\nyields the ordinary differential inequality\n\\begin{equation}\\label{eq:short-area-growth}\n a'(x)\\ge\\frac1{2\\pi^2}\n       \\bigl[2\\sqrt k-\\sqrt{2a(x)(x+2)}\\bigr]_+^2.\n\\end{equation}\nHere the derivative is with respect to $x$, between merger levels.\nThe companion proves the lift estimate and accounts for the nonnegative\narea jumps at mergers; those jumps do not weaken the lower bound.\n\n\\subsection{The recorded rational comparison}\n\\hypertarget{certificate-stopping-time}{}\nAt each level use the largest integer lower bound for the current root\ncount $k$: \\eqref{eq:short-packing}, $k\\ge2x/\\delta(a)$, or the\ncompanion's two further packing estimates. The starting bound\n$a(3/10^5)>10^{-6}$ follows by integrating\n\\eqref{eq:short-area-growth} with $k\\ge2$ and $a\\le1$; it is not an\nextra assumption on the polynomial. The recorded rational calculation\nthen reaches $a>1$ by\n\\[\n X=\\frac{635762889599}{10^{12}}.\n\\]\nIt used $18$ area levels, $126$ certified weighted inequalities and a\nmaximum step $1/400$. On each interval $[u,v]$ of possible root distances, the calculation\nbounds $\\lambda(d)-\\sum_i\\sigma_iw(d,r_i)$ by\n$\\lambda(u)-\\sum_i\\sigma_i\\min(w(u,r_i),w(v,r_i))$; on the tail,\n$\\lambda$ itself suffices. Thus numerical optimisation only proposes\nthe weights. Rational inequalities certify a valid $U$ on the whole\nadmissible half-line, as required by \\eqref{eq:short-packing}.\nThe endpoint minimum is valid because each slice width, as a function\nof $d$, increases and then decreases, or is monotone; it has no interior\ndip. This is an interval bound, not sampling between grid points.\n\nSince $t=\\mu e^x$, the contradiction $a>1$ occurs before level $1$\nprovided that $\\mu e^X<1$. The recorded comparison gives\n$(13/25)e^X<0.982000386<1$. It also gives\n$(529/1000)e^X<0.998996547$, so the ordinary argument supports that\nsharper threshold as well. Neither threshold is claimed\noptimal, and this arithmetic does not formalise the analytic reduction.\n\n\\noindent\\phantomsection\\label{res:constant-factor-path}\nA related ordinary averaging argument, also unformalised in its analytic\nconstruction, gives the following bounds. For a monic polynomial of\ndegree $n\\ge2$, two zero occurrences can be joined in $K_{2\\mu}(f)$\nwith length at most $(71/10)\\mu^{1/n}$; they are distinct in the\nsquarefree case, and a repeated root gives a constant path. When\n$\\mu\\le1/2$, the argument gives a path of length at most $5.7$ in\n$\\Omega_f$. The proof, together with refinements that retain the number\nof zeros in the component or its logarithmic capacity, is given in the\n\\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{constant-factor-proof}\n{inverse-ray averaging argument}.\n\n\\section{An isolated simple critical value}\\label{sec:separation}\n\nSmallness of the least critical value is not the only useful condition.\nAn isolated simple critical value lets us continue the two branches\nmeeting there without encountering further branching. After dividing\nvalues by $v=f(c)\\ne0$, the critical value is $1$ and the roots lie\nover $0$. We seek a disc containing both points but no other critical\nvalue. Its centre $w_0\\in[0,1]$ and radius $S$ must therefore satisfy\n$S>\\max(w_0,1-w_0)$.\n\nThe ordinary construction combines a square-root coordinate, a Bergman\nlength estimate and a component-area bound. These are the contents of\nthe unformalised input \\texttt{DiscSepBergmanArea}; the formal\nconsequences below remain conditional on it.\n\n\\begin{remark}[unformalised isolated-value assertion]\n\\label{res:critical-value-separation}\nLet $f$ be monic of degree $n\\ge3$, let $c$ be a simple critical point, and\nput $v=f(c)\\ne0$.  Fix $w_0\\in[0,1]$ and\n$S>\\max(w_0,1-w_0)$.  Suppose every other critical point $d$ satisfies\n\\[\n  \\left|\\frac{f(d)}v-w_0\\right|\\ge S .\n\\]\nPut $p=w_0(1-w_0)$.  Then two distinct roots are joined inside\n$\\{|f|\\le|v|\\}$ by a curve $\\Gamma$ satisfying\n\\begin{equation}\\label{eq:disk-family-length}\n \\operatorname{length}(\\Gamma)^2\n \\le 2|v|^{2/n}\\Bigl(\\frac{S}{n-1}\\Bigr)^{2/n}\n \\log\\!\\frac{S^2+S+p}{S^2-S+p}.\n\\end{equation}\n\\end{remark}\n\\evidenceremark{Lean assumes \\texttt{DiscSepBergmanArea}: existence of the\nsquare-root connector, its Bergman length bound and the area bound for\nthe two-sheeted component. The complete ordinary argument is in the\n\\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{critical-value-separation-proof}\n{companion's separation proof}.}\n\nFor example, $f(z)=z^3-3a^2z$ with $0<a<1/\\sqrt3$ has zeros\n$0,\\pm\\sqrt3a$. At $c=a$ its critical value is $v=-2a^3$ and the\nother normalised critical value is $-1$. Thus the disc with centre\n$w_0=1$ and radius $4/3$ satisfies the separation hypothesis.\n\n\\begin{remark}[conditional numerical thresholds]\n\\label{res:critical-value-thresholds}\nLet $f$ be monic of degree $n\\ge3$ with all roots in the open unit disc.  If\n$c$ is a simple critical point with $0<|f(c)|<1$ and, for some\n$w_0\\in[0,1]$, every other critical point $d$ satisfies\n\\[\n \\left|\\frac{f(d)}{f(c)}-w_0\\right|\\ge\\frac43,\n\\]\nthen two roots are joined inside $\\{|f|<1\\}$ by a curve of length strictly\nbelow $2$.  In degree three the branch-centred choice $w_0=1$ already works\nwith $4/3$ replaced by $6/5$.\n\\end{remark}\n\\evidenceremark{The geometric conclusion assumes the construction in\nRemark~\\ref{res:critical-value-separation}. Lean checks the numerical\ninequalities for $4/3$ and $6/5$ without that input.}\n\n\\subsection{Square roots and component area}\nThe ordinary argument first removes the scale of $v$, then constructs\na curve, and finally bounds its length. Choose $\\alpha^n=v$ and set\n$P(z)=v^{-1}f(c+\\alpha z)$. This polynomial is monic, with its chosen\ncritical point at $0$ and critical value $1$; Euclidean lengths are\nmultiplied by $|\\alpha|=|v|^{1/n}$ on returning to the original plane.\nThe component $U$ of $P^{-1}(D(w_0,S))$ containing $0$ has degree two.\nTo apply Riemann--Hurwitz, exhaust it by components above regular inner\ndiscs: the outer boundary itself may contain other critical values.\nSince $U$ is simply connected and $1-P$ has only a double zero at $0$\nthere, it has a single-valued analytic square root $\\xi$.\nThis square root removes the simple branching. It is a proper local\nbiholomorphism onto $V=\\{\\xi:|\\xi^2-(1-w_0)|<S\\}$, a star-shaped\nand hence simply connected domain. Thus it is biholomorphic. Its inverse\nmaps $-1$ and $1$ to the two roots and $[-1,1]$ to their connection. Along this interval $P=1-\\xi^2\\in[0,1]$,\nwhich gives the required containment. Let $L$ be the Euclidean length\nof this normalised curve.\n\nPut $a=1-w_0$ and $p=w_0(1-w_0)$. Squaring maps $V$ onto $D(a,S)$.\nA M\\\"obius map from this disc to the unit disc fixing $0$, followed by\ntaking a square root, gives the conformal map\n\\[\n \\zeta=\\xi\\sqrt{\\frac{S}{S^2+a\\xi^2-a^2}}.\n\\]\nChoose the square-root factor positive at $0$. The endpoints\n$\\xi=\\pm1$ then map to $\\zeta=\\pm q$, where $q^2=S/(S^2+p)$.\nThe hypothesis on $S$ gives\n$S^2-S+p=(S-w_0)(S-(1-w_0))>0$, so $0<q<1$.\nThe Bergman segment estimate used in Section~\\ref{sec:constant-factor}\nnow applies to $[-q,q]$ and gives\n\\[\n L^2\\le\\frac2\\pi\\log\\frac{S^2+S+p}{S^2-S+p}\\,\n                    \\operatorname{Area}(U).\n\\]\nThe factor $n-1$ comes from the roots outside $U$, not from P\\'olya's\ninequality alone. For monic $F$ of degree $n$ and regular level $t>0$,\nlet $W$ be a component of $\\{|F|<t\\}$ containing $k<n$ zeros counted\nwith multiplicity. Reflecting the $n-k$ exterior-root factors into the\ndisc gives a Blaschke product $B$ of degree $n-k$ with\n$|B(0)|=\\operatorname{cap}(\\overline W)^n/t$ and $|B'|<n$ on the boundary.\nThe boundary-fibre identity gives\n\\[\n \\frac{n-k}{n}<\n \\sum_{\\substack{B(\\zeta)=w\\\\|\\zeta|=1}}\\frac1{|B'(\\zeta)|}\n =\\frac{1-|B(0)|^2}{|w-B(0)|^2},\\qquad |w|=1.\n\\]\nChoosing $w$ opposite $B(0)$ gives\n$(n-k)/n<(1-|B(0)|)/(1+|B(0)|)$, hence\n$\\operatorname{cap}(\\overline W)^n/t<k/(2n-k)$.\n\nApply this to $F=P-w_0$ on the regular inner components exhausting $U$.\nTheir degree is two, so $k=2$ and $k/(2n-k)=1/(n-1)$.\nHere $k$ counts preimages of $w_0$ under $P$. These need not be the\nroots joined by the curve; the component degree gives both counts.\nP\\'olya's area--capacity inequality \\cite[printed pp.~280--282]{polya1928},\n\\cite[Theorem~6]{crane}, followed by exhaustion, now gives\n$\\operatorname{Area}(U)\\le\\pi(S/(n-1))^{2/n}$.\nThe original curve has length $\\operatorname{length}(\\Gamma)=|v|^{1/n}L$,\nwhich gives \\eqref{eq:disk-family-length}.\nDubinin's relative area inequality \\cite[Theorem~1]{dubinin} has a\ndifferent full-covering hypothesis; the absolute component estimate is\nneeded here.\n\n\\noindent\\phantomsection\\label{res:separation-parent}\nFor the conditional threshold, decrease $S$ to $4/3$.\nSince $p\\ge0$ and $n\\ge3$, the logarithm is at most $\\log7<2$ and\n$(S/(n-1))^{2/n}\\le1$, giving\n$\\operatorname{length}(\\Gamma)<2|v|^{1/n}<2$.\nFor $n=3,w_0=1,S=6/5$, use $(3/5)^{2/3}\\log11<2$.\nThe chosen critical value must be nonzero and have modulus less than\none. Root locations by themselves do not isolate it from the others.\n\\phantomsection\\label{bdry:critical-value-separation}\n\n\\section{Further estimates}\\label{sec:other-results}\n\\hypertarget{supplementary-results}{}\n\n\\subsection{Root distances do not ensure containment}\\label{sec:critical-proximity}\\label{sec:exact-obstructions}\n\\noindent\\phantomsection\\label{res:critical-proximity}\nOrder the root distances at a non-root critical point $c$ as\n$0<d_1\\le\\cdots\\le d_n$. The balance $\\sum_j(c-z_j)^{-1}=0$ and\nthis ordering give\n\\[\n \\frac1{d_1}\\le\\frac{n-1}{d_2},\\qquad\n d_1d_2^{n-1}\\le|f(c)|.\n\\]\nThe ratio bound $1\\le d_2/d_1\\le n-1$ permits the scalar comparison\n\\[\n d_1+d_2\\le2(d_1d_2^{n-1})^{1/n}\\le2|f(c)|^{1/n}.\n\\]\nNeither inequality supplies containment.\n\n\\noindent\\phantomsection\\label{res:two-nearest-roots}\nWhen all zeros lie in the open unit disc, a different estimate holds:\nat every non-root critical point the sum of the distances to the two\nnearest zeros is strictly less than $2$. The companion gives both\n\\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{root-distance-proofs}\n{root-distance estimates}.\nThe nearest pair is chosen among all roots of $f$. It need not be the\npair in the two-root component of the degree-seven example, where\ncontainment restricts which pair can be joined.\n\n\\noindent\\phantomsection\\label{res:straight-no-go}\nEven a short segment from a critical point can leave the lemniscate.\nFor the quintic\n\\[\n (z-a)(z^2+p^2)\\left(z^2+\\frac{902}{901}pz+p^2\\right),\n \\qquad p=\\frac{999}{1000},\\quad a=\\frac{901}{902}p,\n\\]\nall roots lie in the disc and $0$ is a non-root critical point, but its\nunique nearest-root segment leaves the unit lemniscate. Moreover, every\nroot-pair midpoint of $z^3-(99/100)^3$ lies outside it. This cubic still\nadmits the two-segment connections of Theorem~\\ref{res:trinomial-all-degree};\nit is the straight chords that fail. The\n\\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{straight-path-counterexamples}\n{exact counterexamples} include the escape calculations.\n\n\\noindent\\phantomsection\\label{res:sep-or-false}\nNor do the two critical-value criteria cover every polynomial in the\nroot-disc class. For $z^3+3z/100-3/4$, the least critical-value modulus\nexceeds $13/25$, and the two normalised critical values are too close\nfor the isolation criterion.\n\n\\noindent\\phantomsection\\label{res:arity-not-capacity}\nA further cubic has two roots at its first merger but\nnormalised component capacity one at level $2\\mu$. By that level all three roots have merged, so the initial two-root count\ncannot be used at the later level. The component is now the whole filled\nlemniscate, whose capacity is given by\n$\\operatorname{cap}(K_t(f))=t^{1/n}$\n\\cite[Theorem~5.2.5 and p.~153]{ransford1995}.\n\n\n\\noindent\\phantomsection\\label{res:one-root-gamma-false}\nFinally, $z^8-(3/2)z$ has least critical-value modulus $\\mu>1$, yet a\none-root unit-level component has perimeter larger than\n$\\Gamma(1/4)^2/(2\\sqrt\\pi)$; see\n\\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{component-counterexamples}\n{the companion's component examples}. No root-location restriction is imposed.\nThe open question asks for a constant $\\beta$ bounding each component\nperimeter of $K_\\sigma(f)$ by $\\beta\\sigma^{1/n}$, uniformly over\nsquarefree monic $f$ of degree $n\\ge2$ and $0<\\sigma<\\mu$.\nThis level restriction is stronger than a one-root condition on a single\ncomponent. Although \\cite[Proposition~2]{revision2026} reports a Runge\nconstruction of unbounded one-root perimeters, its proof is unavailable\nand we do not use it.\n\n\\subsection{Sparse families and binomial chords}\\label{sec:solved-families}\\label{sec:binomial}\n\\noindent\\phantomsection\\label{res:cubic-fibres}\nLet $f(z)=P((z-h)^q)$, where $q\\ge2$ and $P$ is a monic cubic,\nand suppose all zeros of $f$ lie in the open unit disc. For any root\n$r=y^q$ of $P$, averaging $|h+y\\zeta|^2$ over the $q$th roots of\nunity cancels the cross term and gives $|h|^2+|y|^2<1$. Thus the\nquotient roots satisfy $|r|=|y|^q<1$.\nIf $f$ has two distinct zeros, the companion selects a nonzero root\n$r$ with $[0,r]\\subset\\{|P|<1\\}$. Choose $y^q=r$. Then\n$f(h+ty\\zeta)=P(t^qr)$ for $0\\le t\\le1$, so the whole fibre gives\ncontained segments through $h$. Two distinct fibre points are joined\nwith length $2|y|<2$, using the same average.\n\n\\noindent\\phantomsection\\label{res:primitive-quintic}\nFor $f(z)=z^5+az^4+bz+c$ with open-disc roots, Abel summation\ngives a contained radial segment whenever $|bz_j+c|<1$.\nFor $a\\ne0$, the missing coefficients fix the first three power sums.\nAfter rotation, the companion's harmonic function sums to $5-2|a|$\nover the roots. It is bounded above by $4-2|a|$ on the disc and by\n$2/31$ at roots with $|bz_j+c|\\ge1$. Four such indices would therefore give\n\\[\n 5-2|a|\\le4-2|a|+\\frac8{31}<5-2|a|,\n\\]\na contradiction. Hence two indices satisfy $|bz_j+c|<1$.\nWhen $a=0$, the root equation gives $|bz_j+c|=|z_j|^5<1$ directly.\nCoincident selected root values give the constant path.\nThe companion gives the\n\\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{cubic-fibre-proof}\n{cubic-fibre proof} and the\n\\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{quintic-proof}\n{quintic selection argument}, including repeated root occurrences.\n\n\\noindent\\phantomsection\\label{res:complementary-binomial-chords}\nFor $z^n-r^n$, $0<r<1$, the maximum modulus on an adjacent chord\nof radius $0<s\\le r$ is $r^n+(s\\cos(\\pi/n))^n$.\nThe outer root chord therefore works below\n$r_*=(1+\\cos^n(\\pi/n))^{-1/n}$. At and above this value, take a\nslightly contracted inner radius $s>0$ making that maximum less than one.\nThe radial legs remain contained, and for $n\\ge3$ the total length is\n\\[\n 2(r-s)+2s\\sin(\\pi/n)<2r<2.\n\\]\nThe \\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{binomial-chord-calculation}\n{chord calculation} proves the maximum on the entire segment, not\ngeodesic optimality. For $n=2$, use the diameter.\n\n\\subsection{Collinear roots}\\label{sec:collinear}\\label{bdry:solved-polynomial-families}\n\\noindent\\phantomsection\\label{res:sharp-collinear-root-diameter}\nFor collinear roots of diameter $D$, a Chebyshev alternation argument\ngives two adjacent root occurrences whose segment satisfies\n\\[\n |f|\\le\\frac{(D/2)^n}{2^{n-1}\\cos^n(\\pi/(2n))}.\n\\]\nThe constant is sharp, with equality for the appropriately scaled zeros\nof the Chebyshev polynomial $T_n$. Qualitative existence of a contained root segment was already\nproved in \\cite[Theorem~1]{ehp1958}. For distinct roots, first reduce\nto the real monic case. If every gap maximum exceeded the bound,\nsubtracting the endpoint-normalised monic Chebyshev\npolynomial would give a polynomial of degree at most $n-1$ with $n$\ndistinct zeros: two endpoints and $n-2$ intervening sign changes.\nSee Eremenko and Yuditskii\n\\cite[\\S1 and Theorem~1]{eremenko-yuditskii} for the critical-sequence\nviewpoint. The companion contains the\n\\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{collinear-proof}\n{alternation proof and equality case}.\n\n\\subsection{Critical-value means}\\label{sec:freepoint}\n\\noindent\\phantomsection\\label{res:fp-weighted-all-degree}\nFor $c_j\\in\\overline{\\mathbb D}$ and positive weights of sum\none, set $G(z)=\\prod_j|1-\\bar c_jz|^{w_j}$. Then\n$\\sum_jw_jG(c_j)^2\\le1$, with equality only when all $c_j=0$.\n\nFor interior centres, the nonvanishing factors give an analytic\n$g(z)=1+\\sum_{\\nu\\ge1}a_\\nu z^\\nu$ with $|g|=G$. On\n$\\zeta=e^{i\\theta}$ the weighted Poisson kernel is\n$P=\\sum_jw_j(1-|c_j|^2)/|\\zeta-c_j|^2\n=1-2\\operatorname{Re}(\\zeta g'/g)$.\nWith $dm=d\\theta/(2\\pi)$, subharmonicity gives the inequality below,\nand orthogonality gives the identity:\n\\[\n \\sum_jw_jG(c_j)^2\\le\\int |g|^2P\\,dm\n          =1-\\sum_{\\nu\\ge1}(2\\nu-1)|a_\\nu|^2.\n\\]\nIndeed, $\\int|g|^2\\,dm=1+\\sum|a_\\nu|^2$ and\n$\\int\\zeta g'\\overline g\\,dm=\\sum\\nu|a_\\nu|^2$.\nEach nonconstant coefficient therefore contributes $1-2\\nu<0$. For boundary centres, replace\n$c_j$ by $rc_j$ and let $r\\uparrow1$ with a finite coefficient sum\nfirst; no boundary logarithm is needed. Equality forces every $a_\\nu$\nto vanish. Logarithmic differentiation of $g=1$, grouping repeated\ncentres, then forces all the positive-weight centres to be zero.\n\n\\noindent\\phantomsection\\label{res:fp-to-s}\nTo apply this inequality to a monic polynomial $f$ of degree $n\\ge2$\nwith zeros in the closed unit disc, take its $m=n-1$ critical points,\ncounted with multiplicity, and give them equal weights $1/m$.\nGauss--Lucas puts these points in the closed unit disc. The\nreflected-derivative comparison then gives the pointwise bound\n\\[\n |f(c_j)|^{2/m}\\le\n \\left(\\prod_{k=1}^m|1-\\bar c_kc_j|\\right)^{2/m}=G(c_j)^2.\n\\]\nSumming and using the weighted inequality gives\n$\\sum_j|f(c_j)|^{2/m}\\le m$. For zeros in a closed disc with centre $h$\nand radius $R>0$, apply this to $R^{-n}f(h+Rz)$ and scale back to obtain\n$\\sum_{j=1}^{n-1}|f(c_j)|^{2/(n-1)}\\le(n-1)R^{2n/(n-1)}$.\nIf $R=0$, all critical values vanish. An unformalised refinement, recorded as an ordinary argument, uses\n$g^2$ in the same Poisson integral and gives the proposed fourth-power bound\n\\[\n \\sum_{j=1}^{n-1}|f(c_j)|^{4/(n-1)}\n       \\le(n-1)R^{4n/(n-1)}\n\\]\nfor the same enclosing disc. Equality is attained\nby $(z-h)^n-\\lambda$ with $|\\lambda|=R^n$.\nThe companion supplies the\n\\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{weighted-poisson-proof}\n{weighted identity, reflected-derivative comparison and equality analysis}.\n\nApplying AM--GM to\n$\\sum_j|f(c_j)|^{2/(n-1)}\\le n-1$ in the unit disc gives\nDubinin's sharp critical-value product bound \\cite[Theorem~2]{dubinin2006critical}. With a\nzero prescribed at the disc centre, his Theorem~3 gives a different,\nsharper product estimate and credits earlier work of Tischler\n\\cite[p.~444, as discussed by Dubinin]{tischler1989}. The corresponding marked-zero positive-moment question in\n\\cite{revision2026} remains unresolved here. To turn any of these mean\nbounds into a short-path theorem, one would still have to select a pair\nof roots and control a curve joining them inside the lemniscate.\n\n\\subsection{Successive merger scales}\\label{sec:orlicz}\n\\noindent\\phantomsection\\label{res:orlicz-currency}\nFor a component with $k\\ge1$ roots, let $r\\in(0,1]$ be the earlier\nmerger level divided by the later one. An integral arising in length\nestimates is\n\\[\n I_k(r)=\\int_r^1\n \\frac{dq}{q\\log((1+q^{2/k})/(1-q^{2/k}))}\n       =k\\Phi\\left(\\frac1k\\log\\frac1r\\right),\n \\qquad \\Phi(x)=\\int_0^x\\frac{dt}{\\log\\coth t}.\n\\]\nSubstituting $q=e^{-kt}$ gives the identity; the integrand defining\n$\\Phi$ has limiting value zero at $t=0$. It is increasing, so $\\Phi$\nis strictly convex and $\\Phi(x)/x\\to0$ as $x\\downarrow0$.\nThus, when the two merger levels approach one another ($r\\uparrow1$),\n$I_k(r)$ becomes arbitrarily small relative to $k^{-1}\\log(1/r)$,\neven for fixed $k$. No positive uniform lower factor exists. See the\n\\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{merger-integral-proof}\n{merger-scale calculation}.\n\n\\section{Flow constructions and remaining questions}\\label{sec:open}\n\n\\subsection{Inverse rays and perturbation}\\label{sec:newton}\\label{sec:arguments}\\label{sec:reciprocal}\n\\noindent\\phantomsection\\label{res:value}\nAlong an existing real-interval trajectory $\\dot z=-f(z)/f'(z)$ with\n$f'(z(t))\\ne0$, the chain rule gives $(f\\circ z)'=-f\\circ z$, hence\n$f(z(t))=e^{-(t-t_0)}f(z(t_0))$. This classical identity fixes the\nray of a nonzero value, not the Euclidean length\n$\\int|f(z(t))/f'(z(t))|\\,dt$; see Kozen and Stef\\'ansson\n\\cite[Lemmas~2.1--2.2 and \\S2]{kozen-stefansson1997} on the Newtonian\ngraph of Shub, Tischler and Williams. The companion's local\ncomplex-parameter identities for $(f\\circ z)'$ and $(e^t f(z(t)))'$\nare pointwise statements, distinct from the real-interval integration\nin its \\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{newton-value-proof}\n{value and finite-endpoint theorems}.\n\n\\noindent\\phantomsection\\label{res:ray}\nFor a finite interval $[a,b]$, assume the Newton equation only for\n$a<t<b$ and continuity of $f(z(t))$ up to both endpoints. The interior\nidentity then extends to $f(z(b))=e^{a-b}f(z(a))$, without evaluating\n$f/f'$ at an endpoint. Nonzero endpoint values therefore lie on the\nsame positive ray. Critical values\nwith distinct arguments exclude such a connection; distinct moduli are\ninsufficient.\n\n\\noindent\\phantomsection\\label{res:locus}\nAdding $\\beta$ leaves $f'$ unchanged and sends each critical value $v$\nto $v+\\beta$. For distinct values $a,b$, a common positive ray requires\n$\\beta$ to lie on the real line through $-a$ and $-b$.\nAvoiding these finitely many lines and the points $-v$ therefore makes\nthe distinct values nonzero and ray-separated; equal values remain equal.\nThe formal sources give the\n\\href{https://github.com/wcook04/plectis-erdos/blob/7380b7871687b6bcc41ca0143c61f232e8af6500/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean\\#L107}{one-parameter collision locus}\nand the \\href{https://github.com/wcook04/plectis-erdos/blob/7380b7871687b6bcc41ca0143c61f232e8af6500/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean\\#L162}{finite affine-line avoidance argument}.\nThe \\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{argument-perturbation-proof}\n{companion calculation} describes the forbidden translations; its\nunformalised linear-perturbation argument supplies distinct critical values.\nNeither qualitative step controls the slack in a near-extremal path inequality.\nSeparately, expand $\\log|f|$ about a non-root point and let $q$ be the\ndisplacement divided by the distance to the nearest zero. For $q<1$,\nthe remainder after order $N$ is at most $nq^{N+1}/((N+1)(1-q))$.\nThis local expansion cannot be used through a zero and does not itself\ngive a coefficient-perturbation bound.\n\n\\paragraph{Inverse-branch prerequisites.}\n\\label{res:component-local-covering}\nFor a nonconstant complex polynomial, each component of $\\{|f|<1\\}$\n\\href{https://github.com/wcook04/plectis-erdos/blob/e98130bfe66aa33606e116791080b3ed13e1d68a/evidence/source-prerequisites/8bf96bdcae6b9201c670fff3037c49c70ec6de8d/lean/ErdosProblems/Erdos1041/OutwardSlitDomain.lean\\#L349}{contains a root}.\nThe restriction to that component is\n\\href{https://github.com/wcook04/plectis-erdos/blob/e98130bfe66aa33606e116791080b3ed13e1d68a/evidence/source-prerequisites/8bf96bdcae6b9201c670fff3037c49c70ec6de8d/lean/ErdosProblems/Erdos1041/OutwardSlitDomain.lean\\#L211}{onto the unit disc} and has\n\\href{https://github.com/wcook04/plectis-erdos/blob/e98130bfe66aa33606e116791080b3ed13e1d68a/evidence/source-prerequisites/8bf96bdcae6b9201c670fff3037c49c70ec6de8d/lean/ErdosProblems/Erdos1041/OutwardSlitDomain.lean\\#L233}{finite fibres}.\nOver values avoiding the critical values attained in that component,\nit is a \\href{https://github.com/wcook04/plectis-erdos/blob/e98130bfe66aa33606e116791080b3ed13e1d68a/evidence/source-prerequisites/8bf96bdcae6b9201c670fff3037c49c70ec6de8d/lean/ErdosProblems/Erdos1041/OutwardSlitDomain.lean\\#L250}{covering map}.\nWhen $0$ is not a critical value on the component, choose finitely many\nnonzero slit starts including its critical values, and remove their\noutward rays from the value disc. On the remaining domain, each root\nin the component determines a\n\\href{https://github.com/wcook04/plectis-erdos/blob/e98130bfe66aa33606e116791080b3ed13e1d68a/evidence/source-prerequisites/8bf96bdcae6b9201c670fff3037c49c70ec6de8d/lean/ErdosProblems/Erdos1041/OutwardSlitDomain.lean\\#L359}{unique continuous inverse branch}, taking $0$ to that root.\nThe archived Lean source states these prerequisites and supplies proof\nscripts; a kernel-check receipt for this module is not supplied here.\nThe full conformal\nsheet decomposition, sheet count, monodromy and embedded-tree assertions\nremain ordinary arguments in the companion; the prerequisites give no\nlength estimate.\n\n\\subsection{The failed tree estimate}\\label{sec:gap}\\label{sec:finite}\nFor $z^2-(9/10)^2$, the distance between the roots already forces any\nconnected set containing them to have length at least $9/5$.\nProposition~12 of the March manuscript \\cite{march2026} instead proposes\na bound below $41/25$ plus an arbitrarily small error. Since\n$41/25<9/5$, choosing the error smaller than this gap contradicts the\nlower bound. The spanning-tree estimate is false. At a simple saddle\nthe local level set has four sectors, consistently with the lemniscate\ngraph description in \\cite[Definition~1.2 and Proposition~2.4]{bishop-eremenko-lazebnik}. The\n\\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{cassini-counterexample}\n{Cassini calculation and corrected topological statement} separate the\nmetric failure from the valid sheet decomposition.\n\nThe full topological reconstruction in that discussion remains unformalised.\nCutting outward from critical values to obtain inverse-sheet adjacency\nis classical; see \\cite[\\S1]{dubinin2006critical} and\n\\cite[\\S2]{dubinin2006capacity}. Length control requires additional\nestimates on the actual pieces and their attachment. An uncut Newton orbit space can even fail to be Hausdorff, so it cannot\nbe identified with a finite Reeb graph without further argument. Earlier\ngrid searches in degrees $5,6,8,10$ found no counterexample, but neither\ncertified containment along unexamined edges nor established a statement\nfor all polynomials.\n\n\\subsection{Restricted extremal problems}\nThe companion records the following ordinary compactness argument, which\nhas not been formalised. In the closed lemniscate $K_1(f)$, a finite\ninfimum of connecting path lengths is attained. At fixed degree, compactness of the closed root-disc\nclass and lower semicontinuity of this infimum allow uniform upper bounds\nto pass to coefficient limits. These are statements about the closed\nlemniscate, as proved in the companion's\n\\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{closed-class-limits}\n{compactness argument}. Closed containment survives such limits;\nstrict containment need not. The binomial family approaches length $2$, but the degree-seven example\nrules out $2$ as a universal upper bound. That refuted assertion is not\na remaining open problem. What remains is to determine quantitative\nbounds for restricted classes and as the degree varies.\n\nThe precise unresolved questions about cut-open flow strips,\nattachment lengths and quantitative perturbation are stated in the\ncompanion's\n\\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{remaining-questions}\n{final questions}. The origin-spoke failure for the near-regular-pentagon\nfamily has an ordinary, unformalised proof there. The stronger failure asserted in the research\naddendum \\cite{revision2026} for every critical joining point has no\navailable proof and is not used. Likewise the fixed degree-seven proof\ndoes not establish the whole small-parameter family.\n\n\\section{Related results and verification}\\label{sec:1041-sources}\n\nSeveral nearby results concern different geometric quantities.\nInverse branches, slit domains and capacity also occur in Crane's work\non Smale's mean-value conjecture\n\\cite[Lemma~2.1 and \\S\\S2--4]{crane2007smale}, but its derivative\nnormalisation differs from the monic normalisation used here.\nEremenko and Hayman's bound\n$\\mathcal H^1\\{|p|=1\\}<9.173\\deg p$\n\\cite[Theorem~1]{eremenko-hayman} concerns total boundary length, not\nan internal root-to-root path. Their connectedness result, the\nderivative estimate of Eremenko and Lempert\n\\cite[Theorem~1]{eremenko-lempert}, and its equality analysis\n\\cite[Theorem~A]{eremenko-markov} likewise concern different extremal\nquantities. Kuznetsova and Tkachev\n\\cite[Theorems~1--2]{kuznetsova2003} study length functions on regular\nlevels; log-convexity there does not give a uniform one-root perimeter\nbound at a critical endpoint.\n\nThe hypotheses and endpoints also matter. Dubinin's four-point distortion\ntheorem \\cite[Theorem~1 and Corollary~4]{dubinin2013fourpoint} assumes a\nbound on all critical values, with a different normalisation.\nPendyala's Problem~1120 result\n\\cite[Definition~1.1 and Theorem~1.2]{pendyala2026shortest} joins $0$\nto the unit circle inside a lemniscate intersected with the closed disc.\nNeither the endpoints nor that region restriction are those of\nProblem~1041.\n\n\\paragraph{Verification and reproducibility.}\nThe supplied Lean sources prove the fixed counterexample for preconnected\nsets, the negation and \\texttt{answer(False)} forms of the Formal\nConjectures statement, and the total-variation formulation. They also\nprove the trinomial theorem and the other formally supported estimates\nidentified in the companion's\n\\recordlink{erdos1041-lemniscate-reasoning-surface.pdf}{verification-notes}\n{verification notes}. No fresh Lean build or independent human review\nis reported here, including review of the correspondence with the 1958\nwording. The recorded full rational replay on 29 September 2026\nreproduced the stopping time and $126$ dual certificates; those are\nthe computational results cited here. The analytic assertions in Remarks~\\ref{res:low-critical-thirteen-twentyfifths}--\\ref{res:critical-value-thresholds},\nthe fourth-power mean and the later topological and compactness arguments\nremain outside the formal conclusions. The evidence record retains the\nstatement identifiers and their conditional dependencies.\n\\evidenceparagraph{\\evidencerecordurl}\n\n\\begin{thebibliography}{99}\n\\bibitem{bloom} T.~F. Bloom, \\emph{Erd\\H{o}s Problems}, problem~1041.\n  \\url{https://www.erdosproblems.com/1041}\n\\bibitem{ehp1958} P.~Erd\\H{o}s, F.~Herzog, and G.~Piranian,\n  \\emph{Metric properties of polynomials}, J. Analyse Math. \\textbf{6}\n  (1958), 125--148,\n  doi:\\href{https://doi.org/10.1007/BF02790232}{10.1007/BF02790232}.\n\\bibitem{march2026} \\texttt{shtuka},\n  \\emph{A Short Path Joining Two Zeros Inside a Polynomial Lemniscate},\n  manuscript posted 24 March 2026, 48~pp.\n  \\url{https://shtuka123.github.io/1041/main.pdf}\n\\bibitem{june2026} V.~S. Pendyala,\n  \\emph{A Degree-Four Lemniscate Path Theorem},\n  arXiv:\\href{https://arxiv.org/abs/2606.24875v1}{2606.24875v1} (2026),\n  doi:\\href{https://doi.org/10.48550/arXiv.2606.24875}\n  {10.48550/arXiv.2606.24875}.\n\\bibitem{pendyala2026shortest} V.~S. Pendyala,\n  \\emph{Shortest paths in polynomial lemniscate sublevel sets and a problem\n  of Erd\\H{o}s}, arXiv:\\href{https://arxiv.org/abs/2606.19178v1}{2606.19178v1}\n  (2026),\n  doi:\\href{https://doi.org/10.48550/arXiv.2606.19178}\n  {10.48550/arXiv.2606.19178}.\n\\bibitem{kozen-stefansson1997} D.~Kozen and K.~Stef\\'ansson,\n  \\emph{Computing the Newtonian graph}, J. Symbolic Comput. \\textbf{24}\n  (1997), no.~2, 125--136,\n  doi:\\href{https://doi.org/10.1006/jsco.1997.0118}{10.1006/jsco.1997.0118};\n  authors' copy \\url{https://www.cs.cornell.edu/kozen/Papers/newton.pdf}.\n\\bibitem{crane} E.~Crane,\n  \\emph{The areas of polynomial images and pre-images}, Bull. London Math.\n  Soc. \\textbf{36} (2004), no.~6, 786--792,\n  doi:\\href{https://doi.org/10.1112/S0024609304003509}{10.1112/S0024609304003509};\n  preprint arXiv:\\href{https://arxiv.org/abs/math/0302189v1}{math/0302189v1},\n  whose statement numbers are cited.\n\\bibitem{eks2010} P.~Ebenfelt, D.~Khavinson, and H.~S. Shapiro,\n  \\emph{Two-dimensional shapes and lemniscates}, in \\emph{Complex Analysis and\n  Dynamical Systems IV, Part~1}, Contemp. Math. \\textbf{553}, Amer. Math. Soc.,\n  Providence, RI, 2011, 45--59,\n  doi:\\href{https://doi.org/10.1090/conm/553/10931}{10.1090/conm/553/10931};\n  preprint arXiv:\\href{https://arxiv.org/abs/1003.4567v1}{1003.4567v1},\n  whose statement numbers are cited.\n\\bibitem{eremenko-hayman} A.~Eremenko and W.~Hayman,\n  \\emph{On the length of lemniscates}, Michigan Math. J. \\textbf{46}\n  (1999), no.~2, 409--415,\n  doi:\\href{https://doi.org/10.1307/mmj/1030132418}{10.1307/mmj/1030132418};\n  preprint arXiv:\\href{https://arxiv.org/abs/0805.2295}{0805.2295}.\n\\bibitem{eremenko-yuditskii} A.~Eremenko and P.~Yuditskii,\n  \\emph{Comb functions}, Contemp. Math. \\textbf{578} (2012), 99--118,\n  doi:\\href{https://doi.org/10.1090/conm/578/11472}{10.1090/conm/578/11472};\n  preprint arXiv:\\href{https://arxiv.org/abs/1109.1464v1}{1109.1464v1}.\n\\bibitem{eremenko-lempert} A.~Eremenko and L.~Lempert,\n  \\emph{An extremal problem for polynomials}, Proc. Amer. Math. Soc.\n  \\textbf{122} (1994), no.~1, 191--193,\n  doi:\\href{https://doi.org/10.1090/S0002-9939-1994-1207536-1}\n  {10.1090/S0002-9939-1994-1207536-1}.\n\\bibitem{eremenko-markov} A.~Eremenko,\n  \\emph{A Markov-type inequality for arbitrary plane continua},\n  Proc. Amer. Math. Soc. \\textbf{135} (2007), no.~5, 1505--1510,\n  doi:\\href{https://doi.org/10.1090/S0002-9939-06-08640-0}\n  {10.1090/S0002-9939-06-08640-0};\n  preprint arXiv:\\href{https://arxiv.org/abs/math/0606745v1}{math/0606745v1}.\n\\bibitem{bishop-eremenko-lazebnik} C.~J. Bishop, A.~Eremenko, and\n  K.~Lazebnik, \\emph{On the shapes of rational lemniscates}, Geom. Funct.\n  Anal. \\textbf{35} (2025), no.~2, 359--407,\n  doi:\\href{https://doi.org/10.1007/s00039-025-00704-2}\n  {10.1007/s00039-025-00704-2};\n  preprint arXiv:\\href{https://arxiv.org/abs/2407.14610v1}{2407.14610v1}.\n\\bibitem{dubinin} V.~N. Dubinin, \\emph{Some inequalities for polynomials and\n  rational functions associated with lemniscates}, Zap. Nauchn. Sem. POMI\n  \\textbf{404} (2012), 83--99; English translation, J. Math. Sci. \\textbf{193}\n  (2013), no.~1, 45--54,\n  doi:\\href{https://doi.org/10.1007/s10958-013-1432-4}{10.1007/s10958-013-1432-4}.\n\\bibitem{polya1928} G.~P\\'olya,\n  \\emph{Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach\n  zusammenh\\\"angende Gebiete}, Sitzungsberichte der Preussischen Akademie der\n  Wissenschaften, Physikalisch-Mathematische Klasse (1928), printed\n  pp.~228--232 and 280--282.\n  \\url{https://archive.org/details/sitzungsbericht1928preu}.\n\\bibitem{dubinin2006critical} V.~N. Dubinin,\n  \\emph{Inequalities for critical values of polynomials}, Sb. Math.\n  \\textbf{197} (2006), no.~8, 1167--1176,\n  doi:\\href{https://doi.org/10.1070/SM2006v197n08ABEH003793}{10.1070/SM2006v197n08ABEH003793}.\n\\bibitem{crane2007smale} E.~Crane,\n  \\emph{A bound for Smale's mean value conjecture for complex polynomials},\n  Bull. London Math. Soc. \\textbf{39} (2007), no.~5, 781--791,\n  doi:\\href{https://doi.org/10.1112/blms/bdm063}{10.1112/blms/bdm063};\n  author preprint \\url{https://people.maths.bris.ac.uk/~maetc/SMVCbound.pdf}.\n\\bibitem{dubinin2013fourpoint} V.~N. Dubinin,\n  \\emph{Four-point distortion theorem for complex polynomials},\n  arXiv:\\href{https://arxiv.org/abs/1301.3985v1}{1301.3985v1} (2013).\n\\bibitem{kuznetsova2003} O.~S. Kuznetsova and V.~G. Tkachev,\n  \\emph{Length functions of lemniscates}, Manuscripta Math.\n  \\textbf{112} (2003), 519--538,\n  doi:\\href{https://doi.org/10.1007/s00229-003-0411-3}{10.1007/s00229-003-0411-3};\n  preprint arXiv:\\href{https://arxiv.org/abs/math/0306327}{math/0306327}.\n\\bibitem{tischler1989} D.~Tischler,\n  \\emph{Critical points and values of complex polynomials},\n  J. Complexity \\textbf{5} (1989), no.~4, 438--456.\n\\bibitem{revision2026} W.~Cook / Plectis,\n  \\emph{Three refinements for the lemniscate-path programme}, 16 September 2026,\n  unreviewed research note, Sections~1--3, together with the earlier\n  note \\emph{Structural obstructions}.\n\\bibitem{dubinin2006capacity} V.~N. Dubinin,\n  \\emph{Lemniscates and inequalities for the logarithmic capacities of\n  continua}, Mat. Zametki \\textbf{80} (2006), no.~1, 33--37;\n  English translation, Math. Notes \\textbf{80} (2006), no.~1, 31--35,\n  doi:\\href{https://doi.org/10.1007/s11006-006-0105-8}{10.1007/s11006-006-0105-8}.\n\\bibitem{ransford1995} T.~Ransford,\n  \\emph{Potential Theory in the Complex Plane}, London Mathematical Society\n  Student Texts~28, Cambridge University Press, Cambridge, 1995,\n  doi:\\href{https://doi.org/10.1017/CBO9780511623776}{10.1017/CBO9780511623776}.\n\\end{thebibliography}\n\n\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":64938,"sha256":"sha256:f96e8e7b67243a5bf67e1d0d4a80c8e36422cdda12a12f67d49e3f0181389d2f","cite_as":"Plectis reader corpus, paper:erdos-1041-lemniscate-newton-flow","source_repository":"plectis-erdos","source_path":"paper/1041/erdos-1041-lemniscate-newton-flow.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:088d36d7ba3d3ee0cee2df99e8915927999698f19889f8ea0b8d55b3721ec778","byte_count":6923,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% This is the only copy of the shared preamble.  The eight short notes, the\n% long records assembled from paper/reasoning-parts other than #249 and #257\n% (which carry their own preambles), and the synthesis paper include it with\n% \\input{problem-note-preamble}; paper/Makefile builds from this directory\n% with the directory on the tectonic search path.\n%\n% Shared preamble for the Erdős Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erdős problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{8e58c395e9807ef3374d7365c62cf04d20ee6331}\n\\newcommand{\\commitshort}{9e231ce4371f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-erdos/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-erdos/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\usepackage{longtable}  % the notes' declaration tables; loaded here so the assembled records inherit it\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n% ---------------------------------------------------------------------------\n% Evidence marks.  paper-evidence.tex places \"Lean\" and \"Comparator\" links in\n% the margin beside each result from the paper's generated file\n% paper/evidence/<paper>.tex, which each paper inputs after this preamble.\n% ---------------------------------------------------------------------------\n\\input{paper-evidence}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Production provenance.  The wording is fixed by the corpus provenance\n% contract and reproduced verbatim in every manuscript in this repository, so\n% the papers cannot drift into different accounts of how they were made.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erdős problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every Lean mark in the margin points to propositions that the\n  pinned Lean kernel accepts, with no \\texttt{sorry}, no added axiom, and no\n  unchecked evaluation.  That claim concerns the formal statements only; it\n  says nothing about their mathematical interest, their novelty, or the\n  original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n\n% Compact opening card for the separately stated Comparator theorem selected\n% from each problem note.  The prose supplied by each note owns the\n% contribution and relevance judgements; the shared final sentence states the\n% mechanical assurance boundary once and points to the public evidence owner.\n\\newcommand{\\externalreviewobject}[3]{%\n  \\begin{quote}\\small\n  \\textbf{Contribution.} #1\\par\n  \\textbf{Relation to the open problem.} #2\\par\n  \\textbf{Executable review object.} #3  The repository's\n  external-verification job compares these exact Lean propositions with\n  separately declared challenge statements and an axiom budget, then asks\n  Lean's kernel to check the submitted proofs.  The\n  \\href{https://github.com/wcook04/plectis-erdos/blob/ca0e13f8acf5ccf48506e4bdb870953d3a0856fa/formalization.yaml}%\n  {formalisation manifest} and the commit-bound CI receipt record that check;\n  they do not assess novelty, significance, or whether the original problem is\n  solved.\n  \\end{quote}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  Under\n% Tectonic (XeTeX) the text and mathematics are set in TeX Gyre Pagella from\n% the bundled OpenType fonts, with a Heros sans range for \\mathsf and\n% Inconsolata for monospace; the pdflatex fallback keeps the same Palatino\n% design through newpx.  The A4 text block uses a classical measure with\n% quiet running heads and restrained ink.  Link colours: steel blue for\n% external (Lean-source and web) targets, deep burgundy for internal\n% cross-references and citations.\n\\NeedsTeXFormat{LaTeX2e}\n\\ProvidesPackage{paper-house-style}[2026/07/19 Pagella house style for the papers]\n\n\\RequirePackage{iftex}\n\\RequirePackage{amsmath}\n\\RequirePackage{amsthm}\n\\ifpdftex\n  \\RequirePackage[T1]{fontenc}\n  \\RequirePackage[utf8]{inputenc}\n  \\RequirePackage{amssymb}\n  \\RequirePackage{newpxtext}\n  \\RequirePackage{newpxmath}\n\\else\n  \\RequirePackage{fontspec}\n  \\RequirePackage{unicode-math}\n  \\setmainfont{texgyrepagella}[Extension=.otf,UprightFont=*-regular,\n    BoldFont=*-bold,ItalicFont=*-italic,BoldItalicFont=*-bolditalic]\n  \\setmathfont{texgyrepagella-math.otf}\n  \\setmathfont{texgyreheros-regular.otf}[range=sfup,Scale=MatchLowercase]\n  \\setmonofont{Inconsolatazi4}[Extension=.otf,UprightFont=*-Regular,\n    BoldFont=*-Bold,Scale=MatchLowercase]\n\\fi\n\n\\RequirePackage[a4paper,textwidth=418pt,textheight=636pt,hcentering,\n  top=68pt,headheight=15pt,headsep=20pt,footskip=34pt]{geometry}\n\\RequirePackage{microtype}\n\\linespread{1.045}\n\\clubpenalty=10000 \\widowpenalty=10000 \\tolerance=1600\n\n\\RequirePackage[dvipsnames,table]{xcolor}\n\\definecolor{linkexternal}{HTML}{275D8C}\n\\definecolor{linkinternal}{HTML}{6E2B2B}\n\\definecolor{papertableheader}{HTML}{E3EAF0}\n\\definecolor{papertablerow}{HTML}{F2F5F7}\n\\definecolor{papertablerule}{HTML}{7D8991}\n\\definecolor{paperresultrule}{HTML}{315B72}\n\n% Stable cross-document links between sibling PDFs. 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Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"},{"record_type":"paper_source_dependency","tex_reference":"evidence/erdos-1041-lemniscate-newton-flow","source_path":"paper/evidence/erdos-1041-lemniscate-newton-flow.tex","media_type":"text/x-tex","sha256":"sha256:c60b85cd02a4f493465fbd32c1bccecab7e8d11a0e28a5321ac62a8c2612ec6c","byte_count":4847,"licence":"CC-BY-4.0","body_format":"latex","body":"% Generated by scripts/paper_evidence.py from docs/paper_lean_coverage.json and the\n% Comparator replay receipts under evidence/comparator/.  Do not edit by hand.\n\\newcommand{\\evidencerecordurl}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1041-lemniscate-newton-flow.md}\n\\DeclareResultEvidenceName{res:ani-degree-seven-counterexample}{Theorem 1.2}\n\\DeclareResultEvidenceRecord{res:ani-degree-seven-counterexample}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1041-lemniscate-newton-flow.md\\#res-ani-degree-seven-counterexample}{exact_or_stronger}{pending}\n\\DeclareResultEvidence{res:ani-degree-seven-counterexample}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1041-lemniscate-newton-flow.md\\#res-ani-degree-seven-counterexample}{}\n\\DeclareResultEvidenceName{lem:two-sheet-bottleneck}{Lemma 2.1}\n\\DeclareResultEvidenceRecord{lem:two-sheet-bottleneck}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1041-lemniscate-newton-flow.md\\#lem-two-sheet-bottleneck}{exact_or_stronger}{pending}\n\\DeclareResultEvidence{lem:two-sheet-bottleneck}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean\\#L187}{}\n\\DeclareResultEvidenceName{res:trinomial-all-degree}{Theorem 3.1}\n\\DeclareResultEvidenceRecord{res:trinomial-all-degree}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1041-lemniscate-newton-flow.md\\#res-trinomial-all-degree}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:trinomial-all-degree}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/PaperTrinomialWholeR21.lean\\#L23}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1041-lemniscate-newton-flow.md\\#res-trinomial-all-degree-comparator}\n\\DeclareResultEvidenceName{res:sextic-spoke}{Proposition 3.2}\n\\DeclareResultEvidenceRecord{res:sextic-spoke}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1041-lemniscate-newton-flow.md\\#res-sextic-spoke}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:sextic-spoke}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/PaperCompleteR20/SexticSpokeWhole.lean\\#L9}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1041-lemniscate-newton-flow.md\\#res-sextic-spoke-comparator}\n\\DeclareResultEvidenceName{res:low-critical-thirteen-twentyfifths}{Passage, p. 7}\n\\DeclareResultEvidenceRecord{res:low-critical-thirteen-twentyfifths}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1041-lemniscate-newton-flow.md\\#res-low-critical-thirteen-twentyfifths}{none}{not_applicable}\n\\DeclareResultEvidenceName{res:scaled-low-critical}{Passage, p. 8}\n\\DeclareResultEvidenceRecord{res:scaled-low-critical}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1041-lemniscate-newton-flow.md\\#res-scaled-low-critical}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{res:scaled-low-critical}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1041-lemniscate-newton-flow.md\\#res-scaled-low-critical}{}\n\\DeclareResultEvidenceName{res:critical-value-separation}{Passage, p. 10}\n\\DeclareResultEvidenceRecord{res:critical-value-separation}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1041-lemniscate-newton-flow.md\\#res-critical-value-separation}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{res:critical-value-separation}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1041-lemniscate-newton-flow.md\\#res-critical-value-separation}{}\n\\DeclareResultEvidenceName{res:critical-value-thresholds}{Passage, p. 10}\n\\DeclareResultEvidenceRecord{res:critical-value-thresholds}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1041-lemniscate-newton-flow.md\\#res-critical-value-thresholds}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{res:critical-value-thresholds}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1041-lemniscate-newton-flow.md\\#res-critical-value-thresholds}{}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/1041/erdos-1041-lemniscate-newton-flow.tex","section_outline":["Introduction","A component with a narrow bottleneck","A quadratic model","A length estimate for a double cover","The polynomial","Trinomials and radial segments","A small least critical value","The analytic argument","The recorded rational comparison","An isolated simple critical value","Square roots and component area","Further estimates","Root distances do not ensure containment","Sparse families and binomial chords","Collinear roots","Critical-value means","Successive merger scales","Flow constructions and remaining questions","Inverse rays and perturbation","The failed tree estimate","Restricted extremal problems","Related results and verification"]},"erdos1041-lemniscate-reasoning-surface":{"paper_id":"erdos1041-lemniscate-reasoning-surface","title":"Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections","form":"Reasoning surface","length_class":"long","pages":86,"subject":"Erdős #1041","subject_kind":"problem","problem_id":"erdos_1041","landing_role":"mathematics","question":"How does ani's degree-seven counterexample work, which polynomial families admit short paths, and which critical-value constructions remain unformalised?","precis":"The record gives the rational estimates and geometric proof for ani's degree-seven polynomial: every connected set in its strict unit lemniscate joining two roots has Hausdorff measure greater than two. It develops contained paths for trinomials, collinear roots and two sparse families. Ordinary analytic arguments for critical-value criteria retain their unformalised inputs beside the statements. Critical-value moments, counterexamples to prescribed paths and questions about flow-strip geometry complete the account. The one-polynomial formal statement and correspondence with the historical wording remain distinct.","owns":"the exact geometry of ani's one-polynomial counterexample, proved positive families and critical-value arguments with their unformalised inputs identified; formal-statement status and historical correspondence remain owned by the claims registry","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or the current status of Erdős #1041, including the later one-polynomial Formal Conjectures path-image Hausdorff-length check and its unreviewed historical curve-length correspondence; those distinctions belong to docs/claims.json in the mathematics repository","first_pass":["sec:low-critical-closure","sec:solved-polynomial-families","sec:gap","sec:erdos-1041-complete-family-map"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos1041-lemniscate-reasoning-surface.pdf","hosted_pdf_path":"papers/erdos1041-lemniscate-reasoning-surface.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex","site_html":"maths/papers/erdos1041-lemniscate-reasoning-surface.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1041/erdos1041-lemniscate-reasoning-surface.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1041/erdos1041-lemniscate-reasoning-surface.tex","body_pointer":"papers.records.erdos1041-lemniscate-reasoning-surface.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Long-form reasoning record seeded from the complete problem note.\n% The short note and this record are now independent publication units.\n% This flat manuscript is generated from reviewable parts below.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n% Number remarks locally so their retained labels have distinct destinations.\n% The shared preamble defines an unnumbered remark environment.\n\\let\\remark\\relax\n\\let\\endremark\\relax\n\\newtheorem{remark}[theorem]{Remark}\n\n\\input{evidence/erdos1041-lemniscate-reasoning-surface}\n\\renewcommand{\\commit}{7380b7871687b6bcc41ca0143c61f232e8af6500}\n\\renewcommand{\\commitshort}{7380b7871687}\n\\renewcommand{\\PX}{lean/ErdosProblems}\n\\renewcommand{\\repobase}{https://github.com/wcook04/plectis-erdos/blob/\\commit}\n\\renewcommand{\\externalreviewobject}[3]{%\n  \\begin{quote}\\small\n  \\textbf{Result.} #1\\par\n  \\textbf{Scope.} #2\\par\n  \\textbf{Formal statement.} #3  The repository's\n  external-verification job compares these exact Lean propositions with\n  separately declared challenge statements and an axiom budget, then asks\n  Lean's kernel to check the submitted proofs.  The\n  \\href{https://github.com/wcook04/plectis-erdos/blob/ca0e13f8acf5ccf48506e4bdb870953d3a0856fa/formalization.yaml}%\n  {formalisation manifest} and the commit-bound CI receipt record that check;\n  they do not assess novelty, significance, or whether the original problem is\n  solved.\n  \\end{quote}}\n\\hypersetup{\n  pdftitle={Paths in polynomial lemniscates: a degree-seven counterexample and radial connections, proofs and examples for Erdős Problem 1041},\n  pdfsubject={Erdős Problem Notes: Problem 1041},\n  pdfkeywords={lemniscate, Newton flow, saddle connection, Morse theory, Reeb decomposition, Lean 4},\n}\n\n% The source appendix retains the original commit-pinned links.\n\n\\newcommand{\\dbar}{\\mathbb{D}}\n\n\\runninghead{Paths in polynomial lemniscates: proofs and examples}\n\\title{Paths in Polynomial Lemniscates:\\\\A Degree-Seven Counterexample and Radial Connections}\n\\subtitle{Proofs and examples for Erd\\H{o}s Problem~\\#1041, with a Lean proof for the polynomial constructed by \\texttt{ani}}\n\\author{Will Cook}\n\\date{29 September 2026}\n\n\\begin{document}\n% ---- part core ----\n% Part: core\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\nWe give the rational estimates and geometric proof for the degree-seven\ncounterexample to Erd\\H{o}s Problem~1041 constructed by \\texttt{ani}.\nA double-cover argument gives a Hausdorff-measure lower bound for every\nconnected set joining the two roots in its unique two-root component. For trinomials,\ncollinear roots and two sparse families we describe contained paths.\nWe also retain ordinary analytic arguments for critical-value criteria,\nwith their unformalised inputs identified beside the statements.\nCritical-value moments, counterexamples to prescribed paths and\nquestions about flow-strip geometry complete the record.\n\\end{abstract}\n\nThe \\href{erdos-1041-lemniscate-newton-flow.pdf}{short paper} develops the\ncounterexample and the trinomial argument. Here\nSection~\\ref{sec:counterexample-proof} supplies the exact tests for the\npolynomial, while Sections~\\ref{sec:degree-three} and\n\\ref{sec:solved-polynomial-families} give further explicit paths.\nThe area argument in Section~\\ref{sec:low-critical-closure} and the\nisolated-value construction in Section~\\ref{sec:critical-value-separation}\nare ordinary analytic proofs, still requiring formalisation. Their\nconclusions appear as remarks. The local notes identify the assumptions\nof the existing Lean deductions; Section~\\ref{sec:verification-notes}\ncollects the formal and computational references.\n\n\\section{The question and the degree-seven counterexample}\\label{sec:problem}\n\nFor a monic polynomial $f$, let $\\Omega_f=\\{|f|<1\\}$; the notation\n$E_f$ used in the segment arguments denotes the same open set. We write\n$K_t(f)=\\{|f|\\le t\\}$ for a closed sublevel. Erd\\H{o}s, Herzog and Piranian \\cite[Problem~5, p.~139]{ehp1958} asked:\n\n\\begin{problem}[Erd\\H{o}s \\#1041]\\label{res:problem}\nLet $f(z)=\\prod_{i=1}^{n}(z-z_i)$ be monic of degree $n\\ge2$, with\n$z_i\\in\\dbar$, the open unit disc.  Must two of the roots be joined by a curve of length less than\n$2$ lying in the open lemniscate $\\{z\\in\\mathbb{C}:|f(z)|<1\\}$?\n\\end{problem}\n\nBoundary roots already give an elementary failure of strict containment.\nFor $f(z)=z^2-1$, every continuous path from $-1$ to $1$ meets the\nimaginary axis, where $|f(iy)|=1+y^2\\ge1$. The two endpoints are therefore\nunjoinable in $\\{|f|<1\\}$, although the diameter lies in $\\{|f|\\le1\\}$.\nThis illustrates the distinction between the open and closed lemniscates.\n\nA repeated zero gives a constant path between two occurrences. Thus the\nquestion concerns distinct zeros. The catalogue of Bloom \\cite{bloom}\nrecords the problem and its discussion. Pendyala\n\\cite[Theorem~1 and Lemma~1]{june2026} proved the degree-four case by\nchords or radial segments through the centre of a smallest enclosing disc.\nThe degree-seven construction of\n\\href{https://www.erdosproblems.com/forum/thread/1041\\#post-8861}{\\texttt{ani}}\nhas a different geometry.\n\n\\begin{theorem}[\\texttt{ani}'s counterexample]\\label{res:ani-degree-seven-counterexample-long}\nThe monic polynomial in \\eqref{eq:ani-f-long} has seven distinct roots\nin the open unit disc. Every connected subset of its strict unit\nlemniscate containing two roots has one-dimensional Hausdorff measure\ngreater than $2$.\n\\end{theorem}\n\nFive components contain one zero each; the remaining component\ncontains two, and the polynomial has degree two on it. A quadratic\nestimate near its critical point counts two inverse images, with\nmultiplicity, for each value on an outward slit. Since the component has degree two, these\nare all its inverse images of that value. Hence the entire preimage\nof the slit in the component lies in the small disc where the local\nestimate applies.\n\nOn each of the two inverse sheets, radial projection gives a lower\nbound for the Hausdorff measure of a connected set joining the zeros.\nThe sheets are disjoint, so the two bounds can be added. Their sum\nexceeds $2$.\nSection~\\ref{sec:counterexample-proof} supplies the estimates and proof.\nApplied to a continuous path image, this also excludes a rectifiable\npath of total variation less than $2$.\n\nThe theorem concerns a fixed polynomial. The small-parameter family\nreported by \\texttt{ani} is not proved here, and independent human review\nof correspondence with the 1958 wording is not recorded. The supplied\nformal statements and their scope are listed in\nSection~\\ref{sec:verification-notes}.\n\nFor trinomials the root identity keeps every root-to-origin segment\ninside the lemniscate, giving a short broken line between any pair of\nzeros. The later analytic remarks seek a pair by following sublevel\ncomponents: one argument uses a small least critical-value modulus,\nand another uses an isolated simple critical value. Their conclusions\nhave additional formalisation requirements. A bound on critical-value\nmeans alone makes no choice of a contained path.\n\n\\section{The degree-seven construction}\\label{sec:counterexample-proof}\n\\papersectiontarget{counterexample-proof}\n\nThe construction has two separate quantitative requirements. The two\nroots in the selected component must have combined distance from its\ncritical point greater than $2$, and the preimage of an outward value\nslit must be small enough to preserve that surplus. A narrow neck alone\nwould not suffice: for $p(z)=z^2-r^2$, $0<r<1$, the slit\n$(-1,-r^2]$ has preimage $\\{iy:|y|<\\sqrt{1-r^2}\\}$, while the root\nsegment $[-r,r]$ still has length $2r<2$ and lies in $\\{|p|<1\\}$.\n\nWe first prove the estimate that separates these requirements. We then\nverify root locations, critical values and the membership of the chosen\npair in one component. The final fixed-parameter estimates compare the root-distance\nsurplus with the neck loss, and distinguish both from the radial\ncontraction $1-\\rho=\\varepsilon^8$.\n\n\\subsection{A two-sheeted component}\\label{subsec:two-sheet-bottleneck}\n\\papersectiontarget{two-sheet-bottleneck}\n\nThe quantity $\\delta=1-|p(c)|$ measures the remaining length of the\nslit in the value plane. The local quadratic coefficient converts it\ninto the spatial scale $\\sqrt{\\delta/M}$. The radius $h$ records where\nthat quadratic comparison is valid. Degree two will turn the local\npreimage estimate into a statement about the whole component.\n\n\\begin{lemma}[a bottleneck estimate]\\label{lem:two-sheet-bottleneck-long}\nLet $p$ be a polynomial and $U$ a component of $\\{|p|<1\\}$ on which $p$\nis a proper map of degree two. Suppose its only critical point $c$ is\nsimple and $v=p(c)\\ne0$.\nWrite\n\\[\n p(c+z)-v=z^2 A(z),\\qquad M=|A(0)|,\\qquad \\delta=1-|v|.\n\\]\nSuppose $h>0$, $|A(z)/A(0)-1|\\le1/4$ for $|z|\\le h$, and\n$\\delta<Mh^2/4$. If $a,b$ are the two zeros in $U$, every connected\n$K\\subset U$ containing $a,b$ satisfies\n\\[\n \\mathcal H^1(K)\\ge |a-c|+|b-c|-\\frac83\\sqrt{\\delta/M}.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nCut the value disc along $J=\\{tv:1\\le t<1/|v|\\}$. Its complement is simply\nconnected and contains $0$. The restriction of $p$ above this complement is\nan unramified two-sheeted covering, hence is the disjoint union of two\nbiholomorphic sheets $U_1,U_2$, containing $a,b$ respectively.\n\nPut $r_0=(4/3)\\sqrt{\\delta/M}<2h/3$. On $|z|=r_0$,\n\\[\n |p(c+z)-v|\\ge\\tfrac34Mr_0^2=\\tfrac43\\delta>\\delta.\n\\]\nFor each $w\\in J$, Rouch\\'e's theorem therefore gives two zeros, counted\nwith multiplicity, of $p(c+z)-w$ in $|z|<r_0$. They remain in $U$: as $w$\nmoves out from $v$ along $J$, the zeros continue from $c$ and cannot cross\n$|z|=r_0$. Since the degree on $U$ is two, these are all its preimages of\n$w$. Consequently $p^{-1}(J)\\cap U\\subset B(c,r_0)$.\n\nFor $r_0<t<|a-c|$, the set $U_1\\cap\\{|z-c|>t\\}$ has relative boundary\nin $U$ contained in the circle $|z-c|=t$. If that circle missed $K\\cap U_1$,\nthen $K\\cap U_1\\cap\\{|z-c|>t\\}$ would be both open and closed in $K$.\nIt contains $a$ and excludes $b$, contradicting connectedness. Thus the radial map $z\\mapsto|z-c|$ sends\n$K\\cap U_1$ onto a set containing $(r_0,|a-c|)$ whenever that interval is\nnonempty. This map is $1$-Lipschitz, so\n\\[\n \\mathcal H^1(K\\cap U_1)\\ge (|a-c|-r_0)_+.\n\\]\nThe same argument applies to $b$ and $U_2$. Although the two radial\nintervals can overlap on the real line, the sets $K\\cap U_1$ and\n$K\\cap U_2$ lie in disjoint open subsets of the plane. Hausdorff outer\nmeasure adds across that separation, even when $K$ is not assumed\nmeasurable. Consequently\n\\begin{align*}\n \\mathcal H^1(K)\n &\\ge\\mathcal H^1(K\\cap U_1)+\\mathcal H^1(K\\cap U_2)\\\\\n &\\ge(|a-c|-r_0)_+ + (|b-c|-r_0)_+\\\\\n &\\ge |a-c|+|b-c|-2r_0.\n\\end{align*}\nSubstituting the value of $r_0$ proves the bound. Connectedness supplied\nthe intervals; no path or parametrisation of $K$ was required.\n\\end{proof}\n\nThe \\href{erdos-1041-lemniscate-newton-flow.pdf\\#bottleneck-geometry}\n{diagram in the short paper} distinguishes the value-plane slit from\nthe two radial projections used above.\n\n\\subsection{Polynomial and roots}\\label{subsec:counterexample-polynomial}\n\\papersectiontarget{counterexample-certificate}\n\nThe perturbation of $z^7-1$ is small on the unit circle but has a\nnontrivial shape at $z=\\varepsilon w$: the terms\n$\\varepsilon^4az^3$, $\\varepsilon^5bz^2$ and\n$\\varepsilon^6c_0z$ all become order $\\varepsilon^7$ there.\nThe paired conjugate terms give the antisymmetry used to locate the\nroots. These two features let us control roots near the unit circle\nand critical points near the origin by different tests.\n\nAll constants are rational, so both sets of tests can be made without\nrounding. Set\n\\[\n s=10^{-6},\\qquad \\varepsilon=s^2=10^{-12},\\qquad\n \\rho=1-s^{16}=1-10^{-96},\n\\]\n\\[\n A_0=-\\frac{329507}{1600},\\qquad B_0=\\frac{551827}{800},\\qquad\n C_0=-\\frac{23013813}{32000},\n\\]\nand define\n\\[\n a=A_0-is,\\qquad b=iB_0+\\frac95s,\\qquad\n c_0=-C_0-\\frac{162}{25}is.\n\\]\nThe polynomial is\n\\begin{align}\n F(z)&=z^7-1+\\varepsilon^4(az^3-\\bar a z^4)\n       +\\varepsilon^5(bz^2-\\bar b z^5)\n       +\\varepsilon^6(c_0z-\\bar c_0z^6),\\label{eq:ani-F-long}\\\\\n f(z)&=\\rho^7 F(z/\\rho).\\label{eq:ani-f-long}\n\\end{align}\nThe identity $F(z)=-z^7\\overline{F(1/\\bar z)}$ is useful, but does not by\nitself place the zeros on the unit circle. For that purpose put\n$\\chi(x)=(1+ix)/(1-ix)$. Direct expansion gives\n$(1-ix)^7 F(\\chi(x))=2iH(x)$, where\n\\begin{align*}\n H(x)={}&7x-35x^3+21x^5-x^7\n -\\varepsilon^4(1+x^2)^3(s+A_0x)\\\\\n &+\\varepsilon^5(1+x^2)^2\n       \\bigl[B_0(1-3x^2)+(9s/5)(x^3-3x)\\bigr]\\\\\n &+\\varepsilon^6(1+x^2)\n       \\bigl[C_0(5x-10x^3+x^5)\n          -(162s/25)(1-10x^2+5x^4)\\bigr].\n\\end{align*}\nThe signs at the following rational endpoints are obtained by substitution.\nBoth endpoints in each row are divided by $10^4$.\n\\begin{center}\n\\begin{tabular}{@{}rrrrr@{}}\n\\toprule\n$j$ & Left endpoint & Right endpoint & Left sign & Right sign\\\\\n\\midrule\n0&$-1$&1&$-$&$+$\\\\\n1&4815&4816&$+$&$-$\\\\\n2&12539&12540&$-$&$+$\\\\\n3&43812&43813&$+$&$-$\\\\\n4&$-43813$&$-43812$&$+$&$-$\\\\\n5&$-12540$&$-12539$&$-$&$+$\\\\\n6&$-4816$&$-4815$&$+$&$-$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\nThe seven intervals give distinct real zeros $x_j$ of $H$, so\n$\\zeta_j=\\chi(x_j)$ are all seven zeros of $F$. They have modulus one.\nHence the zeros $b_j=\\rho\\zeta_j$ of $f$ are distinct and lie in the open\nunit disc.\n\n\\subsection{Critical points and local estimates}\\label{subsec:counterexample-critical}\n\nThe critical points cluster at scale $\\varepsilon$. We resolve them by\nwriting $F(\\varepsilon w)=-1+\\varepsilon^7 Q(w)$, where\n\\[\n Q(w)=c_0w+bw^2+aw^3-\\varepsilon\\bar a w^4\n       -\\varepsilon^3\\bar b w^5-\\varepsilon^5\\bar c_0 w^6+w^7.\n\\]\nFor a centre $v_j$, let $T_k(v_j)$ be the coefficient of $z^k$ in\n$Q(v_j+z)$. If $Q(w)=\\sum q_kw^k$, these coefficients are given by the\nfinite formula\n\\[\n T_k(v_j)=\\sum_{\\ell=k}^7{\\ell\\choose k}q_\\ell v_j^{\\ell-k}.\n\\]\nUse the following centres, whose displayed decimals are exact, and radius\n$r=10^{-6}$:\n\\begin{center}\\small\n\\begin{tabular}{@{}rrr@{}}\n\\toprule\n$j$&$\\operatorname{Re}v_j$&$\\operatorname{Im}v_j$\\\\\n\\midrule\n0&$3.1898275786$&$-0.4999999979$\\\\\n1&$-3.1898275845$&$-0.5000000021$\\\\\n2&$0.0000000039$&$0.8232474662$\\\\\n3&$0.0000002274$&$1.8078590858$\\\\\n4&$-0.0000002265$&$1.8838591455$\\\\\n5&$0.0000000011$&$-3.5149656976$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\nThese six discs are disjoint. For a Gaussian rational $z=u+iv$, put\n$N(z)=|u|+|v|$ and $m(z)=\\max(|u|,|v|)$, so\n$m(z)\\le|z|\\le N(z)$. Define\n\\[\n E_j=N(T_1(v_j))+\\sum_{k=2}^{6}(k+1)N(T_{k+1}(v_j))r^k,\n \\qquad L_j=2m(T_2(v_j))r.\n\\]\nThe next table proves $E_j<L_j$ in every disc. Rouch\\'e's theorem, applied\nto the linear term of $Q'$, gives exactly one critical point in each.\n\nTo determine its critical-value modulus, use\n\\[\n \\mathscr H(w)=\\frac{1-\\rho^{14}}{2\\rho^{14}\\varepsilon^7}\n       +\\operatorname{Re}Q(w)-\\frac{\\varepsilon^7}{2}|Q(w)|^2,\n \\qquad |f(\\rho\\varepsilon w)|^2\n       =1-2\\rho^{14}\\varepsilon^7\\mathscr H(w).\n\\]\nFor $|w-v_j|\\le r$, set\n$V_j=\\sum_{k=1}^7N(T_k(v_j))r^k$. Substituting\n$\\operatorname{Re}T_0(v_j)\\pm V_j$ for $\\operatorname{Re}Q(w)$ and\n$N(T_0(v_j))+V_j$ for $|Q(w)|$ gives the following rational bounds.\nAn interval in the last column contains $10^9\\mathscr H(w)$ throughout\nthe whole disc.\n\\begin{center}\n\\begin{tabular}{@{}rrrr@{}}\n\\toprule\n$j$&$10^9 E_j<$&$10^6 L_j>$&$10^9\\mathscr H(w)$\\\\\n\\midrule\n0&423&8340&$[-502,-486]$\\\\\n1&433&8340&$[-502,-486]$\\\\\n2&12&378&$[3556,3558]$\\\\\n3&3&43&$[-75,-74]$\\\\\n4&4&48&$[-864,-863]$\\\\\n5&1298&16812&$[-1566,-1548]$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\nIt follows that $f$ has just one critical point in $\\{|f|<1\\}$, namely\n$c_s=\\rho\\varepsilon w_s$ with $|w_s-v_2|<r$. Its critical value is\nnonzero, since the polynomial has simple roots, and no critical point\nlies on $|f|=1$.\n\nExpanding the same Taylor coefficients throughout $|w-v_2|\\le r$ gives\n\\[\n |T_2(w)|>189.1,\\qquad\n (|T_3(w)|,|T_4(w)|,|T_5(w)|,|T_6(w)|,|T_7(w)|)\n <(190,20,15,6,2).\n\\]\nIn the last coordinate the exact value is $|T_7|=1$. In particular, for\n$h=\\rho\\varepsilon/10$ and\n$f(c_s+z)-f(c_s)=z^2A_s(z)$,\n\\begin{equation}\\label{eq:counterexample-local-bounds}\n |A_s(0)|>180\\rho^5\\varepsilon^5,\n \\qquad\n \\left|\\frac{A_s(z)}{A_s(0)}-1\\right|<\\frac14\\quad(|z|\\le h).\n\\end{equation}\nIndeed the numerator of the relative error is at most\n$190/10+20/100+15/1000+6/10^4+1/10^5<189.1/4$ in scaled\ncoordinates. The positive critical-value bound in the table also gives\n\\[\n 0<\\delta=1-|f(c_s)|\n \\le 1-|f(c_s)|^2\n <\\rho^7\\varepsilon^7\\frac{36}{5\\cdot10^6}.\n\\]\nConsequently\n\\begin{equation}\\label{eq:counterexample-radius}\n \\sqrt{\\delta/|A_s(0)|}<\\frac{\\rho\\varepsilon}{5000},\n \\qquad \\delta<|A_s(0)|h^2/4.\n\\end{equation}\n\n\\subsection{The two roots in the component}\\label{subsec:counterexample-connections}\n\nThe local estimates identify a critical point, but we still have to\nfind the two roots in its component. The connections from $c_s$ to\n$b_3,b_6$ use a small disc and four segments near the critical point,\nshort segments to the root directions, and radial segments out to the\nroots. The following ordinary argument identifies the component\ndirectly; the Lean source uses barriers. Put\n\\begin{align*}\n \\operatorname{Re}w_q={}&\\frac{39226583259917038484346953200647}{10^{40}},\\\\\n \\operatorname{Im}w_q={}&\\frac{8232474661944848953672917849472800078083}{10^{40}},\\\\\n x_3={}&w_q-6+3i,\\qquad x_6=w_q+5-5i,\\\\\n m_3={}&\\frac{-720775094321935300989+347106991294046496381i}{10^{20}},\\\\\n m_6={}&\\frac{498791841486986824420-625465185974423846967i}{10^{20}}.\n\\end{align*}\nThe elementary inequality\n$1-\\rho^{14}\\ge14\\rho^{14}\\varepsilon^8$ shows that\n\\[\n \\mathscr H_0(w)=7\\varepsilon+\\operatorname{Re}Q(w)\n                  -\\frac{\\varepsilon^7}{2}|Q(w)|^2\n \\le\\mathscr H(w).\n\\]\nThus $\\mathscr H_0>0$ suffices for containment. To check a segment from\n$u$ to $v$, expand\n$\\mathscr H_0(u+t(v-u))=\\sum_{k=0}^{14}a_kt^k$. Its Bernstein\ncoefficients of degree $14$ are\n\\[\n \\beta_j=\\sum_{k=0}^j a_k\\frac{{j\\choose k}}{{14\\choose k}},\n \\qquad\n \\mathscr H_0(u+t(v-u))\n   =\\sum_{j=0}^{14}\\beta_j{14\\choose j}t^j(1-t)^{14-j}.\n\\]\nAll weights on the right are nonnegative and sum to one for $0\\le t\\le1$.\nExact substitution gives the lower bounds\n\\begin{center}\n\\begin{tabular}{@{}lr@{}}\n\\toprule\nSegment & Lower bound for every $\\beta_j$\\\\\n\\midrule\n$[w_q,x_3]$ & $3556874/10^{12}$\\\\\n$[x_3,m_3]$ & $640506$\\\\\n$[w_q,x_6]$ & $3556874/10^{12}$\\\\\n$[x_6,m_6]$ & $139485$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\nThe formula for $T_k$ above, followed by the displayed formula for\n$\\beta_j$, specifies these checks using rational arithmetic alone; no\nsampling between endpoints is involved. The same Taylor test at $w_q$,\nwith radius $10^{-8}$, gives one zero of $Q'$ in that disc and\n$\\mathscr H_0>3556874/10^{12}$ throughout it. The disc is contained in\n$|w-v_2|<10^{-6}$, so this zero is $w_s$.\nFor each $j=3,6$, we can therefore follow the segments from $w_s$ to\n$w_q$, then to $x_j$ and $m_j$, staying where $\\mathscr H_0>0$.\nIt remains to connect $m_j$ to the corresponding root.\n\nFor the remaining segments no large polynomial expansion is needed.\nThe Cayley intervals give\n$|m_j/8-\\zeta_j|<1/10$ and $|m_j/8|<101/100$, for $j=3,6$.\nIf $x$ lies between $m_j/8$ and $\\zeta_j$, then\n$|x|\\le101/100$ and $|x-\\zeta_j|<1/10$.\nFrom $F(\\zeta_j)=0$ and $|\\zeta_j|=1$ we have\n$|\\zeta_j^7-1|<10^{-40}$. Telescoping $x^7-\\zeta_j^7$ therefore gives\n\\[\n \\operatorname{Re}x^7\n \\ge1-\\frac7{10}\\left(\\frac{101}{100}\\right)^6-10^{-40}>\\frac14.\n\\]\nThe coefficient bounds\n\\[\n (|q_1|,\\ldots,|q_7|)\n \\le(720,690,206,10^{-9},10^{-30},10^{-55},1)=:(U_1,\\ldots,U_7)\n\\]\nimply\n\\[\n \\sum_{k=1}^6U_k(8.08)^k<160000,\\qquad\n \\operatorname{Re}Q(8x)>350000,\\qquad |Q(8x)|<2500000.\n\\]\nHence $\\mathscr H_0(8x)>0$. This connects $m_j$ to $8\\zeta_j$.\nFinally, for $8\\varepsilon\\le t\\le1$, the root equation gives\n\\[\n F(t\\zeta_j)=t^7-1+\n       \\sum_{k=1}^6\\varepsilon^{7-k}q_k\\zeta_j^k(t^k-t^7),\n\\]\nso\n\\[\n |F(t\\zeta_j)|\\le1-t^7+t^7\\sum_{k=1}^6\\frac{U_k}{8^{7-k}}<1,\n \\qquad \\sum_{k=1}^6\\frac{U_k}{8^{7-k}}<\\frac{3}{40}.\n\\]\nScaling by $\\rho$ joins $\\rho\\varepsilon\\,8\\zeta_j$ to $b_j$ within\n$\\{|f|<1\\}$. The concatenated segments connect $c_s$ to both roots.\n\nEvery component of a polynomial lemniscate at a regular level is simply\nconnected. The component-wise Riemann--Hurwitz formula\n\\cite[proof of Proposition~2.1]{eks2010} says that a component containing\n$k$ zeros has $k-1$ critical points, counted with multiplicity. The critical\npoint table therefore shows that the component $U$ of $c_s$ contains\nexactly $b_3,b_6$, and every other component contains one root.\n\n\\subsection{The length inequality}\\label{subsec:counterexample-length}\n\nIt remains to compare the root distances with the loss at the slit.\nFor a unit complex number $\\zeta$,\n$|\\zeta-\\varepsilon w_s|\\ge1-\\varepsilon\\operatorname{Re}(\\bar\\zeta w_s)$.\nThe Cayley intervals and $|w_s-v_2|<10^{-6}$ give\n\\[\n \\operatorname{Re}\\bigl((\\bar\\zeta_3+\\bar\\zeta_6)w_s\\bigr)\n <-\\frac{143}{1000}.\n\\]\nFor completeness, this last bound can be obtained without trigonometric\nestimates: use the midpoint of each rational interval and the derivative\nbound $|\\chi'(x)|=2/(1+x^2)$ to enclose $\\zeta_j$, then substitute the\ndisc for $w_s$. These enclosures in fact give an upper bound below\n$-0.28635$. Thus\n\\[\n |b_3-c_s|+|b_6-c_s|>\\rho\\left(2+\\frac{143}{1000}\\varepsilon\\right).\n\\]\nLemma~\\ref{lem:two-sheet-bottleneck-long} and\n\\eqref{eq:counterexample-radius} now yield\n\\[\n \\mathcal H^1(K)>\n \\rho\\left(2+\\left(\\frac{143}{1000}-\\frac8{15000}\\right)\n                     \\varepsilon\\right)>2\n\\]\nfor every connected subset of $U$ containing both roots. The last strict\ninequality follows by substituting $\\rho=1-10^{-96}$ and\n$\\varepsilon=10^{-12}$; its excess over $2$ exceeds $1.42\\cdot10^{-13}$.\nA connected set containing any two roots must lie in one component, so\nthis proves the assertion for the whole strict lemniscate. In particular\nit applies to the image of every continuous root-to-root path. Since the\nHausdorff measure of a rectifiable path image is at most its total\nvariation, both interpretations of path length are excluded.\n\nThe construction above fixes $s=10^{-6}$. It establishes no interval of\nparameters for the small-parameter family reported by \\texttt{ani}.\nThe formal files also prove the assertion for arbitrary preconnected sets;\nthe argument here uses the equivalent connected-subset formulation and\nordinary Rouch\\'e and covering arguments. No new Lean build or independent\nhuman review is asserted by this exposition.\n\n\\section{Trinomials}\n\\label{sec:trinomials}\n\nAt a zero of a trinomial, the root equation eliminates the middle\ncoefficient. Along the segment from the origin to that zero, the\nremaining expression is a nonnegative linear combination of two values\nof modulus less than one. The sum of the coefficients is at most one,\nwhich gives the required containment.\n\n\\begin{theorem}[trinomial root connections]\n\\label{res:trinomial-all-degree}\nLet $n,m$ be integers with $1\\le m<n$, and let $f(z)=z^n+az^m+b$ have every\nzero in $\\dbar$.  For every zero $\\zeta$, the segment $[0,\\zeta]$ lies in\n$E_f$.  Consequently any two zeros $\\zeta_1,\\zeta_2$ are joined in $E_f$ by\nthe broken line $\\zeta_1\\to0\\to\\zeta_2$, of length $|\\zeta_1|+|\\zeta_2|<2$.\n\\end{theorem}\n\n\\begin{proof}\nVieta's formula gives $|b|<1$.  At a zero $\\zeta$ the root equation\n$\\zeta^n+a\\zeta^m+b=0$ eliminates the middle coefficient:\n\\begin{equation}\\label{eq:trinomial-cancellation}\n f(t\\zeta)=b(1-t^m)+\\zeta^n(t^n-t^m).\n\\end{equation}\nFor $0\\le t<1$ the two scalar weights $1-t^m$ and $t^m-t^n$ are nonnegative\nand sum to $1-t^n$, so\n\\[\n |f(t\\zeta)|\\le|b|(1-t^m)+|\\zeta|^n(t^m-t^n)<1-t^n\\le1 .\n\\]\nAt $t=1$ the value is zero.  Concatenating two such segments gives the length\nassertion.\n\\end{proof}\n\nNo separate bound on $a$ is needed, since the root equation eliminates\nit before we take absolute values.  The conclusion concerns segments to zeros and\nmakes no assertion that $E_f$ is star-shaped.\n\n\\paragraph{Summation by parts.}\n\\label{subsec:abel-mechanism}\nFor $f(z)=\\sum_{k=0}^nc_kz^k$ and a zero $\\zeta$, put\n$S_j=\\sum_{k=0}^jc_k\\zeta^k$.  Finite summation by parts gives\n\\begin{equation}\\label{eq:abel-control-polygon}\n f(t\\zeta)=\\sum_{j=0}^{n-1}(t^j-t^{j+1})S_j\\qquad(0\\le t\\le1),\n\\end{equation}\nsince the coefficient of $c_k\\zeta^k$ on the right is $t^k-t^n$ and\n$\\sum_{k<n}c_k\\zeta^k=-c_n\\zeta^n$.  The weights\nin~\\eqref{eq:abel-control-polygon} are nonnegative and sum to $1-t^n$, so the\nwhole radial segment lies in the closed unit sublevel set whenever every partial sum lies in the closed\nunit disc. For a trinomial, $S_j=b$ for $j<m$ and $S_j=-\\zeta^n$\nfor $m\\le j<n$; these two values may coincide. This is exactly the\nestimate above. With two intermediate coefficients, the assumption\nthat all zeros lie in the unit disc need not bound every partial sum\nby one. The next\nexample shows that a prescribed radial segment can then escape.\n\n\\begin{example}[an escaping root spoke]\\label{ex:sextic-spoke}\nLet $0<r<1$ satisfy $r^6>320/327$, and set\n\\[\n f_r(z)=z^6+\\tfrac15r^2z^4-\\tfrac15r^4z^2-r^6\n       =(z^2-r^2)\\bigl(z^4+\\tfrac65r^2z^2+r^4\\bigr).\n\\]\nEvery zero has modulus $r$: for the quartic factor put $z=rw$, so that the\nsquared roots solve $v^2+(6/5)v+1=0$, whose two roots are complex conjugates\nof product one.  Nevertheless\n\\[\n f_r(r/2)=-\\frac{327}{320}\\,r^6,\n\\]\nso the segment $[0,r]$ leaves $E_{f_r}$.  This rules out a universal assertion\nabout every origin-to-zero segment.  It exhibits no counterexample to the\nexistence of some short connection.\n\\end{example}\n\n\\paragraph{Sources and scope.}\n\\label{bdry:trinomial}\nLean checks the\n\\href{https://github.com/wcook04/plectis-erdos/blob/0ba585f632fbbbb43af2ee9532d4e83752af3f67/ErdosProblems/Erdos1041/AbelControlPolygon.lean\\#L123}{identity}~\\eqref{eq:abel-control-polygon}, the\n\\href{https://github.com/wcook04/plectis-erdos/blob/0ba585f632fbbbb43af2ee9532d4e83752af3f67/ErdosProblems/Erdos1041/AbelControlPolygon.lean\\#L258}{constant-term bound},\nthe\n\\href{https://github.com/wcook04/plectis-erdos/blob/0ba585f632fbbbb43af2ee9532d4e83752af3f67/ErdosProblems/Erdos1041/AbelControlPolygon.lean\\#L219}{radial estimate},\n\\href{https://github.com/wcook04/plectis-erdos/blob/0ba585f632fbbbb43af2ee9532d4e83752af3f67/ErdosProblems/Erdos1041/AbelControlPolygon.lean\\#L330}{their combination with the length bound}\nand the\n\\href{https://github.com/wcook04/plectis-erdos/blob/0ba585f632fbbbb43af2ee9532d4e83752af3f67/ErdosProblems/Erdos1041/AbelControlPolygon.lean\\#L555}{evaluation} in Example~\\ref{ex:sextic-spoke}.  No novelty claim is made for this elementary trinomial argument.\n\n\\section{A small least critical value}\n\\label{sec:low-critical-closure}\n\\papersectiontarget{low-critical-proof}\n\nWe follow a component from its first merger as the level of $|f|$\nincreases. Under the assumption that it contains no path shorter than\n$2$ between distinct roots, conformal length bounds force the roots\napart. Packing then forces area growth beyond P\\'olya's upper bound.\nThe reduction to this comparison is an ordinary analytic argument;\nthe exact computation at its end does not formalise the reduction.\nWe retain the resulting criterion as an unformalised remark.\nThroughout this section\n\\[\n \\mu=\\min_{f'(c)=0}|f(c)|\n\\]\nis the least critical-value modulus.\n\n\\begin{remark}[unformalised small-critical-value criterion]\n\\label{res:low-critical-thirteen-twentyfifths}\nLet $f$ be squarefree and monic of degree $n\\ge2$ with $\\mu\\le13/25$.  Then\ntwo distinct roots of $f$ are joined inside $\\{|f|<1\\}$ by a rectifiable\ncurve of length strictly below $2$.  No hypothesis is placed on the locations\nof the roots, on the number of roots in any component, or on the capacity of\nany component.\n\\end{remark}\n\\evidenceremark{No Lean proof in the supplied sources establishes this whole assertion.\nThe analytic reduction and the rational comparison below remain separate\nunformalised inputs; checking the final exponential inequality alone\ndoes not check that reduction.}\n\n\nFor $f(z)=z^n-b$ with $0<|b|<1$, the only critical point is $0$ and\n$\\mu=|b|$. Thus this criterion includes $|b|\\le13/25$ but excludes\n$13/25<|b|<1$, although the trinomial theorem gives the required path\nin both ranges. Its value is that it also applies to polynomials with\narbitrary coefficient patterns, including roots outside the unit disc. For example,\n$(z-3)^n-1/2$, $n\\ge2$, has $\\mu=1/2$ and all its roots satisfy\n$|z|>2$. The connection is required to stay in $|f(z)|<1$; the example places\nno restriction on its distance from the origin.\n\n\\begin{remark}[unformalised scale-free consequence]\n\\label{res:low-critical-scale-free}\nEvery squarefree monic $f$ of degree $n\\ge2$ has two distinct roots joined\ninside $\\{|f|<(25/13)\\mu\\}$ by a curve of length below\n$2\\bigl((25/13)\\mu\\bigr)^{1/n}$.\n\\end{remark}\n\\evidenceremark{Lean proves this rescaling under the input\n\\texttt{LowCriticalThirteenTwentyFifths}, namely\nRemark~\\ref{res:low-critical-thirteen-twentyfifths}. That input is the\nunformalised path criterion itself. It has no named published theorem\nas its supplied justification.}\n\n\n\\begin{proof}\nApply Remark~\\ref{res:low-critical-thirteen-twentyfifths} to\n$s^{-n}f(sz)$ with $s=\\bigl((25/13)\\mu\\bigr)^{1/n}$, whose least\ncritical-value modulus is $13/25$, and scale back. Squarefreeness gives\n$\\mu>0$, so the scaling factor is positive.\n\\end{proof}\n\nThe proof has two steps. Failure of the length bound first forces the\nroots to be far apart in the hyperbolic metric of a sublevel component.\nPacking those roots gives a lower bound for their number. Averaging paths\nbetween adjacent boundary points then converts that root count into an\narea-growth inequality. The rational certificate shows that this growth\nwould exceed P\\'olya's area bound before the value level reaches $1$.\n\n\\paragraph{Consequences of having no short path.}\nAssume no two distinct roots are joined inside $\\{|f|<1\\}$ by a curve of\nlength below $2$.  Choose a critical point at level $\\mu$ and two descending\ninverse arcs into distinct one-root components of $\\{|f|<\\mu\\}$; below $\\mu$\neach component maps conformally onto the value disc, and distinct local\ninverse arcs at the first critical point enter distinct components, since two\ninverse images of a nearby regular value in one component would contradict its\ndegree one.  Their union is a compact connected set containing two roots.\nFix a level $t\\in(\\mu,1)$ at which no critical value has modulus $t$,\nso that the level is regular. Let $C_t$ be the component of\n$\\{|f|<t\\}$ containing that set, let $k\\ge2$ be its root count, and put\n$x=\\log(t/\\mu)$ and\n$a=\\operatorname{Area}(C_t)/\\pi$; P\\'olya's inequality \\cite[Theorem~1]{crane}\ngives $a\\le t^{2/n}<1$.\nThe component is a Jordan domain. List its roots as\n$\\zeta_1,\\ldots,\\zeta_k$, choose a Riemann map\n$\\varphi:\\mathbb D\\to C_t$, and write $b_j=\\varphi^{-1}(\\zeta_j)$.\nThe component-wise form of \\cite[(2.2)]{eks2010} is\n\\[\n \\frac{f(\\varphi(z))}{t}\n =e^{i\\theta}\\prod_{j=1}^k\\frac{z-b_j}{1-\\overline{b_j}z}.\n\\]\nIndeed, regularity extends $\\varphi$ across the Jordan boundary. The\nquotient of the two sides without the phase has no zero or pole on the\nclosed disc and has boundary modulus one. The maximum principle,\napplied to that quotient and its reciprocal, makes it a unimodular\nconstant. The product has degree $k$, the root count of this component,\nnot necessarily the full degree $n$. Distances between the $b_j$ below are\nhyperbolic distances in $\\mathbb D$, normalised by\n$d(0,s)=2\\operatorname{artanh}s$ for $0\\le s<1$. Intrinsic distances\nin $C_t$ are infima of Euclidean lengths of curves in that component.\n\nThe length estimate converts area into a bound for the image of a compact real\ninterval $I\\subset(-1,1)$. Here is the coefficient calculation underlying\nthe Bergman-kernel argument. Write\n$\\varphi'(z)=\\sum_{\\ell\\ge0}\\alpha_\\ell z^\\ell$. Injectivity and the\narea formula give\n$\\sum_{\\ell\\ge0}|\\alpha_\\ell|^2/(\\ell+1)=a$.\nTo express length as a linear integral, choose\n$\\omega(x)=\\overline{\\varphi'(x)}/|\\varphi'(x)|$ on $I$; the denominator\nis nonzero because $\\varphi$ is conformal. Cauchy--Schwarz gives\n\\[\n \\operatorname{length}(\\varphi(I))^2\n =\\left|\\sum_{\\ell\\ge0}\\alpha_\\ell\\int_I\\omega(x)x^\\ell\\,dx\\right|^2\n \\le a\\sum_{\\ell\\ge0}(\\ell+1)\n       \\left|\\int_I\\omega(x)x^\\ell\\,dx\\right|^2.\n\\]\nCompactness of $I$ permits termwise integration of the series.\nSince $\\sum_{\\ell\\ge0}(\\ell+1)(xy)^\\ell=(1-xy)^{-2}>0$ for real\n$x,y\\in I$, taking absolute values in the resulting double integral yields\n\\[\n \\operatorname{length}(\\varphi(I))^2\n \\le a\\int_I\\!\\int_I\\frac{dx\\,dy}{(1-xy)^2}.\n\\]\nThis is precisely the positivity of the disc Bergman kernel\n$K(z,w)=\\pi^{-1}(1-z\\overline w)^{-2}$ on the real diameter.\nFor $I=[u,v]\\subset(-1,1)$, direct integration gives\n\\[\n \\int_u^v\\!\\int_u^v\\frac{dx\\,dy}{(1-xy)^2}\n =\\log\\frac{(1-uv)^2}{(1-u^2)(1-v^2)}.\n\\]\nThus $I=[0,r]$ gives\n$\\operatorname{length}(\\varphi([0,r]))^2\\le a\\log(1/(1-r^2))$.\nMoving two preimages to $\\pm s$ and taking $I=[-s,s]$ instead gives\n\\[\n \\operatorname{length}(\\varphi([-s,s]))^2\\le4a\\operatorname{artanh}(s^2).\n\\]\nA pair of roots at hyperbolic distance $d$ may be placed at $\\pm\\tanh(d/4)$.\nTheir connecting curve then has length less than $2$ whenever\n$\\operatorname{artanh}(\\tanh^2(d/4))<1/a$. Under our assumption that no\nsuch curve exists, we therefore have\n\\begin{equation}\\label{eq:lc-separation}\n d(b_i,b_j)\\ \\ge\\ D:=4\\operatorname{artanh}\\sqrt{\\tanh(1/a)},\n \\qquad \\cosh(D/2)=e^{2/a}.\n\\end{equation}\n\nThere is also a point $h$ in the chosen compact connected set whose\nintrinsic distance in $C_t$ from every root is at least $1$.\nOtherwise the open intrinsic balls of radius $1$ about the roots would\ncover that set. Two balls with distinct centres cannot meet: paths from\ntheir centres to a common point would concatenate to length below $2$.\nBut a connected set containing two distinct roots cannot be covered by\nthese disjoint open sets. This contradiction proves the assertion.\nNow choose the Riemann map with $\\varphi(0)=h$, and continue to write\n$b_j=\\varphi^{-1}(\\zeta_j)$. Put $d_j=d(0,b_j)$ and\n$\\lambda(d)=-\\log\\tanh(d/2)$.  The one-root estimate at $h$\ngives $\\tanh(d_j/2)\\ge\\sqrt{1-e^{-1/a}}$. The centre $h$ lies on the\nchosen inverse arcs, so $0<|f(h)|\\le\\mu$; it is not a root because its\nintrinsic distance from every root is at least $1$. The Blaschke identity\nat $0$ gives $|f(h)|/t=\\prod_j|b_j|$, hence\n$\\sum_j\\lambda(d_j)=\\log(t/|f(h)|)\\ge\\log(t/\\mu)=x$.\nTogether these estimates give\n\\begin{equation}\\label{eq:lc-radius-and-budget}\n \\lambda(d_j)\\le\\frac{\\delta(a)}2,\\quad\n \\delta(a)=-\\log\\bigl(1-e^{-1/a}\\bigr),\n \\qquad\n \\sum_{j=1}^{k}\\lambda(d_j)\\ \\ge\\ x .\n\\end{equation}\n\n\\paragraph{A lower bound for the number of roots.}\nEquation~\\eqref{eq:lc-radius-and-budget} already gives\n$k\\ge2x/\\delta(a)$: the required sum is at least $x$, and each root\ncontributes at most $\\delta(a)/2$. Improving this estimate requires the\npairwise separation, not just the distance of each root from the origin.\nTesting only consecutive pairs in angular order loses essential information.\nFor $k=2m$, put $m$ points at hyperbolic radius $d_{\\mathrm{low}}$ and\n$m$ at radius $d_{\\mathrm{low}}+D$, alternating in cyclic order. The\nreverse triangle inequality separates every consecutive pair by at least\n$D$, regardless of the angular gaps, whereas\n\\[\n \\sum_j\\lambda(d_j)\n =m\\bigl(\\lambda(d_{\\mathrm{low}})+\\lambda(d_{\\mathrm{low}}+D)\\bigr)\n \\longrightarrow\\infty\\qquad(m\\to\\infty).\n\\]\nThe even number of points is needed for cyclic alternation. These point\nconfigurations test only the consecutive-pair constraints; they need not\nsatisfy separation for nonconsecutive pairs or arise from a polynomial\ncomponent. To use the missing pairwise information, we impose the packing\nbound on every circle centred at the origin.\n\n\\begin{lemma}[circle-slice packing]\\label{res:circle-slice-packing}\nUnder~\\eqref{eq:lc-separation}, for every $r>0$,\n\\[\n \\sum_{j=1}^{k}w(d_j,r)\\le\\pi,\\qquad\n w(d,r)=\\arccos\\Bigl(\\operatorname{clamp}\n   \\frac{\\cosh d\\cosh r-\\cosh(D/2)}{\\sinh d\\sinh r}\\Bigr),\n\\]\nwhere $\\operatorname{clamp}$ truncates its argument to $[-1,1]$.\n\\end{lemma}\n\n\\begin{proof}\nThe open balls $B_j=B_{\\mathrm{hyp}}(b_j,D/2)$ are pairwise disjoint, since a\ncommon point would force $d(b_i,b_j)<D$.  Fix $r>0$.  By the hyperbolic law of\ncosines the point of the hyperbolic circle of radius $r$ at angle $\\theta$\nlies in $B_j$ exactly when\n$\\cosh d_j\\cosh r-\\sinh d_j\\sinh r\\cos(\\theta-\\theta_j)<\\cosh(D/2)$.\nThe set of angles satisfying this inequality has measure $2w(d_j,r)$:\nclamping gives measure zero for an empty intersection and $2\\pi$ for a\nfull circle, with tangent endpoint sets of measure zero. The intersections\nof the balls with this circle are disjoint, so their angular measures\nsum to at most $2\\pi$.\n\\end{proof}\n\n\\begin{remark}[unformalised application of a bound for the number of roots]\\label{res:dual-arity-floor}\nFix radii $r_1,\\ldots,r_p>0$ and weights $\\sigma_1,\\ldots,\\sigma_p\\ge0$, put\n$\\Sigma=\\sum_i\\sigma_i$ and\n\\[\n U=\\sup_{d\\ge d_{\\mathrm{low}}(a)}\n   \\Bigl[\\lambda(d)-\\sum_i\\sigma_i\\,w(d,r_i)\\Bigr],\n \\qquad\n \\lambda\\bigl(d_{\\mathrm{low}}(a)\\bigr)=\\frac{\\delta(a)}2 .\n\\]\nIf $U>0$, then failure forces $k\\ge(x-\\pi\\Sigma)/U$.\n\\end{remark}\n\\evidenceremark{Lean proves the geometric inequality from separation\n\\eqref{eq:lc-separation} and the individual and summed bounds\n\\eqref{eq:lc-radius-and-budget}. It does not derive those hypotheses\nfrom the absence of a short polynomial-root path. The latter application\nis the unformalised part of this remark.}\n\n\n\\begin{proof}\nBy~\\eqref{eq:lc-radius-and-budget}, Lemma~\\ref{res:circle-slice-packing} and\nthe radius bound,\n\\[\n x\\le\\sum_j\\lambda(d_j)\n  =\\sum_j\\Bigl[\\lambda(d_j)-\\sum_i\\sigma_i w(d_j,r_i)\\Bigr]\n    +\\sum_i\\sigma_i\\sum_j w(d_j,r_i)\n  \\le kU+\\pi\\Sigma. \\qedhere\n\\]\n\\end{proof}\n\nA nonnegative weighted sum of the circle-slice inequalities bounds every\nadmissible configuration at once. The weights give a\nusable bound only when\n$\\Sigma$ and the supremum $U$ have rigorous bounds. In particular, checking\nthe expression on a finite grid does not bound the supremum between grid\npoints. Zero weights recover only the individual bound\n$k\\ge2x/\\delta(a)$; nonzero weights incorporate the circle-packing\ninformation and can improve the result.\nA certificate at area $a'\\ge a$ also applies at $a$. Indeed,\n$d_{\\mathrm{low}}(a)\\ge d_{\\mathrm{low}}(a')$, so the admissible range of\n$d$ only shrinks. Also $D(a)\\ge D(a')$, which enlarges every slice angle\n$w(d,r_i)$ at fixed $d,r_i$. Since the weights are nonnegative, the\nexpression defining $U$ can only decrease. Thus the same certified pair\n$(\\Sigma,U)$ remains valid.\nThe numerical comparison also uses two bounds that do not involve chosen\nweights. They are useful where the circle-slice certificate is weaker.\nWe derive them here so that every root-count input to the comparison is\nvisible.\n\n\\paragraph{Two further root-count bounds.}\nOrder the distances as $d_1\\le\\cdots\\le d_k$. The triangle inequality gives\n$d_j\\ge D-d_1$ for $j\\ge2$. The function $\\lambda$ is decreasing and convex,\nsince $\\lambda'(d)=-1/\\sinh d$ and\n$\\lambda''(d)=\\cosh d/\\sinh^2d>0$. On\n$[d_{\\mathrm{low}},D/2]$, the convex function\n$\\lambda(d)+(k-1)\\lambda(D-d)$ takes its maximum at an endpoint.\nIf $d_1\\ge D/2$, each summand is at most $\\lambda(D/2)$ instead.\n(The definitions give $d_{\\mathrm{low}}<D/2$.) Thus\n\\[\n x\\le\\sum_j\\lambda(d_j)\n \\le\\max\\left\\{\n  \\frac{\\delta(a)}2+(k-1)\\lambda(D-d_{\\mathrm{low}}),\\;\n  k\\lambda(D/2)\\right\\},\n\\]\nand consequently\n\\[\n k\\ge\\min\\left\\{\n  1+\\frac{x-\\delta(a)/2}{\\lambda(D-d_{\\mathrm{low}})},\\;\n  \\frac{x}{\\lambda(D/2)}\\right\\}.\n\\]\nThe minimum is essential: the two expressions arise from alternative\npositions of the nearest root, not from simultaneous restrictions.\n\nFor the second bound, integrate over the disjoint hyperbolic balls of\nradius $D/2$. Put $E=e^{2/a}-1$, so each ball has hyperbolic area $2\\pi E$.\nAt most one contains the origin; its contribution to\n$\\sum_j\\lambda(d_j)$ is at most $\\delta(a)/2$.\nOn every other ball, $-\\log|z|$ is harmonic. To average it, first move\nthe ball's centre to $0$ by a disc automorphism. Hyperbolic area is\ninvariant under this change of coordinates and has radial density\n$4/(1-|z|^2)^2$ there. The ordinary mean-value property on each centred\nEuclidean circle therefore gives hyperbolic area mean\n$\\lambda(d_j)$. If the original ball has $0$ on its boundary, use\nincreasing smaller concentric balls and monotone convergence.\n\nThere are $m=k-1$ nonexceptional balls if one contains the origin, and\n$m=k$ otherwise. Their union has area $2\\pi mE$. Each superlevel set of\n$-\\log|z|$ is a centred ball, so its intersection with any set of this\narea has area at most the smaller of the two areas. The centred ball\nof area $2\\pi mE$ attains that upper bound at every superlevel.\nIntegrating these intersection areas over the superlevel parameter\ntherefore bounds the integral over the union by the integral over that\ncentred ball. Its radius $R$ satisfies $\\cosh R=1+mE$.\nWith $v=\\cosh r-1$, the identity\n$-\\log\\tanh(r/2)=\\tfrac12\\log(1+2/v)$ reduces the radial integral to\n$(2E)^{-1}\\int_0^{mE}\\log(1+2/v)\\,dv$. Evaluating it gives\n\\begin{align*}\n \\frac1{2\\pi E}\\int_{B_{\\mathrm{hyp}}(0,R)}-\\log|z|\\,dA_{\\mathrm{hyp}}\n &=\\frac{\\frac{mE}{2}\\log(1+2/(mE))+\\log(1+mE/2)}{E}\\\\\n &\\le\\frac{1+\\log(1+mE/2)}{E},\n\\end{align*}\nwhere the improper integral is finite because $v\\log v\\to0$ as\n$v\\downarrow0$, and the last inequality uses $\\log(1+u)\\le u$ for $u>0$.\nAdding the possible exceptional contribution and inverting each case yields\n\\[\n k\\ge\\min\\left\\{\n  1+\\frac2E\\bigl(e^{E(x-\\delta(a)/2)-1}-1\\bigr),\\;\n  \\frac2E\\bigl(e^{Ex-1}-1\\bigr)\\right\\}.\n\\]\nAgain the minimum accounts for the two possible cases. This estimate uses\narea rearrangement, whereas the preceding one uses only ordered distances;\nneither replaces the circle-slice inequalities.\n\nThe comparison takes the maximum of $2$, $2x/\\delta(a)$, these two bounds,\nand the certified circle-slice bounds. Each may first be rounded up because\n$k$ is an integer. Any of these estimates derived with an upper bound for\n$a$ remains valid at the actual smaller area: the root configuration still\nsatisfies the weaker separation and radius conditions at that upper bound.\n\n\\paragraph{From the root count to an area contradiction.}\nAt a direction avoiding the finitely many critical-value arguments, lift the\nvalue radius from $0$ to $te^{i\\theta}$ from each of the $k$ roots in $C_t$\nand split each lift at level $\\mu$. Put\n$A_0=\\operatorname{Area}(C_t\\cap\\{|f|<\\mu\\})$. This is the sum of the\nareas of the $k$ one-root components inside $C_t$, not of all the root\ncomponents of $f$; in particular, $0\\le A_0\\le\\pi a$.\nOn each of these components $f/\\mu$ is conformal, since it is a proper\nmap with no critical point onto a simply connected disc. Take\n$\\psi(z)=\\sum_{\\ell\\ge0}\\gamma_\\ell z^\\ell$ to be its inverse, so\n$f(\\psi(z))=\\mu z$ and $\\psi(0)$ is the selected root. The radial curves in\nthis coordinate therefore trace exactly the required value-ray lifts. Cauchy--Schwarz\nin the radial variable and Parseval give\n\\[\n \\operatorname{mean}_\\theta\n   \\Bigl(\\int_0^1|\\psi'(re^{i\\theta})|\\,dr\\Bigr)^2\n \\le\\sum_{\\ell\\ge1}\\frac{\\ell^2|\\gamma_\\ell|^2}{2\\ell-1}\n \\le\\sum_{\\ell\\ge1}\\ell|\\gamma_\\ell|^2\n =\\frac{\\operatorname{Area}(\\text{component})}\\pi.\n\\]\nFirst stop the radial integrals short of the unit circle, then pass to\nthe endpoint by monotone convergence. Thus no smoothness\nof the boundary at level $\\mu$ is assumed. Summing the $k$ square-root\narea bounds gives a mean total low lift length at most $\\sqrt{kA_0/\\pi}$.\nAt a regular intermediate level $u$, let $A_C(u)$ be the area of\n$C_t\\cap\\{|f|<u\\}$ and $P_C(u)$ its total boundary length.\nThe argument principle and the coarea formula give\n\\begin{equation}\\label{eq:lc-perimeter}\n P_C(u)^2\\le2\\pi k\\,u\\,A_C'(u),\n\\end{equation}\nbecause $|dz|=u\\,d(\\arg f)/|f'|$ on the level curve, its total argument\nvariation is $2\\pi k$, and Cauchy--Schwarz applies.  Integrating\n$P_C(u)/(2\\pi u)$ from $\\mu$ to $t$ bounds the mean high lift length by\n$\\sqrt{kx(\\pi a-A_0)/(2\\pi)}$. Cauchy--Schwarz combines the low and high\nbounds as\n\\[\n \\sqrt{\\frac{kA_0}{\\pi}}+\n \\sqrt{\\frac{kx(\\pi a-A_0)}{2\\pi}}\n \\le\\sqrt{ka(1+x/2)}.\n\\]\nChoose a direction whose total lift length is at most\n$\\sqrt{ka(x+2)/2}$. Order its $k$ boundary endpoints cyclically and\njoin each adjacent pair using the two lifts and the intervening boundary\narc. Each path lies in $\\overline{C_t}\\subset\\{|f|<1\\}$ and, under\nfailure, has length at least $2$. Each lift occurs twice in the sum of\nthese lengths, while the boundary occurs once. Therefore\n\\[\n 2k\\le\\sqrt{2ka(x+2)}+P_C(t).\n\\]\nNow let $t=\\mu e^x$ vary, and write\n$a(x)=\\operatorname{Area}(C_{\\mu e^x})/\\pi$ for the component containing\nthe chosen first pair; its root count $k$ may increase at later mergers.\nOn a regular interval, $a'(x)=tA_C'(t)/\\pi$. Applying\n\\eqref{eq:lc-perimeter} at the outer level and rearranging therefore gives\n\\begin{equation}\\label{eq:lc-area-growth}\n a'(x)\\ \\ge\\ \\frac1{2\\pi^2}\n \\Bigl[\\,2\\sqrt k-\\sqrt{2a(x)(x+2)}\\,\\Bigr]_{+}^{2}\n\\end{equation}\non each regular interval for the component containing the selected pair,\nusing the largest of the root-count bounds just derived. At merger levels\nthis component only gains\narea, which strengthens the integrated comparison. P\\'olya's inequality\ngives $a\\le1$ until the level reaches $1$. Thus an integrated lower bound\ngreater than $1$ before $t=1$ contradicts the assumption.\n\nNo positive initial area has to be assumed. While $a\\le1$, for\n$0<x\\le3/10^5$ the bound $k\\ge2$ alone gives\n\\[\n a'(x)\\ge\\frac1{2\\pi^2}\n \\left(2\\sqrt2-\\sqrt{2(2+3/10^5)}\\right)^2>\\frac1{30}\n\\]\nat regular levels. For example, the last inequality follows using\n$\\pi<22/7$, $\\sqrt2>140/99$ and\n$\\sqrt{2(2+3/10^5)}<20001/10000$. Integration, including the\nnonnegative merger jumps, gives $a(3/10^5)>10^{-6}$. This is the\nuniversal starting bound used by the certificate.\n\nThe comparison is integrated cell by cell without assuming that the\nindependently certified lower bounds increase with the table index.\nOn a cell $[x_\\ell,x_r]$, suppose $a(x_\\ell)\\ge a_\\iota$ and choose\na trial upper bound $m$ for the area. Let $g$ be a certified lower bound\nfor the right side of~\\eqref{eq:lc-area-growth} throughout that cell\nunder the trial assumption $a\\le m$. If\n$m<a_\\iota+(x_r-x_\\ell)g$, then $a(x_r)>m$. Indeed, the contrary\nassumption $a(x_r)\\le m$ would imply $a\\le m$ throughout the cell by\nmonotonicity. Integrating the differential inequality, with the\nnonnegative merger jumps, would then give $a(x_r)>m$, a contradiction.\n\n\\paragraph{Certificate.}\nThe comparison gives an upper bound $X$ on the logarithmic time needed\nfor the forced area to exceed $1$. It therefore gives a contradiction\nwhenever $\\mu e^X<1$. The stated constant $13/25$ is a convenient rational\nchoice with room to spare, not an optimality claim. A\n\\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos1041/scripts/check_erdos1041_angular_budget_closure.py}{program} evaluates\nevery accepted inequality in exact rational arithmetic, with directed rounding\non the transcendental evaluations. Its fixed rational brackets for $\\pi$\nand $\\log2$ have different elementary checks: Machin's identity\n$\\pi=16\\arctan(1/5)-4\\arctan(1/239)$ for the former, and\n$\\log2=2\\sum_{j\\ge0}3^{-(2j+1)}/(2j+1)$ for the latter.\nAlternating-series bounds and a geometric tail bound, respectively,\nsuffice to verify them. Its full run uses $18$ area-table levels,\n$14$ radii in each weighted bound,\n$126$ attempted choices of weights, maximum step $1/400$, and the\ninitial lower area $10^{-6}$ at $x=3/10^5$. Only accepted rational\ncertificates enter the comparison; its early steps may be shorter than\n$1/400$. The full run\ncertifies\n\\[\n X=\\frac{635762889599}{1000000000000}<0.6357629,\n \\qquad \\frac{13}{25}e^{X}<1,\n\\]\nwhich gives Remark~\\ref{res:low-critical-thirteen-twentyfifths}.\nThe full replay recorded on 29 September 2026 used\nthe immutable program cited above. It accepted all $126$ dual\ncertificates and reproduced this value of $X$; no new replay is claimed\nfor the present editorial revision. The retained certificate data give the exact radii and nonnegative\nweights, together with their sums and certified upper bounds. The program's Git blob identifier is\n\\texttt{e9b4998884575a31ce0d1f6762a210ae2d617103}.\nThat replay also ran the comparison regression and the\nobserver-containing-circle audit. The comparison check is exact; the geometry audit combines\nan exact full-circle boundary test with floating identity checks.\nNo new per-degree thresholds were computed.\n\nA coarser run gives $X=664373027131/1000000000000$\nand the weaker threshold $51/100$.\nThe floating optimiser proposes weights but supplies no proof. The\nargument uses only the subsequently certified rational bounds on their\nsum and on the supremum $U$.  Each accepted\npair supplies a weight sum $\\Sigma$ and a certified upper bound for the\nsupremum $U$ in Remark~\\ref{res:dual-arity-floor}. For orientation, the\npairs of the coarser run at $a=1$, rounded to six decimal places, are\n$(0.085674,0.045865)$, $(0.126361,0.021829)$, $(0.163157,0.010513)$ and\n$(0.208928,0.004318)$, giving $kU+\\pi\\Sigma$ approximately\n$0.4526$, $0.5716$, $0.6808$ and $0.7945$ at $k=4,8,16,32$, respectively.\nThese rounded pairs do not themselves certify an inequality; the comparison\nuses the unrounded rational values and their directed bounds.\nTo bound the supremum over $d$, the checker subdivides intervals and bounds\neach term in the direction needed for an upper bound. The function\n$\\lambda$ is decreasing. For fixed $r$, the slice angle $w(d,r)$ has no\nstrict interior minimum. Indeed, before clipping the angle to $0$ or $\\pi$,\nits cosine is\n\\[\n H(d)=\\frac{\\cosh d\\cosh r-\\cosh(D/2)}{\\sinh d\\sinh r},\\qquad\n H'(d)=\\frac{\\cosh(D/2)\\cosh d-\\cosh r}{\\sinh^2d\\sinh r}.\n\\]\nThe numerator of $H'$ is increasing. Thus $H$ has at most one interior\nminimum, and $w=\\arccos H$ has at most one interior maximum; clipping\npreserves the endpoint-minimum property. On an interval $[d_-,d_+]$,\ntherefore,\n\\[\n \\lambda(d)-\\sum_i\\sigma_iw(d,r_i)\n \\le\\lambda(d_-)-\\sum_i\\sigma_i\n       \\min\\{w(d_-,r_i),w(d_+,r_i)\\}.\n\\]\nBeyond $d=\\max_i r_i+D/2$, all the slice angles vanish, so only the\ndecreasing function $\\lambda$ remains. These are interval bounds, not\nvalues sampled on a grid.\nThe closing inequality is independently checkable: with\n$X=635762889599/10^{12}$ and\n$e^X\\le\\sum_{j\\le14}X^j/j!+(X^{15}/15!)/(1-X/16)$, rational arithmetic gives\n$(13/25)e^X<0.982000386<1$. The same majorant also gives\n\\[\n \\frac{529}{1000}e^X<0.998996547<1.\n\\]\nThus the full certificate in fact gives the path conclusion for\n$\\mu\\le529/1000$, without changing the geometric argument or its hypotheses.\nThis is an arithmetic consequence of the certified stopping time. We retain $13/25$ in the theorem and its scaling\ncorollaries as the simpler constant.\n\n\\paragraph{Sources and scope.}\n\\label{bdry:low-critical-closure}\nThe analytic chain above is ordinary mathematics.  Its general inputs are the\nRiemann mapping theorem, the Bergman kernel, the argument principle, the\ncoarea formula, and P\\'olya's area inequality\n$\\operatorname{Area}\\{|f|\\le t\\}\\le\\pi t^{2/n}$\n\\cite[printed pp.~280--282]{polya1928}, \\cite[Theorem~1]{crane}.  The exact\nrational certificate is checked by a separate exact-arithmetic program,\nnot by the Lean kernel. The two additional root-count bounds above are\nthe ordered-distance estimate in\n\\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos1041/ClusterSeparationLowCriticalClosure.md}{the earlier distance comparison}\nand the area-rearrangement estimate in\n\\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos1041/HyperbolicPackingArityFloor.md}{the hyperbolic packing argument}.\nTheir derivations above are part of the analytic proof. No independent\nreview of this theorem is recorded. Prior art for\nthe assembled statement is unassessed. No novelty is claimed for the\nslice inequality, which follows directly from disjointness of the balls\nand the hyperbolic law of cosines.  A\n\\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos1041/AngularBudgetLowCriticalClosure.md}{research note}\ncarries the full analytic argument, the table of certified weighted bounds\nand the instructions for running the certificate. The stopping-time test\nrequires $\\mu<e^{-X}$; the stated $13/25$ theorem and the $529/1000$\nconsequence are two rational cutoffs within that range. Other theorems\napply to some polynomials outside it, so failure of this numerical test\ndoes not identify the class left untreated by all the results.\nFixed-degree arguments remain stronger at $n=4$ and $n=5$, where the\ncorresponding thresholds are $61/100$ and $139/250$; from $n=6$ on the\nall-degree constant $13/25$ is the better statement.  These threshold\narguments do not settle the unrestricted existence of a short curve joining two roots.\n\n\\begin{remark}[unformalised scaled path bound]\\label{res:scaled-low-critical-path}\nEvery squarefree monic polynomial $f$ of degree $n\\ge2$ has two distinct\nzeros joined by a rectifiable curve of length less than\n$(5/2)\\mu^{1/n}$ in $\\{|f|<(25/13)\\mu\\}$, where\n$\\mu=\\min_{f'(c)=0}|f(c)|$.\n\\end{remark}\n\\evidenceremark{Above degree two, Lean assumes\n\\texttt{LowCriticalThirteenTwentyFifths}, the criterion in\nRemark~\\ref{res:low-critical-thirteen-twentyfifths}. It checks the\nrescaling and the degree-two case without supplying that missing\nanalytic proof.}\n\n\\begin{proof}\nFor $n\\ge3$, apply Remark~\\ref{res:low-critical-thirteen-twentyfifths}\nto $g(z)=s^{-n}f(sz)$ with $s=((25/13)\\mu)^{1/n}$.\nIts least critical modulus is $13/25$; rescaling gives length less than\n$2(25/13)^{1/n}\\mu^{1/n}<(5/2)\\mu^{1/n}$, since\n$25/13<(5/4)^3$.  For $n=2$, write $f(z)=(z-h)^2-d^2$.\nThe two radial segments through $h$ have total length $2|d|=2\\mu^{1/2}$\nand lie in the closed level $\\mu$, which is inside the stated open level.\n\\end{proof}\nThe next construction retains the root count and capacity of a chosen\ncomponent, at the cost of a larger universal constant.\n\n\\section{A path estimate from area and boundary length}\n\\label{sec:constant-factor}\n\\papersectiontarget{constant-factor-proof}\n\nWe can also average inverse-ray lengths and boundary arcs directly.\nThis ordinary construction keeps both the number of roots and the\ncapacity of the selected component in the estimate. Its formal source\nassumes the construction in degrees above two, so the resulting path\nbounds below are unformalised remarks.\n\nThe scaled packing criterion, Remark~\\ref{res:scaled-low-critical-path},\nhas smaller constants: $5/2<71/10$ for length and $25/13<2$ for the\ncontainment level. The advantage of the direct argument is its explicit\ndependence on component geometry. Its $\\mu\\le1/2$ specialisations are\nalready within the range of Remark~\\ref{res:low-critical-thirteen-twentyfifths}.\nFor a monic degree-$n$ polynomial put\n\\[\n K_t=\\{z:|f(z)|\\le t\\},\\qquad\n \\mu=\\min_{f'(c)=0}|f(c)|,\\qquad \\rho=\\mu^{1/n}.\n\\]\n\n\\begin{remark}[unformalised path bound at level $2\\mu$]\n\\label{res:constant-factor-path}\nFor every monic polynomial $f$ of degree $n\\ge2$, two zero occurrences are\njoined by a possibly degenerate path of length at most\n\\[\n \\frac{71}{10}\\,\\rho\n\\]\ninside $K_{2\\mu}$.  If $f$ is squarefree, their locations are distinct.\nIf $\\mu\\le1/2$, the construction may be chosen inside $\\{|f|<1\\}$ with\nlength at most $5.7$.\n\\end{remark}\n\\evidenceremark{In degrees $n\\ge3$ Lean assumes \\texttt{CFAPathConstruction},\nthe two-parameter path bound (CF) below. Its averaging construction is\nproved only in the ordinary argument. The degenerate case, degree two\nand numerical inequalities have separate formal proofs.}\n\n\nThe constant is the rationally certified specialization of a two-parameter\nbound.  If the selected component contains $k\\ge2$ roots, then for\nevery $r\\in(0,1)$ and $\\lambda>1$ the proof constructs a path in\n$K_{\\lambda\\mu}$ whose length is at most\n\\[\n \\sqrt{\\frac2k}\\left(\n   \\frac{\\sqrt2\\,r}{(1-r)^2}\n   +\\lambda^{1/n}\\left(\n      \\sqrt{\\log(\\lambda/r)}+\n      \\frac{\\pi}{\\sqrt{\\log\\lambda}}\n    \\right)\n \\right)\\rho.                                      \\tag{CF}\n\\]\nFor $n\\ge3$, the choice $\\lambda=2$, $r=3/20$ makes the bracket less\nthan $71/10$: the largest case is $n=3$, where exact rational bounds give\n$66517563/9392500<71/10$.  Degree two has the exact root-segment bound\n$2\\rho$.\n\n\\begin{proof}\nA repeated zero gives the constant path, so assume $f$ is squarefree and\n$\\mu>0$. Degree two is the segment of length $2\\rho$ in $K_\\mu$; assume\n$n\\ge3$. Put $T=\\lambda\\mu$, let $C_T$ be the component of $K_T$\ncontaining a critical point at level $\\mu$, and write\n$A(\\sigma)=\\operatorname{Area}(K_\\sigma\\cap C_T)$.\nWe will first choose a regular level $t$, then a direction in the value\nplane, and finally an adjacent pair of roots. This order ensures that\nthe three estimates below apply to the same path.\n\nAt a regular level $\\sigma$, a component $C'$ containing $k'$ roots has\n$|dz|=\\sigma\\,d\\varphi/|f'|$ on its boundary and total argument variation\n$2\\pi k'$. Cauchy--Schwarz, followed by the coarea formula, gives\n\\[\n \\mathcal H^1(\\partial C')^2\\le2\\pi k'\\,\\sigma A'(\\sigma).\n                                                        \\tag{CF1}\n\\]\nHere $A'$ sums the area derivatives of all the components in $C_T$,\nso it bounds the contribution from $C'$. P\\'olya's inequality\n$A(T)\\le\\pi T^{2/n}$ \\cite[Theorem~1]{crane} and averaging with respect\nto $d\\sigma/\\sigma$ on $(\\mu,T)$ give a regular level $t$ such that\n$tA'(t)\\le\\pi T^{2/n}/\\log\\lambda$. The component $C_t$ containing\nthe chosen first merger has $k\\ge2$ roots and\n\\[\n \\mathcal H^1(\\partial C_t)\n \\le\\pi\\sqrt{2k/\\log\\lambda}\\,T^{1/n}.\n\\]\nAveraging over levels avoids estimating the area derivative at the\ncritical level itself.\n\nChoose a direction avoiding all critical-value arguments. Lift its radial\nsegment from each of the $k$ roots to $\\partial C_t$, splitting each lift\nat value modulus $r\\mu$. Below $\\mu$, the inverse branches of $f/\\mu$\nare univalent. Their derivatives at $0$ are $\\mu/f'(z_i)$, where $z_i$\nranges over the roots. The area formula for their disjoint images gives\n\\[\n \\sum_i\\frac{\\mu^2}{|f'(z_i)|^2}\\le\\rho^2.        \\tag{CF2}\n\\]\nIndeed, the squared derivative at the centre is at most the image area\ndivided by $\\pi$, and the total image area is at most $\\pi\\mu^{2/n}$.\nKoebe distortion and Cauchy--Schwarz therefore bound the sum of the\nlengths below $r\\mu$, uniformly in the direction. Write\n$\\ell_i^{\\rm high}(\\theta)$ for the length of the $i$th piece above\n$r\\mu$ in direction $\\theta$. At an intermediate regular level, apply\n(CF1) to all the lower components contained in $C_t$; their root counts\nsum to $k$. Integrating their total perimeter with measure\n$d\\sigma/(2\\pi\\sigma)$ gives the mean of $\\sum_i\\ell_i^{\\rm high}$.\nCauchy--Schwarz in $\\sigma$, using an area increment at most\n$\\operatorname{Area}(C_t)\\le\\pi T^{2/n}$, gives\n\\[\n \\frac1{2\\pi}\\int_0^{2\\pi}\\sum_i\\ell_i^{\\rm high}(\\theta)\\,d\\theta\n \\le\\sqrt{\\frac{k\\operatorname{Area}(C_t)}{2\\pi}\n                   \\log\\frac{t}{r\\mu}}\n \\le\\sqrt{\\frac k2}\\,T^{1/n}\\sqrt{\\log(\\lambda/r)}.\n\\]\nChoose a direction avoiding the critical-value arguments at which the\nsum is at most this bound. The excluded directions form a finite set\nand hence have measure zero.\n\nOrder its $k$ endpoints cyclically on $\\partial C_t$. Each adjacent pair\nof endpoints gives a path consisting of two lifted segments and the\nintervening boundary arc. In the sum over these $k$ paths, every lifted\nsegment occurs twice and the boundary arcs partition $\\partial C_t$.\nThus some pair has length at most the sum of the following three bounds:\n\\[\n\\begin{aligned}\n \\frac2k\\sum_i\\ell_i^{\\rm low}\n &\\le\\frac{2r\\rho}{\\sqrt{k}(1-r)^2},\\\\\n \\frac2k\\sum_i\\ell_i^{\\rm high}\n &\\le\\sqrt{\\frac2k}\\sqrt{\\log(\\lambda/r)}\\,T^{1/n},\\\\\n \\frac{\\mathcal H^1(\\partial C_t)}k\n &\\le\\sqrt{\\frac2k}\\frac{\\pi T^{1/n}}{\\sqrt{\\log\\lambda}}.\n\\end{aligned}\n\\]\nThe first holds in every direction, the second in the direction just\nchosen, and the third at the previously selected level. Substituting\n$T^{1/n}=\\lambda^{1/n}\\rho$ proves~(CF).\nFor the final assertion, the repeated-zero case is already a constant path,\nand in degree two the root segment has length\n$2\\rho=2\\sqrt\\mu\\le\\sqrt2$.  For $n\\ge3$, take $\\lambda=2$ and $r=3/20$.\nSince $k\\ge2$ and $\\mu\\le1/2$, (CF) gives\n\\[\n \\begin{aligned}\n \\operatorname{length}\n &\\le \\frac{60\\sqrt2}{289}\\mu^{1/n}\n +(2\\mu)^{1/n}\\left(\n   \\sqrt{\\log(40/3)}+\\frac{\\pi}{\\sqrt{\\log2}}\n \\right)\\\\\n &\\le \\frac{60\\sqrt2}{289}+\\sqrt{\\log(40/3)}\n      +\\frac{\\pi}{\\sqrt{\\log2}}\\\\\n &=5.676476\\ldots<5.7.\n \\end{aligned}\n\\]\nHere $2\\mu\\le1$, and the mean-value choice may be taken at a regular level\nstrictly below the top of its positive-measure window.  Thus the entire path\nlies in $\\{|f|<1\\}$.\n\\end{proof}\n\nBefore replacing the right side of (CF) by the constant $71/10$, we can\nretain its dependence on the selected component. Its number $k$ of\nroots occurs in the factor $\\sqrt{2/k}$, and its logarithmic capacity\nbounds the area used in the estimate. Each gives sufficient conditions\nfor the length bound to be less than $2$.\n\nThe number of roots in the first merged component can be much smaller\nthan the degree. At a first merger caused by one simple critical point it\nis $2$, so the following thresholds do not apply. For $z^n-b$, however,\nall $n$ roots merge at the same level, giving $k_0=n$. For example,\n$n\\ge17$ and $0<|b|\\le1/2$ satisfy the first pair of hypotheses. These\nconditions illustrate what the area argument gains from simultaneous\nmergers; the preceding unformalised $13/25$ criterion already covers these three ranges\nwithout the additional lower bound on the number of roots.\n\n\\begin{remark}[unformalised root-count refinement]\n\\label{res:constant-factor-arity}\nLet $f$ be monic with every root in the open unit disc, let $c_*$ be a\ncritical point with $|f(c_*)|=\\mu$, and let $k_0$ be the number of roots,\ncounted with multiplicity, in the component of $K_\\mu$ containing $c_*$.\nThen Erd\\H{o}s~\\#1041 holds for $f$ in each of the three cases\n\\[\n \\mu\\le\\tfrac12\\ \\text{and}\\ k_0\\ge17,\\qquad\n \\mu\\le\\tfrac14\\ \\text{and}\\ k_0\\ge12,\\qquad\n \\mu\\le\\tfrac18\\ \\text{and}\\ k_0\\ge10 .\n\\]\n\\end{remark}\n\\evidenceremark{Lean assumes \\texttt{CFAArityConstruction}, the construction\n(CF) with the stated lower bound on the component's root count. The\nformal rational inequalities do not supply this analytic input.}\n\n\n\\begin{proof}\nIf $\\mu=0$, a repeated zero gives the constant path. Assume henceforth\n$\\mu>0$. Every selected component $C_t$ in the proof of\nRemark~\\ref{res:constant-factor-path} contains the first-merge component, so\n$k\\ge k_0$.  For the first case take $\\lambda=2$, $r=13/100$; then\n$(2\\mu)^{1/n}\\le1$ and $\\rho\\le1$, and the exact bounds\n$\\sqrt2<283/200$, $\\sqrt{\\log(200/13)}<5/3$ and\n$\\pi/\\sqrt{\\log2}<(22/7)/(104/125)=1375/364$ make the bracket in~(CF) at most\n$15668813/2755116$, whose square is $34-12570881068535/7590664173456<34$.\nHence $k_0\\ge17$ gives squared length below $(2/17)\\cdot34=4$.  For the second\ncase take $\\lambda=4$, $r=3/25$: the bracket is below $6075221/1273888$, whose\nsquare is $24-2038665078215/1622790636544<24$, and $2/k_0\\le1/6$ gives squared\nlength below $4$.  For the third take $\\lambda=8$, $r=11/100$: the bracket is\nbelow $55629121/12475575$, whose square is\n$20-18200328379859/155639971580625<20$, and $2/k_0\\le1/5$ again gives squared\nlength below $4$.  In each case $\\lambda\\mu\\le1$, so the freedom in\nthe choice of the regular level $t$ keeps $t<1$ and the containment strict.\n\\end{proof}\n\nThe next criterion uses logarithmic capacity, a measure of the size of a\ncompact set that scales linearly under dilation. It compares the size of\none component with that of the entire polynomial sublevel set. If that\nsublevel set is connected, the ratio $\\kappa$ below is $1$; a cutoff such\nas $\\kappa\\le1/3$ therefore requires the component to have smaller\ncapacity than the full sublevel set. A small number of roots in the\ncomponent does not imply this capacity bound. The ratio requires $\\mu>0$. When $\\mu=0$,\n$f$ has a repeated zero and the constant path already gives the conclusion;\nthere is no capacity ratio to evaluate.\n\n\\begin{remark}[unformalised capacity refinement]\n\\label{res:constant-factor-capacity}\nKeep the hypotheses of Remark~\\ref{res:constant-factor-arity} with\n$0<\\mu\\le1/2$, let $C$ be the component of $\\{|f|<2\\mu\\}$ containing $c_*$, and\nput $\\kappa=\\operatorname{cap}(\\overline C)/(2\\mu)^{1/n}$.  If\n$\\kappa\\le\\tau_{k_0}$, where\n\\[\n \\tau_k=\\frac{\\sqrt{2k}-A}{B},\\qquad\n A=\\frac{283}{3610},\\qquad B=\\frac{52029}{9100},\n\\]\nthen Erd\\H{o}s~\\#1041 holds for $f$. In particular $\\kappa\\le1/3$ suffices for\nevery root count $k_0\\ge2$, and the rational cutoffs\n$2/5,\\,12/25,\\,1/2,\\,7/12,\\,16/25,\\,2/3,\\,7/10$ suffice at\n$k_0=3,\\ldots,9$, rising to $39/40$ at $k_0=16$.\n\\end{remark}\n\\evidenceremark{Lean assumes \\texttt{CFACapacityConstruction}, the averaging\nconstruction with the component area bounded through its capacity.\nIt checks the numerical cutoffs. The construction from the polynomial\nhypotheses remains an additional input.}\n\n\n\\begin{proof}\nRepeat the averaging proof of Remark~\\ref{res:constant-factor-path} with\n$A(\\sigma)=\\operatorname{Area}(K_\\sigma\\cap C)$ for $\\sigma<2\\mu$.\nThis keeps the chosen paths in the open component $C$, even if distinct\ncomponents have touching boundaries at level $2\\mu$. Replace the global\narea input by the component form\n$\\operatorname{Area}(C)\\le\\pi\\operatorname{cap}(\\overline C)^2\n =\\pi\\kappa^2(2\\mu)^{2/n}$ of the area--capacity inequality\n\\cite[Theorem~6]{crane}.  Only the high-lift and boundary terms acquire the\nfactor $\\kappa$; the low-lift term~(CF2) is unchanged.  Taking $\\lambda=2$,\n$r=1/20$, and the exact bounds above together with\n$\\log40<12641/3402<(97/50)^2$, gives\n$\\operatorname{length}<\\sqrt{2/k_0}\\,(A+B\\kappa)$.  The definition of\n$\\tau_{k_0}$ makes the right side at most $2$, and for each displayed rational\n$q_k$ integer arithmetic gives $(A+Bq_k)^2<2k$.  Discarding\n$\\sqrt{2/k_0}\\le1$ altogether gives the uniform cutoff $\\kappa\\le1/3$.\n\\end{proof}\n\nBy the component-capacity formula, $\\kappa=e^{-\\Sigma/n}$ with $\\Sigma$ the sum\nof exterior Green function values at the roots excluded from $C$.  Indeed, let\n$\\Omega$ be the unbounded component of\n$\\widehat{\\mathbb C}\\mathbin{\\backslash}\\overline C$,\nlet $G$ be its Green function with pole at infinity, extended by zero off\n$\\Omega$, and let $g(\\cdot,a)$ be its Green function with pole at $a\\in\\Omega$.\nThe function $\\frac1n\\log\\frac{|f|}{2\\mu}-G+\\frac1n\\sum_a g(\\cdot,a)$, summed\nover the roots $a\\in\\Omega$ with multiplicity, extends harmonically to\n$\\Omega$, is bounded, and vanishes on $\\partial\\Omega\\subseteq\\partial C$, so it\nvanishes identically; its value at infinity is $\\log\\kappa+\\frac1n\\sum_aG(a)$\nby the symmetry $g(\\infty,a)=G(a)$.  Failure of these component-sensitive\ncriteria alone, in the range $\\mu\\le1/2$, forces $k_0\\le16$ and exterior\nGreen sum $\\Sigma<n\\log(40/39)$ when $k_0=16$.  This is a limitation of\nthe area-based criteria.  The low-critical theorem already excludes an actual\ncounterexample throughout that range.\n\nThe proof of Remark~\\ref{res:constant-factor-path} averages over\nlevels in $[\\mu,\\lambda\\mu]$. To keep the path at the first critical\nlevel instead, one would need a perimeter estimate of the following kind.\nThe regular-level argument~(CF1) gives no finite pointwise bound at the\ncritical level: even at a simple critical point, the area derivative\ndiverges logarithmically.\n\nThe addendum \\cite[Proposition~2]{revision2026} reports a\nRunge-approximation construction of monic level-one components with one\nsimple zero and arbitrarily long analytic boundaries. It imposes neither\na unit root disc nor a global lower bound on critical-value moduli.\nIts proof is not available, and\nthe report is not an input to the path estimates here. The question below\nrequires a stronger hypothesis than the existence of one unramified\ncomponent: the level must lie below every critical-value modulus of the\npolynomial. A construction without that global condition cannot by itself\nanswer the question.\n\n\\begin{problem}[perimeter below the smallest critical-value modulus]\\label{prob:conjecture-p}\nLet $f$ be squarefree and monic of degree $n\\ge2$, and put\n$\\mu=\\min_{f'(c)=0}|f(c)|$.  Is there a constant $\\beta$, independent of\n$n,f$ and $0<\\sigma<\\mu$, such that every component $C$ of\n$\\{|f|\\le\\sigma\\}$ satisfies\n\\[\n \\mathcal H^1(\\partial C)\\le\\beta\\sigma^{1/n}?\n\\]\nEquivalently, for each univalent inverse branch defined on $\\mathbb D$\nby $f(\\phi_j(w))=\\sigma w$, is\n\\[\n \\int_0^{2\\pi}|\\phi_j'(e^{it})|\\,dt\\le\\beta\\sigma^{1/n}\n \\qquad(1\\le j\\le n)?\n\\]\nThere is no root-location hypothesis.  If such constants exist, write\n$\\beta_*$ for their infimum.\n\\end{problem}\nAll these components contain one zero. The condition $\\sigma<\\mu$\ncontrols all inverse branches at once; it is stronger than requiring\nonly the selected component to contain one zero. The examples below\ncompare particular constants, not the necessity of this global\nhypothesis for every possible perimeter bound.\nFor $1<R<\\mu/\\sigma$, the area estimate gives\n\\[\n \\frac{1}{2\\pi}\\int_0^{2\\pi}|\\phi_j'(e^{it})|\\,dt\n \\le \\frac{(\\sigma R)^{1/n}}{\\sqrt{R^2-1}}.\n\\]\nIndeed, circular means of $|\\phi_j'|^2$ increase, the image of the annulus\n$1<|w|<R$ has area at least $\\pi(R^2-1)$ times the mean at $1$, and\nP\\'olya's preimage-area bound \\cite[Theorem~1]{crane} bounds that area by\n$\\pi(\\sigma R)^{2/n}$.  The loss as $R\\downarrow1$ belongs to this\nargument and does not prove an endpoint divergence.\n\nFor $d>0$ and $f(z)=z^2-d^2$, one has $\\mu=d^2$ and $\\rho=d$.  For $\\sigma<d^2$ the component\nof $\\{|f|\\le\\sigma\\}$ around $d$ contains one root. Its parametrisations\n$d\\sqrt{1+(\\sigma/d^2)e^{i\\theta}}$ converge uniformly, as $\\sigma\\uparrow d^2$,\nto one loop of Bernoulli's lemniscate $|z^2-d^2|=d^2$, of length $d\\,\\Gamma(1/4)^2/(2\\sqrt\\pi)$ by the substitution\n$\\varphi(w)=d\\sqrt{1+w}$ and $\\Gamma(1/4)\\Gamma(3/4)=\\pi\\sqrt2$.  At\n$\\sigma=d^2$ the two loops meet at the critical point, and the closed sublevel\nset is a single component containing both roots. Lower semicontinuity\nof length under the stated uniform convergence gives\n$\\beta_*\\ge\\Gamma(1/4)^2/(2\\sqrt\\pi)=3.70814935\\ldots$;\nconvergence of the lengths themselves is not required.\nFor $z^n-r^n$, polar parametrisation of one limiting loop gives its\nlength divided by $r$ as\n\\[\n \\frac{2^{1/n}}{n}\\int_{-\\pi/2}^{\\pi/2}\n     (\\cos\\theta)^{1/n-1}\\,d\\theta\n =\\frac{2^{1/n}}{n}\\sqrt\\pi\\,\n     \\frac{\\Gamma(1/(2n))}{\\Gamma(1/(2n)+1/2)}.\n\\]\nThe evaluation is Euler's beta integral\n\\href{https://dlmf.nist.gov/5.12.E1}{(5.12.1)}, in its\n\\href{https://dlmf.nist.gov/5.12.E2}{trigonometric form (5.12.2)}.\nTo verify the claimed decrease with $n$, put $p=1/n$. The expression is\n$2^{p+1}\\sqrt\\pi\\,\\Gamma(1+p/2)/\\Gamma((1+p)/2)$; its logarithmic\nderivative is\n\\[\n \\log2+\\tfrac12\\left(\n    \\frac{\\Gamma'(1+p/2)}{\\Gamma(1+p/2)}-\n    \\frac{\\Gamma'((1+p)/2)}{\\Gamma((1+p)/2)}\\right)>0.\n\\]\nHere $\\Gamma'/\\Gamma$ is increasing on the positive real axis, by the\npositive trigamma series \\href{https://dlmf.nist.gov/5.15.E1}{(5.15.1)}.\nThe expression therefore decreases with $n$ and tends to $2$.\n\nThe comparison with the next example also has an elementary bound,\nindependent of rounded gamma values. One loop of $z^2-d^2$ has\nparametrisation\n\\[\n z(s)=\\sqrt2\\,d\\,\n   \\frac{\\cos s(1+i\\sin s)}{1+\\sin^2s},\n \\qquad -\\pi/2\\le s\\le\\pi/2,\n\\]\nwhose speed is $\\sqrt2\\,d/\\sqrt{1+\\sin^2s}$. Convexity puts\n$(1+x)^{-1/2}$ below its secant on $[0,1]$. Integrating that secant\nat $x=\\sin^2s$ bounds the loop length divided by $d$ by\n$(\\pi/2)(1+\\sqrt2)$, as used below.\n\n\\begin{proposition}[a counterexample to the proposed one-root perimeter bound]\n\\label{res:one-root-gamma-false}\nLet $p(z)=z^8-(3/2)z$ and let $C$ be the component of $\\{|p|\\le1\\}$\ncontaining the origin.  Then $C$ contains exactly one zero and a\nneighbourhood of the closed disc of radius $5/8$, so\n$\\mathcal H^1(\\partial C)>5\\pi/4$.  The constant\n$\\Gamma(1/4)^2/(2\\sqrt{\\pi})$ is at most $(\\pi/2)(1+\\sqrt2)<5\\pi/4$.\n\\end{proposition}\n\nThe quadratic loop is exceeded in degree eight, even with the global\nsubcritical hypothesis. For $p(z)=z^8-(3/2)z$, every critical point $c$\nsatisfies $c^7=3/16$ and $p(c)=-(21/16)c$. Hence\n\\[\n \\mu=\\frac{21}{16}\\left(\\frac3{16}\\right)^{1/7}>1,\n \\qquad \\mu^7=\\frac{3\\cdot21^7}{16^8}>1.\n\\]\nThus $\\sigma=1$ is below every critical-value modulus, not merely a\nregular level for the selected component. The component $C$ of\n$\\{|p|\\le1\\}$ containing the origin lies in\n$\\{|z|<4/5\\}$, because $|p(z)|\\ge6/5-(4/5)^8>1$ on $|z|=4/5$, and it contains a\nneighbourhood of the closed disc of radius $5/8$, because\n$|p(z)|\\le(5/8)^8+15/16<1$ on that disc.  The other roots satisfy\n$|z|^7=3/2$, so $C$ contains exactly one root, and\n$\\mathcal H^1(\\partial C)>5\\pi/4>(\\pi/2)(1+\\sqrt2)\\ge\\Gamma(1/4)^2/(2\\sqrt\\pi)$.  Moreover,\nfor fixed $a>1$ and all sufficiently large $N$, the one-root component of\n$\\{|z^N-az|\\le1\\}$ contains every disc of radius $r<1/a$, and a circle of\nradius between $1/a$ and $1$ isolates it by Rouch\\'e's theorem; letting\n$a\\downarrow1$ shows that every admissible $\\beta$ is at least $2\\pi$.\nFor the latter family all critical values have modulus\n$(1-1/N)a(a/N)^{1/(N-1)}\\to a>1$, so it also gives the necessary bound\n$\\beta_*\\ge2\\pi$ under the corrected global subcritical hypothesis.\nKuznetsova--Tkachev \\cite[Theorems~1--2]{kuznetsova2003} establish\nLaplace-transform and log-convexity properties for regular level-length\nfunctions.  Those results do not provide a uniform bound at the first\ncritical level.  Total lemniscate length and one-branch endpoint length are\ndifferent extremal quantities.\n\nThe preceding examples do not establish a universal perimeter bound.\nThe implication below applies to an individual polynomial; a constant\n$\\beta$ independent of the polynomial and degree would give a uniform\npath bound. The construction uses two components meeting at a first\ncritical point and joins each root to that point at a cost of at most\nhalf its component's perimeter. It does not give the constant $2$ of\nthe historical question.\n\n\\begin{remark}[unformalised consequence of a perimeter bound]\n\\label{res:conjecture-p-consumer}\nLet $f$ be squarefree and monic of degree $n\\ge2$, and put\n$\\mu=\\min_{f'(c)=0}|f(c)|>0$. Suppose $\\beta>0$ satisfies\n\\[\n \\mathcal H^1(\\partial C)\\le\\beta\\sigma^{1/n}\n\\]\nfor every $0<\\sigma<\\mu$ and every component $C$ of $\\{|f|\\le\\sigma\\}$.\nThen two distinct roots of $f$ are joined inside $K_\\mu=\\{|f|\\le\\mu\\}$\nby a rectifiable path of length at most $\\beta\\mu^{1/n}$.\n\\end{remark}\n\\evidenceremark{Lean additionally assumes \\texttt{SubcriticalSplitExists}:\ntwo one-root components whose closures meet at the first critical point,\nwith the required boundary lengths and half-perimeter joins. The\nordinary argument below constructs this datum; it is not formalised\nby the perimeter hypothesis alone.}\n\n\n\\begin{proof}\n\\hypertarget{perimeter-to-path}{}\nPut $\\rho=\\mu^{1/n}$. On each component of $\\{|f|<\\mu\\}$,\n$f$ is a proper unramified covering of the disc $\\{|w|<\\mu\\}$, hence\nunivalent. Equivalently, exhaust by regular levels below $\\mu$ and apply\nthe component count \\cite[proof of Proposition~2.1]{eks2010}.\nAt a critical point $c_*$ with\n$|f(c_*)|=\\mu$, the local degree $d\\ge2$ gives $d$ nearby preimages of\nan inward radial value. Univalence places them in different one-root\ncomponents. Choose two such components $U_a,U_b$, containing roots $a,b$\nand with $c_*$ in both closures; simplicity of $c_*$ is unnecessary.\nFor the boundary passage, take $f(\\phi(w))=\\mu w$ on $\\mathbb D$.\nFix $|w_0|=1$. The finitely many preimages of $\\mu w_0$ have disjoint\nneighbourhoods. Properness of $f$ and connectedness of a sufficiently\nsmall disc cap force $\\phi$ to stay in one of them as $w\\to w_0$.\nAt its preimage $c$, write\n\\[\n f(z)-f(c)=(z-c)^d h(z),\\qquad h(c)\\ne0.\n\\]\nA local analytic $d$th root of $h$ makes $(z-c)h(z)^{1/d}$ a conformal\ncoordinate. In each inverse sector, $\\phi$ is therefore an analytic\nfunction of $(w-w_0)^{1/d}$ and extends continuously to $c$.\nThe finitely many critical boundary values are the only exceptional\npoints; at every other boundary value the ordinary inverse function\ntheorem applies. Moreover, near $e^{i\\theta_0}=w_0$ the boundary\nspeed is $O(|\\theta-\\theta_0|^{1/d-1})$, which is integrable even when\n$d>2$. Thus the extension is continuous on the closed disc and its\nboundary curve is rectifiable. The identity\n$f(\\phi(e^{i\\theta}))=\\mu e^{i\\theta}$ makes the boundary values injective,\nso the curve is Jordan.\n\nAs $r\\uparrow1$, the curves $\\phi(re^{i\\theta})$ converge uniformly to\nthat boundary and have length at most $\\beta(\\mu r)^{1/n}$ by the\nhypothesis. Lower semicontinuity of length gives boundary length at most\n$\\beta\\rho$ for each component.\n\nA bounded rectifiable Jordan domain $U$ has the following elementary\nproperty: any $a\\in U$ can be joined to any $c\\in\\partial U$ in\n$\\overline U$ with length at most $\\mathcal H^1(\\partial U)/2$.\nChoose a line through $a$ but not $c$, and let $[u,v]$ be the closure\nof the connected interval in its intersection with $U$ containing $a$.\nOf the two boundary arcs from $u$ to $v$, write $A$ for the one containing\n$c$ and $A'$ for the other. The paths from $a$ through $u$ or $v$ and\nthen along $A$ to $c$ have total length\n\\[\n |a-u|+|a-v|+\\operatorname{length}(A)\n =|u-v|+\\operatorname{length}(A)\n \\le\\mathcal H^1(\\partial U),\n\\]\nbecause $|u-v|\\le\\operatorname{length}(A')$. One path is at most half\nthis sum. The connected interval ensures containment of the straight\nparts even when $U$ is not convex.\n\nApply this property in $U_a,U_b$ with common boundary point $c_*$.\nConcatenating the two paths gives length at most\n$\\tfrac12(\\mathcal H^1(\\partial U_a)+\\mathcal H^1(\\partial U_b))\n\\le\\beta\\rho$, as required.\nOnly these two component perimeters are used; the subcritical hypothesis\nbounds them by the limiting argument above. Boundary arcs are allowed,\nso containment is in $K_\\mu$, not the open sublevel.\n\\end{proof}\n\n\\paragraph{Sources and scope.}\n\\label{bdry:constant-factor}\nThe inputs of the construction in the proof of\nRemark~\\ref{res:constant-factor-path} are the capacity identity for polynomial\npreimages \\cite[\\S2]{crane}, P\\'olya's area inequality \\cite[Theorem~1]{crane},\nCauchy--Schwarz, the Koebe distortion theorem for univalent maps, and\naveraging over adjacent boundary endpoints.  The public source\nof record is\n\\href{https://github.com/wcook04/plectis-erdos/blob/f214a6b45528dc5eefe20ffadc35f2e981627d4c/research_corpus/Erdos1041/UnconditionalConstantFactorBound.md}{the full area and boundary-length argument}, which\ncarries every rational verification quoted above. Its Section~10 gives\na weaker conditional bound involving an auxiliary parameter and a\nlogarithm; Remark~\\ref{res:conjecture-p-consumer} states the stronger\n$\\beta\\rho$ consequence proved here. That section's proposed numerical\nperimeter constant is refuted by the degree-eight example above and is\nnot assumed in the theorem. The unconditional construction retains\nlevel $2\\mu$ and constant $71/10$.\nThe scaled low-critical corollary has both a smaller constant and a smaller\nlevel.  The proof retained here is ordinary, with component-sensitive\nconsequences on their stated regions; its rational cutoffs are exact integer\ninequalities.  The earlier perimeter samples concerned only a selected component.\nThey do not test the stronger condition that the level lie below every\ncritical value.\nThe existence of a root-to-root curve of length less than $2$ is not established for the unrestricted\nclass by the arguments in this record.\nThe adjacent classical literature on\nlemniscate length concerns the arclength of the level curve $\\{|p|=1\\}$, which\nis Erd\\H{o}s~\\#114.  Fryntov and Nazarov recall its history\n\\cite[Introduction]{fryntovnazarov2008}: Dolzhenko's bound $4\\pi n$,\nPommerenke's bound $74n^2$ of 1961, and Borwein's bound $8\\pi en$\n\\cite{borwein1995}.  Eremenko and Hayman proved $9.173n$\n\\cite[Theorem~1]{eremenkohayman1999}, Fryntov and Nazarov the asymptotically\nsharp $2n+o(n)$ \\cite{fryntovnazarov2008}, and Tao resolved the problem for\nlarge $n$ \\cite{tao2025}.  These results bound the length of a level curve, not the least internal\nroot-pair path.  This distinction identifies the quantity being estimated;\nno claim of novelty follows from the scope or outcome of a literature search.\n\n\\section{Degree three}\n\\label{sec:degree-three}\n\nFor a cubic we can estimate the inverse roots directly after normalising\nat a critical point of least value modulus. The root-disc hypothesis\nmakes that modulus less than one, and two radial segments in the\nnormalised coordinates give a path shorter than $2$.\n\n\n\\begin{theorem}[the cubic case]\n\\label{res:degree-three}\nLet $f(z)=\\prod_{j=1}^{3}(z-z_j)$ with $|z_j|<1$, the roots listed with\nmultiplicity.  Then two listed root occurrences are joined inside\n$\\{|f|<1\\}$ by a polygonal path of length strictly below $2$.  If $f$ is\nsquarefree the two are distinct.\n\\end{theorem}\n\n\\begin{proof}\nMultiple roots give a constant path, so assume $f$ squarefree.\nThe Erd\\H{o}s--Herzog--Piranian component lemma\n\\cite[remark before Problem~5, p.~139]{ehp1958} gives a component of\n$\\{|f|<1\\}$ containing two roots. Join them by a compact path in that\nopen component, and choose a regular value modulus $t<1$ larger than\n$\\max|f|$ on the path. The component of $\\{|f|<t\\}$ containing it has\n$k\\ge2$ roots and hence $k-1\\ge1$ critical points, by the component-wise\nRiemann--Hurwitz count in the proof of \\cite[Proposition~2.1]{eks2010}.\nOne of these critical points has value of modulus below $t<1$. This\nchoice of $t$ avoids assuming that the level $|f|=1$ is regular.\n\nSuppose first that $f'$ has two distinct zeros.  Choose a critical point $c$\nminimising $|f(c)|$, write the other one as $c+\\delta$, and put $v=f(c)$, so\n$0<|v|<1$.  Monicity gives the exact expansion\n$f(c+d)=d^3-\\tfrac32\\delta d^2+v$.  Choose $\\alpha$ with $\\alpha^3=v$ and set\n$b=\\delta/\\alpha$; dividing by $v$ gives\n$f(c+\\alpha w)/v=P_b(w)$ with\n\\[\n P_b(w)=w^3-\\tfrac32bw^2+1 .\n\\]\nAt the other critical point $f(c+\\delta)=v(1-b^3/2)$, so minimality of $|v|$\nis exactly the hypothesis $|1-b^3/2|\\ge1$.\n\nUnder that hypothesis $P_b$ has at least two zeros, with multiplicity, in the\nclosed unit disc.  In the strict region a unit-circle zero $w$ would give\n$b=\\tfrac23(w+w^{-2})$. Write $w^3=e^{2iu}$; substitution gives\n\\[\n |1-b^3/2|^2\n =1+\\frac{64}{729}\\cos^4u\\,(16\\cos^2u-27)\\le1.\n\\]\nSince $16\\cos^2u-27<0$, equality requires $\\cos u=0$, which gives\n$b=0$. Thus the strict region has no unit-circle zero.  The radial deformation $b\\mapsto sb$, $s\\ge1$, stays in\nthe strict region, since $z=b^3/2$ and $|1-z|^2>1$ give\n$|1-s^3z|^2-1=s^3(s^3|z|^2-2\\Re z)>0$.  For large $s$, Rouch\\'e's theorem on\n$|w|=1$ compares $P_{sb}$ with $-\\tfrac32sbw^2$, whose modulus\n$\\tfrac32s|b|$ exceeds $2\\ge|w^3+1|$, so $P_{sb}$ has exactly two zeros in\nthe open unit disc; no zero crosses the circle along the deformation.  For the equality case with $b\\ne0$, the same radial deformation lies in\nthe strict region for every $s>1$, and continuity of the root multiset\nallows $s\\downarrow1$. If $b=0$, then $P_b(w)=w^3+1$ and all three\nroots lie on the unit circle.\n\nIf $P_b(w)=0$ and $0\\le t\\le1$, then\n$P_b(tw)=1-t^2-t^2(1-t)w^3$, so $|w|\\le1$ gives\n$|P_b(tw)|\\le(1-t^2)+t^2(1-t)|w|^3\\le1-t^3\\le1$.  The whole segment from $0$\nto $w$ therefore lies in the closed unit sublevel set.  Selecting two normalized roots $w_1,w_2$ with\n$|w_i|\\le1$, the two segments from $c$ to $c+\\alpha w_i$ lie in\n$\\{|f|\\le|v|\\}\\subset\\{|f|<1\\}$ and have combined length at most\n$2|\\alpha|=2|v|^{1/3}<2$.\n\nIf instead $f'$ has a double zero $c$, then $f(c+d)=d^3+v$. With $\\alpha^3=-v$ and\n$\\omega=e^{2\\pi i/3}$, its roots are\n$c+\\alpha$, $c+\\alpha\\omega$, $c+\\alpha\\omega^2$.  Averaging their squared\nmoduli gives $|c|^2+|\\alpha|^2<1$, so $|\\alpha|<1$. Any two segments from\n$c$ to the roots have total length $2|\\alpha|<2$, and along either segment\nthe value has modulus\n$|v|(1-t^3)<1$ away from the root endpoint.\n\\end{proof}\n\nThe unit level need not be regular, even when all roots lie in the open\nunit disc. For example,\n$Q(z)=(z-1)(z+1)(z+9/10)$ has $Q(1/3)=-148/135<-1$.\nIts minimum $v$ on $[-9/10,1]$ occurs at an interior critical point and\nsatisfies $v<-1$. Set $r=|v|^{-1/3}<1$. The monic cubic\n$r^3Q(z/r)$ has the three distinct roots $-r,-9r/10,r$ in the open unit\ndisc and a critical value equal to $-1$. Its unit level is therefore\nsingular, although the cubic path theorem applies.\n\n\\paragraph{Sources and scope.}\n\\label{bdry:degree-three}\nPendyala \\cite[Theorem~1, pp.~1--3]{june2026} proves the separate quartic\ncase; no historical priority for the cubic theorem is asserted here.\n\n\\section{An isolated simple critical value}\n\\label{sec:critical-value-separation}\n\\papersectiontarget{critical-value-separation-proof}\n\nA disc containing $0$ and one simple critical value selects a component\nof degree two. Taking a square root removes its local branching and\ngives a coordinate in which the proposed connecting curve is a real\nsegment. The ordinary argument\nbelow then bounds its length by area and capacity. Lean assumes this\ncombined analytic construction as \\texttt{DiscSepBergmanArea}; it does\nnot derive it from the polynomial hypotheses. We therefore record the\npath criterion and its numerical consequences as unformalised remarks.\n\nThe simplicity assumption means that the chosen critical point is a simple\nzero of the derivative, so two inverse branches meet there. A multiple\nchosen critical point is excluded; the other critical points need not be\nsimple. Also, separation of roots in the $z$-plane\nis not enough: different critical points can have nearly equal values.\nAn exact quartic example below makes this distinction explicit.\nFor a positive example, take $f(z)=z^3-3a^2z$ with $0<a<1/\\sqrt3$.\nIts roots $0,\\pm\\sqrt3a$ lie in the unit disc. At $c=a$ the critical\nvalue is $v=-2a^3$, and the other normalised critical value is $-1$.\nThus the disc centred at $1$ with radius $4/3$ contains $0$ and $1$\nbut excludes the other critical value. Radius $2$ is allowed as well:\nthe other critical value then lies on the boundary. The proof must\ntherefore allow a nonregular outer level.\n\n\\begin{remark}[unformalised isolated-value criterion]\n\\label{res:critical-value-separation}\nLet $P$ be a polynomial of degree $n\\ge3$ whose leading coefficient has\nmodulus one, with\n\\[\n P(0)=1,\\qquad P'(0)=0,\\qquad P''(0)\\ne0.\n\\]\nFix $w_0\\in[0,1]$ and $S>\\max(w_0,1-w_0)$.  Suppose every other critical\npoint $d\\ne0$ satisfies\n\\[\n |P(d)-w_0|\\ge S.                                      \\tag{4}\n\\]\nPut $p=w_0(1-w_0)$.  The two local solutions of\n$P(Z(\\xi))=1-\\xi^2$, $Z(0)=0$, continue along the real segment to one\ninjective curve $\\Gamma$ joining two roots.  Its endpoints are distinct roots,\n$\\Gamma\\subseteq\\{|P|\\le1\\}$, and\n\\begin{equation*}\\label{eq:disk-family-length}\n \\operatorname{length}(\\Gamma)^2\n \\le 2\\Bigl(\\frac{S}{n-1}\\Bigr)^{2/n}\n \\log\\!\\frac{S^2+S+p}{S^2-S+p}.                       \\tag{5}\n\\end{equation*}\nConsequently this curve has length less than $2$ whenever\n\\begin{equation*}\\label{eq:disk-family-coefficient}\n \\Bigl(\\frac{S}{n-1}\\Bigr)^{2/n}\n \\log\\!\\frac{S^2+S+p}{S^2-S+p}<2.                    \\tag{6}\n\\end{equation*}\n\\end{remark}\n\\evidenceremark{Lean assumes \\texttt{DiscSepBergmanArea}, which includes the\nsquare-root connector, its length estimate and the component-area\nbound. This input includes the combined analytic construction below; a\npublished Bergman inequality alone does not supply it. Its derivation from the stated polynomial\nhypotheses remains unformalised.}\n\n\n\\begin{proof}\nLet $Q=D(w_0,S)$ and let $U$ be the component of $P^{-1}(Q)$ containing\n$0$.  Both $0$ and $1$ lie in $Q$, and (4) says that $0$ is the only critical\npoint in $U$.  A component of a polynomial sublevel set is simply connected: on any\nJordan curve in the component, the maximum principle bounds the same\npolynomial throughout its interior, which must therefore remain in the\ncomponent. Condition (4) permits critical values on $\\partial Q$, so\nwe must not assume that $\\partial U$ is regular. For $1-w_0<S'<S$,\nlet $U'$ be the component of $\\{|P-w_0|<S'\\}$ containing $0$.\nThis level is regular, and $0$ contributes its only unit of ramification.\nThe component count in the proof of \\cite[Proposition~2.1]{eks2010}\ntherefore gives $\\deg(P|_{U'})=2$.\n\nThese nested components exhaust $U$. Indeed, a point of $U$ can be\njoined to $0$ by a compact path in $U$; the maximum of $|P-w_0|$ on\nthat path is strictly below $S$, so the path eventually lies in $U'$.\nFor any fixed regular value in $Q$, all its finitely many preimages in\n$U$ therefore lie in one such $U'$ once $S'$ is sufficiently large.\nThe proper map $P:U\\to Q$ consequently also has degree two. This\nestablishes the count without a smoothness assumption at the outer level.\n\nSet $a=1-w_0$.  On $U$, the function $1-P$ has only its double zero at $0$,\nso it has a single-valued square root\n\\[\n \\xi(z)^2=1-P(z).\n\\]\nTaking this square root removes the double branching at $0$.\nThe map $\\xi$ is proper into\n\\[\n \\widetilde Q=\\{\\xi:|\\xi^2-a|<S\\}.\n\\]\nAway from $0$, the identity $2\\xi\\xi'=-P'$ gives $\\xi'\\ne0$;\nat $0$, $(\\xi'(0))^2=-P''(0)/2\\ne0$. The target is star-shaped about\n$0$: because $S>a$, the convex disc $D(a,S)$ contains $0$, so\n$t^2\\xi^2\\in D(a,S)$ whenever $\\xi\\in\\widetilde Q$ and $0\\le t\\le1$.\nIt is therefore connected and simply connected. A proper local\nbiholomorphism has open and closed image, hence is a covering of this\ntarget. Since $U$ is connected, it is a conformal bijection. Its inverse\n$Z$ continues both local inverse branches through the critical point.  Since $[-1,1]\\subset\\widetilde Q$, the\ncurve $Z([-1,1])$ joins the two distinct points over $P=0$, and\n$P(Z(\\xi))=1-\\xi^2\\in[0,1]$ gives its containment.\n\nWe next map the parameter domain to the unit disc so that the curve\nbecomes a straight interval. This allows its length to be estimated by\nthe area of $U$, without replacing the domain by a larger disc.\nSquaring sends $\\widetilde Q$ onto $D(a,S)$. To map this disc to\n$\\mathbb D$ while fixing $0$, compose $w\\mapsto(w-a)/S$ with the disc\nautomorphism taking $-a/S$ to $0$. The result is the M\\\"obius map\n\\[\n w\\longmapsto\\frac{Sw}{S^2+aw-a^2}.\n\\]\nTaking its square root after substituting $w=\\xi^2$ gives the conformal\nbijection\n\\[\n \\zeta(\\xi)=\\xi\\sqrt{\\frac{S}{S^2+a\\xi^2-a^2}}\n     :\\widetilde Q\\longrightarrow\\mathbb D,\n\\]\nwhere the square-root factor is chosen positive at $0$. It sends\n$[-1,1]$ to $[-q,q]$, with\n\\[\n q^2=\\frac{S}{S^2+p},\\qquad\n S^2-S+p=(S-w_0)(S-(1-w_0))>0.\n\\]\nThe last inequality is exactly where $S>\\max(w_0,1-w_0)$ ensures\n$0<q<1$, as required by the length estimate below.\nFor $\\Phi=Z\\circ\\zeta^{-1}:\\mathbb D\\to U$, the Bergman segment inequality,\nproved by the kernel estimate of Section~\\ref{sec:low-critical-closure} with\nthe double integral taken over $[-q,q]^2$, gives\n\\begin{align*}\n \\operatorname{length}(\\Gamma)^2\n &\\le \\frac2\\pi\\log\\frac{1+q^2}{1-q^2}\n       \\operatorname{Area}(U)\\\\\n &=\\frac2\\pi\\log\\frac{S^2+S+p}{S^2-S+p}\n       \\operatorname{Area}(U).                       \\tag{7}\n\\end{align*}\n\nIt remains to bound the area of this component. The factor $n-1$ comes\nfrom the roots outside it, rather than from P\\'olya's inequality alone.\nThe auxiliary capacity estimate is as follows: for a monic degree-$n$\npolynomial $F$, a regular value $t>0$ of $|F|$, and a component $V$ of\n$\\{|F|<t\\}$ containing $k<n$ zeros counted with multiplicity,\n\\[\n \\frac{\\operatorname{cap}(\\overline V)^n}{t}<\\frac{k}{2n-k}.\n\\]\nMultiplying $F$ by a unimodular constant leaves the estimate unchanged.\nThis is a separate component estimate, not part of the\nRiemann--Hurwitz statement cited above. We give the harmonic-measure\nargument behind it, following\n\\href{https://github.com/wcook04/plectis-erdos/blob/8efbccc235df64a38d83f5dc7b1949e2ad18270d/research_corpus/Erdos1041/problem/ExteriorBlaschkeFibreCapacityGap.md}{Theorem~2 and Corollary~3 of the component-capacity proof}.\n\nLet $m=n-k$ and let $\\psi$ map $|\\zeta|>1$ conformally onto the\nexterior of $\\overline V$, with positive leading coefficient\n$\\operatorname{cap}(\\overline V)$. The $m$ excluded roots have preimages\n$\\xi_j$ with $|\\xi_j|>1$. Put $b_j=1/\\overline{\\xi_j}$ and form\n\\[\n B(z)=\\prod_{j=1}^m\\frac{z-b_j}{1-\\overline{b_j}z}.\n\\]\nReflection in the unit circle shows that $F(\\psi(\\zeta))/t$ equals\n$\\zeta^n\\overline{B(1/\\overline\\zeta)}$ up to a constant of modulus one:\ntheir zeros and pole at infinity agree, and both have modulus one on\nthe boundary. Comparing leading coefficients gives\n\\[\n |B(0)|=\\frac{\\operatorname{cap}(\\overline V)^n}{t}.\n\\]\nRegularity of the level permits analytic continuation of $\\psi$ across\nthe boundary. On the unit circle,\n\\[\n \\frac{d}{d\\theta}\\arg F(\\psi(e^{i\\theta}))\n =n-|B'(e^{i\\theta})|>0.\n\\]\nPositivity follows because a regular polynomial level is mapped locally\nin the positive boundary direction. Thus $|B'|<n$ there.\n\nThe boundary-fibre identity used next is Lemma~1 of the same\ncomponent-capacity proof. Here is its harmonic-measure argument.\nFor a continuous real function $h$ on the unit circle, let $H$ be its\nharmonic extension to the disc. Since $H\\circ B$ is harmonic, the mean\nvalue property and the Poisson formula give\n\\[\n \\frac1{2\\pi}\\int_0^{2\\pi}h(B(e^{i\\theta}))\\,d\\theta\n =H(B(0))\n =\\frac1{2\\pi}\\int_0^{2\\pi}h(e^{i\\phi})\n   \\frac{1-|B(0)|^2}{|e^{i\\phi}-B(0)|^2}\\,d\\phi.\n\\]\nDifferentiating the factors gives the positive angular derivative\n\\[\n \\frac{d}{d\\theta}\\arg B(e^{i\\theta})\n =|B'(e^{i\\theta})|\n =\\sum_{j=1}^m\\frac{1-|b_j|^2}{|e^{i\\theta}-b_j|^2}>0.\n\\]\nChanging variables through the $m$ boundary inverse branches in the\nfirst integral and comparing the continuous densities therefore yields\n\\[\n \\sum_{B(\\zeta)=w}\\frac1{|B'(\\zeta)|}\n =\\frac{1-|B(0)|^2}{|w-B(0)|^2}\\qquad(|w|=1).\n\\]\nIn particular each boundary fibre consists of $m$ distinct points.\nThe Poisson kernel on the right is smallest at the boundary point\nopposite $B(0)$. Choosing $w=-B(0)/|B(0)|$ therefore yields\n\\[\n \\frac mn<\\frac{1-|B(0)|}{1+|B(0)|},\n \\qquad |B(0)|<\\frac{n-m}{n+m}=\\frac{k}{2n-k}.\n\\]\nHere $B(0)\\ne0$ because every excluded-root preimage is finite. This proves\nthe component estimate and explains how the excluded roots enter it.\n\nFor the regular inner components $U'$ above, the degree-two count gives\nexactly two zeros of $P-w_0$, counted with multiplicity. Since $2<n$,\nthe preceding estimate applies with $k=2$ and gives\n\\[\n \\frac{\\operatorname{cap}(\\overline{U'})^n}{S'}\n <\\frac{k}{2n-k}=\\frac1{n-1}.\n\\]\nP\\'olya's area--capacity inequality\n$\\operatorname{Area}(K)\\le\\pi\\operatorname{cap}(K)^2$ \\cite[Theorem~6]{crane}\nthen yields\n$\\operatorname{Area}(U')<\\pi(S'/(n-1))^{2/n}$. The exhaustion proved\nat the start and continuity of area from below give, without assuming\nregularity of the limiting boundary,\n\\[\n \\operatorname{Area}(U)\\le\\pi\\Bigl(\\frac{S}{n-1}\\Bigr)^{2/n}.\n\\]\nSubstitution into (7) proves (5), and (6) makes its right side strictly less\nthan $4$.\n\\end{proof}\n\n\\begin{remark}[unformalised numerical consequences]\n\\label{res:critical-value-thresholds}\nInequality~\\eqref{eq:disk-family-coefficient} holds for every $n\\ge3$,\nevery $w_0\\in[0,1]$, and every $4/3\\le S\\le2$.  Thus, if $f$ is monic with\nroots in the open unit disc, $c$ is a simple critical point with\n$v=f(c)\\ne0$ and $|v|<1$, and\n\\[\n \\left|\\frac{f(d)}v-w_0\\right|\\ge\\frac43\n\\]\nfor every other critical point $d$, then two roots of $f$ are joined inside\n$\\{|f|<1\\}$ by a curve of length strictly below $2$.\n\\end{remark}\n\\evidenceremark{Lean checks the displayed coefficient estimates and numerical\nsteps. Their path conclusions still require \\texttt{DiscSepBergmanArea},\nthe unformalised construction used in\nRemark~\\ref{res:critical-value-separation}.}\n\n\n\\begin{proof}\nFor $S\\ge4/3$ and $p\\ge0$,\n\\[\n \\frac{S^2+S+p}{S^2-S+p}\\le\\frac{S+1}{S-1}\\le7.\n\\]\nAlso $S/(n-1)\\le1$ when $S\\le2$ and $n\\ge3$, while $\\log7<2$.\nThis proves~\\eqref{eq:disk-family-coefficient}.  For the unnormalised\npolynomial, set $z=c+|v|^{1/n}w$. Then $P(w)=f(z)/v$ has leading\ncoefficient $|v|/v$, of modulus one. Scaling the connector multiplies its\nlength by $|v|^{1/n}<1$ and takes $\\{|P|\\le1\\}$ to\n$\\{|f|\\le|v|\\}\\subset\\{|f|<1\\}$. The selected value $v$ need not attain\nthe minimum defining $\\mu$.\n\\end{proof}\n\nTaking $w_0=1$ and $S=2$ gives the earlier criterion\n$|1-f(d)/v|\\ge2$ in every degree $n\\ge3$. If $c$ attains the minimum\nmodulus among \\emph{all} critical values and $v\\ne0$, the\nFekete--resultant bound gives $|v|\\le R^n<1$ when all roots lie in a disc\nof radius $R<1$. Deleting zero critical values before taking the minimum\nwould invalidate that inference. For example, $(z-R)^2(z+R)$ with $0<R<1$ has critical\nvalues $0$ and $32R^3/27$, so its least nonzero critical-value modulus\nexceeds $R^3$. The repeated root already gives a constant path, but it\ndoes not give this nonzero value the required bound. For a critical point\nchosen in another way, $|v|<1$ must remain an explicit hypothesis.\nIn degree three the choice $w_0=1$ already works at\n$S=6/5$, since\n\\[\n \\Bigl(\\frac35\\Bigr)^{2/3}\\log 11<2.\n\\]\nLean checks this sharper cubic constant.\n\nRemark~\\ref{res:critical-value-separation} imposes separation in the\nvalue plane. Separation of the roots or critical points alone does not\nimply its hypothesis.\nTo distinguish these conditions, consider the family\n\\[\n g_\\varepsilon(z)=z^4-\\varepsilon z^3-2z^2+3\\varepsilon z+\\tfrac12,\n \\qquad 0<\\varepsilon\\le\\tfrac1{32}.\n\\]\nIts derivative and two critical values are\n\\[\n g_\\varepsilon'(z)=(z^2-1)(4z-3\\varepsilon),\\qquad\n g_\\varepsilon(\\pm1)=-\\tfrac12\\pm2\\varepsilon.\n\\]\nAt the endpoints of each of the intervals\n$(-3/2,-5/4)$, $(-3/4,-1/2)$, $(1/2,3/4)$ and $(5/4,3/2)$,\nthe values of $g_\\varepsilon$ have opposite signs throughout this parameter\nrange. Each interval therefore contains exactly one root. Together with\nthe critical points $-1,3\\varepsilon/4,1$, these give seven points at\npairwise distances greater than $1/4$, uniformly in $\\varepsilon$.\nNevertheless the two displayed critical values differ by $4\\varepsilon\\to0$.\nThe earlier numerical choice is the exact member $\\varepsilon=1/262144$:\n\\[\n g(z)=z^4-\\frac{z^3}{262144}-2z^2+\\frac{3z}{262144}+\\frac12,\n \\qquad\n g'(z)=4(z+1)(z-3/1048576)(z-1),\n\\]\nfor which that difference is $1/65536$.\nThe monic polynomial $2^{-4}g_\\varepsilon(2z)$ puts every root inside\n$|z|<3/4$ and retains a spatial separation greater than $1/8$.\nFor the critical points $\\pm1/2$ of the scaled polynomial, the ratio\nof the two values is unchanged and obeys\n\\[\n \\frac{g_\\varepsilon(-1)}{g_\\varepsilon(1)}\n =\\frac{1+4\\varepsilon}{1-4\\varepsilon}\\le\\frac97<\\frac43.\n\\]\nThe reciprocal ratio lies between $0$ and $1$. Hence, for any centre\n$w_0\\in[0,1]$, neither selected value satisfies the theorem's\nradius-$4/3$ isolation condition: the other normalised value is less\nthan $4/3$ away. This does not exclude a suitable isolated value at a\n\\emph{different} critical point.\n\n\\paragraph{Sources and scope.}\n\\label{bdry:critical-value-separation}\nThe two-sheeted component, its square-root uniformisation, the Bergman segment\ninequality, the component-capacity estimate, P\\'olya's area--capacity\ninequality and the exhaustion are ordinary arguments. They are collected in\n\\href{https://github.com/wcook04/plectis-erdos/blob/8efbccc235df64a38d83f5dc7b1949e2ad18270d/research_corpus/Erdos1041/problem/DiskFamilyCriticalValueSeparation.md}{the proof using an isolated critical value}, whose capacity step is cited\nseparately above.\n\nThe proof requires both a simple critical point and a disc centred on\n$[0,1]$ that isolates its normalised value. For example, the trinomial\n$z^n-b$ with $n>2$ and $0<|b|<1$ has a short path by\nTheorem~\\ref{res:trinomial-all-degree}, but its only critical point is not\nsimple, so this method does not apply. The restriction is on the chosen\ncritical point and on the distance of the other critical values from the\nchosen disc. The other critical points may be multiple, and their values\nneed not be separated from one another. No sharpness is claimed for\n$S=4/3$, and no assertion is made that every polynomial has a suitable\nisolated simple value. In particular, this is not a proof of the\nunrestricted assertion of Erd\\H{o}s~\\#1041.\n\nA related area comparison is Dubinin's Theorem~1. The\nsource is \\cite{dubinin}, Theorem~1 on printed page~85 of the POMI original.\nIt concerns a holomorphic function that gives a full $n$-fold covering of an\nannulus $t_1<|w|<t_2$, and, in the notation of that theorem, with $E$ its\nexplicitly defined complementary set, it states\n\\[\n \\Bigl(\\frac{t_2}{t_1}\\Bigr)^{2/n}\\le\\frac{m(E\\cup D)}{m(E)}.\n\\]\nThe full covering and the explicit complementary set are hypotheses of the\ntheorem.  Tao cited it on the Erd\\H{o}s Problem~\\#1041 discussion page on\n25 March 2026 for the relative scaling factor $s^{2/n}$ between the areas of\ntwo nested sublevel sets, which is the reading of the displayed inequality in\nwhich $E$ and $E\\cup D$ are those two sublevel sets and the covering\nhypothesis holds.  No step of this note uses that relative inequality.  The\ndisk-family theorem instead uses P\\'olya's absolute area--capacity inequality\non a component containing fewer than $n$ roots, after the\ncomponent-capacity estimate supplies its strict\ncapacity gap.  The absolute sublevel inequality\n$\\operatorname{Area}\\{|P|<T\\}\\le\\pi T^{2/n}$ remains the area input to\nRemark~\\ref{res:low-critical-thirteen-twentyfifths} and to\nRemark~\\ref{res:constant-factor-path}.  Dubinin's theorem is a neighbouring\nresult under a full covering hypothesis and supplies no curve joining two roots.\n\n\\section{Examples separating the geometric hypotheses}\\label{sec:component-examples}\n\\papersectiontarget{component-counterexamples}\n\nThe two critical-value hypotheses leave different restrictions on a\npolynomial. We first give a cubic satisfying neither of them. A second\ncubic shows why the root count at the first merger cannot determine\ncapacity at a later level.\n\n\\begin{proposition}[failure of two critical-value criteria to cover all polynomials]\n\\label{res:sep-or-false}\nLet $f(z)=z^3+(3/100)z-3/4$.  Every root lies in the open unit disc, both\ncritical points are simple, the critical values lie on distinct positive\nrays, $\\mu>13/25$, and\n\\[\n \\bigl|1-f(c_-)/f(c_+)\\bigr|<2/375<2.\n\\]\n\\end{proposition}\n\\begin{proof}\nOn $|z|=1$, $|3z/100-3/4|\\le78/100<1$, so Rouch\\'e's theorem puts\nall three roots in the disc. The critical points are $c_\\pm=\\pm i/10$,\nand $f(c_\\pm)=-3/4\\pm i/500$. They are nonzero, have distinct arguments\nand modulus greater than $3/4$. Moreover\n\\[\n \\left|1-\\frac{f(c_-)}{f(c_+)}\\right|\n =\\frac{1/250}{|f(c_+)|}<\\frac2{375}.\n\\]\nFor any $w_0\\in[0,1]$, the other normalised value is at distance less\nthan $1+2/375<4/3$ from $w_0$, so the radius-$4/3$ hypothesis fails.\n\\end{proof}\n\nA component with two roots immediately after the first merger may\ncontain more roots at a later level. Thus its first root count alone\ndoes not give a capacity saving at level $2\\mu$.\n\n\\begin{proposition}[root count does not force a capacity gap]\n\\label{res:arity-not-capacity}\nLet $g(z)=z^3-(3/400)z-3/32$. All roots lie in the open unit disc and\n$\\mu=187/2000\\le1/2$. The first merger joins two root components, so\n$k_0=2$, but the component at level $2\\mu$ containing that pair has\nnormalised capacity $1$.\n\\end{proposition}\n\\evidenceremark{The capacity clause uses the classical monic-lemniscate\nformula $\\operatorname{cap}\\{|p|\\le t\\}=t^{1/n}$\n\\cite[\\S2]{crane}, \\cite[Theorem~5.2.5]{ransford1995}, with capacity equal to transfinite\ndiameter by Fekete--Szeg\\H{o} \\cite[p.~153]{ransford1995}.\nThis is the named input \\texttt{LemniscateTransfiniteDiameter} in Lean;\nthe root count $k_0=2$ is proved without it.}\n\\begin{proof}\nRouch\\'e's theorem on the unit circle applies because\n$3/400+3/32<1$. The critical points are $\\pm1/20$, with critical values\n$-187/2000$ and $-188/2000$. The first merger therefore has one simple\ncritical point and joins two root components. At level\n$2\\mu=374/2000$, both critical points are inside the filled lemniscate.\nThe component count $n-(n-1)=1$ from Riemann--Hurwitz shows that it is\nconnected. Its capacity is $(2\\mu)^{1/3}$\n\\cite[Theorem~5.2.5]{ransford1995}, so division by that scale gives one.\nThe use of logarithmic capacity agrees with transfinite diameter by the\nFekete--Szeg\\H{o} theorem \\cite[p.~153]{ransford1995}.\n\\end{proof}\n\nThe degree-eight perimeter example, Proposition~\\ref{res:one-root-gamma-false},\nrequires no root-disc hypothesis. It refutes the proposed Gamma constant,\nwhile the existence of a uniform one-root perimeter bound remains open\nin Problem~\\ref{prob:conjecture-p}. The Runge construction reported in\n\\cite[Proposition~2]{revision2026} has no supplied proof and is not used.\n\n\\section{Collinear roots and two sparse polynomial families}\n\\label{sec:solved-polynomial-families}\n\nThe next constructions use the roots or coefficients directly. For\ncollinear roots we compare with a rescaled Chebyshev polynomial. In a\nquintic with two missing coefficients, the resulting moment identities\nselect two radial segments. A cubic composed with a power map provides\na different use of a single contained segment: its rotational copies\njoin through the centre of a fibre.\n\n\\subsection{Collinear roots and Chebyshev comparison}\n\\label{subsec:sharp-collinear-chebyshev}\n\\papersectiontarget{collinear-proof}\n\nWe scale the extreme zeros of $T_n$ to $\\pm1$ and make the polynomial\nmonic. Its endpoint zeros then agree with those of the polynomial under\nstudy, while subtraction lowers the degree. This is the comparison\nneeded for the alternation argument. Put\n\\[\n r_n=\\cos\\frac{\\pi}{2n},\n \\qquad C_n=\\frac{1}{2^{n-1}r_n^n}.\n\\]\n\n\\begin{theorem}[Chebyshev comparison]\n\\label{prop:sharp-collinear-chebyshev-comparator}\nLet $m\\ge0$, let $p\\in\\mathbb R[X]$ be monic of degree $m+2$, and let\n\\[\n -1<c_0<\\cdots<c_m<1,\\qquad |c_i|\\le1.\n\\]\nSuppose $p(-1)=p(1)=0$ and\n$p(c_i)p(c_{i+1})<0$ for $0\\le i<m$.  Then\n\\[\n \\min_{0\\le i\\le m}|p(c_i)|\\le C_{m+2}.\n\\]\n\\end{theorem}\n\nThis theorem is the finite alternation inequality; it does not construct\nthe segment between two roots.\n\nAll real-rooted polynomials, and their images under rotations and\ntranslations, satisfy the collinearity hypothesis. Three noncollinear\nroots already fall outside it. The gain is an explicit sharp bound for\nthe modulus along one adjacent-root segment, not a bound obtained by\nassuming that segment is contained.\n\n\\begin{theorem}[a sharp bound for collinear roots]\n\\label{thm:sharp-collinear-diameter}\nLet $f$ be a monic polynomial of degree $n\\ge2$ whose zero occurrences are\ncollinear, and let $D$ be their diameter.  Some two adjacent zero occurrences\nare joined by a segment of length at most $D$ on which\n\\[\n |f(z)|\\le\n \\frac{(D/2)^n}{2^{n-1}\\cos^n(\\pi/(2n))}.          \\tag{9}\n\\]\nThe constant in \\emph{(9)} is best possible in every degree.  Equality is\nattained by affine images of the zeros of $T_n$ whose extreme zeros have\ndistance $D$.\n\\end{theorem}\n\n\\begin{proof}\nA repeated zero gives the constant path, so assume the zero values are\ndistinct. Let $m$ be the midpoint of the two extreme zeros, and choose\n$\\theta\\in\\mathbb R$ so that their line is $m+e^{i\\theta}\\mathbb R$.\nPut $R=D/2$. Translation by $m$, rotation by $e^{i\\theta}$ and scaling\nby $R$ give the polynomial\n\\[\n q(w)=R^{-n}e^{-in\\theta}f(m+Re^{i\\theta}w),\n\\]\nwhich is monic with real zeros\n$-1=y_1<\\cdots<y_n=1$, and\n$|f(m+Re^{i\\theta}w)|=R^n|q(w)|$.\n\nCompare $q$ with the monic endpoint-normalized Chebyshev polynomial\n\\[\n q_*(x)=\\frac{T_n(r_nx)}{2^{n-1}r_n^n}.\n\\]\nBoth $q$ and $q_*$ vanish at $\\pm1$, and $|q_*|\\le C_n$ on\n$[-1,1]$.  For each gap $[y_i,y_{i+1}]$, choose $c_i$ at which\n$|q|$ is maximal. The maximum lies in the interior: $q$ vanishes at the\nendpoints and nowhere inside the gap. Since every root is simple, the\nsigns at the $n-1$ chosen points alternate. If every gap maximum were\nlarger than $C_n$, then $q-q_*$ would have the same nonzero signs there.\nThe intermediate value theorem would give $n-2$ zeros between consecutive\n$c_i$, in addition to the two zeros at $\\pm1$. These $n$ zeros are distinct,\nbut $q-q_*$ is nonzero and has degree at most $n-1$ because its leading\nterms cancel.\nThis contradiction selects a gap on which $|q|\\le C_n$.  Scaling back proves\n\\emph{(9)}.\n\nFor sharpness, take\n\\[\n y_k=\\frac{\\cos((2k-1)\\pi/(2n))}{r_n},\n \\qquad 1\\le k\\le n.\n\\]\nThese are the zeros of $q_*$, have extremes $\\pm1$, and every adjacent gap\ncontains a scaled Chebyshev extremum where $|q_*|=C_n$.  No smaller universal\nconstant can work.\n\\end{proof}\n\n\\begin{corollary}[collinear Erd\\H{o}s case]\n\\label{cor:collinear-erdos-1041}\nIf the zero occurrences of a monic polynomial of degree $n\\ge2$ lie on one line\nin the open unit disc, two of them are joined by a curve of length strictly\nbelow $2$ inside $\\{|f|<1\\}$.\n\\end{corollary}\n\n\\begin{proof}\nA repeated zero gives the constant path, so assume the zeros are distinct.\nTheir diameter satisfies $0<D<2$. Since\n$\\cos(\\pi/(2n))\\ge1/\\sqrt2$, the defining formula gives\n$C_n\\le2^{1-n/2}\\le1$, with equality possible only at $n=2$.\nThus \\emph{(9)} is strictly below one, and the selected segment has\nlength at most $D<2$.\n\\end{proof}\n\nCorollary~\\ref{cor:collinear-erdos-1041} already follows from a theorem of\nErd\\H{o}s, Herzog and Piranian \\cite[Theorem~1, p.~126]{ehp1958}: if the zeros\nof a monic polynomial of degree $n$ are real, lie in $[-1,1]$ and have\ncentroid in $[0,1]$, then $\\{|f|<1\\}\\cap\\mathbb R$ contains an interval\nholding at least $n/2$ of the zeros.  Apply it to the normalized polynomial\n$q$, replacing it by the monic polynomial $(-1)^nq(-w)$ if its centroid\nis negative. This reverses the root line without changing modulus bounds. For\n$n\\ge3$ two consecutive zeros in that interval are joined by a segment of\nlength at most $D$ inside $\\{|f|<(D/2)^n\\}$, and for $n=2$ the segment between\nthe zeros lies in $\\{|f|\\le(D/2)^2\\}$.\nTheorem~\\ref{thm:sharp-collinear-diameter} adds the sharp level and its\nequality configurations.\n\n\\paragraph{Critical sequences and the extremal configuration.}\n\\hypertarget{collinear-source-comparison}{}\nFor distinct real zeros, each gap between consecutive zeros contains exactly\none zero of $q'$, and $|q|$ attains its maximum over the gap only there.  The\npoints $c_i$ chosen in the proof of Theorem~\\ref{thm:sharp-collinear-diameter}\nare therefore the critical points of $q$, and the gap maxima are the moduli of\nits ordered critical sequence $(q(c_1),\\ldots,q(c_{n-1}))$, with\n$c_1<\\cdots<c_{n-1}$. In this distinct-root setting, the ordered sequence\ndetermines the real polynomial up to an increasing real affine change\nof variable \\cite[Theorem~1]{eremenkoyuditskii2012}. The proof above uses\nonly the alternation count, not this classification theorem.  In this language, \\emph{(9)} says that a\nreal polynomial of degree $n$ with $n$ distinct real zeros, leading\ncoefficient $a$ and zero diameter $D$ has a critical value of modulus at most\n$C_n|a|(D/2)^n$.\n\nThe comparison polynomial $q_*$ is the monic polynomial of least deviation\nfrom zero on $[-1/r_n,1/r_n]$, characterised by the equioscillation conditions\nrecalled in \\cite[\\S1]{eremenkoyuditskii2012}.  All $n-1$ of its critical\nvalues have modulus $C_n$, so every critical point of $q_*$ lies on the level\ncurve $\\{|q_*|=C_n\\}$.  Extremals with this property also occur in the\nlevel-curve length problem, where some extremal polynomial has all its\ncritical points on $\\{|p|=1\\}$ \\cite[Lemma~5]{eremenkohayman1999}, and in the\nsharp bound for $|f'|$ on a connected sublevel set $\\{|f|\\le1\\}$ of a monic\npolynomial, proved by Eremenko and Lempert\n\\cite[Theorem~1]{eremenkolempert1994}, whose\nequality cases are $e^{-in\\theta}T_n(2^{1/n-1}e^{i\\theta}z+b)$ with $\\theta$\nreal \\cite[Theorem~A]{eremenko2007}.\n\n\\paragraph{Scope.}\n\\label{rem:sharp-collinear-formal-boundary}\nThe sharpness assertion concerns the maximum of $|f|$ on the selected adjacent-root\nsegment, and the displayed configurations attain it; no claim is made that\nthey exhaust the equality cases, and no claim is made that the selected\nsegment is a shortest path in $\\{|f|<1\\}$.\n\n\\subsection{Quintics with two missing coefficients}\n\\label{subsec:primitive-sparse-quintic}\n\\papersectiontarget{quintic-proof}\n\nThe missing cubic and quadratic terms of $z^5+az^4+bz+c$ give three\nidentities among the real and radial coordinates of its roots. We first\nuse those identities to select two indices whose radial segments will\nbe contained. For $0<r<2$ and $0\\le s_i\\le1$, $x_i^2\\le s_i$, put\n\\[\n E_i=s_i^4(s_i+r^2+2rx_i).\n\\]\n\n\\begin{theorem}[a consequence of three moment identities]\n\\label{prop:primitive-quintic-two-tail-energy-selector}\nSuppose\n\\[\n \\sum_{i=0}^4x_i=-r,\\qquad\n \\sum_{i=0}^4(2x_i^2-s_i)=r^2,\\qquad\n \\sum_{i=0}^4(4x_i^3-3s_ix_i)=-r^3.              \\tag{10}\n\\]\nThen at least one of the ten pairs $0\\le i<j\\le4$ satisfies\n$E_i<1$ and $E_j<1$.\n\\end{theorem}\n\nThe theorem selects two distinct indices.  It does not assert that the\ncorresponding complex root values are distinct.\n\nThe polynomial must have no $z^3$ or $z^2$ term in the coordinates used\nhere. For example, $z^5+bz+c$ is included, but a general quintic with all\ncoefficients nonzero is not. The root-location assumption alone does not\nimply the three moment identities. Those identities are precisely what\nlets the finite inequality find two radial segments.\n\n\\begin{theorem}[a quintic with two missing coefficients]\n\\label{thm:primitive-quintic-two-tail}\nLet\n\\[\n p(z)=z^5+az^4+bz+c\n\\]\nand suppose its five zero occurrences $w_0,\\ldots,w_4$ lie in the closed unit\ndisc.  At least two distinct indices satisfy\n\\[\n |bw_i+c|\\le1.                                    \\tag{11}\n\\]\nIf $a\\ne0$, two indices can be chosen with strict inequalities.  If $a=0$,\nevery index satisfies \\emph{(11)}, and equality holds exactly when\n$|w_i|=1$.\n\nFor open-disc zeros, two zero occurrences are joined inside $\\{|p|<1\\}$ by a\ncurve of length below $2$: use the two radial spokes through $0$ when their\nvalues are distinct, and the constant path when the selected occurrences\nhave the same value.\n\\end{theorem}\n\n\\begin{proof}\nIf $a=0$, the root equation gives $|bw_i+c|=|w_i|^5$ for every\nindex, proving \\emph{(11)} and its equality clause. For $a\\ne0$, write\n$a=re^{i\\phi}$ with $r>0$ and set $z_i=e^{-i\\phi}w_i$.\nThe missing $z^3$ and $z^2$ coefficients and Newton's identities give\n\\[\n \\sum z_i=-r,\\qquad \\sum z_i^2=r^2,\\qquad\n \\sum z_i^3=-r^3.                                 \\tag{12}\n\\]\nAt a zero,\n\\[\n |bw_i+c|=|w_i|^4|w_i+a|.\n\\]\nThus, for $x_i=\\Re z_i$ and $s_i=|z_i|^2$, the squared modulus $|bw_i+c|^2$ is exactly\n$E_i$ and \\emph{(12)} becomes \\emph{(10)}. The third moment gives\n$r^3=|\\sum_i z_i^3|\\le5<8$, so $0<r<2$ as required by the finite\ninequality.\n\nWe need to rule out four indices with squared modulus at least $1$.\nThe three known power sums determine the sum of any real harmonic\npolynomial of degree at most three over the roots. On the unit circle,\n$s^4(s+r^2+2rx)\\ge1$ reduces to $x\\ge-r/2$. We therefore use the harmonic\nextension of $(-r/2-x)(1-x)^2$, which is nonpositive on that part of the\ncircle. In the variables $x=\\Re z$ and $s=|z|^2$, this extension is\n\\[\n H_r(x,s)=(-r/2-x)(1-x)^2\n +(1-s)\\left(1-\\frac r4-\\frac{3x}{4}\\right).\n\\]\nThe useful feature of this choice is the expansion\n\\[\n H_r(\\Re z,|z|^2)\n =1-\\frac{3r}{4}+\n \\Re\\!\\left[\\left(r-\\frac74\\right)z+\n             \\left(1-\\frac r4\\right)z^2-\\frac{z^3}{4}\\right].\n\\]\nIt is harmonic, and summing this expression with the three moments in\n\\emph{(12)} gives\n\\[\n \\sum_{i=0}^4H_r(x_i,s_i)=5-2r.                  \\tag{13}\n\\]\nOn the unit circle its value is $(-r/2-x)(1-x)^2$, and\n\\[\n 4-2r-(-r/2-x)(1-x)^2\n =(x+1)\\left(\\frac{(2-r)(3-x)}2+(1-x)^2\\right)\\ge0.\n\\]\nThe maximum principle therefore gives $H_r\\le4-2r$ throughout the disc.\nFor an index with $s^4(s+r^2+2rx)\\ge1$, one has $s>0$ and\n$s^{-4}\\ge1+4(1-s)$ by convexity on $(0,1]$. Consequently\n\\[\n 2r(x+r/2)\\ge s^{-4}-s\\ge5(1-s).\n\\]\nPut $d=x+r/2$, $u=1-x$, $A=1-r/4-3x/4$, and\n$P=-u^2+(2r/5)A$.  The assumption $s^4(s+r^2+2rx)\\ge1$ gives $0\\le d<2$.  Since\n$H_r=-du^2+(1-s)A$, it is nonpositive when $A\\le0$; when $A>0$,\nthe preceding bound gives $H_r\\le dP$.  The remaining estimate is the exact\nsum-of-squares identity\n\\[\n \\frac1{31}-P\n =\\frac25\\left(d-\\frac{38}{31}\n               +\\frac14(u-\\frac3{31})\\right)^2\n  +\\frac{31}{40}\\left(u-\\frac3{31}\\right)^2\\ge0,\n\\]\nwhich gives $P\\le1/31$, and hence $H_r(x,s)\\le2/31$ whether $P$ is\npositive or nonpositive.  Thus $H_r\\le2/31$ whenever the squared modulus is at least $1$.\nIf four indices had squared modulus at least $1$, \\emph{(13)} would give\n\\[\n 5-2r\\le4-2r+\\frac8{31}<5-2r,\n\\]\na contradiction. This proves the two-index conclusion when $a\\ne0$.\n\nFor either value of $a$, at a zero $w$,\n\\[\n p(tw)=(1-t)c+(t-t^4)(bw+c)-t^4(1-t)w^5.\n\\]\nFor $0\\le t\\le1$ the three nonnegative weights sum to $1-t^5$.\nThe tail bound, $|w|\\le1$, and $|c|\\le1$ therefore keep each selected\nradial segment in $\\{|p|\\le1\\}$. Under the open-disc hypothesis,\n$|c|<1$. Its coefficient $1-t$ is positive for $0\\le t<1$, so\n\\[\n |p(tw)|<(1-t)+(t-t^4)+t^4(1-t)=1-t^5\\le1.\n\\]\nAt $t=1$ the value is zero. Thus both segments lie in $\\{|p|<1\\}$,\nand their total length is $|w_i|+|w_j|<2$.\n\\end{proof}\n\n\\subsection{Polynomials obtained from a cubic by a power substitution}\n\\label{subsec:translated-cubic-quotient-fibres}\n\\papersectiontarget{cubic-fibre-proof}\n\nFor $f(z)=P((z-h)^q)$, each nonzero root of $P$ gives a regular\n$q$-gon of roots centred at $h$. We exploit this symmetry by lifting\none contained segment for the cubic into two segments through $h$.\nA generic polynomial of degree $3q$ has no such fibre structure.\nThe first theorem supplies the segment to be lifted.\n\n\\begin{theorem}[a contained radial segment for a cubic]\n\\label{lem:cubic-safe-root-spoke}\nIf $r,s,v\\in\\mathbb C$ have modulus below one, at least one\n$u\\in\\{r,s,v\\}$ satisfies\n\\[\n \\left|(tu-r)(tu-s)(tu-v)\\right|\\le1\n \\qquad(0\\le t\\le1).\n\\]\n\\end{theorem}\n\n\\begin{theorem}[a cubic composed with a power map]\n\\label{thm:translated-cubic-quotient-fibres}\nLet $q\\ge2$, $h\\in\\mathbb C$, $P$ be monic cubic, and\n\\[\n f(z)=P((z-h)^q).\n\\]\nIf every zero of $f$ lies in the open unit disc and $f$ has at least two\ndistinct zero values, then two zeros are joined through $h$ by a two-segment\npath of length below $2$ inside $\\{|f|<1\\}$. Equivalently, under the same\nhypotheses the conclusion holds for polynomials of the form\n\\[\n (z-h)^{3q}+A(z-h)^{2q}+B(z-h)^q+C\n\\]\nin every degree $3q\\ge6$.\n\\end{theorem}\n\n\\begin{proof}\nOne nonzero root of $P$ will supply both endpoints of the path through\n$h$. Before selecting it, we check that every zero of $P$ lies in the\nopen unit disc. Fix a $q$th root $y$ of a zero of $P$ and a primitive\n$q$th root of unity $\\zeta$. The full fibre consists of $h+y\\zeta^k$.  Since every fibre point\nlies in the open unit disc,\n\\[\n \\frac1q\\sum_{k=0}^{q-1}|h+y\\zeta^k|^2\n   =|h|^2+|y|^2<1.                                \\tag{14}\n\\]\nThus $|y|<1$, and the corresponding quotient root has modulus\n$|y|^q<1$.  This verifies the open-disc hypothesis for all quotient roots\nbefore selecting a radial segment for the cubic.\nThe identity retains more information than these separate bounds:\n$|h|<1$ and every fibre radius is strictly below $\\sqrt{1-|h|^2}$.\nConsequently, any two contained fibre segments constructed below have total length\nstrictly less than $2\\sqrt{1-|h|^2}$.  The position of the symmetry centre\ntherefore improves the length bound; replacing it by $2$ discards this information.\n\nWrite $P(w)=(w-r)(w-s)(w-v)$ and associate to $r$ the real number\n$A_r=\\Re(r\\overline{s+v})$, cyclically.  The exact sum is\n\\[\n A_r+A_s+A_v=|r+s+v|^2-(|r|^2+|s|^2+|v|^2)>-3,\n\\]\nso one of these numbers, say $A_r$, exceeds $-1$. For $0\\le t\\le1$,\n\\[\n \\frac{|tr-s|^2+|tr-v|^2}{2}\n =t^2|r|^2+\\frac{|s|^2+|v|^2}{2}-tA_r<1+t+t^2.\n\\]\nAM--GM now gives $|tr-s|\\,|tr-v|<1+t+t^2$.\nConsequently, for $0\\le t<1$,\n\\[\n |P(tr)|<(1-t)(1+t+t^2)=1-t^3\\le1.              \\tag{15}\n\\]\nAt $t=1$ the polynomial vanishes. Hence the entire segment, including\nthe origin, lies in the open unit lemniscate.\n\nFor a nonzero quotient root satisfying the segment bound from \\emph{(15)}, two distinct\nfibre points satisfy\n\\[\n f(h+ty\\zeta^k)=P(t^qr),\n\\]\nand the two segments through $h$ have total length\n$2|y|<2\\sqrt{1-|h|^2}\\le2$.  If the selected\nquotient root is zero, choose a nonzero quotient root $s$ and write\n$P(w)=w(w-s)(w-v)$. For $0<t<1$,\n\\[\n |P(ts)|=t(1-t)|s|^2|ts-v|<2t(1-t)\\le\\frac12.\n\\]\nThe values at both endpoints are zero, so this nonzero root supplies\ntwo distinct fibre points. If no nonzero quotient root exists, $f$ has\nonly one distinct zero value, contrary to the hypothesis.\n\\end{proof}\n\n\\paragraph{Scope.}\n\\label{rem:cubic-quotient-formal-boundary}\n\\label{bdry:solved-polynomial-families}\nThe theorem gives length below $2$. Lean does not check the sharper bound\n$2\\sqrt{1-|h|^2}$ obtained in the proof.\n\n\\section{Further families and counterexamples to proposed proof steps}\n\\label{sec:frontier}\n\nSome further uses of the radial and conformal arguments are collected\nhere, together with examples where a prescribed construction fails.\nThe numbered results retain their linked formal statements. The\nadditional analytic arguments appear as unformalised remarks, even\nwhen an ordinary proof is supplied in full. These qualifications specify the available formal evidence without\nchanging the finite calculations.\n\nThe underlying research notes are retained at commit\n\\href{https://github.com/wcook04/plectis-erdos/tree/f214a6b45528dc5eefe20ffadc35f2e981627d4c/research_corpus/Erdos1041}{\\texttt{f214a6b4}}\nand summarised in \\href{https://github.com/wcook04/plectis-erdos/blob/f214a6b45528dc5eefe20ffadc35f2e981627d4c/research_corpus/Erdos1041/FRONTIER.md}{the dated overview of these arguments}.\nNumerical observations are identified separately. These are restricted\nquestions about polynomial classes and constructions; they do not\nsupply a programme for proving the refuted unrestricted assertion.\n\n\\subsection*{Unformalised remark: roots close to a regular polygon}\n\nLet $f(z)=\\prod_{i=1}^n(z-a_i)$, $n\\ge2$, with all $|a_i|\\le1$.\nPut $\\rho_i=|a_i|$ and $D=|\\operatorname{disc}(f)|/n^n$.  The\n\\href{https://github.com/wcook04/plectis-erdos/blob/f214a6b45528dc5eefe20ffadc35f2e981627d4c/research_corpus/Erdos1041/NearFeketeRadialAngularSplit.md}{Vandermonde determinant estimate}\nsays that, when $D\\ge1-\\eta$ and\n$0\\le\\eta\\le1/(80n^2)$,\n\\[\n  1-\\rho_i^2\\le \\frac{n\\eta}{n-1},\\qquad\n  |a_i-a_j|\\ge \\frac{2-2\\sqrt\\eta-n\\eta}{n-1},\n\\]\nwhere the second inequality is for $i\\ne j$. Moreover, the roots lie\nwithin $7\\sqrt\\eta$ of a rotated regular $n$-gon, after a bijection.\nThese estimates come from near equality in Hadamard's determinant\ninequality, as follows.\n\nTake the Vandermonde matrix $V_{im}=a_i^m$, $0\\le m<n$, and let\n$G=VV^*$, $s_i=G_{ii}=\\sum_{m=0}^{n-1}\\rho_i^{2m}$ and\n$H=\\operatorname{diag}(s_i^{-1/2})G\\operatorname{diag}(s_i^{-1/2})$.\nThe normalisation makes $H$ positive semidefinite with diagonal entries\none, and\n\\[\n D=\\det H\\prod_i\\frac{s_i}{n}.\n\\]\nEvery factor is at most one by Hadamard's inequality, so $D\\ge1-\\eta$\nforces each factor to be at least $1-\\eta$. In particular,\n\\[\n (n-1)(1-\\rho_i^2)\\le n-s_i\\le n\\eta.\n\\]\nHadamard--Fischer applied to the $\\{i,j\\}$ principal block gives\n$\\det H\\le1-|H_{ij}|^2$, hence $|G_{ij}|\\le n\\sqrt\\eta$.\nFor $t=a_i\\overline{a_j}$, the finite geometric sum then gives\n\\[\n n\\sqrt\\eta\\ge\\left|\\sum_{m=0}^{n-1}t^m\\right|\n \\ge n-\\frac{n(n-1)}2|1-t|.\n\\]\nTogether with\n$1-t=1-\\rho_j^2+\\overline{a_j}(a_j-a_i)$, this proves the displayed\nseparation bound. These two bounds need only $0\\le\\eta<1$;\nthe smaller range is used for the matching with a regular polygon.\n\nFor that matching, put $\\kappa=n^2\\eta/(n-1)\\le1/80$ and\n$u_i=a_i/\\rho_i$. The radial bound gives\n$\\rho_i^{2n}\\ge1-\\kappa$, while\n$(1-t)G_{ij}=1-t^n$ gives\n$|1-a_i^n\\overline{a_j^n}|\\le2n\\sqrt\\eta$ for $i\\ne j$.\nUsing\n$|1-be^{i\\theta}|^2=(1-b)^2+b|1-e^{i\\theta}|^2$ with\n$b=\\rho_i^n\\rho_1^n\\ge1-\\kappa$, we obtain\n\\[\n |u_i^n-u_1^n|\\le\\frac{2n\\sqrt\\eta}{\\sqrt{1-\\kappa}}.\n\\]\nChoose the nearest $n$th-root phase to $u_i$ among\n$u_1e^{2\\pi i j/n}$. The inequality\n$|e^{i\\theta}-1|\\ge2|\\theta|/\\pi$ for $|\\theta|\\le\\pi$\nshows that its distance from $a_i$ is at most\n\\[\n \\frac{n\\eta}{n-1}+\\frac{\\pi\\sqrt\\eta}{\\sqrt{1-\\kappa}}\n \\le7\\sqrt\\eta.\n\\]\nThe separation bound exceeds $14\\sqrt\\eta$ in the stated range,\nso two roots cannot be assigned the same phase. There are $n$ roots and\n$n$ phases, giving the required bijection. This ordinary argument uses standard determinant inequalities.\nIt has no new Lean declaration, and no priority is claimed for the\nstability estimate.\n\nThe discriminant hypothesis $D\\ge1-\\eta$ with $\\eta\\le1/(80n^2)$\nis stronger than closeness to the unit circle alone. The regular $n$-gon has $D=1$, whereas a configuration with\na repeated root has $D=0$. In particular, radial information alone cannot\nsupply this discriminant bound.\n\nWe next compare moduli pointwise on a specified segment. No ordering\nof complex polynomial values is involved. Write\n$a_k=\\rho_k u_k$, with $|u_k|=1$. Under\n$D\\ge1-\\eta$ and $0\\le\\eta\\le1/(10n^4)$, the same source proves\n\\[\n |f(su_i)|\\le\\prod_k|su_i-u_k|\n \\qquad(0\\le s\\le\\rho_i).\n\\]\nThe elementary step is\n\\[\n |su_i-\\rho_ku_k|^2-|su_i-u_k|^2\n =(1-\\rho_k)\\bigl(2s\\Re(u_i\\overline{u_k})-1-\\rho_k\\bigr).\n\\]\nHere is where the discriminant hypothesis enters. The two bounds just\nproved imply\n\\[\n |a_i-a_k|>\\frac1{n-1}\\ (i\\ne k),\\qquad\n 1-\\rho_k\\le\\frac{n\\eta}{n-1}<\\frac1{(n-1)^2}.\n\\]\nFor $k=i$, the required inequality is $2s\\le1+\\rho_i$, which follows\nfrom $s\\le\\rho_i\\le1$. For $k\\ne i$, it is immediate if\n$\\Re(u_i\\overline{u_k})\\le0$. Otherwise $s\\le\\rho_i$ gives\n\\[\n\\begin{split}\n \\rho_k\\bigl(2s\\Re(u_i\\overline{u_k})-1-\\rho_k\\bigr)\n &\\le\\rho_i^2-\\rho_k-|a_i-a_k|^2\\\\\n &\\le1-\\rho_k-|a_i-a_k|^2<0.\n\\end{split}\n\\]\nMultiplying the factorwise inequalities proves the comparison. This\nargument uses the two elementary bounds, not the polygon-matching range;\nit therefore also covers $n=2$, when $1/(10n^4)>1/(80n^2)$.\nMoving the roots inward reduces this modulus bound, but does not select\nthe angular directions of two contained segments.\n\nFor exact regular-polygon directions, write\n$a_i=\\rho_i\\omega\\zeta^i$, with $|\\omega|=1$ and\n$\\zeta=e^{2\\pi i/n}$. In any degree $n\\ge2$, the sufficient condition\n\\[\n 0<\\rho_k\\le1,\\qquad \\rho_k\\ge2\\cos(2\\pi/n)-1\\quad\\hbox{for every }k\n\\]\nmakes the same factorwise comparison hold. Indeed, for $k\\ne i$ with\npositive cosine, $s\\le1$ gives\n$2s\\Re(\\zeta^{i-k})\\le2\\cos(2\\pi/n)\\le1+\\rho_k$;\nnonpositive cosines and $k=i$ are handled as above. Since\n$\\prod_k(z-\\omega\\zeta^k)=z^n-\\omega^n$, we obtain\n\\[\n |f(s\\,\\omega\\zeta^i)|\\le1-s^n\n \\quad(0\\le s\\le\\rho_i).\n\\]\nEvery root-to-origin segment is therefore contained in $\\{|f|\\le1\\}$.\nIf all $\\rho_i<1$, containment is strict: the displayed bound is below\n$1$ for $s>0$, and $|f(0)|=\\prod_i\\rho_i<1$. Any two roots can then\nbe joined through the origin with length $\\rho_i+\\rho_j<2$.\nFor $2\\le n\\le6$ the lower bound on $\\rho_k$ is nonpositive,\nso every choice of positive radii at most one is allowed. For $n\\ge7$\nthe argument instead requires each radius to be at least\n$2\\cos(2\\pi/n)-1$. This is a sufficient condition for the factorwise\ncomparison, not a claimed necessary condition for a contained path.\nThe favourable index in the general unit-circle averaging identity may\nmove with $s$, however; an existence statement at one level does not\nsupply a fixed pair of roots.\n\n\\subsection*{The exact binomial chord calculation}\n\\hypertarget{binomial-chord-calculation}{}\nFor $n\\ge2$ and $0<|a|<1$, put $r=|a|^{1/n}$ and\n$r_*=(1+\\cos^n(\\pi/n))^{-1/n}$. A rotation reduces $z^n-a$ to\n$z^n-r^n$ without changing lengths or moduli. For $n=2$, $r_*=1$\nand only the outer-diameter case occurs.\n\n\\begin{theorem}[complementary binomial chords]\\label{res:complementary-binomial-chords}\nTwo adjacent zeros of $z^n-a$ can be joined by an explicit polygonal path\ninside $\\{|z^n-a|<1\\}$ of length strictly below $2$.  For $r<r_*$ the\nadjacent-root chord itself works.  For $r\\ge r_*$, two radial legs and an\ninner adjacent crossing chord work after an arbitrarily small radial\ncontraction.  These two constructions meet at $r=r_*$, where the outer chord attains\n$|f|=1$ at its midpoint and therefore lies only in the closed lemniscate.\nOpen containment at and above the switch uses the inner chord\nafter a radial contraction.\n\\end{theorem}\n\nFor $f(z)=z^n-r^n$, $0<r<1$ and $n\\ge2$, the two constructions depend on the maximum over an entire\nchord. The endpoint values alone do not give that maximum. Put\n$\\omega=e^{2\\pi i/n}$ and $c=\\cos(\\pi/n)$. When $n\\ge3$, define\n$r_*=(1+c^n)^{-1/n}$ and $\\varepsilon=(1-r^n)^{1/n}$; in the inner-chord\ncase $r\\ge r_*$, set $t=\\varepsilon/c\\le r$.\n\nWe prove the chord estimate also at the smaller radii needed for the\nsecond construction. For $n\\ge3$, let $0<s\\le r$. A point on the chord\nfrom $s$ to $s\\omega$ can be written\n\\[\n z=\\frac{sc}{\\cos\\theta}e^{i(\\pi/n+\\theta)},\n \\qquad |\\theta|\\le\\pi/n.\n\\]\nConcavity of $\\log\\cos x$ on $[0,\\pi/2)$ gives\n$\\cos(n\\theta/2)\\le\\cos^{n/2}\\theta$ for $0\\le\\theta<\\pi/n$.\nBy symmetry and continuity,\n\\[\n 1+\\cos(n\\theta)\\le2\\cos^n\\theta\n \\qquad (|\\theta|\\le\\pi/n).\n\\]\nSet $u=\\cos^{-n}\\theta\\ge1$. Since\n$z^n=-(sc)^nu e^{in\\theta}$, this inequality gives\n\\[\n \\begin{split}\n |z^n-r^n|^2-\\bigl(r^n+(sc)^n\\bigr)^2\n &\\le (u-1)(sc)^n\\bigl((sc)^n(u+1)-2r^n\\bigr)\\le0.\n \\end{split}\n\\]\nThe last step uses $(sc)^nu\\le s^n\\le r^n$ and\n$(sc)^n\\le r^n$. Equality holds at the midpoint, where $\\theta=0$.\nConsequently the maximum on every such chord is exactly\n$r^n+(sc)^n$. Taking $s=r$ proves the outer threshold; taking\n$s=t=\\varepsilon/c\\le r$ proves the inner bound, and taking\n$s=\\lambda t<t$ makes it strict. Every radial leg between $s$ and\n$r$ has modulus $r^n-|z|^n<1$. Its two legs and crossing chord have\nlength $2r-2s(1-\\sin(\\pi/n))<2r$, which proves the stated length\nformula and shows that contraction preserves the strict length bound.\n\nFor $n=2$, the chord is a diameter and its maximum is $r^2<1$;\nits length is $2r<2$. For $n\\ge3$, the outer chord works exactly when\n$r<r_*$, and its length is $2r\\sin(\\pi/n)$. At $r=r_*$ it reaches\nlevel one at its midpoint. If $r\\ge r_*$, the two radial legs and\ninner crossing have length\n\\[\n 2r-2\\varepsilon\\tan\\left(\\frac\\pi4-\\frac\\pi{2n}\\right)<2r<2.\n\\]\nContracting the crossing radius makes the level bound strict, as\nproved above. The comparison concerns these two constructions only; it makes no\nclaim about shortest paths among all admissible curves.\n\n\\subsection*{Unformalised remark: a bounded-radius concyclic class}\n\nWhen the common root circle is small enough, its self-inversive\npolynomial gives a contained adjacent-root chord. This ordinary\nargument has no complete formal counterpart in the supplied record.\nLet $f$ be monic of degree $n\\ge3$ with distinct zeros on a circle of radius\n$\\rho$.  If $2\\rho^n\\le1$, two adjacent zeros are joined by their straight\nchord, whose length is at most $2\\rho\\sin(\\pi/n)<2$, and the chord lies in\n$\\{|f|<1\\}$.  (If a zero is repeated, the short-connection conclusion is\nimmediate; the distinct case is the substantive one.)\n\nFor each fixed degree $n$, the radius condition is $\\rho\\le2^{-1/n}$.\nIt allows arbitrary spacing on that circle, but does not cover every\nradius $\\rho<1$. No restriction on the circle's centre is imposed.\n\nAfter translating and scaling, let $g$ be the monic polynomial with\nroots $w_1,\\ldots,w_n$ on the unit circle. Such a polynomial is\n\\emph{self-inversive}: $z^n\\overline{g(1/\\overline z)}$\nis a constant multiple of $g(z)$, with the constant of modulus one.\nEquivalently, $e^{-in\\theta/2}g(e^{i\\theta})$ is real up to a fixed phase.\nChoose\n\\[\n c=(-1)^{n+1}\\prod_j w_j,\\qquad q(z)=z^n-c.\n\\]\nThen $g$ and $q$ have the same leading and constant coefficients, and their\nboundary values have a common real phase. If every root gap contained a\npoint with $|g|>|q|$, the real boundary function of $g-q$ would have\nalternating signs at these points. Advancing the angle by $2\\pi$ multiplies\nthat function by $(-1)^n$, so the alternation also holds across the final\ngap. The intermediate value theorem would give $n$ distinct unit-circle\nzeros of $g-q$. This is impossible: its degree is at most $n-1$, and\n$g=q$ cannot satisfy the assumed strict inequalities. Thus some\nadjacent-root arc satisfies $|g|\\le|q|\\le2$. The interior of that arc contains no zero of\n$q$, since $g$ is nonzero there. As consecutive zeros of $q$ are separated\nby angle $2\\pi/n$, the selected arc has angular width at most $2\\pi/n$.\nThis explains why the same selected pair also has the required short chord.\nTo pass from this arc to its chord $[a,b]$, let $\\nu$ be the unit\nnormal pointing into the circular segment bounded by them. On the open\nchord, logarithmic differentiation gives\n\\[\n \\partial_\\nu\\log|g(z)|\n =\\sum_{w_j\\notin\\{a,b\\}}\n   \\frac{\\operatorname{Re}\\bigl(\\nu\\overline{z-w_j}\\bigr)}{|z-w_j|^2}>0.\n\\]\nThe endpoint terms vanish because their directions are tangent to the\nchord. Every remaining numerator is positive because the other roots\nlie strictly on the far side of its line; $n\\ge3$ ensures at least one\nsuch term. The harmonic function $\\log|g|$ has no interior maximum and\ntends to $-\\infty$ at $a,b$. A maximum cannot lie on the open chord\neither, since moving into the circular segment increases the function.\nIt is therefore attained on the arc, and the maximum on the chord is\nstrictly smaller. Scaling back gives $|f|<2\\rho^n\\le1$ throughout the\nchord, with $f=0$ at its endpoints. These are the arguments of\n\\href{https://github.com/wcook04/plectis-erdos/blob/f214a6b45528dc5eefe20ffadc35f2e981627d4c/research_corpus/Erdos1041/ConcyclicAlternation.md}{Theorem~C and Lemma~S, Sections~3--4 of the concyclic note}.\n\nThe argument is an ordinary proof outside Lean.  The\n\\href{https://github.com/wcook04/plectis-erdos/blob/f214a6b45528dc5eefe20ffadc35f2e981627d4c/research_corpus/Erdos1041/scripts/check_erdos1041_concyclic_exact_witness.py}{exact-rational checker}\nchecks finitely many identities used in the proof and configurations, while the\n\\href{https://github.com/wcook04/plectis-erdos/blob/f214a6b45528dc5eefe20ffadc35f2e981627d4c/research_corpus/Erdos1041/scripts/check_erdos1041_concyclic_alternation.py}{numerical checker}\nis regression and stress-test evidence. The arc constant $2$ is attained\nby the regular $n$-gon. Thus the bound obtained by passing through the\narc is $\\rho\\le2^{-1/n}$, although this cutoff tends to $1$ as\n$n\\to\\infty$. This does not assert optimality of the chord bound or\nexclude another contained path at a larger radius. Nor does arc\nsharpness make the pair of radial segments shortest. For example, for\n$g(z)=z^3-1/8$ the chord between $1/2$ and $e^{2\\pi i/3}/2$ has length\n$\\sqrt3/2<1$, whereas the two radii have total length $1$. The binomial\ncalculation above gives a maximum of $9/64<1$ on that chord. The\nlength-$2$ equality at $z^n-1$, proved below, concerns the uncontracted\nclosed-level problem.\n\n\\subsection*{Power substitutions in trinomials}\n\nThe segment identity survives a power substitution. We record the\nlifting argument as an unformalised remark: the source checks the\nsegment inequality and finite length calculation, but not selection\nand lifting of a suitable quotient root. Fix\nintegers $1\\le r<m$ and $q\\ge1$, and consider the translated monic polynomial\n\\[\n f(z)=(z-h)^{qm}+a(z-h)^{qr}+c.\n\\]\nThe strict exponent inequality keeps the leading monomial separate.\nThe algebraic identity below also holds when $r=m$, but then the two\nleading terms merge and the displayed polynomial need not be monic or\nhave degree $qm$.\nWriting $w=(z-h)^q$ reduces the root equation to\n$w^m+aw^r+c=0$.  At such a quotient root the middle coefficient can be\neliminated exactly: for $0\\le u\\le1$,\n\\[\n u^mw^m+au^rw^r+c\n   =(1-u^r)c-(u^r-u^m)w^m.\n\\]\nThis is a nonnegative combination of $c$ and $-w^m$, with weights\n$1-u^r$ and $u^r-u^m$ summing to $1-u^m$. The\n\\lword{Erdos1041/CyclicTrinomialFiberCase.lean}{103}{trinomialRoot_spoke_norm_lt_one_of_norm_lt_one}{formal radial-segment estimate}\ngives strict containment when $|w|<1$ and $|c|<1$.\n\nFor $q\\ge2$ and a nonzero quotient root satisfying these bounds, choose\ntwo distinct solutions $y_1,y_2$ of $y^q=w$. Their common modulus is\n$|w|^{1/q}<1$. To lift a quotient radial segment, set $u=t^q$:\n\\[\n |f(h+ty_i)|\n \\le (1-t^{qm})\\max\\{|c|,|w|^m\\}<1\n \\qquad(0\\le t\\le1).\n\\]\nThus the two lifted segments join $h+y_1$ to $h+y_2$ through $h$\nwith length $2|w|^{1/q}<2$. The construction uses any quotient root\nwith the stated modulus bound; it does not prove that such a root\nexists without an additional hypothesis. A zero quotient root instead\ngives a root of multiplicity at least $q$ at $h$ and hence a constant\npath between two occurrences. The formal source checks the\nfactorization, radial-segment estimate and finite length inequality.\nIt does not establish the existence of a suitable quotient root,\nformalise the selection and lifting just described, or prove\nunrestricted Erd\\H{o}s~\\#1041.\n\nThe radial-segment identity and strict bound are formalised in\n\\lword{Erdos1041/CyclicTrinomialFiberCase.lean}{46}{trinomialRoot_spoke_factorization}{cancellation along a trinomial root segment}\nand \\lword{Erdos1041/CyclicTrinomialFiberCase.lean}{103}{trinomialRoot_spoke_norm_lt_one_of_norm_lt_one}{containment of a trinomial root segment}.\n\n\\subsection*{Unformalised remark: a coefficient condition for four-term polynomials}\n\nOne more monomial introduces a partial sum that need not lie in the\nunit disc. Bounding that sum gives the following ordinary coefficient\ncriterion. Let\n\\[\n  g(w)=w^m+aw^r+bw^s+c,\\qquad m>r>s\\ge1,\n\\]\nassume all roots of $g$ lie in the open unit disk, and let $w_1,w_2$ be roots\nof the two smallest moduli.  If\n\\[\n  |c|+|b|\\,|w_2|^s<1,\n\\]\nthen the complete radial spokes from $w_1$ and $w_2$ to the origin lie in\n$\\{|g|<1\\}$, and their broken line has length strictly below $2$.  The simpler\ncoefficient condition\n\\[\n  |b|+|c|\\le1\n\\]\nmakes every root-to-origin segment contained; the coefficient $a$ has\nno additional restriction beyond the assumed root locations. This coefficient\ncondition includes $b=0$ and $|c|\\le1$, but is not automatic for all\nfour-term polynomials with roots in the disc.\n\nThe mechanism is visible in one identity.  At a root $w$ and for\n$0\\le u\\le1$,\n\\[\n\\begin{split}\n g(uw)={}&(1-u^s)c\n  +(u^s-u^r)(c+bw^s)\n  +(u^r-u^m)(-w^m).\n\\end{split}\n\\]\nHere the root equation has eliminated $aw^r$ before taking absolute\nvalues. The three weights are nonnegative and sum to $1-u^m$.\nThe controlled terms are $c$, $c+bw^s$ and $-w^m$; their moduli are\nstrictly below one under the root-dependent condition. The same\nidentity explains why no bound on $a$ is needed.\nLean checks the \\lword{Erdos1041/CyclicTetranomialCoefficientCase.lean}{27}{tetranomialRoot_spoke_factorization}{factorization}\nand the resulting \\lword{Erdos1041/CyclicTetranomialCoefficientCase.lean}{145}{tetranomialRoot_spoke_norm_lt_one_of_rootBudget}{strict radial-segment theorem} under\nthe exact weak exponent hypotheses $1\\le s\\le r\\le m$; it also checks the\ncoefficient-only corollary above. Its weak bound $|b|+|c|\\le1$ still\ngives strict containment: the root-disc hypothesis implies $|c|<1$,\nand, if $b\\ne0$, $|w|^s<1$ gives\n$|c|+|b||w|^s<|c|+|b|\\le1$. If $b=0$, the required bound is\nsimply $|c|<1$.\n\nAn alternative criterion uses a complex power sum to select two\nindices. Let $S$ index a finite family of roots, $N=|S|\\ge2$, and\n\\[\n  M=\\sum_{i\\in S}w_i^s.\n\\]\nThe \\lword{Erdos1041/TetranomialL2Selector.lean}{33}{sum_normSq_const_add_mul}{squared-modulus identity} gives\n\\[\n \\sum_{i\\in S}|c+bw_i^s|^2\n =N|c|^2+|b|^2\\sum_{i\\in S}|w_i^s|^2\n   +2\\mathop{\\rm Re}(\\overline c\\,bM).\n\\]\nConsequently, if every $|w_i|<1$, $|c|<1$, and\n\\[\n N\\bigl(|b|^2+|c|^2\\bigr)\n   +2\\mathop{\\rm Re}(\\overline c\\,bM)<N-1,\n\\]\nthen two distinct indices $i,j\\in S$ satisfy\n$|c+bw_i^s|<1$ and $|c+bw_j^s|<1$. Indeed, the displayed identity\nbounds the sum of squared moduli strictly below $N-1$. If at most one\nindex had modulus below one, the other $N-1$ terms would each contribute\nat least one. This explains both the threshold $N-1$ and why the\ncriterion selects two indices. Applying the\n\\lword{Erdos1041/TetranomialL2Selector.lean}{208}{exists_two_tetranomialRoot_safeSpokes_of_moment_coeff_budget}{resulting contained-segment theorem} to these inequalities proves that both root segments lie\nstrictly in $\\{|g|<1\\}$.  Retaining the signed cross term permits the inequality to hold for configurations outside the coefficient-only triangle\n$|b|+|c|\\le1$.  The formal hypotheses do not require $i\\mapsto w_i$ to be\ninjective, so distinct indices need not denote distinct root values.\nFor the polynomial consequence, the family must be a submultiset of the\nzeros, respecting their algebraic multiplicities. A repeated zero then\nalready gives a constant path between two occurrences. Repeating a simple\nzero in an arbitrary indexed family cannot supply a second occurrence.\nThe two sufficient conditions are not ordered by strength. For example,\n$g(w)=w^5+19/20$ has all roots in the open unit disc and satisfies the\ncoefficient condition with $b=0$. For its full five-root list, however,\nthe left side of the power-sum condition is\n$5(19/20)^2=361/80>4=N-1$, so that condition fails. Its benefit is the\nuse of cancellation in the cross term, not inclusion of every case\ncovered by the coefficient condition. Conversely,\n\\[\n g(w)=(w^2-1/2)(w^3+4/5)\n     =w^5-\\tfrac12w^3+\\tfrac45w^2-\\tfrac25\n\\]\nhas all roots of modulus $1/\\sqrt2$ or $(4/5)^{1/3}$, hence below\none, but $|b|+|c|=6/5>1$. Its full root list has\n$M=\\sum_iw_i^2=1$: the two quadratic roots contribute $1$, and the\nthree cubic roots contribute $0$. The power-sum left side is\n$5(16/25+4/25)-16/25=84/25<4$. Thus this criterion does cover\nroot-disc polynomials excluded by the coefficient condition.\n\nFor $q\\ge2$ and $f(z)=g((z-h)^q)$, assume also that all roots of $f$\nlie in the open unit disc. If $y^q=w$ and $\\omega=e^{2\\pi i/q}$, the\nroots $h+\\omega^j y$ satisfy\n\\[\n \\frac1q\\sum_{j=0}^{q-1}|h+\\omega^j y|^2=|h|^2+|y|^2<1.\n\\]\nThus every quotient root has $|w|=|y|^q<1$, which supplies the\nhypothesis on the zeros used above, and every lifted segment has length\n$|y|<1$. Two selected spokes join through $h$ with total length below\n$2$; repeated root occurrences instead permit the constant path.\nThe Lean theorem assumes the finite root family, its complex power sum $M$,\nand the displayed inequality. It neither derives that inequality from\narbitrary coefficients nor verifies the family's polynomial\nmultiplicities, the cyclic lifting, or the final path construction.\nCompleteness of the root list is not required for the selection argument:\na submultiset respecting multiplicity suffices.  Without one of the power-sum, root-dependent, or coefficient-only\ninequalities, the tetranomial case is not covered by this argument; this family does not solve\nunrestricted Erd\\H{o}s~\\#1041.\n\n\\subsection*{Unformalised remark: a quartic composed with a power map}\\label{sec:translated-quartic-quotient-fibres}\n\nWe next examine the ordinary lifting argument based on Pendyala's\ndegree-four theorem \\cite[Theorem~1]{june2026} and four-point radial\nlemma \\cite[Lemma~1]{june2026}. The cited quartic result alone does not\nformalise this higher-degree construction. Let $P$ be a monic quartic,\nlet $q\\ge2$ be an integer, and set\n\\[\n  f(z)=P((z-h)^q).\n\\]\nAssume that every root of $f$ lies in the open unit disc.\nA repeated root of $f$ gives the constant path, so suppose $f$ is\nsquarefree. Then the roots of $P$ are distinct and nonzero. For a root\n$w=y^q$ of $P$, all points $h+\\zeta^j y$, with $\\zeta=e^{2\\pi i/q}$,\nare roots of $f$. Hence the open-disc hypothesis gives\n\\[\n \\frac1q\\sum_{j=0}^{q-1}|h+\\zeta^jy|^2=|h|^2+|y|^2<1.\n\\]\nIn particular $|w|^{1/q}=|y|<1$, which is the bound needed for the\nlifted length, and also $|w|<1$. Pendyala's chord-or-radial argument\nsupplies the quotient geometry in two cases.  If two quotient roots satisfy\n$|w_i-w_j|<1$, every point $w$ of the chord $[w_i,w_j]$ lies in the unit disc;\nthe two endpoint factors of $|P(w)|$ have product at most $|w_i-w_j|^2/4$ and\nthe other two factors are below $2$, so $|P(w)|\\le|w_i-w_j|^2<1$.  Otherwise\nthe quotient roots are pairwise at distance at least $1$.  Choosing $R$ with\n$\\max_k|w_k|<R<1$ and applying the four-point radial lemma to the points\n$w_k/R$ gives distinct $i,j$ with $|P(tw_i)|\\le R^4$ and $|P(tw_j)|\\le R^4$ for\n$0\\le t\\le1$, so the radial segments $[0,w_i]$ and $[0,w_j]$ lie in\n$\\{|P|<1\\}$.  The extra issue is metric: a short chord in the\n$w$-plane need not lift isometrically through $y\\mapsto y^q$.\n\nPut $\\alpha=1/q$. On a chord avoiding $0$, choose a continuous\nargument and hence a root lift $y$ with $y^q=w$. Differentiation gives\n$|dy|=\\alpha|w|^{\\alpha-1}|dw|$, independently of the chosen root.\nIf the supporting line has distance $d$ from $0$, and $x$ is distance\nalong that line from the perpendicular foot, then\n\\[\n  \\bigl(\\sqrt{d^2+x^2}\\bigr)^{\\alpha-1}\n    \\le x^{\\alpha-1}\\qquad(x>0).\n\\]\nThe negative exponent is why replacing $|w|$ by $x$ gives an upper\nlength bound. Lean checks \\href{\\repobase/research_corpus/Erdos1041/QuarticQuotientFiberCase.lean\\#L37}{the integral of the power-map derivative}\n\\[\n  \\alpha\\int_0^A x^{\\alpha-1}\\,dx=A^\\alpha.\n\\]\nIf the foot lies on the chord, split there: the two\nintegrals are bounded by the $\\alpha$-powers of the distances from the\nfoot to the endpoints, hence by $|a|^\\alpha+|b|^\\alpha$. If the foot\nlies outside the chord, integrate on one side of it; extending the\ninterval to the foot bounds the result by the $\\alpha$-power of the\nfarther endpoint's modulus. When the chord passes through $0$, choose\na continuous root on each half and join the lifts at $0$; the integral\nis finite because $\\alpha>0$. This proves the chord-lift estimate\n\\[\n  \\operatorname{length}(\\widetilde{[a,b]})\n    \\le |a|^{1/q}+|b|^{1/q}.\n\\]\nThe \\href{\\repobase/research_corpus/Erdos1041/QuarticQuotientFiberCase.lean\\#L59}{strict endpoint-length estimate} also proves that if\n$0\\le a,b<1$, $\\alpha>0$, and a candidate length $L$ is at most\n$a^\\alpha+b^\\alpha$, then $L<2$.  The close-pair chord therefore has a lift of length below $2$.\nIn the other case the quotient arms meet at $0$, so their lifts are radial\nsegments meeting at $h$, with total length\n$|w_i|^{1/q}+|w_j|^{1/q}<2$. In both cases $f=P\\circ((z-h)^q)$\ntransfers containment from the quotient curve. Distinct quotient endpoints\ngive distinct roots of $f$. This proves the path assertion in every degree\n$4q\\ge8$. The fibre average gives a little more: $|h|<1$ and\n$|w_k|^{1/q}<\\sqrt{1-|h|^2}$ for every quotient root. Consequently the same\nconstructed path has length\n\\[\n \\operatorname{length}<2\\sqrt{1-|h|^2}\\le2.\n\\]\nThis refinement concerns the translated power-substitution family, not\narbitrary degree-$4q$ polynomials.\n\nPendyala proves the quartic geometric theorem and its four-point radial\nlemma.  The local Lean module\nchecks the antitone density inequality, its integral, the strict powered\nendpoint-length bound, and the final length estimate.  It does not formalize\nPendyala's geometric lemma, the continuous covering-space construction of the\nroot lift, or the ordinary chord/radial case assembly.  This is a family defined by a power substitution, not a proof of unrestricted Erd\\H{o}s~\\#1041.\n\n\\paragraph{The joining point in the quartic construction.}\nPendyala's proof of \\cite[Theorem~1, pp.~2--3]{june2026} uses a\nsmallest enclosing disc and joins the selected roots through its centre.\nFor the power substitution we instead apply \\cite[Lemma~1]{june2026}\nto the centred disc $|w|\\le R$. That lemma requires only the disc and\npairwise-distance hypotheses, not minimality of the disc. Keeping the\njoining point at $0$ is what makes the radial lift above valid. The\nendpoint-length bound for a single chord would not justify lifting an\narbitrary broken line through a nonzero quotient vertex.\n\n\\subsection*{A mean bound for critical values in every degree}\n\nThe Poisson argument first gives a quadratic mean bound, allowing a\ndisc of any centre and radius and counting all multiplicities. This is\nthe formally supported theorem below. The subsequent fourth-power\nrefinement, and the higher exponents obtained from vanishing complex\npower sums, are unformalised analytic remarks. A critical-value moment\nbound supplies no estimate for the lengths of inverse paths ending at\nthose points.\n\nDubinin \\cite[Theorem~2, (9), pp.~1172--1173]{dubinin2006critical}\nproves the sharp unit-disc critical-value product inequality using the\nresultant identity, the maximum-modulus principle and Schur's Vandermonde\ninequality. The positive-moment estimate below implies his product bound\nby AM--GM. When $n\\ge3$, the product estimate alone cannot control a\npositive moment: a list $T,T^{-1},1,\\ldots,1$ has product one, but its\nsum of $p$th powers is unbounded as $T\\to\\infty$ for every $p>0$.\nThis compares the information in two inequalities for nonnegative lists;\nit does not claim that these lists occur as critical values of polynomials\nin the theorem. In degree two there is just one critical value, and the\nproduct and positive-moment bounds are equivalent.\nFor a zero prescribed at the origin, his Theorem~3 (p.~1174) gives\n\\[\n \\left(\\prod_j|f(c_j)|\\right)^{1/(n-1)}\n \\le(n-1)\\left(\\frac{|f'(0)|^2}{n^n}\\right)^{1/(n-1)}.\n\\]\nDubinin explicitly identifies Tischler's earlier contribution\n\\cite[p.~444, as reported in Dubinin]{tischler1989}.\nHis Theorem~1 uses dissymmetrisation; that is not the proof mechanism of\nTheorem~2.  Schur's original paper \\cite{schur1918} is the historical\nantecedent cited there, not a separately re-proved source in this record.\nA sharp marked-zero positive-moment bound remains a separate question.\nThe addendum \\cite[Section~3]{revision2026} reports a reduction to\nboundary-root configurations, without solving the resulting angular\noptimisation. That unavailable addendum is not a proof dependency of the\ncritical-value theorem below.\n\n\\begin{theorem}[a mean bound for critical values]\n\\label{res:critical-value-budget}\nLet $f$ be monic of degree $n\\ge2$, with roots in a closed disc of radius\n$R\\ge0$.  If $c_1,\\ldots,c_{n-1}$ are its critical points counted with\nmultiplicity, then\n\\begin{equation}\\label{eq:critical-value-quadratic-budget}\n \\sum_{j=1}^{n-1}|f(c_j)|^{2/(n-1)}\n \\le(n-1)R^{2n/(n-1)}.\n\\end{equation}\nConsequently the lower exponents used elsewhere in the record satisfy\n\\begin{equation}\\label{eq:critical-value-power-budget}\n \\sum_{j=1}^{n-1}|f(c_j)|^{1/(n-1)}\n \\le(n-1)R^{n/(n-1)}\n\\end{equation}\nand\n\\[\n \\sum_{j=1}^{n-1}|f(c_j)|^{1/n}\\le(n-1)R.\n\\]\nThe constant is attained by $f(z)=(z-\\tau)^n-\\lambda$ with enclosing\ndisk centred at $\\tau$ and radius $R=|\\lambda|^{1/n}$.\n\\end{theorem}\n\nThe finite inequality behind the theorem concerns arbitrary points of the\ndisk, without asking them to arise as critical points.  Its quadratic form is\n\\[\n \\sum_{j=1}^m\\left(\\prod_{k=1}^m\n       |1-\\overline{c_j}c_k|\\right)^{2/m}\\le m,\n \\qquad |c_j|\\le1.\n\\]\nThe following pointwise bound explains why these finite inequalities\ncontrol critical values in every degree.\n\n\\begin{lemma}[reflected-derivative bound]\\label{res:reflected-critical-value}\nIf $f$ is monic of degree $n\\ge2$ with roots in the closed unit disk,\nand $c_1,\\ldots,c_{n-1}$ list its critical points with multiplicity, then\n\\begin{equation}\\label{eq:critical-reflected-product}\n |f(c_j)|\\le\\prod_{k=1}^{n-1}|1-\\overline{c_j}c_k|.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nFirst put all roots $a_i$ strictly inside the disk. We want a numerator\nthat equals $nf$ at the critical points and can be compared with $f'$\non the circle. Set\n\\[\n N(z)=nf(z)-zf'(z),\\qquad\n G(z)=n\\prod_k(1-\\overline{c_k}z).\n\\]\nThe numerator is the polar derivative at $0$: the conventional polar\nderivative at $\\alpha$ is $nf(z)+(\\alpha-z)f'(z)$; see\n\\href{https://arxiv.org/abs/1303.0068v1}{Rather--Gulzar, Section~1} for this\nnotation. Its leading term cancels, and $N(c_j)=nf(c_j)$.\nReflection gives $|G|=|f'|$ on the circle. Gauss--Lucas puts every\n$c_k$ inside the disc, so $G$ has no zero on its closure. Thus $N/G$\nis a holomorphic quotient to which the maximum-modulus principle applies.  On $|\\zeta|=1$,\n\\[\n \\operatorname{Re}\\frac{\\zeta f'(\\zeta)}{f(\\zeta)}\n =\\sum_i\\left(\\frac12+\n       \\frac{1-|a_i|^2}{2|1-a_i\\bar\\zeta|^2}\\right)\\ge\\frac n2.\n\\]\nWriting the logarithmic derivative as $w$, the identity\n$|n-w|^2-|w|^2=n^2-2n\\operatorname{Re}w\\le0$ gives\n$|N|\\le|G|$ on the circle.  The maximum-modulus principle applied to\n$N/G$ gives the same comparison inside.  At $c_j$, $N(c_j)=nf(c_j)$;\ndividing by $n$ and conjugating each factor proves\n\\eqref{eq:critical-reflected-product}.\nFor closed-disk roots apply this result to\n$f_r(z)=r^nf(z/r)$, $0<r<1$, whose critical points are $rc_k$:\n\\[\n r^n|f(c_j)|\\le\\prod_k|1-r^2\\overline{c_j}c_k|.\n\\]\nLet $r\\uparrow1$.  This also handles boundary critical points and all\nmultiplicities without selecting local branches.\n\\end{proof}\n\nFirst suppose the zeros of $f$ lie in the closed unit disc, and set\n$m=n-1$. Gauss--Lucas puts the critical points in the same disc.\nLemma~\\ref{res:reflected-critical-value} gives the first inequality below;\nthe weighted Poisson inequality, proved next and applied with equal\nweights $1/m$, gives the second:\n\\[\n \\sum_{j=1}^m|f(c_j)|^{2/m}\n \\le\\sum_{j=1}^m\\left(\\prod_{k=1}^m\n             |1-\\overline{c_j}c_k|\\right)^{2/m}\\le m.\n\\]  Applying this to\n$R^{-n}f(h+Rz)$ and scaling back proves\n\\eqref{eq:critical-value-quadratic-budget} for $R>0$; if $R=0$, all critical\nvalues vanish.\n\nThe two displayed consequences are the lower power means of the same\nnonnegative $m$-tuple.  Equivalently, after unit-disc normalization, apply\nthe monotonicity of normalized $L^p$ means from exponent $2/m$ first to\n$1/m$ and then to $1/(m+1)=1/n$.  Scaling back contributes respectively\n$R^{n/m}$ and $R$. These are consequences of the quadratic case;\nthe higher-power argument below strengthens that case without changing\nthese lower-exponent applications.\n\nThe quadratic moment controls the number of large critical values.\nFor $R>0$, write $r_j=|f(c_j)|^{1/n}$. Then\n\\[\n \\sum_j(r_j/R)^{2n/(n-1)}\\le n-1,\\qquad\n \\#\\{j:r_j\\ge tR\\}\\le\\frac{n-1}{t^{2n/(n-1)}}\\quad(t>0).\n\\]\nThe counting bound follows by retaining just the indicated summands.\nThe fourth-power refinement proved below replaces the exponent\n$2n/(n-1)$ here by $4n/(n-1)$, improving the count for $t>1$.\nFor $0<t\\le1$ the trivial bound $n-1$ is at least as good.\nThe binomial equality family has every $r_j=R$, but these distributional\nbounds still leave the geometry of the joining paths to be supplied.\n\n\\paragraph{The weighted Poisson inequality and its equality case.}\n\\hypertarget{weighted-poisson-proof}{}\n\\begin{theorem}[a weighted inequality for points in a disc]\n\\label{res:fp-weighted-all-degree}\nLet $c_1,\\ldots,c_m\\in\\overline{\\mathbb D}$ and let $w_j>0$ satisfy\n$\\sum_j w_j=1$.  Set\n\\[\n G(z)=\\prod_k |1-\\overline{c_k}z|^{w_k}.\n\\]\nThen\n\\[\n \\sum_j w_j G(c_j)^2\\le 1,\n\\]\nwith equality if and only if every $c_j=0$.  Equal weights therefore give\n$\\sum_j(\\prod_k|1-\\overline{c_k}c_j|)^{1/m}\\le m$ for every $m$.\n\\end{theorem}\n\nFor one point $c$ of weight $1$, the quantity to bound is\n$(1-|c|^2)^2\\le1$; equality holds only at $c=0$.\nFor two points $c,-c$ of equal weight, it is $1-|c|^4\\le1$.\nThese examples suggest why a nonzero configuration should lose from equality.\nNow let $w_j\\ge0$, $\\sum_jw_j=1$, and $|c_j|\\le1$, with repetitions allowed.\nWrite $G(z)=\\prod_k|1-\\overline{c_k}z|^{w_k}$, taking a factor of\nexponent zero to be one. We seek an analytic function with modulus $G$:\nits logarithmic derivative will express the weighted Poisson kernel,\nand its Taylor coefficients will measure the loss from equality.\nFor $|c|<1$ and $|\\zeta|=1$, let\n$P_c(\\zeta)=(1-|c|^2)/|\\zeta-c|^2$, and write $dm=dt/(2\\pi)$ on\n$\\zeta=e^{it}$. When every centre is strictly inside the disc, choose\nanalytic logarithms zero at the origin and put\n\\[\n g(z)=\\exp\\sum_jw_j\\log(1-\\overline{c_j}z)\n     =1+\\sum_{\\nu\\ge1}a_\\nu z^\\nu,\\qquad\n P=\\sum_jw_jP_{c_j}.\n\\]\nOn the unit circle $P=1-2\\Re(\\zeta g'/g)$, by logarithmic\ndifferentiation. The Poisson formula reproduces $g(c)$ from its boundary\nvalues and gives $\\int P_c\\,dm=1$ for $|c|<1$. Expanding a square therefore\nyields\n\\[\n \\int |g|^2P_c\\,dm-|g(c)|^2\n   =\\int |g-g(c)|^2P_c\\,dm\\ge0.\n\\]\nThis explains why the same kernel bounds the value at each centre.\nTake $c=c_j$, multiply by $w_j$ and sum to obtain\n\\[\n \\sum_jw_jG(c_j)^2\\le\\int |g|^2P\\,dm\n =1-\\sum_{\\nu\\ge1}(2\\nu-1)|a_\\nu|^2.\n\\]\nTo see the last equality, expand $|g|^2$ and\n$\\zeta g'\\overline g$ and integrate on the circle. Orthogonality leaves\n$1+\\sum_{\\nu\\ge1}|a_\\nu|^2$ and\n$\\sum_{\\nu\\ge1}\\nu|a_\\nu|^2$, respectively. The analytic function $g$ extends to a neighbourhood of the closed disc\nin this interior case,\nwhich justifies the termwise integrations.\nFor closed-disc centres replace $c_j$ by $rc_j$, $0<r<1$.\nThe actual coefficient of degree $\\nu$ becomes $r^\\nu a_\\nu$, where the\nfixed coefficients are defined from the analytic function near zero.\nPass to $r\\uparrow1$ first with an arbitrary finite coefficient sum.\nNonnegative finite deficits then give summability and the infinite inequality;\nno boundary holomorphic logarithm is required. Lean checks the\n\\href{https://github.com/wcook04/plectis-erdos/blob/1d70482583cc9c662cd3b86f66d75c54bb405aa6/lean/ErdosProblems/Erdos1041/WeightedCauchyTransportR10.lean\\#L66}{exact radial transport of each Cauchy coefficient},\nthe\n\\href{https://github.com/wcook04/plectis-erdos/blob/1d70482583cc9c662cd3b86f66d75c54bb405aa6/lean/ErdosProblems/Erdos1041/WeightedBoundaryDeficitR10.lean\\#L108}{finite boundary-deficit inequality},\nand the\n\\href{https://github.com/wcook04/plectis-erdos/blob/1d70482583cc9c662cd3b86f66d75c54bb405aa6/lean/ErdosProblems/Erdos1041/WeightedBoundaryDeficitR10.lean\\#L153}{summable full boundary deficit}.\n\nIf equality holds, all positive-degree coefficients vanish, so $g=1$ near\nzero and $\\sum_jw_j\\overline{c_j}/(1-\\overline{c_j}z)=0$ there.\nGroup equal centres before clearing denominators. A nonzero centre of positive\ntotal weight gives a nonzero pole coefficient, which is impossible. Thus\nequality is equivalent to $c_j=0$ on positive support. In particular,\nstrictly positive weights force all centres zero. A point of weight zero\nis unconstrained: weight one at $0$ and weight zero at $1$ still give\nequality.\nGrouping equal centres is necessary: an argument that assumed they\nwere distinct would not prove the stated equality case.\n\nEqual weights in $\\sum_jw_jG(c_j)^2\\le1$ give the quadratic finite\ninequality above for every $m\\ge1$, with equality only at the zero\nconfiguration. The linear inequality follows separately. For\n$M=\\sum_jw_jG(c_j)$, the variance identity\n\\[\n M^2+\\sum_jw_j(G(c_j)-M)^2=\\sum_jw_jG(c_j)^2\n\\]\ngives $M\\le1$. Equality in this Cauchy--Schwarz step means that the\nvalues $G(c_j)$ are constant at all indices of positive weight; equality\nin the final bound also requires equality in the quadratic inequality. The\nthree-point and four-point specialisations follow with their equality\ncases.  Lean checks\n\\href{https://github.com/wcook04/plectis-erdos/blob/3d6d938d696fed0fb71dd55115a18a73738ff223/lean/ErdosProblems/Erdos1041/PaperWeightedRefinementsR10.lean\\#L17}{the quadratic weighted inequality}.\nThe hypotheses permit repeated centres and zero weights;\nthe equality condition refers only to centres of positive weight.\n\nThis equality condition also identifies all equality cases of\nTheorem~\\ref{res:critical-value-budget} for a fixed containing disc\n$\\overline D(h,R)$ with $R>0$. Equality in the critical-value bound\nforces equality in the quadratic finite inequality after normalisation,\nso every critical point is $0$. Thus the normalised derivative is\n$nz^{n-1}$ and the polynomial is $z^n-\\lambda$, with $|\\lambda|=1$\nforced by equality. Restoring the centre and radius gives exactly\n$f(z)=(z-h)^n-\\lambda$ with $|\\lambda|=R^n$.\nEquality in either lower-power consequence also forces equality in the\nquadratic bound, so it has the same family. For $R=0$ the only possible\npolynomial is $(z-h)^n$. Lean does not check this classification.\n\nThe earlier three-point H\\\"older argument and four-point matching argument\nare not used here. Their scalar identities do not justify the omitted\ninequalities; the proof is the weighted Poisson calculation above.\n\n\\paragraph{Higher powers and vanishing complex power sums.}\nLean checks only the quadratic case; the analytic argument reaches\nfurther. To estimate $G^p$ for $p>0$, we need an\nanalytic function whose squared modulus is $G^p$. Using the same\nlogarithms as above, define\n\\[\n H_p=g^{p/2}=1+\\sum_{\\nu\\ge1}b_\\nu z^\\nu.\n\\]\nFor interior centres, logarithmic differentiation and the Poisson\nmajorant give\n\\begin{align*}\n P&=1-\\frac4p\\Re\\frac{\\zeta H_p'}{H_p},\\\\\n \\sum_jw_jG(c_j)^p\n &\\le\\int |H_p|^2P\\,dm\n =1-\\sum_{\\nu\\ge1}\\left(\\frac{4\\nu}{p}-1\\right)|b_\\nu|^2.\n\\end{align*}\nThe identity follows from the same two orthogonality calculations used\nfor $g$; only the coefficient of the derivative term changes. For\n$0<p\\le4$ every displayed coefficient is nonnegative. Hence\n\\[\n \\sum_jw_jG(c_j)^p\\le1\\qquad(0<p\\le4).\n\\]\nBoundary centres follow by replacing $c_j$ by $rc_j$: the degree-$\\nu$\ncoefficient becomes $r^\\nu b_\\nu$, and the values on the left are\n$G_r(rc_j)=\\prod_k|1-r^2\\overline{c_k}c_j|^{w_k}$.\nFirst retain finitely many nonnegative terms, then let $r\\uparrow1$.\nZero weights can be omitted throughout. This proves the closed-disc\ninequality without differentiating a boundary logarithm. The argument\nis the\n\\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos1041/SharpPowerDiscProducts.md}{power-parameter Poisson identity};\nLean does not check it.\n\nIn particular, equal weights and Lemma~\\ref{res:reflected-critical-value}\ngive, with $m=n-1$,\n\\[\n \\frac1m\\sum_j|f(c_j)|^{4/m}\\le\\frac1m\\sum_jG(c_j)^4\\le1\n\\]\non the unit disc. For roots in $\\overline D(h,R)$ this becomes\n\\[\n \\sum_j|f(c_j)|^{4/(n-1)}\\le(n-1)R^{4n/(n-1)}.\n\\]\nAs before, $R>0$ is handled by $R^{-n}f(h+Rz)$, and $R=0$ by the\nvanishing of all critical values. The constant is attained by\n$(z-h)^n-\\lambda$ with $|\\lambda|=R^n$. Thus the constant is sharp. Optimality of the exponent at each fixed\ndegree is not asserted.\n\nAt $p=4$ the coefficient of $|b_1|^2$ vanishes, so the previous\nquadratic equality proof cannot simply be reused. If equality holds,\nthe finite-deficit bounds imply $H_4=1+b_1z$. If $b_1\\ne0$, the\nrational identity $H_4'/H_4=2g'/g$ shows, by comparing poles and their\nresidues, that all nonzero centres coincide at $a=-\\overline{b_1}$\nand have total weight $1/2$. The other half of the weight is at zero.\nTheir fourth-power average is\n\\[\n \\tfrac12\\bigl(1+(1-|a|^2)^2\\bigr)\n =1-|a|^2+\\tfrac12|a|^4<1\\qquad(0<|a|\\le1),\n\\]\na contradiction. Thus $H_4=1$, and the pole argument already used for\n$g$ forces every centre of positive weight to be zero. Conversely that\nconfiguration gives equality. It follows that the fourth-power\ncritical-value estimate has the same equality family as the quadratic\none for a fixed containing disc.\n\nFor $p>4$, the coefficient $4/p-1$ is negative, so the nonnegative-deficit\nargument no longer applies. To extend its range we can force the first\nTaylor coefficients to vanish. Let $s\\ge1$ be an integer and suppose\n\\[\n \\sum_jw_jc_j^k=0\\qquad(1\\le k<s).\n\\]\nThe expansion\n\\[\n \\log H_p(z)=-\\frac p2\\sum_{k\\ge1}\n       \\frac{\\overline{\\sum_jw_jc_j^k}}{k}z^k\n\\]\nthen has no terms of degree below $s$, so $b_1=\\cdots=b_{s-1}=0$. All remaining\ncoefficient weights are nonnegative for $0<p\\le4s$. The same proof\ntherefore gives $\\sum_jw_jG(c_j)^p\\le1$ throughout that range.\nFor a monic polynomial with roots in $\\overline D(h,R)$, $R>0$,\nthe resulting endpoint is\n\\[\n \\sum_j|f(c_j)|^{4s/(n-1)}\\le(n-1)R^{4sn/(n-1)},\n \\qquad \\sum_j(c_j-h)^k=0\\quad(1\\le k<s).\n\\]\nThere is no cancellation hypothesis when $s=1$. For $s=2$, the condition\nsays that the critical-point centroid is the disc centre. Comparing the\ncoefficients of $f$ and $f'$ shows\n\\[\n \\frac1{n-1}\\sum_jc_j=\\frac1n\\sum_{i=1}^n z_i,\n\\]\nso this is also the root centroid. Recentring can increase the required\nradius: for $0<r<1$, the roots of $(z-r)^2(z+r)$ lie in\n$\\overline D(0,r)$, but their centroid is $r/3$, and a disc centred\nthere needs radius $4r/3$ to contain the root $-r$. Thus the\nexponent-$8/(n-1)$ statement cannot be applied at the old centre with\nthe old radius. The hypotheses require exact cancellation of the moments; upper\nbounds for their moduli are insufficient. Lean does not check the higher-power conclusions in this\nparagraph; the Lean proof of Theorem~\\ref{res:critical-value-budget} has\nexponent $2/(n-1)$.\n\n\\paragraph{Lean proofs of the Poisson and integration steps.}\nThe square-expansion proof above applies whenever $g$ is holomorphic on\nthe disc and continuous on its closure, with $|c|<1$. The Lean\n\\href{https://github.com/wcook04/plectis-erdos/blob/d4fed71423840f70f10edf27b9ad27c22fc4f49a/ErdosProblems/Erdos1041/PoissonNormSquare.lean#L114}{norm-square majorization}\nuses the Poisson formula for a real part and the nonnegativity of a square.\nThe \\href{https://github.com/wcook04/plectis-erdos/blob/d4fed71423840f70f10edf27b9ad27c22fc4f49a/ErdosProblems/Erdos1041/PoissonKernelBridge.lean#L27}{weighted kernel identity}\nidentifies the pointwise Poisson mixture.  Separately,\n\\href{https://github.com/wcook04/plectis-erdos/blob/d4fed71423840f70f10edf27b9ad27c22fc4f49a/ErdosProblems/Erdos1041/CircleSeriesTransport.lean#L49}{absolute series transport}\njustifies termwise circle integration when the terms are circle integrable\nand admit a summable uniform norm bound; its Taylor double-product form\nassumes absolute summability of both coefficient sequences.\n\n\\paragraph{A uniform central region in every degree.}\nFor every $m\\ge1$, the\n\\href{https://github.com/wcook04/plectis-erdos/blob/3be82b1a7340284aea72e9a5c8493cb020843921/ErdosProblems/Erdos1041/FreePointCentralCompletion.lean#L39}{inequality for points in a smaller disc}\nproves\n\\[\n |c_j|\\le\\sqrt{1-e^{-2}}\\quad(1\\le j\\le m)\n \\quad\\Longrightarrow\\quad\n \\sum_{j=1}^m\\left(\\prod_{k=1}^m|1-\\overline c_jc_k|\\right)^{1/m}\\le m.\n\\]\nTo see the mechanism, set\n$h_j=m^{-1}\\sum_k\\log|1-\\overline c_jc_k|$ and\n$D_j=-\\log(1-|c_j|^2)$. Write\n$M_\\nu=m^{-1}\\sum_k c_k^\\nu$ for the normalised complex power sums used in this\ncalculation. The absolutely convergent series for $\\log(1-z)$ gives\n\\[\n h_j=-\\Re\\sum_{\\nu\\ge1}\\frac{\\overline c_j^{\\,\\nu}M_\\nu}{\\nu},\n \\qquad\n -\\frac1m\\sum_jh_j=\\sum_{\\nu\\ge1}\\frac{|M_\\nu|^2}{\\nu}.\n\\]\nIn particular, $\\sum_jh_j\\le0$. Cauchy--Schwarz applied to the first\nseries, using\n$\\sum_{\\nu\\ge1}|c_j|^{2\\nu}/\\nu=D_j$, gives\n$h_j^2\\le(-m^{-1}\\sum_i h_i)D_j$. The radius bound gives $D_j\\le2$.\nPut $r=\\sqrt{-m^{-1}\\sum_i h_i}$. If $r=0$, every $h_j$ vanishes.\nFor $r>0$, convexity bounds $e^h$ by the chord of its graph over\n$[-\\sqrt2r,\\sqrt2r]$. Averaging and using $m^{-1}\\sum_jh_j=-r^2$ gives\n\\[\n \\frac1m\\sum_j e^{h_j}\n \\le\\cosh(\\sqrt2r)-\\frac r{\\sqrt2}\\sinh(\\sqrt2r)\\le1.\n\\]\nFor the last inequality, the function $\\cosh u-(u/2)\\sinh u$ equals\n$1$ at $u=0$ and has derivative $(\\sinh u-u\\cosh u)/2\\le0$ for\n$u\\ge0$. Thus the radius bound supplies the estimates needed for the\ngeometric-mean inequality; no further series hypotheses are imposed.\nThis alternative proof includes the all-zero configuration but requires\nthe smaller radius $\\sqrt{1-e^{-2}}$. The weighted theorem above covers\nthe whole closed unit disc. The next logarithmic argument imposes a\ndifferent sufficient radius condition; neither restriction belongs to\nthe weighted theorem.\n\nFor $m\\ge1$, let $|c_i|^2\\le a<1$, with $a\\ge0$, and put\n$L=-\\log(1-a)$. The\n\\href{https://github.com/wcook04/plectis-erdos/blob/457e0c74b2666f7fb2d825c967e49e439a600b23/ErdosProblems/Erdos1041/FreePointUniformRadius.lean#L84}{uniform-radius theorem}\ngives the geometric-mean inequality\n\\[\n e^L\\le1+2L\\quad\\Longrightarrow\\quad\n \\sum_{i=1}^m\\left(\\prod_{j=1}^m|1-\\overline c_i c_j|\\right)^{1/m}\\le m.\n\\]\nFor $a=1/2$ the scalar hypothesis holds, since\n$2\\le1+2\\log2$; for $a=3/4$ it fails, since $4>1+2\\log4$.\nThus it supplies a sufficient restriction on the squared radii, not a\nnecessary condition for the geometric-mean inequality.\n\nThe proof uses point-dependent logarithmic bounds. Write\n$H_i=m^{-1}\\sum_j\\log|1-\\overline c_i c_j|$ and\n$D_i=-\\log(1-|c_i|^2)$. The preceding logarithmic calculation gives\n$\\sum_iH_i\\le0$ and $H_i^2\\le-(D_i/m)\\sum_jH_j$.\nFor nonnegative upper bounds $M_i\\ge H_i$, Taylor's formula with\nintegral remainder gives\n\\[\n e^{H_i}\\le1+H_i+\\Phi(M_i)H_i^2,\n \\qquad \\Phi(t)=\\int_0^1(1-s)e^{st}\\,ds.\n\\]\nSince $\\Phi(M_i)\\ge0$, summing and using the quadratic bounds yields\n\\[\n \\sum_i e^{H_i}\n \\le m+\\left(1-\\frac1m\\sum_i\\Phi(M_i)D_i\\right)\\sum_jH_j\n \\le m\n\\]\nwhenever $\\sum_i\\Phi(M_i)D_i\\le m$. This is the sufficient condition\nin the\n\\href{https://github.com/wcook04/plectis-erdos/blob/0d34630e1cc9d2b1ac6edfa7cfdba83b31bdc8fe/ErdosProblems/Erdos1041/LogKernelCentralCertificate.lean#L99}{formal inequality with individual upper bounds}.\nTaking $M_i=L$ gives $\\Phi(L)L\\le1$ as a sufficient condition,\nsince $H_i,D_i\\le L$. For $L>0$, integration gives\n$\\Phi(L)=(e^L-1-L)/L^2$; hence this condition is exactly\n$e^L\\le1+2L$. At $L=0$ all centres vanish, so the conclusion is\nimmediate.\n\n\\paragraph{Four points without a matching or numerical split.}\nThe preceding weighted proof with $m=4$ and $w_j=1/4$ gives\n\\[\n \\sum_{j=1}^4\\left(\\prod_{k=1}^4|1-\\overline{c_k}c_j|\\right)^{1/4}\\le4,\n \\qquad |c_j|\\le1,\n\\]\nwith equality if and only if all four centres vanish. Equality in this\nlinear form forces equality in the quadratic inequality, by the variance\nidentity above; Lean proves\n\\href{https://github.com/wcook04/plectis-erdos/blob/e6c2d8f77ac24753c5216a49f4daf7f7388b309f/lean/ErdosProblems/Erdos1041/PaperWeightedRefinementsR10.lean\\#L17-L21}{that equality case}.\nIn particular, the\ninequality is strict as soon as one centre is nonzero, even with repeated\ncentres or centres on the unit circle.\n\nThis four-point consequence needs neither the former $21/25$ split\nnor a perfect-matching H\\\"older step. The scalar stationary-point\ninequalities from that approach are not a proof of its missing steps.\nThe two restricted arguments above remain independent alternatives.\n\nThe mean inequality gives $\\mu\\le R^n$. In the unit-disc normalisation\n$R=1$, this does not force the smaller quintic threshold $\\mu<1/M_5$\nbelow. For $f(z)=z^n-1$, every critical value has modulus one, so no\nuniformly smaller bound holds on the closed-disc class. These modulus bounds still require a separate path selection or\nmetric comparison before they yield an inverse-ray length estimate.\n\n\\subsection*{A sufficient critical-value threshold in degree five}\n\n\\papersectiontarget{root-distance-proofs}\nThe reciprocal balance at a critical point bounds the sum of the two\nnearest root distances. We begin with this formal geometric statement,\nthen distinguish distance from containment in the degree-five argument.\n\n\\begin{theorem}[a geometric-mean bound for distances to a critical point]\n\\label{res:critical-proximity}\nLet $n\\ge2$, let $z_1,\\ldots,z_n,c\\in\\mathbb C$ with $c\\ne z_k$ for every\n$k$, and suppose\n\\[\n  \\sum_{k=1}^n\\frac1{c-z_k}=0.\n\\]\nIf $r>0$ is determined by\n\\[\n  r^n=\\prod_{k=1}^n|c-z_k|,\n\\]\nthen there are distinct indices $i,j$ such that\n\\[\n  |c-z_i|+|c-z_j|\\le2r.\n\\]\n\\end{theorem}\n\\begin{proof}\nRelabel the points so that their distances satisfy\n$d_1\\le d_2\\le\\cdots\\le d_n$. The reciprocal balance and the triangle\ninequality give\n\\[\n \\frac1{d_1}\n =\\left|\\sum_{j=2}^n\\frac1{c-z_j}\\right|\n \\le\\sum_{j=2}^n\\frac1{d_j}\\le\\frac{n-1}{d_2}.\n\\]\nThus $d_2\\le(n-1)d_1$. Also $d_1d_2^{n-1}\\le r^n$, since every\nremaining distance is at least $d_2$. For $t=d_2/d_1\\in[1,n-1]$,\n\\[\n \\frac{d}{dt}\\log\\frac{2t^{(n-1)/n}}{1+t}\n       =\\frac{n-1-t}{nt(1+t)}\\ge0.\n\\]\nThe logarithm vanishes at $t=1$, so\n$d_1+d_2\\le2(d_1d_2^{n-1})^{1/n}\\le2r$.\n\\end{proof}\n\n\\paragraph{The distance bound for the two nearest roots.}\n\\hypertarget{two-nearest-root-distance}{}\n\\begin{corollary}[two nearest roots]\\label{res:two-nearest-roots}\nIf the roots lie in the open unit disc and $c$ is a non-root critical point,\nthe two nearest roots to $c$ have total distance strictly below $2$.\n\\end{corollary}\n\nLet $f$ have degree $n\\ge2$ and all roots in the closed unit disc, and\nlet $c$ be a critical point with $f(c)\\ne0$. List roots with multiplicity.\nIts two nearest roots have\ntotal distance at most $2$. The root-disc hypothesis, rather than a bound\non $|f(c)|$, is what gives this estimate.\n\nRotate so that $c=t\\ge0$, and order the distances\nas $0<d_1\\le d_2\\le\\cdots\\le d_n$. Logarithmic differentiation gives\n$\\sum_j(z_j-c)^{-1}=-f'(c)/f(c)=0$. Dividing\n$|z_j-c|^2+2t\\Re(z_j-c)\\le1-t^2$ by $d_j^2$, then summing and using\nthe reciprocal balance, gives\n\\[\n n\\le(1-t^2)\\sum_jd_j^{-2}.\n\\]\nThus $t<1$ and $d_1\\le\\sqrt{1-t^2}\\le1$. The same reciprocal balance,\nwith the nearest term isolated, gives $d_2\\le(n-1)d_1$.\n\nSuppose $d_1+d_2>2$. Since $d_2\\le1+t$, we have $d_1>1-t$, and hence\n\\[\n 1-t^2<d_1(2-d_1)<d_1d_2.\n\\]\nThe first strict inequality uses that $x(2-x)$ is increasing on $[0,1]$.\nIt follows that\n\\[\n n\\le(1-t^2)\\bigl(d_1^{-2}+(n-1)d_2^{-2}\\bigr)\n <\\frac{d_2}{d_1}+(n-1)\\frac{d_1}{d_2}\\le n.\n\\]\nFor the last step, put $x=d_2/d_1\\in[1,n-1]$ and expand\n$(x-1)(x-(n-1))\\le0$. This contradiction proves the closed-disc bound.\nFor roots in the open unit disc, scaling by a containing radius\n$0<R<1$ gives $d_1+d_2\\le2R<2$.\nA\n\\href{https://github.com/wcook04/plectis-erdos/blob/a729f05c40398663fd586c9da487ad874d898639/research_corpus/Erdos1041/GlobalCriticalTwoNearestBudget.md}{research note} gives the same argument. Lean checks the \\href{https://github.com/wcook04/plectis-erdos/blob/52d6c45ad203ba619cb5fe6ba485c0b5400519ea/lean/ErdosProblems/Erdos1041/PaperCompleteR20/TwoNearestPolynomial.lean\\#L31}{strict distance bound from the critical polynomial}, including the root-disc reduction and the choice of the two nearest indices.\n\nThis argument selects the two nearest roots but does not put their\nsegments inside a polynomial sublevel set. The next calculation supplies\nthat separate containment estimate under a bound on $|f(c)|$.\n\n\\paragraph{Two failures of straight segments.}\n\\papersectiontarget{straight-path-counterexamples}\n\\begin{proposition}[two counterexamples to straight-path assertions]\n\\label{res:straight-no-go}\nThere is a monic quintic with all roots in the open unit disc and a non-root\ncritical point $c$ whose unique nearest root has a point on the straight segment\nto $c$ outside $\\{|f|<1\\}$.  There is also a monic cubic with all roots in the\nopen unit disc such that the midpoint of every pair of distinct roots lies\noutside $\\{|f|<1\\}$.\n\\end{proposition}\n\\begin{proof}\nFor the quintic, put $p=999/1000$ and $a=(901/902)p$, and take\n\\[\n Q(z)=(z-a)(z^2+p^2)\n       \\left(z^2+\\frac{902}{901}pz+p^2\\right).\n\\]\nIts roots are\n\\[\n a,\\quad ip,\\quad-ip,\\quad\n p\\frac{-451+780i}{901},\\quad p\\frac{-451-780i}{901}.\n\\]\nSince $451^2+780^2=901^2$, the last four roots have modulus $p<1$,\nwhile $0<a<p$. Also $Q'(0)=0$ and $Q(0)=-ap^4\\ne0$. Thus $a$ is\nthe uniquely nearest root to the non-root critical point $0$.\nAt one tenth of that segment, exact rational arithmetic gives\n\\[\n |Q(a/10)|=\n \\frac{9a}{10}\\left(\\frac{a^2}{100}+p^2\\right)\n \\left(\\frac{a^2}{100}+\\frac{11p^2}{10}\\right)>1.\n\\]\nFor the cubic, take $g(z)=z^3-(99/100)^3$. If $z,w$ are distinct roots,\nthen $z+w$ is the negative of the third root, so\n\\[\n g\\!\\left(\\frac{z+w}{2}\\right)\n =-\\frac98\\left(\\frac{99}{100}\\right)^3,\n \\qquad \\frac98\\left(\\frac{99}{100}\\right)^3\n =\\frac{8732691}{8000000}>1.\n\\]\nAll three roots have modulus $99/100$.\n\\end{proof}\n\n\\paragraph{Containment of the two selected segments.}\nLet $f$ be monic of degree $n\\ge3$ with roots in the closed unit disc.\nThe following estimate controls the segments from a critical point to\nits two nearest roots. To specify the constant, put\n\\[\n\\begin{split}\n M_n^2=\\max_{\\substack{0\\le t\\le1\\\\4-n\\le y\\le n}}\n &(1-t)^2\\bigl((1-t)^2+2ty+n(n-2)t^2\\bigr)\\\\\n &\\hspace{8mm}\\cdot\n \\left((1-t)^2+\\frac{2t(n-y)}{n-2}\\right)^{n-2}.\n\\end{split}\n\\]\nHere $t$ parametrises the segment. The product estimates below use\n$f'(c)=0$ and the ordering of distances, not the root-disc hypothesis;\nthe latter enters when bounding the combined segment length. To explain\nthe other parameter and the coefficient $n(n-2)$, suppose $f(c)\\ne0$, order the roots\n$a_1,\\ldots,a_n$ by distance from $c$, and set\n$v_j=(a_2-c)/(a_j-c)$. Then $v_2=1$, $|v_j|\\le1$ for $j\\ge2$,\nand $\\sum_jv_j=0$ by $f'(c)=0$. With $y=1-\\Re v_1$, this gives\n\\[\n y=2+\\sum_{j=3}^n\\Re v_j\\in[4-n,n],\\qquad |v_1|\\le n-1.\n\\]\nConsequently\n\\[\n\\begin{split}\n |1-tv_1|^2&\\le(1-t)^2+2ty+n(n-2)t^2,\\\\\n \\frac1{n-2}\\sum_{j=3}^n|1-tv_j|^2\n &\\le(1-t)^2+\\frac{2t(n-y)}{n-2}.\n\\end{split}\n\\]\nThe factor $v_2=1$ contributes $(1-t)^2$. Applying AM--GM to the\nremaining $n-2$ squared factors in\n$f(c+t(a_2-c))/f(c)=\\prod_j(1-tv_j)$ gives the defining expression\nfor $M_n^2$. Thus the maximisation is an explicit bound for the second\nsegment, not an additional optimisation hypothesis about the roots.\n\nFor the nearer segment, put $x_j=(a_1-c)/(a_j-c)$. Now $x_1=1$,\n$|x_j|\\le1$ for every $j$, and $\\sum_jx_j=0$. Hence\n\\[\n \\sum_{j=2}^n|1-tx_j|^2\n \\le (n-1)(1+t^2)+2t,\n\\]\nso AM--GM gives\n\\[\n \\left|\\frac{f(c+t(a_1-c))}{f(c)}\\right|^2\n \\le(1-t)^2\\left((1-t)^2+\\frac{2tn}{n-1}\\right)^{n-1}.\n\\]\nThis bound is at most $M_n^2$ for every $n\\ge3$, not just for\nnumerically tested degrees. Indeed, set $y=n/(n-1)$ in the defining\nmaximum. This value is admissible because\n$n/(n-1)-(4-n)=(n-2)^2/(n-1)\\ge0$. The last factor then has base\n$(1-t)^2+2tn/(n-1)$, and the preceding factor has this same base plus\n$n(n-2)t^2\\ge0$. Their product therefore dominates the displayed\nnearer-segment bound. Thus $|f(z)|\\le M_n|f(c)|$ on both segments, so\n$0<|f(c)|\\le1/M_n$ gives containment in $\\{|f|\\le1\\}$; the\ncritical-point distance estimate gives total length at most $2$.\nIf $f(c)=0$, two root occurrences already coincide at $c$.\nThe quadratic case is the direct root segment treated earlier.\nFor degree five the maximum evaluates to\n\\[\n M_5=(1-t_*)(1+t_*)^3\\sqrt{16t_*^2-4t_*+1},\n \\qquad\n t_*=\\frac{5}{16}+\\frac{3\\sqrt{105}}{80},\n \\qquad\n \\frac{1}{M_5}=0.2760461\\ldots.\n\\]\nThe breakpoint comes from the constraint on $y$, not from a numerical\nsearch. For fixed $0<t<1$, write the two bases in the definition of\n$M_5$ as\n\\[\n A=(1-t)^2+2ty+15t^2,\\qquad\n B=(1-t)^2+\\frac{2t(5-y)}3.\n\\]\nThey are positive for $-1\\le y\\le5$, and\n$\\partial_y\\log(AB^3)=2t(B-A)/(AB)$. Thus the unconstrained maximum\noccurs at $A=B$, or $y=5/4-45t/8$. This reaches the lower endpoint\n$-1$ at $t=2/5$; thereafter the maximum is at $y=-1$.\nFor $0\\le t\\le2/5$ the resulting expression is\n$(1-t)(1+t/2+19t^2/4)^2$, whose derivative is\n\\[\n \\frac{5t(14-19t)(19t^2+2t+4)}{16}\\ge0.\n\\]\nFor $2/5\\le t\\le1$, the square of the resulting expression is\n$(1-t)^2(1+t)^6(16t^2-4t+1)$, whose derivative is\n\\[\n -4t(1-t)(1+t)^5(40t^2-25t-2).\n\\]\nThe quadratic changes sign once on this interval, at $t_*$, so the\nsecond branch increases from the common breakpoint to $t_*$ and then\ndecreases to zero. Since the first branch increases to that breakpoint,\n$t_*$ gives the global maximum. This proves the displayed value without\nsampling either optimisation variable.\nAs an existence criterion this is weaker than the $13/25$ theorem.\nIts different content is the control of the two straight segments from\nthe specified critical point, rather than a curve selected by area\naveraging. For $z^5-b$ it requires $|b|\\le1/M_5$, so it excludes part\nof a binomial family already covered by the trinomial theorem. No claim\nis made that every quintic has a critical value below this level.\n\nA near-regular pentagon illustrates why a prescribed joining point is\nrestrictive. For sufficiently small $\\epsilon>0$, set\n\\[\n P_\\epsilon(z)=z^5+\\epsilon^5z^4+\\epsilon^3z^3\n              -\\epsilon^3z^2-\\epsilon^5z-1,\n \\quad F_\\epsilon(z)=r^5P_\\epsilon(z/r),\\quad r=1-\\epsilon^9.\n\\]\nAll five roots of $F_\\epsilon$ lie in the open unit disc, approach a\nregular pentagon, and have escaping segments from the origin. Here is a\nproof of that assertion, including the effect of the radial contraction.\n\nThe roots of $P_\\epsilon$ are simple perturbations\n$\\zeta_\\epsilon=\\zeta_0+O(\\epsilon^3)$ of the fifth roots of unity.\nThe identity\n\\[\n z^5\\overline{P_\\epsilon(1/\\overline z)}=-P_\\epsilon(z)\n\\]\nmakes the roots invariant under conjugate inversion. Uniqueness of the\nroot near each $\\zeta_0$ therefore forces $|\\zeta_\\epsilon|=1$.\nThe roots $r\\zeta_\\epsilon$ of $F_\\epsilon$ consequently have modulus\n$r<1$. The displacement estimate follows from the implicit function\ntheorem at each simple root of $z^5-1$, since the first coefficient\nperturbations have order $\\epsilon^3$.\n\nFor $\\zeta_0=1,e^{4\\pi i/5},e^{-4\\pi i/5}$, take $t=\\epsilon/4$.\nSubstitution into the polynomial gives\n\\[\n |F_\\epsilon(rt\\zeta_\\epsilon)|^2\n =1+\\left(\\frac{\\Re\\zeta_0^2}{8}-\\frac1{512}\\right)\\epsilon^5\n       +O(\\epsilon^6).\n\\]\nThe coefficient is positive, since\n$\\Re\\zeta_0^2\\ge(\\sqrt5-1)/4$.\nFor the remaining roots, with $\\zeta_0=e^{2\\pi i/5}$ or\n$e^{-2\\pi i/5}$, take $t=\\epsilon^2/4$ instead. Then\n\\[\n |F_\\epsilon(rt\\zeta_\\epsilon)|^2\n =1+\\frac{3\\sqrt5-5}{32}\\epsilon^7+O(\\epsilon^9)>1\n\\]\nfor sufficiently small $\\epsilon$. The two scales are needed because\n$\\Re\\zeta_0^2$ is negative for this second pair; its positive linear\ncontribution dominates at the smaller scale. Multiplication by\n$r^{10}=1+O(\\epsilon^9)$ cannot remove either positive leading term.\nThe remainders are uniform over the five root branches, so one sufficiently\nsmall $\\epsilon$ works for every segment. This is an ordinary asymptotic\nproof; it gives no numerical upper bound for the permitted $\\epsilon$.\n\nThe addendum \\cite[Proposition~1]{revision2026} reports the stronger\nassertion that no critical point of this family supplies two contained\nstraight arms of total length at most $2$. Its proof is not available.\nThe origin calculation above does not prove it: $P_\\epsilon'(0)=-\\epsilon^5$\nshows that the origin is not a critical point. The stronger reported\nassertion is not used in any theorem here. Neither the proved failure at\nthe origin nor the reported critical-point obstruction excludes curved\ninverse images of rays or a different joining point.\n\nA different local question allows a freely chosen, noncritical vertex:\nfor some $\\delta>0$, does every monic quintic with open-disc roots within\n$\\delta$ of a rotated regular pentagon admit two contained straight arms\nthrough some $h\\in\\mathbb C$ of total length below $2$?\nNeither the origin nor a critical point is prescribed in this question.\nThe sufficient low-critical threshold preceding this paragraph is unchanged.\n\n\\paragraph{Area of a connected sublevel set.}\nThe following calculation applies in every degree $n\\ge2$, independently\nof the degree-five segment criterion. Let $f$ be monic of degree $n$ and let $t>0$ with\n$K_t=\\{|f|\\le t\\}$ connected. Translate its root centroid to $0$, so\n$f(z)=z^n+c_{n-2}z^{n-2}+\\cdots$, and let $\\psi$ be the exterior map\nnormalised at infinity with positive leading coefficient. For a regular\nconnected lemniscate, the identity $f(\\psi(\\zeta))=t\\zeta^n$ is\n\\cite[(2.3)]{eks2010} after scaling the value level. The following argument\nalso permits critical boundary points. The exterior Green functions\n$(\\log|f|-\\log t)/n$ and $\\log|\\psi^{-1}|$ agree: both vanish at the\nboundary and have a logarithmic pole of coefficient one at infinity.\nComparison there gives the leading coefficient $t^{1/n}$ for $\\psi$.\nThe quotient $f(\\psi(\\zeta))/(t\\zeta^n)$ has constant modulus one and\nlimit one at infinity, proving the polynomial identity. Its\n$\\zeta^{n-1}$ coefficient forces the constant Laurent coefficient of\n$\\psi$ to vanish. Thus\n\\[\n \\psi(\\zeta)=t^{1/n}\\zeta+\\sum_{k\\ge1}a_k\\zeta^{-k}.\n\\]\n\nGr\\\"onwall's area identity takes the form\n\\[\n \\operatorname{Area}(K_t)=\\pi\\left(t^{2/n}-\n \\sum_{k\\ge1}k|a_k|^2\\right).\n\\]\nOne can obtain it directly without assuming that level $t$ is regular.\nFor $r>1$, Green's area formula on the analytic Jordan curve\n$\\psi(re^{i\\theta})$ gives the enclosed area\n$\\pi(t^{2/n}r^2-\\sum_{k\\ge1}k|a_k|^2r^{-2k})$.\nThese enclosed compact sets decrease to $K_t$ as $r\\downarrow1$;\ncontinuity of area from above and monotone convergence of the series\ngive the identity. Comparing the coefficient of $\\zeta^{n-2}$ in the\nexterior identity now gives\n\\[\n nt^{(n-1)/n}a_1+c_{n-2}t^{(n-2)/n}=0,\n \\qquad a_1=-\\frac{c_{n-2}}{n\\,t^{1/n}}.\n\\]\nThe nonnegative coefficient sum quantifies the loss from P\\'olya's\nupper bound. It does not by itself control an internal path length.\nNo analogous coefficient formula for an individual component of a\ndisconnected sublevel set is established here. The\ndegree-five statements and their evidence classes are recorded in\n\\href{https://github.com/wcook04/plectis-erdos/blob/f214a6b45528dc5eefe20ffadc35f2e981627d4c/research_corpus/Erdos1041/Degree5AssemblyAndSharpenedCuts.md}{the degree-five estimates and their proofs}.\n\n\\subsection*{Numerical tests of selection by the smallest critical value}\n\nWe now ask whether a small critical value selects a short pair of\ninverse arms. The following records test that selection rule and a\nrelated sum over critical points. For a critical point $c$ whose two\ninverse arms over $[0,f(c)]$ reach roots, write $L_f(c)$ for their total\nlength.\nThe first computation tests the rule of choosing the critical point\nwith uniquely smallest value modulus. A\n\\href{https://github.com/wcook04/plectis-erdos/blob/f214a6b45528dc5eefe20ffadc35f2e981627d4c/research_corpus/Erdos1041/MinimalHubArmBudgetRefutation.md}{computational record}\ngives an unnormalised\nquintic $f_0$, a selected critical point $c_0$, and the numerical values\n\\[\n L_{f_0}(c_0)\\approx2.057343275393654508,\n \\qquad R\\approx1.021393477405696164,\n\\]\nwhere $R$ is the radius of a smallest disc containing the roots of $f_0$.\nThe first number is not the length after normalisation to the unit disc.\nIf $h$ is the centre of that disc, set\n\\[\n F(z)=R^{-5}f_0(h+Rz).\n\\]\nThis preserves monicity, sends the roots into the closed unit disc,\nand divides every corresponding curve length by $R$. The reported\nnormalised values are therefore\n\\[\n L_F\\!\\left(\\frac{c_0-h}{R}\\right)\\approx2.01425143287505,\n \\qquad\n \\left|F\\!\\left(\\frac{c_0-h}{R}\\right)\\right|\n \\approx0.899569245.\n\\]\nA further contraction uses $F_s(z)=s^5F(z/s)$, with $0<s<1$: roots and\ncurve lengths are multiplied by $s$, and critical values by $s^5$.\nThus the reported excess would persist for $s$ sufficiently close to $1$,\nwith roots in the open unit disc.\n\nThese observations do not yet constitute a certified counterexample. A small\nresidual at a computed fibre root does not enclose that root or establish\nits membership in the tracked inverse branch. The inscribed-polyline\ncomparison needs both facts before it gives a rigorous lower bound for\nthe intended arms. Another critical point supplies a shorter pair in the reported\ncomputation, which establishes no universal selection rule.\n\nSecond, a\n\\href{https://github.com/wcook04/plectis-erdos/blob/f214a6b45528dc5eefe20ffadc35f2e981627d4c/research_corpus/Erdos1041/SeparatrixAggregateReduction.md}{second computational record}\nreports numerical violations of\n$\\sum_cL_f(c)\\le2(n-1)R$ in degrees four and five. Here $R$ is the radius of\na smallest disc containing the roots. The quartic computation gives\n$\\sum_cL_f(c)=6.006352157\\ldots$ against the proposed bound $6$.\nSeveral differential-equation solvers agree, and an inscribed-polyline\ncalculation using numerically computed fibre roots gives a similar excess.\nThese checks do not provide rigorous enclosures for those roots, their\nbranch assignment or the resulting length lower bound. They suggest failure of the aggregate inequality, but do not certify\nan example or an open set of violating polynomials. Its algebraic factor\n\\[\n \\sum_{k=1}^{n-1}|f(c_k)|^{1/n}\\le(n-1)R\n\\]\nis proved for every $n\\ge2$ in Theorem~\\ref{res:critical-value-budget}.\nThe value estimate alone supplies no bound for the associated path-length\nsum. The algebraic theorem therefore remains useful independently of how\nthe numerical aggregate examples are resolved.\n\nThird, all five origin segments can escape even when the roots are\narbitrarily close to a regular pentagon. The two-scale calculation above\nproves this for the family $F_\\epsilon$ attributed to the earlier note\n\\cite[Proposition~A of the earlier note]{revision2026}; an\nisolated numerical example would not prove the assertion for arbitrarily\nsmall defects. Failure for every critical joining point is a different,\nstronger claim, reported in the addendum but not established by the\norigin calculation. No conclusion about curved inverse-ray paths or an\narbitrary connecting curve follows from either straight-segment assertion.\n\n\\subsection*{Unformalised remark: limits of contained paths at fixed degree}\n\\hypertarget{closed-class-limits}{}\nThe compactness argument linked from the short paper concerns arbitrary\ncurves in a closed lemniscate. We give its ordinary proof here. It has\nnot been formalised, and it gives no attainment statement for paths\nconfined to an open lemniscate.\n\nFix $n\\ge2$, and let $\\mathcal K_n$ be the compact coefficient class of\nmonic degree-$n$ polynomials with roots in the closed unit disc.  Define\n\\[\n \\Lambda(f)=\\inf_\\gamma\\operatorname{length}(\\gamma),\n\\]\nwhere $\\gamma\\subset\\{|f|\\le1\\}$ joins two different listed root\noccurrences. A repeated root permits a constant curve; an empty set of\ncompetitors has infimum $+\\infty$.\n\n\\paragraph{Passing a length bound to a limit.}\nLet $f_j\\to f$ in $\\mathcal K_n$. If the lower limit of\n$\\Lambda(f_j)$ is finite, pass to a subsequence on which these infima\nconverge to it and choose rectifiable curves $\\gamma_j$ with lengths at\nmost $\\Lambda(f_j)+1/j$. Every closed unit sublevel lies in $|z|\\le2$:\nfor $|z|>2$, the root factorization gives\n$|f_j(z)|\\ge(|z|-1)^n>1$. Constant-speed parametrisation on $[0,1]$\ntherefore gives uniformly bounded images and Lipschitz constants.\nArzel\\`a--Ascoli supplies a uniformly convergent subsequence.\n\nTo retain the endpoints with their multiplicities, list the two chosen\noccurrences first among the roots of $f_j$, and list the remaining roots\narbitrarily. Compactness of the product of $n$ closed unit discs gives a\nfurther subsequence on which this entire list converges. The limiting\nfactorization is $f$, so its first two entries remain two occurrences,\neven if their locations coincide. Uniform convergence of the curves and\nof the polynomials on $|z|\\le2$ gives $|f|\\le1$ on the limiting curve\n$\\gamma$. For any partition $0=t_0<\\cdots<t_M=1$,\n\\[\n \\sum_{\\ell=1}^M|\\gamma(t_\\ell)-\\gamma(t_{\\ell-1})|\n =\\lim_j\\sum_{\\ell=1}^M|\\gamma_j(t_\\ell)-\\gamma_j(t_{\\ell-1})|\n \\le\\liminf_j\\operatorname{length}(\\gamma_j).\n\\]\nTaking the supremum over partitions proves the required lower\nsemicontinuity of length and hence\n$\\Lambda(f)\\le\\liminf_j\\Lambda(f_j)$. An infinite lower limit needs no\nargument. With $f$ fixed, the same compactness proof shows that a finite\ninfimum is attained. Thus a bound $\\Lambda\\le2$ on a dense class gives\nan actual path of length at most $2$ throughout $\\mathcal K_n$.\nFor an open-disc polynomial, choose a containing disc of radius\n$0<R<1$ and scale: the resulting path has length at most $2R<2$ and\nlies in $\\{|f|\\le R^n\\}\\subset\\{|f|<1\\}$.\n\nFor $f(z)=z^n-1$, the inequality $|z^n-1|\\le1$ implies\n$2\\Re(z^n)\\ge|z|^{2n}>0$ whenever $z\\ne0$. Thus the punctured sublevel\nlies in $n$ disjoint angular sectors, one per root. A path between\ndifferent roots must pass through zero and has length at least $2$ by\nthe triangle inequality. The two radial segments attain that length.  The corresponding historical assertion on the closed class is\n\\[\n \\boxed{\\Lambda(f)\\le\\Lambda(z^n-1)=2\\qquad(f\\in\\mathcal K_n).}\n\\]\nThe lower-semicontinuity argument is proved in\n\\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos1041/GenericSufficiencyClosure.md}{the compactness proof}.\n\nThe displayed universal inequality is false in degree seven. Indeed,\nif it held throughout $\\mathcal K_7$, the scaling step above would give\na path of length less than $2$ inside the open lemniscate of the\npolynomial in Theorem~\\ref{res:ani-degree-seven-counterexample-long}.\nThat theorem excludes such a path. Thus no estimate on a dense class can\nimply the displayed universal inequality by this compactness argument.\nThe reduction remains valid for separately specified polynomial classes.\n\n\\subsection*{Unformalised remark: Blaschke products and varying degree}\n\\hypertarget{blaschke-product-examples}{}\nPowers of a Blaschke product give critical values close to one another\nwhile also admitting short connectors in a suitable asymptotic regime.\nThe finite certificates and the ordinary limiting arguments have\ndifferent scopes, which we keep separate below.\n\nA research\n\\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos1041/BlaschkePowerCriticalSpectra.md}{note on powers of Blaschke products}\nuses the degree-$2N$ polynomials\n\\[\n F_N(z)=[z(z+b)]^N-(1+bz)^N,\\qquad N\\ge1,\\quad 0<b<1,\n\\]\nand their radial contractions. Its finite rational certificates and\nits asymptotic proofs have different roles. The degree-eight certificate\nexcludes an unconditional critical-value exponent $8/7$. It does not\ncontradict the ordinary exponent $4/7$ proved by the Poisson argument above,\nor the smaller exponent $2/7$ in Theorem~\\ref{res:critical-value-budget}.\nFor $N\\ge2$, the root centroid of $F_N$ is $-b/2$, as its\n$z^{2N-1}$ coefficient is $Nb$. A radial contraction by $r>0$ changes\nthis to $-rb/2$, not zero. Thus the degree-eight example does not satisfy\nthe additional centroid condition that permits numerator $8$ above. The degree-twenty-four certificate gives critical values in\n$|v+1|<1/12$, so $\\mu>11/12$ while every radius-$4/3$ separation test\nfails. These fixed-degree examples do not establish optimality uniformly\nin degree. That conclusion uses the limiting measures in Sections~2--3\nof the note.\n\nFor $b=\\lambda/N$ with fixed $0<\\lambda\\le1$, Section~4 constructs\nconnectors of length at most\n$[\\log(2/\\lambda)+\\pi+o(1)]/N$. The two radial pieces reduce the\noscillatory term; a short circular arc then joins them in a sector where\nthe remaining factor has modulus below one. The unspecified starting\ndegree depends on $\\lambda$. For $b=e^{-\\alpha N}/N$ with fixed\n$\\alpha>0$, the same section instead proves\n$\\Lambda(F_N)\\to2(1-e^{-\\alpha/2})$ for the uncontracted polynomials\n$F_N$ displayed above and the closed-level functional defined earlier.\nThe second assertion needs its own argument: the first does not allow\na varying parameter to be substituted. Both use ordinary asymptotic arguments. Neither follows from the\nfinite certificates, and neither has a formally checked analytic proof. They concern varying degree and do not contradict fixed-degree\ncompactness or prove the unrestricted assertion.\n\n\\subsection*{What the earlier inverse-ray approach would have required}\n\nThe earlier proposed sufficient condition concerned \\emph{curved}\ninverse images of rays, not straight segments.  On a generic ray-separated class let $L_f(c)$ be\nthe length of the two-root descent arc at an admissible critical point,\nwhere $|f(c)|\\le1$.  Then\n\\[\n \\min_{c\\ \\mathrm{admissible}}L_f(c)\\le2\n\\]\nwould imply the existence of a short curve joining two roots by the stated compactness\nargument.  The unrestricted infimum over all contained curves can be\nsmaller; equality of these two optimisation problems is not assumed.\nThe obstruction to two straight segments through a critical point does\nnot by itself prove or disprove this curved-arc estimate.  Examples in which another critical point gives a shorter curve do not\nprove the estimate universally. The critical value bound, containment and\nlength bound must hold for the same curve. The displayed universal\ninequality is a historical sufficient condition, not an open assertion:\ntogether with the stated compactness and scaling argument it would\ncontradict Theorem~\\ref{res:ani-degree-seven-counterexample-long}.\nThe reduction remains useful on separately specified polynomial classes.\n\n\\subsection*{An integral for successive merger scales}\\label{subsec:merger-integral}\n\\papersectiontarget{merger-integral-proof}\nConsider a component containing $k$ roots that later joins a larger\ncomponent as the level of $|f|$ increases. If $r\\in(0,1]$ is the ratio\nof these two merger scales, one logarithmic contribution assigned to each\nof its roots is $k^{-1}\\log(1/r)$. An earlier length estimate instead\ninvolves the integral\n\\[\n I_k(r)=\\int_r^1\n \\frac{dq}{q\\log\\bigl((1+q^{2/k})/(1-q^{2/k})\\bigr)}.\n\\]\nFor nearby merger scales, $r$ is close to $1$. The integral is then much\nsmaller than $k^{-1}\\log(1/r)$. The following change of variables proves\nthat no fixed positive multiple of this logarithmic contribution is a\nuniform lower bound.\n\n\\begin{theorem}[the relation between two merger-scale integrals]\n\\label{res:orlicz-currency}\nDefine\n\\[\n \\Phi(x)=\\int_0^x\\frac{dt}{\\log(\\coth t)}\\qquad(x\\ge0).\n\\]\nAt $t=0$ the integrand is understood by its continuous limiting value $0$\n(equivalently, the integral is improper at that endpoint).\nFor every integer $k\\ge1$ and $0<r\\le1$, with\n$x=k^{-1}\\log(1/r)$,\n\\[\n I_k(r)=k\\Phi(x).                                      \\tag{O1}\n\\]\nThe function $\\Phi$ is increasing and strictly convex on $(0,\\infty)$, and\n\\[\n \\frac{\\Phi(x)}x\\longrightarrow0\\qquad(x\\downarrow0). \\tag{O2}\n\\]\nConsequently, for every fixed $k\\ge1$ and every $c>0$, some $0<r<1$\nsatisfies\n\\[\n I_k(r)<c\\,\\frac1k\\log\\frac1r.                        \\tag{O3}\n\\]\nIn particular, no positive universal constant bounds $I_k(r)$ below\nby that constant times $k^{-1}\\log(1/r)$.\n\\end{theorem}\n\n\\begin{proof}\nSet $q=e^{-kt}$.  Then $dq/q=-k\\,dt$, the endpoints $q=1,r$ become\n$t=0,x$, and\n\\[\n \\frac{1+e^{-2t}}{1-e^{-2t}}=\\coth t,\n\\]\nwhich proves (O1).  Since $\\coth t$, and hence $\\log(\\coth t)$, is strictly\ndecreasing, the positive integrand $1/\\log(\\coth t)$ is strictly increasing.\nThis proves monotonicity and strict convexity.  Moreover\n\\[\n 0\\le\\frac{\\Phi(x)}x\\le\\frac1{\\log(\\coth x)}\\longrightarrow0,\n\\]\nwhich gives (O2); taking $r=e^{-kx}$ gives (O3).\n\\end{proof}\n\n\nAs $r$ approaches $1$, two successive merger scales approach each other.\nThe integral can then be arbitrarily small relative to the logarithmic\ncontribution, even for\nfixed $k$. Any estimate along a chain of mergers must retain this nonlinear\ndependence. Related weighted Jensen estimates and a lemma selecting two\nroots with small total logarithmic contribution are given in a\n\\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos1041/AttachmentAgeLifetimeOrlicz.md}{research note}.\nThose inequalities do not by themselves construct pieces of a contained\npath that meet one another.\n\n\n\\section{The Newton value equation}\\label{sec:newton}\n\\papersectiontarget{newton-value-proof}\n\nAlong a Newton trajectory the polynomial value decays exponentially.\nWe first derive this scalar equation, then use it to exclude a finite\nconnection between critical points whose values lie on different rays.\nFor the classical trajectory and graph statements we follow Kozen and\nStef\\'ansson \\cite[Lemmas~2.1--2.2 and \\S2]{kozen-stefansson1997}.\n\n\nSutherland uses the term Newton flow for $\\dot z=-f(z)/f'(z)$ and\nnotes that its solution curves map under $f$ to radial lines\n\\cite[p.~42]{sutherland1992}. The value equation\n\\cite[Lemma~2.1]{kozen-stefansson1997} determines their orientation and\ntime parametrisation. The global Newtonian graph\n\\cite[\\S2]{kozen-stefansson1997}, described with its endpoint conventions\nbelow, is a separate input to the inverse-sheet discussion in\nSection~\\ref{sec:gap}.\n\nA Newton trajectory is a differentiable curve satisfying\n\\[\n z'(t)=-\\frac{f(z(t))}{f'(z(t))}\n\\]\nwhere $f'(z(t))\\ne0$. Let $I\\subseteq\\mathbb R$ be an interval on\nwhich these assumptions hold;\nput $w(t)=f(z(t))$.  Kozen and Stef\\'ansson record the following identity as\na lemma of Shub, Tischler and Williams \\cite[Lemma~2.1]{kozen-stefansson1997}.\n\n\\begin{theorem}[value equation]\\label{res:value}\nFor a polynomial $f$ and a differentiable curve $z:I\\to\\mathbb C$ on an\ninterval $I$, assume $f'(z(t))\\ne0$ and\n$z'(t)=-f(z(t))/f'(z(t))$ throughout $I$.  Then $w=f\\circ z$ satisfies\n$w'=-w$, and\n\\[\n f(z(t))=e^{-(t-t_0)}f(z(t_0))\\qquad(t,t_0\\in I).\n\\]\n\\end{theorem}\n\n\\noindent The computation is one line: $w'=f'(z)\\,z'=f'(z)\\cdot(-f(z)/f'(z))=-f(z)=-w$.\nEquivalently,\n\\[\n  \\frac{d}{dt}\\Bigl(e^{t}f(z(t))\\Bigr)=0,\n\\]\nand integration on the real interval gives\n\\[\n  f(z(t))=e^{-(t-t_0)}f(z(t_0)),\\qquad t_0\\le t,\\quad t_0,t\\in I .\n\\]\nLean proves the theorem together with this integrated identity.\nThe separate local complex-parameter calculations are the\n\\lword{Erdos1041/NewtonFlowRaySeparation.lean}{50}{newtonFlow_value_hasDerivAt}{value derivative}\nand the\n\\lword{Erdos1041/NewtonFlowRaySeparation.lean}{64}{newtonFlow_scaledValue_hasDerivAt_zero}{scaled derivative}.\nThey require complex differentiability and do not themselves integrate a\ntrajectory on a real interval.\nFor an existing trajectory with nonzero initial value, its value moves\ninward on one positive ray; a zero value has no argument and remains zero.\nThus $|f|<1$ is preserved for later times in that trajectory's existing\ninterval. No global existence or description of a whole ray preimage follows\nfrom this scalar equation.\n\n\\begin{corollary}[ray separation]\\label{res:ray}\nLet $a<b$ and let the value trajectory $t\\mapsto f(z(t))$ be continuous on\n$[a,b]$. Assume the Newton equation and $f'(z(t))\\ne0$ on $(a,b)$ only.\nThen\n\\[\n f(z(b))=e^{a-b}f(z(a)).\n\\]\nIf these endpoint values are nonzero, they lie on one positive ray. Therefore\ncritical points with values on distinct positive rays cannot be endpoints\nof such a finite connection. The trajectory in the $z$ plane need not be radial.\n\\end{corollary}\n\n\\noindent The interior identity passes to the endpoints by continuity, not\nby evaluating $-f/f'$ at a critical endpoint. The corollary constructs no\ntrajectories and supplies no global monodromy theorem.\nThe Lean \\lword{Erdos1041/NewtonFlowRaySeparation.lean}{334}{no_newtonConnection_of_not_samePositiveRay}{ray exclusion}\nassumes the exponential endpoint identity; it does not construct or\nintegrate a connecting trajectory.\nKozen and Stef\\'ansson draw the same ray consequence for the Newtonian graph:\nunder $f$, every edge maps onto a segment of a ray through the origin whose\nendpoints are $0$ or critical values \\cite[\\S2]{kozen-stefansson1997}.\n\n\\section{Separating the rays of critical values}\\label{sec:arguments}\n\\papersectiontarget{argument-perturbation-proof}\n\nA constant perturbation translates the critical values and leaves the\ncritical points fixed. We can therefore separate their positive rays\nby describing the exceptional translation parameters: two nonzero\nvalues collide in argument exactly when their ratio is positive real.\n\nAlong a Newton trajectory the argument of $f$ stays constant while its\nmodulus decreases. By Corollary~\\ref{res:ray}, arranging pairwise distinct\narguments of the nonzero critical values is therefore a sufficient way\nto exclude finite connections between critical points. It is not asserted\nto be necessary. This condition imposes no lower bound on the distances\nbetween critical values and no upper bound on their moduli.\n\nThe weaker conditions really can coexist with a connection, even when\nall roots lie in the open unit disc. Take\n\\[\n f(z)=z^3-\\frac3{16}z+\\frac3{64}.\n\\]\nOn $|z|=1$, the lower terms have total modulus at most $15/64<1$, so\nRouch\\'e's theorem places all three roots inside. The critical points\nare $-1/4$ and $1/4$, with distinct positive values $5/64$ and $1/64$.\nOn the intervening real interval $f$ is positive and strictly decreasing.\nThe equation\n\\[\n f(x(t))=\\frac5{64}e^{-t},\\qquad 0<t<\\log5,\n\\]\ntherefore defines a differentiable path with\n$x'(t)=-f(x(t))/f'(x(t))$. Its endpoint limits are $-1/4$ and $1/4$.\nIt is an actual critical-to-critical Newton trajectory, not merely a\npair of values on one ray. The time convention is that of the value\nequation \\cite[Lemma~2.1]{kozen-stefansson1997}; the singular field is\nnot evaluated at the endpoints.\n\nA common translation of distinct values can separate their arguments.\nThe next calculation describes the translations to avoid.\n\n\\begin{theorem}[ray-collision locus]\\label{res:locus}\nLet $a\\ne b$ be complex.  Every common translation $\\beta$ for which $a+\\beta$\nand $b+\\beta$ lie on the same positive ray has the form\n\\[\n  \\beta=\\frac{ra-b}{1-r},\n  \\qquad r\\in\\mathbb{R}_{>0},\\ r\\ne1 .\n\\]\n\\end{theorem}\n\nIndeed, write $b+\\beta=r(a+\\beta)$ with $r>0$. Since $a\\ne b$,\none has $r\\ne1$, and solving for $\\beta$ gives the displayed formula.\nEquivalently, $\\beta=-a+(a-b)/(1-r)$, so the forbidden translations\nlie on the real affine line through $-a$ and $-b$.\n\n\\noindent A finite union of these loci has empty interior. Adding a constant to\n$f$ translates all critical values by that constant and leaves the critical\npoints unchanged. Section~\\ref{sec:open} records the formal avoidance and\nroot-retention estimates. A constant cannot separate equal critical values;\ncoefficient perturbation and control of the remaining geometric margins\nare separate requirements.\n\n\\subsection*{Local polynomial expansions}\\label{subsec:local-polynomial-expansions}\n\nFor $f(z)=\\prod_j(z-a_j)$ with $a_j\\ne0$, put $r=\\min_j|a_j|$ and\n$p_m=\\sum_j a_j^{-m}$.  On $|z|<r$, the exact expansion\n\\[\n \\log|f(z)|=\\log|f(0)|-\\Re\\sum_{m\\ge1}\\frac{p_m}{m}z^m\n\\]\nexpresses the local level geometry through the reciprocal power sums.  If $q=|z|/r<1$,\nthe tail after $N$ has modulus at most\n$nq^{N+1}/((N+1)(1-q))$; differentiated geometric series give the\ncorresponding first and second derivative bounds.\nA \\href{https://github.com/wcook04/plectis-erdos/blob/f36a98bf3d3e6f65f1074e3b800e3293d5b8a51a/ErdosProblems/Erdos1041/ReciprocalNewtonExpansion.md}{research note}\ngives recurrences for these reciprocal power sums and tests based on the\nfirst two derivatives for a locally contained piecewise path.\nThe assumption $|z|<r$ is essential: this disc contains no zero of $f$,\nso a logarithm exists there. The expansion does not continue through a\nroot and does not select a pair of roots to join. A proposed local path\nmust still satisfy the containment inequalities.\n\n\n\\section{Why the proposed spanning-tree estimate fails}\\label{sec:gap}\n\\papersectiontarget{cassini-counterexample}\n\nThe spanning-tree estimate in \\cite[Proposition~12]{march2026} fails\nalready for a quadratic Cassini lemniscate. That manuscript, posted by\n\\texttt{shtuka} on 24 March 2026, claims the unrestricted statement\n\\cite[Theorem~1, p.~1]{march2026}. The defect was located publicly in the\nproblem's discussion thread: on 25 March 2026 Tao observed that the\ninvocation of Lemma~8 there is unjustified and that the flow lines need not\norganise into connected trees, and on 26 March 2026 the manuscript's author\nagreed that the statement of Proposition~12 itself is incorrect. We give the\nexplicit numerical contradiction first. We then examine the saddle\nmodel and give an unformalised decomposition into inverse sheets,\nincluding the arcs that join them.\n\n\nRecent work on polynomial lemniscates separates component counts from metric\npath questions. Ghosh and Ramachandran record that the open set\n$\\{|f|<1\\}$ has $1+\\#\\{j:|f(c_j)|\\ge1\\}$ components, where\n$c_1,\\ldots,c_{n-1}$ are the critical points listed with multiplicity\n\\cite[Lemma~2.5]{ghosh2023}; the underlying component-wise Riemann--Hurwitz\ncount appears in the proof of \\cite[Proposition~2.1]{eks2010}.  For the\nbinomial family $z^n-a$, the condition $|a|<1$ therefore puts the filled unit\nlemniscate in the connected regime. Connectedness alone gives no path-length bound.\nBishop, Eremenko and Lazebnik describe the possible shapes: every rational\nlemniscate is a lemniscate graph whose vertices are the critical points on the\nlevel set, each of even degree at least four\n\\cite[Definition~1.2 and Proposition~2.4]{bishoperemenkolazebnik2025}, and a\nlemniscate graph is realised by a polynomial lemniscate up to a homeomorphism\nof the plane exactly when it is the boundary of its unbounded face\n\\cite[Corollary~1.6]{bishoperemenkolazebnik2025}.  This topological\ndescription also gives no path-length bound.\n\nThe March manuscript's Proposition~12 claims the following\nestimate.  For $u=-\\log|f|$, a connected component $V$ of $\\{u>c\\}$ carrying\n$m\\ge2$ simple zeros, and the regularity and Morse hypotheses in that proposition, it\nconstructs, for every $\\varepsilon>0$, an embedded spanning tree\n$G_\\varepsilon\\subset V$ with\n\\[\n \\operatorname{len}(G_\\varepsilon)\n \\le\\frac1{2\\pi}\\int_{2\\alpha}^{\\infty}P_V(t)\\,dt+\\varepsilon.\n\\tag{5.1}\\label{eq:prop12-bound}\n\\]\nHere $P_V(t)=\\mathcal H^1(V\\cap\\{u=t\\})$ is the length of the level\nset in $V$, and $\\alpha>0$ is the truncation parameter in the proposed\nestimate. The final theorem depends on this bound.\n\nThe estimate itself is false. Take $f_a(z)=z^2-a^2$ with $a=9/10$\nand choose $2\\alpha=-\\log a$. For $0<c<\\alpha/2$, the component\n$V$ of $\\{-\\log|f_a|>c\\}$ contains $\\{|f_a|\\le a\\}$ and both roots.\nThe only critical point is a nondegenerate saddle at $0$; distinctness\nof critical values is vacuous.\n\nHere the level-length integral can be bounded directly. Put $r=e^{-t}$.\nOn a regular level the inverse parametrisations\n$z=\\pm\\sqrt{a^2+re^{i\\theta}}$ give\n\\[\n P_V(t)=r\\int_0^{2\\pi}|a^2+re^{i\\theta}|^{-1/2}\\,d\\theta.\n\\]\nThe elementary identity\n\\[\n |a^2+re^{i\\theta}|^2\n =(a^2+r)^2\\cos^2(\\theta/2)+(a^2-r)^2\\sin^2(\\theta/2)\n\\]\nimplies\n\\[\n |a^2+re^{i\\theta}|^{-1/2}\n \\le(a^2+r)^{-1/2}|\\cos(\\theta/2)|^{-1/2}.\n\\]\nThe angular factor has mean at most $2$. Indeed, symmetry reduces its\nintegral to $4\\int_0^{\\pi/2}(\\cos s)^{-1/2}\\,ds$, and\n$\\cos s\\ge1-2s/\\pi$ bounds the latter integral by $4\\pi$.\nChanging variables from $t$ to $r$ therefore gives\n\\[\n \\frac1{2\\pi}\\int_{2\\alpha}^{\\infty}P_V(t)\\,dt\n \\le2\\int_0^a\\frac{dr}{\\sqrt{a^2+r}}\n =4\\bigl(\\sqrt{a^2+a}-a\\bigr)<\\frac{41}{25}.\n\\]\nThe isolated critical level is handled by the corresponding improper\nintegral. Every connected set containing both roots has length at least\n$2a=9/5$. Since $9/5-41/25=4/25$, any\n$0<\\varepsilon<4/25$ contradicts~\\eqref{eq:prop12-bound}.  Thus neither a different local saddle neighbourhood nor a perfect\ntopological decomposition can recover the printed coefficient $1/(2\\pi)$.\nThe exact numerical inequality and the resulting contradiction for a\ntree-length bound also have a \\href{https://github.com/wcook04/plectis-erdos/blob/e6c2d8f77ac24753c5216a49f4daf7f7388b309f/research_corpus/Erdos1041/CassiniTreeBudget.lean\\#L117}{Lean proof}, in the repository's\nresearch corpus, a library the checked build does not compile. The\nlevel-length majorant is an explicit analytic input.\n\nAt an interior index-one critical point $p$, the proof invokes a Morse chart\n\\[\n u=u(p)+x^2-y^2\n\\]\nand replaces the saddle by a three-ended neighbourhood having one connected\nlower cross-section and two connected upper cross-sections.  That local model\nis false as written.  Because $p\\in V$ and $V$ is open, a sufficiently small\nclosed disc around $p$ lies entirely in $V$.  In that disc the full Morse chart\nhas four sectors: two components of $u>u(p)$ and two components of $u<u(p)$.\nThe level set $\\{u=u(p)\\}$ is a lemniscate graph with a vertex of degree four\nat $p$ \\cite[Definition~1.2 and Proposition~2.4]{bishoperemenkolazebnik2025}.\nA global component argument cannot delete one local sector from a disc already\ncontained in $V$.\n\nThe four-sector calculation identifies an error in the proof, while the\nCassini example above disproves the printed metric inequality itself.\nPossible alternative constructions include cutting an adjoining regular\nannulus along a separatrix or regular flow arc, retaining a four-pronged\nsaddle neighbourhood, or using the inverse-sheet decomposition below.\nNone can restore~\\eqref{eq:prop12-bound}. A revised estimate would have\nto include the length needed to join the pieces, seek only one short\npair instead of a tree spanning all roots, or use a different global\nmetric inequality.\nThe shorter descriptions of the same three-ended block do not repair the\nfour-sector topology.\n\nThe following ordinary topological argument assumes that the critical\nvalues have pairwise distinct arguments and moduli and that the critical\npoints are simple. Its ingredients are classical: the Newtonian graph of Shub, Tischler and Williams\n\\cite[\\S2]{kozen-stefansson1997}, the component-wise Riemann--Hurwitz count in\nthe proof of \\cite[Proposition~2.1]{eks2010}, and the division of the Riemann\nsurface of the inverse function into sheets by outward critical-value\nslits.  Dubinin describes the sheet adjacency tree directly in\n\\cite[\\S1, pp.~1169--1170]{dubinin2006critical} and uses a related\nnetwork for capacity in \\cite[\\S2, pp.~34--35]{dubinin2006capacity}.\nThe latter paper assumes bounded critical values and controls capacity;\nit does not supply a Euclidean tree-length bound. The remark below\nspecifies the sheets, critical transpositions and descent arcs for the\nstated component.\n\nThe assumptions exclude a component containing a multiple critical point\nor two critical values on the same ray. For $z^n-b$ with $n>2$ and\n$0<|b|<1$, the component containing the origin fails the first condition.\nThey hold, for example, for\n$f(z)=z^3-(3/25)z+1/500$: its critical points are $\\pm1/5$ and its\ncritical values are $-7/500$ and $9/500$. On the unit circle the two\nlower terms have total modulus at most $61/500<1$, so all three roots\nlie in the open unit disc by Rouch\\'e's theorem. In the statement,\n``excellent'' means that the critical levels of the Morse function are\npairwise distinct. This regularity permits a decomposition into sheets;\nit does not bound the total length of the resulting tree.\n\n\\hypertarget{inverse-sheet-component}{}\n\\begin{remark}[unformalised decomposition into inverse sheets]\n\\label{res:attachment-aware-reeb}\nLet $f$ be monic, and let $U$ be a component of $\\{|f|<1\\}$ containing\n$k\\ge2$ roots, counted with multiplicity. Suppose every critical point of\n$f$ in $U$ is simple, its critical value is nonzero, and these critical\nvalues have pairwise distinct arguments and pairwise distinct moduli.\nAll preimages and sheets below are taken inside $U$, and only critical\npoints in $U$ determine the cuts. Then:\n\\begin{enumerate}\n\\item $-\\log|f|:U\\mathbin{\\backslash}f^{-1}(0)\\to(0,\\infty)$ is a proper excellent\nMorse function, with exactly $k-1$ nondegenerate saddles;\n\\item cutting $\\mathbb D\\smallsetminus\\{0\\}$ along the critical-value rays\ndecomposes its preimage in $U$ into conformal strips;\n\\item cutting each ray only from its critical value to the unit circle gives\n$k$ conformal sheets, one per root, whose critical transpositions form a tree;\n\\item for each critical point $c\\in U$, the two inverse lifts of\n$[0,f(c)]$ join two roots through $c$ inside\n$U\\cap\\{|f|\\le|f(c)|\\}$, and the union of these arcs is an embedded\ngeometric realisation of that tree.\n\\end{enumerate}\nSmall neighbourhoods of the saddles can be chosen with diameter\n$O(\\sqrt\\delta)$ at value radius $\\delta$.\n\\end{remark}\n\\evidenceremark{No supplied Lean theorem proves the whole decomposition.\nThe sources check finite ray, level-separation and saddle-scale facts.\nThe linked component-local declarations below also prove covering and\ncontinuous inverse-branch prerequisites. The Morse, conformal strip,\nsheet-count, monodromy and embedded-tree assertions remain ordinary\narguments.}\n\n\\paragraph{Inverse-branch prerequisites.}\n\\label{res:component-local-covering-long}\nFor a nonconstant complex polynomial, each component of $\\{|f|<1\\}$\n\\href{https://github.com/wcook04/plectis-erdos/blob/e98130bfe66aa33606e116791080b3ed13e1d68a/evidence/source-prerequisites/8bf96bdcae6b9201c670fff3037c49c70ec6de8d/lean/ErdosProblems/Erdos1041/OutwardSlitDomain.lean\\#L349}{contains a root}.\nThe restriction to that component is\n\\href{https://github.com/wcook04/plectis-erdos/blob/e98130bfe66aa33606e116791080b3ed13e1d68a/evidence/source-prerequisites/8bf96bdcae6b9201c670fff3037c49c70ec6de8d/lean/ErdosProblems/Erdos1041/OutwardSlitDomain.lean\\#L211}{onto the unit disc} and has\n\\href{https://github.com/wcook04/plectis-erdos/blob/e98130bfe66aa33606e116791080b3ed13e1d68a/evidence/source-prerequisites/8bf96bdcae6b9201c670fff3037c49c70ec6de8d/lean/ErdosProblems/Erdos1041/OutwardSlitDomain.lean\\#L233}{finite fibres}.\nOver values avoiding the critical values attained in that component,\nit is a \\href{https://github.com/wcook04/plectis-erdos/blob/e98130bfe66aa33606e116791080b3ed13e1d68a/evidence/source-prerequisites/8bf96bdcae6b9201c670fff3037c49c70ec6de8d/lean/ErdosProblems/Erdos1041/OutwardSlitDomain.lean\\#L250}{covering map}.\nChoose finitely many nonzero slit starts that include every critical\nvalue attained in the component, and remove the outward rays beginning\nat those starts from the value disc. On the remaining domain, each root\nin the component determines a\n\\href{https://github.com/wcook04/plectis-erdos/blob/e98130bfe66aa33606e116791080b3ed13e1d68a/evidence/source-prerequisites/8bf96bdcae6b9201c670fff3037c49c70ec6de8d/lean/ErdosProblems/Erdos1041/OutwardSlitDomain.lean\\#L359}{unique continuous inverse branch}, taking $0$ to that root.\nThe archived Lean source states these prerequisites and supplies proof\nscripts; a kernel-check receipt for this module is not supplied here.\nThe full conformal\nsheet decomposition, sheet count, monodromy and embedded-tree assertions\nremain ordinary arguments here; the prerequisites give no\nlength estimate.\n\n\n\n\n\\begin{proof}[Proof of Remark~\\ref{res:attachment-aware-reeb}]\nThe map $f:U\\to\\mathbb D$ is proper of degree $k$, and $U$ is simply\nconnected. To apply the count from the proof of\n\\cite[Proposition~2.1]{eks2010} only at a regular level, join all roots and\ncritical points in $U$ by finitely many compact paths in $U$. Choose a\nregular $t<1$ above the maximum of $|f|$ on their union. One component of\n$\\{|f|<t\\}$ then contains exactly the roots and critical points of $U$;\nRiemann--Hurwitz there gives $k-1$ simple critical points. No regularity\nof the level-one boundary is needed. The holomorphic Morse lemma makes\nthese points nondegenerate saddles of $-\\log|f|$, and their distinct\nmoduli separate the critical levels. A compact range in $(0,\\infty)$\nstays away from both the level-one boundary and the deleted roots, proving\nproperness of the Morse function.\n\nRemove $0$ and the rays determined by these $k-1$ values. Each remaining\nsector is simply connected and contains no branch value of\n$f:U\\to\\mathbb D$, even if it contains critical values coming from outside\n$U$. Each component of its preimage in $U$ maps biholomorphically to the\nsector. If the sector has angular range $(\\theta_1,\\theta_2)$, the\nholomorphic coordinate $-\\log f$ maps this component onto the\nsemi-infinite strip\n\\[\n \\{x+iy:x>0,\\ -\\theta_2<y<-\\theta_1\\}.\n\\]\nThus its real coordinate is $-\\log|f|$, while its imaginary coordinate\nis $-\\arg f$. The minus sign in both coordinates is needed for a\nholomorphic map; keeping $+\\arg f$ would reverse orientation.\n\nFor the sheet tree, remove only the outward slits\n$\\{tf(c):1\\le t<1/|f(c)|\\}$, for critical points $c\\in U$.\nThe remaining value domain is star-shaped about $0$, hence simply\nconnected, and its preimage in $U$ splits into $k$ sheets labelled by the\nroots. Local monodromy at each slit endpoint is a transposition of two\nof these sheets. Deleting the finitely many fibres over the branch values\nfrom $U$ leaves a connected domain, so the monodromy action is transitive.\nThe $k-1$ transpositions therefore give a connected graph with $k$ vertices,\nwhich is a tree.\n\nFor $c\\in U$, the inward segment from $f(c)$ to $0$ contains no other\nbranch value of $f:U\\to\\mathbb D$, by the distinct-argument hypothesis.\nDistinct moduli were needed only to separate the Morse levels. Its two\ninverse lifts converge to $c$ by the holomorphic Morse coordinate and\nstart at the two root labels exchanged by its transposition. These labels\nare distinct, as are the roots, because a root in $U$ cannot also be a\ncritical point under the nonzero-critical-value hypothesis. Lifts belonging to\ndifferent critical points cannot cross away from roots, since an intersection\nwould give one nonzero value on two critical rays. Their union is therefore\nan embedded tree. Each edge is rectifiable: at a simple critical endpoint,\nparametrising the value segment by its distance $s$ from the critical value\ngives speed $O(s^{-1/2})$, which is integrable. The root endpoint is regular,\nand the derivative is bounded on compact subsegments between the endpoints.\nThis proves finiteness of each length, not a uniform bound for it.\nThe same local Morse coordinate identifies a value disc of\nradius $\\delta$ with a spatial neighbourhood of diameter\n$O(\\sqrt\\delta)$. The constant and the permitted range of $\\delta$\ndepend on $f$ and the chosen saddle. Indeed, in the coordinate\n$f(z)-f(c)=u^2$, the inverse map has derivative of modulus\n$\\sqrt{2/|f''(c)|}$ at $u=0$. No bound uniform over degenerating\ncritical points is asserted. For a fixed polynomial there are finitely\nmany saddles, so their individual constants have a finite maximum.\n\\end{proof}\n\nThe complete argument is also written out in\n\\href{https://github.com/wcook04/plectis-erdos/blob/a729f05c40398663fd586c9da487ad874d898639/research_corpus/Erdos1041/AttachmentAwareReeb.md}{the proof of the decomposition into inverse sheets}.\nThis remark supplies a topological construction with no useful bound\non the sum of lengths: the canonical inverse-ray tree can already have\nthe Cassini deficit discussed above.\n\nThe component restriction cannot be dropped. For example, take\n\\[\n f(z)=(z^2-4)^2-\\tfrac14.\n\\]\nIts critical points are $0,\\pm2$, with values $63/4,-1/4,-1/4$.\nThe positive roots $\\sqrt{7/2}$ and $\\sqrt{9/2}$ lie in one component\n$U$: on the intervening real interval, $-1/4\\le f\\le0$.\nBut $f(iy)=(y^2+4)^2-1/4>1$ for real $y$, so this component cannot\ncontain the negative roots or $-2$. Its only critical point is the simple\npoint $2$. Thus the theorem applies in $U$, although the other component\nhas the same critical value. This example concerns component topology;\nno root-unit-disc hypothesis is imposed here.\n\nCorollary~\\ref{res:ray} also excludes finite saddle-to-saddle Newton\nconnections inside $U$. The theorem does not treat coincident arguments,\nsimultaneous critical levels or multiple saddles in that component. Nor\ndoes it identify a slit domain with the quotient by all trajectories:\nthe quadratic example following Problem~\\ref{prob:globalflow1041} satisfies\nits hypotheses but has a non-Hausdorff orbit quotient. Boundary tangencies\nand quantitative estimates for joining the strips remain separate questions. The embedded\ntree supplies no length bound for the historical question.\n\n\\section{Numerical path searches}\\label{sec:finite}\n\nThe earlier searches discretised root configurations and tested\ncandidate connections. Their finite scope is recorded here so that a\nnegative search is not mistaken for a general path theorem.\n\n\nA search was run over random monic polynomials with roots in the unit disc.  For\neach sample the region $\\{|f|<1\\}$ was rasterised and shortest grid paths were\ncomputed between every pair of roots.  The reported grid distances were\n\\[\n  \\begin{array}{lrr}\n    \\text{degree } 5, & 500 \\text{ trials}: & 1.1052648928\\\\\n    \\text{degree } 6, & 1500 \\text{ trials}: & 0.8450414343\\\\\n    \\text{degree } 8, & 1500 \\text{ trials}: & 0.6203916714\\\\\n    \\text{degree } 10, & 1500 \\text{ trials}: & 0.4303640486 .\n  \\end{array}\n\\]\nNo counterexample candidate was found in these trials. The reported\ngrid distances are below $2$, but they do not certify lengths of curves\nin the lemniscate. A polygon whose vertices satisfy $|f|<1$ can cross\nthe boundary along an edge. Joining an approximate root to its grid\nvertex introduces another edge requiring the same check.  Turning a candidate into a proof needs a continuous\ncertificate on every edge, for instance strict positivity of the real\npolynomial $1-|f(z(t))|^2$ on $[0,1]$ for each straight edge $z(t)=u+t(v-u)$,\nestablished by exact coefficients or outward-rounded interval arithmetic with\nsubdivision, together with exact root enclosures.  No such certificate is\nattached to the numbers above.  They are grid distances for the sampled\nconfigurations, and the apparent decrease with degree is a property of the\nsample, from which we draw no conjecture.  Even a finite collection of proved instances would not establish\nthe universal statement. These searches do not test \\texttt{ani}'s degree-seven counterexample.\nNumerical tests of a particular path construction should also examine\nnear-degenerate saddles, thin necks, boundary-critical configurations\nand almost-connected separatrices. Any paths found on a grid still\nrequire verification along their entire edges.\n\n\\section{Further questions about topology, length and perturbation}\\label{sec:open}\n\\papersectiontarget{remaining-questions}\n\nThe degree-seven example refutes the unrestricted formulation. The\nquestions below concern the topology and quantitative estimates needed\non restricted classes. They preserve the distinction between a valid\ninverse-sheet decomposition, bounds for its pieces, and a bound for a\ncontained curve assembled from those pieces.\n\n\\subsection*{1. Local saddle geometry and the refuted tree estimate}\n\n\\begin{problem}[corrected local saddle assembly]\\label{prob:saddle1041}\nThe printed $1/(2\\pi)$ spanning-tree estimate of Proposition~12 is false: the\nCassini polynomial $z^2-a^2$ at $a=9/10$ makes the proposed upper bound\nstrictly smaller than the distance between its roots. Any later argument must\ninclude the positive length needed to join the pieces, select only one short\npair instead of spanning every root, or use a different global metric inequality.  A four-pronged or\ncut-annulus local model may still be useful for another estimate, but it cannot\nrecover~\\eqref{eq:prop12-bound}.\n\\end{problem}\n\nThe full-disc model $x^2-y^2$, an annulus in which lower branches rejoin, two\nsaddles joined by a separatrix and simultaneous saddle levels are mandatory\ntests.  Repeating the one-lower/two-upper assertion does not answer the\nproblem.\n\n\\subsection*{2. Cutting a compact region into flow strips}\n\nLet $p$ be a polynomial with simple roots and simple critical points, and\nset $u=-\\log|p|$ away from the roots. Simplicity of the roots makes every\ncritical value nonzero; critical points themselves may lie at $0$.\nFor regular values $c<T$, let $V$ be a component of $\\{u>c\\}$ and assume that\n$\\overline{V\\cap\\{c<u<T\\}}$ is a compact genus-zero surface, its lower\nboundary is one smooth Jordan curve, its upper boundary consists of $m$ smooth\nlevel curves each enclosing one root, all interior critical points are nondegenerate index-one saddles,\nthe normalised gradient field, defined away from the saddles,\n\\[\n X=\\frac{\\nabla u}{|\\nabla u|^2}=-\\frac{p}{p'},\n\\]\nis transverse to the level boundaries, and no maximal $X$-trajectory has two\nsaddle endpoints. The equality uses the identification of a planar vector\nwith a complex number: $\\nabla u=-\\overline{p'/p}$. In particular,\n$du(X)=1$ wherever $X$ is defined.\n\n\\begin{problem}[finite strip decomposition after ray cuts]\\label{prob:reeb1041}\nFor every $\\eta>0$, construct disjoint Morse neighbourhoods $N_j$ with\n$\\sum_j\\operatorname{diam}N_j<\\eta$ and a finite set of complete separatrix or\nregular-flow cuts so that the closure of each remaining component in\nthe cut-open surface is flow-diffeomorphic to a rectangle\n\\[\n [a_S,b_S]\\times[0,1],\\qquad u=t,\n\\]\nwith connected level sections and no uncut annular component.  Give an explicit\nfinite bound for the number of strips in terms of the number $s$ of\nsaddles and the number $m$ of upper boundary curves, or exhibit the minimal missing\nhypothesis or a counterexample.\n\\end{problem}\n\nNo particular formula such as $2s+1$ is presumed. Boundary tangencies,\nsimultaneous levels and branch reunion through an annulus require explicit\ncuts. The closure is taken in the cut-open surface, with its separate\nboundary copies, not after regluing them in the plane. The quadratic\nexample following Problem~\\ref{prob:globalflow1041} explains why replacing\nthis construction by the ordinary orbit quotient would fail.\n\n\\subsection*{3. Length estimates for joining flow strips}\n\nFor a strip $S$, write\n\\[\n \\Gamma_t^S=S\\cap\\{u=t\\},\\qquad\n P_S(t)=\\mathcal H^1(\\Gamma_t^S),\n\\]\nand write\n\\[\n \\Phi_S(t)=\\int_{\\Gamma_t^S}|\\nabla u|\\,ds\n\\]\nfor its transverse flux. On a regular sub-band, $u$ is harmonic and the\nside boundaries are flow lines. The divergence theorem therefore makes\n$\\Phi_S(t)$ independent of $t$. Normalise the transverse measure by\n\\[\n d\\mu_{t_0}(x)=\\frac{|\\nabla u(x)|}{\\Phi_S(t_0)}\\,ds(x).\n\\]\nThe choice of measure is determined by the length calculation. Since\n$u=t$ along the normalised flow, its Euclidean speed is\n$1/|\\nabla u|$. Transporting the flux measure to level $t$ therefore gives\n\\[\n \\begin{aligned}\n \\int_{\\Gamma_{t_0}^S}\\operatorname{len}(\\gamma_x)\\,d\\mu_{t_0}(x)\n &=\\int_{a_S}^{b_S}\\int_{\\Gamma_t^S}\n    \\frac1{|\\nabla u|}\\frac{|\\nabla u|}{\\Phi_S(t_0)}\\,ds\\,dt\\\\\n &=\\frac1{\\Phi_S(t_0)}\\int_{a_S}^{b_S}P_S(t)\\,dt .\n \\end{aligned}\n\\]\nHere $0<\\Phi_S(t_0)<\\infty$ for a nonempty regular strip. Tonelli's\ntheorem applies to the nonnegative integrand. At a critical endpoint\nlevel, exhaustion through regular sub-bands gives the same identity,\nallowing the common value to be infinite. A finite right side gives a\ntrajectory of length at most that mean; an identity with both sides\ninfinite gives no length estimate.\nA strip flux need not be an integer multiple of $2\\pi$. For $f(z)=z$\nand $u=-\\log|z|$, take an annular sector of angular\nwidth $\\theta$. On a circular level arc, $|\\nabla u|=1/r$ and\n$ds=r\\,d\\phi$, so its flux is $\\Phi_S=\\theta$.\nFor $f(z)=z^n$, the same calculation gives $\\Phi_S=n\\theta$. A closed\nregular level curve enclosing $k$ roots has flux $2\\pi k$ by the argument\nprinciple. Restricting it to a strip replaces that full winding by the\nstrip's share of the flux.\nThe averaging measure therefore depends on the width of the strip.\nSumming separate strip estimates cannot replace those widths by the\nwinding number of a closed level curve.  They do not reopen~\\eqref{eq:prop12-bound}.  Cassini already\nexcludes the printed coefficient $1/(2\\pi)$ for any embedded tree that contains\nall $m$ roots: Problem~\\ref{prob:saddle1041} records that no later gluing\nargument can retain that estimate.\n\n\\begin{problem}[strip gluing after the printed coefficient]\\label{prob:metric1041}\nAssuming Problem~\\ref{prob:reeb1041}, do one of the following, for every\n$\\eta>0$.\n\\begin{enumerate}\n\\item Produce an embedded tree $G$ containing all $m$ roots whose length obeys\n  \\begin{equation*}\n \\operatorname{len}(G)\n \\le C\\int_{2\\alpha}^{\\infty}P_V(t)\\,dt+\\eta\n\\tag{6.1}\\label{eq:metric-fanin1041}\n \\end{equation*}\nfor an explicit constant $C>1/(2\\pi)$, uniform over a stated class of\npolynomials and an explicitly restricted range of truncations $\\alpha$.\nThe restrictions must account for attachment cost and for a valid allocation\nof the strip fluxes $\\Phi_S$.\n\\item Connect only one pair of distinct roots, rather than spanning every root.\n\\item Give a different global metric inequality that is not~\\eqref{eq:prop12-bound}.\n\\end{enumerate}\nThe printed coefficient $1/(2\\pi)$ is not an open target.  Better additive\ncontrol of saddle, annular-cut and root-cap cost cannot recover~\\eqref{eq:prop12-bound}.\n\\end{problem}\n\nThe restriction on $\\alpha$ is essential in the first alternative.\nFor a fixed $f(z)=z^2-a^2$, with $0<a<1$, choose\n$0<c<-2\\log a$ and let $V$ be the component of $\\{-\\log|f|>c\\}$\ncontaining both roots. At sufficiently deep levels, the Cassini\nparametrisation used above gives\n\\[\n P_V(t)\\le\\frac{2\\pi e^{-t}}{\\sqrt{a^2-e^{-t}}}.\n\\]\nConsequently $\\int_{2\\alpha}^\\infty P_V(t)\\,dt\\to0$ as\n$\\alpha\\to\\infty$, whereas every spanning tree has length at least\n$2a$. No fixed $C$ can cover arbitrary truncations. This is a\nlimitation of the unrestricted formulation of (6.1), separate from the\nearlier counterexample to the particular coefficient $1/(2\\pi)$.\n\nThe error term assigned to the collar in the printed argument\n\\[\n q=\\frac1{2\\pi}\\int_{\\alpha}^{2\\alpha}P_V(t)\\,dt>0\n\\]\nuses the same excluded coefficient. It cannot be treated as an unused\nerror allowance in an otherwise complete spanning-tree estimate.\nShortest trajectories chosen independently in different strips may\nhave endpoints that do not meet; the lengths needed to join them must\nalso be bounded. Even such a bound would not restore the printed\n$1/(2\\pi)$ tree estimate.\n\n\\subsection*{4. Coefficient perturbation and stability}\n\nOnce a finite critical-value family is injective, Lean\n\\lword{Erdos1041/NewtonFlowRaySeparation.lean}{197}{exists_small_translation_separating_arguments}{proves}\nthat an arbitrarily small translation makes every value nonzero and\npairwise positive-ray separated.  Lean also\n\\lword{Erdos1041/NewtonFlowRaySeparation.lean}{287}{constant_perturbation_roots_in_unitDisk}{proves}\nthe explicit root-retention estimate: for a monic polynomial $f$ of degree\n$n\\ge1$, a constant shift of modulus below $\\varepsilon>0$ keeps all roots\nin the unit disc provided\n\\[\n ((n+1)\\varepsilon)^{1/n}+\\max_{f(b)=0}|b|<1.\n\\]\nAdding a constant leaves the difference of two critical values unchanged,\nso it cannot separate values that were equal to begin with.\n\nFor a constant shift alone, factorisation gives the sharper elementary\nbound\n\\[\n (f+\\beta)(z)=0\\quad\\Longrightarrow\\quad\n \\prod_{f(b)=0}|z-b|=|\\beta|\n \\quad\\Longrightarrow\\quad\n \\min_{f(b)=0}|z-b|\\le|\\beta|^{1/n},\n\\]\nwhere the product counts roots with multiplicity. Hence\n$\\max_{f(b)=0}|b|+|\\beta|^{1/n}<1$ already ensures retention in the\nopen unit disc. The factor $n+1$ in the Lean estimate is not\nneeded for this ordinary constant-shift argument. The coefficient one\nin the displacement bound is optimal, as $f(z)=z^n$ shows. The estimate\ndoes not match labelled root lists or bound the cost of transporting\nan existing curve.\n\nQualitatively, a small linear perturbation also supplies distinct\ncritical values as well as simple critical points. For $\\deg f\\ge2$,\na multiple critical point of $f+\\lambda z$ must satisfy\n$f''(c)=0$ and $\\lambda=-f'(c)$; only finitely many parameters are\nexcluded. Near any other parameter, label the distinct critical points\nby holomorphic functions $c_j(\\lambda)$. Their values\n\\[\n v_j(\\lambda)=f(c_j(\\lambda))+\\lambda c_j(\\lambda)\n \\quad\\hbox{satisfy}\\quad v_j'(\\lambda)=c_j(\\lambda).\n\\]\nThus $v_i-v_j$ cannot vanish identically for $i\\ne j$, and its zeros\nare isolated. Avoiding the finitely many pairs in a sufficiently small\nparameter neighbourhood gives simple critical points with distinct\nvalues. The neighbourhood can be chosen inside any prescribed disc\nabout $\\lambda=0$, proving the required arbitrarily small choice.\n\nIf distinct moduli are also needed for the excellent Morse function,\nthe translation $\\beta$ can avoid them at the same time as the ray\ncollisions. For each distinct pair $v_i,v_j$, the equality\n$|v_i+\\beta|=|v_j+\\beta|$ is the real affine line\n\\[\n 2\\Re\\bigl((v_i-v_j)\\overline\\beta\\bigr)=|v_j|^2-|v_i|^2.\n\\]\nFor ray separation it is enough to avoid, for each pair, the real line\nthrough $-v_i$ and $-v_j$: the ray-collision formula puts every forbidden\ntranslation on that line. Avoid also the finitely many points $-v_j$\nto keep the translated values nonzero. Thus the full excluded set is\ncontained in two real affine lines per pair of critical values,\ntogether with finitely many points. Its complement meets every open\ndisc, so imposing distinct moduli costs no lower bound on the size\nof the perturbation.\nRoot retention is also qualitative: choose a circle $|z|=R<1$\nenclosing the original open-disc roots. If\n$|\\lambda|R+|\\beta|<\\min_{|z|=R}|f(z)|$, Rouch\\'e's theorem\nkeeps all roots of $f+\\lambda z+\\beta$ inside that circle.\nThese ordinary arguments establish genericity and root retention;\nthey do not estimate the collar or the cost of transferring a selected\ncurve. The first two requirements below therefore have qualitative\nproofs, while their simultaneous quantitative use is the remaining task.\n\nFor the next problem, fix a specified valid replacement for the refuted\nmetric inequality. Its slack is the difference between the permitted\nlength and the bound supplied by that replacement. The symbols $q$ and\n$q_g$ refer to this same specified quantity for $f$ and $g$, respectively;\nthey are not the historical collar integral with coefficient $1/(2\\pi)$.\nWithout a choice of replacement inequality, the requirement $q_g>q/2$\nhas no defined geometric content.\n\n\\begin{problem}[two-stage generic perturbation with slack]\\label{prob:perturb1041}\nFor fixed root discs, compact collar $K$, regular levels\n$\\alpha/2,\\alpha,2\\alpha,5\\alpha/2$, and a specified valid replacement metric\ninequality with slack $q>0$ at $f$, prove that an arbitrarily small\n\\[\n g_{\\lambda,\\beta}(z)=f(z)+\\lambda z+\\beta\n\\]\ncan be chosen so that:\n\n\\begin{enumerate}\n\\item $f+\\lambda z$ has simple relevant critical points and injective complex\ncritical values;\n\\item the checked constant translation $\\beta$ makes those values nonzero and\npairwise ray-separated;\n\\item no critical point enters the protected collar or truncation boundary;\n\\item the relevant component remains in $K$, contains exactly the corresponding\nperturbed roots and has slack $q_g>q/2$; and\n\\item root displacement and straight-line transfer back to the original roots\nconsume less than a prescribed fraction of $q/m$.\n\\end{enumerate}\n\nIf the one-coefficient perturbation $\\lambda z$ cannot ensure all five\nproperties, give the smallest additional lower-coefficient direction that can,\nor an explicit obstruction.\n\\end{problem}\n\nThe cited declarations supply finite planar avoidance and the constant\ntranslation. The ordinary argument above additionally supplies a linear\nperturbation with simple critical points and distinct critical values.\nWhat remains for the stated problem is simultaneous control of the\nprotected collar, the component, the quantitative slack and the\nroot-to-root transfer cost. No formalisation of the new genericity\nargument or solution of those stability requirements is claimed.\n\n\\subsection*{5. Newton flow on a compact region between regular levels}\n\nClassical Newton-flow theory describes maximal trajectories and the graph\nthey form. Away from the roots and critical points, every orbit satisfies\n\\[\n p(z(t))=e^{-(t-t_0)}p(z(t_0)),\\qquad\n u(z(t))=u(z(t_0))+t-t_0,\\qquad u=-\\log|p|,\n\\]\nby the lemma of Shub, Tischler and Williams\n\\cite[Lemma~2.1]{kozen-stefansson1997}. For an initial point that is neither\na root nor a critical point, the maximal Newton trajectory has one of\nfour endpoint patterns: each finite-time end is a critical point, an\ninfinite forward end is a root, and an infinite backward end is the\npoint at infinity \\cite[Lemma~2.2]{kozen-stefansson1997}. Stationary\ntrajectories at simple roots are not part of this classification. The\nmaximal nonconstant trajectories with endpoints among the roots and critical\npoints form a connected Newtonian graph with finitely many edges\n\\cite[Definition~2.1 and \\S2]{kozen-stefansson1997}. Critical-to-critical\nedges are included; pairwise distinct arguments exclude them by the value\nequation, not by the definition of the graph.\n\nThe smooth field needed for Morse theory is not $-p/p'$, which is\nundefined at a critical point with nonzero value. On the root-free band the\nEuclidean gradient satisfies\n\\[\n \\nabla u=-\\overline{p'/p}\n   =\\frac{|p'|^2}{|p|^2}\\left(-\\frac{p}{p'}\\right)\n \\qquad(p'\\ne0).\n\\]\nIt has the same oriented nonstationary trajectories as the Newton\nfield, with a different time parameter. At a simple critical point\n$c$, its derivative is the real Hessian of $u$, whose eigenvalues are\n$\\pm|p''(c)/p(c)|$. Thus it extends as a hyperbolic saddle there.\nA finite critical endpoint in Newton time is not a finite arrival at\nan equilibrium of the smooth field. These two parametrisations must be\ndistinguished in a statement about a Morse--Smale flow.\n\n\\begin{problem}[relative global Newton-flow theorem]\\label{prob:globalflow1041}\nLet $p$ be a nonconstant polynomial, let $a<b$ be finite regular values\nof $u=-\\log|p|$, and suppose every critical point in the compact band\n\\[\n \\{z:a\\le u(z)\\le b\\}\n\\]\nis simple, with the critical values in the band on pairwise distinct\npositive rays. Describe the maximal trajectories of the smooth gradient\nfield $\\nabla u$ relative to the lower and upper level boundaries,\neach of which may have several components. Construct a cut-open model\nby regular flow strips, specifying the separate copies of cut boundaries\nand the equivalence relation used for its trajectory parameter space.\nProve that this parameter space is Hausdorff, or identify the obstruction\nto the proposed identifications. The ordinary orbit quotient of the\nuncut band is not required to be Hausdorff. State the boundary convention\nand the comparison with Newton trajectories through the positive change\nof time above, without evaluating $-p/p'$ at its singular points.\n\\end{problem}\n\nThe endpoint classification on this band follows from the cited Newton\nclassification and the time change. Each nonconstant trajectory runs\nfrom the lower boundary to the upper boundary, from the lower boundary\nto a saddle, or from a saddle to the upper boundary. Indeed, Newton time\nequals the change in $u$, so the part of a Newton trajectory in the\nband has bounded Newton-time interval. At either end it must meet a\nregular boundary or tend to a critical point. Ray separation excludes\ntwo different critical endpoints, and strict increase of $u$ excludes\ntwo ends at the same critical point. In smooth-gradient time, regular\nboundary crossings occur in finite time, whereas convergence to a\nsaddle takes infinite time by uniqueness for the smooth differential\nequation. The remaining trajectories are the stationary saddles.\nThis is an ordinary consequence of the classical classification, not\nan additional checked Lean theorem.\n\nThe planar transversality condition also follows from the hypotheses.\nAlong a nonstationary gradient trajectory, the value of $p$ has constant\nargument, so a trajectory between two different saddles would contradict\nray separation. A homoclinic trajectory is excluded\nby the strict increase of $u$. The stable-manifold theorem gives local\nstable and unstable curves tangent to the Hessian eigenspaces. Their\nglobal branches for different saddles therefore do not meet; at one\nsaddle they meet transversely,\nsince their tangent lines are the two distinct Hessian eigenspaces.\nThe regular lower boundary is an entrance and the regular upper boundary\nan exit. With this boundary convention, the gradient satisfies the\nMorse--Smale transversality condition. This does not make the quotient\nby entire trajectories Hausdorff.\n\nFor an explicit obstruction, take $p(z)=z^2-1/4$ and the compact band\n\\[\n \\log2\\le-\\log|p(z)|\\le\\log8.\n\\]\nIts only critical point is the simple saddle at $0$, whose value is\n$-1/4$; both boundary levels are regular and ray separation is vacuous.\nThe positive real separatrix, including its upper-boundary endpoint, is\n\\[\n \\bigl(0,1/(2\\sqrt2)\\bigr],\\qquad\n \\frac{dx}{d\\tau}=\\frac{2x}{1/4-x^2}.\n\\]\nIntegration gives\n$\\tau-\\tau_0=\\tfrac18\\log(x/x_0)-(x^2-x_0^2)/4$, so this\ntrajectory tends to $0$ as smooth-gradient time $\\tau\\to-\\infty$.\nThus this orbit is not closed in the band: its closure also contains the\nstationary orbit $\\{0\\}$. In the quotient topology its image is a\nnonclosed singleton, so the orbit quotient is not even $T_1$, hence not\nHausdorff. Reparametrising the nonstationary trajectories leaves this\nquotient unchanged.\n\nOn an actual cut-open flow rectangle, by contrast, collapsing the\ntrajectories gives the transverse interval. Identifying boundary copies\nis a further quotient and requires a separate separation argument.\nThe \\href{https://arxiv.org/abs/1902.08841v5}{Reeb graph} uses yet another\nequivalence relation: it identifies connected components of level sets,\nnot complete trajectories. Neither that graph nor the embedded Newtonian\ngraph can be substituted for the orbit quotient without specifying a\nnew equivalence relation.\n\nGlobal solution theory and the finite graph are the classical results\ncited above. The relative endpoint classification and transversality are proved\nabove. They do not construct the required finite cut-open space, justify\nits boundary identifications or control the length needed to join the\npieces. The quadratic example settles\nthe uncut-orbit-space question negatively; it does not obstruct a\nconstruction that retains the required separate boundary copies.\n\nThe unrestricted assertion in Erd\\H{o}s~\\#1041 is false by\nTheorem~\\ref{res:ani-degree-seven-counterexample-long}. The positive\nresults do not give a classification of the polynomials that admit a\nshort connection. Their hypotheses and containment levels differ:\nlow critical value, separated simple critical value, and the individually\nproved polynomial families treat separate sufficient conditions.  In particular, the scaled $(5/2)\\mu^{1/n}$ bound is at\nlevel $(25/13)\\mu$, which is smaller than $2\\mu$; it therefore improves\nboth the constant and the level of the $71/10$ construction.  The latter\nis retained for its independent proof and component-sensitive estimates.\nThe critical-value moment does not choose a contained pair.\nThe ordinary generic slit-sheet topology has classical antecedents; the\nearlier proposed sufficient estimate concerned the lengths of particular\ninverse-ray curves. Their minimum need not equal the infimum over all\ncontained curves. The degree-seven counterexample above rules out the corresponding universal\nestimate for these particular curves too, without invalidating conditional\nestimates on smaller classes.\n\n\\paragraph{Adjacent inverse-map methods.}\nCrane \\cite[Lemma~2.1, Theorem~2.2 and \\S\\S3--4]{crane2007smale}\nuses inverse-branch hyperbolic geometry, Dubinin's radial-slit input and\ncapacity for a derivative-normalised Smale ratio.  Dubinin's four-point\nresult \\cite[Theorem~1 and Corollary~4]{dubinin2013fourpoint} assumes\nbounded critical values and estimates distortion.  These are relevant\nmethods, not direct sources of the present positive moment or connector\nconstant, and their normalisations are not interchangeable with a root-disc\nhypothesis.\n\n\\appendix\n% The exact former declaration list is retained in the editorial transfer archive.\n% Part: extended record\n\n\\section{Numerical estimates for the packing inequality}\n\\label{sec:record-circle-slice}\n\nThe circle-slice inequality can be relaxed to finitely many radii.\nThe numerical values below compare that relaxation with specific\nconfigurations. They are historical optimisation measurements; only\nthe directed rational bounds used in\nSection~\\ref{sec:low-critical-closure} enter its certificate.\n\nAt $a=1$ the constants of the failure inequalities have the rounded\nvalues $\\delta\\approx0.4586751$, $D\\approx5.3770730$, and\n$d_{\\mathrm{low}}\\approx2.1700770$; the identity $\\cosh(D/2)=e^{2}$ is exact. The quantity being estimated is\n$\\sum_j\\lambda(d_j)$ for $k$ roots at this fixed area, not a Euclidean\npath length. The four columns compare the earlier bound\n\\[\n \\max\\!\\left\\{\\frac\\delta2+(k-1)\\lambda(D-d_{\\mathrm{low}}),\n                  k\\operatorname{artanh}(e^{-2})\\right\\},\n\\]\nwhere $\\lambda(d)=-\\log\\tanh(d/2)$ as in\nSection~\\ref{sec:low-critical-closure}; the hyperbolic packing bound used\nfor the earlier threshold $2/5$; the linear-programming value\n$M_{\\mathrm{circ}}$ of the relaxation of\nLemma~\\ref{res:circle-slice-packing}; and the largest reported objective\nvalue at a numerically constructed configuration. The last column is a\nsearch result, not a certified feasible value or a proved optimum.\n\n\\begin{center}\n\\begin{tabular}{@{}rrrrr@{}}\n\\toprule\n$k$ & earlier bound & packing & $M_{\\mathrm{circ}}$ & search \\\\\n\\midrule\n2 & 0.3103 & 0.5899 & 0.3102 & 0.3103 \\\\\n3 & 0.4085 & 0.6878 & 0.3893 & 0.3784 \\\\\n4 & 0.5447 & 0.7475 & 0.4450 & 0.4306 \\\\\n5 & 0.6809 & 0.7906 & 0.4846 & 0.4694 \\\\\n6 & 0.8170 & 0.8244 & 0.5155 & 0.4986 \\\\\n7 & 0.9532 & 0.8521 & 0.5408 & 0.5208 \\\\\n8 & 1.0894 & 0.8757 & 0.5622 & 0.5413 \\\\\n10 & 1.3617 & 0.9142 & 0.5972 & 0.5759 \\\\\n14 & 1.9064 & 0.9710 & 0.6487 & 0.6228 \\\\\n20 & 2.7234 & 1.0298 & 0.7022 & 0.6717 \\\\\n30 & 4.0851 & 1.0955 & 0.7624 & 0.7283 \\\\\n100 & 13.617 & 1.2871 & 0.9428 & n/a \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\nBoth right-hand columns come from floating-point searches. In addition,\n$M_{\\mathrm{circ}}$ uses a $2600$-point $d$-grid and a $1400$-point\n$r$-grid. Restricting the allowed $d$ values and checking only finitely\nmany circle constraints have opposite effects on the feasible set.\nWithout a separate continuous feasibility or dual certificate, the\ncomputed value is not a proved lower or upper bound for the continuous\nrelaxation. At $k=2$, the grid value $0.31018$ differs from the analytic\nvalue, approximately $0.310338$. Their proximity supplies no rigorous\nbracket.\n\nThe certified lower bounds for $k$, rounded up to integers, are as\nfollows at $a=1$.\n\n\\begin{center}\n\\begin{tabular}{@{}lrrrrrrrr@{}}\n\\toprule\n$x$ & 0.40 & 0.45 & 0.50 & 0.55 & 0.60 & 0.63 & 0.65 & 0.70 \\\\\n\\midrule\nearlier bound & 3 & 4 & 4 & 5 & 5 & 5 & 5 & 6 \\\\\npacking bound & 2 & 2 & 2 & 2 & 2 & 3 & 3 & 4 \\\\\ncircle-slice bound & 3 & 4 & 6 & 8 & 10 & 12 & 14 & 18 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\nIn one explicit-Euler floating replica with step $10^{-3}$ and a geometric grid\nof $40$ initial areas from $10^{-6}$ to $1$, the largest computed time to reach the area bound\nfalls from $0.89703$ with the earlier packing lower bound for $k$ to\n$0.65503$ with the certified circle-slice bounds, and to $0.63003$ if\nthe relaxation's linear-programming value replaces those bounds.  Shrinking the relaxation by the worst\nmeasured slack factor $1.045$ moves it only to $0.60703$.  These computations\nsuggest that this particular relaxation may stop near\n$e^{-0.607}=0.545$; they do not prove that it does, and they exclude neither a stronger lower bound on the root count nor a larger certified\nrange.\n\nThe search uses only ten or fourteen radii in its two settings and a\ncoarse $a$-grid with upward rounding. The reported loss from the first\nrestriction at $a=1$, $k=8$ is about one per cent; the reported rounding\nloss is comparable. These are empirical estimates, not certified error\nbounds relative to the unrestricted continuous optimum.\n\nThe continuous optimisation remains unevaluated: maximise\n$\\sum_j\\lambda(d_j)$ subject to $d_j\\ge d_{\\mathrm{low}}$ and\n$\\sum_jw(d_j,r)\\le\\pi$ for every $r>0$, allowing the relaxed radial\ndistributions used in the computation. The numerical optimiser spreads\nmass over several radii and suggests logarithmic growth in $k$.\nNeither that growth law nor its coefficient is proved. Evaluating the\nrelaxation would clarify this packing argument; its effect on the final\npath threshold would still need to be established.\n\n% Part: family_catalogue\n\\section{Guide to the results}\\label{sec:erdos-1041-complete-family-map}\n\nThe main constructions have different hypotheses. The table locates\ntheir conclusions and distinguishes the formal results from the\nadditional analytic remarks. The detailed evidence associations are\nin the \\href{\\evidencerecordurl}{evidence record}.\n\\begin{center}\\small\n\\begin{tabular}{p{.24\\linewidth}p{.65\\linewidth}}\n\\toprule\nFamily or mechanism & Conclusion and qualification \\\\\n\\midrule\nDegree seven & The fixed polynomial has no connected two-root subset\nof Hausdorff measure at most $2$ in its strict lemniscate. \\\\\nTrinomials & Every root-to-origin segment is contained; any two roots\nare joined with length below $2$. \\\\\nLow critical values & Unformalised area-growth criterion with cutoff\n$13/25$ and scaled level $(25/13)\\mu$. Its recorded rational comparison\nalso covers $\\mu\\le529/1000$; the analytic reduction remains required. \\\\\nSeparated simple value & Unformalised square-root, Bergman and capacity\nconstruction. The chosen critical point must be simple and its value\nisolated as stated. \\\\\nCollinear and sparse families & Formal segment constructions for the\nspecified root geometry or coefficient patterns. The stronger\n$2\\sqrt{1-|h|^2}$ cyclic-fibre estimate remains an ordinary argument. \\\\\nCritical-value means & Formal exponent $2/(n-1)$. The sharp\n$4/(n-1)$ refinement and higher exponents under complex power-sum\ncancellations remain unformalised. No path selection follows. \\\\\nPerimeter and prescribed paths & The Cassini example refutes the stated\nspanning-tree coefficient. The degree-eight example refutes the proposed\nGamma-function constant, while a uniform perimeter bound is still open.\nThe implication from $\\beta\\sigma^{1/n}$ to a path of length at most\n$\\beta\\rho$ in $K_\\mu$ needs the unformalised first-critical-level split.\nSeveral prescribed straight-segment rules fail. \\\\\n\\bottomrule\n\\end{tabular}\\end{center}\nThe inverse-ray estimate in the earlier note is retained only as a\nhistorical sufficient condition. It does not follow from these value,\nsegment or perimeter calculations.\n\n% Part: back\n\\section{Formal statements and computational checks}\\label{sec:verification-notes}\n\\papersectiontarget{verification-notes}\n\nThe formal sources supplied with these manuscripts have not been rebuilt\nfor this revision. Their recorded statements include the fixed\ncounterexample for\n\\href{https://github.com/wcook04/plectis-erdos/blob/7380b7871687b6bcc41ca0143c61f232e8af6500/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean\\#L283}{preconnected sets},\nthe\n\\href{https://github.com/wcook04/plectis-erdos/blob/7380b7871687b6bcc41ca0143c61f232e8af6500/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean\\#L407}{negation of the Formal Conjectures statement},\nits\n\\href{https://github.com/wcook04/plectis-erdos/blob/7380b7871687b6bcc41ca0143c61f232e8af6500/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean\\#L449}{\\texttt{answer(False)} form},\nand the\n\\href{https://github.com/wcook04/plectis-erdos/blob/f70679d47178bf1174309cb29f5b31748609cc04/lean/ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean\\#L25}{total-variation formulation}.\nThe independent ordinary reconstruction in\nSection~\\ref{sec:counterexample-proof} uses Rouch\\'e's theorem, the\ncomponent degree and Hausdorff measure. Its Cayley signs, critical-disc\nestimates and all four Bernstein tests were recomputed with rational\narithmetic in an earlier revision; the present edit retains those\nfinite calculations. Independent human review of the construction and its\ncorrespondence with the 1958 wording is not recorded.\n\nThe $13/25$ calculation has the recorded full replay described in\nSection~\\ref{sec:low-critical-closure}. Its analytic reduction is an\nordinary proof and remains unformalised. The same distinction applies\nto the fourth-power mean estimate and the longer analytic family\narguments. The numbered statements without a complete formal proof\nhave therefore been retained as remarks, with their original\nmathematical content and local input notes. The following list records\nexactly what the existing formal deductions assume.\n\n\\begin{description}\n\\item[\\ref{res:low-critical-thirteen-twentyfifths}.] No Lean proof covers the whole assertion. Lean checks the\ndegree-two case, and the closing inequality $(13/25)e^X<1$ at the stopping\ntime $X$ certified in the proof. The computation that certifies $X$, done by an\nexact-arithmetic program, and the analytic argument in degrees three and\nhigher are not checked in Lean.\n\n\\item[\\ref{res:low-critical-scale-free}.] The Lean proof assumes\nRemark~\\ref{res:low-critical-thirteen-twentyfifths}.\n\n\\item[\\ref{res:dual-arity-floor}.] The Lean proof takes the separation bound\n\\eqref{eq:lc-separation} and the individual and summed bounds\n\\eqref{eq:lc-radius-and-budget} as hypotheses; their derivation from the\nfailure assumption, given in that section, is not checked in Lean.\n\n\\item[\\ref{res:scaled-low-critical-path}.] The Lean proof assumes\nRemark~\\ref{res:low-critical-thirteen-twentyfifths} in degrees above two;\nthe degree-two case is checked without it.\n\n\\item[\\ref{res:constant-factor-path}.] The Lean proof assumes the two-parameter bound (CF)\nin degrees $n\\ge3$, which the proof obtains by averaging over levels and\nvalue directions. The degenerate case, degree two and the numerical\nconstants are checked without that input.\n\n\\item[\\ref{res:constant-factor-arity}.] The Lean proof assumes the bound (CF) from the proof of\nRemark~\\ref{res:constant-factor-path}, for a component containing at least\n$k_0$ roots; the rational inequalities are checked.\n\n\\item[\\ref{res:constant-factor-capacity}.] The Lean proof assumes the averaging construction in the\nproof of Remark~\\ref{res:constant-factor-path}, rerun with the\narea--capacity inequality for the area of $C$; the rational cutoffs are\nchecked.\n\n\\item[\\ref{res:conjecture-p-consumer}.] The Lean proof assumes the two-component split at the\nfirst critical level constructed in the ordinary argument: two one-root\ncomponents whose closures meet at a critical point, each with boundary\nlength at most $\\beta\\rho$ and each joining its root to that point within\nhalf its perimeter.\n\n\\item[\\ref{res:critical-value-separation}.] The Lean proof assumes the ordinary analytic construction: the connector obtained through the square-root map, its\nBergman length bound (7), and the capacity bound for the area of the\ntwo-sheeted component.\n\n\\item[\\ref{res:critical-value-thresholds}.] The Lean proof assumes the same construction as\nRemark~\\ref{res:critical-value-separation} for the second assertion.\nInequality~\\eqref{eq:disk-family-coefficient} and every numerical step are\nchecked without it.\n\n\\item[\\ref{res:attachment-aware-reeb}.] No Lean proof covers the whole assertion. Lean checks the\nray-disjointness, level-separation and saddle-scale steps of the proof; the\nMorse, monodromy and strip statements are not checked.\n\n\\item[\\ref{res:arity-not-capacity}.] The formal capacity conclusion takes\nas input the classical transfinite-diameter formula for a filled monic\nlemniscate. The root count at the first merger does not require that input.\n\\end{description}\n\nNo supplied proof establishes the reported Runge construction of\nunbounded one-root perimeters or the stronger assertion that every\ncritical joining point fails in the near-pentagon example. Neither is\nused in the positive theorems. The fixed value of the degree-seven\nparameter is not a proof of the reported parameter family.\n\\evidenceparagraph{\\evidencerecordurl}\n\n\\begin{thebibliography}{99}\n\\bibitem{polya1928} G.~P\\'olya,\n  \\emph{Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach\n  zusammenh\\\"angende Gebiete}, Sitzungsberichte der Preussischen Akademie der\n  Wissenschaften, Physikalisch-Mathematische Klasse (1928), printed\n  pp.~228--232 and 280--282.\n  \\url{https://archive.org/details/sitzungsbericht1928preu}.\n\\bibitem{crane} E.~Crane,\n  \\emph{The areas of polynomial images and pre-images}, Bull. London Math.\n  Soc. \\textbf{36} (2004), no.~6, 786--792,\n  doi:\\href{https://doi.org/10.1112/S0024609304003509}{10.1112/S0024609304003509};\n  preprint arXiv:\\href{https://arxiv.org/abs/math/0302189v1}{math/0302189v1},\n  whose statement numbers are cited.\n\\bibitem{bloom} T.~F. Bloom, \\emph{Erd\\H{o}s Problems}, problem~1041.\n  \\url{https://www.erdosproblems.com/1041}\n\\bibitem{ehp1958} P.~Erd\\H{o}s, F.~Herzog, and G.~Piranian,\n  \\emph{Metric properties of polynomials}, J. Analyse Math. \\textbf{6}\n  (1958), 125--148,\n  doi:\\href{https://doi.org/10.1007/BF02790232}{10.1007/BF02790232}.\n\\bibitem{ghosh2023} S.~Ghosh and K.~Ramachandran,\n  \\emph{Number of Components of Polynomial Lemniscates: A Problem of Erd\\H{o}s,\n  Herzog, and Piranian}, J. Math. Anal. 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Shapiro,\n  \\emph{Two-dimensional shapes and lemniscates}, in \\emph{Complex Analysis and\n  Dynamical Systems IV, Part~1}, Contemp. Math. \\textbf{553}, Amer. Math. Soc.,\n  Providence, RI, 2011, 45--59,\n  doi:\\href{https://doi.org/10.1090/conm/553/10931}{10.1090/conm/553/10931};\n  preprint arXiv:\\href{https://arxiv.org/abs/1003.4567v1}{1003.4567v1},\n  whose statement numbers are cited.\n\\bibitem{dubinin2006capacity} V.~N. Dubinin,\n  \\emph{Lemniscates and inequalities for the logarithmic capacities of\n  continua}, Mat. Zametki \\textbf{80} (2006), no.~1, 33--37,\n  doi:\\href{https://doi.org/10.4213/mzm2777}{10.4213/mzm2777}; English\n  translation, Math. 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J. \\textbf{46}\n  (1999), no.~2, 409--415,\n  doi:\\href{https://doi.org/10.1307/mmj/1030132418}{10.1307/mmj/1030132418};\n  preprint arXiv:\\href{https://arxiv.org/abs/0805.2295}{0805.2295}.\n\\bibitem{fryntovnazarov2008} A.~Fryntov and F.~Nazarov,\n  \\emph{New estimates for the length of the Erd\\H{o}s--Herzog--Piranian lemniscate},\n  in \\emph{Linear and Complex Analysis}, Amer. Math. Soc. Transl. Ser.~2\n  \\textbf{226}, Amer. Math. Soc., Providence, RI, 2009, 49--60,\n  doi:\\href{https://doi.org/10.1090/trans2/226/05}{10.1090/trans2/226/05};\n  preprint arXiv:\\href{https://arxiv.org/abs/0808.0717v1}{0808.0717v1}.\n\\bibitem{tao2025} T.~Tao,\n  \\emph{The maximal length of the Erd\\H{o}s--Herzog--Piranian lemniscate in\n  high degree}, arXiv:\\href{https://arxiv.org/abs/2512.12455v1}{2512.12455v1}\n  (2025).\n\\bibitem{eremenkoyuditskii2012} A.~Eremenko and P.~Yuditskii,\n  \\emph{Comb functions}, Contemp. Math. \\textbf{578} (2012), 99--118,\n  doi:\\href{https://doi.org/10.1090/conm/578/11472}{10.1090/conm/578/11472};\n  preprint arXiv:\\href{https://arxiv.org/abs/1109.1464v1}{1109.1464v1}.\n\\bibitem{eremenkolempert1994} A.~Eremenko and L.~Lempert,\n  \\emph{An extremal problem for polynomials}, Proc. Amer. Math. Soc.\n  \\textbf{122} (1994), no.~1, 191--193,\n  doi:\\href{https://doi.org/10.1090/S0002-9939-1994-1207536-1}\n  {10.1090/S0002-9939-1994-1207536-1}.\n\\bibitem{eremenko2007} A.~Eremenko,\n  \\emph{A Markov-type inequality for arbitrary plane continua},\n  Proc. Amer. Math. Soc. \\textbf{135} (2007), no.~5, 1505--1510,\n  doi:\\href{https://doi.org/10.1090/S0002-9939-06-08640-0}\n  {10.1090/S0002-9939-06-08640-0};\n  preprint arXiv:\\href{https://arxiv.org/abs/math/0606745v1}{math/0606745v1}.\n\\bibitem{bishoperemenkolazebnik2025} C.~J. Bishop, A.~Eremenko, and\n  K.~Lazebnik, \\emph{On the shapes of rational lemniscates}, Geom. Funct.\n  Anal. \\textbf{35} (2025), no.~2, 359--407,\n  doi:\\href{https://doi.org/10.1007/s00039-025-00704-2}\n  {10.1007/s00039-025-00704-2};\n  preprint arXiv:\\href{https://arxiv.org/abs/2407.14610v1}{2407.14610v1}.\n\\bibitem{dubinin2006critical} V.~N. Dubinin,\n  \\emph{Inequalities for critical values of polynomials}, Sb. Math.\n  \\textbf{197} (2006), no.~8, 1167--1176,\n  doi:\\href{https://doi.org/10.1070/SM2006v197n08ABEH003793}{10.1070/SM2006v197n08ABEH003793}.\n\\bibitem{crane2007smale} E.~Crane,\n  \\emph{A bound for Smale's mean value conjecture for complex polynomials},\n  Bull. London Math. Soc. \\textbf{39} (2007), no.~5, 781--791,\n  doi:\\href{https://doi.org/10.1112/blms/bdm063}{10.1112/blms/bdm063};\n  author preprint \\url{https://people.maths.bris.ac.uk/~maetc/SMVCbound.pdf}.\n\\bibitem{dubinin2013fourpoint} V.~N. Dubinin,\n  \\emph{Four-point distortion theorem for complex polynomials},\n  arXiv:\\href{https://arxiv.org/abs/1301.3985v1}{1301.3985v1} (2013).\n\\bibitem{kuznetsova2003} O.~S. Kuznetsova and V.~G. Tkachev,\n  \\emph{Length functions of lemniscates}, Manuscripta Math.\n  \\textbf{112} (2003), 519--538,\n  doi:\\href{https://doi.org/10.1007/s00229-003-0411-3}{10.1007/s00229-003-0411-3};\n  preprint arXiv:\\href{https://arxiv.org/abs/math/0306327}{math/0306327}.\n\\bibitem{tischler1989} D.~Tischler,\n  \\emph{Critical points and values of complex polynomials},\n  J. Complexity \\textbf{5} (1989), no.~4, 438--456.\n\\bibitem{revision2026} W.~Cook / Plectis,\n  \\emph{Three refinements for the lemniscate-path programme}, 16 September 2026,\n  unreviewed research note, Sections~1--3, together with the earlier\n  note \\emph{Structural obstructions}.\n\\bibitem{schur1918} I.~Schur,\n  \\emph{\\\"Uber die Verteilung der Wurzeln bei gewissen algebraischen\n  Gleichungen mit ganzzahligen Koeffizienten}, Math. Z. \\textbf{1}\n  (1918), no.~4, 377--402.\n\\bibitem{ransford1995} T.~Ransford,\n  \\emph{Potential Theory in the Complex Plane}, London Mathematical Society\n  Student Texts~28, Cambridge University Press, Cambridge, 1995,\n  doi:\\href{https://doi.org/10.1017/CBO9780511623776}{10.1017/CBO9780511623776}.\n\\end{thebibliography}\n% ---- part extended_record ----\n% ---- part family_catalogue ----\n% ---- part back ----\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":268328,"sha256":"sha256:4bee9a881dbbe9404db6c711ef9ecb60f7ce5b9493b754c4057a8324cc9d95d5","cite_as":"Plectis reader corpus, paper:erdos1041-lemniscate-reasoning-surface","source_repository":"plectis-erdos","source_path":"paper/1041/erdos1041-lemniscate-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:088d36d7ba3d3ee0cee2df99e8915927999698f19889f8ea0b8d55b3721ec778","byte_count":6923,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% This is the only copy of the shared preamble.  The eight short notes, the\n% long records assembled from paper/reasoning-parts other than #249 and #257\n% (which carry their own preambles), and the synthesis paper include it with\n% \\input{problem-note-preamble}; paper/Makefile builds from this directory\n% with the directory on the tectonic search path.\n%\n% Shared preamble for the Erdős Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erdős problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{8e58c395e9807ef3374d7365c62cf04d20ee6331}\n\\newcommand{\\commitshort}{9e231ce4371f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-erdos/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-erdos/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\usepackage{longtable}  % the notes' declaration tables; loaded here so the assembled records inherit it\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n% ---------------------------------------------------------------------------\n% Evidence marks.  paper-evidence.tex places \"Lean\" and \"Comparator\" links in\n% the margin beside each result from the paper's generated file\n% paper/evidence/<paper>.tex, which each paper inputs after this preamble.\n% ---------------------------------------------------------------------------\n\\input{paper-evidence}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Production provenance.  The wording is fixed by the corpus provenance\n% contract and reproduced verbatim in every manuscript in this repository, so\n% the papers cannot drift into different accounts of how they were made.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erdős problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every Lean mark in the margin points to propositions that the\n  pinned Lean kernel accepts, with no \\texttt{sorry}, no added axiom, and no\n  unchecked evaluation.  That claim concerns the formal statements only; it\n  says nothing about their mathematical interest, their novelty, or the\n  original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n\n% Compact opening card for the separately stated Comparator theorem selected\n% from each problem note.  The prose supplied by each note owns the\n% contribution and relevance judgements; the shared final sentence states the\n% mechanical assurance boundary once and points to the public evidence owner.\n\\newcommand{\\externalreviewobject}[3]{%\n  \\begin{quote}\\small\n  \\textbf{Contribution.} #1\\par\n  \\textbf{Relation to the open problem.} #2\\par\n  \\textbf{Executable review object.} #3  The repository's\n  external-verification job compares these exact Lean propositions with\n  separately declared challenge statements and an axiom budget, then asks\n  Lean's kernel to check the submitted proofs.  The\n  \\href{https://github.com/wcook04/plectis-erdos/blob/ca0e13f8acf5ccf48506e4bdb870953d3a0856fa/formalization.yaml}%\n  {formalisation manifest} and the commit-bound CI receipt record that check;\n  they do not assess novelty, significance, or whether the original problem is\n  solved.\n  \\end{quote}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  Under\n% Tectonic (XeTeX) the text and mathematics are set in TeX Gyre Pagella from\n% the bundled OpenType fonts, with a Heros sans range for \\mathsf and\n% Inconsolata for monospace; the pdflatex fallback keeps the same Palatino\n% design through newpx.  The A4 text block uses a classical measure with\n% quiet running heads and restrained ink.  Link colours: steel blue for\n% external (Lean-source and web) targets, deep burgundy for internal\n% cross-references and citations.\n\\NeedsTeXFormat{LaTeX2e}\n\\ProvidesPackage{paper-house-style}[2026/07/19 Pagella house style for the papers]\n\n\\RequirePackage{iftex}\n\\RequirePackage{amsmath}\n\\RequirePackage{amsthm}\n\\ifpdftex\n  \\RequirePackage[T1]{fontenc}\n  \\RequirePackage[utf8]{inputenc}\n  \\RequirePackage{amssymb}\n  \\RequirePackage{newpxtext}\n  \\RequirePackage{newpxmath}\n\\else\n  \\RequirePackage{fontspec}\n  \\RequirePackage{unicode-math}\n  \\setmainfont{texgyrepagella}[Extension=.otf,UprightFont=*-regular,\n    BoldFont=*-bold,ItalicFont=*-italic,BoldItalicFont=*-bolditalic]\n  \\setmathfont{texgyrepagella-math.otf}\n  \\setmathfont{texgyreheros-regular.otf}[range=sfup,Scale=MatchLowercase]\n  \\setmonofont{Inconsolatazi4}[Extension=.otf,UprightFont=*-Regular,\n    BoldFont=*-Bold,Scale=MatchLowercase]\n\\fi\n\n\\RequirePackage[a4paper,textwidth=418pt,textheight=636pt,hcentering,\n  top=68pt,headheight=15pt,headsep=20pt,footskip=34pt]{geometry}\n\\RequirePackage{microtype}\n\\linespread{1.045}\n\\clubpenalty=10000 \\widowpenalty=10000 \\tolerance=1600\n\n\\RequirePackage[dvipsnames,table]{xcolor}\n\\definecolor{linkexternal}{HTML}{275D8C}\n\\definecolor{linkinternal}{HTML}{6E2B2B}\n\\definecolor{papertableheader}{HTML}{E3EAF0}\n\\definecolor{papertablerow}{HTML}{F2F5F7}\n\\definecolor{papertablerule}{HTML}{7D8991}\n\\definecolor{paperresultrule}{HTML}{315B72}\n\n% Stable cross-document links between sibling PDFs. A named destination is\n% more durable than a page number: sections may move while their semantic role\n% remains the same. All public PDFs are copied to one directory at release.\n\\newcommand{\\papersectiontarget}[1]{\\hypertarget{#1}{}}\n\\newcommand{\\papersectionlink}[3]{\\href{#1\\##2}{#3}}\n\n\\RequirePackage{titlesec}\n\\titleformat{\\section}{\\Large\\bfseries}{\\thesection}{0.85em}{}\n\\titlespacing*{\\section}{0pt}{2.6ex plus .8ex minus .3ex}{1.5ex plus .3ex}\n\\titleformat{\\subsection}{\\large\\bfseries}{\\thesubsection}{0.8em}{}\n\\titlespacing*{\\subsection}{0pt}{2.1ex plus .6ex minus .2ex}{1.1ex plus .2ex}\n\\titlespacing*{\\paragraph}{0pt}{1.6ex plus .5ex minus .2ex}{0.85em}\n\n\\RequirePackage{fancyhdr}\n\\newcommand{\\paperrunninghead}{}\n\\newcommand{\\runninghead}[1]{\\renewcommand{\\paperrunninghead}{#1}}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\fancyhead[L]{\\small\\itshape\\paperrunninghead}\n\\fancyhead[R]{\\small\\thepage}\n\\renewcommand{\\headrulewidth}{0pt}\n\\fancypagestyle{plain}{\\fancyhf{}\\renewcommand{\\headrulewidth}{0pt}}\n\n\\RequirePackage{caption}\n\\captionsetup{font=small,labelfont=bf,skip=9pt,width=.92\\linewidth}\n\\RequirePackage{etoolbox}\n\\RequirePackage{pdflscape}\n\\renewcommand{\\arraystretch}{1.20}\n\\newcommand{\\papertablehead}{\\rowcolor{papertableheader}}\n\\newcommand{\\papertablebody}{\\rowcolors{2}{white}{papertablerow}}\n\\AtBeginEnvironment{tabular}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\\AtBeginEnvironment{longtable}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\n% A compact front-matter panel for the result a scanning reader should retain.\n% It is deliberately short and unbreakable: manuscripts should state one\n% result, its evidence, and its limit here, then supply the argument below.\n\\newenvironment{paperresult}[1]{%\n  \\par\\addvspace{0.7em}\n  \\begin{center}\\begin{minipage}{0.94\\linewidth}\n    {\\color{paperresultrule}\\rule{\\linewidth}{0.8pt}}\\par\\vspace{4pt}\n    {\\small\\sffamily\\bfseries\\color{paperresultrule}#1}\\par\\smallskip\n    \\small\\color{black}\n}{%\n    \\par\\vspace{4pt}{\\color{paperresultrule}\\rule{\\linewidth}{0.45pt}}\n    \\end{minipage}\\end{center}\\par\\addvspace{0.5em}\n}\n\n% Shared diagram grammar.  Reader-facing figures use the same restrained\n% hierarchy as the surrounding page: a short eyebrow, a substantial title,\n% one compact explanatory line, and an optional evidence badge.  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For these papers that inference is false, and\n% a declaration reached only in the back matter would leave it standing for the\n% length of the paper.  The note therefore appears on page one, small and\n% visually subordinate to the title and abstract: enough to prevent the false\n% inference, not enough to make the production method the subject.\n%\n% It carries no mark, because it qualifies the whole manuscript rather than any\n% one passage.  \\footnotetext does not advance the footnote counter, and the\n% emptied \\thefootnote is confined to a group, so numbered footnotes elsewhere\n% in the paper are unaffected.\n%\n% The wording is not authored here.  It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"},{"record_type":"paper_source_dependency","tex_reference":"evidence/erdos1041-lemniscate-reasoning-surface","source_path":"paper/evidence/erdos1041-lemniscate-reasoning-surface.tex","media_type":"text/x-tex","sha256":"sha256:9da9d47d30da5719a9568419f21fa0ef6ee2ca6a4b6b5ea20c987cb1766ec2e3","byte_count":24005,"licence":"CC-BY-4.0","body_format":"latex","body":"% Generated by scripts/paper_evidence.py from docs/paper_lean_coverage.json and the\n% Comparator replay receipts under evidence/comparator/.  Do not edit by hand.\n\\newcommand{\\evidencerecordurl}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md}\n\\DeclareResultEvidenceName{res:ani-degree-seven-counterexample-long}{Theorem 1.2}\n\\DeclareResultEvidenceRecord{res:ani-degree-seven-counterexample-long}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-ani-degree-seven-counterexample-long}{exact_or_stronger}{pending}\n\\DeclareResultEvidence{res:ani-degree-seven-counterexample-long}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-ani-degree-seven-counterexample-long}{}\n\\DeclareResultEvidenceName{lem:two-sheet-bottleneck-long}{Lemma 2.1}\n\\DeclareResultEvidenceRecord{lem:two-sheet-bottleneck-long}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#lem-two-sheet-bottleneck-long}{exact_or_stronger}{pending}\n\\DeclareResultEvidence{lem:two-sheet-bottleneck-long}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean\\#L187}{}\n\\DeclareResultEvidenceName{res:trinomial-all-degree}{Theorem 3.1}\n\\DeclareResultEvidenceRecord{res:trinomial-all-degree}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-trinomial-all-degree}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:trinomial-all-degree}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-trinomial-all-degree}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-trinomial-all-degree-comparator}\n\\DeclareResultEvidenceName{res:low-critical-thirteen-twentyfifths}{Passage, p. 10}\n\\DeclareResultEvidenceRecord{res:low-critical-thirteen-twentyfifths}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-low-critical-thirteen-twentyfifths}{none}{not_applicable}\n\\DeclareResultEvidenceName{res:low-critical-scale-free}{Passage, p. 10}\n\\DeclareResultEvidenceRecord{res:low-critical-scale-free}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-low-critical-scale-free}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{res:low-critical-scale-free}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/PaperCompleteR21/LowCriticalScaleTransport.lean\\#L460}{}\n\\DeclareResultEvidenceName{res:circle-slice-packing}{Lemma 4.3}\n\\DeclareResultEvidenceRecord{res:circle-slice-packing}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-circle-slice-packing}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:circle-slice-packing}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-circle-slice-packing}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-circle-slice-packing-comparator}\n\\DeclareResultEvidenceName{res:dual-arity-floor}{Passage, p. 13}\n\\DeclareResultEvidenceRecord{res:dual-arity-floor}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-dual-arity-floor}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{res:dual-arity-floor}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-dual-arity-floor}{}\n\\DeclareResultEvidenceName{res:scaled-low-critical-path}{Passage, p. 17}\n\\DeclareResultEvidenceRecord{res:scaled-low-critical-path}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-scaled-low-critical-path}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{res:scaled-low-critical-path}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/PaperCompleteR21/LowCriticalScaleTransport.lean\\#L491}{}\n\\DeclareResultEvidenceName{res:constant-factor-path}{Passage, p. 18}\n\\DeclareResultEvidenceRecord{res:constant-factor-path}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-constant-factor-path}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{res:constant-factor-path}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-constant-factor-path}{}\n\\DeclareResultEvidenceName{res:constant-factor-arity}{Passage, p. 20}\n\\DeclareResultEvidenceRecord{res:constant-factor-arity}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-constant-factor-arity}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{res:constant-factor-arity}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-constant-factor-arity}{}\n\\DeclareResultEvidenceName{res:constant-factor-capacity}{Passage, p. 21}\n\\DeclareResultEvidenceRecord{res:constant-factor-capacity}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-constant-factor-capacity}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{res:constant-factor-capacity}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-constant-factor-capacity}{}\n\\DeclareResultEvidenceName{res:one-root-gamma-false}{Proposition 5.5}\n\\DeclareResultEvidenceRecord{res:one-root-gamma-false}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-one-root-gamma-false}{exact}{compared}\n\\DeclareResultEvidence{res:one-root-gamma-false}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/PaperCompleteR21/LobeUnconditional.lean\\#L41}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-one-root-gamma-false-comparator}\n\\DeclareResultEvidenceName{res:conjecture-p-consumer}{Passage, p. 24}\n\\DeclareResultEvidenceRecord{res:conjecture-p-consumer}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-conjecture-p-consumer}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{res:conjecture-p-consumer}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-conjecture-p-consumer}{}\n\\DeclareResultEvidenceName{res:degree-three}{Theorem 6.1}\n\\DeclareResultEvidenceRecord{res:degree-three}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-degree-three}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:degree-three}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/PaperCubicCompletion.lean\\#L297}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-degree-three-comparator}\n\\DeclareResultEvidenceName{res:critical-value-separation}{Passage, p. 27}\n\\DeclareResultEvidenceRecord{res:critical-value-separation}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-critical-value-separation}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{res:critical-value-separation}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-critical-value-separation}{}\n\\DeclareResultEvidenceName{res:critical-value-thresholds}{Passage, p. 30}\n\\DeclareResultEvidenceRecord{res:critical-value-thresholds}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-critical-value-thresholds}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{res:critical-value-thresholds}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-critical-value-thresholds}{}\n\\DeclareResultEvidenceName{res:sep-or-false}{Proposition 8.1}\n\\DeclareResultEvidenceRecord{res:sep-or-false}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-sep-or-false}{exact}{compared}\n\\DeclareResultEvidence{res:sep-or-false}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/PaperSeparationCounterexample.lean\\#L189}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-sep-or-false-comparator}\n\\DeclareResultEvidenceName{res:arity-not-capacity}{Proposition 8.2}\n\\DeclareResultEvidenceRecord{res:arity-not-capacity}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-arity-not-capacity}{modulo_named_input}{not_applicable}\n\\DeclareResultEvidence{res:arity-not-capacity}{Lean\\textsuperscript{\\dag}}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-arity-not-capacity}{}\n\\DeclareResultEvidenceName{prop:sharp-collinear-chebyshev-comparator}{Theorem 9.1}\n\\DeclareResultEvidenceRecord{prop:sharp-collinear-chebyshev-comparator}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#prop-sharp-collinear-chebyshev-comparator}{exact}{compared}\n\\DeclareResultEvidence{prop:sharp-collinear-chebyshev-comparator}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/SharpCollinearChebyshev.lean\\#L133}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#prop-sharp-collinear-chebyshev-comparator-comparator}\n\\DeclareResultEvidenceName{thm:sharp-collinear-diameter}{Theorem 9.2}\n\\DeclareResultEvidenceRecord{thm:sharp-collinear-diameter}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#thm-sharp-collinear-diameter}{exact}{compared}\n\\DeclareResultEvidence{thm:sharp-collinear-diameter}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#thm-sharp-collinear-diameter}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#thm-sharp-collinear-diameter-comparator}\n\\DeclareResultEvidenceName{cor:collinear-erdos-1041}{Corollary 9.3}\n\\DeclareResultEvidenceRecord{cor:collinear-erdos-1041}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#cor-collinear-erdos-1041}{exact}{compared}\n\\DeclareResultEvidence{cor:collinear-erdos-1041}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#cor-collinear-erdos-1041}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#cor-collinear-erdos-1041-comparator}\n\\DeclareResultEvidenceName{prop:primitive-quintic-two-tail-energy-selector}{Theorem 9.4}\n\\DeclareResultEvidenceRecord{prop:primitive-quintic-two-tail-energy-selector}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#prop-primitive-quintic-two-tail-energy-selector}{exact}{compared}\n\\DeclareResultEvidence{prop:primitive-quintic-two-tail-energy-selector}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/PrimitiveQuinticInteriorTail.lean\\#L272}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#prop-primitive-quintic-two-tail-energy-selector-comparator}\n\\DeclareResultEvidenceName{thm:primitive-quintic-two-tail}{Theorem 9.5}\n\\DeclareResultEvidenceRecord{thm:primitive-quintic-two-tail}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#thm-primitive-quintic-two-tail}{exact}{compared}\n\\DeclareResultEvidence{thm:primitive-quintic-two-tail}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#thm-primitive-quintic-two-tail}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#thm-primitive-quintic-two-tail-comparator}\n\\DeclareResultEvidenceName{lem:cubic-safe-root-spoke}{Theorem 9.6}\n\\DeclareResultEvidenceRecord{lem:cubic-safe-root-spoke}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#lem-cubic-safe-root-spoke}{exact}{compared}\n\\DeclareResultEvidence{lem:cubic-safe-root-spoke}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/CubicQuotientFiberCase.lean\\#L161}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#lem-cubic-safe-root-spoke-comparator}\n\\DeclareResultEvidenceName{thm:translated-cubic-quotient-fibres}{Theorem 9.7}\n\\DeclareResultEvidenceRecord{thm:translated-cubic-quotient-fibres}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#thm-translated-cubic-quotient-fibres}{exact}{compared}\n\\DeclareResultEvidence{thm:translated-cubic-quotient-fibres}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/PaperCubicFibres.lean\\#L240}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#thm-translated-cubic-quotient-fibres-comparator}\n\\DeclareResultEvidenceName{res:complementary-binomial-chords}{Theorem 10.1}\n\\DeclareResultEvidenceRecord{res:complementary-binomial-chords}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-complementary-binomial-chords}{exact}{compared}\n\\DeclareResultEvidence{res:complementary-binomial-chords}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-complementary-binomial-chords}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-complementary-binomial-chords-comparator}\n\\DeclareResultEvidenceName{res:critical-value-budget}{Theorem 10.2}\n\\DeclareResultEvidenceRecord{res:critical-value-budget}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-critical-value-budget}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:critical-value-budget}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-critical-value-budget}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-critical-value-budget-comparator}\n\\DeclareResultEvidenceName{res:reflected-critical-value}{Lemma 10.3}\n\\DeclareResultEvidenceRecord{res:reflected-critical-value}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-reflected-critical-value}{exact}{compared}\n\\DeclareResultEvidence{res:reflected-critical-value}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/PaperReflectedCompletion.lean\\#L254}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-reflected-critical-value-comparator}\n\\DeclareResultEvidenceName{res:fp-weighted-all-degree}{Theorem 10.4}\n\\DeclareResultEvidenceRecord{res:fp-weighted-all-degree}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-fp-weighted-all-degree}{exact}{compared}\n\\DeclareResultEvidence{res:fp-weighted-all-degree}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-fp-weighted-all-degree}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-fp-weighted-all-degree-comparator}\n\\DeclareResultEvidenceName{res:critical-proximity}{Theorem 10.5}\n\\DeclareResultEvidenceRecord{res:critical-proximity}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-critical-proximity}{exact}{compared}\n\\DeclareResultEvidence{res:critical-proximity}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/CriticalTwoRootProximity.lean\\#L291}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-critical-proximity-comparator}\n\\DeclareResultEvidenceName{res:two-nearest-roots}{Corollary 10.6}\n\\DeclareResultEvidenceRecord{res:two-nearest-roots}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-two-nearest-roots}{exact}{compared}\n\\DeclareResultEvidence{res:two-nearest-roots}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-two-nearest-roots}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-two-nearest-roots-comparator}\n\\DeclareResultEvidenceName{res:straight-no-go}{Proposition 10.7}\n\\DeclareResultEvidenceRecord{res:straight-no-go}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-straight-no-go}{exact}{compared}\n\\DeclareResultEvidence{res:straight-no-go}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/PaperStraightObstructions.lean\\#L178}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-straight-no-go-comparator}\n\\DeclareResultEvidenceName{res:orlicz-currency}{Theorem 10.8}\n\\DeclareResultEvidenceRecord{res:orlicz-currency}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-orlicz-currency}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:orlicz-currency}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-orlicz-currency}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-orlicz-currency-comparator}\n\\DeclareResultEvidenceName{res:value}{Theorem 11.1}\n\\DeclareResultEvidenceRecord{res:value}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-value}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:value}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/PaperCompleteR20/NewtonRealTime.lean\\#L52}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-value-comparator}\n\\DeclareResultEvidenceName{res:ray}{Corollary 11.2}\n\\DeclareResultEvidenceRecord{res:ray}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-ray}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:ray}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/PaperCompleteR20/NewtonRealTime.lean\\#L69}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-ray-comparator}\n\\DeclareResultEvidenceName{res:locus}{Theorem 12.1}\n\\DeclareResultEvidenceRecord{res:locus}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-locus}{exact}{compared}\n\\DeclareResultEvidence{res:locus}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean\\#L107}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-locus-comparator}\n\\DeclareResultEvidenceName{res:attachment-aware-reeb}{Passage, p. 70}\n\\DeclareResultEvidenceRecord{res:attachment-aware-reeb}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1041-lemniscate-reasoning-surface.md\\#res-attachment-aware-reeb}{none}{not_applicable}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/1041/erdos1041-lemniscate-reasoning-surface.tex","section_outline":["The question and the degree-seven counterexample","The degree-seven construction","A two-sheeted component","Polynomial and roots","Critical points and local estimates","The two roots in the component","The length inequality","Trinomials","A small least critical value","A path estimate from area and boundary length","Degree three","An isolated simple critical value","Examples separating the geometric hypotheses","Collinear roots and two sparse polynomial families","Collinear roots and Chebyshev comparison","Quintics with two missing coefficients","Polynomials obtained from a cubic by a power substitution","Further families and counterexamples to proposed proof steps","Unformalised remark: roots close to a regular polygon","The exact binomial chord calculation","Unformalised remark: a bounded-radius concyclic class","Power substitutions in trinomials","Unformalised remark: a coefficient condition for four-term polynomials","Unformalised remark: a quartic composed with a power map","A mean bound for critical values in every degree","A sufficient critical-value threshold in degree five","Numerical tests of selection by the smallest critical value","Unformalised remark: limits of contained paths at fixed degree","Unformalised remark: Blaschke products and varying degree","What the earlier inverse-ray approach would have required","An integral for successive merger scales","The Newton value equation","Separating the rays of critical values","Local polynomial expansions","Why the proposed spanning-tree estimate fails","Numerical path searches","Further questions about topology, length and perturbation","1. Local saddle geometry and the refuted tree estimate","2. Cutting a compact region into flow strips","3. Length estimates for joining flow strips"]},"erdos-1049-rational-base-lambert":{"paper_id":"erdos-1049-rational-base-lambert","title":"Hankel Determinants of Geometric Moments and Rational Lambert Values","form":"Problem note","length_class":"short","pages":20,"subject":"Erdős #1049","subject_kind":"problem","problem_id":"erdos_1049","landing_role":"mathematics","question":"What are the fixed-base Hankel asymptotics, and which rational bases give irrational values?","precis":"Positive geometric moments and a uniform partition-sum bound give the fixed-base asymptotic of Zudilin’s normalised Hankel determinants, including its constant and power correction. A separate calculation of degrees after cancellation in his 2004 forms extends the rational-base irrationality region. It includes every positive integral power of 31/4 and excludes 3/2.","owns":"the fixed-base Hankel asymptotic and rational-base irrationality theorem; the long paper develops exact orders, content and endpoint obstructions, and the remaining problem at 3/2","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #1049, which remains open","first_pass":["sec:problem","sec:geometric-moments","sec:rational-base-irrationality","sec:open"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-1049-rational-base-lambert.pdf","hosted_pdf_path":"papers/erdos-1049-rational-base-lambert.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex","site_html":"maths/papers/erdos-1049-rational-base-lambert.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1049/erdos-1049-rational-base-lambert.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1049/erdos-1049-rational-base-lambert.tex","body_pointer":"papers.records.erdos-1049-rational-base-lambert.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% One of the Erdős Problem Notes: a short standalone treatment of a single\n% problem in the ErdosProblems expansion library.  Build with\n% `tectonic erdos-1049-rational-base-lambert.tex` or `make`.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n% For local PDF links hyperref needs the actual destination after the hash.\n% The shared browser-style \"nameddest=\" prefix otherwise becomes part of /D.\n% Keep this local until Type A can repair the out-of-scope shared link helper.\n\\newcommand{\\shortrecordlink}[3]{\\href{#1\\##2}{#3}}\n\\input{evidence/erdos-1049-rational-base-lambert}\n\\evidenceconcordancenewpage\n\\renewcommand{\\commit}{7380b7871687b6bcc41ca0143c61f232e8af6500}\n\\renewcommand{\\commitshort}{7380b7871687}\n\\renewcommand{\\PX}{lean/ErdosProblems}\n% Sources that postdate \\commit are cited with their lean/ path and resolve at \\latecommit,\n% whose tree keeps the Lean roots under lean/.\n\\newcommand{\\latecommit}{035c414b25fda09ae0d2e56358b8716d321ebd99}\n\\newcommand{\\laterepobase}{https://github.com/wcook04/plectis-erdos/blob/\\latecommit}\n\\hypersetup{\n  pdftitle={Hankel determinants of geometric moments and rational Lambert values (Erdos Problem 1049)},\n  pdfsubject={Erdos Problem Notes: Problem 1049},\n  pdfkeywords={Lambert series, Hankel determinant, geometric moments, rational base, irrationality},\n}\n\\runninghead{Geometric moments and rational Lambert values}\n\\title{Hankel Determinants of Geometric Moments\\\\and Rational Lambert Values}\n\\subtitle{Erd\\H{o}s Problem~\\#1049}\n\\author{Will Cook}\n\\date{September 2026}\n\n\\begin{document}\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\nFor fixed $0<q<1$, we determine the asymptotic of Zudilin's normalised\nHankel determinants as their size $N$ grows, including the constant\nand the factor $N^{-8F(1/q)}$, where $F(t)=\\sum_{n\\ge1}(t^n-1)^{-1}$.\nA general theorem for positive geometric moments separates the node\ngeometry from the weight product; two finite convolutions determine\nits power correction. Independently, Zudilin's 2004 polynomial forms\ngive irrationality of $F(a/b)$ for coprime $a>b\\ge1$ with\n$\\log b/\\log a<0.4056830213840605\\ldots$. This extends the sufficient\nregion of Bundschuh and V\\\"a\\\"an\\\"anen and includes every positive\nintegral power of $31/4$, but not $3/2$.\n\\end{abstract}\n\n\\section{Introduction}\\label{sec:problem}\n\nFix $q\\in(0,1)$. Write\n$(z;q)_m=\\prod_{j=0}^{m-1}(1-zq^j)$ and\n$F(t)=\\sum_{n\\ge1}(t^n-1)^{-1}$ for $t>1$.\nAt the auxiliary parameters $x=z=1$, the normalisation in\n\\cite[(6), \\S4]{zudilin2016} gives\n\\begin{equation}\\label{eq:normalised-moments}\n v_m^*(q)=\\sum_{t\\ge0}q^{(m+1)t}\n \\frac{(q;q)_m^3(q^{t+1};q)_m}{(q^{m+1+t};q)_{m+1}},\n \\qquad V_N^*(q)=\\det(v_{i+j}^*(q))_{0\\le i,j<N}.\n\\end{equation}\nThe sums converge for $0<q<1$. They also define formal series, since\nthe $t$th summand has $q$-order $(m+1)t$.\n\n\\begin{theorem}[the size of $V_N^*$ at a fixed base]\\label{res:sharp-fixed-base}\nFix $0<q<1$ and write $P=(q;q)_\\infty$, $B_N=N(N-1)(2N-1)/6$ and\n$C_N=(N!)^2(N+1)!/2^N$. There is a $K(q)>0$ with\n\\[\n V_N^*(q)\\sim K(q)\\,C_Nq^{B_N}P^{2N}N^{-8F(1/q)}\n \\qquad(N\\to\\infty).\n\\]\n\\end{theorem}\n\nThe constant $K(q)$ is the convergent product in\n\\eqref{eq:fixed-constant}. Zudilin proved\n$\\operatorname{ord}_q V_N^*\\ge B_N$ and the estimate\n$|V_N^*|\\le q^{N^3/3}\\exp(O_q(N^2))$\n\\cite[Lemma~1 and \\S4]{zudilin2016}. Theorem~\\ref{res:sharp-fixed-base} determines the factorial, exponential\nand power factors contained in this estimate. We also recover the exact\nfirst formal term $C_Nq^{B_N}$. The formal calculation fixes $N$ and\nexpands at $q=0$; the asymptotic fixes $q$ and lets $N$ grow. Neither\nlimit can be substituted for the other: multiplication by\n$(1-q)^{N^3}$ preserves the first formal term but changes the\nfixed-base logarithm by a cubic quantity. No uniformity as $q\\to1$\nis asserted.\n\nThe proof compares the moments with constant-weight geometric\nmoments. We write $v_m^*=\\sum_{k\\ge0}a_kq^{(m+1)k}$ with $a_k>0$:\nthe measure has mass $a_kq^k$ at $q^k$. Heine's identity expresses\nthe determinant as a positive sum of squared Vandermondes over\nchoices of $N$ nodes \\cite[\\S2, (2)--(5)]{zudilin2017det}.\nThe first $N$ nodes normalise this sum but do not exhaust it;\nmoving the last node gives a comparable contribution. Partitions\nencode all such displacements. For each fixed partition, only\nfinitely many weight ratios change, at indices tending to infinity,\nso those ratios tend to $1$. A summable bound independent of $N$\nthen makes the whole normalised sum asymptotically independent of\nthe weights. Cauchy's determinant formula evaluates its limit in\nthe constant-weight case.\n\nThe weights retain a different contribution. Their factorisation\ninvolves two-fold and three-fold convolutions of $1/(q;q)_k$,\nwith relative corrections $-2F(1/q)/(k+1)$ and $-6F(1/q)/(k+2)$.\nIn the logarithm of the weight product these give a harmonic sum,\nhence $N^{-8F(1/q)}$; the summable errors affect only the constant.\nTheorem~\\ref{thm:geometric-moments} separates the determinant argument\nfrom this calculation and also treats entries $(1-q^{i+j+1})^{-2}$.\n\nChowla's conjecture, recorded by Erd\\H{o}s\n\\cite[p.~102]{erdos1988}, asks whether $F(t)$ is irrational for every\nrational $t>1$. This is a related arithmetic question, and the\nfixed-base asymptotic alone does not answer it. In\nSection~\\ref{sec:rational-base-irrationality} we instead specialise\nthe polynomial forms of Zudilin's 2004 construction\n\\cite[\\S5]{zudilin2004}. They prove irrationality for coprime\n$a>b\\ge1$ satisfying\n\\[\n \\frac{\\log b}{\\log a}<\\theta^*\n =0.4056830213840605\\ldots,\n\\]\nwhere \\eqref{eq:threshold-constants} defines $\\theta^*$ exactly.\nThe constants are inherited from Zudilin; the degree calculation\nafter polynomial cancellation supplies the rational-base estimate.\nThe resulting region includes every positive integral power of\n$31/4$, which lies outside the earlier sufficient region of\nBundschuh and V\\\"a\\\"an\\\"anen \\cite[Theorem~2]{bv1994}. Neither\nargument here settles $3/2$.\n\nSections~\\ref{sec:geometric-moments}--\\ref{sec:weights} prove\nTheorem~\\ref{res:sharp-fixed-base};\nSection~\\ref{sec:rational-base-irrationality} gives the independent\narithmetic argument. Section~\\ref{sec:open} links further questions\nto the companion. All logarithms are natural, empty products and\ndeterminants equal $1$, and constants in $O_q(\\cdot)$ may depend on\nthe fixed $q$.\n\n\\section{Geometric moment determinants}\\label{sec:geometric-moments}\n\nFor constant weights, the moments are geometric series:\n$M_m=\\sum_{k\\ge0}q^{(m+1)k}=(1-q^{m+1})^{-1}$.\nCauchy's determinant formula evaluates their determinant exactly.\nWriting\n\\[\n P=(q;q)_\\infty,\\qquad\n \\mathcal M(q)=\\prod_{d\\ge1}(1-q^d)^{-d},\\qquad\n B_N=\\sum_{j<N}j^2,\n\\]\nwe obtain\n\\begin{equation}\\label{eq:cauchy-model}\n D_N^{(0)}:=\\det\\left(\\frac1{1-q^{i+j+1}}\\right)_{0\\le i,j<N}\n =\\frac{q^{B_N}\\Delta_N}{\\prod_{0\\le i,j<N}(1-q^{i+j+1})},\n \\qquad\n \\Delta_N=\\prod_{d=1}^{N-1}(1-q^d)^{2(N-d)}.\n\\end{equation}\nFor each fixed $d$, the multiplicity of $1-q^d$ in the denominator\nincreases to $d$, giving the factor $\\mathcal M(q)$ in the limit.\nThe numerator supplies two more copies:\n\\[\n \\frac{\\Delta_N}{P^{2N}}\n =\\prod_{d<N}(1-q^d)^{-2d}\\prod_{d\\ge N}(1-q^d)^{-2N}\n \\longrightarrow\\mathcal M(q)^2.\n\\]\nIndeed, $\\sum d|\\log(1-q^d)|<\\infty$, and the logarithm of the\nsecond product is $O_q(Nq^N)$. Thus\n\\begin{equation}\\label{eq:cauchy-model-limit}\n D_N^{(0)}\\sim\\mathcal M(q)^3q^{B_N}P^{2N}.\n\\end{equation}\nReplacing the constant weights by $a_k$ will contribute\n$\\prod_{k<N}a_k$. The two hypotheses below do different jobs:\nthe ratio limit handles each fixed displacement; the polynomial\nbound permits summation over all displacements.\n\n\\begin{theorem}[geometric moment determinants]\\label{thm:geometric-moments}\nLet $0<q<1$ and $a_k>0$ for $k\\ge0$. Suppose that, for fixed\nconstants $C>0$ and $\\kappa\\ge0$,\n\\begin{equation}\\label{eq:automatic-shift-bound}\n \\frac{a_{k+h}}{a_k}\\longrightarrow1\\quad(k\\to\\infty)\n \\text{ for each fixed }h\\ge0,\n \\qquad\n \\frac{a_{k+h}}{a_k}\\le C(1+h)^\\kappa\\quad(k,h\\ge0).\n\\end{equation}\nFor $M_m=\\sum_{k\\ge0}a_kq^{(m+1)k}$ and\n$D_N=\\det(M_{i+j})_{0\\le i,j<N}$, we have\n\\begin{equation}\\label{eq:geometric-limit}\n D_N\\sim\\mathcal M(q)^3q^{B_N}P^{2N}\\prod_{k=0}^{N-1}a_k\n \\qquad(N\\to\\infty).\n\\end{equation}\n\\end{theorem}\n\nFor every real $s$, the weights $a_k=(k+1)^s$ satisfy\n\\eqref{eq:automatic-shift-bound} with $C=1$ and\n$\\kappa=\\max(s,0)$, and their first $N$ terms have product $(N!)^s$.\nTaking $s=1$ and summing the moments gives\n\\begin{equation}\\label{eq:squared-cauchy-example}\n \\det\\left(\\frac1{(1-q^{i+j+1})^2}\\right)_{0\\le i,j<N}\n \\sim\\mathcal M(q)^3N!q^{B_N}P^{2N}.\n\\end{equation}\nThe polynomial bound excludes $a_k=e^{\\sqrt{k}}$, although its\nadjacent ratio tends to $1$. The companion's\n\\shortrecordlink{erdos1049-rational-base-lambert-reasoning-surface.pdf}\n{long1049-ratio-extension}{ordinary ratio-limit argument}\nallows $a_{k+1}/a_k\\to\\rho\\in(0,q^{-1})$ and, at $\\rho=1$, gives\n\\eqref{eq:geometric-limit} without that bound. This broader claim\nremains unformalised and is not used here: the theorem and its\napplications retain both hypotheses in\n\\eqref{eq:automatic-shift-bound}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:geometric-moments}]\nThe bound $a_h\\le Ca_0(1+h)^\\kappa$ makes\n$\\sum_{k\\ge0}a_kq^k\\delta_{q^k}$ a finite positive measure with\nmoments $M_m$. Apply Cauchy--Binet to its finite truncations.\nThe entries, hence the determinants, converge; the nonnegative\ntuple sums increase. Passing to the limit gives\n\\begin{equation}\\label{eq:heine-geometric}\n D_N=\\sum_{0\\le k_0<\\cdots<k_{N-1}}\n \\prod_{i<N}a_{k_i}q^{k_i}\n \\prod_{i<j<N}(q^{k_i}-q^{k_j})^2.\n\\end{equation}\nThe tuple $k_i=i$ contributes $q^{B_N}\\Delta_N\\prod_{i<N}a_i$.\nIt does not account for the whole asymptotic. For example, replace\n$(0,1,\\ldots,N-2,N-1)$ by $(0,1,\\ldots,N-2,N)$. The ratio of\nits contribution to that of the original tuple is\n\\begin{equation}\\label{eq:last-node-shift}\n q\\frac{a_N}{a_{N-1}}\n \\prod_{d=1}^{N-1}\\left(\\frac{1-q^{d+1}}{1-q^d}\\right)^2\n =q\\frac{a_N}{a_{N-1}}\\left(\\frac{1-q^N}{1-q}\\right)^2\n \\longrightarrow\\frac{q}{(1-q)^2}.\n\\end{equation}\nOnly pairs involving the moved node change, and their product\ntelescopes. At $q=1/2$ the displaced tuple is asymptotically twice\nas large as the normalising tuple.\n\nSubtracting $(0,1,\\ldots,N-1)$ gives the shifts $\\lambda_i=k_i-i$.\nSince $k_{i+1}\\ge k_i+1$, these are nonnegative and weakly increasing;\nreversing their order gives a partition. For $N=4$, for example,\n\\[\n \\underbrace{(0,1,3,5)}_{(k_i)}\n \\longmapsto\\underbrace{(0,0,1,2)}_{(\\lambda_i)}\n \\longmapsto\\underbrace{(2,1,0,0)}_{(\\mu_j)}.\n\\]\nIn general, put $\\mu_j=\\lambda_{N-j}$ for $1\\le j\\le N$, so that\n$\\mu_1\\ge\\cdots\\ge\\mu_N\\ge0$. Division by the contribution of\n$k_i=i$ turns the summand for this tuple into\n\\begin{equation}\\label{eq:partition-summand}\n\\begin{split}\n W_N(\\mu)={}&q^{\\sum_{j\\le N}(2j-1)\\mu_j}\n \\prod_{j\\le N}\\frac{a_{N-j+\\mu_j}}{a_{N-j}}\\\\\n &\\hspace{2mm}\\cdot\\prod_{1\\le j<k\\le N}\n \\left(\\frac{1-q^{k-j+\\mu_j-\\mu_k}}{1-q^{k-j}}\\right)^2.\n\\end{split}\n\\end{equation}\nThe $j$th index from the right enters once through the mass\n$q^{k_i}$ and twice through each of its $j-1$ pairs with later\nindices. This gives the exponent $1+2(j-1)=2j-1$.\n``Right'' refers to increasing indices, not to the nodes $q^{k_i}$,\nwhich decrease towards zero.\n\nSet $W_N(\\mu)=0$ for partitions with more than $N$ positive parts,\ngiving every sum the same index set. A fixed partition with $\\ell$\npositive parts affects only the last $\\ell$ indices of\n$(0,1,\\ldots,N-1)$. These indices tend to infinity while their\ndisplacements stay fixed, so the weight ratios in\n\\eqref{eq:partition-summand} tend to $1$.\n\nThe Vandermonde quotient still contains a growing number of factors,\nbut its dependence on $N$ telescopes. The factors with $j<k\\le\\ell$\nare fixed, and those with $j>\\ell$ equal $1$. For $N\\ge\\ell$ and\neach $j\\le\\ell$, the remaining factors satisfy\n\\begin{equation}\\label{eq:partition-tail}\n \\prod_{k=\\ell+1}^{N}\n \\frac{1-q^{k-j+\\mu_j}}{1-q^{k-j}}\n =\\frac{(q^{N+1-j};q)_{\\mu_j}}{(q^{\\ell+1-j};q)_{\\mu_j}}\n \\longrightarrow\\frac1{(q^{\\ell+1-j};q)_{\\mu_j}}.\n\\end{equation}\nThus each $W_N(\\mu)$ has the same limit as for constant weights.\n\nThe fixed-partition limit does not yet control partitions whose\nlength or parts grow with $N$. For a uniform bound, group the\nVandermonde factors by $j$. Only $j\\le\\ell$ contribute; each group's\ndenominator product is at least $P$ and its numerator product at\nmost $1$. The polynomial shift bound therefore gives\n\\[\n W_N(\\mu)\\le(CP^{-2})^\\ell\n \\prod_{j=1}^\\ell(1+\\mu_j)^\\kappa q^{(2j-1)\\mu_j}.\n\\]\nLet $A=\\sum_{u\\ge1}(1+u)^\\kappa q^{u-1}<\\infty$.\nDropping the ordering of the positive parts, we obtain\n\\begin{align*}\n \\sum_{\\ell(\\mu)=\\ell}\\sup_N W_N(\\mu)\n &\\le(CP^{-2})^\\ell\\prod_{j=1}^\\ell\n       \\sum_{u\\ge1}(1+u)^\\kappa q^{(2j-1)u}\\\\\n &\\le(CP^{-2}A)^\\ell q^{\\ell^2}.\n\\end{align*}\nHere $q^{(2j-1)u}\\le q^{2j-1}q^{u-1}$ and\n$1+3+\\cdots+(2\\ell-1)=\\ell^2$. The size of each part is absorbed\nin $A$; their number is controlled by $q^{\\ell^2}$, which dominates\n$(CP^{-2}A)^\\ell$. Thus the bound is summable over $\\ell$ and\nindependent of $N$.\n\nDominated convergence gives the same limit for\n$\\sum_\\mu W_N(\\mu)$ as for $a_k=1$. Equation~\\eqref{eq:cauchy-model}\nevaluates that limit as $\\mathcal M(q)$, without a separate partition\nidentity. Restoring the normalising tuple supplies the weight\nproduct and, through $\\Delta_N/P^{2N}$, the other two copies of\n$\\mathcal M(q)$. Hence\n\\[\n \\frac{D_N}{D_N^{(0)}\\prod_{k<N}a_k}\\longrightarrow1.\n\\]\nCombining this limit with \\eqref{eq:cauchy-model-limit} proves\nthe theorem.\n\\end{proof}\n\n\\section{The weights of Zudilin's moments}\\label{sec:weights}\n\nWe apply Theorem~\\ref{thm:geometric-moments} by collecting the\ndependence on $m$ in \\eqref{eq:normalised-moments} into the single\nvariable $w=q^{m+1}$. For this purpose, set\n\\begin{equation}\\label{eq:weight-generating-function}\n G_q(w)=\\frac1{(w;q)_\\infty^3}\n \\sum_{t\\ge0}\\frac{w^t}{(q;q)_t}\n       \\frac{(q^tw^2;q)_\\infty}{(q^tw;q)_\\infty^2},\n \\qquad a_k(q)=[w^k]P^4G_q(w).\n\\end{equation}\nThe product identities\n$(q;q)_m=P/(q^{m+1};q)_\\infty$ and\n$(q^a;q)_m=(q^a;q)_\\infty/(q^{a+m};q)_\\infty$\ngive\n\\begin{equation}\\label{eq:positive-moments}\n v_m^*=P^4G_q(q^{m+1})=\\sum_{k\\ge0}a_k(q)q^{(m+1)k}.\n\\end{equation}\nThese identities hold formally in $\\Z[[q,w]]$ and, for fixed $0<q<1$,\nanalytically for $|w|<1$. In the formal identity only finitely many\n$t$ contribute to any $w$-coefficient. For the analytic identity, we use the absence of denominator zeros in\nthat disc and the locally uniform bounds on the product tails.\nThe factor $w^t$ then gives locally uniform convergence of the sum.\nThe signs of the coefficients are not evident from this product.\nThe finite factorisation below gives positive weights and a\nuniform shift bound. Its two convolutions then give the accuracy\nneeded to multiply the first $N$ weights.\n\n\\subsection{Positive weights}\\label{sec:weight-factorisation}\n\nFor $r\\ge1$ and $k\\ge0$, set\n\\[\n R_k^{(r)}(q)=\\sum_{n_1+\\cdots+n_r=k}\n              \\frac{(q;q)_k}{\\prod_{j=1}^r(q;q)_{n_j}},\n\\]\nover $r$ nonnegative parts. These are the multivariate Rogers--Szeg\\H{o} polynomials at unit\narguments, written as sums of Gaussian multinomial polynomials\n\\cite{vinroot2010}. At $q=0$, every Gaussian multinomial equals $1$,\nso they count weak compositions (zero parts are allowed), and\n\\[\n c_k:=R_k^{(2)}(0)R_k^{(3)}(0)=\\frac{(k+1)^2(k+2)}2,\n \\qquad \\prod_{k<N}c_k=C_N.\n\\]\n\n\\begin{samepage}\n\\begin{proposition}[the positive moment weights]\\label{prop:weight-factorisation}\nFor every $k\\ge0$,\n\\[\n a_k=P^4\\frac{R_k^{(2)}R_k^{(3)}}{(q;q)_k}.\n\\]\nAt fixed $0<q<1$ these weights satisfy\n\\[\n P^5c_k\\le a_k\\le P^{-1}c_k,\n \\qquad \\frac{a_{k+h}}{a_k}\\le P^{-6}(1+h)^3\\quad(k,h\\ge0).\n\\]\n\\end{proposition}\n\\end{samepage}\n\n\\begin{proof}\nWe use the Gaussian binomial coefficients\n$\\genfrac{[}{]}{0pt}{}{n}{j}_q=(q;q)_n/((q;q)_j(q;q)_{n-j})$ for\n$0\\le j\\le n$, and put $\\mathcal E(z)=(z;q)_\\infty^{-1}$.\nEuler's expansion gives\n$\\mathcal E(z)^r=\\sum_{n\\ge0}R_n^{(r)}z^n/(q;q)_n$.\nThe desired factorisation multiplies two coefficients at the same\nindex $k$, rather than taking a coefficient of $\\mathcal E^5$.\nWe obtain it by splitting $k=n+j$: the factor $R_k^{(3)}$ stays\nfixed, and summing the two factorial denominators supplies\n$R_k^{(2)}$. To produce this split, use the unnormalised\n$q$-difference operator $\\partial_qf(z)=(f(z)-f(qz))/z$.\nSince $\\partial_qz^j=(1-q^j)z^{j-1}$, it cancels the last factor\nof $(q;q)_j$ and shifts the numerator index:\n\\[\n \\partial_q\\mathcal E=\\mathcal E,\\qquad\n \\partial_q^n\\mathcal E(z)^3\n =\\sum_{j\\ge0}\\frac{R_{n+j}^{(3)}}{(q;q)_j}z^j.\n\\]\nThe $q$-Leibniz rule, obtained by induction from the product rule, is\n\\[\n \\partial_q^n(fg)(z)=\\sum_{j=0}^n\\genfrac{[}{]}{0pt}{}{n}{j}_q\n (\\partial_q^jf)(q^{n-j}z)\\,(\\partial_q^{n-j}g)(z).\n\\]\nLet $h_k(z)=\\sum_{j=0}^k\\genfrac{[}{]}{0pt}{}{k}{j}_q(z;q)_j$.\nApplying the rule to $\\mathcal E\\cdot\\mathcal E$ and using\n$\\mathcal E(q^mz)=(z;q)_m\\mathcal E(z)$, we obtain\n$\\partial_q^k\\mathcal E(z)^2=\\mathcal E(z)^2h_k(z)$\n(for $k=1$, both sides equal $\\mathcal E(z)^2(2-z)$).\nA second application, this time to $\\mathcal E\\cdot\\mathcal E^2$, gives\n\\[\n \\partial_q^n\\mathcal E(z)^3\n =\\mathcal E(z)^3\\sum_{k=0}^n\\genfrac{[}{]}{0pt}{}{n}{k}_q(z;q)_kh_k(z),\n\\]\nand hence\n\\[\n \\sum_{n\\ge0}\\frac{w^n}{(q;q)_n}\\partial_q^n\\mathcal E(z)^3\n =\\mathcal E(w)\\mathcal E(z)^3\n  \\sum_{k\\ge0}\\frac{w^k(z;q)_k}{(q;q)_k}h_k(z).\n\\]\nThe left-hand side is\n$\\sum_{n,j\\ge0}R_{n+j}^{(3)}w^nz^j/((q;q)_n(q;q)_j)$.\nPutting $z=w$ collects all pairs with the same sum $n+j$.\nThe coefficient of $w^k$ is consequently\n$R_k^{(3)}\\sum_{j\\le k}\\bigl((q;q)_j(q;q)_{k-j}\\bigr)^{-1}\n=R_k^{(2)}R_k^{(3)}/(q;q)_k$.\nOn the right-hand side, put $z=w$, expand $h_k$, and write\n$k=j+t$. The identity $(w;q)_{j+t}=(w;q)_j(wq^j;q)_t$ gives\n\\[\n \\mathcal E(w)^4\\sum_{j\\ge0}\\frac{w^j(w;q)_j^2}{(q;q)_j}\n       \\sum_{t\\ge0}\\frac{(wq^j;q)_t}{(q;q)_t}w^t.\n\\]\nThe inner sum is evaluated by the $q$-binomial theorem:\n\\[\n \\sum_{t\\ge0}\\frac{(wq^j;q)_t}{(q;q)_t}w^t\n =\\frac{(w^2q^j;q)_\\infty}{(w;q)_\\infty}.\n\\]\nFinally, $(w;q)_j=(w;q)_\\infty/(wq^j;q)_\\infty$ cancels two\nof the five factors $\\mathcal E(w)$, leaving\n\\[\n \\mathcal E(w)^3\\sum_{j\\ge0}\\frac{w^j}{(q;q)_j}\n       \\frac{(w^2q^j;q)_\\infty}{(wq^j;q)_\\infty^2}\n =G_q(w).\n\\]\nWe may read each identity coefficientwise, or as an absolutely\nconvergent expansion when $|z|$ and $|w|$ are sufficiently small.\n\nFor the bounds, we write $b_k^{(r)}=[z^k](z;q)_\\infty^{-r}$, so that\n$R_k^{(r)}=(q;q)_kb_k^{(r)}$ and\n$a_k=P^4(q;q)_kb_k^{(2)}b_k^{(3)}$. By Euler's expansion,\n$b_k^{(r)}$ is an $r$-fold convolution of $1/(q;q)_n$, whose terms lie\nin $[1,P^{-1}]$. There are $\\binom{k+r-1}{r-1}$ compositions of $k$\ninto $r$ nonnegative parts, and we deduce\n\\[\n \\binom{k+r-1}{r-1}\\le b_k^{(r)}\\le P^{-r}\\binom{k+r-1}{r-1}.\n\\]\nCombining these at $r=2,3$ with $P\\le(q;q)_k\\le1$ and\n$c_k=(k+1)\\binom{k+2}2$ gives $P^5c_k\\le a_k\\le P^{-1}c_k$. The shift bound\nfollows from $c_{k+h}/c_k\\le(1+h)^3$.\n\n\\end{proof}\n\nPositivity holds at each fixed real $q\\in(0,1)$, not coefficientwise\nin the formal $q$-series: $a_0(q)=P^4=1-4q+O(q^2)$.\nThe bounds control shifted ratios, but do not determine the weight\nproduct. A relative error of order $1/k$ accumulates logarithmically;\none of order $1/k^2$ is summable. Computing $a_k/c_k$ to first order\nwill therefore determine the power of $N$ and give the fixed-shift\nlimit.\n\n\\subsection{The coefficient asymptotic}\\label{sec:coefficient-asymptotic}\n\n\\begin{proof}[Proof of Theorem~\\ref{res:sharp-fixed-base}]\nPut $L=F(1/q)$, $e_k=(q;q)_k^{-1}$ and $E=P^{-1}$.\nReplacing every $e_k$ by $E$ gives $E^2(k+1)$ and\n$E^3\\binom{k+2}{2}$, counting nonnegative pairs and triples with\nsum $k$. Terms with one deficit $E-e_i$ give the first correction.\nThat coordinate has two possible positions in a\npair and three in a triple; fixing it at $i$ leaves one pair or\n$k-i+1$ triples. Thus the same total deficit produces a constant\ncorrection in the first convolution and a linear correction in the\nsecond. We calculate it and control its first moment.\n\nWriting $d_k=E-e_k$, the identity\n$e_{k+1}-e_k=q^{k+1}e_{k+1}$ gives\n\\[\n d_k=\\sum_{j>k}q^je_j,\\qquad\n 0\\le d_k\\le\\frac{E q^{k+1}}{1-q}.\n\\]\nThus $\\sum d_k$ and $\\sum k d_k$ converge. To evaluate the former,\nuse Euler's identity\n$\\mathcal E(z)=\\sum_{k\\ge0}e_kz^k=(z;q)_\\infty^{-1}$ for $|z|<1$.\nTonelli's theorem and logarithmic differentiation at $z=q$ give\n\\[\n S:=\\sum_{k\\ge0}d_k\n =\\sum_{j\\ge1}j q^je_j\n =q\\mathcal E'(q)\n =E\\sum_{r\\ge1}\\frac{q^r}{1-q^r}=EL.\n\\]\nIn the double sum, $q^je_j$ occurs once for each $k=0,\\ldots,j-1$,\nwhich explains the factor $j$. Logarithmic differentiation is\njustified by locally uniform convergence on $|z|<1$.\n\nLet $b_k^{(r)}=[z^k]\\mathcal E(z)^r$. The finite-sum formula in\nProposition~\\ref{prop:weight-factorisation} becomes\n\\[\n a_k=P^4(q;q)_k b_k^{(2)}b_k^{(3)}.\n\\]\nWe expand each finite convolution using $e_k=E-d_k$. For two factors,\n\\[\n b_k^{(2)}=E^2(k+1)-2E\\sum_{i=0}^k d_i\n              +\\sum_{i=0}^k d_i d_{k-i}\n =E^2(k+1-2L)+O_q((k+1)q^k).\n\\]\nFor three factors, the terms with one $d_i$ contribute\n$-3E^2\\sum_{i=0}^k(k-i+1)d_i$.\nHere the sum equals $(k+1)S+O_q(1)$ because\n$\\sum i d_i<\\infty$. Terms with two $d_i$ are bounded by\n$3E S^2$, and the term with three is $O_q((k+1)^2q^k)$.\nIt follows that\n\\[\n b_k^{(3)}=E^3\\left(\\binom{k+2}{2}-3L(k+1)\\right)+O_q(1).\n\\]\nDivision by $\\binom{k+2}{2}$ turns $-3L(k+1)$ into the relative\ncorrection $-6L/(k+2)$; the two-fold sum gives $-2L/(k+1)$.\nSince $(q;q)_k=P(1+O_q(q^k))$, the leading constants in $a_k$\ncancel: $P^5E^5=1$. Replacing $(k+2)^{-1}$ by $(k+1)^{-1}$ costs\nonly $O_q((k+1)^{-2})$. The two corrections therefore add, and\nwith $c_k=(k+1)^2(k+2)/2$ we obtain\n\\begin{align*}\n \\frac{a_k}{c_k}\n &=\\left(1-\\frac{2L}{k+1}+O_q(q^k)\\right)\n   \\left(1-\\frac{6L}{k+2}+O_q((k+1)^{-2})\\right)\n   \\left(1+O_q(q^k)\\right)\\\\\n &=1-\\frac{8L}{k+1}+O_q((k+1)^{-2}).\n\\end{align*}\nIn particular $a_{k+1}/a_k\\to1$, and multiplication of finitely\nmany adjacent ratios gives $a_{k+h}/a_k\\to1$ for each fixed $h$.\nThe uniform shift bound follows from\nProposition~\\ref{prop:weight-factorisation}, so\nTheorem~\\ref{thm:geometric-moments} applies.\n\nLet $\\gamma_{\\!E}$ denote Euler's constant and put\n\\begin{equation}\\label{eq:fixed-constant}\n \\mathcal A(q)=e^{-8\\gamma_{\\!E}L}\n \\prod_{k\\ge0}\\left(\\frac{a_k}{c_k}e^{8L/(k+1)}\\right),\n \\qquad K(q)=\\mathcal A(q)\\mathcal M(q)^3.\n\\end{equation}\nFor sufficiently large $k$, $\\log(1+u)=u+O(u^2)$ gives\n$\\log(a_k/c_k)+8L/(k+1)=O_q((k+1)^{-2})$.\nThis tail is summable, and the finitely many initial factors are\npositive by Proposition~\\ref{prop:weight-factorisation}. Thus\n$0<\\mathcal A(q)<\\infty$.\nUsing $\\prod_{k<N}c_k=C_N$, separate the harmonic sum from this\nconvergent remainder:\n\\[\n \\log\\frac{\\prod_{k<N}a_k}{C_N}\n =-8L\\sum_{k<N}\\frac1{k+1}\n  +\\sum_{k<N}\\left(\\log\\frac{a_k}{c_k}+\\frac{8L}{k+1}\\right).\n\\]\nThe second sum tends to $\\log\\mathcal A(q)+8\\gamma_{\\!E}L$.\nUsing $\\sum_{k<N}(k+1)^{-1}=\\log N+\\gamma_{\\!E}+o(1)$,\nthe Euler-constant terms cancel, giving\n$\\prod_{k<N}a_k\\sim\\mathcal A(q)C_NN^{-8L}$.\nSubstitution in \\eqref{eq:geometric-limit} proves the theorem, or\nequivalently\n\\[\n \\log V_N^*=B_N\\log q+\\log C_N+2N\\log P\n             -8L\\log N+\\log K(q)+o(1).\n\\]\n\\end{proof}\n\nThe scales are now separate: the node geometry gives the cubic\nterm $B_N\\log q$; the cubic growth of the weights gives\n$\\log C_N$ of order $N\\log N$; their first relative correction\ngives $-8F(1/q)\\log N$. The other terms are linear and constant.\nIn particular, the power correction does not change the cubic rate.\n\n\\subsection{The first nonzero formal term}\\label{sec:hankel-order}\n\nThe order of a nonzero formal series is the least exponent with\nnonzero coefficient. We now fix $N$ and determine the first term\nin the formal $q$-expansion. The shifted\nterm in \\eqref{eq:last-node-shift} then has an extra factor of $q$.\nMore generally, every nonempty partition has positive displacement\nexponent. Thus a single tuple determines the first formal term,\nalthough all partitions were needed for the fixed-base constant.\n\n\\begin{theorem}[the first nonzero term of the Hankel determinant]\n\\label{res:zudilin-sharp-qorder}\nFor every $N\\ge1$,\n\\[\n \\operatorname{ord}_q V_N^*=\\frac{N(N-1)(2N-1)}6,\n\\]\nand the coefficient of the first nonzero monomial is\n\\[\n [q^{N(N-1)(2N-1)/6}]V_N^*\n   =\\frac{(N!)^2(N+1)!}{2^N}.\n\\]\n\\end{theorem}\n\n\\begin{proof}\nAt $q=0$, the $t=0$ term of $G_q(w)$ is $(1-w^2)/(1-w)^5$, while the\nremaining terms sum to $w/(1-w)^4$. Thus\n\\[\n G_0(w)=\\frac{1+2w}{(1-w)^4},\\qquad a_k(0)=c_k>0.\n\\]\nWe compute modulo $q^{M+1}$ by truncating each moment sum in\n\\eqref{eq:positive-moments} at $k=M$. Since all omitted terms are\ndivisible by $q^{M+1}$, multilinearity leaves the determinant\nunchanged modulo that power. Applying finite Cauchy--Binet and then\nallowing $M$ to increase, we obtain the coefficientwise identity\n\\[\n V_N^*=\\sum_{k_0<\\cdots<k_{N-1}}\n \\left(\\prod_{i<N}a_{k_i}(q)q^{k_i}\\right)\n \\prod_{i<j<N}(q^{k_i}-q^{k_j})^2.\n\\]\nThe order of the summand indexed by $(k_i)$ is\n$\\sum_{i<N}(2N-1-2i)k_i$. Because $k_i\\ge i$ and all the weights\n$2N-1-2i$ are positive, we obtain the unique minimum at $k_i=i$,\nwith order and coefficient\n\\[\n \\sum_{i<N}(2N-1-2i)i=B_N,\\qquad \\prod_{i<N}c_i=C_N.\n\\]\nNo other tuple has this order, so no coefficient from another tuple\ncan cancel it. This argument needs the positive constant terms\n$a_k(0)=c_k$, not nonnegativity of all their formal coefficients.\nFor $N=2$, the pair $(0,1)$ contributes $6q+O(q^2)$,\nand every other increasing pair has order at least two.\n\\end{proof}\n\nThis proves equality in Zudilin's bound\n\\cite[\\S4, pp.~6--7]{zudilin2016}. An alternative argument, which\ncomputes the first nonzero term of every transformed row, is given in\n\\shortrecordlink{erdos1049-rational-base-lambert-reasoning-surface.pdf}\n{long1049-formal-rows}{the companion paper}.\n\n\\section{Rational bases}\n\\label{sec:rational-base-irrationality}\n\nWe now use the 2004 coefficient family. For $U,V\\in\\Z[X]$ of degree\nat most $W$, multiplying $U(a/b)$ and $V(a/b)$ by $b^W$ makes them\nintegers. This is a sufficient clearing factor, not in general the\nsmallest one. Cancelling a common nonconstant polynomial factor first lowers the\ndegree bound. For example, at $3/2$ the pair\n$(X+1)(X+2),(X+1)(X+3)$ gives $(35,45)$ after clearing to degree $2$.\nAfter cancelling $X+1$, clearing to degree $1$ gives $(7,9)$.\nFor Zudilin's forms, let $K_n$ be the original coefficient degree\nand $W_n$ its degree after cancellation; both grow quadratically\nwith $n$. The uncancelled remainder stays between positive constants. The normalising multiplier creates the decay, with\nleading scale $x^{-(K_n-W_n)}$ at fixed $x>1$. Clearing at $x=a/b$\nthen gives\n\\begin{equation}\\label{eq:degree-balance}\n b^{W_n}(a/b)^{-(K_n-W_n)}\n =\\frac{b^{K_n}}{a^{K_n-W_n}}.\n\\end{equation}\nNet degree removal is therefore measured in powers of $a$, while\nthe original degree costs powers of $b$. The remaining logarithmic\nterms are $o(n^2)$, so $K_n/n^2\\to C_1$ and\n$(K_n-W_n)/n^2\\to C_0$ give $C_1\\log b<C_0\\log a$.\nThe proof must establish both the degree estimates and cancellation\nin $\\Z[X]$, before evaluation.\n\nWe use the parameter direction from Zudilin's construction\n\\cite[\\S5, pp.~161--162]{zudilin2004}, together with its thirteen\nintervals and constants $C_1,C_0$. At integer bases his theorem gives\nthe irrationality-exponent bound $C_1/C_0=2.46497868\\ldots$\n\\cite[Thm.~1, p.~154]{zudilin2004}.\nLet $\\psi_1(u)=\\sum_{k\\ge0}(k+u)^{-2}$ for $u>0$, and let\n$\\mathcal I$ consist of the thirteen intervals listed in the proof.\nWe put\n\\begin{equation}\\label{eq:threshold-constants}\n C_1=\\frac{1091}{2},\\qquad\n J=\\sum_{[u,v)\\in\\mathcal I}\\bigl(\\psi_1(u)-\\psi_1(v)\\bigr),\\qquad\n C_0=266-\\frac3{\\pi^2}(225-J).\n\\end{equation}\nThe intervals are disjoint and lie in $[1/14,1)$, so\n$0\\le J\\le\\psi_1(1/14)-\\psi_1(1)<196$.\nTogether with $\\pi>3$, these bounds give $0<C_0<266<C_1/2$.\nThus $\\theta^*=C_0/C_1$ and $\\mu=C_1/C_0$ are positive reciprocal\nconstants. The notation $\\mu_{\\rm irr}(\\xi)$ instead denotes the\nirrationality exponent of a value. The estimates below give a sufficient cutoff; its optimality is unknown.\n\n\n\\begin{theorem}[rational-base region for Zudilin's forms]\n\\label{res:rational-base-threshold}\nLet $a>b\\ge1$ be coprime integers with\n\\[\n\\begin{gathered}\n b^{\\mu}<a,\\qquad\\text{equivalently}\\qquad\n \\frac{\\log b}{\\log a}<\\theta^*,\\\\\n \\theta^*=\\frac{C_0}{C_1}=0.4056830213840605\\ldots,\\\\\n \\mu=\\frac{C_1}{C_0}=2.4649786835749750\\ldots.\n\\end{gathered}\n\\]\nThen $F(a/b)$ is irrational.\n\\end{theorem}\n\n\\noindent Every integer base qualifies, since $b=1$ gives\n$\\log b/\\log a=0$. For each fixed $b>1$, the restriction\n$a>b^{2.46497868\\ldots}$ is much stronger than $a>b$, but still\nadmits infinitely many coprime numerators. It includes $31/4$ and\nits positive integral powers, whose logarithmic ratio is unchanged.\nIt excludes $3/2$, since $\\theta^*<1/2<\\log2/\\log3$.\nAt equality the quadratic exponent vanishes and the estimates give\nno conclusion.\n\n\\begin{proof}[Proof of Theorem~\\ref{res:rational-base-threshold}]\n\\emph{The source forms and polynomial cancellation.}\nFix $n\\ge1$ and set\n\\[\n a_0=14n+1,\\quad a_1=12n+1,\\quad a_2=14n+1,\\quad\\beta=27n+2,\n \\qquad N=15n.\n\\]\nIn this proof $N$ denotes the cyclotomic cutoff. We use the\ncoefficient pair $A_n,B_n$ of the source identities\n\\cite[(8)--(11), pp.~156--157]{zudilin2004}, with $H_n=A_nF-B_n$.\nLet $\\Phi_\\ell$ be the cyclotomic polynomial whose roots are the primitive\n$\\ell$th roots of unity. Put $D_N(X)=\\prod_{\\ell=1}^{N}\\Phi_\\ell(X)$,\n$M_n=266n^2+34n+1$, and\n\\[\n \\Omega_n(X)=\\prod_{\\ell=2}^{N}\\Phi_\\ell(X)^{\\nu_\\ell},\n \\qquad \\nu_\\ell=\\omega(n/\\ell),\n\\]\nwhere the periodic function\n\\[\n\\begin{split}\n\\omega(x)=\\max\\{0,&\\ \\lfloor14x\\rfloor+\\lfloor13x\\rfloor\n                  -\\lfloor12x\\rfloor-\\lfloor15x\\rfloor,\\\\\n                &\\ 2\\lfloor14x\\rfloor-\\lfloor13x\\rfloor\n                  -\\lfloor15x\\rfloor\\}\n\\end{split}\n\\]\nhas period $1$: the coefficients of the floor arguments in each\ndifference have equal sums, $14+13=12+15$ and $2\\cdot14=13+15$.\nOn $[0,1)$ it is zero or one, with support\n\\[\n\\begin{gathered}\n\\,[1/14,1/12),\\ [1/7,1/6),\\ [3/14,1/4),\\ [2/7,1/3),\\\\\n[5/14,2/5),\\ [3/7,7/15),\\ [1/2,8/15),\\ [4/7,3/5),\\\\\n[9/14,2/3),\\ [5/7,11/15),\\ [11/14,4/5),\\\\\n[6/7,13/15),\\ [13/14,14/15).\n\\end{gathered}\n\\]\nThese are the intervals $\\mathcal I$ used to define $J$.\nThe function $\\omega$, these thirteen intervals and the exponents\n$\\nu_\\ell=\\omega(n/\\ell)$ of the source's~(22) are printed at\n\\cite[pp.~161--162]{zudilin2004}.\n\n\nFirst isolate the part that does not create decay. At $x>1$, put\n$q=1/x$. Identity~(9) on p.~156 of the source gives\n\\[\n H_n(x)=\\sum_{t\\ge0}q^{a_0t}\n \\frac{(q^{t+1};q)_{a_1-1}}{(q;q)_{a_1-1}}\n \\frac{(q;q)_{\\beta-a_2-1}}{(q^{a_2+t};q)_{\\beta-a_2}}.\n\\]\nThe denominator length is $\\beta-a_2$, as in the gamma expression~(7)\nand residues~(8) of \\cite[p.~156]{zudilin2004}; the unnumbered display\nof $R(T)$ on that page has length $\\beta-a_2-1$ instead.\nEach finite product lies in $[P,1]$, where $P=(q;q)_\\infty>0$.\nThe $t=0$ term gives the lower bound; summing $q^{a_0t}$ gives\nthe upper bound:\n\\[\n P^2\\le H_n(x)\\le\\frac{P^{-2}}{1-q^{a_0}}.\n\\]\nIn particular $H_n(x)>0$ and $\\log H_n(x)=O_x(1)$.\n\nThe required polynomial integrality is the inclusion in\nZudilin's Lemma~7, display~(23) \\cite[p.~161]{zudilin2004}.\nIts comparison of cyclotomic valuations under six admissible\nparameter permutations gives divisibility in $\\Z[X]$, not merely\nat integer values of $X$.\nIts parameter vector is $n(13,14,12,14,15,13)$, its maximum is $15n$, and\n$\\beta-a_1-a_2=n>0$, which is the positivity condition $s>0$ of the lemma.\nThe conditions~(14) of \\cite[p.~157]{zudilin2004}, $a_1\\le a_2$ and\n$a_1+a_2\\le\\beta\\le a_0+a_2$, hold as well, so the exponent in~(23) is the\ninteger~(16), which equals $M_n$. Lemma~7 therefore gives, for every\n$n\\ge1$,\n\\[\n \\Lambda_n(X)=X^{-M_n}\\frac{D_N(X)}{\\Omega_n(X)}H_n(X)\n            =U_n(X)F(X)-V_n(X),\\qquad U_n,V_n\\in\\mathbb Z[X].\n\\]\nThis is the polynomial conclusion before integer specialisation in the\nsource's~(24), p.~162. The coefficients are the cancelled source coefficients:\n\\[\n U_n=X^{-M_n}(D_N/\\Omega_n)A_n,\\qquad\n V_n=X^{-M_n}(D_N/\\Omega_n)B_n.\n\\]\nIndeed, coefficients of $F$ and $1$ over $\\Q(X)$ are unique because\n$F$ is not a rational function. To see this, we split $F(e^h)$ at $m=1/h$ as $h\\downarrow0$.\nFor $m\\le1/h$, the inequalities\n$y^{-1}-1\\le(e^y-1)^{-1}\\le y^{-1}$ for $0<y\\le1$ give\n$h^{-1}\\sum_{m\\le1/h}m^{-1}+O(h^{-1})$.\nBounding the remaining terms by a geometric tail gives a further\n$O(h^{-1})$, and therefore\n$F(e^h)=h^{-1}\\log(1/h)+O(h^{-1})$, which has no rational-function\npole order at $X=1$. This establishes uniqueness of the coefficient\nfunctions; it assumes nothing about an individual value $F(a/b)$.\nThe cancelled remainder is positive for $x>1$, since $D_N/\\Omega_n$\nis a product of cyclotomic polynomials positive there.\n\n\\emph{The degrees after cancellation.}\nThe Gaussian binomial polynomial\n$\\genfrac{[}{]}{0pt}{}{m}{k}_X=\\prod_{j=1}^k(1-X^{m-k+j})/(1-X^j)$\nis monic of degree $k(m-k)$ for integers $0\\le k\\le m$. The source's~(8) and~(10)\non p.~156 give the coefficient whose degree we need:\n\\[\n A_n(X)=\\sum_{k=a_2}^{\\beta-1}(-1)^{a_1+a_2+k+1}\n X^{a_0k+\\binom{a_1}{2}-\\binom{\\beta-a_2}{2}+\\binom{\\beta-k}{2}}\n \\genfrac{[}{]}{0pt}{}{k-1}{a_1-1}_X\n \\genfrac{[}{]}{0pt}{}{\\beta-a_2-1}{\\beta-k-1}_X.\n\\]\nLet $d_{n,k}$ be the degree of the $k$th summand. Adding the monomial\nexponent and the degrees of the two Gaussian factors gives\n\\[\n \\begin{aligned}\n d_{n,k}\n &=a_0k+\\binom{a_1}{2}-\\binom{\\beta-a_2}{2}+\\binom{\\beta-k}{2}\\\\\n &\\quad +(a_1-1)(k-a_1)+(\\beta-k-1)(k-a_2)\\\\\n &=\\frac{-k^2+80kn+3k-340n^2-26n}{2}.\n \\end{aligned}\n\\]\nHence $d_{n,k+1}-d_{n,k}=40n+1-k>0$ for\n$a_2\\le k\\le\\beta-2=27n$. The last summand alone\ntherefore determines the leading term, giving\n\\[\n K_n:=\\deg A_n=\\frac{1091n^2+81n+2}{2},\\qquad\n W_n:=\\deg U_n=K_n-M_n+\\sum_{\\ell\\le15n}(1-\\nu_\\ell)\\varphi(\\ell).\n\\]\nHere $\\varphi(\\ell)=\\deg\\Phi_\\ell$ is Euler's totient function, and\n$\\nu_1=\\omega(n)=0$. In particular,\n\\[\n K_n-W_n=M_n-\\deg(D_N/\\Omega_n).\n\\]\nThe monomial removes $M_n$ degrees and the surviving cyclotomic\nproduct restores $\\deg(D_N/\\Omega_n)$; their difference is the\nnet removal $K_n-W_n$.\n\nTo determine the other coefficient's degree, fix $n$ and let\n$x\\to\\infty$. The bounds for $H_n$ are uniform for $x\\ge2$, since\n$P\\ge\\prod_{j\\ge1}(1-2^{-j})>0$ and $(1-x^{-a_0})^{-1}\\le2$.\nHence $H_n(x)=O(1)$ and $\\Lambda_n(x)=O(x^{W_n-K_n})$.\nThe unique top summand of $A_n$ has leading coefficient\n$(-1)^{a_1+a_2+\\beta}=(-1)^n$. Since the Gaussian factors and\n$D_N/\\Omega_n$ are monic, $U_n$ has that leading coefficient too.\nMoreover $F(x)=x^{-1}+O(x^{-2})$ and $K_n\\ge2$, so\n\\[\n V_n(x)=U_n(x)F(x)-\\Lambda_n(x)\n       =(-1)^n x^{W_n-1}+O(x^{W_n-2}).\n\\]\nHere $W_n\\ge K_n-M_n=(559n^2+13n)/2>0$. Thus\n$\\deg V_n=W_n-1$, with the same unit leading coefficient as $U_n$.\nFor every prime $p\\mid b$, reduction of the cleared first coordinate gives\n\\[\n b^{W_n}U_n(a/b)\\equiv(-1)^n a^{W_n}\\not\\equiv0\\pmod p.\n\\]\nThus $U_n(a/b)$ has reduced denominator exactly $b^{W_n}$,\nwhich is the least common clearing denominator for the pair.\nAny common integer divisor of the cleared pair is coprime to $b$.\nThis exact cost concerns these coefficients only: common factors\nat other primes and combinations of rows remain separate questions.\n\n\\emph{The quadratic degree limits.}\nWe now keep the base fixed and let $n\\to\\infty$, extracting the\nquadratic part of the net degree removal. These are the cyclotomic\nlimits of \\cite[Lemmas~1--2, p.~155]{zudilin2004}. We use the\nreciprocal intervals, summatory totient estimate and trigamma\nfunction from \\cite[Lemma~1, p.~466]{zudilin2002}, truncating the\nblock sum before passing to the limit.\nThe elementary summatory estimate\n$\\sum_{\\ell\\le y}\\varphi(\\ell)=3y^2/\\pi^2+O(y\\log y)$ gives\n\\[\n \\frac1{n^2}\\sum_{\\ell\\le15n}\\varphi(\\ell)\\longrightarrow\\frac{675}{\\pi^2}.\n\\]\nFor $[u,v)\\in\\mathcal I$, the condition $\\{n/\\ell\\}\\in[u,v)$ is the\ndisjoint union of intervals\n$n/(k+v)<\\ell\\le n/(k+u)$ for $k\\ge0$.\nApply the summatory estimate to finitely many blocks. Instead of\nsumming its errors over all $k$, bound the omitted blocks together:\nfor $k\\ge K\\ge1$, all their indices satisfy $\\ell\\le n/(K+u)$.\nThe bound $\\sum_{\\ell\\le y}\\varphi(\\ell)\\le y^2$ gives\n\\[\n \\frac1{n^2}\\sum_{\\ell\\le n/(K+u)}\\varphi(\\ell)\n \\le\\frac1{(K+u)^2}.\n\\]\nFor $y\\ge1$, sum $\\varphi(\\ell)\\le\\ell$; for $0\\le y<1$ the sum\nis empty. The tail bound is uniform in $n$, so we may let $n$ and\nthen $K$ tend to infinity:\n\\[\n \\frac1{n^2}\\sum_{\\ell\\le15n}\\nu_\\ell\\varphi(\\ell)\n \\longrightarrow\\frac3{\\pi^2}\n \\sum_{[u,v)\\in\\mathcal I}\\sum_{k\\ge0}\n \\left(\\frac1{(k+u)^2}-\\frac1{(k+v)^2}\\right)=\\frac{3J}{\\pi^2}.\n\\]\nHere no contributing index exceeds $14n$, since $u\\ge1/14$.\nConsequently\n\\[\n K_n/n^2\\longrightarrow C_1,\\qquad\n (K_n-W_n)/n^2\\longrightarrow C_0.\n\\]\n\n\\emph{The cleared remainder.}\nIt remains to turn polynomial degrees into values at fixed $x>1$.\nWriting $\\mu_{\\rm Mob}$ for the M\\\"obius function, we have\n\\[\n \\log\\Phi_\\ell(x)-\\varphi(\\ell)\\log x\n =\\sum_{d\\mid\\ell}\\mu_{\\rm Mob}(d)\\log(1-x^{-\\ell/d}).\n\\]\nEach $-\\log(1-x^{-d})$ occurs only for multiples $\\ell$ of $d$,\nat most $15n/d$ times. The total absolute error is therefore at most\n\\[\n 15n\\sum_{d\\ge1}\\frac{-\\log(1-x^{-d})}{d}.\n\\]\nThe series converges because\n$-\\log(1-x^{-d})\\le x^{-d}/(1-x^{-1})$, so the error is $O_x(n)$.\nUsing $0\\le1-\\nu_\\ell\\le1$ and $\\log H_n(x)=O_x(1)$ gives\n\\[\n \\log\\Lambda_n(x)=-(K_n-W_n)\\log x+O_x(n).\n\\]\n\nAt $x=a/b$, the coefficients $b^{W_n}U_n(a/b)$ and\n$b^{W_n}V_n(a/b)$ are integers, and\n\\[\n\\begin{split}\n \\log\\bigl(b^{W_n}\\Lambda_n(a/b)\\bigr)\n &=K_n\\log b-(K_n-W_n)\\log a+O_{a/b}(n)\\\\\n &=\\bigl(C_1\\log b-C_0\\log a\\bigr)n^2+o(n^2).\n\\end{split}\n\\]\nThe quadratic coefficient is negative under the hypothesis.\nIf $F(a/b)=r/s$ with integers $r,s$ and $s>0$, then\n$s b^{W_n}\\Lambda_n(a/b)$ would be a positive integer tending\nto zero. This contradiction uses positivity of the same remainder,\nnot independence of successive forms.\n\\end{proof}\n\n\\begin{corollary}\\label{res:thirtyone-four}\n$F\\bigl((31/4)^r\\bigr)$ is irrational for every integer $r\\ge1$.\n\\end{corollary}\n\\begin{proof}\nThe exact inequalities $31^2<4^5$ and $4^{200}<31^{81}$ give\n$2/5<\\log4/\\log31<81/200$.\nThe first term of each trigamma difference yields\n\\[\n J\\ge\\sum_{[u,v)\\in\\mathcal I}(u^{-2}-v^{-2})\n   =\\frac{2015640690251}{25971865920}.\n\\]\nUsing $\\pi>157/50$ gives the rational lower bound\n\\[\n \\theta^*>\\frac{2359630009523263}{5820307922172744}\n           >\\frac{81}{200}.\n\\]\nTaking a common positive integer power preserves both the logarithmic\nratio and coprimality, so the same criterion applies for every $r\\ge1$.\n\\end{proof}\n\n\\begin{corollary}[an irrationality measure uniform over powers]\n\\label{cor:rational-base-measure}\nFor coprime $a>b\\ge1$ with $\\theta=\\log b/\\log a<\\theta^*$ and every\ninteger $r\\ge1$,\n\\[\n \\mu_{\\rm irr}\\!\\left(F((a/b)^r)\\right)\n \\le\\frac{1-\\theta}{\\theta^*-\\theta}.\n\\]\nHere $\\mu_{\\rm irr}(\\xi)$ is the supremum of the exponents $\\nu$ for which\n$|\\xi-p/q|<q^{-\\nu}$ has infinitely many reduced rational solutions.\nIn particular, $\\mu_{\\rm irr}(F((31/4)^r))<301$ for every $r\\ge1$.\n\\end{corollary}\n\\begin{proof}\nWe use Zudilin's passage from linear forms to an exponent bound\n\\cite[p.~162]{zudilin2004}, at each fixed base $(a/b)^r$.\nTake $U_n,V_n$ and $W_n=\\deg U_n$ from\nTheorem~\\ref{res:rational-base-threshold}. A measure needs more\nthan positive remainders tending to zero: it must bound how small\nthey are relative to their coefficients. The logarithmic remainder\nasymptotic supplies a lower bound; we now bound $U_n$ above.\nIn that construction the coefficient of $F$ before cancellation is a\nsum of $O(n)$ Laurent monomials times two Gaussian binomial polynomials.\nEach Gaussian polynomial has nonnegative coefficients summing to at\nmost $2^{27n+2}$. Thus the sum of the absolute coefficients is\n$\\exp(O(n))$. Its largest exponent is\n$K_n=(1091n^2+81n+2)/2$, so its absolute value at a fixed $x>1$ is at\nmost $x^{K_n}\\exp(O(n))$. The cyclotomic estimate in the same proof\nbounds the normalising multiplier by\n$x^{W_n-K_n}\\exp(O_x(n))$. Multiplication gives\n\\[\n |U_n(x)|\\le x^{W_n}\\exp(O_x(n))\\qquad(x>1\\text{ fixed}).\n\\]\nSet $\\xi=F(a/b)$, $Q_n=b^{W_n}U_n(a/b)$ and\n$P_n=b^{W_n}V_n(a/b)$.  With\n\\[\n \\alpha=(C_1-C_0)\\log a,\\qquad\n \\tau=C_0\\log a-C_1\\log b>0,\n\\]\nwe have $\\log(Q_n\\xi-P_n)=-\\tau n^2+o(n^2)$ and\n$|Q_n|\\le\\exp(\\alpha n^2+o(n^2))$.\n\nHere $q$ denotes an approximation denominator, not the geometric\nbase of Sections~\\ref{sec:geometric-moments} and~\\ref{sec:weights}.\nThe comparison uses the integer $Ap-Bq$, allowing it to vanish.\nFor integers $A,B,p,q$ with $q>0$, if $L=A\\xi-B$ and $2q|L|\\le1$, then\n\\[\n |L|\\le |A|\\,|\\xi-p/q|.\n\\]\nIf $Ap-Bq=0$, this is equality; otherwise\n$1/q\\le |L|+|A|\\,|\\xi-p/q|$ gives the inequality. Thus no rational\n$p/q$ is excluded from the comparison. Fix $0<\\eta<\\tau$ and write\n\\[\n e^{-(\\tau+\\eta)n^2}\\le Q_n\\xi-P_n\\le e^{-(\\tau-\\eta)n^2},\n \\qquad |Q_n|\\le e^{(\\alpha+\\eta)n^2}\n\\]\nfor all sufficiently large $n$. The upper remainder bound determines\nour choice: for a sufficiently large denominator $q$, take\n$n=\\lceil\\sqrt{\\log(2q)/(\\tau-\\eta)}\\rceil$. Then\n$2q(Q_n\\xi-P_n)\\le1$. The preceding integer argument now applies to\n$(A,B)=(Q_n,P_n)$ for every numerator $p$. Also $Q_n\\ne0$: otherwise\n$Q_n\\xi-P_n$ would be an integer strictly between $0$ and $1$.\nThe lower remainder bound and the coefficient bound now give\n\\[\n |\\xi-p/q|\\ge e^{-(\\alpha+\\tau+2\\eta)n^2}\n             =q^{-(\\alpha+\\tau+2\\eta)/(\\tau-\\eta)-o(1)},\n\\]\nsince $n^2=\\log(2q)/(\\tau-\\eta)+O_\\eta(\\sqrt{\\log q}+1)$.\nThe rounding error is $o(\\log q)$, so it does not change the exponent.\nThis choice covers every sufficiently large approximation denominator\n$q$; no monotonicity assumption on the coefficients $Q_n$ is used.\n\nLetting $\\eta\\downarrow0$, we deduce the bound\n$1+\\alpha/\\tau=(1-\\theta)/(\\theta^*-\\theta)$. A common power multiplies\n$\\alpha$ and $\\tau$ by $r$, so the quotient is unchanged, although the\nconstants in the approximation inequality may depend on $r$.\nFor $31/4$, the exact interval bounds in the companion record,\n\\shortrecordlink{erdos1049-rational-base-lambert-reasoning-surface.pdf}{long1049-regionbracket}{Section~2.5},\ngive $\\theta<0.4036982$ and $\\theta^*>0.40568$. The quotient increases\nwith $\\theta$ and decreases with $\\theta^*$ in this range, so\n\\[\n \\frac{1-\\theta}{\\theta^*-\\theta}\n <\\frac{1-0.4036982}{0.40568-0.4036982}\n =\\frac{2981509}{9909}<301.\n\\]\nThe same section gives the rational certificates for these bounds\nand sharper enclosures of the constants.\n\\end{proof}\n\n\\subsection{Comparison with earlier criteria}\nBundschuh and V\\\"a\\\"an\\\"anen's Theorem~2 at $\\alpha=-1$\n\\cite[p.~177]{bv1994} gives irrationality for\n$\\log b/\\log a<\\theta_{\\rm BV}:=1/2-1/\\pi^2$.\nTheir $q$ is the base $a/b>1$, not its reciprocal. Their logarithmic\nderivative is $L_q(z)=\\sum_{j\\ge1}(q^j+z)^{-1}$, so the value at\n$z=\\alpha=-1$ is exactly $F(a/b)$. The rational height is $h(q)=a$,\nand their parameter $\\lambda=\\log h(q)/\\log q$ is\n$1/(1-\\log b/\\log a)$, which gives the displayed cutoff.\nSince $\\pi^2<10$, one has $\\theta_{\\rm BV}<2/5<\\log4/\\log31$, so $31/4$ lies\noutside that sufficient region.  The two sufficient regions differ on\n$[\\theta_{\\rm BV},\\theta^*)$.\nZudilin also notes an extension to noninteger rational bases $p=r/s$\nfor the generalized $q$-logarithm when $\\log|r|>c\\log|s|$, with $c>0$\ncomputable but unspecified \\cite[Sec.~2, p.~4]{zudilin2016}.\nThe value $c=\\mu$ here comes from the 2004 forms\n\\cite[p.~162]{zudilin2004}, not from evaluating the unspecified\nconstant in the 2016 construction.\n\nThe positive-remainder estimates require $x>1$, so they give no\nstatement for negative bases. The series diverges at $x=1$.\nThe companion's height-criterion section also applies Bundschuh\nand V\\\"a\\\"an\\\"anen's cited theorem at $7/2$.\n\n\\section{Further questions}\\label{sec:open}\n\nThe determinant asymptotic still needs a coefficient-denominator\nestimate to give an integer contradiction. The companion's\n\\shortrecordlink{erdos1049-rational-base-lambert-reasoning-surface.pdf}\n{long1049-countermodel}{rational-value model}\nhas the same formal orders, leading coefficients and type of\nfixed-base power correction, but a rational target value. Its\ndenominator calculation shows why clearing need not yield small\nnonzero integer forms. The model and its denominator argument have ordinary proofs only\nand are not used above.\n\nThe approximants' polynomial coefficients pose a separate moment\nproblem. The normalisation matters in Van Assche's little-$q$-Legendre\nconstruction \\cite[\\S3]{vanassche2001} and the multiple-orthogonality\nconstruction of Postelmans and Van Assche\n\\cite[\\S2]{postelmansvanassche2007}. In the 2016 family, the companion\npaper proves positivity of the first matrix of a coefficient pencil\nfor sizes $N\\le8$, with real, interlacing roots below $F(p)$ for\n$p>1$. The shifted minors and coefficientwise positivity are supported\nby the stated finite polynomial computations. The\n\\shortrecordlink{erdos1049-rational-base-lambert-reasoning-surface.pdf}\n{long1049-coefficients}{coefficient-moment section}\ngives the definitions and the precise ranges. Positivity for every\n$N$ and convergence of the largest root remain open. The general\nmoment criteria of Wang and Zhu \\cite{wangzhu2016} and Sokal and\nWalrad \\cite{sw2024}, Berg's factorial-power example\n\\cite[Theorem~5.1]{berg2007}, and the quadrature construction of Golub\nand Welsch \\cite{golubwelsch1969} provide the relevant comparisons.\nUsing a recurrence from another family would require a proof that it\nholds for these coefficients. The root-of-unity method of\nKrattenthaler, Rochev, V\\\"a\\\"an\\\"anen and Zudilin \\cite{krvz2009}, for\nexample, concerns a separate construction.\n\n\\subsection{Congruences and divided forms}\\label{sec:round8-transfer-boundaries}\n\nFor $Q\\in\\Z[X]$ of degree at most $W$, the homogenised value\n$H_W(Q)=2^WQ(3/2)$ is integral. The companion treats its congruences\nat powers of $2$ and $3$, including\n\\shortrecordlink{erdos1049-rational-base-lambert-reasoning-surface.pdf}\n{long1049-arithmetic}{integer rescaling} and\n\\shortrecordlink{erdos1049-rational-base-lambert-reasoning-surface.pdf}\n{long1049-endpoints}{endpoint congruences and signed sums}, with the\nrank-one case. Smith normal form counts the residue image\n\\cite[Theorems~2.3--2.4]{stanley2016}. To obtain a small nonzero\nremainder after division, the count must also distinguish nearby\nreal values from repeated ones.\n\nLet $M$ integral rows $(A_j,B_j)$ have remainders\n$e_j=A_jF(3/2)-B_j$, and let $D$ be a positive integer.\nSuppose the subset sums $\\sum_{j\\in I}(A_j,B_j)$ take $Q$ values in\n$(\\mathbb Z/D\\mathbb Z)^2$, with at most $k$ subsets giving any\nfixed real remainder within a residue class. Put $T=\\sum_j|e_j|$.\nThe least and greatest subset remainders are\n\\[\n \\sum_{e_j<0}e_j\\quad\\text{and}\\quad\\sum_{e_j>0}e_j;\n\\]\ntheir difference is $T$. For a positive integer $n$, divide this\nwhole range into half-open intervals of width $D/n$, starting at\nits least value. At most $\\lfloor nT/D\\rfloor+1$ intervals suffice,\nincluding a last interval when the greatest value is an endpoint.\nIf every residue--interval pair contained only one remainder value,\nit would contain at most $k$ subsets. Hence\n\\begin{equation}\\label{eq:quantitative-selector-budget}\n 2^M>Qk\\left(\\left\\lfloor\\frac{nT}{D}\\right\\rfloor+1\\right)\n\\end{equation}\nforces two different remainders in one residue--interval pair.\nSubtracting the subset sums and dividing by $D$ gives an integral\nlinear form of nonzero absolute value less than $1/n$: the residue\ncondition gives integrality, the interval gives smallness, and the\nmultiplicity bound prevents a zero difference. The\n\\shortrecordlink{erdos1049-rational-base-lambert-reasoning-surface.pdf}\n{long1049-selectors}{quantitative lemma and its proof}\nstate the finite-set result independently of Lambert series.\nDistinct input rows do not rule out equal subset remainders; $k$\nbounds repetitions within a residue class. No family of bounds on\n$Q,T,k$ satisfying \\eqref{eq:quantitative-selector-budget} for\narbitrarily large $n$ at $3/2$ is supplied here.\n\nThe elementary integer-base method provides a different comparison.\nVandehey \\cite{vandehey2013} and Duverney and Tachiya\n\\cite{duverneytachiya2019} use integer-base expansions. At a\nnoninteger rational base the\n\\href{https://github.com/wcook04/plectis-erdos/blob/7380b7871687b6bcc41ca0143c61f232e8af6500/lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean\\#L187}{cleared-tail recurrence}\nhas a growing denominator contribution. The\n\\shortrecordlink{erdos1049-rational-base-lambert-reasoning-surface.pdf}\n{long1049-clearing}{clearing and tail-recurrence calculations}\nin the companion paper include the\n\\href{https://github.com/wcook04/plectis-erdos/blob/7380b7871687b6bcc41ca0143c61f232e8af6500/lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean\\#L155}{coordinatewise obstruction at $3/2$}.\nThey make no assertion about other approximation families. At the\nfunction level, Bell and Smertnig \\cite[Theorem~1.3]{bellsmertnig2026}\nexclude Mahler equations for the divisor generating series, while\nleaving the arithmetic of a single rational argument undecided.\n\n\\paragraph{A construction problem.}\\label{prob:kernel}\nThe unrestricted request for primitive polynomial rows and a signed\ncombination with divided remainder strictly between $0$ and $1/n$\nis equivalent to irrationality of the target; the companion's\n\\shortrecordlink{erdos1049-rational-base-lambert-reasoning-surface.pdf}\n{long1049-open-construction}{section on divided linear forms}\nstates and proves the equivalence. A more specific problem must\nprescribe a family: for the 2004 forms, the question is whether a\nparameter direction improves $\\theta^*$. The\n\\shortrecordlink{erdos1049-rational-base-lambert-reasoning-surface.pdf}\n{long1049-region-extensions}{all-base degree restriction}\nbounds the degree cutoff by $1/2$. It assumes one polynomial family,\nwith common leading constants in its degree, coefficient-height and\nremainder estimates at every fixed real base $x>1$; errors may depend\non $x$. A cutoff above $1/2$ would make remainder decay outpace\ncoefficient growth at a sufficiently large fixed integer base.\nConsecutive integer row determinants would then tend to zero and\neventually vanish, forcing the rational row ratios to stabilise.\nThe remainders would be either zero or bounded away from zero,\ncontradicting nonzero decay. Estimates only at $3/2$ do not permit\nthat choice of base.\n\nUnder the same hypotheses, the\n\\shortrecordlink{erdos1049-rational-base-lambert-reasoning-surface.pdf}\n{long1049-no-decay}{nondecay corollary}\nexcludes decay of the denominator-cleared, undivided forms whenever\n$b<a<b^2$, including $3/2$, without assuming a quadratic degree\nlimit. If that limit exists, their absolute values tend to infinity\nthroughout this strict region. Further division by base-dependent\ncontent and families outside these hypotheses remain unclassified,\nas does equality in the cutoff of\nSection~\\ref{sec:rational-base-irrationality}.\n\n\\appendix\n\\section{Verification and reproducibility}\\label{app:index}\n\nThe \\href{\\evidencerecordurl}{evidence record} lists Lean proofs for\nall seven numbered results and Comparator checks for six, with\ntheir checked revisions. Beside the \\href{\\sourceurl}{main source revision},\nTheorem~\\ref{thm:geometric-moments} uses the later\n\\href{\\laterepobase/lean/ErdosProblems/Erdos1049/PaperCompleteR21/GeometricUniversality.lean\\#L1324}{geometric-moment proof};\nits Comparator association remains pending. Neither the broader\nratio-limit argument nor the rational-value model is a dependency;\nboth have ordinary proofs only. This expository revision changes no\nresult statement and has no fresh Lean or Comparator run or\nindependent review.\n\\evidenceparagraph{\\evidencerecordurl}\n\n\\paragraph{Data availability.}\nThe cited source revision contains the Lean sources, toolchain and\nlibrary manifest. The companion paper supplies the longer algebraic\narguments and the precise scope of the finite computations.\n\n\\paragraph{Funding and competing interests.}\nThis work received no external funding. The author declares no\ncompeting interests.\n\n\\paragraph{Acknowledgements.}\nThe problem numbering follows the catalogue maintained by Thomas\nBloom~\\cite{erdosproblems}.\n\n\\begin{thebibliography}{99}\n\\small\n\\raggedright\n\\setlength{\\itemsep}{2pt}\n\\bibitem{erdos1988} P.~Erd\\H{o}s, \\emph{On the irrationality of certain series:\n  problems and results}, in A.~Baker (ed.), \\emph{New Advances in Transcendence\n  Theory}, Cambridge UP, 1988, pp.~102--109,\n  doi:\\href{https://doi.org/10.1017/CBO9780511897184.009}{10.1017/CBO9780511897184.009}.\n\\bibitem{bv1994} P.~Bundschuh and K.~V\\\"a\\\"an\\\"anen,\n  \\href{https://numdam.org/item/CM_1994__91_2_175_0.pdf}{\\emph{Arithmetical\n  investigations of a certain infinite product}},\n  Compositio Math. \\textbf{91} (1994), no.~2, 175--199.\n\\bibitem{zudilin2002} W.~Zudilin, \\emph{Remarks on irrationality of\n  \\(q\\)-harmonic series}, Manuscripta Math. \\textbf{107} (2002), no.~4,\n  463--477,\n  doi:\\href{https://doi.org/10.1007/s002290200249}{10.1007/s002290200249}.\n\\bibitem{zudilin2004} W.~Zudilin,\n  \\href{https://geodesic.mathdoc.fr/articles/10.4064/aa111-2-4/}{\\emph{Heine's\n  basic transform and a permutation group for \\(q\\)-harmonic series}},\n  Acta Arith. \\textbf{111} (2004), no.~2, 153--164,\n  doi:\\href{https://doi.org/10.4064/aa111-2-4}{10.4064/aa111-2-4}.\n  Page references are to the printed journal pages.\n\\bibitem{zudilin2016} W.~Zudilin,\n  \\href{https://arxiv.org/abs/1601.02688v2}{\\emph{On the irrationality of\n  generalized \\(q\\)-logarithm}}, arXiv:1601.02688; Res. Number Theory\n  \\textbf{2} (2016), Art.~15,\n  doi:\\href{https://doi.org/10.1007/s40993-016-0042-x}{10.1007/s40993-016-0042-x}.\n  Page references are to arXiv:1601.02688v2.\n  The remark that the results extend to non-integer \\(p=r/s\\), \\(|p|>1\\), under\n  an assumption \\(\\log|r|>c\\log|s|\\) for a computable \\(c>0\\), is in Section~2,\n  p.~4, in the paragraph beginning ``Finally, we remark''; no value of \\(c\\) is\n  computed there, and the remark is made for the generalized \\(q\\)-logarithm of\n  that paper.\n\\bibitem{stanley2016} R.~P. Stanley, \\emph{Smith normal form in combinatorics},\n  J. Combin. Theory Ser. A \\textbf{144} (2016), 476--495,\n  doi:\\href{https://doi.org/10.1016/j.jcta.2016.06.013}{10.1016/j.jcta.2016.06.013};\n  arXiv:\\href{https://arxiv.org/abs/1602.00166v1}{1602.00166v1}.\n  Page references are to arXiv:1602.00166v1.\n\\bibitem{zudilin2017det} W.~Zudilin, \\emph{A determinantal approach to\n  irrationality}, Constr. Approx. \\textbf{45} (2017), no.~2, 301--310,\n  doi:\\href{https://doi.org/10.1007/s00365-016-9333-7}{10.1007/s00365-016-9333-7};\n  arXiv:\\href{https://arxiv.org/abs/1507.05697v1}{1507.05697v1}.\n  Page and equation references are to arXiv:1507.05697v1.\n\\bibitem{erdosproblems} T.~F. Bloom,\n  \\href{https://www.erdosproblems.com/1049}{\\emph{Erd\\H{o}s Problem \\#1049}},\n  \\texttt{erdosproblems.com/1049}. Accessed 28 July 2026, when the page\n  displayed ``last edited 28 September 2025''.\n\\bibitem{bellsmertnig2026} J.~Bell and D.~Smertnig,\n  \\href{https://arxiv.org/abs/2603.23456v1}{\\emph{Mahler series with\n  multiplicative coefficient sequences}}, arXiv:2603.23456v1,\n  24 March 2026.  Theorem~1.3 is on pp.~2--3; its stated consequences on\n  p.~3 include that the divisor and totient generating series are not\n  $k$-Mahler for any $k\\ge2$.\n\\bibitem{vandehey2013} J.~Vandehey,\n  \\href{https://arxiv.org/abs/1206.0340v1}{\\emph{On an incomplete argument of\n  Erd\\H{o}s on the irrationality of Lambert series}}, Integers\n  \\textbf{13} (2013), Paper A58. Page references are to\n  arXiv:1206.0340v1 (2012).\n\\bibitem{duverneytachiya2019} D.~Duverney and Y.~Tachiya, \\emph{Refinement of\n  the Chowla--Erd\\H{o}s method and linear independence of certain Lambert\n  series}, Forum Math. \\textbf{31} (2019), no.~6, 1557--1566,\n  doi:\\href{https://doi.org/10.1515/forum-2018-0299}{10.1515/forum-2018-0299}.\n  Page references are to the\n  \\href{https://danielduverney.fr/documents/theorie-des-nombres/DuverneyTachiya190522.pdf}{authors'\n  version}.\n\\bibitem{vanassche2001} W.~Van Assche,\n  \\href{https://arxiv.org/abs/math/0101187v1}{\\emph{Little \\(q\\)-Legendre\n  polynomials and irrationality of certain Lambert series}},\n  Ramanujan J. \\textbf{5} (2001), no.~3, 295--310,\n  doi:\\href{https://doi.org/10.1023/A:1012930828917}{10.1023/A:1012930828917}.\n  Page references are to arXiv:math/0101187v1.\n\\bibitem{postelmansvanassche2007} K.~Postelmans and W.~Van Assche,\n  \\href{https://arxiv.org/abs/math/0604312v1}{\\emph{Irrationality of\n  $\\zeta_q(1)$ and $\\zeta_q(2)$}}, J. Number Theory \\textbf{126} (2007),\n  no.~1, 119--154,\n  doi:\\href{https://doi.org/10.1016/j.jnt.2006.11.011}{10.1016/j.jnt.2006.11.011}.\n  Page references are to arXiv:math/0604312v1 (2006).\n\\bibitem{krvz2009} C.~Krattenthaler, I.~Rochev, K.~V\\\"a\\\"an\\\"anen and\n  W.~Zudilin, \\emph{On the non-quadraticity of values of the\n  \\(q\\)-exponential function and related \\(q\\)-series}, Acta Arith.\n  \\textbf{136} (2009), no.~3, 243--269,\n  doi:\\href{https://doi.org/10.4064/aa136-3-4}{10.4064/aa136-3-4};\n  arXiv:\\href{https://arxiv.org/abs/0812.2921v1}{0812.2921v1}.\n  Page references are to arXiv:0812.2921v1.\n\\bibitem{wangzhu2016} Y.~Wang and B.-X.~Zhu,\n  \\href{https://arxiv.org/abs/1612.04114v1}{\\emph{Log-convex and Stieltjes\n  moment sequences}}, Adv. Appl. Math. \\textbf{81} (2016), 115--127,\n  doi:\\href{https://doi.org/10.1016/j.aam.2016.06.008}{10.1016/j.aam.2016.06.008}.\n  Page references are to arXiv:1612.04114v1.\n\\bibitem{berg2007} C.~Berg, \\emph{On powers of Stieltjes moment sequences, II},\n  J. Comput. Appl. Math. \\textbf{199} (2007), 23--38;\n  arXiv:\\href{https://arxiv.org/abs/math/0412340v1}{math/0412340v1}.\n  Theorem references use the preprint; Theorem~5.1 treats factorial powers.\n\\bibitem{sw2024} A.~D. Sokal and J.~Walrad,\n  \\emph{Continued-fraction characterization of Stieltjes moment sequences\n  with support in $[\\xi,\\infty)$},\n  \\href{https://arxiv.org/abs/2404.12131v1}{arXiv:2404.12131v1}, 2024.\n  The classical Stieltjes criterion is recalled on pp.~1--2.\n\\bibitem{golubwelsch1969} G.~H. Golub and J.~H. Welsch,\n  \\emph{Calculation of Gauss Quadrature Rules}, Math. Comp. \\textbf{23}\n  (1969), no.~106, 221--230,\n  doi:\\href{https://doi.org/10.1090/S0025-5718-69-99647-1}{10.1090/S0025-5718-69-99647-1}.\n\\bibitem{vinroot2010} C.~R. Vinroot,\n  \\href{https://arxiv.org/abs/1011.0984}{\\emph{Multivariate Rogers--Szeg\\H{o}\n  polynomials and flags in finite vector spaces}}, arXiv:1011.0984v1,\n  3 November 2010, abstract accessed 20 September 2026.  Cited for the\n  identification of the sum of all $q$-multinomial coefficients of fixed\n  degree and length with a flag count, and for its recursion generalising the\n  Galois numbers.  The factorisation of the moment weights is proved here.\n\\end{thebibliography}\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":59240,"sha256":"sha256:7d9bf99ea6b7e699659aa4cf7cf6f2bd9edcfcc08b518346472d1df9f7fb22e7","cite_as":"Plectis reader corpus, paper:erdos-1049-rational-base-lambert","source_repository":"plectis-erdos","source_path":"paper/1049/erdos-1049-rational-base-lambert.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:088d36d7ba3d3ee0cee2df99e8915927999698f19889f8ea0b8d55b3721ec778","byte_count":6923,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% This is the only copy of the shared preamble.  The eight short notes, the\n% long records assembled from paper/reasoning-parts other than #249 and #257\n% (which carry their own preambles), and the synthesis paper include it with\n% \\input{problem-note-preamble}; paper/Makefile builds from this directory\n% with the directory on the tectonic search path.\n%\n% Shared preamble for the Erdős Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erdős problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{8e58c395e9807ef3374d7365c62cf04d20ee6331}\n\\newcommand{\\commitshort}{9e231ce4371f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-erdos/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-erdos/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\usepackage{longtable}  % the notes' declaration tables; loaded here so the assembled records inherit it\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n% ---------------------------------------------------------------------------\n% Evidence marks.  paper-evidence.tex places \"Lean\" and \"Comparator\" links in\n% the margin beside each result from the paper's generated file\n% paper/evidence/<paper>.tex, which each paper inputs after this preamble.\n% ---------------------------------------------------------------------------\n\\input{paper-evidence}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Production provenance.  The wording is fixed by the corpus provenance\n% contract and reproduced verbatim in every manuscript in this repository, so\n% the papers cannot drift into different accounts of how they were made.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erdős problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every Lean mark in the margin points to propositions that the\n  pinned Lean kernel accepts, with no \\texttt{sorry}, no added axiom, and no\n  unchecked evaluation.  That claim concerns the formal statements only; it\n  says nothing about their mathematical interest, their novelty, or the\n  original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n\n% Compact opening card for the separately stated Comparator theorem selected\n% from each problem note.  The prose supplied by each note owns the\n% contribution and relevance judgements; the shared final sentence states the\n% mechanical assurance boundary once and points to the public evidence owner.\n\\newcommand{\\externalreviewobject}[3]{%\n  \\begin{quote}\\small\n  \\textbf{Contribution.} #1\\par\n  \\textbf{Relation to the open problem.} #2\\par\n  \\textbf{Executable review object.} #3  The repository's\n  external-verification job compares these exact Lean propositions with\n  separately declared challenge statements and an axiom budget, then asks\n  Lean's kernel to check the submitted proofs.  The\n  \\href{https://github.com/wcook04/plectis-erdos/blob/ca0e13f8acf5ccf48506e4bdb870953d3a0856fa/formalization.yaml}%\n  {formalisation manifest} and the commit-bound CI receipt record that check;\n  they do not assess novelty, significance, or whether the original problem is\n  solved.\n  \\end{quote}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  Under\n% Tectonic (XeTeX) the text and mathematics are set in TeX Gyre Pagella from\n% the bundled OpenType fonts, with a Heros sans range for \\mathsf and\n% Inconsolata for monospace; the pdflatex fallback keeps the same Palatino\n% design through newpx.  The A4 text block uses a classical measure with\n% quiet running heads and restrained ink.  Link colours: steel blue for\n% external (Lean-source and web) targets, deep burgundy for internal\n% cross-references and citations.\n\\NeedsTeXFormat{LaTeX2e}\n\\ProvidesPackage{paper-house-style}[2026/07/19 Pagella house style for the papers]\n\n\\RequirePackage{iftex}\n\\RequirePackage{amsmath}\n\\RequirePackage{amsthm}\n\\ifpdftex\n  \\RequirePackage[T1]{fontenc}\n  \\RequirePackage[utf8]{inputenc}\n  \\RequirePackage{amssymb}\n  \\RequirePackage{newpxtext}\n  \\RequirePackage{newpxmath}\n\\else\n  \\RequirePackage{fontspec}\n  \\RequirePackage{unicode-math}\n  \\setmainfont{texgyrepagella}[Extension=.otf,UprightFont=*-regular,\n    BoldFont=*-bold,ItalicFont=*-italic,BoldItalicFont=*-bolditalic]\n  \\setmathfont{texgyrepagella-math.otf}\n  \\setmathfont{texgyreheros-regular.otf}[range=sfup,Scale=MatchLowercase]\n  \\setmonofont{Inconsolatazi4}[Extension=.otf,UprightFont=*-Regular,\n    BoldFont=*-Bold,Scale=MatchLowercase]\n\\fi\n\n\\RequirePackage[a4paper,textwidth=418pt,textheight=636pt,hcentering,\n  top=68pt,headheight=15pt,headsep=20pt,footskip=34pt]{geometry}\n\\RequirePackage{microtype}\n\\linespread{1.045}\n\\clubpenalty=10000 \\widowpenalty=10000 \\tolerance=1600\n\n\\RequirePackage[dvipsnames,table]{xcolor}\n\\definecolor{linkexternal}{HTML}{275D8C}\n\\definecolor{linkinternal}{HTML}{6E2B2B}\n\\definecolor{papertableheader}{HTML}{E3EAF0}\n\\definecolor{papertablerow}{HTML}{F2F5F7}\n\\definecolor{papertablerule}{HTML}{7D8991}\n\\definecolor{paperresultrule}{HTML}{315B72}\n\n% Stable cross-document links between sibling PDFs. A named destination is\n% more durable than a page number: sections may move while their semantic role\n% remains the same. All public PDFs are copied to one directory at release.\n\\newcommand{\\papersectiontarget}[1]{\\hypertarget{#1}{}}\n\\newcommand{\\papersectionlink}[3]{\\href{#1\\##2}{#3}}\n\n\\RequirePackage{titlesec}\n\\titleformat{\\section}{\\Large\\bfseries}{\\thesection}{0.85em}{}\n\\titlespacing*{\\section}{0pt}{2.6ex plus .8ex minus .3ex}{1.5ex plus .3ex}\n\\titleformat{\\subsection}{\\large\\bfseries}{\\thesubsection}{0.8em}{}\n\\titlespacing*{\\subsection}{0pt}{2.1ex plus .6ex minus .2ex}{1.1ex plus .2ex}\n\\titlespacing*{\\paragraph}{0pt}{1.6ex plus .5ex minus .2ex}{0.85em}\n\n\\RequirePackage{fancyhdr}\n\\newcommand{\\paperrunninghead}{}\n\\newcommand{\\runninghead}[1]{\\renewcommand{\\paperrunninghead}{#1}}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\fancyhead[L]{\\small\\itshape\\paperrunninghead}\n\\fancyhead[R]{\\small\\thepage}\n\\renewcommand{\\headrulewidth}{0pt}\n\\fancypagestyle{plain}{\\fancyhf{}\\renewcommand{\\headrulewidth}{0pt}}\n\n\\RequirePackage{caption}\n\\captionsetup{font=small,labelfont=bf,skip=9pt,width=.92\\linewidth}\n\\RequirePackage{etoolbox}\n\\RequirePackage{pdflscape}\n\\renewcommand{\\arraystretch}{1.20}\n\\newcommand{\\papertablehead}{\\rowcolor{papertableheader}}\n\\newcommand{\\papertablebody}{\\rowcolors{2}{white}{papertablerow}}\n\\AtBeginEnvironment{tabular}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\\AtBeginEnvironment{longtable}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\n% A compact front-matter panel for the result a scanning reader should retain.\n% It is deliberately short and unbreakable: manuscripts should state one\n% result, its evidence, and its limit here, then supply the argument below.\n\\newenvironment{paperresult}[1]{%\n  \\par\\addvspace{0.7em}\n  \\begin{center}\\begin{minipage}{0.94\\linewidth}\n    {\\color{paperresultrule}\\rule{\\linewidth}{0.8pt}}\\par\\vspace{4pt}\n    {\\small\\sffamily\\bfseries\\color{paperresultrule}#1}\\par\\smallskip\n    \\small\\color{black}\n}{%\n    \\par\\vspace{4pt}{\\color{paperresultrule}\\rule{\\linewidth}{0.45pt}}\n    \\end{minipage}\\end{center}\\par\\addvspace{0.5em}\n}\n\n% Shared diagram grammar.  Reader-facing figures use the same restrained\n% hierarchy as the surrounding page: a short eyebrow, a substantial title,\n% one compact explanatory line, and an optional evidence badge.  The styles\n% are installed only when a manuscript loads TikZ.\n\\AtBeginDocument{%\n  \\@ifpackageloaded{tikz}{%\n    \\tikzset{\n      paper box/.style={draw=gray!48, line width=.45pt, rounded corners=3pt,\n        fill=white, align=left, inner xsep=7pt, inner ysep=6pt,\n        font=\\small},\n      paper card/.style={paper box, minimum height=13mm},\n      paper stage/.style={paper box, align=center, minimum height=14mm},\n      paper arrow/.style={-{Stealth[length=2.15mm,width=1.45mm]},\n        line width=.7pt, draw=gray!72!black},\n      paper feedback/.style={paper arrow, densely dashed,\n        draw=Violet!72!black},\n      paper eyebrow/.style={font=\\scriptsize\\sffamily\\bfseries,\n        text=gray!70!black},\n      paper label/.style={font=\\scriptsize\\sffamily,\n        text=gray!72!black, fill=white, inner sep=1.5pt},\n      paper badge/.style={rounded corners=2pt, inner xsep=4pt, inner ysep=2pt,\n        font=\\scriptsize\\sffamily\\bfseries},\n      paper boundary/.style={densely dashed, line width=.55pt,\n        draw=gray!62}\n    }%\n  }{}%\n}\n\n% Front matter: title block with an optional italic subtitle, small-caps\n% author, and a hairline rule; the abstract is set small with a small-caps\n% head and symmetric indents.\n\\newcommand{\\subtitle}[1]{\\gdef\\@papersubtitle{#1}}\n\\renewenvironment{abstract}{%\n  \\vspace{0.1em}\\small\n  \\begin{center}{\\normalsize\\scshape Abstract}\\end{center}\\vspace{-0.5em}\n  \\list{}{\\leftmargin 1.3em\\rightmargin\\leftmargin\\listparindent 0pt\n    \\parsep 0.32em\\itemindent 0pt}\\item\\relax}\n  {\\endlist\\vspace{0.5em}}\n\\def\\@maketitle{%\n  \\newpage\\null\n  \\vspace*{-2.6em}%\n  \\begin{center}%\n    {\\LARGE\\bfseries\\@title\\par}%\n    \\vskip 0.55em\n    {\\large\\itshape\\@papersubtitle\\par}%\n    \\vskip 1.1em\n    {\\scshape\\@author\\textsuperscript{*}\\par}%\n    \\vskip 0.25em\n    {\\small\\@date\\par}%\n    \\vskip 0.8em\n    \\rule{6.5em}{0.4pt}%\n  \\end{center}%\n  \\vskip 0.9em}\n\n% Production provenance, set at the foot of the first page.\n%\n% A reader who meets a single personal byline ordinarily infers that the named\n% author drafted the sentences.  For these papers that inference is false, and\n% a declaration reached only in the back matter would leave it standing for the\n% length of the paper.  The note therefore appears on page one, small and\n% visually subordinate to the title and abstract: enough to prevent the false\n% inference, not enough to make the production method the subject.\n%\n% It carries no mark, because it qualifies the whole manuscript rather than any\n% one passage.  \\footnotetext does not advance the footnote counter, and the\n% emptied \\thefootnote is confined to a group, so numbered footnotes elsewhere\n% in the paper are unaffected.\n%\n% The wording is not authored here.  It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"},{"record_type":"paper_source_dependency","tex_reference":"evidence/erdos-1049-rational-base-lambert","source_path":"paper/evidence/erdos-1049-rational-base-lambert.tex","media_type":"text/x-tex","sha256":"sha256:c4e6515515effaf60879a0a41f057fed821263558b84d7840fcca132c4946e83","byte_count":4960,"licence":"CC-BY-4.0","body_format":"latex","body":"% Generated by scripts/paper_evidence.py from docs/paper_lean_coverage.json and the\n% Comparator replay receipts under evidence/comparator/.  Do not edit by hand.\n\\newcommand{\\evidencerecordurl}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md}\n\\DeclareResultEvidenceName{res:sharp-fixed-base}{Theorem 1.1}\n\\DeclareResultEvidenceRecord{res:sharp-fixed-base}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md\\#res-sharp-fixed-base}{exact}{compared}\n\\DeclareResultEvidence{res:sharp-fixed-base}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperCompleteR21/SharpFixedBaseShort.lean\\#L53}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md\\#res-sharp-fixed-base-comparator}\n\\DeclareResultEvidenceName{thm:geometric-moments}{Theorem 2.1}\n\\DeclareResultEvidenceRecord{thm:geometric-moments}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md\\#thm-geometric-moments}{exact_or_stronger}{pending}\n\\DeclareResultEvidence{thm:geometric-moments}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperCompleteR21/GeometricUniversality.lean\\#L1324}{}\n\\DeclareResultEvidenceName{prop:weight-factorisation}{Proposition 3.1}\n\\DeclareResultEvidenceRecord{prop:weight-factorisation}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md\\#prop-weight-factorisation}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{prop:weight-factorisation}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperCompleteR21/RogersFactorisationAnalytic.lean\\#L1312}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md\\#prop-weight-factorisation-comparator}\n\\DeclareResultEvidenceName{res:zudilin-sharp-qorder}{Theorem 3.2}\n\\DeclareResultEvidenceRecord{res:zudilin-sharp-qorder}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md\\#res-zudilin-sharp-qorder}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{res:zudilin-sharp-qorder}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md\\#res-zudilin-sharp-qorder}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md\\#res-zudilin-sharp-qorder-comparator}\n\\DeclareResultEvidenceName{res:rational-base-threshold}{Theorem 4.1}\n\\DeclareResultEvidenceRecord{res:rational-base-threshold}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md\\#res-rational-base-threshold}{exact}{compared}\n\\DeclareResultEvidence{res:rational-base-threshold}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md\\#res-rational-base-threshold}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md\\#res-rational-base-threshold-comparator}\n\\DeclareResultEvidenceName{res:thirtyone-four}{Corollary 4.2}\n\\DeclareResultEvidenceRecord{res:thirtyone-four}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md\\#res-thirtyone-four}{exact}{compared}\n\\DeclareResultEvidence{res:thirtyone-four}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean\\#L117}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md\\#res-thirtyone-four-comparator}\n\\DeclareResultEvidenceName{cor:rational-base-measure}{Corollary 4.3}\n\\DeclareResultEvidenceRecord{cor:rational-base-measure}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md\\#cor-rational-base-measure}{exact}{compared}\n\\DeclareResultEvidence{cor:rational-base-measure}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md\\#cor-rational-base-measure}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos-1049-rational-base-lambert.md\\#cor-rational-base-measure-comparator}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/1049/erdos-1049-rational-base-lambert.tex","section_outline":["Introduction","Geometric moment determinants","The weights of Zudilin's moments","Positive weights","The coefficient asymptotic","The first nonzero formal term","Rational bases","Comparison with earlier criteria","Further questions","Congruences and divided forms","Verification and reproducibility"]},"erdos1049-rational-base-lambert-reasoning-surface":{"paper_id":"erdos1049-rational-base-lambert-reasoning-surface","title":"Geometric Moments and Rational Lambert Values: Proofs and Further Results","form":"Reasoning surface","length_class":"long","pages":80,"subject":"Erdős #1049","subject_kind":"problem","problem_id":"erdos_1049","landing_role":"mathematics","question":"How are the sharp fixed-base Hankel asymptotic and rational-base irrationality criterion proved, and why do the remaining methods not settle 3/2?","precis":"The record proves the fixed-base Hankel asymptotic for Zudilin's normalised moments and the rational-base irrationality criterion log b/log a<0.4056830213840605…. The determinant proof separates geometric node configurations from positive weights; the irrationality proof instead specialises Zudilin's 2004 polynomial forms after cancellation. Coefficient moments, evaluated lattices and congruence conditions distinguish integer divisibility from nonzero remainder decay. A broader ratio-limit argument and a rational-value model remain unformalised. None of these arguments settles F(3/2).","owns":"the sharp fixed-base Hankel asymptotic and rational-base irrationality proofs, coefficient moments and arithmetic obstructions, with ordinary ratio-limit and rational-value models distinguished from checked results","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #1049, which remains open","first_pass":["long1049:sec:region","long1049:sec:hankel-order","long1049:sec:open"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos1049-rational-base-lambert-reasoning-surface.pdf","hosted_pdf_path":"papers/erdos1049-rational-base-lambert-reasoning-surface.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","site_html":"maths/papers/erdos1049-rational-base-lambert-reasoning-surface.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","body_pointer":"papers.records.erdos1049-rational-base-lambert-reasoning-surface.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Long-form reasoning record seeded from the complete problem note.\n% The short note and this record are now independent publication units.\n% This flat manuscript is generated from reviewable parts below.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n\\input{evidence/erdos1049-rational-base-lambert-reasoning-surface}\n\\renewcommand{\\commit}{7380b7871687b6bcc41ca0143c61f232e8af6500}\n\\renewcommand{\\commitshort}{7380b7871687}\n\\renewcommand{\\PX}{lean/ErdosProblems}\n% Sources that postdate \\commit are cited with their lean/ path and resolve at \\latecommit,\n% whose tree keeps the Lean roots under lean/.\n\\newcommand{\\latecommit}{035c414b25fda09ae0d2e56358b8716d321ebd99}\n\\newcommand{\\laterepobase}{https://github.com/wcook04/plectis-erdos/blob/\\latecommit}\n\\hypersetup{\n  pdftitle={Geometric moments and rational Lambert values: proofs and further results},\n  pdfsubject={Erdős Problem Notes: Problem 1049},\n  pdfkeywords={irrationality, Lambert series, rational base, Padé approximation, Lean 4},\n}\n\n\\runninghead{Geometric moments and rational Lambert values}\n\\title{Geometric Moments and Rational Lambert Values:\\\\Proofs and Further Results}\n\\subtitle{Erd\\H{o}s Problem~\\#1049}\n\\author{Will Cook}\n\\date{September 2026}\n\n\\setlength{\\emergencystretch}{2em}\n\\begin{document}\n% ---- part core ----\n% Part: part core\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\nWe give complete proofs of the fixed-base Hankel asymptotic for\nZudilin's normalised moments and the rational-base irrationality\ncriterion $\\log b/\\log a<0.4056830213840605\\ldots$ for the Lambert\nseries $F(a/b)$. The determinant proof separates geometric node\nconfigurations from their positive weights. The irrationality\nproof instead specialises the polynomial forms of Zudilin's 2004\nconstruction after cancellation. We also examine coefficient\nmoments, evaluated lattices and congruence conditions, distinguishing\ninteger divisibility from nonzero remainder decay. A broader\nratio-limit argument and a rational-value model are retained as\nunformalised remarks. None of these arguments settles $F(3/2)$.\n\\end{abstract}\n\n\\section{Introduction}\\label{long1049:sec:problem}\n\nFor $t>1$, the Lambert series\n\\[\n F(t)=\\sum_{n\\ge1}\\frac1{t^n-1}\n     =\\sum_{n\\ge1}\\frac{\\tau(n)}{t^n}\n\\]\nhas two useful descriptions: as a sum of simple fractions and as\na divisor generating series. Here $\\tau(n)$ counts the positive\ndivisors of $n$. The equality follows by expanding each fraction,\nsince all terms are nonnegative; the bound $\\tau(n)\\le n$ ensures\nconvergence. Chowla conjectured that $F(t)$ is irrational for every\nrational $t>1$, as recorded by Erd\\H{o}s \\cite[p.~102]{erdos1988}\nand in Bloom's catalogue \\cite{erdosproblems}. Erd\\H{o}s proved the\ninteger case in 1948 \\cite{erdos1948}.\n\nTwo constructions of Zudilin enter, for different reasons.\nThe 2016 approximation \\cite{zudilin2016} supplies positive\nmoments, whose Hankel determinants we evaluate asymptotically.\nThe 2004 forms \\cite{zudilin2004} supply polynomial coefficients\nwhose cancellation and rational evaluation give an irrationality\nregion. A denominator estimate for the second construction is not an\nestimate for the first; each coefficient family needs its own\narithmetic analysis. The shorter paper,\n\\emph{Hankel Determinants of Geometric Moments and Rational Lambert\nValues}, begins with the determinant result. Here the polynomial\nforms come first, making their exact arithmetic available for the\nlater comparisons. Readers following only the determinant argument\nmay start with Section~\\ref{long1049:sec:sharp}: its moment calculation\ndoes not use the rational-base transfer.\n\nFor positive weights with fixed-shift ratios tending to $1$ and a\nuniform polynomial bound on those ratios, we prove\n\\[\n \\det\\left(\\sum_{k\\ge0}a_kq^{(i+j+1)k}\\right)_{i,j<N}\n \\sim\\mathcal M(q)^3q^{B_N}P^{2N}\\prod_{k<N}a_k,\n \\qquad 0<q<1,\n\\]\nwhere $P=(q;q)_\\infty$, $B_N=\\sum_{j<N}j^2$ and\n$\\mathcal M(q)=\\prod_{d\\ge1}(1-q^d)^{-d}$.\nHeine's identity expands the determinant into squared Vandermonde\nproducts. We divide by the contribution from the first $N$ nodes\nand index the remaining tuples by partitions. A bound summable\nuniformly in $N$ reduces their limiting sum to the unit-weight\nCauchy determinant. For Zudilin's moments, two finite convolutions\nthen give the correction $N^{-8F(1/q)}$ and its constant.\nWe also determine the first formal term $C_Nq^{B_N}$, where\n$C_N=(N!)^2(N+1)!/2^N$.\n\nThe formal first term fixes the rank and expands at $q=0$;\nthe analytic asymptotic fixes $q$ and lets the rank grow. Neither\ncontrols the coefficient denominators of an irrationality form.\nThe \\hyperref[long1049:thm:geometric-ratio]{ratio-limit remark}\nextends the moment calculation to limiting adjacent-weight ratios,\nwhile Section~\\ref{long1049:sec:calibrated-countermodel} constructs\na rational-value model with the same Hankel features. The latter\nshows why those features alone cannot imply irrationality. Both\narguments are ordinary proofs without Lean proofs in the supplied\nrecord, and neither is used for the checked determinant asymptotic.\n\nFor the 2004 forms, we calculate the degree after cancelling the\ncommon polynomial factors. At a reduced rational base $a/b>1$,\nthe resulting polynomials $U_n,V_n$ have degrees $W_n,W_n-1$ and\nsatisfy\n\\[\n \\log\\!\\left(b^{W_n}\\bigl(U_n(a/b)F(a/b)-V_n(a/b)\\bigr)\\right)\n =\\bigl(C_1\\log b-C_0\\log a\\bigr)n^2+o(n^2).\n\\]\nThe form inside the logarithm is positive. Thus it tends to zero\nwhen $\\log b/\\log a<\\theta^*=C_0/C_1\n=0.4056830213840605\\ldots$. Rationality of $F(a/b)$ would make a\nfixed integer multiple of every such value a positive integer,\ngiving a contradiction. The constants and parameter direction are\nZudilin's; the cancellation and rational evaluation give the region\nused here. It contains $31/4$ and all its positive integral powers.\nBundschuh and V\\\"a\\\"an\\\"anen already treated a smaller rational-base\nregion containing $7/2$ \\cite[Theorem~2, p.~177]{bv1994};\nSection~\\ref{long1049:sec:sevenhalves} checks that specialisation.\nThe value $3/2$\nlies outside both regions and remains unresolved by these arguments.\nIt has the smallest height among reduced noninteger rationals\ngreater than $1$, where the height of $r/s$ is $\\max(r,s)$.\n\nSection~\\ref{long1049:sec:region} gives the rational-base proof and\nits degree restrictions. Section~\\ref{long1049:sec:sharp} contains\nthe determinant arguments and the coefficient-moment calculations.\nThe coefficient pencil is treated for sizes $N\\le8$ and every\n$p>1$; the computational ranges for shifted minors and coefficientwise\npositivity are specified there. Positivity for every $N$ and\nconvergence of the largest root remain open.\nSections~\\ref{long1049:sec:primitive}--\\ref{long1049:sec:endpoints}\napply rescaling and additive congruences to divided forms, including\na quantitative bounded-fibre criterion.\nSections~\\ref{long1049:sec:corridor}--\\ref{long1049:sec:pade} give\nthe clearing, tail and exponent calculations, and\nSection~\\ref{long1049:sec:open} distinguishes unrestricted\nreformulations of irrationality from questions about specified\napproximation families.\n\nAll logarithms are natural, and empty products and determinants\nequal $1$. Constants in $O_x(\\cdot)$ may depend on the fixed base.\nWe introduce the notation for each construction as it is needed.\nAppendix~\\ref{long1049:app:index} identifies the formal sources and\nthe separate finite computations.\n\n\\medskip\n\\noindent\\textbf{Keywords.} irrationality; Lambert series; rational base;\nPad\\'e approximation; Lean~4.\n\\qquad\n\\textbf{MSC 2020.} 11J72 (primary); 11J82, 68V20 (secondary).\n\n\\section{The region \\texorpdfstring{$b^{\\mu}<a$}{b\\textasciicircum mu < a} and the base \\texorpdfstring{$31/4$}{31/4}}\\label{long1049:sec:region}\n\nThe rational-base calculation compares a gain in the remainder\nwith the cost of making its coefficients integral. If the original\ncoefficient has degree $K_n$ and the cancelled coefficient has\ndegree $W_n$, the leading balance is\n\\begin{equation}\\label{long1049:eq:degree-balance}\n b^{W_n}(a/b)^{-(K_n-W_n)}\n =b^{K_n}a^{-(K_n-W_n)}.\n\\end{equation}\nThe polynomial identities and remainder estimates below justify\nthis balance, including its lower-order factors. With\n$K_n/n^2\\to C_1$ and $(K_n-W_n)/n^2\\to C_0$, the cleared\nremainder tends to zero when $C_1\\log b<C_0\\log a$.\nThe degree drop has two effects: it improves the real remainder\nand changes the denominator needed to make its coefficients integral.\nBoth effects concern the same cancelled form. This is why one must\ncalculate the degree before rational evaluation, rather than borrow\na denominator bound from the uncancelled family.\n\nWe evaluate Zudilin's 2004 forms at $a/b$ after cancelling their\ncommon polynomial factors. His direction $(14,12,14;27)$\n\\cite[Sec.~5, pp.~161--162]{zudilin2004} gives the constants below;\nthe comparison between decay and the remaining degree determines\nthe irrationality region:\n\\[\n C_1=(14+12+14)27-\\tfrac12(12^2+14^2+27^2)=\\frac{1091}{2},\n \\qquad\n C_0=266-\\frac{3}{\\pi^{2}}\\,(225-J),\n\\]\n\\[\n J=\\sum_{i=1}^{13}\\bigl(\\psi_1(u_i)-\\psi_1(v_i)\\bigr),\n \\qquad\n \\psi_1(x)=\\sum_{k\\ge0}\\frac1{(k+x)^{2}},\n\\]\nwhere $\\psi_1$ is the trigamma function, used only for positive arguments,\nand the thirteen half-open intervals $[u_i,v_i)$ are\n$[1/14,1/12)$, $[1/7,1/6)$, $[3/14,1/4)$, $[2/7,1/3)$, $[5/14,2/5)$,\n$[3/7,7/15)$, $[1/2,8/15)$, $[4/7,3/5)$, $[9/14,2/3)$, $[5/7,11/15)$,\n$[11/14,4/5)$, $[6/7,13/15)$, $[13/14,14/15)$.\nThe intervals are disjoint and lie in $[1/14,1)$, so\n$0\\le J\\le\\psi_1(1/14)-\\psi_1(1)<196$: in the defining series,\n$\\psi_1(1/14)=196+\\sum_{k\\ge1}(k+1/14)^{-2}<196+\\psi_1(1)$.\nSince $\\pi>3$, it follows that $191<C_0<266<C_1/2$.\nIn particular, the reciprocal constants below are positive without\nusing decimal approximations. The accompanying rational certificate gives\n\\[\n\\begin{aligned}\n77.94318445520543&<J<77.94318449473569,\\\\\n221.30008815898543&<C_0<221.30008817100119,\\\\\n0.40568302137302&<\\theta^*:=C_0/C_1<0.40568302139506.\n\\end{aligned}\n\\]\nPut $\\mu=1/\\theta^*$. These parameter ratios, intervals and constants, including\n$C_1/C_0=2.46497868\\ldots$, occur in \\cite[p.~162]{zudilin2004}; that ratio\nis the integer-base exponent bound in its Theorem~1, not a new optimised\nconstant. The displayed interval endpoints are rounded outwards from\nrational bounds; no digits beyond their certified accuracy are required.\n\nThe proof below needs $J$ in its combinatorial form as well.  Put\n\\[\n \\omega(\\xi)=\\max\\bigl\\{0,\\;\n \\lfloor14\\xi\\rfloor+\\lfloor13\\xi\\rfloor-\\lfloor12\\xi\\rfloor-\\lfloor15\\xi\\rfloor,\\;\n 2\\lfloor14\\xi\\rfloor-\\lfloor13\\xi\\rfloor-\\lfloor15\\xi\\rfloor\\bigr\\},\n\\]\nthe weight that \\cite[p.~162]{zudilin2004} attaches to the six-tuple\n$c=(13,14,12,14,15,13)$ for the chosen parameter ratios; it gives the exponents\n$\\nu_l=\\omega(n/l)$ of the source's~(22) and enters its limit~(26).  Zudilin\nlists the support of $\\omega$ in the same place; the next lemma verifies that\nlist by exact evaluation. Since $14+13=12+15$ and $2\\cdot14=13+15$,\neach floor difference is unchanged by $\\xi\\mapsto\\xi+1$. Thus $\\omega$\nhas period $1$, and the lemma also determines every value $\\omega(n/l)$\nneeded for the cyclotomic exponents.\n\n\\begin{lemma}[the weight is an indicator]\\label{long1049:res:omega-indicator}\nOn $[0,1)$ the function $\\omega$ takes only the values $0$ and $1$, and\n$\\omega=1$ exactly on the union of the thirteen half-open intervals listed above.\n\\end{lemma}\n\n\\begin{proof}\nEach of the three expressions inside the maximum is constant on every interval\nbetween consecutive elements of\n$\\{k/c: c\\in\\{12,13,14,15\\},\\ 0\\le k\\le c\\}$, since those are the only points at\nwhich one of the four floor functions changes.  There are $48$ such intervals in\n$[0,1)$; evaluating $\\omega$ at the midpoint of each gives the value $1$ on the\nthirteen listed half-open intervals and $0$ elsewhere, and the thirteen listed\nintervals are unions of consecutive cells. Each floor function is\nright-continuous, so a cell has the same value at its left endpoint as\nat its midpoint. This also verifies the half-open endpoint convention.\nAll evaluations use exact rational arithmetic.\n\\end{proof}\n\n\\noindent Lemma~\\ref{long1049:res:omega-indicator} and termwise integration of the positive series\n$-\\psi_1'(\\xi)=\\sum_{k\\ge0}2(k+\\xi)^{-3}$ give the two forms of $J$ used below,\n\\[\n J=\\int_0^1\\omega(\\xi)\\sum_{k\\ge0}\\frac{2}{(k+\\xi)^{3}}\\,d\\xi\n  =\\sum_{i=1}^{13}\\bigl(\\psi_1(u_i)-\\psi_1(v_i)\\bigr),\n\\]\nthe first being the integral $\\int_0^1\\omega\\,d(-\\psi_1)$ of\n\\cite[Lemma~2, p.~155, and (26), p.~162]{zudilin2004}.  Since $\\omega$ vanishes\nnear $0$, both are finite.\n\nFor $b=1$ the logarithmic ratio is zero, so the condition includes every\ninteger base. For $b>1$ it requires $a>b^{2.46497868\\ldots}$, substantially\nmore than $a>b$. There are still infinitely many admissible coprime\nnumerators for each fixed denominator. The base $31/4$ is included,\nwhereas $3/2$ is excluded. Taking positive integral powers does not\nchange the ratio. At equality with the cutoff, the estimates give no\nconclusion.\n\n\\begin{theorem}[rational-base region]\\label{long1049:res:region}\nLet $a>b\\ge1$ be coprime integers with\n\\[\n b^{\\mu}<a,\n \\qquad\\text{equivalently}\\qquad\n \\frac{\\log b}{\\log a}<\\theta^{*}=0.40568302138406054\\ldots\n\\]\nThen $F(a/b)=\\sum_{m\\ge1}(( a/b)^{m}-1)^{-1}$ is irrational.\n\\end{theorem}\n\nThe proof has three steps. We cancel the common polynomial factors\nusing Zudilin's integrality argument \\cite[pp.~156--161]{zudilin2004}.\nWe then compute the degree of the remaining polynomials, using his\ncyclotomic limits and the reciprocal-interval method of\n\\cite[Lemma~1, p.~466]{zudilin2002}. Finally, we evaluate at $a/b$\nand show that the positive remainder still decays after multiplication\nby the required power of $b$. These steps occupy\nSections~\\ref{long1049:sec:source-forms}--\\ref{long1049:sec:integer-forms}.\nThe construction, direction and constants are Zudilin's; the rational\nspecialisation and its degree and remainder estimates are proved below.\n\n\\subsection{The polynomial forms and their common factors}\\label{long1049:sec:source-forms}\n\nWe construct $U_n,V_n$ as integral polynomials before evaluating\nthem. Cancelling their common factors at this stage lowers the degree\nand hence the power of $b$ required at a rational base $a/b$.\n\nFor each $n\\ge1$, the coefficient formulas below are rational functions\nof an indeterminate $X$. We evaluate them at real $x>1$ and put $q=x^{-1}$\nwhen using the convergent series. Set\n\\[\n\\begin{gathered}\n a_0=14n+1,\\quad a_1=12n+1,\\quad a_2=14n+1,\\\\\n \\beta_n=27n+2,\\quad N=15n,\\quad M_n=266n^{2}+34n+1 .\n\\end{gathered}\n\\]\nThe symbol $\\beta_n$ is the parameter $b$ of \\cite[Sec.~3, p.~155]{zudilin2004},\nthe exponent in the lower parameter $q^{b}$ of the Heine series~(13) on p.~157;\nthe letter $b$ continues to\ndenote the denominator of the base. In this construction alone,\n$N=15n$ is the cyclotomic cutoff. The exponent $M_n$ is the integer\nof that paper's~(16) on p.~157. Write\n$(z;q)_m=\\prod_{j=0}^{m-1}(1-zq^{j})$ for the $q$-Pochhammer symbol\n(the shifted $q$-factorial). Zudilin's positive remainder is\n\\begin{equation}\\label{long1049:eq:positive-source-form}\n H_n(x)=\\sum_{t\\ge0}\n \\frac{(q^{t+1};q)_{a_1-1}}{(q;q)_{a_1-1}}\\,\n \\frac{(q;q)_{\\beta_n-a_2-1}}{(q^{a_2+t};q)_{\\beta_n-a_2}}\\,q^{a_0t}.\n\\end{equation}\nThe denominator length $\\beta_n-a_2$ agrees with the gamma\nexpression~(7) and residues~(8) of \\cite[p.~156]{zudilin2004}.\nThe unnumbered display of $R(T)$ on that page instead has length\n$\\beta_n-a_2-1$; we use the normalisation fixed by the numbered identities.\nThen \\cite[(8)--(11), pp.~156--157]{zudilin2004} gives $H_n=A_nF-B_n$ with\n$A_n,B_n\\in\\Q(X)$. We need their explicit formulas to compute degrees\nand bound coefficients. Let\n$\\genfrac{[}{]}{0pt}{}{m}{r}_X=\\prod_{j=1}^{r}(1-X^{m-r+j})/(1-X^j)$\nfor $0\\le r\\le m$. This Gaussian binomial, or $q$-binomial, polynomial has degree\n$r(m-r)$ and nonnegative coefficients summing to $\\binom{m}{r}$. For\n$a_2\\le k\\le\\beta_n-1$ put\n\\begin{equation}\\label{long1049:eq:c-explicit}\n c_{n,k}(X)=(-1)^{a_1+a_2+k+1}X^{e_{n,k}}\n \\genfrac{[}{]}{0pt}{}{k-1}{a_1-1}_X\n \\genfrac{[}{]}{0pt}{}{\\beta_n-a_2-1}{\\beta_n-k-1}_X,\n\\end{equation}\n\\[\n e_{n,k}=\\frac{a_1(a_1-1)}2-\\frac{(\\beta_n-a_2)(\\beta_n-a_2-1)}2\n +\\frac{(\\beta_n-k)(\\beta_n-k-1)}2 .\n\\]\nThen\n\\begin{align}\n A_n(X)&=\\sum_{k=a_2}^{\\beta_n-1}c_{n,k}(X)X^{a_0k},\\label{long1049:eq:A-explicit}\\\\\n B_n(X)&=\\sum_{k=a_2}^{\\beta_n-1}c_{n,k}(X)X^{a_0k}\n \\biggl(\\sum_{l=1}^{k-a_1}\\frac1{X^{l}-1}\n +\\sum_{j=1}^{a_0-1}\\frac{X^{-j(k-a_1)}}{X^{j}-1}\\biggr).\\label{long1049:eq:B-explicit}\n\\end{align}\nPut\n\\begin{equation}\\label{long1049:eq:cyclotomic-products}\n D_N(X)=\\prod_{l=1}^{N}\\Phi_l(X),\\qquad\n \\nu_{n,l}=\\omega(n/l),\\qquad\n \\Omega_n(X)=\\prod_{l=2}^{N}\\Phi_l(X)^{\\nu_{n,l}},\n\\end{equation}\nwith $\\Phi_l$ the $l$th cyclotomic polynomial; by\nLemma~\\ref{long1049:res:omega-indicator} every $\\nu_{n,l}$ is $0$ or $1$, and\n$\\nu_{n,l}=0$ for $l>15n$ because the intervals all begin at or above $1/14$.\n\nZudilin's Lemma~7 \\cite[p.~161, (23)]{zudilin2004} applies at the parameters above: the tuple is\n$c=n\\cdot(13,14,12,14,15,13)$ with maximum $m(c)=15n$ and difference $s(c)=n>0$,\nand the source's~(14) on p.~157 holds because $12n+1\\le14n+1$ and\n$26n+2\\le27n+2\\le28n+2$.  Its conclusion is the inclusion\n\\begin{equation}\\label{long1049:eq:integer-polynomial-pair}\n U_n:=X^{-M_n}\\frac{D_N}{\\Omega_n}A_n\\in\\Z[X],\\qquad\n V_n:=X^{-M_n}\\frac{D_N}{\\Omega_n}B_n\\in\\Z[X].\n\\end{equation}\nTwo points about that citation matter.  First, the conclusion used is the one\nin $\\Z[X]$, not integrality at integer arguments: an integer-valued polynomial\nsuch as $X(X-1)/2$ shows that the two statements differ.  The source's proof supplies the polynomial form. Its Lemma~4\n(pp.~157--159) clears the two rational coefficients separately by\nGaussian-binomial identities in $\\Z[X]$. Lemma~5 (p.~160) makes the\nnormalised series invariant under the order-twelve group generated by\nHeine's parameter transformation and the exchange of $a_1$ and $a_2$.\nFor the divisibility argument, however, only its order-six subgroup\npreserving $s>0$ is used: one application of Heine's transformation\nreverses the sign of $s$. Under that subgroup the normalising factorial\nproduct takes the three forms displayed in the source. Comparing their cyclotomic\nvaluations gives the maximum of three terms in $\\omega$.\nLemma~7 (p.~161) removes the resulting common cyclotomic factors by\npolynomial divisibility; the divisor $\\Omega_n$ is monic. These steps\ntake place before the integer specialisation in~(24) on p.~162.\n\nSecond, the module inclusion gives a polynomial coefficient pair,\nand uniqueness identifies it with the displayed cancelled source pair.\nIf two pairs differed, subtracting their identities would express $F$\nas a rational function of $X$. This is impossible: as $h\\downarrow0$,\nsplitting the original Lambert sum $F(e^h)=\\sum_{m\\ge1}(e^{hm}-1)^{-1}$\nat $L=\\lfloor1/h\\rfloor$ gives an initial sum\n$h^{-1}\\sum_{m\\le L}m^{-1}+O(L)$, since\n$y^{-1}-1\\le(e^y-1)^{-1}\\le y^{-1}$ for $0<y\\le1$.\nThe remaining sum is at most\n$e^{-h(L+1)}/((1-e^{-1})(1-e^{-h}))=O(h^{-1})$.  Hence\n$F(e^h)=h^{-1}\\log(1/h)+O(h^{-1})$, so\n$(X-1)F(X)\\to\\infty$ and $(X-1)^{2}F(X)\\to0$ as $X\\downarrow1$, which no\nrational function does. Thus the source's module inclusion yields the\nspecific pair in~\\eqref{long1049:eq:integer-polynomial-pair}, not merely\nsome unspecified polynomial representation of the same function.\nThis functional nonrationality says nothing about an individual value.\nFor $x>1$, write the cancelled remainder as\n\\[\n \\Lambda_n(x)=U_n(x)F(x)-V_n(x)\n =x^{-M_n}\\frac{D_N(x)}{\\Omega_n(x)}H_n(x).\n\\]\n\n\\subsection{Exact degrees}\\label{long1049:sec:degrees}\n\nThe final summand of $A_n$ has strictly larger degree than all the\nothers. We use it to determine the leading coefficient as well as the\ndegree. After cancellation, this will give the least common power\nof the base denominator that clears the pair.\n\nLet $d_{n,k}=\\deg\\bigl(c_{n,k}(X)X^{a_0k}\\bigr)$. The Gaussian factors\nare monic of degrees $(a_1-1)(k-a_1)$ and\n$(\\beta_n-k-1)(k-a_2)$. Adding these to the monomial exponent gives\n\\[\n d_{n,k}=e_{n,k}+a_0k+(a_1-1)(k-a_1)+(\\beta_n-k-1)(k-a_2)\n =\\frac{-k^{2}+80kn+3k-340n^{2}-26n}{2}.\n\\]\nTaking the difference at consecutive indices, we obtain\n\\[\n d_{n,k+1}-d_{n,k}=40n+1-k>0\\qquad(a_2\\le k\\le\\beta_n-2),\n\\]\nthe inequality because $k\\le\\beta_n-2=27n$ and $27n<40n+1$.  The summand of\ntop degree is therefore the one at $k=\\beta_n-1$ alone, no cancellation occurs\nthere, and\n\\begin{equation}\\label{long1049:eq:exact-degree}\n \\begin{gathered}\n K_n:=\\deg A_n=d_{n,\\beta_n-1}=\\frac{1091n^{2}+81n+2}{2},\n \\\\\n W_n:=\\deg U_n=K_n-M_n+\\sum_{l\\le N}\\varphi(l)-\\sum_{2\\le l\\le N}\\nu_{n,l}\\varphi(l),\n\\end{gathered}\n\\end{equation}\nthe second equality because $\\deg D_N=\\sum_{l\\le N}\\varphi(l)$ and\n$\\deg\\Omega_n=\\sum_{2\\le l\\le N}\\nu_{n,l}\\varphi(l)$, and because\n$U_n$ is a polynomial by~\\eqref{long1049:eq:integer-polynomial-pair}, so subtracting\n$M_n$ from the degree of $D_NA_n/\\Omega_n$ is legitimate.  In particular\n$U_n\\ne0$. There is also no integer leading-coefficient factor to\ncancel: the top summand has leading coefficient\n$(-1)^{a_1+a_2+\\beta_n}=(-1)^n$, and both Gaussian factors and\n$D_N/\\Omega_n$ are monic. Thus the leading coefficient of $U_n$ is\n$(-1)^n$.\n\nThe net decrease in degree is\n\\[\n K_n-W_n=M_n-\\deg(D_N/\\Omega_n).\n\\]\nDividing by $X^{M_n}$ lowers the degree by $M_n$, but multiplying\nby the remaining cyclotomic factors increases it by\n$\\deg(D_N/\\Omega_n)$. This is why the remainder estimate below\ninvolves $K_n-W_n$, rather than $M_n$ alone.\n\nFor a reduced fraction $a/b$, the unit leading coefficient also gives\nthe exact denominator of $U_n(a/b)$, not just an upper bound. For every prime $p\\mid b$,\n\\[\n b^{W_n}U_n(a/b)\\equiv(-1)^n a^{W_n}\\not\\equiv0\\pmod p.\n\\]\nNo prime dividing $b$ cancels from the cleared numerator, so the\nreduced denominator is exactly $b^{W_n}$. In particular every common\ninteger divisor of the cleared row is coprime to $b$. This excludes\ncancelling a factor of $b$ from this row, not cancellation at other\nprimes or a saving from a different pair.\n\nThe formal arithmetic checks cover the\n\\lword{Erdos1049/RationalBaseContour.lean}{317}{two_mul_zudilinM}{value of $2M_n$ from the source's formula~(16)},\nthe\n\\lword{Erdos1049/RationalBaseContour.lean}{337}{twoMulZudilinSummandDegree_succ_sub}{degree difference $d_{n,k+1}-d_{n,k}=40n+1-k$},\nits\n\\lword{Erdos1049/RationalBaseContour.lean}{350}{twoMulZudilinSummandDegree_step_pos}{positivity for $a_2\\le k\\le\\beta_n-2$},\nand the\n\\lword{Erdos1049/RationalBaseContour.lean}{344}{twoMulZudilinSummandDegree_top}{top degree $2K_n=1091n^2+81n+2$}.\nThese statements check the polynomial degree calculation, not the\nanalytic estimates used later.\n\nFor the second coefficient, fix $n$ and let $x\\to\\infty$. For $x\\ge2$,\nevery finite product in~\\eqref{long1049:eq:positive-source-form} lies\nbetween $\\prod_{j\\ge1}(1-2^{-j})>0$ and $1$, independently of $t$.\nAlso $\\sum_{t\\ge0}x^{-a_0t}\\le2$. These bounds give $H_n(x)=O(1)$\nfor the whole sum, not just each summand. Also\n$F(x)=x^{-1}+O(x^{-2})$, because $\\tau(1)=1$ and\n$\\sum_{m\\ge2}m x^{-m}=O(x^{-2})$. The monic normalising factor at\ninfinity gives $\\Lambda_n(x)=O(x^{W_n-K_n})$. Since $K_n\\ge2$ and\n$U_n$ has leading coefficient $(-1)^n$,\n\\[\n V_n(x)=U_n(x)F(x)-\\Lambda_n(x)\n       =(-1)^n x^{W_n-1}+O(x^{W_n-2}).\n\\]\nThe degree formula gives $W_n\\ge K_n-M_n=(559n^2+13n)/2>0$.\nPolynomial integrality was established earlier, so the displayed\nasymptotic proves the exact statement\n\\begin{equation}\\label{long1049:eq:V-degree}\n \\deg V_n=W_n-1,\\qquad \\operatorname{lc}V_n=(-1)^n.\n\\end{equation}\nIn particular $V_n$ is not the zero polynomial. Applying the same\nprime-by-prime reduction at its own degree shows that $V_n(a/b)$ has\nreduced denominator exactly $b^{W_n-1}$. Thus $b^{W_n}$ is the least\ncommon clearing denominator, although the second coordinate alone\nneeds one fewer power of $b$. In the common normalisation,\n\\[\n \\gcd\\!\\bigl(b^{W_n}V_n(a/b),b^{W_n}\\bigr)=b.\n\\]\nIndeed $b^{W_n}V_n(a/b)$ is $b$ times an integer coprime to $b$.\nThese facts concern the fixed source pair, not all approximants.\nThe degree calculation fixes $n$ and varies $x$; the decay calculation\nfixes $x$ and varies $n$.\n\n\n\\subsection{The cyclotomic limit}\\label{long1049:sec:cyclotomic-limit}\n\nWe prove the cyclotomic limits of Zudilin\n\\cite[Lemmas~1--2, p.~155]{zudilin2004} by summing over reciprocal\nintervals, following \\cite[Lemma~1, p.~466]{zudilin2002}. The\nsummatory totient estimate handles each finite set of intervals.\nA bound on the remaining small indices permits passage to their\ninfinite union.\n\nWrite $\\Sigma_n=\\sum_{2\\le l\\le N}\\nu_{n,l}\\varphi(l)$.  The elementary\nsummatory estimate is $\\sum_{l\\le y}\\varphi(l)=3\\pi^{-2}y^{2}+O(y\\log(2y))$.\nFix one half-open interval $[u,v)$ of Lemma~\\ref{long1049:res:omega-indicator}.  The condition\n$\\{n/l\\}\\in[u,v)$ holds exactly on the blocks\n\\[\n \\frac{n}{k+v}<l\\le\\frac{n}{k+u},\\qquad k=0,1,2,\\dots,\n\\]\nso the summatory estimate gives, for each fixed $k$,\n$n^{-2}\\sum_{l\\text{ in block }k}\\varphi(l)\\to3\\pi^{-2}\\bigl((k+u)^{-2}-(k+v)^{-2}\\bigr)$.\nFor $L\\ge1$, all blocks with $k\\ge L$ lie below $n/(L+u)$. For every\n$y\\ge0$,\n\\[\n \\sum_{l\\le y}\\varphi(l)\\le y^2.\n\\]\nFor $y<1$ the sum is empty; otherwise it is at most\n$\\lfloor y\\rfloor(\\lfloor y\\rfloor+1)/2\\le y^2$. Hence the normalised\ncontribution of the omitted blocks is at most $(L+u)^{-2}$, uniformly\nin $n$. This is the bound needed to pass from finitely many blocks to\nall of them. Taking $n\\to\\infty$ first and then $L\\to\\infty$ justifies the\ninfinite block sum, and summing the thirteen half-open intervals gives\n$n^{-2}\\Sigma_n\\to3J/\\pi^{2}$.  With $\\sum_{l\\le15n}\\varphi(l)=3\\pi^{-2}225n^{2}+O(n\\log n)$,\nequation~\\eqref{long1049:eq:exact-degree} yields\n\\begin{equation}\\label{long1049:eq:degree-limits}\n \\frac{K_n}{n^{2}}\\to C_1,\n \\qquad\n \\frac{K_n-W_n}{n^{2}}=\\frac{M_n-\\sum_{l\\le N}\\varphi(l)+\\Sigma_n}{n^{2}}\n \\to266-\\frac{3}{\\pi^{2}}(225-J)=C_0,\n\\end{equation}\nand hence $W_n/n^{2}\\to C_1-C_0$.  These are the rational-base degree limits\nbuilt on Zudilin's cyclotomic limits \\cite[Lemmas~1--2, p.~155; (26),\np.~162]{zudilin2004}.\n\nFor the real size estimate, fix $x>1$. The M\\\"obius product formula\nand reindexing the divisors give\n\\[\n \\sum_{l\\le N}|\\log\\Phi_l(x)-\\varphi(l)\\log x|\n \\le \\sum_{d\\le N}\\lfloor N/d\\rfloor[-\\log(1-x^{-d})]\n \\le N\\sum_{d\\ge1}\\frac{-\\log(1-x^{-d})}{d}.\n\\]\nThe bound $-\\log(1-x^{-d})\\le x^{-d}/(1-x^{-1})$ shows that the\nseries on the right is finite for each fixed $x>1$.\nSince $\\nu_{n,l}\\in\\{0,1\\}$, this proves the bound\n\\begin{equation}\\label{long1049:eq:cyclotomic-size}\n \\log\\frac{D_N(x)}{\\Omega_n(x)}\n =\\Bigl(\\sum_{l\\le N}\\varphi(l)-\\Sigma_n\\Bigr)\\log x+O_x(N).\n\\end{equation}\nHere $N=15n$, so the error is $O_x(n)=o(n^2)$. The bracket equals\n$W_n-K_n+M_n$ by~\\eqref{long1049:eq:exact-degree}. No uniform constant as\n$x\\downarrow1$ is asserted.\n\n\\subsection{Positive linear forms with integer coefficients}\\label{long1049:sec:integer-forms}\n\nWe now specialise at $x=a/b>1$. Put $q=1/x$ and $P_q=(q;q)_\\infty$.\nEvery finite $q$-Pochhammer product in\n\\eqref{long1049:eq:positive-source-form} has factors $1-q^j$ with $j\\ge1$,\nso it lies between $P_q$ and $1$. Each of the two ratios therefore lies\nbetween $P_q$ and $P_q^{-1}$. Summing the resulting geometric bounds gives\n\\[\n P_q^{2}\\le H_n(x)\\le\\frac{P_q^{-2}}{1-q^{a_0}}\\le\\frac{P_q^{-2}}{1-q},\n\\]\nwhence $H_n(x)>0$ and $\\log H_n(x)=O_x(1)$ uniformly in $n$.\nThe cyclotomic factors are positive on $(1,\\infty)$, so the cancelled\nremainder $\\Lambda_n(x)$ is positive as well, by\n\\eqref{long1049:eq:integer-polynomial-pair}.\nBy~\\eqref{long1049:eq:exact-degree} and~\\eqref{long1049:eq:V-degree} both polynomial degrees are\nat most $W_n$, so\n\\[\n \\widehat U_n=b^{W_n}U_n(a/b),\\qquad \\widehat V_n=b^{W_n}V_n(a/b)\n\\]\nare integers, and $\\widehat\\Lambda_n:=b^{W_n}\\Lambda_n(a/b)\n=\\widehat U_nF(a/b)-\\widehat V_n$ is positive and lies in $\\Z F(a/b)+\\Z$.  This\nfollows by multiplying each polynomial value by $b^{W_n}$.\nThe integer coefficients and the earlier cancellation of a common\ncyclotomic factor are separate inputs.\nCombining the size estimates,\n\\[\n \\begin{aligned}\n \\log\\widehat\\Lambda_n\n &=W_n\\log b-M_n\\log x+\\Bigl(\\sum_{l\\le N}\\varphi(l)-\\Sigma_n\\Bigr)\\log x+O_x(n)\\\\\n &=K_n\\log b-(K_n-W_n)\\log a+o(n^{2}),\n\\end{aligned}\n\\]\nthe second equality by $\\log x=\\log a-\\log b$ and~\\eqref{long1049:eq:exact-degree}.\nWith~\\eqref{long1049:eq:degree-limits} this proves\n\\begin{equation}\\label{long1049:eq:final-limit}\n \\lim_{n\\to\\infty}\\frac{\\log\\widehat\\Lambda_n}{n^{2}}=C_1\\log b-C_0\\log a .\n\\end{equation}\n\n\\begin{proof}[Proof of Theorem~\\ref{long1049:res:region}]\nThe hypothesis $\\log b/\\log a<\\theta^{*}=C_0/C_1$ makes the right side\nof~\\eqref{long1049:eq:final-limit} negative, so $\\widehat\\Lambda_n>0$ and\n$\\widehat\\Lambda_n\\to0$.  Suppose $F(a/b)=P/Q$ with integers $P$ and $Q\\ge1$.\nThen $Q\\widehat\\Lambda_n=P\\widehat U_n-Q\\widehat V_n$ is a positive integer for\nevery $n$ and tends to $0$, which is impossible.\n\\end{proof}\n\n\\noindent Above the threshold~\\eqref{long1049:eq:final-limit} says that these same\nhomogenised forms grow like $e^{cn^{2}}$ with $c>0$, and at equality the limit\ndecides nothing.  Both statements are about the displayed family.\n\n\\noindent The hypothesis $b\\ge1$ admits $b=1$, where the statement reduces to\nthe known integer-base theorem. Coprimality fixes the reduced\nrepresentation of the base. It is used in the exact-denominator\nassertion above, but is not needed for the sufficient irrationality\nimplication once the displayed logarithmic inequality holds.\nNegative bases are excluded,\nbecause the proof uses positivity for real bases greater than $1$.\n\n\\begin{sloppypar}\n\\begin{theorem}[the base $31/4$ outside the Bundschuh--V\\\"a\\\"an\\\"anen\nregion]\\label{long1049:res:31over4}\n$F(31/4)$ is irrational, and so is $F\\bigl((31/4)^{r}\\bigr)$ for every integer\n$r\\ge1$.  Here\n\\[\n \\frac{\\log4}{\\log31}=0.4036981731641997\\ldots<\\frac{81}{200}<\\theta^{*},\n\\]\nwhile\n\\[\n 4^{\\mu}=30.483515\\ldots<31<4^{\\mu_{\\mathrm{BV}}}=32.369642\\ldots\n\\]\nwith\n$\\mu_{\\mathrm{BV}}=2\\pi^{2}/(\\pi^{2}-2)=2.508284761994\\ldots$, so $31/4$ lies\noutside the region $\\log b/\\log a<1/2-1/\\pi^{2}=0.3986788163576622\\ldots$ of\n\\cite[Thm.~2, p.~177]{bv1994} and inside the region of\nTheorem~\\ref{long1049:res:region}.\n\\end{theorem}\n\\end{sloppypar}\n\n\\begin{proof}\nThe comparison $\\log4/\\log31<81/200$ is the integer certificate\n$4^{200}<31^{81}$, which Lean\n\\lword{Erdos1049/ZudilinHeightRegion.lean}{26}{thirtyoneFour_power_certificate}{checks},\nand Lean also checks that it places\n\\lword{Erdos1049/ZudilinHeightRegion.lean}{45}{thirtyoneFour_mem_zudilinHeightRegion}{$31/4$ in the region $\\log b/\\log a<81/200$}.\nThe ratio $\\log b/\\log a$ is invariant under $(a,b)\\mapsto(a^{r},b^{r})$,\nwhich places\n\\lword{Erdos1049/ZudilinHeightRegion.lean}{59}{thirtyoneFour_power_mem_zudilinHeightRegion}{every\npower of $31/4$} in the same region, and $(31^{r},4^{r})$ are coprime.\nLean also checks that\n\\lword{Erdos1049/ZudilinHeightRegion.lean}{91}{thirtyoneFour_outside_bundschuhVaananenHeightRegion}{$31/4$ lies outside the earlier region}.\nThat exclusion also has a two-line rational certificate: $31^{2}<4^{5}$ gives\n$\\log4/\\log31>2/5$, and $\\pi^{2}<10$ gives $1/2-1/\\pi^{2}<2/5$, so\n\\[\n \\frac12-\\frac1{\\pi^{2}}<\\frac25<\\frac{\\log4}{\\log31}<\\frac{81}{200}<\\theta^{*}.\n\\]\nThe remaining comparison $81/200<\\theta^{*}$ is proved in\nSection~\\ref{long1049:sec:regionbracket}.  Theorem~\\ref{long1049:res:region} then applies.\n\\end{proof}\n\n\\begin{corollary}[an irrationality measure uniform over powers]\n\\label{long1049:cor:rational-base-measure}\nFor coprime $a>b\\ge1$ with $\\theta=\\log b/\\log a<\\theta^*$ and every\ninteger $r\\ge1$,\n\\[\n \\mu_{\\rm irr}\\!\\left(F((a/b)^r)\\right)\n \\le\\frac{1-\\theta}{\\theta^*-\\theta}.\n\\]\nHere $\\mu_{\\rm irr}(\\xi)$ is the supremum of the exponents $\\nu$ for which\n$|\\xi-p/q|<q^{-\\nu}$ has infinitely many reduced rational solutions.\nIn particular, $\\mu_{\\rm irr}(F((31/4)^r))<301$ for every $r\\ge1$.\n\\end{corollary}\n\\begin{proof}\nFor the irrationality exponent, bound $A_n$ in~\\eqref{long1049:eq:A-explicit}:\nits $O(n)$ Laurent monomials multiply two Gaussian binomial polynomials,\neach with nonnegative coefficients summing to at most $2^{27n+2}$.\nThus its absolute coefficient sum is $\\exp(O(n))$, and its largest\nexponent $K_n$ gives $|A_n(x)|\\le x^{K_n}\\exp(O(n))$ for fixed $x>1$.\nThe cyclotomic estimate in Section~\\ref{long1049:sec:integer-forms}\nbounds the multiplier $x^{-M_n}D_N(x)/\\Omega_n(x)$ by\n$x^{W_n-K_n}\\exp(O_x(n))$. Therefore\n\\[\n |U_n(x)|\\le x^{W_n}\\exp(O_x(n))\\qquad(x>1\\text{ fixed}).\n\\]\nSet $\\xi=F(a/b)$, $Q_n=b^{W_n}U_n(a/b)$ and\n$P_n=b^{W_n}V_n(a/b)$.  With\n\\[\n \\alpha=(C_1-C_0)\\log a,\\qquad\n \\tau=C_0\\log a-C_1\\log b>0,\n\\]\nwe have $\\log(Q_n\\xi-P_n)=-\\tau n^2+o(n^2)$ and\n$|Q_n|\\le\\exp(\\alpha n^2+o(n^2))$.\n\nFor integers $A,B,p,q$ with $q>0$, if $L=A\\xi-B$ and $2q|L|\\le1$, then\n\\[\n |L|\\le |A|\\,|\\xi-p/q|.\n\\]\nWhen $Ap-Bq=0$ this is equality.  Otherwise the nonzero integer $Ap-Bq$\ngives $1/q\\le |L|+|A|\\,|\\xi-p/q|$, which proves the inequality.\nFix $0<\\eta<\\tau$. For all sufficiently large $n$, the estimates above give\n\\[\n e^{-(\\tau+\\eta)n^2}\\le Q_n\\xi-P_n\\le e^{-(\\tau-\\eta)n^2},\n \\qquad |Q_n|\\le e^{(\\alpha+\\eta)n^2}.\n\\]\nFor every sufficiently large denominator $q$, the upper remainder bound\nsuggests choosing\n$n=\\lceil\\sqrt{\\log(2q)/(\\tau-\\eta)}\\rceil$. Then\n$2q(Q_n\\xi-P_n)\\le1$, so the preceding integer argument applies to\n$(A,B)=(Q_n,P_n)$ for every numerator $p$. Also $Q_n\\ne0$: otherwise\n$Q_n\\xi-P_n$ would be an integer strictly between $0$ and $1$.\nThe lower remainder bound and the coefficient bound now give\n\\[\n |\\xi-p/q|\\ge e^{-(\\alpha+\\tau+2\\eta)n^2}\n             =q^{-(\\alpha+\\tau+2\\eta)/(\\tau-\\eta)-o(1)},\n\\]\nsince $n^2=\\log(2q)/(\\tau-\\eta)+O_\\eta(\\sqrt{\\log q}+1)$.\n\nLet $\\eta\\downarrow0$.  The resulting bound is\n$1+\\alpha/\\tau=(1-\\theta)/(\\theta^*-\\theta)$.\nTaking a common power multiplies $\\alpha,\\tau$ by $r$, leaving this quotient\nunchanged; the constants in the approximation inequality may depend on $r$.\nFor $31/4$, the exact rational interval calculation below gives\n\\[\n\\begin{gathered}\n 0.40568302137302<\\theta^*<0.40568302139506,\\\\\n 0.40369817316419<\\frac{\\log4}{\\log31}<0.40369817316420.\n\\end{gathered}\n\\]\nFor each trigamma difference, sum $k=0,\\ldots,255$ exactly and bound the\nremaining decreasing positive summand $f_{u,v}(x)=(x+u)^{-2}-(x+v)^{-2}$\nby its integral from $256$ to infinity and that integral plus $f_{u,v}(256)$.\nBound $\\pi$ with Machin's identity and alternating arctangent series,\nand logarithms with the positive $\\operatorname{arctanh}$ series and a\ngeometric tail bound. Outward rational interval operations then place\n$(1-\\rho)/(\\theta^*-\\rho)$, where $\\rho=\\log4/\\log31$,\nbetween $300.4269130$ and $300.4269164$, hence below $301$.\nAll endpoints are rational. For the bound $301$ alone, the coarser\ninequalities $\\rho<0.4036982$ and $\\theta^*>0.40568$ suffice:\n\\[\n \\frac{1-\\rho}{\\theta^*-\\rho}\n <\\frac{1-0.4036982}{0.40568-0.4036982}\n =\\frac{2981509}{9909}<301.\n\\]\nHere the quotient is increasing in $\\rho$ and decreasing in $\\theta^*$\nbecause $\\rho<\\theta^*<1$. The finer enclosure also certifies the\ndisplayed decimal approximation. Lean checks the coarser calculation,\nincluding $2981509/9909<301$, and proves enclosures of $\\theta^*$ and\n$\\rho$ sharper than the finer ones displayed above\n(Section~\\ref{long1049:sec:regionbracket} and\nTheorem~\\ref{long1049:res:31over4}); the $256$-summand calculation is an\nexact computation outside Lean.\n\\end{proof}\n\n\\subsection{The rational bracket around \\texorpdfstring{$\\theta^{*}$}{theta*}}\\label{long1049:sec:regionbracket}\n\\papersectiontarget{long1049-regionbracket}\n\nRational bounds for the trigamma differences locate $\\theta^*$\nwithout a numerical approximation to the full sums. Their first\nterms already place $31/4$ inside the region and $3/2$ outside it.\nA sharper enclosure then gives the irrationality-exponent bound\n$301$.\n\nFor the lower bound, keep only the $k=0$ term of each of the thirteen\ndifferences $\\psi_1(u_i)-\\psi_1(v_i)$.  All later terms are positive, so\n\\[\n J\\ \\ge\\ \\sum_{i=1}^{13}\\Bigl(\\frac1{u_i^{2}}-\\frac1{v_i^{2}}\\Bigr)\n  =\\frac{2015640690251}{25971865920}\n  >\\frac{776}{10}.\n\\]\nSince $\\pi>157/50$ and $225-J<225$,\n\\[\n C_0>266-\\frac{3\\,(225-776/10)}{(157/50)^{2}}=\\frac{5451134}{24649}\n >\\frac{88371}{400}=\\frac{81}{200}\\,C_1,\n\\]\nso $81/200<\\theta^{*}$.  For the upper bound the thirteen half-open intervals are\ndisjoint and ordered and $\\psi_1$ is positive and decreasing, so the sum\ntelescopes below its first term:\n\\[\n J<\\psi_1(1/14)=196+\\sum_{k\\ge1}\\frac1{(k+1/14)^{2}}<196+\\frac{\\pi^{2}}6<198<225,\n\\]\nwhence $C_0<266$ and $\\theta^{*}<532/1091<1/2$.  So\n\\[\n \\frac{81}{200}<\\theta^{*}<\\frac12 .\n\\]\n\nLean checks the definitions of $C_0$, $C_1$ and $\\theta^*$ together\nwith the\n\\lword{Erdos1049/RationalBaseContour.lean}{159}{zudilinJ_ge}{lower bound on $J$},\n\\lword{Erdos1049/RationalBaseContour.lean}{180}{zudilinC0_gt}{rational lower bound on $C_0$},\nand the two comparisons\n\\lword{Erdos1049/RationalBaseContour.lean}{201}{eightyOne_twoHundredths_lt_zudilinContour}{$81/200<\\theta^*$}\nand\n\\lword{Erdos1049/RationalBaseContour.lean}{251}{zudilinContour_lt_half}{$\\theta^*<1/2$}.\nIt also checks that\n\\lword{Erdos1049/RationalBaseContour.lean}{266}{thirtyoneFour_mem_zudilinContourRegion}{$31/4$ belongs to the region},\nthat \\lword{Erdos1049/RationalBaseContour.lean}{279}{thirtyoneFour_power_mem_zudilinContourRegion}{every positive integral power does too},\nand that \\lword{Erdos1049/RationalBaseContour.lean}{296}{threeHalves_outside_zudilinContourRegion}{$3/2$ is excluded}.\nThese comparisons use the defined constant; the analytic estimates belong\nto the proof of Theorem~\\ref{long1049:res:region}, which Lean also checks.\nLean also proves\n$0.40568302138406054100\\le\\theta^*\\le0.40568302138406054104$ and\nenclosures of $J$ and $C_0$ (\\href{\\evidencerecordurl}{evidence record},\nTheorem~\\ref{long1049:res:region}). The sharper\ninterval in the preceding proof comes from the stated\nexact-arithmetic certificate with $256$ summands per trigamma\ndifference. It does not rely on the historical thousand-term decimal\nevaluation reported in Section~\\ref{long1049:sec:receipts}.\n\nThe shorter interval above suffices for the bound $301$ but does not\ncertify all digits printed in the cutoff. Lean's enclosure of $\\theta^*$\nimplies the following bounds, and a separate integer-arithmetic\ncalculation reproduces them independently of Lean:\n\\[\n\\begin{aligned}\n0.4056830213840605403&<\\theta^*<0.4056830213840605417,\\\\\n2.4649786835749750334&<\\mu<2.4649786835749750415.\n\\end{aligned}\n\\]\nHere is the complete error bound used in that calculation. For each of the thirteen\nintervals, put $f_{u,v}(k)=(k+u)^{-2}-(k+v)^{-2}$.\nRound each of the first $M=65536$ positive rational summands down to a\nmultiple of $10^{-50}$. Their true sum is between this rounded sum and\nthat sum plus $M10^{-50}$. Since $f_{u,v}$ is positive and decreasing,\nthe remaining tail lies between its integral from $M$ to infinity\nand that integral plus $f_{u,v}(M)$. The integral is\n$(M+u)^{-1}-(M+v)^{-1}$. Summing these rational intervals and using\nMachin's alternating-series bounds for $\\pi$ gives the displayed\nbounds on $\\theta^*$ and its reciprocal. The coefficient\n$225-J$ is positive, so the interval endpoints for $C_0$ have the\nclaimed order. This verifies the displayed decimal truncations; \\href{https://github.com/wcook04/plectis-erdos/blob/4beca4ad0c00a626def9c73f2420e20ebf8d1968/research/experiments/erdos1049/threshold_digits.py}{the program} and\nits \\href{https://github.com/wcook04/plectis-erdos/blob/4beca4ad0c00a626def9c73f2420e20ebf8d1968/research/experiments/erdos1049/receipts/threshold-digits.json}{recorded result} are public. The proofs use the exact definitions and never these decimal values.\n\n\\subsection{Earlier criteria and changes of direction}\\label{long1049:sec:region-extensions}\n\\papersectiontarget{long1049-region-extensions}\n\nZudilin's parameters and constants give a rational-base condition\nthat can be compared directly with the earlier height criterion.\nWe make that comparison before varying the parameter direction.\nA single family valid at every real base greater than $1$ is subject\nto the degree restriction proved below.\n\n\\paragraph{Attribution.}\nBundschuh and V\\\"a\\\"an\\\"anen proved the irrationality of $F(a/b)$ on the\nregion $\\log b/\\log a<1/2-1/\\pi^{2}=0.3986788163576622\\ldots$\n\\cite[Thm.~2, p.~177; hypotheses pp.~175--176]{bv1994}, and every base of\nTheorem~\\ref{long1049:res:region} with $b\\le3$ is already theirs.  Their printed\nhypothesis for $\\alpha=-1$ is $\\lambda<(1/2+1/\\pi^{2})^{-1}$ with\n$\\lambda=\\log h(q)/\\log|q|$. Here the source uses $q$ for the base,\nnot its reciprocal, and\n\\[\n E_q(z)=\\prod_{j\\ge1}(1+zq^{-j}),\\qquad\n L_q(z)=\\frac{E_q'(z)}{E_q(z)}=\\sum_{j\\ge1}\\frac1{q^j+z}.\n\\]\nFor $q=a/b>1$, $E_q(-1)>0$ and $L_q(-1)=F(a/b)$; in particular\n$\\alpha=-1$ is not one of the excluded zeros $-q^j$.\nSince $a,b$ are coprime and $a>b>0$, the rational height is $h(q)=a$.\nThus $\\lambda=\\log a/\\log(a/b)=1/(1-\\log b/\\log a)$, and the\nprinted hypothesis is equivalent to the displayed inequality.\nDuverney proved another, strictly smaller, region for the same series\n\\cite[Th\\'eor\\`eme~2, p.~174]{duverney1996}.  Zudilin remarked that his\ngeneralized $q$-logarithm results can be given at non-integer rational bases\nunder an assumption $\\log|r|>c\\log|s|$ for a computable $c>0$, without computing\na value \\cite[Sec.~2, p.~4]{zudilin2016}.  Zudilin's 2004 paper supplies the forms and the exponent\n$\\mu$ under the standing hypothesis $p=1/q\\in\\Z\\mathbin{\\backslash}\\{0,\\pm1\\}$, and it\nstates no rational-base result \\cite[Sec.~2, p.~154; Thm.~1]{zudilin2004}.  The\ncontribution here is the rational specialisation of the 2004 forms, with the\ndenominator accounting and the limit passage carried out in full, which\nidentifies the printed $\\mu$ as an admissible $c$ for $F$ itself, together with\nthe region that constant defines and its application to $31/4$.\n\n\\paragraph{Where the earlier criterion stops.}\nBoth regions are cut out by the same quantity.  The published cutoff\n$1/2-1/\\pi^{2}$ of \\cite[Thm.~2, p.~177]{bv1994} is the reciprocal of\n$\\mu_{\\mathrm{BV}}=2\\pi^{2}/(\\pi^{2}-2)$, the constant Van Assche later\nrecovered as an integer-base irrationality-exponent bound for $F(p)$\n\\cite[Thm.~1, p.~10]{vanassche2001}, and the cutoff $\\theta^{*}$ of\nTheorem~\\ref{long1049:res:region} is the reciprocal of the smaller integer-base bound\n$\\mu=C_1/C_0$ printed at \\cite[p.~162]{zudilin2004}.  For Zudilin's family this reciprocal relation uses both the exact degree\nlimit $d=C_1-C_0$ and the stronger evaluation bound\n$|U_n(p)|\\le p^{W_n}\\exp(O_p(n))$ proved in\nCorollary~\\ref{long1049:cor:rational-base-measure}, together with decay\nexponent $\\sigma=C_0$. A general exponential coefficient-height bound\nwould contribute an additional term to the integer-base exponent\nestimate; it cannot simply be discarded. The discussion following\nTheorem~\\ref{long1049:res:archcap} makes that distinction explicit.\nThe rational-base proof clears the denominators of the already\ncancelled polynomials. It does not infer value irrationality merely by\ntaking the reciprocal of a published exponent bound.  The bases\ngained are exactly the strip $s^{\\mu}<r\\le s^{\\mu_{\\mathrm{BV}}}$, which is empty\nfor $s=2$ and $s=3$ and is first occupied at $s=4$ by the single coprime numerator\n$31$.  The integer comparisons and rational bracket in\nSection~\\ref{long1049:sec:regionbracket} establish membership in the region.\nThe irrationality conclusion in Theorem~\\ref{long1049:res:31over4} also uses\nthe constructed forms and their proved analytic estimates. This application\ndoes not improve either inherited integer-base exponent bound.\n\n\\paragraph{What a larger region would require.}\nAt $3/2$ the logarithmic parameter is\n\\[\n \\frac{\\log2}{\\log3}=0.6309297535714574\\ldots,\n\\]\nwhich exceeds $\\theta^*$ by $0.2252467\\ldots$.\nTheorem~\\ref{long1049:res:region} gives no conclusion when\n$\\log b/\\log a\\ge\\theta^*$. The same argument can apply to another family\nonce its polynomial integrality, degrees and fixed-base remainder\nestimates are proved. A smaller ratio of the resulting constants\n$C_1/C_0$ would enlarge the reciprocal-cutoff region. For example,\n$C_1/C_0\\le2.4234$ would give a cutoff greater than $0.4126$ and admit\n$29/4$. An integer-base irrationality-exponent bound alone does not\nsupply the polynomial data needed for this specialisation.\nThe existing formal proofs establish irrationality and measure bounds\nfor the constructed forms; they do not supply a different family\nwhose divided remainder is small and nonzero at $3/2$.\n\nThe next restriction concerns one polynomial family whose estimates\nhold at every fixed $x>1$, with the same leading constants but\npossibly base-dependent error terms. This quantifier is essential:\nthe proof chooses a large integer base at which remainder decay\nwould outpace coefficient growth. Estimates only at $3/2$, or for\na different family at each base, do not permit that choice. The\n2004 family satisfies the hypotheses. The conclusion concerns\nforms in $F(x)$ and $1$, not simultaneous forms in several\nindependent target values. Here height means the largest absolute\ncoefficient. To compare it with the sum of the absolute coefficients,\nnote that for a polynomial of degree at most $d$,\n\\[\n \\max_j|p_j|\\le\\sum_j|p_j|\\le(d+1)\\max_j|p_j|.\n\\]\nFor a nonzero polynomial of degree $d=O(n^2)$, the logarithms of\nthese norms differ by $O(\\log n)$. They have the same leading\nquadratic growth rate, but the literal zero-height conditions are\nnot identical: $1+X$ has maximum coefficient $1$ and coefficient sum\n$2$. In particular, when $h=0$, the displayed bound\n$hn^2(1+o(1))$ is zero, not an arbitrary $o(n^2)$ term.\nThe proof below also works with the additive bound $hn^2+o(n^2)$;\nits evaluation estimate absorbs the $O(\\log n)$ difference.\nAlternatively, either norm convention implies the hypothesis for the\nother norm after replacing $h$ by any larger positive constant.\nThe degree conclusion is independent of that replacement.\nIn the rational-base conclusions, write $a/b$ with integers $a>b\\ge1$.\nThe estimates hold without coprimality; a reduced representation gives\nthe smaller denominator-clearing factor.\n\n\\begin{theorem}[a degree restriction for estimates valid at every base]\\label{long1049:res:archcap}\nLet $(U_n,V_n)\\in\\Z[x]^{2}$ be a sequence such that, for constants\n$\\sigma,\\delta>0$ and $h\\ge0$ independent of $n$ and of the base,\n\\begin{enumerate}\n\\item $\\Lambda_n(x):=U_n(x)F(x)-V_n(x)\\ne0$ for every real $x>1$;\n\\item $\\deg U_n,\\deg V_n\\le\\delta n^{2}(1+o(1))$;\n\\item the coefficient heights satisfy\n\\[\n \\log\\max\\bigl(H(U_n),H(V_n)\\bigr)\\le hn^{2}(1+o(1)),\n\\]\nwhere $H(P)$ is the\nlargest absolute value of a coefficient of $P$;\n\\item the remainders satisfy\n\\[\n \\log|\\Lambda_n(x)|=-\\sigma n^{2}\\log x\\,(1+o(1))\n\\]\nfor every real $x>1$.\n\\end{enumerate}\nPut $d_n:=\\max(\\deg U_n,\\deg V_n)$.  Then $\\sigma\\le\\delta$, and for every fixed\nrational base $a/b>1$,\n\\[\n \\limsup_{n\\to\\infty}n^{-2}\\log\\bigl|b^{d_n}\\Lambda_n(a/b)\\bigr|\n \\le\\delta\\log b-\\sigma\\log(a/b).\n\\]\nConsequently the homogenised forms tend to zero whenever\n$\\log b/\\log a<\\sigma/(\\sigma+\\delta)$, a sufficient region whose cutoff is at\nmost $1/2$.  If the actual degrees satisfy $d_n/n^{2}\\to d$, then $d\\ge\\sigma$\nand the limit exists and equals $d\\log b-\\sigma\\log(a/b)$; in that case the\nforms tend to zero below $\\log b/\\log a=\\sigma/(\\sigma+d)$ and their absolute\nvalues tend to infinity above it, so the exact-degree case has no decaying\nhomogenised forms at $3/2$.\n\\end{theorem}\n\n\\begin{proof}\nThe coefficient-height bound controls evaluation at an integer base.\nFor every fixed $p>1$,\n\\[\n |U_n(p)|,|V_n(p)|\n \\le (d_n+1)\\max(H(U_n),H(V_n))p^{d_n}\n \\le \\exp\\bigl((h+\\delta\\log p)n^2+o(n^2)\\bigr).\n\\]\nHere $\\log(d_n+1)=o(n^2)$ by the degree bound. This is where the\ncoefficient-height hypothesis is needed.\n\nSuppose $\\sigma>\\delta$. To make remainder decay beat coefficient growth,\nchoose an integer $p\\ge2$ with $(\\sigma-\\delta)\\log p>h$ and put\n$u_n=U_n(p)$, $v_n=V_n(p)$ and $\\ell_n=u_nF(p)-v_n$.\nThe integer determinant eliminates $F(p)$; the evaluation bound\nand hypothesis~(4) give\n\\[\n u_nv_{n+1}-u_{n+1}v_n=u_{n+1}\\ell_n-u_n\\ell_{n+1}=o(1).\n\\]\nIndeed, each product on the right has absolute value at most\n$\\exp((h+(\\delta-\\sigma)\\log p)n^2+o(n^2))$.\nThe expression on the left is an integer, so it is eventually zero.\nAlso $u_n\\ne0$ eventually, since $u_n=0$ would make $\\ell_n=-v_n$\na nonzero integer of absolute value less than $1$.\nThus $v_n/u_n$ is eventually a fixed rational number $r$.\nIf $F(p)=r$, then $\\ell_n=0$; otherwise\n$|\\ell_n|=|u_n|\\,|F(p)-r|\\ge|F(p)-r|$.\nBoth contradict the nonzero remainders tending to zero. Hence\n$\\sigma\\le\\delta$.\n\nFor a fixed rational base $a/b>1$, taking logarithms gives\n\\[\n n^{-2}\\log|b^{d_n}\\Lambda_n(a/b)|\n =\\frac{d_n}{n^2}\\log b-\\sigma\\log(a/b)+o(1).\n\\]\nHypothesis~(2) yields the stated upper limit and sufficient region.\nSince $\\sigma\\le\\delta$, its cutoff is at most $1/2$.\nIf $d_n/n^2\\to d$, apply the preceding integer-base argument with\n$d+\\varepsilon$ in place of $\\delta$ for every $\\varepsilon>0$ to obtain\n$d\\ge\\sigma$. The displayed identity then gives the exact limit.\nIts sign is that of $(\\sigma+d)\\log b-\\sigma\\log a$, which proves\nboth assertions away from equality. At $3/2$ this sign is positive\nbecause $\\log2/\\log3>1/2\\ge\\sigma/(\\sigma+d)$.\n\\end{proof}\n\n\\papersectiontarget{long1049-no-decay}\n\\begin{corollary}[nondecay when $b<a<b^2$]\n\\label{long1049:cor:no-decay-below-square}\nUnder the hypotheses of Theorem~\\ref{long1049:res:archcap}, for positive integers\n$a,b$ with $b<a<b^2$, the undivided forms $b^{d_n}\\Lambda_n(a/b)$ do not\ntend to zero.  No limit of $d_n/n^2$ is assumed.\n\\end{corollary}\n\\begin{proof}\nWrite $x=a/b$ and suppose $C_n=b^{d_n}\\Lambda_n(x)\\to0$.\nSince $C_n\\ne0$ and $|C_n|\\le1$ eventually, logarithms and the\nremainder asymptotic at $x$ give the degree bound\n\\[\n d_n\\log b\\le-\\log|\\Lambda_n(x)|\n       =\\sigma\\log x\\,n^2+o(n^2).\n\\]\nSet $c=\\sigma\\log x/\\log b$.  Since $1<x<b$, we have $0<c<\\sigma$,\nand the displayed inequality gives $d_n\\le(c+\\varepsilon)n^2$ eventually\nfor each $\\varepsilon>0$. Apply Theorem~\\ref{long1049:res:archcap} to\nthe same polynomial family with degree upper rate $c$: its height bound,\nnonvanishing and all-base remainder asymptotics are unchanged. This gives\n$\\sigma\\le c$, contradicting $c<\\sigma$.\n\\end{proof}\n\n\nThis strengthens the failure of a sufficient-cutoff test to an exclusion\nof decay under the stated all-base hypotheses.  It does not supply an\nactual-degree limit or assert divergence.  If $d_n/n^2\\to d$, the separate\nexact-degree result gives the stronger conclusion\n$|b^{d_n}\\Lambda_n(a/b)|\\to\\infty$ in the same strict region.\nEquality $a=b^2$ remains unclassified.  The corollary concerns the undivided\nforms; it does not exclude base-dependent content division or other\nirrationality methods outside its hypotheses.\n\n\n\\noindent A degree bound alone would not supply the evaluation\nestimate: even degree-zero polynomials can have arbitrarily large\ninteger coefficients. This explains the height hypothesis (3),\nwithout establishing that it cannot be weakened or omitted.\nAt $\\log b/\\log a=\\sigma/(\\sigma+d)$, the normalised logarithm\nis zero and the hypotheses decide neither behaviour.\nTheorem~\\ref{long1049:res:archcap} is a restriction on the stated\nfamilies, not on all Pad\\'e constructions.\n\nThere is also a distinction between the degree cutoff and an\nirrationality-exponent estimate at a fixed integer base $p\\ge2$.\nThe hypotheses above give\n\\[\n |U_n(p)|\\le\\exp((h+\\delta\\log p)n^2+o(n^2)),\\qquad\n |\\Lambda_n(p)|=\\exp(-\\sigma\\log p\\,n^2+o(n^2)).\n\\]\nApply the integer-form argument in the proof of\nCorollary~\\ref{long1049:cor:rational-base-measure}, with coefficient\ngrowth rate $h+\\delta\\log p$ and remainder decay rate $\\sigma\\log p$.\nIt gives\n\\[\n \\mu_{\\rm irr}(F(p))\\le\n 1+\\frac{\\delta}{\\sigma}+\\frac{h}{\\sigma\\log p}.\n\\]\nThus the reciprocal $\\sigma/(\\sigma+\\delta)$ of the degree expression\n$1+\\delta/\\sigma$ is not, under these hypotheses alone, the reciprocal\nof the exponent bound furnished by that argument. To obtain the latter\nidentification it suffices to prove the stronger evaluated estimate\n$|U_n(p)|\\le\\exp(\\delta\\log p\\,n^2+o(n^2))$. This is separate from\nbounding the polynomial's coefficients by $\\exp(O(n^2))$.\n\nFor the family of Section~\\ref{long1049:sec:source-forms} the fourth hypothesis is the\nsize estimate proved there and the second is~\\eqref{long1049:eq:exact-degree}.  The third\nis proved next, so Theorem~\\ref{long1049:res:archcap} applies to that family.\n\n\\begin{lemma}[coefficient heights of the constructed polynomials]\\label{long1049:res:sourceheight}\nThere is a constant $h$ with\n$\\log\\max\\bigl(H(U_n),H(V_n)\\bigr)\\le hn^{2}$ for every $n\\ge1$, where $U_n$ and\n$V_n$ are the polynomials of~\\eqref{long1049:eq:integer-polynomial-pair}.\n\\end{lemma}\n\n\\begin{proof}\nFor a polynomial or Laurent polynomial $P$, write $\\|P\\|$ for the sum\nof the absolute values of its coefficients. Then\n$\\|PQ\\|\\le\\|P\\|\\,\\|Q\\|$; for an ordinary polynomial,\n$H(P)\\le\\|P\\|$. This convention includes the Laurent monomials in\n$c_{n,k}$.\n\nBy Lemma~\\ref{long1049:res:omega-indicator} every $\\nu_{n,l}$ is $0$ or $1$, so\n$D_N/\\Omega_n=\\prod_{l\\in S}\\Phi_l$ over a subset $S\\subseteq\\{1,\\dots,N\\}$.\nA monic polynomial with roots $\\zeta_1,\\ldots,\\zeta_d$ on the unit circle\nhas coefficient norm at most $2^d$: expanding its linear factors bounds\nthe sum of the absolute coefficients by\n$\\prod_{j=1}^d(1+|\\zeta_j|)=2^d$. Hence $\\|\\Phi_l\\|\\le2^{\\varphi(l)}$ and\n\\[\n \\Bigl\\|\\frac{D_N}{\\Omega_n}\\Bigr\\|\\le2^{\\sum_{l\\le N}\\varphi(l)}\\le2^{225n^{2}} .\n\\]\nFor $A_n$, the Gaussian binomial $\\genfrac{[}{]}{0pt}{}{m}{r}_X$ has nonnegative\ncoefficients summing to $\\binom{m}{r}$, so\n$\\|c_{n,k}\\|\\le\\binom{k-1}{a_1-1}\\binom{\\beta_n-a_2-1}{\\beta_n-k-1}\\le2^{2\\beta_n}$\nby~\\eqref{long1049:eq:c-explicit}, and summing the at most $\\beta_n$ terms\nof~\\eqref{long1049:eq:A-explicit} gives $\\|A_n\\|\\le\\beta_n2^{2\\beta_n}$.  Hence\n$\\|U_n\\|\\le\\|D_N/\\Omega_n\\|\\,\\|A_n\\|\\le e^{O(n^{2})}$ and\n$H(U_n)\\le e^{O(n^{2})}$.\n\nFor $V_n$, bound it on the circle $|z|=2$ and use Cauchy's estimate\n$H(V_n)\\le\\max_{|z|=2}|V_n(z)|$: each coefficient satisfies\n$|v_i|\\le2^{-i}\\max_{|z|=2}|V_n(z)|$. The bound also holds for the\nzero polynomial.  On that circle\n$|z^{l}-1|\\ge2^{l}-1\\ge1$, so each of the at most $\\beta_n+a_0$ inner terms\nof~\\eqref{long1049:eq:B-explicit} has modulus at most $1$; also\n$|c_{n,k}(z)z^{a_0k}|\\le\\|c_{n,k}\\|\\,2^{K_n}$ and\n$|D_N(z)/\\Omega_n(z)|\\le\\|D_N/\\Omega_n\\|\\,2^{225n^{2}}\\le e^{O(n^{2})}$, while\n$|z^{-M_n}|\\le1$. There are $O(n)$ outer summands, each with $O(n)$\ninner terms. After multiplication by the normalising factor, each\nterm is bounded by $e^{O(n^2)}$, since $K_n=O(n^2)$. Summing the\n$O(n^2)$ terms preserves this bound. Hence\n$\\max_{|z|=2}|V_n(z)|\\le e^{O(n^2)}$ and $H(V_n)\\le e^{O(n^2)}$.\n\\end{proof}\n\n\\noindent With Lemma~\\ref{long1049:res:sourceheight}, Zudilin's family satisfies all four\nhypotheses of Theorem~\\ref{long1049:res:archcap}, with exact degree limit $d=C_1-C_0$\nby~\\eqref{long1049:eq:degree-limits}, and decay exponent $\\sigma=C_0$ because\n$\\log\\Lambda_n(x)=-(K_n-W_n)\\log x+o(n^{2})$ for each fixed real $x>1$ by the\nsize estimate of Section~\\ref{long1049:sec:integer-forms}.  For it $\\sigma/(\\sigma+d)=\\theta^{*}$ and\n$(\\sigma+d)/\\sigma=\\mu$. The stronger bound\n$|U_n(p)|\\le p^{W_n}\\exp(O_p(n))$ from\nCorollary~\\ref{long1049:cor:rational-base-measure} removes the extra height\nterm at each integer base. Within this family the rational-base\nthreshold is therefore the reciprocal of the proved integer-base\nirrationality-exponent bound.\n\n\\paragraph{Examples of rational bases.}\nAmong reduced $a/b$ with $1\\le b<a\\le60$, the region of\nTheorem~\\ref{long1049:res:region} has $137$ members, of which $78$ are\nnon-integral. The two largest admitted logarithmic ratios belong to\n$53/5$ and $31/4$, respectively $0.000313$ and $0.001985$ below\n$\\theta^*$; the smallest excluded ratio is that of $52/5$, namely\n$0.4073243836\\ldots$, which exceeds the cutoff by $0.001641$.\nA calculation with rational bounds for logarithms and for the cutoff\nchecks all $1101$ reduced fractions in the stated range, these rounded\nmargins and the four fixed-denominator comparisons below\n(\\href{https://github.com/wcook04/plectis-erdos/blob/4beca4ad0c00a626def9c73f2420e20ebf8d1968/research/experiments/erdos1049/rational_base_examples.py}{program}, \\href{https://github.com/wcook04/plectis-erdos/blob/4beca4ad0c00a626def9c73f2420e20ebf8d1968/research/experiments/erdos1049/receipts/rational-base-examples.json}{result}). The\nbases in this region and outside the region of \\cite{bv1994} are those in the strip\n$s^{\\mu}<r\\le s^{\\mu_{\\mathrm{BV}}}$, which is empty for $s=2$ and $s=3$, is\n$\\{31\\}$ for $s=4$, and is $\\{53,54,56\\}$ for $s=5$.  The strip is infinite:\nits width $s^{\\mu_{\\mathrm{BV}}}-s^{\\mu}$ eventually exceeds $s$, so for every\nlarge enough denominator it contains an integer congruent to $1$ modulo $s$.\nHence $31/4$ is the member of the strip with least denominator and least numerator.\n\n\\paragraph{A certified finite parameter search.}\nThe parameter ratios $(14,12,14;27)$ are Zudilin's \\cite[p.~162]{zudilin2004}.\nThey maximise $C_0/C_1$ in the following finite box: positive integer\nparameters $(\\alpha_0,\\alpha_1,\\alpha_2;\\beta)$ with every entry at most\n$30$, greatest common divisor $1$, and\n\\[\n \\alpha_1\\le\\alpha_2,\\qquad\n \\alpha_1+\\alpha_2<\\beta\\le\\alpha_0+\\alpha_2.\n\\]\nThese are the source's parameter inequalities\n\\cite[Sec.~5, p.~161]{zudilin2004}, with a bound imposed for enumeration.\nThe definitions of $C_0,C_1$ and their floor function are reproduced in\nSection~\\ref{long1049:sec:open}. There are exactly $37{,}533$ tuples.\nExactly two attain the maximum: $(14,12,14;27)$ and $(15,12,13;26)$.\nBoth give $\\theta^*$, although they are distinct primitive directions.\n\nThe enumeration and comparisons are certified by rational intervals.\nThe fractions $j/c$ with $1\\le c\\le30$ partition $[0,1)$ into $278$\nhalf-open intervals. Every floor function occurring in the search is\nconstant on each interval, and the first interval contributes zero. On each remaining interval $[u,v)$,\nthe integral is a sum of positive differences\n$(k+u)^{-2}-(k+v)^{-2}$, weighted by $0$ or $1$. With $M=256$, the tail\nlies between\n\\[\n (M+u)^{-1}-(M+v)^{-1}\n \\quad\\hbox{and}\\quad\n (M+u)^{-1}-(M+v)^{-1}+(M+u)^{-2}-(M+v)^{-2}.\n\\]\nThis follows by comparing the positive decreasing summand with its\nintegral. Outward rounding of rational numbers and Machin's bounds for\n$\\pi$ give, for each of the two maximisers,\n\\[\n 0.40568302137302<C_0/C_1<0.40568302139506.\n\\]\nEvery other tuple has $C_0/C_1<0.40563943278333$. The intervals therefore\nseparate these two candidates from all other tuples, without relying\non floating-point ordering.\n\nTheir exact equality is not inferred from overlapping intervals.\nWrite $\\omega_A,\\omega_B$ for their step functions in the order just\nlisted, and let $J_A,J_B$ be their integrals against $d(-\\psi_1)$\non $[0,1]$. The floor formulas give\n\\[\n \\omega_B(u)-\\omega_A(u)\n =\\lfloor13u\\rfloor+\\lfloor15u\\rfloor-2\\lfloor14u\\rfloor.\n\\]\nFor every positive integer $c$, telescoping the $c$ intervals and\nsplitting the defining trigamma series into residue classes gives\n\\[\n \\int_0^1\\lfloor cu\\rfloor\\,d(-\\psi_1(u))\n =\\sum_{j=1}^{c-1}\\psi_1(j/c)-(c-1)\\psi_1(1)\n =\\frac{c(c-1)\\pi^2}{6}.\n\\]\nHence $J_B-J_A=\\pi^2/3$. Both directions have $m=15$ and\n$C_1=1091/2$, while the quadratic part of $C_0$ before the $3/\\pi^2$ correction is\n$266$ for $A$ and $265$ for $B$. Its decrease by $1$ exactly cancels\nthe increase $3(J_B-J_A)/\\pi^2=1$. Thus $C_0$ also agrees.\n\nA calculation compares every tuple of the box with interval bounds,\nand a second implementation repeats it with $512$ summands before the\ntail and a different enumeration order; both are in\n\\href{https://github.com/wcook04/plectis-erdos/blob/4beca4ad0c00a626def9c73f2420e20ebf8d1968/research/experiments/erdos1049/parameter_box.py}{one program}, with its \\href{https://github.com/wcook04/plectis-erdos/blob/4beca4ad0c00a626def9c73f2420e20ebf8d1968/research/experiments/erdos1049/receipts/parameter-box.json}{recorded result}.\nThe calculation proves optimality only in the displayed finite box. No\nbound on a direction outside that box follows, and this calculation\nis not needed for the irrationality theorem.\n\nThe two-coordinate forms in $1$ and $F(p)$ should be distinguished\nfrom simultaneous approximation to two target values. Postelmans and Van Assche prove the $\\Q$-linear independence\nof $1,\\zeta_q(1),\\zeta_q(2)$ for $q=1/p$ with integer $p\\ge2$\n\\cite[Thm.~1.3, p.~3]{postelmansvanassche2007}.\nTheir confluent multiple little $q$-Jacobi construction has two\northogonality conditions; its common integer normalisation and\nnonvanishing argument are in Section~6, especially (6.1)--(6.4).\nThat theorem treats two target values and retains the integer\ninverse-base restriction. It supplies neither the coefficient pairs\nnor the all-base estimates assumed in Theorem~\\ref{long1049:res:archcap};\nit is not an application at $3/2$.\n\n\\subsection{Finite calculations}\\label{long1049:sec:receipts}\n\nThe polynomial reconstruction below determines the forms for\n$n=1,2,3,4$. Together with the exact comparison of all directions in\nthe preceding finite box, it provides small cases against which to\ncheck the general formulas. The two calculations are available,\nalong with the other finite calculations, in the\n\\href{https://github.com/wcook04/plectis-erdos/blob/f5c9e560df38cd32e9b49473a409384aab00d9ae/research/experiments/erdos1049/README.md}{reproduction guide}, and a\n\\href{https://github.com/wcook04/plectis-erdos/blob/f5c9e560df38cd32e9b49473a409384aab00d9ae/research/experiments/erdos1049/replay.py}{single script}\nreruns them and compares each with its recorded result. The historical\nhigh-precision evaluations were not rerun. None of these finite\ncalculations proves a statement for all indices, and none is a Lean proof. The\ntests for $N\\le8$ and the rational interval certificate are described\nseparately.\n\nThe \\href{https://github.com/wcook04/plectis-erdos/blob/f5c9e560df38cd32e9b49473a409384aab00d9ae/research/experiments/erdos1049/README.md}{public reproduction guide} gives the command and dependency pin for\n\\href{https://github.com/wcook04/plectis-erdos/blob/39a0a2078c8f9fbe7fc24f912654815cc24685df/research/experiments/erdos1049/direction_search.py}{the direction-search program}.\nRun it with \\texttt{--bound 30} and the output path named there; adding\n\\texttt{--check} compares the result without replacing the\n\\href{https://github.com/wcook04/plectis-erdos/blob/39a0a2078c8f9fbe7fc24f912654815cc24685df/research/experiments/erdos1049/receipts/direction-search-bound30.json}{stored search results}.\nThe stored record specifies mpmath~1.3.0 at thirty decimal digits.  Its exact enumeration\ncount is $37{,}533$, but its ordering by $C_0/C_1$ is numerical and has no\ninterval certificate in that historical record. The separate rational\ncertificate above verifies the maximisers in the stated finite box;\nthe archived program itself still uses numerical comparisons.\n\nAn exact integer calculation (\\href{https://github.com/wcook04/plectis-erdos/blob/4beca4ad0c00a626def9c73f2420e20ebf8d1968/research/experiments/erdos1049/source_polynomials.py}{program}, \\href{https://github.com/wcook04/plectis-erdos/blob/4beca4ad0c00a626def9c73f2420e20ebf8d1968/research/experiments/erdos1049/receipts/source-polynomials.json}{every coefficient}) reconstructs the forms of\nSection~\\ref{long1049:sec:source-forms}\ndirectly in $\\Z[X]$ from the displayed source formulas. It checks\ndivision by $\\Omega_n$ and $X^{M_n}$ with zero remainder and computes\nall coefficients of $U_n,V_n$. The exact degrees are\n\\[\n\\begin{array}{c|rrrr}\n n&1&2&3&4\\\\\\hline\n K_n&587&2264&5032&8891\\\\\n W_n&333&1315&2944&5220\\\\\n \\deg D_N&72&278&628&1102\\\\\n \\deg\\Omega_n&25&94&219&380.\n\\end{array}\n\\]\nIn each case $\\deg V_n=W_n-1$ and both leading coefficients are\n$(-1)^n$. Independently expanding $F(1/q)=\\sum_{j\\ge1}\\tau(j)q^j$\nchecks the vanishing coefficients below order $K_n-W_n$ in\n$U_n(1/q)F(1/q)-V_n(1/q)$ and its next twelve coefficients against\nthe positive source expression. Exact rational evaluation at\n$31/4$, $3$ and $7/2$ also checks homogenisation; at the two noninteger\nbases, the power $b^{W_n-1}$ fails to clear $U_n(a/b)$, as predicted\nby the general leading-coefficient argument.\n\nA separate historical numerical report evaluated the remainder identity\nat those three bases to relative accuracy below $10^{-39}$ at $n=3$.\nIt reported positivity of $\\widehat\\Lambda_n$ and agreement with the\nbounds on $H_n$ in Section~\\ref{long1049:sec:integer-forms}. These\nhigh-precision real evaluations were not reproduced by the exact\npolynomial calculation just described.\n\nThe historical constant evaluation used $\\omega$, the thirteen intervals\nof Lemma~\\ref{long1049:res:omega-indicator} and forty-digit trigamma values.\nIt reported\n\\[\n J=77.94318447500909\\ldots,\\qquad C_0=221.30008816500502\\ldots,\n\\]\nand $C_1/C_0=2.46497868357497\\ldots$, agreeing with the source's printed\nprecision \\cite[p.~162]{zudilin2004}. The rational interval estimate stated earlier, rather than these\nrounded digits, supports the numerical inequality.\n\nThe main term $K_n\\log b-(K_n-W_n)\\log a$ at $31/4$ is negative\nfor every $1\\le n\\le400$, as verified by exact rational logarithm\nenclosures and exact totient sums (\\href{https://github.com/wcook04/plectis-erdos/blob/4beca4ad0c00a626def9c73f2420e20ebf8d1968/research/experiments/erdos1049/main_term_31_4.py}{program}, \\href{https://github.com/wcook04/plectis-erdos/blob/4beca4ad0c00a626def9c73f2420e20ebf8d1968/research/experiments/erdos1049/receipts/main-term-31-4.json}{result}). Dividing by $n^2$ gives, to three\ndecimal places, $-58.478,-10.280,-4.297,-3.863$ at\n$n=1,10,100,400$. This finite sign check does not assert negativity at\nevery index. The proved limit is\n$C_1\\log4-C_0\\log31=-3.718\\ldots$; the elementary convergence-rate bound\nis $O(\\log^2(n+2)/n)$.\nIndeed, summing the $O(y\\log(2y))$ endpoint errors over the floor blocks\nwith $k\\le n$ gives $O(n\\log^2(n+2))$; the remaining main-term tail is\n$O(1)$ since its summands are $O(n^2/k^3)$.  The linear terms of $M_n$\nadd $O(n)$.  The stronger rate $O(1/n)$ does not follow from these estimates.  Only the limit enters Theorem~\\ref{long1049:res:region}.\n\nFor orientation, the all-rank order and leading-coefficient formulas of\nTheorem~\\ref{long1049:res:zudilin-sharp-qorder} give the following first seven values:\n\\[\n\\begin{array}{c|rrrrrrr}\nN&1&2&3&4&5&6&7\\\\\\hline\n\\operatorname{ord}&0&1&5&14&30&55&91\\\\\n\\operatorname{lc}&1&6&108&4320&324000&40824000&8001504000.\n\\end{array}\n\\]\n\n\n\\section{Power comparisons and Hankel determinants}\\label{long1049:sec:sharp}\n\nThree integer comparisons will be used to estimate residue counts\nand scalar exponents. The first gives $\\log3/\\log2<65/41$ and shows\nthat $65$ is the least integer $q$ with $3^{41}<2^q$; the inequality\n$2^{129}<3^{82}$ checks the failure of the subset count at depth $41$\none row earlier. They determine numerical thresholds, leaving the\ncoefficient and remainder hypotheses to be checked for any family\nof approximants. Readers interested first in the determinant\nasymptotic may begin at Section~\\ref{long1049:sec:hankel-order}.\n\n\\begin{theorem}[sharp power bracket]\\label{long1049:res:powerbracket}\nOne has\n\\[\n 2^{64}<3^{41}<2^{65}.\n\\]\nConsequently\n\\[\n \\frac{41}{65}<\\frac{\\log2}{\\log3},\n \\qquad\n \\frac{\\log3}{\\log2}<\\frac{65}{41}.\n\\]\n\\end{theorem}\n\n\\begin{proof}\nDirect evaluation gives\n\\[\n \\begin{aligned}\n 2^{64}&=18446744073709551616,\\\\\n 3^{41}&=36472996377170786403,\\\\\n 2^{65}&=36893488147419103232.\n \\end{aligned}\n\\]\nTaking logarithms of the upper bound gives $41\\log3<65\\log2$.\nDividing this inequality by $65\\log3$ and by $41\\log2$, respectively,\ngives the two stated logarithmic bounds; both divisors are positive.\n\\end{proof}\n\nThe upper power bound gives $\\log2/\\log3>41/65$, a sharper lower\nbound than $81/200$. In particular,\n$41/65-2/5=3/13$. Combining this with\n$1/2-1/\\pi^2<2/5$ gives the first gap below; the same logarithmic\nlower bound is then compared with the rectangular expression.\n\nFor $\\rho\\ge0$ and $\\sigma\\ge1+\\rho$, define\n\\[\n \\Theta_{\\mathrm{HP}}(\\rho,\\sigma)=\n \\frac{(1+\\rho^2)/2+\\sigma-3\\sigma^2/\\pi^2}\n {(1+\\rho)^2/2+\\sigma(1+\\rho)+(1+\\rho^2)/2+\\sigma}.\n\\]\nThe denominator is positive on this domain.\n\n\\begin{corollary}[gaps between the stated logarithmic thresholds]\\label{long1049:res:sharpgaps}\nFor every $\\rho,\\sigma\\in\\R$ with $0\\le\\rho$ and $1+\\rho\\le\\sigma$,\n\\[\n \\frac3{13}<\\frac{\\log2}{\\log3}\n   -\\left(\\frac12-\\frac1{\\pi^2}\\right),\n \\qquad\n \\frac3{13}<\\frac{\\log2}{\\log3}-\\Theta_{\\mathrm{HP}}(\\rho,\\sigma),\n\\]\nwhere $\\Theta_{\\mathrm{HP}}$ is the rectangular exponent threshold.  Moreover\n\\[\n \\frac{\\log3/\\log2-1}{3}<\\frac8{41}.\n\\]\n\\end{corollary}\n\n\\begin{proof}\nThe first inequality follows from\n$41/65-2/5=3/13$, Theorem~\\ref{long1049:res:powerbracket}, and\n$1/2-1/\\pi^2<2/5$. The polynomial calculation in\nSection~\\ref{long1049:sec:open} proves\n$\\Theta_{\\mathrm{HP}}(\\rho,\\sigma)\\le1/2-1/\\pi^2$ on the stated\ndomain, which gives the second inequality.  The final inequality is a direct\nrearrangement of $\\log3/\\log2<65/41$.\n\\end{proof}\n\nThe next inequalities compare two proposed savings in the denominator\nexponent with the exponent $4N^3-3N^2$ before division. Here $E$ denotes\nthe exponent saved. The statement assumes the bounds $E\\le N^3-N$ or\n$E\\le2N^3-N$; it does not derive them for a polynomial family. Under either\nbound, the saving is less than $39/41$ of the original exponent. Thus\nthese bounds alone cannot justify the reduction required by this model.\n\n\\begin{theorem}[bounds for two proposed degree savings]\\label{long1049:res:chargeceilings}\nFor every integer $N>0$,\n\\[\n 41(N^3-N)<39(4N^3-3N^2).\n\\]\nFor every integer $N\\ge2$,\n\\[\n 41(2N^3-N)<39(4N^3-3N^2).\n\\]\nHence the same strict inequalities hold with the left side replaced by\n$41E$ whenever, respectively, $E\\le N^3-N$ or $E\\le2N^3-N$.\n\\end{theorem}\n\n\\begin{proof}\nAfter subtraction, the first inequality is\n\\[\n N(115N^2-117N+41)>0,\n\\]\nwhose quadratic factor is $115(N-1)^2+113(N-1)+39$, positive for\n$N\\ge1$. The second becomes\n\\[\n N(74N^2-117N+41)>0.\n\\]\nIts quadratic factor is $74(N-2)^2+179(N-2)+103$, positive for\n$N\\ge2$. The assertions for $E$ follow by monotonicity.\n\\end{proof}\n\nAn application of these degree bounds must separately\nprove that its proposed divisor satisfies one of the stated bounds.\nThese inequalities do not rule out additional factors caused by\ncancellation in a determinant, a different choice of integral forms,\nor another proof of irrationality at $3/2$.\n\n\\subsection{The sharp \\texorpdfstring{$q$}{q}-order of the normalised Hankel\ndeterminant}\\label{long1049:sec:hankel-order}\n\\papersectiontarget{long1049-formal-rows}\n\nWe begin with the formal expansion of Zudilin's normalised Hankel\ndeterminant at $x=z=1$. Determining its first nonzero term is a\ncalculation in power series in $q$; an interpretation as a divisor\nafter integer evaluation would additionally require an integral\nnormalisation and a divisibility argument. Zudilin constructs the\ndeterminant by combining the Pad\\'e-type approximations of Bundschuh\nand Zudilin \\cite{bundschuhzudilin2008} with B\\'ezivin's method as developed\nin \\cite{krvz2009}; see \\cite[Sec.~2, p.~3]{zudilin2016}.  His passage from the\n$q$-order to the size of the determinant borrows the proofs of\n\\cite[Lemma~2, p.~12, and Prop.~3, p.~13]{krvz2009}, as he notes in\n\\cite[Sec.~4, p.~7]{zudilin2016}.\n\nThe source writes $\\ell_p(x,z)=x\\sum_{r\\ge1}z^r/(p^r-x)$, so\n$\\ell_p(1,1)=F(p)$. Here $p=q^{-1}$ and $x,z$ are auxiliary\nparameters, not the base. Write $N$ for the Hankel rank. With $q$ a\nformal variable, the moments and determinant in question are\n\\[\n v_m^*=\\sum_{t\\ge0}q^{(m+1)t}\n       \\frac{(q;q)_m^3(q^{t+1};q)_m}{(q^{m+t+1};q)_{m+1}},\n \\qquad V_N^*=\\det(v_{i+j}^*)_{0\\le i,j<N}.\n\\]\nAll product denominators have constant term $1$, so their inverses\nexist in $\\Z[[q]]$. The order of a nonzero series is the least\nexponent of $q$ with a nonzero coefficient.\n\n\\begin{theorem}[the first nonzero term at every rank]\\label{long1049:res:zudilin-sharp-qorder}\nFor every rank $N$, the normalised Hankel determinant $V_N^{*}$ of\n\\cite[Sec.~4, (6), p.~6]{zudilin2016}, evaluated at $x=z=1$, has\n\\[\n \\operatorname{ord}_q V_N^{*}=\\frac{N(N-1)(2N-1)}{6},\n \\qquad\n \\text{leading coefficient}\\quad\\frac{(N!)^{2}(N+1)!}{2^{N}} .\n\\]\n\\end{theorem}\n\n\\noindent The source proves the order inequality and also gives\n$|V_N^*(q)|\\le |q|^{N^3/3}\\exp(O(N^2))$, referring to additional estimates\nfor the passage from formal order to analytic size\n\\cite[Sec.~4, p.~7]{zudilin2016}.\nEquality in the order bound rules out a further initial power of $q$ at\n$x=z=1$; it does not itself prove a fixed-$q$ estimate.\nFor $0<q<1$, the positive-measure argument below gives both positivity\nand a two-sided comparison with the leading term, with logarithmic error\n$O_q(N)$. It refines the subleading control, not the cubic exponent of $q$.\n\n\\paragraph{The row identity.}\nWe choose row operations whose first surviving coefficients can be\ncomputed explicitly. The auxiliary lemma applies to more general\nproducts than the moments above; its use here is to identify the\ncoefficient that remains after each row operation.\n\nWork over $A=\\Z[[q]]$. Let $H(X)=1+\\sum_{s\\ge1}a_s(q)X^s$.\nFor nonnegative integers $m,t$, put\n\\[\n W_m(t)=q^{(m+1)t}\\prod_{r=1}^{m}H(q^{r}),\n \\qquad\n D_j=\\prod_{r=0}^{j-1}(I-q^{r}\\mathcal N),\n \\qquad\n E(m,j)=mj-\\frac{j(j-1)}2,\n\\]\nwhere $\\mathcal N$ is the backward shift $(\\mathcal Nf)_m=f_{m-1}$ in the index\n$m$. Thus $D_j$ is Zudilin's backward-difference operator in\n\\cite[Sec.~4, (7), p.~6]{zudilin2016}.  Write $\\bar H=H\\bmod q$ and\n$h_r=[X^{r}]\\bar H(X)^{-1}$, with $h_r=0$ for $r<0$ and $h_0=1$.\n\nThe lemma allows arbitrary coefficients $a_s(q)\\in\\Z[[q]]$; the only\nnormalisation imposed on $H$ is its constant term $1$. It therefore\ncovers the product series used below, but not an unnormalised series\nwith a different constant term. All statements here are coefficientwise\nidentities of formal series, with no analytic convergence assumption.\nFor a general $H$, the coefficient in the lemma can vanish. The\napplication below computes it for the chosen products and proves the\nrequired nonvanishing.\n\n\\begin{lemma}[a coefficient of each transformed row]\\label{long1049:res:allrowinitial}\nFor $m\\ge j\\ge0$ one has $D_jW_m(t)\\in q^{E(m,j)}A$ and\n\\[\n \\bigl[q^{E(m,j)}\\bigr]D_jW_m(t)=(-1)^{j}h_{j-t}.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nLet $e_u$ denote a formal basis vector indexed by $u\\ge0$, and suppress\nthe argument $q$ in $a_s(q)$. We use the transition rule\n\\[\n \\Phi e_0=\\sum_{s\\ge1}a_se_{s-1},\n \\qquad\n \\Phi e_u=(q^{u}-1)e_{u-1}+q^{u}\\sum_{s\\ge1}a_se_{u+s-1}\\quad(u>0).\n\\]\nIn $\\Phi^je_t$, a coefficient means the sum of the weights of length-$j$\npaths from $t$ to the specified endpoint. Each step lowers its index by\nat most one. A path ending at $u$ therefore visits only indices at most\n$\\max(t,u+j)$, so each coefficient is a finite sum. No action on the\nalgebraic direct sum is assumed.\n\nFor $m\\ge1$, a direct expansion gives\n\\[\n W_m(u)-W_{m-1}(u)=q^m\\sum_v(\\Phi e_u)_vW_{m-1}(v),\n\\]\nsince both sides equal\n\\[\n q^{mu}\\prod_{r=1}^{m-1}H(q^r)\\bigl(q^uH(q^m)-1\\bigr).\n\\]\nFor the induction step take $m\\ge j+1$ and note that\n$E(m,j)-E(m-1,j)=j$. Applying $I-q^j\\mathcal N$ to $q^{E(m,j)}W_{m-j}(u)$ therefore gives\n$q^{E(m,j)}(W_{m-j}(u)-W_{m-j-1}(u))$.\nThe preceding identity contributes one further factor $q^{m-j}$, and\n$E(m,j)+m-j=E(m,j+1)$. Induction, starting with $D_0=I$, gives\n\\begin{equation}\\label{long1049:eq:state-expansion}\n D_jW_m(t)=q^{E(m,j)}\\sum_u(\\Phi^{j}e_t)_uW_{m-j}(u).\n\\end{equation}\nThe endpoint sum is also locally finite:\n$\\operatorname{ord}W_{m-j}(u)=(m-j+1)u$, so only finitely many endpoints\ncontribute to any fixed power of $q$.\n\nModulo $q$ the operator simplifies: $\\bar\\Phi e_u=-e_{u-1}$ for $u>0$, and\n$\\bar\\Phi e_0=\\sum_{s\\ge1}a_s(0)e_{s-1}$.  Put $c_{j,t}=(\\bar\\Phi^{j}e_t)_0$.\nThen $c_{j+1,t}=-c_{j,t-1}$ for $t>0$ and\n$c_{j+1,0}=\\sum_{s=1}^{j+1}a_s(0)c_{j,s-1}$, the sum terminating because a state\nabove $j$ cannot reach $0$ in $j$ steps.  Now $c_{0,t}=\\delta_{t,0}=h_{-t}$, and\nif $c_{j,t}=(-1)^{j}h_{j-t}$ for all $t$ then $c_{j+1,t}=(-1)^{j+1}h_{j+1-t}$ for\n$t>0$ at once, while for $t=0$ the identity $\\bar H\\bar H^{-1}=1$ gives\n$h_{j+1}=-\\sum_{s\\ge1}a_s(0)h_{j+1-s}$ and hence\n$c_{j+1,0}=(-1)^{j}\\sum_{s\\ge1}a_s(0)h_{j+1-s}=(-1)^{j+1}h_{j+1}$.  So\n$c_{j,t}=(-1)^{j}h_{j-t}$ for all $j,t$.  Since\n$\\operatorname{ord}W_{m-j}(u)=(m-j+1)u$ and $m\\ge j$, in\n\\eqref{long1049:eq:state-expansion} only the state $u=0$ contributes at degree $E(m,j)$,\nwhich gives both assertions.\n\\end{proof}\n\n\\noindent Apply the lemma to a summand of the normalised remainder:\n\\[\n T_{m,t}=q^{(m+1)t}\\,(q;q)_m^{3}\\,\\frac{(q^{t+1};q)_m}{(q^{m+t+1};q)_{m+1}} .\n\\]\nSet\n\\[\n H_t(X)=\\frac{(1-X)^{3}(1-q^{t}X)^{2}}{(1-q^{t}X^{2})(1-q^{t+1}X^{2})}.\n\\]\nThen $T_{m,t}=(1-q^{t+1})^{-1}W_m^{H_t}(t)$, by the telescoping identity\n\\[\n \\prod_{r=1}^{m}H_t(q^r)=\n \\frac{(q;q)_m^3(1-q^{t+1})(q^{t+1};q)_m}{(q^{m+t+1};q)_{m+1}}.\n\\]\nThe two denominator products contribute the consecutive factors\n$1-q^{t+2},\\ldots,1-q^{t+2m+1}$.\nReducing modulo $q$ gives $\\bar H_t=(1-X)^{3}$ for $t>0$, so\n$h^{(t)}_r=\\binom{r+2}{2}$, and $\\bar H_0=(1-X)^{4}/(1+X)$, so\n$h^{(0)}_r=(r+1)(r+2)(2r+3)/6$. The scalar factor\n$(1-q^{t+1})^{-1}$ has constant term $1$. The rows are $v_m^*=\\sum_{t\\ge0}T_{m,t}$. In $D_j$ only the shifts\n$m,m-1,\\ldots,m-j$ occur. Since $m\\ge j$ and\n$\\operatorname{ord}T_{m-r,t}=(m-r+1)t\\ge(m-j+1)t$, only finitely many\n$t$ can affect any fixed coefficient after any of these shifts.\nThis justifies interchanging the sum and $D_j$, and then extracting its\ncoefficient at degree $E(m,j)$. Terms with $t>j$ contribute\n$h^{(t)}_{j-t}=0$, so Lemma~\\ref{long1049:res:allrowinitial} at\n$m=j+\\ell$, where $E(j+\\ell,j)=j(j+1)/2+j\\ell$, gives\n\\begin{equation}\\label{long1049:eq:row-initial}\n D_jv_{j+\\ell}^*=(-1)^{j}\\frac{(j+1)^{2}(j+2)}2\\,q^{\\,j(j+1)/2+j\\ell}\n +O\\bigl(q^{\\,j(j+1)/2+j\\ell+1}\\bigr)\n \\qquad(j,\\ell\\ge0).\n\\end{equation}\nThe coefficient is obtained by summing\n\\[\n h^{(0)}_j+\\sum_{t=1}^{j}h^{(t)}_{j-t}\n =\\frac{(j+1)(j+2)(2j+3)}6+\\binom{j+2}3\n =\\frac{(j+1)^{2}(j+2)}2 .\n\\]\n\n\\begin{proof}[Proof of Theorem~\\ref{long1049:res:zudilin-sharp-qorder}]\nThe operators $D_j$ act by lower unitriangular row operations, so they leave\n$\\det(v_{i+j}^*)_{0\\le i,j<N}$ unchanged.  By~\\eqref{long1049:eq:row-initial} the entry in\nrow $j$ and column $\\ell$ has order $j(j+1)/2+j\\ell$. For $N=2$,\nthe matrix of entry orders is\n\\[\n \\begin{pmatrix}0&0\\\\1&2\\end{pmatrix}.\n\\]\nThe off-diagonal product is the unique term of order one. Its\npermutation sign cancels the negative leading coefficient of the second\nrow, giving $6q+O(q^2)$. At arbitrary rank the same argument selects\nthe reversed permutation. For a permutation $\\varsigma$, the corresponding\nLeibniz term has weight $\\sum_j\\bigl(j(j+1)/2+j\\varsigma(j)\\bigr)$.\nBy the rearrangement inequality, $\\sum_jj\\varsigma(j)$ is uniquely\nminimised by the reversal\n$\\varsigma(j)=N-1-j$, the values $j$ being distinct. The minimum weight is\n$\\sum_{j<N}j^{2}=N(N-1)(2N-1)/6$, so exactly one Leibniz term attains it and no\ncancellation is possible there. The sign of the reversal is\n$(-1)^{N(N-1)/2}$, which cancels $\\prod_{j<N}(-1)^{j}$, and the surviving\ncoefficient is\n\\[\n \\prod_{j=0}^{N-1}\\frac{(j+1)^{2}(j+2)}2=\\frac{(N!)^{2}(N+1)!}{2^{N}} .\n\\]\n\\end{proof}\n\n\\paragraph{Formal order and analytic size are different questions.}\nTheorem~\\ref{long1049:res:zudilin-sharp-qorder} fixes the first nonzero power of $q$ and\nits coefficient at each fixed rank.  It does not control the value at a fixed\nrational $q$ as the rank grows.  For the rest of this subsection, write $B_N=\\sum_{j<N}j^2$ and\n$C_N=(N!)^2(N+1)!/2^N$ for the order and leading coefficient.\nThe integer polynomials $f_N(q)=C_Nq^{B_N}(1-q)^{N^3}$ have exactly\nthese same two quantities, whereas $f_N(2/3)=C_N(2/3)^{B_N}3^{-N^{3}}$ carries a further\ncubic exponential factor that neither datum sees.  The following separate\npositive-measure argument supplies the fixed-base estimate for $V_N^*$;\nit is not inferred from the formal order.\n\n\\paragraph{A separate positive-measure estimate.}\nWe seek positive moment weights comparable to $(k+1)^2(k+2)/2$,\nwith constants depending only on $q$. In the determinant expansion\nthese constants give factors exponential in $N$, so they do not\nchange its cubic exponent of $q$.\nFor fixed $0<q<1$ write $P=(q;q)_\\infty$, $Q=(\\sqrt q;q)_\\infty$,\n$T=(-1;q)_\\infty^2$, and\n\\[\n G_q(w)=\\frac1{(w;q)_\\infty^3}\\sum_{t\\ge0}\\frac{w^t}{(q;q)_t}\n          \\frac{(q^tw^2;q)_\\infty}{(q^tw;q)_\\infty^2},\n \\qquad \\gamma_k=[w^k]G_q(w),\\qquad c_k=\\frac{(k+1)^2(k+2)}2.\n\\]\nThe choice $w=q^{m+1}$ collects all dependence on the moment index.\nIndeed, the three finite products in the $t$th remainder summand become\n\\[\n (q;q)_m=\\frac{P}{(w;q)_\\infty},\\quad\n (q^{t+1};q)_m=\\frac{P}{(q;q)_t(q^tw;q)_\\infty},\\quad\n (q^{m+t+1};q)_{m+1}\n =\\frac{(q^tw;q)_\\infty}{(q^tw^2;q)_\\infty}.\n\\]\nTheir substitution gives $v_m^*=P^4G_q(q^{m+1})$ term by term.\nFor fixed $q$ and $0<r<1$, the $t$th summand on $|w|\\le r$ satisfies\n\\[\n \\left|\\frac{w^t(q^tw^2;q)_\\infty}\n {(q;q)_t(w;q)_\\infty^3(q^tw;q)_\\infty^2}\\right|\n \\le \\frac{(-r^2;q)_\\infty}{P(r;q)_\\infty^5}\\,r^t.\n\\]\nIndeed, $(q;q)_t\\ge P$, each denominator product in $w$ has modulus\nat least $(r;q)_\\infty>0$, and the numerator product has modulus at\nmost $(-r^2;q)_\\infty$. The bound is independent of $t$, so the sum\nconverges uniformly and absolutely on each such disk. The individual\nproducts converge there as well, since their tails are bounded by\ngeometric series in $q$. Thus $G_q$ is holomorphic for $|w|<1$,\nand its Taylor series may be evaluated at $w=q^{m+1}$ to obtain the\nmoment expansion. For formal series, the corresponding substitution is\njustified coefficient by coefficient, as in the short paper.\nThe $q$-binomial theorem \\cite[Eq.~17.2.37]{dlmf}\n\\[\n \\frac{(Aw;q)_\\infty}{(w;q)_\\infty}\n   =\\sum_{j\\ge0}\\frac{(A;q)_j}{(q;q)_j}w^j\\quad(0\\le A\\le1)\n\\]\nfollows by comparing coefficients in\n$(1-w)R(w)=(1-Aw)R(qw)$ with $R(0)=1$; the series converges for $|w|<1$\nsince its coefficients are at most $P^{-1}$.  They are nonnegative, and\nat least $(A;q)_\\infty$ if $A<1$.  For $A=1$ the series equals $1$.\nFactor the numerator of the $t$th term using\n\\[\n (q^tw^2;q)_\\infty=(q^{t/2}w;q)_\\infty(-q^{t/2}w;q)_\\infty\n (q^{(t+1)/2}w;q)_\\infty(-q^{(t+1)/2}w;q)_\\infty.\n\\]\nAfter pairing each positive-argument factor with one denominator,\nthe $t$th summand of $G_q$ becomes\n\\[\n\\begin{aligned}\n &\\frac{w^t}{(q;q)_t}\n  \\frac{(q^{t/2}w;q)_\\infty}{(w;q)_\\infty}\n  \\frac{(q^{(t+1)/2}w;q)_\\infty}{(w;q)_\\infty}\\\\\n &\\qquad{}\\times\n  \\frac{(-q^{t/2}w;q)_\\infty(-q^{(t+1)/2}w;q)_\\infty}\n       {(w;q)_\\infty(q^tw;q)_\\infty^2}.\n\\end{aligned}\n\\]\nThe $q$-binomial identity makes the two ratios nonnegative\ncoefficientwise; the remaining product also has nonnegative coefficients.\nWhen $t=0$, the first ratio is $1$ and every coefficient of the second\nis at least $Q$. The last fraction has coefficients at least those\nof $(1-w)^{-3}$. Thus the $t=0$ term alone bounds $\\gamma_k$ below by\n$Q\\binom{k+3}3$.\n\nFor the upper bound, if $R$ has nonnegative coefficients and\n$R(1)\\le C<\\infty$, convolution with a nondecreasing sequence $d_k$\nis bounded by $Cd_k$. At $t=0$, bound the second ratio coefficientwise\nby $P^{-1}(1-w)^{-1}$. After extracting $(1-w)^{-3}$ from the last\nfraction, its remaining factor has value at $w=1$ at most $TP^{-3}$.\nThis gives $TP^{-4}\\binom{k+3}3$.\nFor $t\\ge1$, both ratios are bounded by $P^{-1}(1-w)^{-1}$.\nExtracting the one factor $(1-w)^{-1}$ from the last fraction leaves\na factor with value at $1$ at most $TP^{-3}$; also $(q;q)_t^{-1}\\le P^{-1}$.\nThe $t$th summand is therefore bounded coefficientwise by\n$TP^{-6}w^t(1-w)^{-3}$. Summing $t\\ge1$ gives\n$TP^{-6}\\binom{k+2}3$ as the bound for its $k$th coefficient. Consequently\n\\[\n \\frac Q3 c_k\\le\\gamma_k\\le T(P^{-4}+P^{-6})c_k.\n\\]\nThus $\\sum_{k\\ge0}P^4\\gamma_kq^k\\delta_{q^k}$ is a finite positive measure\non $[0,1]$, with infinitely many distinct support points and moments\n$v_m^*(q)$. In standard terminology, $(v_m^*(q))_{m\\ge0}$ is a\nHausdorff moment sequence. This statement fixes $q$; the measure is\nnot a representing measure for the coefficient sequence $s_m(p)$\nconsidered in the next subsection.\nA nonzero polynomial of degree less than $N$ cannot vanish at all of its\nfirst $N$ atoms; the associated Gram matrix is positive definite.\nTruncate the measure to its first $K+1$ atoms. At fixed rank $N$, each\nmatrix entry converges as $K\\to\\infty$, and the determinant converges\nbecause it is a polynomial in those entries. Finite Cauchy--Binet and\nmonotone convergence of the nonnegative tuple sums therefore give\nHeine's expansion, whose integral form is reproved in\n\\cite[Sec.~2, (2)--(5), pp.~2--3]{zudilin2017det}:\n\\[\n V_N^*=\\sum_{k_0<\\cdots<k_{N-1}}\n \\prod_i(P^4\\gamma_{k_i}q^{k_i})\\prod_{i<j}(q^{k_i}-q^{k_j})^2.\n\\]\nRetaining the tuple $k_i=i$ and using\n$\\prod_{d=1}^{N-1}(1-q^d)^{2(N-d)}\\ge P^{2N}$ gives the lower bound\nbelow. For the upper bound put $k_i=i+\\lambda_i$, where the\n$\\lambda_i$ are nonnegative and nondecreasing. Factoring the smaller\npower from each Vandermonde difference gives the exact exponent\n\\[\n \\sum_i k_i+2\\sum_{i<j}k_i\n =B_N+\\sum_{i=0}^{N-1}(2N-1-2i)\\lambda_i.\n\\]\nEvery weight $2N-1-2i$ is at least $1$. Since $0<q<1$, the remaining\npower of $q$ is at most $q^{\\sum_i\\lambda_i}$. Also\n$c_{i+\\lambda}/c_i\\le(\\lambda+1)^3$. Discard the remaining\nVandermonde factors, each bounded by $1$, and enlarge the nonnegative\nsum by dropping the ordering restriction on the $\\lambda_i$.\nIt now factors into $N$ copies of $\\sum_{\\lambda\\ge0}(\\lambda+1)^3q^\\lambda$,\nwhich gives\n\\[\n (P^6Q/3)^N C_Nq^{B_N}\\le V_N^*(q)\n \\le\\left[P^4T(P^{-4}+P^{-6})\n          \\frac{1+4q+q^2}{(1-q)^4}\\right]^N C_Nq^{B_N}.\n\\]\nBoth constants are positive and finite, so $V_N^*(q)>0$ and\n$\\log(V_N^*(q)/(C_Nq^{B_N}))=O_q(N)$, including $N=0$ under the empty\ndeterminant convention.\nHere $N\\to\\infty$ with $q$ fixed; the constants are not uniform as\n$q\\uparrow1$. The formal-order calculation instead fixes $N$ and expands\nat $q=0$. No joint uniform limit is asserted. The measure depends on $q$, not on the moment index or rank: one\nmeasure represents the whole sequence at that base. This is an\nanalytic representation, not a denominator factor for the 2004\npolynomial forms.\nSection~\\ref{long1049:sec:sharp-fixed-base} replaces the band $O_q(N)$ by an\nexact asymptotic with an explicit positive constant, using the same weights.\n\n\\paragraph{Formal verification and computations.}\nThe positive-measure argument, including its generating-function\nidentification and infinite determinant expansion, is an ordinary proof\nand is not formalised. The formal-series argument has a checked\n\\lword{Erdos1049/AdelicHeightBridge.lean}{200}{hankelAssociatedCoeff_eq_reciprocal}{recurrence for the leading coefficients}.\nThe Lean proof of Theorem~\\ref{long1049:res:zudilin-sharp-qorder} uses\nthe first nonzero term of every transformed row, which\nfor row $1$ has order exactly $l+1$ and coefficient exactly $-6$ in every\ncolumn.  Independently of the proof, the order and the\nleading coefficient were computed exactly for $1\\le N\\le7$\n(\\href{https://github.com/wcook04/plectis-erdos/blob/f5c9e560df38cd32e9b49473a409384aab00d9ae/research/experiments/erdos1049/hankel_qorder.py}{program}, \\href{https://github.com/wcook04/plectis-erdos/blob/f5c9e560df38cd32e9b49473a409384aab00d9ae/research/experiments/erdos1049/receipts/hankel-qorder.json}{result}), giving orders\n$0,1,5,14,30,55,91$ and leading coefficients\n$1,6,108,4320,324000,40824000,8001504000$, each matching the closed forms.\n\n\\paragraph{Earlier work and the unresolved value at $3/2$.}\nThe antecedent is the inequality of \\cite[Sec.~4, p.~7]{zudilin2016}; the equality and\nthe leading coefficient are proved here.\nLemma~\\ref{long1049:res:allrowinitial} and~\\eqref{long1049:eq:row-initial}\ngive that row calculation.  Nothing in\nthis subsection decides the arithmetic nature of $F(3/2)$.\n\n\n\\subsection{The size of \\texorpdfstring{$V_N^*$}{V(N)} at a fixed base}\n\\label{long1049:sec:sharp-fixed-base}\n\nThe preceding bounds leave an error of order $N$ in the logarithm.\nWe now determine its linear, logarithmic and constant terms. There\nare two calculations: compare every node configuration with the\nfirst $N$ nodes, then multiply the resulting moment weights.\nThe first is independent of the particular weights under the\nhypotheses below; the second produces the Lambert-value correction\nfor $V_N^*$. Throughout this subsection $0<q<1$ is fixed, and we write\n\\[\n P=(q;q)_\\infty,\\qquad\n \\mathcal M(q)=\\prod_{d\\ge1}(1-q^d)^{-d},\\qquad\n L=F(1/q)=\\sum_{r\\ge1}\\frac{q^r}{1-q^r},\n\\]\nwith $B_N=\\sum_{j<N}j^2$ and $C_N=\\prod_{k<N}c_k=(N!)^2(N+1)!/2^N$ as above.\nBoth products converge to positive finite values, and the abbreviation $L$ is\nused only in this subsection; it is the value at $q$ of the divisor series\n$\\mathcal L$ of Section~\\ref{long1049:sec:open}.\nThe estimates in this subsection concern real determinants at fixed\n$q$. Denominator valuations for the 2004 forms were treated separately\nin Section~\\ref{long1049:sec:region}.\n\nFor $a_k=1$, the moments are $(1-q^{m+1})^{-1}$, so Cauchy's\ndeterminant formula gives the model calculation\n\\begin{equation}\\label{long1049:eq:cauchy-model}\n D_N^{(0)}:=\\det\\left(\\frac1{1-q^{i+j+1}}\\right)_{0\\le i,j<N}\n =\\frac{q^{B_N}\\Delta_N}{\\prod_{0\\le i,j<N}(1-q^{i+j+1})},\n \\qquad\n \\Delta_N=\\prod_{d=1}^{N-1}(1-q^d)^{2(N-d)}.\n\\end{equation}\nThe denominator has limiting multiplicity $d$ at $1-q^d$, while\n\\[\n \\frac{\\Delta_N}{P^{2N}}\n =\\prod_{d<N}(1-q^d)^{-2d}\\prod_{d\\ge N}(1-q^d)^{-2N}\n \\longrightarrow\\mathcal M(q)^2.\n\\]\nThe convergence follows from $\\sum d|\\log(1-q^d)|<\\infty$ and the\nbound $O_q(Nq^N)$ for the logarithm of the second product. Hence\n\\begin{equation}\\label{long1049:eq:cauchy-model-limit}\n D_N^{(0)}\\sim\\mathcal M(q)^3q^{B_N}P^{2N}.\n\\end{equation}\nTwo copies of $\\mathcal M(q)$ come from the squared Vandermonde\nand one from the Cauchy denominator. We shall compare general\nweights with this exact model. Fixed displacements of the last\nfew node indices have weight ratios tending to $1$; the polynomial\nbound in the theorem controls the sum over all displacements.\n\n\\begin{theorem}[determinants of geometric moments]\n\\label{long1049:thm:geometric-universality}\nLet $a_k>0$ satisfy, for fixed constants $C$ and $\\kappa$,\n\\[\n \\frac{a_{k+h}}{a_k}\\longrightarrow1\\quad(k\\to\\infty)\n \\text{ for each fixed }h\\ge0,\n \\qquad\n \\frac{a_{k+h}}{a_k}\\le C(1+h)^\\kappa\\quad(k,h\\ge0).\n\\]\nPut $M_m=\\sum_{k\\ge0}a_kq^{(m+1)k}$ and $D_N=\\det(M_{i+j})_{0\\le i,j<N}$. Then\n\\[\n D_N\\sim\\mathcal M(q)^3\\,q^{B_N}P^{2N}\\prod_{k=0}^{N-1}a_k\n \\qquad(N\\to\\infty).\n\\]\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{long1049:thm:geometric-universality}]\nWe may increase $C$ to at least $1$ and $\\kappa$ to at least $0$.\nThe bound $a_h\\le Ca_0(1+h)^\\kappa$ makes\n$\\sum_{k\\ge0}a_kq^k\\delta_{q^k}$ a finite positive measure with\nmoments $M_m$. Apply Cauchy--Binet to finite truncations. Entrywise\nconvergence gives convergence of their determinants, and monotone\nconvergence applies to the nonnegative tuple sums. Hence\n\\begin{equation}\\label{long1049:eq:polynomial-heine}\n D_N=\\sum_{0\\le k_0<\\cdots<k_{N-1}}\n \\prod_{i<N}a_{k_i}q^{k_i}\n \\prod_{i<j<N}(q^{k_i}-q^{k_j})^2.\n\\end{equation}\nThe tuple $k_i=i$ contributes $q^{B_N}\\Delta_N\\prod_{i<N}a_i$.\nIt does not account for the whole asymptotic. For example, replace\n$(0,1,\\ldots,N-2,N-1)$ by $(0,1,\\ldots,N-2,N)$. The ratio of\nits contribution to that of the original tuple is\n\\begin{equation}\\label{long1049:eq:last-node-shift}\n q\\frac{a_N}{a_{N-1}}\n \\prod_{d=1}^{N-1}\\left(\\frac{1-q^{d+1}}{1-q^d}\\right)^2\n =q\\frac{a_N}{a_{N-1}}\\left(\\frac{1-q^N}{1-q}\\right)^2\n \\longrightarrow\\frac{q}{(1-q)^2}.\n\\end{equation}\nOnly pairs involving the moved node change, so the product\ntelescopes. At $q=1/2$ the limiting ratio is $2$: the displaced tuple\ncontributes twice as much asymptotically as $(0,1,\\ldots,N-1)$.\n\nSubtracting $(0,1,\\ldots,N-1)$ gives the shifts $\\lambda_i=k_i-i$.\nSince $k_{i+1}\\ge k_i+1$, these are nonnegative and weakly increasing;\nreversing their order gives a partition. For $N=4$, for example,\n\\[\n \\underbrace{(0,1,3,5)}_{(k_i)}\n \\longmapsto\\underbrace{(0,0,1,2)}_{(\\lambda_i)}\n \\longmapsto\\underbrace{(2,1,0,0)}_{(\\mu_j)}.\n\\]\nIn general, put $\\mu_j=\\lambda_{N-j}$ for $1\\le j\\le N$, so that\n$\\mu_1\\ge\\cdots\\ge\\mu_N\\ge0$. Division by the contribution of\n$k_i=i$ turns the summand for this tuple into\n\\begin{equation}\\label{long1049:eq:polynomial-partition}\n\\begin{split}\n W_N(\\mu)={}&q^{\\sum_{j\\le N}(2j-1)\\mu_j}\n \\prod_{j\\le N}\\frac{a_{N-j+\\mu_j}}{a_{N-j}}\\\\\n &\\hspace{2mm}\\cdot\\prod_{1\\le j<k\\le N}\n \\left(\\frac{1-q^{k-j+\\mu_j-\\mu_k}}{1-q^{k-j}}\\right)^2.\n\\end{split}\n\\end{equation}\nWe order the indices increasingly, so their geometric nodes decrease\ntowards zero. The $j$th index from the right contributes once through\n$q^{k_i}$ and twice through each of its $j-1$ pairs with later\nindices in the squared Vandermonde. Its displacement therefore has exponent\n$1+2(j-1)=2j-1$.\n\nSet $W_N(\\mu)=0$ when $\\mu$ has more than $N$ positive parts,\nso that every sum has the same index set. Fix a partition with\n$\\ell$ positive parts. It affects only the last $\\ell$ indices\nof $(0,1,\\ldots,N-1)$: for instance, $(2,1)$ affects the last two\nfor every $N\\ge2$. Those indices tend to infinity, so the weight\nratios in \\eqref{long1049:eq:polynomial-partition} tend to $1$ by the fixed-shift hypothesis.\n\nThe Vandermonde quotient still contains a growing number of factors,\nbut its dependence on $N$ telescopes. The factors with $j<k\\le\\ell$\nare fixed, and those with $j>\\ell$ equal $1$. For $N\\ge\\ell$ and\neach $j\\le\\ell$, the remaining factors satisfy\n\\begin{equation}\\label{long1049:eq:partition-tail}\n \\prod_{k=\\ell+1}^{N}\n \\frac{1-q^{k-j+\\mu_j}}{1-q^{k-j}}\n =\\frac{(q^{N+1-j};q)_{\\mu_j}}{(q^{\\ell+1-j};q)_{\\mu_j}}\n \\longrightarrow\\frac1{(q^{\\ell+1-j};q)_{\\mu_j}}.\n\\end{equation}\nThus each $W_N(\\mu)$ has the same limit as for constant weights.\n\nTo pass this limit through the sum over partitions, we need a bound\nindependent of $N$ that remains summable as both the number and size\nof the parts grow. Only factors with $j\\le\\ell$ can differ from $1$.\nFor each such $j$, the denominator product in the Vandermonde\nquotient is at least $P$, and its numerator product is at most $1$.\nTogether with the polynomial shift bound, this gives\n\\[\n W_N(\\mu)\\le(CP^{-2})^\\ell\n \\prod_{j=1}^\\ell(1+\\mu_j)^\\kappa q^{(2j-1)\\mu_j}.\n\\]\nLet $A=\\sum_{u\\ge1}(1+u)^\\kappa q^{u-1}<\\infty$.\nDropping the ordering of the positive parts, we obtain\n\\begin{align*}\n \\sum_{\\ell(\\mu)=\\ell}\\sup_N W_N(\\mu)\n &\\le(CP^{-2})^\\ell\\prod_{j=1}^\\ell\n       \\sum_{u\\ge1}(1+u)^\\kappa q^{(2j-1)u}\\\\\n &\\le(CP^{-2}A)^\\ell q^{\\ell^2}.\n\\end{align*}\nHere $q^{(2j-1)u}\\le q^{2j-1}q^{u-1}$ and\n$1+3+\\cdots+(2\\ell-1)=\\ell^2$. Since $q<1$, the bound\n$(CP^{-2}A)^\\ell q^{\\ell^2}$ is summable over $\\ell$,\nindependently of $N$.\n\nDominated convergence gives the same limiting partition sum as\nfor $a_k=1$. By \\eqref{long1049:eq:cauchy-model}, this sum tends\nto $\\mathcal M(q)$, the reciprocal of the limiting Cauchy\ndenominator. Together with $\\Delta_N/P^{2N}\\to\\mathcal M(q)^2$,\nit accounts for all three copies of $\\mathcal M(q)$. Restoring\nthe principal tuple gives\n\\[\n \\frac{D_N}{D_N^{(0)}\\prod_{k<N}a_k}\\longrightarrow1.\n\\]\nThe model asymptotic \\eqref{long1049:eq:cauchy-model-limit} now proves\nthe theorem. The comparison explains why the normalised limit\nretains the geometric spacing of the nodes but loses the individual\nweights.\n\\end{proof}\n\nWe now return to $a_k=P^4\\gamma_k$. A factorisation into two\nfinite Gaussian multinomial sums supplies both the shift hypotheses\nand the first correction to $a_k/c_k$. The shift hypotheses allow\napplication of the determinant theorem; the correction determines\nthe product $\\prod_{k<N}a_k$.\n\n\\begin{proposition}[the moment weights as a product of two finite sums]\n\\label{long1049:prop:rogers-factorisation}\nFor $r\\ge1$ and $k\\ge0$ set\n\\[\n R_k^{(r)}(q)=\\sum_{n_1+\\cdots+n_r=k}\n \\frac{(q;q)_k}{\\prod_{j\\le r}(q;q)_{n_j}},\n\\]\nthe sum being over compositions into $r$ nonnegative parts, so that\n$R_k^{(r)}$ is the sum of all $q$-multinomial coefficients of degree $k$ and\nlength $r$, equivalently the multivariate Rogers--Szeg\\H{o} polynomial at unit\narguments~\\cite{vinroot2010}. These are polynomial identities in the\nindeterminate $q$; at a prime power, $r=2$ gives the Galois number counting\nthe subspaces of a $k$-dimensional space over the field of that order, and\n$r=3$ counts the flags $V_1\\subseteq V_2$ in that space. The source is cited\nfor those identifications; the factorisation below is proved here. Then\n\\[\n \\gamma_k(q)=\\frac{R_k^{(2)}(q)R_k^{(3)}(q)}{(q;q)_k}.\n\\]\nThe product $R_k^{(2)}R_k^{(3)}$ lies in $\\Z_{\\ge0}[q]$, has degree\n$\\lfloor k^2/4\\rfloor+\\lfloor k^2/3\\rfloor$, and has coefficient sum $6^k$.\nConsequently $(q;q)_k\\gamma_k\\in\\Z[q]$, and the weights\n$a_k=P^4\\gamma_k$ satisfy\n\\[\n P^5c_k\\le a_k\\le P^{-1}c_k\n \\qquad(k\\ge0),\n\\]\ntogether with $a_{k+h}/a_k\\le P^{-6}(1+h)^3$, so the hypotheses of\nTheorem~\\ref{long1049:thm:geometric-universality} hold with $\\kappa=3$.\n\\end{proposition}\n\n\\begin{proof}\nWrite $\\mathcal E(z)=(z;q)_\\infty^{-1}$. Euler's expansion gives\n$\\mathcal E(z)^r=\\sum_{n\\ge0}R_n^{(r)}z^n/(q;q)_n$.\nWe seek the product $R_k^{(2)}R_k^{(3)}$ at a common index $k$.\nSplitting $k=n+j$ keeps $R_k^{(3)}$ fixed while the sum over\n$((q;q)_n(q;q)_j)^{-1}$ supplies $R_k^{(2)}/(q;q)_k$.\nThe operator $\\partial_qf(z)=(f(z)-f(qz))/z$ produces the needed\nindex shift: $\\partial_qz^j=(1-q^j)z^{j-1}$ cancels the last\nfactor of $(q;q)_j$. Thus\n\\[\n \\partial_q\\mathcal E=\\mathcal E,\\qquad\n \\partial_q^n\\mathcal E(z)^3\n =\\sum_{j\\ge0}\\frac{R_{n+j}^{(3)}}{(q;q)_j}z^j.\n\\]\nThe $q$-Leibniz rule, obtained by induction from the product rule, is\n\\[\n \\partial_q^n(fg)(z)=\\sum_{j=0}^n\\genfrac{[}{]}{0pt}{}{n}{j}_q\n (\\partial_q^jf)(q^{n-j}z)\\,(\\partial_q^{n-j}g)(z).\n\\]\nWrite $h_k(z)=\\sum_{j=0}^k\\genfrac{[}{]}{0pt}{}{k}{j}_q(z;q)_j$. Applying the\nrule to $\\mathcal E\\cdot\\mathcal E$ and using\n$\\mathcal E(q^mz)=(z;q)_m\\mathcal E(z)$ gives\n$\\partial_q^k\\mathcal E(z)^2=\\mathcal E(z)^2h_k(z)$; at $k=1$ both sides are\n$\\mathcal E(z)^2(2-z)$. Applying it once more, to\n$\\mathcal E\\cdot\\mathcal E^2$, gives\n\\[\n \\partial_q^n\\mathcal E(z)^3\n =\\mathcal E(z)^3\\sum_{k=0}^n\\genfrac{[}{]}{0pt}{}{n}{k}_q(z;q)_kh_k(z),\n\\]\nHence\n\\[\n \\sum_{n\\ge0}\\frac{w^n}{(q;q)_n}\\partial_q^n\\mathcal E(z)^3\n =\\mathcal E(w)\\mathcal E(z)^3\n  \\sum_{k\\ge0}\\frac{w^k(z;q)_k}{(q;q)_k}h_k(z).\n\\]\nThe left-hand side is\n$\\sum_{n,j\\ge0}R_{n+j}^{(3)}w^nz^j/((q;q)_n(q;q)_j)$.\nOn setting $z=w$, all pairs with the same sum contribute to a\nsingle coefficient. Thus the coefficient of $w^k$ is\n$R_k^{(3)}\\sum_{j\\le k}\\bigl((q;q)_j(q;q)_{k-j}\\bigr)^{-1}\n=R_k^{(2)}R_k^{(3)}/(q;q)_k$.\nOn the right-hand side, put $z=w$, expand $h_k$, and write\n$k=j+t$. The identity $(w;q)_{j+t}=(w;q)_j(wq^j;q)_t$ gives\n\\[\n \\mathcal E(w)^4\\sum_{j\\ge0}\\frac{w^j(w;q)_j^2}{(q;q)_j}\n       \\sum_{t\\ge0}\\frac{(wq^j;q)_t}{(q;q)_t}w^t.\n\\]\nThe inner sum is evaluated by the $q$-binomial theorem:\n\\[\n \\sum_{t\\ge0}\\frac{(wq^j;q)_t}{(q;q)_t}w^t\n =\\frac{(w^2q^j;q)_\\infty}{(w;q)_\\infty}.\n\\]\nFinally, $(w;q)_j=(w;q)_\\infty/(wq^j;q)_\\infty$ cancels two\nof the five factors $\\mathcal E(w)$, leaving\n\\[\n \\mathcal E(w)^3\\sum_{j\\ge0}\\frac{w^j}{(q;q)_j}\n       \\frac{(w^2q^j;q)_\\infty}{(wq^j;q)_\\infty^2}\n =G_q(w).\n\\]\nEvery step holds coefficientwise, and absolutely for $|w|$ small.\n\nA $q$-multinomial coefficient has nonnegative integral coefficients and degree\n$\\sum_{i<j}n_in_j$. At $q=1$ it equals the corresponding ordinary\nmultinomial coefficient. Summing over compositions gives coefficient sum $r^k$ for\n$R_k^{(r)}$ and hence $6^k$ for the product. The degree is maximised by\nbalanced parts, which gives $\\lfloor k^2/4\\rfloor$ for $r=2$ and\n$\\lfloor k^2/3\\rfloor$ for $r=3$; no leading cancellation occurs because all\ncoefficients are nonnegative.\n\nFor the bounds, write $b_k^{(r)}=[z^k](z;q)_\\infty^{-r}$, so that\n$R_k^{(r)}=(q;q)_kb_k^{(r)}$ and\n$\\gamma_k=(q;q)_kb_k^{(2)}b_k^{(3)}$. Euler's expansion exhibits $b_k^{(r)}$ as\nan $r$-fold convolution of the sequence $1/(q;q)_n$, whose terms lie in\n$[1,P^{-1}]$; the number of compositions of $k$ into $r$ nonnegative parts is\n$\\binom{k+r-1}{r-1}$, so\n\\[\n \\binom{k+r-1}{r-1}\\le b_k^{(r)}\\le P^{-r}\\binom{k+r-1}{r-1}.\n\\]\nCombining these at $r=2,3$ with $P\\le(q;q)_k\\le1$ and\n$c_k=(k+1)\\binom{k+2}2$ gives $P^5c_k\\le a_k\\le P^{-1}c_k$. The shift bound\nfollows from $c_{k+h}/c_k\\le(1+h)^3$.\n\nIt remains to show that $a_{k+h}/a_k\\to1$ for each fixed $h$.\nThe two convolutions grow at least linearly and quadratically;\nit suffices to bound their increments by a constant and a linear\nfunction, respectively. Put $e_n=1/(q;q)_n$. Since $(q;q)_{n+1}=(1-q^{n+1})(q;q)_n$, these terms satisfy\n$1=e_0\\le e_1\\le\\cdots\\le P^{-1}$, and the convolutions read\n$b_k^{(2)}=\\sum_{i\\le k}e_ie_{k-i}$ and\n$b_k^{(3)}=\\sum_{i\\le k}e_ib_{k-i}^{(2)}$. Hence\n\\[\n \\begin{aligned}\n b_{k+1}^{(2)}-b_k^{(2)}\n &=\\sum_{i\\le k}e_i\\bigl(e_{k+1-i}-e_{k-i}\\bigr)+e_{k+1},\\\\\n b_{k+1}^{(3)}-b_k^{(3)}\n &=\\sum_{i\\le k}e_i\\bigl(b_{k+1-i}^{(2)}-b_{k-i}^{(2)}\\bigr)+e_{k+1}.\n \\end{aligned}\n\\]\nEvery bracket is nonnegative. In the first line the brackets add up to\n$e_{k+1}-1<P^{-1}$ and each $e_i\\le P^{-1}$, so\n$0\\le b_{k+1}^{(2)}-b_k^{(2)}\\le P^{-2}+P^{-1}\\le2P^{-2}$. Inserting this\nbound into the $k+1$ terms of the second line gives\n$0\\le b_{k+1}^{(3)}-b_k^{(3)}\\le2(k+1)P^{-3}+P^{-1}\\le3(k+1)P^{-3}$.\nAgainst the lower bounds $b_k^{(2)}\\ge k+1$ and\n$b_k^{(3)}\\ge(k+1)(k+2)/2$ displayed above, the ratio\n$b_{k+1}^{(r)}/b_k^{(r)}$ lies in $[1,1+2P^{-2}/(k+1)]$ for $r=2$ and in\n$[1,1+6P^{-3}/(k+2)]$ for $r=3$. Both ratios tend to $1$, and therefore so\ndoes $b_{k+h}^{(r)}/b_k^{(r)}$, a product of $h$ consecutive ratios.\nFinally $(q;q)_{k+h}/(q;q)_k=\\prod_{j=1}^h(1-q^{k+j})\\to1$, and\n$a_{k+h}/a_k=\\gamma_{k+h}/\\gamma_k$ is the product of these three ratios.\nThe Lean proof obtains the\n\\href{\\laterepobase/lean/ErdosProblems/Erdos1049/PaperCompleteR21/RogersFactorisationAnalytic.lean\\#L1097}{fixed-shift limit}\nby the same steps.\n\\end{proof}\n\nThe factorisation proves positivity at each fixed real $q$, not\ncoefficientwise positivity in $q$: for example, $a_0=P^4$ has\n$q$-coefficient $-4$. The cubic factors $c_k$ multiply to $C_N$,\nbut $a_k/c_k\\to1$ alone would not determine the product of $N$ weights. The first\nrelative correction has order $1/k$ and accumulates as a harmonic\nsum; the remaining $O_q(k^{-2})$ terms determine a finite constant.\nThe two convolutions supply this first correction explicitly.\n\n\\begin{theorem}[the size of $V_N^*$ at a fixed base]\n\\label{long1049:thm:sharp-fixed-base}\nLet $\\gamma_{\\!E}$ be Euler's constant. The product\n\\[\n \\mathcal A(q)=e^{-8\\gamma_{\\!E}L}\n \\prod_{k\\ge0}\\left(\\frac{a_k}{c_k}\\,e^{8L/(k+1)}\\right)\n\\]\nconverges to a positive finite value. With $K(q)=\\mathcal A(q)\\mathcal M(q)^3$,\n\\[\n V_N^*(q)\\sim K(q)\\,C_Nq^{B_N}P^{2N}N^{-8L},\n\\]\nequivalently\n\\[\n \\log V_N^*(q)=B_N\\log q+\\log C_N+2N\\log P-8L\\log N+\\log K(q)+o(1).\n\\]\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{long1049:thm:sharp-fixed-base}]\nPut $L=F(1/q)$, $e_k=(q;q)_k^{-1}$ and $E=P^{-1}$.\nReplacing every $e_k$ by $E$ in the two convolution sums gives\n$E^2(k+1)$ and $E^3\\binom{k+2}{2}$. The factors $k+1$ and\n$\\binom{k+2}{2}$ count the nonnegative pairs and triples with sum $k$,\nrespectively.\nFor a fixed index $i$, there is one pair $(i,k-i)$ and there are\n$k-i+1$ triples whose first coordinate is $i$. Thus a summable\ndeficit in the sequence $e_i$ produces a constant correction in the\ntwo-fold sum and a linear correction in the three-fold sum.\nWe calculate that deficit and bound terms with two or more deficits.\n\nWriting $d_k=E-e_k$, the identity\n$e_{k+1}-e_k=q^{k+1}e_{k+1}$ gives\n\\[\n d_k=\\sum_{j>k}q^je_j,\\qquad\n 0\\le d_k\\le\\frac{E q^{k+1}}{1-q}.\n\\]\nThus $\\sum d_k$ and $\\sum k d_k$ converge. To evaluate the former,\nuse Euler's identity\n$\\mathcal E(z)=\\sum_{k\\ge0}e_kz^k=(z;q)_\\infty^{-1}$ for $|z|<1$.\nTonelli's theorem and logarithmic differentiation at $z=q$ give\n\\[\n S:=\\sum_{k\\ge0}d_k\n =\\sum_{j\\ge1}j q^je_j\n =q\\mathcal E'(q)\n =E\\sum_{r\\ge1}\\frac{q^r}{1-q^r}=EL.\n\\]\nIn the double sum, $q^je_j$ occurs once for each $k=0,\\ldots,j-1$,\nwhich explains the factor $j$. Logarithmic differentiation is\njustified by locally uniform convergence on $|z|<1$.\n\nLet $b_k^{(r)}=[z^k]\\mathcal E(z)^r$. The finite-sum formula in\nProposition~\\ref{long1049:prop:rogers-factorisation} becomes\n\\[\n a_k=P^4(q;q)_k b_k^{(2)}b_k^{(3)}.\n\\]\nWe expand each finite convolution using $e_k=E-d_k$. For two factors,\n\\[\n b_k^{(2)}=E^2(k+1)-2E\\sum_{i=0}^k d_i\n              +\\sum_{i=0}^k d_i d_{k-i}\n =E^2(k+1-2L)+O_q((k+1)q^k).\n\\]\nFor three factors, the terms with one $d_i$ contribute\n$-3E^2\\sum_{i=0}^k(k-i+1)d_i$.\nHere the sum equals $(k+1)S+O_q(1)$ because\n$\\sum i d_i<\\infty$. Terms with two $d_i$ are bounded by\n$3E S^2$, and the term with three is $O_q((k+1)^2q^k)$.\nIt follows that\n\\[\n b_k^{(3)}=E^3\\left(\\binom{k+2}{2}-3L(k+1)\\right)+O_q(1).\n\\]\nDividing the correction $-3L(k+1)$ by the leading term\n$\\binom{k+2}{2}$ gives the relative correction $-6L/(k+2)$.\nThe two-fold convolution contributes $-2L/(k+1)$ in the same way.\nSince $(q;q)_k=P(1+O_q(q^k))$, the leading constants in\n$a_k=P^4(q;q)_k b_k^{(2)}b_k^{(3)}$ multiply to $P^5E^5=1$.\nReplacing $(k+2)^{-1}$ by $(k+1)^{-1}$ changes only the\n$O_q((k+1)^{-2})$ error. Thus the coefficients $2L$ and $6L$ add,\nand with $c_k=(k+1)^2(k+2)/2$ we obtain\n\\begin{align*}\n \\frac{a_k}{c_k}\n &=\\left(1-\\frac{2L}{k+1}+O_q(q^k)\\right)\n   \\left(1-\\frac{6L}{k+2}+O_q((k+1)^{-2})\\right)\n   \\left(1+O_q(q^k)\\right)\\\\\n &=1-\\frac{8L}{k+1}+O_q((k+1)^{-2}).\n\\end{align*}\nIn particular $a_{k+1}/a_k\\to1$, and multiplication of finitely\nmany adjacent ratios gives $a_{k+h}/a_k\\to1$ for each fixed $h$.\nThe uniform shift bound follows from\nProposition~\\ref{long1049:prop:rogers-factorisation}, so\nTheorem~\\ref{long1049:thm:geometric-universality} applies.\n\nFor large $k$, taking logarithms preserves the first-order term and\nleaves the summable error\n$\\log(a_k/c_k)+8L/(k+1)=O_q((k+1)^{-2})$.\nThe finitely many earlier factors are positive, so the regularised\nproduct also has a positive finite limit: $0<\\mathcal A(q)<\\infty$.\nSince $\\prod_{k<N}c_k=C_N$, the harmonic contribution separates as\n\\[\n \\log\\frac{\\prod_{k<N}a_k}{C_N}\n =-8L\\sum_{k<N}\\frac1{k+1}\n  +\\sum_{k<N}\\left(\\log\\frac{a_k}{c_k}+\\frac{8L}{k+1}\\right).\n\\]\nThe second sum tends to $\\log\\mathcal A(q)+8\\gamma_{\\!E}L$.\nUsing $\\sum_{k<N}(k+1)^{-1}=\\log N+\\gamma_{\\!E}+o(1)$,\nthe Euler-constant terms cancel, giving\n$\\prod_{k<N}a_k\\sim\\mathcal A(q)C_NN^{-8L}$.\nTheorem~\\ref{long1049:thm:geometric-universality} now gives the\nstated determinant asymptotic. Taking logarithms yields\n\\[\n \\log V_N^*=B_N\\log q+\\log C_N+2N\\log P\n             -8L\\log N+\\log K(q)+o(1).\n\\]\n\\end{proof}\n\nThe convolution proof explains $8L$ as $2L+6L$, one contribution\nfrom each finite sum. An alternative reads the same correction from\nthe generating function near $w=1$. Its pole of order four accounts\nfor the cubic growth of $a_k$; the next principal-part coefficient\naccounts for the relative term of order $1/k$.\n\\begin{proof}[An alternative coefficient calculation]\nWrite $\\Pi(w)=(qw;q)_\\infty$, so that $(w;q)_\\infty=(1-w)\\Pi(w)$, and for\n$t\\ge1$ put\n\\[\n f_t(w)=\\frac{(q^tw^2;q)_\\infty}{(q;q)_t(q^tw;q)_\\infty^2},\n \\qquad\n \\mathcal S(w)=\\sum_{t\\ge1}w^t\\bigl(f_t(w)-P^{-1}\\bigr).\n\\]\nOn each compact subset of $|w|<q^{-1}$ the bracket is $O_q(q^t)$ uniformly, so\n$\\mathcal S$ is analytic there. Separating the summand $t=0$, whose\ndenominator carries the factor $(1-w)^2$, and subtracting the constant part of\nthe remaining summands gives\n\\[\n \\mathcal D(w):=P^4(1-w)^4G_q(w)\n =\\frac{P^4}{\\Pi(w)^3}\n \\left((1+w)\\frac{(qw^2;q)_\\infty}{\\Pi(w)^2}+\\frac wP\n       +(1-w)\\mathcal S(w)\\right).\n\\]\nHence $\\mathcal D$ is analytic on a disc of radius greater than $1$.\n\nAt $w=1$ one has $\\Pi(1)=P$ and $\\Pi'(1)/\\Pi(1)=-L$, since\n$\\Pi'/\\Pi=-\\sum_{j\\ge1}q^j/(1-q^jw)$. Also $(qw^2;q)_\\infty$ equals $P$ at\n$w=1$, and the logarithmic derivative of $(qw^2;q)_\\infty/\\Pi(w)^2$ at $w=1$ is\n$-2L+2L=0$. Therefore the bracket has value $3/P$ and derivative $(2-L)/P$\nat $w=1$; here $\\mathcal S(1)=L/P$, because\n$f_t(1)=\\bigl(P(1-q^t)\\bigr)^{-1}$ and\n$\\sum_{t\\ge1}\\bigl((1-q^t)^{-1}-1\\bigr)=L$. Combining with the factor\n$P^4\\Pi^{-3}$, whose logarithmic derivative at $1$ is $3L$, gives\n\\[\n \\mathcal D(1)=3,\\qquad\n \\mathcal D'(1)=3\\left(3L+\\frac{2-L}3\\right)=2+8L.\n\\]\nThe first two principal-part terms are determined by $\\mathcal D(1)$\nand $\\mathcal D'(1)$. The terms from its second and third derivatives\nhave coefficients of order at most $k$; after the full principal part\nis subtracted, the remainder is analytic on a larger disc and has\nexponentially decreasing coefficients. Keeping only the first two\nterms therefore gives\n\\[\n a_k=3\\binom{k+3}3-(2+8L)\\binom{k+2}2+O_q(k),\n \\qquad\n \\frac{a_k}{c_k}=1-\\frac{8L}{k+1}+O_q\\bigl((k+1)^{-2}\\bigr),\n\\]\nthe second line because $3\\binom{k+3}3=c_k(k+3)/(k+1)$ and\n$\\binom{k+2}2=c_k/(k+1)$.\n\nThe logarithmic error is again summable. The product argument in\nthe first proof therefore gives $0<\\mathcal A(q)<\\infty$ and,\nusing $\\sum_{k<N}(k+1)^{-1}=\\log N+\\gamma_{\\!E}+o(1)$,\n\\[\n \\prod_{k<N}a_k\\sim\\mathcal A(q)\\,C_NN^{-8L}.\n\\]\nProposition~\\ref{long1049:prop:rogers-factorisation} supplies the uniform shift\nbound and the fixed-shift limit, so\nTheorem~\\ref{long1049:thm:geometric-universality} applies to $(a_k)$ and gives\n$V_N^*\\sim\\mathcal M(q)^3q^{B_N}P^{2N}\\prod_{k<N}a_k$. Substituting the product\nasymptotic proves both displays.\n\n\\end{proof}\n\n\\paragraph{Comparison with the cubic bound.}\nThe estimate identifies the terms $2N\\log P$, $-8F(1/q)\\log N$\nand $\\log K(q)$ within the previous $O_q(N)$ error. It leaves the\ncubic coefficient unchanged and has no effect on prime-power\nvaluations. For the denominator estimate at $3/2$, the difference\nof the relevant exponents is\n\\[\n \\frac{39}{41}\\bigl(4N^3-3N^2\\bigr)-\\bigl(2N^3-N\\bigr)\n =\\frac{N(74N^2-117N+41)}{41}>0\\qquad(N\\ge2),\n\\]\nthe positivity being immediate from\n$74N^2-117N+41=74(N-2)^2+179(N-2)+103$. Its leading size is $(74/41)N^3$. A logarithmic correction cannot\noffset this cubic gap in the stated denominator model. The\nfactorisation refines the moment weights, not the denominator of\nan irrationality form; any such form still needs an estimate for\nits own rational coefficients.\n\n\n\\subsubsection*{An extension with limiting weight ratio}\n\\papersectiontarget{long1049-ratio-extension}\n\nFor $0<\\rho<q^{-1}$ set\n\\[\n \\mathcal M(\\rho;q)=\\prod_{d\\ge1}(1-\\rho q^d)^{-d},\n \\qquad \\mathcal M(1;q)=\\mathcal M(q).\n\\]\nA limiting adjacent-weight ratio supplies an exponential shift\nbound, replacing the polynomial bound above. The condition\n$\\rho q<1$ makes that bound summable and changes one copy of\nthe product constant from $\\mathcal M(1;q)$ to $\\mathcal M(\\rho;q)$.\nThe ordinary proof below has no Lean proof in the supplied record;\nthis unformalised remark is not used in the checked fixed-base argument.\n\n\\phantomsection\n\\begin{remark}[limiting weight ratios; unformalised]\\label{long1049:thm:geometric-ratio}\nLet $0<q<1$ and $0<\\rho<q^{-1}$. Suppose that $a_k>0$ for $k\\ge0$ and\n\\[\n \\frac{a_{k+1}}{a_k}\\longrightarrow\\rho\\qquad(k\\to\\infty).\n\\]\nDefine\n\\[\n M_m=\\sum_{k\\ge0}a_kq^{(m+1)k},\\qquad\n D_N=\\det(M_{i+j})_{0\\le i,j<N}.\n\\]\nThen, as $N\\to\\infty$,\n\\begin{equation}\\label{long1049:eq:ratio-limit}\n D_N\\sim \\mathcal M(1;q)^2\\mathcal M(\\rho;q)\\,\n q^{B_N}P^{2N}\\prod_{k=0}^{N-1}a_k.\n\\end{equation}\n\\end{remark}\n\n\\begin{proof}[Ordinary argument for the ratio-limit remark]\nChoose $\\widehat\\rho$ with $\\rho<\\widehat\\rho<q^{-1}$.\nThere is a $K$ such that $a_{j+1}/a_j\\le\\widehat\\rho$ for $j\\ge K$.\nWith\n\\[\n C=\\prod_{j<K}\\max\\left(1,\\frac{a_{j+1}}{\\widehat\\rho a_j}\\right),\n\\]\nwe obtain, by multiplying adjacent ratios,\n\\begin{equation}\\label{long1049:eq:ratio-shift-bound}\n \\frac{a_{k+h}}{a_k}\\le C\\widehat\\rho^{\\,h}\\qquad(k,h\\ge0).\n\\end{equation}\nThis also gives $a_h\\le Ca_0\\widehat\\rho^{\\,h}$, so every moment\nconverges because $q\\widehat\\rho<1$.\nFor finitely many nodes,\nCauchy--Binet expands their moment determinant as a sum of squared\nVandermonde products. We let the number of nodes tend to infinity.\nEntrywise convergence gives the limit of the determinant, while\npositivity gives the limit of the sum. Hence\n\\begin{equation}\\label{long1049:eq:ratio-heine}\n D_N=\\sum_{0\\le k_0<\\cdots<k_{N-1}}\n \\prod_{i<N}a_{k_i}q^{k_i}\n \\prod_{i<j<N}(q^{k_i}-q^{k_j})^2.\n\\end{equation}\nThe tuple $k_i=i$ contributes\n\\[\n q^{B_N}\\Delta_N\\prod_{i<N}a_i,\\qquad\n \\Delta_N=\\prod_{d=1}^{N-1}(1-q^d)^{2(N-d)}.\n\\]\nWrite $k_i=i+\\lambda_i$ and reverse the displacements:\n$\\mu_j=\\lambda_{N-j}$ for $1\\le j\\le N$. Then\n$\\mu_1\\ge\\cdots\\ge\\mu_N\\ge0$. After division by the least-tuple\ncontribution, the summand becomes\n\\begin{equation}\\label{long1049:eq:ratio-partition}\n\\begin{split}\n W_N(\\mu)={}&q^{\\sum_{j\\le N}(2j-1)\\mu_j}\n \\prod_{j\\le N}\\frac{a_{N-j+\\mu_j}}{a_{N-j}}\\\\\n &\\hspace{2mm}\\cdot\\prod_{1\\le j<k\\le N}\n \\left(\\frac{1-q^{k-j+\\mu_j-\\mu_k}}{1-q^{k-j}}\\right)^2.\n\\end{split}\n\\end{equation}\nFor partitions of length greater than $N$, set $W_N(\\mu)=0$.\nAll the sums now have the same countable index set.\n\nSuppose that $\\mu$ has $\\ell$ positive parts. Only the factors with\n$j\\le\\ell$ can differ from $1$. For a fixed such $j$, the denominators\nin the Vandermonde quotient multiply to at least $P$, and its\nnumerators are at most $1$. We therefore have a bound independent\nof $N$\n\\[\n W_N(\\mu)\\le (CP^{-2})^\\ell\n \\prod_{j=1}^{\\ell}\\widehat\\rho^{\\,\\mu_j}q^{(2j-1)\\mu_j}.\n\\]\nWe sum this majorant by first fixing $\\ell$ and then dropping the\nordering condition on the positive parts. This gives\n\\begin{align*}\n \\sum_{\\ell(\\mu)=\\ell}\\sup_N W_N(\\mu)\n &\\le(CP^{-2})^\\ell\\prod_{j=1}^{\\ell}\n        \\sum_{u\\ge1}\\widehat\\rho^{\\,u} q^{(2j-1)u}\\\\\n &\\le C_*^{\\ell}q^{\\ell^2},\\qquad\n C_*=\\frac{C\\widehat\\rho P^{-2}}{1-\\widehat\\rho q}.\n\\end{align*}\nHere $\\sum_{j\\le\\ell}(2j-1)=\\ell^2$ and\n$1-\\widehat\\rho q^{2j-1}\\ge1-\\widehat\\rho q>0$.\nThe series $\\sum_{\\ell\\ge0}C_*^{\\ell}q^{\\ell^2}$ converges.\n\nFor fixed $h$, telescoping the adjacent ratios shows that\n$a_{k+h}/a_k\\to \\rho^h$. Thus, for each fixed partition, the weight\nratios in \\eqref{long1049:eq:ratio-partition} have the same limit as for\n$a_k=\\rho^k$. The Vandermonde quotient is independent of the\nweights, and its convergence follows from the same finite identity\n\\eqref{long1049:eq:partition-tail}. Dominated convergence\ntherefore reduces the limit of $\\sum_\\mu W_N(\\mu)$ to the case\n$a_k=\\rho^k$.\n\nIn that case $M_m=(1-\\rho q^{m+1})^{-1}$, and Cauchy's determinant gives\n\\[\n D_N=\\frac{\\rho^{N(N-1)/2}q^{B_N}\\Delta_N}\n {\\prod_{0\\le i,j<N}(1-\\rho q^{i+j+1})}.\n\\]\nThe multiplicity of $1-\\rho q^d$ in the denominator increases to $d$,\nso the normalised sum tends to $\\mathcal M(\\rho;q)$. Finally,\n\\[\n \\frac{\\Delta_N}{P^{2N}}\n =\\prod_{d<N}(1-q^d)^{-2d}\\prod_{d\\ge N}(1-q^d)^{-2N}\n \\longrightarrow\\mathcal M(1;q)^2,\n\\]\nsince the logarithm of the second product is $O_q(Nq^N)$.\nCombining the two limits proves \\eqref{long1049:eq:ratio-limit}.\n\\end{proof}\n\nFor example, $a_k=\\rho^k(k+1)^s$, with $s\\in\\mathbb R$, satisfies\nthe ratio hypothesis, and\n$\\prod_{k<N}a_k=\\rho^{N(N-1)/2}(N!)^s$.\nFor $s=1$ the moments are $(1-\\rho q^{m+1})^{-2}$, giving\n\\[\n \\det\\left((1-\\rho q^{i+j+1})^{-2}\\right)_{i,j<N}\n \\sim\\mathcal M(q)^2\\mathcal M(\\rho;q)\n       q^{B_N}P^{2N}\\rho^{N(N-1)/2}N!.\n\\]\nThe choice $a_k=\\rho^k\\exp(\\sqrt{k})$ also satisfies the ratio\nhypothesis, since $\\sqrt{k+1}-\\sqrt{k}\\to0$.\nFor $\\rho=1$, its relative growth at $k=0$ exceeds every polynomial in the shift.\nThus this argument extends beyond the polynomial-bound hypotheses of\nTheorem~\\ref{long1049:thm:geometric-universality}.\n\n\n\n\\subsection{Positive Hankel determinants and a rational special value}\n\\label{long1049:sec:calibrated-countermodel}\n\\papersectiontarget{long1049-countermodel}\n\nWe construct a generating function with the rational special value\n$1$, although its moment determinants have the same formal orders,\nleading coefficients and type of fixed-base power correction as\nZudilin's. This shows why those determinant properties alone do not\nestablish irrationality. The construction uses neither Zudilin's\nrecurrence nor his polynomial factorisation or cyclotomic divisors,\nand gives no conclusion about $F(3/2)$. The arguments below have\nordinary proofs only and are stated as unformalised remarks.\n\nWe choose the tail of the weight sequence to obtain the determinant\nasymptotic, and its first terms to fix the special value. A telescoping\ndifference of the moments will then give the value $1$. The cut-off\nbelow separates these two choices while leaving room to keep every\nweight positive.\n\nPut\n\\[\n c_k=\\frac{(k+1)^2(k+2)}2,\\qquad\n d_k=\\frac{(k+1)(k+2)}2,\\qquad\n R(w)=\\sum_{k\\ge0}c_kw^k=\\frac{1+2w}{(1-w)^4}.\n\\]\nFor $0<q\\le2/3$ define, with a single fixed cut-off $32$,\n\\begin{equation}\\label{long1049:eq:calibrated-weights}\n a_k(q)=\\begin{cases}\n c_k(1-q)^{32},&1\\le k<32,\\\\\n c_k-12q d_k,&k\\ge32,\n \\end{cases}\n \\qquad a_0(q)=1-\\sum_{k\\ge1}a_k(q)q^k.\n\\end{equation}\nThe weight generating function $A_q(w)=\\sum_{k\\ge0}a_k(q)w^k$\nis rational in $w$. The value $G_q(q)$ concerns a different\ngenerating function, formed from the moments below. To obtain $A_q$, put\n\\[\n T_q(w)=R(w)-\\frac{12q}{(1-w)^3}\n            -\\sum_{k=0}^{31}(c_k-12qd_k)w^k,\n\\]\nso that $A_q(w)$ equals\n\\[\n a_0(q)+(1-q)^{32}\\sum_{k=1}^{31}c_kw^k+T_q(w),\\quad\n a_0(q)=1-(1-q)^{32}\\sum_{k=1}^{31}c_kq^k-T_q(q).\n\\]\nSet $\\mu_m(q)=A_q(q^{m+1})$, $\\nu_m(q)=\\mu_m(q)-q\\mu_{m+1}(q)$,\nand $G_q(z)=\\sum_{m\\ge0}\\nu_m(q)z^m$.\nThe choice of $a_0$ makes $\\mu_0=1$, while the factor $q$ in\n$\\nu_m$ makes the sum defining $G_q(q)$ telescope. Independently,\nfor $k\\ge32$ the tail satisfies $a_k/c_k=1-12q/(k+1)$;\nat $q=2/3$ this becomes $1-8/(k+1)$, which produces the required\npower $N^{-8}$ in the product of the weights. The proof below checks\npositivity, convergence and both determinant claims.\n\n\\phantomsection\n\\begin{remark}[a rational special value]\n\\label{long1049:thm:calibrated-model}\nAll the weights in~\\eqref{long1049:eq:calibrated-weights} are positive\nfor $0<q\\le2/3$. The two moment sequences $(\\mu_m)$ and $(\\nu_m)$\nhave positive Hankel determinants of every size, $G_q$ is not a rational\nfunction, and\n$G_q(q)=1$.  Their formal Hankel determinants both begin with\n\\[\n C_Nq^{B_N},\\qquad B_N=\\sum_{j<N}j^2,\\quad\n C_N=\\frac{(N!)^2(N+1)!}{2^N}\\qquad(N\\ge1).\n\\]\nAt the fixed base $q_0=2/3$, with\n$P(q)=\\prod_{j\\ge1}(1-q^j)$ and\n$\\mathcal M(q)=\\prod_{j\\ge1}(1-q^j)^{-j}$,\nthere is an explicit $\\mathcal A>0$ such that\n\\[\n \\det(\\mu_{i+j}(q_0))_{0\\le i,j<N}\n \\sim\\mathcal A\\mathcal M(q_0)^3 C_Nq_0^{B_N}\n             P(q_0)^{2N}N^{-8\\xi},\\qquad \\xi=G_{q_0}(q_0)=1.\n\\]\n\\end{remark}\n\nThis countermodel and its denominator calculation have ordinary proofs\nbelow; they have not been Lean-checked or externally peer reviewed.\n\n\\begin{proof}\nFor $k\\ge32$ we have $a_k(q)=d_k(k+1-12q)>0$, and the smaller-index\nweights are plainly positive.  Write\n$M(q)=(1-q)^{32}\\sum_{k=1}^{31}c_kq^k$.\nThe generating function for the $c_k$ gives\n$R(q)-1\\le6q(1-q)^{-4}$, whence\n$M(q)\\le6q(1-q)^{28}\\le6/29<1/4$.\nFor $k\\ge32$,\n\\[\n \\frac{d}{dq}\\bigl(d_k(k+1-12q)q^k\\bigr)\n   =d_k(k+1)q^{k-1}(k-12q)>0.\n\\]\nDirect substitution into the rational expression for $T_q$ gives\n\\[\n T_{2/3}(2/3)\n   =\\frac{24794846199808}{205891132094649}<\\frac14.\n\\]\nTherefore $1/2<a_0(q)=1-M(q)-T_q(q)<1$.\n\nThe moments have positive atomic representations\n\\[\n \\mu_m=\\sum_{k\\ge0}a_kq^k(q^k)^m,\\qquad\n \\nu_m=\\sum_{k\\ge0}a_kq^k(1-q^{k+1})(q^k)^m.\n\\]\nInfinitely many distinct atoms have positive mass.  The Gram\nquadratic form of a nonzero polynomial of degree below $N$ is strictly\npositive for either measure, proving positive definiteness of every\n$N$-by-$N$ Hankel matrix.  Summing the geometric series in $m$ gives\n\\[\n G_q(z)=\\sum_{k\\ge0}\\frac{a_kq^k(1-q^{k+1})}{1-zq^k}.\n\\]\nEach $z=q^{-k}$ is a genuine pole: its residue is nonzero and the other\nterms are analytic there.  Hence $G_q$ is not rational.  Telescoping\nshows $G_q(q)=\\mu_0(q)=A_q(q)=1$ and, for\n$S_{n-1}=\\sum_{m<n}q^m\\nu_m$, gives\n\\[\n 1-S_{n-1}=q^n\\mu_n\\in(0,q^n]\n \\qquad(n\\ge0).\n\\]\n\nEvery $a_k(q)$ belongs to $\\Z[[q]]$ and has constant term $c_k$,\nincluding $a_0$, for which the defining sum is coefficientwise finite.\nThe moments $\\mu_m$ and $\\nu_m$ also lie in $\\Z[[q]]$, but their\nconstant terms are $1$: only $a_0$ contributes at $q=0$. Their\nrational expressions have no poles for $|q|<1$, so these formal\nseries converge at $q_0$.\nFor $H_N(q)=\\det(\\mu_{i+j}(q))_{i,j<N}$, the Gram determinant expansion is\n\\[\n H_N(q)=\\sum_{0\\le k_0<\\cdots<k_{N-1}}\n     \\left(\\prod_{i<N}a_{k_i}(q)q^{k_i}\\right)\n     \\prod_{i<j}(q^{k_i}-q^{k_j})^2.\n\\]\nIt holds first for finitely many atoms and then coefficientwise.\nThe lowest possible degree of one summand is\n$\\sum_{i<N}(2N-1-2i)k_i$.  Its unique minimum is attained at\n$k_i=i$ and equals $B_N$; the coefficient is\n$\\prod_{i<N}c_i=C_N$.  Replacing $a_k$ by\n$a_k(1-q^{k+1})$ proves the same leading term for $(\\nu_m)$.\n\nAt $q_0$ set $\\varepsilon=3^{-32}$ and $b_0=a_0(q_0)>0$.  For $k\\ge32$,\n$a_k(q_0)/c_k=(k-7)/(k+1)$; all these ratios lie in\n$[\\varepsilon,1]$.  Hence for fixed $h$,\n$a_{k+h}/a_k\\to1$ as $k\\to\\infty$, and uniformly\n$a_{k+h}/a_k\\le3^{32}(1+h)^3$.\nThese are precisely the weight hypotheses of\nTheorem~\\ref{long1049:thm:geometric-universality}, whose conclusion is\n\\[\n H_N(q_0)\\sim\\mathcal M(q_0)^3q_0^{B_N}P(q_0)^{2N}\n                  \\prod_{k<N}a_k(q_0).\n\\]\nThe finite product has the exact form, for $N\\ge32$,\n\\[\n \\frac{\\prod_{k<N}a_k(q_0)}{C_N}\n =b_0\\varepsilon^{31}\\frac{32!}{24!}\\frac{(N-8)!}{N!}.\n\\]\nThus $\\mathcal A=b_0\\varepsilon^{31}32!/24!$, and\n$N^8(N-8)!/N!\\to1$.  The displayed exponent is exactly $-8\\xi$\nbecause $\\xi=1$; it is asserted at this one fixed base, not as a\nuniform special-value formula for this family.\n\\end{proof}\n\nThe following calculation is retained with its ordinary proof. It has\nno Lean proof in the supplied record.\n\n\\phantomsection\n\\begin{remark}[exact denominators of the calibrated moments]\n\\label{long1049:thm:calibrated-denominators}\nLet $q_0=2/3$ and $\\delta_n=3^{n+1}-2^{n+1}$.  For every $n\\ge1$,\nthe reduced denominators are\n\\[\n \\operatorname{den}\\mu_n(q_0)=3^{31n+62}\\delta_n^4,\n \\qquad\n \\operatorname{den}(q_0^n\\mu_n(q_0))=3^{32n+62}\\delta_n^4.\n\\]\nThe least integer clearing both rational coefficients in\n$\\mu_n=q_0^{-n}\\xi-q_0^{-n}S_{n-1}$ is\n$2^n3^{31n+62}\\delta_n^4$.  At $n=0$ the moment and tail equal one.\n\\end{remark}\n\n\\begin{proof}\nAt $q_0$, split the rational seed into a pole part and a polynomial:\n\\[\n A_{q_0}(w)=\\frac{-7+10w}{(1-w)^4}+Q(w),\\qquad\n Q(w)=p_0+\\sum_{k=1}^{31}p_kw^k,\n\\]\nwhere $p_0=b_0+7$ and\n$p_k=(3^{-32}-1)c_k+8d_k$.  These coefficients have only powers\nof three in their denominators.  The finite formulas give\n$v_3(p_0)\\ge-63$, $v_3(p_k)\\ge-32$ for $1\\le k\\le30$, and\n$v_3(p_{31})=-31$: in the last calculation $c_{31}=16896$ has\n$v_3(c_{31})=1$ and the other numerator terms are multiples of\n$3^{32}$.  At $w=(2/3)^{n+1}$, $n\\ge1$, the term\n$p_{31}w^{31}$ has valuation $-31n-62$, uniquely lower than all\nother terms of $Q(w)$.  The pole part equals\n\\[\n \\frac{3^{3n+3}(-7\\cdot3^{n+1}+10\\cdot2^{n+1})}{\\delta_n^4}\n\\]\nand has positive $3$-adic valuation.  Hence\n$v_3(\\mu_n)=-31n-62$.\n\nEvery prime $\\ell\\mid\\delta_n$ differs from $2,3$.\nIts numerator in the displayed pole part is congruent to\n$3\\cdot2^{n+1}\\pmod{\\delta_n}$ and is coprime to $\\delta_n$.\nThe polynomial part is $\\ell$-integral, so it cannot cancel the\npole part's $\\ell$-denominator of order $4v_\\ell(\\delta_n)$.\nThere are no other denominator primes.  Multiplying by\n$q_0^n=2^n/3^n$ adds exactly $n$ factors of three, proving the second\nformula.  In the affine error, $q_0^{-n}=3^n/2^n$ has denominator\n$2^n$, coprime to the odd denominator of $\\mu_n$.  Subtraction does\nnot reduce either negative valuation, proving the stated least\ncommon denominator.\n\\end{proof}\n\nEach individual $\\mu_n$ has logarithmic height linear in $n$:\n$\\log\\operatorname{den}\\mu_n=(35n+66)\\log3+o(1)$.\nThe determinant has a different scale.\n\nThe following calculation is retained with its ordinary proof. It has\nno Lean proof in the supplied record.\n\n\\phantomsection\n\\begin{remark}[determinant height and row denominators]\n\\label{long1049:prop:calibrated-height}\nFor $D_N=\\operatorname{den}H_N(q_0)$ in lowest terms,\n\\[\n \\liminf_{N\\to\\infty}\\frac{\\log D_N}{N^3}\n       \\ge\\frac{\\log(3/2)}3,\n \\qquad \\log D_N=O(N^3).\n\\]\nAlready for $N=2$, multiplying $H_2=\\mu_2-\\mu_1^2$ by\n$\\operatorname{den}\\mu_1\\operatorname{den}\\mu_2$ leaves reduced\ndenominator $625$.\n\\end{remark}\n\n\\begin{proof}\nThe positive Heine expansion above gives a bound independent of the\nasymptotic theorem.  Since $0<a_k(q_0)\\le c_k$, write $k_i=i+s_i$.\nDropping the ordering restriction and bounding\n$c_{i+s_i}/c_i\\le(1+s_i)^3$ and each residual Vandermonde factor by\none gives\n\\[\n 0<H_N(q_0)\\le C_Nq_0^{B_N}\n   \\prod_{j\\ge0}\\sum_{s\\ge0}(s+1)^3q_0^{(2j+1)s}.\n\\]\nFor $A(t)=\\sum_{s\\ge0}(s+1)^3t^s=(1+4t+t^2)/(1-t)^4$,\n$\\log A(t)\\le A(t)-1\\le8t/(1-t)^4$.  The product is therefore at\nmost $\\exp(3888/5)$.  Since $D_NH_N$ is a positive integer,\n\\[\n \\log D_N\\ge B_N\\log(3/2)-\\log C_N-3888/5.\n\\]\nHere $B_N\\sim N^3/3$ and $\\log C_N=O(N\\log N)$.\nConversely,\n\\[\n E_N=3^{31N(N-1)+62N}\\prod_{0\\le i,j<N}\\delta_{i+j}^{4}\n\\]\nclears every term in the determinant expansion, by the preceding\ndenominator formula.  Thus $D_N\\mid E_N$; using\n$\\log\\delta_m\\le(m+1)\\log3$ and\n$\\sum_{i,j<N}(i+j+1)=N^3$ gives the asserted cubic upper bound.\n\nFor the assertion about $H_2$,\n$\\operatorname{den}\\mu_1=3^{93}5^4$ and\n$\\operatorname{den}\\mu_2=3^{124}19^4$.\nAt five, $v_5(\\mu_2)=0$ and $v_5(\\mu_1^2)=-8$,\nwhile the proposed product contributes only $5^4$.\nAt three and nineteen that product clears the denominator, and no\nother prime occurs.  The remaining denominator is therefore $5^4=625$.\n\\end{proof}\n\nZudilin's determinantal criterion retains a divisibility condition on\nits successive clearing integers~\\cite[Props.~1--2]{zudilin2017detv2};\nthe calculation for $H_2$ rules out replacing it with their numerical\nordering, not the criterion itself.  The usual integer-polynomial\nheight criterion likewise needs a common coefficient denominator for the\n\\emph{same} polynomial being evaluated.  The model's exact partial\nerrors alone cannot supply one cheaply.  In fact, with\n$\\alpha_m=q_0^{-m}$ and $\\beta_m=\\alpha_m-\\mu_m(q_0)$, the pencil\n$\\det(\\alpha_{i+j}X-\\beta_{i+j})_{i,j<N}$ is affine in $X$ because\n$(\\alpha_{i+j})$ has rank one.  At $X=1$ it is the positive rational\n$H_N(q_0)$; after primitive integral clearing its value is a positive\ninteger.  Matching Hankel order, positivity, fixed-base asymptotic and\na small \\emph{rational} value thus leaves the rational-base denominator\nproblem exactly where it was.  The Christoffel-type reweighting\n$a_kq^k\\mapsto a_kq^k(1-q^{k+1})$ is standard moment machinery, not a\nnew transform~\\cite[Intro., (3)--(4)]{kozhanvaktnas2024}; this example's claim is\nthe explicit combination and its arithmetic obstruction.\n\n\\subsection{Coefficient moments and cyclotomic content}\n\\label{long1049:sec:coefficient-questions}\n\\papersectiontarget{long1049-coefficients}\nWe now study the coefficient sequence, not the remainder sequence.\nThe identity $v_m^*=\\alpha_mF(p)-\\beta_m$ does not transfer a\nrepresenting measure from $(v_m^*)$ to $(\\alpha_m)$: subtraction\nmixes two coefficient sequences. Their positivity, spectral\nproperties and cyclotomic factors therefore require separate\ncalculations. Fix $p>1$ and define, using Gaussian binomial\npolynomials,\n\\[\n \\begin{aligned}\n R_m(p)&=\\sum_{k=0}^m(-1)^{m+k}p^{k(k+1)/2}\n          \\genfrac{[}{]}{0pt}{}{m}{k}_p\\genfrac{[}{]}{0pt}{}{m+k}{k}_p,\\\\\n s_m(p)&=([m]_p!)^3R_m(p),\n \\end{aligned}\n\\]\nwhere $[m]_p!=\\prod_{j=1}^m(1+p+\\cdots+p^{j-1})$.\nEach $R_m$ is the signed value at $x=p^{m+1}$ of the little\n$q$-Legendre polynomial of degree $m$, with $q=p^{-1}$; this\nidentification is discussed in the related-work section.\nThe polynomial normalisation of~\\cite[Sec.~3, p.~5]{zudilin2016}, at\n$x=z=1$, gives\n\\[\n \\begin{aligned}\n \\alpha_m&=p[p(p-1)^3]^m s_m(p),\\\\\n \\beta_m&=\\left[\\alpha_m\\sum_{j\\ge1}\\tau(j)p^{-j}\\right]_+-1,\\\\\n v_m^*&=\\alpha_mF(p)-\\beta_m.\n \\end{aligned}\n\\]\nThe brackets mean the polynomial part at infinity. Thus\n$\\alpha_0=p$ and $\\beta_0=0$. This fixes the normalisation before any\ncontent or positivity test.\n\n\\paragraph{A positive expansion is not a moment representation.}\nSubstituting $x=p^{m+1}$ into Van Assche's alternative expansion\n\\cite[(16), p.~4]{vanassche2001} gives the identity\n\\[\n R_m(p)=\\sum_{k=0}^m\\genfrac{[}{]}{0pt}{}{m}{k}_p\\genfrac{[}{]}{0pt}{}{m+k}{k}_p\n p^{(m-k)(m-k+1)/2}\\prod_{j=m-k+1}^{m}(p^j-1).\n\\]\nIndeed $(qx;q)_k=(-1)^k\\prod_{j=m-k+1}^m(p^j-1)$, so the two signs cancel.\nThis proves $R_m(p)>0$ for $p>1$, and nonnegative coefficients in the\nvariable $t=p-1$; it does not prove Hankel positivity. In fact,\n\\[\n \\det(R_{i+j}(p))_{i,j<3}=-36(p-1)^2+O((p-1)^3)\n \\quad(p\\downarrow1).\n\\]\nA geometric factor $c\\rho^m$ with $c,\\rho>0$ would change a Hankel\nmatrix only by a positive scalar and an invertible diagonal congruence,\nso it would preserve positive definiteness. The factor $([m]_p!)^3$\nis not geometric: its values at $m=0,1,2$ are $1,1,(1+p)^3$.\nIt cannot be discarded in testing the moment condition. Nor does its\npositivity alone prove that multiplication by it repairs the failed\ncondition for $R_m$. At $p=1$, $s_m(1)=(m!)^3$ is the moment sequence of a\nproduct of three independent unit exponential variables, since their\n$m$th moments multiply. This is only the polynomial endpoint; $F(1)$\ndiverges. Berg's Theorem~5.1 states that $(m!)^c$ is Stieltjes indeterminate\nfor $c>2$ \\cite{berg2007}. Thus a measure exists at the endpoint but is not\nunique. Neither uniqueness nor a canonical deformation is a premise of\nthe question at $p>1$.\n\n\\paragraph{Finite certificates and an all-rank degree check.}\nPut $D_{N,h}(p)=\\det(s_{i+j+h}(p))_{0\\le i,j<N}$, with $D_{0,h}=1$.\nExact polynomial calculations give strictly positive coefficients in\n$t=p-1$ for $D_{N,h}(1+t)$ when $h=0,1$ and $1\\le N\\le8$.\nThus sixteen determinant polynomials are positive for every real $p\\ge1$.\nTheir degrees, in increasing rank, are\n\\[\n\\begin{array}{c|rrrrrrrr}\nh=0&0&10&54&156&340&630&1050&1624\\\\\nh=1&2&26&96&236&470&822&1316&1976.\n\\end{array}\n\\]\nThe calculation uses exact arithmetic in $\\Z[p]$. Starting from\n$D_{0,h}=1$ and $D_{1,h}=s_h$, the Desnanot--Jacobi identity gives\n\\[\n D_{N,h}=\n \\frac{D_{N-1,h}D_{N-1,h+2}-D_{N-1,h+1}^{\\,2}}{D_{N-2,h+2}}\n \\qquad(N\\ge2).\n\\]\nThe degree argument below shows that each denominator is nonzero.\nThe computation checks that every division has zero remainder, then\nsubstitutes $p=1+t$ and tests every coefficient. All $8824$ coefficients\nin the sixteen polynomials are strictly positive. A\n\\href{https://github.com/wcook04/plectis-erdos/blob/4beca4ad0c00a626def9c73f2420e20ebf8d1968/computations/finite_coefficients_generate.py}{program}\ncomputes the\n\\href{https://github.com/wcook04/plectis-erdos/blob/4beca4ad0c00a626def9c73f2420e20ebf8d1968/computations/finite_coefficients.json}{full lists}.\nAn\n\\href{https://github.com/wcook04/plectis-erdos/blob/4beca4ad0c00a626def9c73f2420e20ebf8d1968/computations/check_finite_coefficients.py}{independent checker}\nevaluates each stored polynomial at more integer bases than its degree and\ncompares the values with fraction-free determinants of the integer Hankel\nmatrices, which identifies the stored list with $D_{N,h}(1+t)$ in $\\Z[t]$.\nAs a separate check of the normalisation, direct integer determinants\nat $p=1,2,3,5,11$ agree with evaluations of all sixteen polynomials.\nThose evaluations are checks, not the proof of polynomial positivity.\nThis is a finite computer-algebra calculation and gives no all-rank result.\n\nFor the eight unshifted determinants, positivity is also a\n\\href{\\laterepobase/lean/ErdosProblems/Erdos1049/PaperR20/HankelKroneckerCertificate.lean\\#L458}{Lean theorem},\nproved by a different certificate. Put $T=2^{2929}$. An explicit bound\nshows that every coefficient of $D_{N,0}(1+t)$, $N\\le8$, has absolute\nvalue less than $T/2$. An integer polynomial with this property is\ndetermined by its value at $t=T$, and its coefficients are the base-$T$\ndigits of that value whenever every digit is less than $T/2$. The Lean\nkernel computes the eight integers $D_{N,0}(1+T)$ exactly and checks this\ndigit condition and that the lowest digit is nonzero. Every coefficient\nof $D_{N,0}(1+t)$ is therefore nonnegative and the constant term is\npositive, so $D_{N,0}(p)>0$ for every real $p\\ge1$. This is the positivity\nused in Proposition~\\ref{long1049:res:finite-pencil}. The strict\npositivity of every coefficient, and the shifted determinants $D_{N,1}$,\nrest on the computation above.\nNeither the finite computation nor the Lean proof yields an all-rank\ncoefficient-positivity theorem.\n\nThe degrees admit a separate all-rank proof. The Gaussian binomial\n$\\genfrac{[}{]}{0pt}{}{u}{v}_p$ is monic of degree $v(u-v)$. In the signed sum for $R_m$,\nthe degree increases by $2m-k$ from the $k$th to the $(k+1)$st term.\nThe unique maximal term is $k=m$, with positive leading coefficient.\nHence $R_m$ is monic of degree $(3m^2+m)/2$, and $s_m$ is monic of\ndegree $3m^2-m$. In the determinant, the only permutation-dependent\npart of the degree is $6\\sum_i i\\sigma(i)$; strict rearrangement makes\nthe identity its unique maximum. Therefore\n\\[\n \\deg D_{N,h}=\\sum_{i=0}^{N-1}\\bigl(3(2i+h)^2-(2i+h)\\bigr),\n \\qquad \\operatorname{lc}D_{N,h}=1.\n\\]\nMore generally the same argument works for a fixed minor with distinct\nincreasing row and column indices. Each such minor is positive for all\nsufficiently large $p$, with a threshold that may depend on the minor.\nAt $p=1$, both leading Hankel families are positive definite by the\ninfinite-support product measure, so each fixed rank is also positive in\nsome neighbourhood of $1$. Neither argument provides a neighbourhood or\nlarge-base threshold uniform over every rank.\n\nA Stieltjes moment representation requires positive semidefiniteness of\nboth Hankel families at all ranks. The positive-definite formulation\nfor nondegenerate sequences is stated in\n\\cite[Lemma~2.1, p.~4]{wangzhu2016}; finite support requires allowing\nsemidefinite matrices. The necessity of both conditions is visible\nfrom their quadratic forms:\nif a positive measure $\\mu$ on $[0,\\infty)$ satisfies\n$s_m=\\int_0^\\infty x^m\\,d\\mu(x)$, and\n$P(x)=\\sum_{i=0}^{N-1}c_ix^i$ has real coefficients, then\n\\[\n \\sum_{i,j=0}^{N-1}c_ic_j s_{i+j+h}\n =\\int_0^\\infty x^hP(x)^2\\,d\\mu(x)\\ge0,\n \\qquad h=0,1.\n\\]\nThe shifted test records nonnegative support, not merely positivity\nof the measure. For example, the positive point mass at $-1$ has\nmoments $(-1)^m$. Its unshifted quadratic form is $P(-1)^2$, whereas\nits shifted form is $-P(-1)^2$. Thus a Hamburger moment sequence,\nwhich permits support anywhere on $\\R$, need not be a Stieltjes\nmoment sequence. The two tests above do not assert a measure for the\nentire coefficient sequence considered here.\nCoefficientwise total positivity would be stronger than the sixteen\ncertificates above. Positive production matrices and path constructions\ncan supply such a mechanism in other families\n\\cite[Sec.~2]{lrz2017}\\cite[Thms.~9.7--9.8]{psz2023}; no such matrix for\nthis moving-degree sequence has been established here.\n\n\\paragraph{A finite spectral consequence.}\nWrite $A_N=(\\alpha_{i+j})_{0\\le i,j<N}$ and\n$B_N=(\\beta_{i+j})_{0\\le i,j<N}$. We study the matrix pencil\n$YA_N-B_N$, where $Y$ is a scalar variable, through the roots of its\ndeterminant.\nLet $D=\\operatorname{diag}(1,p(p-1)^3,\\ldots,[p(p-1)^3]^{N-1})$.\nThen $A_N=pD(s_{i+j})D$. For $p>1$ this is an invertible positive diagonal\ncongruence. The remainder representation already proves\n$F(p)A_N-B_N=(v^*_{i+j})>0$ at every rank.\n\n\\begin{proposition}[finite coefficient positivity and pencil roots]\n\\label{long1049:res:finite-pencil}\nFor every real $p>1$ and $1\\le N\\le8$, $A_N$ is positive definite and\nall roots of $\\det(YA_N-B_N)$ are real and strictly less than $F(p)$.\nThe roots at consecutive ranks $N,N+1\\le8$ interlace non-strictly.\n\\end{proposition}\n\\evidenceremark{The Lean proof checks all three conclusions for every\nreal $p>1$ and every rank $N\\le8$; it states nothing beyond rank eight.}\n\\begin{proof}\nThe leading principal minors of $(s_{i+j})_{i,j<N}$ are exactly\n$D_{k,0}(p)$ for $1\\le k\\le N$; their coefficient positivity proved above, together with\nSylvester's criterion, gives positive definiteness. The diagonal congruence gives\n$A_N>0$. The real symmetric matrix $A_N^{-1/2}B_NA_N^{-1/2}$ has these\npencil roots as eigenvalues, and $F(p)I-A_N^{-1/2}B_NA_N^{-1/2}>0$ puts\nthem strictly below $F(p)$. The minimum--maximum principle for\n$x^{\\mathsf T}B_Nx/(x^{\\mathsf T}A_Nx)$ on nested coordinate subspaces\ngives non-strict interlacing. These are subspaces for the original\npencils; the separately conjugated symmetric matrices need not be\nprincipal submatrices of one another.\n\\end{proof}\nProposition~\\ref{long1049:res:finite-pencil} uses only the eight\nunshifted certificates $D_{k,0}$, not an all-rank coefficient measure.\nThe shifted certificates $D_{k,1}$ are used below for a truncated\nStieltjes moment representation and the first fifteen continued-fraction\ncoefficients, not for this application of Sylvester's criterion.\nAn infinite-support coefficient measure would extend the pencil argument\nto all ranks. At the certified ranks, the largest root is a nondecreasing\nlower bound for $F(p)$; convergence to $F(p)$, simple roots, strict\ninterlacing and denominator control are not established. At $p=1$ the\ncongruence degenerates and $F$ diverges, so the proposition excludes that\nendpoint.\n\nThe pencil is different from the Jacobi matrix of the coefficient\nmoments. This is already visible at rank two. The defining formulas give\n\\[\n \\beta_0=0,\\qquad \\beta_1=p^3(p-1)^2(p+2)>0\\quad(p>1).\n\\]\nConsequently $\\det B_2=-\\beta_1^2<0$. Since $A_2>0$, the product of the\ntwo real pencil roots is $\\det B_2/\\det A_2<0$: one root is negative\nand the other positive. In contrast, the finite Stieltjes construction\nbelow has only positive nodes. Positivity of the coefficient Hankel\nmatrix therefore does not identify these two spectral constructions.\n\n\\paragraph{Formal continued fractions and finite quadrature.}\nWrite $D_n=D_{n,0}$, $E_n=D_{n,1}$, with $D_0=E_0=1$.\nAll these determinant polynomials are nonzero by the all-rank degree\ncalculation above. The classical Stieltjes continued fraction is thus\ndefined over the rational-function field $\\Q(p)$, with the convention\n\\[\n \\sum_{m\\ge0}s_m(p)z^m\n =\\cfrac{1}{1-\\cfrac{\\lambda_1z}{1-\\cfrac{\\lambda_2z}{1-\\cdots}}},\n\\]\n\\[\n \\lambda_{2n-1}=\\frac{E_nD_{n-1}}{D_nE_{n-1}},\\quad\n \\lambda_{2n}=\\frac{D_{n+1}E_{n-1}}{E_nD_n}\\quad(n\\ge1).\n\\]\nHere $s_0=1$, and the identity is in $\\Q(p)[[z]]$. The classical\nmoment correspondence is recalled in \\cite[pp.~1--2]{sw2024}.\nSpecialising a ratio at a fixed real $p$ requires its displayed\ndenominators to be nonzero. The sixteen certificates ensure this and\n$\\lambda_1,\\ldots,\\lambda_{15}>0$ for every $p\\ge1$; they do not ensure\nit at all indices. Seeking positive formulas for all these ratios is\ntherefore a specific version of the coefficient-moment question.\nThe generating series has radius of convergence zero for every\n$p\\ge1$, so the displayed identity must be read formally. At $p=1$\nthis follows from $s_m(1)=(m!)^3$. For $p>1$, the $k=0$ term of the\npositive expansion for $R_m$ and the inequality\n$[m]_p!\\ge p^{m(m-1)/2}$ give\n\\[\n s_m(p)\\ge p^{2m^2-m}.\n\\]\nThus $s_m(p)^{1/m}\\to\\infty$. This concerns the moment power series;\nit does not decide convergence of the continued fraction at an\nindividual nonzero value of $z$.\n\nThe same certificates also give a finite representing measure:\nfor each fixed $p\\ge1$, eight positive atoms reproduce exactly\n$s_0,\\ldots,s_{15}$. The construction uses only the two certified\n$8\\times8$ matrices, not positivity at higher ranks. Set\n\\[\n H_0=(s_{i+j})_{i,j<8},\\qquad H_1=(s_{i+j+1})_{i,j<8},\\qquad\n T=H_0^{-1}H_1,\n\\]\nand give $\\R^8$ the inner product\n$\\langle u,v\\rangle=u^{\\mathsf T}H_0v$.\nThe operator $T$ is self-adjoint and positive definite because\n$H_0T=H_1$ is symmetric and positive definite. If $e_0,\\ldots,e_7$\nare the coordinate vectors, the Hankel identities give\n$Te_j=e_{j+1}$ for $0\\le j<7$. Hence $e_0$ is cyclic: its first\neight iterates form a basis. The spectral theorem now gives eight\ndistinct positive eigenvalues and a positive weight at each, namely\nthe squared norm of the corresponding projection of $e_0$.\nCyclicity ensures that no projection vanishes and that no eigenspace\nhas dimension greater than one.\n\nLet $\\nu$ be the resulting atomic measure. For $0\\le m\\le14$, choose\n$i,j\\le7$ with $i+j=m$. Self-adjointness gives\n\\[\n \\int x^m\\,d\\nu(x)=\\langle T^m e_0,e_0\\rangle\n =\\langle e_i,e_j\\rangle=s_m.\n\\]\nFor the last moment,\n\\[\n \\int x^{15}\\,d\\nu(x)=\\langle T^7e_0,T T^7e_0\\rangle\n =e_7^{\\mathsf T}H_1e_7=s_{15}.\n\\]\nThe weights sum to $s_0=1$. Hence the measure agrees with the moment\nfunctional on every polynomial of degree at most $15$; this is the\ndegree range certified for the eight-node quadrature obtained here.\nIn an orthonormal polynomial basis it is the classical Jacobi construction;\nGolub and Welsch describe the computation of its nodes and weights\n\\cite[Sec.~2, (2.6); Sec.~4]{golubwelsch1969}.\nNothing here identifies the subsequent moments of $\\nu$ with\n$s_{16},s_{17},\\ldots$. This finite construction must also be distinguished\nfrom the coefficient pencil $(A_N,B_N)$.\nFor rational modifications of an already known moment functional, see\nKrattenthaler \\cite[Thm.~1; Sec.~6]{krattenthaler2021}; such a modification\nproducing the entire sequence $s_m$ has not been identified.\n\n\\paragraph{Testing for additional cyclotomic factors.}\nIndependently of the moment question, let $v_{\\Phi_d}$ denote the\nmultiplicity of the cyclotomic factor $\\Phi_d$ in a nonzero polynomial\nin $\\Q[p]$, and set $v_{\\Phi_d}(0)=\\infty$.\nFor a $2\\times2$ matrix with entry valuations $0,2,2,5$, the two\ndeterminant products have valuations $5$ and $4$. The lower one is\nunique, so the determinant has valuation $4$. A tie is different:\nall four entries of\n$\\left(\\begin{smallmatrix}1&1\\\\1&1+\\Phi_d\\end{smallmatrix}\\right)$\nhave valuation zero, but its determinant is $\\Phi_d$.\nAt general rank, we must find the least total valuation and then test\nwhether the terms attaining it cancel. For $N,d\\ge1$, define\n\\[\n \\begin{aligned}\n \\mathcal H_N(Y;p)&=\\det(\\alpha_{i+j}Y-\\beta_{i+j})_{i,j<N},\\\\\n t_{m,d}&=\\min(v_{\\Phi_d}(\\alpha_m),v_{\\Phi_d}(\\beta_m)),\\\\\n e_{N,d}&=\\min_{\\sigma\\in S_N}\\sum_{i=0}^{N-1}t_{i+\\sigma(i),d}.\n \\end{aligned}\n\\]\nThe last expression is a minimum-cost assignment: row $i$ is matched\nto column $\\sigma(i)$ at cost $t_{i+\\sigma(i),d}$.\nThe valuation of a polynomial in $Y$ means the minimum valuation of\nits coefficients in $\\Q[p]$. Each determinant term is\ndivisible by $\\Phi_d^{e_{N,d}}$. Set\n\\[\n \\Gamma_{N,d}(Y)=\n \\sum_{\\substack{\\sigma\\in S_N\\\\\\sum_i t_{i+\\sigma(i),d}=e_{N,d}}}\n \\operatorname{sgn}(\\sigma)\n \\prod_i\\overline{\\Phi_d^{-t_{i+\\sigma(i),d}}\n        (\\alpha_{i+\\sigma(i)}Y-\\beta_{i+\\sigma(i)})},\n\\]\nwhere the bar is reduction in $(\\Q[p]/(\\Phi_d))[Y]$.\nThen $v_{\\Phi_d}\\mathcal H_N=e_{N,d}$ exactly when\n$\\Gamma_{N,d}\\ne0$. This isolates the cancellation that a genuine\nadditional factor would require.\nThe assignment bound records the smallest valuation available among\nthe determinant terms. Excess divisibility is possible only if their\nleading residues cancel. To test that cancellation by one determinant,\nwe distribute the minimum valuation between rows and columns.\nThe integer costs are finite, since every $\\alpha_m$ is nonzero. Minimum-cost assignment duality gives\ninteger row and column potentials with\n\\[\n u_i+v_j\\le t_{i+j,d},\\qquad\n \\sum_i u_i+\\sum_jv_j=e_{N,d}.\n\\]\nHere is a direct construction, so their existence is not an additional\nhypothesis. Choose a minimum-cost permutation $\\sigma$. On the column\nindices put a directed edge from $\\sigma(i)$ to $j$ of length\n$t_{i+j,d}-t_{i+\\sigma(i),d}$. A negative directed cycle would improve\n$\\sigma$ by reassigning the corresponding rows along that cycle;\nthere is therefore no such cycle. Add a new vertex with an edge of\nlength zero to every column. The shortest-path distances $v_j$ are\nfinite integers and satisfy\n\\[\n v_j-v_{\\sigma(i)}\\le t_{i+j,d}-t_{i+\\sigma(i),d}.\n\\]\nPutting $u_i=t_{i+\\sigma(i),d}-v_{\\sigma(i)}$ gives the inequalities\nabove, with equality on the chosen assignment and hence equality of\ntotals.\n\nDivide row $i$ by $\\Phi_d^{u_i}$ and column $j$ by $\\Phi_d^{v_j}$\nover $\\Q(p)$. The inequalities ensure that every resulting entry is\nin $\\Q[p,Y]$, even if some potentials are negative. Reduce these\nentries modulo $\\Phi_d$. Only those with\n$u_i+v_j=t_{i+j,d}$ survive. The determinant of the reduced matrix is\n$\\Gamma_{N,d}$, because every non-minimal permutation vanishes on\nreduction. Thus the valuation exceeds $e_{N,d}$ exactly when the\nreduced matrix is singular over $(\\Q[p]/(\\Phi_d))(Y)$.\nA kernel relation valid at just one value of $Y$ is insufficient:\nthe determinant must vanish as a polynomial in $Y$. Conversely, one\nnonzero value $\\Gamma_{N,d}(Y_0)$ proves equality with the assignment\nbound. Neither assignment duality nor the existence of the potentials\nforces the reduced determinant to vanish.\n\n\nAn exact calculation outside Lean gives the following monic contents.\nHere $\\operatorname{cont}_Y$ means the monic gcd in $\\Q[p]$ of the\ncoefficients in $Y$:\n\\[\n\\begin{array}{c|l}\nN&\\operatorname{cont}_Y\\mathcal H_N\\\\\\hline\n1&p\\\\\n2&p^5(p-1)^4\\\\\n3&p^{14}(p-1)^{15}(p+1)^4\\\\\n4&p^{30}(p-1)^{32}(p+1)^8(p^2+p+1)^4\\\\\n5&p^{55}(p-1)^{55}(p+1)^{19}(p^2+1)^4(p^2+p+1)^8.\n\\end{array}\n\\]\nThese factorisations attain the assignment bound for every $d$ at\nthese five ranks. The residue certificates verify equality for\n$d\\le8$; the displayed complete factorisations contain no other\ncyclotomic factors, so the nonnegative assignment bound is zero at\nall remaining $d$. The power of $p$ is not cyclotomic content.\n\nFor the table, an exact\n\\href{https://github.com/wcook04/plectis-erdos/blob/4beca4ad0c00a626def9c73f2420e20ebf8d1968/research/experiments/erdos1049/cyclotomic_content.py}{program}\ncomputes the full polynomials\n$\\mathcal H_N(Y_0;p)$ at $Y_0=0,\\ldots,N$ by fraction-free\nelimination, checking every polynomial division. Their monic gcd in $\\Q[p]$ is $\\operatorname{cont}_Y\\mathcal H_N$.\nA divisor of all coefficients divides every evaluated polynomial;\nconversely, the $N+1$ values recover those coefficients by rational\ninterpolation in $Y$. This proves the reverse divisibility over\n$\\Q[p]$, where the interpolation denominators are units. The calculation includes every\nfactor, not just a prescribed list of cyclotomic candidates.\n\nThe same program produces a nonzero residue witness for every pair\n\\[\n 1\\le N\\le8,\\qquad 1\\le d\\le\\max(8,2N-2).\n\\]\nThere are $76$ such pairs. For each, the certificate gives an optimal\nassignment, integer dual potentials, an integer $Y_0\\in\\{0,\\ldots,N\\}$\nand the nonzero residue $\\Gamma_{N,d}(Y_0)$. The signed\nminimum-cost subset recurrence and the determinant of the reduced\nmatrix give the same residue. A separate check computes the full\npolynomial $\\mathcal H_N(Y_0;p)$ and verifies directly that division\nby $\\Phi_d^{e_{N,d}}$ leaves exactly this nonzero residue.\nAt $N=2,d=1$, for example, the entry valuations are\n$\\left(\\begin{smallmatrix}0&2\\\\2&5\\end{smallmatrix}\\right)$,\n$e_{2,1}=4$, and $\\Gamma_{2,1}(0)=-9$.\n\nThe source coefficients, determinant-value polynomials and witnesses\nare computed by that program, which also repeats the direct\ndeterminant checks; its \\href{https://github.com/wcook04/plectis-erdos/blob/4beca4ad0c00a626def9c73f2420e20ebf8d1968/research/experiments/erdos1049/receipts/cyclotomic-content.json}{recorded result} lists every witness. At ranks six to eight\nthese certificates cover only the stated cyclotomic-index window;\ncomplete coefficient contents are computed only through rank five.\n\nThese computations distinguish systematic common factors from\ncancellation beyond entry valuations, a distinction also important in\nfraction-free matrix decompositions \\cite[Sec.~2; Thm.~6]{middeke2021}.\nIn particular the gcd argument at the points $0,\\ldots,N$ is over\n$\\Q[p]$: the rational Vandermonde inverse is not generally integral, so\nit must not be used to assert the corresponding coefficient gcd in $\\Z[p]$.\n\n\nProposition~4 of Krattenthaler--Rochev--V\\\"a\\\"an\\\"anen--Zudilin\n\\cite[pp.~14--15; (4.10), p.~17]{krvz2009} obtains cyclotomic factors\nthrough root-of-unity annihilation for a different first-order tail\nrecurrence. No corresponding recurrence has been established for this\nmoving-diagonal pencil. The finite witnesses prove equality with the assignment bound at\nthese $76$ pairs. They do not exclude excess at larger ranks or, at\nranks six to eight, cyclotomic indices outside the stated window. Whether minimum-valuation residues ever cancel\nsystematically, and whether $s_m$ is a Stieltjes sequence, remain separate\nquestions. The remainder measure answers neither.\n\n\\section{Rescaling integer rows}\\label{long1049:sec:primitive}\n\\papersectiontarget{long1049-arithmetic}\n\nFor a single irrationality form, the essential conditions are\nintegral coefficients and a nonzero remainder tending to zero.\nDividing a nonzero integer row $(U,V)$ by $\\gcd(|U|,|V|)$\npreserves integrality and divides its remainder by the same amount;\nthe resulting row is primitive. A useful content division removes a factor already shared by the\ntwo entries. Multiplying a row first merely introduces a factor\nthat also multiplies its remainder. For two rows, the same factors\nmultiply their determinant. The identities below separate this\nrescaling from genuine cancellation. A height estimate is needed\nfor an irrationality measure, not for the elementary single-form\ncriterion. For a real target $S$, put\n\\[\n L_S(U,V)=US-V,\n \\qquad\n \\Delta\\bigl((U_n,V_n),(U_m,V_m)\\bigr)=U_nV_m-U_mV_n.\n\\]\nHere \\(L_S(U,V)\\) is the \\emph{error} of the integer linear form,\nnot the rational approximation error $|S-V/U|$ when $U\\ne0$.\nThe determinant eliminates \\(S\\) exactly:\n\\[\n\\Delta=U_mL_S(U_n,V_n)-U_nL_S(U_m,V_m).\n\\]\n\nIf this integer determinant does not vanish, then\n\\[\n 1\\le|\\Delta|\n  \\le|U_m|\\,\\bigl|L_S(U_n,V_n)\\bigr|+|U_n|\\,\\bigl|L_S(U_m,V_m)\\bigr|,\n\\]\nand the two errors cannot both be smaller than\n\\(1/(|U_n|+|U_m|)\\).  The next theorem compares the divisor introduced by rescaling with\nthe resulting change in the determinant's absolute value.\n\n\\begin{theorem}[rescaling rows and their determinant]\\label{long1049:res:content}\nLet \\(S\\) be real, let \\((U_n,V_n)\\) and \\((U_m,V_m)\\) be pairs of integers, and\nlet \\(c_n,c_m\\) be integers.  Then\n\\[\n L_S(c_nU_n,c_nV_n)=c_nL_S(U_n,V_n),\n\\]\n\\[\n \\Delta\\bigl(c_n(U_n,V_n),c_m(U_m,V_m)\\bigr)\n =c_nc_m\\Delta\\bigl((U_n,V_n),(U_m,V_m)\\bigr),\n\\]\nand consequently\n\\[\n \\left|\\Delta\\bigl(c_n(U_n,V_n),c_m(U_m,V_m)\\bigr)\\right|\n =|c_n|\\,|c_m|\\,\n   \\left|\\Delta\\bigl((U_n,V_n),(U_m,V_m)\\bigr)\\right|.\n\\]\nIn particular \\(c_nc_m\\) divides the scaled determinant. Multiplication\nby these scalars therefore introduces a divisor whose absolute value is\nexactly the factor multiplying the determinant's absolute value.\n\\end{theorem}\n\n\\begin{proof}\nExpand the linear form and use the bilinearity of the determinant.\nThe original integer determinant is the quotient witnessing divisibility;\nit need not be primitive.\n\\end{proof}\n\n\\begin{example}\\label{long1049:ex:content}\nTake \\((U_n,V_n)=(1,2)\\) and \\((U_m,V_m)=(3,5)\\), so that\n\\(\\Delta=1\\cdot5-3\\cdot2=-1\\).  Multiplying the first row by \\(c_n=6\\) and the\nsecond by \\(c_m=10\\) gives the rows \\((6,12)\\) and \\((30,50)\\), whose\ndeterminant is \\(6\\cdot50-30\\cdot12=-60\\).  That determinant is now divisible\nby \\(60\\), which looks like a local gain of \\(60\\); and its absolute value has\nrisen from \\(1\\) to \\(60\\), which is a cost of exactly the same size.\n\\end{example}\n\n\\noindent Lean checks the error identity in\n\\lword{Erdos1049/RationalPadeArithmetic.lean}{84}{rationalPadeError_mul}{error\nscaling}, the determinant identity in\n\\lword{Erdos1049/RationalPadeArithmetic.lean}{94}{rationalPadeExteriorDet_mul_contents}{content\nfactorisation}, the exact absolute-height identity in\n\\lword{Erdos1049/RationalPadeArithmetic.lean}{104}{natAbs_rationalPadeExteriorDet_mul_contents}{absolute\ndeterminant scaling}, the divisor statement in\n\\lword{Erdos1049/RationalPadeArithmetic.lean}{116}{contentProduct_dvd_rationalPadeExteriorDet}{content-product\ndivisibility}, and the elimination identity in\n\\lword{Erdos1049/RationalPadeArithmetic.lean}{124}{rationalPadeExteriorDet_cast_eq}{exterior\ndeterminant identity}.\n\nThe identities allow zero scalars, but cancelling the scalar factors\nrequires them to be nonzero. Multiplication alone gives no improvement\nin the comparison between a divisor and the determinant's absolute\nvalue. The theorem neither asserts that the determinant is nonzero nor\nestimates the original approximation error. Cancelling polynomial\nfactors before evaluation, finding factors common to several minors,\nand adding different rows remain separate operations.\n\n\\section{Congruences after evaluation at \\texorpdfstring{$3/2$}{3/2}}\\label{long1049:sec:endpoints}\n\\papersectiontarget{long1049-endpoints}\n\nWe use congruences modulo powers of $2$ and $3$ to find combinations\nof evaluated polynomials with prescribed divisibility. The argument\napplies to arbitrary integral polynomials, including the forms of\n\\cite[Sections~3--5]{zudilin2004}. The denominator-clearing map must be the same for every row that\nwill be added. We therefore fix a common truncation index $W\\ge0$\nand, for $P(X)=\\sum_i p_iX^i\\in\\Z[X]$, put\n\\[\n H_W(P)=\\sum_{i=0}^{W}p_i\\,3^i2^{W-i}.\n\\]\nThis is the\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{98}{homEvalThreeTwo}{denominator-cleared evaluation at $3/2$}.  Each term \\(p_iX^i\\) with \\(i\\le W\\) contributes\n\\(p_i3^i2^{W-i}\\). Keep the index \\(W\\) fixed when polynomials are\nadded. When \\(W\\ge\\deg P\\), this is exactly the cleared numerator, since\n\\[\n \\sum_{i=0}^{W}p_i\\,3^i2^{W-i}=2^{W}\\sum_{i=0}^{W}p_i\\left(\\tfrac32\\right)^{i}\n =2^{W}P\\!\\left(\\tfrac32\\right).\n\\]\n\nHere a unit coefficient in $\\Z$ means $1$ or $-1$. Thus a monic\npolynomial of degree $W$ with constant term $\\pm1$ satisfies both unit\nconditions below. Those conditions are sufficient, not necessary:\nthe congruences themselves only require that the relevant coefficient\nnot be divisible by the relevant prime.\n\n\\begin{theorem}[endpoint residues]\\label{long1049:res:endpoints}\nLet \\(P=\\sum_ip_iX^i\\in\\Z[X]\\) and let \\(W\\ge0\\).  Then\n\\[\n H_W(P)\\equiv p_0\\,2^W\\pmod 3,\\qquad\n H_W(P)\\equiv p_W\\,3^W\\pmod 2.\n\\]\nConsequently a unit constant coefficient prevents divisibility by \\(3\\), and\na unit coefficient at index \\(W\\) prevents divisibility by \\(2\\).\n\\end{theorem}\n\n\\begin{proof}\nModulo \\(3\\), every summand with \\(i>0\\) vanishes; modulo \\(2\\), every summand\nwith \\(i<W\\) vanishes.  The remaining powers are units in the corresponding\nresidue fields.\n\\end{proof}\n\nSince \\(2^{W}\\) is invertible modulo \\(3\\) and \\(3^{W}\\) is invertible modulo\n\\(2\\), the two congruences say more than the stated consequence: divisibility\nof \\(H_W(P)\\) by \\(3\\) is decided by \\(p_0\\) alone, and divisibility by \\(2\\) by\n\\(p_W\\) alone.  The rest of the coefficient vector is invisible to both primes.\nThe coefficient \\(p_W\\) is the top coefficient of \\(P\\) exactly when\n\\(\\deg P=W\\), and is zero when \\(\\deg P<W\\). The identity\n\\(H_W(P)=2^{W}P(3/2)\\) is guaranteed when \\(\\deg P\\le W\\).\nWithout a degree bound, increasing the truncation index gives\n\\[\n H_{W+1}(P)=2H_W(P)+p_{W+1}3^{W+1},\n\\]\nso the truncated sum doubles precisely when the extra coefficient is zero.\n\n\\begin{example}\\label{long1049:ex:endpoints}\nTake \\(W=2\\).  The three polynomials below differ only at an endpoint.\n\\begin{center}\n\\begin{tabular}{@{}llcc@{}}\n\\toprule\n\\papertablehead\n\\(P\\) & \\(H_2(P)\\) & \\(3\\mid H_2(P)\\) & \\(2\\mid H_2(P)\\)\\\\\n\\midrule\n\\(X^2+1\\)  & \\(1\\cdot4+0\\cdot6+1\\cdot9=13\\) & no  & no \\\\\n\\(X^2+3\\)  & \\(3\\cdot4+0\\cdot6+1\\cdot9=21\\) & yes & no \\\\\n\\(2X^2+1\\) & \\(1\\cdot4+0\\cdot6+2\\cdot9=22\\) & no  & yes \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\nThe first has both endpoints equal to \\(1\\) and its evaluation, \\(13\\), is\ndivisible by neither prime; as a check, \\(2^{2}\\bigl((3/2)^2+1\\bigr)=13\\).  The\nsecond and third show that each hypothesis is used: spoiling the constant\nendpoint admits \\(3\\), and spoiling the top endpoint admits \\(2\\).\n\\end{example}\n\n\\noindent Formal proofs cover the congruences\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{203}{homEvalThreeTwo_mod_three}{modulo $3$} and\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{320}{homEvalThreeTwo_mod_two}{modulo $2$}, together with the consequences for a\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{338}{three_not_dvd_homEvalThreeTwo_of_const_unit}{unit constant coefficient} and a\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{356}{two_not_dvd_homEvalThreeTwo_of_top_unit}{unit coefficient of $X^W$}.\n\nIn the following proposition, a unit top endpoint means that the\ncoefficient at index $W$ is $1$ or $-1$, with no degree bound imposed.\nThe two conditions apply to different entries of the polynomial pair.\n\n\\begin{proposition}[common divisor]\\label{long1049:res:commonmult}\nLet \\(U,V\\in\\Z[X]\\) and let \\(W\\ge0\\).  If \\(U\\) has unit top endpoint, \\(V\\)\nhas unit constant endpoint, and an integer\n\\(c\\) divides both \\(H_W(U)\\) and \\(H_W(V)\\), then\n\\[\n 2\\nmid c\\qquad\\text{and}\\qquad 3\\nmid c.\n\\]\n\\end{proposition}\n\\begin{proof}\nIf $2\\mid c$, then $2\\mid H_W(U)$, contrary to the unit coefficient\nat index $W$ and Theorem~\\ref{long1049:res:endpoints}. Similarly,\n$3\\mid c$ would imply $3\\mid H_W(V)$, contrary to the unit constant\ncoefficient.\n\\end{proof}\n\n\\noindent Lean checks this\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{398}{commonMultiplier_not_two_not_three_of_endpoint_units}{common-divisor\nexclusion}.  The proposition does not say that the two evaluations are coprime;\nit says only that every common divisor is coprime to $6$. It gives\nno restriction on the other prime factors of that divisor.\n\n\\begin{example}\\label{long1049:ex:commonmult}\nTake \\(W=2\\), \\(U=X^2+3\\) and \\(V=5X^2+1\\).  The top endpoint of \\(U\\) and the\nconstant endpoint of \\(V\\) are both \\(1\\), and\n\\[\n H_2(U)=3\\cdot4+1\\cdot9=21,\\qquad H_2(V)=1\\cdot4+5\\cdot9=49 .\n\\]\nHere $\\gcd(21,49)=7$. The endpoint assumptions therefore allow a\nnontrivial common divisor, but that divisor is coprime to $6$.\n\\end{example}\n\n\\begin{corollary}[limits of rescaling and common-divisor cancellation at $3/2$]\\label{long1049:res:nomult}\nUnder the endpoint hypotheses of Proposition~\\ref{long1049:res:commonmult},\nevery common divisor of the unscaled evaluations $H_W(U)$ and $H_W(V)$\nis coprime to $6$. Multiplying two integer rows by nonzero integers\n$c_n,c_m$ multiplies their determinant by $c_nc_m$ and its absolute\nvalue by $|c_nc_m|$. Cancelling this introduced scalar factor therefore\nleaves the original comparison between divisor and determinant size\nunchanged.\n\\end{corollary}\n\n\\begin{proof}\nThe common-divisor assertion is Proposition~\\ref{long1049:res:commonmult}.\nBy Theorem~\\ref{long1049:res:content}, rescaling the two rows by\n$c_n,c_m$ multiplies their errors by $c_n,c_m$, respectively, and\ntheir $2\\times2$ determinant by $c_nc_m$. For nonzero scalars, the\nintroduced divisor has the same absolute value as the introduced\nfactor in that determinant. Dividing out that factor restores the\noriginal determinant, including its original divisor-to-size comparison.\n\\end{proof}\n\nOne further consequence of Theorem~\\ref{long1049:res:endpoints} is worth stating,\nbecause it bears on the most natural way one might hope to import an existing\ndenominator reduction.  Write $\\Phi_m$ for the $m$th cyclotomic polynomial and,\nfor coprime $a>b\\ge1$, put $\\Phi_m(a,b)=b^{\\varphi(m)}\\Phi_m(a/b)$ for its\nhomogenisation, with exponent $\\varphi(m)=\\deg\\Phi_m$.\n\n\\begin{proposition}[coprimality of homogenised cyclotomic values]\n\\label{long1049:res:cyclounit}\nLet $a>b\\ge1$ with $\\gcd(a,b)=1$ and let $m\\ge1$.  Then\n$\\gcd(\\Phi_m(a,b),ab)=1$.  In particular $\\gcd(\\Phi_m(3,2),6)=1$ for every $m$.\n\\end{proposition}\n\n\\begin{proof}\nThe polynomial $\\Phi_m$ is monic and $\\Phi_m(0)=\\pm1$. The endpoint\nargument in Theorem~\\ref{long1049:res:endpoints} also works at $(a,b)$:\nmodulo a prime dividing $b$, only $a^{\\varphi(m)}$ survives, and\nmodulo a prime dividing $a$, only $\\Phi_m(0)b^{\\varphi(m)}$ survives.\nCoprimality of $a$ and $b$ makes each surviving term nonzero.\n\\end{proof}\n\nRhin and Viola's factorial-coset quotients~\\cite{rhinviola1996}\nreduce denominators of rational forms in $\\zeta(2)$. Zudilin's\n$q$-analogue has an order-twelve symmetry group for the normalised\nseries; its denominator cancellation uses the order-six subgroup\npreserving the required sign condition\n\\cite[Sec.~4, pp.~159--161]{zudilin2004}.\nProposition~\\ref{long1049:res:cyclounit} applies to these cyclotomic\nfactors: each homogenised value at $(3,2)$, and hence any product of\nthem, is divisible by neither $2$ nor $3$. It makes no such assertion\nabout an arbitrary integer or factorial factor, which can contain\nboth primes. Cancelling a common cyclotomic factor can still reduce\nthe real absolute values. That reduction is distinct from producing\nthe powers of $2$ or $3$ sought in Section~\\ref{long1049:sec:open}.\n\n\\noindent Scalar multiplication can introduce powers of $2$ and $3$,\nbut does not improve this comparison. Under the endpoint hypotheses,\nthe unscaled pair has no common factor at either prime. Neither\nobservation shows that a gain at these primes is necessary for a proof\nby linear forms at $3/2$.\n\nThe operation studied next is different: take an integer combination\nof several rows and ask that the combination be divisible where the\nindividual rows are not. The proposed divisor must come from\ncancellation between rows, not from multiplying a row by that divisor.\nThe endpoint congruences are the first case of a divisibility condition that can be imposed\nto any depth, and it is that condition, read additively, which is counted\nbelow.\n\nWe first raise the two congruences to prime powers.  Fix depths \\(R,S\\ge0\\).\nFor $P\\in\\Z[X]$, let $J_{3,R}(P)$ and $J_{2,S}(P)$ be the residues of\n$H_W(P)$ modulo $3^R$ and $2^S$, respectively.\nTheorem~\\ref{long1049:res:endpoints} computes the depth-one residues.\nAt depth $R$, terms with $i\\ge R$ are invisible modulo $3^R$;\nat depth $S$, terms with $W-i\\ge S$ are invisible modulo $2^S$.\nThus deeper congruences inspect longer segments of the two ends\nof the coefficient list, always at the same fixed $W$.\n\n\\begin{lemma}[a residue and divisibility by $3^R$]\\label{res:bottomjet}\nVanishing of the\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{191}{bottomJet3_eq_zero_iff_dvd}{residue modulo $3^R$}\nis exactly divisibility by the corresponding power of three:\n$J_{3,R}(P)=0$ if and only if $3^R\\mid H_W(P)$.\n\\end{lemma}\n\\begin{proof}\nBy definition $J_{3,R}(P)$ is the image of the integer $H_W(P)$ in\n$\\mathbb Z/3^R\\mathbb Z$. The kernel of this quotient map is\n$3^R\\mathbb Z$, which proves the equivalence, including $R=0$.\n\\end{proof}\nThe same argument gives\n$J_{2,S}(P)=0\\iff2^S\\mid H_W(P)$ for $S\\ge0$.\n\nThe \\emph{vector of four residues} of a coefficient pair\n\\((U,V)\\) is then the quadruple\n\\[\n \\bigl(J_{3,R}(U),J_{3,R}(V),J_{2,S}(U),J_{2,S}(V)\\bigr)\n \\in(\\Z/3^R\\Z)^2\\times(\\Z/2^S\\Z)^2 ,\n\\]\ntwo residues for each of the two primes, one from each entry of the pair.  By\nthe displayed criteria it vanishes exactly when \\(3^R\\) divides both specialised\nentries and \\(2^S\\) divides both.\n\nInstead of requiring each input row to have a common divisor, we seek\na small integer combination whose two entries are both divisible by\n$3^R2^S$. Since $H_W$ is additive, the four residues of the combination\nare the corresponding sums of the input residues. This allows a\npigeonhole argument on subset sums.\n\n\\begin{theorem}[equal residues for two subset sums]\\label{long1049:res:jetkernel}\nFix a truncation index \\(W\\) and depths \\(R,S\\), and let \\((U_j,V_j)_{j<M}\\) be any \\(M\\)\npairs of integral polynomials. Represent each subset of\n\\(\\{0,\\dots,M-1\\}\\) by its indicator vector in \\(\\{0,1\\}^M\\).\nIf the \\(2^M\\) subsets outnumber the possible residue vectors in\n\\[\n (\\Z/3^R\\Z)^2\\times(\\Z/2^S\\Z)^2,\n\\]\nthen two distinct subsets have the same residue vector.  Subtracting their\nindicator vectors gives a nonzero coefficient vector in\n\\(\\{-1,0,1\\}^M\\) cancelling all four residues.  The target has exact cardinality\n\\[\n (3^R)^2(2^S)^2.\n\\]\nIn particular, if \\(R>0\\) and \\(4R+2S\\le M\\), such a collision exists.\n\\end{theorem}\n\n\\begin{proof}\nSend each subset to the sum of the residue vectors of its members.\nThe pigeonhole principle gives two distinct subsets with the same\nresidue vector. The number of possible vectors is the product of\nthe four moduli.  For \\(R>0\\),\n\\[\n (3^R)^2(2^S)^2<(4^R)^2(2^S)^2=2^{4R+2S}\\le2^M,\n\\]\nwhich proves the stated sufficient threshold.\n\\end{proof}\n\nThe power bracket improves the generic coefficient $4R$ when the depth\n$R$ is a positive integer multiple of $41$. All depths and row counts\nin the following corollary, including $T$, are integers.\n\n\\begin{corollary}[the exact count at depth $41$]\\label{long1049:res:rankfortyone}\nLet $T>0$.  At modulus $3^{R}$ with $R=41T$, any family of\n$M\\ge130T+2S$ integral polynomial pairs has two distinct binary vectors\nwith the same residue vector.  For $T=1$ the coefficient $130$ is exact for\nthis counting argument:\n\\[\n 2^{129+2S}<\\bigl| (\\Z/3^{41}\\Z)^2\\times\n                       (\\Z/2^S\\Z)^2\\bigr|.\n\\]\nNo exact-optimality assertion is made here for $T>1$.\n\\end{corollary}\n\n\\begin{proof}\nThe upper power inequality gives\n\\[\n (3^{41T})^2(2^S)^2\n <(2^{65})^{2T}(2^S)^2=2^{130T+2S}\\le2^M,\n\\]\nso Theorem~\\ref{long1049:res:jetkernel} applies.  For $T=1$, direct integer\nevaluation gives $2^{129}<3^{82}$; multiplying by $(2^S)^2$ gives the\ndisplayed reverse count at $129+2S$.\n\\end{proof}\n\n\\noindent The formal statements check the\n\\lword{Erdos1049/AdelicHeightBridge.lean}{1800}{exists_distinct_binary_selectors_same_fourJet_of_rank_41}{sufficient count $130T+2S$}\nand, when $T=1$, the\n\\lword{Erdos1049/AdelicHeightBridge.lean}{1813}{fourJet_card_gt_two_pow_of_rank_41}{failure of the count at $129+2S$}.\n\nFor all depths at once the ambient-cardinality inequality $2^{M}>3^{2R}2^{2S}$\nholds exactly when\n\\[\n M\\ \\ge\\ \\bigl\\lfloor2R\\log_23+2S\\bigr\\rfloor+1,\n\\]\nby the definition of the floor function; irrationality of $\\log_2 3$\nis not needed for this strict-inequality reformulation.\nAt $R=41$ this is $130+2S$, and\nat $R=41\\cdot31$ it is $4029+2S$, one below the uniform bound $4030+2S$ of\nCorollary~\\ref{long1049:res:rankfortyone}; the corollary trades exactness for a\ncertificate that is a single integer comparison.  Here $41$ is the exponent of $3$ in the modulus, not the number of\ninput rows; at that depth the counting bound is $130+2S$ rows.\n\nThe count produces two distinct subsets, not necessarily two\ndistinct row sums or remainder values. Repeated rows already\nillustrate the first failure. To prevent the second, we bound how\nmany subsets can have any specified remainder. The following\nfinite-set statement isolates that requirement.\n\n\\begin{theorem}[equal residues with different values]\\label{long1049:res:boundedfibre}\nLet $A$ and $B$ be finite sets, let $f:A\\to B$, and let $g:A\\to C$ be any\nmap into a set $C$.  Suppose every fibre of $g$ has at most $k$ elements.  If\n\\[\n |B|k<|A|,\n\\]\nthen there exist distinct $x,y\\in A$ such that\n\\[\n f(x)=f(y)\\qquad\\hbox{and}\\qquad g(x)\\ne g(y).\n\\]\nThus, when $f$ records the four residues and $g$ records the real\nremainder, a bound on the multiplicities of equal remainders guarantees\na pair with equal residues and different remainders.\n\\end{theorem}\n\n\\begin{proof}\nIf every pair in a common $f$-fibre also had the same $g$-value, each\n$f$-fibre would lie in one $g$-fibre and hence have size at most $k$.\nSumming over the at most $|B|$ fibres of $f$ would give\n$|A|\\le |B|k$, contrary to the hypothesis.\n\\end{proof}\n\nIf $g$ is injective, the hypothesis holds with $k=1$. It need not hold\nwith a useful small $k$ for subset sums: repeated rows produce\nmany equal sums, even when every row is primitive. The theorem assumes\na bound on every fibre of $g$, not merely on the fibres of $(f,g)$.\nIt gives a nonzero difference but no upper bound on its real size.\nTo bound the size as well, we partition the range of $g$ into small\nintervals. The count now uses residue class, interval and multiplicity.\nOnly repetitions within one residue class matter, so the quantitative\nversion assumes a bound for fibres of $(f,g)$ rather than for all\nfibres of $g$.\n\n\\papersectiontarget{long1049-selectors}\n\\begin{theorem}[equal residues with close, distinct values]\\label{res:boundedfibre}\nLet $A,B,J$ be finite sets and let $f:A\\to B$, $g:A\\to\\mathbb R$ and\n$\\iota:A\\to J$.  Suppose that each simultaneous fibre of $(f,g)$ has at\nmost $k$ elements and that, for some $\\delta>0$,\n\\[\n \\iota(x)=\\iota(y)\\quad\\Longrightarrow\\quad |g(x)-g(y)|<\\delta.\n\\]\nIf $|B||J|k<|A|$, then some distinct $x,y\\in A$ satisfy\n\\[\n f(x)=f(y),\\qquad 0<|g(x)-g(y)|<\\delta.\n\\]\n\\end{theorem}\n\\begin{proof}\nSuppose that no such pair exists. Two points in a fibre of\n$(f,\\iota)$ have equal $f$-values and $g$-values less than $\\delta$\napart, so their $g$-values must be equal. Thus each fibre of\n$(f,\\iota)$ lies in a fibre of $(f,g)$ and contains at most $k$\npoints. Summing over its at most $|B||J|$ fibres gives\n$|A|\\le |B||J|k$, a contradiction.\n\\end{proof}\n\nFor an application, let $M$ integral coefficient rows $(A_j,B_j)$ have\nreal remainders $e_j$ at the same target. Let $D$ be a positive integer,\nand suppose that the coefficient-pair subset sums $\\sum_{j\\in I}(A_j,B_j)$\ntake $Q$ values in $(\\mathbb Z/D\\mathbb Z)^2$.\nPut $T=\\sum_j|e_j|$. The least and greatest\nsubset remainders are respectively\n\\[\n \\sum_{e_j<0}e_j\\quad\\text{and}\\quad\\sum_{e_j>0}e_j,\n\\]\nso the interval between them has length $T$. Divide it into half-open\nintervals of width $D/n$, using a final interval for the right\nendpoint if necessary. At most $\\lfloor nT/D\\rfloor+1$ intervals\nare needed. If each simultaneous residue-and-remainder fibre has\nsize at most $k$, the inequality\n\\[\n 2^M>Qk\\bigl(\\lfloor nT/D\\rfloor+1\\bigr)\n\\]\ntherefore gives equal residues with remainder difference strictly\nbetween $0$ and $D/n$ in absolute value. Subtracting the two subset sums\nand dividing both coefficients by $D$ gives an integral form\nof nonzero absolute value less than $1/n$. For a prescribed family,\nthe number $Q$, the width $T$ and the multiplicity bound $k$ must\nall be justified. We have no such bounds giving the required forms\nfor the Zudilin family at $3/2$.\n\n\nThe next improvement needs a much stronger hypothesis than\nprimitivity: adjacent determinants must vanish modulo the chosen\nmodulus. Unit multiples of one unimodular row satisfy it. Arbitrary\nprimitive rows do not; for example, $(1,0)$ and $(0,1)$ have determinant\n$1$. Under this hypothesis the rows lie in the free rank-one submodule\ngenerated by a unimodular row. A change of coordinates makes their\nsecond coordinates zero, so only one residue coordinate must be\ncounted.\n\n\\begin{theorem}[vanishing minors and a residue count]\n\\label{long1049:res:plucker-collapse}\nLet $R_0$ be a commutative ring and let $w_n=(A_n,B_n)\\in R_0^2$.  Suppose\nthat each row is unimodular, meaning that $u_nA_n+v_nB_n=1$ for some\n$u_n,v_n\\in R_0$, and that every adjacent minor vanishes:\n\\[\n A_nB_{n+1}-B_nA_{n+1}=0\\qquad(n\\ge0).\n\\]\nThen every pairwise minor $A_iB_j-B_iA_j$ vanishes.  In particular, take\n$R_0=\\Z/(2^S3^R)\\Z$ with $R>0$.  If $S+2R\\le k$, there are two distinct\nbinary vectors $s,t\\in\\{0,1\\}^{k}$ such that\n\\[\n \\sum_{i<k}s_iw_i=\\sum_{i<k}t_iw_i.\n\\]\nThus $S+2R$ rows suffice. The ambient two-coordinate argument gives\nthe sufficient bound $2S+4R$.\n\\end{theorem}\n\n\\begin{proof}\nIf $(a,b)$ is unimodular, say $ua+vb=1$, and $ay-bx=0$, then\n\\[\n (x,y)=(ux+vy)(a,b).\n\\]\nIndeed, the first coordinate follows by replacing $ay$ with $bx$, and the\nsecond by the reverse substitution.  Apply this identity to consecutive rows:\neach next row is a scalar multiple of the current one.  Induction places the\nentire tail on the line through $w_0$ and proves the pairwise-minor assertion.\nRight multiplication by\n\\[\n \\begin{pmatrix}u_0&-B_0\\\\v_0&A_0\\end{pmatrix}\n\\]\nhas determinant $u_0A_0+v_0B_0=1$ and sends $w_0$ to $(1,0)$.\nAll transformed subset sums therefore have second coordinate zero\nand occupy at most $2^S3^R$ values.  Finally\n\\[\n 2^S3^R<2^S4^R=2^{S+2R}\\le2^k,\n\\]\nso the pigeonhole principle gives two distinct subsets with the same sum.\n\\end{proof}\n\n\\noindent No particular coordinate needs to be invertible: modulo six,\n$(2,3)$ is unimodular because $-2+3=1$, although neither entry is a unit.\nSome nondegeneracy is essential. The rows $(1,0),(0,0),(0,1)$ have\nzero adjacent minors but outer minor one.\nFor the counting conclusion, vanishing is required only in the quotient\nring. The integer rows $(1,0),(1,6)$ used in Section~\\ref{long1049:sec:open}\nare independent over $\\Q$, with determinant $6$, but coincide modulo $6$.\nThus dependence modulo the modulus neither follows from primitivity nor\nimplies dependence of the integer rows.\n\n\\noindent This conditional theorem is stronger than the ambient count of possible residue vectors only\nafter its minor-vanishing hypothesis has been established.  No such all-tail\nhypothesis is proved here for an actual $q$-Ap\\'ery or Zudilin family, and the\ntheorem says nothing about whether the difference of the subset sums has\nnonzero analytic remainder.\n\nThe count in Corollary~\\ref{long1049:res:rankfortyone} is sharp at\n$T=1$ for comparison with the full residue space. To obtain different\nremainders, Theorem~\\ref{long1049:res:boundedfibre} additionally needs\na bound on how often the same real value occurs. Such a bound is not\nproved here for the $q$-Ap\\'ery or Zudilin remainder family.\n\n\\noindent Lean checks the\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{131}{fourJetSignature_card}{cardinality of the residue space}, the\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{142}{exists_distinct_binary_selectors_same_fourJet}{pigeonhole argument for the residue map} and the\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{161}{exists_distinct_binary_selectors_same_fourJet_of_rank}{sufficient row count for equal residues}. The pigeonhole argument gives\ndivisibility without an independence hypothesis. Further information\nabout the family is needed to ensure that the nonzero difference of the\nindicator vectors gives a nonzero polynomial pair and a nonzero analytic\nremainder. None of these statements supplies such a family or proves\neither nonvanishing conclusion.\n\n\\begin{example}\\label{long1049:ex:jetcount}\nAt depths \\(R=S=1\\) the space of four residue coordinates is\n\\((\\Z/3\\Z)^2\\times(\\Z/2\\Z)^2\\), of cardinality \\(9\\cdot4=36\\), and the threshold\nreads \\(M\\ge4\\cdot1+2\\cdot1=6\\). With six pairs there are \\(2^{6}=64\\)\nsubsets and only \\(36\\) possible residue vectors. Two subsets therefore\nhave equal residues. The difference of their indicator vectors is a\nnonzero vector in \\(\\{-1,0,1\\}^{6}\\) whose sum vanishes in all four coordinates.\n\\end{example}\n\nThe following elementary comparison concerns only the two scalar\nexponents of Zudilin's construction; it makes no assertion about\nthe coefficient polynomials or their real remainders.\n\n\\begin{theorem}[a restriction on the two scalar exponents]\\label{long1049:res:scalar}\nLet \\(C_1>0\\).  If \\(C_0\\le0\\) or \\(2C_0\\le C_1\\), then\n\\[\n C_0\\log 3-C_1\\log 2<0.\n\\]\nIn the positive branch $C_0>0$ and $2C_0\\le C_1$, the stronger estimate is\n\\[\n C_0\\log3-C_1\\log2< -\\frac{17}{41}C_0\\log2.\n\\]\n\\end{theorem}\n\n\\begin{proof}\nIf $C_0\\le0$, then $C_0\\log3-C_1\\log2\\le-C_1\\log2<0$.\nIf $C_0>0$ and $2C_0\\le C_1$, use $3^{41}<2^{65}$ to obtain\n\\[\n C_0\\log3-C_1\\log2\n \\le C_0(\\log3-2\\log2)\n <-\\frac{17}{41}C_0\\log2<0.\n\\]\n\\end{proof}\n\n\\noindent Written multiplicatively, the conclusion is \\(3^{C_0}<2^{C_1}\\).\nLean checks the\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{428}{three_two_scalar_margin_neg}{inequality}\nand the\n\\lword{Erdos1049/AdelicHeightBridge.lean}{1763}{three_two_scalar_margin_lt_explicit}{$17/41$ margin}\nof the positive branch.\nThe scope of this elementary inequality matters.  The primary Zudilin\ntheorem~\\cite[Theorem~1, p.~154; Sec.~5, pp.~161--162]{zudilin2004} supplies an integer-base\nirrationality-exponent estimate on its parameter cone, and the elementary\ninequality \\(\\mu\\ge2\\) then forces \\(2C_0\\le C_1\\) whenever \\(C_0>0\\); Lean\nchecks that implication separately.  The primary 2004 theorem is stated for an integer inverse base.\nThe rational specialisation proved earlier in this record has separate\nproofs for the constructed forms. The scalar statement in this paragraph is\nonly the displayed inequality; it neither constructs a new family at\n$3/2$ nor formalises a universal exclusion of linear-form methods.\n\n\\section{Failure of the stated clearing conditions at \\texorpdfstring{$3/2$}{3/2}}\\label{long1049:sec:corridor}\n\\papersectiontarget{long1049-clearing}\n\nAt an integer base, congruences for divisor coefficients can produce\nlong blocks of zero digits. We recall that argument before comparing\nit with the coordinatewise clearing conditions below.\n\n\\paragraph{The integer-base argument.}\nFor an integer $b\\ge2$, Erd\\H{o}s uses the expansion\n$F(b)=\\sum_{n\\ge1}\\tau(n)b^{-n}$ and the Chinese remainder theorem to\nproduce arbitrarily long blocks of divisor coefficients with prescribed\npowers of $b$. After controlling carries from the tail, he obtains\narbitrarily long blocks of zero digits and proves that the expansion\ndoes not terminate. An eventually periodic expansion with arbitrarily\nlong zero blocks must terminate, so these two facts imply irrationality\n\\cite[pp.~63--66]{erdos1948}.\nPositivity of the original summands alone would not prove nontermination;\nfor example, $\\sum_{n\\ge1}(b-1)b^{-n}=1$.\n\n\\paragraph{Direct clearing of a partial sum.}\nA more naive attempt is to suppose $F(b)=p/q$, cut at $N$, and multiply the\nremaining identity by $qb^N$.  This does not by itself trap a positive integer\nbelow $1$: the first uncleared tail contribution is\n$q\\tau(N+1)/b$.  The clearing conditions below describe one bounded-window version\nof this attempt. They fail at $3/2$.\n\nWrite $\\beta=r/s$ for the base and $c(n)$ for the coefficient of $\\beta^{-n}$,\nso that $c=\\tau$ in the case at hand.  The term $c(n)\\beta^{-n}$ is\n$c(n)s^{n}/r^{n}$.  Clearing the\npower of $r$ leaves the numerator factor $s^{n}$ in place.  That factor is\ninvisible when $s=1$ and grows geometrically when $s\\ge2$.  At $3/2$ it is\n$2^{n}$.\n\n\nWe return to the attempt to clear successive terms of the series.\nWe retain just two necessary demands: the last coordinate in the\nwindow must clear, and the coefficient-free lower bound from the\nfirst omitted positive term must be smaller than $1$. Even these\nweaker tests are incompatible at\n$3/2$. The argument below uses six natural-number parameters;\nit excludes the stated clearing scheme, not other approximations\nto $F(3/2)$.\n\n\\begin{definition}[clearing conditions]\\label{long1049:def:corridor}\nFor natural numbers $a,b,N,K,Q,D$, the clearing conditions are\n\\[\n a>0,\\qquad Q>0,\\qquad D>0,\\qquad D\\le N+K,\\qquad\n a^{K}\\mid QD,\\qquad Q\\,b^{\\,N+K+1}<a^{\\,K+1}.\n\\]\n\\end{definition}\n\n\\noindent The reading is: $a$ and $b$ are the numerator and denominator of the\nbase, playing the roles of $r$ and $s$ in the partial-sum calculation\nabove, so that\n$(a,b)=(3,2)$ is the case of interest;\n$N$ is the shift, $K$ is the width of the cleared window, $Q$ is the\naccumulated clearing factor, and $D$ is the final coefficient being cleared.\nThe bound $D\\le N+K$ is the only property of the coefficient used; for the\ndivisor-counting coefficient it holds because $\\tau(n)\\le n$.  The divisibility $a^{K}\\mid QD$ tests the final coordinate of the cleared\nwindow; it does not by itself clear every earlier coordinate. For a\nreduced base $a/b$, the denominator of\n$Qa^N D(b/a)^{N+K}$ is cleared exactly when $a^K\\mid QD$, since\n$a$ and $b$ are coprime. For example, at $a=2$, $b=1$,\n$N=21$, $K=3$ and $Q=1$, the final term clears because\n$2^3\\mid\\tau(24)=8$, but the preceding scaled term is\n$\\tau(23)/2^2=1/2$ and is not integral.\n\nThe final inequality is a necessary smallness test, not a sufficient\nbound for the entire tail. For $a>b\\ge1$ and $Q>0$, positivity and\n$\\tau(m)\\ge1$ give\n\\[\n Q a^N\\sum_{m\\ge N+K+1}\\tau(m)(b/a)^m\n >\\frac{Qb^{N+K+1}}{a^{K+1}}.\n\\]\nA tail smaller than $1$ must therefore satisfy the final inequality.\nThe divisibility gives $a^K\\le Q(N+K)$, while the smallness test requires\n$Qb^{N+K+1}<a^{K+1}$. The same $Q$ must satisfy both. These\nconditions are defined in Lean by the\n\\lword{Erdos1049/RationalBaseLambert.lean}{113}{CoordinatewiseCorridor}{six clearing conditions}.\n\n\\begin{theorem}[a necessary inequality for clearing]\\label{long1049:res:corridorbound}\nIf $(a,b,N,K,Q,D)$ satisfies the clearing conditions, then\n\\[\n b^{\\,N+K+1}<a\\,(N+K).\n\\]\n\\end{theorem}\n\n\\begin{proof}\nSince $QD>0$ and $a^{K}\\mid QD$, we have\n$a^{K}\\le QD$, and $D\\le N+K$ gives\n$a^{K}\\le Q(N+K)$.  Hence\n\\[\n Q\\,b^{\\,N+K+1}<a^{\\,K+1}=a^{K}\\cdot a\\le Q(N+K)\\cdot a,\n\\]\nand cancelling the positive factor $Q$ gives the claim.\n\\end{proof}\n\nThe necessary inequality in Theorem~\\ref{long1049:res:corridorbound}\nshows the difference between these clearing tests at integer and\nnoninteger rational bases.  At $b=1$ it reads $1<a(N+K)$, which holds for every\n$a\\ge2$ and every nonempty window; this necessary inequality imposes no\nobstruction.  The other clearing conditions still have to be checked.\nAt $b\\ge2$ the left side is exponential in $N+K$ and the right side is linear,\nso the conditions can hold only for bounded $N+K$ at each fixed base.  At $(a,b)=(3,2)$ the\nnecessary inequality already fails when $N=K=1$.\n\n\\begin{example}\\label{long1049:ex:corridor}\nTake the smallest window with $N,K\\ge1$, namely $N=K=1$, and numerator $a=3$.  At $b=1$ the tuple\n$(3,1,1,1,3,1)$ satisfies the conditions: $D=1\\le2$, the divisibility reads $3\\mid3$, and\nthe last inequality reads $3\\cdot1^{3}=3<3^{2}=9$. At $b=2$ no choice works.\nThe last inequality becomes $Q\\cdot2^{3}<3^{2}$, which forces $Q=1$; the divisibility\nthen reads $3\\mid D$, and the only candidates are $D=1$ and $D=2$, neither\ndivisible by $3$.  This is the proof of\nTheorem~\\ref{long1049:res:corridorbound} in miniature: it derives $a^{K}\\le Q(N+K)$,\nhere $3\\le2$.\n\\end{example}\n\nFeasibility of these necessary conditions is not a successful tail\nestimate. For $(a,b,N,K,Q,D)=(2,1,1,1,1,2)$ all six conditions hold,\nand $D=\\tau(N+K)$. Nevertheless,\n\\[\n 2\\sum_{m\\ge3}\\frac{\\tau(m)}{2^m}\n >2\\left(\\frac2{2^3}+\\frac3{2^4}+\\frac2{2^5}\\right)=1.\n\\]\nThe failure at $3/2$ is useful because it already occurs at the weaker,\nnecessary tests; their feasibility at an integer base proves no bound\nfor the full remainder.\n\n\\begin{proposition}\\label{long1049:res:exp}\nFor every natural number $x\\ge2$ we have $3x<2^{\\,x+1}$.\n\\end{proposition}\n\n\\begin{proof}\nInduct on $x$, starting from $6<8$ at $x=2$.  For the step, $2^{x+1}\\ge2^{2}>3$ when\n$x\\ge1$, so $3(x+1)=3x+3<2^{x+1}+2^{x+1}=2^{x+2}$.\n\\end{proof}\n\n\\begin{theorem}[failure of the stated clearing conditions at $3/2$]\\label{long1049:res:nocorridor}\nFor all $N\\ge1$ and $K\\ge1$ and all natural $Q,D$, the tuple\n$(3,2,N,K,Q,D)$ does not satisfy the clearing conditions.\n\\end{theorem}\n\n\\begin{proof}\nThe clearing conditions would give $2^{\\,N+K+1}<3(N+K)$ by\nTheorem~\\ref{long1049:res:corridorbound}, contradicting Proposition~\\ref{long1049:res:exp}\napplied to $x=N+K\\ge2$.\n\\end{proof}\n\nTheorem~\\ref{long1049:res:nocorridor} excludes the divisibility pattern and\nnecessary tail test in Definition~\\ref{long1049:def:corridor} at $3/2$.\nIt gives no conclusion about the rationality of $F(3/2)$ or about\nclearing schemes with different conditions.\n\n\\section{Successive scaled remainders}\\label{long1049:sec:tail}\n\nConsecutive partial sums differ by one term. After clearing the\npowers of a rational base, this identity gives a recurrence for\nthe scaled differences from a rational number. We derive it for\narbitrary coefficients; the differences are tails only when the\nseries converges to the chosen rational number.\n\nLet $r,s,B,\\xi$ be rationals with $r\\ne0$ and let $c:\\N\\to\\Q$ be arbitrary.\nDefine the prefix and the scaled remainder by\n\\[\n P_N=\\sum_{m=0}^{N-1}\\frac{c(m+1)\\,s^{\\,m+1}}{r^{\\,m+1}},\n \\qquad\n U_N=B\\,r^{N}\\bigl(\\xi-P_N\\bigr).\n\\tag{$\\ast$}\\label{long1049:eq:tailstate}\n\\]\nThus $P_N$ is the partial sum through index $N$, and $U_N$ is its\nscaled difference from $\\xi$. These are the\n\\lword{Erdos1049/RationalBaseLambert.lean}{168}{rationalBasePrefixQ}{rational-base partial sum} and the\n\\lword{Erdos1049/RationalBaseLambert.lean}{181}{rationalBaseClearedTailQ}{scaled remainder}.\n\n\\begin{theorem}[recurrence for the scaled remainder]\\label{long1049:res:tailrec}\nLet $r,s,B,\\xi\\in\\Q$ with $r\\ne0$, let $c:\\N\\to\\Q$, and let $P_N$ and $U_N$ be as\nin~\\eqref{long1049:eq:tailstate}.  Then for every $N$,\n\\[\n U_{N+1}=r\\,U_N-B\\,c(N+1)\\,s^{\\,N+1}.\n\\]\n\\end{theorem}\n\n\\begin{proof}\nExpanding\n$P_{N+1}=P_N+c(N+1)s^{N+1}/r^{N+1}$ and $r^{N+1}=r^{N}\\cdot r$ in the\ndefinition of $U_{N+1}$ and clearing the denominator $r^{N+1}$, which is\nnonzero, gives the identity.\n\\end{proof}\n\nFor a reduced positive base $r/s>1$, the forcing term\n$Bc(N+1)s^{N+1}$ contains the denominator power absent at integer\nbases. Its size gives a useful bound on two consecutive remainders,\nalthough it need not bound each remainder separately.\n\n\\begin{theorem}[the forcing term]\\label{long1049:res:forcing}\nLet $s,B$ be natural numbers and $c:\\N\\to\\N$.\n\\begin{enumerate}\n\\item If $s\\ge2$, $B\\ge1$ and $c(N+1)\\ge1$, then\n$2^{\\,N+1}\\le B\\,c(N+1)\\,s^{\\,N+1}$.\n\\item If $s=1$, then $B\\,c(N+1)\\,s^{\\,N+1}=B\\,c(N+1)$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nFor the first part,\n$2^{N+1}\\le s^{N+1}=1\\cdot s^{N+1}\\le Bc(N+1)s^{N+1}$,\nusing $B\\,c(N+1)\\ge1$.  The second part is the definition with $s=1$.\n\\end{proof}\n\nAt $s=1$ the forcing term is $Bc(N+1)$, so it grows only as fast as the\ncoefficient; for the divisor function this is $O(N^{\\varepsilon})$ for every\n$\\varepsilon>0$. This comparison does not itself construct a bounded\nsequence of remainders.  At $s\\ge2$ the same\nterm is at least $2^{N+1}$ whenever the coefficient is nonzero.\n\n\\begin{example}\\label{long1049:ex:forcing}\nTake $B=1$, $c=\\tau$ and $N=9$, so that the coefficient is $\\tau(10)=4$.  At\n$s=1$ the forcing term is $4$.  At $s=2$ it is $4\\cdot2^{10}=4096$, and part~(1)\nof Theorem~\\ref{long1049:res:forcing} already guarantees at least $2^{10}=1024$ without\nknowing the coefficient at all.\n\\end{example}\n\nA large forcing term cannot be balanced by two small consecutive\nremainders. For a natural index $N$, with $B,s$ positive integers,\n$s\\ge2$ and $c(N+1)\\ge1$, the recurrence and triangle inequality give\n\\[\n 2^{N+1}\\le Bc(N+1)s^{N+1}\n =|rU_N-U_{N+1}|\n \\le(1+|r|)\\max\\{|U_N|,|U_{N+1}|\\}.\n\\]\nIf these assumptions hold at every index, the full sequence is unbounded.\nThey do not imply $|U_N|\\to\\infty$ under the stated algebraic hypotheses. For example, take $r=-2$, $s=2$,\n$B=1$, $c(n)=1$ and $\\xi=0$. Directly from the defining partial sums,\n\\[\n U_{2j}=0,\\qquad U_{2j+1}=-2^{2j+1}\\qquad(j\\ge0).\n\\]\nThe forcing term grows exponentially while every even remainder is\nzero. This is an algebraic example, not a convergent Lambert series: it\nshows that the adjacent-pair bound does not control every subsequence.\nIt concerns the displayed normalisation only and supplies no small\nnonzero integer forms.\n\n\\section{The height criterion at \\texorpdfstring{$7/2$}{7/2}}\\label{long1049:sec:sevenhalves}\n\nThe base $7/2$ already lies in the irrationality region of\nBundschuh and V\\\"a\\\"an\\\"anen\n\\cite[Theorem~2, p.~177; hypotheses pp.~175--176]{bv1994}.\nWe check their height condition in their notation, with $q$\ndenoting the base. At $\\alpha=-1$ it reads\n\\[\n \\lambda<\\left(\\frac12+\\frac1{\\pi^2}\\right)^{-1}.\n\\]\nAt $q=7/2$ the Archimedean parameter is\n$\\lambda=\\log 7/\\log(7/2)$.  The criterion therefore applies once the following\nstrict inequality is checked.\n\n\\begin{theorem}[the $7/2$ height condition]\\label{long1049:res:sevenhalves}\n\\[\n \\frac{\\log 7}{\\log(7/2)}\n <\n \\left(\\frac12+\\frac1{\\pi^2}\\right)^{-1}.\n\\]\n\\end{theorem}\n\\begin{proof}\nSince $2^{18}=262144<823543=7^7$, we have\n$\\log2/\\log7<7/18$. Also $\\pi>3$ gives\n$1/2+1/\\pi^2<11/18$. Hence\n\\[\n \\frac{\\log7}{\\log(7/2)}\n =\\frac1{1-\\log2/\\log7}<\\frac{18}{11}\n <\\left(\\frac12+\\frac1{\\pi^2}\\right)^{-1}.\n\\]\nEvery denominator is positive, so the reciprocal inequalities have\nthe displayed directions.\n\\end{proof}\n\n\\noindent Numerically the two sides are $1.5533\\ldots$ and $1.6630\\ldots$, so\nthe condition holds with a margin of about $0.11$.  The Lean proof factors the\nestimate through the explicit\n\\lword{Erdos1049/RationalBaseLambert.lean}{44}{BundschuhVaananenHeightRegion}{height-region\npredicate} and the\n\\lword{Erdos1049/RationalBaseLambert.lean}{32}{sevenHalves_power_certificate}{integer\ncertificate} $2^{18}<7^7$, the resulting\n\\lword{Erdos1049/RationalBaseLambert.lean}{48}{sevenHalves_log_ratio_lt}{logarithmic\nratio bound} $\\log2/\\log7<7/18$, the\n\\lword{Erdos1049/RationalBaseLambert.lean}{64}{one_div_pi_sq_lt_one_nine}{$\\pi$-bound}\n$1/\\pi^2<1/9$, and the\n\\lword{Erdos1049/RationalBaseLambert.lean}{73}{sevenHalves_bundschuhVaananen_margin}{strict\nmargin}\n\\[\n \\frac{\\log2}{\\log7}<\\frac12-\\frac1{\\pi^2}.\n\\]\nThe\n\\lword{Erdos1049/RationalBaseLambert.lean}{83}{sevenHalves_archimedean_height_condition}{final\nheight inequality} then rewrites\n$\\log(7/2)=\\log7-\\log2$ and closes the displayed condition.\n\nThis formalises the complete elementary parameter check at $q=7/2$; it does\nnot formalise Bundschuh and V\\\"a\\\"an\\\"anen's analytic irrationality theorem,\nwhose proof occupies pp.~189--193 of the source.  The conclusion that\n$F(7/2)$ is irrational is consequently cited from that theorem, not claimed as\na Lean theorem here.\n\nThe criterion does not cover $3/2$, and the results of\nSections~\\ref{long1049:sec:corridor} and~\\ref{long1049:sec:tail} give no evidence either way\nabout whether $F(3/2)$ is irrational.\n\n\\subsection{The \\texorpdfstring{$81/200$}{81/200} logarithmic region}\n\nConsider the rational-height region\n\\[\n \\frac{\\log b}{\\log a}<\\frac{81}{200}\n \\qquad(a>b>0).\n\\]\nThe number $81/200$ is the rational lower bound on $\\theta^*$\nproved earlier. The bracket\n$81/200<\\theta^{*}<1/2$ in Section~\\ref{long1049:sec:regionbracket}\nputs this entire smaller region inside\nTheorem~\\ref{long1049:res:region}. Membership of the smaller region is\nan integer comparison.  The Lean module of\nthis library treats the displayed inequality as a definition and proves\nelementary memberships and exclusions.  In particular,\n\\[\n \\frac25\n <\\frac{\\log4}{\\log31}\n <\\frac{81}{200}\n <\\frac{\\log2}{\\log3}.\n\\]\nThe first two comparisons come respectively from $31^2<4^5$ and\n$4^{200}<31^{81}$; the last comes from $3^{81}<2^{200}$.  Lean checks the\n\\lword{Erdos1049/ZudilinHeightRegion.lean}{65}{twoFifths_lt_thirtyoneFour_log_ratio}{lower\nbound}.  Consequently $31/4$\n\\lword{Erdos1049/ZudilinHeightRegion.lean}{45}{thirtyoneFour_mem_zudilinHeightRegion}{lies in the enlarged region},\nand so does\n\\lword{Erdos1049/ZudilinHeightRegion.lean}{59}{thirtyoneFour_power_mem_zudilinHeightRegion}{every positive power $(31/4)^r$}.\nThe same base\n\\lword{Erdos1049/ZudilinHeightRegion.lean}{91}{thirtyoneFour_outside_bundschuhVaananenHeightRegion}{lies strictly outside}\nthe earlier Bundschuh--V\\\"a\\\"an\\\"anen region, so the $81/200$ inequality defines a strict set-theoretic enlargement of the\nearlier recorded logarithmic region.  Combined with the bracket\n$81/200<\\theta^{*}$, these memberships are exactly the finite input to\nTheorem~\\ref{long1049:res:31over4}, which is where the irrationality of $F(31/4)$ and of\nevery $F\\bigl((31/4)^{r}\\bigr)$ is proved.\n\nIt still does not approach $3/2$.  The exact comparison\n\\[\n 3^{81}<2^{200}\n \\quad\\Longrightarrow\\quad\n \\frac{81}{200}<\\frac{\\log2}{\\log3}\n\\]\nis Lean-checked, as is the conclusion that $3/2$ belongs to neither height\nregion\n(\\lword{Erdos1049/ZudilinHeightRegion.lean}{105}{eightyOneTwoHundredths_lt_threeHalves_log_ratio}{comparison with $\\log2/\\log3$},\n\\lword{Erdos1049/ZudilinHeightRegion.lean}{122}{threeHalves_outside_zudilinHeightRegion}{$3/2$ fails the condition}).\nThus $81/200$ is the larger of the two explicitly defined cutoffs used below,\nwhile a cutoff that includes $3/2$ must be strictly larger than\n$\\log2/\\log3\\approx0.6309$.\n\n\\section{A denominator-exponent model for Pad\\'e-type\nforms}\\label{long1049:sec:pade}\n\nFor a sum of Pad\\'e-type terms, a common denominator must clear\neach summand. The two exponent expressions below are bounded by\n$(3n^2-n)/2$, as their differences factor over $\\Z$. We double all\nexponents to avoid fractions and write $\\widetilde E_n=3n^2-n$.\nThese are algebraic comparisons only: no approximation family is\nspecified, so integral coefficients and a nonzero remainder estimate\nwould still be needed for an arithmetic application.\n\n\\begin{proposition}[exponent model: summand bound and exact gap]\\label{long1049:res:pade}\nLet $\\widetilde{E}_n=3n^{2}-n$ and put\n\\[\n \\widetilde{P}(n,k)=2\\bigl(k(n-k)+nk\\bigr)+k(k-1),\n\\]\n\\[\n \\widetilde{Q}(n,m)=2(n^{2}-n)+j^{2}+2jm+j-m^{2}+3m,\n \\qquad j=n-m-1 .\n\\]\nThen, for integers $n,k,m$:\n\\begin{enumerate}\n\\item if $0\\le k\\le n$, then $\\widetilde{P}(n,k)\\le\\widetilde{E}_n$, and the\ngap factors as\n$\\widetilde{E}_n-\\widetilde{P}(n,k)=(n-k)(3n-k-1)$;\n\\item $\\widetilde{E}_n-\\widetilde{Q}(n,m)=2\\bigl(n+m(m-1)\\bigr)$ identically;\n\\item if $n\\ge0$ and $m\\ge1$, then $\\widetilde{Q}(n,m)\\le\\widetilde{E}_n$.\n\\end{enumerate}\n\\end{proposition}\n\n\\noindent Part~(1) is the\n\\lword{Erdos1049/RationalPadeArithmetic.lean}{30}{rationalPadePSummandDenExpTwice_le}{summand\nexponent bound}, part~(2) the\n\\lword{Erdos1049/RationalPadeArithmetic.lean}{52}{rationalPadeQMaxDenExpTwice_gap}{exact\ngap identity}, and part~(3) the\n\\lword{Erdos1049/RationalPadeArithmetic.lean}{60}{rationalPadeQMaxDenExpTwice_le}{maximal\nexponent bound}.  The gap in~(2) is an identity in $\\Z[n,m]$, so~(3) follows\nfrom $m(m-1)\\ge0$ and $n\\ge0$. The range $m\\ge1$ is sufficient,\nnot necessary for this polynomial inequality: for integer $m$, one\nalso has $m(m-1)\\ge0$ when $m\\le0$. No claim about a summand of an\nunspecified approximation is needed.\n\n\\begin{example}\\label{long1049:ex:pade}\nAt $n=2$ the proposed doubled exponent is $\\widetilde{E}_2=10$, and\n$\\widetilde{P}(2,k)$ takes the values $0,6,10$ at $k=0,1,2$.  The three gaps are\n$10,4,0$, matching the factorisation $(2-k)(5-k)$ of part~(1); the summand at\n$k=2$ is the one that saturates the proposed denominator.  For part~(2), at\n$m=1$ we have $j=0$ and $\\widetilde{Q}(2,1)=6$, with gap\n$4=2\\bigl(2+1\\cdot0\\bigr)$.\n\\end{example}\n\nThese are inequalities between polynomials in the exponents. They\nestablish that the proposed exponent $\\widetilde{E}_n$ dominates the two\ndisplayed summand exponent expressions, and nothing further.  Positivity of the remainder, its rate of\ndecay, and the comparison of that rate against the denominator height are the\nanalytic obligations, and none of them is treated here, so nothing in this\nsection is an irrationality measure.\n\n\\section{Complements and further questions}\\label{long1049:sec:open}\n\\papersectiontarget{long1049-open-construction}\n\nAn unrestricted construction of small nonzero integer forms is\nanother formulation of irrationality, not yet a method of proving\nit. The next problem remains equivalent to irrationality at $3/2$\neven with its quadratic degree bounds, and even after imposing\nthe exponential coefficient-height bound discussed below.\nWe prove the equivalence before asking what a specified\napproximation family can supply.\n\n\\begin{problem}[a divided linear form with small nonzero remainder]\\label{long1049:prob:kernel}\nExhibit an integer constant \\(C\\ge1\\) and, for every sufficiently large\npositive integer \\(n\\), positive integers \\(W_n,R_n,S_n,M_n\\) such that\n\\[\n n^2\\le W_n,R_n,S_n\\le Cn^2,\n \\qquad 4R_n+2S_n\\le M_n\\le Cn^2,\n\\]\ntogether with polynomial pairs\n\\((U_{n,j},V_{n,j})\\in\\Z[X]^2\\) for \\(0\\le j<M_n\\), each of degree at most\nthe common degree bound \\(W_n\\), whose specialised integer rows are\nprimitive:\n\\[\n \\gcd\\!\\bigl(H_{W_n}(U_{n,j}),H_{W_n}(V_{n,j})\\bigr)=1.\n\\]\nFind a nonzero vector\n\\(\\lambda^{(n)}\\in\\{-1,0,1\\}^{M_n}\\) for which, on putting\n\\[\n U_n=\\sum_{j<M_n}\\lambda^{(n)}_jU_{n,j},\n \\qquad V_n=\\sum_{j<M_n}\\lambda^{(n)}_jV_{n,j},\n\\]\nthe pair \\((U_n,V_n)\\) is not \\((0,0)\\), all four residues vanish,\n\\[\n J_{3,R_n}(U_n)=J_{3,R_n}(V_n)=0,\n \\qquad J_{2,S_n}(U_n)=J_{2,S_n}(V_n)=0,\n\\]\nwhere every residue in this display is formed using the common degree bound \\(W_n\\),\nand the resulting divided integer linear form\n\\[\n A_n=\\frac{H_{W_n}(U_n)}{3^{R_n}2^{S_n}},\n \\qquad\n B_n=\\frac{H_{W_n}(V_n)}{3^{R_n}2^{S_n}},\n \\qquad\n \\rho_n=A_nF(3/2)-B_n\n\\]\nsatisfies the explicit analytic condition\n\\[\n 0<|\\rho_n|<\\frac1n.\n\\]\n\\end{problem}\n\nThe residue equations make \\(A_n,B_n\\) integers.  A solution would prove\nirrationality: if \\(F(3/2)=a/b\\) in lowest terms, every nonzero \\(\\rho_n\\) has\nabsolute value at least \\(1/b\\), contradicting the displayed bound for\n\\(n>b\\).  Theorem~\\ref{long1049:res:jetkernel} supplies only a nonzero signed relation\nwith the four residue equations once the pairs and size inequality are present; it\ndoes not supply primitive input rows, a nonzero combined polynomial pair, or\nthe nonvanishing and decay of \\(\\rho_n\\).\n\n\\paragraph{Why the choice of family matters.}\nThe converse can start from an arbitrary small integer form and\nbuild the polynomial rows afterwards. Pigeonhole approximation\nsupplies the small form; B\\'ezout's identity lifts its coordinates\nto polynomial evaluations. Thus the degree and residue conditions\ndo not constrain how the approximation was obtained. Let $\\xi$ be\nany irrational real number. Comparing the fractional parts\nof $0,\\xi,\\ldots,(n+1)\\xi$ in $n+1$ equal half-open intervals gives\nintegers $A\\ge1$ and $B$ with\n\\[\n A\\le n+1,\\qquad 0<|A\\xi-B|<\\frac1{n+1}<\\frac1n.\n\\]\nSet $w=n^2$, $W_n=R_n=S_n=w$, $M_n=6w$, and $D=6^w$.\nThe three integer rows\n\\[\n (DA,1),\\qquad (1,DB-1),\\qquad (-1,0)\n\\]\nare primitive and sum to $(DA,DB)$. Pad them to $M_n$ rows with\n$(1,0)$, and use the coefficient vector $(1,1,1,0,\\ldots,0)$.\n\nTo realise every coordinate as a polynomial evaluation of degree at\nmost $w$, choose integers $u,v$ such that $u2^w+v3^w=1$. For any\ninteger $h$, the polynomial $h(u+vX^w)$ satisfies\n$H_w(h(u+vX^w))=h$. Lift all row coordinates this way.\nThe combined polynomial pair is nonzero because its first evaluation\nis $DA\\ne0$. Its two evaluations are divisible by $3^w2^w$, and its\ndivided remainder is $A\\xi-B$. All the conditions of\nProblem~\\ref{long1049:prob:kernel}, with $F(3/2)$ replaced by $\\xi$,\nhold with $C=6$.\n\nEven a coefficient-height bound of the form $\\exp(O_\\xi(n^2))$ would\nnot exclude these lifts. Choose $0\\le v<2^w$, which gives $|u|<3^w$.\nSince $|B|\\le(n+1)|\\xi|+1$, every lifted coefficient has absolute value\nat most $(|\\xi|+2)(n+1)18^w$.\nTogether with the preceding rationality contradiction, this proves that\nthe unrestricted construction problem for a real target $\\xi$ is\nequivalent to the irrationality of $\\xi$. It does not establish either\nassertion for $F(3/2)$.\n\nThe short paper refers to this formulation and to its equivalence with\nirrationality.\nTo pose a more specific question, one must replace the free choice of\npolynomial pairs by an actual set given by coefficient formulas,\nadmissible parameters and permitted normalisations. The restriction must exclude the freedom to choose the rows from\nan already good approximation. Merely giving the resulting family\na name, an explicit formula or an exponential height bound does\nnot do so: the lifts above have all three. For a fixed indexed family, the distinction between all\nsufficiently large indices and a sparse subsequence can matter.\nArbitrary padding and reindexing remove it in the unrestricted problem.\n\n\nCorollary~\\ref{long1049:res:nomult} gives no improvement from integer\nrescaling and, under its endpoint hypotheses, no common factor $2$ or\n$3$ in the unscaled evaluations. It makes no general assertion about\npolynomial cancellation before specialisation, combinations of rows,\ndeterminant-specific divisibility or other approximation families.\n\nA prescribed family must supply nonvanishing and decay for the same\nsigned combinations. A residue collision alone supplies neither;\nseparate examples of the two properties would not suffice either.\nA coefficient-height estimate is needed when the approximation\nargument calls for it, not as an extra hypothesis of the elementary\nirrationality criterion above.\n\n\\subsection{A congruence relation with nonzero remainder}\n\nFixing the approximation family is not enough: the residue map\nmust also use a fixed normalisation. Polynomial coefficient content,\ncontent of an evaluated row, and content of a combined row are\nthree different quantities. We keep them separate as follows.\nFor each $n$, let $W_n$ be a common degree bound and let\n\\[\n (U_{n,j},V_{n,j},\\mathcal R_{n,j})_{j<M_n}\n\\]\nbe the chosen forms, with $U_{n,j},V_{n,j}\\in\\Z[X]$ of degree at most\n$W_n$ and\n\\[\n \\mathcal R_{n,j}(t)=U_{n,j}(t)F(t)-V_{n,j}(t)\\qquad(t>1).\n\\]\nThis identity uses the same normalisation as the evaluated rows and\nall subsequent residue equations. Fix all divisions before forming the\nresidue map. Require each\npolynomial pair to have coefficient gcd $1$, by dividing its common\ninteger coefficient factor if necessary. This is different from making\nthe two evaluated integers coprime, as required in\nProblem~\\ref{long1049:prob:kernel}. The two operations need not preserve\nthe same polynomial family. Coefficient content $1$ alone gives no\ncoprimality with $6$ after evaluation: at $W=1$, the pairs $(X,3)$\nand $(2X-1,2)$ have coefficient content $1$ but give the integer rows\n$(3,6)$ and $(4,4)$, respectively.\n\nThe gcd\n\\[\n \\gcd\\!\\bigl(H_{W_n}(U_{n,j}),H_{W_n}(V_{n,j})\\bigr)\n\\]\nof one evaluated row can differ from the gcd of the final sum. Neither\nis divided out without also dividing the corresponding remainder and\nmeasuring the height after that division. If, in addition, the\ncoefficient of $X^{W_n}$ in $U_{n,j}$ and the constant coefficient of\n$V_{n,j}$ are both units, Proposition~\\ref{long1049:res:commonmult}\nmakes this row gcd coprime to $6$. These endpoint assumptions are\nextra conditions, not consequences of coefficient primitivity.\nDividing by such a gcd preserves divisibility by every power of $2$\nand $3$ for that individual row. It need not preserve a relation\nwith a fixed coefficient vector when different rows are divided by different\ncontents. For example, at $W=1$ the pairs $(X-1,X-1)$ and $(X+1,X+1)$\ngive rows $(1,1)$ and $(5,5)$. Their sum is zero modulo $6$, whereas\nthe sum of their primitive normalisations is $(2,2)$, not zero modulo\n$6$. Both pairs satisfy the two unit conditions just stated. Each division is invertible modulo $6$, but different rows are\nmultiplied by different units, so their sum need not stay zero.\nThe residue map must therefore be formed again after row-by-row\nnormalisation.\n\nDividing a specialised row by its content need not preserve the chosen\npolynomial family, and lifting the divided row back is a constrained problem. For fixed $W$, the map $P\\mapsto H_W(P)$ on integer polynomials of\ndegree at most $W$ is surjective onto $\\Z$, since $3^{W}$ and $2^{W}$ are\ncoprime, so a lift always exists. The issue is not an arbitrary exponential\nheight bound, as the construction above shows. It is whether the lift\nbelongs to the chosen approximation family and satisfies that family's\nremainder identity and quantitative height estimate. With $W=1$,\nfor instance, $(X+1,X+1)$ specialises to $(5,5)$, whose primitive\nnormalisation $(1,1)$ lifts to $(X-1,X-1)$, but not to an integer scalar\nmultiple of the original pair.  Any lift used below is therefore supplied together with its\ndegree bound, its height bound and its exact remainder.\n\nFor target depths $R_n,S_n$, let $J_n(\\lambda)$ be the vector of four residues of\nthe pair $\\sum_j\\lambda_j(U_{n,j},V_{n,j})$, and define\n\\[\n \\mathcal C_n=\n \\{\\lambda\\in\\{-1,0,1\\}^{M_n}\\mathbin{\\backslash}\\{0\\}:J_n(\\lambda)=0\\},\n\\]\n\\[\n\\begin{aligned}\n K_n^{\\mathrm{poly}}\n   &=\\left\\{\\lambda\\in\\{-1,0,1\\}^{M_n}:\n       \\sum_j\\lambda_jU_{n,j}=0,\\ \\sum_j\\lambda_jV_{n,j}=0\\right\\},\\\\\n K_n^{\\mathrm{rem}}\n   &=\\left\\{\\lambda\\in\\{-1,0,1\\}^{M_n}:\n       \\sum_j\\lambda_j\\mathcal R_{n,j}(3/2)=0\\right\\}.\n\\end{aligned}\n\\]\nThe second set consists of signed relations whose remainder vanishes\nat $3/2$. It also contains every relation with an identically zero\nremainder function. These are sets of restricted coefficient vectors,\nnot assertions that the signed cube is a vector space.\n\nThe exact pigeonhole condition is\n\\begin{equation}\n 2^{M_n}>3^{2R_n}2^{2S_n},\n \\qquad\\text{equivalently}\\qquad\n M_n>2R_n\\log_2 3+2S_n.\n\\label{long1049:eq:exact-jet-threshold}\n\\end{equation}\nThus the least integer number of rows satisfying this counting test is\n\\[\n \\left\\lfloor2R\\log_2 3+2S\\right\\rfloor+1.\n\\]\nThe checked condition $M\\ge4R+2S$ for $R>0$ is a convenient sufficient\ncorollary, not the exact threshold. Applying Cauchy--Schwarz to the\nsizes of the residue classes shows that there are at least\n\\begin{equation}\n \\frac12\\left(\\frac{2^{2M_n}}{3^{2R_n}2^{2S_n}}-2^{M_n}\\right)\n\\label{long1049:eq:collision-count}\n\\end{equation}\nunordered pairs of distinct subsets with equal residues. It would\ntherefore suffice to prove that fewer than this many pairs give a zero\npolynomial combination or a zero remainder at $3/2$.\n\nTheorem~\\ref{long1049:res:jetkernel} supplies only\n$\\mathcal C_n\\ne\\varnothing$; it does not rule out zero polynomial pairs\nor zero real remainders. For a useful approximation family, bounding\nthe multiplicities in~\\eqref{long1049:eq:collision-count} remains a\npossible way to obtain nonvanishing. Without restrictions on the\nfamily, however, nonvanishing alone is elementary.\n\n\\begin{example}[nonvanishing without decay]\\label{long1049:prob:escape}\nFor $n\\ge1$, set $W_n=R_n=S_n=n^2$ and $M_n=6n^2$. Take\n\\[\n\\begin{aligned}\n (U_{n,0},V_{n,0})&=((1+2^{W_n})X^{W_n},1),\\\\\n (U_{n,j},V_{n,j})&=(X^{W_n},1)\\qquad(1\\le j<M_n),\n\\end{aligned}\n\\]\nand define $\\mathcal R_{n,j}(t)=U_{n,j}(t)F(t)-V_{n,j}(t)$ for $t>1$.\nAll pairs have coefficient gcd $1$ and primitive evaluated rows. The\ncoefficient vector $(1,-1,0,\\ldots,0)$ belongs to\n$\\mathcal C_n\\mathbin{\\backslash}\n(K_n^{\\mathrm{poly}}\\cup K_n^{\\mathrm{rem}})$, but the divided\ninteger form equals $F(3/2)$ for every $n$.\n\\end{example}\n\nIndeed, the evaluated rows are\n$(3^{W_n}+6^{W_n},2^{W_n})$ and $(3^{W_n},2^{W_n})$; their first\ncoordinates are odd, so both rows are primitive. Their difference is\n$(6^{W_n},0)$, while the polynomial difference is\n$(2^{W_n}X^{W_n},0)$. Thus all four residues vanish. The unscaled remainder difference\nis $3^{W_n}F(3/2)>0$. Clearing the evaluation denominators multiplies\nit by $2^{W_n}$, giving $6^{W_n}F(3/2)$. Dividing this cleared\ninteger form by $3^{R_n}2^{S_n}=6^{W_n}$ leaves the same positive\nconstant $F(3/2)$. The counting condition\n\\eqref{long1049:eq:exact-jet-threshold} holds because $\\log_2 3<2$.\n\nExample~\\ref{long1049:prob:escape} meets the two nonvanishing\nrequirements in Problem~\\ref{long1049:prob:kernel} without any\nirrationality assumption. What it fails is the smallness condition.\nThe useful question is therefore to obtain nonvanishing and decay in\nthe same prescribed approximation family, not merely to exhibit some\nfamily with a nonzero congruence relation.\n\n\\paragraph{The range of subset sums.}\nThe residue condition and the real-error condition can be counted\ntogether. Within each residue class, subdivide the range of subset\nremainders into intervals of width $D/n$; a bound on repeated\nremainder values then forces a nonzero small difference. Fix a\npositive integer $n$. For $M$ integer rows $(A_j,B_j)$ and a modulus\n$D\\ge1$, set $e_j=A_jF(3/2)-B_j$. For each attained residue vector $b$,\nlet $T_b$ be the difference between the largest and smallest remainders\n$\\sum_j\\varepsilon_je_j$ with $\\varepsilon_j\\in\\{0,1\\}$ and\n$\\sum_j\\varepsilon_j(A_j,B_j)\\equiv b\\pmod D$. Let the integer $k_b$\nbound the number of subsets attaining any one remainder value in that\nresidue class. Then\n\\[\n 2^M>\\sum_b k_b\\left(\\left\\lfloor\\frac{nT_b}{D}\\right\\rfloor+1\\right)\n\\]\nproduces two subsets with equal residues and distinct remainders\nless than $D/n$ apart. Within each residue fibre, subtract the least\nremainder, multiply by $n/D$, and take floors. These integer indices range\nfrom $0$ to $\\lfloor nT_b/D\\rfloor$, including a separate final index\nwhen $T_b$ is a positive integral multiple of $D/n$. Equal indices\ngive a remainder difference strictly less than $D/n$. If each index\ncorresponded to only one remainder value, it would account for at most\n$k_b$ subsets, contradicting the displayed inequality. Subtracting the\ntwo subset sums and dividing their coefficients by $D$ gives an integer\nform with nonzero absolute value less than $1/n$. The common width bound\n$T=\\sum_j|e_j|$ and a common multiplicity bound $k$ give the coarser\ncount $Qk(\\lfloor nT/D\\rfloor+1)$, where $Q$ is the number of attained\nresidue vectors. All rows must have the same normalisation. Primitive\nrows can still have repeated subset sums, and positivity need not\nsurvive subtraction.\n\n\\paragraph{The lattice after evaluation.}\nFor example, $(1,0),(1,6)$ generate $\\Z\\oplus6\\Z$. They give two\nresidues modulo $2$. Within that lattice, the vectors whose two\ncoordinates are even form $2\\Z\\oplus6\\Z$; dividing them by $2$ gives\n$\\Z\\oplus3\\Z$, of index $3$. These are two different counts: the number of attainable residues\nbefore division, and the index of the lattice left after division.\nSmith normal form computes both for any rank-two lattice. The\nfirst count describes congruences available before division; the\nsecond describes the integrality restriction that survives it.\nNeither count estimates the real remainders of the rows.\n\nLet primitive integer rows $u_j\\in\\Z^2$ span a rank-two lattice $L$, and\nlet $g$ be the gcd of their $2\\times2$ minors. Since at least one row is\nprimitive, the Smith invariants are $1,g$\n\\cite[Thms.~2.3--2.4, p.~3]{stanley2016}. Hence, for $D\\ge1$,\n\\[\n |\\operatorname{im}(L\\longrightarrow(\\Z/D\\Z)^2)|\n =\\frac{D^2}{\\gcd(g,D)},\\qquad\n [\\Z^2:(L\\cap D\\Z^2)/D]=\\frac{g}{\\gcd(g,D)}.\n\\]\nIndeed an integral unimodular change of coordinates takes $L$ to\n$\\Z\\times g\\Z$; reduction modulo $D$ and intersection with $D\\Z^2$\ngive the two formulas. This is Smith normal form over $\\Z$, not over\n$\\Z[p]$. It measures the actual image, rather than the ambient residue space. If two independent divided rows have coordinate height at most\n$H$ and remainders of absolute value at most $\\varepsilon$, their nonzero\ninteger determinant gives the necessary inequality\n$2H\\varepsilon\\ge g/\\gcd(g,D)$. That is not a necessary condition for\nproducing a single nonzero form.\nIf the specialised rows are not primitive, let $d_1\\mid d_2$ be the\npositive Smith invariants instead. For $D\\ge1$ the corresponding formulas are\n\\[\n\\begin{aligned}\n |\\operatorname{im}(L\\bmod D)|\n &=\\frac{D^2}{\\gcd(d_1,D)\\gcd(d_2,D)},\\\\\n [\\Z^2:(L\\cap D\\Z^2)/D]\n &=\\frac{d_1d_2}{\\gcd(d_1,D)\\gcd(d_2,D)}.\n\\end{aligned}\n\\]\nThey follow by applying the one-dimensional calculation to each summand\n$d_i\\Z$. Polynomial coefficient content alone does not determine either\ninteger Smith invariant after specialisation.\n\n\\paragraph{The lattice spanned by the actual tails.}\nThe formulas above apply to any rank-two lattice. For the rows that the\napproximation problem actually supplies, the invariant can be computed\noutright, and it is the smallest one compatible with the base. Let $a>b\\ge1$ be\ncoprime, write the partial sum $S_m(b/a)=\\sum_{r\\le m}(b/a)^r/(1-(b/a)^r)$ in\nlowest terms as $P_m/Q_m$ with $Q_m>0$, and set $P_0=0$, $Q_0=1$. The primitive\ninteger row of the $m$th tail is $(Q_m,P_m)$, which represents\n$Q_mF(a/b)-P_m$.\n\n\\begin{proposition}[the prefix lattice of the tails]\n\\label{long1049:res:tail-lattice}\nEvery $Q_m$ is coprime to $ab$, and $b$ divides every $P_m$. For every prefix\ncontaining $m=0$ and $m=1$,\n\\[\n \\operatorname{span}_{\\Z}\\{(Q_m,P_m):0\\le m<M\\}=\\Z\\times b\\Z\n \\qquad(M\\ge2),\n\\]\nso the Smith invariants are $1,b$, the gcd of the $2\\times2$ minors is $b$, and\nthe image modulo $D$ has cardinality $D^2/\\gcd(b,D)$. Multiplying an individual\ntail by a nonzero rational weight leaves its primitive row unchanged up to\nsign.\n\\end{proposition}\n\n\\begin{proof}\nA common denominator of $S_m(b/a)$ is a product of the integers $a^r-b^r$ for\n$r\\le m$, each coprime to $ab$; over that denominator every summand has\nnumerator divisible by $b^r$. Reduction to lowest terms removes no prime factor\nof $b$, which gives the first two assertions. The rows at $m=0,1$ are $(1,0)$\nand $(a-b,b)$, and subtracting $a-b$ copies of the first from the second gives\n$(0,b)$; every later row already lies in $\\Z\\times b\\Z$ by the first\nassertions. The Smith and image statements are the specialisation of the\nformulas above at $g=b$. Finally the rational line through a primitive pair\nmeets $\\Z^2$ in exactly the multiples of that pair, so clearing denominators in\na nonzero rational multiple and dividing by the coordinate gcd returns the same\npair up to sign.\n\\end{proof}\n\nAt $3/2$ the first three partial sums are $0$, $2$ and $14/5$, with rows\n$(1,0)$, $(1,2)$ and $(5,14)$. Their span is already $\\Z\\times2\\Z$. There is no\nthree-adic collapse, and the only local saving is the single factor of two that\nthe image formula displays. For a consecutive window containing $m=s$ and $m=s+1$ and no\nearlier row, every pairwise minor is divisible by $b^{s+1}$.\nThe gcd has valuation $(s+1)v_p(b)$ at each prime $p\\mid b$ and\nvaluation zero at each prime $p\\mid a$. The adjacent-row identity\n\\[\n Q_sP_{s+1}-P_sQ_{s+1}\n =\\frac{Q_sQ_{s+1}b^{s+1}}{a^{s+1}-b^{s+1}}\n\\]\nattains both, since its denominator is a unit at every prime dividing $ab$.\n\nRescaling an individual tail and then making its row primitive\nreturns the same row up to sign. Thus it preserves the lattice\nspanned by the full family. Integer combinations remain in that\nlattice, but a selected family of combinations need not span all\nof it and can have a larger minor gcd. At $3/2$, write\n$u_0=(1,0)$ and $u_1=(1,2)$. The primitive rows $u_0$ and\n$3u_1-2u_0=(1,6)$ span $\\Z\\times6\\Z$, with minor gcd $6$,\nalthough $u_0,u_1$ span $\\Z\\times2\\Z$. The selection replaces the\noriginal lattice by an index-three sublattice, accounting for the\nchange from $2$ to $6$. Containment therefore preserves divisibility\nby the original gcd, not equality of gcds or endpoint valuations. The prefix and window equalities above\nconcern the full indicated families.\n\nRational combinations can also leave the original lattice: the\nhalf-sum of $u_0,u_1$ is the primitive row $(1,1)$, whose minor\nagainst $u_0$ is $1$. Coefficient denominators must then be accounted\nfor; likewise, primitive normalisation after taking combinations\nrequires a new lattice calculation. These operations may change\nthe minor gcd in either direction. None by itself controls the\nreal error: a useful selected family must still have nonzero\nremainders small enough after the proposed division.\n\n\n\n\\subsection{Estimating the divided remainder}\n\nOnce a construction is fixed, let $H_n\\ge1$ be its height, $L_n\\ne0$\nits undivided remainder, and $D_n=3^{R_n}2^{S_n}$ its proved divisor,\ncounted once. We ask for the additional height and remainder estimate\nbelow. With a different divisor, the two logarithmic terms should be\nreplaced by $\\log D_n$. The estimate must use the height of the objects\nthat enter the comparison with a nonzero integer; scalar and\nexterior-determinant heights cannot be interchanged.\nFor scalar forms, integral coefficients and $0<|L_n|/D_n\\to0$\nalready suffice for irrationality, without a height factor. The\nproposed estimate implies that decay because $H_n\\ge1$, although it\nis not necessary for the scalar criterion. Allowing arbitrarily\nsmall positive heights would lose this implication.\n\n\\begin{problem}[an additional height and remainder estimate]\nProve the explicit estimate\n\\begin{equation}\n \\limsup_{n\\to\\infty}\n \\frac{\\log H_n+\\log|L_n|-R_n\\log3-S_n\\log2}{n^2}<0.\n\\label{long1049:eq:negative-margin}\n\\end{equation}\nEvery denominator, row content and final-combination content must already be\nincluded in $H_n$ and $L_n$.\n\\end{problem}\n\nNonvanishing alone does not address~\\eqref{long1049:eq:negative-margin}.  Conversely,\na formal decay estimate cannot supply a nonzero form if every\ncombination with the required residues has zero remainder.  A proposed divisor saving must be compared with the height and remainder\nof the same explicitly normalised objects.\n\n\\subsection{A restriction on Mahler functional equations}\n\nFor the divisor generating series\n$\\mathcal L(z)=\\sum_{n\\ge1}\\tau(n)z^n$, the classification of Bell and\nSmertnig excludes a $k$-Mahler equation for every $k\\ge2$\n\\cite[Thm.~1.3 and the consequences on p.~3]{bellsmertnig2026}.\nWe first explain its application to $\\tau$. We then prove the weaker\nsimultaneous $2$/$3$ statement by combining the Adamczewski--Bell theorem\nwith the functional nonrationality from\nSection~\\ref{long1049:sec:source-forms}.\nThe cited 2026 works have the status of the identified preprints.\nThese functional statements leave the arithmetic nature of an\nindividual rational-base value undecided.\n\nTo apply the classification, note that $\\tau$ is multiplicative:\nfor coprime integers $m,n$, each divisor of $mn$ has a unique\nfactorisation into a divisor of $m$ and a divisor of $n$.\nSuppose that $\\mathcal L$ were $k$-Mahler. The classification would give a prime $p$, an integer\n$r\\ge0$ and an eventually periodic function $\\chi$ with\n$\\tau(m)=m^r\\chi(m)$ whenever $p\\nmid m$. For a prime $\\ell\\ne p$,\nthis forces\n\\[\n \\chi(\\ell^j)=\\frac{j+1}{\\ell^{jr}}\\qquad(j\\ge0).\n\\]\nIf $r=0$, these values are unbounded. If $r>0$, they are nonzero\nand tend to zero. Both alternatives contradict the finite range of\nan eventually periodic function.\n\n\nHere the ambient space is $\\Q((z))$, the field of formal Laurent\nseries, viewed as a vector space over $\\Q(z)$. Stability means that\nsubstituting $z^k$ for $z$ sends each member of the subspace back into\nthat subspace; this substitution is not a $\\Q(z)$-linear map.\n\n\\begin{proposition}[no finite simultaneous $2/3$-system]\\label{long1049:res:nomahler}\nLet\n\\[\n \\mathcal L(z)=\\sum_{n\\ge1}\\frac{z^n}{1-z^n}.\n\\]\nThere is no finite-dimensional $\\Q(z)$-vector space that contains $\\mathcal L$\nand is stable under both $z\\mapsto z^2$ and $z\\mapsto z^3$.\n\\end{proposition}\n\n\\begin{proof}\nSuppose $V$ were such a space, of dimension $d$.  Stability under $z\\mapsto z^2$\nplaces the $d+1$ elements $\\mathcal L(z),\\mathcal L(z^2),\\dots,\\mathcal\nL(z^{2^{d}})$ in $V$, so they are linearly dependent over $\\Q(z)$; clearing\ndenominators gives polynomials $P_0,\\dots,P_d$, not all zero, with\n$\\sum_{i=0}^{d}P_i(z)\\mathcal L(z^{2^i})=0$.\n\nLinear dependence need not give a nonzero coefficient of the\nunshifted function, as a Mahler equation requires. One cannot simply\nreplace $z$ by $z^{1/2^j}$, since the polynomial coefficients need\nnot have exponents divisible by $2^j$. Instead, separate their\nexponent classes. Let $j$ be the least index with $P_j\\ne0$. Write each polynomial uniquely as\n$P_i(z)=\\sum_{r=0}^{2^j-1}z^rQ_{i,r}(z^{2^j})$.\nEvery series $\\mathcal L(z^{2^i})$ with $i\\ge j$ has exponents\ndivisible by $2^j$, so the relation splits by exponent residues modulo\n$2^j$. Choose $r$ with $Q_{j,r}\\ne0$ and put $u=z^{2^j}$. The\ncorresponding relation is\n\\[\n \\sum_{i=j}^{d}Q_{i,r}(u)\\mathcal L(u^{2^{i-j}})=0,\n\\]\nwith a nonzero coefficient of $\\mathcal L(u)$. Thus $\\mathcal L$ is\n$2$-Mahler. The same argument with $3$ in place of $2$ makes it\n$3$-Mahler. Since $2$ and\n$3$ are multiplicatively independent, a theorem of Adamczewski and Bell\n\\cite[Thm.~1.1, p.~6]{adamczewskibell2013} then forces $\\mathcal L$ to be a rational\nfunction. Since $\\mathcal L(z)=F(1/z)$ for $0<z<1$, this contradicts\nthe nonrationality of $F$ proved in Section~\\ref{long1049:sec:source-forms}.\nRivin's periodic-coefficient corollary\n\\cite[Cor.~6.4, p.~9]{rivin2026} gives the same nonrationality conclusion.\n\\end{proof}\n\nProposition~\\ref{long1049:res:nomahler} uses no property of the point\n$2/3$: the obstruction is functional and appears before regularity at a particular point is considered.\nFor this scalar function, Bell and Smertnig's single-base classification\nalready implies the proposition. The proof above instead derives it\nfrom simultaneous closure and elementary functional nonrationality,\nusing the Adamczewski--Bell theorem.  A\nconstruction using additional functions or functional relations must specify\nthose functions and its closure conditions; the single-base statement is not\nan obstruction to every approximation method. The classification and the\nAdamczewski--Bell theorem are cited, not proved here; the needed\nnonrationality has the elementary proof given earlier.\n\nThe Lean proof of Proposition~\\ref{long1049:res:nomahler}\n(\\href{\\laterepobase/lean/ErdosProblems/Erdos1049/PaperCompleteR21/SimultaneousMahlerSystemUnconditional.lean\\#L298}{no finite simultaneous $2/3$-system})\nuses neither cited theorem, and it uses only stability under\n$z\\mapsto z^2$. As in the first step of the proof above, that stability\ngives polynomials $P_0,\\dots,P_d$, not all zero, with\n$\\sum_{i=0}^{d}P_i(z)\\mathcal L(z^{2^i})=0$, and this relation also holds\nbetween the values of these functions on the open unit disc. If $\\zeta$ is\na primitive $n$th root of unity, then\n$(1-r)\\mathcal L(r\\zeta)/\\log(1/(1-r))\\to1/n$ as $r\\uparrow1$.\nThese limits, taken at roots of unity of suitable orders, force every\n$P_i$ to vanish. The same argument excludes, for every $k\\ge2$, a\nfinite-dimensional space containing $\\mathcal L$ and stable under\n$z\\mapsto z^k$ alone\n(\\href{\\laterepobase/lean/ErdosProblems/Erdos1049/PaperCompleteR21/SimultaneousMahlerSystemUnconditional.lean\\#L263}{single-base statement}).\n\n\\subsection{A limitation of the rectangular exponent model}\n\nWe conclude the comparison of exponent models by bounding\n$\\Theta_{\\mathrm{HP}}$, defined in Section~\\ref{long1049:sec:sharp}.\nFor $\\rho\\ge0$ and $\\sigma\\ge1+\\rho$ its denominator\n\\[\n \\frac{(1+\\rho)^2}{2}+\\sigma(1+\\rho)+\\frac{1+\\rho^2}{2}+\\sigma\n\\]\nis positive. Put $u=\\sigma-1-\\rho\\ge0$. Multiplying the difference\n$\\Theta_{\\mathrm{HP}}(\\rho,\\sigma)-(1/2-1/\\pi^2)$ by\n$2\\pi^2$ times this denominator gives exactly\n\\[\n -\\pi^2\\rho^2-\\pi^2\\rho u-2\\pi^2\\rho-2\\rho^2-10\\rho u\n -4\\rho-6u^2-8u.\n\\]\nEvery term is nonpositive. The original difference has the same sign, and\nit vanishes exactly when $\\rho=0$ and $\\sigma=1$\n(\\lword{Erdos1049/HermitePadeNoGo.lean}{48}{hpClearedGap_expansion}{exact expansion},\n\\lword{Erdos1049/HermitePadeNoGo.lean}{58}{hpClearedGap_nonpos}{nonpositivity},\n\\lword{Erdos1049/HermitePadeNoGo.lean}{75}{hpClearedGap_eq_zero_iff}{equality case}).\nEquivalently, within that model the displayed threshold never exceeds the\nclassical one-function margin, with equality only at the classical endpoint\n(\\lword{Erdos1049/HermitePadeNoGo.lean}{103}{rectangular_hp_threshold_le_classical}{bound},\n\\lword{Erdos1049/HermitePadeNoGo.lean}{126}{rectangular_hp_threshold_eq_classical_iff}{equality}).\nThis bounds only the displayed exponent model. Applying it to an\napproximation family would require the polynomial construction,\nintegrality and asymptotic estimates connecting that family to\n$\\Theta_{\\mathrm{HP}}$.\nThe rational cutoff $81/200$ discussed above satisfies\n\\[\n \\frac{81}{200}<\\frac{\\log2}{\\log3}.\n\\]\nFor a sufficient criterion of the form $\\log b/\\log a<T$, including\n$3/2$ requires $T>\\log2/\\log3$, not merely an improvement on\n$81/200$. This numerical target applies to that form of criterion,\nnot to every irrationality argument.\n\nA concrete optimisation question is available within the 2004\nconstruction itself. Write a source parameter direction as\n$\\alpha=(\\alpha_0,\\alpha_1,\\alpha_2;\\beta)$, with positive integer entries satisfying\n\\[\n \\alpha_1\\le\\alpha_2,\\qquad\n \\alpha_1+\\alpha_2<\\beta\\le\\alpha_0+\\alpha_2,\\qquad\n \\gcd(\\alpha_0,\\alpha_1,\\alpha_2,\\beta)=1.\n\\]\nThe source parameters are $a_j=\\alpha_jn+1$ and $b=\\beta n+2$;\nhere $b$ is not a rational-base denominator. The gcd condition is a\nnormalisation of the parameter direction, not a primitivity assertion\nabout evaluated rows. Its invariance under dilation is checked below.\nFor these directions set\n\\[\n \\begin{gathered}\n c_{00}=\\alpha_0+\\alpha_1+\\alpha_2-\\beta,\\quad\n c_{01}=\\alpha_0,\\quad c_{11}=\\alpha_1,\\quad c_{21}=\\alpha_2,\\\\\n c_{12}=\\beta-\\alpha_1,\\quad c_{22}=\\beta-\\alpha_2,\\qquad\n m=\\max(c_{00},c_{01},c_{11},c_{21},c_{12},c_{22}),\n \\end{gathered}\n\\]\nand define the periodic step function\n\\[\n \\omega_\\alpha(u)=\\max\\left\\{\\begin{aligned}\n &0,\\\\[-2pt]\n &\\lfloor c_{21}u\\rfloor+\\lfloor c_{22}u\\rfloor\n       -\\lfloor c_{11}u\\rfloor-\\lfloor c_{12}u\\rfloor,\\\\[-2pt]\n &\\lfloor c_{01}u\\rfloor+\\lfloor c_{21}u\\rfloor\n       -\\lfloor c_{00}u\\rfloor-\\lfloor c_{12}u\\rfloor\n \\end{aligned}\\right\\}.\n\\]\nZudilin's constants in (25) and (26), already used in the finite\ndirection scan, are\n\\[\n \\begin{aligned}\n C_1(\\alpha)&=(\\alpha_0+\\alpha_1+\\alpha_2)\\beta\n               -\\frac{\\alpha_1^2+\\alpha_2^2+\\beta^2}{2},\\\\\n C_0(\\alpha)&=\\frac{\\alpha_1^2}{2}+\\alpha_0\\alpha_1\n            +(\\beta-\\alpha_2)(\\alpha_2-\\alpha_1)\\\\\n &\\hspace{8mm}-\\frac3{\\pi^2}\n       \\left(m^2-\\int_0^1\\omega_\\alpha(u)\\,d(-\\psi_1(u))\\right).\n \\end{aligned}\n\\]\nHere $\\psi_1$ is the trigamma function used earlier. The step function\nis bounded, nonnegative, periodic with period $1$, and zero near zero,\nso the integral is finite.\n\nThe gcd normalisation does not change the ratio being optimised.\nIndeed, periodicity and\n$-\\psi_1'(u)=2\\sum_{j\\ge0}(u+j)^{-3}$ give, by Tonelli's theorem,\n\\[\n \\int_0^1\\omega_\\alpha(u)\\,d(-\\psi_1(u))\n   =\\int_0^\\infty\\frac{2\\omega_\\alpha(u)}{u^3}\\,du.\n\\]\nFor any positive integer $k$, the floor formulas give\n$\\omega_{k\\alpha}(u)=\\omega_\\alpha(ku)$. Substitution in the last integral\ntherefore multiplies it by $k^2$. All other terms in $C_0$ and $C_1$\nare quadratic in the direction, including $m^2$. Hence\n$C_i(k\\alpha)=k^2C_i(\\alpha)$ for $i=0,1$, so passing to a primitive\ndirection preserves both positivity and $C_0/C_1$.\nLet $\\mathcal A$ be exactly the displayed primitive directions with\n$C_0(\\alpha)>0$ and $C_1(\\alpha)>0$.\n\n\\begin{problem}[optimising the published parameter directions]\nDetermine\n\\[\n \\sup_{\\alpha\\in\\mathcal A}\\frac{C_0(\\alpha)}{C_1(\\alpha)},\n\\]\nor improve its bounds. In particular, does an admissible direction give\na ratio strictly greater than the value $\\theta^*$ in\nTheorem~\\ref{long1049:res:region}, attained at $(14,12,14;27)$?\nA finite numerical scan does not establish optimality over $\\mathcal A$.\n\\end{problem}\n\nThe immediate bounds are $\\theta^*\\le\\sup_{\\mathcal A}C_0/C_1\\le1/2$.\nIndeed, at any integer base $p\\ge2$ the source gives\n$\\mu_{\\rm irr}(F(p))\\le C_1(\\alpha)/C_0(\\alpha)$, whereas pigeonhole\napproximation gives $\\mu_{\\rm irr}(F(p))\\ge2$.\nThus even the optimal reciprocal criterion in this class cannot exceed\n$1/2$, which is below $\\log2/\\log3$. Optimising these directions cannot\ninclude $3/2$ by that criterion. For other noninteger bases, a new\ndirection would still require the polynomial degree and coefficient-height\nestimates used in the rational-base transfer; the integer-base source\nalone does not supply that application.\nThe unrestricted question of which rational bases give irrational values\nis not settled here and is not reduced to any one of these problems.\n\n\\section*{Related work and comparison of constructions}\nThe relevant comparisons require agreement of both target and\nconstruction. A theorem about the Lambert function need not decide\na special value; a theorem at integer bases need not survive\nrational specialisation; and a shared orthogonal-polynomial kernel\nneed not give the same recurrence on different diagonals.\nThe following examples distinguish these three questions.\n\n\\paragraph{Functional nonrationality is not value irrationality.}\nAt the level of functions, Rivin proves that if a sequence \\(\\gamma\\) and its\ndivisor-sum sequence are both eventually linearly recurrent, then \\(\\gamma\\)\nis finitely supported\n\\cite[Theorem~1.1, p.~2; proof pp.~6--7]{rivin2026}.\nHis periodic-coefficient corollary therefore shows that\n\\[\n  \\sum_{n\\ge1}\\frac{z^n}{1-z^n}\n\\]\nis not a rational function~\\cite[Corollary~6.4, p.~9]{rivin2026}.\nThis is an exact structural statement about the function underlying\nProblem~\\#1049, but it gives no irrationality statement for a special value\nat \\(z=1/t\\): a nonrational function may take rational values at particular\nrational points.\n\n\\paragraph{Pad\\'e and orthogonal-polynomial constructions.}\nIn 1991\nBorwein proved the irrationality of shifted series\n$\\sum_{n\\ge1}(t^{n}+w)^{-1}$ at integer bases $t\\ge2$, for every nonzero\nrational $w$ with $w\\ne-t^m$ for all $m\\ge1$, by Pad\\'e approximation rather\nthan by digit clearing; his estimates also show that these values are not\nLiouville numbers~\\cite[Thm.~4, pp.~257--258]{borwein1991}.  In 2001 Van\nAssche recovered the integer-base irrationality and the bound\n$\\mu_{\\rm irr}(F(p))\\le 2\\pi^2/(\\pi^2-2)=2.50828\\ldots$ using little $q$-Legendre\nPad\\'e approximants\n~\\cite[Thm.~1, p.~10; proof pp.~10--11]{vanassche2001}.  His more general\nTheorem~3 proves irrationality of\n$\\sum_{k\\ge1}(cp^k-1)^{-1}$ for an integer $p>1$ and a fixed nonzero\nrational $c$ with $cp^k\\ne1$ for every $k\\ge1$\n~\\cite[Thm.~3, p.~14]{vanassche2001}; it does not cover a\nrational noninteger base $t$ in $F(t)$, because the multiplier needed to\nwrite $t^k$ over an integer base varies with $k$.\n\n\\paragraph{Related series and their different recurrences.}\nThe same Lambert value was already the target of Amdeberhan and\nZeilberger's $q$-WZ construction~\\cite{az1998}.  Here $p>1$ is the integer-base parameter, $q=p^{-1}$, and\n$P_n(x\\mid q)$ denotes the little $q$-Legendre polynomial used by these\nauthors, normalised by $P_n(0\\mid q)=1$.  The two constructions\nshare that bivariate little-$q$-Legendre Pad\\'e kernel, but Van Assche's\ndiagonal does not satisfy the Amdeberhan--Zeilberger scalar recurrence.\nTheir Theorems~1 and~2 do prove, respectively, the irrationality of the\nnon-alternating $h_p(1)$ and alternating $q$-logarithm values for their stated\ninteger parameters, with reported irrationality measure $4.80$; those are\nexternal integer-parameter results and do not transfer to the rational\nnoninteger base $3/2$ studied here.\n\nAmdeberhan--Zeilberger use the moving diagonal\n$P_n(p^{n+1}\\mid p^{-1})$, whereas Van Assche uses\n$P_n(p^n\\mid p^{-1})$.  Van Assche also records, citing\nBorwein's 1992 Lemma~2, the neighbouring evaluation\n$P_{n-1}(c p^{n+1}\\mid p^{-1})$\n\\cite[(23) and the following paragraph, p.~6]{vanassche2001};\nsee \\cite[Lemma~2, p.~143]{borwein1992}.\nThe shared kernel therefore does not license transfer of recurrence,\nendpoint, lattice, or valuation claims between the diagonals.  Indeed, if\n$A_n(p)=P_n(p^n\\mid p^{-1})$, exact substitution at $n=0$ into the\nAmdeberhan--Zeilberger operator \\cite[Sec.~1.5, p.~2]{az1998} leaves\n\\[\n -p(p-1)^2(p+1)(p^5+2p^4+2p^3+2p^2+2),\n\\]\nwhich is nonzero for every real $p>1$.  One nonzero residual is decisive for\nnon-transfer of that recurrence.\n\nLean checks the\n\\lword{Erdos1049/QAperyDiagonalNonEquivalence.lean}{67}{vanAsscheQAperyResidualAtZero_eq}{exact\nresidual factorisation} and its\n\\lword{Erdos1049/QAperyDiagonalNonEquivalence.lean}{94}{vanAsscheQAperyResidualAtZero_ne_zero}{nonvanishing\nfor $p>1$}.  These are finite statements at $n=0$; they do not supply a\nrecurrence for either diagonal.  No recurrence, endpoint, lattice, or\nvaluation statement for one diagonal is used for the other; the displayed\nexact residual is the sole claim made here about their incompatibility.\n\nCoussement and Smet use little $q$-Jacobi polynomials for further Lambert\nseries whose base parameter is the reciprocal of an integer\n\\cite[Thms.~1.2--1.3, p.~2; Sec.~2]{coussementsmet2007}, and Koizumi and\nYokoi identify one of their three-parameter Ap\\'ery-type approximations with\nthe Coussement--Smet Pad\\'e approximants\n\\cite[Sec.~7, Prop.~7.1, p.~33]{koizumiyokoi2026}.\nThat identification includes the source's moving evaluation point\n$z_n=p_2^{n+1}$ and its normalization; it is not an identification of\nall little-$q$-Jacobi diagonals or their scalar recurrences.\n\nAt a noninteger rational base, convergence of the rational approximants\nis not enough. If $P_n/Q_n\\to\\xi$, the cleared error is\n$Q_n\\xi-P_n=Q_n(\\xi-P_n/Q_n)$, whose size also depends on $Q_n$.\nFor example, $1/n\\to0$, but the corresponding cleared error is always\n$-1$. Thus convergence alone does not give the small integer linear\nforms required for irrationality.\n\nIn 2013 Vandehey proved that\n$\\sum_{n\\ge1}d(n)a_n/b^n$ is irrational whenever $b>1$ is an integer and\n$(a_n)$ ranges in a finite integer alphabet excluding zero; taking\n$a_n=(-1)^n$ completes the elementary digit method for integer bases\n$b\\le-2$~\\cite[Thm.~1.2, p.~2]{vandehey2013}.  His companion theorem permits\na finite alphabet of nonnegative integers containing zero, provided the\ncoefficient sequence is not eventually zero\n\\cite[Thm.~1.1, p.~2]{vandehey2013}.  All\nof these statements require an integer base. Luca and Tachiya's periodic-coefficient theorem\nlikewise proves irrationality at every integer base of absolute value greater\nthan one; their full-support example strengthens this to joint linear\nindependence for finite families of iterated divisor functions\n\\cite[Theorem~A, p.~139; Example~1, p.~140]{lucatachiya2017}.  Duverney and\nTachiya refine the Chowla--Erd\\H{o}s method to linear independence results for\nLambert series at an integer base $q$ with $|q|>1$\n\\cite[Thms.~1.1--1.2, pp.~2--3]{duverneytachiya2019}; their Corollary~1.1\ngeneralises Vandehey's Theorem~1.2 \\cite[p.~3]{duverneytachiya2019}.  Each of these results assumes an integer base.\nFor the underlying small-integral-tail criterion, see also Duverney\n\\cite[Thm.~2, p.~4 of the author manuscript]{duverney2011}.\nWe use that elementary criterion only as methodological context, not his\nseparate square-index linear-independence argument.\n\n\nIn 2004 Zudilin obtained the uniform bound\n$\\mu_{\\rm irr}(F(p))\\le2.46497868\\ldots$ for every integer $p\\ge2$,\nby way of Heine's basic transform and a permutation group\n~\\cite[Thm.~1, p.~154; Secs.~4--5, pp.~159--162]{zudilin2004}.\nHis source theorem also covers negative integer inverse bases; our\nnotation $F(t)$ and positive-remainder arguments are restricted to $t>1$.\nThe ordinary-hypergeometric antecedent is Rhin and Viola's $S_5$ action and\ntwelve-coset denominator reduction for rational forms in\n$\\zeta(2)$~\\cite[Sec.~3, pp.~38--42; Sec.~4, pp.~46--51]{rhinviola1996}.\n\nThese sources suggest looking for common polynomial factors before\nspecialisation. The congruence lemmas in\nSection~\\ref{long1049:sec:endpoints} describe what the first and last\ncoefficients can imply at $3/2$. Their hypotheses have not been proved\nthere for either cited coefficient family.\n\nThe neighbouring problem of Lambert\nsubseries $\\sum_{n\\in A}(t^{n}-1)^{-1}$ over a restricted index set $A$ is\ntreated by Kova\\v{c} and Tao~\\cite[Sec.~2.1.2 and Thm.~2.3, pp.~4--5]{kovactao2024}.\n\nThe rational non-integer\nprogress relevant here is instead the height criterion of Bundschuh and\nV\\\"a\\\"an\\\"anen~\\cite[Thm.~2, p.~177]{bv1994}, which gives the strongest explicit\nnumerical threshold for this value among the previously published sources\ncompared here; Theorem~\\ref{long1049:res:region} enlarges the region that\ncriterion covers.  Zudilin's later Pad\\'e-and-Hankel treatment of the generalized\n$q$-logarithm also comments on non-integral rational bases: it remarks that its results can be\ngiven for non-integer $p=r/s$ with $|p|>1$, under an assumption\n$\\log|r|>c\\log|s|$ for some computable\n$c>0$~\\cite[Sec.~2, p.~4]{zudilin2016}.  No value of $c$ is computed there.  That remark\nannounces the shape of a rational-base extension for the generalized\n$q$-logarithm of that paper.  Section~\\ref{long1049:sec:region} proves an explicit extension for\n$F$ using the different, 2004 construction. Its polynomial cancellation\nand denominator calculation show that\n$c=\\mu=2.464978683574975\\ldots$, the 2004 bound quoted above, is admissible,\nand gives the resulting region and its application to $31/4$.\n\nA third published rational-base region for this same series is Duverney's\nTh\\'eor\\`eme~2, whose hypothesis is\n$\\log|s|/\\log|r|<\\frac13\\bigl(1-3/\\pi^{2}\\bigr)=0.2320\\ldots$\n\\cite[Th\\'eor\\`eme~2, p.~174]{duverney1996}; his Th\\'eor\\`eme~1 for general\nnumerators is weaker still.  Both cutoffs lie strictly inside the Bundschuh and\nV\\\"a\\\"an\\\"anen region.  Matala-aho, V\\\"a\\\"an\\\"anen and Zudilin's combined\ntreatment of $q$-logarithms keeps the integer hypothesis $p=1/q\\in\\Z\\mathbin{\\backslash}\\{0,\\pm1\\}$\nthroughout, and states in print that its methods do not sharpen the\n$q$-harmonic case of~\\cite{zudilin2004}\n\\cite[abstract p.~879; introduction p.~880]{matalaaho2006}.  So none of the\nprinted regions for $F$ compared here reaches $\\log4/\\log31$.\n\n\n\\pdfbookmark[1]{Statements and declarations}{statements-declarations}\n\\subsection*{Statements and declarations}\n\n\\paragraph{Funding and competing interests.} This work received no external\nfunding.  The author declares no competing interests.\n\n\\paragraph{Acknowledgements.} The problem numbering and status follow the\nErd\\H{o}s Problems catalogue maintained by Thomas Bloom~\\cite{erdosproblems}.\n\n\n\\appendix\n\\section{Verification and reproducibility}\\label{long1049:app:index}\n\nThe \\href{\\evidencerecordurl}{evidence record} identifies the supplied\nLean declarations and recorded Comparator comparisons;\nno fresh execution or independent review of this revision is\nreported. Theorem~\\ref{long1049:thm:geometric-universality}\nretains its checked polynomial shift-bound hypotheses, and its\nproof does not use the \\hyperref[long1049:thm:geometric-ratio]{ratio-limit remark}.\nThat broader argument, the\n\\hyperref[long1049:thm:calibrated-model]{rational-value model}, its\n\\hyperref[long1049:thm:calibrated-denominators]{denominator calculation}\nand the \\hyperref[long1049:prop:calibrated-height]{height bound}\nhave ordinary proofs here but no Lean proofs in the supplied record. The unformalised material\nis kept separate from the checked fixed-base theorem.\nThe \\href{\\sourceurl}{source revision} includes the Lean sources,\ntoolchain and library manifest.\n\\evidenceparagraph{\\evidencerecordurl}\n\n\\paragraph{Formal verification and computations.}\nThe printed proofs of Theorems~\\ref{long1049:res:region}\nand~\\ref{long1049:res:31over4} use Zudilin's construction: his Lemma~7 in its polynomial reading, together with the\ninputs of that lemma's own proof, namely his Lemma~3, the exponent $M(a;b)$\nof~(16) proved by his Lemma~4, the group stability of his Lemma~5, and the\nidentity~(9)--(11), all in \\cite[pp.~156--161]{zudilin2004}, and his direction and\nconstants \\cite[p.~162]{zudilin2004}.  Proved here are\nthe nonrationality of $F$ as a function, the identification of the\npolynomial coefficients in Lemma~7, the exact degrees of $U_n$ and $V_n$,\npositivity, the Archimedean size estimate, and the limit\n$(K_n-W_n)/n^{2}\\to C_0$.  That last limit rests on Zudilin's Lemmas~1 and~2\n\\cite[p.~155]{zudilin2004}, whose method goes back to\n\\cite[Lemma~1, p.~466]{zudilin2002}; Section~\\ref{long1049:sec:cyclotomic-limit}\nproves them by that method with an explicit truncation estimate.  Lean\nproves both theorems and the measure bound of\nCorollary~\\ref{long1049:cor:rational-base-measure}; it constructs the\npolynomials and proves their remainder estimates.  The Lean sources also\ncontain the finite comparisons, the rational bracket in\nSection~\\ref{long1049:sec:regionbracket} and the degree identities in\nSection~\\ref{long1049:sec:degrees}. Appendix~\\ref{long1049:app:index}\ndistinguishes these proofs from computer algebra.\n\n\\paragraph{Statements for every rank or every base.}\nLean constructs the cancelled polynomial forms of\nSection~\\ref{long1049:sec:region} and proves irrationality in the\nstated region, irrationality at every positive integral power of $31/4$\nand the irrationality-exponent bound. It proves the first nonzero term\nof every transformed row of the 2016 determinant, deduces the order and\nleading coefficient of the determinant at every rank, and proves the\nfixed-base asymptotic of Section~\\ref{long1049:sec:sharp-fixed-base}.\nBoth constructions therefore supply their mathematical objects; neither\nis assumed to exist. Proposition~\\ref{long1049:res:nomahler} is checked\nwith no additional hypothesis; its Lean proof uses stability under\n$z\\mapsto z^2$ alone, without the theorem of Adamczewski and\nBell~\\cite[Thm.~1.1, p.~6]{adamczewskibell2013} cited in the printed\nproof, as explained after that proof.\n\n\\paragraph{Statements through rank eight.}\nProposition~\\ref{long1049:res:finite-pencil} is checked in Lean for\nevery real $p>1$ and every rank $N\\le8$. The Lean proof includes the\nidentity $F(p)A_N-B_N=(v^*_{i+j})$, the positivity of the eight unshifted\ndeterminants $D_{k,0}$ by the certificate described in\nSection~\\ref{long1049:sec:coefficient-questions}, the reality of the\npencil roots, their position below $F(p)$ and their interlacing. It has\nno all-rank coefficient-positivity or root-convergence conclusion.\n\n\\paragraph{Computations outside Lean.}\nThe shifted determinants $D_{k,1}$ and the strict positivity of every\ncoefficient rest on the sixteen-polynomial computation of\nSection~\\ref{long1049:sec:coefficient-questions}. The reconstruction of\n$U_n,V_n$ for $n\\le4$ in Section~\\ref{long1049:sec:receipts}, the five\ncomplete polynomial contents and the seventy-six cyclotomic residue\nwitnesses have exact reproduction scripts and direct determinant checks.\nSeparate exact computations certify the rational interval bounds and the\nfinite parameter search. The historical high-precision real remainder\nevaluations are separate numerical reports, not reproduced by these exact\ncomputations. None of these finite computations is a Lean declaration or\nis needed for the proof of Theorem~\\ref{long1049:res:region}.\n\n\\paragraph{The Lambert identity and the integer case.}\nThe cited \\emph{Formal Conjectures} file contains unproved declarations\nfor the conjecture and the integer-base irrationality theorem, as well\nas a proved identity between the two series~\\cite{formalconjectures1049}.\nFor $t>1$, that identity is an equality of convergent series. The file\nalso treats a different range using Lean's convention that an\nunsummable series has totalised sum zero; that convention is not used\nin the present argument. The proved identity should therefore not be\nconfused with a proved irrationality statement. The rational-base\nformal proofs used here are those described above.\n% Part: part family_catalogue\n% Part: part back\n\\begin{thebibliography}{99}\n\\small\n\\raggedright\n\\setlength{\\itemsep}{2pt}\n\\bibitem{erdos1948} P.~Erd\\H{o}s,\n  \\href{https://users.renyi.hu/~p_erdos/1948-04.pdf}{\\emph{On arithmetical\n  properties of Lambert series}},\n  J. Indian Math. Soc. (N.S.) \\textbf{12} (1948), 63--66.\n\\bibitem{erdos1988} P.~Erd\\H{o}s, \\emph{On the irrationality of certain series:\n  problems and results}, in A.~Baker (ed.), \\emph{New Advances in Transcendence\n  Theory}, Cambridge UP, 1988, pp.~102--109,\n  doi:\\href{https://doi.org/10.1017/CBO9780511897184.009}{10.1017/CBO9780511897184.009}.\n\\bibitem{borwein1991} P.~B. Borwein, \\emph{On the irrationality of\n  $\\sum1/(q^{n}+r)$},\n  J. Number Theory \\textbf{37} (1991), no.~3, 253--259,\n  doi:\\href{https://doi.org/10.1016/S0022-314X(05)80041-1}{10.1016/S0022-314X(05)80041-1}.\n\\bibitem{borwein1992} P.~B. Borwein, \\emph{On the irrationality of certain\n  series}, Math. Proc. Cambridge Philos. Soc. \\textbf{112} (1992), no.~1,\n  141--146,\n  doi:\\href{https://doi.org/10.1017/S030500410007081X}{10.1017/S030500410007081X}.\n  Van Assche cites Lemma~2 for the neighbouring little-$q$-Legendre\n  evaluation used above.\n\\bibitem{az1998} T.~Amdeberhan and D.~Zeilberger, \\emph{$q$-Ap\\'ery\n  irrationality proofs by $q$-WZ pairs}, Adv. Appl. Math. \\textbf{20} (1998),\n  no.~2, 275--283,\n  doi:\\href{https://doi.org/10.1006/aama.1997.0565}{10.1006/aama.1997.0565};\n  arXiv:\\href{https://arxiv.org/abs/math/9804122v1}{math/9804122v1}.\n  Page references are to arXiv:math/9804122v1.\n\\bibitem{bv1994} P.~Bundschuh and K.~V\\\"a\\\"an\\\"anen,\n  \\href{https://numdam.org/item/CM_1994__91_2_175_0.pdf}{\\emph{Arithmetical\n  investigations of a certain infinite product}},\n  Compositio Math. \\textbf{91} (1994), no.~2, 175--199.\n\\bibitem{zudilin2002} W.~Zudilin, \\emph{Remarks on irrationality of\n  \\(q\\)-harmonic series}, Manuscripta Math. \\textbf{107} (2002), no.~4,\n  463--477,\n  doi:\\href{https://doi.org/10.1007/s002290200249}{10.1007/s002290200249}.\n\\bibitem{zudilin2004} W.~Zudilin,\n  \\href{https://geodesic.mathdoc.fr/articles/10.4064/aa111-2-4/}{\\emph{Heine's\n  basic transform and a permutation group for \\(q\\)-harmonic series}},\n  Acta Arith. \\textbf{111} (2004), no.~2, 153--164,\n  doi:\\href{https://doi.org/10.4064/aa111-2-4}{10.4064/aa111-2-4}.\n  Page references are to the printed journal pages.\n\\bibitem{zudilin2016} W.~Zudilin,\n  \\href{https://arxiv.org/abs/1601.02688v2}{\\emph{On the irrationality of\n  generalized \\(q\\)-logarithm}}, arXiv:1601.02688; Res. Number Theory\n  \\textbf{2} (2016), Art.~15,\n  doi:\\href{https://doi.org/10.1007/s40993-016-0042-x}{10.1007/s40993-016-0042-x}.\n  Page references are to arXiv:1601.02688v2.\n  The remark that the results extend to non-integer \\(p=r/s\\), \\(|p|>1\\), under\n  an assumption \\(\\log|r|>c\\log|s|\\) for a computable \\(c>0\\), is in Section~2,\n  p.~4, in the paragraph beginning ``Finally, we remark''; no value of \\(c\\) is\n  computed there, and the remark is made for the generalized \\(q\\)-logarithm of\n  that paper.\n\\bibitem{zudilin2017det} W.~Zudilin, \\emph{A determinantal approach to\n  irrationality}, Constr. Approx. \\textbf{45} (2017), no.~2, 301--310,\n  doi:\\href{https://doi.org/10.1007/s00365-016-9333-7}{10.1007/s00365-016-9333-7};\n  arXiv:\\href{https://arxiv.org/abs/1507.05697v1}{1507.05697v1}.\n  Page and equation references are to arXiv:1507.05697v1.\n\\bibitem{zudilin2017detv2} W.~Zudilin, \\emph{A determinantal approach to\n  irrationality}, arXiv:\\href{https://arxiv.org/abs/1507.05697v2}{1507.05697v2}.\n  Propositions 1--2 are cited from v2; the preceding Heine-expansion\n  locators retain their separately cited v1 edition.\n\\bibitem{bundschuhzudilin2008} P.~Bundschuh and W.~Zudilin, \\emph{Rational\n  approximations to a \\(q\\)-analogue of \\(\\pi\\) and some other \\(q\\)-series},\n  in H.~P. Schlickewei, K.~Schmidt and R.~F. Tichy (eds.), \\emph{Diophantine\n  Approximation}, Dev. Math. \\textbf{16}, Springer, 2008, pp.~123--139,\n  doi:\\href{https://doi.org/10.1007/978-3-211-74280-8_6}{10.1007/978-3-211-74280-8\\_6}.\n\\bibitem{krvz2009} C.~Krattenthaler, I.~Rochev, K.~V\\\"a\\\"an\\\"anen and\n  W.~Zudilin, \\emph{On the non-quadraticity of values of the\n  \\(q\\)-exponential function and related \\(q\\)-series}, Acta Arith.\n  \\textbf{136} (2009), no.~3, 243--269,\n  doi:\\href{https://doi.org/10.4064/aa136-3-4}{10.4064/aa136-3-4};\n  arXiv:\\href{https://arxiv.org/abs/0812.2921v1}{0812.2921v1}.\n  Page references are to arXiv:0812.2921v1.\n\\bibitem{duverney1996} D.~Duverney,\n  \\href{https://numdam.org/item/JTNB_1996__8_1_173_0.pdf}{\\emph{\\`A propos de la\n  s\\'erie $\\sum_{n\\ge1}x^{n}/(q^{n}-1)$}},\n  J. Th\\'eor. Nombres Bordeaux \\textbf{8} (1996), no.~1, 173--181.\n  Th\\'eor\\`eme~2 on p.~174 gives the rational-base region\n  $\\log|s|/\\log|r|<\\frac13(1-3/\\pi^{2})=0.2320\\ldots$ for this series;\n  Th\\'eor\\`eme~1, for a general numerator, is weaker.\n\\bibitem{matalaaho2006} T.~Matala-aho, K.~V\\\"a\\\"an\\\"anen and W.~Zudilin,\n  \\href{https://doi.org/10.1090/S0025-5718-05-01812-0}{\\emph{New irrationality\n  measures for \\(q\\)-logarithms}}, Math. Comp. \\textbf{75} (2006), no.~254,\n  879--889, doi:10.1090/S0025-5718-05-01812-0.  The hypothesis\n  \\(p=1/q\\in\\Z\\mathbin{\\backslash}\\{0,\\pm1\\}\\) is carried in the abstract on p.~879 and in\n  both theorem statements on p.~880, where the authors also record that their\n  methods do not sharpen the \\(q\\)-harmonic case of~\\cite{zudilin2004}.\n\\bibitem{adamczewskibell2013} B.~Adamczewski and J.~P. Bell,\n  \\href{https://arxiv.org/abs/1303.2019v1}{\\emph{A problem about Mahler\n  functions}}, Ann. Sc. Norm. Super. Pisa Cl. Sci. \\textbf{17} (2017), no.~4,\n  1301--1355; arXiv:\\href{https://arxiv.org/abs/1303.2019v1}{1303.2019v1}, 2013.\n  Theorem~1.1 on p.~6 of arXiv:1303.2019v1: over a field of characteristic\n  zero, a power series is both \\(k\\)- and \\(\\ell\\)-Mahler for multiplicatively\n  independent \\(k,\\ell\\) if and only if it is a rational function.  Bell and\n  Smertnig cite it as Theorem~1.3 of the journal version\n  \\cite[Thm.~2.5, p.~5]{bellsmertnig2026}.\n\\bibitem{rhinviola1996} G.~Rhin and C.~Viola,\n  \\href{https://geodesic.mathdoc.fr/articles/10.4064/aa-77-1-23-56/}\n  {\\emph{On a permutation group related to $\\zeta(2)$}},\n  Acta Arith. \\textbf{77} (1996), no.~1, 23--56,\n  doi:10.4064/aa-77-1-23-56.\n\\bibitem{vanassche2001} W.~Van Assche,\n  \\href{https://arxiv.org/abs/math/0101187v1}{\\emph{Little \\(q\\)-Legendre\n  polynomials and irrationality of certain Lambert series}},\n  Ramanujan J. \\textbf{5} (2001), no.~3, 295--310,\n  doi:\\href{https://doi.org/10.1023/A:1012930828917}{10.1023/A:1012930828917}.\n  Page references are to arXiv:math/0101187v1.\n\\bibitem{coussementsmet2007} J.~Coussement and C.~Smet, \\emph{Irrationality\n  proof of certain Lambert series using little \\(q\\)-Jacobi polynomials},\n  J. Comput. Appl. Math. \\textbf{233} (2009), no.~3, 680--690,\n  doi:\\href{https://doi.org/10.1016/j.cam.2009.02.036}{10.1016/j.cam.2009.02.036};\n  arXiv:\\href{https://arxiv.org/abs/math/0701345v1}{math/0701345v1}.\n  Page references are to arXiv:math/0701345v1.\n\\bibitem{koizumiyokoi2026} J.~Koizumi and A.~Yokoi, \\emph{Ap\\'ery-type\n  approximations and irrationality measures for certain \\(q\\)-series},\n  arXiv:\\href{https://arxiv.org/abs/2608.26918v1}{2608.26918v1},\n  27 August 2026.\n\\bibitem{vandehey2013} J.~Vandehey,\n  \\href{https://arxiv.org/abs/1206.0340v1}{\\emph{On an incomplete argument of\n  Erd\\H{o}s on the irrationality of Lambert series}}, Integers\n  \\textbf{13} (2013), Paper A58. Page references are to\n  arXiv:1206.0340v1 (2012).\n\\bibitem{lucatachiya2017} F.~Luca and Y.~Tachiya,\n  \\href{https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2014-14.pdf}\n  {\\emph{Linear independence results for the values of divisor functions\n  series}}, RIMS K\\^oky\\^uroku No.~2014 (2017), 138--150.\n  Theorem~A on p.~139 restates the periodic-coefficient irrationality theorem;\n  Example~1 on p.~140 gives the divisor-function specialization.\n\\bibitem{duverneytachiya2019} D.~Duverney and Y.~Tachiya, \\emph{Refinement of\n  the Chowla--Erd\\H{o}s method and linear independence of certain Lambert\n  series}, Forum Math. \\textbf{31} (2019), no.~6, 1557--1566,\n  doi:\\href{https://doi.org/10.1515/forum-2018-0299}{10.1515/forum-2018-0299}.\n  Page references are to the\n  \\href{https://danielduverney.fr/documents/theorie-des-nombres/DuverneyTachiya190522.pdf}{authors'\n  version}.\n\\bibitem{rivin2026} I.~Rivin,\n\\href{https://arxiv.org/abs/2604.25151v1}{\\emph{Zero Coefficients of Rational\n  Power Series and Rational Lambert Series}}, arXiv:2604.25151v1,\n  28 April 2026.\n  Theorem~1.1 is on p.~2 and proved on pp.~6--7; the periodic-coefficient\n  Corollary~6.4 is on p.~9.\n\\bibitem{kovactao2024} V.~Kova\\v{c} and T.~Tao,\n  \\href{https://arxiv.org/abs/2406.17593v4}{\\emph{On several irrationality\n  problems for Ahmes series}}, Acta Math. Hungar. \\textbf{175} (2025), no.~2,\n  572--608,\n  doi:\\href{https://doi.org/10.1007/s10474-025-01528-0}{10.1007/s10474-025-01528-0};\n  arXiv:\\href{https://arxiv.org/abs/2406.17593v4}{2406.17593v4}.\n  Page references are to arXiv:2406.17593v4.\n\\bibitem{erdosproblems} T.~F. Bloom,\n  \\href{https://www.erdosproblems.com/1049}{\\emph{Erd\\H{o}s Problem \\#1049}},\n  \\texttt{erdosproblems.com/1049}. Accessed 28 July 2026, when the page\n  displayed ``last edited 28 September 2025''.\n\\bibitem{bellsmertnig2026} J.~Bell and D.~Smertnig,\n  \\href{https://arxiv.org/abs/2603.23456v1}{\\emph{Mahler series with\n  multiplicative coefficient sequences}}, arXiv:2603.23456v1,\n  24 March 2026.  Theorem~1.3 is on pp.~2--3; its stated consequences on\n  p.~3 include that the divisor and totient generating series are not\n  $k$-Mahler for any $k\\ge2$.\n\\bibitem{dlmf} F.~W.~J. Olver et al.\\ (eds.), \\emph{NIST Digital Library of\n  Mathematical Functions},\n  \\href{https://dlmf.nist.gov/17.2.E37}{\\texttt{dlmf.nist.gov/17.2.E37}},\n  Eq.~17.2.37 in \\S17.2(iii), accessed 15 September 2026.\n\\bibitem{formalconjectures1049} The Formal Conjectures Authors,\n  \\href{https://github.com/google-deepmind/formal-conjectures/blob/f776d2f2039351b00737ffcafb9d7d7666e1d9af/FormalConjectures/ErdosProblems/1049.lean}\n  {\\emph{FormalConjectures.ErdosProblems.\\texttt{1049}}},\n  Lean source at commit \\texttt{f776d2f}, 2026, accessed 28 July 2026.  The\n  irrationality declarations are unproved; the Lambert-series identity is\n  proved.\n\\bibitem{postelmansvanassche2007} K.~Postelmans and W.~Van Assche,\n  \\href{https://arxiv.org/abs/math/0604312v1}{\\emph{Irrationality of\n  $\\zeta_q(1)$ and $\\zeta_q(2)$}}, J. Number Theory \\textbf{126} (2007),\n  no.~1, 119--154,\n  doi:\\href{https://doi.org/10.1016/j.jnt.2006.11.011}{10.1016/j.jnt.2006.11.011}.\n  Page references are to arXiv:math/0604312v1 (2006).\n\\bibitem{kozhanvaktnas2024} R.~Kozhan and M.~Vaktn\\\"{a}s,\n  \\emph{Christoffel transform and multiple orthogonal polynomials},\n  arXiv:\\href{https://arxiv.org/abs/2407.13946v1}{2407.13946v1} (2024).\n  The moment-transform comparison uses the introduction, (3)--(4), of v1.\n\\bibitem{krattenthaler2021} C.~Krattenthaler,\n  \\href{https://arxiv.org/abs/2103.03969v1}{\\emph{A determinant identity for\n  moments of orthogonal polynomials that implies Uvarov's formula for the\n  orthogonal polynomials of rationally related densities}},\n  arXiv:2103.03969v1 (2021). Page references are to this version.\n\\bibitem{wangzhu2016} Y.~Wang and B.-X.~Zhu,\n  \\href{https://arxiv.org/abs/1612.04114v1}{\\emph{Log-convex and Stieltjes\n  moment sequences}}, Adv. Appl. Math. \\textbf{81} (2016), 115--127,\n  doi:\\href{https://doi.org/10.1016/j.aam.2016.06.008}{10.1016/j.aam.2016.06.008}.\n  Page references are to arXiv:1612.04114v1.\n\\bibitem{stanley2016} R.~P. Stanley, \\emph{Smith normal form in combinatorics},\n  J. Combin. Theory Ser. A \\textbf{144} (2016), 476--495,\n  doi:\\href{https://doi.org/10.1016/j.jcta.2016.06.013}{10.1016/j.jcta.2016.06.013};\n  arXiv:\\href{https://arxiv.org/abs/1602.00166v1}{1602.00166v1}.\n  Page references are to arXiv:1602.00166v1.\n\\bibitem{duverney2011} D.~Duverney,\n  \\href{https://doi.org/10.1063/1.3630035}{\\emph{Arithmetical functions and\n  irrationality of Lambert series}}, AIP Conf. Proc. \\textbf{1385} (2011),\n  5--16, doi:10.1063/1.3630035. Theorem and page references are to the\n  12-page author manuscript.\n\\bibitem{berg2007} C.~Berg, \\emph{On powers of Stieltjes moment sequences, II},\n  J. Comput. Appl. Math. \\textbf{199} (2007), 23--38;\n  arXiv:\\href{https://arxiv.org/abs/math/0412340v1}{math/0412340v1}.\n  Theorem references use the preprint; Theorem~5.1 treats factorial powers.\n\\bibitem{sw2024} A.~D. Sokal and J.~Walrad,\n  \\emph{Continued-fraction characterization of Stieltjes moment sequences\n  with support in $[\\xi,\\infty)$},\n  \\href{https://arxiv.org/abs/2404.12131v1}{arXiv:2404.12131v1}, 2024.\n  The classical Stieltjes criterion is recalled on pp.~1--2.\n\\bibitem{lrz2017} H.~Liang, J.~Remmel and S.~Zheng,\n  \\emph{Stieltjes moment sequences of polynomials},\n  \\href{https://arxiv.org/abs/1710.05795v1}{arXiv:1710.05795v1}, 2017.\n\\bibitem{psz2023} M.~P\\'etr\\'eolle, A.~D. Sokal and B.-X.~Zhu,\n  \\emph{Lattice paths and branched continued fractions: An infinite sequence\n  of generalizations of the Stieltjes--Rogers and Thron--Rogers polynomials,\n  with coefficientwise Hankel-total positivity}, Mem. Amer. Math. Soc.\n  \\textbf{291} (2023), no.~1450,\n  doi:\\href{https://doi.org/10.1090/memo/1450}{10.1090/memo/1450};\n  \\href{https://arxiv.org/abs/1807.03271v2}{arXiv:1807.03271v2}.\n  Theorem and page references use that preprint version.\n\\bibitem{middeke2021} J.~Middeke, D.~J. Jeffrey and C.~Koutschan,\n  \\emph{Common Factors in Fraction-Free Matrix Decompositions},\n  Math. Comput. Sci. \\textbf{15} (2021), no.~4, 589--608,\n  doi:\\href{https://doi.org/10.1007/s11786-020-00495-9}{10.1007/s11786-020-00495-9};\n  \\href{https://arxiv.org/abs/2005.12380v1}{arXiv:2005.12380v1}.\n  Section and theorem references use the preprint.\n\\bibitem{golubwelsch1969} G.~H. Golub and J.~H. Welsch,\n  \\emph{Calculation of Gauss Quadrature Rules}, Math. Comp. \\textbf{23}\n  (1969), no.~106, 221--230,\n  doi:\\href{https://doi.org/10.1090/S0025-5718-69-99647-1}{10.1090/S0025-5718-69-99647-1}.\n\\bibitem{vinroot2010} C.~R. Vinroot,\n  \\href{https://arxiv.org/abs/1011.0984}{\\emph{Multivariate Rogers--Szeg\\H{o}\n  polynomials and flags in finite vector spaces}}, arXiv:1011.0984v1,\n  3 November 2010, abstract accessed 20 September 2026.  Cited for the\n  identification of the sum of all $q$-multinomial coefficients of fixed\n  degree and length with a flag count, and for its recursion generalising the\n  Galois numbers.  The factorisation of the moment weights is proved here.\n\\end{thebibliography}\n% ---- part family_catalogue ----\n% ---- part back ----\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":258584,"sha256":"sha256:1f04580702ed8f8cb553b617b264f9654eb525828225c9ea81204f1db65b18bc","cite_as":"Plectis reader corpus, paper:erdos1049-rational-base-lambert-reasoning-surface","source_repository":"plectis-erdos","source_path":"paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:088d36d7ba3d3ee0cee2df99e8915927999698f19889f8ea0b8d55b3721ec778","byte_count":6923,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% This is the only copy of the shared preamble.  The eight short notes, the\n% long records assembled from paper/reasoning-parts other than #249 and #257\n% (which carry their own preambles), and the synthesis paper include it with\n% \\input{problem-note-preamble}; paper/Makefile builds from this directory\n% with the directory on the tectonic search path.\n%\n% Shared preamble for the Erdős Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erdős problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{8e58c395e9807ef3374d7365c62cf04d20ee6331}\n\\newcommand{\\commitshort}{9e231ce4371f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-erdos/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-erdos/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\usepackage{longtable}  % the notes' declaration tables; loaded here so the assembled records inherit it\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n% ---------------------------------------------------------------------------\n% Evidence marks.  paper-evidence.tex places \"Lean\" and \"Comparator\" links in\n% the margin beside each result from the paper's generated file\n% paper/evidence/<paper>.tex, which each paper inputs after this preamble.\n% ---------------------------------------------------------------------------\n\\input{paper-evidence}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Production provenance.  The wording is fixed by the corpus provenance\n% contract and reproduced verbatim in every manuscript in this repository, so\n% the papers cannot drift into different accounts of how they were made.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erdős problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every Lean mark in the margin points to propositions that the\n  pinned Lean kernel accepts, with no \\texttt{sorry}, no added axiom, and no\n  unchecked evaluation.  That claim concerns the formal statements only; it\n  says nothing about their mathematical interest, their novelty, or the\n  original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n\n% Compact opening card for the separately stated Comparator theorem selected\n% from each problem note.  The prose supplied by each note owns the\n% contribution and relevance judgements; the shared final sentence states the\n% mechanical assurance boundary once and points to the public evidence owner.\n\\newcommand{\\externalreviewobject}[3]{%\n  \\begin{quote}\\small\n  \\textbf{Contribution.} #1\\par\n  \\textbf{Relation to the open problem.} #2\\par\n  \\textbf{Executable review object.} #3  The repository's\n  external-verification job compares these exact Lean propositions with\n  separately declared challenge statements and an axiom budget, then asks\n  Lean's kernel to check the submitted proofs.  The\n  \\href{https://github.com/wcook04/plectis-erdos/blob/ca0e13f8acf5ccf48506e4bdb870953d3a0856fa/formalization.yaml}%\n  {formalisation manifest} and the commit-bound CI receipt record that check;\n  they do not assess novelty, significance, or whether the original problem is\n  solved.\n  \\end{quote}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  Under\n% Tectonic (XeTeX) the text and mathematics are set in TeX Gyre Pagella from\n% the bundled OpenType fonts, with a Heros sans range for \\mathsf and\n% Inconsolata for monospace; the pdflatex fallback keeps the same Palatino\n% design through newpx.  The A4 text block uses a classical measure with\n% quiet running heads and restrained ink.  Link colours: steel blue for\n% external (Lean-source and web) targets, deep burgundy for internal\n% cross-references and citations.\n\\NeedsTeXFormat{LaTeX2e}\n\\ProvidesPackage{paper-house-style}[2026/07/19 Pagella house style for the papers]\n\n\\RequirePackage{iftex}\n\\RequirePackage{amsmath}\n\\RequirePackage{amsthm}\n\\ifpdftex\n  \\RequirePackage[T1]{fontenc}\n  \\RequirePackage[utf8]{inputenc}\n  \\RequirePackage{amssymb}\n  \\RequirePackage{newpxtext}\n  \\RequirePackage{newpxmath}\n\\else\n  \\RequirePackage{fontspec}\n  \\RequirePackage{unicode-math}\n  \\setmainfont{texgyrepagella}[Extension=.otf,UprightFont=*-regular,\n    BoldFont=*-bold,ItalicFont=*-italic,BoldItalicFont=*-bolditalic]\n  \\setmathfont{texgyrepagella-math.otf}\n  \\setmathfont{texgyreheros-regular.otf}[range=sfup,Scale=MatchLowercase]\n  \\setmonofont{Inconsolatazi4}[Extension=.otf,UprightFont=*-Regular,\n    BoldFont=*-Bold,Scale=MatchLowercase]\n\\fi\n\n\\RequirePackage[a4paper,textwidth=418pt,textheight=636pt,hcentering,\n  top=68pt,headheight=15pt,headsep=20pt,footskip=34pt]{geometry}\n\\RequirePackage{microtype}\n\\linespread{1.045}\n\\clubpenalty=10000 \\widowpenalty=10000 \\tolerance=1600\n\n\\RequirePackage[dvipsnames,table]{xcolor}\n\\definecolor{linkexternal}{HTML}{275D8C}\n\\definecolor{linkinternal}{HTML}{6E2B2B}\n\\definecolor{papertableheader}{HTML}{E3EAF0}\n\\definecolor{papertablerow}{HTML}{F2F5F7}\n\\definecolor{papertablerule}{HTML}{7D8991}\n\\definecolor{paperresultrule}{HTML}{315B72}\n\n% Stable cross-document links between sibling PDFs. 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It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"},{"record_type":"paper_source_dependency","tex_reference":"evidence/erdos1049-rational-base-lambert-reasoning-surface","source_path":"paper/evidence/erdos1049-rational-base-lambert-reasoning-surface.tex","media_type":"text/x-tex","sha256":"sha256:40598a8029bfe0c8db72229ad40d1ffef53b110185e6238bffaff99aecfb0062","byte_count":28225,"licence":"CC-BY-4.0","body_format":"latex","body":"% Generated by scripts/paper_evidence.py from docs/paper_lean_coverage.json and the\n% Comparator replay receipts under evidence/comparator/.  Do not edit by hand.\n\\newcommand{\\evidencerecordurl}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md}\n\\DeclareResultEvidenceName{long1049:res:omega-indicator}{Lemma 2.1}\n\\DeclareResultEvidenceRecord{long1049:res:omega-indicator}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-omega-indicator}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:omega-indicator}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperOmegaIndicatorR7.lean\\#L1297}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-omega-indicator-comparator}\n\\DeclareResultEvidenceName{long1049:res:region}{Theorem 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2.4}\n\\DeclareResultEvidenceRecord{long1049:cor:rational-base-measure}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-cor-rational-base-measure}{exact}{compared}\n\\DeclareResultEvidence{long1049:cor:rational-base-measure}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-cor-rational-base-measure}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-cor-rational-base-measure-comparator}\n\\DeclareResultEvidenceName{long1049:res:archcap}{Theorem 2.5}\n\\DeclareResultEvidenceRecord{long1049:res:archcap}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-archcap}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long1049:res:archcap}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-archcap}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-archcap-comparator}\n\\DeclareResultEvidenceName{long1049:cor:no-decay-below-square}{Corollary 2.6}\n\\DeclareResultEvidenceRecord{long1049:cor:no-decay-below-square}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-cor-no-decay-below-square}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long1049:cor:no-decay-below-square}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperNoDecayR9.lean\\#L69}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-cor-no-decay-below-square-comparator}\n\\DeclareResultEvidenceName{long1049:res:sourceheight}{Lemma 2.7}\n\\DeclareResultEvidenceRecord{long1049:res:sourceheight}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-sourceheight}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:sourceheight}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-sourceheight}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-sourceheight-comparator}\n\\DeclareResultEvidenceName{long1049:res:powerbracket}{Theorem 3.1}\n\\DeclareResultEvidenceRecord{long1049:res:powerbracket}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-powerbracket}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:powerbracket}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperFiniteAssembliesR7.lean\\#L24}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-powerbracket-comparator}\n\\DeclareResultEvidenceName{long1049:res:sharpgaps}{Corollary 3.2}\n\\DeclareResultEvidenceRecord{long1049:res:sharpgaps}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-sharpgaps}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:sharpgaps}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperFiniteAssembliesR7.lean\\#L34}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-sharpgaps-comparator}\n\\DeclareResultEvidenceName{long1049:res:chargeceilings}{Theorem 3.3}\n\\DeclareResultEvidenceRecord{long1049:res:chargeceilings}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-chargeceilings}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:chargeceilings}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperFiniteAssembliesR7.lean\\#L44}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-chargeceilings-comparator}\n\\DeclareResultEvidenceName{long1049:res:zudilin-sharp-qorder}{Theorem 3.4}\n\\DeclareResultEvidenceRecord{long1049:res:zudilin-sharp-qorder}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-zudilin-sharp-qorder}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:zudilin-sharp-qorder}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-zudilin-sharp-qorder}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-zudilin-sharp-qorder-comparator}\n\\DeclareResultEvidenceName{long1049:res:allrowinitial}{Lemma 3.5}\n\\DeclareResultEvidenceRecord{long1049:res:allrowinitial}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-allrowinitial}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:allrowinitial}{Lean}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-allrowinitial}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-allrowinitial-comparator}\n\\DeclareResultEvidenceName{long1049:thm:geometric-universality}{Theorem 3.6}\n\\DeclareResultEvidenceRecord{long1049:thm:geometric-universality}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-thm-geometric-universality}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long1049:thm:geometric-universality}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperCompleteR21/GeometricUniversality.lean\\#L1324}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-thm-geometric-universality-comparator}\n\\DeclareResultEvidenceName{long1049:prop:rogers-factorisation}{Proposition 3.7}\n\\DeclareResultEvidenceRecord{long1049:prop:rogers-factorisation}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-prop-rogers-factorisation}{exact}{compared}\n\\DeclareResultEvidence{long1049:prop:rogers-factorisation}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperCompleteR21/RogersFactorisationAnalytic.lean\\#L1312}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-prop-rogers-factorisation-comparator}\n\\DeclareResultEvidenceName{long1049:thm:sharp-fixed-base}{Theorem 3.8}\n\\DeclareResultEvidenceRecord{long1049:thm:sharp-fixed-base}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-thm-sharp-fixed-base}{exact}{compared}\n\\DeclareResultEvidence{long1049:thm:sharp-fixed-base}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperCompleteR21/SharpFixedBase.lean\\#L990}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-thm-sharp-fixed-base-comparator}\n\\DeclareResultEvidenceName{long1049:thm:geometric-ratio}{Remark 1, p. 35}\n\\DeclareResultEvidenceRecord{long1049:thm:geometric-ratio}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-thm-geometric-ratio}{none}{not_applicable}\n\\DeclareResultEvidenceName{long1049:thm:calibrated-model}{Remark 2, p. 38}\n\\DeclareResultEvidenceRecord{long1049:thm:calibrated-model}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-thm-calibrated-model}{none}{not_applicable}\n\\DeclareResultEvidenceName{long1049:thm:calibrated-denominators}{Remark 3, p. 39}\n\\DeclareResultEvidenceRecord{long1049:thm:calibrated-denominators}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-thm-calibrated-denominators}{none}{not_applicable}\n\\DeclareResultEvidenceName{long1049:prop:calibrated-height}{Remark 4, p. 40}\n\\DeclareResultEvidenceRecord{long1049:prop:calibrated-height}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-prop-calibrated-height}{none}{not_applicable}\n\\DeclareResultEvidenceName{long1049:res:finite-pencil}{Proposition 3.9}\n\\DeclareResultEvidenceRecord{long1049:res:finite-pencil}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-finite-pencil}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long1049:res:finite-pencil}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperR20/FinitePencilProposition.lean\\#L43}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-finite-pencil-comparator}\n\\DeclareResultEvidenceName{long1049:res:content}{Theorem 4.1}\n\\DeclareResultEvidenceRecord{long1049:res:content}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-content}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:content}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperFiniteAssembliesR7.lean\\#L59}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-content-comparator}\n\\DeclareResultEvidenceName{long1049:res:endpoints}{Theorem 5.1}\n\\DeclareResultEvidenceRecord{long1049:res:endpoints}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-endpoints}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:endpoints}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperFiniteAssembliesR7.lean\\#L74}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-endpoints-comparator}\n\\DeclareResultEvidenceName{long1049:res:commonmult}{Proposition 5.3}\n\\DeclareResultEvidenceRecord{long1049:res:commonmult}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-commonmult}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:commonmult}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean\\#L398}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-commonmult-comparator}\n\\DeclareResultEvidenceName{long1049:res:nomult}{Corollary 5.5}\n\\DeclareResultEvidenceRecord{long1049:res:nomult}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-nomult}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:nomult}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperFiniteAssembliesR7.lean\\#L220}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-nomult-comparator}\n\\DeclareResultEvidenceName{long1049:res:cyclounit}{Proposition 5.6}\n\\DeclareResultEvidenceRecord{long1049:res:cyclounit}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-cyclounit}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:cyclounit}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean\\#L302}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-cyclounit-comparator}\n\\DeclareResultEvidenceName{res:bottomjet}{Lemma 5.7}\n\\DeclareResultEvidenceRecord{res:bottomjet}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#res-bottomjet}{exact}{compared}\n\\DeclareResultEvidence{res:bottomjet}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean\\#L191}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#res-bottomjet-comparator}\n\\DeclareResultEvidenceName{long1049:res:jetkernel}{Theorem 5.8}\n\\DeclareResultEvidenceRecord{long1049:res:jetkernel}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-jetkernel}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:jetkernel}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperFiniteAssembliesR7.lean\\#L248}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-jetkernel-comparator}\n\\DeclareResultEvidenceName{long1049:res:rankfortyone}{Corollary 5.9}\n\\DeclareResultEvidenceRecord{long1049:res:rankfortyone}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-rankfortyone}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:rankfortyone}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperFiniteAssembliesR7.lean\\#L149}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-rankfortyone-comparator}\n\\DeclareResultEvidenceName{long1049:res:boundedfibre}{Theorem 5.10}\n\\DeclareResultEvidenceRecord{long1049:res:boundedfibre}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-boundedfibre}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:boundedfibre}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/AdelicHeightBridge.lean\\#L1832}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-boundedfibre-comparator}\n\\DeclareResultEvidenceName{res:boundedfibre}{Theorem 5.11}\n\\DeclareResultEvidenceRecord{res:boundedfibre}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#res-boundedfibre}{exact}{compared}\n\\DeclareResultEvidence{res:boundedfibre}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/QuantitativeSelectorEscape.lean\\#L79}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#res-boundedfibre-comparator}\n\\DeclareResultEvidenceName{long1049:res:plucker-collapse}{Theorem 5.12}\n\\DeclareResultEvidenceRecord{long1049:res:plucker-collapse}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-plucker-collapse}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:plucker-collapse}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperFiniteAssembliesR7.lean\\#L269}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-plucker-collapse-comparator}\n\\DeclareResultEvidenceName{long1049:res:scalar}{Theorem 5.14}\n\\DeclareResultEvidenceRecord{long1049:res:scalar}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-scalar}{exact_or_stronger}{compared}\n\\DeclareResultEvidence{long1049:res:scalar}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperFiniteAssembliesR7.lean\\#L177}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-scalar-comparator}\n\\DeclareResultEvidenceName{long1049:res:corridorbound}{Theorem 6.2}\n\\DeclareResultEvidenceRecord{long1049:res:corridorbound}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-corridorbound}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:corridorbound}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean\\#L121}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-corridorbound-comparator}\n\\DeclareResultEvidenceName{long1049:res:exp}{Proposition 6.4}\n\\DeclareResultEvidenceRecord{long1049:res:exp}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-exp}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:exp}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean\\#L142}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-exp-comparator}\n\\DeclareResultEvidenceName{long1049:res:nocorridor}{Theorem 6.5}\n\\DeclareResultEvidenceRecord{long1049:res:nocorridor}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-nocorridor}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:nocorridor}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean\\#L155}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-nocorridor-comparator}\n\\DeclareResultEvidenceName{long1049:res:tailrec}{Theorem 7.1}\n\\DeclareResultEvidenceRecord{long1049:res:tailrec}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-tailrec}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:tailrec}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean\\#L187}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-tailrec-comparator}\n\\DeclareResultEvidenceName{long1049:res:forcing}{Theorem 7.2}\n\\DeclareResultEvidenceRecord{long1049:res:forcing}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-forcing}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:forcing}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperFiniteAssembliesR7.lean\\#L186}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-forcing-comparator}\n\\DeclareResultEvidenceName{long1049:res:sevenhalves}{Theorem 8.1}\n\\DeclareResultEvidenceRecord{long1049:res:sevenhalves}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-sevenhalves}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:sevenhalves}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean\\#L83}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-sevenhalves-comparator}\n\\DeclareResultEvidenceName{long1049:res:pade}{Proposition 9.1}\n\\DeclareResultEvidenceRecord{long1049:res:pade}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-pade}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:pade}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperFiniteAssembliesR7.lean\\#L201}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-pade-comparator}\n\\DeclareResultEvidenceName{long1049:res:tail-lattice}{Proposition 10.3}\n\\DeclareResultEvidenceRecord{long1049:res:tail-lattice}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-tail-lattice}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:tail-lattice}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperCompleteR21/TailPrefixLattice.lean\\#L475}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-tail-lattice-comparator}\n\\DeclareResultEvidenceName{long1049:res:nomahler}{Proposition 10.5}\n\\DeclareResultEvidenceRecord{long1049:res:nomahler}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-nomahler}{exact}{compared}\n\\DeclareResultEvidence{long1049:res:nomahler}{Lean}{https://github.com/wcook04/plectis-erdos/blob/436f55ebdafa67e4af0fff79f621c13f2ded12bf/lean/ErdosProblems/Erdos1049/PaperCompleteR21/SimultaneousMahlerSystemUnconditional.lean\\#L298}{https://github.com/wcook04/plectis-erdos/blob/846d57d3f9926696332d782eb232aaf3cf803a99/evidence/erdos1049-rational-base-lambert-reasoning-surface.md\\#long1049-res-nomahler-comparator}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","section_outline":["Introduction","The polynomial forms and their common factors","Exact degrees","The cyclotomic limit","Positive linear forms with integer coefficients","Earlier criteria and changes of direction","Finite calculations","Power comparisons and Hankel determinants","An extension with limiting weight ratio","Positive Hankel determinants and a rational special value","Coefficient moments and cyclotomic content","Rescaling integer rows","Successive scaled remainders","A denominator-exponent model for Pad\\'e-type forms","Complements and further questions","A congruence relation with nonzero remainder","Estimating the divided remainder","A restriction on Mahler functional equations","A limitation of the rectangular exponent model","Related work and comparison of constructions","Statements and declarations","Verification and reproducibility"]},"optimal-sparse-perturbations":{"paper_id":"optimal-sparse-perturbations","title":"Reading Eight Erdős Problems Together","form":"Cross-problem paper","length_class":"cross_problem","pages":38,"subject":"Eight Erdős problems read together","subject_kind":"synthesis","problem_id":null,"landing_role":"mathematics","question":"When do sparse integer digits fill intervals of values, when do they force irrationality or transcendence, and what does that show about the methods used across the Erdős problems?","precis":"One account of the mathematics developed across the programmes: an exact tail-capacity criterion under eventual congruences, its sharp factorial support-gap threshold, rational and irrational Lambert subsums across bases, and the limits of finite greedy tests and other irrationality methods. Full arguments, exact counterexamples, unsuccessful routes and attribution are retained together. The principal proofs are ordinary mathematics; cited Lean ingredients have their own stated scope. Historical novelty and independent expert review are not established.","owns":"the synthesis exposition and ordinary proofs across the covered problems: capacity and congruence constructions, Lambert subsums, rational-point counts, method limits and their research record","not_authority_for":"a solution to any original Erdős target, historical novelty, independent expert review, or a full Lean proof of the analytic capacity criterion or Lambert-chain theorem","first_pass":["sec:map","capacity:sec:criterion","sec:results","sec:questions"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/optimal-sparse-perturbations.pdf","hosted_pdf_path":"papers/optimal-sparse-perturbations.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/synthesis/optimal-sparse-perturbations.tex","site_html":"maths/papers/optimal-sparse-perturbations.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/synthesis/optimal-sparse-perturbations.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/synthesis/optimal-sparse-perturbations.tex","body_pointer":"papers.records.optimal-sparse-perturbations.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated cross-problem paper. Build with `make -C paper optimal-sparse-perturbations.pdf`.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n\\usepackage{xurl}\n\\hypersetup{\n  pdftitle={Reading eight Erdős problems together},\n  pdfsubject={Derivative interpolation, congruences and limits of irrationality methods},\n  pdfkeywords={irrationality, Lambert series, subsum sets, achievement sets, Mahler's method, Farey fractions},\n}\n\n% Immutable source revision for the formal ingredients cited in this paper.\n\\renewcommand{\\commit}{8d6596fbde2c4aacf946adec4b0226ba1a97545d}\n\\renewcommand{\\commitshort}{8d6596fbde2c}\n\n\\runninghead{Reading eight Erd\\H{o}s problems together}\n\\title{Reading Eight Erd\\H{o}s Problems Together}\n\\subtitle{Subsum sets, congruences and limits of irrationality methods}\n\\author{Will Cook}\n\\date{22 September 2026}\n\n\\begin{document}\n\\maketitle\n\\noteprovenance\n\\begin{abstract}\nLet $c>0$ and let $d\\ge1$ be an integer. For entire functions\n$f(z)=\\sum_{n\\ge1}e_nz^n/n!$ with nonnegative integer\ncoefficients $e_n\\le n^c$ eventually, each eventually divisible by every fixed\ninteger, the vectors $(f(1),\\ldots,f^{(d-1)}(1))$ have Hausdorff dimension\n$\\min(c,d)$. They contain an open set exactly when $c>d$; at $c=d$ they are\nfull-dimensional, null and meagre. A nonpolynomial such function with all\nthese derivatives rational exists exactly when $c>d$. Multiplication and\ndivision by powers of $z-1$ give the matching construction and obstruction.\nWe also study series values under restrictions on supports and congruences,\nusing questions from seven irrationality problems of Erd\\H{o}s. Let $Q_n$ be\npositive integers with $Q_n\\mid Q_{n+1}$ and $Q_{n+1}\\ge2Q_n$, and let integer\nallowances $F_n\\ge0$ satisfy $\\sum F_n/Q_n<\\infty$. Nonnegative integer digits\nbounded by $F_n$ eventually and eventually divisible by each fixed integer\nrepresent every nonnegative real exactly when\n$Q_N\\sum_{n>N}F_n/Q_n\\to\\infty$; otherwise their values form a null meagre set.\nThe positive construction also imposes eventual divisibility of cumulative\ndigit sums. For factorial denominators and allowances $n^c$, with $c>0$,\non a fixed infinite support, interval filling requires successive gaps to be eventually\nstrictly smaller than $c$.\nThe number-field Subspace Theorem gives transcendence at every real\nalgebraic base greater than one for bounded positive integer weighted\nLambert sums on infinite supports admitting separated divisibility cuts.\nThis includes every divisibility chain and explicit hosts of unbounded\ndivisibility width and divergent reciprocal sum. In contrast, every rational base between one and two admits\nrational values on infinite supports. At base two, a fixed-depth rational count and an\nexact late rejection separate finite survival from membership. Full proofs,\nmethod obstructions and the unsuccessful approaches are included. We also\nclassify the shift families that detect irrationality in every integer-digit\ndyadic recurrence: at every threshold $0<c<\\tau$, precisely those containing\na multiple of each positive integer suffice, where $\\tau$ is the Thue--Morse\nconstant in a cited sharp bound. Factorial shifts suffice; power-of-two\nshifts do not. For polynomially selected shifts, the condition becomes a\nroot modulo every integer, or a unit root modulo every integer when the\npolynomial argument must be prime. The main\narguments are ordinary proofs, with formal ingredients and external results\nidentified separately; no original Erd\\H{o}s problem is settled here.\n\\end{abstract}\n\n\\section{From individual problems to reusable questions}\\label{sec:map}\nSeven of the eight programmes ask whether a series is irrational:\n$\\sum(n!-1)^{-1}$ in \\#68, reciprocal sums of near-Sylvester sequences in\n\\#243, $\\sum\\varphi(n)2^{-n}$ in \\#249, $\\sum p_n2^{-n}$ in \\#251,\n$\\sum_{n\\in S}(2^n-1)^{-1}$ in \\#257, reciprocal running least common\nmultiples in \\#269, and $\\sum(t^n-1)^{-1}$ at rational $t$ in \\#1049\n\\cite{erdosgraham1980,erdos1988}. The eighth, \\#1041, concerns polynomial\nlemniscates. It supplies no premise for the series arguments below.\n\nWhat can the freedom to choose digits preserve, and when does arithmetic\nremove that freedom? Sparse corrections in the \\#251 paper motivate the\ncapacity criterion of Section~\\ref{capacity:sec:criterion}. Its proof makes\none interval of possible continuations work for every cumulative residue.\nThe factorial examples show that the arithmetic restrictions change a sharp\nsupport-gap threshold. Factorial denominators $n!$ in this construction are\nnot the denominators $n!-1$ in \\#68.\n\nSection~\\ref{capacity:sec:jets-intro} asks whether a single factorial digit\nsequence can prescribe several derivatives independently. Theorems\n\\ref{capacity:thm:jets} and~\\ref{capacity:thm:rational-jets} give the sharp\nthreshold and the dimension of the attainable vectors. The proof uses the\nscalar capacity theorem first, then a carry that preserves lower derivatives;\nSections~\\ref{capacity:sec:jets-proof} and~\\ref{capacity:sec:rational-jets}\ngive the argument and the rationality obstruction.\n\nThe common Lambert series in \\#257 and \\#1049 gives a second comparison in\nSection~\\ref{sec:results}: a base below two permits rational subsums, while\non every support with separated divisibility cuts the sum is transcendental\nat every real algebraic base greater than one, even with bounded positive\ninteger weights. Section~\\ref{sec:separated-cuts} gives the proof and a\nnon-chain host. Neither\ninterval filling nor a measure estimate decides whether one specified\nrational is a subsum at base two. Sections~\\ref{sec:base2} and\n\\ref{sec:misread} explain that obstruction, including the exact computation\nthat disproved a proposed stopping rule. Section~\\ref{sec:second} retains the\nother method limits without claiming that they have one common cause.\n\nSection~\\ref{sec:dyadic-shifts} follows a different transfer. A difference of\ntwo dyadic tail states is itself an integer-digit dyadic orbit. The resulting\nirrationality criterion leads to a question about which shift lengths need\ntesting; the answer depends on divisibility rather than the size or density\nof the chosen family.\n\n\\section{Prescribing a value and its derivatives}\\label{capacity:sec:jets-intro}\nAn integer factorial expansion can represent a real number while its digits\neventually vanish modulo every fixed integer. Can the same digits prescribe\nseveral real quantities independently? Consider their exponential generating\nfunction. Its derivatives at $1$ are different weighted sums of the same\ndigits, so this is simultaneous prescription, not several independent expansions.\n\nFor $c>0$, let $\\mathcal H_c$ consist of the entire functions\n\\[\n \\begin{gathered}\n f(z)=\\sum_{n\\ge1}e_n\\frac{z^n}{n!},\\qquad e_n\\in\\Z_{\\ge0},\\\\\n e_n\\le n^c\\ \\hbox{eventually},\\qquad\n (\\forall q\\ge1)\\ q\\mid e_n\\ \\hbox{eventually}.\n \\end{gathered}\n\\]\nPolynomial coefficient growth implies locally uniform convergence of every\nderivative. Put $J_d(f)=(f(1),f'(1),\\ldots,f^{(d-1)}(1))$ and\n$\\mathcal J_{d,c}=\\{J_d(f):f\\in\\mathcal H_c\\}\\subseteq\\R^d$.\nThese are Hurwitz functions: all derivatives at $0$ are integers. Our\nrestrictions concern those integers; the prescribed derivatives are at $1$.\n\n\\begin{theorem}[Dimension and the sharp interpolation threshold]\n\\label{capacity:thm:jets}\nFor every $c>0$ and integer $d\\ge1$,\n\\[\n \\dim_{\\mathrm H}\\mathcal J_{d,c}=\\min(c,d).\n\\]\nIf $c>d$, this set contains a nonempty open subset of $\\R^d$.\nIf $c\\le d$, it has $d$-dimensional Lebesgue measure zero and is meagre;\nthese conclusions already follow from eventual evenness of the coefficients.\nThe dimension lower bound and the open-set conclusion can both be realised\nwith $e_n\\le n^c$ at every position, an arbitrarily long initial zero segment,\nand congruence cutoffs common to the constructed family. The open set can\nbe realised using only functions with eventually positive coefficients.\n\\end{theorem}\n\n\\begin{theorem}[Rational derivatives force polynomiality]\n\\label{capacity:thm:rational-jets}\nLet $d\\ge1$, and let\n\\[\n f(z)=\\sum_{n\\ge0}a_n\\frac{z^n}{n!},\\qquad a_n\\in\\Z,\\quad |a_n|=O(n^d).\n\\]\nAssume that, for each $q\\ge1$, $q\\mid a_n$ eventually.\nIf $f(1),f'(1),\\ldots,f^{(d-1)}(1)$ are all rational, then $f$ is a polynomial.\nConsequently, a nonpolynomial member of $\\mathcal H_c$ with all these\nderivatives rational exists if and only if $c>d$.\n\\end{theorem}\n\nThus if a nonpolynomial $f\\in\\mathcal H_2$ has rational $f(1)$, its derivative\n$f'(1)$ is irrational. Increasing the allowance to $n^{2+\\varepsilon}$\npermits both to be rational, for every $\\varepsilon>0$. At $c=2$, the\nattainable pairs nevertheless form a full-dimensional null set.\nWe do not assert that every vector is attained:\nnonnegative coefficients impose inequalities such as $f'(1)\\ge f(1)$.\n\nThe identity behind the construction is\n\\[\n (z-1)\\sum_{n\\ge0}b_n\\frac{z^n}{n!}\n =-b_0+\\sum_{n\\ge1}(n b_{n-1}-b_n)\\frac{z^n}{n!}.\n \\tag{J}\\label{capacity:eq:jet-carry}\n\\]\nAdding the left-hand side to a function leaves its value at $1$ unchanged\nand changes its derivative there by\n$\\sum b_n/n!$. Multiplication by $(z-1)^k$ leaves the first $k$ entries of\nthe derivative vector unchanged, at a cost of $k$ powers of $n$ in the\ncoefficient allowance. The scalar construction of\nTheorem~\\ref{capacity:thm:capacity} supplies the free quantity.\nFor necessity, eventual evenness reduces the number of prefixes by an\nexponential factor, which forces measure zero even at $c=d$.\nSection~\\ref{capacity:sec:jets-proof} proves the theorem, including its\ndimension statement at the critical exponent. In the reverse direction,\ndivision by $(z-1)^d$ converts vanishing derivatives into bounded integer\ncoefficients. Eventual divisibility then forces them to vanish; this proves\nTheorem~\\ref{capacity:thm:rational-jets} in\nSection~\\ref{capacity:sec:rational-jets}.\n\n\n\\section{An exact capacity criterion}\\label{capacity:sec:criterion}\nA factorial expansion usually permits a digit of size about $n$ at position\n$n$. If more digits are allowed, can most positions be discarded? What changes\nif each fixed integer must eventually divide every digit?\n\nLet $Q_n$ be positive integers, $n\\ge1$, such that $Q_n\\mid Q_{n+1}$ and\n$Q_{n+1}\\ge2Q_n$. Let $F_n\\in\\Z_{\\ge0}$ satisfy\n$\\sum_nF_n/Q_n<\\infty$, and put\n\\[\n U_N=Q_N\\sum_{n>N}\\frac{F_n}{Q_n}.\n \\tag{C}\\label{capacity:eq:capacity}\n\\]\nDefine $E_Q(F)$ to be the set of sums $\\sum_{n\\ge1}e_n/Q_n$, allowing\nnonnegative integer digits $e_n\\le F_n$ eventually and requiring, for each\nfixed $q\\ge1$, that $q\\mid e_n$ eventually. The cutoffs and finite initial\ndigits may depend on the represented value.\n\n\\begin{theorem}\\label{capacity:thm:capacity}\nIf $U_N\\to\\infty$, then $E_Q(F)=[0,\\infty)$; otherwise $E_Q(F)$ is null\nand meagre. In particular interval filling is equivalent to $U_N\\to\\infty$.\nIn the positive case, every nonnegative target may be represented with\n$e_n=0$ before any prescribed cutoff, at most one exception to $e_n\\le F_n$,\nand\n\\[\n q\\mid e_n,\\qquad q\\mid\\sum_{k<n}e_k\n \\quad\\text{for all sufficiently large }n,\n\\]\nwith congruence cutoffs depending on $q$ but independent of the target.\nIf the allowance must hold at every position, a nondegenerate interval is\nstill represented with both congruences and a common cutoff for each $q$.\n\\end{theorem}\nThe criterion compares \\emph{all} future capacity with the prefix lattice.\nIt requires no monotonicity, regular variation or gap bound on the allowances,\nand no upper bound on the denominator ratios. It does not require the $Q_n$\nto clear every rational denominator. Section~\\ref{capacity:sec:capacity-proof} proves\nthe theorem using the same residue-feedback mechanism already present in\nthe public Lean repository. The capacity estimates and complete criterion\nare ordinary proofs; the abstract feedback endpoint is kernel-checked\nseparately. Both geometric and factorial denominators satisfy the hypotheses.\n\n\\subsection{The sharp support threshold}\nLet $S\\subseteq\\N_{\\!>0}$ and $c>0$. Define $E(S,c)$ to be the set of sums\n\\[\n x=\\sum_{n\\ge1}\\frac{e_n}{n!},\\qquad e_n\\in\\Z_{\\ge0},\\quad\n e_n=0\\ (n\\notin S),\\quad e_n\\le n^c\\text{ eventually},\n \\tag{1}\\label{capacity:eq:class}\n\\]\nsubject to\n\\[\n \\text{for every }q\\ge1,\\qquad q\\mid e_n\\text{ for all sufficiently large }n.\n \\tag{2}\\label{capacity:eq:congruence}\n\\]\nThe cutoffs and the finite initial digits in this definition may depend on $x$.\nAll series converge, since finitely many unrestricted digits do not affect\nconvergence. A permitted position need not carry a nonzero digit.\n\n\\begin{theorem}\\label{capacity:thm:main}\nIf $S$ is finite, $E(S,c)$ is countable. If $S$ is infinite, enumerate it as\n$n_0<n_1<\\cdots$.\n\\begin{enumerate}\n\\item For $0<c\\le1$, $E(S,c)$ is null and meagre.\n\\item For $c>1$, $E(S,c)$ contains a nondegenerate interval if and only if\n\\[\n n_j-n_{j-1}<c\\qquad\\text{for all sufficiently large }j.\n \\tag{3}\\label{capacity:eq:gaps}\n\\]\nIf this condition fails, $E(S,c)$ is null and meagre.\n\\end{enumerate}\nWhen \\eqref{capacity:eq:gaps} holds, one may require $e_n=0$ before any prescribed\ncutoff and impose both\n\\[\n q\\mid e_n,\\qquad q\\mid\\sum_{k<n}e_k\n \\tag{4}\\label{capacity:eq:both}\n\\]\nfor all sufficiently large $n$, with cutoffs depending on $q$ but independent\nof $x$ throughout the constructed interval.\n\\end{theorem}\nThe theorem is an ordinary mathematical proof. The associated formal sources\ncheck the general digit-feedback construction and, separately, the\ncommon-divisor carry obstruction of Section~\\ref{capacity:sec:carry}; they do not\nformalise this gap classification or its measure argument. The source and\nattribution account is in Section~\\ref{capacity:sec:sources}.\n\nFor example, with allowance $n^2$, every second position suffices without\ncongruences. Under \\eqref{capacity:eq:congruence}, infinitely many omitted positions\nalready force a null set. With allowance $n^{2.01}$, every second position\nagain suffices, even under \\eqref{capacity:eq:both}. Thus the strict inequality in\n\\eqref{capacity:eq:gaps} is essential.\n\n\\begin{corollary}\\label{capacity:cor:density}\nFor $c>1$, the least possible asymptotic density of a fixed permitted support\nthat fills an interval under \\eqref{capacity:eq:congruence} is\n$1/(\\lceil c\\rceil-1)$. Every such support has lower density at least this\nvalue, and an arithmetic progression attains it. A density bound alone is\nnot sufficient: even rare gaps of length $\\lceil c\\rceil$ prevent interval\nfilling if they occur infinitely often.\n\\end{corollary}\n\\begin{proof}\nPut $d=\\lceil c\\rceil-1$. Eventual gaps at most $d$ give\n$\\liminf_{N\\to\\infty}|S\\cap[1,N]|/N\\ge1/d$. An arithmetic progression of\nstep $d$ satisfies the theorem.\n\\end{proof}\n\n\\subsection{Proof of the capacity criterion}\\label{capacity:sec:capacity-proof}\nThe necessity and sufficiency use different parts of the expansion. Prefixes\nsupply an arithmetic obstruction; a common interval of possible future\nremainders makes the residue choices compatible with exact representation.\n\n\\subsubsection{Bounded capacity along a subsequence}\nIf $U_N$ does not tend to infinity, there are a constant $B>0$ and arbitrarily\nlarge $N$ with $U_N\\le B$. Choose one integer $q>B$. Fix a cutoff $K$ after\nwhich the allowance and divisibility by $q$ hold, and fix the finite prefix,\nof value $y_0$. The set $V$ of resulting sums is compact by summability and\nthe finite choices at each remaining position. For every such $N>K$,\n\\[\n V\\subseteq y_0+\\frac q{Q_N}\\Z+[0,B/Q_N].\n\\]\nIndeed the scaled prefix after $K$ is a multiple of $q$ and the remaining\ntail is at most $U_N/Q_N$. These grids leave holes inside every fixed open\ninterval for all sufficiently large selected $N$, so $V$ is nowhere dense.\nFor a bounded interval $J$ they also give\n$|V\\cap J|\\le(B/q)|J|+2B/Q_N$.\nPassing to the subsequence and covering $V$ by finitely many intervals with\ntotal length at most $|V|+\\varepsilon$ gives\n$|V|\\le(B/q)(|V|+\\varepsilon)$, hence $|V|=0$.\nA countable union over the cutoffs and integer prefixes covers $E_Q(F)$.\n\n\\subsubsection{One continuation interval for every cumulative residue}\nSuppose $U_N\\to\\infty$ and put $h_N=\\inf_{k\\ge N}U_k$. For sufficiently\nlarge $n$, let $M_n$ be the largest factorial at most\n$\\min\\{n,\\sqrt{h_{\\lfloor n/2\\rfloor}}\\}$, and set the finitely many\nearlier moduli equal to one. These positive integers are nested under\ndivisibility, tend to infinity, and are eventually divisible by each fixed\ninteger. The delayed lower envelope controls the whole future cost of\nrounding, even when the denominator ratios are unbounded. Indeed,\n$Q_N/Q_k\\le2^{N-k}$, so for sufficiently large $N$,\n\\[\n \\begin{aligned}\n C_N:=Q_N\\sum_{k>N}\\frac{M_k}{Q_k}\n &\\le\\sqrt{h_N}\\sum_{k=N+1}^{2N}2^{N-k}\n       +\\sum_{k>2N}k\\,2^{N-k}\\\\\n &\\le\\sqrt{h_N}+(2N+2)2^{-N}.\n \\end{aligned}\n\\]\nThe first bound uses $\\lfloor k/2\\rfloor\\le N$ when $k\\le2N$; the second\nuses $M_k\\le k$. Since $U_N\\ge h_N\\to\\infty$ and\n$M_N\\le\\sqrt{h_N}$, this proves $U_N-2C_N\\ge M_N$ for every sufficiently\nlarge $N$. Start beyond this point and any prescribed cutoff.\n\nKeep only the active positions $S=\\{n:F_n\\ge2M_n\\}$ after this cutoff.\nDefine two common tail bounds\n\\[\n \\alpha_N=\\sum_{\\substack{n>N\\\\n\\in S}}\\frac{M_n}{Q_n},\\qquad\n \\beta_N=\\sum_{\\substack{n>N\\\\n\\in S}}\\frac{F_n-M_n}{Q_n}.\n\\]\nBoth are nonnegative and tend to zero. Discarding an inactive position loses\nless than $2M_n$; reserving both margins at an active position loses exactly\n$2M_n$. Thus the preceding estimate gives the decisive overlap inequality\n\\[\n Q_N(\\beta_N-\\alpha_N)\\ge U_N-2C_N\\ge M_N.\n \\tag{O}\\label{capacity:eq:overlap}\n\\]\nIn particular the active set is infinite and each continuation interval\nhas positive width.\n\nChoose a target in $[\\alpha_{N_0},\\beta_{N_0}]$ and set all preceding digits\nto zero. Suppose the current unweighted cumulative sum is $C$ and the\nremaining target before position $n$ lies in\n$[\\alpha_{n-1},\\beta_{n-1}]$. At an inactive position choose zero.\nAt an active position, the permitted digits in the residue class\n$e\\equiv-C\\pmod{M_n}$, $0\\le e\\le F_n$, form a nonempty arithmetic\nprogression of step $M_n$. Its least digit is less than $M_n$ and its\ngreatest digit exceeds $F_n-M_n$. By (O), the intervals\n\\[\n \\frac e{Q_n}+[\\alpha_n,\\beta_n]\n\\]\nfor these digits overlap. Their union contains\n\\[\n \\left[\\frac{M_n}{Q_n}+\\alpha_n,\n       \\frac{F_n-M_n}{Q_n}+\\beta_n\\right]\n   =[\\alpha_{n-1},\\beta_{n-1}].\n\\]\nChoose a digit whose interval contains the remaining target. The next\nremainder stays between $\\alpha_n$ and $\\beta_n$, and these bounds tend to\nzero, proving exact representation. This is precisely the common-interval\nfeedback construction; no target-independent ordinary block sum is required.\n\nAfter an active position $n$, the cumulative sum is divisible by $M_n$.\nEach later active digit is divisible by the previous active modulus,\nbecause both adjacent cumulative sums are; the moduli are nested.\nFor a fixed $q$, take one active position with $q\\mid M_n$. After that\nposition, all individual digits and all preceding cumulative sums are\ndivisible by $q$. This cutoff is common to all targets in the interval.\n\n\\subsubsection{From one interval to every nonnegative target}\nThe finite-exception convention in $E_Q(F)$ strengthens the conclusion.\nGiven $y>0$, choose $N$ arbitrarily late with $\\alpha_N<y$ and put\n\\[\n p=M_N\\left\\lfloor\\frac{Q_N(y-\\alpha_N)}{M_N}\\right\\rfloor.\n\\]\nThen $p\\ge0$, $M_N\\mid p$, and (O) gives\n$y-p/Q_N\\in[\\alpha_N,\\beta_N]$. Set $e_N=p$ and all earlier digits to\nzero. Run the same continuation construction after $N$, with initial\ncumulative sum $C=p$; its interval covering worked for every $C$.\nAll later digits obey their allowances. The zero target uses zero digits.\n\nThe congruence cutoffs can also be common to all $y$. For fixed $q$, choose\nan active position $j$ with $q\\mid M_j$. If the target's exceptional index\n$N$ precedes $j$, the repair at $j$ gives the required divisibilities after\n$j$. If $N\\ge j$, all preceding digits are zero, $q\\mid M_N\\mid p$, and\nall subsequent repairs preserve divisibility by $q$. Thus $j+1$ is a valid\ncutoff for every target. The allowance-exception index itself may depend\non the target. This completes Theorem~\\ref{capacity:thm:capacity}.\n\nThe growth hypothesis has content. If repeated denominators are allowed,\ntake $Q_n=2^{\\lfloor\\sqrt n\\rfloor}$ and $F_n=1$. The allowance series\nconverges, while $U_N\\ge\\lfloor\\sqrt N\\rfloor$ by counting the next complete\ndenominator block. Yet eventual divisibility by two forces all sufficiently\nlate digits to vanish, so the attainable set is countable. Imposing all\ncumulative congruences as well leaves only zero. Thus nestedness alone does\nnot justify the criterion.\n\n\\subsection{Why a large gap prevents interval filling}\\label{capacity:sec:negative}\nThe decisive obstruction is local. At a position $N$, the whole preceding\nfactorial sum lies on a lattice of spacing $1/N!$. Eventual divisibility\nwidens that spacing to $q/N!$, up to a fixed translation. After a sufficiently\nlong gap, all remaining permitted digits reach only a bounded multiple of\n$1/N!$. Choosing one fixed $q$ larger than that bound leaves holes at\narbitrarily small scales.\n\nHere are the details, including the target-dependent cutoffs in (1)--(2).\nPut $r=\\lceil c\\rceil$. Suppose infinitely many successive $S$-gaps have\nlength at least $r$, and let $N$ run through the positions immediately before\nthese gaps. For large $k$,\n\\[\n \\frac{(k+1)^c/(k+1)!}{k^c/k!}\n =\\frac{(1+1/k)^c}{k+1}\\le\\frac12.\n\\]\nConsequently, for sufficiently large such $N$,\n\\begin{align}\n N!\\sum_{\\substack{k>N\\\\k\\in S}}\\frac{k^c}{k!}\n &\\le 2N!\\frac{(N+r)^c}{(N+r)!}\\\\\n &\\le 2^{c+1}N^{c-r}\\le B,\\qquad B=2^{c+1}.\n \\tag{5}\\label{capacity:eq:tailbound}\n\\end{align}\nIn the second inequality we used $N\\ge r$ and\n$(N+1)\\cdots(N+r)\\ge N^r$.\n\nThe bounded capacity along this subsequence invokes the necessity argument\nof Theorem~\\ref{capacity:thm:capacity}, with allowances\n$F_n=\\lfloor n^c\\rfloor$ on $S$ and zero elsewhere. It gives nullity and\nmeagreness even with target-dependent congruence and allowance cutoffs.\n\nIf $c\\le1$, every support gap is at least $r=1$, so this proves the first\npart of Theorem~\\ref{capacity:thm:main}. If $c>1$ and (3) fails, it proves the negative\npart. Notice that infrequent large gaps are enough; average digit counts do\nnot detect this obstruction.\n\n\\subsection{A digit that also corrects the cumulative residue}\nThe following construction supplies the positive direction. It is useful\nbeyond factorial weights.\n\n\\begin{lemma}\\label{capacity:lem:feedback}\nLet $w_j>0$, let positive integers $M_j$ satisfy $M_{j-1}\\mid M_j$, and let\n$F_j\\ge0$. Suppose $M_jw_j\\to0$ and, for $j\\ge1$,\n\\[\n 2M_j\\le M_{j-1}\\frac{w_{j-1}}{w_j},\\qquad\n 2M_{j-1}\\frac{w_{j-1}}{w_j}\\le F_j.\n \\tag{7}\\label{capacity:eq:step}\n\\]\nEvery $y\\in[M_0w_0,2M_0w_0]$ has a representation\n$y=\\sum_{j\\ge1}b_jw_j$ with integers $0\\le b_j\\le F_j$ such that\n\\[\n M_j\\mid\\sum_{i\\le j}b_i,\\qquad M_{j-1}\\mid b_j.\n\\]\n\\end{lemma}\n\\begin{proof}\nStart with residual $R_0=y$ and cumulative sum $C_0=0$. Suppose\n$M_{j-1}w_{j-1}\\le R_{j-1}\\le2M_{j-1}w_{j-1}$ and\n$M_{j-1}\\mid C_{j-1}$. Put $u=R_{j-1}/w_j$ and let\n$v$ be the least nonnegative residue of $-C_{j-1}$ modulo $M_j$. Choose\n\\[\n b_j=v+M_j\\left\\lfloor\\frac{u-M_j-v}{M_j}\\right\\rfloor.\n \\tag{8}\\label{capacity:eq:selector}\n\\]\nSince $u\\ge2M_j$ and $0\\le v<M_j$, this integer is nonnegative.\nThe floor identity gives $M_j\\le u-b_j<2M_j$; also $b_j\\le u\\le F_j$.\nThus $R_j=R_{j-1}-b_jw_j$ lies in\n$[M_jw_j,2M_jw_j)$, while $C_j=C_{j-1}+b_j$ is divisible by $M_j$.\nThe incoming modulus divides both $C_{j-1}$ and $M_j$, hence also $b_j$.\nFinally $R_j\\to0$, so the partial sums converge to $y$.\n\\end{proof}\nThe residue in (8) may depend on $y$. Only the schedule of moduli needs to\nbe common to all targets. Requiring target-independent ordinary block sums\nwould impose an unnecessary restriction on the construction.\n\n\\subsection{The positive direction and the arithmetic cost}\nSuppose $c>1$ and the support gaps are eventually at most\n$d=\\lceil c\\rceil-1$. Discard a finite prefix and write\n\\[\n r_j=\\frac{n_j!}{n_{j-1}!},\\qquad F_j=n_j^c,\\qquad w_j=\\frac1{n_j!}.\n\\]\nThen $r_j\\ge n_j\\to\\infty$, while $r_j\\le n_j^d$ and therefore\n$F_j/r_j\\ge n_j^{c-d}\\to\\infty$. Define\n\\[\n h_j=\\inf_{k\\ge j}\\min\\left\\{\\frac{\\sqrt{r_k}}2,\n                                  \\frac{F_k}{2r_k}\\right\\}.\n\\]\nThis lower envelope is nondecreasing and tends to infinity. Begin sufficiently\nlate that $h_1\\ge1$, set $M_0=1$, and let $M_j$ be the largest factorial at\nmost $h_j$. These moduli are nested and eventually divisible by every fixed\ninteger. Moreover,\n\\[\n 2M_j\\le\\sqrt{r_j}\\le r_jM_{j-1},\\qquad\n 2r_jM_{j-1}\\le F_j.\n\\]\nThe second inequality uses $M_{j-1}\\le h_j$, also valid for $j=1$.\nFinally,\n$M_j/n_j!\\le\\sqrt{r_j}/(2n_j!)\\le1/(2\\sqrt{n_j!})\\to0$.\nLemma~\\ref{capacity:lem:feedback} fills $[1/n_0!,2/n_0!]$ using the positions $n_j$,\n$j\\ge1$. Put zero digits elsewhere. Once $q\\mid M_J$, all later digits and\ncumulative sums at the original integer indices have the divisibilities in\n(4). This proves the positive direction, including common cutoffs.\n\nFor comparison, remove condition (2) and call the resulting attainable set\n$E_0(S,c)$.\n\\begin{proposition}\\label{capacity:prop:unrestricted}\nFor an infinite support and $c\\ge1$, $E_0(S,c)$ contains an interval exactly\nwhen its successive gaps are eventually at most $\\lfloor c\\rfloor$.\nOtherwise it is null and meagre. For $0<c<1$ it is always null and meagre.\n\\end{proposition}\n\\begin{proof}\nFor necessity repeat Section~\\ref{capacity:sec:negative} with\n$r=\\lfloor c\\rfloor+1$. The bound in (5) now tends to zero. With lattice\nspacing $1/N!$, it is eventually bounded by $1/(2N!)$, so the same compact-set\nargument applies with $q=1$, $B=1/2$.\nFor sufficiency, $r_j\\le n_j^c$ along the support. The ordinary mixed-radix\nexpansion using digits $0\\le b_j<r_j$ fills $[0,1/n_0!]$. To see this\ndirectly, the maximum tail after $n_j$ is $1/n_j!$, since\n$(r_k-1)/n_k!=1/n_{k-1}!-1/n_k!$. The adjacent digit intervals therefore\nmeet, and their lengths tend to zero.\n\\end{proof}\nThus the difference between the two gap thresholds occurs exactly at integer\n$c$. At $c=m\\ge2$, imposing eventual congruences raises the least support\ndensity from $1/m$ to $1/(m-1)$. At $c=1$ it destroys interval filling entirely.\nWith quadratic allowances, deleting only the positions $2^k$ destroys\ninterval filling, although the permitted support still has density one.\nThis is the simplest example of why support density loses decisive information.\nThese statements concern intervals of values on a common permitted support;\nthey are not lower bounds for representing one specially chosen value.\n\n\\subsection{A common-divisor test for irrationality}\\label{capacity:sec:carry}\nThe distinction between small and large allowances also appears directly in\nrationality. Let $b_n\\ge2$ be integers, put $Q_0=1$ and\n$Q_n=b_1\\cdots b_n$, and consider a Cantor series.\n\\begin{theorem}\\label{capacity:thm:gcd}\nSuppose $0\\le a_n\\le A$ and $e_n\\ge0$ are integers, $e_n\\le Cb_n$\neventually, and $a_n+e_n$ is nonzero infinitely often. If\n\\[\n \\limsup_{n\\to\\infty}\\gcd(b_n,e_n)=\\infty,\n\\]\nthen $\\sum_{n\\ge1}(a_n+e_n)/Q_n$ is irrational.\n\\end{theorem}\n\\begin{proof}\nFor large $n$, the scaled tail satisfies\n\\[\n 0<Q_n\\sum_{k>n}\\frac{a_k+e_k}{Q_k}\\le A+2C.\n\\]\nIndeed, the bounded $a_k$ contribute at most\n$A\\sum_{j\\ge1}2^{-j}=A$, and $e_k\\le Cb_k$ contributes at most\n$C\\sum_{j\\ge1}2^{-(j-1)}=2C$.\nIf the total were $p/D$, the numbers\n\\[\n T_n=DQ_n\\left(\\frac pD-\\sum_{k\\le n}\\frac{a_k+e_k}{Q_k}\\right)\n\\]\nwould be positive integers eventually bounded by $D(A+2C)$. No hypothesis\nthat $D\\mid Q_n$ is needed: the extra factor $D$ clears it. The recurrence\n\\[\n b_nT_{n-1}=D(a_n+e_n)+T_n\n\\]\nimplies $\\gcd(b_n,e_n)\\mid Da_n+T_n$. The integer on the right is positive\nand eventually at most $D(2A+2C)$, contradicting the unbounded gcd.\n\\end{proof}\nFor factorial denominators, $b_n=n$, and eventual divisibility of $e_n$ by\neach fixed integer makes $\\gcd(n,e_n)$ arbitrarily large arbitrarily late:\nchoose a late multiple of that integer. Hence bounded nonnegative digits that\nare nonzero infinitely often cannot be rationalised by nonnegative $O(n)$\ncorrections satisfying (2). Conversely any allowance $F(n)$ with\n$F(n)/n\\to\\infty$ permits interval filling under (4): apply the positive\nconstruction with $r_n=n$, replacing $n^c$ by $F(n)$.\n\nThe arithmetic hypothesis cannot be replaced by $b_n\\to\\infty$, even if both\ncongruences (4) are retained. Set $a_n=1$, $e_1=2$, and\n$e_n=(n+1)!-n!$ for $n\\ge2$, and let $b_n=e_n+2$.\nThen $\\sum_{k\\le n}e_k=(n+1)!$, so (4) holds, and $e_n<b_n$. Nevertheless\n\\[\n \\sum_{n\\ge1}\\frac{a_n+e_n}{Q_n}\n =\\sum_{n\\ge1}\\frac{b_n-1}{Q_n}=1.\n\\]\nHere $\\gcd(b_n,e_n)\\le2$. This example separates rapid denominator growth\nfrom the arithmetic obstruction used in Theorem~\\ref{capacity:thm:gcd}.\n\n\n\\subsection{Why a small allowance along a subsequence is insufficient}\nWithout eventual congruences, the allowance $F(n)=n-1$ for $n\\ge2$ gives the\n  full interval $[0,1]$, whereas $F(n)=o(n)$ as $n\\to\\infty$ gives a null\n  attainable set.  The latter conclusion does not follow from small\n  allowances merely along a subsequence, as the example below shows.\n\nFor the nullity assertion, eventually $F(n)+1\\le n/2$ and $F(n)\\le n$.\nThus the number of prefixes through $N$ is at most $C2^{-N}N!$ for a fixed\n$C$, while the capacity after $N$ is at most\n$\\sum_{n>N}n/n!\\le2/N!$.  The covering bound of Theorem~\\ref{thm:dichotomy}(i), allowing\nzero-capacity levels to be omitted, tends to zero.\n\nFor the subsequence counterexample put $F(2k)=0$ and\n$F(2k+1)=(2k)(2k+1)-1$ for $k\\ge1$, with $F(1)=0$.  Then\n\\[\n \\frac{F(2k+1)}{(2k+1)!}\n =\\frac1{(2k-1)!}-\\frac1{(2k+1)!}.\n\\]\nThe total capacity is $1$, and the capacity after each permitted index\n$2k+1$ is exactly $1/(2k+1)!$, equal to the spacing between its choices.\nThe interval criterion therefore gives every value in $[0,1]$, even though\n$F(n)/n=0$ at every even index.  The exact telescoping identities and sample\ngreedy expansions are reproduced by the script cited in\nSection~\\ref{sec:misread}; the interval conclusion follows from this argument,\nnot from the samples.\n\n\n\\section{Proof of the derivative interpolation theorem}\\label{capacity:sec:jets-proof}\nWrite $(n)_r=n(n-1)\\cdots(n-r+1)$, with $(n)_0=1$ and $(n)_r=0$ for\n$r>n$. Then $f^{(r)}(1)=\\sum_n(n)_r e_n/n!$.\n\n\\subsection{Nullity at and below the critical exponent}\nHere only eventual evenness is needed. Fix a cutoff $K$ and a prefix before\n$K$, and allow every continuation with $0\\le e_n\\le n^c$ and $2\\mid e_n$\nfor $n\\ge K$. The derivative vectors form a compact set $V$.\nFor all sufficiently large $n$, there are at most\n\\[\n \\lfloor n^c/2\\rfloor+1\\le(3/4)n^c\n\\]\nchoices at position $n$. Thus the number of prefixes through $N$ is at most\n$C(3/4)^N(N!)^c$. Every remaining derivative tail has norm at most\n$C_{c,d}N^{c+d-1}/(N+1)!$, by comparison with its first term. Balls of this\nradius, one for each prefix, cover $V$ with total $d$-dimensional volume\nat most\n\\[\n C'(3/4)^N(N!)^{c-d}N^{d(c+d-1)}.\n\\]\nFor $c\\le d$ this tends to zero. Hence $V$ is null and, being compact,\nnowhere dense. Countably many cutoffs and integer prefixes cover the\nattainable set. This proves both conclusions, including the critical case.\n\n\\subsection{Lifting a scalar interval to an open set}\nFirst allow signed integer coefficients. For every $\\eta>0$ and cutoff\n$N\\ge1$, consider series supported on $n\\ge N$ with $|e_n|\\le\\eta n^c$\nand the same eventual divisibilities. We prove by induction on $d$ that\ntheir derivative vectors contain an open set whenever $c>d$.\nFor $d=1$, apply Theorem~\\ref{capacity:thm:capacity} to\n$F_n=\\lfloor\\eta n^c\\rfloor$ and $Q_n=n!$, starting after $N$.\nThe first tail term makes $U_N\\to\\infty$ when $c>1$; the all-position\nversion gives an interval with common congruence cutoffs.\n\nFor the induction step, use dimension $d-1$ and exponent $c-1$ for\n$B(z)=\\sum b_nz^n/n!$ with $|b_n|\\le(\\eta/4)n^{c-1}$ and $b_n=0$\nbefore $N$. The higher derivative coordinates of $(z-1)B$ form an open\nset $W\\subseteq\\R^{d-1}$, because\n\\[\n ((z-1)B)^{(r)}(1)=rB^{(r-1)}(1),\\qquad r\\ge1.\n\\]\nChoose $y_0+[-2\\rho,2\\rho]^{d-1}\\subset W$. Choose a later cutoff $M$\nso that every $A$ supported on $n\\ge M$ with\n$0\\le a_n\\le(\\eta/2)n^c$ has $|A^{(r)}(1)|<\\rho$ for $1\\le r<d$.\nThis follows uniformly from the convergent derivative tails. The scalar\ntheorem gives a nondegenerate interval $I$ of values $A(1)$.\nFor each $x\\in I$ choose one such $A$. For every\n$y\\in y_0+(-\\rho,\\rho)^{d-1}$, a suitable $B$ cancels the higher\ncoordinates of $A$ and gives $y$. Hence the attainable vectors contain\n\\[\n I^\\circ\\times\\bigl(y_0+(-\\rho,\\rho)^{d-1}\\bigr).\n\\]\nNo continuous choice of $A$ is assumed: uniform smallness of all its higher\nderivatives makes the same cube work for every $x$.\nThe coefficients of $A+(z-1)B$ obey the bound, since\n$n|b_{n-1}|+|b_n|\\le(\\eta/2)n^c$; divisibility holds term by term.\nThe finitely many families in the induction have common cutoffs for each $q$.\n\nTo restore nonnegativity, choose nested positive integers $m_n\\to\\infty$\nsuch that every fixed integer eventually divides $m_n$ and\n$\\log m_n=o(\\log n)$. One choice is the largest factorial at most\n$\\log(n+3)$. For large $n$,\n\\[\n h_n=m_n\\left\\lfloor\\frac{n^c}{2m_n}\\right\\rfloor,\n \\qquad n^c/3\\le h_n\\le n^c/2.\n\\]\nSet earlier $h_n$ to zero and start the signed construction after that point.\nTake $\\eta=1/4$. Adding $\\sum h_nz^n/n!$ translates the open set and\nputs every sufficiently late coefficient between $n^c/12$ and $3n^c/4$.\nAll coefficients lie between $0$ and $n^c$, and each function is nonpolynomial.\n\n\\subsection{The upper bound for Hausdorff dimension}\nFix a finite prefix and an allowance cutoff. Through index $N$ there are\nat most $\\prod_{n\\le N}(\\lfloor n^c\\rfloor+1)$ continuations, whose\nlogarithm is $c\\log(N!)+O(N)$. Each derivative tail has norm at most\n$O_{c,d}(N^{c+d-1}/(N+1)!)$. For any $s>c$, the sum of the $s$-th powers\nof the diameters of these covering balls tends to zero. Countably many\nprefixes and cutoffs preserve this bound; the ambient bound is $d$.\n\n\\subsection{A separated construction attaining the dimension}\nFor $0\\le k<d$ with $k<c$, set $\\alpha_k=\\min(1,c-k)$. Use only these\nindices. Their sum is $s=\\min(c,d)$. Let $\\varepsilon=2^{-d-6}$ and use\nthe same $m_n$. Independently for every $n$ and $k$, choose\n\\[\n b_n^{(k)}\\in\\left\\{0,m_n,2m_n,\\ldots,\n m_n\\left\\lfloor\\frac{\\varepsilon n^{\\alpha_k}}{m_n}\\right\\rfloor\\right\\}.\n \\tag{D}\\label{capacity:eq:jet-digits}\n\\]\nAll digits vanish before a large cutoff. Put\n\\[\n f(z)=\\sum_n h_n\\frac{z^n}{n!}\n       +\\sum_k(z-1)^k\\sum_n b_n^{(k)}\\frac{z^n}{n!}.\n \\tag{F}\\label{capacity:eq:jet-code}\n\\]\nThe coefficient added at position $n$ is\n\\[\n \\sum_k\\sum_{j=0}^k(-1)^{k-j}\\binom{k}{j}(n)_j b_{n-j}^{(k)},\n\\]\nwith negative indices interpreted as zero. Its absolute value is at most\n$\\varepsilon\\sum_{k<d}2^k n^c<n^c/4$, since $j+\\alpha_k\\le c$.\nThus $f\\in\\mathcal H_c$, with the allowance everywhere and common\ncongruence cutoffs. Local uniform convergence makes the image of this\nproduct of finite digit sets compact.\n\nThe crucial step is to rule out cancellation between different codes.\nSuppose two codes first differ at index $n$, and let $g$ be their difference.\nFor $0\\le r<d$, take the Taylor coefficients of $z^{-n}g(z)$ at $1$:\n\\[\n [u^r](1+u)^{-n}g(1+u)\n =\\frac{\\Delta b_n^{(r)}}{n!}\n   +\\sum_{m>n}\\sum_{k\\le r}\n       \\frac{\\Delta b_m^{(k)}}{m!}\\binom{m-n}{r-k}.\n \\tag{T}\\label{capacity:eq:moving-taylor}\n\\]\nUnused coordinates are zero. The binomial coefficient now depends on\n$m-n$, not on $m$. Since $|\\Delta b_m^{(k)}|\\le\\varepsilon m$, the\nabsolute tail, multiplied by $n!$, is at most\n\\[\n \\varepsilon\\sum_{t\\ge1}\\frac{2^t}{(t-1)!}\n =2e^2\\varepsilon<18\\varepsilon<\\frac12.\n\\]\nHere $n!/(n+t-1)!\\le1/(t-1)!$ and\n$\\sum_{k\\le r}\\binom{t}{r-k}\\le2^t$. Some $\\Delta b_n^{(r)}$ is a\nnonzero integer. Therefore one coordinate on the left of\n\\eqref{capacity:eq:moving-taylor} has absolute value at least $1/(2n!)$.\nThe linear map from $J_d(g)$ to those coordinates has norm at most\n$C_dn^{d-1}$: its entries are\n$(-1)^j\\binom{n+j-1}{j}/(r-j)!$, $0\\le j\\le r<d$.\nDistinct codes first differing at $n$ consequently have separation at least\n\\[\n \\delta_n=\\frac{1}{2C_dn^{d-1}n!}\n \\tag{S}\\label{capacity:eq:jet-separation}\n\\]\nin the supremum norm of $\\R^d$.\n\nGive all choices in \\eqref{capacity:eq:jet-digits} uniform independent\nprobabilities. If $A_N$ counts prefixes through $N$, then\n\\[\n \\log A_N=s\\log(N!)+o(N\\log N),\n\\]\nbecause $\\log m_n=o(\\log n)$ and each used $\\alpha_k$ is positive.\nThe separation makes the coding injective. A ball of radius less than\n$\\delta_N/2$ meets at most one length-$N$ cylinder and has measure at most\n$A_N^{-1}$. For a small radius $r$, choose $N$ with\n$\\delta_{N+1}/2\\le r<\\delta_N/2$. Since\n$\\log(1/\\delta_{N+1})/\\log(N!)\\to1$, for every $t<s$ this implies\n$\\mu(B(x,r))\\le C_t r^t$. Summing this inequality over any ball cover\ngives positive $t$-dimensional Hausdorff content. The dimension is at least\n$s$, completing Theorem~\\ref{capacity:thm:jets}.\n\n\\section{The arithmetic threshold for rational derivatives}\n\\label{capacity:sec:rational-jets}\nWe prove Theorem~\\ref{capacity:thm:rational-jets}. The coefficient estimate\nand the divisibilities are both needed, including at the critical exponent.\n\nSuppose the first $d$ derivatives at $1$, starting with the value, are rational.\nSubtract their Taylor polynomial\n\\[\n P(z)=\\sum_{r=0}^{d-1}\\frac{f^{(r)}(1)}{r!}(z-1)^r\n\\]\nand choose a positive integer $D$ clearing its coefficient denominators.\nThen $g=D(f-P)=\\sum_{k\\ge0}g_kz^k/k!$ has integer coefficients,\n$|g_k|=O(k^d)$, and the same eventual divisibilities. Moreover\n$g^{(r)}(1)=0$ for $0\\le r<d$. Thus $B(z)=g(z)/(z-1)^d$ is entire.\nExpansion at zero gives\n\\[\n B(z)=\\sum_{n\\ge0}b_n\\frac{z^n}{n!},\\qquad\n b_n=(-1)^d\\sum_{k=0}^n\\binom{n-k+d-1}{d-1}\\frac{n!}{k!}g_k.\n \\tag{Q}\\label{capacity:eq:quotient-prefix}\n\\]\nIn particular $b_n$ is an integer. Fix $q$, and choose $K$ so that\n$q\\mid g_k$ for $k\\ge K$. Those terms in \\eqref{capacity:eq:quotient-prefix}\nare divisible by $q$. For each of the finitely many $k<K$, the factor\n$n!/k!$ is divisible by $q$ once $n$ is sufficiently large. Hence\n$q\\mid b_n$ eventually as well.\n\nThe vanishing derivatives imply\n$\\sum_{k\\ge0}R(k)g_k/k!=0$ for every polynomial $R$ of degree less than\n$d$, because the falling factorials of orders $0,\\ldots,d-1$ form a basis.\nApply this to $R(k)=\\binom{n-k+d-1}{d-1}$, interpreted as a polynomial.\nIts values at $k=n+1,\\ldots,n+d-1$ vanish, and at $k=n+j$, $j\\ge d$,\nthey equal $(-1)^{d-1}\\binom{j-1}{d-1}$. The complementary tail of\n\\eqref{capacity:eq:quotient-prefix} therefore gives\n\\[\n b_n=n!\\sum_{j\\ge d}\\binom{j-1}{d-1}\\frac{g_{n+j}}{(n+j)!}.\n \\tag{R}\\label{capacity:eq:quotient-tail}\n\\]\nAll series here converge absolutely. For large $n$, the absolute value is\nbounded by $C$ times the sum of\n\\[\n v_j=\\binom{j-1}{d-1}\\frac{n!(n+j)^d}{(n+j)!},\\qquad j\\ge d.\n\\]\nTheir successive ratios satisfy\n\\[\n \\frac{v_{j+1}}{v_j}\n =\\frac{j}{j-d+1}\\left(\\frac{n+j+1}{n+j}\\right)^d\\frac1{n+j+1}\n \\le\\frac{d2^d}{n+1}.\n\\]\nFor sufficiently large $n$ this is at most $1/2$, whereas\n$v_d=n!(n+d)^d/(n+d)!$ is bounded. Thus $(b_n)$ is bounded.\nChoose an integer $q$ larger than its eventual absolute bound. Eventual\ndivisibility by this $q$ forces $b_n=0$ eventually. It follows that $B$,\n$g$ and $f$ are polynomials.\n\nFor the converse, when $c>d$, Theorem~\\ref{capacity:thm:jets} gives an open\nset of vectors realised by nonpolynomial members of $\\mathcal H_c$.\nEvery nonempty open subset of $\\R^d$ meets $\\mathbb Q^d$. This completes\nthe proof of the sharp threshold.\n\n\\begin{remark}[Why the congruences cannot be omitted]\nThe function $(z-1)^de^z$ has its first $d$ derivatives zero at $1$.\nIts integer factorial coefficients are\n\\[\n p_d(n)=\\sum_{j=0}^d(-1)^{d-j}\\binom dj(n)_j\n       =n^d-\\frac{d(d+1)}2n^{d-1}+O_d(n^{d-2})\n\\]\nfor $d\\ge2$; for $d=1$ they are $n-1$.\nThus $0\\le p_d(n)\\le n^d$ eventually. Replacing the finitely many earlier\ncoefficients, including the constant coefficient, by zero adds a rational\npolynomial. The resulting nonpolynomial function has nonnegative integer\ncoefficients bounded by $n^d$ everywhere and all the specified derivatives\nrational. It fails the eventual divisibility assumption. Full dimension\nand even the growth bound alone therefore do not supply the arithmetic\nconclusion.\n\\end{remark}\n\n\n\\section{Lambert subsums across bases}\\label{sec:results}\n\nFor real $t>1$ and a set $S$ of\npositive integers put\n\\[\n X_S(t)=\\sum_{n\\in S}\\frac{1}{t^n-1},\\qquad\n w_n=\\frac{1}{t^n-1},\\qquad R_N=\\sum_{n>N}w_n .\n\\]\nProblem \\#257 asks whether $X_S(t)$ is irrational for every infinite $S$ at\nevery integer $t\\ge2$ \\cite[p.~62]{erdosgraham1980}; Problem \\#1049 asks about\n$X_{\\N_{>0}}(t)$ at rational $t$.\n\nThe comparison behind everything is stated for weights with multiplicities.\nLet $u_n>0$ and integers $D_n\\ge1$ satisfy $\\sum_nD_nu_n<\\infty$, and put\n\\[\n V=\\Bigl\\{\\sum_{n\\ge1}\\varepsilon_nu_n:\\ \\varepsilon_n\\in\\{0,1,\\dots,D_n\\}\\Bigr\\},\n \\qquad C_N=\\sum_{n>N}D_nu_n .\n\\]\n\n\\begin{theorem}[choices against contraction]\\label{thm:dichotomy}\n\\textup{(i)} For every $N$, the Lebesgue measure of $V$ is at most\n$C_N\\prod_{n\\le N}(D_n+1)$.  If $\\liminf_NC_N\\prod_{n\\le N}(D_n+1)=0$ then $V$\nis null.\n\n\\textup{(ii)} If $u_n\\le C_n$ for every $n>N_0$, then for each choice of\n$\\varepsilon_1,\\dots,\\varepsilon_{N_0}$ the set $V$ contains the interval\n$[\\mu,\\mu+C_{N_0}]$, where $\\mu=\\sum_{n\\le N_0}\\varepsilon_nu_n$.\n\n\\textup{(iii)} For $u_n=\\beta^{-n}$ with real $\\beta>1$ and $D_n=D$, the bound in\n\\textup{(i)} tends to $0$ exactly when $D+1<\\beta$, and the hypothesis of\n\\textup{(ii)} holds exactly when $D+1\\ge\\beta$.\n\\end{theorem}\n\nPart (ii) is Kakeya's covering argument and part (i) is the standard covering\nbound; both are classical \\cite{hornich1941,nitecki2013,fridy1966}, and\nKova\\v{c} and Tao give a scalar reciprocal-choice covering lemma\n\\cite[Lemma~5.1]{kovactao2024}; their higher-dimensional approximation lemma is\nLemma~7.2 of the same paper.  We claim no novelty for\nTheorem~\\ref{thm:dichotomy}.  Its use here is to say which side each problem\nlies on.\n\n\\begin{theorem}[the subseries across bases]\\label{thm:bases}\nLet $t>1$ be real.\n\n\\textup{(a)} If $t<2$, let $N_0\\ge0$ be least with $t^{-n}\\le2-t$ for all\n$n>N_0$.  Then the set of values $X_S(t)$ is the union of the intervals\n$[X_F(t),X_F(t)+R_{N_0}]$ over $F\\subseteq\\{1,\\dots,N_0\\}$, where\n$X_F(t)=\\sum_{n\\in F}w_n$; in particular it contains $[0,R_{N_0}]$.  If moreover\n$t=a/b$ in lowest terms, then every rational in $[0,R_{N_0}]$ whose reduced\ndenominator shares a prime factor with $ab$ equals $X_S(t)$ for some $S$, and\nevery such $S$ is infinite.  There is an infinite $S\\subseteq\\{2,3,\\dots\\}$\nwith $X_S(3/2)=1/2$.\n\n\\textup{(b)} If $t=2$, every value has exactly one $S$, and the set of values\nis a Cantor set of Lebesgue measure $1$ inside $[0,E]$, where\n$E=\\sum_{n\\ge1}(2^n-1)^{-1}=1.6066951524\\ldots$.\n\n\\textup{(c)} If $t>2$, the set of values is null.\n\\end{theorem}\n\nPart (b) is proved in the companion note on \\#257, Section~7, from Hornich's\ntheorem as proved by Nitecki \\cite{plectis257,hornich1941,nitecki2013}; Kova\\v{c}\nand Tao record the strict inequality $w_N>R_N$ and the Cantor-set conclusion for\nevery fixed base $t\\ge2$ \\cite[Remark~4.1]{kovactao2024}.  Parts (a) and (c) follow\nfrom Theorem~\\ref{thm:dichotomy} in a few lines.  We have not found part (a)\nstated for non-integer bases and it may be known.  It shows that the statement\nasked in \\#257 is false at every rational base below $2$, so any proof at base\n$2$ must use more than the shape of the series.\n\n\\begin{theorem}[divisibility chains]\\label{thm:chains}\nLet $S=\\{n_1<n_2<\\cdots\\}$ be infinite with\n$n_j\\mid n_{j+1}$ for every $j$.\n\n\\textup{(a)} If $t=a/b>1$ is rational in lowest terms and $a^2>b^3$, then\n$X_S(t)$ is irrational.\n\n\\textup{(b)} If the integer ratios $n_{j+1}/n_j$ are eventually periodic,\nthen $X_S(t)$ is transcendental for every algebraic real $t>1$.\n\\end{theorem}\n\nFor example, the chain $1,2,6,12,36,72,\\ldots$ has alternating ratios\n$2,3$. Part (b) proves\n\\[\n \\sum_{k\\ge0}\\left(\\frac1{(4/3)^{6^k}-1}\n                  +\\frac1{(4/3)^{2\\cdot6^k}-1}\\right)\n \\quad\\hbox{is transcendental},\n\\]\nalthough $4^2<3^3$. Repeated blocks of ratios, rather than a stronger\ntail estimate, supply the functional equation used in this case.\nPart (b) is a direct corollary of the classical Mahler value theorem cited\nbelow; no historical novelty is claimed for this specialisation.\n\nThe hypothesis $a^2>b^3$ says $\\log b/\\log a<2/3$.  It holds for every integer\nbase, for $3/2$, $5/2$ and $7/3$, and fails for $4/3$ and $5/4$.  At integer\nbases part (a) is contained in the theorem of Erd\\H{o}s on supports with\n$\\sum_{n\\in S}1/n<\\infty$ \\cite{erdos1968}, proved in full in\n\\cite[Theorem~2]{plectis257}.  At base $3/2$ it sits inside the regime of\nTheorem~\\ref{thm:bases}(a): rational values occur there, and exact divisibility\nstill forces irrationality.  For comparison, the companion note on \\#1049 proves the irrationality of the\nfull sum $X_{\\N_{>0}}(a/b)$ when $\\log b/\\log a<0.4056830213840605\\ldots$, using Zudilin's\nlinear forms \\cite[Theorem~1]{plectis1049}, \\cite{zudilin2004}, and proves that\nthe sufficient cutoff supplied by one integer-polynomial family with common\nleading degree, coefficient-height and decay bounds at every fixed real base\n$x>1$ is at most $1/2$ \\cite[Theorem~5]{plectis1049}.  This restriction does not\nexclude stronger estimates at a particular base or a different choice of\nfamily there.  Thin supports reach further than the full sum\nbecause the denominators divide one another.\n\nAt base $2$ the greedy rule characterises membership and supplies finite\ncertificates of nonmembership: starting from $r_0=x$, take\nindex $n$ when $r_{n-1}\\ge w_n$ and subtract.  Since $w_n>R_n$, a real\n$x\\in[0,E]$ is a subsum if and only if no remainder falls strictly between\n$R_n$ and $w_n$; we say $x$ is \\emph{rejected at step $n$} when that happens\nfirst at index $n$.\n\n\\begin{theorem}[the fixed-depth rational count]\\label{thm:forced}\nFix $N\\ge1$.  Among the reduced fractions $p/q\\in(0,E]$ with $q\\le Q$, the\nproportion not rejected in the first $N$ steps tends to $2^NR_N/E$ as\n$Q\\to\\infty$, with error $O(2^N\\log Q/Q)$.  Consequently the upper limit of\nthe proportion that are subsums is at most $1/E=0.62239\\ldots$.\n\\end{theorem}\n\nThe limit $2^NR_N/E$ does not depend on whether \\#257 is true.  Agreement with\nthis fixed-depth limiting proportion therefore does not establish membership.\nAn exact computation in \\cite{plectisinvestigation} first read a surviving\nshare near $62\\%$ as evidence that most such fractions are subsums;\nTheorem~\\ref{thm:forced} is the correction.  Section~\\ref{sec:base2} states\nwhat remains.\n\n\\paragraph{Evidence.} The proofs of Theorems~\\ref{thm:dichotomy}--\\ref{thm:forced}\nare ordinary proofs. Theorem~\\ref{thm:chains} uses Nishioka's theorem for\nMahler systems as an external premise in the eventually periodic case, hence\nin both parts. Results quoted from the problem papers retain the evidence\nclass given at each use.\n\n\\subsection{Choices against contraction}\\label{sec:dichotomy}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:dichotomy}]\n(i) A choice of $\\varepsilon_1,\\dots,\\varepsilon_N$ fixes\n$\\sum_{n\\le N}\\varepsilon_nu_n$, and the rest of the sum lies in $[0,C_N]$.\nSo $V$ is covered by at most $\\prod_{n\\le N}(D_n+1)$ intervals of length $C_N$.\n\n(ii) Subtracting $\\mu$, it suffices to reach every $y\\in[0,C_{N_0}]$ with the\nlevels $n>N_0$.  Put $y_{N_0}=y$ and, for $n>N_0$, let\n$\\varepsilon_n=\\min(D_n,\\lfloor y_{n-1}/u_n\\rfloor)$ and\n$y_n=y_{n-1}-\\varepsilon_nu_n$.  If $0\\le y_{n-1}\\le C_{n-1}=D_nu_n+C_n$ then\n$0\\le y_n\\le C_n$: when $\\varepsilon_n=D_n$ this is a subtraction, and when\n$\\varepsilon_n<D_n$ it holds because $y_n<u_n\\le C_n$.  Since $C_n\\to0$, the\nsum of the $\\varepsilon_nu_n$ is $y$.\n\n(iii) Here $C_n=D\\beta^{-n}/(\\beta-1)$.  The product in (i) is a constant\nmultiple of $((D+1)/\\beta)^N$, and $u_n\\le C_n$ says $\\beta-1\\le D$.\n\\end{proof}\n\nThe reading that matters for irrationality is immediate.  Suppose a family of\nseries has values $x_0+\\sum\\varepsilon_nu_n$ with the $\\varepsilon_n$ free as in\n(ii).  The values then fill an interval, which contains rationals and\nirrationals, so a property that all members of the family share implies\nneither.  A proof of irrationality for one member has to use something that\ndistinguishes it inside the family.  Two constructions in the companion notes\nare families of this kind.\n\n\\emph{Problem \\#251.}  Proposition~1 of \\cite{plectis251} (ordinary proof),\napplied in its Corollary~2 to the prime gaps, changes the gaps on a set of upper\nBanach density zero, by nonnegative integers below any prescribed function\ntending to infinity, keeping every congruence modulo every fixed $q$ from some\npoint on, and reaches every real in an interval.  The resulting positions\n$P_n$ satisfy $P_n\\sim n\\log n$ and are not asserted to be prime.  In the notation above the $j$-th block has $u_j=M_j2^{-n_j-2}$ and\n$D_j=2^{s_j}-1$, and the inequality verified there is $u_j\\le C_j$.  The\nlemma is credited there to \\cite{fridy1966}, \\cite[Lemma~4]{crmarickovac2025}\nand \\cite[Lemma~5.1]{kovactao2024}.  The note also proves that a bounded\nallowance is impossible under the stated congruences.  Thus every allowance\ntending to infinity suffices, while no bounded allowance does.  The specified\ngrowth, eventual fixed-modulus congruences and empirical distributions of\nunnormalised blocks therefore do not suffice to prove\n$\\sum p_n2^{-n}$ irrational; neither primality nor every quantitative\ncorrelation is preserved.\n\n\\emph{Problem \\#249.}  Section~5 of \\cite{plectis249} gives an integer sequence\n$c$ with $c(n)=\\varphi(n)$ for odd $n$, $|c(n)-\\varphi(n)|\\le2$ for even $n$,\n$0\\le c(n)\\le n$ and $\\sum c(n)2^{-n}=5/4$ (Lean-checked there).  Changing even\nindices by at most $2$ is the case $u_n=2^{-n}$ for even $n$, $D_n=4$, where\n$C_n\\ge\\tfrac43 2^{-n}>u_n$.\n\nThe same comparison appears in Kova\\v{c} and Tao's theorem that for integers\n$2\\le t_1<\\dots<t_m$ with $\\sum1/(t_k-1)>1$ there are sets $S_k$, one of them\ninfinite, with $\\sum_kX_{S_k}(t_k)$ rational \\cite[Theorem~2.3]{kovactao2024}:\nmerging the weights of the $m$ bases, the sum of all weights below a given\nweight $u$ is at least $(1-O(u))\\,u\\sum1/(t_k-1)$, which exceeds $u$ once $u$\nis small.\n\n\\subsection{The subseries across bases}\\label{sec:bases}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:bases}]\n(a) Since $R_n\\ge\\sum_{k>n}t^{-k}=t^{-n}/(t-1)$ and\n$w_n=t^{-n}/(1-t^{-n})$, the inequality $w_n\\le R_n$ holds as soon as\n$t^{-n}\\le2-t$.  Theorem~\\ref{thm:dichotomy}(ii) with $D_n=1$ gives, for\neach $F\\subseteq\\{1,\\dots,N_0\\}$, the interval $[X_F(t),X_F(t)+R_{N_0}]$, and every\nvalue $X_S(t)$ lies in the one with $F=S\\cap[1,N_0]$.  For $t=a/b$ a finite subsum is\n$\\sum_{n\\in F}b^n/(a^n-b^n)$, whose denominator divides $\\prod(a^n-b^n)$ and is\ncoprime to $ab$.  A rational whose denominator is not coprime to $ab$ is\ntherefore never a finite subsum, and inside the interval it is a subsum.  At\n$t=3/2$ the inequality $t^{-n}\\le1/2$ holds for $n\\ge2$ and $R_1>1/2$.\n\n(c) For $t>2$, $2^NR_N\\le2^Nt^{-N}/((t-1)(1-t^{-1}))\\to0$, and\nTheorem~\\ref{thm:dichotomy}(i) applies.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:chains}]\nFor part (a), write $\\rho=b/a$, $D_j=a^{n_j}-b^{n_j}$ and\n$S_j=\\sum_{i\\le j}b^{n_i}/(a^{n_i}-b^{n_i})$.  Because $n_i\\mid n_j$ for\n$i\\le j$, each $a^{n_i}-b^{n_i}$ divides $D_j$, so $D_jS_j$ is an integer.  The\ntail satisfies\n\\[\n 0<X_S(t)-S_j\\le\\sum_{n\\ge n_{j+1}}\\frac{\\rho^n}{1-\\rho^n}\n \\le\\frac{\\rho^{\\,n_{j+1}}}{(1-\\rho)^2}.\n\\]\nEach ratio $n_{j+1}/n_j$ is an integer at least $2$.\n\nSuppose the ratio is at least $3$ for infinitely many $j$.  For those $j$,\n\\[\n 0<D_j\\bigl(X_S(t)-S_j\\bigr)\\le\\frac{a^{n_j}\\rho^{3n_j}}{(1-\\rho)^2}\n =\\frac{1}{(1-\\rho)^2}\\Bigl(\\frac{b^3}{a^2}\\Bigr)^{n_j}\\longrightarrow0\n\\]\nwhen $a^2>b^3$.  If $X_S(t)=p/q$, then $qD_j(X_S(t)-S_j)$ is a positive\ninteger for every $j$, a contradiction.\n\nOtherwise the ratio is $2$ from some index on, so part (b) completes the\nproof of (a).\n\nFor part (b), discard a finite prefix and write the repeating ratio block as\n$r_1,\\ldots,r_\\ell$, with every $r_i\\ge2$. Put\n\\[\n Q=\\prod_{i=1}^{\\ell}r_i,\\qquad e_0=1,\\qquad\n e_i=\\prod_{h=1}^i r_h\\quad(1\\le i<\\ell).\n\\]\nWith $d$ the first exponent after that prefix, the remaining exponents are exactly\n$d e_iQ^k$ with $0\\le i<\\ell$ and $k\\ge0$. Indeed, one complete block\nmultiplies an exponent by $Q$, and its intermediate positions multiply it\nby the $e_i$. These exponents are distinct since\n$1=e_0<\\cdots<e_{\\ell-1}<Q$. Define\n\\[\n G(z)=\\sum_{k\\ge0}\\sum_{i=0}^{\\ell-1}\n          \\frac{z^{e_iQ^k}}{1-z^{e_iQ^k}},\\qquad\n R(z)=\\sum_{i=0}^{\\ell-1}\\frac{z^{e_i}}{1-z^{e_i}}.\n\\]\nThe series for $G$ converges normally on compact subsets of $|z|<1$:\non $|z|\\le r<1$ its absolute sum is at most\n$\\ell(1-r)^{-1}\\sum_{k\\ge0}r^{Q^k}<\\infty$. Its Taylor coefficients\nare integers, and removing the $k=0$ block gives\n$G(z)=R(z)+G(z^Q)$.\nFor the one-term block $(2)$ this is the function\n$g(z)=\\sum_{k\\ge0}z^{2^k}/(1-z^{2^k})$ used below.\n\nThe function $G$ is transcendental over $\\mathbb C(z)$. To see this,\nfix a primitive root of unity $\\zeta$ of order $Q^j$, with $j\\ge1$,\nand approach it along $r\\zeta$ as $r\\uparrow1$. For every $k\\ge j$,\nall terms in the $k$th block are positive real numbers. Already the term\n$k=j,i=0$ tends to $+\\infty$. Among the finitely many earlier terms,\nthose with $\\zeta^{e_iQ^k}=1$ are also positive, and all the others remain\nbounded. Thus $\\operatorname{Re}G(r\\zeta)\\to+\\infty$.\nThe infinitely many distinct orders $Q^j$ give infinitely many singularities,\nwhereas an algebraic function over $\\mathbb C(z)$ has only finitely many.\n\nWe use Nishioka's value theorem for Mahler systems\n\\cite{nishioka1996}, in the precise form quoted by Adamczewski and Faverjon\n\\cite[Theorem~1.1, p.~3]{adamczewskifaverjon2016}.  At an algebraic regular\npoint it equates the transcendence degree of the function values with that\nof the functions over $\\overline{\\Q}(z)$.  Here the system is\n\\[\n \\begin{pmatrix}G(z)\\\\1\\end{pmatrix}\n =\\begin{pmatrix}1&R(z)\\\\0&1\\end{pmatrix}\n  \\begin{pmatrix}G(z^Q)\\\\1\\end{pmatrix}.\n\\]\nFor algebraic real $t>1$, the point $\\alpha=t^{-d}$ is algebraic and lies\nin $(0,1)$. The matrix and its inverse have poles only at roots of unity,\nso none of $\\alpha^{Q^k}$ is a pole and $\\alpha$ is regular.\nNishioka's theorem gives\n$\\operatorname{trdeg}_{\\overline{\\Q}}(G(\\alpha),1)=1$.\nThe discarded finite sum is algebraic, so adding it to $G(\\alpha)$\nproves part (b).\n\\end{proof}\n\nThe accompanying exact coefficient probe checks the proposed functional\nequation through degree $100{,}000$ for five ratio blocks, including $(2,3)$.\nFor that alternating chain, the doubling equation\n$G(z)-G(z^2)=z/(1-z)$ already fails at degree four.\nThe probe also rejects the $(2,3)$ block's equation on an explicit\nnonperiodic ratio word.\nThe probe is in\n\\href{https://github.com/wcook04/plectis-erdos/blob/01de6282/research/experiments/interestingness/periodic_chain_probe.py}\n{\\path{research/experiments/interestingness/periodic_chain_probe.py}}.\nThose finite checks test the formulas; the block decomposition above proves\nthe equation at every degree.\n\n\\begin{theorem}[divisibility chains at every rational base]\\label{thm:chain-transcendence}\nLet $a>b\\ge1$ be coprime integers and let $S=\\{n_1<n_2<\\cdots\\}$ be infinite\nwith $n_j\\mid n_{j+1}$ for every $j$. Then $X_S(a/b)$ is transcendental.\n\\end{theorem}\n\nTheorem~\\ref{thm:chain-transcendence} strengthens the rational-base conclusion\nof Theorem~\\ref{thm:chains}. It removes the hypothesis $a^2>b^3$ from part (a)\nand strengthens the conclusion there from irrationality to transcendence. In\nparticular every divisibility chain at bases $4/3$ and $5/4$ has a\ntranscendental sum, including the chains at base $4/3$ whose ratios are $2$\nexcept for infinitely many $3$s, the first case left open by\nTheorem~\\ref{thm:chains}(a). At rational bases it also gives part (b) without\nMahler's method; part (b) also covers irrational algebraic bases. The proof of Theorem~\\ref{thm:chains}(a) compares one partial\nsum with the whole tail, and at base $4/3$ with ratio $n_{j+1}/n_j=2$ that\ncomparison fails: $D_j(X_S(4/3)-S_j)$ grows like $(9/4)^{n_j}$. The proof below\nkeeps the first few terms of the tail as separate coordinates of an integer\nvector. Those coordinates are products of powers of $a$ and $b$, and by the\nproduct formula they contribute nothing to the product of absolute values over\nthe archimedean place and the primes dividing $ab$. Only one linear form and\nthe coordinate carrying the partial sum remain, and the Subspace Theorem\napplies.\n\n\\paragraph{Evidence.} Ordinary proof, with the $p$-adic Subspace Theorem as an\nexternal premise, awaiting specialist review; it is not checked in Lean. Exact finite checks of the\nformulas used below are in \\path{research/experiments/chain_transcendence/} of\nthe repository. They check the bookkeeping and form no part of the proof.\n\n\\begin{proof}\nAs in the proof of Theorem~\\ref{thm:chains}, write $\\rho=b/a$,\n$D_j=a^{n_j}-b^{n_j}$ and $S_j=\\sum_{i\\le j}b^{n_i}/(a^{n_i}-b^{n_i})$; then\n$D_jS_j$ is a positive integer. Suppose that $x=X_S(a/b)$ is algebraic.\n\n\\emph{The tail.} Fix $j$ and put $N=n_j$ and $M=n_{j+1}$. Each $n_l$ with\n$l>j$ is a multiple of $M$, and $\\rho^n/(1-\\rho^n)=\\sum_{i\\ge1}\\rho^{in}$, so\n\\[\n x-S_j=\\sum_{k\\ge1}c_j(k)\\,\\rho^{kM},\\qquad c_j(k)=\\#\\{l>j:\\ n_l\\mid kM\\}.\n\\]\nSince $n_{j+1}=M$ and $n_{j+s}\\ge2^{s-1}M$, we have $1\\le c_j(k)\\le1+\\log_2k$.\nFor an integer $K\\ge1$ put $E_j=\\sum_{k\\ge K}c_j(k)\\rho^{kM}$. Then\n$0<E_j\\le C_K\\rho^{KM}$, where $C_K=\\sum_{i\\ge0}(1+\\log_2(K+i))\\rho^{i}$ does\nnot depend on $j$.\n\n\\emph{Choice of $K$.} The ratios $r_j=n_{j+1}/n_j$ are integers at least $2$.\nChoose an infinite set $J_0$ of indices on which either $r_j$ equals a\nconstant $r$, or $r_j\\to\\infty$. In the first case fix $K\\ge1$ with\n$Kr\\log(a/b)>\\log a$; at base $4/3$ with $r=2$ the least such $K$ is $3$. In the second\ncase put $K=1$. The vector $(c_j(1),\\dots,c_j(K-1))$ takes only finitely many\nvalues, so it equals a fixed $(c_1,\\dots,c_{K-1})$ for all $j$ in an infinite\nset $J\\subseteq J_0$.\n\n\\emph{Integer points and one linear form.} For $j\\in J$ put $H=a^{N+(K-1)M}$\nand\n\\[\n Y_j=\\bigl(H,\\ H\\rho^{N},\\ a^{(K-1)M}D_jS_j,\\ H\\rho^{M},\\ H\\rho^{N+M},\\ \\dots,\\\n H\\rho^{(K-1)M},\\ H\\rho^{N+(K-1)M}\\bigr)\\in\\Z^{2K+1}.\n\\]\nEvery coordinate except the third is an integer whose prime factors divide\n$ab$; for example $H\\rho^{N+kM}=a^{(K-1-k)M}b^{N+kM}$. For\n$y=(y_1,\\dots,y_{2K+1})$ put\n\\[\n L(y)=xy_1-xy_2-y_3-\\sum_{k=1}^{K-1}c_k\\,(y_{2k+2}-y_{2k+3}).\n\\]\nSince $a^{(K-1)M}D_jS_j=H(1-\\rho^N)S_j$, the expansion of the tail gives\n\\[\n L(Y_j)=H(1-\\rho^N)\\Bigl(x-S_j-\\sum_{k=1}^{K-1}c_k\\rho^{kM}\\Bigr)\n =H(1-\\rho^N)E_j .\n\\]\n\n\\emph{The product over places.} Let $\\Sigma$ consist of the archimedean place\nand the primes dividing $ab$, with $|p|_p=p^{-1}$. At the archimedean place\ntake the $2K+1$ forms $L$ and $y_i$ for $i\\ne3$; at each prime $p\\mid ab$ take\nthe $2K+1$ coordinate forms. The coordinate forms are linearly independent,\nand so are the forms at the archimedean place, because the coefficient of\n$y_3$ in $L$ is $-1$. By the product formula,\n$\\prod_{v\\in\\Sigma}|y|_v=1$ for every nonzero integer $y$ whose prime factors\ndivide $ab$. Hence, in the product of $|F(Y_j)|_v$ over all $v\\in\\Sigma$ and all\nforms $F$ chosen at $v$, only $L$ and the third coordinate at the primes remain:\n\\[\n \\Pi_j=|L(Y_j)|\\prod_{p\\mid ab}\\bigl|a^{(K-1)M}D_jS_j\\bigr|_p\n \\le HE_j\\,a^{-(K-1)M}\\le C_K\\,a^{N}\\rho^{KM},\n\\]\nby $|D_jS_j|_p\\le1$ and $\\prod_{p\\mid ab}|a|_p=a^{-1}$. Also\n$\\|Y_j\\|=\\max_i|Y_{j,i}|\\le\\max(1,x)\\,a^{N+(K-1)M}$. In the first case\n$M=rN$ and $a^N\\rho^{KM}=a^{-\\delta N}$ with $\\delta=Kr\\log(a/b)/\\log a-1>0$,\nso $\\Pi_j\\le\\|Y_j\\|^{-\\varepsilon}$ for all large $j\\in J$, where\n$\\varepsilon=\\delta/(2+2(K-1)r)$. In the second case $K=1$,\n$\\|Y_j\\|\\le\\max(1,x)\\,a^N$ and $a^N\\rho^M\\le a^{-N}$ once\n$r_j\\log(a/b)\\ge2\\log a$, so $\\Pi_j\\le\\|Y_j\\|^{-1/2}$ for all large $j\\in J$.\n\n\\emph{The Subspace Theorem.} We use Schmidt's Subspace Theorem in the\n$p$-adic form due to Schlickewei \\cite{schmidt1972,schlickewei1976}, as stated\nin \\cite[Theorem~2.3]{bilu2008}. Let $\\Sigma$ be a finite set of places of $\\Q$\ncontaining the archimedean one and, for each $v\\in\\Sigma$, let\n$F_{v,1},\\dots,F_{v,m}$ be linearly independent linear forms in $m$ variables\nwith algebraic coefficients. Then for every $\\varepsilon>0$ the vectors\n$y\\in\\Z^m$ with\n$\\prod_{v\\in\\Sigma}\\prod_{i=1}^m|F_{v,i}(y)|_v\\le\\|y\\|^{-\\varepsilon}$ lie in\nfinitely many proper linear subspaces of $\\Q^m$. Here every form has rational\ncoefficients except possibly $L$, at the archimedean place, whose coefficients\nlie in $\\Q(x)$ with $x$ real algebraic. The vectors $Y_j$ are pairwise distinct, so there are\n$\\lambda\\in\\Q^{2K+1}\\smallsetminus\\{0\\}$ and an infinite $J'\\subseteq J$ with\n$\\lambda\\cdot Y_j=0$ for every $j\\in J'$.\n\n\\emph{No fixed relation.} Divide $\\lambda\\cdot Y_j=0$ by $H$ and put\n$z=\\rho^{n_j}$, which tends to $0$ along $J'$. In the first case\n$\\rho^{kM}=z^{kr}$, and substituting $S_j=x-\\sum_{k<K}c_kz^{kr}-E_j$ gives\n$Q(z)=\\lambda_3(1-z)E_j$ for the polynomial\n\\[\n Q(z)=\\lambda_1+\\lambda_3x+(\\lambda_2-\\lambda_3x)z\n +\\sum_{k=1}^{K-1}\\bigl((\\lambda_{2k+2}-\\lambda_3c_k)z^{kr}\n +(\\lambda_{2k+3}+\\lambda_3c_k)z^{1+kr}\\bigr).\n\\]\nBecause $r\\ge2$, the exponents $0$, $1$, $kr$ and $1+kr$ with $1\\le k<K$ are\ndistinct and smaller than $Kr$. The polynomial $Q$ does not depend on $j$ and\n$|Q(z)|\\le|\\lambda_3|C_Kz^{Kr}$ along $J'$, so every coefficient of $Q$\nvanishes; otherwise its lowest nonzero term would dominate as $z\\to0$. Then\n$\\lambda_3(1-z)E_j=0$, and $E_j>0$ gives $\\lambda_3=0$. The coefficients of $Q$\nare now $\\lambda_1$, $\\lambda_2$, $\\lambda_{2k+2}$ and $\\lambda_{2k+3}$, so\n$\\lambda=0$, a contradiction. In the second case the same substitution gives\n$\\lambda_1+\\lambda_3x+(\\lambda_2-\\lambda_3x)z=\\lambda_3(1-z)E_j$ with\n$0<E_j\\le C_1z^{r_j}\\le C_1z^2$, and the same comparison gives $\\lambda=0$.\nTherefore $x$ is transcendental.\n\\end{proof}\n\nThe last step uses that the ratios of the chain are at least $2$. The identity\n\\[\n \\sum_{j\\ge0}\\frac{z^{2^j}}{1-z^{2^{j+1}}}=\\frac{z}{1-z}\\qquad(|z|<1)\n\\]\nshows what happens without this. Let $\\mu_j$ be its $j$th partial sum and\n$w=z^{2^{j+1}}$. Then $(1-w)\\mu_j$ is a polynomial in $z$ with integer\ncoefficients of degree less than $2^{j+1}$, so at $z=b/a$ it becomes an integer\nafter multiplication by $a^{2^{j+1}}$, as $D_jS_j$ does above. The tail,\nhowever, begins at the exponent of $w$ itself: $(1-w)(z/(1-z)-\\mu_j)=w$\nexactly. At $z=b/a$ this is a fixed linear relation between $a^{2^{j+1}}$,\n$b^{2^{j+1}}$ and the cleared partial sum, the situation that the last step\nexcludes when the ratios are at least $2$, and the value is rational.\n\nThe same proof applies when the terms of $X_S(a/b)$ carry bounded positive\ninteger weights, since the tail coefficients $c_j(k)$ then remain positive\nintegers bounded in terms of $k$.\n\nThe argument is an instance of the method of Corvaja and Zannier, who apply\nthe Subspace Theorem when a fixed linear combination of numbers composed of\nfinitely many fixed primes approximates an integer\n\\cite[Theorem~4]{corvajazannier2002}. Here those numbers are products of powers\nof $a$ and $b$, and divisibility along the chain supplies the integer\n$a^{(K-1)M}D_jS_j$. Two older methods bear on parts of\nTheorem~\\ref{thm:chain-transcendence}. When the ratios $r_j$ are unbounded,\nRoth's theorem \\cite{roth1955} suffices: once $r_j\\log(a/b)\\ge3\\log a$, the\nrational $S_j$, of height at most $\\max(1,X_S(a/b))\\,a^{n_j}$, satisfies\n$0<X_S(a/b)-S_j\\le Ca^{-3n_j}$ with $C$ independent of $j$, and irrationality\nfollows as in the proof of Theorem~\\ref{thm:chains}. When the ratios are bounded, the extension of\nMahler's method to chains of functional equations by Loxton and van der\nPoorten \\cite[Theorem~4, p.~38]{loxtonvanderpoorten1977} applies to\n$\\sum_h\\varphi(\\alpha^{n_h})$ with $\\varphi(z)=z/(1-z)$ at algebraic $\\alpha$\nwith $0<|\\alpha|<1$, provided the functions\n$f_k(z)=\\sum_{h\\ge k}\\varphi(z^{n_h/n_k})$ satisfy their hypothesis of strong\ntranscendence, a transcendence condition uniform in $k$. That hypothesis fails\nfor unbounded ratios, since $(1-z)f_k(z)-z$ vanishes at $0$ to order\n$n_{k+1}/n_k$. We have not checked it for bounded ratios; if it holds, their\ntheorem gives such chains at every real algebraic base greater than $1$. We did\nnot find Theorem~\\ref{thm:chain-transcendence} stated in the literature. The\nsearch, with locators, is recorded in\n\\path{research/experiments/chain_transcendence/README.md} and ended on\n26 September 2026.\n\nThe preceding integer-vector argument uses a rational base. The next\nnumber-field argument controls the conjugates explicitly and extends the\nconclusion to every real algebraic base greater than one. It also permits\nnon-chain blocks with separated divisibility cuts. Arbitrary supports,\nincluding the full support at $3/2$, remain outside its hypotheses.\n\n\\subsection{Divisibility cuts at every algebraic base}\n\\label{sec:separated-cuts}\n\nA support need not be a chain for the prefix-clearing argument to work.\nWhat it needs is a place to cut: every earlier exponent divides one integer\n$L$, while every later exponent is a multiple of a larger integer $M$.\nThe prefix then becomes a polynomial of degree at most $L$, and the tail\nhas a bounded number of possible initial coefficient patterns in powers of\n$t^{-M}$. Keeping several of those powers as separate coordinates pays for\nthe height of the prefix, including all its algebraic conjugates.\n\n\\begin{theorem}[Lambert sums across divisibility cuts]\n\\label{thm:separated-cuts}\nLet $H$ be an infinite subset of the positive integers. Suppose that positive\nintegers $L_j,M_j$ satisfy\n\\[\n L_j\\longrightarrow\\infty,\\qquad M_j>L_j,\\qquad M_j-L_j\\longrightarrow\\infty,\n\\]\nand, for every $j$ and $n\\in H$,\n\\[\n n\\le L_j\\ \\Longrightarrow\\ n\\mid L_j,\n \\qquad n>L_j\\ \\Longrightarrow\\ M_j\\mid n.\n\\]\nFor every infinite $B\\subseteq H$, every bounded family of positive integer\nweights $(w_n)_{n\\in B}$, and every real algebraic $t>1$, the sum\n\\[\n \\sum_{n\\in B}\\frac{w_n}{t^n-1}\n\\]\nis transcendental.\n\\end{theorem}\n\nThese hypotheses hold for every increasing divisibility chain, taking\n$L_j=n_j$ and $M_j=n_{j+1}$. They also allow arbitrarily large sets of\npairwise incomparable exponents in one block, as the example below shows.\nThey do not hold for the full positive support. The theorem therefore does\nnot settle the rationality of the full Lambert series at $3/2$.\n\nThe proof uses the number-field Subspace Theorem in the normalization of\nEvertse and Ferretti~\\cite[\\S1.1, (1.1); \\S2.1]{evertseferretti2013}.\nIt extends the preceding argument based on the method of Corvaja and\nZannier~\\cite[Theorem~4]{corvajazannier2002}. This is an ordinary proof;\nneither a Lean proof of the transcendence conclusion nor historical\npriority is asserted.\n\n\\begin{proof}\nWrite $\\rho=t^{-1}\\in(0,1)$ and $1\\le w_n\\le W$. The series converges by\ncomparison with a geometric series. Suppose its value $x$ is algebraic,\nand put $K=\\Q(\\rho,x)$, $d=[K:\\Q]$. Use absolute values $|\\cdot|_v$\nnormalized by the product formula. At the specified real embedding $v_0$,\n$|a|_{v_0}=|a|^{1/d}$. Let $S$ contain the archimedean places and every finite\nplace where $\\rho$ is not a unit. Then $\\rho$ is an $S$-unit. Write\n$h=\\log H(\\rho)>0$, where $H$ is the multiplicative projective height.\n\nFor one cut, abbreviate $L=L_j$, $M=M_j$, and set\n\\[\n s_j=\\sum_{\\substack{n\\in B\\\\n\\le L}}\n       \\frac{w_n\\rho^n}{1-\\rho^n},\n \\qquad z_j=(1-\\rho^L)s_j=P_j(\\rho).\n\\]\nFor $n\\mid L$,\n\\[\n (1-X^L)\\frac{X^n}{1-X^n}=X^n+X^{2n}+\\cdots+X^L.\n\\]\nThus $P_j\\in\\Z[X]$, its degree is at most $L$, and its coefficient sum is\nat most $B_j=WL^2$. Consequently, at a finite place,\n$|z_j|_v\\le\\max(1,|\\rho|_v)^L$; at an archimedean place the additional\nfactor is $B_j^{d_v/d}$, where $d_v$ is its local degree. Since the\narchimedean exponents sum to one,\n\\begin{equation}\\label{eq:cuts-conjugates}\n \\prod_{v\\in S\\smallsetminus\\{v_0\\}}|z_j|_v\\le B_j H(\\rho)^L.\n\\end{equation}\nAll coordinates used below are $S$-integers. Discard finitely many cuts\nso that the prefix is nonempty.\n\nEvery remaining exponent is a multiple of $M$, so\n\\[\n x-s_j=\\sum_{k\\ge1}c_j(k)\\rho^{kM},\\qquad\n c_j(k)=\\sum_{\\substack{n\\in B,\\ n>L\\\\ n\\mid kM}}w_n.\n\\]\nWriting $n=Mr$ shows $0\\le c_j(k)\\le W\\tau(k)\\le Wk$.\nChoose a fixed integer $R\\ge2$ such that\n$\\gamma=R(-\\log\\rho)/d-h>0$. On an infinite subsequence the vector\n$(c_j(1),\\ldots,c_j(R-1))$ is constant; denote it by $(c_1,\\ldots,c_{R-1})$.\nThe remaining tail satisfies\n\\[\n 0<E_j:=x-s_j-\\sum_{k=1}^{R-1}c_k\\rho^{kM}\n \\le C_R\\rho^{RM},\\qquad\n C_R=W\\left(\\frac{R}{1-\\rho}+\\frac{\\rho}{(1-\\rho)^2}\\right).\n\\]\nStrict positivity follows from the infinite positive support, even if\nsome of the frozen coefficients are zero.\n\nConsider the $2R+1$ coordinates\n\\[\n Y_j=(1,\\rho^L,z_j,\\rho^M,\\rho^{L+M},\\ldots,\n          \\rho^{(R-1)M},\\rho^{L+(R-1)M}).\n\\]\nPut $D=L+(R-1)M$. The coordinates $1,\\rho^D$ give the lower height bound;\nthe polynomial estimate for $z_j$ gives the upper bound:\n\\begin{equation}\\label{eq:cuts-height}\n H(\\rho)^D\\le H(Y_j)\\le B_j H(\\rho)^D.\n\\end{equation}\nAt every place of $S$ use the coordinate linear forms, except that at $v_0$\nreplace the $z$-coordinate form by\n\\[\n F(Y)=xY_0-xY_1-Y_z-\\sum_{k=1}^{R-1}c_k(Y_{k,0}-Y_{k,1}).\n\\]\nIts coefficient of $Y_z$ is $-1$, so the forms remain independent, and\n$F(Y_j)=(1-\\rho^L)E_j$. Every other coordinate is an $S$-unit and its\nproduct of absolute values over $S$ is one. Hence\n\\[\n \\prod_{v\\in S}\\prod_i|F_{v,i}(Y_j)|_v\n \\le C_R^{1/d} B_j\\exp(-\\gamma M).\n\\]\nSince $L<M$, $\\log B_j=o(M)$ and\n$\\log H(Y_j)\\le(Rh+1)M$ eventually. The last product is at most\n$H(Y_j)^{-\\epsilon}$ for some fixed $\\epsilon>0$.\nOutside $S$ the local vector norm is exactly one, because the coordinates\nare integral and the first is one. Dividing the displayed product by\n$H(Y_j)^{2R+1}$ therefore gives precisely the normalized hypothesis of\nthe Subspace Theorem. Infinitely many of the $Y_j$ lie in one proper\n$K$-linear subspace, and hence in one fixed nonzero hyperplane.\n\nSubstitute\n\\[\n z_j=(1-\\rho^L)\\left(x-\\sum_{k=1}^{R-1}c_k\\rho^{kM}-E_j\\right)\n\\]\nin that hyperplane equation. It becomes a fixed linear combination of\nthe monomials with exponents\n\\[\n 0,L,M,L+M,\\ldots,(R-1)M,L+(R-1)M\n\\]\nequal to a fixed multiple of $(1-\\rho^L)E_j$. Successive exponent gaps\nare $L$ or $M-L$, both tending to infinity; the remainder starts at $RM$,\nalso a gap $M-L$ beyond the last exponent. If any monomial coefficient\nwere nonzero, divide by the first such monomial and let $j$ tend to\ninfinity. All other terms tend to zero, a contradiction. Thus all these\ncoefficients vanish. The strict inequality $E_j>0$ then forces the\nhyperplane's $z$ coefficient to vanish, and substitution forces every\nother coefficient to vanish. This contradicts the chosen nonzero\nhyperplane and proves the theorem.\n\\end{proof}\n\n\\begin{corollary}[A host of unbounded divisibility width]\n\\label{cor:cuts-wide-host}\nDefine\n\\[\n \\begin{gathered}\n N_0=1,\\quad m_j=2^{N_j},\\quad\n L_j=N_j\\operatorname{lcm}(1,\\ldots,m_j),\\quad N_{j+1}=2L_j,\\\\\n H_* = \\bigcup_{j\\ge0}\\{N_j,2N_j,\\ldots,m_jN_j\\}.\n \\end{gathered}\n\\]\nThen $\\sum_{n\\in H_*}1/n=\\infty$, and $H_*$ is not contained in any\nfinite union of divisibility chains. Nevertheless every infinite subset\nof $H_*$, with bounded positive integer weights, has a transcendental\nLambert sum at every real algebraic base greater than one.\n\\end{corollary}\n\n\\begin{proof}\nEvery exponent in the first $j+1$ blocks divides $L_j$; all later\nexponents are multiples of $N_{j+1}=2L_j$. The blocks are disjoint and\nordered. These are the required cuts, with $M_j=2L_j$. Dyadic grouping\ngives $\\sum_{k\\le2^{N_j}}1/k\\ge N_j/2$, so each block contributes at\nleast $1/2$ to the reciprocal sum. In the upper half of the $j$th block,\ntwo different coefficients have ratio less than two and neither divides\nthe other. The resulting antichains have unbounded cardinality. A union\nof finitely many chains cannot contain them. Apply the theorem.\n\\end{proof}\n\nThis example also lies in the one-prime weighted class used for \\#257.\nFor $h(n)=2^{v_2(n)}$ and every real $t>1$, its weighted mass obeys\n\\[\n \\sum_{n\\in H_*}\\frac{h(n)}{n(t^{h(n)}-1)}\n \\le \\frac{2t^2}{(t-1)^3}.\n\\]\nIndeed, $h_j=h(N_j)$ at least doubles. For $n=N_jk$, the inequality\n$t^{h_jh(k)}-1\\ge h(k)(t^{h_j}-1)$ bounds one block by\n$[h_j/(t^{h_j}-1)]\\sum_{k\\le2^{N_j}}1/(N_jk)$, which is at most\n$2h_j/(t^{h_j}-1)$. Summing over distinct positive integers $h_j$ and\nusing $t^r-1\\ge(t-1)t^{r-1}$ proves the displayed bound. Thus the new\nconclusion here is algebraic-base transcendence on every infinite subset,\nnot merely another instance of the existing integer-base irrationality\ncriterion.\n\nFor a smaller example take\n$H=\\bigcup_{j\\ge0}\\{12^j,2\\cdot12^j,3\\cdot12^j\\}$, with cuts\n$L_j=6\\cdot12^j$, $M_j=12^{j+1}$. Every infinite thinning satisfies the\ntheorem, including the terms $2\\cdot12^p,3\\cdot12^p$ for prime $p$ at\nbase $3/2$. The theorem is hereditary under thinning because the same\ncuts continue to work. Arbitrary supports do not have this property;\nthe earlier rational subsums below base two remain a necessary warning.\n\n\n\\subsection{Base two}\\label{sec:base2}\n\n\\subsubsection{Removed intervals and finite translations}\\label{sec:lemmas}\n\n\nFor the remainder of this subsection, take $t=2$: $w_n=(2^n-1)^{-1}$,\n$R_N=\\sum_{n>N}w_n$, $g_n=w_n-R_n>0$, $X_F=\\sum_{n\\in F}w_n$ for finite $F$,\nand $\\mathcal A$ is the set of all subsums.\n\n\\begin{lemma}[the removed intervals]\\label{lem:gaps}\n$[0,E]\\smallsetminus\\mathcal A$ is the disjoint union, over finite nonempty $F$, of\nthe open intervals $(X_F-g_{\\max F},\\,X_F)$.  Their total length is\n$\\sum_n2^{n-1}g_n=E-1$, and\n$g_n=\\sum_{j\\ge2}\\frac{2^j-2}{2^j-1}2^{-jn}=\\tfrac23 4^{-n}+\\tfrac67 8^{-n}+\\cdots$.\n\\end{lemma}\n\n\\begin{proof}\nFix the digits $\\varepsilon_1,\\dots,\\varepsilon_{n-1}$ and let\n$s=\\sum_{i<n}\\varepsilon_iw_i$.\nThe values with these digits lie in $[s,s+R_{n-1}]$, and they split into\n$[s,s+R_n]$ and $[s+w_n,s+w_n+R_n]$.  The interval between them is\n$(s+R_n,s+w_n)$.  Its right end is $X_F$ with\n$F=\\{i<n:\\varepsilon_i=1\\}\\cup\\{n\\}$ and its length is $g_n$.  Every finite\nnonempty $F$ arises once.  The total length is $E$ minus the measure of\n$\\mathcal A$, which is $1$ \\cite[Section~7]{plectis257}.  The series for $g_n$\nfollows from $w_n=\\sum_{j\\ge1}2^{-jn}$ and\n$R_n=\\sum_{j\\ge1}2^{-jn}/(2^j-1)$.\n\\end{proof}\n\n\\begin{lemma}[translation by a finite subsum]\\label{lem:translate}\nLet $F$ be finite with largest element $n$ and let $0\\le x\\le R_n$.  The greedy\nrule applied to $X_F+x$ selects exactly $F$ among the indices up to $n$ and then\nagrees with the greedy rule applied to $x$.  In particular $X_F+x\\in\\mathcal A$\nif and only if $x\\in\\mathcal A$, and $X_F+x$ is rejected at a step $m>n$ exactly\nwhen $x$ is.\n\\end{lemma}\n\n\\begin{proof}\nAt an index $i\\le n$ the remainder is $X_{F\\cap[i,n]}+x$.  If $i\\in F$ it is at\nleast $w_i$ and the index is taken.  If $i\\notin F$ it is at most\n$R_i-R_n+x\\le R_i<w_i$, so the index is skipped and nothing is rejected.  After\nindex $n$ the remainder is $x$.\n\\end{proof}\n\nTwo consequences.  The map $x\\mapsto x+1$ preserves reduced denominators, so at\nevery step $n\\ge2$ the number of fractions of height at most $Q$ rejected at\nthat step is even; every saved table has this property.  The condition is\n$x\\le R_{\\max F}$.  It is not enough that $X_F+x\\le E$: $x=1/2$ is not rejected\nthrough step $160$, while $1/2+1/3=5/6$ lies in $(R_1,1)$ and is rejected at\nstep $1$.  Counts of rejections should therefore be taken modulo these\ntranslations.  At height $200$ the four fractions rejected at step $12$ are\n$46/183$ and its translates by $1/3$, $1$ and $4/3$, one event.\n\nA finite subsum has odd denominator, since $\\prod_{n\\in F}(2^n-1)$ is odd.  So a\nrational with even denominator is never a finite subsum, and if it is a subsum\nat all its set $S$ is infinite.\n\n\n\\subsubsection{Fixed-depth rational counts}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:forced}]\nFixing the first $N$ digits gives $2^N$ closed intervals of length $R_N$.\nThey are pairwise disjoint because $w_n>R_n$, and a real in $[0,E]$ is not\nrejected in the first $N$ steps exactly when it lies in their union $K_N$,\nwhich has measure $2^NR_N$.  For an interval $I$ of length $\\ell$, the number\nof integers $p$ coprime to $q$ with $p/q\\in I$ is\n$\\varphi(q)\\ell+O(2^{\\omega(q)})$ by inclusion and exclusion.  Summing over\n$q\\le Q$ with $\\sum_{q\\le Q}\\varphi(q)=3Q^2/\\pi^2+O(Q\\log Q)$ and\n$\\sum_{q\\le Q}2^{\\omega(q)}=O(Q\\log Q)$ gives $3Q^2\\ell/\\pi^2+O(Q\\log Q)$.\nApply this to the $2^N$ intervals of $K_N$ and to $(0,E]$ and divide.  The\nsecond statement follows because every subsum lies in every $K_N$ and\n$2^NR_N\\to1$.\n\\end{proof}\n\nThe intervals removed at step $n$ are explicit.  Put $g_n=w_n-R_n$, so that\n$g_n=\\tfrac23 4^{-n}+O(8^{-n})$.  For a finite nonempty $F$ with largest element\n$n$, the interval $(X_F(2)-g_n,\\,X_F(2))$ is removed at step $n$, and\n\\[\n [0,E]\\smallsetminus\\mathcal A=\\bigsqcup_{F\\ne\\varnothing}\\bigl(X_F(2)-g_{\\max F},\\,X_F(2)\\bigr),\n \\qquad\\sum_{n\\ge1}2^{n-1}g_n=E-1,\n\\]\nwhere $\\mathcal A$ is the set of subsums.  By Theorem~\\ref{thm:bases}(b) a\nrational with an infinite $S$ is exactly a counterexample to \\#257 at base $2$.\nSo \\#257 at base $2$ holds if and only if every rational in $[0,E]$ that is not\na finite subsum lies strictly between $X_F(2)-g_{\\max F}$ and $X_F(2)$ for some\nfinite nonempty $F$.\nThis is a one-sided question of approximation by the countable set of finite\nsubsums, with an error that shrinks like $4^{-\\max F}$.\n\nUnder \\#257 the only fractions of height at most $Q$ that are subsums are the\nfinite subsums, $40$ of the $19{,}653$ fractions with $2\\le q\\le200$.  A model\nthat treats later remainders as equidistributed gives the opposite extreme, a\nproportion tending to $1/E$, because the shares $2^{n-1}g_n/E$ of the removed\nintervals are summable.  That model asserts that \\#257 fails for a\npositive proportion of all rationals.  The fixed-depth limiting proportion does\nnot distinguish this claim from its negation.  An arithmetic argument, or a\ncount with separately justified estimates as both depth and height grow, is needed.\nMeasure does not decide either.  Boes, Darst and Erd\\H{o}s construct symmetric\nCantor sets of every measure in $[0,1)$ that contain essentially no rationals\n\\cite{boesdarsterdos1981}.\n\nThe exact computation in \\cite{plectisinvestigation} agrees with\nTheorem~\\ref{thm:forced} step by step.  Among the $19{,}653$ reduced fractions\nwith $2\\le q\\le200$, the numbers rejected at steps $1$, $2$ and $7$ are $4809$,\n$1470$ and $32$, against $4811$, $1467$ and $32$ from the measures of the\nremoved intervals.  The measure-based main term for the number rejected at step $n$ is\n$2^{n-1}g_n\\sum_{2\\le q\\le Q}\\varphi(q)$, asymptotically $(3Q^2/\\pi^2)2^{n-1}g_n$,\nwhich falls below $1$ near $n=2\\log_2Q-2\\log_2\\pi$.  This is not a\ndeterministic cutoff: the error in Theorem~\\ref{thm:forced} does not justify\nsuch an extrapolation.  For example, $189/388$ is first rejected at step $17$,\nbeyond this scale for $Q=388$.  Its selected indices before rejection are\n$F=\\{2,3,7,9,10,14,15,16\\}$, and exact arithmetic gives\n\\[\n R_{17}\\le\\frac{196609}{25769803776}\n <\\frac{189}{388}-X_F(2)\n =\\frac{9291822600689}{1217890317075045460}\n <\\frac1{131071}=w_{17}.\n\\]\nSection~\\ref{sec:misread} supplies the earlier-step checks.  Deeper computation can\ntherefore produce new exclusion certificates; absence of a later rejection\nstill does not prove membership.  No rational is known to have an infinite\n$S$ at base $2$.\n\n\\begin{problem}\\label{prob:membership}\nDecide whether $1/2$ is a subsum of $\\sum(2^n-1)^{-1}$.  By\n\\cite[Theorem~7]{plectis257} this holds if and only if the integer remainders\nof the greedy rule fail to increase at infinitely many steps.\n\\end{problem}\n\nThe rational-point counting papers examined here concern null Cantor sets\nsuch as the middle-third set \\cite{rstw2019,chowvarjuyu2024}.  We did not locate\na theorem settling the present positive-measure subsum problem.\nProblem~\\ref{prob:membership} asks about one explicit rational point.\n\n\n\\section{Which dyadic shifts detect irrationality?}\\label{sec:dyadic-shifts}\n\nConsider a real sequence satisfying\n\\[\n T_{N+1}=2T_N-g_{N+1},\\qquad g_{N+1}\\in\\Z.\n\\]\nWrite $\\|x\\|=\\operatorname{dist}(x,\\Z)$. Iterating the recurrence shows that\n$T_N=2^NT_0-a_N$ for some integers $a_N$, and hence\n% paper-text-number: eq:dyadic-shift-identity=2\n\\begin{equation}\\label{eq:dyadic-shift-identity}\n \\|T_{N+h}-T_N\\|=\\|2^N(2^h-1)T_0\\|.\n\\end{equation}\nFor $H\\subseteq\\Z_{>0}$ and $c\\in\\R$, say that $H$ detects $T$ at threshold\n$c$ when\n% paper-text-number: eq:dyadic-restricted-predicate=3\n\\begin{equation}\\label{eq:dyadic-restricted-predicate}\n \\forall h\\in H\\;\\forall N_0\\;\\exists N\\ge N_0:\\quad\n \\|T_{N+h}-T_N\\|\\ge c.\n\\end{equation}\nThe witnessing index may depend on the shift.\n\nThe Lean-checked transfer from the \\#251 tail classifier and the \\#269\nbounded-radix escape theorem gives the implication from irrationality in\n\\eqref{eq:dyadic-restricted-predicate} for every positive shift when\n$c\\le1/3$; the \\href{\\repobase/research/experiments/premise_exchange/README.md\\#how-the-question-closed}{research record}\nkeeps the original $1/31$ experiment and the later sharp-constant argument\ndistinct. Dubickas's theorem, in the form stated by Akiyama and\nKaneko~\\cite{akiyamakaneko}, gives\n\\[\n \\limsup_{N\\to\\infty}\\|2^N\\xi\\|\\ge\\tau\n \\quad(\\xi\\notin\\Q),\\qquad\n \\tau=\\sum_{n\\ge0}\\frac{t_n}{2^{n+1}}=0.412454\\ldots,\n\\]\nwhere $t_n$ is the parity of the binary digit sum of $n$. This cited input\nextends the implication to every $c<\\tau$. The endpoint argument below is\nordinary mathematics; the sharp bound itself is not formalised here.\n\nFor fixed $c$ and $H$, call \\eqref{eq:dyadic-restricted-predicate} the\nselected-shift test. The next theorem asks when that test detects\nirrationality \\emph{for every} integer-digit dyadic recurrence, rather than\nfor one chosen orbit.\n\n\\begin{theorem}[Restricted dyadic shifts]\\label{thm:restricted-dyadic-shifts}\nFix $c\\in\\R$ and $H\\subseteq\\Z_{>0}$. The following conditions are\nequivalent:\n\\begin{enumerate}\n\\item For every integer-digit dyadic recurrence $T$, the initial value $T_0$\nis irrational if and only if $T$ passes the selected-shift test.\n\\item The parameters satisfy $0<c<\\tau$, and every positive integer $d$\ndivides some shift $h\\in H$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nSuppose first that these two conditions hold. For irrational $T_0$ and\nfixed $h>0$, $(2^h-1)T_0$ is irrational. Equation\n\\eqref{eq:dyadic-shift-identity} and Dubickas's bound give indices as late\nas desired with distance at least $c$. Conversely, let $T_0=p/q$ with\n$q=2^sr$ and $r$ odd. Some $d>0$ satisfies $2^d\\equiv1\\pmod r$; take\n$d=1$ if $r=1$. Choose $h\\in H$ divisible by $d$. For every $N\\ge s$,\n$2^N(2^h-1)p/q$ is an integer, so \\eqref{eq:dyadic-restricted-predicate}\nfails.\n\nFor necessity of the divisibility condition, suppose no member of $H$ is\ndivisible by some $d\\ge2$. Put $L=3d$, $q=2^L-1$, and\n$T_N=\\{2^N/q\\}$. This bounded rational orbit has digits in $\\{0,1\\}$.\nFor a tested shift $h$, let $r\\in\\{1,\\ldots,L-1\\}$ be its residue modulo\n$L$. Since $2^L\\equiv1\\pmod q$, the distance sequence for $h$ is periodic\nand agrees with that for $r$. The sequences for $r$ and $L-r$ agree up to\na cyclic shift, because\n$2^r(2^{L-r}-1)\\equiv-(2^r-1)\\pmod q$. We may therefore use\n$m=\\max(r,L-r)\\ge L/2\\ge3$. At the index $N=L-m-1$, one distance is\n\\[\n v=\\frac{2^{L-1}-2^{L-m-1}}{2^L-1}\n   =\\frac{1-2^{-m}}{2(1-2^{-L})},\n \\qquad \\frac7{16}\\le v<\\frac12.\n\\]\nIt recurs every $L$ indices. The first six Thue--Morse digits are\n$011010$, so $\\tau<27/64<7/16$. Thus this rational orbit passes every\ntested shift at every $0<c<\\tau$.\n\nIf $c\\le0$, the zero orbit passes. If $H$ is empty, every orbit passes.\nFinally, for $c\\ge\\tau$ and any $h_0\\in H$, take\n$T_0=\\tau/(2^{h_0}-1)$ and $g_N=0$. The cited sharp-bound construction\nidentifies $\\tau$ as irrational~\\cite{akiyamakaneko}. The strict endpoint\ninequality $\\|2^N\\tau\\|<\\tau$ for every $N\\ge1$ follows from the\nThue--Morse shift argument just below. Equation\n\\eqref{eq:dyadic-shift-identity} makes the test fail at $h_0$.\n\\end{proof}\n\nHere is the strict endpoint step used in the proof. Write the Thue--Morse\nword as $t=011010\\ldots$, its bitwise complement as $\\bar t$, and let\n$\\mu$ be the order-preserving substitution $0\\mapsto01$, $1\\mapsto10$.\nThe word $t$ is fixed by $\\mu$. An odd-indexed suffix of $t$ starts with\n$001$ or $010$ when its first bit is $0$, both strictly below the prefix\n$011$ of $t$. When its first bit is $1$, it starts with $101$ or $110$,\nboth strictly above the prefix $100$ of $\\bar t$. An even-indexed suffix\nis the image under $\\mu$ of a shorter suffix, so induction and order\npreservation give the same strict comparisons at every positive index.\nThe binary value of each suffix is therefore below $\\tau$ or above\n$1-\\tau$, according to its first bit. This proves\n$\\|2^N\\tau\\|<\\tau$ for $N\\ge1$. The\n\\href{\\repobase/research/experiments/premise_exchange/README.md\\#how-the-question-closed}{research record}\nretains the longer historical calculation. Related\nextremal-word constructions appear in Allouche, Clarke and\nSidorov~\\cite{alloucheclarkesidorov}, whose published bibliography points to\nearlier work of Allouche and Cosnard. The linked research record gives the\nlonger nearest-integer calculation; no historical priority for the\nspecific formulation above is asserted.\n\nThe factorial family $H=\\{j!:j\\ge1\\}$ satisfies the divisibility condition:\n$d\\mid d!$. The power-of-two family does not, since none of its members is\ndivisible by $3$. An especially small counterexample for the latter is the\nrational orbit $T_N=\\{2^N/7\\}$: for each tested shift its distances cycle\nthrough $1/7$, $2/7$, and $3/7$, so it passes at every $c<\\tau$.\nNo finite shift family suffices. Conversely, excluding all multiples of a\nlarge $d$ leaves a family of density $1-1/d$ that fails the test; factorial\nshifts have density zero and succeed. This criterion does not establish\nirrationality for the actual prime-gap tail in \\#251.\n\nDivisor coverage also implies that $H\\cap d\\Z_{>0}$ is unbounded for every\n$d>0$: apply coverage to the multiples $kd$ as $k$ grows. Consequently\ndeleting finitely many shifts from a working family preserves the criterion.\n\n\\subsection{Polynomially selected shifts}\\label{sec:polynomial-shifts}\n\nFor a polynomial $P\\in\\Z[x]$ with positive leading coefficient, set\n\\[\n H_P=\\{P(n):n\\ge0,\\ P(n)>0\\},\\qquad\n H_P^{\\mathrm{prime}}=\\{P(p):p\\text{ prime},\\ P(p)>0\\}.\n\\]\nThe argument of $P$ in the second family is prime; this is different from\nchecking whether $P$ has a root modulo every prime.\n\n\\begin{corollary}[Polynomial shift families]\\label{cor:polynomial-shifts}\nFix $0<c<\\tau$. The $H_P$-selected test detects irrationality for every\ninteger-digit dyadic recurrence if and only if $P$ has a root modulo every\npositive integer. The $H_P^{\\mathrm{prime}}$-selected test has this property\nif and only if, for every positive integer $d$, there is a root $r$ of $P$\nmodulo $d$ with $\\gcd(r,d)=1$.\n\\end{corollary}\n\n\\begin{proof}\nBy Theorem~\\ref{thm:restricted-dyadic-shifts}, each assertion reduces to\nwhether every $d>0$ divides a member of the selected family. If\n$P(r)\\equiv0\\pmod d$, all sufficiently large integers $n\\equiv r\\pmod d$\ngive positive multiples $P(n)$ of $d$. This proves the first assertion in\nboth directions.\n\nFor the prime-argument family, a root $r$ coprime to $d$ gives arbitrarily\nlarge primes $p\\equiv r\\pmod d$ by Dirichlet's theorem, and hence positive\nmultiples $P(p)$ of $d$. Conversely, suppose there is no unit root modulo\nsome $d$. Every prime $p$ for which $d\\mid P(p)$ then satisfies\n$\\gcd(p,d)>1$, so $p$ is one of the finitely many prime divisors of $d$.\nThus $H_P^{\\mathrm{prime}}\\cap d\\Z$ is bounded. Divisor coverage would make\nthis intersection unbounded, since for every $k>0$ it supplies a member\ndivisible by $kd$. This is a contradiction.\n\\end{proof}\n\nThe first condition is the usual intersectivity condition~\\cite{leintersective}.\nThe unit-root condition for prime arguments is $P$-intersectivity, also\ncalled intersectivity of the second kind~\\cite[\\S1.2]{ricepintersective}. For\nexample, $P(n)=n^2$ works with integer arguments, while $n^2+1$ fails\nmodulo $3$. At prime arguments, $P(p)=p$ fails already modulo $6$, while\n$p-1$ and $p^2-1$ work: the residue $1$ is a unit root modulo every $d$.\n\nIntersectivity need not come from an integer root. Mishra lists\n$F(x)=(x^2-13)(x^2-17)(x^2-221)$ as a polynomial with a root modulo\nevery positive integer but no rational root~\\cite{mishraquadratic}. In\nfact, it also has a \\emph{unit} root modulo every positive integer. For\nodd primes other than $13$ and $17$, at least one of $13,17,221$ is a\nnonzero quadratic residue, since $221=13\\cdot17$; its root lifts to every\nprime power. Modulo powers of $13$, use $x^2-17$ with $x\\equiv2\\pmod{13}$;\nmodulo powers of $17$, use $x^2-13$ with $x\\equiv8\\pmod{17}$. At powers\nof $2$, the unit $17\\equiv1\\pmod8$ has a square root. The Chinese\nremainder theorem supplies unit roots modulo arbitrary $d$. Thus both\n$H_F$ and $H_F^{\\mathrm{prime}}$ pass the criterion, without relying on a\nsingle global root.\n\nPrime moduli alone do not suffice for the first condition. Let\n$Q(x)=(x^2-2)(x^2-3)(x^2-6)$. It has a root modulo every prime: for\nodd primes not dividing $6$, if neither $2$ nor $3$ is a square, their\nproduct $6$ is; the primes $2$ and $3$ are immediate. But $Q(n)\\equiv4$\nwhen $n$ is even and $Q(n)\\equiv6$ when $n$ is odd, modulo $8$.\nTherefore neither $H_Q$ nor its prime-argument subfamily contains a multiple\nof $8$. L\\^e's cited arXiv v1 introduction lists this $Q$ as\nintersective~\\cite[p.~1]{leintersective}; the modulo-$8$ calculation\ncorrects that example, without affecting the local-root criterion stated\nthere. The failure is visible without the general counterexample\nconstruction: take the rational orbit\n$T_N=\\{2^N/255\\}$. Since $2^8\\equiv1\\pmod{255}$, for every positive\nshift $h=Q(n)$, indices $N\\equiv3\\pmod8$ when $h\\equiv4\\pmod8$ give\n$\\|T_{N+h}-T_N\\|=120/255$, and indices $N\\equiv1\\pmod8$ when\n$h\\equiv6\\pmod8$ give $126/255$. Both distances exceed\n$7/16>\\tau$, so this rational orbit passes every $Q$-selected test at\n$0<c<\\tau$. The modular root and orbit calculations are ordinary proofs;\nthis polynomial extension is not claimed as Lean checked.\n\n\\section{Limits on methods}\\label{sec:second}\\label{sec:inventory}\n\n\nStatus is as stated in each note: L for checked in Lean there, O for an\nordinary proof there, C for cited there.\n\n\\begin{longtable}{p{0.10\\linewidth}p{0.20\\linewidth}p{0.52\\linewidth}p{0.06\\linewidth}}\n\\hline\nNote & Location & Statement & \\\\\n\\hline\n\\#68 \\cite{plectis68} & Section 6 & Under the displayed cancellation hypotheses, the integer-gap comparison fails at cutoffs $N=D+O(1)$ as the cancellation cutoff $D\\to\\infty$.  Small tails alone do not give the strict comparison. & O \\\\\n\\#243 \\cite{plectis243} & Proposition 17 & A counterexample has errors that are eventually nonzero, relatively small, with unbounded negative parts.  This necessary profile is formalised; the comparison with scalar profiles and the need for denominator compatibility are ordinary discussion. & L, O \\\\\n\\#249 \\cite{plectis249} & Section 5 & A rational series with the totient's values at odd indices, within $2$ at even indices, and sum $5/4$.  Positive tail differences need not be nonintegral. & L \\\\\n\\#249 \\cite{plectis249} & Theorem 9 & Every admissible rank-one quotient stays more than $21/320$ from its target. & L \\\\\n\\#251 \\cite{plectis251} & Proposition 1, Corollary 2 & For every allowance $f(n)\\to\\infty$, sparse nonnegative corrections reach every real in an interval while every fixed modulus eventually divides both the corrections and their cumulative sums.  A bounded allowance is impossible under these congruences.  Sources: \\cite{fridy1966}, \\cite[Lemma~4]{crmarickovac2025}, \\cite[Lemma~5.1]{kovactao2024}, \\cite{vandoornkovac2025}. & O \\\\\n\\#257 \\cite{plectis257} & Section 6 & The small-displacement quantity stays above $1/2$ at full support, where the value is irrational \\cite{erdos1948}.  No proof covering full support can rest on it. & L, O \\\\\n\\#257 \\cite{plectis257} & Section 3 & Every positive divisor cover costs at least $e$ times the mean of $\\log^+$ of its multiplicity.  The averaging method cannot reach the prime support, where irrationality is known at base $2$ \\cite{taoteravainen2025}. & O \\\\\n\\#269 \\cite{plectis269} & Theorem 1 & Nonsingular minors of every order: no finite sum of products separates one exponent from the other two.  Fan posted the two-prime separation \\cite{fan2026comment}; the three-prime statement is the note's. & L \\\\\n\\#1049 \\cite{plectis1049} & Theorem 5 & One family of nonzero integer-polynomial linear forms with common leading degree, coefficient-height and decay bounds at every fixed real $x>1$ has $\\sigma\\le\\delta$; the sufficient cutoff $\\sigma/(\\sigma+\\delta)$ supplied by those estimates is at most $1/2$.  Ordinary proof, with the contradiction step and the comparison with $1/2$ checked in Lean. & O, L \\\\\n\\#1049 \\cite{plectis1049} & Theorem 7 & The stated clearing conditions cannot be met at base $3/2$. & L \\\\\n\\hline\n\\end{longtable}\n\nTheorem~\\ref{thm:dichotomy}(ii) explains interval filling in the \\#251 construction.\nThe \\#249 countermodel is a separate explicit construction with its own\npreserved identities.  The \\#1049 restriction does not exclude stronger\nbase-specific estimates, different families at different bases, or forms\ninvolving several target values.\nThe others bound a method.  We tried to state one inequality that covers\n\\#1049 Theorem 5 and the cover cost of \\#257, a cost of clearing denominators\nagainst the decay gained, and did not find a formulation that survives both\nsets of hypotheses.  We do not claim the rows share a cause.\n\n\n\\section{The computation and its limits}\\label{sec:misread}\n\nThe computation \\cite{plectisinvestigation} runs the greedy rule in exact\narithmetic on every reduced fraction in $(0,E]$ with $2\\le q\\le Q$.\n\n\\emph{First reading.}  At $Q=36$, $382$ of $633$ fractions are not rejected\nthrough step $160$ and $14$ are finite subsums, a share $(382+14)/633=0.6256$\nclose to $1/E=0.6224$.  This was read as evidence that about $62\\%$ of rationals are\nsubsums, hence that \\#257 is false.  The reading is wrong.  By Theorem~\\ref{thm:forced} above the share at any fixed depth tends to $2^NR_N/E$ whatever the truth of\n\\#257, because fractions equidistribute over the $2^N$ intervals that survive\n$N$ steps.\n\n\\emph{An incorrect stopping rule.}  A subsequent interpretation went too far\nin the opposite direction, asserting that survival after about $2\\log_2Q-3.3$\nsteps was forced.  The measure-based main term for the number rejected at step $n$ is\n$2^{n-1}g_n\\sum_{2\\le q\\le Q}\\varphi(q)$, asymptotically\n$(3Q^2/\\pi^2)2^{n-1}g_n$, which is below $1$ for $n>2\\log_2Q-2\\log_2\\pi$, about\nstep $12$ at $Q=200$.  A main term below $1$ does not make the actual count\nzero.  Theorem~\\ref{thm:forced} is a fixed-depth asymptotic, and its error\n$O(2^N\\log Q/Q)$ does not justify an extrapolation to $N\\sim2\\log_2Q$.\nLate rejections remain exact nonmembership certificates; the $12{,}218$\nfractions not rejected through step $60$ have only that finite-depth status.\nAt $N=12$ the observed share is $0.62372$ against\n$2^{12}R_{12}/E=0.62245$.\n\n\\emph{An exact rejection at step $17$.}  The witness $189/388$, recorded in the\nearlier investigation's Desk B report, contradicts the proposed stopping rule:\n$2\\log_2 388-2\\log_2\\pi$ is about $13.9$.  The selected indices through step $16$\nare $F=\\{2,3,7,9,10,14,15,16\\}$.  At each skipped earlier index $n$, exact\nrational arithmetic gives a remainder at most $2^{-n}<R_n$, so no rejection\nhas yet occurred.  The remaining value is\n\\[\n r=\\frac{189}{388}-X_F\n  =\\frac{9291822600689}{1217890317075045460}.\n\\]\nSince $1/(2^k-1)=2^{-k}+4^{-k}/(1-2^{-k})$ and\n$1/(1-2^{-k})\\le2$ for $k\\ge1$, summing gives\n\\[\n R_n\\le2^{-n}+\\frac{2}{3\\cdot4^n}.\n\\]\nAt $n=17$, the exact comparison is\n\\[\n R_{17}\\le\\frac{196609}{25769803776}<r<\\frac1{131071}=w_{17}.\n\\]\nThe same rejection occurs for $577/388=1+189/388$ by\nLemma~\\ref{lem:translate}.  The independent reproduction, including every\nearlier skipped step, is in\n\\href{\\repobase/research/experiments/sparse_interpolation/late_rejection.py}\n{\\path{research/experiments/sparse_interpolation/late_rejection.py}}.\nThis certificate shows that deeper computation can add exclusions.  It does\nnot convert survival to any finite depth into a membership certificate.\n\nWhat survives is the agreement itself.  At $Q=200$ the counts at steps $1$ to\n$9$ are $4809$, $1470$, $600$, $268$, $132$, $66$, $32$, $8$, $6$, against\n$4811$, $1467$, $604$, $277$, $133$, $65$, $32$, $16$, $8$ from the measures of\nLemma~\\ref{lem:gaps}; steps $10$ and $11$ have none against $4$ and $2$, and\nstep $12$ has $4$ against $1$.  The late counts fluctuate more than independent\nevents would, because rejections arrive in the families of\nLemma~\\ref{lem:translate}.\n\n\\section{Eliminated ideas}\\label{sec:eliminated}\n\n\\begin{enumerate}\n\\item \\emph{If the reachable values form a null set, no rational value is\n  reachable.}  False.  Take the binary series with digit $1$ everywhere except\n  digit $0$ at positions $n_1<n_2<\\cdots$, and allow each of those digits to be\n  changed to $1$.  The reachable values form a null set when the positions are\n  sparse, by Theorem~\\ref{thm:dichotomy}(i), yet changing all of them gives $\\sum2^{-n}=1$.\n  So no count of choices against contraction, and no depth depending only on\n  sparsity, excludes a particular rational.\n\\item \\emph{Every infinite subset of a host with null subsum set has an\n  irrational sum.}  A host is a set $B$ of allowed indices, and its subsum set\n  is $\\{X_S(2):S\\subseteq B\\}$.  For hosts chosen without reference to the target this is\n  open and is a form of \\#257 itself.  As a universal statement it cannot be a\n  route: the support of any rational subsum with infinite $S$ would be such a\n  host.  The subsum set of a host $B$ has positive measure exactly when the\n  complement of $B$ is finite, by Theorem~\\ref{thm:dichotomy}(i) and the measure at full\n  support.\n\\item \\emph{The share of surviving fractions as evidence.}  Section~\\ref{sec:misread}.\n\\item \\emph{A wrong locator.}  A draft of Theorem~\\ref{thm:chains}(b) cited Theorem~\\ref{thm:dichotomy} of\n  \\cite{adamczewskifaverjon2026} as Nishioka's theorem.  That theorem says a\n  value of a Mahler function at an algebraic point is rational or\n  transcendental, which cannot prove irrationality.  The proof in Section~\\ref{sec:bases} uses\n  Nishioka's value theorem as quoted in\n  \\cite[Theorem~1.1, p.~3]{adamczewskifaverjon2016}, applied there to the\n  two-dimensional system for $(g,1)$ with regular points in $(0,1)$.\n\\item \\emph{Algebraic independence for \\#1049.}  With $g$ as in Section~\\ref{sec:bases},\n  $\\sum_{n\\ge1}(t^n-1)^{-1}=\\sum_{m\\ \\mathrm{odd}}g(t^{-m})$, and each $g(t^{-m})$\n  is transcendental for rational $t>1$.  This gives nothing for the infinite\n  sum: limits of transcendental numbers take every value.  No applicable\n  value theorem for this decomposition is supplied here.\n\\end{enumerate}\n\n\\section{What was already known}\\label{sec:known}\n\n\\begin{itemize}\n\\item The covering argument of Theorem~\\ref{thm:dichotomy}(ii) goes back to Kakeya; see\n  \\cite{hornich1941,nitecki2013,fridy1966}.  Its use to build rational series\n  inside a class defined by soft data is the method of Kova\\v{c} and Tao\n  \\cite{kovactao2024}, of Crmari\\'{c} and Kova\\v{c} \\cite{crmarickovac2025} and\n  of van Doorn and Kova\\v{c} \\cite{vandoornkovac2025}.\n\\item That the subsums of $\\sum(t^n-1)^{-1}$ form a Cantor set at fixed\n  $t\\ge2$ is \\cite[Remark~4.1]{kovactao2024}.\n\\item A closed set of positive measure can contain essentially no rationals\n  \\cite{boesdarsterdos1981}, so the heuristic of Section~\\ref{sec:base2} cannot be\n  a consequence of measure.\n\\item The rational-point counting papers examined here concern null Cantor\n  sets \\cite{rstw2019,chowvarjuyu2024}.  We did not locate a theorem settling\n  the present positive-measure subsum problem.  The searches were made on\n  20 September 2026 and are listed in the repository record.\n\\item One identity with a consequence.  Since\n  $\\sum_{n\\ge1}\\mu(n)/(b^n-1)=1/b$, the sums of $(b^n-1)^{-1}$ over squarefree\n  $n$ with an even, respectively odd, number of prime factors are\n  $\\tfrac12(X_{\\mathrm{sf}}(b)\\pm1/b)$.  Duverney and Tachiya prove that\n  $X_{\\mathrm{sf}}(2^j)$ is irrational \\cite[Corollary~1.2 and\n  Example~1.1]{duverneytachiya}, as quoted in \\cite[Section~6]{plectis257}, so\n  both sums are irrational at every base $2^j$.  Both supports have divergent\n  reciprocal sums.  The identity at base $2$ is derived in\n  \\cite[Section~5]{plectis249}; the corollary may be known.\n\\end{itemize}\n\n\n\\section{Verification and sources}\\label{capacity:sec:sources}\nThe starting point is the sparse perturbation construction accompanying\nErd\\H{o}s Problem \\#251 in this repository, especially its\n\\href{erdos-251-prime-gap-dyadic-series.pdf}{short paper} and\n\\href{\\repobase/lean/ErdosProblems/Erdos251/ResidueFeedbackCore.lean}{\\texttt{ResidueFeedbackCore.lean}}.\nThe latter already proves residue-dependent selection and an abstract infinite\nsum endpoint. An operator-supplied review supplied the form of\nLemma~\\ref{capacity:lem:feedback}, the sharp exponential support constants, and the\nlinear/superlinear factorial contrast. Those ingredients are credited to that\nreview, not presented as discoveries of this paper. The extensions developed here are the exact capacity criterion on arbitrary\nstrict integer divisibility chains, the factorial support classification and\nits integer-exponent comparison, the common-divisor formulation, and the\nderivative dimension and rationality thresholds proved above.\n\nThe derivative theorem combines residue feedback from \\#251 with the\ntreatment of factorial carries in \\#68. Identity~\\eqref{capacity:eq:jet-carry}\ninterprets a carry as multiplication by a polynomial vanishing at the\nevaluation point. This produces the independent higher derivative instead of\nassuming its availability. The moving coordinates in\n\\eqref{capacity:eq:moving-taylor} make the full-dimensional critical\nconstruction possible; the counting obstruction proves its measure is zero.\nConversely, division by the same vanishing polynomial preserves integer\nfactorial coefficients and eventual divisibility. The complementary tail\nformula then proves rational-derivative rigidity at precisely the same\nthreshold. These are explicit\ntransfers between representations, not evidence for a general improvement in\nautomated discovery. The other programmes motivated comparison of rank,\narithmetic and analytic obstructions; their endpoints are not premises here.\n\nHurwitz functions and interpolation by vanishing polynomials are classical.\nWaldschmidt's survey \\cite[Sections 2--3]{waldschmidt2022} describes growth\nand multipoint derivative questions. Here we fix a polynomial bound on the\ninteger Taylor coefficients and study the dimension and interior of a finite\nreal derivative image, as well as rationality of finitely many derivatives\nunder eventual coefficient divisibility. This last hypothesis is restrictive:\nit is not an unrestricted rational-value theorem for Hurwitz functions.\nFor the classical scalar dimension formula, see\nWegmann \\cite[Satz 4, p.~625]{wegmann1968}, who credits \\v{S}al\\'at.\nOur additional task is to separate a joint derivative image after several\ncarry constructions have been added. Neither the definition of a Hurwitz\nfunction nor the mass-distribution argument is new. The precise dimension\nformula and the two sharp thresholds are proved here; historical priority\nremains unestablished.\n\nAirey, Mance and Vandehey already use digit sets eventually divisible by every\nfixed integer while retaining asymptotically full digit entropy\n\\cite[Section 6, p.~1321]{amv2015}. Their theorem concerns normality and\nHausdorff dimension for chosen Cantor bases. Here a fixed divisibility chain and arbitrary summable allowances are given,\nand the conclusion distinguishes interval filling from nullity and meagreness. These are elementary arguments in the\nclassical theory of Cantor series and achievement sets; historical novelty of\nthe exact classification is not established. Classical interval\ncovering is background, rather than a contribution claimed here. A literature\ncomparison and the reproducible checks are recorded in\n\\href{\\repobase/research/experiments/sparse_interpolation/README.md}{the accompanying research record}.\nThe work was developed with AI assistance and mathematical cross-checking by\nseparate agent passes; that does not constitute independent expert review.\n\nThe formal module\n\\href{\\repobase/lean/ErdosProblems/Synthesis/CongruenceInterpolation.lean}{\\texttt{CongruenceInterpolation.lean}}\nuses the existing feedback module and states the common-divisor obstruction\nfor a real carry recurrence. The analytic identification of that recurrence\nwith the Cantor series, and the capacity and support classifications, are\nthe ordinary proofs above. The module\n\\href{\\repobase/lean/ErdosProblems/Synthesis/FeedbackContinuation.lean}{\\texttt{FeedbackContinuation.lean}}\nreuses the existing interval-feedback endpoint and proves eventual individual\nand cumulative divisibility from nested cofinal moduli; choosing the moduli\nand continuation intervals remains part of the ordinary proof. See the research record for the exact build status and source\nrevision. No claim about any of the eight Erd\\H{o}s programmes changes;\nin particular factorial denominators $n!$ here are not $n!-1$ from \\#68.\n\n\nThe accompanying module\n\\href{https://github.com/wcook04/plectis-erdos/blob/2bd14835b940b3c695a9cc8828c7e2967d65c053/lean/ErdosProblems/Synthesis/FactorialJet.lean}{\\texttt{FactorialJet.lean}}\nchecks the finite\nfactorial-carry identity and its first weighted version, including endpoint\nterms, and preservation of divisibility. It does not formalise\nTheorems~\\ref{capacity:thm:jets} and~\\ref{capacity:thm:rational-jets}, their\nlimits or dimension proof. The\nexact-arithmetic script \\texttt{jets.py} checks the formulas at specified\nfinite degrees, both quotient formulas, and independently checks the uniform tail majorant in\n\\eqref{capacity:eq:moving-taylor}. These finite degree tests are not the\nproof for every $d$.\n\nThe capacity criterion already covers non-power and oscillating allowances.\n\\section{Further questions}\\label{sec:questions}\n\n\n\\begin{enumerate}\n\\item Is $1/2$ a subsum of $\\sum(2^n-1)^{-1}$?  The exact obligation is in\n  \\cite[Theorem~7]{plectis257}.  By Lemma~\\ref{lem:gaps} the general question is\n  one-sided approximation of a rational by finite subsums $X_F$ to within\n  $g_{\\max F}$.\n\\item Does the count of fractions of height at most $Q$ rejected at step $n$\n  stay close to $(3Q^2/\\pi^2)2^{n-1}g_n$ in the joint range\n  $n\\le(2-\\varepsilon)\\log_2Q$, counted modulo the translations of\n  Lemma~\\ref{lem:translate}?  A persistent excess would be the first sign of an\n  arithmetic mechanism for \\#257.\n\\item Theorem~\\ref{thm:separated-cuts} settles every divisibility chain at\n  every real algebraic base $t>1$, and includes non-chain hosts of unbounded\n  width. Which supports lacking separated cuts admit comparable control of\n  the cleared prefix height and the initial tail patterns? The full support\n  at $3/2$ still has neither conclusion nor such a transfer here.\n\\item Is there one inequality behind \\#1049 Theorem 5 and the cover cost of\n  \\#257?\n\\end{enumerate}\n\nThe capacity criterion already covers non-power and oscillating allowances.\nFor factorial gaps of fixed length $m$, a bounded multiple of $n^m$ still\ngives the lattice obstruction, whereas $n^mL(n)$ with $L(n)\\to\\infty$\npermits interval filling. A further question concerns the null case:\nwhat finer tail data determine its Hausdorff dimension? The criterion itself\ndoes not separate dimension zero from full-dimensional null sets. Outside\ninteger divisibility chains the prefix lattice changes, so no corresponding\nnecessity is asserted here.\n\\begin{thebibliography}{99}\n\\bibitem{leintersective} T.~H. L\\^e,\n\\emph{Intersective polynomials and the primes},\n\\href{https://arxiv.org/abs/0910.1880v1}{arXiv:0910.1880v1}.\n\n\\bibitem{ricepintersective} A.~Rice,\n\\emph{S\\'ark\\\"ozy's theorem for $P$-intersective polynomials},\n\\href{https://arxiv.org/abs/1111.6559}{arXiv:1111.6559}.\n\n\\bibitem{mishraquadratic} B.~Mishra,\n\\emph{Polynomials consisting of quadratic factors with roots modulo any\npositive integer}, Amer. Math. Monthly \\textbf{129} (2022), 178--182,\n\\href{https://arxiv.org/abs/2102.08379}{arXiv:2102.08379}.\n\\bibitem{akiyamakaneko} S.~Akiyama and H.~Kaneko, \\emph{Multiplicative analogue\nof Markoff--Lagrange spectrum and Pisot numbers}, Adv.\\ Math.\\ 380 (2021),\n107547, p.~3, reporting the theorem of A.~Dubickas,\n\\href{https://arxiv.org/abs/1911.06170v6}{arXiv:1911.06170v6}.\n\n\\bibitem{alloucheclarkesidorov} J.-P.~Allouche, M.~Clarke and N.~Sidorov,\n\\emph{Periodic unique beta-expansions: the Sharkovski\\u{\\i} ordering},\nErgodic Theory Dynam.\\ Systems 29 (2009), 1055--1074,\n\\href{https://doi.org/10.1017/S0143385708000746}{doi:10.1017/S0143385708000746}.\n\n\\bibitem{erdosgraham1980} P.~Erd\\H{o}s and R.~L. Graham,\n  \\href{https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf}{\\emph{Old\n  and New Problems and Results in Combinatorial Number Theory}},\n  Monogr. Enseign. Math. 28, Geneva, 1980.\n\n\\bibitem{erdos1988} P.~Erd\\H{o}s, \\emph{On the irrationality of certain series:\n  problems and results}, in A.~Baker (ed.), \\emph{New Advances in Transcendence\n  Theory}, Cambridge UP, 1988, pp.~102--109,\n  doi:\\href{https://doi.org/10.1017/CBO9780511897184.009}{10.1017/CBO9780511897184.009}.\n\n\\bibitem{erdos1968} P.~Erd\\H{o}s,\n  \\href{https://users.renyi.hu/~p_erdos/1969-09.pdf}{\\emph{On the\n  irrationality of certain series}}, Math.\\ Student 36 (1968), 222--226\n  (issued 1969).\n\n\\bibitem{hornich1941}\nH.~Hornich,\n\\emph{\\\"Uber beliebige Teilsummen absolut konvergenter Reihen},\nMonatshefte f\\\"ur Mathematik und Physik \\textbf{49} (1941), 316--320.\n\\href{https://doi.org/10.1007/BF01707309}{doi:10.1007/BF01707309}.\n\n\\bibitem{nitecki2013} Z.~Nitecki,\n  \\href{https://arxiv.org/abs/1106.3779v2}{\\emph{Subsum sets: intervals, Cantor\n  sets, and Cantorvals}}, arXiv:1106.3779v2 (2013).\n\n\\bibitem{fridy1966} John A. Fridy, \\href{https://www.fq.math.ca/Scanned/4-3/fridy.pdf}{\\emph{Generalized bases for the real numbers}}.\n  The Fibonacci Quarterly \\textbf{4} (1966), no.~3, 193--201.\n\n\\bibitem{crmarickovac2025} T.~Crmari\\'{c} and V.~Kova\\v{c},\n  \\href{https://arxiv.org/abs/2504.18712v1}\n  {\\emph{On the irrationality of certain super-polynomially decaying series}},\n  Colloq. Math. \\textbf{179} (2025), 55--68,\n  doi:\\href{https://doi.org/10.4064/cm9628-5-2025}{10.4064/cm9628-5-2025}.\n  Theorem locators follow arXiv:2504.18712v1.\n\n\\bibitem{kovactao2024} Vjekoslav Kova\\v{c} and Terence Tao, \\href{https://arxiv.org/abs/2406.17593v4}{\\emph{On several irrationality problems for Ahmes series}}.\n  Acta Mathematica Hungarica \\textbf{175} (2025), 572--608, doi:\\href{https://doi.org/10.1007/s10474-025-01528-0}{10.1007/s10474-025-01528-0}; arXiv:\\href{https://arxiv.org/abs/2406.17593v4}{2406.17593v4}. Statement and section numbers refer to arXiv version 4.\n\n\\bibitem{zudilin2004} W.~Zudilin,\n  \\href{https://geodesic.mathdoc.fr/articles/10.4064/aa111-2-4/}{\\emph{Heine's\n  basic transform and a permutation group for \\(q\\)-harmonic series}},\n  Acta Arith. \\textbf{111} (2004), no.~2, 153--164,\n  doi:\\href{https://doi.org/10.4064/aa111-2-4}{10.4064/aa111-2-4}.\n\n\\bibitem{taoteravainen2025} T.~Tao and J.~Ter\\\"av\\\"ainen,\n  \\href{https://arxiv.org/abs/2512.01739v2}{\\emph{Quantitative correlations and\n  some problems on prime factors of consecutive integers}}, arXiv:2512.01739v2\n  (submitted December 2025, revised April 2026).\n\n\\bibitem{fan2026comment} Steve Fan,\n  \\emph{Comment on Erd\\H{o}s Problem \\#269, thread 269, post 7218} (2026),\n  \\href{https://www.erdosproblems.com/forum/thread/269#post-7218}{source}. 26 June 2026.\n\n\\bibitem{mahler1929} K.~Mahler, \\emph{Arithmetische Eigenschaften der L\\\"osungen\n  einer Klasse von Funktionalgleichungen}, Math.\\ Ann.\\ \\textbf{101} (1929),\n  342--367.\n\n\\bibitem{adamczewskifaverjon2016} B.~Adamczewski and C.~Faverjon,\n  \\href{https://arxiv.org/abs/1508.07158v2}{\\emph{M\\'ethode de Mahler : relations\n  lin\\'eaires, transcendance et applications aux nombres automatiques}},\n  arXiv:1508.07158v2 (2016), Theorem~1.1, p.~3.\n\n\\bibitem{nishioka1996} Ku.~Nishioka, \\emph{Mahler Functions and Transcendence},\n  Lecture Notes in Math.\\ 1631, Springer, Berlin, 1996.\n\n\\bibitem{schmidt1972} W.~M. Schmidt, \\emph{Norm form equations}, Ann.\\ of\n  Math.\\ (2) \\textbf{96} (1972), 526--551.\n\n\\bibitem{schlickewei1976} H.~P. Schlickewei, \\emph{Die $p$-adische\n  Verallgemeinerung des Satzes von Thue--Siegel--Roth--Schmidt}, J.\\ Reine\n  Angew.\\ Math.\\ \\textbf{288} (1976), 86--105.\n\n\\bibitem{bilu2008} Yu.~F. Bilu, \\emph{The many faces of the subspace theorem\n  [after Adamczewski, Bugeaud, Corvaja, Zannier\\dots]}, S\\'eminaire Bourbaki,\n  Vol.\\ 2006/2007, Exp.\\ No.~967, Ast\\'erisque \\textbf{317} (2008), 1--38;\n  \\href{https://arxiv.org/abs/0907.2098}{arXiv:0907.2098}. Theorem~2.3 is the\n  statement used here.\n\n\\bibitem{corvajazannier2002} P.~Corvaja and U.~Zannier, \\emph{Some new\n  applications of the subspace theorem}, Compositio Math.\\ \\textbf{131}\n  (2002), 319--340,\n  doi:\\href{https://doi.org/10.1023/A:1015594913393}{\\nolinkurl{10.1023/A:1015594913393}}.\n\n\\bibitem{loxtonvanderpoorten1977} J.~H. Loxton and A.~J. van der Poorten,\n  \\emph{Arithmetic properties of certain functions in several variables III},\n  Bull.\\ Austral.\\ Math.\\ Soc.\\ \\textbf{16} (1977), 15--47,\n  doi:\\href{https://doi.org/10.1017/S0004972700022978}{\\nolinkurl{10.1017/S0004972700022978}}.\n\n\\bibitem{roth1955} K.~F. Roth, \\emph{Rational approximations to algebraic\n  numbers}, Mathematika \\textbf{2} (1955), 1--20.\n\n\\bibitem{boesdarsterdos1981} D.~Boes, R.~Darst and P.~Erd\\H{o}s,\n  \\href{https://www.tandfonline.com/doi/abs/10.1080/00029890.1981.11995266}{\\emph{Fat,\n  symmetric, irrational Cantor sets}}, Amer.\\ Math.\\ Monthly \\textbf{88} (1981),\n  340--341.\n\n\\bibitem{rstw2019} A.~Rahm, N.~Solomon, T.~Trauthwein and B.~Weiss,\n  \\href{https://arxiv.org/abs/1909.01198}{\\emph{The distribution of rational\n  numbers on Cantor's middle thirds set}}, arXiv:1909.01198.\n\n\\bibitem{chowvarjuyu2024} S.~Chow, P.~Varj\\'u and H.~Yu,\n  \\href{https://arxiv.org/abs/2402.18395}{\\emph{Counting rationals and\n  diophantine approximation in missing-digit Cantor sets}}, arXiv:2402.18395.\n\n\\bibitem{plectis68} W.~Cook,\n  \\href{erdos-68-factorial-denominator-irrationality.pdf}{\\emph{Two\n  incomparable denominator exclusions for $\\sum_{n\\ge2}(n!-1)^{-1}$}},\n  Erd\\H{o}s Problem Notes, 2026.\n\n\\bibitem{plectis243} W.~Cook,\n  \\href{erdos-243-reciprocal-tail-rigidity.pdf}{\\emph{Bounded\n  increments and rational reciprocal sums}}, Erd\\H{o}s Problem Notes, 2026.\n\n\\bibitem{plectis249} W.~Cook,\n  \\href{erdos-249-binary-totient-series.pdf}{\\emph{Bases and\n  integral relations for the $k$-kernel of Euler's totient}}, Erd\\H{o}s Problem\n  Notes, 2026.\n\n\\bibitem{plectis251} W.~Cook,\n  \\href{erdos-251-prime-gap-dyadic-series.pdf}{\\emph{Sparse\n  congruence-preserving perturbations of dyadic series}}, Erd\\H{o}s Problem\n  Notes, 2026.\n\n\\bibitem{plectis257} W.~Cook,\n  \\href{erdos-257-mersenne-support-subseries.pdf}{\\emph{Weighted\n  support criteria for reciprocal Mersenne subseries}}, Erd\\H{o}s Problem Notes,\n  2026.\n\n\\bibitem{plectis269} W.~Cook,\n  \\href{erdos-269-three-prime-running-lcm.pdf}{\\emph{No\n  finite separable representation at three prime generators}}, Erd\\H{o}s Problem\n  Notes, 2026.\n\n\\bibitem{plectis1049} W.~Cook,\n  \\href{erdos-1049-rational-base-lambert.pdf}{\\emph{Zudilin's\n  forms at rational bases and the exact normalised Hankel order}}, Erd\\H{o}s\n  Problem Notes, 2026.\n\n\\bibitem{plectisinvestigation} W.~Cook,\n  \\emph{Choices, contraction and rational membership}, exact computation with saved outputs in \\texttt{research/experiments/choices\\_contraction/} of the repository,\n  2026.\n\n\\bibitem{erdos1948} P.~Erd\\H{o}s, \\emph{On arithmetical properties of Lambert\n  series}, J.~Indian Math.\\ Soc.\\ 12 (1948), 63--66.\n\n\\bibitem{vandoornkovac2025} Wouter van Doorn and Vjekoslav Kova\\v{c}, \\href{https://arxiv.org/abs/2509.24971v3}{\\emph{Lacunary sequences whose reciprocal sums represent all rational numbers in an interval}}.\n  Acta Arithmetica \\textbf{223} (2026), 275--295, doi:\\href{https://doi.org/10.4064/aa251001-13-1}{10.4064/aa251001-13-1}; arXiv:\\href{https://arxiv.org/abs/2509.24971v3}{2509.24971v3}. Statement numbers refer to arXiv version 3.\n\n\\bibitem{duverneytachiya} D.~Duverney and Y.~Tachiya,\n  \\href{https://danielduverney.fr/documents/theorie-des-nombres/DuverneyTachiya190522.pdf}\n  {\\emph{Refinement of the Chowla--Erd\\H{o}s method and linear independence\n  of certain Lambert series}},\n  Forum Math.\\ 31 (2019), no.~6, 1557--1566,\n  \\href{https://doi.org/10.1515/forum-2018-0299}{DOI}.  Page numbers refer to\n  the linked author preprint.\n\n\\bibitem{adamczewskifaverjon2026} Boris Adamczewski and Colin Faverjon,\n  \\emph{Mahler's method in several variables and finite automata},\n  Annals of Mathematics \\textbf{204}, no.~2 (2026), 455--533,\n  \\href{https://doi.org/10.4007/annals.2026.204.2.1}{doi:\\nolinkurl{10.4007/annals.2026.204.2.1}}.\n  Locators refer to the\n  \\href{https://faverjon.perso.math.cnrs.fr/AdamczewskiFaverjon_MahlerFiniteAutomata.pdf}{68-page author manuscript}.\n\n\\bibitem{amv2015} D.~Airey, B.~Mance and J.~Vandehey,\n\\emph{Normality preserving operations for Cantor series expansions and\nassociated fractals. II}, New York J. Math. \\textbf{21} (2015), 1311--1326.\n\\url{https://nyjm.albany.edu/j/2015/21-60v.pdf}.\n\\bibitem{waldschmidt2022} M.~Waldschmidt,\n\\emph{Integer-valued functions, Hurwitz functions, and related topics: a survey},\nin K.~Kur\\c{s}ung\\\"oz and A.~Zeytin (eds.), \\emph{Number Theory: Proceedings\nof the Journ\\'ees Arithm\\'etiques 2019}, De Gruyter, 2022, pp.~61--82.\n\\url{https://doi.org/10.1515/9783110761115-005}.\n\\bibitem{wegmann1968} H.~Wegmann,\n\\emph{Die Hausdorffsche Dimension von Mengen reeller Zahlen, die durch\nZifferneigenschaften einer Cantorentwicklung charakterisiert sind},\nCzechoslovak Math. J. \\textbf{18} (1968), 622--632.\n\\url{https://dml.cz/handle/10338.dmlcz/100861}.\n\\bibitem{evertseferretti2013} J.-H. Evertse and R. Ferretti,\n  \\emph{A further improvement of the quantitative Subspace Theorem},\n  Ann. of Math. \\textbf{177} (2013), 513--590.\n  Normalizations refer to \\S1.1, (1.1), and \\S2.1 of the\n  \\href{https://pub.math.leidenuniv.nl/~evertsejh/10-subspace.pdf}{3 May 2012 author version}.\n\n\\end{thebibliography}\n\n\\section*{Declaration of generative AI use}\nThe mathematics and text were developed with large language model agents\nunder Will Cook's direction. The first-page disclosure states his review\nboundary. The earlier synthesis record reports separate agent proof checks,\nindependent reruns of the exact computations, and source checks on 20 September\n2026; the failed citation is retained in Section~\\ref{sec:eliminated}.\nThis consolidation preserves those arguments and their evidence classes.\nIt is not an independent mathematical review.\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":120067,"sha256":"sha256:83e0788b73c2144a6a86fbd5b0e09b3d4e6a2ce999ade2ba3fb5d36437be9eaf","cite_as":"Plectis reader corpus, paper:optimal-sparse-perturbations","source_repository":"plectis-erdos","source_path":"paper/synthesis/optimal-sparse-perturbations.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:088d36d7ba3d3ee0cee2df99e8915927999698f19889f8ea0b8d55b3721ec778","byte_count":6923,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% This is the only copy of the shared preamble.  The eight short notes, the\n% long records assembled from paper/reasoning-parts other than #249 and #257\n% (which carry their own preambles), and the synthesis paper include it with\n% \\input{problem-note-preamble}; paper/Makefile builds from this directory\n% with the directory on the tectonic search path.\n%\n% Shared preamble for the Erdős Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erdős problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{8e58c395e9807ef3374d7365c62cf04d20ee6331}\n\\newcommand{\\commitshort}{9e231ce4371f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-erdos/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-erdos/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\usepackage{longtable}  % the notes' declaration tables; loaded here so the assembled records inherit it\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n% ---------------------------------------------------------------------------\n% Evidence marks.  paper-evidence.tex places \"Lean\" and \"Comparator\" links in\n% the margin beside each result from the paper's generated file\n% paper/evidence/<paper>.tex, which each paper inputs after this preamble.\n% ---------------------------------------------------------------------------\n\\input{paper-evidence}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Production provenance.  The wording is fixed by the corpus provenance\n% contract and reproduced verbatim in every manuscript in this repository, so\n% the papers cannot drift into different accounts of how they were made.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erdős problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every Lean mark in the margin points to propositions that the\n  pinned Lean kernel accepts, with no \\texttt{sorry}, no added axiom, and no\n  unchecked evaluation.  That claim concerns the formal statements only; it\n  says nothing about their mathematical interest, their novelty, or the\n  original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n\n% Compact opening card for the separately stated Comparator theorem selected\n% from each problem note.  The prose supplied by each note owns the\n% contribution and relevance judgements; the shared final sentence states the\n% mechanical assurance boundary once and points to the public evidence owner.\n\\newcommand{\\externalreviewobject}[3]{%\n  \\begin{quote}\\small\n  \\textbf{Contribution.} #1\\par\n  \\textbf{Relation to the open problem.} #2\\par\n  \\textbf{Executable review object.} #3  The repository's\n  external-verification job compares these exact Lean propositions with\n  separately declared challenge statements and an axiom budget, then asks\n  Lean's kernel to check the submitted proofs.  The\n  \\href{https://github.com/wcook04/plectis-erdos/blob/ca0e13f8acf5ccf48506e4bdb870953d3a0856fa/formalization.yaml}%\n  {formalisation manifest} and the commit-bound CI receipt record that check;\n  they do not assess novelty, significance, or whether the original problem is\n  solved.\n  \\end{quote}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-evidence","source_path":"paper/paper-evidence.tex","media_type":"text/x-tex","sha256":"sha256:1ba1b3a112445f6fee7794f2083c29a60fb102d239d5e799808a36b9a5e562e3","byte_count":6036,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Consolidated links to the formal evidence for each result.\n%\n% Each paper inputs its own generated file paper/evidence/<paper>.tex, written by\n% scripts/paper_evidence.py from the coverage ledger and the Comparator replay\n% receipts.  That file holds one line per result,\n%\n%   \\DeclareResultEvidence{<label>}{<mark>}{<Lean URL>}{<Comparator URL>}\n%\n% plus its record, statement name and evidence status. A result's label binds\n% its identity; its Lean and Comparator links appear once in the final\n% verification concordance. Missing and duplicate result labels are errors.\n% Do not edit the per-paper files by hand.\n\n% Companion-PDF links use the same readable external-link colour in grayscale.\n\\hypersetup{filecolor=linkexternal}\n\n\\makeatletter\n\\newcommand*{\\ev@all}{}\n\\newcommand*{\\ev@rows}{}\n\\newif\\ifev@second\n\\newif\\ifev@newpage\n\\newcommand{\\evidenceconcordancenewpage}{\\ev@newpagetrue}\n\\newcommand{\\DeclareResultEvidenceName}[2]{%\n  \\expandafter\\gdef\\csname ev@name@#1\\endcsname{#2}}\n\\newcommand{\\DeclareResultEvidence}[4]{%\n  \\ifcsname ev@map@#1\\endcsname\n    \\PackageError{paper-evidence}{Evidence declared twice for label #1}%\n      {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n  \\else\n    \\expandafter\\gdef\\csname ev@map@#1\\endcsname{\\ev@marks{#1}{#2}{#3}{#4}}%\n    \\expandafter\\gdef\\csname ev@inline@#1\\endcsname{\\ev@inline{#1}{#2}{#3}{#4}}%\n    \\g@addto@macro\\ev@all{\\do{#1}}%\n  \\fi}\n% Generated record targets also cover statements without a complete Lean proof.\n% They let an authored inline reference use the very same result identity as\n% the heading links, rather than copying a source pin or a declaration coordinate.\n\\newcommand{\\DeclareResultEvidenceRecord}[4]{%\n  \\expandafter\\gdef\\csname ev@record@#1\\endcsname{#2}%\n  \\expandafter\\gdef\\csname ev@leanstatus@#1\\endcsname{#3}%\n  \\expandafter\\gdef\\csname ev@cmpstatus@#1\\endcsname{#4}%\n  \\g@addto@macro\\ev@rows{\\ev@row{#1}}}\n\\newcommand{\\ev@inline}[4]{%\n  \\begingroup\\normalfont\\scriptsize\n  \\href{#3}{#2}%\n  \\if\\relax\\detokenize{#4}\\relax\n    \\ifcsname ev@cmpstatus@#1\\endcsname\n      \\edef\\ev@status{\\csname ev@cmpstatus@#1\\endcsname}%\n      \\def\\ev@pending{pending}%\n      \\ifx\\ev@status\\ev@pending\n        \\enspace\\textperiodcentered\\enspace\n        \\href{\\csname ev@record@#1\\endcsname-comparator}{Comparator pending}%\n      \\fi\n    \\fi\n  \\else\n    \\enspace\\textperiodcentered\\enspace\\href{#4}{Comparator}%\n  \\fi\n  \\endgroup}\n\\newcommand{\\resultevidence}[1]{%\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    \\ifcsname ev@record@#1\\endcsname\n      \\href{\\csname ev@record@#1\\endcsname}{evidence record}%\n    \\else\n      \\PackageError{paper-evidence}{No evidence record for #1}%\n        {Use the label of a result recorded by scripts/paper_evidence.py.}%\n    \\fi\n  \\fi}\n\\newcommand{\\ev@marks}[4]{%\n  \\ifcsname ev@placed@#1\\endcsname\\else\n    \\expandafter\\gdef\\csname ev@placed@#1\\endcsname{}%\n    \\if\\relax\\detokenize{#3}\\relax\n      \\PackageError{paper-evidence}{No Lean target for #1}%\n        {Regenerate the paper's evidence file with scripts/paper_evidence.py.}%\n    \\else\n      % The label establishes the result identity; evidence is printed once\n      % in the verification concordance, away from the mathematical argument.\n    \\fi\n  \\fi}\n% Labels establish identities without adding links to theorem headings.\n\\newcommand{\\ev@hook}[1]{%\n  \\ifcsname ev@map@#1\\endcsname\\csname ev@map@#1\\endcsname\\fi}\n\\AtBeginDocument{%\n  \\let\\ev@label\\label\n  \\renewcommand{\\label}[1]{\\ev@label{#1}\\ev@hook{#1}}%\n  \\ifdefined\\compatlabel\n    \\let\\ev@compatlabel\\compatlabel\n    \\renewcommand{\\compatlabel}[1]{\\ev@compatlabel{#1}\\ev@hook{#1}}%\n  \\fi}\n\\AtEndDocument{%\n  \\def\\do#1{\\ifcsname ev@placed@#1\\endcsname\\else\n    \\PackageError{paper-evidence}{The evidence mark for #1 was never placed}%\n      {Its label does not occur in this paper; regenerate the evidence file.}\\fi}%\n  \\ev@all\n  \\ifx\\ev@rows\\@empty\\else\n    \\ifev@newpage\\clearpage\\fi\n    \\section*{Verification concordance}\n    \\addcontentsline{toc}{section}{Verification concordance}\n    \\begingroup\\small\n    \\begin{longtable}{@{}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{\\hspace{.03\\linewidth}}p{.16\\linewidth}@{\\hspace{.02\\linewidth}}p{.30\\linewidth}@{}}\n    \\textbf{Statement} & \\textbf{Formal proof} & \\textbf{Statement} & \\textbf{Formal proof}\\\\\\hline\n    \\endhead\n    \\ev@rows\n    \\ifev@second & \\\\ \\fi\n    \\end{longtable}\n    \\endgroup\n  \\fi}\n\\newcommand{\\ev@row}[1]{%\n  \\hypertarget{verification-result.#1}{}%\n  {\\scriptsize\\hyperref[#1]{\\csname ev@name@#1\\endcsname}} &\n  \\ifcsname ev@inline@#1\\endcsname\n    \\csname ev@inline@#1\\endcsname\n  \\else\n    {\\scriptsize\\href{\\csname ev@record@#1\\endcsname}{Verification record}}%\n  \\fi\n  \\ifev@second\\global\\ev@secondfalse\\\\\n  \\else\\global\\ev@secondtrue &\\fi}\n\\makeatother\n\n% Once per paper: what the marks mean.  #1 is the URL of the paper's evidence\n% record.\n\\newcommand{\\evidenceparagraph}[1]{%\n  \\par\\smallskip\\noindent\n  The verification concordance lists the formal proofs by statement.\n  \\emph{Lean} links to the supporting declarations; a dagger identifies a\n  proof that assumes a named input. \\emph{Comparator} links to a recorded\n  kernel check against a separately written statement; \\emph{pending} means\n  that this comparison has not been recorded. The\n  \\href{#1}{verification record} gives the precise correspondence,\n  dependencies and reproducible checks. A row with only a record link has\n  no complete formal proof recorded.}\n\n% A plain remark after a result, for what its mark alone cannot say: the input\n% a conditional Lean proof assumes, or what is and is not checked for a result\n% without a mark.  The skip after the result is replaced by one that cannot\n% break, so the remark never starts a page away from its result.\n\\newcommand{\\evidenceremark}[1]{%\n  \\par\\removelastskip\\nobreak\\vspace{0.3\\baselineskip}%\n  \\noindent{\\small #1\\par}\\vspace{0.15\\baselineskip}}\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  Under\n% Tectonic (XeTeX) the text and mathematics are set in TeX Gyre Pagella from\n% the bundled OpenType fonts, with a Heros sans range for \\mathsf and\n% Inconsolata for monospace; the pdflatex fallback keeps the same Palatino\n% design through newpx.  The A4 text block uses a classical measure with\n% quiet running heads and restrained ink.  Link colours: steel blue for\n% external (Lean-source and web) targets, deep burgundy for internal\n% cross-references and citations.\n\\NeedsTeXFormat{LaTeX2e}\n\\ProvidesPackage{paper-house-style}[2026/07/19 Pagella house style for the papers]\n\n\\RequirePackage{iftex}\n\\RequirePackage{amsmath}\n\\RequirePackage{amsthm}\n\\ifpdftex\n  \\RequirePackage[T1]{fontenc}\n  \\RequirePackage[utf8]{inputenc}\n  \\RequirePackage{amssymb}\n  \\RequirePackage{newpxtext}\n  \\RequirePackage{newpxmath}\n\\else\n  \\RequirePackage{fontspec}\n  \\RequirePackage{unicode-math}\n  \\setmainfont{texgyrepagella}[Extension=.otf,UprightFont=*-regular,\n    BoldFont=*-bold,ItalicFont=*-italic,BoldItalicFont=*-bolditalic]\n  \\setmathfont{texgyrepagella-math.otf}\n  \\setmathfont{texgyreheros-regular.otf}[range=sfup,Scale=MatchLowercase]\n  \\setmonofont{Inconsolatazi4}[Extension=.otf,UprightFont=*-Regular,\n    BoldFont=*-Bold,Scale=MatchLowercase]\n\\fi\n\n\\RequirePackage[a4paper,textwidth=418pt,textheight=636pt,hcentering,\n  top=68pt,headheight=15pt,headsep=20pt,footskip=34pt]{geometry}\n\\RequirePackage{microtype}\n\\linespread{1.045}\n\\clubpenalty=10000 \\widowpenalty=10000 \\tolerance=1600\n\n\\RequirePackage[dvipsnames,table]{xcolor}\n\\definecolor{linkexternal}{HTML}{275D8C}\n\\definecolor{linkinternal}{HTML}{6E2B2B}\n\\definecolor{papertableheader}{HTML}{E3EAF0}\n\\definecolor{papertablerow}{HTML}{F2F5F7}\n\\definecolor{papertablerule}{HTML}{7D8991}\n\\definecolor{paperresultrule}{HTML}{315B72}\n\n% Stable cross-document links between sibling PDFs. A named destination is\n% more durable than a page number: sections may move while their semantic role\n% remains the same. All public PDFs are copied to one directory at release.\n\\newcommand{\\papersectiontarget}[1]{\\hypertarget{#1}{}}\n\\newcommand{\\papersectionlink}[3]{\\href{#1\\##2}{#3}}\n\n\\RequirePackage{titlesec}\n\\titleformat{\\section}{\\Large\\bfseries}{\\thesection}{0.85em}{}\n\\titlespacing*{\\section}{0pt}{2.6ex plus .8ex minus .3ex}{1.5ex plus .3ex}\n\\titleformat{\\subsection}{\\large\\bfseries}{\\thesubsection}{0.8em}{}\n\\titlespacing*{\\subsection}{0pt}{2.1ex plus .6ex minus .2ex}{1.1ex plus .2ex}\n\\titlespacing*{\\paragraph}{0pt}{1.6ex plus .5ex minus .2ex}{0.85em}\n\n\\RequirePackage{fancyhdr}\n\\newcommand{\\paperrunninghead}{}\n\\newcommand{\\runninghead}[1]{\\renewcommand{\\paperrunninghead}{#1}}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\fancyhead[L]{\\small\\itshape\\paperrunninghead}\n\\fancyhead[R]{\\small\\thepage}\n\\renewcommand{\\headrulewidth}{0pt}\n\\fancypagestyle{plain}{\\fancyhf{}\\renewcommand{\\headrulewidth}{0pt}}\n\n\\RequirePackage{caption}\n\\captionsetup{font=small,labelfont=bf,skip=9pt,width=.92\\linewidth}\n\\RequirePackage{etoolbox}\n\\RequirePackage{pdflscape}\n\\renewcommand{\\arraystretch}{1.20}\n\\newcommand{\\papertablehead}{\\rowcolor{papertableheader}}\n\\newcommand{\\papertablebody}{\\rowcolors{2}{white}{papertablerow}}\n\\AtBeginEnvironment{tabular}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\\AtBeginEnvironment{longtable}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\n% A compact front-matter panel for the result a scanning reader should retain.\n% It is deliberately short and unbreakable: manuscripts should state one\n% result, its evidence, and its limit here, then supply the argument below.\n\\newenvironment{paperresult}[1]{%\n  \\par\\addvspace{0.7em}\n  \\begin{center}\\begin{minipage}{0.94\\linewidth}\n    {\\color{paperresultrule}\\rule{\\linewidth}{0.8pt}}\\par\\vspace{4pt}\n    {\\small\\sffamily\\bfseries\\color{paperresultrule}#1}\\par\\smallskip\n    \\small\\color{black}\n}{%\n    \\par\\vspace{4pt}{\\color{paperresultrule}\\rule{\\linewidth}{0.45pt}}\n    \\end{minipage}\\end{center}\\par\\addvspace{0.5em}\n}\n\n% Shared diagram grammar.  Reader-facing figures use the same restrained\n% hierarchy as the surrounding page: a short eyebrow, a substantial title,\n% one compact explanatory line, and an optional evidence badge.  The styles\n% are installed only when a manuscript loads TikZ.\n\\AtBeginDocument{%\n  \\@ifpackageloaded{tikz}{%\n    \\tikzset{\n      paper box/.style={draw=gray!48, line width=.45pt, rounded corners=3pt,\n        fill=white, align=left, inner xsep=7pt, inner ysep=6pt,\n        font=\\small},\n      paper card/.style={paper box, minimum height=13mm},\n      paper stage/.style={paper box, align=center, minimum height=14mm},\n      paper arrow/.style={-{Stealth[length=2.15mm,width=1.45mm]},\n        line width=.7pt, draw=gray!72!black},\n      paper feedback/.style={paper arrow, densely dashed,\n        draw=Violet!72!black},\n      paper eyebrow/.style={font=\\scriptsize\\sffamily\\bfseries,\n        text=gray!70!black},\n      paper label/.style={font=\\scriptsize\\sffamily,\n        text=gray!72!black, fill=white, inner sep=1.5pt},\n      paper badge/.style={rounded corners=2pt, inner xsep=4pt, inner ysep=2pt,\n        font=\\scriptsize\\sffamily\\bfseries},\n      paper boundary/.style={densely dashed, line width=.55pt,\n        draw=gray!62}\n    }%\n  }{}%\n}\n\n% Front matter: title block with an optional italic subtitle, small-caps\n% author, and a hairline rule; the abstract is set small with a small-caps\n% head and symmetric indents.\n\\newcommand{\\subtitle}[1]{\\gdef\\@papersubtitle{#1}}\n\\renewenvironment{abstract}{%\n  \\vspace{0.1em}\\small\n  \\begin{center}{\\normalsize\\scshape Abstract}\\end{center}\\vspace{-0.5em}\n  \\list{}{\\leftmargin 1.3em\\rightmargin\\leftmargin\\listparindent 0pt\n    \\parsep 0.32em\\itemindent 0pt}\\item\\relax}\n  {\\endlist\\vspace{0.5em}}\n\\def\\@maketitle{%\n  \\newpage\\null\n  \\vspace*{-2.6em}%\n  \\begin{center}%\n    {\\LARGE\\bfseries\\@title\\par}%\n    \\vskip 0.55em\n    {\\large\\itshape\\@papersubtitle\\par}%\n    \\vskip 1.1em\n    {\\scshape\\@author\\textsuperscript{*}\\par}%\n    \\vskip 0.25em\n    {\\small\\@date\\par}%\n    \\vskip 0.8em\n    \\rule{6.5em}{0.4pt}%\n  \\end{center}%\n  \\vskip 0.9em}\n\n% Production provenance, set at the foot of the first page.\n%\n% A reader who meets a single personal byline ordinarily infers that the named\n% author drafted the sentences.  For these papers that inference is false, and\n% a declaration reached only in the back matter would leave it standing for the\n% length of the paper.  The note therefore appears on page one, small and\n% visually subordinate to the title and abstract: enough to prevent the false\n% inference, not enough to make the production method the subject.\n%\n% It carries no mark, because it qualifies the whole manuscript rather than any\n% one passage.  \\footnotetext does not advance the footnote counter, and the\n% emptied \\thefootnote is confined to a group, so numbered footnotes elsewhere\n% in the paper are unaffected.\n%\n% The wording is not authored here.  It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/synthesis/optimal-sparse-perturbations.tex","section_outline":["From individual problems to reusable questions","Prescribing a value and its derivatives","An exact capacity criterion","The sharp support threshold","Proof of the capacity criterion","Bounded capacity along a subsequence","One continuation interval for every cumulative residue","From one interval to every nonnegative target","Why a large gap prevents interval filling","A digit that also corrects the cumulative residue","The positive direction and the arithmetic cost","A common-divisor test for irrationality","Why a small allowance along a subsequence is insufficient","Proof of the derivative interpolation theorem","Nullity at and below the critical exponent","Lifting a scalar interval to an open set","The upper bound for Hausdorff dimension","A separated construction attaining the dimension","The arithmetic threshold for rational derivatives","Lambert subsums across bases","Choices against contraction","The subseries across bases","Divisibility cuts at every algebraic base","Base two","Removed intervals and finite translations","Fixed-depth rational counts","Which dyadic shifts detect irrationality?","Polynomially selected shifts","Limits on methods","The computation and its limits","Eliminated ideas","What was already known","Verification and sources","Further questions","Declaration of generative AI use"]},"writing-a-good-mathematical-paper":{"paper_id":"writing-a-good-mathematical-paper","title":"Writing a Good Mathematical Paper","form":"Compact writing guide","length_class":"system","pages":2,"subject":"Exposition method","subject_kind":"system","problem_id":null,"landing_role":"method","question":"How should an AI write a clear and mathematically faithful paper?","precis":"A self-contained guide to explaining a mathematical argument: read nearby primary literature, state the exact hypotheses and conclusion, connect proof steps, introduce notation, attribute borrowed ideas, and inspect the rendered paper. The guide offers writing practice rather than mathematical certification.","owns":"the exposition methodology: a compact general guide for writing a mathematically faithful paper","not_authority_for":"a mathematical result, independent review, or a measured improvement in writing","first_pass":[],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/writing-a-good-mathematical-paper.pdf","hosted_pdf_path":"papers/writing-a-good-mathematical-paper.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/exposition/writing-a-good-mathematical-paper.tex","site_html":"maths/papers/writing-a-good-mathematical-paper.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/exposition/writing-a-good-mathematical-paper.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/exposition/writing-a-good-mathematical-paper.tex","body_pointer":"papers.records.writing-a-good-mathematical-paper.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n\\documentclass[10pt]{article}\n\\usepackage{paper-house-style}\n\\usepackage{xurl}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,citecolor=linkinternal]{hyperref}\n\\hypersetup{pdftitle={Writing a Good Mathematical Paper},pdfauthor={Will Cook},pdflang={en-GB},bookmarksnumbered=true}\n\\runninghead{Writing a good mathematical paper}\n\\title{Writing a Good Mathematical Paper}\n\\subtitle{A compact guide for authors and AI assistants}\n\\author{Will Cook}\n\\date{7 October 2026}\n\\emergencystretch=1.5em\n\\newcommand{\\instruction}[2]{\\par\\medskip\\noindent\\textbf{#1.} #2\\par}\n\\begin{document}\n\\maketitle\n\\paperprovenance\n\\papersectiontarget{compact-writing-guide}\nTest the draft for its intended reader: state the contribution, reconstruct\nthe decisive inference and apply the result. Begin where one of these tasks\nfails; use the relevant instruction, not every instruction.\n\n\\instruction{1. Decide what the paper contributes}{\nChoose a coherent principal result; say what it enables or clarifies.\nCompared with the nearest prior work, what changes in the assumptions or\nconclusion? State what remains open. For a conditional result, give familiar\ncases its hypothesis admits or excludes, where known, and show where the proof\nuses it. Distinguish contribution from proof mechanism; the abstract needs\nboth. Neither theorem count nor generality establishes importance.}\n\n\\instruction{2. Read the kind of mathematics you are writing}{\nRead nearby papers for terminology, hypotheses and attribution; read complete\nlocal arguments for explanation. Match the task: a parameter choice, limit or\nestimate. Check the version and passage. Identify what justifies the source's\nnext step and the corresponding fact in your proof; explain it in your own words.}\n\n\\instruction{3. Check the statement before polishing its prose}{\nCheck domains, quantifiers, signs, endpoints, constants and both directions\nof claimed equivalences. State what is fixed and what later choices may depend\non. Keep assumptions with the claim; compare titles, abstracts and conclusions\nwith it. When properties are needed together, prove they hold at the same\nwitness: the even and odd integers are both unbounded but disjoint.\nRefer a discovered proof gap for mathematical review.}\n\n\\instruction{4. Organise by logical dependence}{\nGive the question, result and proof idea before technical detail accumulates.\nFor each major step, say what it uses, what it proves and where that conclusion\nis needed. Introduce prerequisites when their purpose is visible; use examples\nto explain unfamiliar operations. Let headings name objects, results or\noperations; make assumptions and proofs findable from the theorem. Logical\norder need not reproduce discovery order.}\n\n\\instruction{5. Explain the hard transition}{\nWork backwards from the next conclusion: what must be controlled, and how\naccurately? State the requirement before estimating or choosing a parameter.\nTo certify $x\\in(1/2,1)$, place an enclosure wholly inside that interval;\n$[1/2,1]$ does not suffice. For a small-integer contradiction, justify\nintegrality, nonvanishing and $|N|<1$ separately. Explain which fact supplies\neach requirement; a repeated technique may have a different job.}\n\n\\instruction{6. Separate construction from completion}{\nJustify each choice and preserved condition, then prove attainment.\nFor example, $r_n=x-S_n$, $0\\le r_n\\le\\varepsilon_n$ and\n$\\varepsilon_n\\to0$ give $S_n\\to x$; repeated choice alone does not.\nA classification needs construction and exhaustiveness; an enclosure alone\nneed not establish attainment. State what stays fixed; justify convergence of\nthe whole summand, including growing products. To interchange limit and sum,\ncheck the theorem's hypotheses; dominated convergence needs a summable majorant\nindependent of the limiting parameter.}\n\n% Keep the final reading paragraph with the fourteen instructions.\n\\newpage\\enlargethispage{3\\baselineskip}\n\\instruction{7. Make examples and figures do work}{\nChoose a small instance that exposes a definition, invariant, obstruction or\nnecessary hypothesis; a simpler case may hide it. Show what changes and what\nis preserved. When reindexing a sum, check multiplicities, not just values.\nInclude relevant endpoints and say what remains unproved. A finite check\nsupports an infinite claim only through a proved reduction. Check that a\nwitness belongs to the claimed object. Label a diagram's objects, maps and\nconventions; keep its explanation nearby.}\n\n\\instruction{8. Use the field's words and purposeful notation}{\nKeep established terms when their definitions fit, such as ``Stieltjes moment\nsequence''. Replace private labels by the actual condition; $u_n/2^n\\to0$\nis weaker than subexponential growth. Define symbols by their role before use\nand keep that role stable. Remove abbreviations whose translation costs more\nthan their reuse saves. Translate borrowed indices and normalisations.\nDistinguish a sum, its partial sums and its remainders, and a value from its enclosure.}\n\n\\instruction{9. Make sentences express the argument}{\nReplace ``We proceed with the estimate'' by its reason: ``The terms are\nnonnegative, so dropping the ordering restriction gives an upper bound.'' Let ``since'' supply a reason\nand ``hence'' mark a deduction. Give each ``this'' an unmistakable referent.\nAfter ``similarly'', check that the same hypotheses apply. Punctuate displays\nas parts of sentences. Keep parallel claims parallel and conditions beside\ntheir consequences. Remove filler, not mathematical qualifications.}\n\n\\instruction{10. Credit the exact input at its use}{\nDistinguish applied theorems, adapted constructions, historical attribution,\nterminology and expository models. Check hypotheses, version\nand passage; identify the input used here. State what an adaptation retains and\nchanges. Credit supplied ideas and corrections locally. A writing analogy\nsupplies no proof. Report only reading and checks performed.}\n\n\\instruction{11. State what has actually been established}{\nSeparate scope from evidence: conditional implications may have ordinary or\nformal proofs. Match formal support to the exact statement, definitions and\nhypotheses. Check coverage of the steps joining lemmas. Missing formalisation\ndoes not invalidate an ordinary proof. Keep unproved inputs beside dependent\nresults. Label computations and conjectures; give computations' finite scope\nlocally and reproduction details once. For one-sided tests, failure may be inconclusive.\nDetecting each fixed input need not give a uniform stopping bound or the\nrequired witnesses.}\n\n\\instruction{12. Preserve useful detail and informative failures}{\nKeep omitted detail recoverable by section or theorem references; check their\ndestinations and reconcile statements, assumptions and evidence in both directions.\nFor a failed route, retain the attempted implication, assumptions, witness and\nmissing ingredient. Name the assertion a counterexample refutes; a failed method\nneed not refute the theorem. Put build commands and source identifiers in the\nverification record. Do not invent discovery history.}\n\n\\instruction{13. Review changes as claims, then as prose}{\nCompare meanings before fluency, including words beside unchanged formulas.\nCheck earlier criticisms against the current source. Correct overclaims and\nstale underclaims without losing hypotheses or credit. Accept, repair, reject\nor defer each proposal for a stated reason; preserve useful rejected revisions.\nTest a local improvement's limits before promoting it to general advice.}\n\n\\instruction{14. Read the delivered paper from a cold start}{\nRender the final sources; inspect formulas, captions, page breaks, bibliography\nand reference destinations. Read without supplying steps from memory. Can the\nreader reconstruct the hard inference, apply the result and identify an\nexcluded case, where known? Where reconstruction stops, record what is\nestablished and what the next step needs. A successful build is not a comprehension test.\nRevise that transition; preserve what works.}\n\n\\par\\medskip\\noindent\\small\n\\textit{Further reading.}\nHalmos, \\href{https://doi.org/10.5169/seals-43857}{\\emph{How to write mathematics}}, \\S\\S3--5 (1970);\nKnuth et al., \\href{https://cs.stanford.edu/~knuth/klr.html}{\\emph{Mathematical Writing}}, \\S\\S1--3 (1988);\nGowers, \\href{https://gowers.wordpress.com/2007/10/19/my-favourite-pedagogical-principle-examples-first/}{\\emph{Examples first!}} (2007);\nTao, \\href{https://terrytao.wordpress.com/advice-on-writing-papers/}{\\emph{On writing}}.\nThe \\href{https://github.com/wcook04/plectis-erdos/blob/main/paper/exposition/writing-mathematics-from-reviewed-revisions.pdf}{\\emph{long companion}}\ngives worked cases (\\S\\S2--4), revision practice (\\S5) and genre transfer (\\S6).\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":8862,"sha256":"sha256:e96a5d3154eabc36ecd618ca3590672486f1be6d71868377ef1fc6b4c38b7452","cite_as":"Plectis reader corpus, paper:writing-a-good-mathematical-paper","source_repository":"plectis-erdos","source_path":"paper/exposition/writing-a-good-mathematical-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  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Reader-facing figures use the same restrained\n% hierarchy as the surrounding page: a short eyebrow, a substantial title,\n% one compact explanatory line, and an optional evidence badge.  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For these papers that inference is false, and\n% a declaration reached only in the back matter would leave it standing for the\n% length of the paper.  The note therefore appears on page one, small and\n% visually subordinate to the title and abstract: enough to prevent the false\n% inference, not enough to make the production method the subject.\n%\n% It carries no mark, because it qualifies the whole manuscript rather than any\n% one passage.  \\footnotetext does not advance the footnote counter, and the\n% emptied \\thefootnote is confined to a group, so numbered footnotes elsewhere\n% in the paper are unaffected.\n%\n% The wording is not authored here.  It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/exposition/writing-a-good-mathematical-paper.tex","section_outline":[]},"writing-mathematics-from-reviewed-revisions":{"paper_id":"writing-mathematics-from-reviewed-revisions","title":"Writing Mathematics from the Literature and Reviewed Revisions","form":"Editorial method companion","length_class":"system","pages":21,"subject":"Exposition method","subject_kind":"system","problem_id":null,"landing_role":"method","question":"How can reading nearby papers and reviewed revisions improve the explanation of a particular mathematical argument?","precis":"A method for reading nearby mathematical prose, examining reviewed revisions, and retaining editorial lessons with their source, native decision and limit. The examples come from eight Erdős-problem papers. This documented practice has not been evaluated as an intervention in reader understanding.","owns":"the exposition methodology: reading primary mathematical prose, reviewing specific revisions, and retaining bounded editorial lessons","not_authority_for":"the example mathematics, independent mathematical review, or a measured improvement in reader understanding","first_pass":["sec:reading","sec:revisions","sec:review","sec:practice"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/writing-mathematics-from-reviewed-revisions.pdf","hosted_pdf_path":"papers/writing-mathematics-from-reviewed-revisions.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/exposition/writing-mathematics-from-reviewed-revisions.tex","site_html":"maths/papers/writing-mathematics-from-reviewed-revisions.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/exposition/writing-mathematics-from-reviewed-revisions.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/exposition/writing-mathematics-from-reviewed-revisions.tex","body_pointer":"papers.records.writing-mathematics-from-reviewed-revisions.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n\\documentclass[11pt]{article}\n\\usepackage{paper-house-style}\n\\usepackage{booktabs,tabularx,array,xurl}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,citecolor=linkinternal]{hyperref}\n\\hypersetup{pdftitle={Writing Mathematics from the Literature and Reviewed Revisions},pdfauthor={Will Cook},pdflang={en-GB},bookmarksnumbered=true}\n\\runninghead{Writing mathematics from reviewed revisions}\n\\title{Writing Mathematics from the Literature\\\\and Reviewed Revisions}\n\\subtitle{Worked examples in explaining proofs and reviewing revisions}\n\\author{Will Cook}\n\\date{7 October 2026}\n\\emergencystretch=1.8em\n\\newcommand{\\caseid}[1]{\\texttt{#1}}\n\\newcommand{\\corpus}{\\url{https://github.com/wcook04/plectis-erdos}}\n\\begin{document}\n\\maketitle\n\\paperprovenance\n\\begin{abstract}\nA fluent sentence can leave the decisive inference unexplained. This guide\nstarts with that difficulty: locate the step a reader cannot reconstruct,\nstudy a relevant argument in the literature, and explain the step using the\nmanuscript's own proof. Worked cases show how a remainder establishes\nattainment, how a target interval determines an error threshold, and why\nconvergence of each summand is not yet convergence of an infinite sum.\nOther cases expose changes of scope hidden in ordinary words and distinguish\nwhat a system records, checks and demonstrates. Drawn from revisions of eight\nErd\\H{o}s-problem papers and their supporting systems manuscript, the examples\nsupply reasons and limits for the compact companion's instructions.\nThe guide documents editorial practice, not new mathematical results or\nmeasured improvements in reader understanding.\n\\end{abstract}\n\\input{exposition/parts/reading.tex}\n\\input{exposition/parts/revisions.tex}\n\\input{exposition/parts/review.tex}\n\\input{exposition/parts/practice.tex}\n\\input{exposition/parts/references.tex}\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":1951,"sha256":"sha256:96b7f3e3bad868197b0207ec66ac4496bc205193b49dc6ec8033df8de4d27d5e","cite_as":"Plectis reader corpus, paper:writing-mathematics-from-reviewed-revisions","source_repository":"plectis-erdos","source_path":"paper/exposition/writing-mathematics-from-reviewed-revisions.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"exposition/parts/reading.tex","source_path":"paper/exposition/parts/reading.tex","media_type":"text/x-tex","sha256":"sha256:4e80a17ad9d57281ad2acbf11fe63bf004c81fff8900885b7b152d27cfbaf352","byte_count":10802,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n\\section{Learning to explain a particular argument}\\label{sec:reading}\n\\papersectiontarget{exposition-reading}\n\nSuppose a proof chooses digits one at a time to represent a target. Showing\nthat another choice is always possible does not yet identify the infinite\nsum. The missing information is what remains of the target:\n\\[\n r_n=x-S_n,\\qquad 0\\le r_n\\le\\varepsilon_n,\\qquad\n \\varepsilon_n\\longrightarrow0\n\\]\nso the partial sums $S_n$ converge to $x$. Replacing ``we repeat'' by\n``we continue inductively'' would explain no more. The remainder bound connects\nthe finite choices to the infinite sum; the name of the iteration does not.\n\nTo find such an explanation, locate the last fact already established and\nthe first conclusion the reader is asked to accept. Ask what connects them.\nHere the connection is a remainder bound tending to zero, not merely a\nremainder that stays bounded. A source facing the same task can suggest how\nto expose this distinction. Section~\\ref{sec:remainder} develops the comparison;\nthe new manuscript must still justify its own covering and remainder bounds.\n\nHalmos discusses audience, organisation and examples~\\cite[\\S\\S3--4]{halmos}.\nKnuth, Larrabee and Roberts discuss introducing symbols, connecting sentences\nand explaining why a step is taken~\\cite[\\S1]{knuth}. Gowers considers where\nexamples should precede unfamiliar abstractions~\\cite{gowers}; Tao cautions\nagainst sacrificing usefulness to excessive optimisation~\\cite{tao}.\nThis advice identifies questions. A nearby argument shows how an author\nanswered them in a particular setting.\n\nThe cases come from revisions of papers on Erd\\H{o}s problems. Read them by\nthe difficulty at hand: construction and attainment in\nSection~\\ref{sec:remainder}; the accuracy needed from an estimate and the\nmeaning of a failed test in Section~\\ref{sec:certificate}; limits through\ninfinite sums in Section~\\ref{sec:worked-explanations}; changes of meaning in\nSection~\\ref{sec:review}. Section~\\ref{sec:practice} explains how to review a\nrevision and keep its useful lesson without making every case a new rule.\n\n\\subsection{Choose sources by the work the prose must do}\n\nChoose the reader before choosing the amount of explanation. An expert may\nknow a lemma but not why it is used here; a neighbouring specialist may need\nits statement. Tao's advice distinguishes the author's familiarity from the\nreader's knowledge~\\cite{tao-detail}. State the assumed background, then explain\nwhat it does not supply.\n\nSubject and genre matter too. An irrationality note may turn an assumed\nrational sum into a positive integer smaller than one. A geometric paper\nmay first need to identify the component being measured and the map preserving\nits size. A survey compares viewpoints; a short theorem paper may develop\none argument. A useful source should illuminate the particular task, not\nmerely have an admired style.\n\nThe introduction must answer both ``what changes with this result?'' and\n``why does the proof work?'' Compare assumptions and conclusions with the\nclosest prior result: a stronger conclusion under stronger assumptions need\nnot be a stronger theorem. Then identify the obstacle and the step that\novercomes it. A list of section titles does neither. Nor does calling a hypothesis ``natural'' explain its\nforce. For a conditional theorem, say which familiar cases satisfy or fail\nthe hypothesis, where known, and where the proof uses it. One example shows\napplicability, not the breadth of the admissible class. Identify any unresolved\npremise so that a reduction does not read as though that premise were proved.\n\nRead a close subject paper for vocabulary and attribution, and, when useful,\na structurally similar proof for the explanation of a construction or estimate.\nA remote paper cannot settle the target field's conventions, and frequent usage\ndoes not excuse a mismatched definition.\n\nUse the same comparison at the scale of the proof. A roadmap should explain\nhow its stages fit, not just name them~\\cite[\\S4]{halmos}. In\nSection~\\ref{sec:estimate}, finitely many tail values give repetition; recovery\nis needed to make repeated values force repeated blocks. ``We estimate the\ntails and finish the proof'' hides that dependency.\n\nRead the original passage in an identified version, including the statement,\nhypotheses and argument around the sentence. An abstract may establish a term,\nbut seldom shows how the proof is explained. Record a section, theorem or\nequation locator and a page; numbering can change between versions. Keep\nsource digests in the reading record. They identify the supplied bytes,\nwhereas a public link gives the reader a route to the work.\n\nFor a difficult passage, reconstruct the local argument before borrowing its\npresentation. Knuth, Larrabee and Roberts give a useful worked\ncomparison~\\cite[\\S\\S2--3, pp.~7--8]{knuth}. Their example considers vectors\n$c+kb$ whose coordinates are nonincreasing for every nonnegative integer $k$.\nFix the vectors $c,b$ and coordinates $i<j$. Rearranging the coordinate\ninequality gives\n\\[\n                 c_i-c_j\\ge k(b_j-b_i)\\qquad(k\\ge0).\n\\]\nThe left side is fixed. If $b_j-b_i$ were positive, the right side would grow\nwithout bound; hence $b_j\\le b_i$. The source compares a proof choosing a\nparticular $k$ for a contradiction with a direct proof using this dependence.\nThe improvement exposes what stays fixed while $k$ grows. The words ``for\narbitrarily large $k$'' make the displayed inequality decisive; they do not\nmerely describe a limiting procedure. This is a reason for this proof's order,\nnot a reason to ban contradiction arguments.\n\nNow ask which fact in the target proof could make its own conclusion equally\nunavoidable. In the opening construction, that fact is a remainder bound tending\nto zero. When the target proof contains no such fact, the comparison has exposed\na mathematical obligation; it has not supplied an explanation that can safely\nbe inserted.\n\nSource availability, a recipient's reading declaration and a later reviewer's\npassage inspection are separate facts. Later inspection can verify a locator\nor expose a poor analogy; it cannot establish what the earlier recipient read.\n\n\\subsection{Read sentences as mathematical relations}\n\nRead each explanatory phrase as part of the argument. A theorem must state\nthe objects and assumptions needed for its conclusion; inside the proof,\nrecall an assumption when it explains a step. ``By positivity'' leaves work\nfor the reader unless the positive quantity and its use are clear. In the\ntail estimate of Section~\\ref{sec:certificate}, positivity of the omitted\ntail gives the lower inequality, whereas its upper bound gives the other.\nNaming those two jobs is more useful than adding ``clearly'' to the conclusion.\n\nConnective words deserve the same attention. ``Since'' introduces a reason;\n``hence'' asserts that the conclusion follows; ``provided that'' makes a\ncondition visible; ``it remains to prove'' identifies unfinished work. These\nwords are not interchangeable devices for varying a paragraph. A revision\nthat changes ``provided that'' to ``hence'' can change the logical force of\nthe sentence while leaving every displayed formula intact.\n\nParallel syntax helps compare claims; a subordinate clause keeps a condition\nbeside its consequence. Sentence length alone settles neither choice: a short\nsentence may hide its antecedent, while a longer one makes the dependency clear.\nStudy that relation, then write original prose for the new argument.\n\nNotation should receive this close reading as well. Ask when a symbol first\nbecomes necessary, what repeated work it saves and which nearby quantities\nthe reader might confuse with it. A new name for a standard object creates an\nadditional translation. Removing an established term such as ``Stieltjes moment\nsequence'' can instead sever a useful connection to the literature~\\cite{wangzhu}.\nCheck both the name and its referent: in Section~\\ref{sec:formula-choice}, the\ndivisibility condition concerns a future index, not its offset from the present\none. Familiar words can misidentify an object as easily as private terminology.\n\nGive space to the reason hardest to recover, not the longest calculation.\nState the required accuracy before the estimate and the selecting constraint\nbefore the parameter; routine substitutions can then be brief. A one-line\nlimit may nevertheless need a paragraph when the number of factors grows\n(Section~\\ref{sec:fixed-partition}). Expose the dependency, not an author's cadence.\n\nA repeated technique may serve a different purpose later in the proof. In the\nR11 revision, the \\#243 paper uses finite differences for two distinct purposes. A fourth\ndifference tends to zero and is integer-valued, so it eventually vanishes;\nthis gives eventual polynomial behaviour. Later, a positive constant third\ndifference bounds a divisibility chain of positive greatest common divisors,\nso that chain stabilises~\\cite[finite-difference argument]{paper243}. Naming\nthe technique twice would not distinguish these conclusions. The added\nsentence identifies the second job. This is a local instance of Knuth et\nal.'s advice to tell the reader why a step is taken~\\cite[\\S1, item~12]{knuth},\nnot a requirement to explain every routine calculation again.\n\n\n\\subsection{Use examples without delaying the result}\n\nAn example earns its place when it prepares a particular inference. Choose\nits size for that job: a table in which every height occurs once would conceal\nthe multiplicity issue in Section~\\ref{sec:multiplicity}. In the opening\nconstruction, one choice can show how the remainder stays admissible, but not\nthat it tends to zero. An example may reveal a classification's coordinates;\nthe proof must still show that every admissible object has them. State which\ntask the example performs.\n\nPlacement depends on what is unfamiliar. Gowers's follow-up to ``examples\nfirst'' qualifies the recommendation by audience and genre, and considers\nreaders who approach a text nonlinearly~\\cite{gowers-followup}. A brief example\ncan precede a locally unfamiliar definition while the headline theorem remains\nearly and easy to find. An example of a familiar definition may instead belong\nafter the statement, at the point where it explains a new use.\n\nThis distinction also governs repetition. Knuth, Larrabee and Roberts recommend\ncomplementary descriptions of important objects~\\cite[\\S1, item~11]{knuth}.\nIn Section~\\ref{sec:certificate}, the inequalities tell the reader what to\ncheck; the enclosure explains why those checks suffice. Removing either\nwould lose a different use. Repeating the conclusion in new adjectives would\nadd neither. Judge repetition by the work the second account makes possible.\n"},{"record_type":"paper_source_dependency","tex_reference":"exposition/parts/revisions.tex","source_path":"paper/exposition/parts/revisions.tex","media_type":"text/x-tex","sha256":"sha256:258e6e2b80bb7f40d2a20e3d52babccaa18e1d464c16752b8e00d9d791de3413","byte_count":15933,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n\\section{Explain the choices in a proof}\\label{sec:revisions}\n\\papersectiontarget{exposition-revisions}\n\nThese cases concern attainment and the accuracy an inference requires.\nThey were accepted in the short-paper R6 round of 30 September 2026, not the\nearlier September series with the same round numbers. The records retain\nthe passages at commit \\texttt{18cedaddedd1}, its full identity and source\ndigests~\\cite{ledger}.\n\n\\subsection{From repeated choice to an attained infinite sum}\\label{sec:remainder}\n\nThe \\#251 paper contains a construction in which a target $x$ is represented\nby chosen digits $d_j$ and weights $w_j$. A finite remainder $\\rho_j$ is kept\nbetween $0$ and a bound $F_j$. The old passage began:\n\\begin{quote}\nGiven $x\\in[0,F_0]$, we choose $d_0$ so that\n$x-d_0w_0\\in[0,F_1]$ and repeat this choice for each successive remainder.\n\\end{quote}\nThis leaves two questions: why is the next digit available, and why does\ncontinuing the choices represent the whole target? The reviewed addition\nanswers them separately:\n\\begin{quote}\nThe covering just proved makes this possible at every stage. The invariant\n$0\\le\\rho_j\\le F_j$ and $F_j\\to0$ then give\n\\end{quote}\nand continues with the limiting identity. The first sentence says why the\nconstruction can continue. The second says why its partial sums reach the\nchosen target: the part still missing is forced to zero. The record is\n\\caseid{r6-251-style\\_rules-04}; its full identifier includes the series date.\n\nCrmari\\'c and Kova\\v{c}'s proof of Lemma~4(a), in the first version of their\npaper on sums of reciprocals, provides a close model~\\cite[pp.~4--5]{crmaric}.\nThey keep a finite remainder in an interval and use convergence to obtain\nthe represented value. What transfers is the separation of two questions:\ncan the next choice be made, and does the resulting sequence attain the target?\n\nThe \\#251 construction must justify its covering, invariant and $F_j\\to0$,\nas well as sparsity and arithmetic constraints. Neither a finite digit\nexample nor the literary comparison supplies those proofs. The integration\npreserved these obligations and the unresolved status of the original\nprime-gap problem.\n\n\\subsection{State how accurate the estimate must be}\\label{sec:estimate}\n\nIn the \\#269 argument, a rationality assumption places normalized tails\n$Y_a$ in the finite set\n\\[\n             (K^{-1}\\mathbb Z)\\cap(0,1).\n\\]\nHere $K\\ge1$ is the integer denominator in the rationality assumption.\nFiniteness forces two tails to agree; it does not force their successors to\nagree. The accepted revision makes the remaining task explicit: recover the\nnext block and tail from the present tail. Only then does equality propagate.\n\nHan\\v{c}l and Tijdeman begin their proof of Theorem~2.1 with an integral\nscaled remainder obtained from rationality~\\cite[p.~373]{hancl}. The \\#269\nrevision likewise identifies the arithmetic object before estimating it;\nrecovery is its own additional task, not supplied by the cited proof.\n\nThe next difficulty is quantitative. The maps\n$G_{53}(t)=(9+t)/30$ and $G_5(t)=(3+t)/10$ encode two possible blocks of\nprime-power jumps. Their images overlap on the initial tail range $[0,1]$, so a value in the\noverlap does not identify its block. Both maps are increasing: to separate\ntheir images, compare the upper end of one with the lower end of the other.\nWith the lower bound $3/10$ already established, the revision states the\nrequired accuracy first:\n\\begin{quote}\nWith that lower endpoint, separating these two images requires an upper\nendpoint $U$ satisfying\n\\[\n G_{53}(U)<G_5(3/10),\\qquad\n \\frac{9+U}{30}<\\frac{33}{100},\\qquad U<\\frac9{10}.\n\\]\nThis specifies the improvement needed from the arithmetic sequence.\n\\end{quote}\nThe case is \\caseid{r6-269-style\\_rules-03}. The threshold is sufficient for\nthis argument, not necessary for every possible method. The return's Knuth\nanalogy concerns presentation; the separation inequality supplies the mathematics.\n\nThe source obtains the required improvement from the spacing of powers of $5$.\nAt a block containing such a power the tail is at most $7/15$. Every three\nconsecutive blocks contain at least one such block; moving backwards at most\ntwice, each step is bounded by $t\\mapsto(1+t)/2$. This gives $U=13/15<9/10$, for the actual\nprime-power sequence, not arbitrary block words. The source separates all five\nblock images on this interval. The image identifies the block; its affine\ninverse determines the next tail. Equality can now propagate~\\cite[proof of the distinct-height theorem]{paper269}.\n\n\\subsection{An example must reveal the relevant difference}\\label{sec:multiplicity}\n\nThe same manuscript distinguishes summing over smooth integers from summing\nover the distinct values of their running least common multiple. Here the\nallowed prime factors are $2,3,5$; a positive integer with no other prime\nfactor is called $5$-smooth. For $x\\ge1$ its running least common multiple is\n\\[\n H(x)=\\operatorname{lcm}\\{u\\le x:u\\text{ is }5\\text{-smooth}\\}\n     =2^{\\lfloor\\log_2x\\rfloor}\n      3^{\\lfloor\\log_3x\\rfloor}\n      5^{\\lfloor\\log_5x\\rfloor}.\n\\]\nThe formula selects the largest power of each allowed prime not exceeding\n$x$~\\cite[definition of running least common multiples]{paper269}. For example,\n$H(16)=16\\cdot9\\cdot5=720$. These are the quantities called heights in the\nrevision. The seven $5$-smooth integers in $[16,32)$ give:\n\\begin{center}\n\\begin{tabular}{@{}lll@{}}\n\\toprule\n$5$-smooth integers $u$ & $H(u)$ & Sum of $1/H(u)$\\\\\n\\midrule\n$16,18,20,24$ & $720$ & $4/720$\\\\\n$25$ & $3600$ & $1/3600$\\\\\n$27,30$ & $10800$ & $2/10800$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\nThe map from a smooth integer to its height is not one-to-one. Summing over\nsmooth integers counts every visit to a height; summing over distinct heights\ncounts it once. The multiplicities $4$ and $2$ are the sizes of the corresponding\nfibres. They explain why the sums cannot be identified merely because the\nsame heights occur in both.\nThe record \\caseid{r6-269-style\\_rules-04} also retains an endpoint warning:\nthe distinct-height argument uses $(16,32]$, whereas the table uses\n$[16,32)$. Matching the endpoint conventions is a separate task from\npreserving multiplicity.\n\nThe repeated heights make this table useful: they expose the counting\nconvention. Halmos's concrete cases and Gowers's examples-first discussion\nsuggest that choice~\\cite[\\S4]{halmos}\\cite{gowers}; they prescribe no order\nfor every reader. The infinite-series theorem still needs its proof.\n\n\n\\section{From a formula to the sentence that explains it}\\label{sec:worked-explanations}\n\nThese cases answer a reader's question from an existing formula. They come\nfrom eight additional editorial returns (\\#1049 is labelled R9), checked\nagainst current manuscript sources. They add no theorem, formal coverage\nor release status.\n\n\\subsection{How does the target determine the finite test?}\\label{sec:certificate}\n\nA certificate is easier to understand when the reader sees the set it must\ncertify before seeing its arithmetic form. In the \\#251 paper, two tail\ndifferences satisfy\n\\[\n D'=2D-\\delta,\\qquad \\delta=2s,\\qquad s\\in\\{-1,1\\}.\n\\]\nThe already proved local criterion asks for $1/2<sD<1$; it then gives\n$-1<sD'<0$. The finite calculation supplies integers $A,Q,B$ with $Q>0$,\n$B\\ge0$ and $|QD-A|\\le B$~\\cite[section on finite separation]{paper251}.\nIt is tempting to write only that two ensuing inequalities certify the target.\nThe missing explanation is the enclosure itself:\n\\[\n sD\\in\\left[\\frac{sA-B}{Q},\\frac{sA+B}{Q}\\right].\n\\]\nMultiplication by either sign preserves the error bound. The lower endpoint\nmust exceed $1/2$, and the upper endpoint must be below $1$. Thus the entire\nclosed enclosure lies in $(1/2,1)$ exactly when\n\\[\n             2sA-Q>2B,\\qquad Q-sA>B.\n\\]\nThe two inequalities express the two endpoint gaps. The factor $2$ in the\nfirst comes from the lower endpoint $1/2$, not an arbitrary safety margin.\nStrictness matters because the boundary values give $sD'=-1$ or $0$.\n\nNow separate a failed enclosure from a failed claim. Take $D=3/4$, $s=1$,\n$Q=4$, $A=3$ and $B=1$. The error bound is valid, but the enclosure\n$[1/2,1]$ touches both forbidden endpoints. Neither strict test passes,\nalthough $D$ lies in the target interval. With these same $D,Q,A$, the sharper\nbound $B=0$ would pass both tests. The failed enclosure therefore need not\nmean a failed claim. The enclosure even has width less than one and still\ncontains an integer: its position, not just its width, matters.\n\nThe \\#68 remainder has a related, one-sided geometry. It is\n$A_N+M(S-H_N)$, where $S$ is the series sum, $H_N$ its $N$th partial sum,\nand $M>0$ scales the strictly positive omitted tail. To put this remainder\nbetween consecutive integers, it suffices to prove\n\\[\n             0<M(S-H_N)<\\lfloor A_N\\rfloor+1-A_N.\n\\]\nThe left inequality puts the remainder above $\\lfloor A_N\\rfloor$; the right\nputs it below $\\lfloor A_N\\rfloor+1$~\\cite[section on nonintegrality]{paper68}.\nA tail smaller than one could still cross the next integer; it must be\nsmaller than this particular gap. At an integral $A_N$ the gap is $1$. Replacing the strict successor by\n$\\lceil A_N\\rceil$ would make it zero. The endpoint distinguishes two\nsuperficially interchangeable conventions.\n\n\\subsection{Why was this base or kernel chosen?}\\label{sec:formula-choice}\n\nA change of notation should reveal a constraint or save an operation. In one\n\\#249 comparison construction, only the even-indexed coefficients may change.\nWriting $e_n$ for the coefficient changes, their contribution to a binary series is\n\\[\n                 \\sum_{m\\ge1} e_{2m}\\,2^{-2m}\n                 =\\sum_{m\\ge1}e_{2m}\\,4^{-m}.\n\\]\nThe permitted support has selected base four. Stating this before the digit\nconstruction explains the choice and shows immediately why the odd coefficients\nremain untouched~\\cite[comparison-sequence construction]{paper249long}.\nThe identity alone says nothing about which corrections can be attained under\nadditional coefficient bounds; those are separate assertions of the construction.\n\nThe \\#257 divisor kernel admits an equally direct introduction. Fix integers\n$N\\ge0$ and $d\\ge1$, and let $B>1$. If $r\\ge1$ satisfies $d\\mid N+r$,\nthen its possible values are\n\\[\n       d-(N\\bmod d),\\quad 2d-(N\\bmod d),\\quad\\ldots.\n\\]\nConsequently\n\\[\n \\sum_{\\substack{r\\ge1\\\\d\\mid N+r}}B^{-r}\n =\\frac{B^{N\\bmod d}}{B^d-1}.\n\\]\nThis is the weight of the future positions at which $d$ divides the index.\nIt explains the kernel before averaging is introduced\n\\cite[finite means for shifted divisor tails]{paper257}. Divisibility applies\nto $N+r$, not to $r$ alone. When $d\\mid N$, the first positive offset is $d$,\nso the same formula applies without adding an unwanted offset zero. In that paper $B=2^\\alpha$, with $0<\\alpha\\le1$; estimates\nmust retain their dependence on $B-1$ as $\\alpha$ becomes small.\n\nIn both examples the operation determines the notation. Even-indexed\nbinary weights become base-four weights; offsets to future indices divisible\nby a fixed divisor form the progression giving the kernel. State that\nreason before naming the formula. Tao's notation advice supports making\nimportant dependencies visible and translating borrowed conventions~\\cite{tao-notation};\nthe support and divisibility conditions supply the mathematical reasons here.\n\n\\subsection{What remains to justify after an index is fixed?}\\label{sec:fixed-partition}\n\nFixing a summation index need not leave a fixed finite product. In the \\#1049\nmoment-determinant argument, a summand is indexed by a partition\n$\\mu_1\\ge\\cdots\\ge\\mu_\\ell>0$, with $\\mu_k=0$ for $k>\\ell$.\nThe base $q\\in(0,1)$ and the partition are fixed while $N\\to\\infty$.\nThe ratios of shifted weights tend to one, but the Vandermonde quotient\nstill contains factors indexed by $k$ up to $N$\n\\cite[proof for geometric moment determinants]{paper1049}.\n\nFor $N\\ge\\ell$ and fixed $1\\le j\\le\\ell$, cancel the common factors in\nthe part with $k>\\ell$:\n\\[\n \\prod_{k=\\ell+1}^{N}\n       \\frac{1-q^{k-j+\\mu_j}}{1-q^{k-j}}\n =\\frac{\\displaystyle\\prod_{r=0}^{\\mu_j-1}(1-q^{N+1-j+r})}\n        {\\displaystyle\\prod_{r=0}^{\\mu_j-1}(1-q^{\\ell+1-j+r})}.\n\\]\nThe original product has $N-\\ell$ factors, but each endpoint product has the\nfixed length $\\mu_j$. Its numerator tends to $1$; its denominator is fixed\nand positive. The remaining factors have $j,k\\le\\ell$ and are independent\nof $N$, so the whole quotient converges for this fixed partition. The\ntelescoping formula, proposed in the later return, justifies what factorwise\nconvergence in a growing product would not.\n\nPassing the limit through the sum requires a second argument. Extend the\nsummand by zero when $\\ell>N$, fixing the index set. The source's nonnegative\nmajorant is independent of $N$. Dropping the ordering of the positive parts\ngives an upper bound by separate sums for each part. The $j$th sum is at most\na fixed constant times $q^{2j-1}$: geometric decay controls its polynomial\nweight. Multiplying gives a bound for the total majorant at length $\\ell$:\n\\[\n                         K^\\ell q^{\\ell^2},\n\\]\nwhere $K>0$ may depend on the fixed base and the fixed weight-bound constants.\nHere $\\ell^2=1+3+\\cdots+(2\\ell-1)$ is the minimum total displacement exponent\nfor $\\ell$ positive parts. Their sizes have been summed out; their number\nremains. The ratio of successive bounds is $Kq^{2\\ell+1}\\to0$, so the sum over\n$\\ell$ converges and dominated convergence applies.\n\nThe conclusions are distinct: the whole summand converges for a fixed partition;\na summable bound controls all partitions uniformly in $N$. Neither asserts\nuniformity as $q\\uparrow1$. The fixed base is part of the statement.\n\n\\subsection{Does a complete test provide the required witnesses?}\\label{sec:detection}\n\nIn the \\#249 test, fix a positive integer shift $h$ and an index $N\\ge0$,\nand write $D$ for that particular real tail difference. At each positive integer\ndepth $L$, the paper computes an integer $A_L$ and proves\n\\[\n       |2^LD-A_L|\\le B_L,\\qquad B_L=N+h+L+2.\n\\]\nAn integer $D$ would put a multiple of $2^L$ in this closed enclosure. Hence\none can certify nonintegrality by checking\n\\[\n       B_L<(A_L\\bmod 2^L)<2^L-B_L,\n\\]\nwhere the residue is chosen in $\\{0,\\ldots,2^L-1\\}$\n\\cite[section on binary totient tails]{paper249}. The two strict margins\nkeep the enclosure away from the neighbouring multiples of $2^L$.\n\nThe test also detects every nonintegral fixed $D$ at some depth. Its gap\n$\\|D\\|_{\\mathbb R/\\mathbb Z}$ from the nearest integer is then a positive\nconstant. The source proves that detection follows once\n\\[\n                 2^L\\|D\\|_{\\mathbb R/\\mathbb Z}>2B_L.\n\\]\nThe factor $2$ pays for two distances: from $2^LD$ to the computed centre\n$A_L$, then from that centre to an enclosure endpoint. Each is at most $B_L$;\nthe displayed gap keeps the whole enclosure away from multiples of $2^L$.\nSince $2^L$ grows exponentially and $B_L$ linearly, a suitable depth exists.\nThus the test has a converse, not merely a sufficient condition.\n\nThe converse has two limits. An integral value never triggers the test;\nfailure at the depths tried does not certify integrality. Nor does detecting\na fixed nonintegral value supply the inputs needed for irrationality: for\nevery positive shift and every cutoff, some later index must have a nonintegral\ndifference. Existence of such an index is a separate question; increasing depth\nonly refines the test of a fixed index. Preserve the converse without mistaking\nit for the missing existence statement.\n\nWhenever a proof offers arbitrarily accurate certificates, state which object\nstays fixed as the accuracy improves. If the object changes too, its distance\nfrom the forbidden set may shrink, and a new comparison is needed. This is the\nsame discipline that made the fixed-base limit in\nSection~\\ref{sec:fixed-partition} readable.\n"},{"record_type":"paper_source_dependency","tex_reference":"exposition/parts/review.tex","source_path":"paper/exposition/parts/review.tex","media_type":"text/x-tex","sha256":"sha256:6dac232d3cb954231d52165102e3b80703a9b33840f2a6d6269f7815fc06a31c","byte_count":11243,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n\\section{Review changes without changing their meaning}\\label{sec:review}\n\\papersectiontarget{exposition-review}\n\nA smoother sentence can make a different claim. Compare the proposal with\nthe current statement and proof before judging its style: what may vary,\nwhich objects must be shared, and which direction of implication is justified?\nThe following cases show how a changed noun, verb or omitted condition can\nalter those answers.\n\nRecord whether the proposal was accepted, revised, rejected or left pending\na named check, retaining the exact passages and the decision's source. A\ntext match locates a change; it does not establish equivalence. A missing\nargument requires mathematical review, and acceptance of a local revision\ndoes not adopt a universal writing rule.\n\n\\subsection{Replace a local term without losing its definition}\n\nIn the R7 \\#249 revision, a private growth adjective was replaced by its\ndefining condition:\n\\begin{quote}\n\\emph{Before:} tempered integer carry orbit.\\par\n\\emph{Selected revision:} integer carry sequence $u$ with\n$u(N)/2^N\\to0$.\n\\end{quote}\nThe gain is specific. A reader no longer has to recover a private definition\nto know the required rate. It would be incorrect to replace this condition by\nthe more familiar phrase ``subexponential growth'', which says something\nstronger. Other revisions retained standard terms such as upper Banach\ndensity because those terms denote the properties the paper actually uses.\nThe test is definitional: substitute the proposed term's meaning back into\nthe claim, then check that the same sequences qualify.\n\nRemoving a shorthand can also remove the visible source of its assumptions.\nIn \\#243, ``positive exact state'' bundled recurrence and positivity\nconditions. A proposed replacement referred to the same assumptions after\nthat definition had disappeared. The integrating reviewer instead stated the\nrecurrences for the first implication, then explicitly added natural-number\ndomains and $a_n>1$, $C_n>0$ for the eventual consequence. In a second passage,\nthe reviewer removed ``earlier'' from ``earlier decreases'': the original\nqualification did not restrict the time at which decreases could occur.\n\nThese R7 selections are not the original proposals or later formal-correspondence\nand rendering checks. The replacement must carry the same assumptions and\nquantifiers, not merely read more easily.\n\n\\subsection{Check the ordinary words that carry scope}\n\nAn unchanged formula can acquire a stronger claim from its surrounding prose.\nThe R11 \\#251 revision replaces ``uses all late indices'' by\n``permits changes at every index after the prefix'' in a comparison\nconstruction~\\cite[comparison construction]{paper251long}. The second describes allowed positions, not a promise that\nevery one is changed. In the sparse construction, the containing set $S$ and\ntarget interval are fixed before the target is chosen; the correction depends\non that target, and its support may occupy only part of $S$\n\\cite[sparse-perturbation proposition]{paper251}.\n\nThe same revision replaces the claim that corrections can ``grow arbitrarily\nslowly'' by an eventual upper bound chosen in advance. Given any prescribed\nfunction tending to infinity, the corrections can eventually be bounded by\nit. This does not say that the corrections themselves tend to infinity.\nRead the choice order as an instruction: prescribe the bound first, then\nconstruct corrections subject to it. This exposes what the short phrases\nobscured. Expand the expression whose ordinary reading changes that order\nor turns permission into a requirement.\n\n\\subsection{Rejecting an attractive universal rule}\n\nEarlier reviews of the \\#243 manuscript illustrate why historical advice\nneeds its original setting. One round asked for a cubic irrationality example\nnear the beginning. A subsequent round explicitly retained the live\nbounded-negative result as the paper's lead and rejected an invariant rule\nthat an irrationality result must always come first. Later editorial routes\ntreated page targets as subordinate to the proof's dependencies.\n\nThe retained lesson is to choose a coherent principal contribution and explain\nits relation to the named problem. It does not prescribe a theorem category\nor a fixed page on which every proof must finish. A conditional reduction\ncan be the useful result; an unconditional statement can be too weak to carry\nthe paper. The choice requires comparing what the results actually say. The\nhistory record preserves the successive recommendations instead of combining\nthem into contradictory commands.\n\nThis distinction also protects the longer research record. Shortening a\npaper can be appropriate when the subordinate details have a complete,\nchecked destination. Regenerating the long record from the shortened version\nwould lose the very argument that made the shorter presentation possible.\nThe two documents must be compared in both directions: a stronger hypothesis,\na repaired endpoint or a newly explained limiting step may appear first in\neither one.\n\n\\subsection{Distinguish clarification from a proof repair}\n\nAn earlier \\#251 review found more than an unclear sentence. A condensed\nsparse-construction proof used a buffer quantity without assigning it a\nsequence coordinate or including it in the value, size and capacity accounting.\nThe original returned memorandum did include those coordinates and their\ncontribution to the weighted sum. The accepted ordinary repair restored the\nassignments and the accounting across all indices.\n\nThe distinction matters for both mathematics and attribution. The condensed\nproof needed repair; the recoverable original showed that the loss had occurred\nduring shortening or integration, not in that construction. This repair was\nundertaken in its own mathematical review. An exposition-only pass should\nidentify such a defect and refer it, not silently supply the missing proof.\nA later sharp companion construction was still unreviewed at its recorded\ndisposition. Sharing a bundle with the repaired proof did not give it the\nsame status.\n\nOther cases expose similar changes of meaning in small amounts of prose.\nIn \\#1049, language suggesting that a degree bound was inevitable was repaired\nto state a sufficient inequality and preserve the qualification contributed\nby a remaining factor. In \\#1041, the revision retained the fixed-polynomial\nscope instead of turning it into a freely varying family. For mixed\nirrationality criteria in \\#257, two unbounded sets of candidate indices\nmay be disjoint: the even and odd integers provide an elementary example.\nThe proof needs one index satisfying both conditions. The source adds the\ntwo nonnegative, normalised errors on the same finite distribution. Their\nsum has mean below one, so it is below one at some sampled index.\nNonnegativity puts both errors below one there; no independence is needed~\\cite[common-index argument]{paper257}.\nThese are mathematical checks even when the edit is presented as compression or style.\n\nCompare the Lean proposition with the prose, including definitions,\nhypotheses and quantifiers. A checked ingredient need not cover the assembled\nargument. An ordinary proof may lack complete formalisation. Keep named inputs\nand pending comparisons visible. The systems paper's records preserve these\ndistinctions~\\cite{systems}; describing them does not confer proof status.\n\nA late-September prose pass removed remarks naming mathematical inputs beside\nthe dependent results; the integrating agent restored them. Details of running\na check could recede, but the dependencies could not. Calling both kinds of\nsentence workflow commentary had erased a mathematical qualification.\n\n\\subsection{Read explanatory words as mathematical claims}\\label{sec:words-as-claims}\n\nTest a mathematical adjective by substituting its definition. A ``convex\ncombination'' requires nonnegative weights whose sum is one. Checking only\ntheir signs leaves half the condition untested. In \\#1041, Abel summation\nexpresses $f(t\\zeta)$ as\n\\[\n f(t\\zeta)=\\sum_{j=0}^{n-1}(t^j-t^{j+1})S_j,\n \\qquad S_j=\\sum_{k=0}^{j}a_k\\zeta^k,\n\\]\nwhere $n\\ge1$, $f(z)=\\sum_{k=0}^n a_kz^k$ and $f(\\zeta)=0$.\nFor $0\\le t<1$, the nonnegative coefficients sum to $1-t^n$.\nThus it is $f(t\\zeta)/(1-t^n)$ that the calculation directly presents as a\nconvex combination of the $S_j$~\\cite[section on trinomials]{paper1041}.\nAt $t=1$ this quotient is undefined, and the proof instead uses $f(\\zeta)=0$.\nNaming the normalised quantity checks the definition and the endpoint in the\nsame sentence. A root-distance calculation or a statement about segments to\nzeros must likewise retain the particular geometric object it controls.\n\nA shorter proof can also be clearer when its reason is already available.\nThe \\#243 arithmetic normal form is\n\\[\n                 Q(n)=m\\binom{n+2}{3}+c,\n                 \\qquad m\\in\\mathbb Z_{>0},\n\\]\nand $Q(n)$ is known to be integral at every sufficiently large integer $n$.\nAt any such nonnegative $n$, the binomial coefficient is integral, so\n$c=Q(n)-m\\binom{n+2}{3}$ is integral\n\\cite[reduction and the constant term]{paper243}.\nThe proposed explanation uses exactly those established facts. Replacing them\nby ``$Q$ is an integer polynomial'' would be unsafe: an integer-valued polynomial\ncan have nonintegral coefficients, as $n(n-1)/2$ shows. The useful simplification\nnames the available arithmetic structure.\n\nIn the \\#257 finite average, $L,d,T$ are positive integers. The residues of\n$L,2L,3L,\\ldots$ modulo $d$ repeat with period $d/\\gcd(L,d)$; the sample\nuses only the first $T$ terms. The inequality $d\\le LT$ does not assert\nthat one whole period occurs. For $L=3,d=5,T=2$, it holds while the period\nhas length five. Split the sample into complete periods, possibly none,\nand one remainder. The kernel weights of Section~\\ref{sec:formula-choice}\nare nonnegative, so that remainder contributes no more than a whole period.\nAllowing one extra period supplies the error term after division by $T$\n\\cite[finite means for shifted divisor tails]{paper257}.\nThe example refutes the claim of a complete sampled period, not the estimate.\n\n\\subsection{Correct the source without rewriting its history}\n\nThe R6 \\#1049 return attached a reader-motivation specimen from Knuth's notes\nto \\S1, item~18. Inspection of the original places the passage in item~12,\non printed page~3 (PDF page~5). Item~18 concerns a different matter. The\npublic lesson record retains both the reported locator and the correction.\n\nA search hit may be a table of contents, another occurrence or an adjacent\ncolumn. Inspect the original passage, then correct the locator without crediting\nthe earlier return with that correction. Useful prose can survive a citation\nrepair; its usefulness does not excuse the error.\n\nThe same care applies to the role of the source. A paper may supply a theorem,\nhistorical attribution, established terminology or an example of composition.\nThose uses call for different claims. A paragraph about proof pacing does not\nvalidate the proof being paced. A literature specimen establishes that an\nauthor made a particular choice, not that this choice is universally preferable.\n"},{"record_type":"paper_source_dependency","tex_reference":"exposition/parts/practice.tex","source_path":"paper/exposition/parts/practice.tex","media_type":"text/x-tex","sha256":"sha256:6a1971fdb963bb14a67b33d68c4f5d80d2ac6e124f7c635e680b5487126c0553","byte_count":16339,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n\\section{Keeping a useful practice small}\\label{sec:practice}\n\\papersectiontarget{exposition-practice}\n\nRevise with the two-page \\emph{Writing a Good Mathematical Paper}~\\cite{compact};\nuse this companion for examples and limits. The writing skill~\\cite{skill}\nand its records retain the detailed cases. Locate the present difficulty,\nuse the relevant advice and leave successful passages alone. Rereading every\npast return is not a prerequisite.\n\n\\subsection{Revise in an order that preserves the argument}\\label{sec:revision-order}\n\nStart with the full statement, proof and closest prior results. State the\ncontribution, decisive inference and boundary in a few sentences, then compare\nthem with the abstract and introduction. Trace each advertised gain to the\nprecise statement that supplies it. Preserve sufficient-only conditions, shared\nwitnesses and fixed parameters under compression. Agreement between summaries\ndoes not help when all inherit the same overclaim.\n\nNext mark the point where a reader must supply a consequential step. Write\ndown what the preceding passage establishes and what the next passage needs.\nIn the certificate case, the first supplies an error bound and the second\nneeds containment in an open interval. The missing explanation is the comparison\nwith both endpoint gaps, not more description of the computation. A nearby\nproof can suggest how to present that comparison.\n\nFind the answer in the manuscript's own argument. If it is established elsewhere,\nbring the relevant fact or a precise reference to this point. If it is not\nestablished, record a mathematical issue rather than insert ``therefore''.\nWhen it is established, write the connecting sentence with its conditions.\nThen check whether it makes the inference or merely renames it: ``the estimate\nis sufficient'' still leaves the certificate's two endpoint comparisons unexplained.\nThe source models the explanation; the local proof supplies its warrant.\n\nReview the revised paragraph with its predecessor and successor. Definitions\nmust arrive before they are used, and the paragraph's conclusion must supply\nwhat the next one needs. Read displayed formulas as parts of grammatical\nsentences, checking punctuation and the referent of each symbol. Conrad's\nexamples illustrate how unspecified variables and loose quantification can\nmake ordinary-looking prose say the wrong thing~\\cite[\\S\\S1--2]{conrad}.\nA sentence-level pass is valuable after the dependency structure is sound;\nit cannot substitute for that earlier work.\n\nFinally, remove repetition that performs no further work. Deriving a formula\nand showing how to apply it are different tasks. Test a cut by reading the\nremaining transition: would the reader now have to rediscover a parameter\nchoice, hypothesis or inference? Keep that explanation; cut its duplicate. Stop when another change resolves no identifiable\ndifficulty, improves no useful reading route and corrects no inaccuracy.\nThis is a reason to preserve an effective passage, not a ban on substantial\nrevision when the difficulty requires it.\n\n\\subsection{Keep the short paper and long record useful separately}\\label{sec:short-long}\n\nBefore moving a passage, decide what each document must let its reader do.\nThe short paper needs an intelligible proof of its principal result or a clearly\nlabelled sketch with a precise full-proof destination. The long record must\npreserve the omitted derivation, assumptions and role in the argument. A\n``details'' link is insufficient when its destination proves a different\nstatement or skips the difficult step.\n\nCompare the shared assertions in both directions. A repaired endpoint in the\nshort paper must reach the long proof; an assumption exposed in the long proof\nmust reach the short statement. After a move, follow the link to the actual\npassage. Match its hypotheses to the claim, and its notation to the short\npaper's quantities; a correct proof of a nearby statement is not the missing\nproof. Retain alternatives and counterexamples that explain scope or a failed\nmethod. Material that serves only the revision history can remain in the\neditorial record rather than interrupting the mathematical argument.\n\n\\subsection{Put verification details where they can be checked}\n\nThe mathematical argument should state its hypotheses and explain the\ninference. An exact computation may be part of that inference; give its\nmathematical input, output and finite scope at the point of use. Put the\nsoftware version, source identifiers, replay commands and review history in\none verification and reproducibility section or appendix. Refer there from\na theorem when its evidence class matters. This keeps proof status available\nwithout making a reader decode repository machinery between proof steps.\n\nState shared limitations once, repeating a condition wherever its omission\nwould make a dependent claim read unconditionally. Editorial disclaimers\nneed not follow every equation. Cite an external result where it is used;\nlocate an omitted proof by section or theorem, not merely by filename or\n``long record''.\n\n\\subsection{How the practice developed}\n\nThe practice developed by comparing proposed revisions with the arguments\nthey were meant to explain. The proposals were not a consistent programme.\nEarlier \\#243 reviews recommended a cubic result and a bounded negative\nresult as the lead at different stages. The useful lesson was to choose the\ncontribution that gives the current paper a coherent argument, not to preserve\na permanent ranking of result types.\n\nIntegration also distinguished delivery from acceptance. In the later second\nround, only one of nine candidates was accepted although the returns had\npassed transport checks. Third-round integration restored seventeen named-input\nremarks removed during compression. Intact files did not ensure that a\nrevision preserved the mathematics or credit. Comparing short and long versions\nalso recovered omitted detail, as in the \\#251 buffer-coordinate case.\n\nLater returns made source models and before-and-after passages more explicit.\nThe finite-remainder explanation survived review; a Knuth locator required\ncorrection; a broad author--reader analogy did not justify a particular rule\nabout estimates. The records retain those different outcomes rather than\ncounting every proposal as progress.\n\nThe nine R11 returns received editorial acceptance; accepted R12 revisions\nrefine existing advice, not nine new instructions. The \\#269 case separates a\nshort interval from one excluding an integer; the systems case separates a\nrecorded rationale from its adequacy~\\cite{systems-r12}. These dispositions\nrecord editorial acceptance, not reader benefit. Later criticism must be\nchecked against the current source: an earlier defect may already be repaired.\n\n\\subsection{From one case to a bounded amendment}\n\nMost accepted changes should become examples of existing guidance. Before\nadding a rule, try the existing instruction on the case: what decision does\nit fail to settle? Add only that missing distinction, with its circumstance,\naction, evidence and limit. A page-one request\ncan become a reminder to remove unnecessary delay without becoming a page-one\nquota. A failed proof can become an explanation of a missing hypothesis without\nbecoming a claim that every alternative approach is impossible.\n\nAn edit summary without the accepted words establishes neither verbatim\nacceptance nor rejection. Keep the proposal, source and decision for later\ninspection; do not resolve the correspondence by assumption. Nor does including\na source claim a review of all its mathematics.\n\nFor a failure, record the attempted implication, assumptions, witness and\nmissing information. A failed estimate does not refute the theorem; an unrun\ncheck is not a failed one. An unavailable source limits review, not the truth\nof its claim. Preserve what the failed route actually ruled out.\n\n\\subsection{Keep a revision recoverable}\n\nKeep the manuscript, the guidance used and the source passages needed for\nreview at identified versions. A manifest of paths and byte digests identifies\nthe supplied files; it does not establish which passages anyone read. Retain\nthe proposed words, their reason and limit, and the decision reached against\nthe current argument. In a cumulative revision, check each earlier improvement\nagainst the new wording: an intact archive preserves history, not necessarily\nthe improvement in the manuscript.\n\nA later correction to a citation or proof does not become an achievement of\nthe earlier return. Preserve both versions rather than silently replacing\nits history. Adopt a general lesson only through a separate review of its\nscope. This is revision of documents and guidance, not training of model\nweights.\n\n\\subsection{Read the page that will be read}\n\nRender the sources that will be delivered, recording the toolchain used.\nAfter changing a figure, inspect it at the size in which it appears. Follow a\nreference to the intended theorem or long-form argument, rather than merely\nchecking that a target file exists. After reflow, inspect the affected pages\nand their neighbours: a useful explanation can be separated from its figure,\nor a small heading can be stranded above a page break. Bibliographic labels,\ncaptions, interval endpoints and cross-document links belong to this review.\n\nRead the result along three routes. First, scan the title, abstract,\nintroduction and main statements: recover the contribution and its boundary\nwithout importing private knowledge. Second, reconstruct the proof: identify\nwhere each hypothesis is used and explain the difficult inference in your own\nwords. Third, try to use a result: check its assumptions on an example, identify\nan excluded case where known, and find the cited input needed for an application.\n\nAsk a question that requires using the explanation. In\nSection~\\ref{sec:certificate}, why are there two inequalities, and can a failed\ntest become successful without changing $D$? The target is $(1/2,1)$; each\ninequality protects an endpoint. For $D=3/4$, replacing the enclosure $[1/2,1]$\nby a sharper one can change the verdict without changing the value. The reader\nmust distinguish value from enclosure, not merely repeat ``compare both endpoints''.\nA difficulty with that task identifies a passage to revisit, not a numerical\nscore for understanding.\n\nAsk an independent reader where reconstruction stopped and what seemed\nestablished. An author's cold-start pass is useful self-review; describing\nthis procedure does not imply that an independent reader took part.\n\nThe checks answer limited questions. A successful build establishes that the\nsource rendered under the recorded toolchain. A passage comparison helps\nestablish what changed. A source inspection supports a citation's locator\nand role. Neither those checks nor a lower rate of phrases flagged by a style\ndetector establishes improved comprehension. A reported style comparison must\nidentify the versions, prose denominator and genre, and retain regressions\nalongside gains. A claim of improved comprehension would require evidence\nfrom readers performing a specified task on identified versions, with the\ncomparison and its limitations recorded.\n\nThe immediate editorial test is whether the revision exposes a warranted\nrelation and the record identifies what changed and was accepted. That test\ncan favour a longer explanation in one proof and a shorter one in another;\nmeasured reader benefit remains a separate question.\n\n\n\\section{Transfer the method across research genres}\\label{sec:genre-transfer}\n\nA theorem needs a proof under its stated hypotheses. For a system, distinguish\nwhat is proposed, implemented and tested. A performance claim needs the\nimplementation, workload, comparator and observed result; an artefact build\ndoes not establish unrestricted performance or usability.\n\nLevin and Redell distinguish implemented, proposed and theoretical systems:\nevaluation criteria depend on the paper's class~\\cite{levin-redell}. SIGPLAN's\nempirical-evaluation checklist aids judgement; it is not a universal\nscore~\\cite{sigplan-evaluation}. Read a nearby system's opening and evaluation,\nthen trace the target task: what each component receives, changes and passes\non, and who reviews it. A component list leaves those relations unexplained.\nThese sources guide presentation; they do not verify a particular system.\n\nThe 6 October systems revision makes the task concrete before listing record\nfields~\\cite[\\S\\S1,3; Appendix~B]{systems-current}. Its\nhypothetical editor announces a solution to \\#257 while the unchanged theorem\nstill restricts the exponent set. A successful proof check would not justify\nthat introduction. One record links a paper statement to formal supports;\nanother links explanatory prose to sources. The reviewer must compare the\nproposed passage with those sources. This example motivates the design; it is\nnot an observed trial. The exact mathematical condition remains recoverable\nin the case's own paper and the systems appendix; the introduction need not\nreproduce its derivation.\n\nA fair comparison names the closest corresponding object, not just a common\naim. Prove2Me separates immutable theorem statements from submitted proofs and\nfixes a human-audited mission core. Its milestones link source statements to\nattested formalisations~\\cite[\\S\\S3--4]{prove2me}. The R12 systems revision\ncompares a milestone with its coverage record, which follows a particular\npaper occurrence and lists the registered formal supports. Listing supports\ndoes not compose their proofs. A separate record binds explanatory prose to\nsources~\\cite[related work]{systems-r12}. This difference of focus establishes\nneither exclusivity, priority nor an advantage of repository hosting over a\nservice.\n\nKeep the operations distinct: builders regenerate views; release checks test\nconsistency. The passage checker requires matching text and source digests\nand a nonempty rationale, not an adequate rationale~\\cite[\\S3.2; Appendix~A]{systems-current}.\n\n\nThe repository's systems manuscript gives a concrete example. It reports nine\nrejections among ten deliberately false, author-selected edits in a historical\ntrial. After one edit escaped, a follow-up checked the intact baseline and\nthat escaped edit against a repair; the other nine edits were not rerun\n\\cite[recorded observations]{systems}. The first report describes those ten\ntrials, not a general $90\\%$ reliability rate. The second describes the tested\nrepair, not a post-repair ten-out-of-ten result. The manuscript also records\nmissing original logs and the absence of an independent comparison. A repair\nthat catches one escaped edit has passed that regression check; its effects\non unrerun cases remain unmeasured.\n\nIn a scientific exposition, identify the study or derivation, the population\nor model, the reported result and the author's synthesis. State when there\nis no new experiment. Check each inference against the primary study: an\nassociation is not automatically causation, and a model prediction is not\nan observation.\n\nKeep the condition and comparator beside each empirical claim. The two reports\nabove concern different trials; a distant limitation cannot undo a sentence\nthat combines them. Use parallel clauses to make their different scopes visible.\n\n\\section*{Availability and authorship}\nThe writing skill, source records, lesson history and this paper's LaTeX source\nare maintained in the public repository~\\cite{ledger,skill}. The accompanying\neditorial return documents the new-return examples separately from the accepted\nhistorical cases. The public writing method and its review ledger record\nselected guidance, with source-reading and acceptance boundaries preserved.\nThe worked revisions were made across papers on eight Erd\\H{o}s problems.\nSome agents worked from frozen packets; others integrated changes against the\nlive repository. Their editorial judgements do not measure how independent\nmathematical readers understood the papers. The method has not been tested\nfor improved reader comprehension.\nWill Cook built and directed the research infrastructure and reviewed\nthe claims when he could. AI agents did most of the research and drafting.\nCook did not independently verify every claim.\n"},{"record_type":"paper_source_dependency","tex_reference":"exposition/parts/references.tex","source_path":"paper/exposition/parts/references.tex","media_type":"text/x-tex","sha256":"sha256:d7a4c97fb1544e9504e134a70df96ed97785740fd14ca328f4646a4fba0f9356","byte_count":8148,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n\\begin{thebibliography}{99}\n% Keep each reference entry together after prose reflow.\n\\interlinepenalty=10000\n\\bibitem{halmos}\nP.~R. Halmos, How to write mathematics, \\emph{L'Enseignement Math\\'ematique}\n(2) \\textbf{16} (1970), 123--152. Cited edition: the scanned article,\n\\S\\S3--4. \\url{https://doi.org/10.5169/seals-43857}.\n\n\\bibitem{knuth}\nD.~E. Knuth, T.~Larrabee and P.~M. Roberts, \\emph{Mathematical Writing},\nStanford Computer Science report STAN-CS-88-1193 (1988), notes from the\n1987 course; book edition, Mathematical Association of America, 1989.\nLocators here refer to \\S\\S1--3 of the report; the worked exercise and\nproof comparison are on printed pp.~7--8 (PDF pp.~9--10).\n\\url{https://cs.stanford.edu/~knuth/klr.html}.\n\n\\bibitem{gowers}\nT.~Gowers, My favourite pedagogical principle: examples first,\n19 October 2007, author's essay.\n\\url{https://gowers.wordpress.com/2007/10/19/my-favourite-pedagogical-principle-examples-first/}.\n\n\\bibitem{tao}\nT.~Tao, Don't overoptimise, author's writing advice,\nsource snapshot inspected 30 September 2026.\n\\url{https://terrytao.wordpress.com/advice-on-writing-papers/dont-overoptimise/}.\n\n\\bibitem{crmaric}\nT.~Crmari\\'c and V.~Kova\\v{c}, On the irrationality of certain\nsuper-polynomially decaying series, arXiv:2504.18712v1 (25 April 2025).\nCited passage: Lemma~4(a), pp.~4--5.\n\\url{https://arxiv.org/abs/2504.18712v1}.\n\n\\bibitem{hancl}\nJ.~Han\\v{c}l and R.~Tijdeman, On the irrationality of Cantor and Ahmes series,\n\\emph{Publicationes Mathematicae Debrecen} \\textbf{65}, no.~3--4\n(2004), 371--380. \\url{https://doi.org/10.5486/PMD.2004.3254}.\n\n\\bibitem{wangzhu}\nY.~Wang and B.-X. Zhu, Log-convex and Stieltjes moment sequences,\n\\emph{Advances in Applied Mathematics} \\textbf{81} (2016), 115--127.\nCited source: arXiv:1612.04114v1, \\S1, pp.~2--3.\n\\url{https://arxiv.org/abs/1612.04114v1}.\n\n\\bibitem{ledger}\nW.~Cook, \\href{https://github.com/wcook04/plectis-erdos/tree/main/docs/papers/exposition-method}{Literature and reviewed-revision guide},\nwith lesson, history, return and source records, 30 September 2026.\n\n\\bibitem{skill}\nW.~Cook, \\href{https://github.com/wcook04/plectis-erdos/blob/main/skills/public-mathematical-writing/SKILL.md}{Public mathematical writing}, repository skill.\n\n\\bibitem{compact}\nW.~Cook, \\href{https://github.com/wcook04/plectis-erdos/blob/main/paper/exposition/writing-a-good-mathematical-paper.pdf}{Writing a Good Mathematical Paper},\ncompact guide, 1 October 2026.\n\n\\bibitem{systems}\nW.~Cook, \\href{https://github.com/wcook04/plectis-erdos/blob/main/paper/systems/claim-faithful-publication-systems-paper.tex}{Publishing Mathematical Results from a Lean\nRepository}, 30 September 2026.\n\\bibitem{gowers-followup}\nT.~Gowers, Examples first II, 24 October 2007, author's follow-up essay.\n\\url{https://gowers.wordpress.com/2007/10/24/examples-first-ii/}.\n\n\\bibitem{tao-detail}\nT.~Tao, Give appropriate amounts of detail, author's writing advice,\ninspected 30 September 2026.\n\\url{https://terrytao.wordpress.com/advice-on-writing-papers/give-appropriate-amounts-of-detail/}.\n\n\\bibitem{tao-notation}\nT.~Tao, Use good notation, author's writing advice,\ninspected 30 September 2026.\n\\url{https://terrytao.wordpress.com/advice-on-writing-papers/use-good-notation/}.\n\n\\bibitem{conrad}\nK.~Conrad, Advice on Mathematical Writing, author's ten-page handout,\n\\S\\S1--2, inspected 30 September 2026.\n\\url{https://kconrad.math.uconn.edu/blurbs/proofs/writingtips.pdf}.\n\n\\bibitem{paper68}\nW.~Cook, Integer Linear Forms for a Factorial Reciprocal Series,\n\\#68 short paper, supplied source dated 30 September 2026.\n\\url{https://github.com/wcook04/plectis-erdos/blob/main/paper/68/erdos-68-factorial-denominator-irrationality.tex}.\n\n\\bibitem{paper243}\nW.~Cook, Reciprocal Sums and the Sylvester Recurrence,\n\\#243 short paper, supplied source dated 30 September 2026.\n\\url{https://github.com/wcook04/plectis-erdos/blob/main/paper/243/erdos-243-reciprocal-tail-rigidity.tex}.\n\n\\bibitem{paper249}\nW.~Cook, Integral Relations among Totient Sections,\n\\#249 short paper, supplied source dated 30 September 2026.\n\\url{https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos-249-binary-totient-series.tex}.\n\n\\bibitem{paper249long}\nW.~Cook, \\#249 companion research record, supplied source dated\n30 September 2026, comparison-sequence construction.\n\\url{https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos249-totient-reasoning-surface.tex}.\n\n\\bibitem{paper251}\nW.~Cook, Sparse Congruence-Preserving Perturbations of Dyadic Series,\n\\#251 short paper, supplied source dated 30 September 2026.\n\\url{https://github.com/wcook04/plectis-erdos/blob/main/paper/251/erdos-251-prime-gap-dyadic-series.tex}.\n\n\\bibitem{paper257}\nW.~Cook, Irrationality criteria for Lambert subseries,\n\\#257 short paper, supplied source dated 30 September 2026.\n\\url{https://github.com/wcook04/plectis-erdos/blob/main/paper/257/erdos-257-mersenne-support-subseries.tex}.\n\n\\bibitem{paper269}\nW.~Cook, Distinct running least common multiples,\n\\#269 short paper, manuscript snapshot, 1 October 2026.\n\\url{https://github.com/wcook04/plectis-erdos/blob/main/paper/269/erdos-269-three-prime-running-lcm.tex}.\n\n\\bibitem{paper1041}\nW.~Cook, Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections,\n\\#1041 short paper, supplied source dated 30 September 2026.\n\\url{https://github.com/wcook04/plectis-erdos/blob/main/paper/1041/erdos-1041-lemniscate-newton-flow.tex}.\n\n\\bibitem{paper1049}\nW.~Cook, Hankel Determinants of Geometric Moments and Rational Lambert Values,\n\\#1049 short paper, supplied source dated 30 September 2026.\n\\url{https://github.com/wcook04/plectis-erdos/blob/main/paper/1049/erdos-1049-rational-base-lambert.tex}.\n\n\\bibitem{levin-redell}\nR.~Levin and D.~D.~Redell, How (and How Not) to Write a Good Systems Paper,\n\\emph{ACM SIGOPS Operating Systems Review} 17 (1983), 35--40;\nauthorised copy hosted by USENIX.\n\\url{https://www.usenix.org/guidelines-authors}.\n\n\\bibitem{sigplan-evaluation}\nACM SIGPLAN, Empirical Evaluation Guidelines, committee guidance and FAQ\n(updated 2018; inspected 1 October 2026).\n\\url{https://sigplan-www.sigplan.hosting.acm.org/Resources/EmpiricalEvaluation/}.\n\n\\bibitem{paper251long}\nW.~Cook, \\#251 companion research record, R11 manuscript snapshot\nsupplied 4 October 2026, comparison construction and sparse perturbations.\n\\url{https://github.com/wcook04/plectis-erdos/blob/main/paper/251/erdos251-prime-gap-reasoning-surface.tex}.\n\n\\bibitem{prove2me}\nS.~Chen, K.~Marwaha, X.~Lu, H.~Yuen and T.~Peng,\n\\emph{Prove2Me: An Open Collaborative Platform for Scaling Math Formalization},\narXiv:2608.28433v2, 31 August 2026. Cited passages: \\S\\S3--4.\n\\url{https://arxiv.org/abs/2608.28433v2}.\n\n\\bibitem{systems-r12}\nW.~Cook, \\emph{A Repository-Based System for Research and Publication},\n30 September 2026; R12 manuscript snapshot supplied 5 October 2026.\n\\url{https://github.com/wcook04/plectis-erdos/blob/main/paper/systems/claim-faithful-publication-systems-paper.tex}.\n\n\\bibitem{systems-current}\nW.~Cook, \\emph{A Repository-Based System for Research and Publication},\nsource revision accepted 6 October 2026; publication gates pending.\n\\url{https://github.com/wcook04/plectis-erdos/blob/main/paper/systems/claim-faithful-publication-systems-paper.tex}.\n\n\\end{thebibliography}\n\n\\noindent\\small\\textit{Source inspection and reproducibility.}\nHistorical mathematical examples retain their stated revision identities;\nthe current-source review uses the 6 October 2026 packet. The new systems\nexample uses that day's accepted source revision, not the earlier R12 snapshot.\nThe return identifies exact source bytes, inspected passages and decisions,\nseparating earlier reading declarations from fresh inspection. Knuth's proof\ncomparison and Halmos's audience and organisation advice were rechecked;\nthe retained Prove2Me comparison uses version~2, \\S\\S3--4, without new reading\ncredit. Repository links locate manuscript families, not immutable versions.\nSource availability does not imply full reading, and earlier acceptance does\nnot accept this revision.\n"},{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  Under\n% Tectonic (XeTeX) the text and mathematics are set in TeX Gyre Pagella from\n% the bundled OpenType fonts, with a Heros sans range for \\mathsf and\n% Inconsolata for monospace; the pdflatex fallback keeps the same Palatino\n% design through newpx.  The A4 text block uses a classical measure with\n% quiet running heads and restrained ink.  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It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/exposition/writing-mathematics-from-reviewed-revisions.tex","section_outline":[]},"claim-faithful-publication-systems":{"paper_id":"claim-faithful-publication-systems","title":"A Repository-Based System for Research and Publication","form":"Research architecture","length_class":"system","pages":25,"subject":"Systems paper","subject_kind":"system","problem_id":null,"landing_role":"method","question":"How can questions, proofs, computations and failed approaches stay linked to reviewed claims and papers, and what can publication checks miss that mathematical review must catch?","precis":"A correct Lean theorem can be misrepresented when its hypotheses disappear from a paper. This account follows a weighted-support criterion for Erdős Problem 257 through its mathematical argument, two registered formal supports and written explanation. It describes source inspection, publication checks, editorial revision and contributions in a repository covering eight problems. A historical test rejected nine false edits but missed a false completion claim; the paper distinguishes that record from the limited follow-up check. Source-byte agreement and repository adoption do not establish mathematical meaning, independent review or recognition by the community. The paper retains its mathematical antecedents and compares the recorded relations with blueprints and Prove2Me, without claiming comparative benefit.","owns":"the research architecture: problem-centred collections, the path from a proof to a public claim, and the contribution cycle","not_authority_for":"the mathematics it uses as examples, which the problem papers own, or the novelty and significance of those results","first_pass":["sec:intro","sec:example","sec:paper","sec:checks","sec:limits"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/claim-faithful-publication-systems-paper.pdf","hosted_pdf_path":"papers/claim-faithful-publication-systems-paper.pdf","hosted_tex_path":"papers/source/plectis-erdos/paper/systems/claim-faithful-publication-systems-paper.tex","site_html":"maths/papers/claim-faithful-publication-systems.html","github_current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/systems/claim-faithful-publication-systems-paper.tex","github_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/systems/claim-faithful-publication-systems-paper.tex","body_pointer":"papers.records.claim-faithful-publication-systems.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n% SYSTEMS_PAPER_VERSION 2\n\\documentclass[10pt]{article}\n\\usepackage{paper-house-style}\n\\usepackage{needspace,textcomp,tikz,booktabs,array,float,placeins,tabularx,xurl}\n\\usetikzlibrary{arrows.meta,positioning,calc,fit,matrix}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,citecolor=linkinternal]{hyperref}\n\\hypersetup{pdftitle={A Repository-Based System for Research and Publication},pdfauthor={Will Cook},\n pdfsubject={Architecture, evidence and iteration in a Lean research repository},pdflang={en-GB},\n bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-erdos}\n\\newcommand{\\repolink}[2]{\\href{\\repobase/blob/992cf60a27efd8c34f437e347c413bb781d75c32/#1}{\\texttt{#2}}}\n% Prose links retain the same pinned destination without printing a file identifier.\n\\newcommand{\\reporef}[2]{\\href{\\repobase/blob/992cf60a27efd8c34f437e347c413bb781d75c32/#1}{#2}}\n% Records, derived views and review decisions have distinct forms.\n\\tikzset{\n w/.style={text width=#1cm},h/.style={minimum height=#1cm},\n box/.style={draw=black!35,fill=white,inner sep=6pt,align=left,font=\\small},\n rec/.style={box,draw=linkexternal!70,fill=linkexternal!3},\n gate/.style={box,draw=linkinternal,fill=linkinternal!5,line width=.8pt},\n view/.style={box,dashed},\n flow/.style={-{Stealth[length=2mm]},draw=black!65,rounded corners=3pt},\n link/.style={draw=linkexternal!70,line width=.6pt},\n note/.style={font=\\small,align=left,inner sep=0pt},\n tag/.style={font=\\footnotesize,align=center,fill=white,inner sep=2pt}\n}\n\\emergencystretch=1.7em\n\\runninghead{A system for research and publication}\n\\title{A Repository-Based System for Research and Publication}\n\\subtitle{Architecture, evidence and iteration in a Lean research repository}\n\\author{Will Cook}\n\\date{5 October 2026}\n% BEGIN generated_pipeline_counts\n\\newcommand{\\LedgerRows}{689}\n\\newcommand{\\LedgerExact}{498}\n\\newcommand{\\LedgerSpecialising}{142}\n\\newcommand{\\LedgerNamedInput}{23}\n\\newcommand{\\LedgerNoLean}{26}\n\\newcommand{\\LedgerWithLean}{663}\n\\newcommand{\\LedgerFormal}{640}\n\\newcommand{\\LedgerCompared}{616}\n\\newcommand{\\LedgerQueued}{24}\n\\newcommand{\\LeanModules}{1,903}\n\\newcommand{\\Claims}{160}\n\\newcommand{\\ClaimObligations}{19}\n\\newcommand{\\ClaimStatuses}{eight}\n\\newcommand{\\JournalHead}{d3ad85269386}\n\\newcommand{\\Rounds}{7}\n\\newcommand{\\RoundsSealed}{2}\n\\newcommand{\\Returns}{25}\n\\newcommand{\\ReturnsAdmitted}{18}\n\\newcommand{\\ReturnsUnreviewed}{7}\n\\newcommand{\\ProblemWorlds}{8}\n\\newcommand{\\RegisteredPapers}{25}\n\\newcommand{\\ShortPapers}{8}\n\\newcommand{\\LongRecords}{8}\n\\newcommand{\\HistoricalEdits}{ten}\n\\newcommand{\\HistoricalRejected}{nine}\n\\newcommand{\\HistoricalEscaped}{one}\n\\newcommand{\\HistoricalNotRerun}{nine}\n% END generated_pipeline_counts\n% BEGIN generated_semantic_coverage_macros\n\\newcommand{\\SemanticAuthoredTheoremLike}{152,386}\n\\newcommand{\\SemanticAuthoredInterpreted}{139,818}\n\\newcommand{\\SemanticDirectEvidence}{3,330}\n\\newcommand{\\SemanticContextual}{136,488}\n\\newcommand{\\SemanticStructuralOnly}{12,568}\n\\newcommand{\\SemanticAuthoredPercent}{91.8}\n% END generated_semantic_coverage_macros\n\\begin{document}\n\\maketitle\n% BEGIN audited_body\n% SENTENCE r4.provenance\n\\paperprovenance\n% END SENTENCE r4.provenance\n\\begin{abstract}\n% SENTENCE r4.abstract.01\nPlectis is a repository-based research environment for retaining mathematical work across sessions and carrying selected results into publication.\n% END SENTENCE r4.abstract.01\n% SENTENCE r4.abstract.02\nThe public prototype brings questions, sources, proofs, computations and failed approaches together in eight problem collections.\n% END SENTENCE r4.abstract.02\n% SENTENCE r4.abstract.03\nIts publication workflow records the formal-support status of paper statements and binds explanatory passages to cited sources, while keeping these records distinct.\n% END SENTENCE r4.abstract.03\n% SENTENCE r4.abstract.04\nResearchers and agents choose the work; Python tools retrieve evidence, record Lean probes and regenerate views. Maintainers decide what to adopt, and a journal retains entered contributions and review decisions.\n% END SENTENCE r4.abstract.04\n% SENTENCE r4.abstract.05\nA worked revision shows how an explanation can lose a hypothesis while its formal proofs remain valid.\n% END SENTENCE r4.abstract.05\n% SENTENCE r4.abstract.06\nA historical author-run test rejected \\HistoricalRejected{} of \\HistoricalEdits{} false edits and accepted one false completion claim. Reader benefit, discovery rate and adoption by independent laboratories remain unmeasured.\n% END SENTENCE r4.abstract.06\n\\end{abstract}\n\\section{Introduction}\\label{sec:intro}\n\\papersectiontarget{systems-lifecycle}\n% SENTENCE r4.intro1.01\nResearch continues across sessions, contributors and drafts. The next researcher needs more than the latest result: which question remains open, why an approach failed, and what justifies the claims already published. Even a preserved formal proof leaves room for a later explanation to drop a hypothesis or describe an unresolved step as complete. A usable research record must retain these distinctions as the work changes.\n% END SENTENCE r4.intro1.01\n\n% SENTENCE r4.intro1.02\nPlectis supports this work in a public repository. Questions, prior sources, computations, proofs and unresolved steps remain available alongside proposed contributions and their review. The prototype is a local research environment: a researcher or agent chooses what to investigate and which tools to invoke. Broader autonomous discovery and services for other scientific fields remain proposals (Section~\\ref{sec:limits}).\n% END SENTENCE r4.intro1.02\n\n% SENTENCE r4.intro1.03\nThe architecture separates authored records, derived views and adoption decisions (Figure~\\ref{fig:lifecycle}). A successful proof probe can remain a session result; a revised explanation can remain a proposal. Acceptance updates the source records, after which builders regenerate their indexes and summaries.\n% END SENTENCE r4.intro1.03\n\n% SENTENCE r4.intro2.01\nErd\\H{o}s Problems supplies the questions and their public discussion, while Lean and mathlib supply the formal language and library~\\cite{erdosproblems,lean4,mathlib}.\n% END SENTENCE r4.intro2.01\n% SENTENCE r4.intro2.02\nFormalisation blueprints link a written argument to named Lean declarations~\\cite{leanblueprint,leanarchitect}, and the Equational Theories Project maintains shared problem records~\\cite{etp}. Prove2Me combines human-audited missions, source-linked milestones and separately submitted proofs, with intermediate results available for other agents to reuse~\\cite[Sections~3--4]{prove2me}.\n% END SENTENCE r4.intro2.02\n\n% SENTENCE r4.intro2.03\nThe question is how to keep a publication aligned with its changing research record. Assertions, formal supports, explanations and acceptance decisions need separate identities and links. Checks detect specified disagreements; preserving the mathematics remains a review task.\n% END SENTENCE r4.intro2.03\n\n% SENTENCE r4.intro2.04\nSection~\\ref{sec:world} describes the architecture; Sections~\\ref{sec:example}--\\ref{sec:paper} follow an explanation change through checking and review. Contributions, observations and related systems follow. Appendices~\\ref{app:repro}--\\ref{app:source-map} cover inspection, the worked proof, scope and attribution examples, and references grouped by purpose.\n% END SENTENCE r4.intro2.04\n\n% SENTENCE r4.fig.lifecycle\n\\begin{figure}[H]\\centering\n\\begin{tikzpicture}\n\\node[note,anchor=west]at(-6.7,.7){\\textsc{Authored sources}};\n\\node[note,anchor=west]at(2.6,.7){\\textsc{Derived views}};\n\\node[rec,w=6.2,h=1.35](r)at(-3.4,-.4){\\textbf{Research records}\\\\Sources, proofs, computations,\\\\failed approaches and open questions};\n\\node[rec,w=6.2,h=1.35](p)at(-3.4,-2.4){\\textbf{Publication records}\\\\Manuscript; claim and coverage records;\\\\passage bindings and rationales};\n\\node[view,w=3.7,h=1.35](i)at(4.6,-.4){\\textbf{Research views}\\\\Indexes, maps and\\\\evidence summaries};\n\\node[view,w=3.7,h=1.35](v)at(4.6,-2.4){\\textbf{Reading views}\\\\Rendered papers\\\\and source links};\n\\draw[flow](r)--(i);\\draw[flow](p)--(v);\n\\node[tag]at(.8,.05){builders};\\node[tag]at(.8,-1.95){builders};\n\\node[gate,w=6.2,h=1.1](a)at(-3.4,-4.75){\\textbf{Review proposed changes}\\\\Check meaning, scope, evidence and credit.};\n\\node[draw=linkexternal!25,fit=(r)(p),inner sep=3pt](sources){};\n\\draw[flow,draw=linkinternal](a)--node[tag,right]{accepted edits}(sources);\n\\node[note,w=3.7](q)at(4.6,-4.75){Readers and agents\\\\choose further work\\\\and propose changes.};\n\\draw[flow](v)--(q);\\draw[flow](q)--(a);\n\\end{tikzpicture}\n\\caption{Authored records and derived views. Accepted edits change the relevant sources; builders regenerate views. Tool success alone does not adopt a claim.}\n\\label{fig:lifecycle}\\end{figure}\n% END SENTENCE r4.fig.lifecycle\n\\section{A repository organised around problems}\\label{sec:world}\n\\papersectiontarget{systems-scaling}\n% SENTENCE r8.architecture.records\n\\subsection{Stored records and derived views}\nFiles store the research state: Lean and manuscript arguments, selected claims and obligations, evidence, entered returns and decisions. Indexes, coordinates and status summaries derive from them. Freshness checks detect disagreement; builders regenerate views. Neither adopts a claim. The \\repolink{docs/ARCHITECTURE.md}{architecture guide} identifies each view's source, where repairs belong.\n% END SENTENCE r8.architecture.records\n\n% SENTENCE r12.architecture.correspondence\nThe coverage ledger follows a labelled \\emph{paper occurrence}, identified by location and statement digest, and lists its registered Lean supports and evidence status. A stronger formal result needs an authored relation note bound to both statements. The passage ledger records explanatory text, cited spans, evidence class and rationale. The first asks what supports the statement; the second exposes the explanation's sources for review. Neither generates prose from a proof.\n% END SENTENCE r12.architecture.correspondence\n\n% SENTENCE r4.world1.01\n\\subsection{Finding the relevant work}\nA problem collection retains proofs, counterexamples, exact experiments, corrected references and failed approaches, so later work can inspect them before choosing another route.\n% END SENTENCE r4.world1.01\n% SENTENCE r4.world1.02\nFour views provide entry points without requiring a full Lean build (Figure~\\ref{fig:world}).\n% END SENTENCE r4.world1.02\n\n% SENTENCE r4.world1.03\nA \\emph{declaration} is a named Lean definition, theorem or lemma. The declaration index locates it; the dependency index follows the declarations used by selected results.\n% END SENTENCE r4.world1.03\n% SENTENCE r4.world1.04\nThe interpretation graph adds authored accounts of declarations and their connections. The claim registry selects assertions for public reporting and records their status and remaining obligations.\n% END SENTENCE r4.world1.04\n\n% SENTENCE r4.fig.world\n\\begin{figure}[htbp]\n\\centering\\small\n\\begin{minipage}{\\linewidth}\n\\textbf{For each problem:} the question and literature, Lean statements and proofs,\ncomputations and certificates, claims and open obligations, failed approaches,\nand the paper with its supporting record.\n\\par\\medskip\n\\begin{tabularx}{\\linewidth}{@{}>{\\raggedright\\arraybackslash}p{3.25cm}>{\\raggedright\\arraybackslash}X@{}}\n\\toprule\nView available without a Lean build & Question it answers\\\\\n\\midrule\nDeclaration index & Where is the declaration? Source locations are exhaustive within the inventory.\\\\[3pt]\nDependency index & Which declarations does this result use? Exact edges are available for selected starting declarations.\\\\[3pt]\nInterpretation graph & How has the work been interpreted? Authored accounts are selective and largely contextual, not individual proof reviews.\\\\[3pt]\nClaim registry & What is being claimed, with which status and open obligations? Only selected assertions have claim records.\\\\\n\\bottomrule\n\\end{tabularx}\n\\par\\medskip\nAcross the corpus: \\ProblemWorlds{} problems, \\LeanModules{} Lean modules and\n\\RegisteredPapers{} registered papers. The claim registry contains \\Claims{}\nclaims in \\ClaimStatuses{} statuses, with \\ClaimObligations{} open obligations.\n\\end{minipage}\n\\caption{Contents of a problem collection and the four ways to inspect the recorded work. The counts refer to the whole corpus. Source locations, dependency edges, interpretations and claim statuses answer different questions.}\n\\label{fig:world}\n\\end{figure}\n% END SENTENCE r4.fig.world\n\n% SENTENCE r4.world2.01\nA reader can start from a claim, locate its declarations, follow the recorded dependencies and request the source lines relevant to a question.\n% END SENTENCE r4.world2.01\n% SENTENCE r4.world2.02\nDeclaration locations cover the inventory; dependency edges are exact for selected starts. Interpretations are selective and largely contextual, not records of individual proof review.\n% END SENTENCE r4.world2.02\n% SENTENCE r4.world2.03\nLookup locates source without recording execution. A Lean check requires a focused build and inspection of its result.\n\\FloatBarrier\n% END SENTENCE r4.world2.03\n\n\n% SENTENCE r8.architecture.tools\n\\subsection{Operations on the records}\nThe entry points below act on the same repository records. They support a research iteration; their order of use is chosen by the researcher or agent.\n\\begin{center}\\small\n\\begin{tabularx}{\\linewidth}{@{}>{\\raggedright\\arraybackslash}p{2.55cm}>{\\raggedright\\arraybackslash}X@{}}\n\\toprule\nComponent & Input, output and responsibility\\\\\n\\midrule\nTask routing & \\repolink{scripts/agent_entry.py}{agent\\_entry.py} maps a stated task to instructions, relevant files and commands. It routes a request rather than choosing the research question.\\\\[3pt]\nEvidence retrieval & \\repolink{scripts/query_corpus.py}{query\\_corpus.py} returns problem, claim, declaration and paper-source records from the indexes.\\\\[3pt]\nProof sessions & \\repolink{scripts/proof_workbench.py}{proof\\_workbench.py} stores notes and submitted Lean probes, obtains verdicts from Lean, and binds a session claim to an accepted probe.\\\\[3pt]\nStatement evidence & \\repolink{scripts/paper_evidence.py}{paper\\_evidence.py} locates each registered support at the recorded revision and checks its source and verification bindings; \\repolink{scripts/paper_claim_evidence.py}{paper\\_claim\\_evidence.py} derives status and gap reports from those records.\\\\[3pt]\nEditorial exchange & \\repolink{scripts/short_paper_writer.py}{short\\_paper\\_writer.py} audits manuscripts and packages proposed revisions without applying or approving the return.\\\\[3pt]\nResearch history & \\repolink{scripts/research_record.py}{research\\_record.py} verifies the journal of entered returns and dispositions and reports the outstanding milestones of recorded outputs.\\\\[3pt]\nDerived views & \\repolink{scripts/refresh_projections.py}{refresh\\_projections.py} runs builders in dependency order; \\repolink{scripts/check_release.py}{check\\_release.py} checks release contracts and freshness.\\\\\n\\bottomrule\n\\end{tabularx}\n\\end{center}\nA separate self-audit, \\repolink{scripts/systems_paper_evidence.py}{systems\\_paper\\_evidence.py}, checks this systems manuscript's text and cited source bindings. These are implemented interfaces, not reports that every operation has been executed successfully.\n% END SENTENCE r8.architecture.tools\n\n% SENTENCE r9.architecture.state\nAn optional proof session illustrates the boundary between producing evidence and adopting a result. It stores notes, exact Lean inputs, process-derived receipts and a closing outcome. Its claim command requires an accepted probe and appends a session claim; it does not alter the reviewed public claim registry. The journal separately checks the sequence and hash chain of entered returns and dispositions. Later work can therefore inspect recorded inputs and decisions without the original conversation.\n% END SENTENCE r9.architecture.state\n\n% SENTENCE r8.architecture.roles\n\\subsection{Responsibilities and adoption}\nA researcher chooses the question and the evidence needed to answer it. An agent can inspect files, run tools and prepare a change under the public instructions; a contributor can also send an ordinary argument or explanation without a clone or an agent. The maintainer reviews scope, attribution and unresolved obligations before adoption. These are responsibilities, not necessarily different people or agents: contributor and reviewer may coincide. The \\repolink{docs/agents/README.md}{agent quickstart} and \\repolink{CONTRIBUTING.md}{contribution guide} describe setup and review.\n% END SENTENCE r8.architecture.roles\n\n% SENTENCE r8.architecture.iteration\nAn iteration retrieves relevant work, investigates a question and returns an argument, experiment, correction or manuscript change with its evidence and limits. Review can adopt, revise, defer or reject the proposal. Accepted changes update the authored records; builders regenerate dependent views and release checks test consistency. Structured returns explicitly entered into the journal retain their dispositions. Ordinary issues and pull requests can reach review without a journal session (Section~\\ref{sec:contribute}).\n% END SENTENCE r8.architecture.iteration\n\n\\section{Worked case: reviewing an explanation}\\label{sec:example}\\label{sec:predigestion}\n\n\n% SENTENCE r8.case.system-role\nThe worked case comes from Erd\\H{o}s Problem~257, which asks whether every infinite subseries of $1+1/3+1/7+1/15+\\cdots$ is irrational. The theorem used here proves irrationality under a weighted summability condition on the retained exponents, with the same conclusion for every infinite subset and every integer base at least two. Dropping that condition would claim the unresolved general case. Appendix~\\ref{app:worked-proof} introduces the series, its antecedents and the proof.\n% END SENTENCE r8.case.system-role\n\n% SENTENCE r8.case.proof-role\nConsider a contributor revising that explanation. The coverage row \\texttt{res:weighted-support} in \\repolink{docs/paper_lean_coverage.json}{paper\\_lean\\_coverage.json} fixes the printed statement and lists two Lean supports. The explanation here has a separate \\repolink{docs/systems_paper_sentences.json}{passage record}, citing the ordinary proof and a formal declaration. The claim-evidence builder derives status or gap reports; the passage checker tests text and source bindings.\n\nNeither Lean proof need change when the explanation loses its weighted hypothesis. The prose edit makes its passage binding stale; rebinding can remove that warning without repairing the claim. Review must compare the wording with the cited statements. Accepted edits update the explanation and its record before publication views are regenerated. Figure~\\ref{fig:hypothesis} compares the three versions. This is an illustration, not a new test run.\n% END SENTENCE r8.case.proof-role\n\n% SENTENCE r15.fig.hypothesis\n\\begin{figure}[htbp]\\centering\n\\begin{tikzpicture}\n\\node[rec,w=12.8](f)at(0,0){\\textbf{Formal supports held fixed throughout}\\\\The statements retain their weighted hypothesis.};\n\\foreach \\x/\\n in {-4.6/1,0/2,4.6/3}{\n \\node[tag]at(\\x,-1){\\textsc{Version \\n}};}\n\\node[box,w=3.75,h=2.6](a)at(-4.6,-2.65){\\textbf{Faithful explanation}\\\\[4pt]\nUnder the weighted condition, irrationality holds for every infinite subset and every integer base at least two.};\n\\node[gate,w=3.75,h=2.6](b)at(0,-2.65){\\textbf{Hypothesis removed}\\\\[4pt]\nEvery infinite subseries is irrational.\\\\[4pt]\n\\emph{The unresolved general case is now claimed.}};\n\\node[gate,w=3.75,h=2.6](c)at(4.6,-2.65){\\textbf{Binding refreshed}\\\\[4pt]\nThe unsupported wording remains; the passage record is updated to match the new bytes.};\n\\node[note,w=3.75,anchor=north]at(-4.6,-4.35){Recorded wording agrees with its sources.};\n\\node[note,w=3.75,anchor=north]at(0,-4.35){Prose-only edit:\\\\passage binding is stale.};\n\\node[note,w=3.75,anchor=north]at(4.6,-4.35){Binding warning can disappear; the claim remains unproved.};\n\\draw[black!30](-6.7,-5.55)--(6.7,-5.55);\n\\node[note,w=12.8,anchor=north]at(0,-5.8){\\textbf{Review question:} does the explanation retain the hypothesis and quantifiers of the cited statements? Valid formal proofs and matching source hashes do not answer it.};\n\\end{tikzpicture}\n\\caption{Hypothesis loss with unchanged formal supports. Rebinding can restore byte consistency while leaving the overclaim intact. This is an illustration, not a new test or historical edit~8.}\n\\label{fig:hypothesis}\\end{figure}\\FloatBarrier\n% END SENTENCE r15.fig.hypothesis\n\n\n\\section{Proof and publication checks}\\label{sec:checks}\\label{sec:graph}\n\\papersectiontarget{systems-trust}\n% SENTENCE r4.checks1.01\nThe two formal supports in the example overlap (Figure~\\ref{fig:theorem}). One gives fixed-base irrationality and the all-base conclusion for every infinite subset; the other states the fixed-base hereditary conclusion explicitly. The \\emph{registered support set} is the list attached to this paper occurrence, not the dependency closure of those proofs. The publication audit checks the full registered list even when one declaration suffices for a particular clause. That checks the record; it adds no mathematical hypothesis.\n% END SENTENCE r4.checks1.01\n% SENTENCE r4.checks1.02\nThe resolver follows registered named inputs and checks source revisions. A named input is an additional assumed result and must remain visible. By contrast, the weighted condition in the example is a hypothesis inside the proved statement. Revising either the statement or its source requires checking the corresponding records again; evidence for one revision cannot silently support another.\n% END SENTENCE r4.checks1.02\n\n% SENTENCE r4.checks1.03\nFigure~\\ref{fig:checks} separates the objects checked mechanically from the questions that remain for review.\n% END SENTENCE r4.checks1.03\n\n% SENTENCE r4.fig.checks\n\\begin{figure}[htbp]\\centering\n\\begin{tikzpicture}\n\\node[note,w=5.9](s)at(-3.5,0){\\textbf{Labelled paper statement}};\n\\node[note,w=5.9](p)at(3.5,0){\\textbf{Explanatory passage}};\n\\node[rec,w=5.9,h=1.3](c)at(-3.5,-1.65){\\textbf{Coverage record}\\\\Statement digest; registered supports;\\\\correspondence relation};\n\\node[rec,w=5.9,h=1.3](r)at(3.5,-1.65){\\textbf{Passage record}\\\\Text digest; source spans;\\\\evidence class and rationale};\n\\node[box,w=5.9,h=1.1](l)at(-3.5,-3.75){\\textbf{Registered Lean declarations}\\\\Full registered list, including overlaps};\n\\node[box,w=5.9,h=1.1](e)at(3.5,-3.75){\\textbf{Cited source excerpts}\\\\Declared revisions and source digests};\n\\draw[link](s)--(c);\\draw[link](c)--(l);\n\\draw[link](p)--(r);\\draw[link](r)--(e);\n\\node[note,w=5.9,anchor=north]at(-3.5,-4.65){\\textbf{Separate formal checks}\\\\Lean: proof against formal statement.\\\\Comparator: selected implementation against a separate challenge and axiom policy.};\n\\node[note,w=5.9,anchor=north]at(3.5,-4.65){\\textbf{What the binding checks establish}\\\\Recorded text and source bytes agree.\\\\A nonempty rationale is required;\\\\its meaning is not checked.};\n\\node[gate,w=12.8,anchor=north]at(0,-6.65){\\textbf{Review across both relationships}\\\\Do the hypotheses, quantifiers, scope and attribution match? Does the explanation justify the claim? Novelty, significance and acceptance require further judgement.};\n\\end{tikzpicture}\n\\caption{Two correspondence records. Lines are recorded links, not implications. Binding and formal checks do not establish the paper's intended meaning. Review responsibilities do not imply independent review.}\n\\label{fig:checks}\\end{figure}\n% END SENTENCE r4.fig.checks\n\n% SENTENCE r4.checks2.01\nLean verifies that a proof establishes the formal statement written in the source; it does not verify whether that statement captures the intended mathematics or whether the paper describes it well.\n% END SENTENCE r4.checks2.01\n% SENTENCE r4.checks2.02\nComparator adds a separately stated challenge and a check of its selected Lean implementation under the permitted axioms~\\cite{leanfrocomparator,nanodalib}.\n% END SENTENCE r4.checks2.02\n\n% SENTENCE r4.checks2.03\nEach marked passage in this systems paper has a text record, specified source lines, an evidence class and a rationale. A passage may contain several sentences or a figure; it is a unit of source checking, not a grammatical category.\n% END SENTENCE r4.checks2.03\n\n% SENTENCE r4.checks2.04\nFor the appendix\\textquotesingle s conclusion about infinite subsets, the sources explain why restriction preserves the weighted condition. The passage checker compares cryptographic hashes of the marked text and declared sources, rejects missing records or unbound passages, and requires a nonempty rationale. It does not assess whether the rationale justifies the text.\n% END SENTENCE r4.checks2.04\n\n% SENTENCE r4.checks2.05\nA matching digest establishes agreement with recorded bytes. It does not establish that the source entails the passage.\n% END SENTENCE r4.checks2.05\n% SENTENCE r4.checks2.06\nIn the worked revision, review must check the hypothesis and quantifiers even if both Lean links still resolve. Historical edit~8 tested a different failure: an unresolved step was described as complete (Table~\\ref{tab:mutations}).\n% END SENTENCE r4.checks2.06\n\n% SENTENCE r4.checks3.01\nExecution is a separate question. A matching declaration type and reachable source link do not identify a successful continuous-integration run for that module.\n% END SENTENCE r4.checks3.01\n% SENTENCE r4.checks3.02\nThe probe interface takes verdicts from the specified Lean process; callers cannot submit their own verdict through it.\n% END SENTENCE r4.checks3.02\n% SENTENCE r4.checks3.03\nAn actor with filesystem access can still alter the implementation or stored record.\n% END SENTENCE r4.checks3.03\n% SENTENCE r4.checks3.04\nThe protocol distinguishes contributor and reviewer roles but does not technically force a second independent mathematician. No independent human mathematical review of the corpus is recorded.\n% END SENTENCE r4.checks3.04\n\n\n\\section{Publication and revision}\\label{sec:paper}\\label{sec:short}\\label{sec:long}\\label{sec:production}\n% SENTENCE r4.paper1.01\nPublication selects from the research record and explains the selected result. The short paper states its conditions, makes the decisive step intelligible and directs the reader to the full argument.\n% END SENTENCE r4.paper1.01\n% SENTENCE r4.paper1.02\nA dependency graph can locate the ingredients; it cannot choose their explanatory order. That choice requires reading the argument.\n% END SENTENCE r4.paper1.02\n% SENTENCE r4.paper1.03\nThe companion retains the complete proof and subsidiary cases. Correspondence links take a short-paper claim to the passage that supports it.\n% END SENTENCE r4.paper1.03\n% SENTENCE r4.paper1.04\nThe assembler inlines authored companion sections into a flat manuscript with their cross-references; it supplies no missing proof step.\n% END SENTENCE r4.paper1.04\n% SENTENCE reader.guidance\nThe reading guide invites readers to try an example, reconstruct a step or vary a hypothesis. The reader chooses the amount of help; an explaining agent is instructed to respect the requested depth. These authoring choices are not measurements of understanding.\n% END SENTENCE reader.guidance\n\n\\subsection{Permissions and tasks}\n\n% SENTENCE r14.assignment.authority\nAn integrating agent may change the live checkout; an advisory agent returns a proposal against supplied sources. The repository calls these permissions Type A and Type B.\n% END SENTENCE r14.assignment.authority\n\n% SENTENCE r14.assignment.jobs\nPermission to edit does not determine the task. Either assignment can concern research, proof or claim audit, literature and attribution, or exposition.\n% END SENTENCE r14.assignment.jobs\n\n% SENTENCE r14.assignment.products\nThe expected product differs: a candidate argument, a source-located finding, an attribution proposal or a revised manuscript. Each needs the corresponding review.\n% END SENTENCE r14.assignment.products\n\n% SENTENCE r14.assignment.exposition\nExposition preserves established hypotheses, conclusions and evidence status. A suspected mathematical defect is returned for a separate audit or research assignment.\n% END SENTENCE r14.assignment.exposition\n\n\\subsection{From frozen inputs to an accepted revision}\n\n% SENTENCE r13.paper.01\nThe writer first identifies the result, its antecedents, the difficult explanatory step and the unresolved questions, with source locations.\n% END SENTENCE r13.paper.01\n\n% SENTENCE r13.paper.02\nSelection remains an authorial decision; the manuscript checker tests source consistency and declared rules.\n% END SENTENCE r13.paper.02\n\n% SENTENCE r13.paper.03\nThe exchange freezes selected manuscripts, required inputs, coverage records and audit rules under a manifest of their file digests. Writer and integrating reviewer then work from identifiable starting bytes, including any selected uncommitted changes.\n% END SENTENCE r13.paper.03\n\n% SENTENCE r13.paper.04\nAn advisory model may work from that packet; a tool-enabled agent may prepare a proposal in a checkout. The packet identifies editable manuscript paths and read-only evidence and tools. Both routes return proposed changes for the same review of mathematical scope, wording and credit.\n% END SENTENCE r13.paper.04\n\n% SENTENCE r13.fig.refinement\n\\begin{figure}[H]\\centering\n\\begin{tikzpicture}\n\\node[rec,w=3.7,h=1.3](f)at(-4.6,0){\\textbf{Frozen inputs}\\\\Manuscripts, evidence\\\\and audit rules};\n\\node[box,w=3.7,h=1.3](p)at(-4.6,-2.2){\\textbf{Proposed revisions}\\\\Wording, organisation,\\\\examples and figures};\n\\node[box,w=3.7,h=1.5](a)at(-4.6,-4.5){\\textbf{Return audit}\\\\Packet identity, permitted paths and statement changes; temporary copy};\n\\node[gate,w=3.2,h=1.9](r)at(0,-4.5){\\textbf{Review decision}\\\\Meaning, scope,\\\\wording and credit};\n\\node[rec,w=3.7,h=1.5](i)at(4.6,-4.5){\\textbf{Integrate accepted edits}\\\\Apply accepted edits;\\\\update source bindings};\n\\node[view,w=3.7,h=1.3](v)at(4.6,-6.9){\\textbf{Render and inspect}\\\\PDFs, evidence views\\\\and reading links};\n\\draw[flow](f)--(p);\\draw[flow](p)--(a);\\draw[flow](a)--(r);\n\\draw[flow,draw=linkinternal](r)--node[tag,above]{accept}(i);\n\\draw[flow](i)--(v);\n\\draw[flow,dashed](r.north)--(0,-2.2)--node[tag,above]{revise}(p.east);\n\\draw[flow](r.south)--(0,-6.15)node[note,below]{Decline; record reason.};\n\\node[note,w=3.7,anchor=north]at(4.6,-7.95){Record accepted changes,\\\\lessons and limitations.};\n\\end{tikzpicture}\n\\caption{Only accepted edits enter live sources. The return audit neither applies edits nor executes returned programs. Review may request revision or decline; one agent may perform several responsibilities.}\n\\label{fig:refinement}\\end{figure}\n% END SENTENCE r13.fig.refinement\n\n% SENTENCE r13.paper.05\nThe return checker verifies packet identity and permitted paths, accounts for statement changes, and checks retained labels, citation keys and bibliography entries. It audits the proposed manuscript in a temporary copy of the frozen sources, leaving the live repository unchanged.\n% END SENTENCE r13.paper.05\n\n% SENTENCE r13.paper.06\nIt neither applies changes nor executes returned programs. A declared statement change remains an explicit review obligation.\n% END SENTENCE r13.paper.06\n\n% SENTENCE r13.paper.07\nThe reviewer then checks the whole statement, its hypotheses and implication directions against all registered supports. An ordinary argument keeps its actual evidence class wherever formalisation is incomplete.\n% END SENTENCE r13.paper.07\n\n% SENTENCE r13.paper.08\nAfter acceptance, the integrating reviewer edits the owning sources and updates affected evidence bindings. For an assembled companion, the authored sections are edited before rebuilding the flat paper. Integration then regenerates evidence and navigation and includes inspection of the compiled pages, figures and proof links.\n% END SENTENCE r13.paper.08\n\n\\subsection{Retaining improvements}\n\n% SENTENCE r13.paper.09\nA reusable writing lesson is a separate proposal, with an example and limits. Adopting it requires a reviewed change to the writing contract.\n% END SENTENCE r13.paper.09\n\n% SENTENCE r13.paper.10\nThe packet and audit stages in Figure~\\ref{fig:refinement} are implemented by \\repolink{scripts/short_paper_writer.py}{short\\_paper\\_writer.py} and \\repolink{scripts/paper_refinement.py}{paper\\_refinement.py}. A return can change the paper without changing the rules for later writers. Accepted lessons revise documents and instructions, not model weights; the procedure itself measures neither reader benefit nor autonomous discovery.\n% END SENTENCE r13.paper.10\n\n\\section{An open route for contributions}\\label{sec:contribute}\\label{sec:loop}\\label{sec:cycle}\n\\papersectiontarget{systems-public}\n\\papersectiontarget{systems-job-lifecycle}\n% SENTENCE r4.contribute1.01\nThe same review boundary applies to contributions beyond exposition. An idea or reference needs no clone; a patch identifies the public commit on which it was prepared.\n% END SENTENCE r4.contribute1.01\n% SENTENCE r4.contribute1.02\nAn executable return includes changed files, commands and results, resource use, surviving limitations, and credit for the people and tools involved.\n% END SENTENCE r4.contribute1.02\n% SENTENCE r4.contribute1.03\nFigure~\\ref{fig:contribute} follows both entry routes through triage and adoption.\n% END SENTENCE r4.contribute1.03\n% SENTENCE r4.contribute1.04\nThe protocol asks reviewers to reproduce executable evidence at the submitted base before reconciling it with current main, and to credit a substantive integration repair separately.\n% END SENTENCE r4.contribute1.04\n\n% SENTENCE r4.fig.contribute\n\\begin{figure}[H]\\centering\n\\begin{tikzpicture}\n\\node[box,w=5.9](a)at(-3.5,0){\\textbf{Idea, reference or review}\\\\State the question and proposal.\\\\No checkout is required.};\n\\node[box,w=5.9](b)at(3.5,0){\\textbf{Executable result or patch}\\\\Identify base, changes, commands,\\\\results, resource use, limits and credit.};\n\\node[gate,w=12.8](r)at(0,-2.15){\\textbf{Triage and review}\\\\Reproduce executable evidence at its submitted base before reconciling with main.\\\\Decide whether to adopt, request revision or decline.};\n\\draw[flow](a)--(a|-r.north);\\draw[flow](b)--(b|-r.north);\n\\node[rec,w=6.1,h=1.8](yes)at(-3.4,-4.85){\\textbf{Adopted source changes}\\\\Mathematics or methods\\\\[3pt]Retain evidence, limits, role credit\\\\and correction history.};\n\\node[note,w=2.6](rev)at(2.1,-4.85){\\textbf{Revise}\\\\Return the review\\\\obligations.};\n\\node[note,w=2.4](no)at(5.2,-4.85){\\textbf{Decline}\\\\Record the\\\\reason.};\n\\draw[flow](yes|-r.south)--node[tag,left]{adopt}(yes);\n\\draw[flow,dashed](rev|-r.south)--(rev);\n\\draw[flow](no|-r.south)--(no);\n\\end{tikzpicture}\n\\caption{Two entry routes and three dispositions. Adopted work includes mathematics and methods. Only explicitly entered returns become journal events; adoption establishes neither independent review nor wider acceptance.}\n\\label{fig:contribute}\\end{figure}\n% END SENTENCE r4.fig.contribute\n\n\\papersectiontarget{systems-research-loop}\n\\papersectiontarget{systems-coupled-goals}\n% SENTENCE r4.contribute2.01\nFor explicitly entered returns, the journal preserves an append-only, hash-linked sequence of dispositions. A later correction can withdraw an earlier endorsement without erasing that history. Ordinary issues and pull requests need not become journal events.\n% END SENTENCE r4.contribute2.01\n% SENTENCE r4.contribute2.02\nResearch investigates the question; stewardship compares a return with prior work, groups related results, reconciles affected papers and identifies what to investigate next.\n% END SENTENCE r4.contribute2.02\n% SENTENCE r4.contribute2.03\nThe researcher or agent chooses that next task. The public journal records entered decisions and does not schedule research.\n% END SENTENCE r4.contribute2.03\n% SENTENCE r4.contribute2.04\nTo decide what was learned, a later reviewer must compare the return with earlier work and inspect its evidence; a recorded disposition alone cannot supply that assessment.\n% END SENTENCE r4.contribute2.04\n\n\\papersectiontarget{systems-mathloop}\n% SENTENCE r4.contribute3.01\nThe repository's proposed route from a local result to an upstream library includes a literature and library comparison, a proof of the extension and a second useful instance before expert submission~\\cite{mathlibcontrib}.\n% END SENTENCE r4.contribute3.01\n% SENTENCE r4.contribute3.02\nThis route is not automated, and the supplied record reports no Mathlib contribution from it.\n% END SENTENCE r4.contribute3.02\n% SENTENCE r4.contribute3.03\nCredit distinguishes an idea from its implementation, formalisation and exposition. CRediT supplies a vocabulary for roles, not an authorship decision~\\cite{credit}.\n% END SENTENCE r4.contribute3.03\n% SENTENCE r4.contribute3.04\nAn attributed dependency or correction can record later use; it implies neither a scalar impact score nor a financial contract.\n% END SENTENCE r4.contribute3.04\n% SENTENCE r4.contribute3.05\nOpening another problem collection still requires assembling a sourced question, explicit status, inspection routes and a review owner.\n% END SENTENCE r4.contribute3.05\n\n\\section{Recorded observations}\\label{sec:instance}\\label{sec:evaluation}\n\\subsection{Inventory and formal-support status}\n% SENTENCE r4.obs1.01\nThe frozen inventory contains \\LedgerRows{} paper occurrences, of which \\LedgerFormal{} have exact or specialising Lean support; \\LedgerNamedInput{} depend on a named input and \\LedgerNoLean{} have no recorded Lean support.\n% END SENTENCE r4.obs1.01\n% SENTENCE r4.obs1.02\nOf the \\LedgerFormal{} exact-or-specialising occurrences, \\LedgerCompared{} are recorded as compared and \\LedgerQueued{} as queued.\n% END SENTENCE r4.obs1.02\n% SENTENCE r4.obs1.03\nThese are occurrence counts, not counts of distinct theorems or independently reviewed proofs. Exact support matches the printed statement; specialising support yields it by specialization. Named-input entries retain an additional assumed result. An entry without Lean support may have an ordinary proof; neither of these last two groups is counted as fully covered.\n% END SENTENCE r4.obs1.03\n\n\\subsection{Historical false-edit trial}\n% SENTENCE r4.obs2.01\nThe historical publication-evidence record reports that \\HistoricalRejected{} of the \\HistoricalEdits{} deliberately false edits were rejected and \\HistoricalEscaped{} escaped.\n% END SENTENCE r4.obs2.01\n% SENTENCE r4.obs2.02\nThe edits were authored by the checker's author, the original run logs were not retained, and the other \\HistoricalNotRerun{} edits were not rerun against the extended checklist.\n% END SENTENCE r4.obs2.02\n% SENTENCE r4.obs2.03\nThe edits were applied separately, with the baseline restored between trials.\n% END SENTENCE r4.obs2.03\n% SENTENCE r4.obs2.04\nThe escaped edit changed a description of an unresolved step from ``does not supply\\textquotesingle\\textquotesingle\\ to ``completes\\textquotesingle\\textquotesingle, and the publication checks accepted it.\n% END SENTENCE r4.obs2.04\n\n% SENTENCE r4.table.mutations\n\\begin{table}[htbp]\n\\centering\\small\n\\begin{tabular}{@{}lp{6.0cm}p{3.5cm}l@{}}\n\\toprule\nEdit & Change & First detector & Outcome\\\\\n\\midrule\nedit~1 & Conditional result labelled proved & Projection freshness & rejected\\\\\nedit~2 & Open-boundary clause deleted & Boundary wording & rejected\\\\\nedit~3 & Declaration coordinate moved & Source coordinates & rejected\\\\\nedit~4 & Paper source link retargeted & Source coordinates & rejected\\\\\nedit~5 & Claim assigned to two families & Registry structure & rejected\\\\\nedit~6 & Module count hand-edited & Projection freshness & rejected\\\\\nedit~7 & \\texttt{native\\_decide} introduced & Proof-trust policy & rejected\\\\\n\\rowcolor{black!6}\nedit~8 & ``does not supply\\textquotesingle\\textquotesingle\\ changed to ``completes\\textquotesingle\\textquotesingle & None & \\textbf{escaped}\\\\\nedit~9 & Open-problem title mangled & Paper anchoring & rejected\\\\\nedit~10 & Orientation byte budget exceeded & Byte budget & rejected\\\\\n\\bottomrule\n\\end{tabular}\n\\par\\medskip\n\\begin{minipage}[t]{.45\\linewidth}\n\\textbf{Historical trial: edit 8}\\\\\nThe false completion clause passed all recorded checks.\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{.45\\linewidth}\n\\textbf{Later witness: baseline and edit 8}\\\\\nBaseline accepted, edit 8 rejected; the other \\HistoricalNotRerun{} edits were not rerun.\n\\end{minipage}\n\\caption{The \\HistoricalEdits{} historical edits and their recorded first detectors; ``proof-trust policy'' refers to this repository's permitted proof mechanisms, not to a general unsoundness claim about \\texttt{native\\_decide}.}\n\\label{tab:mutations}\n\\end{table}\n% END SENTENCE r4.table.mutations\n% SENTENCE r4.obs3.01\nThe follow-up tested the intact baseline and only the escaped edit against an added check. It supplies no post-repair ten-of-ten result.\n% END SENTENCE r4.obs3.01\n% SENTENCE r4.obs3.02\nIn that recorded follow-up, the post-repair witness accepts the current README and rejects a test copy containing the false clause.\n% END SENTENCE r4.obs3.02\n% SENTENCE r4.obs3.03\nThe trial uses one corpus and the checker's own selection of errors, with no manual-review control or ordinary continuous-integration comparator.\n% END SENTENCE r4.obs3.03\n% SENTENCE r4.obs3.04\nThe study locates a coverage boundary. Its nine rejections do not estimate how reliable the checker is.\n% END SENTENCE r4.obs3.04\n\n\\subsection{Local integration and unperformed reader evaluation}\n% SENTENCE r4.obs4.01\nOn 23 September 2026, a maintainer-operated agent accepted one architecture handoff into a local integration history after focused checks of return routing and architecture packaging.\n% END SENTENCE r4.obs4.01\n% SENTENCE r4.obs4.02\nThe contributor and reviewer were the same agent. The receipt records no public pull request, human review or independent outside clone replay.\n% END SENTENCE r4.obs4.02\n\n% SENTENCE r4.obs5.01\nA proposed reader study would hold the mathematical task fixed and compare an ordinary repository, an information-equivalent static briefing and the navigation tools.\n% END SENTENCE r4.obs5.01\n% SENTENCE r4.obs5.02\nIt would measure understanding and severe errors alongside preparation, use, review and repair costs, retaining missing returns, timeouts and disagreements.\n% END SENTENCE r4.obs5.02\n% SENTENCE r4.obs5.03\nNo independent writing comparison, blind grading run or cold-reader experiment is reported, so there is no comparative reader result.\n% END SENTENCE r4.obs5.03\n% SENTENCE r4.obs5.04\nThe records above document inventory, a checker failure and one local integration; general reliability, reader understanding and mathematical transfer remain unmeasured.\n% END SENTENCE r4.obs5.04\n\n\\section{Related work}\\label{sec:related}\\label{app:credit}\n% SENTENCE r4.related1.01\nThe closest comparisons concern the objects a system records and the work it lets contributors perform. Lean and mathlib provide formal statements, proofs and a maintained library; blueprints connect declarations to exposition~\\cite{lean4,mathlib,leanblueprint}. Prove2Me provides a more direct comparison for agent-assisted collaboration. Appendix~\\ref{app:source-map} retains the wider source map.\n% END SENTENCE r4.related1.01\n% SENTENCE r4.related1.02\n\\paragraph{Prove2Me.}\nWe use its published version~2, dated 31 August 2026; the first version appeared on 28 August~2026~\\cite{prove2me}. Its basic unit is an immutable theorem statement, stored separately from the proofs submitted for it. A proof is checked against the target type in a pinned verification environment. A \\emph{proof-sketch} derives a target using imported theorem statements, some of which may remain open. Contributors can prove those children separately; the target becomes verified when the required children are verified. Immutability makes those local proofs composable without changing their targets~\\cite[Sections~3.1--3.2 and~4.2]{prove2me}.\n% END SENTENCE r4.related1.02\n\n% SENTENCE r11.related.prove2me-audit\nHuman review is part of Prove2Me\\textquotesingle s design. A mission fixes a human-audited core of goals, definitions and milestones; each milestone pairs a source statement with its attested canonical formalisation. For read-back, an independent agent receives the Lean declaration and dependent definitions without the original source; a human compares its mathematical rendering with that source. Natural-language accounts accompany statements and proofs, statement descriptions support library search, and imports give contributors reuse credit. The discussion record also includes a disproved statement followed by a corrected replacement~\\cite[Sections~3.3 and~4.3--4.5]{prove2me}. Explanation, reuse, correction and shared agent work are therefore established parts of this comparison.\n% END SENTENCE r11.related.prove2me-audit\n\n% SENTENCE r4.related1.03\nFor correspondence with a source, the closest analogue is Prove2Me\\textquotesingle s milestone: both designs identify the mathematical statement whose formalisation is being reviewed. Our coverage record follows a particular paper occurrence and can list several overlapping formal supports, with a relation note explaining their connection to the printed claim. It is not a proof-sketch: listing those supports does not compose proof terms or discharge open lemmas. A separate passage record binds the explanation to sources and rationale, while entered contribution records retain revision decisions and role credit. The focus here is the consistency and review of these publication records across revisions. This difference of focus establishes neither priority for persistent research records nor a measured advantage over Prove2Me.\n% END SENTENCE r4.related1.03\n\n% SENTENCE r4.related1.04\nPolymath, BOINC and GIMPS supply precedents for small mathematical contributions and donated compute~\\cite{polymath,boinc,gimps}, while the Carleson blueprint shows formalisation organised into publicly claimable tasks~\\cite{carleson}.\n% END SENTENCE r4.related1.04\n\n\n% SENTENCE r8.related.systems\nThe regeneration tools address the familiar build-system problem of keeping derived outputs consistent with their inputs~\\cite{buildsystems}. They run an explicit ordered list of builders and then check the resulting views; this is not a general incremental scheduler. Source bindings and the journal add provenance for assertions and entered returns at particular revisions. Together these mechanisms make dependencies and changes available for inspection. Their effect on research efficiency and semantic correctness needs separate evaluation.\n% END SENTENCE r8.related.systems\n\n\n\\section{Limits}\\label{sec:limits}\\label{sec:conclusion}\n% SENTENCE r4.limits1.01\nThe prototype establishes a set of implemented record and review operations, with the limited observations in Section~\\ref{sec:evaluation}. It reports no comparative formalisation result or autonomous-optimisation experiment.\n% END SENTENCE r4.limits1.01\n% SENTENCE r4.limits1.02\nTransfer to unseen mathematics and understanding by independent readers require direct evaluation.\n% END SENTENCE r4.limits1.02\n% SENTENCE r4.limits1.03\nIn particular, no controlled comparison shows that the contribution process converts compute into useful mathematics more effectively or increases discovery rate.\n% END SENTENCE r4.limits1.03\n% SENTENCE r4.limits1.04\nThere is also no evidence of inexpensive adoption by a new project or an advantage over an agent using a good README.\n% END SENTENCE r4.limits1.04\n% SENTENCE r4.limits1.05\nFull scans and initial builds still have costs, and an open corpus cannot rule out prior model exposure to its contents.\n% END SENTENCE r4.limits1.05\n\n% SENTENCE r4.limits2.01\nGovernance remains maintainer-centred. Review capacity is scarce and credit is contestable; openness alone does not equalise resources or resolve appeals.\n% END SENTENCE r4.limits2.01\n% SENTENCE r4.limits2.02\nRepository adoption cannot confer acceptance by the wider mathematical community.\n% END SENTENCE r4.limits2.02\n% SENTENCE r4.limits2.03\nReturned code is untrusted, and expensive or privileged continuous-integration jobs must not execute fork code with repository secrets~\\cite{githubsecurity}.\n% END SENTENCE r4.limits2.03\n% SENTENCE r4.limits2.04\nA volunteer-compute service and extensions to physical science remain proposals requiring their own validation, resource and safety governance.\n% END SENTENCE r4.limits2.04\n\n% SENTENCE r14.agmai.intro\nThe Advisory Group on Mathematics and Artificial Intelligence at IAS\nrecommends distinguishing papers fully understood by a responsible\nmathematician from AI output that its human prompters do not understand,\nand asks AI laboratories to support subsequent community-led\nunderstanding~\\cite{agmai2026}. Its 29 September 2026 statement gives\nconcrete release expectations for attribution, exposition, formalisation\nand disclosure of the research process.\n% END SENTENCE r14.agmai.intro\n\n% SENTENCE r14.agmai.limits\nSeveral of these proposed release practices have counterparts in the\nrecords described here~\\cite[Section 2.B, Step I]{agmai2026}: the short\npaper and companion paper explain and attribute the argument, the paper-to-Lean\nledger records formal correspondence and its limits, and the files prepared for formal comparison identify its inputs.\nThose records make the result inspectable; they do not certify a\nmathematician's understanding. The architecture alone does not establish\ncomplete per-result model, prompt, time and cost disclosure, an account of\ncampaign-wide selection and unsuccessful attempts, or an independent\nscholarly deposit. The further recommendations on funding community-led\nunderstanding and equitable model access address AI laboratories; this\nprototype does not demonstrate those institutional outcomes.\n% END SENTENCE r14.agmai.limits\n\n% SENTENCE r4.limits2.05\nThe contribution is an inspectable record from which to recover a claim's conditions, argument and limits, then submit a correction or extension against identified sources. Checks preserve specified links; review decides what the work justifies. Revision requires both.\n% END SENTENCE r4.limits2.05\n\n% SENTENCE r4.layout.endbody\n\\FloatBarrier\n% END SENTENCE r4.layout.endbody\n\\appendix\n\\section{Reading the materials and reproducing checks}\\label{app:repro}\n\n% SENTENCE r14p3.repro.purpose\nThe materials support reading a result, inspecting its records and rerunning checks. Reading requires no Lean installation; proof replay requires the recorded sources and matching Lean environment. The routes below separate document checks from reproduced proofs.\n% END SENTENCE r14p3.repro.purpose\n\n\\subsection{Read a result and follow its sources}\n\n% SENTENCE r4.repro4.04\nThe \\reporef{docs/READING_GUIDE.md}{reading guide} routes readers through one problem or across the papers, including \\emph{Reading Eight Erd\\H{o}s Problems Together}~\\cite{synthesis}. Read a result's short paper, then follow its argument, attribution and formal-support references. Agents can use \\reporef{AGENTS.md}{agent instructions} and the \\reporef{docs/agents/AGENT_WORKBENCH.md}{workbench's} optional proof sessions. Corrections follow the \\reporef{CONTRIBUTING.md}{contribution guide} and \\reporef{docs/research-commons/CREDIT_POLICY.md}{credit policy}.\n% END SENTENCE r4.repro4.04\n\n\\subsection{Inspect the records and rerun document checks}\n\n% SENTENCE r4.repro3.02\nThe \\reporef{docs/publication_evidence.json}{experiment record} contains the ten-edit matrix and limited follow-up in Section~\\ref{sec:checks}. The \\reporef{docs/publication_contract.json}{publication-check specification} identifies required checks. These structured files trace evidence; they do not replace the paper's explanation.\n% END SENTENCE r4.repro3.02\n\n% SENTENCE r4.repro3.01\nFrom a complete public checkout, run:\n\\begin{quote}\\small\n\\texttt{python3 scripts/systems\\_paper\\_evidence.py}\\\\\n\\texttt{python3 scripts/build\\_systems\\_paper\\_counts.py}\n\\end{quote}\nThe first checks passage-to-source links and content hashes; the second compares the generated inventory with its records. Success establishes agreement, not the correctness of an explanation.\n% END SENTENCE r4.repro3.01\n\n% SENTENCE r4.repro3.03\nFor proof replay, the \\reporef{lean/Erdos249257.lean}{Lean entry file} and \\reporef{.github/workflows/lean.yml}{CI workflow} identify build entry points. The \\reporef{docs/claims.json}{claim register} lists curated assertions; \\reporef{scripts/check_release.py}{release checks} cover wider publication requirements. Use the result's recorded revisions and environment.\n% END SENTENCE r4.repro3.03\n\n% SENTENCE r4.repro3.04\nA writing packet contains manuscripts, selected records and inspection tools, but no Lean tree. It supports document checks, not a full Lean, Comparator or release rerun.\n% END SENTENCE r4.repro3.04\n\n\\subsection{Locate the formal supports of the worked example}\n\n% SENTENCE r4.repro2.01\nFor the weighted criterion in Appendix~\\ref{app:worked-proof}, the \\reporef{docs/paper_lean_coverage.json}{paper-to-Lean coverage record} identifies two formal supports: a weighted criterion and a version whose conclusion explicitly covers every infinite subset. Figure~\\ref{fig:theorem} shows the recorded links. The exact declaration names are provided for readers locating them in Lean.\\footnote{Both are in namespace \\nolinkurl{ErdosProblems.Erdos257.PaperCompleteR8}: \\nolinkurl{divisibilityWeightedClaim} and \\nolinkurl{finitePrimeWeighted_fixedBase_hereditary}. The coverage row is \\texttt{res:weighted-support}; the weighted comparison entries are \\texttt{PalomarCorpus/E257} and \\texttt{E257bc}.} These entries support the stated conclusion; they do not assert line-by-line agreement between the ordinary proof and the Lean proofs.\n% END SENTENCE r4.repro2.01\n\n% SENTENCE r4.repro2.02\nThe recorded Comparator runs are the \\href{https://github.com/wcook04/plectis-erdos-lean/actions/runs/35544127144}{first comparison run} and the \\href{https://github.com/wcook04/plectis-erdos-lean/actions/runs/35624228171}{second comparison run}. Comparator checks a formal result against a separately stated challenge. These records concern the specified formal statements, not the historical attribution or the correctness of this paper's explanation.\n% END SENTENCE r4.repro2.02\n\n% SENTENCE r4.example2.02\nThe reciprocal-summability criterion attributed to Erd\\H{o}s in Appendix~\\ref{app:worked-proof} has its own \\href{\\repobase/blob/7308896010e6b527d283dfdd0687a717a6183fb6/lean/Erdos249257/AllBaseReciprocalSupportIrrationality.lean}{formal proof}, also recorded in the second comparison run.\\footnote{The declaration is \\nolinkurl{Erdos249257.irrational_erdosSupportSeries_of_summable_reciprocal}; its comparison entry is \\texttt{PalomarCorpus/E257av}, recorded as compared.}\n% END SENTENCE r4.example2.02\n\n% SENTENCE r4.repro2.03\nBoth runs are recorded as compared. Palomar entries are prepared, not submitted; comparison does not establish registry acceptance.\n% END SENTENCE r4.repro2.03\n\n% SENTENCE r4.fig.theorem\n\\begin{figure}[htbp]\\centering\n\\begin{tikzpicture}\n\\node[rec,w=12.8](s)at(0,0){\\textbf{Weighted criterion: the conditional statement}\\\\\n\\emph{If the weighted sum in~\\eqref{eq:worked-condition} is finite}, then irrationality holds for every infinite subset and every integer base at least two.};\n\\node[rec,w=12.8](r)at(0,-1.9){\\textbf{Coverage row \\texttt{res:weighted-support}}\\\\Exact or stronger support; both entries below are registered and audited.};\n\\node[box,w=5.9,h=1.6](a)at(-3.5,-4){\\textbf{Weighted criterion}\\\\Fixed-base irrationality; all-base\\\\conclusion for every infinite subset.\\\\\\reporef{lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean}{Formal proof}};\n\\node[box,w=5.9,h=1.6](b)at(3.5,-4){\\textbf{Hereditary criterion}\\\\Explicit conclusion for every infinite\\\\subset at a fixed base.\\\\\\reporef{lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedHereditaryClaim.lean}{Formal proof}};\n\\draw[link](s)--(r);\n\\draw[link](r.south)--(0,-2.85);\n\\draw[link](-3.5,-2.85)--(3.5,-2.85);\n\\draw[link](-3.5,-2.85)--(a.north);\\draw[link](3.5,-2.85)--(b.north);\n\\node[note,w=12.8,anchor=north]at(0,-5.2){\\textbf{Recorded comparisons:} compared. \\quad\\textbf{Palomar submission:} prepared, not submitted.};\n\\end{tikzpicture}\n\\caption{Registered supports have overlapping roles; lines are not proof dependencies. Auditing the full list adds no hypothesis to the weighted statement. Recorded comparison does not establish registry acceptance.}\n\\label{fig:theorem}\\end{figure}\\FloatBarrier\n% END SENTENCE r4.fig.theorem\n\n\\subsection{Interpret the recorded inventory}\n\n% SENTENCE r4.repro4.01\nThe recorded inventory counts occurrences of paper statements, rather than distinct new theorems. Its \\LedgerRows{} entries comprise \\LedgerExact{} with exact-or-stronger formal support, \\LedgerSpecialising{} obtained by specialising a formal result, \\LedgerNamedInput{} linked to a named formal ingredient only, and \\LedgerNoLean{} with no registered Lean support. The last two categories do not certify the whole paper statement.\n% END SENTENCE r4.repro4.01\n\n% SENTENCE r4.repro4.02\nThe contribution journal records \\Rounds{} rounds and \\Returns{} returned contributions at the recorded journal revision.\\footnote{Journal revision: \\texttt{\\JournalHead}.} Of the rounds, \\RoundsSealed{} are sealed, meaning no further arrivals are expected. Of the returns, \\ReturnsAdmitted{} have been admitted for integration and \\ReturnsUnreviewed{} remain unreviewed; admission does not itself record integration. Earlier private-history rounds do not constitute a complete public journal.\n% END SENTENCE r4.repro4.02\n\n% SENTENCE r4.repro4.03\nExplanatory text is recorded for \\SemanticAuthoredInterpreted{} of the \\SemanticAuthoredTheoremLike{} authored theorem-like declarations (\\SemanticAuthoredPercent\\%). Most of these descriptions provide surrounding context. This inventory measures the presence of recorded interpretations, not individual proof review or demonstrated reader understanding.\n% END SENTENCE r4.repro4.03\n\n\\subsection{Identify the version and the earlier design accounts}\n\n% SENTENCE r4.repro1.02\nRepository links in this paper identify fixed file versions, so that a reader can inspect the material cited even if the working repository later changes. Unless a link states otherwise, its revision is \\texttt{992cf60a27ef}; the Comparator runs identify their own inputs. No new Lean or Comparator run was performed for this revision.\n% END SENTENCE r4.repro1.02\n\n% SENTENCE r4.repro1.01\nFor archival identification, the initial unified manuscript was prepared from the frozen packet at \\texttt{ced96e58b337}, a local integration commit rather than public main. The 29 September state report lists it under the earlier title \\emph{Problem-Sized Lean Worlds}.\n% END SENTENCE r4.repro1.01\n\n% SENTENCE r4.repro1.03\nThis paper supersedes the two earlier design accounts described below. Cite a mathematical result by the paper and version actually used; cite repository software using its \\reporef{CITATION.cff}{software citation record} at the chosen commit or release tag.\n% END SENTENCE r4.repro1.03\n\n% SENTENCE r8.repro.historical-crosswalk\n\\emph{\\reporef{paper/systems/cold-clone-to-proof-receipt.tex}{From a Cold Clone to a Proof Receipt}} (August 2026) describes the declaration, dependency, interpretation and claim views and the optional recorded proof session. Its counts and audit figures belong to that snapshot; Section~\\ref{sec:world} describes the current task entry and inspected implementation.\n\n\\emph{\\reporef{paper/systems/open-source-mathematics-strategy.tex}{From Spare Compute to Cumulative Mathematics}} (September 2026) describes contribution roles, reproduction against the original inputs, subsequent integration and role credit. Section~\\ref{sec:contribute} gives the current procedure. The earlier paper's volunteer-compute services, inexpensive adoption by new laboratories and increased discovery rate remain proposals or unmeasured outcomes. These manuscripts explain the design's history; they are not additional current specifications.\n% END SENTENCE r8.repro.historical-crosswalk\n\n\\section{The mathematical example: a criterion for irrationality}\\label{app:worked-proof}\n\n% SENTENCE r8.appendix.proof-scope\nSection~\\ref{sec:example} used a theorem whose explanation could become false by losing a hypothesis. Following the Problem~257 paper~\\cite{paper257}, this appendix introduces the series and earlier work, states the weighted criterion, gives an example of its reach, and explains the averaging proof. Each part identifies information that a faithful summary must retain. The formal-support records are in Appendix~\\ref{app:repro}; the calculation is optional for readers interested only in the system.\n% END SENTENCE r8.appendix.proof-scope\n\n\\subsection{The question and its antecedents}\n\n% SENTENCE r4.example1.01\nBegin with the series\n\\[\n 1+\\frac13+\\frac17+\\frac1{15}+\\cdots\n =\\sum_{a=1}^{\\infty}\\frac1{2^a-1}.\n\\]\nAn infinite subseries retains the terms whose exponents belong to an infinite set $A$ of positive integers. We call $A$ the \\emph{support}: it specifies which terms are present. To allow other integer bases $b\\ge2$, write\n\\[\n X_A(b)=\\sum_{a\\in A}\\frac1{b^a-1}.\n\\]\nThese series converge by comparison with a geometric series. The question concerns their arithmetic value: can an infinite choice of retained terms have a rational sum?\n% END SENTENCE r4.example1.01\n\n% SENTENCE r4.example1.05\nProblem~257 asks whether $X_A(2)$ is irrational for every infinite support~$A$. The weighted theorem below gives a sufficient condition on~$A$; it does not settle that unrestricted question~\\cite{paper257,erdosproblems}.\n% END SENTENCE r4.example1.05\n\n% SENTENCE r4.example2.01\nErd\\H{o}s proved the all-base irrationality conclusion for pairwise coprime exponents satisfying $\\sum_{a\\in A}1/a<\\infty$. He also stated that coprimality could be removed, without supplying the details~\\cite[pp.~222, 226]{erdos1968}. The Problem~257 paper gives a direct proof of that stated extension, as well as the weighted criterion used here~\\cite{paper257}.\n% END SENTENCE r4.example2.01\n\n% SENTENCE r4.example2.03\nDuverney and Tachiya provide a methodological antecedent: they choose an arithmetic progression by congruences and use averaging to control the remaining contribution~\\cite[Section~2, (2.3)--(2.9)]{duverneytachiya}. The Problem~257 paper credits that selection method. Its weighted proof uses an additional average over progression lengths, which recovers a factor needed to sum the error bounds. The sources thus supply a classical criterion, an averaging method and the weighted extension, respectively.\n\\par\\Needspace{16\\baselineskip}\n% END SENTENCE r4.example2.03\n\n\\subsection{A weighted condition on the retained exponents}\n\n% SENTENCE r4.example1.02\nThe weighted condition gives a smaller contribution to exponents divisible by large powers of a fixed set of primes. For $a=2^km$ with $m$ odd, it replaces the reciprocal weight $1/(2^km)$ by $1/[m(2^{2^k}-1)]$. More generally, for a finite nonempty set $P$ of primes, the \\emph{$P$-part} of~$a$ is\n$h_P(a)=\\prod_{p\\in P}p^{v_p(a)}$, where $v_p(a)$ is the exponent of~$p$ in the prime factorisation of~$a$. Thus $h_{\\{2\\}}(2^km)=2^k$.\n\\par\\Needspace{7\\baselineskip}\n% END SENTENCE r4.example1.02\n\n% SENTENCE r4.example1.03\nThe weighted-support theorem of the Problem~257 paper states that, if\n\\begin{equation}\\label{eq:worked-condition}\n W_{2,P}(A):=\\sum_{a\\in A}\\frac{h_P(a)}{a(2^{h_P(a)}-1)}<\\infty,\n\\end{equation}\nthen $X_B(b)$ is irrational for every integer $b\\ge2$ and every infinite $B\\subseteq A$~\\cite[Theorem~1.2]{paper257}.\n% END SENTENCE r4.example1.03\n\n% SENTENCE r4.example1.04\nThe conclusion is \\emph{hereditary}: it applies to every infinite subset of~$A$. For a fixed base~$b$, write $W_{b,P}(A)$ for the sum in~\\eqref{eq:worked-condition} with $2$ replaced by~$b$. The fixed-base version assumes $W_{b,P}(A)<\\infty$ and gives the same hereditary conclusion at that base. We prove this version below and recover all integer bases from the base-two hypothesis.\n% END SENTENCE r4.example1.04\n\n\\subsection{Why the weighted condition reaches further}\n\n% SENTENCE r4.example3.01\nTo see why the weight matters, compare it with the older condition $\\sum_{a\\in A}1/a<\\infty$. Both are tests on the exponents; neither is needed merely for convergence of~$X_A(b)$. The source paper gives\n\\[\n A_\\star=\\{2^km:k\\ge1,\\ m\\text{ odd},\\ 1\\le m\\le2^{2^k}\\},\\qquad P=\\{2\\}.\n\\]\nThis set is arranged in disjoint layers according to the power $2^k$ dividing an exponent. Within layer~$k$, reciprocal summation discounts each odd $m$ by $2^{-k}$; weighted summation discounts it by $(2^{2^k}-1)^{-1}$. The second discount is much smaller.\n% END SENTENCE r4.example3.01\n\n% SENTENCE r4.example3.02\nWrite $S_r=\\sum_{1\\le m\\le2^r,\\ m\\ \\mathrm{odd}}1/m$ for the sum of odd reciprocals up to $2^r$; layer~$k$ uses $r=2^k$. Grouping these integers into intervals $[2^j,2^{j+1})$ gives $r/4\\le S_r\\le r$ for integers $r\\ge2$.\n% END SENTENCE r4.example3.02\n\n% SENTENCE r4.example3.03\nThe reciprocal contribution of each disjoint layer is at least $2^{-k}S_{2^k}\\ge1/4$. Its weighted contribution satisfies\n\\[\n \\frac{S_{2^k}}{2^{2^k}-1}\\le\\frac{2^k}{2^{2^k}-1}\\le2^{1-k},\n\\]\nso the layer contributions have a convergent geometric majorant in the weighted sum.\n% END SENTENCE r4.example3.03\n\n% SENTENCE r4.example3.04\nThus $\\sum_{a\\in A_\\star}1/a$ diverges, whereas $W_{2,\\{2\\}}(A_\\star)$ converges. The weighted criterion therefore applies to some supports excluded by the reciprocal-summability condition. This is the source paper's ordinary calculation, with no separate entry in the paper-to-Lean ledger~\\cite[calculation following the weighted proof]{paper257}.\n% END SENTENCE r4.example3.04\n\n\\subsection{The proof by two averages}\n\n% SENTENCE r14p3.proof.overview\nFix an integer base $b\\ge2$, an infinite support~$A$ and a finite nonempty set~$P$ with $W_{b,P}(A)<\\infty$. The proof has two parts. Rationality would force a certain positive remainder to be at least the reciprocal of a fixed denominator. Averaging will make that remainder smaller. The main difficulty is that one average leaves an error bound that cannot be summed using the hypothesis; a second average over powers of two supplies the missing factor.\n% END SENTENCE r14p3.proof.overview\n\n% SENTENCE r4.example4.01\n\\Needspace{9\\baselineskip}\n\\paragraph{The lower bound under rationality.}\nSuppose that $X_A(b)=p/q$ with integers $p,q$ and $q>0$. For a positive integer $N$, multiplication by $b^N-1$ separates an integer contribution from each summand, leaving\n\\begin{equation}\\label{eq:worked-displacement}\n \\Delta_{b,A}(N)=\\sum_{a\\in A}\\frac{b^{N\\bmod a}-1}{b^a-1}\n =(b^N-1)X_A(b)-J_{b,A}(N),\\qquad N\\ge1,\n\\end{equation}\nwhere $J_{b,A}(N)=\\sum_{a\\in A,\\,a\\le N}\\sum_{j=1}^{\\lfloor N/a\\rfloor}b^{N-ja}$ is an integer. Here $N\\bmod a$ is the remainder on dividing $N$ by~$a$. The symbol $\\Delta$ denotes the positive remainder after this particular integer is subtracted, not necessarily a fractional part in $[0,1)$.\n% END SENTENCE r4.example4.01\n\n% SENTENCE r4.example4.02\nEvery summand is nonnegative, and one is positive because the infinite set~$A$ contains an exponent larger than~$N$. Hence $q\\Delta_{b,A}(N)$ is a positive integer, and\n\\(\\Delta_{b,A}(N)\\ge1/q\\).\n% END SENTENCE r4.example4.02\n\n% SENTENCE r4.example4.03\nWe obtain a contradiction by finding a multiple $N=tQ$ with smaller remainder. Divisibility helps because the summand indexed by $a$ vanishes whenever $a\\mid Q$.\n% END SENTENCE r4.example4.03\n\n% SENTENCE r4.example4.04\nChoose a finite nonempty $F\\subset A$ outside which the weighted sum is less than $\\varepsilon$, and make $Q$ a multiple of every element of~$F$. Those terms then disappear for every~$t$; we must control the remaining terms on average.\n% END SENTENCE r4.example4.04\n\n% SENTENCE r4.example4.05\n\\paragraph{Averaging over a residue period.}\nWith $b=2$, $Q=4$ and $a=6$, the residues $tQ\\bmod a$ are $4,2,0$, and the corresponding mean contribution to $\\Delta$ is\n\\[\n \\frac13\\left(\\frac{15}{63}+\\frac3{63}+0\\right)=\\frac2{21}\\le\\frac19.\n\\]\nFor a general exponent $a$, put $g=\\gcd(a,Q)$. The residues run through the multiples of~$g$ in a cycle of length $a/g$. Replacing each numerator $b^r-1$ by $b^r$ bounds the sum over one cycle by $1/(b^g-1)$. There are at most $T/(a/g)$ complete cycles and one unfinished cycle among the first $T$ terms. Dividing their total by~$T$ gives\n\\begin{equation}\\label{eq:worked-orbit}\n \\frac1T\\sum_{t=1}^T\\frac{b^{tQ\\bmod a}-1}{b^a-1}\n \\le \\underbrace{\\frac{g}{a(b^g-1)}}_{\\text{complete cycles}}\n    +\\underbrace{\\frac1{T(b^g-1)}}_{\\text{unfinished cycle}}.\n\\end{equation}\n% END SENTENCE r4.example4.05\n\n% SENTENCE r4.example4.06\nThe first term has the shape of the weighted hypothesis once suitable prime powers have been put into~$Q$. The second has lost the factor $1/a$. Summability of the first therefore does not justify summing the unfinished-cycle bounds over all exponents. This is the difficulty that a single progression average leaves unresolved.\n% END SENTENCE r4.example4.06\n\n% SENTENCE r4.example4.07\n\\paragraph{Averaging over dyadic lengths.}\nThe source proof averages over the $M$ lengths $T=2^j$, $M\\le j<2M$; these powers-of-two lengths are called dyadic. Thus $M$ counts the lengths being averaged, whereas $T$ counts the terms in one progression average. For nonnegative weights $\\alpha_a$ indexed by positive integers, with $\\sum_a\\alpha_a/a<\\infty$, it uses\n\\begin{equation}\\label{eq:worked-dyadic}\n \\sum_{j=M}^{2M-1}2^{-j}\\sum_{a\\le Q2^j}\\alpha_a\n \\le 2Q\\sum_a\\frac{\\alpha_a}{a}.\n\\end{equation}\nIndeed, a fixed $a$ enters only when $2^j\\ge a/Q$, and the sum of the admissible $2^{-j}$ is at most $2Q/a$. This recovers the missing factor before summing over~$a$. Dividing by the number $M$ of lengths then reduces the cost from $2Q$ to $2Q/M$.\n% END SENTENCE r4.example4.07\n\n% SENTENCE r6.example.balance\n\\paragraph{Choosing the modulus and the averaging lengths.}\nThe modulus must cancel $F$ while making the remaining exponents amenable to one of two estimates. Fix a positive common multiple $L$ of~$F$, put $p_*=\\max P$, and, for $H\\ge2p_*$, set\n\\[\n Q=L\\prod_{p\\in P}p^{\\lfloor\\log_p H\\rfloor},\\qquad\n G=\\lfloor H/p_*\\rfloor.\n\\]\nIf $h_P(a)\\le H$, then $h_P(a)\\mid Q$. Since $n/(b^n-1)$ decreases for positive integers~$n$, the complete-cycle term in~\\eqref{eq:worked-orbit} is bounded by $h_P(a)/[a(b^{h_P(a)}-1)]$. Its sum outside~$F$ is less than $\\varepsilon$.\n\nIf $h_P(a)>H$, then $\\gcd(a,Q)\\ge G$: either every $P$-prime-power component of $h_P(a)$ is at most $H$, so $h_P(a)\\mid Q$, or one exceeds $H$, and its truncated component in $Q$ is greater than $H/p_*\\ge G$. Summing~\\eqref{eq:worked-orbit} over these exponents up to $QT$ costs at most\n\\[\n \\frac{G(1+\\log(QT))+Q}{b^G-1}:\n\\]\nthe harmonic sum supplies $1+\\log(QT)$, and at most $QT$ unfinished-cycle terms supply~$Q$.\n\nIncreasing $M$ therefore has two effects. It reduces the first group's unfinished-cycle cost $Q/M$, but increases the second group's cost through $\\log T<2M\\log2$. We need both $Q/M\\to0$ and $GM/b^G\\to0$, not merely a long average. With $F$ and $L$ fixed, $Q\\le LH^{|P|}$ and $G=H/p_*+O(1)$. The choice $M=\\lfloor b^{G/2}\\rfloor$ meets both requirements.\n\nThe source proof combines these estimates into the finite mean\n\\begin{equation}\\label{eq:worked-mean}\n\\begin{aligned}\n &\\frac1M\\sum_{j=M}^{2M-1}\\frac1{2^j}\n       \\sum_{t=1}^{2^j}\\Delta_{b,A}(tQ)\\\\\n &\\qquad\\le \\varepsilon+\\frac{2QW_{b,P}(A)}M\n       +\\frac{G(1+\\log Q+2M\\log2)+Q}{b^G-1}+4\\,2^{-M}.\n\\end{aligned}\n\\end{equation}\nThe terms bound, respectively, the weighted tail over complete cycles, its unfinished cycles, the exponents with large gcd, and the exponents beyond $QT$. The second term uses~\\eqref{eq:worked-dyadic} with $\\alpha_a=\\mathbf1_A(a)/(b^{h_P(a)}-1)$; the last uses the geometric tail bound $4/T$. After $\\varepsilon$, $F$ and $L$ have been fixed, every term after $\\varepsilon$ tends to zero as $H\\to\\infty$~\\cite[Section~2, weighted-support proof]{paper257}.\n% END SENTENCE r6.example.balance\n\n% SENTENCE r4.example4.08\nThe left side of~\\eqref{eq:worked-mean} is an average with nonnegative weights summing to one: each length has weight $1/M$, distributed equally among its $2^j$ terms. For large $H$, at least one displacement is therefore less than $2\\varepsilon$. Taking $0<\\varepsilon<1/(2q)$ contradicts the lower bound $1/q$. This argument gives no rate of decay in~$N$. Passing to an infinite $B\\subseteq A$ only decreases the weighted sum, and $b^{h_P(a)}-1\\ge2^{h_P(a)}-1$ gives the all-base conclusion from the binary hypothesis~\\cite[Section~2]{paper257}.\n% END SENTENCE r4.example4.08\n\n% SENTENCE r14p3.proof.return\nThis is the information the worked explanation must preserve: the restriction on the support, the distinction from earlier criteria, and the reason for the second average. The formal-support links identify proofs of the stated conclusion. They do not by themselves explain these choices or prevent a summary from dropping the hypothesis.\n\\par\\Needspace{16\\baselineskip}\n% END SENTENCE r14p3.proof.return\n\n\\section{Attribution and scope in three further examples}\\label{app:attribution}\n\n% SENTENCE r14p3.attribution.purpose\nThe following cases distinguish a result's exact scope, its dependence on earlier work and its formal verification. They illustrate further information a faithful account must preserve, rather than steps in the preceding proof or additional systems experiments.\n% END SENTENCE r14p3.attribution.purpose\n\n\\subsection{Counting repeated terms changes the question}\n\n% SENTENCE r4.paper2.01\nA running least common multiple is the least common multiple of the terms seen so far in an increasing sequence. Its value can repeat. For integers with prime factors only $2$, $3$ and $5$, the source paper's example begins $1,2,6,12,60,60,120,360,360,\\ldots$~\\cite{paper269}. A reciprocal sum counting every occurrence contains $1/60$ twice; one counting distinct values contains it once.\n% END SENTENCE r4.paper2.01\n\n% SENTENCE r4.paper2.02\nThe Problem~269 record proves a result for distinct running least common multiples. The original question counts them with multiplicity, so the result does not automatically answer it~\\cite{paper269}.\n% END SENTENCE r4.paper2.02\n\n% SENTENCE r4.related2.02\nThe same record credits Fan's earlier public two-prime result despite an independently reported derivation, and separately identifies the relevant Hecke--Mahler series literature~\\cite{paper269,fan269,loxtonvdp1977,bugeaudlaurent2023}. Independent derivation does not remove the obligation to cite prior work.\n% END SENTENCE r4.related2.02\n\n\\subsection{An explicit description and a stronger antecedent}\n\n% SENTENCE r4.related2.01\nThe Euler totient $\\varphi(n)$ counts the integers from $1$ to $n$ coprime to~$n$. The Problem~249 paper studies its subsequences $\\varphi(k^j n+r)$, where $k\\ge2$ is fixed, $j\\ge0$ and $0\\le r<k^j$. For each level~$j$, it describes the integer linear relations and gives an explicit basis~\\cite{paper249}. Coons's non-$k$-regularity theorem says that no finite set generates these subsequences over all levels. Martin's stronger separation theorem already implies the independence needed for the basis; the explicit reductions and integral relations are a different contribution. The direct proof uses the Chinese remainder theorem and Dirichlet's theorem on primes in arithmetic progressions, following an argument in Yazdani that Yazdani credits to Shallit~\\cite{coons,martin,yazdani}.\n% END SENTENCE r4.related2.01\n\n% SENTENCE r4.paper2.04\nReorganising a proof, making its consequences explicit and formalising it are distinguishable contributions. None makes an inherited ingredient new.\n% END SENTENCE r4.paper2.04\n\n\\subsection{A construction, its formalisation and the historical question}\n\n% SENTENCE r4.related2.03\nProblem~1041 concerns connecting roots of a polynomial by short curves in a strict lemniscate: for a complex polynomial $f$, this is the region $|f(z)|<1$. The degree-seven counterexample construction is due to the forum contributor ani~\\cite{aniforum,erdos1041}. The recorded Lean result refutes the exact statement in the Formal Conjectures repository; independent human review of its correspondence with the curve-length wording in the 1958 paper has not been recorded~\\cite{formalconjectures,fcpr,ehp1958}. The construction, its formalisation and that historical correspondence are separate claims.\n% END SENTENCE r4.related2.03\n\n% SENTENCE r4.repro1041.01\nReaders inspecting the formal result can follow the \\reporef{lean/ErdosProblems/Erdos1041/Counterexample/Assembly.lean}{counterexample assembly} and the \\reporef{lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean}{Hausdorff-length proofs}.\\footnote{The four declarations, all in namespace \\nolinkurl{Erdos1041.Counterexample}, are \\nolinkurl{erdos1041_counterexample}, \\nolinkurl{erdos1041_counterexample_hausdorff}, \\nolinkurl{erdos1041_hausdorff_negation} and \\nolinkurl{erdos1041_hausdorff_answer_false}.}\n% END SENTENCE r4.repro1041.01\n\n% SENTENCE r4.repro1041.02\nThe coverage row records exact-or-stronger Lean support for these statements.\n% END SENTENCE r4.repro1041.02\n\n% SENTENCE r4.related2.04\nComparator remains pending for this result, so its recorded Lean support should not be described as a completed formal comparison.\n% END SENTENCE r4.related2.04\n\n% SENTENCE r4.related2.05\nThe other seven motivating targets remain unresolved in this work.\n% END SENTENCE r4.related2.05\n\n% SENTENCE r4.paper2.03\nA revised statement must retain its scope, attribution and registered supports. Sharpening, merging or moving it requires renewed review of its relation to every support and named input.\n% END SENTENCE r4.paper2.03\n\n% SENTENCE r6.layout.credit\n\\clearpage\n% END SENTENCE r6.layout.credit\n\\section{A reading map for the wider literature}\\label{app:source-map}\n\n% SENTENCE r14p3.sources.purpose\nSection~\\ref{sec:related} compares the closest systems; Appendices~\\ref{app:worked-proof} and~\\ref{app:attribution} explain the mathematical antecedents used in the examples. Figure~\\ref{fig:credit} is a guide to the remaining references and to sources cited by the two historical design accounts in Appendix~\\ref{app:repro}. Its groups indicate why a source is cited, not that every listed method is implemented here or that every work has been evaluated against this system.\n% END SENTENCE r14p3.sources.purpose\n\n\n% SENTENCE r4.fig.credit\n\\begin{figure}[H]\n\\centering\\small\n\\begin{tabularx}{\\linewidth}{@{}>{\\raggedright\\arraybackslash}p{3.8cm}>{\\raggedright\\arraybackslash}X@{}}\n\\toprule\nArea & Reason for citation\\\\\n\\midrule\nFormal foundations & Lean and mathlib; library maintenance and contributor practice~\\cite{lean4,mathlib,growingmathlib,mathlibcontrib}\\\\\nSource and statement checks & Comparator, kernel replay, vacuity, linters and property tests~\\cite{leanfrocomparator,nanodalib,beyondvacuity,batteries,plausible}\\\\\nFormal-to-prose tools & Blueprints, source documentation, translation and refactoring~\\cite{leanblueprint,leanarchitect,hattori,leanmarathon,proofrefactor,p3docagent}\\\\\nRetrieval and interaction & Premises, proof states and exact declaration neighbourhoods~\\cite{leandojo,pantograph,leanexplore,leansearchv2,theoremgraph,leanatlas}\\\\\nProof automation & Premise selection and tactic search~\\cite{leanhammer,aesop,judgementday}\\\\\nLibrary structure & Declaration networks and repository knowledge graphs~\\cite{mathlibnetwork,huch,repokg}\\\\\nAssumptions and reuse & Proof mining, generalisation, lemma mining and theory exploration~\\cite{kohlenbach,gandhi,lemmamining,hipster}\\\\\nGrowing proof knowledge & Library learning, evolving procedures and candidate statements~\\cite{dreamcoder,legoprover,dreamprover,proofevolve,leanconjecturer,circuitprover}\\\\\nResearch agents & Executable objectives, proof cost and evolving tasks~\\cite{alphaevolve,alphaevolvemath,patel,dreamrsi,berlotattwell}\\\\\nAgent collaboration & Bounties, planner/worker state, retrieval and compiler feedback~\\cite{agenthunt,openprover,oprover,prove2me,axle,etp}\\\\\nEvaluation & Research budgets, proof engineering and documentation tasks~\\cite{rebench,p6-swd,p6-ape,p6-poiroux,documentationtest,henkel,openproofcorpus}\\\\\nValidity and intended meaning & Research case studies, interactive proving and public accounting~\\cite{aletheia,autonomousmath,lihai,taowiki}\\\\\nPublic infrastructure & Problem sources, formal conjectures, verification and collaboration~\\cite{erdosproblems,formalconjectures,palomar,carleson,polymath,boinc,gimps,credit,githubsecurity}\\\\\nWriting and maintenance & Mathematical exposition, build dependencies and provenance~\\cite{halmos,knuth,buildsystems,dekleer,workflowprovenance}\\\\\nUnderstanding and reuse & Explanation, mathematical understanding, reuse and research possibilities~\\cite{kra,cohn,sanderson,essays,taomining,litt,tao2026}\\\\\nDifferent community views & Declarations, dissent, benchmarks, research infrastructure and pace~\\cite{fieldsdecl,chu,greenletter,gowers,ringer,koukoulopoulos,antieau}\\\\\nMathematical antecedents & Weighted support, totient ranks and arithmetic separation~\\cite{paper257,erdos1968,duverneytachiya,paper249,coons,martin,yazdani}\\\\\nOther mathematical records & Running LCM, lemniscates and cross-problem connections~\\cite{paper269,fan269,loxtonvdp1977,bugeaudlaurent2023,erdosgraham,akiyamakaneko,ehp1958,erdos1041,aniforum,fcpr,synthesis}\\\\\n\\bottomrule\n\\end{tabularx}\n\\caption{Reading map for this paper and its two historical predecessors. The groups record uses of sources, not a performance ranking or a list of implemented components. Appendices~\\ref{app:worked-proof} and~\\ref{app:attribution} distinguish mathematical antecedents, refinements and independent rediscovery.}\n\\label{fig:credit}\n\\end{figure}\n% END SENTENCE r4.fig.credit\n\n% END audited_body\n\\clearpage\n\\begin{thebibliography}{999}\n\\bibitem{erdosproblems} T.~F.~Bloom, \\emph{Erd\\H{o}s problems},\n\\href{https://www.erdosproblems.com}{erdosproblems.com}, accessed September 2026.\n\n\\bibitem{paper257} W.~Cook, \\emph{Weighted and Covered Supports for Mersenne Subseries}, short paper on Erd\\H{o}s Problem~257, September 2026,\n\\repolink{paper/257/erdos-257-mersenne-support-subseries.pdf}{PDF}.\n\n\\bibitem{erdos1968} P.~Erd\\H{o}s, \\emph{On the irrationality of certain\nseries}, Math.\\ Student 36 (1968), 222--226 (issued 1969),\n\\href{https://users.renyi.hu/~p_erdos/1969-09.pdf}{scan}.\n\n\\bibitem{duverneytachiya} D.~Duverney and Y.~Tachiya, \\emph{Refinement of the\nChowla--Erd\\H{o}s method and linear independence of certain Lambert series},\nForum Math.\\ 31 (2019), 1557--1566,\n\\href{https://doi.org/10.1515/forum-2018-0299}{DOI}.\n\n\\bibitem{kra} B.~Kra, \\emph{Deep theorems were scarce and difficult and so\nbecame an effective mechanism to identify deep thought. 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Tao, \\emph{Mathematics in the age of AI}, 2026,\n\\href{https://arxiv.org/abs/2608.16753}{arXiv:2608.16753}.\n\n\\bibitem{boinc}\nD. P. Anderson, \\emph{BOINC: A Platform for Volunteer Computing}, Journal of\nGrid Computing 18 (2020), 99--122,\n\\href{https://doi.org/10.1007/s10723-019-09497-9}{DOI}.\n\n\\bibitem{gimps}\nGreat Internet Mersenne Prime Search, \\emph{GIMPS}, project documentation and\ndiscovery-credit record, \\href{https://www.mersenne.org/}{mersenne.org},\naccessed August 2026.\n\n\\bibitem{polymath}\nPolymath Project, \\emph{General polymath rules},\n\\href{https://polymathprojects.org/general-polymath-rules/}{project rules},\naccessed August 2026.\n\n\\bibitem{mathlibcontrib}\nLean community, \\emph{Contributing to mathlib},\n\\href{https://leanprover-community.github.io/contribute/index.html}{contributor guide},\naccessed August 2026.\n\n\\bibitem{palomar}\nPalomar Registry, \\emph{About Palomar} and \\emph{Contribution policy},\n\\href{https://palomar-registry.org/about}{registry documentation} and\n\\href{https://github.com/PalomarRegistry/PalomarPolicy/blob/main/CONTRIBUTING.md}{submission standard},\naccessed August 2026.\n\n\\bibitem{henkel}\nJ. Henkel, \\emph{The Mathematician's Assistant: Integrating AI into Research\nPractice}, 2025,\n\\href{https://arxiv.org/abs/2508.20236}{arXiv:2508.20236}.\n\n\\bibitem{openproofcorpus}\nJ. Dekoninck et al., \\emph{The Open Proof Corpus: A Large-Scale Study of\nLLM-Generated Mathematical Proofs}, 2025,\n\\href{https://arxiv.org/abs/2506.21621}{arXiv:2506.21621}.\n\n\\bibitem{credit}\nNISO, \\emph{CRediT: Contributor Roles Taxonomy},\n\\href{https://credit.niso.org/contributor-roles-defined/}{role definitions},\naccessed August 2026.\n\n\\bibitem{githubsecurity}\nGitHub, \\emph{Preventing pwn requests}, GitHub Actions security guidance,\n\\href{https://docs.github.com/en/actions/reference/security/secure-use}{documentation},\naccessed August 2026.\n\n\\bibitem{documentationtest}\nM. Yuan et al., \\emph{End-to-End Testing of Open-Source Hardware Documentation\nDeveloped in Large Collaborations}, 2023,\n\\href{https://arxiv.org/abs/2309.05942}{arXiv:2309.05942}.\n\n\\end{thebibliography}\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":105065,"sha256":"sha256:b6c54d522ad39c0c9d59efdaeb12aa892ff6493d2fe4ccc2a3f9d8e6cdcb3f8f","cite_as":"Plectis reader corpus, paper:claim-faithful-publication-systems","source_repository":"plectis-erdos","source_path":"paper/systems/claim-faithful-publication-systems-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:554ed85b5315bd936a4d09d58f8727547c5bb1b299f9684eb6ef720b329759a2","byte_count":7909,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared house typography for the repository's rendered papers.  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A named destination is\n% more durable than a page number: sections may move while their semantic role\n% remains the same. All public PDFs are copied to one directory at release.\n\\newcommand{\\papersectiontarget}[1]{\\hypertarget{#1}{}}\n\\newcommand{\\papersectionlink}[3]{\\href{#1\\##2}{#3}}\n\n\\RequirePackage{titlesec}\n\\titleformat{\\section}{\\Large\\bfseries}{\\thesection}{0.85em}{}\n\\titlespacing*{\\section}{0pt}{2.6ex plus .8ex minus .3ex}{1.5ex plus .3ex}\n\\titleformat{\\subsection}{\\large\\bfseries}{\\thesubsection}{0.8em}{}\n\\titlespacing*{\\subsection}{0pt}{2.1ex plus .6ex minus .2ex}{1.1ex plus .2ex}\n\\titlespacing*{\\paragraph}{0pt}{1.6ex plus .5ex minus .2ex}{0.85em}\n\n\\RequirePackage{fancyhdr}\n\\newcommand{\\paperrunninghead}{}\n\\newcommand{\\runninghead}[1]{\\renewcommand{\\paperrunninghead}{#1}}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\fancyhead[L]{\\small\\itshape\\paperrunninghead}\n\\fancyhead[R]{\\small\\thepage}\n\\renewcommand{\\headrulewidth}{0pt}\n\\fancypagestyle{plain}{\\fancyhf{}\\renewcommand{\\headrulewidth}{0pt}}\n\n\\RequirePackage{caption}\n\\captionsetup{font=small,labelfont=bf,skip=9pt,width=.92\\linewidth}\n\\RequirePackage{etoolbox}\n\\RequirePackage{pdflscape}\n\\renewcommand{\\arraystretch}{1.20}\n\\newcommand{\\papertablehead}{\\rowcolor{papertableheader}}\n\\newcommand{\\papertablebody}{\\rowcolors{2}{white}{papertablerow}}\n\\AtBeginEnvironment{tabular}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\\AtBeginEnvironment{longtable}{%\n  \\papertablebody\n  \\arrayrulecolor{papertablerule}%\n  \\setlength{\\extrarowheight}{1.4pt}%\n}\n\n% A compact front-matter panel for the result a scanning reader should retain.\n% It is deliberately short and unbreakable: manuscripts should state one\n% result, its evidence, and its limit here, then supply the argument below.\n\\newenvironment{paperresult}[1]{%\n  \\par\\addvspace{0.7em}\n  \\begin{center}\\begin{minipage}{0.94\\linewidth}\n    {\\color{paperresultrule}\\rule{\\linewidth}{0.8pt}}\\par\\vspace{4pt}\n    {\\small\\sffamily\\bfseries\\color{paperresultrule}#1}\\par\\smallskip\n    \\small\\color{black}\n}{%\n    \\par\\vspace{4pt}{\\color{paperresultrule}\\rule{\\linewidth}{0.45pt}}\n    \\end{minipage}\\end{center}\\par\\addvspace{0.5em}\n}\n\n% Shared diagram grammar.  Reader-facing figures use the same restrained\n% hierarchy as the surrounding page: a short eyebrow, a substantial title,\n% one compact explanatory line, and an optional evidence badge.  The styles\n% are installed only when a manuscript loads TikZ.\n\\AtBeginDocument{%\n  \\@ifpackageloaded{tikz}{%\n    \\tikzset{\n      paper box/.style={draw=gray!48, line width=.45pt, rounded corners=3pt,\n        fill=white, align=left, inner xsep=7pt, inner ysep=6pt,\n        font=\\small},\n      paper card/.style={paper box, minimum height=13mm},\n      paper stage/.style={paper box, align=center, minimum height=14mm},\n      paper arrow/.style={-{Stealth[length=2.15mm,width=1.45mm]},\n        line width=.7pt, draw=gray!72!black},\n      paper feedback/.style={paper arrow, densely dashed,\n        draw=Violet!72!black},\n      paper eyebrow/.style={font=\\scriptsize\\sffamily\\bfseries,\n        text=gray!70!black},\n      paper label/.style={font=\\scriptsize\\sffamily,\n        text=gray!72!black, fill=white, inner sep=1.5pt},\n      paper badge/.style={rounded corners=2pt, inner xsep=4pt, inner ysep=2pt,\n        font=\\scriptsize\\sffamily\\bfseries},\n      paper boundary/.style={densely dashed, line width=.55pt,\n        draw=gray!62}\n    }%\n  }{}%\n}\n\n% Front matter: title block with an optional italic subtitle, small-caps\n% author, and a hairline rule; the abstract is set small with a small-caps\n% head and symmetric indents.\n\\newcommand{\\subtitle}[1]{\\gdef\\@papersubtitle{#1}}\n\\renewenvironment{abstract}{%\n  \\vspace{0.1em}\\small\n  \\begin{center}{\\normalsize\\scshape Abstract}\\end{center}\\vspace{-0.5em}\n  \\list{}{\\leftmargin 1.3em\\rightmargin\\leftmargin\\listparindent 0pt\n    \\parsep 0.32em\\itemindent 0pt}\\item\\relax}\n  {\\endlist\\vspace{0.5em}}\n\\def\\@maketitle{%\n  \\newpage\\null\n  \\vspace*{-2.6em}%\n  \\begin{center}%\n    {\\LARGE\\bfseries\\@title\\par}%\n    \\vskip 0.55em\n    {\\large\\itshape\\@papersubtitle\\par}%\n    \\vskip 1.1em\n    {\\scshape\\@author\\textsuperscript{*}\\par}%\n    \\vskip 0.25em\n    {\\small\\@date\\par}%\n    \\vskip 0.8em\n    \\rule{6.5em}{0.4pt}%\n  \\end{center}%\n  \\vskip 0.9em}\n\n% Production provenance, set at the foot of the first page.\n%\n% A reader who meets a single personal byline ordinarily infers that the named\n% author drafted the sentences.  For these papers that inference is false, and\n% a declaration reached only in the back matter would leave it standing for the\n% length of the paper.  The note therefore appears on page one, small and\n% visually subordinate to the title and abstract: enough to prevent the false\n% inference, not enough to make the production method the subject.\n%\n% It carries no mark, because it qualifies the whole manuscript rather than any\n% one passage.  \\footnotetext does not advance the footnote counter, and the\n% emptied \\thefootnote is confined to a group, so numbered footnotes elsewhere\n% in the paper are unaffected.\n%\n% The wording is not authored here.  It is fixed by the corpus provenance\n% contract and reproduced verbatim in each manuscript, so that the three papers\n% cannot drift into three different accounts of how they were made.\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{*}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n% Corpus-wide production disclosure.  It appears in small print on page one so\n% that the byline cannot be mistaken for a claim of unaided authorship.  The\n% longer form is repeated in each paper's declarations section.\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and AI use.}\nWill Cook built and directed the research infrastructure and reviewed the claims\nwhen he could. AI agents did most of the research and drafting. Cook did not\nindependently verify every claim.}}\n\n\\endinput\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-erdos/main/paper/systems/claim-faithful-publication-systems-paper.tex","section_outline":["Introduction","A repository organised around problems","Stored records and derived views","Finding the relevant work","Operations on the records","Responsibilities and adoption","Worked case: reviewing an explanation","Proof and publication checks","Publication and revision","Permissions and tasks","From frozen inputs to an accepted revision","Retaining improvements","An open route for contributions","Recorded observations","Inventory and formal-support status","Historical false-edit trial","Local integration and unperformed reader evaluation","Related work","Limits","Reading the materials and reproducing checks","Read a result and follow its sources","Inspect the records and rerun document checks","Locate the formal supports of the worked example","Interpret the recorded inventory","Identify the version and the earlier design accounts","The mathematical example: a criterion for irrationality","The question and its antecedents","A weighted condition on the retained exponents","Why the weighted condition reaches further","The proof by two averages","Attribution and scope in three further examples","Counting repeated terms changes the question","An explicit description and a stronger antecedent","A construction, its formalisation and the historical question","A reading map for the wider literature"]},"plectis-public-system":{"paper_id":"plectis-public-system","title":"Plectis: What a Stranger Can Check","form":"Evidence boundary","length_class":"system","pages":22,"subject":"Systems paper","subject_kind":"system","problem_id":null,"landing_role":"boundary","question":"What may a stranger conclude from public evidence when the author chose both what to publish and what counts as a pass?","precis":"The author publishes selected code, fixed inputs, expected results, decision rules, and run records while the larger system stays private. What can a stranger conclude after rerunning them? A matching result shows repeatability for one stated public case. It does not establish where the code came from, whether the expected answer is correct, whether the published selection resembles the private whole, or whether the private system works reliably. The contribution is a method for making one selected disagreement local, not a score.","owns":"the evidence boundary: what a passing public run does and does not establish","not_authority_for":"the private system's internal state, which is not public, and the mathematical results it cites as evidence","first_pass":["sec:problem","sec:early-example","sec:distinctions","sec:stronger"],"first_pass_stated_by_paper":false,"publication_state":"active","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/plectis-public-system.pdf","hosted_pdf_path":"papers/plectis-public-system.pdf","hosted_tex_path":"papers/source/plectis/paper/plectis-public-system.tex","site_html":"maths/papers/plectis-public-system.html","github_current_url":"https://github.com/wcook04/practical-formal-verification/blob/main/paper/plectis-public-system.tex","github_permalink_url":"https://github.com/wcook04/practical-formal-verification/blob/f2ed2582e6de24c71b45dfc30dae33863e2ab539/paper/plectis-public-system.tex","body_pointer":"papers.records.plectis-public-system.body","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n\\documentclass[11pt]{article}\n\\usepackage{paper-house-style}\n\\usepackage{booktabs,array,enumitem}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\usepackage{fancyvrb}\n\\usepackage{tikz}\n\\usetikzlibrary{positioning,arrows.meta,calc}\n\\usepackage{float}\n\\setlist{nosep,leftmargin=*}\n\\emergencystretch=2em\n\\newcommand{\\code}[1]{\\texttt{#1}}\n\\newcommand{\\component}[1]{\\nolinkurl{#1}}\n\\newcommand{\\repo}{\\href{https://github.com/wcook04/practical-formal-verification}{\\nolinkurl{github.com/wcook04/practical-formal-verification}}}\n\n% Shared figure styles: black mechanism, dashed unavailable/unestablished\n% material, and two accents used with one meaning each across all figures:\n% blue for what a stranger can do, amber for words and choices the project\n% itself supplies.\n\\colorlet{strangerblue}{RoyalBlue}\n\\colorlet{authoramber}{BurntOrange!85!black}\n\\tikzset{\n  part/.style={draw=black!75, line width=0.6pt, rounded corners=2pt,\n    align=center, inner sep=4pt, font=\\footnotesize, fill=gray!8},\n  words/.style={part, fill=white, draw=authoramber!75},\n  ghost/.style={draw=black!45, dashed, line width=0.6pt, rounded corners=2pt,\n    align=center, inner sep=4pt, font=\\footnotesize, text=black!60, fill=none},\n  act/.style={font=\\scriptsize\\itshape, text=strangerblue, align=center},\n  gnote/.style={font=\\scriptsize, text=black!60, align=center},\n  flow/.style={-{Stealth[length=2.2mm]}, line width=0.6pt, draw=black!75},\n  softflow/.style={-{Stealth[length=2.0mm]}, line width=0.6pt, draw=black!50, dashed},\n  authlink/.style={softflow, draw=authoramber!70},\n  actflow/.style={-{Stealth[length=2.0mm]}, line width=0.6pt, draw=strangerblue},\n  flabel/.style={font=\\scriptsize, inner sep=2pt},\n  gaplabel/.style={font=\\scriptsize, fill=white, inner sep=2.2pt, align=center,\n    text=authoramber},\n}\n\n% These values are checked against the public registries by\n% scripts/check_public_system_paper.py.  Do not update them from memory.\n\\newcommand{\\snapshotcommit}{57ffb7f5830cc446a2cbd8083d5d80bcab2af563}\n\\newcommand{\\snapshotshort}{57ffb7f5830c}\n\\newcommand{\\componentcount}{88}\n\\newcommand{\\familycount}{7}\n\\newcommand{\\entrycount}{2}\n\\newcommand{\\mappingcount}{12}\n\\newcommand{\\mathcount}{20}\n\\newcommand{\\safetycount}{20}\n\\newcommand{\\researchcount}{9}\n\\newcommand{\\publiccopycount}{20}\n\\newcommand{\\recoverycount}{5}\n\\newcommand{\\leanfilecount}{58}\n\\newcommand{\\copiedsourcecount}{21}\n\\newcommand{\\refactorcount}{27}\n\\newcommand{\\runtimecount}{22}\n\\newcommand{\\validatorcount}{18}\n\n\\hypersetup{\n  bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdftitle={Plectis: What a Stranger Can Check},\n  pdfauthor={Will Cook},\n  pdfsubject={Public evidence from a private system: what a stranger can check, what the author can only report, and what remains unestablished},\n  pdflang={en-GB},\n  pdfkeywords={research software, claim verification, public evidence, reproducibility, software artifacts, agent-assisted development},\n}\n\\runninghead{Plectis: what a stranger can check}\n\\title{Plectis: What a Stranger Can Check}\n\\subtitle{What public evidence can and cannot show about a private system}\n\\author{Will Cook}\n\\date{July 2026; updated September 2026}\n\n\\begin{document}\n\\maketitle\n\\provenancenote{\\emph{Authorship and generative-system use.} Every word of this\nmanuscript was generated by agents based on large language models operating\nwithin Will Cook's private research system for artificial intelligence. Cook\nset the project's objectives and acceptance criteria and directed the agent\nwork. This version was revised by agents; the public record does not establish\npersonal review or approval of this build. The agents are production tools,\nnot authors. See \\emph{Statements and declarations}.}\n\n\\begin{abstract}\nThis paper treats one named version of the public Plectis software toolkit as\na single case in a bounded descriptive analysis of tests for claims about a\nlarger private workbench. The author releases selected code, fixed inputs,\nexpected results, decision rules, and run records while the larger system remains\nprivate.  What can a stranger conclude after rerunning those tests?  The answer\nis narrow.  The repository lets a reader inspect and rerun its published\nprocedures.  A matching result shows repeatability for a stated public case; it\ndoes not establish where the code came from, whether the expected answer is\ncorrect, whether the published selection resembles the private whole, or\nwhether the private system works reliably.\n\nThe contribution is a method for making one selected disagreement local: the\nregistered components (separately testable parts) place a prose claim beside a\nfixed test case, runnable procedure, expected result, decision rule, run\nrecord, and explicit inference limit.  The repository calls this package a\nclaim--evidence--limit contract.  It is an executable-disclosure case study,\nnot an evaluation of reviewer performance.  A worked component calculates an\naudio level from five small sample arrays.  A reader can repeat the calculation and,\nfor one row, derive the expected number independently of the implementation.\nEven there, the run cannot show whether the pass rule tests the claim as\nwritten, how the code behaves on untested inputs, or whether the chosen formula\nis the right account of audio level.  I chose the published fragments, cases,\nanswers, rules, and limits.  The paper therefore treats the repository as an\nauthor-curated test collection, not as independent validation of the private\nsystem.  I report that its published material was copied or adapted from that\nsystem and that I developed it by directing coding tools based on artificial\nintelligence (AI).  The argument does not depend on that account or on the\nclaimed origin of the published material.\n\\end{abstract}\n\n\\section{The problem}\n\\label{sec:problem}\n\nI have a larger private software system that I cannot responsibly publish:\nit contains personal records, credentials, live browser and account\ninformation, my working notes, and unfinished tools. I would still like\nstrangers to be able to test some claims about it instead of accepting my\nsay-so. I use \\emph{answerable} in a deliberately local sense: a claim is\nanswerable when a stranger can run a public test that bears on it, see how the\nresult is judged, and locate a disagreement in a named input, rule, or output.\nThis is not a new synonym for truth, reproducibility, verification, or\nindependent review. For the rest of the paper I use the plainer phrase\n\\emph{publicly testable claim}. The question is what conclusions such a test\ncan support when the larger system and the unpublished pool from which the\ntest was selected remain invisible.\n\nPlectis is my working answer: a public software repository, stored at\n\\repo. A \\emph{repository} is an organised project folder, usually kept\nwith its revision history; to \\emph{clone} a repository is to copy that\nfolder and history to another computer. Everything this paper says a\nstranger can check is in that repository. This paper describes how its\nclaims are exposed and where the evidence stops.\n\nThis paper examines the public software collection only. The companion\nmathematics repository is a separate public release with work on eight\nErd\\H{o}s problem programmes. It credits ani with an explicit degree-seven\npolynomial for which every path through the strict lemniscate joining two\ndistinct roots has image of one-dimensional Hausdorff measure greater than~$2$.\nLean checks the resulting refutation of the Formal Conjectures path-image\nformulation of \\#1041, as well as the earlier total-variation statement;\nthe \\href{https://github.com/wcook04/plectis-erdos/blob/a25cb360bef8dd818dde14b5fb752244304af354/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean\\#L393}{path-image declaration}\nstates the checked scope. Correspondence with the 1958 curve-length wording\nstill awaits independent human review, and the other seven target problems\nremain unresolved there. The recorded walkthroughs on the project\nwebsite show the private interface. These objects share a reported production\nhistory, but each supports different conclusions and must be assessed from its\nown evidence. The audio example below does not test any mathematical result.\nReaders interested in the breadth of the private system should start from the\narchitecture documentation rather than from that small example.\n\nAs a current public companion to this dated software case, the mathematics\nrepository's \\href{https://github.com/wcook04/plectis-erdos/blob/main/paper/257/erdos-257-mersenne-support-subseries.pdf}{\\#257 short paper}\nand \\href{https://github.com/wcook04/plectis-erdos/blob/main/paper/257/erdos257-mersenne-reasoning-surface.pdf}{longer record}\nprove a Lean-checked theorem for an infinite exponent set $A$ and a finite\nnonempty prime set $P$. Write $h_P(a)=\\prod_{p\\in P}p^{v_p(a)}$, where\n$v_p(a)$ counts factors of $p$ in $a$. If\n$\\sum_{a\\in A}h_P(a)/[a(2^{h_P(a)}-1)]<\\infty$, then\n$\\sum_{a\\in A}(b^a-1)^{-1}$ is irrational at every integer base $b\\ge2$.\nThe condition holds for some supports with divergent $\\sum_{a\\in A}1/a$\nand passes to infinite subsets. Irrationality for arbitrary infinite\nsupports remains open.\nThe audio example below does not test this theorem.\n\nThe companion publishes short problem papers, longer reasoning records, and\nlinked Lean source. Its \\href{https://github.com/wcook04/plectis-erdos/blob/main/research/experiments/choices_contraction/README.md}{exact-rational experiment}\ngives a separate, bounded way to test one \\#257 question. Its\n\\href{https://github.com/wcook04/plectis-erdos/blob/main/docs/EXTERNAL_VERIFICATION.md}{Comparator}\nchecks selected formal statements, while\n\\href{https://github.com/wcook04/plectis-erdos/blob/main/docs/verification/PALOMAR_QUALIFICATION.md}{Palomar qualification}\nis repository-local review of candidates for possible external submission;\nneither is an external verdict on novelty or the historical problem wording.\nReaders can propose a proof-workflow or other infrastructure change through\nthe companion's \\href{https://github.com/wcook04/plectis-erdos/blob/main/docs/research-commons/ARCHITECTURE_CONTRIBUTIONS.md}{architecture guide},\nor a toolkit change through \\href{https://github.com/wcook04/practical-formal-verification/blob/main/CONTRIBUTING.md}{this repository's contribution guide}.\nThose mathematics artifacts and invitations are outside the dated software\ncollection examined here; they do not establish independent use or validation\nof the private system.\n\nThe main result is a boundary, not a score. Public code, inputs, rules, and\nrun records can make a selected claim easier to challenge. They cannot by\nthemselves establish three missing premises: that the public code came from\nthe private system, that the project's expected answers and pass rules are\ncorrect, or that the selected cases represent the unseen whole. The audio\nexample in Section~\\ref{sec:run} shows all three limits in one elementary\ncalculation. The later inventory records what was published; it is not\nevidence of breadth, quality, or independence.\nRuneson and H\\\"ost likewise treat configuration-managed project documents as\narchival data while warning that records created for another purpose may omit\nneeded information and be difficult to assess without complementary sources\n\\cite[\\S4.4, p.~148]{runeson2009}.\nThe repository therefore lets a reader test claims about the public objects it\nactually exposes, within the limits stated for each test.\n\n\\paragraph{Who made the decisions.}\nCook directed the private system's construction through AI coding agents. He\nset project objectives and review criteria and accepted or rejected agent\noutput in the development process. The public record does not show how closely\nhe reviewed each present case, expected answer, decision rule, or sentence.\nNo other person worked on the system. This account explains the\nconcentration of control, but it remains testimony about origin: Plectis\ncannot independently prove how the private work was produced, and the public\nrepository does not prove the story.  The same evidential limit would hold if\nevery line had been written by hand.  One process can give an implementation\nand its check the same mistake; this paper compares no development methods and\nestimates no error rates.\n\nThe boundary of the evidence can be stated before any command is run:\n\n\\begin{center}\n\\small\n\\begin{tabular}{>{\\raggedright\\arraybackslash}p{0.24\\linewidth}>{\\raggedright\\arraybackslash}p{0.67\\linewidth}}\n\\toprule\n\\textbf{Status} & \\textbf{What belongs in it} \\\\\n\\midrule\nPublicly checkable & The contents of a named public version, the commands listed for its components, the files those commands write, and the results of its public tests. \\\\\nReported here & The private system's history, the role of AI agents in producing it, and my claim that the published material came from the private system. \\\\\nNot established here & Whether the public selection represents the private whole, whether the published code works beyond its fixed inputs, whether the project's own pass rules are correct or even test the stated claims, and whether the private system works reliably at all. \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\n\\paragraph{What I examined.} The paper examines one author-curated software\ncollection. At the named repository version, I read the list\nand plain descriptions of all \\componentcount{} components, counted them by\nproject group and by the kind of public evidence offered, traced the worked\ncomponent from input to limit, and reran the appendix's public commands. I did\nnot compare every pass rule with its prose claim or derive fresh expected\nanswers; Section~\\ref{sec:stronger} reserves those checks for an outside\nevaluator. The paper defines the component contract, then follows\none ordinary component from input to limit. Five distinctions separate what a\npassing run shows from conclusions it may appear to support. The final\nsections identify the choices an evaluator would need to make independently\nof me.\n\nThis is a descriptive analysis of one public repository, not a statistical\nstudy. Runeson and H\\\"ost define the case as the object of study and its units\nof analysis as items contained within it\n\\cite[\\S2.5, p.~138]{runeson2009}. They also note that this boundary depends\non the study context and research goals \\cite[\\S3.1, p.~139]{runeson2009}. I\nborrow only those two terms: the named repository version is the single case,\nand each component is a \\emph{unit of analysis} (one item examined inside that\ncase). This paper does not otherwise claim a full case-study design. No subset\nof the components is used to estimate performance or represent the private\nsystem.\n\nReading this paper requires no programming. Reproducing its runs requires\nbasic command-line use, a terminal (a text window for commands), Git (for\ncopying and versioning repositories), Python 3.11 or later, and \\code{pip}\n(for installing Python software). The appendix gives the exact commands,\nenvironment details, and installation notes. It is a dated record of one\nverification pass at the version named there.\n\n\\section{One public test before the general design}\n\\label{sec:early-example}\n\nConsider four decimal audio samples:\n\\[\n0,\\quad 0.05,\\quad -0.05,\\quad 0.1.\n\\]\nThe published procedure squares them, averages the squares, takes the square\nroot, and multiplies by eight.  Their squared average is \\(0.00375\\), so the\nstated rule gives \\(8\\sqrt{0.00375}=0.489897\\ldots\\).  The public program\nproduces \\(0.489897949\\).  Anyone with a calculator can reproduce the expected\nnumber without trusting the program.\n\nThat agreement establishes something real and small: this public program\nagrees with the stated formula on this test case.  It does not establish that\nthe formula is the right account of audio level, that the public code came\nfrom the private system, that other inputs behave similarly, or that important\nrisks were selected for testing.  Those conclusions require additional\npremises that the observed match does not contain.\n\nThe paper asks the same six questions of every published item:\n\\begin{enumerate}\n\\item What sentence is offered for belief?\n\\item What public object or procedure was checked?\n\\item Who supplied the interpretation or expected result?\n\\item What does the observed match establish?\n\\item What remains unestablished?\n\\item Which independently controlled evidence would strengthen it?\n\\end{enumerate}\nSection~\\ref{sec:run} gives the full command, all five audio cases, and the\nsaved run record.  We now generalise from this example to the project's common\nrecord format.\n\n\\section{The component contract}\n\\label{sec:contract}\n\nA \\emph{component} in Plectis is one registered, testable part of the\nproject. Its \\emph{fixture} is a fixed sample input: in ordinary testing\nlanguage, a test case. A \\emph{receipt} is the saved record of a run. The\nrepository supplies one from an earlier run; rerunning can write another.\nA \\emph{validator} is the program that applies a component's pass and refusal\nrules. In testing terminology, the rule and expected result together act as\nthe test oracle: they say what should count as correct for that case. A\n\\emph{commit} is a named, saved version of the repository. These are Plectis's\ninternal names for familiar objects---test case, test oracle, run record, and\nsaved software version---and the paper uses the familiar terms whenever the\ninternal filename is irrelevant.\n\nHere \\emph{evidence} means the public input, procedure, output, and judgement\nrule offered in support of a claim. Calling them evidence does not establish\ntheir adequacy. \\emph{Contract} means only that the claim, those materials,\nand a limit are stated together so readers can check their fit. Nothing in\nthese labels grades the importance of what a component does.\n\nAt commit \\code{\\snapshotshort} the registry, the component list read by the\nprograms, contains \\componentcount{} entries. Every entry is required to\nconnect the same things: a claim stated in prose; a fixture; a command; the\noutput files the command writes; a stored receipt from a prior run; an\nexplicit rule for what counts as a pass and what must be refused; and a\nstated limit on what a pass establishes. Figure~\\ref{fig:contract} lays the\nrequired parts out, together with the fact about them that the rest of this\npaper keeps returning to: every part is supplied by me, and every part is\nexposed to a stranger.\n\n\\begin{figure}[t]\n\\centering\n\\makebox[\\linewidth][c]{\\begin{tikzpicture}[x=1cm,y=1cm]\n% words row\n\\node[words] (claim) at (3.45,2.05) {The claim, stated in prose};\n\\node[words] (limit) at (13.0,2.05) {The stated limit};\n% mechanism row\n\\node[part] (fix) at (0,0) {Fixed input\\\\(the fixture)};\n\\node[part] (code) at (3.45,0) {Public code};\n\\node[part] (out) at (6.55,0) {Fresh output\\\\files};\n\\node[part, text width=2.9cm] (val) at (9.55,0) {Comparison, pass rule, refusal rule\\\\(the validator)};\n\\node[part] (verdict) at (13.0,0) {Pass, or\\\\declared refusal};\n\\node[part] (rec) at (9.55,-2.0) {Fresh receipt\\\\(saved run record)};\n% flows\n\\draw[flow] (fix) -- node[flabel, above=1pt]{command} (code);\n\\draw[flow] (code) -- node[flabel, above=1pt]{writes} (out);\n\\draw[flow] (out) -- (val);\n\\draw[flow] (val) -- (verdict);\n\\draw[flow] (val) -- node[flabel, right]{records} (rec);\n\\draw[authlink] (code) -- node[flabel, right, align=left]{said to\\\\do this} (claim);\n\\draw[authlink] (verdict) -- node[flabel, right, align=left]{read under\\\\this ceiling} (limit);\n% stranger actions\n% top-anchored so the action row reads as one aligned row despite\n% differing line counts\n\\node[act, anchor=north] (a1) at (0,-2.95) {change the input};\n\\node[act, anchor=north] (a2) at (3.45,-2.95) {read it;\\\\rerun it};\n\\node[act, anchor=north] (a3) at (6.55,-2.95) {regenerate\\\\them};\n\\node[act, anchor=north] (a4) at (9.55,-2.95) {compare fresh\\\\with stored;\\\\dispute the rule};\n\\node[act, anchor=north] (a5) at (13.0,-2.95) {hold every\\\\conclusion\\\\to the limit};\n\\draw[RoyalBlue!45, dotted, line width=0.6pt] (a1) -- (fix);\n\\draw[RoyalBlue!45, dotted, line width=0.6pt] (a2) -- (code);\n\\draw[RoyalBlue!45, dotted, line width=0.6pt] (a3) -- (out);\n\\draw[RoyalBlue!45, dotted, line width=0.6pt] (a4) -- (rec);\n\\draw[RoyalBlue!45, dotted, line width=0.6pt] (a5) -- (verdict);\n\\end{tikzpicture}}\n\\caption{How to read the component contract, at commit\n\\code{\\snapshotshort}. Read the middle row from left to right: a command applies\npublic code to a fixed input and the code writes output. The validator judges it\nunder the published pass and refusal rules, then writes a fresh receipt recording\nthe run. The stored receipt from the earlier run is available for comparison; it\ndoes not determine the new verdict. The two\nboxes above are words rather than mechanisms: the project states the claim and\nthe limit on what a pass may support. Every part is public, and the project\nsupplied every part. The italic blue row names what a stranger can do without\nasking anyone; amber marks the project's words and choices. Across the figures,\na dashed outline marks something outside what the public run establishes. This\nfigure has no dashed boxes because none of its parts is hidden. In\nFigures~\\ref{fig:boundary} and~\\ref{fig:gaps}, dashes mark private material or\nconclusions not established by the observed run. Section~\\ref{sec:run} follows\none component through this chain.}\n\\label{fig:contract}\n\\end{figure}\n\nThe repository cannot by itself support my assertions about the unseen private\nsystem. A component does something narrower: it gives a related public claim a\nprocedure that can fail in front of a stranger. The procedure\ndoes not make the claim true or establish the published fragment's origin. It\nchanges the form disagreement can take: a reader who doubts a component's\nclaim can point at a particular input, a particular line of the validator, or a\nparticular number in the output, instead of arguing with my description of\nsoftware they cannot see. A component that fails is still answerable in\nthis sense, and a component can be answerable and trivial; answerability\nconcerns the form a claim takes, not its truth and not its importance.\n\nA component may also end in a declared refusal rather than an answer. The\nworked example below refuses an input format it does not support. Components\nthat depend on an external tool, such as the Lean proof checker (a program\nthat verifies mathematical proofs), must record a refusal with a stated limit\nwhen the tool is absent, rather than reporting success.\n\\emph{Runnable} therefore means that the operation and its declared failure\nboundary are present in public form; it does not mean that every optional\ndependency exists on every machine. A refusal is informative only when it is\ndeclared in advance and distinguishable from a breakdown: a predeclared\nrefusal marks the edge of what the component claims, while an unexpected\ncrash is a defect. The two must not be read as the same kind of event. One\nhonesty condition on refusals cannot be discharged from inside the\nrepository at all: the public record shows each boundary as declared at the\nnamed commit, and it cannot show whether the boundary was drawn before or\nafter an awkward case was first met. A boundary drawn afterwards turns a\nfailure into a designed refusal in the telling. Saving the version before\nan outside evaluation begins, as Section~\\ref{sec:stronger} proposes, is\nwhat would make that ordering checkable.\n\nThe idea does not depend on software. The contract needs a claim in words; a\ndeterminate public case; a procedure a stranger can\ncause to run; an observable result; a rule connecting result to claim; a\ndeclared boundary of application; and a stated ceiling on inference. That\nthe case here is a file of numbers, the procedure a command, and the record a\ntext file is a convenience of this example. The same contract could be\nsatisfied in another field by a laboratory protocol with declared rejection\nconditions, or by a sealed evaluation service with published scoring rules.\nI make no claim about those settings; the point is only that the contract\nis a shape for exposing claims to strangers, and this repository is one\nexample. Read simply as software, it is a test suite with a registry and\nunusually complete paperwork. That is a fair description. The paper asks\nwhat such a test suite lets a stranger challenge, and which conclusions\nstill lie beyond it.\n\nThis local notion overlaps with reproducibility and transparency but is not\na synonym for either. A run can repeat perfectly with no stated claim attached,\nand a claim can be answerable while still awaiting a good test. The design\nalso begins from partial disclosure: it offers one limited point of contact,\nnot a view of the hidden system. A component exposes the published fragment\nto challenge. It does not make the private whole answerable.\n\nAnswerability is not the same property as computational reproducibility,\nalthough the component contract supplies materials needed to test the\nlatter. Under the convention used by the National Academies, computational\nreproducibility means obtaining consistent results using the same input\ndata, computational steps, methods, code, and conditions of analysis\n\\cite[conclusion 3-1]{nasem2019}. The report separately cautions that exact\nreproducibility does not guarantee computational correctness: undetected\nerroneous code can reproduce the same erroneous result\n\\cite[p.~9]{nasem2019}.\nIn software testing, a \\emph{test oracle} is a procedure used to distinguish\ncorrect from incorrect behaviour. The challenge of making that distinction is\ncalled the test oracle problem \\cite[p.~507]{barr2015}. Barr et al.\\ distinguish\na partial test oracle \\(D\\) from a conceptual ground-truth oracle that always\ngives the right answer, modelled as a total oracle \\(G\\), while observing that\nsuch ground truth is unknowable except in trivial cases\n\\cite[Defs.~2.4 and 2.6--2.8, pp.~509--510]{barr2015}. Soundness requires\n\\(D(a)\\Rightarrow G(a)\\), and completeness requires\n\\(G(a)\\Rightarrow D(a)\\). A passing validator therefore applies its published\nrule; it does not by itself establish that the rule is sound. The risk\nthat a curated collection disproportionately exposes favourable cases is\nanalogous to selective publication and the file-drawer problem\n\\cite[p.~638]{rosenthal1979}.\nAn \\emph{assurance case} links system claims to arguments and evidence. The\nObject Management Group's Structured Assurance Case Metamodel (SACM), version\n2.3, defines such a case as a collection of auditable claims, arguments, and\nevidence. It separately represents an artifact citation and the\n\\emph{AssertedEvidence} relationship said to connect that artifact to a claim;\nthe relationship is itself an assertion by the case's author. Representing an\nargument is therefore not the same as establishing its validity\n\\cite[secs. 1.1, 4, 7.3, 11.9--11.15]{omgsacm2023}. The GSN Community Standard\nstates the boundary more directly: GSN documents an asserted chain of\nreasoning and its evidential references, but the notation itself establishes\nneither the truth of the argument nor the sufficiency of its support\n\\cite[\\S0:3.2, p.~11; \\S\\S0:4.11--0:4.13, p.~15]{gsn}. Plectis is not an\nimplementation of that standard or of GSN, and makes no claim that the private\nsystem is safe. Its validator applies a published rule and records a result;\nit neither models nor independently justifies the evidential link from that\nresult to the prose claim.\nThe Association for Computing Machinery (ACM) calls digital research materials\nsuch as code, scripts, and data\n\\emph{artifacts}. Its \\emph{Artifacts Evaluated---Functional} badge means\nthose materials passed an independent audit for documentation, consistency,\ncompleteness, ability to run, and evidence of verification and validation.\nThe NISO recommended practice separates object review, reproduction with the\nauthor's objects, and replication by a new study\n\\cite[secs.~2.2--2.4]{nisobadging2021}. ACM's harmonised current names are\n\\emph{Results Reproduced} and \\emph{Results Replicated}; its badges are\nindependent, and results validation requires a peer-reviewed report\n\\cite{acmartifact}. Plectis claims neither status: its receipts are\nauthor-generated, not independent review or validation. These standards are\ncomparison points, not statuses awarded here. This paper claims no new theory of\nassurance, testing, or reproducibility; it reads each public procedure against\nits claim and stated limit.\n\n\\section{One run, examined}\n\\label{sec:run}\n\nThe component \\component{batch8_audio_level_rms_port} computes a loudness\nlevel from short arrays of audio samples. I chose it because it is ordinary.\nIt needs only a square root and no knowledge of the private system; its claim\nis small enough that every part of the evidence can be seen at once.\n\nThe reported origin is this. The private system contains a small macOS\nrecording application, and one of that application's functions reduces a\nshort stretch of audio samples held in memory to a level between zero and\none. The public\ncomponent re-expresses that calculation in Python. The component's code and\nits stored receipts name the private file it claims to re-express\n(\\component{AudioLevelMonitor.swift}); a stranger cannot open that file, so\nthe name is a report, not evidence. What the stranger can check is\neverything else.\n\nThe declared calculation begins with root-mean-square (the \\code{rms} in\nthe component's name): square each sample, average the squares, and take\nthe square root. It then applies the project's gain of eight and caps the\nresult at one. Signed sixteen-bit integer samples are first divided by\n32{,}767, the largest positive value in that representation, to bring them\nonto the same scale. An empty input must produce zero. An unrecognised\nformat name must be refused rather than guessed at.\n\nThe component's public command runs it against its fixture and writes fresh\noutput outside the repository (the backslashes split one long command across\nlines):\n\n\\begin{Verbatim}[fontsize=\\small]\nPYTHONPATH=src python3 -m plectis \\\n  batch8-audio-level-rms-port run \\\n  --input fixtures/first_wave/batch8_audio_level_rms_port/input \\\n  --out /tmp/plectis-audio-rms \\\n  --acceptance-out /tmp/plectis-audio-rms-acceptance.json\n\\end{Verbatim}\n\nAt commit \\code{\\snapshotshort}, the run produced:\n\n\\begin{center}\n\\small\n\\begin{tabular}{>{\\raggedright\\arraybackslash}p{0.21\\linewidth}>{\\raggedright\\arraybackslash}p{0.26\\linewidth}>{\\raggedleft\\arraybackslash}p{0.15\\linewidth}>{\\raggedleft\\arraybackslash}p{0.15\\linewidth}>{\\raggedright\\arraybackslash}p{0.09\\linewidth}}\n\\toprule\n\\textbf{Case} & \\textbf{Input} & \\textbf{Expected} & \\textbf{Observed} & \\textbf{Result} \\\\\n\\midrule\nDecimal samples & 0.0,\\ 0.05,\\ $-0.05$,\\ 0.1 & 0.489897949 & 0.489897949 & pass \\\\\nSixteen-bit integer samples & 0,\\ 1638,\\ $-1638$,\\ 3277 & 0.489892970 & 0.489892970 & pass \\\\\nValues above the cap & 1.0,\\ $-1.0$,\\ 0.5 & 1.000000000 & 1.000000000 & pass \\\\\nEmpty input & (none) & 0 & 0 & pass \\\\\nUnknown format & \\code{pcm24} & refusal & refusal & pass \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\nHere \\emph{pass} means ``refused as expected''; \\code{pcm24} was not accepted.\nBecause the inputs and the rule are both printed above, the first row can be\nchecked without the repository or a computer: the squares of the four\nsamples average to 0.00375, whose square root is $0.061237\\ldots$, and eight\ntimes that is $0.489897\\ldots$. A reader with a calculator can recompute this\nexpected value from the stated formula alone. For this one case the reader\ncan supply the test oracle: elementary arithmetic gives an expected number\nderived independently of the program's output, but not of the project's\nformula.\n\nAlmost nothing else in the collection is like that, which is why the example\nmatters. Elsewhere the project usually supplies the implementation, the\ninput, the expected values, and the validator through the same\nauthor-controlled process. A test\nassembled that way can pass for four different reasons: the implementation\nand the expectation are both correct; the two share a mistake; the chosen\ncases happen to sit where the implementation behaves; or the validator\nchecks a weaker property than the prose claim suggests. A matching result\ntherefore establishes repeatability under that public test. Independent\nevidence bearing on correctness needs an expected result justified apart from\nthe implementation. Arithmetic supplies that for the first row only after\naccepting the project's formula. Fresh inputs alone do not supply one.\n\nThe example suggests a question for any software test: where did the expected\nvalue come from, and who supplied it? The answer changes what a match can\nsupport:\n\n\\begin{center}\n\\small\n\\begin{tabular}{>{\\raggedright\\arraybackslash}p{0.44\\linewidth}>{\\raggedright\\arraybackslash}p{0.47\\linewidth}}\n\\toprule\n\\textbf{Who or what supplies the expected answer} & \\textbf{What a matching result then supports} \\\\\n\\midrule\nThe project itself: implementation, cases, expectations, and rule from one author-controlled process. & Repeatability under the project's own test; the four readings above all stay open. \\\\\n\\addlinespace\nThe reader derives a result from stated premises, as in the decimal-samples row. & Agreement with the stated formula on that case, not proof that the formula is right. \\\\\n\\addlinespace\nAn established external tool or reference. & Whatever the reference itself warrants, on the cases run, within the reference's own limits. \\\\\n\\addlinespace\nAn evaluator who chooses inputs and independently justifies expected results after the version is saved. & Independent evidence bearing on correctness for those cases, within the oracle's own justification and limits. \\\\\n\\addlinespace\nThe world: an outcome observed apart from any program. & Evidence about a real-world outcome, where one exists. \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\nMost components in the collection sit in the first position, except where\na reader promotes a case to the second by recomputing it, as the worked\nexample's decimal-samples row invites. Components that invoke a named\nexternal tool occupy a hybrid of the first and third positions: the project\nstill chooses the case and the rule, while the tool supplies a result that\ninherits its own limits. The evaluations in Section~\\ref{sec:stronger} describe\nways to move consequential choices out of the project's control.\n\n\\component{batch8_audio_level_rms_port_validation_receipt.json} is the stored\nreference for the table. It sits under\n\\component{receipts/first_wave/batch8_audio_level_rms_port/}, so a fresh run is\ncompared against a fixed record rather than against memory. A stored record by\nitself does not prove that the stated command produced it; its use is that\nanyone can generate a fresh one and compare the two.\n\nThe limit is part of the component. This run checks one public Python\ncalculation over synthetic sample arrays (arrays composed for the test, not\ncaptured from a device). It does not run the original application, start an\naudio session, request microphone permission, or capture sound, and it\ncannot establish that the named private file is where the calculation came\nfrom. Nor does the table show that the task was difficult (it was not),\nthat the component is useful elsewhere, or that the remaining entries\nresemble it. What it shows is that one assertion now has a form in which a\nstranger can rerun it, dispute its rule, perturb its input, or replace my\nexpected value with a better-derived one.\n\nThe calculator check derives one expected number without the implementation.\nIt shows agreement with the project-supplied formula on that case, not that\nthe formula is the right audio rule. It does not establish origin, reach, or\nrisks beyond the declared scans. Its correctness evidence is limited to that\ncalculation. The next section treats these five gaps separately.\n\n\\section{Five distinctions}\n\\label{sec:distinctions}\n\nFive labels organise the gaps in Figure~\\ref{fig:gaps}: \\emph{origin}, where\nthe material came from; \\emph{correctness}, whether an answer is right;\n\\emph{meaning}, whether a rule tests its claim; \\emph{reach}, whether shown\ncases stand for unshown ones; and \\emph{risk}, what the declared checks may\nmiss. These short labels correspond to familiar questions about provenance,\ntest-oracle validity, claim--test fidelity, representativeness, and residual\nrisk. They are not five new kinds of evidence. Each subsection separates what\nthe public run directly shows from the additional premise needed for a stronger\nconclusion.\n\n\\subsection*{Public execution versus where the code came from}\n\nA reader can establish what a named public commit contains and what its\ncommands do. No public run establishes where the material came from.\nPlectis records that boundary. Each copied file has a claimed source and a\nSHA-256 message digest, used here as a fingerprint: a 256-bit value calculated\nfrom its contents and used to detect change\n\\cite[abstract, p.~iii; \\S1 and Fig.~1, p.~3; \\S6.2, p.~21]{nistfips1804}.\nTwo different files can in principle have the same digest: FIPS 180-4 treats\nfinding such a pair as computationally infeasible, not mathematically\nimpossible \\cite[p.~iv]{nistfips1804}. For this comparison, the relevant question is\nwhether one can start with a specified file and find a different file with the\nsame fingerprint. The National Institute of Standards and Technology (NIST)\ncalls this second-preimage resistance and expects an approved hash function to\nmake such a search computationally infeasible. That differs from collision\nresistance, which asks only for any pair of different inputs that match\n\\cite[security-strength table]{nisthashfunctions}. A match therefore supports a\nnarrow claim: the public file's contents agree with the version represented in\nits public record. But I control both sides, so that agreement is evidence of\ninternal consistency, not of origin. The same holds for the\naudio component's named source file: the name is an entry I wrote. Origin\nclaims could only be strengthened by something a public repository cannot\nsupply about itself, such as commitments published before the fact, or a\ntrusted person permitted to examine both sides. The term \\emph{provenance}\nmeans an object's origin and history. Here, that origin remains reported\nrather than publicly established. FIPS 180-4 itself similarly cautions that\nconformance to the standard does not assure that a particular implementation\nis secure \\cite[qualification 12, p.~v]{nistfips1804}.\n\n\\subsection*{Repeatability versus correctness}\n\nA run that yields the same checked values and verdict as the stored receipt\nshows that the public code, on the supplied input, in a sufficiently similar\nenvironment, produces the same result again. Correctness is a different\nproperty: it needs a standard for\nwhat the result should have been, and a standard is only worth something if\nit does not merely restate the implementation. The worked example makes the\ndifference concrete. The first row's expected number comes from arithmetic\napplied to the project's formula, not from the implementation. That supports\nagreement with the formula on one case; it does not validate the formula.\nMost components have only the project's own expected values and validator.\nPassing tests anywhere in Plectis should be read at that strength and no more.\n\n\\subsection*{A validator's rule versus the stated claim}\n\nA pass is a verdict from the validator, and a validator checks exactly the\nproperty written into it, which can be narrower than the claim as worded. A\ncomponent whose prose speaks of handling a class of inputs may in fact\nverify that one output file exists and matches a stored shape; a rule can\nhold while the sentence it stands under goes untested. The two previous\ndistinctions do not cover this gap: a result can be repeatable, and correct\nas far as the checked property goes, while the checked property is beside\nthe stated point. Plectis makes the gap inspectable rather than closed.\nEvery validator is public, so a reader can put the rule beside the prose\nand object where the rule is weaker, and the review in\nSection~\\ref{sec:conclusion} makes exactly that comparison its central step.\nSince the same author-controlled project produced both the claims and the\nrules, agreement between them records one judgement twice, and no count of\npassing components can substitute for reading one rule against one sentence.\n\n\\subsection*{Selected cases versus general behaviour}\n\nSelection acts at two levels. Within a component, the fixture covers only a\nfew possible inputs. Across the collection, the published components are my\nselection from a private whole.\n\n\\begin{samepage}\nI chose which mechanisms\ncould be given public form, which inputs to include, which expected values to\nstore, and which pass rules to enforce. Nothing in the public record shows\nhow many candidates there were, what was excluded beyond the privacy rules,\nor whether the collection resembles ordinary work in the private system. A\ncollection can consist entirely of genuine items and still give an\nunrepresentative picture of what it was drawn from. This is analogous to\nselective publication and the file-drawer problem, not the same process\n\\cite[p.~638]{rosenthal1979}.\nFigure~\\ref{fig:boundary} draws the situation: public checking reaches\neverything to the right of the boundary, and the pool the selection was\ndrawn from stays on the left.\n\\end{samepage}\n\n\\begin{figure}[H]\n\\centering\n\\makebox[\\linewidth][c]{\\begin{tikzpicture}[x=1cm,y=1cm]\n% private region\n\\draw[black!55, dashed, line width=0.7pt, rounded corners=3pt] (0,-0.1) rectangle (5.8,3.7);\n\\node[font=\\footnotesize, align=center] at (2.9,3.3) {The private system\\\\[-1pt]{\\scriptsize (not published)}};\n% mechanism squares: 5 x 3 grid, four of them selected (darker)\n\\foreach \\x in {0.75,1.80,2.85,3.90,4.95}\n  \\foreach \\y in {0.55,1.35,2.15}\n    \\draw[black!35, fill=gray!25] (\\x-0.14,\\y-0.14) rectangle (\\x+0.14,\\y+0.14);\n\\foreach \\sx/\\sy in {1.80/0.55,2.85/1.35,0.75/2.15,4.95/1.35}\n  \\draw[authoramber!80!black, fill=authoramber!55] (\\sx-0.14,\\sy-0.14) rectangle (\\sx+0.14,\\sy+0.14);\n\\node[font=\\small, text=black!45] at (5.42,1.35) {$\\cdots$};\n\\node[font=\\small, text=black!45] at (2.85,0.22) {$\\cdots$};\n% boundary\n\\draw[black!70, dashed, line width=0.8pt] (7.15,-1.95) -- (7.15,4.15);\n\\node[font=\\scriptsize, text=black!70, above] at (7.15,4.2) {the publication boundary};\n% selection arrow; the label plaque interrupts the boundary line symmetrically\n\\draw[flow, draw=authoramber] (5.8,1.8) -- (8.55,1.8);\n\\node[flabel, fill=white, align=center, inner sep=2.5pt, text=authoramber] at (7.15,2.44) {I chose and\\\\adapted these};\n% public box\n\\draw[black!75, line width=0.7pt, rounded corners=3pt, fill=gray!6] (8.55,0.55) rectangle (13.35,3.05);\n\\node[font=\\footnotesize, align=center] at (10.95,1.8) {The public repository\\\\at commit \\code{\\snapshotshort}:\\\\\\componentcount{} components in \\familycount{} groups};\n% stranger\n\\node[act, text width=2.0cm] (str) at (14.75,1.8) {a stranger:\\\\clone it, run it,\\\\compare, dispute};\n\\draw[actflow] (str.west) -- (13.35,1.8);\n% notes\n\\node[gnote, text width=5.6cm, anchor=north] at (2.9,-0.40) {outside public evidence: how many candidates there were, what was excluded and why, whether these resemble ordinary private work};\n\\node[gnote, text width=4.6cm, anchor=north] at (10.95,-0.40) {inside public evidence: files, commands, records, and tests of the published fragments};\n\\end{tikzpicture}}\n\\caption{The selection problem. Each square stands for a private\nmechanism; the darker ones were given public form. Each public component\nhas the parts shown in Figure~\\ref{fig:contract}. The grid trails off\nbecause its true extent is exactly what the public side cannot know. The\ncount of squares on the left, the\nrule that picked the darker ones, and the resemblance between the two sides\nare all invisible from the right. The collection supplies directly\ncheckable evidence about the published files and their behaviour under\nnamed tests, but no basis by itself for generalising to the hidden whole.}\n\\label{fig:boundary}\n\\end{figure}\n\nThe registry's count should be read in that light. A registry of\n\\componentcount{} entries defines exactly what is published. A later\nevaluation would need that list to draw a sample and report failure rates\nagainst a fixed total. The count is an inventory, not a score, and\neven the unit of counting is a choice I control, since one mechanism could\nhave been registered as several components, or several as one.\n\n\\subsection*{Risk reduction versus guarantee}\n\nThe publication process runs checks whose honest description is that they\nreduce specific risks. Automated scans search copied material for specified\npatterns of credentials, personal information, and private file paths, and\nrecord what was omitted or rejected; a scan establishes that the scanner\ndid not find the patterns it was designed to find, and nothing stronger.\nPages such as the component map are generated from the registry, and tests\ncompare page and registry, which prevents the particular failure of prose\ndrifting away from records; a generator and its records can still be wrong\ntogether. The documented \\code{tour} run takes the public checkout as its\nexplicit input and does not automatically discover a neighbouring private\nrepository. The release-export contamination check is an explicit\nexception: when a copy of the private project folder is available, it may\ncompare the contents of public source files with private files to detect\nleakage. None of this\namounts to a guarantee that no secret, no identifying detail, and no\ninconsistency survives. Publication remains a judgement about acceptable\nrisk, and I made it.\n\nOrigin evidence and privacy pull in opposite directions. Establishing\nprovenance would require someone to examine private\nmaterial, and the privacy constraint that motivates the whole design forbids\nexactly that examination in public. Some claims are therefore permanently in\nthe reported category, short of a confidential review. The honest treatment\nis to label them, not to dress them up as public proof through bookkeeping\nthat one maintainer controls.\n\n\\medskip\n\nThe same mistake lies behind all five: treating an observation as if it proved\nsomething never tested. A broader conclusion needs an added assumption. For\norigin, one must assume\nthat the public object came from the private one. For correctness, one must\nknow that the expected answer was right; for meaning, that the rule tested\nwhat the claim said. For reach, one must assume that the chosen cases stand\nfor the unchosen; for risk, that the scan covered the relevant risks. The\nobservation itself supplies none of these assumptions. The contract makes\neach one explicit. Once named, a gap can be disputed, assigned a cost to\naddress, or deliberately left open. An unnamed gap is easy to cross without\nnoticing. These five are not exhaustive. Two more gaps appeared earlier:\nobserving a refusal today does not show when it was drawn\n(Section~\\ref{sec:contract}), and observing a number at one commit does not\ndescribe the next (the appendix). These are simply the five gaps this\ncollection most needs a reader to hold.\n\n\\begin{figure}[t]\n\\centering\n\\makebox[\\linewidth][c]{\\begin{tikzpicture}[x=1cm,y=1cm]\n\\node[part, text width=3.2cm] (obs) at (1.8,0) {Observed: at one commit, in one environment, a fresh run matches the stored record};\n\\draw[black!55, dashed, line width=0.7pt] (4.55,-3.75) -- (4.55,3.75);\n\\node[font=\\scriptsize, text=black!60, rotate=90] at (4.3,2.55) {edge of the observation};\n\\node[ghost, text width=4.6cm] (g1) at (13.1,2.9)  {\\ldots so it came from the private system as described};\n\\node[ghost, text width=4.6cm] (g2) at (13.1,1.45) {\\ldots so the result is what it should have been};\n\\node[ghost, text width=4.6cm] (g3) at (13.1,0)    {\\ldots so the test supports the claim as written};\n\\node[ghost, text width=4.6cm] (g4) at (13.1,-1.45){\\ldots so it behaves like this on inputs not shown};\n\\node[ghost, text width=4.6cm] (g5) at (13.1,-2.9) {\\ldots so no risk of that kind remains};\n% fanned starts so the five tails do not pile on one point; all lines are\n% drawn first, then all label plaques, so no later line strikes a label\n\\draw[softflow] ([yshift=22pt]obs.east) -- (g1.west);\n\\draw[softflow] ([yshift=11pt]obs.east) -- (g2.west);\n\\draw[softflow] (obs.east) -- (g3.west);\n\\draw[softflow] ([yshift=-11pt]obs.east) -- (g4.west);\n\\draw[softflow] ([yshift=-22pt]obs.east) -- (g5.west);\n\\path ([yshift=22pt]obs.east) -- (g1.west) node[gaplabel, pos=0.50]{origin: evidence about where it came from};\n\\path ([yshift=11pt]obs.east) -- (g2.west) node[gaplabel, pos=0.60]{correctness: an answer derived separately};\n\\path (obs.east) -- (g3.west) node[gaplabel, pos=0.50]{meaning: a rule that tests the words};\n\\path ([yshift=-11pt]obs.east) -- (g4.west) node[gaplabel, pos=0.60]{reach: cases I did not choose};\n\\path ([yshift=-22pt]obs.east) -- (g5.west) node[gaplabel, pos=0.50]{risk: checks beyond\\\\the listed patterns};\n\\end{tikzpicture}}\n\\caption{Five conclusions a matching run invites and does not license.\nEach dashed arrow needs a further assumption that the observation itself\ncannot supply. From top to bottom, the tags follow the order discussed above:\norigin (provenance), correctness, meaning, reach, and risk. The contract's\ncontribution is not to cross these gaps but to make them nameable, one\ncomponent at a time.}\n\\label{fig:gaps}\n\\end{figure}\n\n\\section{What the collection contains}\n\\label{sec:collection}\n\nThis section is a dated inventory, not a result. The categories are useful for\nfinding a test and for seeing which kind of limitation accompanies it. Their\nnames and counts do not measure importance, correctness, independence, or\ncoverage of the private system.\n\nAt commit \\code{\\snapshotshort} the \\componentcount{} components fall into\n\\familycount{} groups:\n\n\\begin{center}\n\\small\n\\begin{tabular}{>{\\raggedright\\arraybackslash}p{0.38\\linewidth}r>{\\raggedright\\arraybackslash}p{0.47\\linewidth}}\n\\toprule\n\\textbf{Group} & \\textbf{Count} & \\textbf{What its components do} \\\\\n\\midrule\nGetting started & \\entrycount & Show a new reader where to begin and guide a first inspection. \\\\\nProject mapping & \\mappingcount & Map files and written rules, find relevant material, and route a question to the part of the project that owns it. \\\\\nComputer-checked mathematics & \\mathcount & Check records and fixed examples from formal mathematics, including small Lean statements and one exact finite calculation. \\\\\nAI reliability and safety tests & \\safetycount & Replay fixed cases involving hostile instructions, software boundaries, stored information, tool use, and benchmark rules. \\\\\nResearch-method tests & \\researchcount & Exercise fixed cases involving replication, conflicting forecasts, explaining model behaviour, simulation, and laboratory safety. \\\\\nPublic-copy maintenance & \\publiccopycount & Check copied source, generated pages, publication boundaries, and disagreement between records and derived files. \\\\\nWork recovery & \\recoverycount & Record unfinished changes and test how work resumes after interruption. \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\nThe groups do not share a subject. Their only unity is the contract of\nSection~\\ref{sec:contract}: each item became a small public test. The collection\nis organised by how claims are checked, not by a common scientific question.\n\nThe registry also classifies what kind of evidence each component supplies.\nIt calls this classification a \\emph{route}. A route says whether the\npublic component preserves source text, re-creates behaviour on fixed cases,\nrecords a run, or checks a declared rule. The four routes below cover all\n\\componentcount{} entries. Each leaves a different question open:\n\n\\begin{center}\n\\small\n\\begin{tabular}{>{\\raggedright\\arraybackslash}p{0.20\\linewidth}r>{\\raggedright\\arraybackslash}p{0.30\\linewidth}>{\\raggedright\\arraybackslash}p{0.32\\linewidth}}\n\\toprule\n\\textbf{Route} & \\textbf{Count} & \\textbf{What it preserves} & \\textbf{What it cannot show} \\\\\n\\midrule\nRecord-checked copied source & \\copiedsourcecount & The exact text of non-secret source files, checked against a public import record. & That the recorded origin is real; record and file share one maintainer. \\\\\nPublic reimplementation & \\refactorcount & One calculation's behaviour on fixed cases. & That it came from the private system; the private text is absent. \\\\\nDirect or external-tool run & \\runtimecount & A local computation, or a named external tool's result, captured with its context. & Behaviour where the tool or environment differs; the result is as strong as the tool and the case. \\\\\nContract checker & \\validatorcount & Whether a public input satisfies a declared structural rule. & Whether the declared rule is the right rule to require. \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\nCopied source preserves text but demonstrates little behaviour; a\nreimplementation does the converse; an external-tool run inherits the tool's\nlimits; and a contract checker is worth exactly the rule its author chose.\nTo inspect one item, use the public component map,\n\\component{ORGANS.md} (an inherited filename). Follow it rather than guessing,\nbecause some components share a page. The audio row points to\n\\component{paper_modules/batch8_audio_level_rms_port.md}, which in turn names\nthe fixture, receipts, route, and limit.\n\nTwo cautions govern the tables. First, components are not independent\nwitnesses: they share code, fixture style, authorship, and judgement.\n\\componentcount{} entries are registered claims, not \\componentcount{}\nconfirmations. Pages, entries, and receipts generated from the same underlying\nrecords repeat one witness rather than supply several. Machine checks can catch\ntheir accidental disagreement. They cannot create an independent witness.\nThe four routes impose four different limits on what may be concluded; they\nare not four independent methods confirming one claim.\n\nSecond, names such as computer-checked mathematics and AI reliability and\nsafety tests borrow gravity from serious fields. Read each row as the weakest\nproperty its tests actually check---often that a fixed case replays or a record\nhas a stated shape. The names help find components; they do not weigh them.\n\n\\section{The author's hand}\n\\label{sec:hand}\n\nAll five distinctions return to control. At the commit this paper describes,\nI hold every consequential choice on both sides of every test.\nI chose which private mechanisms were candidates, which were selected, and\nhow finely each was divided into components, and so what the count counts.\nI chose which of the four routes each took, which inputs went into the fixtures, what\nthe expected values are, what each validator checks, and where each\nrefusal boundary sits. And I chose which group each entry is filed under,\nwhich failures this paper explains and in what tone, which single\ncomponent the worked example honours, and\nwhere the publication boundary lies. Somebody had to make each of these\nchoices, and the privacy constraint meant it was me. None of that is\nmisconduct.\n\nBut that concentration of choice permits a failure mode, and it involves\nno false statement anywhere. Select the mechanisms that show well. Divide\nthe favourable ones finely. Narrow each claim until its fixed cases satisfy\nit. Give awkward material the least demanding check. Declare\nas refusals whatever would otherwise fail. File\nsmall replays under weighty names. Then let the counts, the categories, and\nthe machinery of records accumulate into an impression of scale and\nmaturity that no individual sentence asserts. A document can be cautious\nsentence by sentence while its arrangement still overstates, because a\nreader's impression forms on what is counted, named, worked through, and\nleft out, as much as on what is claimed. This paper is inside its own\narrangement, so declaring the risk does not discharge it; nothing in the\ntext can establish that the text is not doing what this paragraph\ndescribes.\n\nMaking disagreement local can also displace the larger question. A reader\ncan settle an argument about one input or one rule while leaving untouched\nwhether the selection is fair, whether the whole is sound, or what any of it\nis for. An objection about selection is not answered by the successful\nexecution of any number of selected items. The registry hosts the small\narguments; it does not settle the large ones.\n\nTwo corrections need no cooperation from me. First, read any component at the\nweakest property its validator checks, as Section~\\ref{sec:collection}\nproposes. Second, treat every count as an inventory, and every agreement among\nmy records as one witness. A third point concerns what no reader can inspect:\na private mechanism is easiest to publish when it can be separated into a\nsmall task, gives the same output from the same input, and runs in isolation.\nA collection like this\ntherefore over-represents pure calculation, replay, and record-checking, and\nunder-represents stateful, entangled, long-running behaviour. The\npublished picture tilts towards what publishes well before any question of\nhonesty arises.\n\nThe National Aeronautics and Space Administration (NASA) defines independence\nfor independent verification and validation (IV\\&V), a formal software check,\nthrough three separations: technical (the evaluator did not develop the system);\nmanagerial (a separate organisation chooses scope, methods, and schedule); and\nfinancial (an independent group controls the budget without adverse financial\npressure) \\cite[\\S4.4.1.2, p.~48]{nasa8739}.\nThese are criteria for NASA IV\\&V, not a universal test. Plectis has undergone\nno IV\\&V. No other outside evaluation report is cited or included, and I know\nof none; an unreported private rerun would remain invisible. I use the NASA\ncriteria only for the paper's five gaps: evidence about a choice depends less\non me once the choice is no longer mine. Section~\\ref{sec:stronger} lists\nevaluations that address those dependencies.\n\n\\section{What stronger evidence would look like}\n\\label{sec:stronger}\n\nA passing run shows that named public code produced a stated result from a\nsupplied input in a stated environment. It does not establish private\nprovenance, representativeness, correctness beyond self-supplied criteria,\nusefulness, security, originality, or the private system's reliability. The\ncomputer-checked mathematics label covers different checks: proof-work records,\nsmall public statements checked by Lean, and one exact finite calculation. The\nlabel itself supplies no theorem; each entry supports only its own public claim\nand stated limit. At the named commit, the repository's evidence still came\nfrom the project. It includes no outside evaluator's report; I know of none,\nalthough a private rerun could be unreported.\n\nFive forms of independent evaluation could add evidence that the public\nrepository cannot supply by itself. These are not the four publication routes\nin Section~\\ref{sec:collection}. A publication route classifies evidence already\nin the registry; the evaluations below describe new work by someone outside the\nproject. The order is only for exposition. It ranks nothing, and no evaluation\nis a prerequisite for another. Figure~\\ref{fig:ladder} shows which choice each\nevaluation transfers and which claim it still leaves open. The common question\nis who makes the choice that matters for that claim.\n\n\\begin{figure}[H]\n\\centering\n\\makebox[\\linewidth][c]{\\begin{tikzpicture}[x=1cm,y=1cm]\n% uniform box heights and bottom/top-anchored labels keep every dotted\n% leader the same length\n% a whisper of stranger-blue on the evaluations: each is evidence that lives\n% outside the author's hands\n\\node[part, text width=2.15cm, minimum height=1.52cm, fill=strangerblue!8] (s1) at (1.2,0) {Another person repeats the run};\n\\node[part, text width=2.15cm, minimum height=1.52cm, fill=strangerblue!8] (s2) at (4.3,0) {Evaluator chooses the cases};\n\\node[part, text width=2.15cm, minimum height=1.52cm, fill=strangerblue!8] (s3) at (7.4,0) {New inputs, answers worked out separately};\n\\node[part, text width=2.15cm, minimum height=1.52cm, fill=strangerblue!8] (s4) at (10.5,0) {Outside task, ordinary alternative};\n\\node[part, text width=2.15cm, minimum height=1.52cm, fill=strangerblue!8] (s5) at (13.6,0) {Confidential private-source review};\n\\node[act, text width=2.25cm, anchor=south] (b1) at (1.2,1.12) {who runs it};\n\\node[act, text width=2.25cm, anchor=south] (b2) at (4.3,1.12) {which cases count};\n\\node[act, text width=2.25cm, anchor=south] (b3) at (7.4,1.12) {what the answer should have been};\n\\node[act, text width=2.25cm, anchor=south] (b4) at (10.5,1.12){what counts as useful};\n\\node[act, text width=2.25cm, anchor=south] (b5) at (13.6,1.12){whether the origin account holds};\n\\node[gnote, text width=2.25cm, anchor=north] (n1) at (1.2,-1.12){does not by itself establish correctness};\n\\node[gnote, text width=2.25cm, anchor=north] (n2) at (4.3,-1.12){cannot reach the private pool};\n\\node[gnote, text width=2.25cm, anchor=north] (n3) at (7.4,-1.12){tested cases only, not generality};\n\\node[gnote, text width=2.25cm, anchor=north] (n4) at (10.5,-1.12){advantage only where compared};\n\\node[gnote, text width=2.25cm, anchor=north] (n5) at (13.6,-1.12){moves trust to the reviewer};\n\\foreach \\i in {1,...,5}{\n  \\draw[RoyalBlue!45, dotted, line width=0.6pt] (b\\i.south) -- (s\\i.north);\n  \\draw[black!30, dotted, line width=0.6pt] (s\\i.south) -- (n\\i.north);}\n\\end{tikzpicture}}\n\\caption{Five forms of independent evaluation. Above each evaluation, in italic\nblue, the choice that leaves my hands; below it, in grey, what that evaluation\nstill cannot show. The evaluations are not cumulative stages. At the commit\nthis paper describes, the repository records none, and I know of none.}\n\\label{fig:ladder}\n\\end{figure}\n\n\\emph{Independent repetition} means a person with no involvement in the\nproject reruns the supplied commands in their own environment and publishes\nwhat happened, including failures. That would reduce dependence on my\nmachine and test whether the published instructions suffice elsewhere. It\nwould not by itself establish correctness.\n\nYuan et al. report a related end-to-end exercise in which two student teams\nworked from public documentation for scientific hardware designed more than a\ndecade earlier~\\cite{yuan2023}. Their case concerns hardware documentation in\nlarge collaborations, not software-claim contracts, and supplies no evidence\nabout Plectis. It is a precedent for treating an outsider's attempt to complete\nthe documented task as evidence about accessibility, including the failures\nand ambiguities that the project's authors did not anticipate.\n\n\\emph{Evaluator-controlled selection} means the evaluator chooses\ncomponents from the full registry, without substitution when one proves\nawkward, and original failures stay on the record even if I later repair\nthem. The units themselves are open to the same challenge: an evaluator\nwho finds that two entries are one dependent mechanism, or that a\nboundary excludes the difficult part of a task, is disputing the registry,\nnot misusing it. That addresses curation within the public collection. It\ncannot address whether the collection represents the private system,\nbecause the private pool stays invisible.\n\n\\emph{Fresh inputs with independently derived expectations} can move\nrepeatability towards evidence bearing on correctness: inputs chosen after\nthe version is saved, and expected results worked out by a method that\ndoes not pass through my code, whether independent calculation, a second\nimplementation, or an external reference. The result would support claims\nabout those cases only to the extent that the derivation is credible and\nindependent and the cases are adequate; general correctness still would not\nfollow.\n\n\\emph{Comparison on outside tasks} is what a claim of advantage over\nordinary alternatives would need: someone bringing a task that was not\ndesigned around the project's own categories and comparing the component\nagainst an ordinary alternative, which for several components would be a\nshort conventional script. Nothing in this paper makes that comparison.\n\n\\emph{A provenance review}, among the five evaluations shown here, bears most\ndirectly on the origin story: a trusted person examining selected private\nmaterial against the public collection under confidentiality. Prior public\ncommitments could provide another kind of provenance evidence. Even\ncompleted, this review would\ntransfer trust rather than remove it: public readers would be relying on\nthe reviewer's access, competence, and report instead of on mine, and the\nreview itself could not be published. The privacy cost may reasonably\nnever be paid. If it is not, the origin story remains testimony, and this\npaper is written so that its argument does not depend on it.\n\n\\subsection*{A falsifiable comparison}\n\nThe next study should compare two presentations of the same frozen public\nitems.  The baseline would provide ordinary documentation, code, outputs, and\ntests.  The comparison condition would additionally provide the explicit\nclaim, test case, expected result, decision rule, run record, and inference\nlimit used here.  Evaluators who did not design the items would choose from the\nfrozen public list, inspect deliberately mismatched claim--rule pairs, use\nfresh inputs where an expected result can be justified independently, and\npublish failures without author substitution.\n\nThe useful observations would be whether readers detect a mismatch, how often\nthey wrongly accuse a matching item, how long they take to locate the point of\ndisagreement, whether they reproduce the run, and how well their confidence\ntracks those outcomes.  That study would measure reviewer behaviour on the\npublic items.  It would still not measure the private system, the unseen\nselection pool, or correctness beyond the independently justified cases.  No\nsuch comparison is reported in this paper.\n\\subsection*{From an open question to a falsifiable handoff}\n\\label{sec:hypothesis-handoff}\n\nThe five evaluations still leave an outsider to infer what the project\nexpects, reconstruct credible alternatives, and decide which observation\nwould matter.  ``Evaluate Plectis'' transfers that framing work; a confident\nprediction can make the project's own expectation look like evidence.\n\nThe current repository, beyond the historical snapshot examined above,\ntherefore includes a small, read-only\n\\emph{hypothesis-handoff} format.  A packet names one declared known gap and\nrefuses to treat it as a complete inventory of unknown questions.  It then\nnames the current wall, a tentative leading hypothesis, evidence for it,\nevidence against or still missing, serious alternatives, and discriminators\nwhose possible outcomes point to named hypotheses and have stated\ninterpretations.  It also names an exact expert return, decisive and route-only effects,\nrepository-relative landing targets with validators, and the order by which a\nreturn could enter an authority record and public release.  Run\n\\begin{center}\n\\begin{minipage}{0.94\\linewidth}\n\\begin{Verbatim}[fontsize=\\small]\nplectis hypothesis-handoff --input packet.json --format text\n\\end{Verbatim}\n\\end{minipage}\n\\end{center}\nvalidates that structure and renders it as a briefing.  It does not call a\nmodel, infer probabilities, judge an expert, or change a claim.  This later\ninterface is not evidence for the historical snapshot analysis.\n\nThe worked public packet asks whether evaluator-selected cases would expose a\ndifferent failure profile from the author-selected fixtures.  Its working\nhypothesis is that they would expose more failures and a broader taxonomy.  The\nreasons are visible: I selected the existing cases and rules, and a later cold\nroute exposed one real wiring defect.  The counterweight is equally visible:\nthere is no independent comparison corpus, and one defect estimates no rate.\nTwo alternatives remain live: the current fixtures may approximate the\nindependent profile, or environment and dependency failures may dominate the\ndifference.  A preregistered case-selection rule and a failure taxonomy fixed\nbefore outcomes are inspected would separate those accounts better than\nanother author-selected example.\n\nThis format does not make the leading hypothesis independent.  It is the\nproject's expected answer, in the same sense that Section~\\ref{sec:run}'s\nexpected output is project-supplied.  Its contribution is different: it makes\nthe prior, alternatives, and update rule inspectable before an expert answer is\nknown.  A reader can reject the leading hypothesis, add a missing alternative,\nor dispute a discriminator without first reverse-engineering the project's\nprivate reasoning.  If an answer arrives, the packet also prevents the answer\nfrom becoming a public conclusion merely because it came from an expert.  The\nreturn remains advisory until its evidence is reproduced or independently\nchecked, its intended meaning is reviewed, the authoritative record is\nupdated, and the declared release checks pass.\n\nThe useful claim is therefore narrow.  A hypothesis handoff can reduce the\ncost of informed disagreement and preserve what would change the project's\nmind.  It cannot establish that the option set is complete, that the leading\nhypothesis is well calibrated, that the requested evidence is obtainable, or\nthat any expert will agree.  Those are precisely the kinds of finding the\nhandoff is meant to make easier to return.\n\n\\subsection*{How the record should age}\n\nThe evaluations above may expose failures. Repairs should add facts, not erase\nthem. The appendix keeps three failures at the pinned commit although a later\ncommit repairs one; that repair does not subtract the first fact. If criticism\nshows a validator too weak or a claim too broad, record the revision's date and\ncause instead of making the project appear always to have meant the narrower\nclaim. A test rewritten after its outcome is known is a different test and may\nbe better. The known case can show whether the same failure returns; fresh\nconfirmation requires new cases whose answers were not used to design the\nrepair. Disappointment is in scope: if an evaluation finds a component trivial\nor no better than a short ordinary script, that conclusion belongs on the\nrecord with the same standing as a pass. Any outside evaluator's report should\nremain under that evaluator's control, cited rather than absorbed. If the\nrepository later falls short of this paragraph, the shortfall is itself a\nfinding, and Section~\\ref{sec:conclusion} says where findings go.\n\nThese evaluations do not pair one-for-one with the distinctions. They address\nthe runner, public-case selection, tested-case correctness, usefulness, and origin.\nNone repairs a validator--claim mismatch, proves representativeness, or turns\nrisk reduction into a guarantee.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nPlectis makes selected public claims testable; it does not verify the private\nsystem as a whole. Its fixed cases, procedures, outputs, rules, and stated\nlimits let a stranger repeat a result and identify exactly what they dispute.\nA pass is no stronger than the test oracle---the rule and expected answer---or\nthan the basis for believing that oracle.\n\nThe worked audio calculation demonstrates the hierarchy. Repeating the stored\nresult supports computational reproducibility for that public case. Deriving\none expected number by arithmetic adds evidence that the implementation agrees\nwith the stated formula on that case. Neither step establishes that the formula\nis the right audio rule, that the code came from the private system, or that the\nselected case represents it.\n\nMore author-controlled components enlarge the inventory without changing who\nchose the cases or rules. Moving a relevant choice out of the author's hands\ncan strengthen evidence about that choice; it does not repair every gap.\n\n\\appendix\n\\small\n\\setlength{\\parskip}{0pt}\n\n\\section{Dated reproduction record}\n\\label{sec:appendix}\n\nExcept where a paragraph gives a later date, this appendix records one run on\n17~July~2026 under macOS 15.6.1 on Apple silicon (arm64) and Python 3.13.7.\nThe repository version was commit \\code{\\snapshotcommit}, abbreviated\n\\code{\\snapshotshort} elsewhere. Later commits or machines may differ; the body\ndoes not depend on them.\n\n\\paragraph{Later public-entry check.} A maintainer-run \\code{make ci} in\nSeptember 2026, after commit \\code{c69f9afb32} and before\n\\code{09794d4cbb}, reported 795 passing tests and five skips, followed by\npassing public smoke and package-install checks. The package check installed\nthe source into a fresh environment, moved the source tree out of the import\npath, and ran the installed \\code{plectis} command against its packaged\nhypothesis-handoff example. It checked that the command accepted the packet\nwithout changing the input. The current first-action command, \\code{plectis\ncomprehend --first-action}, is also part of that installed check. This later\nrun was performed within the project, not by an independent evaluator; it does\nnot amend the pinned July results or verify the companion repository's\nmathematics. Changes after that run require fresh checks.\n\n\\paragraph{A first review.} Use the appendix's no-install block and select the\nversion analysed here. The tour describes the checkout but does not choose a\ncomponent claim. Ask the repository to find the audio example with\n\\begin{Verbatim}[fontsize=\\small]\nPYTHONPATH=src python3 -m plectis comprehend \\\n  --first-action \"audio level calculation\" --format text\n\\end{Verbatim}\nThe response identifies the audio component, but do not run its first suggested\ncommand: it writes to the repository's saved receipt paths. Run the paper's exact\nno-install command near the start of Section~\\ref{sec:run} instead; it writes only\nto \\code{/tmp} and leaves the checkout unchanged. Under\n\\component{/tmp/plectis-audio-rms/}, open\n\\component{batch8_audio_level_rms_port_result.json};\nJSON is a plain-text field-and-value format. Open \\code{exercise}. In each\n\\component{reference_cases} item, compare \\code{expected\\_level} with\n\\code{observed\\_level}. The top-level \\component{anti_claim} says what a pass\ndoes not establish; \\component{claim_ceiling} inside \\code{exercise} gives the\nproject's strongest conclusion. The project wrote both cautions; no independent\nreviewer checked their limits. To challenge rather than repeat the supplied case,\nmake a disposable copy of the fixture input:\n\\begin{Verbatim}[fontsize=\\scriptsize]\ncp -R fixtures/first_wave/batch8_audio_level_rms_port/input/. /tmp/plectis-audio-rms-input\n\\end{Verbatim}\nIn the copied \\component{batch8_audio_level_rms_port_probe_manifest.json},\nchange the first case's second sample from \\code{0.05} to \\code{0.06}, but\nleave \\code{expected\\_level} unchanged. Then run:\n\\begin{Verbatim}[fontsize=\\scriptsize]\nPYTHONPATH=src python3 -m plectis batch8-audio-level-rms-port run \\\n  --input /tmp/plectis-audio-rms-input --out /tmp/plectis-audio-rms-probe \\\n  --acceptance-out /tmp/plectis-audio-rms-check.json; echo $?\n\\end{Verbatim}\nThe final number printed should be \\code{1}; in the acceptance file,\n\\code{status} should be \\code{blocked} and \\code{accepted} should be\n\\code{false}. This means the validator caught the deliberate mismatch; the\nexercise did not crash. Report any different result in\n\\code{CONTRIBUTING.md}. Treat a pass as evidence only for the published rule.\n\n\\paragraph{Names.} Repository: \\repo. Formerly Microcosm, Plectis retains\npackage \\code{microcosm\\_core}, page \\code{ORGANS.md} (``organs'' means\ncomponents), and the \\code{microcosm} alias. This paper uses command\n\\code{plectis}.\n\n\\paragraph{Getting the code and running without installing.}\n\n\\begin{Verbatim}[fontsize=\\small]\ngit clone https://github.com/wcook04/practical-formal-verification\ncd plectis\ngit checkout 57ffb7f5830c\nPYTHONPATH=src python3 -m plectis tour --format text .\n\\end{Verbatim}\n\nThe final full stop means ``the current folder''. \\code{git checkout} pins the\nexact state described here; omit it to run the newest version, whose dated\nnumbers may differ.\n\nThe clean-clone tour read 5{,}018 files and chose the README-first order\n(internally \\component{readme_onboarding_route}) from five reading orders. It\nwrote 21 generated files under \\code{.microcosm/} without changing the examined\nsource. Those files can be rebuilt; the README's different count is another\ndated run, not a stable project size.\n\n\\paragraph{Installing.}\n\n\\begin{Verbatim}[fontsize=\\small]\npython3 -m pip install .\nplectis tour --format text .\n\\end{Verbatim}\n\nNo runtime library is required; \\code{pip} may fetch \\code{setuptools} to\nbuild. No check proves the absence of network or model calls, so inspect the\nsource.\n\n\\paragraph{The worked example.} A fresh Section~\\ref{sec:run} rerun reproduced\nall displayed passes and values and overwrote \\component{/tmp/plectis-audio-rms}.\nA stale field reports a copied private module, while the saved import record\nreports none because public code replaced it; the discrepancy remains.\n\n\\paragraph{A second Python version.} On 18~July~2026, a clean macOS clone under\nPython 3.12.7 matched the tour and example. Its offline install used only the\nmachine's \\code{setuptools} 80.9.0 (minimum 77.0.3). Only versions 3.12.7 and\n3.13.7 were tested.\n\n\\paragraph{Project-wide checks.} The project-wide Lean trust scan checks\n\\leanfilecount{} files for six text patterns but does not run Lean or verify\nproofs: unfinished placeholders; custom axioms\n(unproved assumptions); compiled calculations whose acceptance\nrelies on more than Lean's small proof-checking core;\n\\code{partial} definitions (runnable but not unfoldable in proofs);\n\\code{unsafe} definitions (barred from theorems); and removed computation\nlimits. No pattern appeared. The scan does not show whether files compile,\nproofs are valid, or statements express their authors' intent\n\\cite{leanvalidation}.\nSeparate components run Lean on small test projects; their results apply only\nto those projects.\nRegistry and trust checks passed; the \\leanfilecount{} files are distinct from\nthe \\mathcount{} mathematics components.\nThe limited \\code{make test} run covered 32 public test files, not every test\nfile: 356 of 361 tests passed, two skipped, and three failed. The\nfailures were an \\code{AGENTS.md} line-break expectation; outdated line or byte\ncounts in copied-source manifests; and root-layout rules that did not yet allow\nthe \\code{paper} directory or root PDF; the last was later repaired without\naltering this pinned run. Exact failure names, environment, and counts are in\n\\component{paper/public-test-receipt.json}. \\code{make ci} would add smoke and\npackage-install checks, but was not green because \\code{make test} failed. These\nchecks cover only the public checkout; this is my run, not an independent\nrepetition.\nThe paper's own\n\\component{scripts/check_public_system_paper.py} fails if counts, pinned facts,\nthe example, cautions, or citations disagree. It checks historical facts against\nthe pinned commit separately from current prose; with one maintainer, that is\ninternal consistency, not external truth.\n\n\\section*{Statements and declarations}\n\n\\paragraph{Declaration of generative AI use.}\nEvery word of this manuscript was generated by agents based on large language\nmodels operating within Will Cook's private research system for artificial\nintelligence. The formal proofs and repository software were likewise drafted\nand revised by the agents through that system under Cook's direction. Cook set\nthe project's objectives and acceptance criteria and directed the agent work.\nThe public record does not establish that he personally reviewed every claim\nor approved this specific version. Generative systems are production tools,\nnot authors, and supply no independent authority.\n\n\\paragraph{Authority boundary.}\nThe account of private production is testimony about origin, not independent\nevidence about the system's correctness, representativeness, or reliability.\nThe public commands and records establish only the cases and limits stated in\nthis paper.\n\n\\begin{thebibliography}{13}\n\\fontsize{7.5pt}{8.3pt}\\selectfont\n\\raggedright\n\\setlength{\\itemsep}{0pt}\n\\setlength{\\parsep}{0pt}\n\\setlength{\\parskip}{0pt}\n\n\\bibitem{runeson2009}\nPer Runeson and Martin H\\\"ost. ``Guidelines for Conducting and Reporting Case\nStudy Research in Software Engineering.'' \\emph{Empirical Software\nEngineering}, 14(2):131--164, 2009.\n\\href{https://doi.org/10.1007/s10664-008-9102-8}{doi:10.1007/s10664-008-9102-8}.\n\n\\bibitem{nasem2019}\nNational Academies of Sciences, Engineering, and Medicine.\n\\emph{Reproducibility and Replicability in Science}. Washington, DC:\nThe National Academies Press, 2019.\n\\href{https://doi.org/10.17226/25303}{doi:10.17226/25303}.\n\n\\bibitem{barr2015}\nEarl T. Barr, Mark Harman, Phil McMinn, Muzammil Shahbaz, and Shin Yoo.\n``The Oracle Problem in Software Testing: A Survey.''\n\\emph{IEEE Transactions on Software Engineering}, 41(5):507--525, 2015.\n\\href{https://doi.org/10.1109/TSE.2014.2372785}{doi:10.1109/TSE.2014.2372785}.\n\n\\bibitem{rosenthal1979}\nRobert Rosenthal. ``The File Drawer Problem and Tolerance for Null Results.''\n\\emph{Psychological Bulletin}, 86(3):638--641, 1979.\n\\href{https://doi.org/10.1037/0033-2909.86.3.638}{doi:10.1037/0033-2909.86.3.638}.\n\n\\bibitem{omgsacm2023}\nObject Management Group.\n\\href{https://www.omg.org/spec/SACM/2.3/PDF}{\\emph{Structured Assurance Case Metamodel (SACM)}},\n2.3, formal/23-05-08, October 2023.\n\n\\bibitem{gsn}\nSCSC Assurance Case Working Group (ACWG).\n\\emph{Goal Structuring Notation Community Standard, Version 3}.\nSCSC-141C, May 2021.\n\\href{https://scsc.uk/scsc-141c}{SCSC-141C}.\n\n\\bibitem{nisobadging2021}\nNational Information Standards Organization.\n\\href{https://doi.org/10.3789/niso-rp-31-2021}{\\emph{Reproducibility Badging\nand Definitions}},\nNISO RP-31-2021, approved 15 Jan. 2021.\n\n\\bibitem{acmartifact}\nAssociation for Computing Machinery.\n\\href{https://www.acm.org/publications/policies/artifact-review-and-badging-current}{``Artifact Review and Badging---Current,''}\nv1.1, 24 Aug. 2020; accessed 18 July 2026.\n\n\\bibitem{nistfips1804}\nNational Institute of Standards and Technology. \\emph{Secure Hash Standard},\nFIPS PUB 180-4, 2015.\n\\href{https://doi.org/10.6028/NIST.FIPS.180-4}{NIST FIPS 180-4}.\n\n\\bibitem{nisthashfunctions}\nNational Institute of Standards and Technology.\n\\href{https://csrc.nist.gov/projects/hash-functions}{``Hash Functions.''}\nUpdated 9 September 2024. Accessed 18 July 2026.\n\n\\bibitem{nasa8739}\nNational Aeronautics and Space Administration.\n\\emph{Software Assurance and Software Safety Standard},\nNASA-STD-8739.8B, Section 4.4.1.2, p.~48, 2022.\n\\href{https://standards.nasa.gov/standard/NASA/NASA-STD-87398}{NASA-STD-8739.8B}.\n\n\\bibitem{yuan2023}\nMelinda Yuan et al. ``End-to-End Testing of Open-Source Hardware Documentation\nDeveloped in Large Collaborations.'' 2023.\n\\href{https://arxiv.org/abs/2309.05942}{arXiv:2309.05942}.\n\n\\bibitem{leanvalidation}\nLean Project. \\emph{Lean Language Reference} sections\n\\href{https://lean-lang.org/doc/reference/latest/ValidatingProofs/}{``Validating a Lean Proof''} and\n\\href{https://lean-lang.org/doc/reference/latest/Definitions/Recursive-Definitions/#partial-and-unsafe-definitions}{``Partial and Unsafe Definitions.''}\nAccessed 18 July 2026.\n\n\\end{thebibliography}\n\n\\end{document}\n","body_format":"latex","source_embedded":true,"byte_count":83596,"sha256":"sha256:b7193e6bece8bae9c183388ea97736cff7c67e45f3fffcb549dd5c551b596c92","cite_as":"Plectis reader corpus, paper:plectis-public-system","source_repository":"plectis","source_path":"paper/plectis-public-system.tex","manuscript_source_commit":"f2ed2582e6de24c71b45dfc30dae33863e2ab539","formal_source_ref":null,"dependencies":[{"record_type":"paper_source_dependency","tex_reference":"paper-house-style.sty","source_path":"paper/paper-house-style.sty","media_type":"text/x-tex","sha256":"sha256:a60029db8fbb4ec42f5dea1b9d0ad76eeca1f8c69eaa381592534fc3fcae7095","byte_count":3134,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared typography for the rendered Plectis papers.  Under Tectonic (XeTeX)\n% the text and mathematics use TeX Gyre Pagella, with Heros for sans-serif\n% mathematics and Inconsolata for monospace.  The A4 text block follows a\n% classical measure with quiet running heads and restrained link colours.\n\\NeedsTeXFormat{LaTeX2e}\n\\ProvidesPackage{paper-house-style}[2026/07/28 Pagella house style for Plectis papers]\n\n\\RequirePackage{iftex}\n\\RequirePackage{amsmath}\n\\RequirePackage{amsthm}\n\\ifpdftex\n  \\RequirePackage[T1]{fontenc}\n  \\RequirePackage[utf8]{inputenc}\n  \\RequirePackage{amssymb}\n  \\RequirePackage{newpxtext}\n  \\RequirePackage{newpxmath}\n\\else\n  \\RequirePackage{fontspec}\n  \\RequirePackage{unicode-math}\n  \\setmainfont{texgyrepagella}[Extension=.otf,UprightFont=*-regular,\n    BoldFont=*-bold,ItalicFont=*-italic,BoldItalicFont=*-bolditalic]\n  \\setmathfont{texgyrepagella-math.otf}\n  \\setmathfont{texgyreheros-regular.otf}[range=sfup,Scale=MatchLowercase]\n  \\setmonofont{Inconsolatazi4}[Extension=.otf,UprightFont=*-Regular,\n    BoldFont=*-Bold,Scale=MatchLowercase]\n\\fi\n\n\\RequirePackage[a4paper,textwidth=418pt,textheight=636pt,hcentering,\n  top=68pt,headheight=15pt,headsep=20pt,footskip=34pt]{geometry}\n\\RequirePackage{microtype}\n\\linespread{1.045}\n\\clubpenalty=10000 \\widowpenalty=10000 \\tolerance=1600\n\n\\RequirePackage[dvipsnames]{xcolor}\n\\definecolor{linkexternal}{HTML}{275D8C}\n\\definecolor{linkinternal}{HTML}{6E2B2B}\n\n\\RequirePackage{titlesec}\n\\titleformat{\\section}{\\Large\\bfseries}{\\thesection}{0.85em}{}\n\\titlespacing*{\\section}{0pt}{2.6ex plus .8ex minus .3ex}{1.5ex plus .3ex}\n\\titleformat{\\subsection}{\\large\\bfseries}{\\thesubsection}{0.8em}{}\n\\titlespacing*{\\subsection}{0pt}{2.1ex plus .6ex minus .2ex}{1.1ex plus .2ex}\n\\titlespacing*{\\paragraph}{0pt}{1.6ex plus .5ex minus .2ex}{0.85em}\n\n\\RequirePackage{fancyhdr}\n\\newcommand{\\paperrunninghead}{}\n\\newcommand{\\runninghead}[1]{\\renewcommand{\\paperrunninghead}{#1}}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\fancyhead[L]{\\small\\itshape\\paperrunninghead}\n\\fancyhead[R]{\\small\\thepage}\n\\renewcommand{\\headrulewidth}{0pt}\n\\fancypagestyle{plain}{\\fancyhf{}\\renewcommand{\\headrulewidth}{0pt}}\n\n\\RequirePackage{caption}\n\\captionsetup{font=small,labelfont=bf,skip=9pt,width=.92\\linewidth}\n\\renewcommand{\\arraystretch}{1.12}\n\n\\newcommand{\\subtitle}[1]{\\gdef\\@papersubtitle{#1}}\n\\renewenvironment{abstract}{%\n  \\vspace{0.1em}\\small\n  \\begin{center}{\\normalsize\\scshape Abstract}\\end{center}\\vspace{-0.5em}\n  \\list{}{\\leftmargin 1.3em\\rightmargin\\leftmargin\\listparindent 0pt\n    \\parsep 0.32em\\itemindent 0pt}\\item\\relax}\n  {\\endlist\\vspace{0.5em}}\n\\def\\@maketitle{%\n  \\newpage\\null\n  \\vspace*{-2.6em}%\n  \\begin{center}%\n    {\\LARGE\\bfseries\\@title\\par}%\n    \\vskip 0.55em\n    {\\large\\itshape\\@papersubtitle\\par}%\n    \\vskip 1.1em\n    {\\scshape\\@author\\par}%\n    \\vskip 0.25em\n    {\\small\\@date\\par}%\n    \\vskip 0.8em\n    \\rule{6.5em}{0.4pt}%\n  \\end{center}%\n  \\vskip 0.9em}\n\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\n\\endinput\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis/main/paper/plectis-public-system.tex","section_outline":["The problem","One public test before the general design","The component contract","One run, examined","Five distinctions","Public execution versus where the code came from","Repeatability versus correctness","A validator's rule versus the stated claim","Selected cases versus general behaviour","Risk reduction versus guarantee","What the collection contains","The author's hand","What stronger evidence would look like","A falsifiable comparison","From an open question to a falsifiable handoff","How the record should age","Conclusion","Dated reproduction record","Statements and declarations"]}}},"problem_path_index":{"schema":"plectis_reviewer_problem_path_index_v0","role":"one_row_per_covered_erdos_problem","reading_rule":"Every covered Erdős problem remains open. A Lean module path is a route into source, not a solution claim. Comparator checks a separately declared statement; it does not assess novelty or close the problem.","problem_count":8,"problem_ids":["erdos_68","erdos_243","erdos_249","erdos_251","erdos_257","erdos_269","erdos_1041","erdos_1049"],"problems":{"erdos_68":{"problem_id":"erdos_68","erdos_number":68,"short_title":"The factorial-denominator series","question":"Is the series sum_{n >= 2} 1/(n! - 1) irrational?","status":"open","library_root":"ErdosProblems.lean","directory":"lean/ErdosProblems/Erdos68","directory_github":"https://github.com/wcook04/plectis-erdos/tree/main/lean/ErdosProblems/Erdos68","claim_registry_status":"registered_declarations","site":{"problem_page":"maths/problems/erdos_68.html","maths_library":"maths/index.html","results":"maths/documents/results.html","source_map":"maths/documents/source-map.html"},"papers":[{"paper_id":"erdos-68-factorial-denominator-irrationality","form":"Problem note","length_class":"short","pages":9,"title":"Integer Linear Forms for a Factorial Reciprocal Series","precis":"For S=∑_{n≥2}(n!−1)^{-1}, the paper classifies finitely supported integer vectors that cancel prescribed initial weighted sums in forms MS+k. An integral basis, one Diophantine equation and a finite gcd determine the possible moments M. At cancellation depth four, the least positive moment is 1380. Irrationality by these forms still requires a nonintegral form with q dividing M for every positive q.","site_html":"maths/papers/erdos-68-factorial-denominator-irrationality.html","hosted_pdf_path":"papers/erdos-68-factorial-denominator-irrationality.pdf","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-68-factorial-denominator-irrationality.pdf","body_pointer":"papers.records.erdos-68-factorial-denominator-irrationality.body"},{"paper_id":"erdos68-factorial-reasoning-surface","form":"Reasoning surface","length_class":"long","pages":51,"title":"Factorial Linear Forms and Denominators: Detailed Proofs and Rationality Criteria","precis":"The record proves liminf (log L_N)/(N^{3/2} log N) ≥ 2√2/3 for L_N=lcm(2!−1,…,N!−1), by comparing a terminal block of denominators with pairwise gcds rather than an external multiplicity theorem. Specialising a Louwsma–Martino valuation, it exhibits primes 139 and 2593 absent from all later reduced denominators. Coefficient constructions give integer forms MS+k; at fixed moment the remainder changes only by an integer. Rationality criteria use factorial digits and the next integer above a scaled partial sum; the digit argument also recovers the irrationality of e. Two exact calculations give q∤299999! and q≥2^39990>10^12038. 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The unrestricted Erdős–Graham question remains open.","site_html":"maths/papers/erdos-243-reciprocal-tail-rigidity.html","hosted_pdf_path":"papers/erdos-243-reciprocal-tail-rigidity.pdf","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-243-reciprocal-tail-rigidity.pdf","body_pointer":"papers.records.erdos-243-reciprocal-tail-rigidity.body"},{"paper_id":"erdos243-reciprocal-tail-reasoning-surface","form":"Reasoning surface","length_class":"long","pages":61,"title":"Reciprocal Sums and the Sylvester Recurrence: Further Results and Proofs","precis":"Integer finite differences exclude nonintegral regular rates and reduce the cubic rate to a polynomial. A square condition in the cubic field and congruences modulo seven then prove irrationality. 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Every section is an explicit integer multiple of a retained one, and these reductions generate all integral relations, with unique coefficients. The proof combines the local formula for Euler's totient with a nonsingular evaluation matrix obtained from the Chinese remainder theorem and Dirichlet's theorem. The basis theorem does not decide the irrationality of S=∑ φ(n)2^{-n}.","site_html":"maths/papers/erdos-249-binary-totient-series.html","hosted_pdf_path":"papers/erdos-249-binary-totient-series.pdf","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-249-binary-totient-series.pdf","body_pointer":"papers.records.erdos-249-binary-totient-series.body"},{"paper_id":"erdos249-totient-reasoning-surface","form":"Reasoning surface","length_class":"long","pages":163,"title":"The Binary Totient Series","precis":"The dyadic totient sections through level e≥1 have an explicit rational basis of size 2^e+1. 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prime-gap dyadic series","question":"Is the dyadic series of consecutive primes irrational? 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One set of upper Banach density zero supports the corrections, which satisfy any prescribed bound tending to infinity and eventually preserve each fixed coefficient and partial-sum congruence. For prime gaps with logarithmic corrections, block distributions remain close for lengths o(log log X), and cumulative positions remain asymptotic to n log n. Those positions need not be prime; the original irrationality question remains open.","site_html":"maths/papers/erdos-251-prime-gap-dyadic-series.html","hosted_pdf_path":"papers/erdos-251-prime-gap-dyadic-series.pdf","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-251-prime-gap-dyadic-series.pdf","body_pointer":"papers.records.erdos-251-prime-gap-dyadic-series.body"},{"paper_id":"erdos251-prime-gap-reasoning-surface","form":"Reasoning surface","length_class":"long","pages":47,"title":"Prime-Gap Dyadic Series: Perturbations, Exact Criteria and Certificates","precis":"A convergent dyadic series with nonnegative integer coefficients can be made rational by sparse nonnegative corrections of upper Banach density zero, eventually bounded by any f(n)→∞, while preserving every fixed modulus and, for a logarithmic allowance, short-block empirical distributions. For prime gaps the cumulative positions remain asymptotic to n log n; no primality conclusion is asserted. Summation by parts reduces the prime series to the gap series, and the tails T_N=∑_{j≥1} g_{N+j} 2^{-j} satisfy T_{N+1}=2T_N−g_{N+1}. That recurrence characterises rationality and yields finite tests with explicit remainder bounds. An exact finite certificate shows that a rational value of the prime series would have denominator at least 2^{39998}>10^{12040}. The bound does not establish irrationality.","site_html":"maths/papers/erdos251-prime-gap-reasoning-surface.html","hosted_pdf_path":"papers/erdos251-prime-gap-reasoning-surface.pdf","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos251-prime-gap-reasoning-surface.pdf","body_pointer":"papers.records.erdos251-prime-gap-reasoning-surface.body"}],"lean":{"modules":[{"module":"ErdosProblems.Erdos251.PrimeGapDyadicTail","path":"lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","present":true,"theorem_count":92,"declaration_count":116,"github":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","tree":null}],"module_count":1,"directory":"lean/ErdosProblems/Erdos251","directory_github":"https://github.com/wcook04/plectis-erdos/tree/main/lean/ErdosProblems/Erdos251","library_root":"ErdosProblems.lean","claims_json":"lean/claims.json","problems_json":"lean/problems.json","results_md":"lean/RESULTS.md","source_map_md":"lean/human-source/docs/SOURCE_MAP.md","clone":"https://github.com/wcook04/plectis-erdos"},"comparator":{"what_comparator_is":"Comparator is a LeanProver-org tool that checks whether a submitted proof proves a statement declared separately from it. 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The checked local consumer then excludes eventual integrality of the h-shift."}],"claim_registry_status":"registered_declarations"},"registered_claim_ids":["erdos_251","prime_gap_irrationality_equivalence","prime_gap_unboundedness_and_nonperiodicity"],"registered_claim_count":3},"erdos_257":{"problem_id":"erdos_257","erdos_number":257,"short_title":"Reciprocal sums over infinite exponent supports","question":"Is the sum of 1/(2^n-1) over every infinite set of positive exponents irrational?","status":"open","library_root":"ErdosProblems.lean","directory":"lean/ErdosProblems/Erdos257","directory_github":"https://github.com/wcook04/plectis-erdos/tree/main/lean/ErdosProblems/Erdos257","claim_registry_status":"programme_only","site":{"problem_page":"maths/problems/erdos_257.html","maths_library":"maths/index.html","results":"maths/documents/results.html","source_map":"maths/documents/source-map.html"},"papers":[{"paper_id":"erdos-257-mersenne-support-subseries","form":"Problem note","length_class":"short","pages":26,"title":"Irrationality criteria for Lambert subseries","precis":"The series ∑_{a∈A}(b^a−1)^{-1} is irrational for every integer b≥2 and infinite A contained in E∪V when E satisfies a finite-prime weighted summability condition and V has a summable positive divisor cover. The two classes are incomparable, and their union reaches supports in neither class. A common average over dyadic lengths produces one index with both displacements small. Arbitrary infinite supports at base two remain unresolved.","site_html":"maths/papers/erdos-257-mersenne-support-subseries.html","hosted_pdf_path":"papers/erdos-257-mersenne-support-subseries.pdf","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-257-mersenne-support-subseries.pdf","body_pointer":"papers.records.erdos-257-mersenne-support-subseries.body"},{"paper_id":"erdos257-mersenne-reasoning-surface","form":"Reasoning surface","length_class":"long","pages":152,"title":"Reciprocal Mersenne Subseries","precis":"A weighted summability criterion proves irrationality of reciprocal Mersenne subseries: if h is the part of a supported on a finite nonempty prime set P, every infinite A with ∑ h(a)/(a(b^{h(a)}−1)) finite has irrational ∑ (b^a−1)^{-1}. The base-two condition gives the conclusion at every integer base for every infinite subset, and permits an example with divergent reciprocal sum, while excluding full support and all odd exponents. The proof averages over a finite part of the support and over dyadic observation lengths. Erdős's weaker reciprocal-summable theorem is proved in full, and a common window extends both arguments to unions of weighted and positive-cover supports. Finite-denominator and achievement-set calculations formulate tests for the targets 1/2 and 1/21; those tests do not decide either target or the universal question. Prime support at base 2 and squarefree support at power-of-two bases are attributed as existing work.","site_html":"maths/papers/erdos257-mersenne-reasoning-surface.html","hosted_pdf_path":"papers/erdos257-mersenne-reasoning-surface.pdf","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos257-mersenne-reasoning-surface.pdf","body_pointer":"papers.records.erdos257-mersenne-reasoning-surface.body"},{"paper_id":"erdos257-expository-sums-never-fractions","form":"Expository paper","length_class":"short","pages":16,"title":"Why certain sums of 1/(2^n-1) are irrational","precis":"Erdős asked whether infinitely many of the fractions 1, 1/3, 1/7, 1/15, …, each of the form 1/(2^n−1) and each used at most once, can ever add up to a fraction. The paper proves that the chosen fractions add up to an irrational number whenever 1/n, added up over the chosen exponents n, is finite; for the exponents 2, 4, 8, 16, … this makes 1/3+1/15+1/255+1/65535+… irrational. Sections 1 to 8 explain the proof with examples; Appendix A supplies supporting details and formal-proof correspondence. The finite reconstruction here is an ordinary proof; the linked Lean and Comparator receipts establish the formal theorem and its own proof. For an arbitrary infinite choice the question is still open.","site_html":"maths/papers/erdos257-expository-sums-never-fractions.html","hosted_pdf_path":"papers/erdos257-expository-sums-never-fractions.pdf","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos257-expository-sums-never-fractions.pdf","body_pointer":"papers.records.erdos257-expository-sums-never-fractions.body"}],"lean":{"modules":[{"module":"ErdosProblems.Erdos257.MersenneSubseriesRigidity","path":"lean/ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean","present":true,"theorem_count":23,"declaration_count":28,"github":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean","tree":null}],"module_count":1,"directory":"lean/ErdosProblems/Erdos257","directory_github":"https://github.com/wcook04/plectis-erdos/tree/main/lean/ErdosProblems/Erdos257","library_root":"ErdosProblems.lean","claims_json":"lean/claims.json","problems_json":"lean/problems.json","results_md":"lean/RESULTS.md","source_map_md":"lean/human-source/docs/SOURCE_MAP.md","clone":"https://github.com/wcook04/plectis-erdos"},"comparator":{"what_comparator_is":"Comparator is a LeanProver-org tool that checks whether a submitted proof proves a statement declared separately from it. 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Complete ordinary arguments for general distinct-running-LCM sums and prime-power subseries are retained as remarks awaiting formalisation.","site_html":"maths/papers/erdos269-running-lcm-reasoning-surface.html","hosted_pdf_path":"papers/erdos269-running-lcm-reasoning-surface.pdf","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos269-running-lcm-reasoning-surface.pdf","body_pointer":"papers.records.erdos269-running-lcm-reasoning-surface.body"},{"paper_id":"erdos-269-three-prime-running-lcm","form":"Problem note","length_class":"short","pages":16,"title":"Distinct running least common multiples","precis":"A five-map argument proves irrationality of the reciprocal sum over distinct running least common multiples of the 5-smooth integers. The spacing of powers of five separates the tail images, so rationality would force a periodic jump sequence. 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The proof confines a slit preimage near a critical point and applies radial projection on its two inverse sheets. For monic trinomials with zeros in the open unit disc, the root equation instead keeps every root-to-origin segment inside the lemniscate.","site_html":"maths/papers/erdos-1041-lemniscate-newton-flow.html","hosted_pdf_path":"papers/erdos-1041-lemniscate-newton-flow.pdf","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-1041-lemniscate-newton-flow.pdf","body_pointer":"papers.records.erdos-1041-lemniscate-newton-flow.body"},{"paper_id":"erdos1041-lemniscate-reasoning-surface","form":"Reasoning surface","length_class":"long","pages":86,"title":"Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections","precis":"The record gives the rational estimates and geometric proof for ani's degree-seven polynomial: every connected set in its strict unit lemniscate joining two roots has Hausdorff measure greater than two. It develops contained paths for trinomials, collinear roots and two sparse families. Ordinary analytic arguments for critical-value criteria retain their unformalised inputs beside the statements. Critical-value moments, counterexamples to prescribed paths and questions about flow-strip geometry complete the account. The one-polynomial formal statement and correspondence with the historical wording remain distinct.","site_html":"maths/papers/erdos1041-lemniscate-reasoning-surface.html","hosted_pdf_path":"papers/erdos1041-lemniscate-reasoning-surface.pdf","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos1041-lemniscate-reasoning-surface.pdf","body_pointer":"papers.records.erdos1041-lemniscate-reasoning-surface.body"}],"lean":{"modules":[{"module":"ErdosProblems.Erdos1041.Counterexample.HausdorffLength","path":"lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","present":true,"theorem_count":9,"declaration_count":10,"github":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","tree":null},{"module":"ErdosProblems.Erdos1041.Counterexample.CatalogueAdapter","path":"lean/ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean","present":true,"theorem_count":2,"declaration_count":2,"github":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean","tree":null},{"module":"ErdosProblems.Erdos1041.NewtonFlowRaySeparation","path":"lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","present":true,"theorem_count":25,"declaration_count":28,"github":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","tree":null}],"module_count":3,"directory":"lean/ErdosProblems/Erdos1041","directory_github":"https://github.com/wcook04/plectis-erdos/tree/main/lean/ErdosProblems/Erdos1041","library_root":"ErdosProblems.lean","claims_json":"lean/claims.json","problems_json":"lean/problems.json","results_md":"lean/RESULTS.md","source_map_md":"lean/human-source/docs/SOURCE_MAP.md","clone":"https://github.com/wcook04/plectis-erdos"},"comparator":{"what_comparator_is":"Comparator is a LeanProver-org tool that checks whether a submitted proof proves a statement declared separately from it. 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It includes every positive integral power of 31/4 and excludes 3/2.","site_html":"maths/papers/erdos-1049-rational-base-lambert.html","hosted_pdf_path":"papers/erdos-1049-rational-base-lambert.pdf","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-1049-rational-base-lambert.pdf","body_pointer":"papers.records.erdos-1049-rational-base-lambert.body"},{"paper_id":"erdos1049-rational-base-lambert-reasoning-surface","form":"Reasoning surface","length_class":"long","pages":80,"title":"Geometric Moments and Rational Lambert Values: Proofs and Further Results","precis":"The record proves the fixed-base Hankel asymptotic for Zudilin's normalised moments and the rational-base irrationality criterion log b/log a<0.4056830213840605…. The determinant proof separates geometric node configurations from positive weights; the irrationality proof instead specialises Zudilin's 2004 polynomial forms after cancellation. Coefficient moments, evaluated lattices and congruence conditions distinguish integer divisibility from nonzero remainder decay. A broader ratio-limit argument and a rational-value model remain unformalised. None of these arguments settles F(3/2).","site_html":"maths/papers/erdos1049-rational-base-lambert-reasoning-surface.html","hosted_pdf_path":"papers/erdos1049-rational-base-lambert-reasoning-surface.pdf","public_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos1049-rational-base-lambert-reasoning-surface.pdf","body_pointer":"papers.records.erdos1049-rational-base-lambert-reasoning-surface.body"}],"lean":{"modules":[{"module":"ErdosProblems.Erdos1049.RationalBaseLambert","path":"lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean","present":true,"theorem_count":11,"declaration_count":16,"github":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean","tree":null},{"module":"ErdosProblems.Erdos1049.RationalPadeArithmetic","path":"lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean","present":true,"theorem_count":14,"declaration_count":19,"github":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean","tree":null},{"module":"ErdosProblems.Erdos1049.ZudilinConeArithmetic","path":"lean/ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean","present":true,"theorem_count":25,"declaration_count":36,"github":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean","tree":null},{"module":"ErdosProblems.Erdos1049.HermitePadeNoGo","path":"lean/ErdosProblems/Erdos1049/HermitePadeNoGo.lean","present":true,"theorem_count":5,"declaration_count":10,"github":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/HermitePadeNoGo.lean","tree":null},{"module":"ErdosProblems.Erdos1049.ZudilinHeightRegion","path":"lean/ErdosProblems/Erdos1049/ZudilinHeightRegion.lean","present":true,"theorem_count":12,"declaration_count":13,"github":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/ZudilinHeightRegion.lean","tree":null},{"module":"ErdosProblems.Erdos1049.QAperyDiagonalNonEquivalence","path":"lean/ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean","present":true,"theorem_count":4,"declaration_count":11,"github":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean","tree":null}],"module_count":6,"directory":"lean/ErdosProblems/Erdos1049","directory_github":"https://github.com/wcook04/plectis-erdos/tree/main/lean/ErdosProblems/Erdos1049","library_root":"ErdosProblems.lean","claims_json":"lean/claims.json","problems_json":"lean/problems.json","results_md":"lean/RESULTS.md","source_map_md":"lean/human-source/docs/SOURCE_MAP.md","clone":"https://github.com/wcook04/plectis-erdos"},"comparator":{"what_comparator_is":"Comparator is a LeanProver-org tool that checks whether a submitted proof proves a statement declared separately from it. A pass is not independent peer review and does not close the Erdős problem.","site_page":"maths/documents/external-verification.html","hosted_markdown":"lean/human-source/docs/EXTERNAL_VERIFICATION.md","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/EXTERNAL_VERIFICATION.md","glossary":"docs/glossary.html#glossary-comparator","fixture_site_page":"maths/documents/comparator-1049.html","fixture_hosted_markdown":"lean/human-source/docs/COMPARATOR_1049_NUMERICAL_HEIGHT.md","fixture_github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/COMPARATOR_1049_NUMERICAL_HEIGHT.md","digest_lanes":{"reviewed_core_lane":{"authority":"docs/claims.json","library_root":"Erdos249257.lean","reviewed":true,"registered_claim_count":9,"registered_claim_ids_pointer":"/mathematical_assertions/problem_digest/erdos_1049/registered_claim_ids","what_this_is":"curated claims the release reviews and publishes with a status. The registry covers the Erdos249257 core library only, which is the joint #249/#257 programme."},"problem_expansion_lane":{"authority":"docs/problems.json","library_root":"ErdosProblems.lean","directory":"lean/ErdosProblems/Erdos1049","reviewed":false,"claim_registry_status":"registered_declarations","declared_module_count":6,"modules_pointer":"/mathematical_assertions/problem_digest/erdos_1049/modules","what_this_is":"exact Lean propositions in this problem's own directory. They are kernel-checked and they are deliberately not promoted into reviewed public claims; reporting one as a registered claim is the specific error this split exists to prevent."},"read_this_first":"The three lanes are different bodies of source with different authorities: a reviewed registry, a kernel-checked expansion library, and a written manuscript. A problem can have evidence in any of them, or in all three. Say which lane an answer comes from, and never upgrade one into another.","manuscript_lane":{"authority":"manuscript_argument","reviewed":false,"source_lane":"manuscript_result","manuscript_ids":["erdos-1049-rational-base-lambert","erdos1049-rational-base-lambert-reasoning-surface"],"argued_result_count":4,"posed_problem_count":0,"rows_pointer":"mathematical_assertions.assertions filtered by source_lane == 'manuscript_result' and problem_ids containing 'erdos_1049'","what_this_is":"results the manuscripts prove that no registered claim carries. Proved in prose at a quoted line, not by a Lean declaration; a formalisation is neither implied nor denied. For the six problems the claim registry does not cover, this is where the authored mathematics is."}}},"headline":{"erdos_number":1049,"question":"For which rational bases is the corresponding series irrational? The first resistant explicit base is three halves.","status":"open","what_is_checked":["Lean checks irrationality and the stated irrationality-exponent bound for rational bases in the exact Zudilin contour region, including every positive integral power of 31/4. The contour excludes 3/2.","For p>1 and N≤8, Lean checks positive definite coefficient matrices, real finite-pencil roots below F(p), and non-strict interlacing at adjacent ranks. Neither all-rank positivity nor irrationality at 3/2 follows.","That a literal coordinatewise transfer of the integer-base clearing argument forces a power-versus-linear inequality, and that this inequality is impossible at base three halves.","The exact denominator-cleared rational-base tail recurrence, including its forcing term.","That for every noninteger base with denominator at least two the forcing magnitude grows at least exponentially, while at denominator one it collapses to a bounded expression.","The complete elementary Archimedean height inequality used by the published Bundschuh-Väänänen criterion at base seven halves; the analytic irrationality theorem itself remains external.","Polynomial bounds on the doubled denominator exponents in the homogenised Padé construction, together with an exact gap identity.","The exact exterior-determinant elimination identity, and that rowwise contents scale the Padé errors, exterior determinant, and absolute determinant height by the same factors.","The exact Heine-Zudilin cone endpoint exponent arithmetic used to identify the bottom and top coefficient endpoints.","That homogeneous evaluation at (3,2) sees only the constant endpoint modulo three and the top endpoint modulo two, so unit endpoints obstruct common local factors.","That under unit top and constant endpoint hypotheses, a common multiplier of the two specialised coefficient channels is divisible by neither two nor three; the missing local gain must therefore survive primitive normalisation.","That vanishing of the higher bottom and top endpoint jets is exactly divisibility by the corresponding powers of three and two.","That the source integer-base ratio inequality implies a strictly negative scalar product-formula margin at three halves.","That the four-jet target has exact cardinality (3^R)^2(2^S)^2 and, for positive R, any family with n at least 4R+2S forces two distinct binary selectors to collide; their {-1,0,1} difference cancels all four endpoint jets.","That the rectangular Hermite-Pade threshold never improves on the classical one in the checked exponent model, with equality characterised exactly.","The exact 81/200 height region: 31/4 and every positive power lie inside it and outside the earlier Bundschuh-Vaananen region, while 3/2 lies outside both.","At n = 0, the Amdeberhan-Zeilberger q-Apery operator applied to Van Assche's moving diagonal has residual -p(p-1)^2(p+1)(p^5+2p^4+2p^3+2p^2+2), which is strictly negative and nonzero for every real p > 1."],"what_is_not_checked":["The universal rational-base conjecture, including the case 3/2, remains open. The checked 31/4 power family settles one region and leaves that conjecture untouched.","The published analytic height theorem, which is cited and used as an external result and is not formalised here.","The all-scale Heine-Zudilin coefficient construction at rational base three halves; the primary source assumes an integer base.","A sufficiently large primitive-normalised, non-collapsed coefficient family meeting the checked n >= 4R+2S threshold at the required quadratic depths.","That one of the resulting bounded-coefficient collision vectors has a nonzero polynomial pair, lies outside the analytic remainder nullspace, and has sufficient local gain.","The elementary Padé pilot checks exponent arithmetic only; it lacks remainder positivity or decay and does not qualify the separate Zudilin contour result.","The n=0 q-Apéry residual gives no general recurrence, lattice, valuation, denominator, or irrationality result; it only blocks the named scalar transfer."],"open_obligations":[{"id":"three_halves_growing_rank_endpoint_jet_kernel","statement":"After rowwise primitive normalisation, exhibit at least 4R_n+2S_n coefficient pairs at positive quadratic depths from a non-collapsed deformation; preserve the two- and three-adic gain; then show that one checked {-1,0,1} collision has a nonzero polynomial pair outside the analytic remainder nullspace, or prove an exact rank obstruction."}],"claim_registry_status":"registered_declarations"},"registered_claim_ids":["calibrated_rational_hankel_countermodel","coefficient_pencil_finite_rank","erdos_1049","rational_base_cleared_tail_recurrence","rational_base_forcing_growth","three_halves_coordinatewise_corridor_no_go","van_assche_qapery_diagonal_no_go","zudilin_all_rank_hankel_order","zudilin_rational_base_region"],"registered_claim_count":9}}},"mathematical_assertions":{"schema":"plectis_mathematical_assertions_v2","status":"available","role":"citable_mathematical_records","why_this_exists":"Every mathematical statement this corpus publishes, as one flat collection of rows carrying content, status, boundary and routes, so a reader can cite one statement rather than a paper. 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Calling every row atomic would be a claim about their shape that the source does not support.","reference_pins":{"release":"0.10.0","release_tag":"v0.10.0","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","formal_source_public_tag":null,"lean_toolchain":"leanprover/lean4:v4.29.1","ref_rule":"Every declaration row carries both. permalink_url pins formal_source_ref above and is the one to cite -- it will not move. current_url follows the default branch and may drift as the library is edited; use it to read the live library, never to cite."},"permalink_resolution":{"verified":true,"pinned_ref":"2a8c1667566020d123a081b6165e3ee94550566d","pinned_ref_available":true,"declarations_checked":490,"resolved_at_pin":490,"unresolved_at_pin":[],"line_lands_off_declaration":[],"what_this_proves":"every permalink names a declaration that exists at the pinned commit, at the line the registry gives. It does not prove the theorem is correct -- no Lean kernel ran during this build."},"evidence_resolution_legend":{"declaration_level":"the corpus binds this statement to named Lean declarations with module and line; the strongest binding available","module_level_core_library":"bound to modules of the reviewed Erdos249257 core library, not to individual declarations","module_level_expansion_library":"bound to modules of an ErdosProblems expansion lane. Nothing binds an individual sentence to a declaration -- the problem index publishes no such mapping, and inferring one would be a more confident citation than the source supports. At problem granularity there is more: the index names the theorems each note is required to rest on, carried under problem_digest[<problem>].required_note_declarations with pinned links. So the right answer to 'what backs this' is declaration-level for the problem and module-level for the sentence.","paper_level_only":"resolves to a manuscript section, not to Lean source","statement_only":"no source binding is published for this row","open_by_construction":"nothing is claimed, so nothing backs it","boundary_declaration":"a declared limit on what is claimed","negative_statement_no_evidence_needed":"an explicit statement about what was not checked"},"problem_binding_legend":{"open_proposition_target":"the registry lists this claim against an open proposition that names the problem. The corpus's own voice, and the strongest binding available","paper_owner":"the claim's paper anchor is defined in exactly one manuscript, and that manuscript treats one numbered problem","core_programme_joint":"the claim's only anchor is the retired combined #249/#257 exposition, which treats both together. Bound to both rather than guessed apart","problem_index_membership":"the statement is published inside that problem's index row","unbound":"no published source signal binds this row to a problem"},"counts":{"assertions":414,"by_kind":{"registered_claim":160,"remaining_open_proposition":19,"non_claim":6,"problem_checked":83,"problem_not_checked":40,"open_obligation":27,"paper_posed_problem":6,"paper_argued_result":73},"by_problem_binding_method":{"core_programme_joint":14,"gateway_section":17,"open_proposition_target":112,"paper_owner":7,"programme_representative":10},"declaration_rows":490,"declaration_rows_with_permalink":490,"declaration_rows_with_lean_signature":490,"declaration_permalinks_verified_at_pin":490,"problems_indexed":8,"registered_claims_bound_to_one_problem":146,"registered_claims_bound_jointly":14,"problem_module_rows":43,"paper_anchors_resolved":145,"paper_anchors_by_resolution_rule":{"T0_unique":123,"T1a_declaration_citation_elimination":3,"registry_supplied_composite_anchor":19},"paper_anchors_tie_broken":0,"claims_without_a_paper_label":34,"paper_anchors_ambiguous":0,"paper_anchors_unresolved":0,"labels_shared_across_manuscripts":44,"coordinate_conflicts":0,"by_source_lane":{"registry_claim":160,"registry_open":19,"registry_boundary":6,"problem_index_checked":83,"problem_index_negative":40,"problem_index_open":27,"manuscript_problem":6,"manuscript_result":73}},"shared_label_watch":{"labels":["app:index","app:sources","bdry:critical-value-separation","bdry:solved-polynomial-families","eq:canonical-gap","eq:disk-family-length","res:arity-not-capacity","res:canonicalmersenne","res:complementary-binomial-chords","res:constant-factor-path","res:critical-proximity","res:critical-value-separation","res:critical-value-thresholds","res:fp-weighted-all-degree","res:fulldepth","res:gap-nonperiodic","res:irr-equivalence","res:locus","res:low-critical-thirteen-twentyfifths","res:one-root-gamma-false","res:orlicz-currency","res:problem","res:rankonefloor","res:ray","res:sep-or-false","res:straight-no-go","res:trinomial-all-degree","res:two-nearest-roots","res:value","sec:arguments","sec:carry","sec:constant-factor","sec:finite","sec:frontier","sec:gap","sec:intro","sec:limits","sec:map","sec:newton","sec:open","sec:problem","sec:rank","sec:reduction","sec:results"],"why_this_is_here":"These LaTeX labels are defined in more than one manuscript. Only the ones a claim actually cites can currently mislead a reader, but the rest are the same defect waiting for a claim to reference them, so they are listed rather than filtered out."},"reviewed_vs_unreviewed":"reviewed=true rows come from the curated claims registry. reviewed=false rows come from two different places and the difference is load-bearing: the per-problem index, whose declarations are kernel-checked and which states plainly that checking them does not promote them into reviewed public claims; and the manuscripts, whose results are proved by written argument with no Lean declaration asserted to cover them. source_lane says which, on every row. Do not report an unreviewed row as a registered claim, and do not read a manuscript row as a formalised one.","coordinate_conflicts":[],"semantic_prior_art_join":{"status":"available","source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","source_path":"docs/semantic_corpus.json.gz","state_vocabulary":["not_assessed","known_classical","formalisation_only","routine_corollary_or_specialisation","candidate_new_statement","external_review_pending","externally_supported_as_new","prior_art_found"],"registered_claims_with_semantic_edges":123,"joined_edge_count":241,"joined_state_counts":{"candidate_new_statement":1,"formalisation_only":17,"known_classical":15,"not_assessed":156,"prior_art_found":2,"routine_corollary_or_specialisation":50},"boundary":"These are source-authored semantic-node states joined by exact local id or declaration overlap. candidate_new_statement means only that the source marks a candidate; it is not an external novelty verdict."},"prior_art_owner_material":{"paper/erdos249-257-main-paper.tex":{"status":"historical_source_available","active_paper":false,"source_path":"paper/archive/erdos249-257-main-paper.tex","source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/archive/erdos249-257-main-paper.tex","body_sha256":"sha256:60dd9a22f4335145ce3917490d31ee84a6751ff1fe2e7235ecb292426599a570","bibliography_latex":"\\begin{thebibliography}{99}\n\\small\n\\raggedright\n\\setlength{\\itemsep}{0pt}\n\\bibitem{erdos1948} P.~Erd\\H{o}s,\n  \\href{https://users.renyi.hu/~p_erdos/1948-04.pdf}{\\emph{On arithmetical properties of Lambert series}},\n  J. Indian Math. Soc. (N.S.) \\textbf{12} (1948), 63--66.\n\\bibitem{erdos1968} P.~Erd\\H{o}s,\n  \\href{https://users.renyi.hu/~p_erdos/1969-09.pdf}{\\emph{On the irrationality of certain series}},\n  Math. Student \\textbf{36} (1968), 222--226 (issued 1969).  The theorem is\n  on p.~222; the article does not print the claimed extension beyond\n  pairwise-coprime support.\n\\bibitem{erdosgraham1980} P.~Erd\\H{o}s and R.~L. Graham,\n  \\href{https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf}{\\emph{Old and New Problems and Results in Combinatorial Number Theory}},\n  Monogr. Enseign. Math. 28, Geneva, 1980, pp.~61--62.\n\\bibitem{erdos1988} P.~Erd\\H{o}s, \\emph{On the irrationality of certain series: problems and\n  results}, in A.~Baker (ed.), \\emph{New Advances in Transcendence Theory},\n  Cambridge UP, 1988, pp.~102--109,\n  doi:\\href{https://doi.org/10.1017/CBO9780511897184.009}{10.1017/CBO9780511897184.009}.\n\\bibitem{borwein1992} P.~B. Borwein, \\emph{On the irrationality of certain series},\n  Math. Proc. Cambridge Philos. Soc. \\textbf{112} (1992), no.~1, 141--146,\n  doi:\\href{https://doi.org/10.1017/S030500410007081X}{10.1017/S030500410007081X}.\n\\bibitem{apostol} T.~M. Apostol,\n  \\href{https://doi.org/10.1007/978-1-4757-5579-4}{\\emph{Introduction to Analytic Number Theory}},\n  Springer, 1976.\n\\bibitem{merca2017} M.~Merca, \\emph{The Lambert series factorization theorem},\n  Ramanujan J. \\textbf{44} (2017), no.~2, 417--435,\n  doi:\\href{https://doi.org/10.1007/s11139-016-9856-3}{10.1007/s11139-016-9856-3}.\n\\bibitem{mercaschmidt} M.~Merca and M.~D. Schmidt, \\emph{Generating special arithmetic\n  functions by Lambert series factorizations}, Contrib. Discrete Math. \\textbf{14}\n  (2019), no.~1, 31--45,\n  doi:\\href{https://doi.org/10.55016/ojs/cdm.v14i1.62425}{10.55016/ojs/cdm.v14i1.62425}.\n\\bibitem{kakeya1914} S.~Kakeya, \\emph{On the set of partial sums of an infinite series},\n  Proc. Tokyo Math.-Phys. Soc., 2nd ser. \\textbf{7} (1914), no.~14, 250--251,\n  doi:\\href{https://doi.org/10.11429/ptmps1907.7.14_250}{10.11429/ptmps1907.7.14\\_250}.\n\\bibitem{farey1816} J.~Farey,\n  \\href{https://doi.org/10.1080/14786441608628487}{\\emph{On a curious property of vulgar fractions}},\n  Philos. Mag. \\textbf{47} (1816), no.~217, 385--386.\n\\bibitem{stern1858} M.~Stern,\n  \\href{https://doi.org/10.1515/crll.1858.55.193}{\\emph{Ueber eine zahlentheoretische Funktion}},\n  J. Reine Angew. Math. \\textbf{55} (1858), 193--220.\n\\bibitem{alloucheshallit} J.-P. Allouche and J. Shallit,\n  \\href{https://cs.uwaterloo.ca/~shallit/Papers/as0.pdf}{\\emph{The ring of\n  $k$-regular sequences}}, Theoret. Comput. Sci. \\textbf{98} (1992), no.~2,\n  163--197,\n  doi:\\href{https://doi.org/10.1016/0304-3975(92)90001-V}\n  {10.1016/0304-3975(92)90001-V}.  Definitions~1.1 and~2.1 are on\n  pp.~2--3 of the linked author preprint; those definitions are unchanged\n  in the published version.\n\\bibitem{coons2010} M.~Coons, \\emph{(Non)automaticity of number theoretic\n  functions}, J. Th\\'eor. Nombres Bordeaux \\textbf{22} (2010), no.~2,\n  339--352, doi:\\href{https://doi.org/10.5802/jtnb.718}{10.5802/jtnb.718}.\n  Theorem~3.2, pp.~348--349, is the totient nonregularity result; it appears\n  as Theorem~3.3 on pp.~8--9 of \\href{https://arxiv.org/abs/0810.3709}{the\n  arXiv preprint}.\n\\bibitem{bellsmertnig2026} J.~Bell and D.~Smertnig,\n  \\emph{Mahler series with multiplicative coefficient sequences}, 2026,\n  \\href{https://doi.org/10.48550/arXiv.2603.23456}{arXiv:2603.23456}.\n  Theorem~1.3 is on pp.~2--3; the non-\\(k\\)-Mahler consequence for Euler's\n  totient and Corollary~1.4 are recorded on p.~3.\n\\bibitem{nesterenko} Yu.~V. Nesterenko, \\emph{Modular functions and transcendence questions},\n  Sb. Math. \\textbf{187} (1996), no.~9, 1319--1348,\n  doi:\\href{https://doi.org/10.1070/SM1996v187n09ABEH000158}{10.1070/SM1996v187n09ABEH000158}.\n  Theorem~1 and Corollary~2 on p.~1320 are stated in the Ramanujan\n  \\(q\\)-series variables \\(P(q),Q(q),R(q)\\); the latter is the\n  algebraic-\\(q\\) specialization used here.\n\\bibitem{postelmansvanassche} K.~Postelmans and W.~Van Assche,\n  \\href{https://arxiv.org/abs/math/0604312v1}{\\emph{Irrationality of $\\zeta_q(1)$ and $\\zeta_q(2)$}},\n  J. Number Theory \\textbf{126} (2007), no.~1, 119--154,\n  doi:\\href{https://doi.org/10.1016/j.jnt.2006.11.011}{10.1016/j.jnt.2006.11.011}.\n  The linear-independence statement is Theorem~1.3 in the published version\n  and Theorem~3 on p.~4 of the linked arXiv preprint; its preprint proof is\n  Section~7, pp.~30--34.\n\\bibitem{lucatachiya2017} F.~Luca and Y.~Tachiya,\n  \\href{https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2014-14.pdf}\n  {\\emph{Linear independence results for the values of divisor functions\n  series}}, RIMS K\\^oky\\^uroku No.~2014 (2017), 138--150.\n  Theorem~A on p.~139 explicitly restates their earlier Theorem~1.1 for\n  nonzero purely periodic integer weights.  Theorem~1 is on p.~139, its\n  examples are on p.~140, and its proof is on pp.~149--150.\n\\bibitem{duverneytachiya} D.~Duverney and Y.~Tachiya,\n  \\href{https://danielduverney.fr/documents/theorie-des-nombres/DuverneyTachiya190522.pdf}\n  {\\emph{Refinement of the Chowla--Erd\\H{o}s method and linear independence\n  of certain Lambert series}}, Forum Math. \\textbf{31} (2019), no.~6,\n  1557--1566,\n  doi:\\href{https://doi.org/10.1515/forum-2018-0299}{10.1515/forum-2018-0299}.\n  Corollary~1.2 and Example~1.1 are on p.~4 of the linked author preprint;\n  the proof is on pp.~9--11.\n\\bibitem{kovactao} V.~Kova\\v{c} and T.~Tao,\n  \\href{https://arxiv.org/abs/2406.17593v4}{\\emph{On several irrationality problems for\n  Ahmes series}}, Acta Math. Hungar. \\textbf{175} (2025), no.~2, 572--608,\n  doi:\\href{https://doi.org/10.1007/s10474-025-01528-0}{10.1007/s10474-025-01528-0}.\n\\bibitem{wang2026} H.~Wang,\n  \\href{https://arxiv.org/abs/2606.24972v2}{\\emph{Positive dyadic density for\n  rational weighted binary expansions}}, arXiv:2606.24972v2\n  (revised 15 July 2026),\n  doi:\\href{https://doi.org/10.48550/arXiv.2606.24972}{10.48550/arXiv.2606.24972}.\n\\bibitem{crandall} R.~Crandall,\n  \\href{https://math.colgate.edu/~integers/m23/m23.pdf}\n  {\\emph{The googol-th bit of the Erd\\H{o}s--Borwein constant}},\n  Integers \\textbf{12} (2012), electronic article \\#A23 (open 18-page\n  version); hardcopy no.~5, 811--840,\n  doi:\\href{https://doi.org/10.1515/integers-2012-0007}{10.1515/integers-2012-0007}.\n  The remote-bit result is in \\S5, open-PDF pp.~6--7; the finite-prefix\n  computation is in \\S6, pp.~7--9; and the infinite-\\(11\\) question is in\n  \\S7, p.~9.\n\\bibitem{campbell} J.~M. Campbell,\n  \\href{https://arxiv.org/abs/2605.24160v1}{\\emph{On the binary digits of the\n  Erd\\H{o}s--Borwein constant}}, arXiv:2605.24160v1 (22 May 2026),\n  doi:\\href{https://doi.org/10.48550/arXiv.2605.24160}{10.48550/arXiv.2605.24160}.\n  Theorem~1 is on p.~12 and its proof is on pp.~12--24; the AGP special\n  case is Lemma~1 and Remark~1 on p.~5.\n\\bibitem{alfordgranvillepomerance} W.~R. Alford, A.~Granville, and C.~Pomerance,\n  \\emph{There are infinitely many Carmichael numbers}, Ann. of Math. (2)\n  \\textbf{139} (1994), no.~3, 703--722,\n  doi:\\href{https://doi.org/10.2307/2118576}{10.2307/2118576}.\n  The progression property used here is (0.3) on p.~705; Theorem~2.1\n  (p.~712, proof pp.~712--715) supplies the underlying\n  exceptional-modulus estimate.\n\\bibitem{taoteravainen} T.~Tao and J.~Ter\\\"av\\\"ainen,\n  \\href{https://arxiv.org/abs/2512.01739v2}{\\emph{Quantitative correlations and\n  some problems on prime factors of consecutive integers}}, arXiv:2512.01739v2\n  (revised 25 April 2026),\n  doi:\\href{https://doi.org/10.48550/arXiv.2512.01739}{10.48550/arXiv.2512.01739}.\n  Theorem~1.3 is on p.~4 and its proof is Section~5, pp.~44--56.\n\\bibitem{lean} L.~de Moura and S.~Ullrich,\n  \\href{https://doi.org/10.1007/978-3-030-79876-5_37}\n  {\\emph{The Lean~4 theorem prover and programming language}},\n  in \\emph{CADE-28}, LNCS 12699 (2021), 625--635,\n  doi:\\href{https://doi.org/10.1007/978-3-030-79876-5_37}\n  {10.1007/978-3-030-79876-5\\_37}.\n\\bibitem{mathlib} The mathlib Community,\n  \\href{https://doi.org/10.1145/3372885.3373824}\n  {\\emph{The Lean mathematical library}},\n  in \\emph{CPP 2020}, ACM, 2020, 367--381,\n  doi:\\href{https://doi.org/10.1145/3372885.3373824}\n  {10.1145/3372885.3373824}.  The article describes a December 2019\n  Lean~3-era snapshot; the repository lock owns the current revision.\n\\bibitem{erdosproblems} T.~F. Bloom,\n  \\href{https://www.erdosproblems.com/249}{\\emph{Erd\\H{o}s Problem \\#249}} and\n  \\href{https://www.erdosproblems.com/257}{\\emph{Erd\\H{o}s Problem \\#257}}\\unskip,\n  accessed 28 July 2026.  The \\#249 page displays ``last edited 28 September\n  2025'' and cites \\texttt{[ErGr80, p.~61]} and\n  \\texttt{[Er88c, p.~102]}; the \\#257 page displays ``last edited 15 April\n  2026'' and cites \\texttt{[Er68d]}, \\texttt{[ErGr80, p.~62]} and\n  \\texttt{[Er88c, p.~105]}.  Both pages explicitly describe OPEN as the\n  website owner's present assessment rather than a literature-completeness\n  guarantee.\n\n\\end{thebibliography}","boundary":"Historical narrative owner at the formal-source revision. It is not an active paper and is carried only to resolve contribution-family provenance and bibliography references."}},"problem_edge_contract":{"relation_vocabulary":["direct_target","direct_special_case","shared_programme_context","supporting_or_contextual"],"basis_vocabulary":{"problem_binding":"derived from the row's authored problem_binding","claim_prose":"the claim sentence explicitly names the problem","source_family":"the source-authored contribution family identifies the named #257 support special cases"},"coarse_binding_rule":"problem_ids is retained for programme navigation; direct answer routes select only direct_target and direct_special_case edges."},"coordinate_conflict_rule":"Where the claims registry and the declaration atlas disagree about a line, both numbers are carried and the row is flagged line_disputed. Neither authority is silently preferred; the module and declaration name still identify the theorem unambiguously.","how_to_query":{"load":"import json; p = json.load(open(PACKET))","all_rows":"rows = p['mathematical_assertions']['assertions']","strongest_unconditional":"[r for r in rows if r['status'] == 'unconditional progress']","one_problem_everything":"[r for r in rows if 'erdos_249' in (r.get('problem_ids') or [])]","registered_claims_for_one_problem":"[r for r in rows if r['assertion_kind'] == 'registered_claim' and 'erdos_249' in (r.get('problem_ids') or [])]","direct_claims_for_one_problem":"[r for r in rows if r['assertion_kind'] == 'registered_claim' and any(e.get('problem_id') == 'erdos_249' and e.get('relation') in ('direct_target', 'direct_special_case') for e in (r.get('problem_edges') or []))]","what_a_claim_does_not_establish":"[d['statement'] for r in rows if r['assertion_id'] == 'claim:dyadic_totient_certificate_interface' for d in (r.get('does_not_establish') or [])]","what_is_not_claimed":"[r for r in rows if r['assertion_kind'] in ('non_claim', 'problem_not_checked')]","what_remains_open":"[r for r in rows if r['assertion_kind'] in ('remaining_open_proposition', 'open_obligation')]","citable_links_for_a_row":"[l['url'] for r in rows if r['assertion_id'] == 'claim:dyadic_totient_certificate_interface' for l in (r.get('links') or []) if l.get('cite_this')]","checked_but_not_reviewed":"[r for r in rows if r.get('reviewed') is False and r['assertion_kind'] == 'problem_checked']","resolve_a_paper_anchor":"[r['paper_anchor'] for r in rows if r.get('paper_label') == 'thm:kkernelrank' and r.get('paper_anchor')]","modules_backing_a_problem_row":"p['mathematical_assertions']['problem_digest']['erdos_243']['modules']"},"join_rule":"Rows carry their own content, status and boundary. Anything that belongs to a whole problem rather than to one sentence -- its modules, its papers, its registry posture -- is carried once in problem_module_index and reached through the row's problem_ids or problem_join pointer.","contribution_families":[{"id":"classical_full_support_and_named_257_families","claim_ids":["eb_full_support","eb_constant","support_families","squarefree_support_irrationality","prime_support_irrationality","universal_257"],"status_summary":"Classical formalisation, cited special cases, and the open universal target.","prior_art_posture":"Erdős owns full support and the pairwise-coprime family; Duverney–Tachiya own squarefree-support irrationality at every power-of-two base; Tao–Teräväinen own the base-2 prime-support theorem. The cited squarefree theorem does not cover integer bases that are not powers of two, and the cited prime-power extension is remark-only. No priority claim is made for the formalised families.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim eb_full_support","consumer_or_open_obligation":"Establishes the classical benchmark and leaves remaining_open.universal_257_all_infinite_supports.","view_decision":"gateway_headline"},{"id":"periodic_weighted_lambert_theorems","claim_ids":["periodic_nonnegative_weight_irrationality","periodic_signed_weight_dichotomy","period_four_signed_residue_obstruction"],"status_summary":"One classical/broader antecedent, one exact signed dichotomy, and one proved residue obstruction.","prior_art_posture":"Luca–Tachiya prove the broader eventually-periodic rational-coefficient theorem by another method; the signed endpoint formalised here is only irrational-or-b-power-denominator.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim periodic_nonnegative_weight_irrationality","consumer_or_open_obligation":"Supports structured #257 families; unconditional signed irrationality would require excluding base termination.","view_decision":"gateway_summary_reference_detail"},{"id":"orthogonal_petal_and_divisor_layers","claim_ids":["orthogonal_petal_sunflower_reduction","composite_dilation_defect_identity","mixed_prime_power_layer_identity"],"status_summary":"One conditional support-family reduction and two exact local divisor identities.","prior_art_posture":"Classical divisor algebra is background; no combined selector theorem is identified, and absence of a match is not a novelty claim.","primary_narrative_owner":"docs/claims.json::claims","source_route":"python3 scripts/query_corpus.py --claim orthogonal_petal_sunflower_reduction","consumer_or_open_obligation":"The uniform tail-selection predicate for orthogonal-petal bouquets remains open.","view_decision":"living_reference_only"},{"id":"generic_binary_carry_rigidity","claim_ids":["tail_orbit_rigidity","boolean_mobius_carry","reciprocal_mass_bound","unbounded_carry_states","sublog_zero_windows"],"status_summary":"Exact rationality characterisations and proved necessary consequences.","prior_art_posture":"Han Wang supplies the closest weighted-binary carry recurrence; the abstraction, converse, and downstream constraints make no priority claim.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim tail_orbit_rigidity","consumer_or_open_obligation":"Constrains any rational #257 counterexample without settling universal irrationality.","view_decision":"gateway_appendix_reference_detail"},{"id":"achievement_geometry_and_half_target_interface","claim_ids":["campbell_shift_synchronization_boundary","greedy_achievement_geometry","half_greedy_two_thirds_band","half_second_channel_reduction","mobius_sign_support_no_go","twenty_one_quotient_greedy_frontier"],"status_summary":"Classical geometry formalised, exact local bands, a conditional second channel, and scoped no-go boundaries.","prior_art_posture":"Kovač–Tao supply the strict-tail subsum geometry; the classical Lambert–Möbius identity is the baseline for the scoped sign-support no-go; Crandall and Campbell concern adjacent binary-digit questions and are not proof inputs. No priority claim is made for the sign separation.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim greedy_achievement_geometry","consumer_or_open_obligation":"Decide whether 1/2 belongs to the achievement set; positive membership refutes universal #257.","view_decision":"gateway_headline"},{"id":"half_carry_compactness_consumers","claim_ids":["half_trapping_return_carry_interface","half_carry_reachability_compactness","cofinal_strip_return_closure","terminal_only_cofinal_closure","terminal_scaled_vanishing_closure","half_cylinder_finite_shadow"],"status_summary":"Conditional compactness consumers from progressively weaker finite-word or carry supplies.","prior_art_posture":"No executable seam/closure antecedent is recorded beyond the global achievement-set geometry.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim half_carry_reachability_compactness","consumer_or_open_obligation":"Supply canonical seams, strip returns, terminal witnesses, or scaled-vanishing words cofinally.","view_decision":"gateway_appendix_future_257_companion"},{"id":"dyadic_prefix_block_safety_interfaces","claim_ids":["dyadic_prefix_block_safety_interfaces"],"status_summary":"Exact reduced-denominator transport, block-safety and primitive-overshoot equivalences, conditional half-membership consumers, and explicit failures of naive induction.","prior_art_posture":"Exact rational and greedy-block formalisation; no priority claim is made for the assembled source-level package.","primary_narrative_owner":"docs/SOURCE_MAP.md","source_route":"python3 scripts/query_corpus.py --claim dyadic_prefix_block_safety_interfaces","consumer_or_open_obligation":"Supply actual block-safety, raw-margin, or primitive-overshoot witnesses cofinally; no such unbounded producer is proved. Until the qualified #257 companion has a citable low-duplication theorem spine, this package remains source-map material rather than another gateway inventory.","view_decision":"source_map_now_qualified_future_257_companion_source_only"},{"id":"rewind_windows_and_protected_seams","claim_ids":["fixed_coefficient_rewind_contraction","selected_half_carry_window_compactness","realized_rewind_phase_identity","localized_protected_seam_compactness","mixed_cofinal_half_carry_fan_in","rewind_seam_operational_bridge","selected_window_depth18_base"],"status_summary":"Conditional rewind/window consumers plus one verified depth-18 base.","prior_art_posture":"No theorem-level match is recorded; any novelty posture remains provisional.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim selected_half_carry_window_compactness","consumer_or_open_obligation":"Produce repeated object-level coefficient agreement or protected seams cofinally.","view_decision":"gateway_appendix_future_257_companion"},{"id":"fatal_gap_and_last_producer_spine","claim_ids":["fatal_gap_right_tail_classification","final_middle_cell_escape","final_middle_neg_two_phase_sieve","half_membership_seam_classification","seam_producer_carry_alignment","concrete_producer_carry_coordinates","last_producer_tail_escape_reduction","middle_producer_cardinality_reduction","fixed_tail_survival_socket","producer_carry_future_tail_socket","integer_greedy_first_wrap_reduction","concrete_seam_adapter"],"status_summary":"Exact half-membership/nonmembership classifications, proved branch eliminations, a rank-26 phase sieve for the middle -2 cell, and explicit conditional tail consumers.","prior_art_posture":"No exact local seam or last-producer antecedent is recorded beyond strict-tail achievement-set background.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim half_membership_seam_classification","consumer_or_open_obligation":"Eliminate the 412 surviving classes modulo 2730 in the middle -2 cell, then handle the middle -1 and right-tail branches or prove future-tail escape directly.","view_decision":"gateway_spine_future_257_companion_detail"},{"id":"largest_skip_and_reset_deficit_frontier","claim_ids":["largest_skip_gap_propagation","boundary_pulse_normalization","largest_skip_induction_reduction","quarter_band_endpoint_cells","reset_deficit_escape_geometry","upper_reset_band_certificates_through_thirty"],"status_summary":"Conditional largest-skip propagation, exact pulse geometry, and finite reset-band evidence.","prior_art_posture":"No close theorem-level antecedent is recorded.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim largest_skip_gap_propagation","consumer_or_open_obligation":"Supply a global first-crossing or anti-concentration theorem preventing persistent right-branch pinning.","view_decision":"gateway_appendix_future_257_companion"},{"id":"suffix_cylinders_and_feedback_boundary","claim_ids":["selected_suffix_cylinder_propagation","selected_suffix_pre_feedback_run","half_divisor_unit_drop","rewind_boundary_pair_obstruction"],"status_summary":"An exact conditional child-gap recurrence with a cofinal half-value consequence, verified finite stages through depth 52, an exact unit drop, and a concrete scalar-seam obstruction.","prior_art_posture":"Kovač–Tao supply the surrounding strict-tail achievement-set geometry. No exact antecedent for this finite carry recurrence is recorded, and no novelty claim is made.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim selected_suffix_cylinder_propagation","consumer_or_open_obligation":"Establish the child-gap inequalities at arbitrarily large depths, or otherwise construct cofinal full suffix-cylinder stages.","view_decision":"future_257_companion_reference_stages"},{"id":"mersenne_lambert_classical_ladder","claim_ids":["mersenne_lambert_ladder","sigma_transcendence"],"status_summary":"Classical identities formalised and one transcendental rung cited only.","prior_art_posture":"Apostol, Merca, and Merca–Schmidt own the transform background; Nesterenko owns the cited transcendence theorem.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim mersenne_lambert_ladder","consumer_or_open_obligation":"Provides the conceptual bridge between the classical #257 benchmark and the open #249 constant.","view_decision":"gateway_spine"},{"id":"probabilistic_gcd_and_stern_brocot_geometry","claim_ids":["totient_coprimality_probability","squared_lambert_gcd_moments","stern_brocot_cylinder_law","stern_brocot_run_fibonacci_stability"],"status_summary":"Exact coprimality and gcd-moment identities, a Stern–Brocot cylinder law with uniform geometric convergence, and source-level Fibonacci/continuant run stability.","prior_art_posture":"Classical Lambert-series, convolution, Pillai gcd-sum, probability, and Stern–Brocot background; Postelmans–Van Assche own the cited irrationality theorem for the constant-weight squared-Lambert value. No priority claim is made for this assembled formal package.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --route probabilistic_gcd_geometry","consumer_or_open_obligation":"Recasts #249 and exposes independent gcd-distribution and continued-fraction structure; the run package has no endpoint consumer, and irrationality remains open.","view_decision":"gateway_supporting_theorems_and_living_reference_detail"},{"id":"erdos249_quantitative_headline","claim_ids":["denominator_exclusion","erdos_249"],"status_summary":"One unconditional quantitative theorem and the open irrationality claim.","prior_art_posture":"Farey is historical language only; no exact antecedent for the explicit bound is identified, without asserting novelty.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim denominator_exclusion","consumer_or_open_obligation":"Leaves remaining_open.erdos_249_irrationality.","view_decision":"gateway_headline"},{"id":"exact_certificate_equivalence_and_deposits","claim_ids":["certificate_reduction","certificate_completeness","certified_kill_instances","diagonal_pincer_t64_primality_closure","diagonal_pincer_t64_endpoint_certificate"],"status_summary":"Exact equivalence and completeness with bounded verified deposits.","prior_art_posture":"Classical Lambert and Möbius identities underlie the construction; no exact certificate-equivalence antecedent is recorded, without a novelty conclusion.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim certificate_completeness","consumer_or_open_obligation":"Produce certificates at unbounded parameters.","view_decision":"gateway_headline_finite_matrix"},{"id":"diagonal_pincer_and_denominator_survival","claim_ids":["diagonal_pincer_reduction","diagonal_fresh_loss_reduction","actual_foreign_residue_projection","squared_mersenne_enclosure","mersenne_shadow_denominator"],"status_summary":"Exact diagonal targets, sharp enclosures and denominators, and conditional avoidance projections.","prior_art_posture":"Classical Möbius/Lambert algebra only; no theorem-specific antecedent is identified.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim diagonal_pincer_reduction","consumer_or_open_obligation":"Supply unbounded full-target avoidance, fresh-loss separation, or the missing analytic complement bridge.","view_decision":"gateway_supporting_spine"},{"id":"fresh_loss_local_arithmetic","claim_ids":["fresh_prime_deficit_decomposition","power_two_bit_lift","parity_socket_obstruction","flexible_odd_window_reduction","flexible_odd_window_affine_frontier","canonical_jump_slack_reduction"],"status_summary":"Exact local arithmetic, conditional band/slack reductions, and scoped parity ceilings.","prior_art_posture":"No close statement comparison is recorded; novelty remains unasserted.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim canonical_jump_slack_reduction","consumer_or_open_obligation":"Prove cofinal nonnegative slack or a correlated multi-rank band supply.","view_decision":"gateway_appendix_reference_detail"},{"id":"alternative_249_algebraic_boundaries","claim_ids":["primitive_coordinate_index_obstruction","primitive_unit_gap_ceiling","mersenne_tail_atom_boundary","adelic_height_obstruction","square_crt_correction_suppression","signed_dyadic_moment_substrate","dyadic_totient_basis","dyadic_totient_certificate_interface","all_base_totient_kernel_index_reduction","all_base_totient_kernel_conditional_rank","residual_gauge_obstruction","totient_carry_kernel_anti_compression","totient_residue_series_classification","totient_residue_series_integer_radix","totient_residue_series_dilations"],"status_summary":"Exact finite obstructions and interfaces, a constructive dyadic basis, and an all-base arithmetic/spanning layer. The reusable rank wrapper is conditional; AllBaseTotientKernel.lean separately proves its independence premise and the complete rank unconditionally. The residue-series claims classify the rational totient residue series and prove the dilated series linearly independent.","prior_art_posture":"Classical Euler/Möbius, CRT, determinant, and Lambert tools; Coons already implies full-kernel infinite rank, and Martin supplies a broader external affine-independence antecedent. The Lean all-base proof uses CRT and Dirichlet independently. Erdős 1957 (Lemma 1) and Kaneko–Suzuki–Tachiya (Theorem 3) already give the irrationality of the totient residue series. No #249 irrationality endpoint or priority claim.","primary_narrative_owner":"docs/claims.json::claims","source_route":"python3 scripts/query_corpus.py --route lambert_obstruction_interfaces","consumer_or_open_obligation":"No rationality-to-finite-rank bridge or #249 irrationality endpoint is proved. The all-base independence premise is discharged in AllBaseTotientKernel.lean; the reusable conditional interface remains separately recorded.","view_decision":"gateway_selected_boundaries_and_living_reference_detail"},{"id":"transport_curvature_finite_difference_architecture","claim_ids":["transport_curvature_reductions","affine_transport_annihilation","transport_certificate_instances"],"status_summary":"Exact affine annihilation, conditional finite-difference consumers, and two verified windows.","prior_art_posture":"Finite differences and affine moments are general techniques; no matching combined theorem was identified, and that absence is not a novelty claim.","primary_narrative_owner":"docs/claims.json::claims","source_route":"python3 scripts/query_corpus.py --claim affine_transport_annihilation","consumer_or_open_obligation":"Produce an unbounded supply of a named transport certificate.","view_decision":"registry_only_after_one_sided_companion_retirement"},{"id":"harmonic_and_phase_reductions","claim_ids":["first_harmonic_certificate_interface","first_harmonic_pivot_decomposition","first_harmonic_good_index_reduction","adjacent_phase_separation_reduction","pivot_anti_reconstruction_reduction","strict_prime_tail_orbit_gap","totient_v8_conditional_peripheral_assembly","totient_v8_finite_arithmetic_threshold","totient_v8_generic_centering_countermodel","totient_v8_one_finite_first_harmonic_instance","totient_v8_peripheral_finite_budget_bridge"],"status_summary":"Conditional harmonic and phase consumers, including an exact four-term pivot decomposition and budget, an exact window-pair equivalence, and the two-branch tail-orbit phase-density and natural-prime strict-gap producers that feed the same certificate socket.","prior_art_posture":"No matching combined theorem is identified; all novelty language remains provisional.","primary_narrative_owner":"docs/claims.json::claims","source_route":"python3 scripts/query_corpus.py --claim pivot_anti_reconstruction_reduction","consumer_or_open_obligation":"Supply a cofinal first-harmonic pivot budget, a cofinal 603X/1000 first-harmonic bound on the good indices, adjacent-pair separator, anti-reconstruction producer, or one of the two tail-orbit strict-gap antecedents (late-block nonpositive phase density, or cofinally many natural primes with tail-orbit real part below 9/10).","view_decision":"registry_only_after_one_sided_companion_retirement"},{"id":"transport_strategy_no_go_boundaries","claim_ids":["fixed_precision_transport_no_go","lcm_factor_ideal_anchor_pulse_no_go"],"status_summary":"Two unconditional scoped countermodels to fixed local precision and finite shift algebra.","prior_art_posture":"No matching theorem is identified; the record explicitly refuses to infer novelty from that.","primary_narrative_owner":"docs/claims.json::claims","source_route":"python3 scripts/query_corpus.py --claim fixed_precision_transport_no_go","consumer_or_open_obligation":"Future arguments must use growing precision, global correlation, nonlinear arithmetic, or actual fresh-channel structure.","view_decision":"registry_only_after_one_sided_companion_retirement"},{"id":"actual_lcm_orbit_and_sign_frontier","claim_ids":["actual_lcm_orbit_nonintegrality_frontier","actual_lcm_positive_corridor_top_edge"],"status_summary":"An exact actual-LCM cofinal non-integrality reformulation and an unconditional positive-corridor/top-edge boundary.","prior_art_posture":"The actual-LCM tail equivalence and sign/carry analysis are source-specific; no novelty or priority claim is made.","primary_narrative_owner":"paper/249/erdos-249-binary-totient-series.tex","source_route":"python3 scripts/query_corpus.py --claim actual_lcm_orbit_nonintegrality_frontier","consumer_or_open_obligation":"Produce a cofinal actual-LCM non-integrality supply or independently exclude the top-edge boundary.","view_decision":"registry_only_after_one_sided_companion_retirement"},{"id":"factorial_quotient_band_breakpoint","claim_ids":["erdos_68","channel_quotient_band_breakpoint"],"status_summary":"An open factorial-denominator endpoint plus an unconditional finite quotient-band factorisation and breakpoint under channel cancellation.","prior_art_posture":"The quotient-band identity and its breakpoint consequence are source-specific formal results; no novelty, priority, or significance claim is made.","primary_narrative_owner":"paper/68/erdos-68-factorial-denominator-irrationality.tex","source_route":"python3 scripts/query_corpus.py --claim channel_quotient_band_breakpoint","consumer_or_open_obligation":"The finite-family result supplies no cancelling family, simultaneous-channel control, residual estimate, or cofinal non-unit carry for the still-open #68 endpoint.","view_decision":"problem_note_reference_detail"},{"id":"three_prime_residue_escape_consumers","claim_ids":["erdos_269","three_prime_finite_residue_contradiction","three_prime_cofinal_window_carry_extinction","weighted_phase_carry_finite_observer","three_prime_exact_modulus_pinning","three_prime_octic_window_equivalence","three_prime_finite_prefix_completion_boundary","three_prime_bounded_certificate_safety"],"status_summary":"An open three-prime #269 endpoint, a finite least-positive-residue contradiction, a conditional cofinal-window extinction consumer, and the weighted-phase carry split with its finite realised-span observer.","prior_art_posture":"The residue classifier and bounded-carry consumer are source-specific formal results; no novelty, priority, significance, or three-prime irrationality claim is made.","primary_narrative_owner":"paper/269/erdos-269-three-prime-running-lcm.tex","source_route":"python3 scripts/query_corpus.py --claim three_prime_finite_residue_contradiction","consumer_or_open_obligation":"The actual rationality-to-positive-reduced-carry bridge is established. The denominator-dependent cofinal local-window escape remains open, so the consumers do not settle the three-prime problem; the observer supplies no two-dimensional representation and excludes none.","view_decision":"problem_note_reference_detail"},{"id":"prime_gap_tail_shift_frontier","claim_ids":["erdos_251","prime_gap_unboundedness_and_nonperiodicity","prime_gap_irrationality_equivalence","prime_gap_identity_tao"],"status_summary":"An open prime-index endpoint, exact coefficient constraints, a Lean-formalised prime/gap irrationality equivalence under an elementary polynomial prime bound, and Tao's already-known identity recorded as cited only.","prior_art_posture":"The prime-to-gap identity is Tao's (erdosproblems.com/251, 7 October 2025). This checkout formalises a version that discharges summability without the prime-number theorem. No novelty, priority, significance, or irrationality claim is made.","primary_narrative_owner":"paper/251/erdos-251-prime-gap-dyadic-series.tex","source_route":"python3 scripts/query_corpus.py --claim prime_gap_unboundedness_and_nonperiodicity","consumer_or_open_obligation":"The exact prime-gap coefficient facts do not supply the cofinal non-integrality or small-shift escape required for Erdős #251.","view_decision":"problem_note_reference_detail"},{"id":"conditional_half_countermodel_frontiers","claim_ids":["terminal_scaled_vanishing_half_countermodel","cofinal_cylinder_half_countermodel"],"status_summary":"Two conditional routes to an infinite rational half-valued Mersenne support: terminal scaled-vanishing and cofinal full-cylinder stages.","prior_art_posture":"These are source-specific conditional consumers; no existence, novelty, priority, or significance claim is made for either antecedent.","primary_narrative_owner":"paper/257/erdos-257-mersenne-support-subseries.tex","source_route":"python3 scripts/query_corpus.py --claim terminal_scaled_vanishing_half_countermodel","consumer_or_open_obligation":"Construct or refute the corresponding terminal-scaled or cofinal-cylinder antecedent; neither route settles half-membership or universal Erdős #257 without that antecedent.","view_decision":"future_257_companion_reference_stages"},{"id":"newton_ray_separation_frontier","claim_ids":["erdos_1041","newton_ray_separation_consumer"],"status_summary":"Finite Newton-flow ray-separation and perturbative root-retention inputs, alongside the historical short-connection question whose correspondence with the separately checked total-variation counterexample awaits independent review.","prior_art_posture":"The finite ray and perturbation statements are source-specific formal results. Their checked hypotheses do not imply unrestricted connection; no novelty, priority, significance or independent human-review claim is made.","primary_narrative_owner":"paper/1041/erdos-1041-lemniscate-newton-flow.tex","source_route":"python3 scripts/query_corpus.py --claim newton_ray_separation_consumer","consumer_or_open_obligation":"These finite conditional consumers do not construct a universal short connector. The exact Formal Conjectures path-image-length statement and the total-variation formulation are refuted by the separate degree-seven result; historical correspondence and valid positive-family hypotheses remain distinct questions.","view_decision":"problem_note_reference_detail"},{"id":"reciprocal_tail_rigidity_frontier","claim_ids":["erdos_243","bounded_rise_fresh_modulus_barrier","eventually_periodic_negative_no_go","bounded_negative_exclusion","cubic_survivor_phase_density","cubic_rate_irrationality_unconditional","nonintegral_regular_rate_irrationality"],"status_summary":"An open eventual-Sylvester-recurrence endpoint plus Lean-checked irrationality theorems for the stated cubic rate and every nonintegral regular rate above one, unconditional bounded-rise and eventually-periodic negative-orbit obstructions, and a conditional cofinally bounded negative-magnitude exclusion.","prior_art_posture":"These are source-specific formal results; no novelty, priority, significance, or #243 solution claim is made, and normalised vanishing, mixed-sign, and unbounded-negative boundaries remain explicit.","primary_narrative_owner":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","source_route":"python3 scripts/query_corpus.py --claim bounded_rise_fresh_modulus_barrier","consumer_or_open_obligation":"The cubic and nonintegral-rate theorems apply only under their respective quantitative asymptotics. The three no-go consumers remove fresh-modulus avoidance, eventually-periodic negative tails, and the cofinally bounded negative branch under normalised vanishing; the original eventual-recurrence assertion still requires control of the mixed-sign unbounded orbit regime.","view_decision":"problem_note_reference_detail"},{"id":"finite_period_noncollapse_frontier","claim_ids":["finite_period_noncollapse"],"status_summary":"An unconditional finite-support denominator noncollapse theorem for Mersenne-type sums, with the infinite-support irrationality boundary still open.","prior_art_posture":"The finite-support order theorem is a source-specific formal result; no novelty, priority, significance, or universal #257 solution claim is made.","primary_narrative_owner":"paper/257/erdos-257-mersenne-support-subseries.tex","source_route":"python3 scripts/query_corpus.py --claim finite_period_noncollapse","consumer_or_open_obligation":"The finite order identity supplies denominator growth for finite partial sums, but no unbounded support certificate is supplied and the universal infinite-support #257 endpoint remains open.","view_decision":"problem_note_reference_detail"},{"id":"rational_base_coordinatewise_boundaries","claim_ids":["erdos_1049","three_halves_coordinatewise_corridor_no_go","rational_base_cleared_tail_recurrence","rational_base_forcing_growth","van_assche_qapery_diagonal_no_go"],"status_summary":"An open universal rational-base endpoint, a base-3/2 coordinatewise exclusion, exact cleared-tail dynamics and forcing bounds, and a finite q-Apéry diagonal non-transfer result.","prior_art_posture":"These are source-specific formal arithmetic results and an open target; no novelty, priority, significance, or irrationality claim is made for the new declarations.","primary_narrative_owner":"paper/1049/erdos-1049-rational-base-lambert.tex","source_route":"python3 scripts/query_corpus.py --claim three_halves_coordinatewise_corridor_no_go","consumer_or_open_obligation":"The corridor exclusion covers only its stated six-condition architecture, while the recurrence and forcing bound supply no rationality bridge or bounded-state contradiction; irrationality at 3/2 and the universal #1049 assertion remain open.","view_decision":"problem_note_reference_detail"},{"id":"zudilin_rational_base_and_hankel_theorems","claim_ids":["zudilin_rational_base_region","zudilin_all_rank_hankel_order","coefficient_pencil_finite_rank"],"status_summary":"Formalized rational-base irrationality and measure in the contour region, all-rank normalized Hankel order and coefficient formulas, and the coefficient-pencil proposition through rank eight.","prior_art_posture":"Zudilin supplies the cited 2004 polynomial-form construction and 2016 Hankel sequence. The formalizations preserve their hypotheses and make no novelty, priority or independent human-review claim.","primary_narrative_owner":"paper/1049/erdos-1049-rational-base-lambert.tex","source_route":"python3 scripts/query_corpus.py --claim zudilin_rational_base_region","consumer_or_open_obligation":"The contour-region theorems include all positive powers of 31/4. The coefficient-pencil result is finite-rank only; its all-rank extension, irrationality at base 3/2, and universal rational-base irrationality remain open.","view_decision":"problem_note_reference_detail"},{"id":"weighted_and_cover_support_criteria","claim_ids":["finite_prime_weighted_support","strengthened_positive_cover_support","mixed_weighted_cover_support","reciprocal_summable_support"],"status_summary":"Unconditional hereditary support criteria checked in Lean; universal Erdős #257 remains open.","prior_art_posture":"The formal statements preserve their weighted-summability and positive-cover assumptions. No novelty, priority, or independent human-review claim is made.","primary_narrative_owner":"paper/257/erdos-257-mersenne-support-subseries.tex","source_route":"python3 scripts/query_corpus.py --claim finite_prime_weighted_support","consumer_or_open_obligation":"remaining_open.universal_257_all_infinite_supports","view_decision":"problem_note_reference_detail"},{"id":"degree_seven_total_variation_counterexample","claim_ids":["ani_degree_seven_total_variation_counterexample"],"status_summary":"Ani's explicit degree-seven polynomial refutes the exact Formal Conjectures path-image-length statement by a Lean-checked one-dimensional Hausdorff-measure bound; the total-variation formulation is also refuted.","prior_art_posture":"Formalisation of ani’s existing construction; no novelty or independent human review of historical correspondence is claimed.","primary_narrative_owner":"paper/1041/erdos-1041-lemniscate-newton-flow.tex","source_route":"python3 scripts/query_corpus.py --claim ani_degree_seven_total_variation_counterexample","consumer_or_open_obligation":"Independent human review of correspondence with the 1958 curve-length wording remains open; the exact Formal Conjectures Hausdorff-length statement is checked and refuted.","view_decision":"problem_note_reference_detail"},{"id":"separated_cut_algebraic_lambert_transcendence_family","claim_ids":["separated_cut_algebraic_lambert_transcendence"],"status_summary":"Reviewed ordinary proof; full infinite statement not formalised and independent human review not recorded.","prior_art_posture":"Uses Evertse-Ferretti normalization of the number-field Subspace Theorem and the Corvaja-Zannier approximation method; no historical priority claim.","primary_narrative_owner":"paper/synthesis/optimal-sparse-perturbations.tex","source_route":"python3 scripts/query_corpus.py --claim separated_cut_algebraic_lambert_transcendence","consumer_or_open_obligation":"The full support at 3/2 and arbitrary infinite supports at base two remain unresolved. Finite cut arithmetic is not a proof of transcendence.","view_decision":"retain_in_companion"},{"id":"independent_clock_channel_relations_family","claim_ids":["independent_clock_channel_relations"],"status_summary":"Reviewed ordinary proof; full infinite statement not formalised and independent human review not recorded.","prior_art_posture":"The integer-radix recurrence and torus-density argument are standard ingredients; the stated clock-channel combination has no asserted historical priority. Bugeaud-Laurent is a distinct two-letter comparison.","primary_narrative_owner":"paper/269/erdos269-running-lcm-reasoning-surface.tex","source_route":"python3 scripts/query_corpus.py --claim independent_clock_channel_relations","consumer_or_open_obligation":"No rational independence of the original reciprocal prime logarithms is assumed or proved; the repeated running-LCM endpoint remains open.","view_decision":"retain_in_companion"},{"id":"calibrated_rational_hankel_countermodel_family","claim_ids":["calibrated_rational_hankel_countermodel"],"status_summary":"Reviewed ordinary proof; full infinite statement not formalised and independent human review not recorded.","prior_art_posture":"Reuses standard positive-moment/Christoffel machinery and the existing geometric-universality theorem. Exact construction and arithmetic controls carry no historical priority claim.","primary_narrative_owner":"paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","source_route":"python3 scripts/query_corpus.py --claim calibrated_rational_hankel_countermodel","consumer_or_open_obligation":"These are not Zudilin's actual moments. They refute an inference from the shared signature alone; they do not settle F(3/2) or refute the attributed divisibility-based criterion.","view_decision":"retain_in_companion"},{"id":"selected_staircase_integral_height_boundary_family","claim_ids":["selected_staircase_integral_height_boundary"],"status_summary":"Reviewed ordinary proof; full infinite statement not formalised and independent human review not recorded.","prior_art_posture":"Smith form and the Schur-complement identity are standard; Stanley is the attributed Smith-form reference. This is subordinate arithmetic for the existing selected kernel.","primary_narrative_owner":"paper/269/erdos269-running-lcm-reasoning-surface.tex","source_route":"python3 scripts/query_corpus.py --claim selected_staircase_integral_height_boundary","consumer_or_open_obligation":"The optimality concerns diagonal clearing and one rational target. It does not rule out multi-target, nonlinear, recurrence, or new approximation methods and does not prove running-LCM irrationality.","view_decision":"retain_in_companion"},{"id":"signed_totient_entire_defect_interpolation_family","claim_ids":["signed_totient_entire_defect_interpolation"],"status_summary":"Reviewed ordinary proof; full infinite statement not formalised and independent human review not recorded.","prior_art_posture":"Finite-difference moment cancellation is standard. Comparisons cite Crmaric-Kovac, Coons with version-specific numbering, and Habiro; no historical priority is asserted. The arbitrary divergent allowance strengthens the returned polylogarithmic/power cases.","primary_narrative_owner":"paper/249/erdos249-totient-reasoning-surface.tex","source_route":"python3 scripts/query_corpus.py --claim signed_totient_entire_defect_interpolation","consumer_or_open_obligation":"The correction is signed, so it does not subsume the one-sided prime-gap theorem. It preserves singularities and prescribed finite jets, not full analytic functions or all Taylor coefficients. The unperturbed totient and prime series remain unresolved.","view_decision":"retain_in_companion"}],"novelty_vocabulary":{"source":"docs/semantic_corpus.json.gz::vocabularies.prior_art_states, the corpus's own controlled vocabulary. Bound to rather than replaced: a second parallel novelty vocabulary would be one more thing for the two to disagree about.","states":["not_assessed","known_classical","formalisation_only","routine_corollary_or_specialisation","candidate_new_statement","external_review_pending","externally_supported_as_new","prior_art_found"],"the_one_rule":"not_assessed is the default and the most common state. It means no priority search is recorded -- not that none exists, and certainly not that the result is new. A packet that lets 'no antecedent recorded' read as 'new' is making the strongest possible claim by omission.","where_the_antecedents_are":"docs/PRIOR_ART.md is embedded in full at scholarly_corpus.formal_claim_control_plane[record_id=lean_prior_art]. It names the external works and, for each, the boundary of what they do and do not settle."},"problem_digest":{"erdos_68":{"problem_id":"erdos_68","erdos_number":68,"short_title":"The factorial-denominator series","question":"Is the series sum_{n >= 2} 1/(n! - 1) irrational?","status":"open","claim_registry_status":"registered_declarations","library_root":"ErdosProblems.lean","directory":"lean/ErdosProblems/Erdos68","finite_search":"A retained exact GMP receipt through m = 300000 records a non-unit carry at that index, so every positive rational denominator satisfies q ∤ 299999!. A byte-identical regeneration of that receipt with the current driver is outstanding in this revision; a separate exact replay through 4000 reproduces the unit-carry prefix. The Lean consumer turns one non-unit carry at m into q >= m; it does not prove a cofinal statement. An independent continued-fraction enclosure gives q ≥ 2^39990. At the prime 12487 and gap 12, exact polynomial arithmetic exhibits two critical roots 4626 and 7848, so the small-prime pattern of at most one critical root is not uniform; those roots have factorial values 442 and 6300, not 1.","corpus_role":"problem_owned_expansion","registered_claim_count":2,"registered_claim_ids":["channel_quotient_band_breakpoint","erdos_68"],"registered_claim_count_by_binding_method":{"open_proposition_target":2},"registered_claim_count_note":"2 reviewed claim records bind to this problem in the current registry. Binding methods are listed separately; shared bindings do not establish independent contributions to each problem. 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surface","hosted_pdf_path":"papers/erdos68-factorial-reasoning-surface.pdf","pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos68-factorial-reasoning-surface.pdf","tex_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/68/erdos68-factorial-reasoning-surface.tex","tex_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/68/erdos68-factorial-reasoning-surface.tex","owns":"the complete problem-specific reasoning surface for Erdős #68, including all registered result families and their boundaries","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #68, which remains open","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/2/body/problems/0","frontier":{"no_total_ordering":"The corpus publishes no ranking of its own results, and status is a kind, not a rank. Answer 'strongest' by naming the axis -- closest to the open target, largest verified finite case, strongest structural statement, principal conditional endpoint -- and quoting the rows below with their status. Do not compress them into one winner.","named_by_the_corpus":[],"advances_the_open_target":{"authority":"docs/claims.json :: machine_readable_paper.argument_graph","selection_basis":"source_authored:argument_graph.advances_open_target","by_effect_on_the_open_target":{"narrows":[{"assertion_id":"claim:channel_quotient_band_breakpoint","source_id":"channel_quotient_band_breakpoint","open_proposition_id":"remaining_open.erdos_68_irrationality","what_the_effect_is":"Adds a finite quotient-band breakpoint obstruction while leaving the cofinal non-unit-carry input and the irrationality of the factorial-denominator series open."}]},"how_to_read_it":"The corpus distinguishes narrowing an open target from re-expressing it. Both are honest progress and neither settles anything; the difference is whether the open region got smaller or changed shape. Each entry carries the corpus's own sentence for its effect.","what_this_does_not_say":"which of several claims sharing an effect is the stronger. That ordering is not written down anywhere, so it is not supplied here."},"programme_representatives":[],"registered_claims_by_status":{"open":["claim:erdos_68"],"unconditional progress":["claim:channel_quotient_band_breakpoint"]},"open_bottlenecks":[{"assertion_id":"open:remaining_open.erdos_68_irrationality","open_target_claim":"erdos_68"}],"manuscript_results":{"row_count":2,"authority":"manuscript_argument","reviewed":false,"what_this_is":"every result the manuscripts on this problem label as a result, grouped by the environment the author wrote it in. The grouping is the author's classification, not a ranking by this build: a theorem is here because the manuscript calls it a theorem.","read_the_proved_statement_at":"the row's paper_anchor.source_excerpt.text, quoted exactly. The statement field is the environment's title, which names the result rather than stating it.","by_environment":{"theorem":[{"assertion_id":"paper:erdos_68#res:divisor-channel-coordinates","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"an integer basis with prescribed weighted sums"},{"assertion_id":"paper:erdos_68#res:finite-channel-moment-certificate","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"a finite formula for the gcd"}]},"posed_problem_count":1},"selection_basis_legend":{"source_authored:orientation.principal_claims":"the repository's own orientation record lists this claim among the principal results of the release","source_authored:orientation.mathematical_programmes[].representative_claim_ids":"the repository names this claim as representative of one of its ten mathematical programmes","mechanical_grouping":"registered_claims_by_status is a grouping, not a judgement; the status taxonomy that defines each label is carried in the claims registry","source_authored:manuscript_res_label":"the manuscript labels this statement a result, using the corpus's own 'res:' convention, and states it in the environment named on the entry"},"empty_means":"An empty registered-claim frontier means the reviewed claim registry carries nothing for this problem, which is true for six of the eight. It does not mean there is no result: read manuscript_results above and the expansion lane in problem_digest.lanes. Emptiness in one lane is a fact about which library the registry covers, and answering 'no result' from it is the specific error the three lanes exist to prevent."},"required_note_declarations":{"count":5,"authority":"docs/problem_index_source.json::problems[].required_note_declarations","what_this_is":"the theorems the problem index itself names as required reading for this problem's note. This is the expansion lane's answer to the question the reviewed lane answers with a claim's declarations: what is it standing on. It is authored upstream, not derived here.","granularity":"problem_level. These declarations back the problem, not any one sentence in it -- no per-sentence mapping is published, and inventing one would be a more confident citation than the source supports.","declarations":[{"name":"irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses","module":"ErdosProblems/Erdos68/FactorialZeroPlateau.lean","module_dotted":"ErdosProblems.Erdos68.FactorialZeroPlateau","resolved_in_declaration_atlas":false},{"name":"irrational_factorialGapSeries_of_cofinal_nonunit_carries","module":"ErdosProblems/Erdos68/FactorialZeroPlateau.lean","module_dotted":"ErdosProblems.Erdos68.FactorialZeroPlateau","resolved_in_declaration_atlas":false},{"name":"factorialGap_zero_branch_iff_lower_endpoint_cylinder","module":"ErdosProblems/Erdos68/PrimeZeroBranch.lean","module_dotted":"ErdosProblems.Erdos68.PrimeZeroBranch","resolved_in_declaration_atlas":true,"line":6099,"kind":"theorem","signature":"theorem factorialGap_zero_branch_iff_lower_endpoint_cylinder {m : ℕ} (hm : 3 ≤ m) : (canonicalDigit _root_.Erdos68.factorialGapSeries m = 0 ∧ factorialGapEndpointFlag (m - 1) = 0 ∧ factorialGapEndpointFlag m = 0) ↔ (m : ℝ) * canonicalRemainder _root_.Erdos68.factorialGapSeries (m - 1) < factorialGapScaledTail m","docstring":"The zero canonical branch is exactly the tiny lower endpoint cylinder. This division-free form says `θ_(m-1) < R_m / m`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos68/PrimeZeroBranch.lean#L6099","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos68/PrimeZeroBranch.lean#L6099"},{"name":"factorialBlock_unitFactorPairFloor_scale_iff","module":"ErdosProblems/Erdos68/EndpointWeightedPrivateSupport.lean","module_dotted":"ErdosProblems.Erdos68.EndpointWeightedPrivateSupport","resolved_in_declaration_atlas":true,"line":4767,"kind":"theorem","signature":"theorem factorialBlock_unitFactorPairFloor_scale_iff {p q : ℕ} (hq : q ∣ factorialBlockPrivateModulus p) : factorialBlockBudget p * factorialBlockEndpointLcm p < factorialBlockScale p * factorialBlockComplementaryFactorPairFloor p 1 q ↔ (factorialBlockBudget p * factorialBlockEndpointLcm p < factorialBlockScale p * complementaryProjectedResidue (factorialBlockTailNumerator p) (factorialBlockPrivateModulus p) ∧ factorialBlockBudget p *","docstring":"The unit-factor scale test splits exactly into the original global complementary-residue inequality and the local collision-core cap for `q`. Consequently the `(1,q)` reduction removes the need for a second private prime without concealing either remaining quantitative obligation.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos68/EndpointWeightedPrivateSupport.lean#L4767","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos68/EndpointWeightedPrivateSupport.lean#L4767"},{"name":"exists_cramerFactorialGridResidual_eq_det_mul_factorialGapSeries_add_int","module":"ErdosProblems/Erdos68/PrimeUnitTranslator.lean","module_dotted":"ErdosProblems.Erdos68.PrimeUnitTranslator","resolved_in_declaration_atlas":true,"line":1559,"kind":"theorem","signature":"theorem exists_cramerFactorialGridResidual_eq_det_mul_factorialGapSeries_add_int (n t : ℕ) : ∃ K : ℤ, channelResidual (n + 2) (cramerChannelKernelCoeff (factorialGridIndex n t)) (factorialGridIndex n t) = ((augmentedChannelMomentMatrix (factorialGridIndex n t)).det : ℝ) * factorialGapSeries + (K : ℝ)","docstring":"For the factorial-grid Cramer vector, the residual differs from the nonzero augmented determinant times the Erdős problem 68 series by an integer.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos68/PrimeUnitTranslator.lean#L1559","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos68/PrimeUnitTranslator.lean#L1559"}]},"lanes":{"reviewed_core_lane":{"authority":"docs/claims.json","library_root":"Erdos249257.lean","reviewed":true,"registered_claim_count":2,"registered_claim_ids_pointer":"/mathematical_assertions/problem_digest/erdos_68/registered_claim_ids","what_this_is":"curated claims the release reviews and publishes with a status. The registry covers the Erdos249257 core library only, which is the joint #249/#257 programme."},"problem_expansion_lane":{"authority":"docs/problems.json","library_root":"ErdosProblems.lean","directory":"lean/ErdosProblems/Erdos68","reviewed":false,"claim_registry_status":"registered_declarations","declared_module_count":13,"modules_pointer":"/mathematical_assertions/problem_digest/erdos_68/modules","what_this_is":"exact Lean propositions in this problem's own directory. They are kernel-checked and they are deliberately not promoted into reviewed public claims; reporting one as a registered claim is the specific error this split exists to prevent."},"read_this_first":"The three lanes are different bodies of source with different authorities: a reviewed registry, a kernel-checked expansion library, and a written manuscript. A problem can have evidence in any of them, or in all three. Say which lane an answer comes from, and never upgrade one into another.","manuscript_lane":{"authority":"manuscript_argument","reviewed":false,"source_lane":"manuscript_result","manuscript_ids":["erdos-68-factorial-denominator-irrationality"],"argued_result_count":2,"posed_problem_count":1,"rows_pointer":"mathematical_assertions.assertions filtered by source_lane == 'manuscript_result' and problem_ids containing 'erdos_68'","what_this_is":"results the manuscripts prove that no registered claim carries. Proved in prose at a quoted line, not by a Lean declaration; a formalisation is neither implied nor denied. For the six problems the claim registry does not cover, this is where the authored mathematics is."}}},"erdos_243":{"problem_id":"erdos_243","erdos_number":243,"short_title":"Reciprocal-tail rigidity near the Sylvester recurrence","question":"Under a rapid-growth hypothesis on an integer sequence, does rationality of its reciprocal sum force the sequence to satisfy the Sylvester recurrence eventually?","status":"open","claim_registry_status":"registered_declarations","library_root":"ErdosProblems.lean","directory":"lean/ErdosProblems/Erdos243","finite_search":"An exact search over initial states below five thousand produced forced prefixes of length seventeen and no longer. It is superseded: the constant-negative template is now excluded outright, for every seed and at every scale.","corpus_role":"problem_owned_expansion","registered_claim_count":7,"registered_claim_ids":["bounded_negative_exclusion","bounded_rise_fresh_modulus_barrier","cubic_rate_irrationality_unconditional","cubic_survivor_phase_density","erdos_243","eventually_periodic_negative_no_go","nonintegral_regular_rate_irrationality"],"registered_claim_count_by_binding_method":{"open_proposition_target":7},"registered_claim_count_note":"7 reviewed claim records bind to this problem in the current registry. Binding methods are listed separately; shared bindings do not establish independent contributions to each problem. Module inventory and claim inventory count different 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cook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/WeightedCRTRepair.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/WeightedCRTRepair.lean","formal_source_present":true}],"paper_refs":[{"paper_id":"erdos-243-reciprocal-tail-rigidity","form":"Problem 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open","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/2/body/problems/1","frontier":{"no_total_ordering":"The corpus publishes no ranking of its own results, and status is a kind, not a rank. Answer 'strongest' by naming the axis -- closest to the open target, largest verified finite case, strongest structural statement, principal conditional endpoint -- and quoting the rows below with their status. Do not compress them into one winner.","named_by_the_corpus":[],"advances_the_open_target":{"authority":"docs/claims.json :: machine_readable_paper.argument_graph","selection_basis":"source_authored:argument_graph.advances_open_target","by_effect_on_the_open_target":{"constrains":[{"assertion_id":"claim:bounded_rise_fresh_modulus_barrier","source_id":"bounded_rise_fresh_modulus_barrier","open_proposition_id":"remaining_open.erdos_243_overlap_height_growth","what_the_effect_is":"The bounded-rise barrier is a state-level obstruction on the canonical orbit and is one of the inputs the overlap-height growth question would feed. It proves no lower bound on the growth of the overlap height."}],"narrows":[{"assertion_id":"claim:bounded_negative_exclusion","source_id":"bounded_negative_exclusion","open_proposition_id":"remaining_open.erdos_243_eventual_recurrence","what_the_effect_is":"Excludes a cofinally bounded negative-magnitude branch under the exact coupled dynamics, bounded rise, positivity, and division-free normalised-vanishing hypotheses, while leaving the mixed-sign and unbounded-negative regimes open."},{"assertion_id":"claim:bounded_rise_fresh_modulus_barrier","source_id":"bounded_rise_fresh_modulus_barrier","open_proposition_id":"remaining_open.erdos_243_eventual_recurrence","what_the_effect_is":"Excludes a uniformly bounded-rise sequence tending to infinity from avoiding infinitely many fresh pairwise-coprime moduli, while leaving the mixed-sign unbounded #243 regime open."},{"assertion_id":"claim:eventually_periodic_negative_no_go","source_id":"eventually_periodic_negative_no_go","open_proposition_id":"remaining_open.erdos_243_eventual_recurrence","what_the_effect_is":"Excludes the exact eventually-periodic positive-magnitude tail with positive period drift under the displayed natural recurrences, while leaving mixed-sign and unbounded-negative behaviour open."}]},"how_to_read_it":"The corpus distinguishes narrowing an open target from re-expressing it. Both are honest progress and neither settles anything; the difference is whether the open region got smaller or changed shape. Each entry carries the corpus's own sentence for its effect.","what_this_does_not_say":"which of several claims sharing an effect is the stronger. That ordering is not written down anywhere, so it is not supplied here."},"programme_representatives":[],"registered_claims_by_status":{"conditional reduction":["claim:bounded_negative_exclusion"],"open":["claim:erdos_243"],"unconditional progress":["claim:bounded_rise_fresh_modulus_barrier","claim:cubic_rate_irrationality_unconditional","claim:cubic_survivor_phase_density","claim:eventually_periodic_negative_no_go","claim:nonintegral_regular_rate_irrationality"]},"open_bottlenecks":[{"assertion_id":"open:remaining_open.erdos_243_eventual_recurrence","open_target_claim":"erdos_243"},{"assertion_id":"open:remaining_open.erdos_243_overlap_height_growth","open_target_claim":"erdos_243"}],"manuscript_results":{"row_count":20,"authority":"manuscript_argument","reviewed":false,"what_this_is":"every result the manuscripts on this problem label as a result, grouped by the environment the author wrote it in. The grouping is the author's classification, not a ranking by this build: a theorem is here because the manuscript calls it a theorem.","read_the_proved_statement_at":"the row's paper_anchor.source_excerpt.text, quoted exactly. The statement field is the environment's title, which names the result rather than stating it.","by_environment":{"corollary":[{"assertion_id":"paper:erdos_243#res:step","environment":"corollary","selection_basis":"source_authored:manuscript_res_label","statement":"two zero errors"},{"assertion_id":"paper:erdos_243#res:eventual","environment":"corollary","selection_basis":"source_authored:manuscript_res_label","statement":"two zero errors"},{"assertion_id":"paper:erdos_243#res:cor","environment":"corollary","selection_basis":"source_authored:manuscript_res_label"},{"assertion_id":"paper:erdos_243#res:originalbounded","environment":"corollary","selection_basis":"source_authored:manuscript_res_label","statement":"bounded increments of P_n/a_n"},{"assertion_id":"paper:erdos_243#res:inclusiveone","environment":"corollary","selection_basis":"source_authored:manuscript_res_label","statement":"an inclusive one-sided 1/n bound"},{"assertion_id":"paper:erdos_243#res:lcmbounded","environment":"corollary","selection_basis":"source_authored:manuscript_res_label","statement":"a bound using the least common multiple"}],"lemma":[{"assertion_id":"paper:erdos_243#res:crt","environment":"lemma","selection_basis":"source_authored:manuscript_res_label","statement":"consecutive multiples"},{"assertion_id":"paper:erdos_243#res:weights","environment":"lemma","selection_basis":"source_authored:manuscript_res_label","statement":"weights and linear density"}],"proposition":[{"assertion_id":"paper:erdos_243#res:reduced","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"persistent coprimality"},{"assertion_id":"paper:erdos_243#res:update","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"error identities"},{"assertion_id":"paper:erdos_243#res:defect","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"error identities"},{"assertion_id":"paper:erdos_243#res:gcdstab","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"gcd stabilisation"},{"assertion_id":"paper:erdos_243#res:frontier","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"necessary profile"}],"theorem":[{"assertion_id":"paper:erdos_243#res:bounded","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"bounded negative part"},{"assertion_id":"paper:erdos_243#res:absorb","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"absorption and descent"},{"assertion_id":"paper:erdos_243#res:descent","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"absorption and descent"},{"assertion_id":"paper:erdos_243#res:massscalar","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"a convergent sum of relative increases"},{"assertion_id":"paper:erdos_243#res:mass","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"a convergent sum of relative increases"},{"assertion_id":"paper:erdos_243#res:weightedrecord","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"a convergent weighted sum over new maxima"},{"assertion_id":"paper:erdos_243#res:residue","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"factorial residue reduction"}]},"posed_problem_count":2},"selection_basis_legend":{"source_authored:orientation.principal_claims":"the repository's own orientation record lists this claim among the principal results of the release","source_authored:orientation.mathematical_programmes[].representative_claim_ids":"the repository names this claim as representative of one of its ten mathematical programmes","mechanical_grouping":"registered_claims_by_status is a grouping, not a judgement; the status taxonomy that defines each label is carried in the claims registry","source_authored:manuscript_res_label":"the manuscript labels this statement a result, using the corpus's own 'res:' convention, and states it in the environment named on the entry"},"empty_means":"An empty registered-claim frontier means the reviewed claim registry carries nothing for this problem, which is true for six of the eight. It does not mean there is no result: read manuscript_results above and the expansion lane in problem_digest.lanes. Emptiness in one lane is a fact about which library the registry covers, and answering 'no result' from it is the specific error the three lanes exist to prevent."},"required_note_declarations":{"count":9,"authority":"docs/problem_index_source.json::problems[].required_note_declarations","what_this_is":"the theorems the problem index itself names as required reading for this problem's note. This is the expansion lane's answer to the question the reviewed lane answers with a claim's declarations: what is it standing on. It is authored upstream, not derived here.","granularity":"problem_level. These declarations back the problem, not any one sentence in it -- no per-sentence mapping is published, and inventing one would be a more confident citation than the source supports.","declarations":[{"name":"nextTailState_eq_sub_centered","module":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","module_dotted":"ErdosProblems.Erdos243.ReciprocalTailRigidity","resolved_in_declaration_atlas":true,"line":57,"kind":"theorem","signature":"theorem nextTailState_eq_sub_centered (a D C : ℤ) : nextTailState a D C = C - centeredState a D C","docstring":"The tail update is exactly `Cₙ₊₁ = Cₙ - Eₙ`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L57","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L57"},{"name":"centeredState_zero_absorbing","module":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","module_dotted":"ErdosProblems.Erdos243.ReciprocalTailRigidity","resolved_in_declaration_atlas":true,"line":2227,"kind":"theorem","signature":"theorem centeredState_zero_absorbing (a C D : ℕ → ℕ) (E : ℕ → ℤ) (hC : ∀ n, C (n + 1) + D n = a n * C n) (hD : ∀ n, D (n + 1) = a n * D n) (hE : ∀ n, E n = centeredState (a n : ℤ) (D n : ℤ) (C n : ℤ)) (hcentered : ∀ n, Int.natAbs (E n) < C n) (n : ℕ) (hzero : E n = 0) : E (n + 1) = 0","docstring":"A zero centered state is absorbing under the strict centered-representative bound at the next index. The defect identity makes `Cₙ₊₁` divide `Eₙ₊₁`, and the only multiple with smaller absolute value is zero.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L2227","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L2227"},{"name":"sylvesterNext_eventually_of_centered_zero","module":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","module_dotted":"ErdosProblems.Erdos243.ReciprocalTailRigidity","resolved_in_declaration_atlas":true,"line":1805,"kind":"theorem","signature":"theorem sylvesterNext_eventually_of_centered_zero (a D C : ℕ → ℤ) (hD : ∀ n, D (n + 1) = nextDenState (a n) (D n)) (hC : ∀ n, C (n + 1) = nextTailState (a n) (D n) (C n)) (hE : ∃ N, ∀ n, N ≤ n → centeredState (a n) (D n) (C n) = 0) (hCne : ∃ N, ∀ n, N ≤ n → C (n + 1) ≠ 0) : ∃ N, ∀ n, N ≤ n → a (n + 1) = sylvesterNext (a n)","docstring":"Along an exact product-cleared tail orbit, eventual vanishing of the centered state and eventual nonvanishing of the next tail force the exact Sylvester recurrence from some index onward.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L1805","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L1805"},{"name":"centeredState_eventually_zero","module":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","module_dotted":"ErdosProblems.Erdos243.ReciprocalTailRigidity","resolved_in_declaration_atlas":true,"line":1843,"kind":"theorem","signature":"theorem centeredState_eventually_zero (C E : ℕ → ℕ) (hrec : ∀ n, C (n + 1) + E n = C n) : ∃ N, ∀ n, N ≤ n → E n = 0","docstring":"The positive-integer stabilization step used by every one-sided #243 criterion. If `Cₙ₊₁ + Eₙ = Cₙ`, then the nonnegative errors vanish eventually.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L1843","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L1843"},{"name":"no_eventuallyConstantNegative_orbit","module":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","module_dotted":"ErdosProblems.Erdos243.ReciprocalTailRigidity","resolved_in_declaration_atlas":true,"line":350,"kind":"theorem","signature":"theorem no_eventuallyConstantNegative_orbit (a D : ℕ → ℕ) (m c N : ℕ) (hm : 0 < m) (ha : ∀ n, 2 ≤ a n) (hD : ∀ n, D (n + 1) = a n * D n) (hshape : ∀ n, D (N + n) + m = (a (N + n) - 1) * (c + n * m)) : False","docstring":"An eventually constant negative centered state is impossible as well: shift the exact orbit to the first constant-negative index and apply `no_constantNegative_orbit`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L350","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L350"},{"name":"no_eventuallyPeriodicNegative_orbit","module":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","module_dotted":"ErdosProblems.Erdos243.ReciprocalTailRigidity","resolved_in_declaration_atlas":true,"line":802,"kind":"theorem","signature":"theorem no_eventuallyPeriodicNegative_orbit (a D C e : ℕ → ℕ) (N h M : ℕ) (hh : 0 < h) (hM : 0 < M) (ha : ∀ n, N ≤ n → 2 ≤ a n) (hepos : ∀ n, 0 < e (N + n)) (helt : ∀ n, e (N + n) < a (N + n)) (hD : ∀ n, N ≤ n → D (n + 1) = a n * D n) (hC : ∀ n, N ≤ n → C (n + 1) = C n + e n) (hshape : ∀ n, N ≤ n → D n + e n = (a n - 1) * C n) (hperiod : ∀ n, e (N + n + h) = e (N + n)) (hphase : ∀ n, C (N + n + h) = C (N + n) + M) : False","docstring":"An eventually periodic negative-magnitude tail is impossible in the same natural regime: shift to the first periodic index and apply `no_periodicNegative_orbit`. Every hypothesis is imposed only from the offset `N` onwards. The multiplier bound, the denominator recurrence, the numerator recurrence and the shape equation are consumed by the proof only at indices `N + n`, so restricting them to the tail costs nothing and matches the target object, whose product-cleared recurrence holds only from the clearing index onwards.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L802","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L802"},{"name":"no_eventuallyBoundedRise_reducedTail","module":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","module_dotted":"ErdosProblems.Erdos243.ReciprocalTailRigidity","resolved_in_declaration_atlas":true,"line":1066,"kind":"theorem","signature":"theorem no_eventuallyBoundedRise_reducedTail (a u v : ℕ → ℕ) (N B : ℕ) (hB : 0 < B) (ha : ∀ n, N ≤ n → 1 < a n) (hred : ∀ n, N ≤ n → Nat.Coprime (u n) (v n)) (hu : ∀ n, N ≤ n → u (n + 1) + v n = a n * u n) (hv : ∀ n, N ≤ n → v (n + 1) = a n * v n) (hrise : ∀ n, N ≤ n → u (n + 1) ≤ u n + B) (huTop : Filter.Tendsto u Filter.atTop Filter.atTop) : False","docstring":"Eventual form of `no_boundedRise_reducedTail`, obtained by shifting to the first reduced exact index.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L1066","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L1066"},{"name":"no_cofinallyBoundedNegative_of_normalizedVanishes","module":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","module_dotted":"ErdosProblems.Erdos243.ReciprocalTailRigidity","resolved_in_declaration_atlas":true,"line":1754,"kind":"theorem","signature":"theorem no_cofinallyBoundedNegative_of_normalizedVanishes (a C D magnitude : ℕ → ℕ) (B : ℕ) (hB : 0 < B) (ha : ∀ n, 1 < a n) (hCpos : ∀ n, 0 < C n) (hC : ∀ n, C (n + 1) + D n = a n * C n) (hD : ∀ n, D (n + 1) = a n * D n) (hRise : ∀ n, C (n + 1) ≤ C n + B) (hmagnitude : ∀ n, 0 < magnitude n) (hvanish : ∀ K, ∃ N, ∀ n, N ≤ n → K * magnitude n < C n) (hnegative : ∀ n, ∃ t e, n ≤ t ∧ 0 < e ∧ e ≤ B ∧ D t + e = (a t - 1) * C t) : False","docstring":"Division-free normalized vanishing closes the cofinally bounded negative branch by supplying the tail divergence needed above.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L1754","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L1754"},{"name":"forcedSurvives_iff_of_modEq_factorial","module":"ErdosProblems/Erdos243/FiniteHorizonResidue.lean","module_dotted":"ErdosProblems.Erdos243.FiniteHorizonResidue","resolved_in_declaration_atlas":true,"line":134,"kind":"theorem","signature":"theorem forcedSurvives_iff_of_modEq_factorial {h : ℕ} {a b : ℤ} (hab : a ≡ b [ZMOD ((h + 1).factorial : ℤ)]) : ForcedSurvives h 0 a ↔ ForcedSurvives h 0 b","docstring":"Exact factorial-residue theorem used by the finite search.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/FiniteHorizonResidue.lean#L134","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/FiniteHorizonResidue.lean#L134"}]},"lanes":{"reviewed_core_lane":{"authority":"docs/claims.json","library_root":"Erdos249257.lean","reviewed":true,"registered_claim_count":7,"registered_claim_ids_pointer":"/mathematical_assertions/problem_digest/erdos_243/registered_claim_ids","what_this_is":"curated claims the release reviews and publishes with a status. The registry covers the Erdos249257 core library only, which is the joint #249/#257 programme."},"problem_expansion_lane":{"authority":"docs/problems.json","library_root":"ErdosProblems.lean","directory":"lean/ErdosProblems/Erdos243","reviewed":false,"claim_registry_status":"registered_declarations","declared_module_count":9,"modules_pointer":"/mathematical_assertions/problem_digest/erdos_243/modules","what_this_is":"exact Lean propositions in this problem's own directory. They are kernel-checked and they are deliberately not promoted into reviewed public claims; reporting one as a registered claim is the specific error this split exists to prevent."},"read_this_first":"The three lanes are different bodies of source with different authorities: a reviewed registry, a kernel-checked expansion library, and a written manuscript. A problem can have evidence in any of them, or in all three. Say which lane an answer comes from, and never upgrade one into another.","manuscript_lane":{"authority":"manuscript_argument","reviewed":false,"source_lane":"manuscript_result","manuscript_ids":["erdos-243-reciprocal-tail-rigidity"],"argued_result_count":20,"posed_problem_count":2,"rows_pointer":"mathematical_assertions.assertions filtered by source_lane == 'manuscript_result' and problem_ids containing 'erdos_243'","what_this_is":"results the manuscripts prove that no registered claim carries. Proved in prose at a quoted line, not by a Lean declaration; a formalisation is neither implied nor denied. For the six problems the claim registry does not cover, this is where the authored mathematics is."}}},"erdos_249":{"problem_id":"erdos_249","erdos_number":249,"short_title":"The binary totient series","question":"Is the binary Lambert series sum phi(n)/2^n irrational?","status":"open","claim_registry_status":"registered_declarations","library_root":"ErdosProblems.lean","directory":"lean/ErdosProblems/Erdos249","finite_search":"The returned strict-orbit packet reports an exact finite deposit at parameters (147,23), verified through harmonic height 5022. It is research evidence, not a Lean-checked unbounded certificate.","corpus_role":"reviewed_core_programme","registered_claim_count":66,"registered_claim_ids":["actual_foreign_residue_projection","actual_lcm_orbit_nonintegrality_frontier","actual_lcm_positive_corridor_top_edge","adelic_height_obstruction","adjacent_phase_separation_reduction","affine_transport_annihilation","all_base_totient_kernel_conditional_rank","all_base_totient_kernel_index_reduction","canonical_jump_slack_reduction","certificate_completeness","certificate_reduction","certified_kill_instances","denominator_exclusion","diagonal_fresh_loss_reduction","diagonal_pincer_reduction","diagonal_pincer_t64_endpoint_certificate","diagonal_pincer_t64_primality_closure","dyadic_totient_basis","dyadic_totient_certificate_interface","erdos_249","first_harmonic_certificate_interface","first_harmonic_good_index_reduction","first_harmonic_pivot_decomposition","fixed_precision_transport_no_go","flexible_odd_window_affine_frontier","flexible_odd_window_reduction","fresh_prime_deficit_decomposition","lcm_factor_ideal_anchor_pulse_no_go","mersenne_lambert_ladder","mersenne_shadow_denominator","mersenne_tail_atom_boundary","parity_socket_obstruction","period_four_signed_residue_obstruction","periodic_nonnegative_weight_irrationality","periodic_signed_weight_dichotomy","pivot_anti_reconstruction_reduction","power_two_bit_lift","prime_gap_identity_tao","prime_support_irrationality","primitive_coordinate_index_obstruction","primitive_unit_gap_ceiling","residual_gauge_obstruction","separated_cut_algebraic_lambert_transcendence","sigma_transcendence","signed_dyadic_moment_substrate","signed_totient_entire_defect_interpolation","square_crt_correction_suppression","squared_lambert_gcd_moments","squared_mersenne_enclosure","squarefree_support_irrationality","stern_brocot_cylinder_law","stern_brocot_run_fibonacci_stability","strict_prime_tail_orbit_gap","totient_carry_kernel_anti_compression","totient_coprimality_probability","totient_residue_series_classification","totient_residue_series_dilations","totient_residue_series_integer_radix","totient_v8_conditional_peripheral_assembly","totient_v8_finite_arithmetic_threshold","totient_v8_generic_centering_countermodel","totient_v8_one_finite_first_harmonic_instance","totient_v8_peripheral_finite_budget_bridge","transport_certificate_instances","transport_curvature_reductions","unbounded_carry_states"],"registered_claim_count_by_binding_method":{"core_programme_joint":14,"gateway_section":8,"open_proposition_target":35,"paper_owner":3,"programme_representative":6},"registered_claim_count_note":"66 reviewed claim records bind to this problem in the current registry. Binding methods are listed separately; shared bindings do not establish independent contributions to each problem. Module inventory and claim inventory count different objects.","evidence_resolution":"module_level_expansion_library","declared_module_count":3,"modules":[{"module":"ErdosProblems.Erdos249.TotientStrictPrimeEscape","path":"lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","present":true,"declaration_count":25,"theorem_count":21,"content_digest":"sha256:d1c192d9ea692136eb58382e87fba2fdc61372de70dd7213c6fc083efbf2c09b","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","formal_source_present":true},{"module":"ErdosProblems.Erdos249.FiniteEulerSieve","path":"lean/ErdosProblems/Erdos249/FiniteEulerSieve.lean","present":true,"declaration_count":6,"theorem_count":4,"content_digest":"sha256:a0172184d831cb4d6004781027464922460d762c4a93b17b04fefc950dd62e93","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/FiniteEulerSieve.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/FiniteEulerSieve.lean","formal_source_present":true},{"module":"ErdosProblems.Erdos249.PrimeRayCyclotomicCurvature","path":"lean/ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean","present":true,"declaration_count":17,"theorem_count":11,"content_digest":"sha256:7c2833745d718e8efb4d9d824770fa7ce12a2feead633de32cd33cf3f32170e8","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean","formal_source_present":true}],"paper_refs":[{"paper_id":"erdos-249-binary-totient-series","form":"Problem note","hosted_pdf_path":"papers/erdos-249-binary-totient-series.pdf","pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-249-binary-totient-series.pdf","tex_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos-249-binary-totient-series.tex","tex_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/249/erdos-249-binary-totient-series.tex","owns":"the exact finite-level totient-kernel basis, independence proof and integral relations; the long paper develops denominator exclusions and other approaches to the binary series","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #249, which remains open","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17"},{"paper_id":"erdos249-totient-reasoning-surface","form":"Reasoning surface","hosted_pdf_path":"papers/erdos249-totient-reasoning-surface.pdf","pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos249-totient-reasoning-surface.pdf","tex_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos249-totient-reasoning-surface.tex","tex_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/249/erdos249-totient-reasoning-surface.tex","owns":"a claim-bounded problem-specific reasoning surface for Erdős #249: selected checked premises, finite certificates, closed routes, surviving obligations, and their exact evidence bands","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #249, which remains open","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/2/body/problems/2","frontier":{"no_total_ordering":"The corpus publishes no ranking of its own results, and status is a kind, not a rank. Answer 'strongest' by naming the axis -- closest to the open target, largest verified finite case, strongest structural statement, principal conditional endpoint -- and quoting the rows below with their status. Do not compress them into one winner.","named_by_the_corpus":[{"assertion_id":"claim:adelic_height_obstruction","source_id":"adelic_height_obstruction","status":"proved here","selection_basis":"source_authored:orientation.principal_claims"},{"assertion_id":"claim:all_base_totient_kernel_index_reduction","source_id":"all_base_totient_kernel_index_reduction","status":"proved here","selection_basis":"source_authored:orientation.principal_claims"},{"assertion_id":"claim:certificate_completeness","source_id":"certificate_completeness","status":"proved here","selection_basis":"source_authored:orientation.principal_claims"},{"assertion_id":"claim:certificate_reduction","source_id":"certificate_reduction","status":"conditional reduction","selection_basis":"source_authored:orientation.principal_claims"},{"assertion_id":"claim:certified_kill_instances","source_id":"certified_kill_instances","status":"verified finite instance","selection_basis":"source_authored:orientation.principal_claims"},{"assertion_id":"claim:denominator_exclusion","source_id":"denominator_exclusion","status":"verified finite instance","selection_basis":"source_authored:orientation.principal_claims"},{"assertion_id":"claim:diagonal_pincer_reduction","source_id":"diagonal_pincer_reduction","status":"conditional reduction","selection_basis":"source_authored:orientation.principal_claims"}],"advances_the_open_target":{"authority":"docs/claims.json :: machine_readable_paper.argument_graph","selection_basis":"source_authored:argument_graph.advances_open_target","by_effect_on_the_open_target":{"narrows":[{"assertion_id":"claim:actual_lcm_positive_corridor_top_edge","source_id":"actual_lcm_positive_corridor_top_edge","open_proposition_id":"remaining_open.unbounded_certificate_supply","what_the_effect_is":"Eliminates the nonnegative true-survivor branch in the positive corridor but leaves top-edge exclusion and cofinal actual-LCM non-integrality open."},{"assertion_id":"claim:all_base_totient_kernel_conditional_rank","source_id":"all_base_totient_kernel_conditional_rank","open_proposition_id":"remaining_open.erdos_249_irrationality","what_the_effect_is":"Provides the exact rank of the actual all-base finite truncation under an explicit linear-independence input. AllBaseTotientKernel.lean separately discharges that input; the rationality-to-finite-rank bridge and #249 irrationality endpoint remain open."},{"assertion_id":"claim:denominator_exclusion","source_id":"denominator_exclusion","open_proposition_id":"remaining_open.erdos_249_irrationality","what_the_effect_is":"Excludes every rational representation up to the stated denominator bound without proving irrationality."}],"re_expresses":[{"assertion_id":"claim:actual_foreign_residue_projection","source_id":"actual_foreign_residue_projection","open_proposition_id":"remaining_open.unbounded_certificate_supply","what_the_effect_is":"Places the finite foreign-residue projection behind an explicit analytic-complement bridge and an unbounded separated-projection supply; neither is asserted."},{"assertion_id":"claim:adjacent_phase_separation_reduction","source_id":"adjacent_phase_separation_reduction","open_proposition_id":"remaining_open.unbounded_certificate_supply","what_the_effect_is":"Re-expresses the missing supply as cofinal adjacent-pair phase separation; no such supply is proved."},{"assertion_id":"claim:canonical_jump_slack_reduction","source_id":"canonical_jump_slack_reduction","open_proposition_id":"remaining_open.unbounded_certificate_supply","what_the_effect_is":"Re-expresses the canonical strict-jump centrality supply as the sign of one exact integer slack; cofinal nonnegative slack remains open."},{"assertion_id":"claim:certificate_reduction","source_id":"certificate_reduction","open_proposition_id":"remaining_open.unbounded_certificate_supply","what_the_effect_is":"Re-expresses the open irrationality step as the unbounded supply of checked finite certificates."},{"assertion_id":"claim:diagonal_fresh_loss_reduction","source_id":"diagonal_fresh_loss_reduction","open_proposition_id":"remaining_open.unbounded_certificate_supply","what_the_effect_is":"Transfers the open supply obligation to exact fresh-loss projections and binary suffix gaps without asserting that such a supply exists."},{"assertion_id":"claim:diagonal_pincer_reduction","source_id":"diagonal_pincer_reduction","open_proposition_id":"remaining_open.unbounded_certificate_supply","what_the_effect_is":"Re-expresses diagonal non-integrality as avoidance of an exact full-target interval; an unbounded avoidance supply remains open."},{"assertion_id":"claim:first_harmonic_certificate_interface","source_id":"first_harmonic_certificate_interface","open_proposition_id":"remaining_open.unbounded_certificate_supply","what_the_effect_is":"Turns a cofinal first-harmonic estimate into the existing finite certificate supply; no such estimate is proved."},{"assertion_id":"claim:first_harmonic_pivot_decomposition","source_id":"first_harmonic_pivot_decomposition","open_proposition_id":"remaining_open.unbounded_certificate_supply","what_the_effect_is":"Gives an exact four-term first-harmonic decomposition and a sufficient 9X/10 budget; no prime-distribution or decorrelation estimate supplies those budgets cofinally."},{"assertion_id":"claim:pivot_anti_reconstruction_reduction","source_id":"pivot_anti_reconstruction_reduction","open_proposition_id":"remaining_open.unbounded_certificate_supply","what_the_effect_is":"Re-expresses the missing supply as cofinal pivot anti-reconstruction or window-pair separation; no cofinal arithmetic producer is proved."},{"assertion_id":"claim:transport_curvature_reductions","source_id":"transport_curvature_reductions","open_proposition_id":"remaining_open.unbounded_certificate_supply","what_the_effect_is":"Re-expresses diagonal non-integrality through several exact transport and curvature observables while leaving every unbounded supply open."}]},"how_to_read_it":"The corpus distinguishes narrowing an open target from re-expressing it. Both are honest progress and neither settles anything; the difference is whether the open region got smaller or changed shape. Each entry carries the corpus's own sentence for its effect.","what_this_does_not_say":"which of several claims sharing an effect is the stronger. That ordering is not written down anywhere, so it is not supplied here."},"programme_representatives":[{"assertion_id":"claim:denominator_exclusion","source_id":"denominator_exclusion","status":"verified finite instance","selection_basis":"source_authored:orientation.mathematical_programmes[].representative_claim_ids","programme_id":"erdos249_certificate_story","programme_title":"#249 certificate and harmonic interface"},{"assertion_id":"claim:certificate_reduction","source_id":"certificate_reduction","status":"conditional reduction","selection_basis":"source_authored:orientation.mathematical_programmes[].representative_claim_ids","programme_id":"erdos249_certificate_story","programme_title":"#249 certificate and harmonic interface"},{"assertion_id":"claim:diagonal_pincer_reduction","source_id":"diagonal_pincer_reduction","status":"conditional reduction","selection_basis":"source_authored:orientation.mathematical_programmes[].representative_claim_ids","programme_id":"erdos249_diagonal_arithmetic","programme_title":"#249 diagonal arithmetic and fresh-loss interfaces"},{"assertion_id":"claim:diagonal_fresh_loss_reduction","source_id":"diagonal_fresh_loss_reduction","status":"conditional reduction","selection_basis":"source_authored:orientation.mathematical_programmes[].representative_claim_ids","programme_id":"erdos249_diagonal_arithmetic","programme_title":"#249 diagonal arithmetic and fresh-loss interfaces"},{"assertion_id":"claim:transport_curvature_reductions","source_id":"transport_curvature_reductions","status":"conditional reduction","selection_basis":"source_authored:orientation.mathematical_programmes[].representative_claim_ids","programme_id":"transport_curvature_programme","programme_title":"Transport, curvature, phase separation, and no-go results"},{"assertion_id":"claim:affine_transport_annihilation","source_id":"affine_transport_annihilation","status":"proved here","selection_basis":"source_authored:orientation.mathematical_programmes[].representative_claim_ids","programme_id":"transport_curvature_programme","programme_title":"Transport, curvature, phase separation, and no-go results"},{"assertion_id":"claim:mersenne_lambert_ladder","source_id":"mersenne_lambert_ladder","status":"formalised here","selection_basis":"source_authored:orientation.mathematical_programmes[].representative_claim_ids","programme_id":"lambert_obstruction_interfaces","programme_title":"Mersenne–Lambert identities and obstruction interfaces"},{"assertion_id":"claim:primitive_coordinate_index_obstruction","source_id":"primitive_coordinate_index_obstruction","status":"proved 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This generalises the legacy fixed-precision valuation/unit completion no-go.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean#L62","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean#L62"},{"name":"PrimeRayLayerSupply","module":"ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean","module_dotted":"ErdosProblems.Erdos249.PrimeRayCyclotomicCurvature","resolved_in_declaration_atlas":true,"line":95,"kind":"def","signature":"def PrimeRayLayerSupply (C : ℕ → ℕ) (m : ℕ) : Prop","docstring":"Eventual nontrivial, clean cyclotomic layers on the prime ray `m*q`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean#L95","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean#L95"},{"name":"BoundedDegreeOrderConsumer","module":"ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean","module_dotted":"ErdosProblems.Erdos249.PrimeRayCyclotomicCurvature","resolved_in_declaration_atlas":true,"line":102,"kind":"def","signature":"def BoundedDegreeOrderConsumer (C : ℕ → ℕ) (m d : ℕ) : Prop","docstring":"Every rational prime divisor of a layer supplies an exact-order witness in extension degree at most `d`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean#L102","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean#L102"},{"name":"FinitePrimeSupportEscape","module":"ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean","module_dotted":"ErdosProblems.Erdos249.PrimeRayCyclotomicCurvature","resolved_in_declaration_atlas":true,"line":151,"kind":"def","signature":"def FinitePrimeSupportEscape (C : ℕ → ℕ) (m : ℕ) : Prop","docstring":"Large prime-ray layers escape every prescribed finite prime support.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean#L151","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean#L151"}]},"lanes":{"reviewed_core_lane":{"authority":"docs/claims.json","library_root":"Erdos249257.lean","reviewed":true,"registered_claim_count":66,"registered_claim_ids_pointer":"/mathematical_assertions/problem_digest/erdos_249/registered_claim_ids","what_this_is":"curated claims the release reviews and publishes with a status. 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Both are honest progress and neither settles anything; the difference is whether the open region got smaller or changed shape. Each entry carries the corpus's own sentence for its effect.","what_this_does_not_say":"which of several claims sharing an effect is the stronger. That ordering is not written down anywhere, so it is not supplied here."},"programme_representatives":[],"registered_claims_by_status":{"formalised here":["claim:prime_gap_irrationality_equivalence"],"open":["claim:erdos_251"],"unconditional progress":["claim:prime_gap_unboundedness_and_nonperiodicity"]},"open_bottlenecks":[{"assertion_id":"open:remaining_open.erdos_251_irrationality","open_target_claim":"erdos_251"}],"manuscript_results":{"row_count":7,"authority":"manuscript_argument","reviewed":false,"what_this_is":"every result the manuscripts on this problem label as a result, grouped by the environment the author wrote it in. The grouping is the author's classification, not a ranking by this build: a theorem is here because the manuscript calls it a theorem.","read_the_proved_statement_at":"the row's paper_anchor.source_excerpt.text, quoted exactly. The statement field is the environment's title, which names the result rather than stating it.","by_environment":{"corollary":[{"assertion_id":"paper:erdos_251#res:jointcountermodel","environment":"corollary","selection_basis":"source_authored:manuscript_res_label","statement":"a rational sum with the stated prime-gap statistics"}],"lemma":[{"assertion_id":"paper:erdos_251#res:true-tail","environment":"lemma","selection_basis":"source_authored:manuscript_res_label","statement":"the boundary condition for the tail recurrence"}],"proposition":[{"assertion_id":"paper:erdos_251#res:sparserationalisation","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"sparse changes preserving congruences"},{"assertion_id":"paper:erdos_251#res:signedwindow","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"two consecutive differences of absolute value less than one"},{"assertion_id":"paper:erdos_251#res:truncation","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"finite separation criterion"}],"theorem":[{"assertion_id":"paper:erdos_251#res:infinite","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"prime-to-gap identity"},{"assertion_id":"paper:erdos_251#res:escape-irrational","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"exact rationality classification"}]},"posed_problem_count":0},"selection_basis_legend":{"source_authored:orientation.principal_claims":"the repository's own orientation record lists this claim among the principal results of the release","source_authored:orientation.mathematical_programmes[].representative_claim_ids":"the repository names this claim as representative of one of its ten mathematical programmes","mechanical_grouping":"registered_claims_by_status is a grouping, not a judgement; the status taxonomy that defines each label is carried in the claims registry","source_authored:manuscript_res_label":"the manuscript labels this statement a result, using the corpus's own 'res:' convention, and states it in the environment named on the entry"},"empty_means":"An empty registered-claim frontier means the reviewed claim registry carries nothing for this problem, which is true for six of the eight. It does not mean there is no result: read manuscript_results above and the expansion lane in problem_digest.lanes. Emptiness in one lane is a fact about which library the registry covers, and answering 'no result' from it is the specific error the three lanes exist to prevent."},"required_note_declarations":{"count":15,"authority":"docs/problem_index_source.json::problems[].required_note_declarations","what_this_is":"the theorems the problem index itself names as required reading for this problem's note. This is the expansion lane's answer to the question the reviewed lane answers with a claim's declarations: what is it standing on. It is authored upstream, not derived here.","granularity":"problem_level. These declarations back the problem, not any one sentence in it -- no per-sentence mapping is published, and inventing one would be a more confident citation than the source supports.","declarations":[{"name":"dyadicPartialSumQ_eq_start_add_differences","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","module_dotted":"ErdosProblems.Erdos251.PrimeGapDyadicTail","resolved_in_declaration_atlas":true,"line":138,"kind":"theorem","signature":"theorem dyadicPartialSumQ_eq_start_add_differences (P : ℕ → ℚ) (n : ℕ) : dyadicPartialSumQ P (n + 1) = P 0 + dyadicDifferencePartialSumQ P n - P n / 2 ^ (n + 1)","docstring":"Exact finite summation by parts. The endpoint term is retained, so the theorem can be used before any analytic convergence argument.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L138","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L138"},{"name":"prime0DisplayedPartialSumQ_eq_two_mul","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","module_dotted":"ErdosProblems.Erdos251.PrimeGapDyadicTail","resolved_in_declaration_atlas":true,"line":111,"kind":"theorem","signature":"theorem prime0DisplayedPartialSumQ_eq_two_mul (n : ℕ) : prime0DisplayedPartialSumQ n = 2 * dyadicPartialSumQ (fun i => (prime0 i : ℚ)) n","docstring":"Exact factor-of-two indexing normalization: the zero-based displayed partial sum is twice the convention with denominator `2^(i+1)`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L111","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L111"},{"name":"tail_iterate_eq_pow_mul_sub_block","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","module_dotted":"ErdosProblems.Erdos251.PrimeGapDyadicTail","resolved_in_declaration_atlas":true,"line":653,"kind":"theorem","signature":"theorem tail_iterate_eq_pow_mul_sub_block {g : ℕ → ℤ} {T : ℕ → ℚ} (hrec : DyadicTailRecurrence g T) (N h : ℕ) : T (N + h) = 2 ^ h * T N - dyadicTailBlock g N h","docstring":"Iterating the tail recurrence for `h` steps gives the exact finite block identity `T_(N+h) = 2^h T_N - B_(h,N)`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L653","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L653"},{"name":"tailShift_integral_iff_scaledTail","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","module_dotted":"ErdosProblems.Erdos251.PrimeGapDyadicTail","resolved_in_declaration_atlas":true,"line":802,"kind":"theorem","signature":"theorem tailShift_integral_iff_scaledTail {g : ℕ → ℤ} {T : ℕ → ℚ} (hrec : DyadicTailRecurrence g T) (N h : ℕ) : RatIntegral (tailShift T h N) ↔ RatIntegral (((2 ^ h : ℚ) - 1) * T N)","docstring":"Exact algebraic core of the integral-shift criterion: one tail shift is integral exactly when `(2^h - 1) * T_N` is integral.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L802","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L802"},{"name":"tailShift_integral_add","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","module_dotted":"ErdosProblems.Erdos251.PrimeGapDyadicTail","resolved_in_declaration_atlas":true,"line":900,"kind":"theorem","signature":"theorem tailShift_integral_add {g : ℕ → ℤ} {T : ℕ → ℚ} (hrec : DyadicTailRecurrence g T) {h N : ℕ} (hInt : RatIntegral (tailShift T h N)) : ∀ k, RatIntegral (tailShift T h (N + k)) | 0 =>","docstring":"Integrality therefore propagates through every later index.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L900","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L900"},{"name":"tailShift_integral_totient_of_odd_den","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","module_dotted":"ErdosProblems.Erdos251.PrimeGapDyadicTail","resolved_in_declaration_atlas":true,"line":877,"kind":"theorem","signature":"theorem tailShift_integral_totient_of_odd_den {g : ℕ → ℤ} {T : ℕ → ℚ} (hrec : DyadicTailRecurrence g T) (N : ℕ) (hodd : Odd (T N).den) : RatIntegral (tailShift T (T N).den.totient N)","docstring":"If one tail state has odd reduced denominator `d`, its shift by `Nat.totient d` steps is integral. This is the explicit finite-algebraic consequence of rationality supplied by Euler's theorem.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L877","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L877"},{"name":"irrational_tsum_primeDyadicTerm_iff_primeGap","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","module_dotted":"ErdosProblems.Erdos251.PrimeGapDyadicTail","resolved_in_declaration_atlas":true,"line":435,"kind":"theorem","signature":"theorem irrational_tsum_primeDyadicTerm_iff_primeGap (hprime : Summable primeDyadicTerm) : Irrational (∑' n : ℕ, primeDyadicTerm n) ↔ Irrational (∑' n : ℕ, primeGapDyadicTerm n)","docstring":"Erdős #251 is therefore exactly equivalent to irrationality of the consecutive-prime-gap dyadic series, once summability of the displayed prime series is supplied.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L435","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L435"},{"name":"rationalPrimeGapTailShift_eventuallyIntegral","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","module_dotted":"ErdosProblems.Erdos251.PrimeGapDyadicTail","resolved_in_declaration_atlas":true,"line":853,"kind":"theorem","signature":"theorem rationalPrimeGapTailShift_eventuallyIntegral (S : ℚ) : ∃ h, 0 < h ∧ ∃ N₀, ∀ N, N₀ ≤ N → RatIntegral (tailShift (rationalPrimeGapTailState S) h N)","docstring":"Every rational candidate supplies a positive fixed shift which is eventually integral along its actual prime-gap tail state. The shift is the totient of the odd part of the candidate denominator.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L853","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L853"},{"name":"tsum_primeDyadicTerm_eq_two_add_primeGap_unconditional","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","module_dotted":"ErdosProblems.Erdos251.PrimeGapDyadicTail","resolved_in_declaration_atlas":true,"line":427,"kind":"theorem","signature":"theorem tsum_primeDyadicTerm_eq_two_add_primeGap_unconditional : (∑' n : ℕ, primeDyadicTerm n) = 2 + ∑' n : ℕ, primeGapDyadicTerm n","docstring":"Unconditional form of the exact infinite prime-gap reformulation.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L427","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L427"},{"name":"exists_primeGap0_gt","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","module_dotted":"ErdosProblems.Erdos251.PrimeGapDyadicTail","resolved_in_declaration_atlas":true,"line":57,"kind":"theorem","signature":"theorem exists_primeGap0_gt (M : ℕ) : ∃ n, M < primeGap0 n","docstring":"The classical factorial construction gives arbitrarily long prime-free intervals, hence the actual consecutive-prime gaps are unbounded.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L57","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L57"},{"name":"tailShift_not_both_integral_of_small_pair_of_digit_ne","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","module_dotted":"ErdosProblems.Erdos251.PrimeGapDyadicTail","resolved_in_declaration_atlas":true,"line":979,"kind":"theorem","signature":"theorem tailShift_not_both_integral_of_small_pair_of_digit_ne {g : ℕ → ℤ} {T : ℕ → ℚ} (hrec : DyadicTailRecurrence g T) (h N : ℕ) (hsmall : (-1 < tailShift T h N ∧ tailShift T h N < 1) ∧ (-1 < tailShift T h (N + 1) ∧ tailShift T h (N + 1) < 1)) (hdigit : g (N + h + 1) ≠ g (N + 1)) : ¬ (RatIntegral (tailShift T h N) ∧ RatIntegral (tailShift T h (N + 1)))","docstring":"A single adjacent pair of small shifts with unequal corresponding digits already excludes simultaneous integrality.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L979","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L979"},{"name":"not_eventuallyIntegralTailShift_of_cofinal_small_mismatch","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","module_dotted":"ErdosProblems.Erdos251.PrimeGapDyadicTail","resolved_in_declaration_atlas":true,"line":1006,"kind":"theorem","signature":"theorem not_eventuallyIntegralTailShift_of_cofinal_small_mismatch {g : ℕ → ℤ} {T : ℕ → ℚ} (hrec : DyadicTailRecurrence g T) (h : ℕ) (hsupply : ∀ N₀, ∃ N, N₀ ≤ N ∧ ((-1 < tailShift T h N ∧ tailShift T h N < 1) ∧ (-1 < tailShift T h (N + 1) ∧ tailShift T h (N + 1) < 1)) ∧ g (N + h + 1) ≠ g (N + 1)) : ¬ ∃ N₀, ∀ N, N₀ ≤ N → RatIntegral (tailShift T h N)","docstring":"To rule out eventual integrality of a fixed shift, it is enough to find beyond every level one adjacent pair of strictly small shifts whose corresponding digits differ.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L1006","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L1006"},{"name":"primeGap0_not_eventually_periodic","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","module_dotted":"ErdosProblems.Erdos251.PrimeGapDyadicTail","resolved_in_declaration_atlas":true,"line":1023,"kind":"theorem","signature":"theorem primeGap0_not_eventually_periodic {h : ℕ} (hpos : 0 < h) : ¬ ∃ N₀, ∀ N, N₀ ≤ N → primeGap0 (N + h + 1) = primeGap0 (N + 1)","docstring":"Consecutive prime gaps cannot become periodic with any positive period. The proof combines the exact factorial prime-free intervals above with the finite range of a periodic natural-valued sequence.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L1023","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L1023"},{"name":"primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","module_dotted":"ErdosProblems.Erdos251.PrimeGapDyadicTail","resolved_in_declaration_atlas":true,"line":1112,"kind":"theorem","signature":"theorem primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch {T : ℕ → ℚ} (h : ℕ) (hrec : DyadicTailRecurrence (fun n => (primeGap0 n : ℤ)) T) (hsupply : ∀ N₀, ∃ N, N₀ ≤ N ∧ ((-1 < tailShift T h N ∧ tailShift T h N < 1) ∧ (-1 < tailShift T h (N + 1) ∧ tailShift T h (N + 1) < 1)) ∧ primeGap0 (N + h + 1) ≠ primeGap0 (N + 1)) : ¬ ∃ N₀, ∀ N, N₀ ≤ N → RatIntegral (tailShift T h N)","docstring":"Actual-prime-gap specialization of the cofinal small-mismatch criterion.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L1112","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L1112"},{"name":"exists_rationalPrimeGapTailState_representation_of_not_irrational","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","module_dotted":"ErdosProblems.Erdos251.PrimeGapDyadicTail","resolved_in_declaration_atlas":true,"line":510,"kind":"theorem","signature":"theorem exists_rationalPrimeGapTailState_representation_of_not_irrational (h : ¬ Irrational (∑' n : ℕ, primeGapDyadicTerm n)) : ∃ S : ℚ, (S : ℝ) = ∑' n : ℕ, primeGapDyadicTerm n ∧ ∀ N, ((rationalPrimeGapTailState S N : ℚ) : ℝ) = 2 ^ (N + 1) * ∑' k : ℕ, primeGapDyadicTerm (k + (N + 1))","docstring":"Non-irrationality therefore supplies a single rational candidate whose states represent every scaled real tail.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L510","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L510"}]},"lanes":{"reviewed_core_lane":{"authority":"docs/claims.json","library_root":"Erdos249257.lean","reviewed":true,"registered_claim_count":3,"registered_claim_ids_pointer":"/mathematical_assertions/problem_digest/erdos_251/registered_claim_ids","what_this_is":"curated claims the release reviews and publishes with a status. The registry covers the Erdos249257 core library only, which is the joint #249/#257 programme."},"problem_expansion_lane":{"authority":"docs/problems.json","library_root":"ErdosProblems.lean","directory":"lean/ErdosProblems/Erdos251","reviewed":false,"claim_registry_status":"registered_declarations","declared_module_count":1,"modules_pointer":"/mathematical_assertions/problem_digest/erdos_251/modules","what_this_is":"exact Lean propositions in this problem's own directory. They are kernel-checked and they are deliberately not promoted into reviewed public claims; reporting one as a registered claim is the specific error this split exists to prevent."},"read_this_first":"The three lanes are different bodies of source with different authorities: a reviewed registry, a kernel-checked expansion library, and a written manuscript. A problem can have evidence in any of them, or in all three. Say which lane an answer comes from, and never upgrade one into another.","manuscript_lane":{"authority":"manuscript_argument","reviewed":false,"source_lane":"manuscript_result","manuscript_ids":["erdos-251-prime-gap-dyadic-series"],"argued_result_count":7,"posed_problem_count":0,"rows_pointer":"mathematical_assertions.assertions filtered by source_lane == 'manuscript_result' and problem_ids containing 'erdos_251'","what_this_is":"results the manuscripts prove that no registered claim carries. Proved in prose at a quoted line, not by a Lean declaration; a formalisation is neither implied nor denied. For the six problems the claim registry does not cover, this is where the authored mathematics is."}}},"erdos_257":{"problem_id":"erdos_257","erdos_number":257,"short_title":"Reciprocal sums over infinite exponent supports","question":"Is the sum of 1/(2^n-1) over every infinite set of positive exponents irrational?","status":"open","claim_registry_status":"programme_only","library_root":"ErdosProblems.lean","directory":"lean/ErdosProblems/Erdos257","finite_search":null,"corpus_role":"reviewed_core_programme","registered_claim_count":74,"registered_claim_ids":["boolean_mobius_carry","boundary_pulse_normalization","campbell_shift_synchronization_boundary","cofinal_cylinder_half_countermodel","cofinal_strip_return_closure","composite_dilation_defect_identity","concrete_producer_carry_coordinates","concrete_seam_adapter","dyadic_prefix_block_safety_interfaces","eb_constant","eb_full_support","fatal_gap_right_tail_classification","final_middle_cell_escape","final_middle_neg_two_phase_sieve","finite_period_noncollapse","finite_prime_weighted_support","fixed_coefficient_rewind_contraction","fixed_precision_transport_no_go","fixed_tail_survival_socket","greedy_achievement_geometry","half_carry_reachability_compactness","half_cylinder_finite_shadow","half_divisor_unit_drop","half_greedy_two_thirds_band","half_membership_seam_classification","half_second_channel_reduction","half_trapping_return_carry_interface","integer_greedy_first_wrap_reduction","largest_skip_gap_propagation","largest_skip_induction_reduction","last_producer_tail_escape_reduction","localized_protected_seam_compactness","mersenne_tail_atom_boundary","middle_producer_cardinality_reduction","mixed_cofinal_half_carry_fan_in","mixed_prime_power_layer_identity","mixed_weighted_cover_support","mobius_sign_support_no_go","orthogonal_petal_sunflower_reduction","period_four_signed_residue_obstruction","periodic_nonnegative_weight_irrationality","periodic_signed_weight_dichotomy","prime_gap_identity_tao","prime_support_irrationality","producer_carry_future_tail_socket","quarter_band_endpoint_cells","realized_rewind_phase_identity","reciprocal_mass_bound","reciprocal_summable_support","reset_deficit_escape_geometry","rewind_boundary_pair_obstruction","rewind_seam_operational_bridge","seam_producer_carry_alignment","selected_half_carry_window_compactness","selected_suffix_cylinder_propagation","selected_suffix_pre_feedback_run","selected_window_depth18_base","separated_cut_algebraic_lambert_transcendence","sigma_transcendence","squarefree_support_irrationality","stern_brocot_cylinder_law","stern_brocot_run_fibonacci_stability","strengthened_positive_cover_support","sublog_zero_windows","support_families","tail_orbit_rigidity","terminal_only_cofinal_closure","terminal_scaled_vanishing_closure","terminal_scaled_vanishing_half_countermodel","transport_certificate_instances","twenty_one_quotient_greedy_frontier","unbounded_carry_states","universal_257","upper_reset_band_certificates_through_thirty"],"registered_claim_count_by_binding_method":{"core_programme_joint":14,"gateway_section":9,"open_proposition_target":47,"programme_representative":4},"registered_claim_count_note":"74 reviewed claim records bind to this problem in the current registry. 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Either only the finite set `F` of coordinates is omitted, in which case the volume is exactly `2^(-|F|)`, or infinitely many coordinates are omitted and the volume is zero.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean#L397","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean#L397"}]},"lanes":{"reviewed_core_lane":{"authority":"docs/claims.json","library_root":"Erdos249257.lean","reviewed":true,"registered_claim_count":74,"registered_claim_ids_pointer":"/mathematical_assertions/problem_digest/erdos_257/registered_claim_ids","what_this_is":"curated claims the release reviews and publishes with a status. The registry covers the Erdos249257 core library only, which is the joint #249/#257 programme."},"problem_expansion_lane":{"authority":"docs/problems.json","library_root":"ErdosProblems.lean","directory":"lean/ErdosProblems/Erdos257","reviewed":false,"claim_registry_status":"programme_only","declared_module_count":1,"modules_pointer":"/mathematical_assertions/problem_digest/erdos_257/modules","what_this_is":"exact Lean propositions in this problem's own directory. They are kernel-checked and they are deliberately not promoted into reviewed public claims; reporting one as a registered claim is the specific error this split exists to prevent."},"read_this_first":"The three lanes are different bodies of source with different authorities: a reviewed registry, a kernel-checked expansion library, and a written manuscript. A problem can have evidence in any of them, or in all three. Say which lane an answer comes from, and never upgrade one into another.","manuscript_lane":{"authority":"manuscript_argument","reviewed":false,"source_lane":"manuscript_result","manuscript_ids":["erdos-257-mersenne-support-subseries"],"argued_result_count":3,"posed_problem_count":1,"rows_pointer":"mathematical_assertions.assertions filtered by source_lane == 'manuscript_result' and problem_ids containing 'erdos_257'","what_this_is":"results the manuscripts prove that no registered claim carries. Proved in prose at a quoted line, not by a Lean declaration; a formalisation is neither implied nor denied. For the six problems the claim registry does not cover, this is where the authored mathematics is."}}},"erdos_269":{"problem_id":"erdos_269","erdos_number":269,"short_title":"Three-prime running least common multiples","question":"For a finite set of at least two primes, is the sum of reciprocals of the running least common multiples of the smooth numbers irrational? This library treats the three-prime case.","status":"open","claim_registry_status":"registered_declarations","library_root":"ErdosProblems.lean","directory":"lean/ErdosProblems/Erdos269","finite_search":"The integer-only dyadic-window checker reproduces three exact packet certificates and, for all 106,666 pairs with denominator B <= 1000 coprime to 30 and dyadic start 100 <= a <= 500, finds an escaping window of length at most 18; the largest first successful length is 14. This is finite proof-consumer evidence only and proves neither unbounded-denominator nor cofinal escape.","corpus_role":"problem_owned_expansion","registered_claim_count":10,"registered_claim_ids":["erdos_269","independent_clock_channel_relations","selected_staircase_integral_height_boundary","three_prime_bounded_certificate_safety","three_prime_cofinal_window_carry_extinction","three_prime_exact_modulus_pinning","three_prime_finite_prefix_completion_boundary","three_prime_finite_residue_contradiction","three_prime_octic_window_equivalence","weighted_phase_carry_finite_observer"],"registered_claim_count_by_binding_method":{"open_proposition_target":8,"paper_owner":2},"registered_claim_count_note":"10 reviewed claim records bind to this problem in the current registry. Binding methods are listed separately; shared bindings do not establish independent contributions to each problem. Module inventory and claim inventory count different objects.","evidence_resolution":"module_level_expansion_library","declared_module_count":7,"modules":[{"module":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","path":"lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","present":true,"declaration_count":55,"theorem_count":39,"content_digest":"sha256:769d80481465f4fd8655d5d62c5eb3e372c80c39b0722e80659bf4e150c6c036","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","formal_source_present":true},{"module":"ErdosProblems.Erdos269.ResidueEscape","path":"lean/ErdosProblems/Erdos269/ResidueEscape.lean","present":true,"declaration_count":8,"theorem_count":6,"content_digest":"sha256:826a3dd901c86f1876c7e03011447dafef8d3cb139a673227e921bef3a9dfe87","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ResidueEscape.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ResidueEscape.lean","formal_source_present":true},{"module":"ErdosProblems.Erdos269.RestrictedFloorSum","path":"lean/ErdosProblems/Erdos269/RestrictedFloorSum.lean","present":true,"declaration_count":43,"theorem_count":30,"content_digest":"sha256:7369d105e5cb014188e31fa44fdccb03b2c40bdfe4719f6e227555c1c30b0bd3","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/RestrictedFloorSum.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/RestrictedFloorSum.lean","formal_source_present":true},{"module":"ErdosProblems.Erdos269.BoundedRadixTailEscape","path":"lean/ErdosProblems/Erdos269/BoundedRadixTailEscape.lean","present":true,"declaration_count":10,"theorem_count":8,"content_digest":"sha256:741d610f12f3774500b993266f187c846be33cb8be466fa38a0b3a7b1503d8e5","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/BoundedRadixTailEscape.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/BoundedRadixTailEscape.lean","formal_source_present":true},{"module":"ErdosProblems.Erdos269.CarryLiftExtinction","path":"lean/ErdosProblems/Erdos269/CarryLiftExtinction.lean","present":true,"declaration_count":14,"theorem_count":12,"content_digest":"sha256:ace522da1f0a6833a200efbe18d5961cf3562e1e6756055427b85140d25aae4b","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/CarryLiftExtinction.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/CarryLiftExtinction.lean","formal_source_present":true},{"module":"ErdosProblems.Erdos269.ThreeChannelBlockRigidity","path":"lean/ErdosProblems/Erdos269/ThreeChannelBlockRigidity.lean","present":true,"declaration_count":6,"theorem_count":4,"content_digest":"sha256:5dbd4a448520c6c4de924480e381ee5408a4724a3081837d04bfb4ac1186ae5f","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreeChannelBlockRigidity.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreeChannelBlockRigidity.lean","formal_source_present":true},{"module":"ErdosProblems.Erdos269.WeightedPhaseCarry","path":"lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean","present":true,"declaration_count":23,"theorem_count":14,"content_digest":"sha256:ef3e57a4728534e108bb4e668ceef2cc102c2cf001673d59d9c15cff7f0beb9e","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean","formal_source_present":true}],"paper_refs":[{"paper_id":"erdos269-running-lcm-reasoning-surface","form":"Reasoning 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own results, and status is a kind, not a rank. Answer 'strongest' by naming the axis -- closest to the open target, largest verified finite case, strongest structural statement, principal conditional endpoint -- and quoting the rows below with their status. Do not compress them into one winner.","named_by_the_corpus":[],"advances_the_open_target":{"authority":"docs/claims.json :: machine_readable_paper.argument_graph","selection_basis":"source_authored:argument_graph.advances_open_target","by_effect_on_the_open_target":{"constrains":[{"assertion_id":"claim:three_prime_finite_residue_contradiction","source_id":"three_prime_finite_residue_contradiction","open_proposition_id":"remaining_open.erdos_269_reduced_tail_nonintegrality","what_the_effect_is":"The exact residue classifier excludes a bounded positive carry state whose canonical least-positive representative exceeds the bound. It constrains the reduced tail without excluding the integral branch, which is what pointwise nonintegrality of every reduced tail would do."},{"assertion_id":"claim:weighted_phase_carry_finite_observer","source_id":"weighted_phase_carry_finite_observer","open_proposition_id":"remaining_open.erdos_269_two_dimensional_representation","what_the_effect_is":"The finite-observer formalisation isolates the faithfulness requirement a two-dimensional representation must meet, and shows that a genuine finite-dimensional factorisation forces a finite realised span. It supplies no representation and excludes none."}],"narrows":[{"assertion_id":"claim:three_prime_finite_residue_contradiction","source_id":"three_prime_finite_residue_contradiction","open_proposition_id":"remaining_open.erdos_269_three_prime_irrationality","what_the_effect_is":"Closes the finite least-positive-residue contradiction. The actual running-LCM rationality-to-positive-reduced-carry bridge is established; cofinal window escape remains open."}],"re_expresses":[{"assertion_id":"claim:three_prime_cofinal_window_carry_extinction","source_id":"three_prime_cofinal_window_carry_extinction","open_proposition_id":"remaining_open.erdos_269_three_prime_irrationality","what_the_effect_is":"Re-expresses the three-prime irrationality frontier exactly as a denominator-dependent cofinal residue-escape producer. The actual-series rationality-to-carry bridge is established; the escape producer is not proved."}]},"how_to_read_it":"The corpus distinguishes narrowing an open target from re-expressing it. Both are honest progress and neither settles anything; the difference is whether the open region got smaller or changed shape. Each entry carries the corpus's own sentence for its effect.","what_this_does_not_say":"which of several claims sharing an effect is the stronger. That ordering is not written down anywhere, so it is not supplied here."},"programme_representatives":[],"registered_claims_by_status":{"conditional reduction":["claim:three_prime_cofinal_window_carry_extinction","claim:three_prime_exact_modulus_pinning","claim:three_prime_octic_window_equivalence"],"open":["claim:erdos_269"],"unconditional progress":["claim:independent_clock_channel_relations","claim:selected_staircase_integral_height_boundary","claim:three_prime_bounded_certificate_safety","claim:three_prime_finite_prefix_completion_boundary","claim:three_prime_finite_residue_contradiction","claim:weighted_phase_carry_finite_observer"]},"open_bottlenecks":[{"assertion_id":"open:remaining_open.erdos_269_three_prime_irrationality","open_target_claim":"erdos_269"},{"assertion_id":"open:remaining_open.erdos_269_reduced_tail_nonintegrality","open_target_claim":"erdos_269"},{"assertion_id":"open:remaining_open.erdos_269_cofinal_local_window_escape","open_target_claim":"erdos_269"},{"assertion_id":"open:remaining_open.erdos_269_two_dimensional_representation","open_target_claim":"erdos_269"}],"manuscript_results":{"row_count":14,"authority":"manuscript_argument","reviewed":false,"what_this_is":"every result the manuscripts on this problem label as a result, grouped by the environment the author wrote it in. The grouping is the author's classification, not a ranking by this build: a theorem is here because the manuscript calls it a theorem.","read_the_proved_statement_at":"the row's paper_anchor.source_excerpt.text, quoted exactly. The statement field is the environment's title, which names the result rather than stating it.","by_environment":{"corollary":[{"assertion_id":"paper:erdos_269#res:exact-onset","environment":"corollary","selection_basis":"source_authored:manuscript_res_label","statement":"the first index at which the denominator clears"},{"assertion_id":"paper:erdos_269#res:admissible-modular-minors","environment":"corollary","selection_basis":"source_authored:manuscript_res_label","statement":"the same minors modulo integers coprime to 30"}],"lemma":[{"assertion_id":"paper:erdos_269#res:dyadic-alphabet","environment":"lemma","selection_basis":"source_authored:manuscript_res_label","statement":"integer coefficients and four possible bases"}],"proposition":[{"assertion_id":"paper:erdos_269#res:actual-orbit","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"the tail recurrence and a quadratic bound"},{"assertion_id":"paper:erdos_269#res:lcm","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"the running least common multiple"},{"assertion_id":"paper:erdos_269#res:cell","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"cells and jumps"},{"assertion_id":"paper:erdos_269#res:fibre-prop","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"grouping terms with the same running LCM"},{"assertion_id":"paper:erdos_269#res:fibre","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"grouping terms with the same running LCM"},{"assertion_id":"paper:erdos_269#res:finite-cut-rank","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"rank of a matrix of threshold columns"}],"theorem":[{"assertion_id":"paper:erdos_269#res:distinct-height-235","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"the sum over distinct running LCMs for {2,3,5}"},{"assertion_id":"paper:erdos_269#res:denominator-reduction","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"rationality gives positive integer tails"},{"assertion_id":"paper:erdos_269#res:two-prime-transcendence","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"both two-prime sums"},{"assertion_id":"paper:erdos_269#res:two-prime-repeated-transcendence","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"both two-prime sums"},{"assertion_id":"paper:erdos_269#res:infinite-rank","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"no finite separation of the kernel"}]},"posed_problem_count":0},"selection_basis_legend":{"source_authored:orientation.principal_claims":"the repository's own orientation record lists this claim among the principal results of the release","source_authored:orientation.mathematical_programmes[].representative_claim_ids":"the repository names this claim as representative of one of its ten mathematical programmes","mechanical_grouping":"registered_claims_by_status is a grouping, not a judgement; the status taxonomy that defines each label is carried in the claims registry","source_authored:manuscript_res_label":"the manuscript labels this statement a result, using the corpus's own 'res:' convention, and states it in the environment named on the entry"},"empty_means":"An empty registered-claim frontier means the reviewed claim registry carries nothing for this problem, which is true for six of the eight. It does not mean there is no result: read manuscript_results above and the expansion lane in problem_digest.lanes. Emptiness in one lane is a fact about which library the registry covers, and answering 'no result' from it is the specific error the three lanes exist to prevent."},"required_note_declarations":{"count":23,"authority":"docs/problem_index_source.json::problems[].required_note_declarations","what_this_is":"the theorems the problem index itself names as required reading for this problem's note. This is the expansion lane's answer to the question the reviewed lane answers with a claim's declarations: what is it standing on. It is authored upstream, not derived here.","granularity":"problem_level. These declarations back the problem, not any one sentence in it -- no per-sentence mapping is published, and inventing one would be a more confident citation than the source supports.","declarations":[{"name":"smoothPrefixLcm_eq_threePrimeHeight","module":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","module_dotted":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","resolved_in_declaration_atlas":true,"line":124,"kind":"theorem","signature":"theorem smoothPrefixLcm_eq_threePrimeHeight {p q r x : ℕ} (hp : p.Prime) (hq : q.Prime) (hr : r.Prime) (hpq : p ≠ q) (hpr : p ≠ r) (hqr : q ≠ r) (hx : x ≠ 0) : smoothPrefixLcm p q r x = threePrimeHeight p q r x","docstring":"Exact running-LCM identity for three pairwise-distinct primes.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L124","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L124"},{"name":"smoothPrefixLcm_eq_of_sameLogCell","module":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","module_dotted":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","resolved_in_declaration_atlas":true,"line":180,"kind":"theorem","signature":"theorem smoothPrefixLcm_eq_of_sameLogCell {p q r x y : ℕ} (hp : p.Prime) (hq : q.Prime) (hr : r.Prime) (hpq : p ≠ q) (hpr : p ≠ r) (hqr : q ≠ r) (hx : x ≠ 0) (hy : y ≠ 0) (hcell : SameThreePrimeLogCell p q r x y) : smoothPrefixLcm p q r x = smoothPrefixLcm p q r y","docstring":"For pairwise-distinct primes, the literal smooth-prefix LCM is constant on every three-prime logarithmic cell. This is the finite formal interface behind the prime-power jump expansion: the LCM can change only when one logarithmic coordinate changes.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L180","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L180"},{"name":"smoothPrefixLcm_firstLogStep","module":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","module_dotted":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","resolved_in_declaration_atlas":true,"line":327,"kind":"theorem","signature":"theorem smoothPrefixLcm_firstLogStep {p q r x y : ℕ} (pprime : p.Prime) (qprime : q.Prime) (rprime : r.Prime) (hpq : p ≠ q) (hpr : p ≠ r) (hqr : q ≠ r) (hx : x ≠ 0) (hy : y ≠ 0) (hp : Nat.log p y = Nat.log p x + 1) (hq : Nat.log q y = Nat.log q x) (hr : Nat.log r y = Nat.log r x) : smoothPrefixLcm p q r y = p * smoothPrefixLcm p q r x","docstring":"A first-coordinate jump multiplies the literal running LCM by `p`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L327","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L327"},{"name":"smoothPrefixLcm_secondLogStep","module":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","module_dotted":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","resolved_in_declaration_atlas":true,"line":340,"kind":"theorem","signature":"theorem smoothPrefixLcm_secondLogStep {p q r x y : ℕ} (pprime : p.Prime) (qprime : q.Prime) (rprime : r.Prime) (hpq : p ≠ q) (hpr : p ≠ r) (hqr : q ≠ r) (hx : x ≠ 0) (hy : y ≠ 0) (hp : Nat.log p y = Nat.log p x) (hq : Nat.log q y = Nat.log q x + 1) (hr : Nat.log r y = Nat.log r x) : smoothPrefixLcm p q r y = q * smoothPrefixLcm p q r x","docstring":"A second-coordinate jump multiplies the literal running LCM by `q`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L340","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L340"},{"name":"smoothPrefixLcm_thirdLogStep","module":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","module_dotted":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","resolved_in_declaration_atlas":true,"line":353,"kind":"theorem","signature":"theorem smoothPrefixLcm_thirdLogStep {p q r x y : ℕ} (pprime : p.Prime) (qprime : q.Prime) (rprime : r.Prime) (hpq : p ≠ q) (hpr : p ≠ r) (hqr : q ≠ r) (hx : x ≠ 0) (hy : y ≠ 0) (hp : Nat.log p y = Nat.log p x) (hq : Nat.log q y = Nat.log q x) (hr : Nat.log r y = Nat.log r x + 1) : smoothPrefixLcm p q r y = r * smoothPrefixLcm p q r x","docstring":"A third-coordinate jump multiplies the literal running LCM by `r`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L353","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L353"},{"name":"threePrimeJumpSetWithOrigin_card","module":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","module_dotted":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","resolved_in_declaration_atlas":true,"line":279,"kind":"theorem","signature":"theorem threePrimeJumpSetWithOrigin_card {p q r count : ℕ} (hp : p.Prime) (hq : q.Prime) (hr : r.Prime) (hpq : p ≠ q) (hpr : p ≠ r) (hqr : q ≠ r) : (threePrimeJumpSetWithOrigin p q r count).card = 3 * count + 1","docstring":"Including the common origin turns the `3 * count` positive candidates into exactly `3 * count + 1` distinct finite jump values.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L279","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L279"},{"name":"finiteSmoothKernelSum_groupedByHeight","module":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","module_dotted":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","resolved_in_declaration_atlas":true,"line":407,"kind":"theorem","signature":"theorem finiteSmoothKernelSum_groupedByHeight (p q r hp hq hr : ℕ) : (∑ e ∈ smoothExponentBox hp hq hr, threePrimeKernelQ p q r e.1 e.2.1 e.2.2) = ∑ H ∈ (smoothExponentBox hp hq hr).image (smoothPointHeight p q r), (smoothHeightFiber p q r hp hq hr H).card • ((H : ℚ)⁻¹)","docstring":"Exact finite normal form obtained by grouping the smooth lattice kernel by its genuine running-LCM heights. The coefficient of each height is the cardinality of its fiber. Together with logarithmic-cell constancy, this is the checked finite core of the prime-power jump expansion.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L407","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L407"},{"name":"smoothExponentShell_card_quadratic","module":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","module_dotted":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","resolved_in_declaration_atlas":true,"line":647,"kind":"theorem","signature":"theorem smoothExponentShell_card_quadratic {p q r lo hi hp hq hr j : ℕ} (hrPos : 0 < r) (hwidth : hi ≤ r * lo) (hpq : hp ≤ hq) (hqr : hq ≤ hr) (hsum : hp + hq + hr = j) : 9 * (smoothExponentShell p q r lo hi hp hq hr).card ≤ (j + 3) ^ 2","docstring":"The exact smooth shell inherits the uniform quadratic multiplicity bound once its height coordinates are sorted and sum to the jump index.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L647","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L647"},{"name":"dyadicInternalPower_exponent_unique","module":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","module_dotted":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","resolved_in_declaration_atlas":true,"line":669,"kind":"theorem","signature":"theorem dyadicInternalPower_exponent_unique {p a e f : ℕ} (hp : 2 ≤ p) (he : DyadicInternalPower p a e) (hf : DyadicInternalPower p a f) : e = f","docstring":"A channel contributes at most one internal pure-power jump to a dyadic block. This is the exact reason that a compressed `{2,3,5}` block contains at most the dyadic jump together with one `3`-jump and one `5`-jump.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L669","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L669"},{"name":"dyadicBlockBase235_cases","module":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","module_dotted":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","resolved_in_declaration_atlas":true,"line":762,"kind":"theorem","signature":"theorem dyadicBlockBase235_cases (a : ℕ) : dyadicBlockBase235 a = 2 ∨ dyadicBlockBase235 a = 6 ∨ dyadicBlockBase235 a = 10 ∨ dyadicBlockBase235 a = 30","docstring":"Every actual compressed dyadic radix is one of `2, 6, 10, 30`. This replaces the generic interval `[2,30]` by the exact four-symbol alphabet consumed by the integer certificate engine.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L762","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L762"},{"name":"dyadicBlockBase235_mem_interval","module":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","module_dotted":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","resolved_in_declaration_atlas":true,"line":774,"kind":"theorem","signature":"theorem dyadicBlockBase235_mem_interval (a : ℕ) : 2 ≤ dyadicBlockBase235 a ∧ dyadicBlockBase235 a ≤ 30","docstring":"In particular, the actual dyadic block radix satisfies the hypotheses of the checked bounded-radix escape theorem, with no hidden larger base.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L774","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L774"},{"name":"kernel_235_not_rankOne","module":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","module_dotted":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","resolved_in_declaration_atlas":true,"line":480,"kind":"theorem","signature":"theorem kernel_235_not_rankOne : threePrimeKernelQ 2 3 5 0 0 0 * threePrimeKernelQ 2 3 5 1 1 0 ≠ threePrimeKernelQ 2 3 5 1 0 0 * threePrimeKernelQ 2 3 5 0 1 0","docstring":"The smallest exact witness that the three-prime kernel is not rank one. The two sides are `1/60` and `1/12`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L480","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L480"},{"name":"kernel_235_minor_eq_neg_one_fifteen","module":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","module_dotted":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","resolved_in_declaration_atlas":true,"line":904,"kind":"theorem","signature":"theorem kernel_235_minor_eq_neg_one_fifteen : threePrimeKernelQ 2 3 5 0 0 0 * threePrimeKernelQ 2 3 5 1 1 0 - threePrimeKernelQ 2 3 5 1 0 0 * threePrimeKernelQ 2 3 5 0 1 0 = -(1 / 15 : ℚ)","docstring":"The determinant of the smallest `{2,3,5}` kernel rectangle is exactly `-1/15`. In particular, the corresponding `2 × 2` matrix has rank two over the rationals; this is stronger than merely recording that it is not rank one.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L904","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L904"},{"name":"tailStateStep_eq","module":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","module_dotted":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","resolved_in_declaration_atlas":true,"line":491,"kind":"theorem","signature":"theorem tailStateStep_eq (base digit state next : ℤ) : next = tailStateStep base digit state ↔ next = base * state - base * digit","docstring":null,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L491","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L491"},{"name":"no_bounded_positive_state_of_residue_escape","module":"ErdosProblems/Erdos269/ResidueEscape.lean","module_dotted":"ErdosProblems.Erdos269.ResidueEscape","resolved_in_declaration_atlas":true,"line":76,"kind":"theorem","signature":"theorem no_bounded_positive_state_of_residue_escape {C bound residue c : ℕ} (hcpos : 0 < c) (hcbound : c ≤ bound) (hescape : ResidueEscapesWindow C bound residue) (hmod : c % C = residue % C) : False","docstring":"No positive state bounded by `bound` can represent, modulo `C`, a residue in the canonical positive range that lies above `bound`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ResidueEscape.lean#L76","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ResidueEscape.lean#L76"},{"name":"no_bounded_positive_int_state_of_leastPositiveResidue","module":"ErdosProblems/Erdos269/ResidueEscape.lean","module_dotted":"ErdosProblems.Erdos269.ResidueEscape","resolved_in_declaration_atlas":true,"line":110,"kind":"theorem","signature":"theorem no_bounded_positive_int_state_of_leastPositiveResidue {C bound : ℕ} {x c : ℤ} (hC : 0 < C) (hcpos : 0 < c) (hcbound : Int.natAbs c ≤ bound) (hescape : bound < leastPositiveResidue C x) (hmod : Int.ModEq C c x) : False","docstring":"Integer-valued form of the finite contradiction. A positive integer bounded by `bound` cannot be congruent to an integer whose least positive residue lies above that bound.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ResidueEscape.lean#L110","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ResidueEscape.lean#L110"},{"name":"CofinalLocalWindowEscape","module":"ErdosProblems/Erdos269/RestrictedFloorSum.lean","module_dotted":"ErdosProblems.Erdos269.RestrictedFloorSum","resolved_in_declaration_atlas":true,"line":629,"kind":"def","signature":"def CofinalLocalWindowEscape (b m : ℕ → ℕ) (shortBound : ℕ → ℕ → ℕ) : Prop","docstring":"The exact remaining producer after local-window compression. It is kept as a named proposition, not asserted as a theorem.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/RestrictedFloorSum.lean#L629","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/RestrictedFloorSum.lean#L629"},{"name":"no_positive_reducedCarry_of_cofinalLocalWindowEscape","module":"ErdosProblems/Erdos269/RestrictedFloorSum.lean","module_dotted":"ErdosProblems.Erdos269.RestrictedFloorSum","resolved_in_declaration_atlas":true,"line":645,"kind":"theorem","signature":"theorem no_positive_reducedCarry_of_cofinalLocalWindowEscape (b m : ℕ → ℕ) (shortBound : ℕ → ℕ → ℕ) (hescape : CofinalLocalWindowEscape b m shortBound) (B : ℕ) (hBpos : 0 < B) (hBcoprime : Nat.Coprime B 30) (d : ℕ → ℤ) (hrec : ∀ n, d (n + 1) = (b n : ℤ) * d n - (B : ℤ) * (m n : ℤ)) (hpos : ∀ n, 0 < d n) (hbound : ∀ n, Int.natAbs (d n) ≤ shortBound B n) : False","docstring":"The denominator-dependent local-window producer is exactly strong enough to rule out a positive reduced carry with the matching short bound. The coprimality hypothesis belongs to the still-open producer; the checked consumer uses only the emitted escaping window.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/RestrictedFloorSum.lean#L645","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/RestrictedFloorSum.lean#L645"},{"name":"boundedRadix_zero_or_cofinal_far","module":"ErdosProblems/Erdos269/BoundedRadixTailEscape.lean","module_dotted":"ErdosProblems.Erdos269.BoundedRadixTailEscape","resolved_in_declaration_atlas":true,"line":206,"kind":"theorem","signature":"theorem boundedRadix_zero_or_cofinal_far (P : ℕ → ℕ) (c : ℕ → ℤ) (x : ℕ → ℝ) (hPlo : ∀ a, 2 ≤ P a) (hPhi : ∀ a, P a ≤ 30) (hrec : ∀ a, x (a + 1) = (P a : ℝ) * x a - (c a : ℝ)) : (∃ a : ℕ, ∃ z : ℤ, x a = (z : ℝ)) ∨ ∀ a₀, ∃ a, a₀ ≤ a ∧ FarFromIntegers (x a) ((1 : ℝ) / 31)","docstring":"A real affine orbit with integral digits and radices in `[2,30]` either has an integral state or returns cofinally often to distance at least `1/31` from every integer.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/BoundedRadixTailEscape.lean#L206","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/BoundedRadixTailEscape.lean#L206"},{"name":"no_bounded_carryLift_of_blockNull_twoAnchors","module":"ErdosProblems/Erdos269/CarryLiftExtinction.lean","module_dotted":"ErdosProblems.Erdos269.CarryLiftExtinction","resolved_in_declaration_atlas":true,"line":238,"kind":"theorem","signature":"theorem no_bounded_carryLift_of_blockNull_twoAnchors (D : ℤ) (base digit z carry : ℕ → ℤ) (jumpBase : ℕ → Prime235) (hcarry : ∀ n, carry (n + 1) = base n * carry n - D * digit n) (hchannel : Function.Surjective jumpBase) (hnull : ChannelBlockNull jumpBase (carryLiftPerturbation base digit z)) {n23 n25 : ℕ} (h23start : jumpBase n23 = .two) (h23end : jumpBase (n23 + 1) = .three)","docstring":"Uniformly bounded nonzero lift errors are therefore impossible.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/CarryLiftExtinction.lean#L238","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/CarryLiftExtinction.lean#L238"},{"name":"channelBlockNull_iff_channelPotential","module":"ErdosProblems/Erdos269/ThreeChannelBlockRigidity.lean","module_dotted":"ErdosProblems.Erdos269.ThreeChannelBlockRigidity","resolved_in_declaration_atlas":true,"line":59,"kind":"theorem","signature":"theorem channelBlockNull_iff_channelPotential {ι G : Type*} [AddCommGroup G] (jumpBase : ℕ → ι) (ε : ℕ → G) (hbase : Function.Surjective jumpBase) : ChannelBlockNull jumpBase ε ↔ ∃ C : ι → G, ∀ N, ε N = C (jumpBase (N + 1)) - C (jumpBase N)","docstring":"If every channel occurs and all complete same-channel blocks have zero signed sum, then the word is exactly a channel coboundary. Conversely every channel coboundary has zero sum on such blocks. The surjectivity hypothesis is an explicit assumption; this theorem does not construct a channel word.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ThreeChannelBlockRigidity.lean#L59","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ThreeChannelBlockRigidity.lean#L59"},{"name":"carry_eq_residueDigit_add_coboundary","module":"ErdosProblems/Erdos269/WeightedPhaseCarry.lean","module_dotted":"ErdosProblems.Erdos269.WeightedPhaseCarry","resolved_in_declaration_atlas":true,"line":109,"kind":"theorem","signature":"theorem carry_eq_residueDigit_add_coboundary (B : ℤ) (hB : 0 < B) (base carry digit : ℕ → ℤ) (hrec : ∀ n, carry (n + 1) = base n * carry n - B * digit n) : let residue","docstring":"Rationality's exact carry recurrence decomposes into a finite residue digit and an integral coboundary. No recurrence or finite-state property of the quotient coordinate is assumed or concluded.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean#L109","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean#L109"},{"name":"finite_realisedSpan_of_factorisation","module":"ErdosProblems/Erdos269/WeightedPhaseCarry.lean","module_dotted":"ErdosProblems.Erdos269.WeightedPhaseCarry","resolved_in_declaration_atlas":true,"line":334,"kind":"theorem","signature":"theorem finite_realisedSpan_of_factorisation {𝕜 State Func E : Type*} [DivisionRing 𝕜] [AddCommGroup State] [Module 𝕜 State] [AddCommGroup Func] [Module 𝕜 Func] [AddCommGroup E] [Module 𝕜 E] [FiniteDimensional 𝕜 E] (realise : State →ₗ[𝕜] Func) (C : State →ₗ[𝕜] E) (readout : E →ₗ[𝕜] Func) (hfactor : realise = readout.comp C) : FiniteDimensional 𝕜 (LinearMap.range realise)","docstring":"An explicit factorization through a finite-dimensional observer forces the realised symbolic span to be finite-dimensional. A scalar value alone does not provide this factorization.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean#L334","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean#L334"}]},"lanes":{"reviewed_core_lane":{"authority":"docs/claims.json","library_root":"Erdos249257.lean","reviewed":true,"registered_claim_count":10,"registered_claim_ids_pointer":"/mathematical_assertions/problem_digest/erdos_269/registered_claim_ids","what_this_is":"curated claims the release reviews and publishes with a status. The registry covers the Erdos249257 core library only, which is the joint #249/#257 programme."},"problem_expansion_lane":{"authority":"docs/problems.json","library_root":"ErdosProblems.lean","directory":"lean/ErdosProblems/Erdos269","reviewed":false,"claim_registry_status":"registered_declarations","declared_module_count":7,"modules_pointer":"/mathematical_assertions/problem_digest/erdos_269/modules","what_this_is":"exact Lean propositions in this problem's own directory. They are kernel-checked and they are deliberately not promoted into reviewed public claims; reporting one as a registered claim is the specific error this split exists to prevent."},"read_this_first":"The three lanes are different bodies of source with different authorities: a reviewed registry, a kernel-checked expansion library, and a written manuscript. A problem can have evidence in any of them, or in all three. Say which lane an answer comes from, and never upgrade one into another.","manuscript_lane":{"authority":"manuscript_argument","reviewed":false,"source_lane":"manuscript_result","manuscript_ids":["erdos-269-three-prime-running-lcm"],"argued_result_count":14,"posed_problem_count":0,"rows_pointer":"mathematical_assertions.assertions filtered by source_lane == 'manuscript_result' and problem_ids containing 'erdos_269'","what_this_is":"results the manuscripts prove that no registered claim carries. Proved in prose at a quoted line, not by a Lean declaration; a formalisation is neither implied nor denied. For the six problems the claim registry does not cover, this is where the authored mathematics is."}}},"erdos_1041":{"problem_id":"erdos_1041","erdos_number":1041,"short_title":"Short connections inside polynomial lemniscates","question":"Must two roots of a monic polynomial in the open unit disc be joined by a sub-two-length curve inside its unit lemniscate? Ani’s degree-seven example refutes the exact Formal Conjectures statement; correspondence with the 1958 wording awaits human review.","status":"formal statement refuted","claim_registry_status":"registered_declarations","library_root":"ErdosProblems.lean","directory":"lean/ErdosProblems/Erdos1041","finite_search":"Raster tests through degree ten suggest candidate bounds only; they are separate from the exact degree-seven Lean counterexample.","corpus_role":"problem_owned_expansion","registered_claim_count":3,"registered_claim_ids":["ani_degree_seven_total_variation_counterexample","erdos_1041","newton_ray_separation_consumer"],"registered_claim_count_by_binding_method":{"open_proposition_target":2,"paper_owner":1},"registered_claim_count_note":"3 reviewed claim records bind to this problem in the current registry. Binding methods are listed separately; shared bindings do not establish independent contributions to each problem. Module inventory and claim inventory count different objects.","evidence_resolution":"module_level_expansion_library","declared_module_count":3,"modules":[{"module":"ErdosProblems.Erdos1041.Counterexample.HausdorffLength","path":"lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","present":true,"declaration_count":10,"theorem_count":9,"content_digest":"sha256:069ea76e27304bd34e020cf8aab0fcc9200c495ff892127612e8afe571e3fcb4","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","formal_source_present":true},{"module":"ErdosProblems.Erdos1041.Counterexample.CatalogueAdapter","path":"lean/ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean","present":true,"declaration_count":2,"theorem_count":2,"content_digest":"sha256:1f807ef3e453c12f382d4ce7a4f422397e1e2dee33ba87d68323273a1860a600","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean","formal_source_present":true},{"module":"ErdosProblems.Erdos1041.NewtonFlowRaySeparation","path":"lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","present":true,"declaration_count":28,"theorem_count":25,"content_digest":"sha256:9a2bc4dcafece55cd76b0bfe1a6b1dc8e1ebabf570c7ea92f9e53fe7629464fa","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","formal_source_present":true}],"paper_refs":[{"paper_id":"erdos-1041-lemniscate-newton-flow","form":"Problem note","hosted_pdf_path":"papers/erdos-1041-lemniscate-newton-flow.pdf","pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-1041-lemniscate-newton-flow.pdf","tex_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1041/erdos-1041-lemniscate-newton-flow.tex","tex_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1041/erdos-1041-lemniscate-newton-flow.tex","owns":"ani's degree-seven counterexample and the trinomial connection theorem; the long paper develops critical-value estimates, other subclasses and subsidiary formal checks","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or the current status of Erdős #1041, including the later one-polynomial Formal Conjectures path-image Hausdorff-length check and its unreviewed historical curve-length correspondence; those distinctions belong to docs/claims.json in the mathematics repository","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17"},{"paper_id":"erdos1041-lemniscate-reasoning-surface","form":"Reasoning surface","hosted_pdf_path":"papers/erdos1041-lemniscate-reasoning-surface.pdf","pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos1041-lemniscate-reasoning-surface.pdf","tex_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1041/erdos1041-lemniscate-reasoning-surface.tex","tex_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1041/erdos1041-lemniscate-reasoning-surface.tex","owns":"the exact geometry of ani's one-polynomial counterexample, proved positive families and critical-value arguments with their unformalised inputs identified; formal-statement status and historical correspondence remain owned by the claims registry","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or the current status of Erdős #1041, including the later one-polynomial Formal Conjectures path-image Hausdorff-length check and its unreviewed historical curve-length correspondence; those distinctions belong to docs/claims.json in the mathematics repository","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/2/body/problems/6","frontier":{"no_total_ordering":"The corpus publishes no ranking of its own results, and status is a kind, not a rank. Answer 'strongest' by naming the axis -- closest to the open target, largest verified finite case, strongest structural statement, principal conditional endpoint -- and quoting the rows below with their status. Do not compress them into one winner.","named_by_the_corpus":[],"advances_the_open_target":{"authority":"docs/claims.json :: machine_readable_paper.argument_graph","selection_basis":"source_authored:argument_graph.advances_open_target","by_effect_on_the_open_target":{"narrows":[{"assertion_id":"claim:newton_ray_separation_consumer","source_id":"newton_ray_separation_consumer","open_proposition_id":"remaining_open.erdos_1041_lemniscate_connection","what_the_effect_is":"Removes same-ray Newton-flow connections under the exact exponential-decay hypothesis and supplies arbitrarily small finite-family ray-separating translations, while leaving the global short-curve topology and length argument open."}]},"how_to_read_it":"The corpus distinguishes narrowing an open target from re-expressing it. Both are honest progress and neither settles anything; the difference is whether the open region got smaller or changed shape. Each entry carries the corpus's own sentence for its effect.","what_this_does_not_say":"which of several claims sharing an effect is the stronger. That ordering is not written down anywhere, so it is not supplied here."},"programme_representatives":[],"registered_claims_by_status":{"formal statement refuted":["claim:erdos_1041"],"formalised here":["claim:ani_degree_seven_total_variation_counterexample"],"unconditional progress":["claim:newton_ray_separation_consumer"]},"open_bottlenecks":[{"assertion_id":"open:remaining_open.erdos_1041_lemniscate_connection","open_target_claim":"erdos_1041"}],"manuscript_results":{"row_count":19,"authority":"manuscript_argument","reviewed":false,"what_this_is":"every result the manuscripts on this problem label as a result, grouped by the environment the author wrote it in. The grouping is the author's classification, not a ranking by this build: a theorem is here because the manuscript calls it a theorem.","read_the_proved_statement_at":"the row's paper_anchor.source_excerpt.text, quoted exactly. The statement field is the environment's title, which names the result rather than stating it.","by_environment":{"corollary":[{"assertion_id":"paper:erdos_1041#res:two-nearest-roots","environment":"corollary","selection_basis":"source_authored:manuscript_res_label","statement":"two nearest roots"}],"lemma":[{"assertion_id":"paper:erdos_1041#res:circle-slice-packing","environment":"lemma","selection_basis":"source_authored:manuscript_res_label","statement":"circle-slice packing"},{"assertion_id":"paper:erdos_1041#res:reflected-critical-value","environment":"lemma","selection_basis":"source_authored:manuscript_res_label","statement":"reflected-derivative bound"}],"proposition":[{"assertion_id":"paper:erdos_1041#res:sextic-spoke","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"failure of a prescribed radial segment"},{"assertion_id":"paper:erdos_1041#res:one-root-gamma-false","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"a counterexample to the proposed one-root perimeter bound"},{"assertion_id":"paper:erdos_1041#res:sep-or-false","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"failure of two critical-value criteria to cover all polynomials"},{"assertion_id":"paper:erdos_1041#res:arity-not-capacity","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"root count does not force a capacity gap"},{"assertion_id":"paper:erdos_1041#res:straight-no-go","environment":"proposition","selection_basis":"source_authored:manuscript_res_label","statement":"two counterexamples to straight-path assertions"}],"theorem":[{"assertion_id":"paper:erdos-1041-lemniscate-newton-flow#res:trinomial-all-degree","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"all-degree monic trinomials"},{"assertion_id":"paper:erdos_1041#res:ani-degree-seven-counterexample-long","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"ani's counterexample"},{"assertion_id":"paper:erdos1041-lemniscate-reasoning-surface#res:trinomial-all-degree","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"trinomial root connections"},{"assertion_id":"paper:erdos_1041#res:degree-three","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"the cubic case"},{"assertion_id":"paper:erdos_1041#res:complementary-binomial-chords","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"complementary binomial chords"},{"assertion_id":"paper:erdos_1041#res:critical-value-budget","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"a mean bound for critical values"},{"assertion_id":"paper:erdos_1041#res:fp-weighted-all-degree","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"a weighted inequality for points in a disc"},{"assertion_id":"paper:erdos_1041#res:critical-proximity","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"a geometric-mean bound for distances to a critical point"},{"assertion_id":"paper:erdos_1041#res:orlicz-currency","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"the relation between two merger-scale integrals"},{"assertion_id":"paper:erdos_1041#res:value","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"value equation"},{"assertion_id":"paper:erdos_1041#res:locus","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"ray-collision locus"}]},"posed_problem_count":2},"selection_basis_legend":{"source_authored:orientation.principal_claims":"the repository's own orientation record lists this claim among the principal results of the release","source_authored:orientation.mathematical_programmes[].representative_claim_ids":"the repository names this claim as representative of one of its ten mathematical programmes","mechanical_grouping":"registered_claims_by_status is a grouping, not a judgement; the status taxonomy that defines each label is carried in the claims registry","source_authored:manuscript_res_label":"the manuscript labels this statement a result, using the corpus's own 'res:' convention, and states it in the environment named on the entry"},"empty_means":"An empty registered-claim frontier means the reviewed claim registry carries nothing for this problem, which is true for six of the eight. It does not mean there is no result: read manuscript_results above and the expansion lane in problem_digest.lanes. Emptiness in one lane is a fact about which library the registry covers, and answering 'no result' from it is the specific error the three lanes exist to prevent."},"required_note_declarations":{"count":8,"authority":"docs/problem_index_source.json::problems[].required_note_declarations","what_this_is":"the theorems the problem index itself names as required reading for this problem's note. This is the expansion lane's answer to the question the reviewed lane answers with a claim's declarations: what is it standing on. It is authored upstream, not derived here.","granularity":"problem_level. These declarations back the problem, not any one sentence in it -- no per-sentence mapping is published, and inventing one would be a more confident citation than the source supports.","declarations":[{"name":"newtonFlow_value_hasDerivAt","module":"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","module_dotted":"ErdosProblems.Erdos1041.NewtonFlowRaySeparation","resolved_in_declaration_atlas":true,"line":50,"kind":"theorem","signature":"theorem newtonFlow_value_hasDerivAt {f f' z : ℂ → ℂ} {t : ℂ} (hf : HasDerivAt f (f' (z t)) (z t)) (hz : HasDerivAt z (newtonFlowVector (f (z t)) (f' (z t))) t) (hcritical : f' (z t) ≠ 0) : HasDerivAt (fun s => f (z s)) (-f (z t)) t","docstring":"Along a curve tangent to the Newton field, the polynomial value has derivative equal to its own negative. This is the local chain-rule form of `f(z(t)) = exp(-t) f(z(0))`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean#L50","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean#L50"},{"name":"newtonFlow_scaledValue_hasDerivAt_zero","module":"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","module_dotted":"ErdosProblems.Erdos1041.NewtonFlowRaySeparation","resolved_in_declaration_atlas":true,"line":64,"kind":"theorem","signature":"theorem newtonFlow_scaledValue_hasDerivAt_zero {f f' z : ℂ → ℂ} {t : ℂ} (hf : HasDerivAt f (f' (z t)) (z t)) (hz : HasDerivAt z (newtonFlowVector (f (z t)) (f' (z t))) t) (hcritical : f' (z t) ≠ 0) : HasDerivAt (fun s => Complex.exp s * f (z s)) 0 t","docstring":"Multiplication by `exp(t)` turns the Newton-flow value equation into a quantity with zero derivative.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean#L64","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean#L64"},{"name":"translated_samePositiveRay_parameterization","module":"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","module_dotted":"ErdosProblems.Erdos1041.NewtonFlowRaySeparation","resolved_in_declaration_atlas":true,"line":107,"kind":"theorem","signature":"theorem translated_samePositiveRay_parameterization {a b shift : ℂ} (hab : a ≠ b) (hray : SamePositiveRay (a + shift) (b + shift)) : ∃ r : ℝ, 0 < r ∧ r ≠ 1 ∧ shift = ((r : ℂ) * a - b) / ((1 - r : ℝ) : ℂ)","docstring":"If two distinct values become positively ray-aligned after the same translation, then the translation belongs to an explicit one-real-parameter forbidden locus. For a finite set of critical values, ray separation is therefore a finite planar avoidance problem.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean#L107","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean#L107"},{"name":"exists_small_translation_separating_arguments","module":"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","module_dotted":"ErdosProblems.Erdos1041.NewtonFlowRaySeparation","resolved_in_declaration_atlas":true,"line":197,"kind":"theorem","signature":"theorem exists_small_translation_separating_arguments {ι : Type*} [Fintype ι] (c : ι → ℂ) (hc : Function.Injective c) {ε : ℝ} (hε : 0 < ε) : ∃ shift : ℂ, ‖shift‖ < ε ∧ (∀ i, c i + shift ≠ 0) ∧ ∀ i j, i ≠ j → ¬ SamePositiveRay (c i + shift) (c j + shift)","docstring":"An injective finite complex family admits an arbitrarily small common translation after which every value is nonzero and all positive-ray arguments are distinct.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean#L197","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean#L197"},{"name":"constant_perturbation_roots_in_unitDisk","module":"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","module_dotted":"ErdosProblems.Erdos1041.NewtonFlowRaySeparation","resolved_in_declaration_atlas":true,"line":287,"kind":"theorem","signature":"theorem constant_perturbation_roots_in_unitDisk (f : ℂ[X]) (hf : f.Monic) (hdeg : 0 < f.natDegree) (hsplit : f.Splits) {ρ ε : ℝ} (hρ : 0 ≤ ρ) (hroots : ∀ b ∈ f.roots, ‖b‖ ≤ ρ) (hε : 0 < ε) (hmargin : ((f.natDegree + 1) * ε) ^ (f.natDegree : ℝ)⁻¹ + ρ < 1) {shift : ℂ} (hshift : ‖shift‖ < ε) : ∀ a : ℂ, (f + C shift).eval a = 0 → ‖a‖ < 1","docstring":"Under an explicit strict margin, a small constant perturbation preserves the property that every polynomial root lies in the open unit disk.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean#L287","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean#L287"},{"name":"no_newtonConnection_of_not_samePositiveRay","module":"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","module_dotted":"ErdosProblems.Erdos1041.NewtonFlowRaySeparation","resolved_in_declaration_atlas":true,"line":334,"kind":"theorem","signature":"theorem no_newtonConnection_of_not_samePositiveRay {startValue endValue : ℂ} {time : ℝ} (hrays : ¬ SamePositiveRay startValue endValue) (hconnection : endValue = (Real.exp (-time) : ℂ) * startValue) : False","docstring":"Distinct critical-value rays exclude a Newton-flow connection. The topological repair uses this with all critical values chosen on pairwise distinct rays.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean#L334","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean#L334"},{"name":"erdos1041_ani_degree_seven","module":"ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean","module_dotted":"ErdosProblems.Erdos1041.Counterexample.CatalogueAdapter","resolved_in_declaration_atlas":true,"line":25,"kind":"theorem","signature":"theorem erdos1041_ani_degree_seven : ∃ (p : ℂ[X]), p.Monic ∧ p.natDegree = 7 ∧ (∀ z, p.IsRoot z → ‖z‖ < 1) ∧ p.roots.Nodup ∧ ∀ z₁ z₂, p.IsRoot z₁ → p.IsRoot z₂ → z₁ ≠ z₂ → ∀ γ : ℝ → ℂ, ContinuousOn γ (Set.Icc 0 1) → γ 0 = z₁ → γ 1 = z₂ → (∀ τ ∈ Set.Icc (0 : ℝ) 1, ‖p.eval (γ τ)‖ < 1) → (2 : ℝ≥0∞) < eVariationOn γ (Set.Icc 0 1)","docstring":"Existential wrapper matching `FormalConjectures.ErdosProblems.erdos_1041.variants.ani_degree_seven`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean#L25","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean#L25"},{"name":"erdos1041_hausdorff_negation","module":"ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","module_dotted":"ErdosProblems.Erdos1041.Counterexample.HausdorffLength","resolved_in_declaration_atlas":true,"line":407,"kind":"theorem","signature":"theorem erdos1041_hausdorff_negation : ¬ ∀ (n : ℕ) (f : ℂ[X]), n ≥ 2 → f.natDegree = n → f.Monic → f.rootSet ℂ ⊆ Metric.ball 0 1 → ∃ (z₁ z₂ : ℂ) (h : ({z₁, z₂} : Multiset ℂ) ≤ f.roots) (γ : Path z₁ z₂), Set.range γ ⊆ { z : ℂ | ‖f.eval z‖ < 1 } ∧ fcLength (Set.range γ) < 2","docstring":"Negation of Formal Conjectures' `Erdos1041.erdos_1041`. Its section variables `n`, `f`, `hn`, `hnum`, `h_monic`, `h` are quantified in their declared order, and path length is `length (Set.range γ)`, the one-dimensional Hausdorff measure of the image. The polynomial `f` of `Defs.lean` refutes it.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L407","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L407"}]},"lanes":{"reviewed_core_lane":{"authority":"docs/claims.json","library_root":"Erdos249257.lean","reviewed":true,"registered_claim_count":3,"registered_claim_ids_pointer":"/mathematical_assertions/problem_digest/erdos_1041/registered_claim_ids","what_this_is":"curated claims the release reviews and publishes with a status. The registry covers the Erdos249257 core library only, which is the joint #249/#257 programme."},"problem_expansion_lane":{"authority":"docs/problems.json","library_root":"ErdosProblems.lean","directory":"lean/ErdosProblems/Erdos1041","reviewed":false,"claim_registry_status":"registered_declarations","declared_module_count":3,"modules_pointer":"/mathematical_assertions/problem_digest/erdos_1041/modules","what_this_is":"exact Lean propositions in this problem's own directory. They are kernel-checked and they are deliberately not promoted into reviewed public claims; reporting one as a registered claim is the specific error this split exists to prevent."},"read_this_first":"The three lanes are different bodies of source with different authorities: a reviewed registry, a kernel-checked expansion library, and a written manuscript. A problem can have evidence in any of them, or in all three. Say which lane an answer comes from, and never upgrade one into another.","manuscript_lane":{"authority":"manuscript_argument","reviewed":false,"source_lane":"manuscript_result","manuscript_ids":["erdos-1041-lemniscate-newton-flow","erdos1041-lemniscate-reasoning-surface"],"argued_result_count":19,"posed_problem_count":2,"rows_pointer":"mathematical_assertions.assertions filtered by source_lane == 'manuscript_result' and problem_ids containing 'erdos_1041'","what_this_is":"results the manuscripts prove that no registered claim carries. Proved in prose at a quoted line, not by a Lean declaration; a formalisation is neither implied nor denied. For the six problems the claim registry does not cover, this is where the authored mathematics is."}}},"erdos_1049":{"problem_id":"erdos_1049","erdos_number":1049,"short_title":"Lambert-type series at rational bases","question":"For which rational bases is the corresponding series irrational? The first resistant explicit base is three halves.","status":"open","claim_registry_status":"registered_declarations","library_root":"ErdosProblems.lean","directory":"lean/ErdosProblems/Erdos1049","finite_search":null,"corpus_role":"problem_owned_expansion","registered_claim_count":9,"registered_claim_ids":["calibrated_rational_hankel_countermodel","coefficient_pencil_finite_rank","erdos_1049","rational_base_cleared_tail_recurrence","rational_base_forcing_growth","three_halves_coordinatewise_corridor_no_go","van_assche_qapery_diagonal_no_go","zudilin_all_rank_hankel_order","zudilin_rational_base_region"],"registered_claim_count_by_binding_method":{"open_proposition_target":8,"paper_owner":1},"registered_claim_count_note":"9 reviewed claim records bind to this problem in the current registry. Binding methods are listed separately; shared bindings do not establish independent contributions to each problem. Module inventory and claim inventory count different objects.","evidence_resolution":"module_level_expansion_library","declared_module_count":6,"modules":[{"module":"ErdosProblems.Erdos1049.RationalBaseLambert","path":"lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean","present":true,"declaration_count":16,"theorem_count":11,"content_digest":"sha256:593c3b3e776820de229bad00c7f374bb39416099de60250b375d52cfa5f6e56a","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean","formal_source_present":true},{"module":"ErdosProblems.Erdos1049.RationalPadeArithmetic","path":"lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean","present":true,"declaration_count":19,"theorem_count":14,"content_digest":"sha256:c7b54971556577d73bbc563dcf14eaf62aef8d31e1450c4e7337028e46013690","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean","formal_source_present":true},{"module":"ErdosProblems.Erdos1049.ZudilinConeArithmetic","path":"lean/ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean","present":true,"declaration_count":36,"theorem_count":25,"content_digest":"sha256:9001917590300c4f10a049af0dc56192f9db95f2700b9cfac5ac88b31505b7d0","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean","formal_source_present":true},{"module":"ErdosProblems.Erdos1049.HermitePadeNoGo","path":"lean/ErdosProblems/Erdos1049/HermitePadeNoGo.lean","present":true,"declaration_count":10,"theorem_count":5,"content_digest":"sha256:812f2ca2ea8b18f8ce8e0a68ae14897aa851fba2e050a07881d839168182f2cc","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/HermitePadeNoGo.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/HermitePadeNoGo.lean","formal_source_present":true},{"module":"ErdosProblems.Erdos1049.ZudilinHeightRegion","path":"lean/ErdosProblems/Erdos1049/ZudilinHeightRegion.lean","present":true,"declaration_count":13,"theorem_count":12,"content_digest":"sha256:bdb9ffaeab12363ed91a6385a6df8ef8e931003ad7862096ed28fc09aa3669fa","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/ZudilinHeightRegion.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/ZudilinHeightRegion.lean","formal_source_present":true},{"module":"ErdosProblems.Erdos1049.QAperyDiagonalNonEquivalence","path":"lean/ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean","present":true,"declaration_count":11,"theorem_count":4,"content_digest":"sha256:8094ead680a8a3be3e7ad2c6c2fbc89cd51c9f84a360bfb46f91f6fc4b90cda1","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean","formal_source_present":true}],"paper_refs":[{"paper_id":"erdos-1049-rational-base-lambert","form":"Problem note","hosted_pdf_path":"papers/erdos-1049-rational-base-lambert.pdf","pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-1049-rational-base-lambert.pdf","tex_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1049/erdos-1049-rational-base-lambert.tex","tex_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1049/erdos-1049-rational-base-lambert.tex","owns":"the fixed-base Hankel asymptotic and rational-base irrationality theorem; the long paper develops exact orders, content and endpoint obstructions, and the remaining problem at 3/2","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #1049, which remains open","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17"},{"paper_id":"erdos1049-rational-base-lambert-reasoning-surface","form":"Reasoning surface","hosted_pdf_path":"papers/erdos1049-rational-base-lambert-reasoning-surface.pdf","pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos1049-rational-base-lambert-reasoning-surface.pdf","tex_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","tex_permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","owns":"the sharp fixed-base Hankel asymptotic and rational-base irrationality proofs, coefficient moments and arithmetic obstructions, with ordinary ratio-limit and rational-value models distinguished from checked results","not_authority_for":"the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #1049, which remains open","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/2/body/problems/7","frontier":{"no_total_ordering":"The corpus publishes no ranking of its own results, and status is a kind, not a rank. Answer 'strongest' by naming the axis -- closest to the open target, largest verified finite case, strongest structural statement, principal conditional endpoint -- and quoting the rows below with their status. Do not compress them into one winner.","named_by_the_corpus":[],"advances_the_open_target":{"authority":"docs/claims.json :: machine_readable_paper.argument_graph","selection_basis":"source_authored:argument_graph.advances_open_target","by_effect_on_the_open_target":{"narrows":[{"assertion_id":"claim:rational_base_cleared_tail_recurrence","source_id":"rational_base_cleared_tail_recurrence","open_proposition_id":"remaining_open.erdos_1049_irrationality","what_the_effect_is":"Records the exact rational-base cleared-tail identity and its denominator-base forcing term without deriving it from rationality or deciding any #1049 endpoint."},{"assertion_id":"claim:rational_base_forcing_growth","source_id":"rational_base_forcing_growth","open_proposition_id":"remaining_open.erdos_1049_irrationality","what_the_effect_is":"Bounds the forcing term below by 2^(N+1) for s ≥ 2 and records its exact s = 1 collapse, without proving a bounded-state contradiction or deciding any #1049 endpoint."},{"assertion_id":"claim:three_halves_coordinatewise_corridor_no_go","source_id":"three_halves_coordinatewise_corridor_no_go","open_proposition_id":"remaining_open.erdos_1049_irrationality","what_the_effect_is":"Excludes the literal six-condition coordinatewise clearing corridor at base 3/2 while leaving irrationality, rationality, and all other proof architectures open."},{"assertion_id":"claim:van_assche_qapery_diagonal_no_go","source_id":"van_assche_qapery_diagonal_no_go","open_proposition_id":"remaining_open.erdos_1049_irrationality","what_the_effect_is":"Rules out the selected Van Assche diagonal as a solution of the Amdeberhan–Zeilberger recurrence at n = 0 for p > 1, without establishing a general recurrence or deciding any #1049 endpoint."}]},"how_to_read_it":"The corpus distinguishes narrowing an open target from re-expressing it. Both are honest progress and neither settles anything; the difference is whether the open region got smaller or changed shape. Each entry carries the corpus's own sentence for its effect.","what_this_does_not_say":"which of several claims sharing an effect is the stronger. That ordering is not written down anywhere, so it is not supplied here."},"programme_representatives":[],"registered_claims_by_status":{"formalised here":["claim:zudilin_rational_base_region"],"open":["claim:erdos_1049"],"unconditional progress":["claim:calibrated_rational_hankel_countermodel","claim:coefficient_pencil_finite_rank","claim:rational_base_cleared_tail_recurrence","claim:rational_base_forcing_growth","claim:three_halves_coordinatewise_corridor_no_go","claim:van_assche_qapery_diagonal_no_go","claim:zudilin_all_rank_hankel_order"]},"open_bottlenecks":[{"assertion_id":"open:remaining_open.erdos_1049_irrationality","open_target_claim":"erdos_1049"}],"manuscript_results":{"row_count":4,"authority":"manuscript_argument","reviewed":false,"what_this_is":"every result the manuscripts on this problem label as a result, grouped by the environment the author wrote it in. The grouping is the author's classification, not a ranking by this build: a theorem is here because the manuscript calls it a theorem.","read_the_proved_statement_at":"the row's paper_anchor.source_excerpt.text, quoted exactly. The statement field is the environment's title, which names the result rather than stating it.","by_environment":{"corollary":[{"assertion_id":"paper:erdos_1049#res:thirtyone-four","environment":"corollary","selection_basis":"source_authored:manuscript_res_label"}],"lemma":[{"assertion_id":"paper:erdos_1049#res:bottomjet","environment":"lemma","selection_basis":"source_authored:manuscript_res_label","statement":"a residue and divisibility by 3^R"}],"theorem":[{"assertion_id":"paper:erdos_1049#res:sharp-fixed-base","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"the size of V_N^* at a fixed base"},{"assertion_id":"paper:erdos_1049#res:boundedfibre","environment":"theorem","selection_basis":"source_authored:manuscript_res_label","statement":"equal residues with close, distinct values"}]},"posed_problem_count":0},"selection_basis_legend":{"source_authored:orientation.principal_claims":"the repository's own orientation record lists this claim among the principal results of the release","source_authored:orientation.mathematical_programmes[].representative_claim_ids":"the repository names this claim as representative of one of its ten mathematical programmes","mechanical_grouping":"registered_claims_by_status is a grouping, not a judgement; the status taxonomy that defines each label is carried in the claims registry","source_authored:manuscript_res_label":"the manuscript labels this statement a result, using the corpus's own 'res:' convention, and states it in the environment named on the entry"},"empty_means":"An empty registered-claim frontier means the reviewed claim registry carries nothing for this problem, which is true for six of the eight. It does not mean there is no result: read manuscript_results above and the expansion lane in problem_digest.lanes. Emptiness in one lane is a fact about which library the registry covers, and answering 'no result' from it is the specific error the three lanes exist to prevent."},"required_note_declarations":{"count":12,"authority":"docs/problem_index_source.json::problems[].required_note_declarations","what_this_is":"the theorems the problem index itself names as required reading for this problem's note. This is the expansion lane's answer to the question the reviewed lane answers with a claim's declarations: what is it standing on. It is authored upstream, not derived here.","granularity":"problem_level. These declarations back the problem, not any one sentence in it -- no per-sentence mapping is published, and inventing one would be a more confident citation than the source supports.","declarations":[{"name":"threeHalves_no_coordinatewiseCorridor","module":"ErdosProblems/Erdos1049/RationalBaseLambert.lean","module_dotted":"ErdosProblems.Erdos1049.RationalBaseLambert","resolved_in_declaration_atlas":true,"line":155,"kind":"theorem","signature":"theorem threeHalves_no_coordinatewiseCorridor {N K Q digit : ℕ} (hN : 1 ≤ N) (hK : 1 ≤ K) : ¬ CoordinatewiseCorridor 3 2 N K Q digit","docstring":"The literal coordinatewise Erdős corridor cannot occur at base `3/2` once both the shift and cleared window are nonempty.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean#L155","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean#L155"},{"name":"rationalBaseClearedTailQ_succ","module":"ErdosProblems/Erdos1049/RationalBaseLambert.lean","module_dotted":"ErdosProblems.Erdos1049.RationalBaseLambert","resolved_in_declaration_atlas":true,"line":187,"kind":"theorem","signature":"theorem rationalBaseClearedTailQ_succ {r s B F : ℚ} {coeff : ℕ → ℚ} (hr : r ≠ 0) (N : ℕ) : rationalBaseClearedTailQ r s B F coeff (N + 1) = r * rationalBaseClearedTailQ r s B F coeff N - B * coeff (N + 1) * s ^ (N + 1)","docstring":"Exact rational-base recurrence. 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The gap factors as `(n-k)(3n-k-1)`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean#L30","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean#L30"},{"name":"rationalPadeQMaxDenExpTwice_gap","module":"ErdosProblems/Erdos1049/RationalPadeArithmetic.lean","module_dotted":"ErdosProblems.Erdos1049.RationalPadeArithmetic","resolved_in_declaration_atlas":true,"line":52,"kind":"theorem","signature":"theorem rationalPadeQMaxDenExpTwice_gap (n m : ℤ) : rationalPadeDenExpTwice n - rationalPadeQMaxDenExpTwice n m = 2 * (n + m * (m - 1))","docstring":"Exact doubled gap identity for the rational `Q`-summand maximum: `2(Eₙ-Fmax) = 2(n + m(m-1))`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean#L52","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean#L52"},{"name":"rationalPadeExteriorDet_cast_eq","module":"ErdosProblems/Erdos1049/RationalPadeArithmetic.lean","module_dotted":"ErdosProblems.Erdos1049.RationalPadeArithmetic","resolved_in_declaration_atlas":true,"line":124,"kind":"theorem","signature":"theorem rationalPadeExteriorDet_cast_eq (S : ℝ) (Un Vn Um Vm : ℤ) : (rationalPadeExteriorDet Un Vn Um Vm : ℝ) = (Um : ℝ) * rationalPadeError S Un Vn - (Un : ℝ) * rationalPadeError S Um Vm","docstring":"Exact cancellation of the target value in the exterior determinant.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean#L124","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean#L124"},{"name":"rationalPadeExteriorDet_mul_contents","module":"ErdosProblems/Erdos1049/RationalPadeArithmetic.lean","module_dotted":"ErdosProblems.Erdos1049.RationalPadeArithmetic","resolved_in_declaration_atlas":true,"line":94,"kind":"theorem","signature":"theorem rationalPadeExteriorDet_mul_contents (cn cm Un Vn Um Vm : ℤ) : rationalPadeExteriorDet (cn * Un) (cn * Vn) (cm * Um) (cm * Vm) = cn * cm * rationalPadeExteriorDet Un Vn Um Vm","docstring":"Rowwise contents factor from the exterior determinant exactly. Thus any local divisor coming only from independent contents of the two rows is paid for by the same product in the Archimedean determinant height.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean#L94","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean#L94"},{"name":"natAbs_rationalPadeExteriorDet_mul_contents","module":"ErdosProblems/Erdos1049/RationalPadeArithmetic.lean","module_dotted":"ErdosProblems.Erdos1049.RationalPadeArithmetic","resolved_in_declaration_atlas":true,"line":104,"kind":"theorem","signature":"theorem natAbs_rationalPadeExteriorDet_mul_contents (cn cm Un Vn Um Vm : ℤ) : Int.natAbs (rationalPadeExteriorDet (cn * Un) (cn * Vn) (cm * Um) (cm * Vm)) = Int.natAbs cn * Int.natAbs cm * Int.natAbs (rationalPadeExteriorDet Un Vn Um Vm)","docstring":"The Archimedean determinant height acquires exactly the absolute product of the two row contents—no less and no more.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean#L104","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean#L104"},{"name":"bottomJet3_eq_zero_iff_dvd","module":"ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean","module_dotted":"ErdosProblems.Erdos1049.ZudilinConeArithmetic","resolved_in_declaration_atlas":true,"line":191,"kind":"theorem","signature":"theorem bottomJet3_eq_zero_iff_dvd (R W : ℕ) (P : Polynomial ℤ) : bottomJet3 R W P = 0 ↔ ((3 ^ R : ℕ) : ℤ) ∣ homEvalThreeTwo W P","docstring":"Vanishing of the bottom jet is exactly divisibility of the homogeneous evaluation by the requested power of `3`.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean#L191","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean#L191"},{"name":"thirtyoneFour_power_mem_zudilinHeightRegion","module":"ErdosProblems/Erdos1049/ZudilinHeightRegion.lean","module_dotted":"ErdosProblems.Erdos1049.ZudilinHeightRegion","resolved_in_declaration_atlas":true,"line":59,"kind":"theorem","signature":"theorem thirtyoneFour_power_mem_zudilinHeightRegion (r : ℕ) (hr : 0 < r) : ZudilinHeightRegion (31 ^ r) (4 ^ r)","docstring":"Every positive power of `31 / 4` satisfies the same height threshold.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/ZudilinHeightRegion.lean#L59","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/ZudilinHeightRegion.lean#L59"},{"name":"threeHalves_outside_zudilinHeightRegion","module":"ErdosProblems/Erdos1049/ZudilinHeightRegion.lean","module_dotted":"ErdosProblems.Erdos1049.ZudilinHeightRegion","resolved_in_declaration_atlas":true,"line":122,"kind":"theorem","signature":"theorem threeHalves_outside_zudilinHeightRegion : ¬ ZudilinHeightRegion 3 2","docstring":"The height region defined above does not contain `3 / 2`. This is a method boundary, not a rationality or irrationality theorem for the corresponding Lambert value.","current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/ZudilinHeightRegion.lean#L122","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/ZudilinHeightRegion.lean#L122"}]},"lanes":{"reviewed_core_lane":{"authority":"docs/claims.json","library_root":"Erdos249257.lean","reviewed":true,"registered_claim_count":9,"registered_claim_ids_pointer":"/mathematical_assertions/problem_digest/erdos_1049/registered_claim_ids","what_this_is":"curated claims the release reviews and publishes with a status. 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The catalogue sum for three or more primes is unresolved in this release: for {2,3,5} the rationality-to-positive-reduced-carry bridge is checked, and the remaining endpoint is cofinal local-window escape, equivalently exclusion of the integral branch.","status":"open","status_meaning":"Not proved by this release","readme_headline":false,"problem_ids":["erdos_269"],"problem_erdos_numbers":[269],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_269","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_269","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"registry_statement_only","evidence_resolution":"statement_only","declarations":[],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. 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The irrationality of Π remains open: these facts do not supply the required cofinal escape for the actual prime gaps.","status":"open","status_meaning":"Not proved by this release","readme_headline":false,"problem_ids":["erdos_251"],"problem_erdos_numbers":[251],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_251","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_251","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"registry_statement_only","evidence_resolution":"statement_only","declarations":[],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. 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This checkout formalises a version that discharges summability without the prime-number theorem. No novelty, priority, significance, or irrationality claim is made.","primary_narrative_owner":"paper/251/erdos-251-prime-gap-dyadic-series.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/24","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. 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The proof combines the exact factorial prime-free intervals above with the finite range of a periodic natural-valued sequence.","generated_certificate":false}],"paper_label":"res:gap-nonperiodic","paper_anchor":{"paper_label":"res:gap-nonperiodic","owner_source":"paper/251/erdos251-prime-gap-reasoning-surface.tex","source_line":1082,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":1082,"anchor_title":"prime gaps do not become periodic","anchor_title_tex":"prime gaps do not become periodic","section_title":"A local certificate, and one actual pair","section_path":[{"level":"section","title":"A local certificate, and one actual pair","title_tex":"A local certificate, and one actual pair","line":1040}],"source_excerpt":{"text":"\\begin{proposition}[prime gaps do not become periodic]\\label{long251:res:gap-nonperiodic}\\label{res:gap-nonperiodic}\nFor every positive $h$, the actual consecutive-prime-gap sequence is not\neventually periodic with period $h$.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":53825,"source_char_end":54069,"start_line":1082,"end_line":1085,"truncated":false,"sha256":"sha256:32f0010508efeff87f79c9a572338c27d97f5511243c833abd1eae2a49fe8476","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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This formalises known summation-by-parts; the public attribution of that identity is the cited-only row prime_gap_identity_tao.","status":"formalised here","status_meaning":"Known mathematics represented and checked in Lean","readme_headline":false,"problem_ids":["erdos_251"],"problem_erdos_numbers":[251],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_251","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_251","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos251-prime-gap-reasoning-surface","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"summable_primeDyadicTerm","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","line":379,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L379","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L379","kind":"theorem","signature":"theorem summable_primeDyadicTerm : Summable primeDyadicTerm","docstring":"The normalized prime series converges unconditionally, using the explicit elementary polynomial bound above.","generated_certificate":false},{"name":"tsum_primeDyadicTerm_eq_two_add_primeGap_unconditional","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","line":427,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L427","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L427","kind":"theorem","signature":"theorem tsum_primeDyadicTerm_eq_two_add_primeGap_unconditional : (∑' n : ℕ, primeDyadicTerm n) = 2 + ∑' n : ℕ, primeGapDyadicTerm n","docstring":"Unconditional form of the exact infinite prime-gap reformulation.","generated_certificate":false},{"name":"irrational_tsum_primeDyadicTerm_iff_primeGap","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","line":435,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L435","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L435","kind":"theorem","signature":"theorem irrational_tsum_primeDyadicTerm_iff_primeGap (hprime : Summable primeDyadicTerm) : Irrational (∑' n : ℕ, primeDyadicTerm n) ↔ Irrational (∑' n : ℕ, primeGapDyadicTerm n)","docstring":"Erdős #251 is therefore exactly equivalent to irrationality of the consecutive-prime-gap dyadic series, once summability of the displayed prime series is supplied.","generated_certificate":false}],"paper_label":"res:irr-equivalence","paper_anchor":{"paper_label":"res:irr-equivalence","owner_source":"paper/251/erdos251-prime-gap-reasoning-surface.tex","source_line":791,"anchor_kind":"environment","anchor_unit":"corollary","anchor_unit_line":791,"anchor_title":"exact irrationality reformulation","anchor_title_tex":"exact irrationality reformulation","section_title":"The infinite identity and the irrationality equivalence","section_path":[{"level":"section","title":"Summation by parts, with the endpoint retained","title_tex":"Summation by parts, with the endpoint retained","line":695},{"level":"subsection","title":"The infinite identity and the irrationality equivalence","title_tex":"The infinite identity and the irrationality equivalence","line":753}],"source_excerpt":{"text":"\\begin{corollary}[exact irrationality reformulation]\\label{long251:res:irr-equivalence}\\label{res:irr-equivalence}\n$\\Pi$ is irrational if and only if $S=\\sum_{n\\ge0}g_n2^{-(n+1)}$ is irrational.\nThe corresponding zero-based series with denominator $2^n$ equals $4+2S$ and\nhas the same irrationality status.\n\\end{corollary}","media_type":"text/x-tex","source_char_start":39219,"source_char_end":39541,"start_line":791,"end_line":795,"truncated":false,"sha256":"sha256:e7b976b76bfc8a21d3fbd4236ee0e3f36f7f63f09368bbb8158da9d153f53e9b","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/251/erdos251-prime-gap-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/251/erdos251-prime-gap-reasoning-surface.tex#L791","owner_problem_erdos_number":251,"owner_paper_id":"erdos251-prime-gap-reasoning-surface","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos251-prime-gap-reasoning-surface.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos251-prime-gap-reasoning-surface.pdf","resolution_status":"resolved","resolution_rule":"T1a_declaration_citation_elimination","owner_confidence":"decided","candidate_sources":["paper/251/erdos-251-prime-gap-dyadic-series.tex","paper/251/erdos251-prime-gap-reasoning-surface.tex"],"why_this_owner":"several manuscripts define it, but only one cites any of the claim's Lean declarations through \\lword/\\mword. Citing proves relevance; not citing proves only that that file does not cite it, so this rule eliminates and never selects","alternate_anchors":[{"paper_label":"res:irr-equivalence","owner_source":"paper/251/erdos-251-prime-gap-dyadic-series.tex","source_line":510,"anchor_kind":"environment","anchor_unit":"corollary","anchor_unit_line":510,"anchor_title":"exact irrationality reformulation","anchor_title_tex":"exact irrationality reformulation","section_title":"The prime series and its actual tails","section_path":[{"level":"section","title":"The prime series and its actual tails","title_tex":"The prime series and its actual tails","line":483}],"source_excerpt":{"text":"\\begin{corollary}[exact irrationality reformulation]\\label{res:irr-equivalence}\nThe prime-value dyadic series is irrational if and only if the prime-gap\ndyadic series is. Both series converge by the polynomial bound proved above;\nneither side is proved irrational.\n\\end{corollary}","media_type":"text/x-tex","source_char_start":23177,"source_char_end":23457,"start_line":510,"end_line":514,"truncated":false,"sha256":"sha256:8d62b444efad816e4d0698b367776031a899a1af8c6a41a9384de2832e0162cd","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/251/erdos-251-prime-gap-dyadic-series.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/251/erdos-251-prime-gap-dyadic-series.tex#L510","owner_problem_erdos_number":251,"owner_paper_id":"erdos-251-prime-gap-dyadic-series","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-251-prime-gap-dyadic-series.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-251-prime-gap-dyadic-series.pdf","rejected_because":"cites none of this claim's Lean declarations"}]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_251_irrationality","statement":"Prove that Π = ∑_{n≥1} p_n/2ⁿ is irrational; the exact tail-shift equivalence and prime-gap constraints do not supply the required cofinal escape.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/2"}],"links":[{"kind":"lean_declaration_pinned","label":"summable_primeDyadicTerm","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L379","cite_this":true},{"kind":"lean_declaration_pinned","label":"tsum_primeDyadicTerm_eq_two_add_primeGap_unconditional","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L427","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_tsum_primeDyadicTerm_iff_primeGap","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L435","cite_this":true},{"kind":"paper_pdf","label":"erdos251-prime-gap-reasoning-surface","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos251-prime-gap-reasoning-surface.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/4","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"prime_gap_irrationality_equivalence"}}],"expect":"object"},"prior_art":{"family_id":"prime_gap_tail_shift_frontier","status_summary":"An open prime-index endpoint, exact coefficient constraints, a Lean-formalised prime/gap irrationality equivalence under an elementary polynomial prime bound, and Tao's already-known identity recorded as cited only.","prior_art_posture":"The prime-to-gap identity is Tao's (erdosproblems.com/251, 7 October 2025). This checkout formalises a version that discharges summability without the prime-number theorem. No novelty, priority, significance, or irrationality claim is made.","primary_narrative_owner":"paper/251/erdos-251-prime-gap-dyadic-series.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/24","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z31::prime_and_gap_irrationality_are_exactly_equivalent","prior_art_state":"not_assessed"}]}},"alternate_anchor_fidelity":[{"anchor_role":"alternate","paper_id":"erdos-251-prime-gap-dyadic-series","paper_label":"res:irr-equivalence","owner_source":"paper/251/erdos-251-prime-gap-dyadic-series.tex","owner_status":"registry_active","relationship_to_formal_source":"compatible_duplicate_or_weaker","render_policy":"secondary_exposition","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/251/erdos-251-prime-gap-dyadic-series.tex#L510","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."}],"source_lane":"registry_claim"},{"assertion_id":"claim:prime_gap_identity_tao","assertion_kind":"registered_claim","source_id":"prime_gap_identity_tao","reviewed":true,"granularity":"single_registered_claim","label":"Tao's already-known prime-to-gap identity","statement":"Tao posted the prime-to-gap identity by summation by parts (erdosproblems.com/251, 7 October 2025). That identity is already-known mathematics, not a theorem of this checkout, and is not the public lead; the supported lead is the coefficient-only countermodel and sparse rationalising perturbation.","status":"cited only","status_meaning":"External theorem used for positioning, not formalised","readme_headline":false,"problem_ids":["erdos_249","erdos_257"],"problem_erdos_numbers":[249,257],"problem_binding":{"method":"core_programme_joint","warrant":"this claim belongs to the reviewed Erdos249257 core library, which is the joint #249/#257 programme. No source surface -- not its remaining-open propositions, not its anchor's enclosing gateway section, not the orientation programme map -- assigns it to one rather than both. Both are listed because both are correct, not because the binding is unknown.","separability":"not_separable_from_published_source"},"problem_edges":[{"problem_id":"erdos_249","relation":"shared_programme_context","basis":"problem_binding"},{"problem_id":"erdos_257","relation":"shared_programme_context","basis":"problem_binding"}],"programme_ids":["joint_249_257_gateway"],"publication_edges":[],"evidence_authority":"registry_statement_only","evidence_resolution":"statement_only","declarations":[],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[],"links":[],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/5","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"prime_gap_identity_tao"}}],"expect":"object"},"prior_art":{"family_id":"prime_gap_tail_shift_frontier","status_summary":"An open prime-index endpoint, exact coefficient constraints, a Lean-formalised prime/gap irrationality equivalence under an elementary polynomial prime bound, and Tao's already-known identity recorded as cited only.","prior_art_posture":"The prime-to-gap identity is Tao's (erdosproblems.com/251, 7 October 2025). 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Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:erdos_1041","assertion_kind":"registered_claim","source_id":"erdos_1041","reviewed":true,"granularity":"single_registered_claim","label":"Erdős #1041: short connections inside polynomial lemniscates","statement":"Using ani's explicit monic degree-seven polynomial, Lean proves that every preconnected subset of its strict unit lemniscate containing two distinct roots has one-dimensional Hausdorff measure greater than 2. This refutes the exact Formal Conjectures path-image-length statement and its total-variation counterpart. The reported family is not formalised, and independent human review of correspondence with the 1958 wording is not recorded.","status":"formal statement refuted","status_meaning":"Lean refutes the exact Formal Conjectures statement; correspondence with the original wording awaits independent review","readme_headline":false,"problem_ids":["erdos_1041"],"problem_erdos_numbers":[1041],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_1041","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_1041","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"erdos1041_counterexample_hausdorff","module":"ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","line":283,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L283","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L283","kind":"theorem","signature":"theorem erdos1041_counterexample_hausdorff : ∀ z₁ z₂, f.IsRoot z₁ → f.IsRoot z₂ → z₁ ≠ z₂ → ∀ K : Set ℂ, IsPreconnected K → z₁ ∈ K → z₂ ∈ K → K ⊆ Omega f → (2 : ℝ≥0∞) < μH[1] K","docstring":"Every preconnected subset of the strict lemniscate of `f` that contains two distinct roots has one-dimensional Hausdorff measure greater than two.","generated_certificate":false},{"name":"erdos1041_hausdorff_negation","module":"ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","line":407,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L407","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L407","kind":"theorem","signature":"theorem erdos1041_hausdorff_negation : ¬ ∀ (n : ℕ) (f : ℂ[X]), n ≥ 2 → f.natDegree = n → f.Monic → f.rootSet ℂ ⊆ Metric.ball 0 1 → ∃ (z₁ z₂ : ℂ) (h : ({z₁, z₂} : Multiset ℂ) ≤ f.roots) (γ : Path z₁ z₂), Set.range γ ⊆ { z : ℂ | ‖f.eval z‖ < 1 } ∧ fcLength (Set.range γ) < 2","docstring":"Negation of Formal Conjectures' `Erdos1041.erdos_1041`. Its section variables `n`, `f`, `hn`, `hnum`, `h_monic`, `h` are quantified in their declared order, and path length is `length (Set.range γ)`, the one-dimensional Hausdorff measure of the image. The polynomial `f` of `Defs.lean` refutes it.","generated_certificate":false},{"name":"erdos1041_hausdorff_answer_false","module":"ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","line":449,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L449","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L449","kind":"theorem","signature":"theorem erdos1041_hausdorff_answer_false : False ↔ ∀ (n : ℕ) (f : ℂ[X]), n ≥ 2 → f.natDegree = n → f.Monic → f.rootSet ℂ ⊆ Metric.ball 0 1 → ∃ (z₁ z₂ : ℂ) (h : ({z₁, z₂} : Multiset ℂ) ≤ f.roots) (γ : Path z₁ z₂), Set.range γ ⊆ { z : ℂ | ‖f.eval z‖ < 1 } ∧ fcLength (Set.range γ) < 2","docstring":"The `answer(False) ↔ …` form of the same statement, which is how Formal Conjectures records a question settled in the negative (`answer(False)` elaborates to `False`).","generated_certificate":false},{"name":"erdos1041_ani_degree_seven","module":"ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean","line":25,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean#L25","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean#L25","kind":"theorem","signature":"theorem erdos1041_ani_degree_seven : ∃ (p : ℂ[X]), p.Monic ∧ p.natDegree = 7 ∧ (∀ z, p.IsRoot z → ‖z‖ < 1) ∧ p.roots.Nodup ∧ ∀ z₁ z₂, p.IsRoot z₁ → p.IsRoot z₂ → z₁ ≠ z₂ → ∀ γ : ℝ → ℂ, ContinuousOn γ (Set.Icc 0 1) → γ 0 = z₁ → γ 1 = z₂ → (∀ τ ∈ Set.Icc (0 : ℝ) 1, ‖p.eval (γ τ)‖ < 1) → (2 : ℝ≥0∞) < eVariationOn γ (Set.Icc 0 1)","docstring":"Existential wrapper matching `FormalConjectures.ErdosProblems.erdos_1041.variants.ani_degree_seven`.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.erdos_1041_lemniscate_connection","statement":"Independently review whether the 1958 curve-length wording matches the exact Formal Conjectures path-image-length statement, whose negation is now Lean-checked using ani's explicit degree-seven polynomial. 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Their checked hypotheses do not imply unrestricted connection; no novelty, priority, significance or independent human-review claim is made.","primary_narrative_owner":"paper/1041/erdos-1041-lemniscate-newton-flow.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/26","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. 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Therefore\ncritical points with values on distinct positive rays cannot be endpoints\nof such a finite connection. The trajectory in the $z$ plane need not be radial.\n\\end{corollary}","media_type":"text/x-tex","source_char_start":202536,"source_char_end":203016,"start_line":4291,"end_line":4301,"truncated":false,"sha256":"sha256:f208143259915dddb547845d5bfa9273d65a6e3c35514cf661c793747116e97c","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Citing proves relevance; not citing proves only that that file does not cite it, so this rule eliminates and never selects","alternate_anchors":[{"paper_label":"res:ray","owner_source":"paper/1041/erdos-1041-lemniscate-newton-flow.tex","source_line":1021,"anchor_kind":"inline","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Inverse rays and perturbation","section_path":[{"level":"section","title":"Flow constructions and remaining questions","title_tex":"Flow constructions and remaining questions","line":1005},{"level":"subsection","title":"Inverse rays and perturbation","title_tex":"Inverse rays and perturbation","line":1007}],"source_excerpt":{"text":"\\noindent\\phantomsection\\label{res:ray}\nFor a finite interval $[a,b]$, assume the Newton equation only for\n$a<t<b$ and continuity of $f(z(t))$ up to both endpoints. The interior\nidentity then extends to $f(z(b))=e^{a-b}f(z(a))$, without evaluating\n$f/f'$ at an endpoint. Nonzero endpoint values therefore lie on the\nsame positive ray. Critical values\nwith distinct arguments exclude such a connection; distinct moduli are\ninsufficient.","media_type":"text/x-tex","source_char_start":49040,"source_char_end":49475,"start_line":1021,"end_line":1028,"truncated":false,"sha256":"sha256:0e7bc72ee54a867240a9c332a7e0f4b074c368b7c5b735ca58c15dc6fc622038","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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The checked one-dimensional Hausdorff-measure bound settles that formal statement negatively; the reported small-parameter family is not formalised.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/8"}],"links":[{"kind":"lean_declaration_pinned","label":"exists_small_translation_separating_arguments","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean#L197","cite_this":true},{"kind":"lean_declaration_pinned","label":"samePositiveRay_of_real_exp_decay","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean#L325","cite_this":true},{"kind":"lean_declaration_pinned","label":"no_newtonConnection_of_not_samePositiveRay","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean#L334","cite_this":true},{"kind":"paper_pdf","label":"erdos1041-lemniscate-reasoning-surface","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos1041-lemniscate-reasoning-surface.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/7","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"newton_ray_separation_consumer"}}],"expect":"object"},"prior_art":{"family_id":"newton_ray_separation_frontier","status_summary":"Finite Newton-flow ray-separation and perturbative root-retention inputs, alongside the historical short-connection question whose correspondence with the separately checked total-variation counterexample awaits independent review.","prior_art_posture":"The finite ray and perturbation statements are source-specific formal results. 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In particular, the Erd\\H{o}s--Borwein constant $E=\\sum_{n\\ge1}1/(2^n-1)$\nis irrational.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":31681,"source_char_end":31938,"start_line":610,"end_line":614,"truncated":false,"sha256":"sha256:5b0fd94db36dae3cf95c8a7ae0d61337ba1a27825f9cfc7df235da8fa6390cc8","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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The cited squarefree theorem does not cover integer bases that are not powers of two, and the cited prime-power extension is remark-only. No priority claim is made for the formalised families.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/0","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"known_classical":1},"edges":[{"semantic_node_id":"Z01::erdos_borwein_full_support_irrationality","prior_art_state":"known_classical"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:full","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L610","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:eb_constant","assertion_kind":"registered_claim","source_id":"eb_constant","reviewed":true,"granularity":"single_registered_claim","label":"The Erdős–Borwein constant E is irrational","statement":"E = ∑_{n≥1} 1/(2ⁿ − 1) is irrational, as the base-2 corollary of the full-support theorem.","status":"formalised here","status_meaning":"Known mathematics represented and checked in Lean","readme_headline":true,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"programme_representative","warrant":"orientation.json lists this claim among the representative claims of a mathematical programme whose remaining open propositions all name one problem"},"problem_edges":[{"problem_id":"erdos_257","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":["structured_support_families"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"irrational_erdosBorwein_series","module":"Erdos249257/CertificateKernel.lean","line":8335,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CertificateKernel.lean#L8335","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L8335","kind":"theorem","signature":"theorem irrational_erdosBorwein_series : Irrational (∑' k : ℕ, (1 : ℝ) / ((2 : ℝ) ^ (k + 1) - 1))","docstring":"**The Erdős–Borwein constant is irrational.** Base-2 instance of the full-support theorem: `∑_{n≥1} 1/(2^n - 1)` is irrational.","generated_certificate":false}],"paper_label":"res:full","paper_anchor":{"paper_label":"res:full","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":610,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":610,"anchor_title":"full-support irrationality","anchor_title_tex":"full-support irrationality","section_title":"Full support: the Erdős--Borwein constant, every base","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Full support: the Erdős--Borwein constant, every base","title_tex":"Full support: the Erd\\H{o}s--Borwein constant, every base","line":608}],"source_excerpt":{"text":"\\begin{theorem}[full-support irrationality]\\label{res:full}\nFor every integer $b\\ge2$, the series $\\displaystyle\\sum_{n\\ge1}\\frac{1}{b^n-1}$ is\nirrational. In particular, the Erd\\H{o}s--Borwein constant $E=\\sum_{n\\ge1}1/(2^n-1)$\nis irrational.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":31681,"source_char_end":31938,"start_line":610,"end_line":614,"truncated":false,"sha256":"sha256:5b0fd94db36dae3cf95c8a7ae0d61337ba1a27825f9cfc7df235da8fa6390cc8","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L610","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"irrational_erdosBorwein_series","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L8335","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/9","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"eb_constant"}}],"expect":"object"},"prior_art":{"family_id":"classical_full_support_and_named_257_families","status_summary":"Classical formalisation, cited special cases, and the open universal target.","prior_art_posture":"Erdős owns full support and the pairwise-coprime family; Duverney–Tachiya own squarefree-support irrationality at every power-of-two base; Tao–Teräväinen own the base-2 prime-support theorem. The cited squarefree theorem does not cover integer bases that are not powers of two, and the cited prime-power extension is remark-only. No priority claim is made for the formalised families.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/0","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"known_classical":1},"edges":[{"semantic_node_id":"Z01::erdos_borwein_full_support_irrationality","prior_art_state":"known_classical"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:full","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L610","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:support_families","assertion_kind":"registered_claim","source_id":"support_families","reviewed":true,"granularity":"single_registered_claim","label":"Named infinite-support irrationality families toward #257","statement":"For every base b ≥ 2, ∑_{n∈A} 1/(bⁿ − 1) is irrational for factorial support, power-of-two support, multiples, pairwise-coprime supports with summable reciprocals, eventually-periodic supports, residue classes, and the odd numbers. Proved families, not the universal statement.","status":"formalised here","status_meaning":"Known mathematics represented and checked in Lean","readme_headline":true,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"erdos257_family_factorial_instance","module":"Erdos249257/CertificateKernel.lean","line":6082,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CertificateKernel.lean#L6082","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L6082","kind":"theorem","signature":"theorem erdos257_family_factorial_instance : Irrational (∑' k, (1 : ℝ) / ((2 : ℝ) ^ (Nat.factorial (k + 1)) - 1))","docstring":"**Erdős #257 family, factorial instance at base 2.** The literal series `∑ 1 / (2^(n!) - 1)` over the infinite support `A = {n! : n ≥ 1}` is irrational. This is an instance of the statement family of Erdős problem #257 (which asks the same for *every* infinite support); the universal problem remains open and is not claimed.","generated_certificate":false},{"name":"erdos257_family_two_pow_instance","module":"Erdos249257/CertificateKernel.lean","line":6090,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CertificateKernel.lean#L6090","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L6090","kind":"theorem","signature":"theorem erdos257_family_two_pow_instance : Irrational (∑' k, (1 : ℝ) / ((2 : ℝ) ^ (2 ^ k) - 1))","docstring":"**Erdős #257 family, geometric instance at base 2.** The literal series `∑ 1 / (2^(2^n) - 1)` over the infinite support `A = {2^n : n ≥ 0}` is irrational. An instance of the #257 statement family; the universal problem remains open and is not claimed.","generated_certificate":false},{"name":"irrational_erdosSupportSeries_pairwise_coprime","module":"Erdos249257/CertificateKernel.lean","line":10776,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CertificateKernel.lean#L10776","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L10776","kind":"theorem","signature":"theorem irrational_erdosSupportSeries_pairwise_coprime (b : ℕ) (A : Set ℕ) (hb : 2 ≤ b) (hA : A.Infinite) (hpair : A.Pairwise Nat.Coprime) (hsum : Summable (Set.indicator A fun a : ℕ => (1 : ℝ) / a)) : Irrational (erdosSupportSeries b A)","docstring":"**Erdős's pairwise-coprime support theorem (1968), machine-checked.** For every base `b ≥ 2` and every infinite pairwise-coprime support `A ⊆ ℕ` with summable reciprocals, `∑_{a ∈ A} 1/(b^a - 1)` is irrational. The first non-dilation support class of Erdős #257 to submit to the support-coefficient calculus: this is NOT valuation forcing — small support elements join the CRT modulus at residue zero, adaptive block sizes absorb their deterministic first-block hits, and summable reciprocals control the strays.","generated_certificate":false}],"paper_label":"res:support","paper_anchor":{"paper_label":"res:support","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":686,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":686,"anchor_title":"irrationality for structured supports","anchor_title_tex":"irrationality for structured supports","section_title":"Named infinite-support cases","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Named infinite-support cases","title_tex":"Named infinite-support cases","line":680}],"source_excerpt":{"text":"\\begin{theorem}[irrationality for structured supports]\\label{res:support}\nFor every integer $b\\ge2$, the series $\\sum_{n\\in A} 1/(b^n-1)$ is irrational for\neach of the following infinite supports $A$:\n\\begin{enumerate}[label=\\textup{(\\alph*)},leftmargin=2em]\n\\item factorial support $A=\\{n! : n\\ge1\\}$, giving $\\sum 1/(b^{n!}-1)$;\n\\item power-of-two support $A=\\{2^n : n\\ge0\\}$, giving $\\sum 1/(b^{2^n}-1)$;\n\\item multiples $A=d\\Npos=\\{dk:k\\ge1\\}$ with $d\\ge1$;\n\\item pairwise-coprime supports with summable reciprocals (Erd\\H{o}s's\ncondition~\\cite[Theorem, p.~222]{erdos1968});\n\\item eventually-periodic supports, residue classes, and the odd numbers.\n\\end{enumerate}\n\\end{theorem}","media_type":"text/x-tex","source_char_start":35918,"source_char_end":36600,"start_line":686,"end_line":697,"truncated":false,"sha256":"sha256:22efd6308f05dd8af2334720468d5f52826987bfbc992ec22907876431a94134","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L686","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"erdos257_family_factorial_instance","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L6082","cite_this":true},{"kind":"lean_declaration_pinned","label":"erdos257_family_two_pow_instance","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L6090","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_erdosSupportSeries_pairwise_coprime","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L10776","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/10","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"support_families"}}],"expect":"object"},"prior_art":{"family_id":"classical_full_support_and_named_257_families","status_summary":"Classical formalisation, cited special cases, and the open universal target.","prior_art_posture":"Erdős owns full support and the pairwise-coprime family; Duverney–Tachiya own squarefree-support irrationality at every power-of-two base; Tao–Teräväinen own the base-2 prime-support theorem. The cited squarefree theorem does not cover integer bases that are not powers of two, and the cited prime-power extension is remark-only. No priority claim is made for the formalised families.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/0","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"known_classical":1,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z01::lcm_gap_support_instances","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z01::pairwise_coprime_support_irrationality","prior_art_state":"known_classical"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:support","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L686","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:channel_quotient_band_breakpoint","assertion_kind":"registered_claim","source_id":"channel_quotient_band_breakpoint","reviewed":true,"granularity":"single_registered_claim","label":"Quotient-band channel breakpoint for factorial moments","statement":"For a finite integer coefficient family whose indices all lie in one quotient band k·d ≤ i_j < (k+1)·d, the factorial moment is exactly (d!)^k times the d-channel numerator. In the first band d ≤ i_j < 2d, channel cancellation forces the factorial moment to vanish; with all indices at least d and a nonzero moment, some index is therefore at least 2d.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_68"],"problem_erdos_numbers":[68],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_68","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_68","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos68-factorial-reasoning-surface","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"factorialMoment_eq_factorial_pow_mul_channelNumerator_band","module":"ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean","line":71,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean#L71","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean#L71","kind":"theorem","signature":"theorem factorialMoment_eq_factorial_pow_mul_channelNumerator_band {ι : Type*} [Fintype ι] (coeff : ι → ℤ) (index : ι → ℕ) (d k : ℕ) (hlo : ∀ j, k * d ≤ index j) (hhi : ∀ j, index j < (k + 1) * d) : factorialMoment coeff index = (d.factorial ^ k : ℤ) * channelNumerator coeff index d","docstring":"If all support indices lie in one quotient band for channel `d`, the factorial moment is `(d!)^k` times that channel numerator.","generated_certificate":false},{"name":"factorialMoment_eq_zero_of_channelNumerator_eq_zero_band","module":"ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean","line":91,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean#L91","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean#L91","kind":"theorem","signature":"theorem factorialMoment_eq_zero_of_channelNumerator_eq_zero_band {ι : Type*} [Fintype ι] (coeff : ι → ℤ) (index : ι → ℕ) (d k : ℕ) (hlo : ∀ j, k * d ≤ index j) (hhi : ∀ j, index j < (k + 1) * d) (hchannel : channelNumerator coeff index d = 0) : factorialMoment coeff index = 0","docstring":"Channel cancellation inside one quotient band forces the corresponding factorial moment to vanish.","generated_certificate":false},{"name":"factorialMoment_eq_factorial_mul_channelNumerator_firstBand","module":"ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean","line":101,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean#L101","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean#L101","kind":"theorem","signature":"theorem factorialMoment_eq_factorial_mul_channelNumerator_firstBand {ι : Type*} [Fintype ι] (coeff : ι → ℤ) (index : ι → ℕ) (d : ℕ) (hlo : ∀ j, d ≤ index j) (hhi : ∀ j, index j < 2 * d) : factorialMoment coeff index = (d.factorial : ℤ) * channelNumerator coeff index d","docstring":"In the first quotient band, the factorial moment is `d!` times the `d`-channel numerator.","generated_certificate":false},{"name":"exists_index_ge_two_mul_of_factorialMoment_ne_zero_of_channel_eq_zero","module":"ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean","line":130,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean#L130","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean#L130","kind":"theorem","signature":"theorem exists_index_ge_two_mul_of_factorialMoment_ne_zero_of_channel_eq_zero {ι : Type*} [Fintype ι] [Nonempty ι] (coeff : ι → ℤ) (index : ι → ℕ) (d : ℕ) (hlo : ∀ j, d ≤ index j) (hchannel : channelNumerator coeff index d = 0) (hmoment : factorialMoment coeff index ≠ 0) : ∃ j, 2 * d ≤ index j","docstring":"If every support index is at least `d`, channel cancellation and a nonzero factorial moment force some support index to be at least `2d`.","generated_certificate":false}],"paper_label":"long68:res:bandbreakpoint","paper_anchor":{"paper_label":"long68:res:bandbreakpoint","owner_source":"paper/68/erdos68-factorial-reasoning-surface.tex","source_line":606,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":606,"anchor_title":"constant values of the floor in the weights","anchor_title_tex":"constant values of the floor in the weights","section_title":"Support bounds and nonintegrality","section_path":[{"level":"section","title":"Integer coefficient vectors and cancellation","title_tex":"Integer coefficient vectors and cancellation","line":130},{"level":"subsection","title":"Support bounds and nonintegrality","title_tex":"Support bounds and nonintegrality","line":563}],"source_excerpt":{"text":"\\begin{theorem}[constant values of the floor in the weights]\\label{long68:res:bandbreakpoint}\nLet $d\\ge2$ and $k\\ge0$, and suppose every supported index $i$ satisfies\n$kd\\le i<(k+1)d$.  Then $M(c)=(d!)^k\\chn dc$.  In particular, cancellation\non the interval $d\\le i<2d$ forces $M(c)=0$; and if every supported index\nis at least $d$ while $M(c)\\ne0$ and $\\chn dc=0$, then some supported index\nis at least $2d$.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":28340,"source_char_end":28763,"start_line":606,"end_line":612,"truncated":false,"sha256":"sha256:3f37baa9cbdc3bb24571b424d945f4e68d7785c9629877aa714c8a907a9d35be","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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The integer carry form and exact residue classifier are checked in Lean.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_269"],"problem_erdos_numbers":[269],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_269","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_269","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-269-three-prime-running-lcm","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"no_bounded_positive_state_of_residue_escape","module":"ErdosProblems/Erdos269/ResidueEscape.lean","line":76,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ResidueEscape.lean#L76","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ResidueEscape.lean#L76","kind":"theorem","signature":"theorem no_bounded_positive_state_of_residue_escape {C bound residue c : ℕ} (hcpos : 0 < c) (hcbound : c ≤ bound) (hescape : ResidueEscapesWindow C bound residue) (hmod : c % C = residue % C) : False","docstring":"No positive state bounded by `bound` can represent, modulo `C`, a residue in the canonical positive range that lies above `bound`.","generated_certificate":false},{"name":"no_bounded_positive_int_state_of_leastPositiveResidue","module":"ErdosProblems/Erdos269/ResidueEscape.lean","line":110,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ResidueEscape.lean#L110","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ResidueEscape.lean#L110","kind":"theorem","signature":"theorem no_bounded_positive_int_state_of_leastPositiveResidue {C bound : ℕ} {x c : ℤ} (hC : 0 < C) (hcpos : 0 < c) (hcbound : Int.natAbs c ≤ bound) (hescape : bound < leastPositiveResidue C x) (hmod : Int.ModEq C c x) : False","docstring":"Integer-valued form of the finite contradiction. A positive integer bounded by `bound` cannot be congruent to an integer whose least positive residue lies above that bound.","generated_certificate":false},{"name":"leastPositiveResidue_eq_natAbs_of_pos_le_modEq","module":"ErdosProblems/Erdos269/ResidueEscape.lean","line":138,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/ResidueEscape.lean#L138","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/ResidueEscape.lean#L138","kind":"theorem","signature":"theorem leastPositiveResidue_eq_natAbs_of_pos_le_modEq {C : ℕ} (hC : 0 < C) {x c : ℤ} (hcpos : 0 < c) (hcle : Int.natAbs c ≤ C) (hmod : Int.ModEq C c x) : leastPositiveResidue C x = Int.natAbs c","docstring":"A positive integer already lying in the canonical interval is exactly the least positive representative of its congruence class.","generated_certificate":false}],"paper_label":"res:consumer","paper_anchor":{"paper_label":"res:consumer","owner_source":"paper/269/erdos-269-three-prime-running-lcm.tex","source_line":525,"anchor_kind":"environment","anchor_unit":"lemma","anchor_unit_line":525,"anchor_title":"least positive residues","anchor_title_tex":"least positive residues","section_title":"A residue criterion","section_path":[{"level":"section","title":"A residue criterion","title_tex":"A residue criterion","line":471}],"source_excerpt":{"text":"\\begin{lemma}[least positive residues]\\label{res:consumer}\nIf $d$ is a positive integer with $d\\le K$ and $d\\equiv -BF\\pmod W$,\nwhere $W\\ge1$, then $\\operatorname{lpr}_W(-BF)\\le K$.\n\\end{lemma}","media_type":"text/x-tex","source_char_start":24113,"source_char_end":24306,"start_line":525,"end_line":528,"truncated":false,"sha256":"sha256:ddbe8e62ea6db9785659a228f59c1cc350f58728e5995071d54549caaf8e9185","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"routine_corollary_or_specialisation":2},"edges":[{"semantic_node_id":"Z17::n269_residue_escape_contradiction","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z32::least_positive_residue_excludes_bounded_positive_state","prior_art_state":"routine_corollary_or_specialisation"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:three_prime_cofinal_window_carry_extinction","assertion_kind":"registered_claim","source_id":"three_prime_cofinal_window_carry_extinction","reviewed":true,"granularity":"single_registered_claim","label":"Conditional cofinal-window extinction of bounded carries for #269","statement":"If the denominator-dependent cofinal local-window residue escape holds, then for every positive multiplier B coprime to 30 there is no positive integral sequence satisfying the exact cleared recurrence d_{n+1}=b_n d_n−B m_n together with the corresponding short bound. The same contradiction applies to an explicitly absorbed nonzero common-factor carry after exact cancellation. For the actual {2,3,5} series, the rationality-to-positive-reduced-carry bridge is established separately; the cofinal escape producer remains open.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_269"],"problem_erdos_numbers":[269],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_269","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_269","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-269-three-prime-running-lcm","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"no_positive_reducedCarry_of_cofinalLocalWindowEscape","module":"ErdosProblems/Erdos269/RestrictedFloorSum.lean","line":645,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/RestrictedFloorSum.lean#L645","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/RestrictedFloorSum.lean#L645","kind":"theorem","signature":"theorem no_positive_reducedCarry_of_cofinalLocalWindowEscape (b m : ℕ → ℕ) (shortBound : ℕ → ℕ → ℕ) (hescape : CofinalLocalWindowEscape b m shortBound) (B : ℕ) (hBpos : 0 < B) (hBcoprime : Nat.Coprime B 30) (d : ℕ → ℤ) (hrec : ∀ n, d (n + 1) = (b n : ℤ) * d n - (B : ℤ) * (m n : ℤ)) (hpos : ∀ n, 0 < d n) (hbound : ∀ n, Int.natAbs (d n) ≤ shortBound B n) : False","docstring":"The denominator-dependent local-window producer is exactly strong enough to rule out a positive reduced carry with the matching short bound. The coprimality hypothesis belongs to the still-open producer; the checked consumer uses only the emitted escaping window.","generated_certificate":false},{"name":"no_positive_absorbedCarry_of_cofinalLocalWindowEscape","module":"ErdosProblems/Erdos269/RestrictedFloorSum.lean","line":689,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/RestrictedFloorSum.lean#L689","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/RestrictedFloorSum.lean#L689","kind":"theorem","signature":"theorem no_positive_absorbedCarry_of_cofinalLocalWindowEscape (b m : ℕ → ℕ) (shortBound : ℕ → ℕ → ℕ) (hescape : CofinalLocalWindowEscape b m shortBound) (B : ℕ) (hBpos : 0 < B) (hBcoprime : Nat.Coprime B 30) (c d : ℕ → ℤ) (smoothFactor : ℤ) (hsmooth : smoothFactor ≠ 0) (hfactor : ∀ n, c n = smoothFactor * d n) (hrec : ∀ n, c (n + 1) = (b n : ℤ) * c n - (smoothFactor * (B : ℤ)) * (m n : ℤ)) (hpos : ∀ n, 0 < d n) (hbound : ∀ n, Int.natAbs (d n) ≤ shortBound B n) : False","docstring":"A hypothetical carry recurrence with an absorbed nonzero common factor is already contradictory once its reduced denominator satisfies the same cofinal local-window escape premise. This is only a conditional bridge from absorbed integral carries to the existing reduced-carry consumer; it proves no irrationality statement without the separate actual-series carry bridge and a producer for `CofinalLocalWindowEscape`.","generated_certificate":false}],"paper_label":"res:windowconsumer","paper_anchor":{"paper_label":"res:windowconsumer","owner_source":"paper/269/erdos-269-three-prime-running-lcm.tex","source_line":547,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":547,"anchor_title":"a residue criterion for irrationality","anchor_title_tex":"a residue criterion for irrationality","section_title":"A residue criterion","section_path":[{"level":"section","title":"A residue criterion","title_tex":"A residue criterion","line":471}],"source_excerpt":{"text":"\\begin{theorem}[a residue criterion for irrationality]\\label{res:windowconsumer}\nThe number $S$ is irrational if and only if\n\\begin{equation}\\label{eq:escape}\n \\begin{gathered}\n \\text{for every }B\\ge1\\text{ with }\\gcd(B,30)=1\n \\text{ and every }a_0\\ge1,\\\\\n \\text{there are }\\ell\\ge a_0,\\ h\\ge1\\text{ such that}\n \\operatorname{lpr}_{W_{\\ell,h}}(-BF_{\\ell,h})>K(B,\\ell+h).\n \\end{gathered}\n\\end{equation}\n\\end{theorem}","media_type":"text/x-tex","source_char_start":25040,"source_char_end":25455,"start_line":547,"end_line":557,"truncated":false,"sha256":"sha256:6e88aecc42ca139ca95d9ff7f80540657081ecf8f2f2c390ee61b1b01ebdbcb3","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/269/erdos-269-three-prime-running-lcm.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/269/erdos-269-three-prime-running-lcm.tex#L547","owner_problem_erdos_number":269,"owner_paper_id":"erdos-269-three-prime-running-lcm","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-269-three-prime-running-lcm.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-269-three-prime-running-lcm.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/269/erdos-269-three-prime-running-lcm.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_269_three_prime_irrationality","statement":"Determine irrationality of the running-LCM reciprocal sum for finite prime sets of size at least three. 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Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z32::cofinal_escape_rules_out_matching_positive_reduced_carry","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:three_prime_exact_modulus_pinning","assertion_kind":"registered_claim","source_id":"three_prime_exact_modulus_pinning","reviewed":true,"granularity":"single_registered_claim","label":"Exact-modulus pinning for an actual #269 window","statement":"If an actual finite {2,3,5} window has a least-positive forcing residue no larger than the exact carry cap, then the scaled normalized launch lies within that cap divided by the actual window modulus of an integer. Irrationality plus a separately supplied exact-modulus decay estimate therefore yields cofinal local-window escape. The decay estimate and irrationality input are not proved by this claim.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_269"],"problem_erdos_numbers":[269],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_269","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_269","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-269-three-prime-running-lcm","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"exact_modulus_pinning","module":"ErdosProblems/Erdos269/R12/ExactModulus.lean","line":45,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/R12/ExactModulus.lean#L45","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/R12/ExactModulus.lean#L45","kind":"theorem","signature":"theorem exact_modulus_pinning {x y K : ℝ} {W F r : ℤ} (hW : 0 < W) (hwin : y = (W : ℝ) * x - (F : ℝ)) (hcong : W ∣ F + r) (hy0 : 0 ≤ y) (hyK : y ≤ K) (hr0 : (0 : ℝ) ≤ r) (hrK : (r : ℝ) ≤ K) : ∃ k : ℤ, |x - (k : ℝ)| ≤ K / (W : ℝ)","docstring":"Exact pinning with an arbitrary congruent representative. Positivity of W and both lower endpoint bounds are indispensable hypotheses.","generated_certificate":false},{"name":"actual_exact_modulus_pinning","module":"ErdosProblems/Erdos269/R12/ExactModulus.lean","line":88,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/R12/ExactModulus.lean#L88","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/R12/ExactModulus.lean#L88","kind":"theorem","signature":"theorem actual_exact_modulus_pinning (B lo len K : ℕ) (hB : 0 < B) (hK : B * bridgeWidth (lo + len) ≤ K) (hres : leastPositiveResidue (Int.natAbs (actualWindowBase lo len)) (-((B : ℤ) * actualWindowForcing lo len)) ≤ K) : ∃ k : ℤ, |(B : ℝ) * trueNormalizedState lo - (k : ℝ)| ≤ (K : ℝ) / (actualWindowBase lo len : ℝ)","docstring":"Actual-source pinning; no source window equality is left as a premise.","generated_certificate":false}],"paper_label":"res:windowconsumer","paper_anchor":{"paper_label":"res:windowconsumer","owner_source":"paper/269/erdos-269-three-prime-running-lcm.tex","source_line":547,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":547,"anchor_title":"a residue criterion for irrationality","anchor_title_tex":"a residue criterion for irrationality","section_title":"A residue criterion","section_path":[{"level":"section","title":"A residue criterion","title_tex":"A residue criterion","line":471}],"source_excerpt":{"text":"\\begin{theorem}[a residue criterion for irrationality]\\label{res:windowconsumer}\nThe number $S$ is irrational if and only if\n\\begin{equation}\\label{eq:escape}\n \\begin{gathered}\n \\text{for every }B\\ge1\\text{ with }\\gcd(B,30)=1\n \\text{ and every }a_0\\ge1,\\\\\n \\text{there are }\\ell\\ge a_0,\\ h\\ge1\\text{ such that}\n \\operatorname{lpr}_{W_{\\ell,h}}(-BF_{\\ell,h})>K(B,\\ell+h).\n \\end{gathered}\n\\end{equation}\n\\end{theorem}","media_type":"text/x-tex","source_char_start":25040,"source_char_end":25455,"start_line":547,"end_line":557,"truncated":false,"sha256":"sha256:6e88aecc42ca139ca95d9ff7f80540657081ecf8f2f2c390ee61b1b01ebdbcb3","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/269/erdos-269-three-prime-running-lcm.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/269/erdos-269-three-prime-running-lcm.tex#L547","owner_problem_erdos_number":269,"owner_paper_id":"erdos-269-three-prime-running-lcm","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-269-three-prime-running-lcm.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-269-three-prime-running-lcm.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/269/erdos-269-three-prime-running-lcm.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_269_cofinal_local_window_escape","statement":"Prove that for every B ≥ 1 coprime to 30 and every a₀ ≥ 1 there are ℓ ≥ a₀ and h ≥ 1 with lpr_{W(ℓ,h)}(−B·F(ℓ,h)) > K(B, ℓ+h), where K is the defined long-paper cap.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/12"}],"links":[{"kind":"lean_declaration_pinned","label":"exact_modulus_pinning","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/R12/ExactModulus.lean#L45","cite_this":true},{"kind":"lean_declaration_pinned","label":"actual_exact_modulus_pinning","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/R12/ExactModulus.lean#L88","cite_this":true},{"kind":"paper_pdf","label":"erdos-269-three-prime-running-lcm","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-269-three-prime-running-lcm.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/14","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"three_prime_exact_modulus_pinning"}}],"expect":"object"},"prior_art":{"family_id":"three_prime_residue_escape_consumers","status_summary":"An open three-prime #269 endpoint, a finite least-positive-residue contradiction, a conditional cofinal-window extinction consumer, and the weighted-phase carry split with its finite realised-span observer.","prior_art_posture":"The residue classifier and bounded-carry consumer are source-specific formal results; no novelty, priority, significance, or three-prime irrationality claim is made.","primary_narrative_owner":"paper/269/erdos-269-three-prime-running-lcm.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/23","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z17::n269_exact_window_modulus_pinning","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:three_prime_octic_window_equivalence","assertion_kind":"registered_claim","source_id":"three_prime_octic_window_equivalence","reviewed":true,"granularity":"single_registered_claim","label":"Conditional o(8^a) equivalence band for #269","statement":"For a cap G which dominates the long-paper cap for every positive denominator and satisfies G(B,n)/8^n -> 0 for each positive B, cofinal local-window escape is equivalent to irrationality of the three-prime paper series. Both the dominance and little-o hypotheses are explicit; the theorem proves neither side unconditionally.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_269"],"problem_erdos_numbers":[269],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_269","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_269","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos269-running-lcm-reasoning-surface","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"octic_cap_is_exactly_beaten","module":"ErdosProblems/Erdos269/R12/OcticWindowBand.lean","line":19,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/R12/OcticWindowBand.lean#L19","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/R12/OcticWindowBand.lean#L19","kind":"theorem","signature":"theorem octic_cap_is_exactly_beaten (G : ℕ → ℕ → ℕ) (hsmall : ∀ B, 0 < B → Tendsto (fun n : ℕ => (G B n : ℝ) / (8 : ℝ) ^ n) atTop (𝓝 0)) : ∀ B lo : ℕ, 0 < B → ∀ ε : ℝ, 0 < ε → ∃ len : ℕ, 0 < len ∧ ((max (G B (lo + len)) (B * bridgeWidth (lo + len)) : ℕ) : ℝ) / (actualWindowBase lo len : ℝ) < ε","generated_certificate":false},{"name":"octic_window_band","module":"ErdosProblems/Erdos269/R12/OcticWindowBand.lean","line":77,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/R12/OcticWindowBand.lean#L77","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/R12/OcticWindowBand.lean#L77","kind":"theorem","signature":"theorem octic_window_band (G : ℕ → ℕ → ℕ) (hdom : ∀ B a, 0 < B → longPaperCap B a ≤ G B a) (hsmall : ∀ B, 0 < B → Tendsto (fun n : ℕ => (G B n : ℝ) / (8 : ℝ) ^ n) atTop (𝓝 0)) : CofinalLocalWindowEscape dyadicBlockBase235 dyadicOrderedBlockDigit235 G ↔ Irrational paperSeries235","docstring":"A genuine strengthening of the long-paper band. The universal cap and limit hypotheses are explicit; the conclusion is an equivalence, not an unconditional proof of either side.","generated_certificate":false}],"paper_label":"long269:res:actual-escape-endpoint","paper_anchor":{"paper_label":"long269:res:actual-escape-endpoint","owner_source":"paper/269/erdos269-running-lcm-reasoning-surface.tex","source_line":2737,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":2737,"anchor_title":"a residue criterion for every dominating bound of size o(8^a)","anchor_title_tex":"a residue criterion for every dominating bound of size $o(8^a)$","section_title":"A residue criterion and the bounds it allows","section_path":[{"level":"section","title":"A residue criterion and the bounds it allows","title_tex":"A residue criterion and the bounds it allows","line":2636}],"source_excerpt":{"text":"\\begin{theorem}[a residue criterion for every dominating bound of size $o(8^a)$]\\label{long269:res:actual-escape-endpoint}\n\\label{long269:res:lead-escape-equivalence}\\label{long269:res:windowconsumer}\nLet $G:\\N_{>0}\\times\\N\\to\\N$ satisfy $K(B,a)\\le G(B,a)$ for all $B$ and $a$,\nand $G(B,a)/8^{a}\\to0$ as $a\\to\\infty$ for each fixed $B$.  Then\n\\[\n \\Esc(G)\\quad\\Longleftrightarrow\\quad S\\notin\\Q .\n\\]\nBoth $K$ of~\\eqref{long269:eq:actual-bound} and $K_0(B,a)=90B(a+1)^{2}$ satisfy these hypotheses, so $\\Esc(K)$, $\\Esc(K_0)$ and irrationality of $S$ are mutually\nequivalent.  By contrast, $\\Esc(0)$ holds automatically, since every\nleast positive residue is at least $1$; its truth alone therefore\nprovides no contradiction to an integral tail.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":131648,"source_char_end":132404,"start_line":2737,"end_line":2748,"truncated":false,"sha256":"sha256:9c2ba0745c7fe1ef77d46588d7a87adaf998b6ede8144d648485c9023d322253","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/269/erdos269-running-lcm-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/269/erdos269-running-lcm-reasoning-surface.tex#L2737","owner_problem_erdos_number":269,"owner_paper_id":"erdos269-running-lcm-reasoning-surface","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos269-running-lcm-reasoning-surface.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos269-running-lcm-reasoning-surface.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/269/erdos269-running-lcm-reasoning-surface.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_269_cofinal_local_window_escape","statement":"Prove that for every B ≥ 1 coprime to 30 and every a₀ ≥ 1 there are ℓ ≥ a₀ and h ≥ 1 with lpr_{W(ℓ,h)}(−B·F(ℓ,h)) > K(B, ℓ+h), where K is the defined long-paper cap.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/12"},{"open_proposition_id":"remaining_open.erdos_269_three_prime_irrationality","statement":"Determine irrationality of the running-LCM reciprocal sum for finite prime sets of size at least three. For {2,3,5}, the rationality-to-carry bridge is established and cofinal local-window escape remains open.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/7"}],"links":[{"kind":"lean_declaration_pinned","label":"octic_cap_is_exactly_beaten","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/R12/OcticWindowBand.lean#L19","cite_this":true},{"kind":"lean_declaration_pinned","label":"octic_window_band","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/R12/OcticWindowBand.lean#L77","cite_this":true},{"kind":"paper_pdf","label":"erdos269-running-lcm-reasoning-surface","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos269-running-lcm-reasoning-surface.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/15","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"three_prime_octic_window_equivalence"}}],"expect":"object"},"prior_art":{"family_id":"three_prime_residue_escape_consumers","status_summary":"An open three-prime #269 endpoint, a finite least-positive-residue contradiction, a conditional cofinal-window extinction consumer, and the weighted-phase carry split with its finite realised-span observer.","prior_art_posture":"The residue classifier and bounded-carry consumer are source-specific formal results; no novelty, priority, significance, or three-prime irrationality claim is made.","primary_narrative_owner":"paper/269/erdos-269-three-prime-running-lcm.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/23","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z17::n269_octic_window_equivalence_band","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:three_prime_finite_prefix_completion_boundary","assertion_kind":"registered_claim","source_id":"three_prime_finite_prefix_completion_boundary","reviewed":true,"granularity":"single_registered_claim","label":"Finite #269 radix prefixes admit an altered rational completion","statement":"Every finite positive integral radix prefix with nonnegative integral digits extends to a positive rational orbit by changing the future to radix 2 and digit 1. In particular, the actual word through any finite horizon has such an abstract completion. This is a countermodel to finite-prefix-only inference and is not a rational model of the actual infinite smooth-number series.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_269"],"problem_erdos_numbers":[269],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_269","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_269","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"finite_prefix_has_positive_rational_completion","module":"ErdosProblems/Erdos269/R12/PrefixExtension.lean","line":75,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/R12/PrefixExtension.lean#L75","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/R12/PrefixExtension.lean#L75","kind":"theorem","signature":"theorem finite_prefix_has_positive_rational_completion (b d : ℕ → ℤ) (N : ℕ) (hb : ∀ i, i < N → 0 < b i) (hd : ∀ i, i < N → 0 ≤ d i) : ∃ b' d' : ℕ → ℤ, ∃ X : ℕ → ℚ, (∀ i, i < N → b' i = b i ∧ d' i = d i) ∧ (∀ i, 0 < b' i ∧ 0 ≤ d' i ∧ 0 < X i) ∧ (∀ i, X (i + 1) = (b' i : ℚ) * X i - (d' i : ℚ)) ∧ (∀ i, N ≤ i → b' i = 2 ∧ d' i = 1 ∧ X i = 1)","docstring":"The completion preserves every supplied transition, not just its length.","generated_certificate":false},{"name":"actual_prefix_has_abstract_rational_completion","module":"ErdosProblems/Erdos269/R12/PrefixExtension.lean","line":105,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/R12/PrefixExtension.lean#L105","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/R12/PrefixExtension.lean#L105","kind":"theorem","signature":"theorem actual_prefix_has_abstract_rational_completion (N : ℕ) : ∃ b' d' : ℕ → ℤ, ∃ X : ℕ → ℚ, (∀ i, i < N → b' i = (dyadicBlockBase235 i : ℤ) ∧ d' i = (dyadicOrderedBlockDigit235 i : ℤ)) ∧ (∀ i, 0 < b' i ∧ 0 ≤ d' i ∧ 0 < X i) ∧ (∀ i, X (i + 1) = (b' i : ℚ) * X i - (d' i : ℚ)) ∧ (∀ i, N ≤ i → b' i = 2 ∧ d' i = 1 ∧ X i = 1)","docstring":"Even an arbitrarily long ACTUAL finite prefix has an abstract rational completion. No assertion identifies that altered continuation with the actual infinite 2,3,5 source.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.erdos_269_three_prime_irrationality","statement":"Determine irrationality of the running-LCM reciprocal sum for finite prime sets of size at least three. For {2,3,5}, the rationality-to-carry bridge is established and cofinal local-window escape remains open.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/7"}],"links":[{"kind":"lean_declaration_pinned","label":"finite_prefix_has_positive_rational_completion","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/R12/PrefixExtension.lean#L75","cite_this":true},{"kind":"lean_declaration_pinned","label":"actual_prefix_has_abstract_rational_completion","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/R12/PrefixExtension.lean#L105","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/16","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"three_prime_finite_prefix_completion_boundary"}}],"expect":"object"},"prior_art":{"family_id":"three_prime_residue_escape_consumers","status_summary":"An open three-prime #269 endpoint, a finite least-positive-residue contradiction, a conditional cofinal-window extinction consumer, and the weighted-phase carry split with its finite realised-span observer.","prior_art_posture":"The residue classifier and bounded-carry consumer are source-specific formal results; no novelty, priority, significance, or three-prime irrationality claim is made.","primary_narrative_owner":"paper/269/erdos-269-three-prime-running-lcm.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/23","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z17::n269_finite_prefix_completion_boundary","prior_art_state":"routine_corollary_or_specialisation"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:three_prime_bounded_certificate_safety","assertion_kind":"registered_claim","source_id":"three_prime_bounded_certificate_safety","reviewed":true,"granularity":"single_registered_claim","label":"Size-authenticated bounded certificate checks for #269","statement":"The bounded width and cumulative-array checkers expose their size hypotheses, are sound on every accessed entry, compose across adjacent ranges, and carry explicit regressions showing that default-zero array access, half-height cancellation, and finite coverage cannot certify an infinite/cofinal conclusion. No paper numerical table is executed or validated by these safety theorems alone.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_269"],"problem_erdos_numbers":[269],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_269","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_269","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"widthCheckBoundedR12_sound","module":"ErdosProblems/Erdos269/CertificateSafetyR12.lean","line":14,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L14","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L14","kind":"theorem","signature":"theorem widthCheckBoundedR12_sound {p q r lo n : ℕ} {data : Array ℕ} (h : widthCheckBoundedR12 p q r data lo n = true) : lo + n ≤ data.size ∧ ∀ i, i < n → lo + i < data.size ∧ data[lo + i]?.getD 0 = pairCountChecked q r (p ^ (lo + i))","generated_certificate":false},{"name":"widthCheckBoundedR12_actual","module":"ErdosProblems/Erdos269/CertificateSafetyR12.lean","line":25,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L25","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L25","kind":"theorem","signature":"theorem widthCheckBoundedR12_actual {p q r lo n : ℕ} {data : Array ℕ} (hq : 1 < q) (hr : 1 < r) (h : widthCheckBoundedR12 p q r data lo n = true) (i : ℕ) (hi : i < n) : lo + i < data.size ∧ data[lo + i]?.getD 0 = (strictSmoothPairs q r (p ^ (lo + i))).card","generated_certificate":false},{"name":"widthCheckBoundedR12_append","module":"ErdosProblems/Erdos269/CertificateSafetyR12.lean","line":34,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L34","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L34","kind":"theorem","signature":"theorem widthCheckBoundedR12_append {p q r lo n m : ℕ} {data : Array ℕ} (h₁ : widthCheckBoundedR12 p q r data lo n = true) (h₂ : widthCheckBoundedR12 p q r data (lo + n) m = true) : widthCheckBoundedR12 p q r data lo (n + m) = true","generated_certificate":false},{"name":"cumulativeCheckBoundedR12_sound","module":"ErdosProblems/Erdos269/CertificateSafetyR12.lean","line":54,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L54","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L54","kind":"theorem","signature":"theorem cumulativeCheckBoundedR12_sound {w c : Array ℕ} {n : ℕ} (h : cumulativeCheckBoundedR12 w c n = true) : (0 < n ∧ n ≤ w.size ∧ n ≤ c.size) ∧ c[0]?.getD 0 = 0 ∧ ∀ e : Fin (n - 1), c[e.val + 1]?.getD 0 = c[e.val]?.getD 0 + w[e.val + 1]?.getD 0","generated_certificate":false},{"name":"width_shape_regressions_R12","module":"ErdosProblems/Erdos269/CertificateSafetyR12.lean","line":64,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L64","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L64","kind":"theorem","signature":"theorem width_shape_regressions_R12 : widthCheck 2 3 5 #[] 0 1 = true ∧ widthCheckBoundedR12 2 3 5 #[] 0 1 = false ∧ widthCheckBoundedR12 2 3 5 #[0] 0 1 = true ∧ widthCheckBoundedR12 2 3 5 #[0] 0 2 = false ∧ widthCheckBoundedR12 2 3 5 #[] 0 0 = true","docstring":"Default-zero acceptance is not evidence of an array entry's existence.","generated_certificate":false},{"name":"factor_two_cannot_be_cancelled_R12","module":"ErdosProblems/Erdos269/CertificateSafetyR12.lean","line":73,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L73","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L73","kind":"theorem","signature":"theorem factor_two_cannot_be_cancelled_R12 : (2 : ℚ) * (1 / 2) = 1 ∧ ¬ ∃ z : ℤ, (1 / 2 : ℚ) = (z : ℚ)","docstring":"Full-height clearing cannot be replaced by half-height clearing.","generated_certificate":false},{"name":"finite_coverage_not_cofinal_R12","module":"ErdosProblems/Erdos269/CertificateSafetyR12.lean","line":84,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L84","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L84","kind":"theorem","signature":"theorem finite_coverage_not_cofinal_R12 (U : ℕ) : (∀ a : ℕ, a ≤ U → a ≤ U) ∧ ¬ (∀ onset : ℕ, ∃ a : ℕ, onset ≤ a ∧ a ≤ U)","docstring":"A counterexample to the abstract finite-to-cofinal implication. This predicate is NOT claimed to be the actual digit word or actual escape set.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.erdos_269_cofinal_local_window_escape","statement":"Prove that for every B ≥ 1 coprime to 30 and every a₀ ≥ 1 there are ℓ ≥ a₀ and h ≥ 1 with lpr_{W(ℓ,h)}(−B·F(ℓ,h)) > K(B, ℓ+h), where K is the defined long-paper cap.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/12"}],"links":[{"kind":"lean_declaration_pinned","label":"widthCheckBoundedR12_sound","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L14","cite_this":true},{"kind":"lean_declaration_pinned","label":"widthCheckBoundedR12_actual","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L25","cite_this":true},{"kind":"lean_declaration_pinned","label":"widthCheckBoundedR12_append","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L34","cite_this":true},{"kind":"lean_declaration_pinned","label":"cumulativeCheckBoundedR12_sound","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L54","cite_this":true},{"kind":"lean_declaration_pinned","label":"width_shape_regressions_R12","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L64","cite_this":true},{"kind":"lean_declaration_pinned","label":"factor_two_cannot_be_cancelled_R12","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L73","cite_this":true},{"kind":"lean_declaration_pinned","label":"finite_coverage_not_cofinal_R12","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/CertificateSafetyR12.lean#L84","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/17","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"three_prime_bounded_certificate_safety"}}],"expect":"object"},"prior_art":{"family_id":"three_prime_residue_escape_consumers","status_summary":"An open three-prime #269 endpoint, a finite least-positive-residue contradiction, a conditional cofinal-window extinction consumer, and the weighted-phase carry split with its finite realised-span observer.","prior_art_posture":"The residue classifier and bounded-carry consumer are source-specific formal results; no novelty, priority, significance, or three-prime irrationality claim is made.","primary_narrative_owner":"paper/269/erdos-269-three-prime-running-lcm.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/23","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z17::n269_bounded_certificate_safety","prior_art_state":"routine_corollary_or_specialisation"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:periodic_nonnegative_weight_irrationality","assertion_kind":"registered_claim","source_id":"periodic_nonnegative_weight_irrationality","reviewed":true,"granularity":"single_registered_claim","label":"Eventually-periodic nonnegative weighted Lambert irrationality","statement":"For every integer base b ≥ 2, an eventually-periodic nonnegative rational coefficient sequence with a positive value at a positive index in its periodic region has irrational series ∑ γ(a)/(b^a − 1). This includes eventually-periodic natural-number weights and indicator supports. Luca–Tachiya prove a broader periodic rational-coefficient theorem by different methods; no priority claim is made here.","status":"formalised here","status_meaning":"Known mathematics represented and checked in Lean","readme_headline":false,"problem_ids":["erdos_249","erdos_257"],"problem_erdos_numbers":[249,257],"problem_binding":{"method":"core_programme_joint","warrant":"the claim's only paper anchor is owned by the retired combined #249/#257 exposition, which treats both problems together and publishes nothing that separates them. Bound to both rather than guessed apart.","separability":"not_separable_from_published_source"},"problem_edges":[{"problem_id":"erdos_249","relation":"shared_programme_context","basis":"problem_binding"},{"problem_id":"erdos_257","relation":"shared_programme_context","basis":"problem_binding"}],"programme_ids":["joint_249_257_gateway"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"irrational_weightedErdosSeries_periodic","module":"Erdos249257/CertificateKernel.lean","line":12698,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CertificateKernel.lean#L12698","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L12698","kind":"theorem","signature":"theorem irrational_weightedErdosSeries_periodic (b m : ℕ) (w : ℕ → ℕ) (hb : 2 ≤ b) (hm : 0 < m) (hper : ∀ n : ℕ, w (n + m) = w n) (hpos : ∃ a : ℕ, 0 < a ∧ 0 < w a) : Irrational (weightedErdosSeries b w)","docstring":"**Periodic Nat-weight irrationality (wave-11 core).** For every base `b ≥ 2` and every `m`-periodic weight `w : ℕ → ℕ` (`m ≥ 1`) with a positive value at a positive index, `∑_a w(a)/(b^a - 1)` is irrational. At indicator weights this recovers the wave-10 periodic support theorem; at `w ≡ 1` the full-support theorem.","generated_certificate":false},{"name":"irrational_weightedErdosSeries_eventuallyPeriodic","module":"Erdos249257/CertificateKernel.lean","line":12725,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CertificateKernel.lean#L12725","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L12725","kind":"theorem","signature":"theorem irrational_weightedErdosSeries_eventuallyPeriodic (b m N₀ : ℕ) (w : ℕ → ℕ) (hb : 2 ≤ b) (hm : 0 < m) (hper : ∀ n : ℕ, N₀ ≤ n → w (n + m) = w n) (hpos : ∃ a : ℕ, N₀ ≤ a ∧ 0 < a ∧ 0 < w a) : Irrational (weightedErdosSeries b w)","docstring":"**Eventually periodic Nat-weight irrationality (wave-11 frontier theorem, Nat form).** For every base `b ≥ 2`: a weight that is `m`-periodic from some threshold `N₀` on, with a positive value at a positive index inside the periodic region, has an irrational weighted series. The purely periodic shift `w_pure(n) = w(n + m·N₀)` agrees with `w` above the threshold, so the two series differ by the rational finite prefix difference.","generated_certificate":false},{"name":"irrational_ratWeightSeries_eventuallyPeriodic","module":"Erdos249257/CertificateKernel.lean","line":12811,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CertificateKernel.lean#L12811","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L12811","kind":"theorem","signature":"theorem irrational_ratWeightSeries_eventuallyPeriodic (b m N₀ : ℕ) (γ : ℕ → ℚ) (hb : 2 ≤ b) (hm : 0 < m) (hγ0 : ∀ n : ℕ, 0 ≤ γ n) (hper : ∀ n : ℕ, N₀ ≤ n → γ (n + m) = γ n) (hpos : ∃ a : ℕ, N₀ ≤ a ∧ 0 < a ∧ 0 < γ a) : Irrational (∑' a : ℕ, ((γ a : ℝ)) / ((b : ℝ) ^ a - 1))","docstring":"**Nonnegative rational periodic Lambert coefficients (wave-11 frontier theorem, ℚ form).** For every base `b ≥ 2`: an eventually periodic NONNEGATIVE rational coefficient sequence `γ` with a positive value at a positive index inside its periodic region has `∑_a γ(a)/(b^a - 1)` irrational. One common denominator `D` over the prefix-plus-period window clears `γ` to an eventually periodic Nat weight; the series scales by `1/D`, and irrationality survives nonzero rational scaling. Mixed-sign rational periodic coefficients (the full Luca–Tachiya statement) are NOT claimed.","generated_certificate":false}],"paper_label":"res:periodicnonnegative","paper_anchor":{"paper_label":"res:periodicnonnegative","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":3843,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":3842,"anchor_title":"eventually-periodic nonnegative weights","anchor_title_tex":"eventually-periodic nonnegative weights","section_title":"Periodic weights.","section_path":[{"level":"section","title":"Exact remaining problems and further questions","title_tex":"Exact remaining problems and further questions","line":3819},{"level":"subsection","title":"Weighted Lambert comparison families","title_tex":"Weighted Lambert comparison families","line":3831},{"level":"paragraph","title":"Periodic weights.","title_tex":"Periodic weights.","line":3839}],"source_excerpt":{"text":"\\begin{theorem}[eventually-periodic nonnegative weights]\n\\label{res:periodicnonnegative}\nFor every $b\\ge2$, an eventually-periodic nonnegative rational weight that is\nnot identically zero on its eventual period gives an irrational series.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":202318,"source_char_end":202570,"start_line":3842,"end_line":3846,"truncated":false,"sha256":"sha256:fa440ebca22e2cd251380e5716d256a6ddcdbec5e40adc86fff25fa033721b78","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3843","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"irrational_weightedErdosSeries_periodic","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L12698","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_weightedErdosSeries_eventuallyPeriodic","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L12725","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_ratWeightSeries_eventuallyPeriodic","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L12811","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/18","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"periodic_nonnegative_weight_irrationality"}}],"expect":"object"},"prior_art":{"family_id":"periodic_weighted_lambert_theorems","status_summary":"One classical/broader antecedent, one exact signed dichotomy, and one proved residue obstruction.","prior_art_posture":"Luca–Tachiya prove the broader eventually-periodic rational-coefficient theorem by another method; the signed endpoint formalised here is only irrational-or-b-power-denominator.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/1","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z01::periodic_weight_irrationality","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:periodicnonnegative","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3843","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:periodic_signed_weight_dichotomy","assertion_kind":"registered_claim","source_id":"periodic_signed_weight_dichotomy","reviewed":true,"granularity":"single_registered_claim","label":"Periodic signed-weight irrational-or-terminating dichotomy","statement":"For every integer base b ≥ 2 and every periodic integer weight w, the signed series ∑ w(a)/(b^a − 1) is irrational or some power of b times its value is an integer. Excluding that terminating alternative implies irrationality. The terminating alternative is not excluded in general, so the full mixed-sign periodic-coefficient theorem is not claimed.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249","erdos_257"],"problem_erdos_numbers":[249,257],"problem_binding":{"method":"core_programme_joint","warrant":"the claim's only paper anchor is owned by the retired combined #249/#257 exposition, which treats both problems together and publishes nothing that separates them. Bound to both rather than guessed apart.","separability":"not_separable_from_published_source"},"problem_edges":[{"problem_id":"erdos_249","relation":"shared_programme_context","basis":"problem_binding"},{"problem_id":"erdos_257","relation":"shared_programme_context","basis":"problem_binding"}],"programme_ids":["joint_249_257_gateway"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"irrational_or_bpow_mul_eq_intCast_intWeightedErdosSeries_periodic","module":"Erdos249257/CertificateKernel.lean","line":14175,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CertificateKernel.lean#L14175","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L14175","kind":"theorem","signature":"theorem irrational_or_bpow_mul_eq_intCast_intWeightedErdosSeries_periodic (b m : ℕ) (w : ℕ → ℤ) (hb : 2 ≤ b) (hm : 0 < m) (hper : ∀ n : ℕ, w (n + m) = w n) : Irrational (intWeightedErdosSeries b w) ∨ ∃ (k : ℕ) (z : ℤ), (b : ℝ) ^ k * intWeightedErdosSeries b w = (z : ℝ)","docstring":"**The wave-13 dichotomy, Lambert form** — UNCONDITIONAL, mixed signs allowed: for every base `b ≥ 2` and EVERY `m`-periodic signed integer weight `w`, the signed weighted Erdős series `∑_a w(a)/(b^a − 1)` is irrational OR base-`b` terminating (`b^k·x ∈ ℤ` for some `k`). This discharges the entire protected-certificate-supply obligation except on the terminating escape set; excluding that escape for general periodic signed weights is the remaining open frontier and is NOT claimed.","generated_certificate":false},{"name":"irrational_intWeightedErdosSeries_periodic_of_bpow_mul_ne_intCast","module":"Erdos249257/CertificateKernel.lean","line":14200,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CertificateKernel.lean#L14200","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L14200","kind":"theorem","signature":"theorem irrational_intWeightedErdosSeries_periodic_of_bpow_mul_ne_intCast (b m : ℕ) (w : ℕ → ℤ) (hb : 2 ≤ b) (hm : 0 < m) (hper : ∀ n : ℕ, w (n + m) = w n) (hnt : ∀ (k : ℕ) (z : ℤ), (b : ℝ) ^ k * intWeightedErdosSeries b w ≠ (z : ℝ)) : Irrational (intWeightedErdosSeries b w)","docstring":"**Non-termination socket.** A periodic signed weight whose series never terminates in base `b` has an irrational series. Compared with the wave-12 protected-block socket this hypothesis is a single condition on the VALUE rather than a certificate family at every precision — the residual analytic content of the mixed-sign Luca–Tachiya statement lives entirely inside it.","generated_certificate":false}],"paper_label":"res:periodicsigned","paper_anchor":{"paper_label":"res:periodicsigned","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":3857,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":3857,"anchor_title":"periodic signed-weight dichotomy","anchor_title_tex":"periodic signed-weight dichotomy","section_title":"Periodic weights.","section_path":[{"level":"section","title":"Exact remaining problems and further questions","title_tex":"Exact remaining problems and further questions","line":3819},{"level":"subsection","title":"Weighted Lambert comparison families","title_tex":"Weighted Lambert comparison families","line":3831},{"level":"paragraph","title":"Periodic weights.","title_tex":"Periodic weights.","line":3839}],"source_excerpt":{"text":"\\begin{theorem}[periodic signed-weight dichotomy]\\label{res:periodicsigned}\nFor a periodic integer weight \\(\\gamma\\), either \\(L_b(\\gamma)\\) is irrational\nor \\(b^kL_b(\\gamma)\\in\\Z\\) for some \\(k\\ge0\\).  In the rational branch its\ndenominator therefore divides a power of \\(b\\); the theorem does not select a branch.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":203249,"source_char_end":203578,"start_line":3857,"end_line":3861,"truncated":false,"sha256":"sha256:f60fe91e63b6b3ca9f916085ed636fd5e77275d978eae56cd905ba2d2b5de4d6","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3857","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"irrational_or_bpow_mul_eq_intCast_intWeightedErdosSeries_periodic","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L14175","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_intWeightedErdosSeries_periodic_of_bpow_mul_ne_intCast","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L14200","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/19","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"periodic_signed_weight_dichotomy"}}],"expect":"object"},"prior_art":{"family_id":"periodic_weighted_lambert_theorems","status_summary":"One classical/broader antecedent, one exact signed dichotomy, and one proved residue obstruction.","prior_art_posture":"Luca–Tachiya prove the broader eventually-periodic rational-coefficient theorem by another method; the signed endpoint formalised here is only irrational-or-b-power-denominator.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/1","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z01::signed_weight_dichotomy","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:periodicsigned","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3857","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:period_four_signed_residue_obstruction","assertion_kind":"registered_claim","source_id":"period_four_signed_residue_obstruction","reviewed":true,"granularity":"single_registered_claim","label":"Period-four signed residue-selection obstruction","statement":"For the period-four signed weight 1,0,−1,0, the associated signed divisor coefficient vanishes for every n congruent to 3 modulo 4. A whole arithmetic progression can therefore carry zero coefficient, so residue-blind selection cannot force a protected nonzero residue for every periodic signed weight.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249","erdos_257"],"problem_erdos_numbers":[249,257],"problem_binding":{"method":"core_programme_joint","warrant":"the claim's only paper anchor is owned by the retired combined #249/#257 exposition, which treats both problems together and publishes nothing that separates them. Bound to both rather than guessed apart.","separability":"not_separable_from_published_source"},"problem_edges":[{"problem_id":"erdos_249","relation":"shared_programme_context","basis":"problem_binding"},{"problem_id":"erdos_257","relation":"shared_programme_context","basis":"problem_binding"}],"programme_ids":["joint_249_257_gateway"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"intWeightedCoeff_periodFourSignWeight_eq_zero_of_mod_four_eq_three","module":"Erdos249257/CertificateKernel.lean","line":14226,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CertificateKernel.lean#L14226","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L14226","kind":"theorem","signature":"theorem intWeightedCoeff_periodFourSignWeight_eq_zero_of_mod_four_eq_three (n : ℕ) (hn : n % 4 = 3) : intWeightedCoeff periodFourSignWeight n = 0","docstring":"**The residue-selection obstruction, machine-checked.** For the period-4 sign weight the divisor involution `d ↦ n/d` pairs opposite weights on every `n ≡ 3 mod 4` (the product of the two residues is `3 mod 4`, so exactly one factor is `1` and the other `3`), hence the signed coefficient vanishes IDENTICALLY on that residue class. Zero sets of periodic signed coefficients can therefore contain full arithmetic progressions: no residue-blind selection over a cofactor progression can force a nonzero protected residue for every periodic signed weight, which is why wave 13 lands a dichotomy rather than a protected-certificate supply.","generated_certificate":false}],"paper_label":"res:periodfourobstruction","paper_anchor":{"paper_label":"res:periodfourobstruction","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":3864,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":3864,"anchor_title":"period-four example","anchor_title_tex":"period-four example","section_title":"Periodic weights.","section_path":[{"level":"section","title":"Exact remaining problems and further questions","title_tex":"Exact remaining problems and further questions","line":3819},{"level":"subsection","title":"Weighted Lambert comparison families","title_tex":"Weighted Lambert comparison families","line":3831},{"level":"paragraph","title":"Periodic weights.","title_tex":"Periodic weights.","line":3839}],"source_excerpt":{"text":"\\begin{proposition}[period-four example]\\label{res:periodfourobstruction}\nFor the nonzero period-four weight \\(\\gamma=1,0,-1,0,\\ldots\\),\n$c_\\gamma(n)=0$ whenever \\(n\\equiv3\\pmod4\\).\n\\end{proposition}","media_type":"text/x-tex","source_char_start":203741,"source_char_end":203940,"start_line":3864,"end_line":3867,"truncated":false,"sha256":"sha256:496031caad93a6ed5f68d6cd86fc410af976a992e9c0dc1576db0ac4b7366f4b","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3864","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"intWeightedCoeff_periodFourSignWeight_eq_zero_of_mod_four_eq_three","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L14226","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/20","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"period_four_signed_residue_obstruction"}}],"expect":"object"},"prior_art":{"family_id":"periodic_weighted_lambert_theorems","status_summary":"One classical/broader antecedent, one exact signed dichotomy, and one proved residue obstruction.","prior_art_posture":"Luca–Tachiya prove the broader eventually-periodic rational-coefficient theorem by another method; the signed endpoint formalised here is only irrational-or-b-power-denominator.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/1","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"formalisation_only":1},"edges":[{"semantic_node_id":"Z01::period_four_sign_weight_zero_progression","prior_art_state":"formalisation_only"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:periodfourobstruction","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3864","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:orthogonal_petal_sunflower_reduction","assertion_kind":"registered_claim","source_id":"orthogonal_petal_sunflower_reduction","reviewed":true,"granularity":"single_registered_claim","label":"Conditional orthogonal-petal sunflower reduction","statement":"For a finite-core orthogonal-petal bouquet at base 2, the exact uniform tail-selection predicate implies a forced-carry certificate supply and hence irrationality of the associated support series. Lean formalises the carry conversion and an explicit infinite alternating-core, squared-prime-petal bouquet whose elements have three prime factors with multiplicity. The tail-selection predicate itself is not proved, so this does not add an unconditional #257 support family.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"sunflowerForcedCarrySupply_of_orthogonalPetalBouquet","module":"Erdos249257/SupportSunflowerDichotomy.lean","line":531,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SupportSunflowerDichotomy.lean#L531","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SupportSunflowerDichotomy.lean#L531","kind":"theorem","signature":"theorem sunflowerForcedCarrySupply_of_orthogonalPetalBouquet {A : Set ℕ} (hB : OrthogonalPetalBouquet A) (hselection : SunflowerForcedSlotTailSelection A) : SunflowerForcedCarrySupply A","docstring":"Exact bouquet composition endpoint. 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Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z10::mixed_prime_power_layer_algebra","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z10::prime_power_layer_fixtures","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:mixedlayer","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1460","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:dyadic_prefix_block_safety_interfaces","assertion_kind":"registered_claim","source_id":"dyadic_prefix_block_safety_interfaces","reviewed":true,"granularity":"single_registered_claim","label":"Dyadic-prefix, block-safety, and primitive-overshoot interfaces","statement":"Inside a hypothetical dyadic-rational support value, a reduced finite positive prefix transports exactly to a residual rational support value and wrap-ratio certificate. For the half-greedy orbit, reduced and raw margins, three-depth take-run excess, primitive-lattice cap violation, support-coefficient tail gaps, and gcd-scaled predecessor overshoot are connected by exact identities and equivalences; cofinal supplies of the named witnesses imply that 1/2 belongs to the Mersenne achievement set. Checked fixtures show that safe brackets with numerator monotonicity and a factor-11 predecessor rule are not inductive. No cofinal witness supply is proved, so neither half-membership nor universal Erdős #257 follows.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"finiteFragment_dyadicPrefixCompression","module":"Erdos249257/DyadicPrefixCompression.lean","line":1636,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DyadicPrefixCompression.lean#L1636","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L1636","kind":"theorem","signature":"theorem finiteFragment_dyadicPrefixCompression (A : Set ℕ) (F : Finset ℕ) (p r c D : ℕ) (hF : F.Nonempty) (h0 : 0 ∉ F) (hFA : (↑F : Set ℕ) ⊆ A) (hwhole : erdosSupportSeries 2 A = (p : ℝ) / ((2 ^ c : ℕ) : ℝ)) (hprefix : finiteErdosSum F 2 = (r : ℚ) / (D : ℚ)) (hDactual : D = (finiteErdosSum F 2).den) (hrD : r.Coprime D) (hDgtOne : 1 < D) (hle : 2 ^ c * r ≤ p * D) (hNpos : 0 < dyadicResidualNumerator p r c D) (hResidual : ∃ a : ℕ, 0 < a ∧ a ∈ A \\ (↑F : Set ℕ)) : DyadicPrefixCompressionCertificate A F p r c D","docstring":"Generic finite-fragment T12 transport. The prefix certificate names the actual reduced denominator and numerator; the existing noncollapse theorem supplies the exact order, while T9 supplies the analytic disjunction.","generated_certificate":false},{"name":"halfGreedyThreeDepthTakeRunExcessBound_iff_skippedExcessBound","module":"Erdos249257/DyadicPrefixCompression.lean","line":2176,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DyadicPrefixCompression.lean#L2176","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L2176","kind":"theorem","signature":"theorem halfGreedyThreeDepthTakeRunExcessBound_iff_skippedExcessBound : HalfGreedyThreeDepthTakeRunExcessBound ↔ HalfGreedySkippedExcessBound","docstring":"The triple-depth take-run socket is not merely sufficient: after the forced-block theorem, it is exactly equivalent to the original skipped-rank excess producer. This records the endpoint of what uniform run geometry can remove from the arithmetic obligation.","generated_certificate":false},{"name":"greedyHalf_badSkipCapViolation_iff_primitiveLattice","module":"Erdos249257/DyadicPrefixCompression.lean","line":2188,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DyadicPrefixCompression.lean#L2188","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L2188","kind":"theorem","signature":"theorem greedyHalf_badSkipCapViolation_iff_primitiveLattice (n : ℕ) : (halfDyadicCap (n + 1) < greedyMersenneRemainder (1 / 2 : ℝ) n ∧ greedyMersenneRemainder (1 / 2 : ℝ) n < mersenneWeight (n + 1) ∧ halfTwoChannelCap (n + 1) < greedyMersenneRemainder (1 / 2 : ℝ) n) ↔ (0 < halfGreedyNextDyadicExcessNumerator n ∧ 2 * halfGreedyNextDyadicExcessNumerator n < halfGreedyResidualDisplayedNumerator n ∧ (halfGreedyPrefixDenominator n : ℤ) < 3 * (((2 ^ (n + 1) : ℕ) : ℤ) * halfGreedyNextDyadicExcessNumerator n))","docstring":"A skipped, dyadically unsafe, two-channel-cap-violating half-prefix is exactly a point of the primitive integer lattice cut out by the three strict inequalities below. The raw support coordinates introduced above retain the additional common-numerator information that reduction erases.","generated_certificate":false},{"name":"halfGreedyPrefixSupportTailGap_eq_excess_div_denominator","module":"Erdos249257/DyadicPrefixCompression.lean","line":2281,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DyadicPrefixCompression.lean#L2281","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L2281","kind":"theorem","signature":"theorem halfGreedyPrefixSupportTailGap_eq_excess_div_denominator (n : ℕ) : halfGreedyPrefixSupportTailGap n = (halfGreedyNextDyadicExcessNumerator n : ℝ) / (halfGreedyPrefixDenominator n : ℝ)","docstring":"The support-tail gap is not a new scalar relaxation: it is exactly the primitive dyadic excess ratio `Eₙ / Dₙ`. Its extra content is the decomposition of that ratio as an integer carry minus the true finite-support coefficient tail.","generated_certificate":false},{"name":"greedyHalf_badSkipCapViolation_iff_supportCoeffTailGap","module":"Erdos249257/DyadicPrefixCompression.lean","line":2319,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DyadicPrefixCompression.lean#L2319","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L2319","kind":"theorem","signature":"theorem greedyHalf_badSkipCapViolation_iff_supportCoeffTailGap (n : ℕ) : (halfDyadicCap (n + 1) < greedyMersenneRemainder (1 / 2 : ℝ) n ∧ greedyMersenneRemainder (1 / 2 : ℝ) n < mersenneWeight (n + 1) ∧ halfTwoChannelCap (n + 1) < greedyMersenneRemainder (1 / 2 : ℝ) n) ↔ ((1 : ℝ) / (3 * (2 : ℝ) ^ (n + 1)) < halfGreedyPrefixSupportTailGap n ∧ halfGreedyPrefixSupportTailGap n < (1 : ℝ) / ((2 : ℝ) ^ (n + 1) - 1))","generated_certificate":false},{"name":"rawSafe_but_not_gcdScaled_fixture","module":"Erdos249257/DyadicPrefixCompression.lean","line":3450,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DyadicPrefixCompression.lean#L3450","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L3450","kind":"theorem","signature":"theorem rawSafe_but_not_gcdScaled_fixture : Nat.gcd 7 15 = 1 ∧ BlockDyadicSafeAt 7 15 2 ∧ 7 ≤ 2 * 15 - 7 * (2 ^ 2 - 1) ∧ ¬ Nat.gcd 15 (2 ^ 2 - 1) * 7 ≤ 2 * 15 - 7 * (2 ^ 2 - 1)","docstring":"The gcd-scaled producer is strictly stronger than raw dyadic safety, even for coprime displayed coordinates.","generated_certificate":false},{"name":"factorEleven_predecessorOvershoot_not_inductive_fixture","module":"Erdos249257/DyadicPrefixCompression.lean","line":3472,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DyadicPrefixCompression.lean#L3472","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L3472","kind":"theorem","signature":"theorem factorEleven_predecessorOvershoot_not_inductive_fixture : BlockDyadicSafeAt 39 539 4 ∧ 11 * 39 ≤ 2 * 539 - 39 * (2 ^ 4 - 1) ∧ BlockTakeAt 39 539 5 ∧ rawTakenResidualNumerator 39 539 5 = 131 ∧ rawTakenResidualDenominator 539 5 = 16709 ∧ Nat.gcd 131 16709 = 1 ∧ Nat.gcd 16709 (2 ^ (9 - 1) - 1) = 1 ∧ 10 * 9 ≤ (2 ^ (9 - 1) - 1) / Nat.gcd 16709 (2 ^ (9 - 1) - 1) ∧ IsFirstBlockTakeAfter 131 16709 5 9 ∧ 2 * 16709 - 131 * (2 ^ (9 - 1) - 1) = 13 ∧ ¬ BlockDyadicSafeAt 131 16709 (9 - 1) ∧ ¬ Nat.gcd 16709 (2 ^ (9 - 1) - 1) * 131 ≤","docstring":"Exact one-step falsifier for generic propagation of the sharp empirical factor `11`. The source `(39,539)` is predecessor-safe and has overshoot `493 ≥ 11*39`; taking rank `5` gives the reduced state `(131,16709)`, whose first next take is rank `9`, but its predecessor overshoot is only `13`. The target fresh quotient is already `255 ≥ 10*9`, so its size alone does not repair the failed margin.","generated_certificate":false},{"name":"unsafeSkipGcdOvershootSupply_iff_primitive","module":"Erdos249257/DyadicPrefixCompression.lean","line":4521,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DyadicPrefixCompression.lean#L4521","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L4521","kind":"theorem","signature":"theorem unsafeSkipGcdOvershootSupply_iff_primitive : HalfGreedyUnsafeSkipGcdOvershootSupply ↔ HalfGreedyUnsafeSkipPrimitiveOvershootSupply","docstring":"The gcd-scaled and primitive-quotient actual-orbit producers are exactly equivalent, not merely one-way sufficient conditions.","generated_certificate":false},{"name":"half_mem_mersenneAchievementSet_of_unsafeSkipPrimitiveOvershootSupply","module":"Erdos249257/DyadicPrefixCompression.lean","line":4562,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DyadicPrefixCompression.lean#L4562","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L4562","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_of_unsafeSkipPrimitiveOvershootSupply (hsupply : HalfGreedyUnsafeSkipPrimitiveOvershootSupply) : (1 / 2 : ℝ) ∈ mersenneAchievementSet","docstring":"The exact primitive quotient inequality on each putative unsafe skip is a complete arithmetic endpoint for half-membership.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"finiteFragment_dyadicPrefixCompression","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L1636","cite_this":true},{"kind":"lean_declaration_pinned","label":"halfGreedyThreeDepthTakeRunExcessBound_iff_skippedExcessBound","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L2176","cite_this":true},{"kind":"lean_declaration_pinned","label":"greedyHalf_badSkipCapViolation_iff_primitiveLattice","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L2188","cite_this":true},{"kind":"lean_declaration_pinned","label":"halfGreedyPrefixSupportTailGap_eq_excess_div_denominator","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L2281","cite_this":true},{"kind":"lean_declaration_pinned","label":"greedyHalf_badSkipCapViolation_iff_supportCoeffTailGap","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L2319","cite_this":true},{"kind":"lean_declaration_pinned","label":"rawSafe_but_not_gcdScaled_fixture","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L3450","cite_this":true},{"kind":"lean_declaration_pinned","label":"factorEleven_predecessorOvershoot_not_inductive_fixture","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L3472","cite_this":true},{"kind":"lean_declaration_pinned","label":"unsafeSkipGcdOvershootSupply_iff_primitive","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L4521","cite_this":true},{"kind":"lean_declaration_pinned","label":"half_mem_mersenneAchievementSet_of_unsafeSkipPrimitiveOvershootSupply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DyadicPrefixCompression.lean#L4562","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/24","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"dyadic_prefix_block_safety_interfaces"}}],"expect":"object"},"prior_art":{"family_id":"dyadic_prefix_block_safety_interfaces","status_summary":"Exact reduced-denominator transport, block-safety and primitive-overshoot equivalences, conditional half-membership consumers, and explicit failures of naive induction.","prior_art_posture":"Exact rational and greedy-block formalisation; no priority claim is made for the assembled source-level package.","primary_narrative_owner":"docs/SOURCE_MAP.md","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/6","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":6},"edges":[{"semantic_node_id":"Z10::finite_fragment_residual_transport","prior_art_state":"not_assessed"},{"semantic_node_id":"Z10::greedy_decisions_are_sign_conditions_on_excess_numerator","prior_art_state":"not_assessed"},{"semantic_node_id":"Z10::greedy_remainder_in_support_coefficient_coordinates","prior_art_state":"not_assessed"},{"semantic_node_id":"Z10::half_membership_from_gcd_overshoot_supply","prior_art_state":"not_assessed"},{"semantic_node_id":"Z10::half_membership_from_take_run_excess_bounds","prior_art_state":"not_assessed"},{"semantic_node_id":"Z10::raw_dyadic_safety_is_strictly_weaker_than_gcd_overshoot","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:half_trapping_return_carry_interface","assertion_kind":"registered_claim","source_id":"half_trapping_return_carry_interface","reviewed":true,"granularity":"single_registered_claim","label":"Half-trapping reverse-carry interface","statement":"Scalar-relation-invariant rational-linear channels have no nontrivial square determinant at any rank. Separately, a common coefficient word with a 1/0 seam and a common following overlap forces the terminal reverse-carry difference to be a power of two times an odd integer; a common nonnegative real carry bound yields the corresponding dyadic spacing inequality. The module neither supplies reverse-carry presentations nor a predecessor overlap, so it is not a greedy separation or irrationality theorem.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"relationInvariantLinearChannels_det_eq_zero","module":"Erdos249257/HalfTrappingReturnCarry.lean","line":42,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfTrappingReturnCarry.lean#L42","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfTrappingReturnCarry.lean#L42","kind":"theorem","signature":"theorem relationInvariantLinearChannels_det_eq_zero {V ι : Type*} [AddCommGroup V] [Module ℚ V] [Fintype ι] [DecidableEq ι] [Nontrivial ι] (ev : V →ₗ[ℚ] ℚ) (channel : ι → V →ₗ[ℚ] ℚ) (hker : ∀ j : ι, LinearMap.ker ev ≤ LinearMap.ker (channel j)) (he : ∃ e : V, ev e = 1) (row : ι → V) : Matrix.det (fun i j : ι ↦ channel j (row i)) = 0","docstring":"If every channel kills every relation killed by scalar evaluation, then the evaluation matrix has rank at most one. Hence every square minor indexed by a nontrivial finite type has determinant zero. This is the all-rank determinant consequence of `AdelicHeightObstruction.linearDescender_eq_smul_eval`.","generated_certificate":false},{"name":"overlappingReverseCarryWords_carryDifference_eq_twoPow_mul_odd","module":"Erdos249257/HalfTrappingReturnCarry.lean","line":124,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfTrappingReturnCarry.lean#L124","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfTrappingReturnCarry.lean#L124","kind":"theorem","signature":"theorem overlappingReverseCarryWords_carryDifference_eq_twoPow_mul_odd (left right : ReverseCarryWord) (seam length : ℕ) (hcoeffSeam : left.coeff seam = right.coeff seam) (hleftSeam : left.bit seam = 1) (hrightSeam : right.bit seam = 0) (hcoeffOverlap : ∀ j < length, left.coeff (seam + 1 + j) = right.coeff (seam + 1 + j)) (hbitOverlap : ∀ j < length, left.bit (seam + 1 + j) = right.bit (seam + 1 + j)) : ∃ z : ℤ, Odd z ∧ carryDifference left right (seam + 1 + length) = (2 : ℤ) ^ length * z","docstring":"**Nonlinear carry-transfer core.** A common coefficient at the seam, a `1/0` seam bit, and a common coefficient/output word for `length` further positions force the terminal carry difference to be `2^length` times an odd integer.","generated_certificate":false},{"name":"overlappingReverseCarryWords_twoPow_le_realBound","module":"Erdos249257/HalfTrappingReturnCarry.lean","line":191,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfTrappingReturnCarry.lean#L191","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfTrappingReturnCarry.lean#L191","kind":"theorem","signature":"theorem overlappingReverseCarryWords_twoPow_le_realBound (left right : ReverseCarryWord) (seam length : ℕ) (bound : ℝ) (hcoeffSeam : left.coeff seam = right.coeff seam) (hleftSeam : left.bit seam = 1) (hrightSeam : right.bit seam = 0) (hcoeffOverlap : ∀ j < length, left.coeff (seam + 1 + j) = right.coeff (seam + 1 + j)) (hbitOverlap : ∀ j < length, left.bit (seam + 1 + j) = right.bit (seam + 1 + j)) (hleftNonneg : 0 ≤ left.carry (seam + 1 + length)) (hrightNonneg : 0 ≤ right.carry (seam + 1 + length)) (hleftBound : (left.carry (seam + 1 + length) : ℝ) ≤ bound) (hrightBound : (right.carry (seam + 1 + length) : ℝ) ≤ bound) : (2 : ℝ) ^ length ≤ bound","docstring":"Combined overlap-and-smallness theorem. This is the precise abstract same-scale obstruction used by the trapping-return route.","generated_certificate":false}],"paper_label":"res:halftrapping","paper_anchor":{"paper_label":"res:halftrapping","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1476,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Rank-one obstruction and reverse-carry spacing","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Conditional criteria and local inputs for structured supports","title_tex":"Conditional criteria and local inputs for structured supports","line":1407},{"level":"subsubsection","title":"Rank-one obstruction and reverse-carry spacing","title_tex":"Rank-one obstruction and reverse-carry spacing","line":1476}],"source_excerpt":{"text":"\\subsubsection{Rank-one obstruction and reverse-carry spacing}\\label{res:halftrapping}","media_type":"text/x-tex","source_char_start":75355,"source_char_end":75441,"start_line":1476,"end_line":1476,"truncated":false,"sha256":"sha256:7fcbe89f434b222a480bb1617ea50ae6ebeed5fce76807f02afc6ac0f2d2eb11","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1476","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"relationInvariantLinearChannels_det_eq_zero","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfTrappingReturnCarry.lean#L42","cite_this":true},{"kind":"lean_declaration_pinned","label":"overlappingReverseCarryWords_carryDifference_eq_twoPow_mul_odd","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfTrappingReturnCarry.lean#L124","cite_this":true},{"kind":"lean_declaration_pinned","label":"overlappingReverseCarryWords_twoPow_le_realBound","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfTrappingReturnCarry.lean#L191","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/25","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"half_trapping_return_carry_interface"}}],"expect":"object"},"prior_art":{"family_id":"half_carry_compactness_consumers","status_summary":"Conditional compactness consumers from progressively weaker finite-word or carry supplies.","prior_art_posture":"No executable seam/closure antecedent is recorded beyond the global achievement-set geometry.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/5","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. 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Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:half_carry_reachability_compactness","assertion_kind":"registered_claim","source_id":"half_carry_reachability_compactness","reviewed":true,"granularity":"single_registered_claim","label":"Half-carry reachability and compactness","statement":"At an unresolved even seam, the scalar transition reaches exactly every integer terminal carry other than the one value 2*δ-c. If canonical seam coordinates are supplied at every sufficiently large even depth, then admissible finite half-carry words occur cofinally; compactness yields an infinite support A with erdosSupportSeries 2 A = 1/2. The canonical seam-supply hypothesis is not proved.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":["half_carry_compactness_programme"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"evenSeamReachable_iff","module":"Erdos249257/HalfCarryReachability.lean","line":235,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarryReachability.lean#L235","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryReachability.lean#L235","kind":"theorem","signature":"theorem evenSeamReachable_iff (δ c k : ℤ) : EvenSeamReachable δ c k ↔ k ≠ 2 * δ - c","docstring":"**Exact one-hole theorem.** The unresolved half-divisor transition reaches every integer except `2δ-c`; there are no other holes.","generated_certificate":false},{"name":"cofinalAdmissibility_of_canonicalEvenSeamSupply","module":"Erdos249257/HalfCarryReachability.lean","line":1028,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarryReachability.lean#L1028","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryReachability.lean#L1028","kind":"theorem","signature":"theorem cofinalAdmissibility_of_canonicalEvenSeamSupply (hcanonical : HalfCarryCanonicalEvenSeamSupply) : HalfCarryCofinalAdmissibility","docstring":"An exact one-hole even seam always leaves at least one of carries `1` and `2`, so canonical seam supply gives admissible words at cofinally many levels.","generated_certificate":false},{"name":"exists_infinite_support_half_of_canonicalEvenSeamSupply","module":"Erdos249257/HalfCarryReachability.lean","line":1056,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarryReachability.lean#L1056","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryReachability.lean#L1056","kind":"theorem","signature":"theorem exists_infinite_support_half_of_canonicalEvenSeamSupply (hcanonical : HalfCarryCanonicalEvenSeamSupply) : ∃ A : Set ℕ, A.Infinite ∧ erdosSupportSeries 2 A = (1 : ℝ) / 2","docstring":"**One-hole seam consumer.** Canonical even-seam supply alone gives cofinally many nonempty admissible levels. The cofinal compactness consumer then produces an infinite support. Neither seam escape nor a separate odd-level full-strip theorem is needed for this existence conclusion.","generated_certificate":false}],"paper_label":"res:halfcarryreachability","paper_anchor":{"paper_label":"res:halfcarryreachability","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1493,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Half-carry reachability","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Conditional criteria and local inputs for structured supports","title_tex":"Conditional criteria and local inputs for structured supports","line":1407},{"level":"subsubsection","title":"Half-carry reachability","title_tex":"Half-carry reachability","line":1493}],"source_excerpt":{"text":"\\subsubsection{Half-carry reachability}\\label{res:halfcarryreachability}","media_type":"text/x-tex","source_char_start":76525,"source_char_end":76597,"start_line":1493,"end_line":1493,"truncated":false,"sha256":"sha256:313b8df7161b65e00fd67fc89fbd5ec84397cf72069b8ce8cf62b4da662e2b89","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1493","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"evenSeamReachable_iff","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryReachability.lean#L235","cite_this":true},{"kind":"lean_declaration_pinned","label":"cofinalAdmissibility_of_canonicalEvenSeamSupply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryReachability.lean#L1028","cite_this":true},{"kind":"lean_declaration_pinned","label":"exists_infinite_support_half_of_canonicalEvenSeamSupply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryReachability.lean#L1056","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/26","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"half_carry_reachability_compactness"}}],"expect":"object"},"prior_art":{"family_id":"half_carry_compactness_consumers","status_summary":"Conditional compactness consumers from progressively weaker finite-word or carry supplies.","prior_art_posture":"No executable seam/closure antecedent is recorded beyond the global achievement-set geometry.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/5","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2},"edges":[{"semantic_node_id":"Z12::canonical_even_seam_supply_route","prior_art_state":"not_assessed"},{"semantic_node_id":"Z12::even_seam_one_hole_theorem","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:halfcarryreachability","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1493","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:cofinal_strip_return_closure","assertion_kind":"registered_claim","source_id":"cofinal_strip_return_closure","reviewed":true,"granularity":"single_registered_claim","label":"Cofinal square-root-strip return closure","statement":"The binary-scaled uncentred greedy half carry is antitone, and the square-root strip envelope divided by 2^N tends to zero. Therefore cofinal returns of the canonical greedy carry to that strip force the scaled carry to tend to zero and yield membership of 1/2 in the Mersenne achievement set. The cofinal return hypothesis is not proved.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":["half_carry_compactness_programme"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"greedy_integerHalfCarry_scaled_tendsto_zero_of_cofinalStripReturn","module":"Erdos249257/CofinalStripReturn.lean","line":84,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CofinalStripReturn.lean#L84","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CofinalStripReturn.lean#L84","kind":"theorem","signature":"theorem greedy_integerHalfCarry_scaled_tendsto_zero_of_cofinalStripReturn (hreturn : GreedyHalfCarryCofinalStripReturn) : Tendsto (fun N : ℕ ↦ (integerHalfCarry (greedyMersenneSupport (1 / 2 : ℝ)) N : ℝ) / (2 : ℝ) ^ N) atTop (nhds 0)","docstring":"Cofinal strip returns force the entire scaled uncentred greedy carry to zero. Antitonicity is the fan-in: after one late return, every still later scaled state is no larger.","generated_certificate":false},{"name":"greedy_half_infinite_of_cofinalStripReturn","module":"Erdos249257/CofinalStripReturn.lean","line":130,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CofinalStripReturn.lean#L130","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CofinalStripReturn.lean#L130","kind":"theorem","signature":"theorem greedy_half_infinite_of_cofinalStripReturn (hreturn : GreedyHalfCarryCofinalStripReturn) : (greedyMersenneSupport (1 / 2 : ℝ)).Infinite ∧ erdosSupportSeries 2 (greedyMersenneSupport (1 / 2 : ℝ)) = (1 : ℝ) / 2","docstring":"Cofinal returns to the square-root strip already give an infinite canonical greedy support with exact Mersenne support-series value `1/2`.","generated_certificate":false},{"name":"half_mem_mersenneAchievementSet_of_cofinalStripReturn","module":"Erdos249257/CofinalStripReturn.lean","line":157,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CofinalStripReturn.lean#L157","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CofinalStripReturn.lean#L157","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_of_cofinalStripReturn (hreturn : GreedyHalfCarryCofinalStripReturn) : (1 / 2 : ℝ) ∈ mersenneAchievementSet","docstring":"Achievement-set form of the same closure lemma.","generated_certificate":false}],"paper_label":"res:cofinalstripreturn","paper_anchor":{"paper_label":"res:cofinalstripreturn","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1532,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":1532,"anchor_title":"arbitrarily late strip-return criterion","anchor_title_tex":"arbitrarily late strip-return criterion","section_title":"Arbitrarily late strip returns.","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Conditional criteria and local inputs for structured supports","title_tex":"Conditional criteria and local inputs for structured supports","line":1407},{"level":"subsubsection","title":"Half-carry reachability","title_tex":"Half-carry reachability","line":1493},{"level":"paragraph","title":"Arbitrarily late strip returns.","title_tex":"Arbitrarily late strip returns.","line":1518}],"source_excerpt":{"text":"\\begin{proposition}[arbitrarily late strip-return criterion]\\label{res:cofinalstripreturn}\nLet \\(\\Gamma_M\\) be the uncentred integer carry of the canonical greedy expansion\nof \\(1/2\\).  If\n\\[\n \\forall N\\in\\N\\;\\exists M\\geq N,\\qquad\n \\Gamma_M\\leq 2\\lfloor\\sqrt{M+1}\\rfloor+4,\n\\]\nthen \\(1/2\\in\\mathcal A\\), and its canonical greedy support is infinite.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":78892,"source_char_end":79260,"start_line":1532,"end_line":1540,"truncated":false,"sha256":"sha256:fd2eb4bcafba0ea63dd1ff17fcea7ff3e3863eb5bedf72e757a95189dcb3c4e7","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1532","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"greedy_integerHalfCarry_scaled_tendsto_zero_of_cofinalStripReturn","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CofinalStripReturn.lean#L84","cite_this":true},{"kind":"lean_declaration_pinned","label":"greedy_half_infinite_of_cofinalStripReturn","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CofinalStripReturn.lean#L130","cite_this":true},{"kind":"lean_declaration_pinned","label":"half_mem_mersenneAchievementSet_of_cofinalStripReturn","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CofinalStripReturn.lean#L157","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/27","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"cofinal_strip_return_closure"}}],"expect":"object"},"prior_art":{"family_id":"half_carry_compactness_consumers","status_summary":"Conditional compactness consumers from progressively weaker finite-word or carry supplies.","prior_art_posture":"No executable seam/closure antecedent is recorded beyond the global achievement-set geometry.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/5","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z12::cofinal_strip_return_yields_half","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:cofinalstripreturn","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1532","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:terminal_only_cofinal_closure","assertion_kind":"registered_claim","source_id":"terminal_only_cofinal_closure","reviewed":true,"granularity":"single_registered_claim","label":"Terminal-only cofinal square-root-strip closure","statement":"A normalized finite word with terminal half carry in the square-root strip has support-series value within (4M+12)/2^M of 1/2. Thus cofinally many such terminal-only witnesses, with neither coherent histories nor earlier-row bounds, force 1/2 into the Mersenne achievement set and yield an infinite support with support-series value 1/2. The cofinal terminal-witness supply is not proved.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"dist_half_erdosSupportSeries_wordSupport_le_of_terminalOnlyWitness","module":"Erdos249257/TerminalOnlyCofinal.lean","line":68,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TerminalOnlyCofinal.lean#L68","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TerminalOnlyCofinal.lean#L68","kind":"theorem","signature":"theorem dist_half_erdosSupportSeries_wordSupport_le_of_terminalOnlyWitness {M : ℕ} (hM : 1 ≤ M) {a : HalfWord M} (hone : ∀ h : 1 < M + 1, a ⟨1, h⟩ = false) (hcarry : |(integerHalfCarry (wordSupport a) (M - 1) : ℝ)| ≤ (halfStripBound M : ℝ)) : dist (erdosSupportSeries 2 (wordSupport a)) (1 / 2 : ℝ) ≤ (4 * (M : ℝ) + 12) / (2 : ℝ) ^ M","docstring":"Pointwise approximation estimate. A terminal-only strip witness at positive depth gives a finite support value within `(4*M + 12) / 2^M` of `1/2`. The constant combines the discrete terminal strip with the existing analytic support-coefficient-tail bound.","generated_certificate":false},{"name":"half_mem_mersenneAchievementSet_of_cofinalTerminalOnlyStrip","module":"Erdos249257/TerminalOnlyCofinal.lean","line":134,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TerminalOnlyCofinal.lean#L134","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TerminalOnlyCofinal.lean#L134","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_of_cofinalTerminalOnlyStrip (hcofinal : HalfCarryCofinalTerminalOnlyStrip) : (1 / 2 : ℝ) ∈ mersenneAchievementSet","docstring":"**Terminal-only cofinal consumer.** Cofinal square-root-strip terminal witnesses force `1/2` into the Mersenne achievement set. Unlike `exists_infinite_support_half_of_cofinalAdmissibility`, this theorem assumes neither coherent witnesses nor strip control at any earlier row.","generated_certificate":false},{"name":"exists_infinite_support_half_of_cofinalTerminalOnlyStrip","module":"Erdos249257/TerminalOnlyCofinal.lean","line":192,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TerminalOnlyCofinal.lean#L192","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TerminalOnlyCofinal.lean#L192","kind":"theorem","signature":"theorem exists_infinite_support_half_of_cofinalTerminalOnlyStrip (hcofinal : HalfCarryCofinalTerminalOnlyStrip) : ∃ A : Set ℕ, A.Infinite ∧ erdosSupportSeries 2 A = (1 : ℝ) / 2","docstring":"The terminal-only cofinal producer yields an infinite normalized support summing exactly to `1/2`. Infinitude follows from the existing odd- denominator obstruction for finite Boolean supports.","generated_certificate":false}],"paper_label":"res:terminalonlycofinal","paper_anchor":{"paper_label":"res:terminalonlycofinal","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1568,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":1568,"anchor_title":"terminal-only unbounded-depth criterion","anchor_title_tex":"terminal-only unbounded-depth criterion","section_title":"Terminal-only unbounded-depth approximation.","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Conditional criteria and local inputs for structured supports","title_tex":"Conditional criteria and local inputs for structured supports","line":1407},{"level":"subsubsection","title":"Half-carry reachability","title_tex":"Half-carry reachability","line":1493},{"level":"paragraph","title":"Terminal-only unbounded-depth approximation.","title_tex":"Terminal-only unbounded-depth approximation.","line":1549}],"source_excerpt":{"text":"\\begin{proposition}[terminal-only unbounded-depth criterion]\\label{res:terminalonlycofinal}\nNormalised terminal witnesses at unbounded depths in the square-root strip imply that\n$1/2$ belongs to the Mersenne achievement set, and hence that an infinite\nsupport has support-series value $1/2$.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":80545,"source_char_end":80854,"start_line":1568,"end_line":1572,"truncated":false,"sha256":"sha256:53b638aaef860bbb37babd4f6639048f03ef1af1b4874d938f5859abe4395f23","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1568","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"dist_half_erdosSupportSeries_wordSupport_le_of_terminalOnlyWitness","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TerminalOnlyCofinal.lean#L68","cite_this":true},{"kind":"lean_declaration_pinned","label":"half_mem_mersenneAchievementSet_of_cofinalTerminalOnlyStrip","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TerminalOnlyCofinal.lean#L134","cite_this":true},{"kind":"lean_declaration_pinned","label":"exists_infinite_support_half_of_cofinalTerminalOnlyStrip","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TerminalOnlyCofinal.lean#L192","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/28","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"terminal_only_cofinal_closure"}}],"expect":"object"},"prior_art":{"family_id":"half_carry_compactness_consumers","status_summary":"Conditional compactness consumers from progressively weaker finite-word or carry supplies.","prior_art_posture":"No executable seam/closure antecedent is recorded beyond the global achievement-set geometry.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/5","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2},"edges":[{"semantic_node_id":"Z12::terminal_only_cofinal_yields_half","prior_art_state":"not_assessed"},{"semantic_node_id":"Z12::terminal_only_strip_error_bound","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:terminalonlycofinal","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1568","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:terminal_scaled_vanishing_closure","assertion_kind":"registered_claim","source_id":"terminal_scaled_vanishing_closure","reviewed":true,"granularity":"single_registered_claim","label":"Scaled terminal-vanishing closure","statement":"A normalized sequence of finite terminal words whose depths tend to infinity and whose terminal carries divided by the corresponding powers of 2 tend to zero has support-series values converging to 1/2. Closedness of the Mersenne achievement set then gives membership of 1/2 and an infinite support with that value. The prior cofinal square-root-strip condition maps into this weaker socket, but no scaled-vanishing sequence is constructed.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"dist_half_erdosSupportSeries_wordSupport_le_carry_add_tail","module":"Erdos249257/TerminalOnlyScaledVanishing.lean","line":24,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TerminalOnlyScaledVanishing.lean#L24","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TerminalOnlyScaledVanishing.lean#L24","kind":"theorem","signature":"theorem dist_half_erdosSupportSeries_wordSupport_le_carry_add_tail {M : ℕ} (hM : 1 ≤ M) {a : HalfWord M} (hone : ∀ h : 1 < M + 1, a ⟨1, h⟩ = false) : dist (erdosSupportSeries 2 (wordSupport a)) (1 / 2 : ℝ) ≤ (|(integerHalfCarry (wordSupport a) (M - 1) : ℝ)| + (2 * Real.sqrt (M : ℝ) + 4)) / (2 : ℝ) ^ M","docstring":"A terminal word is close to `1/2` by the sum of two normalized errors: its absolute terminal carry and the universal support-coefficient tail.","generated_certificate":false},{"name":"half_mem_mersenneAchievementSet_of_terminalScaledVanishing","module":"Erdos249257/TerminalOnlyScaledVanishing.lean","line":165,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TerminalOnlyScaledVanishing.lean#L165","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TerminalOnlyScaledVanishing.lean#L165","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_of_terminalScaledVanishing (S : HalfTerminalOnlyScaledVanishingSequence) : (1 / 2 : ℝ) ∈ mersenneAchievementSet","docstring":"**Scaled-terminal consumer.** Vanishing of the normalized terminal carry already puts `1/2` in the Mersenne achievement set.","generated_certificate":false},{"name":"exists_infinite_support_half_of_terminalScaledVanishing","module":"Erdos249257/TerminalOnlyScaledVanishing.lean","line":221,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TerminalOnlyScaledVanishing.lean#L221","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TerminalOnlyScaledVanishing.lean#L221","kind":"theorem","signature":"theorem exists_infinite_support_half_of_terminalScaledVanishing (S : HalfTerminalOnlyScaledVanishingSequence) : ∃ A : Set ℕ, A.Infinite ∧ erdosSupportSeries 2 A = (1 : ℝ) / 2","docstring":"The scaled-terminal producer yields an infinite support summing exactly to `1/2`.","generated_certificate":false}],"paper_label":"res:terminalscaledvanishing","paper_anchor":{"paper_label":"res:terminalscaledvanishing","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1596,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":1596,"anchor_title":"scaled terminal-vanishing criterion","anchor_title_tex":"scaled terminal-vanishing criterion","section_title":"Scaled terminal vanishing.","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Conditional criteria and local inputs for structured supports","title_tex":"Conditional criteria and local inputs for structured supports","line":1407},{"level":"subsubsection","title":"Half-carry reachability","title_tex":"Half-carry reachability","line":1493},{"level":"paragraph","title":"Scaled terminal vanishing.","title_tex":"Scaled terminal vanishing.","line":1581}],"source_excerpt":{"text":"\\begin{proposition}[scaled terminal-vanishing criterion]\\label{res:terminalscaledvanishing}\nA normalised terminal-word sequence with depths tending to infinity and\nvanishing binary-scaled terminal carries yields an infinite support with\nsupport-series value $1/2$.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":82123,"source_char_end":82405,"start_line":1596,"end_line":1600,"truncated":false,"sha256":"sha256:16ec0924f0d553527db73128459c14c533c5d4308bf3f2ab57951011ee5a0985","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1596","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"dist_half_erdosSupportSeries_wordSupport_le_carry_add_tail","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TerminalOnlyScaledVanishing.lean#L24","cite_this":true},{"kind":"lean_declaration_pinned","label":"half_mem_mersenneAchievementSet_of_terminalScaledVanishing","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TerminalOnlyScaledVanishing.lean#L165","cite_this":true},{"kind":"lean_declaration_pinned","label":"exists_infinite_support_half_of_terminalScaledVanishing","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TerminalOnlyScaledVanishing.lean#L221","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/29","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"terminal_scaled_vanishing_closure"}}],"expect":"object"},"prior_art":{"family_id":"half_carry_compactness_consumers","status_summary":"Conditional compactness consumers from progressively weaker finite-word or carry supplies.","prior_art_posture":"No executable seam/closure antecedent is recorded beyond the global achievement-set geometry.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/5","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2},"edges":[{"semantic_node_id":"Z12::terminal_only_strip_error_bound","prior_art_state":"not_assessed"},{"semantic_node_id":"Z12::terminal_scaled_vanishing_yields_half","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:terminalscaledvanishing","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1596","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:half_cylinder_finite_shadow","assertion_kind":"registered_claim","source_id":"half_cylinder_finite_shadow","reviewed":true,"granularity":"single_registered_claim","label":"Finite shadows of the half cylinder","statement":"For an admissible depth-N half-carry word with support A, terminal carry K_N, and future tail T_N, the residual is exactly 1/2 - X_A = (K_N - T_N)/2^N. At a first even seam the sole missing carry is H = 2 delta - c, so the integer strip [1,B] is fully reachable exactly when H lies outside it. The rank-three fixture A = {2,3} has residual 1/42 < 1/32 but its depth-six strip [1,8] contains H = 3, so raw dyadic safety does not force seam escape. The sole conditional membership input is the unproved governed frozen-margin producer, which requires future coefficient rows to cover the centred carry within J <= k+1 after every skipped rank k+1.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"halfStripAdmissible_residual_eq","module":"Erdos249257/HalfCylinderFiniteShadow.lean","line":55,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderFiniteShadow.lean#L55","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderFiniteShadow.lean#L55","kind":"theorem","signature":"theorem halfStripAdmissible_residual_eq {N : ℕ} (hN : 1 ≤ N) (a : HalfWord N) (ha : HalfStripAdmissible N a) : (1 : ℝ) / 2 - erdosSupportSeries 2 (wordSupport a) = ((integerHalfCarry (wordSupport a) (N - 1) : ℤ) - binaryCoeffTail (supportCoeff (wordSupport a)) N) / (2 : ℝ) ^ N","docstring":"**Exact finite-shadow identity.** For an admissible word at positive depth `N`, the discrepancy of its finite Mersenne support sum from `1/2` is the terminal integer half carry minus the future-multiple coefficient tail, scaled by `2^N`.","generated_certificate":false},{"name":"evenSeam_fullIntegerStrip_iff_escape","module":"Erdos249257/HalfCylinderFiniteShadow.lean","line":824,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderFiniteShadow.lean#L824","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderFiniteShadow.lean#L824","kind":"theorem","signature":"theorem evenSeam_fullIntegerStrip_iff_escape {B δ c : ℤ} : (∀ k : ℤ, 1 ≤ k → k ≤ B → EvenSeamReachable δ c k) ↔ 2 * δ - c < 1 ∨ B < 2 * δ - c","docstring":"The landed one-hole theorem makes seam escape exactly equivalent to full scalar reachability of the integer strip.","generated_certificate":false},{"name":"half_mem_mersenneAchievementSet_of_governedFrozenMarginProducer","module":"Erdos249257/HalfCylinderFiniteShadow.lean","line":1275,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderFiniteShadow.lean#L1275","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderFiniteShadow.lean#L1275","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_of_governedFrozenMarginProducer (hproducer : HalfGreedyGovernedFrozenMarginProducer) : (1 / 2 : ℝ) ∈ mersenneAchievementSet","docstring":"The governed first-passage condition implies exact half-membership. The condition itself remains unproved.","generated_certificate":false}],"paper_label":"res:halfcylindershadow","paper_anchor":{"paper_label":"res:halfcylindershadow","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1618,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Finite residuals for the half-value problem","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Finite-to-unbounded-depth criteria: windows, seams, and compactness","title_tex":"Finite-to-unbounded-depth criteria: windows, seams, and compactness","line":1609},{"level":"subsubsection","title":"Finite residuals for the half-value problem","title_tex":"Finite residuals for the half-value problem","line":1618}],"source_excerpt":{"text":"\\subsubsection{Finite residuals for the half-value problem}\\label{res:halfcylindershadow}","media_type":"text/x-tex","source_char_start":83460,"source_char_end":83549,"start_line":1618,"end_line":1618,"truncated":false,"sha256":"sha256:ceca3351a99237e611d029bde0e93f95645f0e293d33709f52a1622b64d35262","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1618","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"halfStripAdmissible_residual_eq","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderFiniteShadow.lean#L55","cite_this":true},{"kind":"lean_declaration_pinned","label":"evenSeam_fullIntegerStrip_iff_escape","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderFiniteShadow.lean#L824","cite_this":true},{"kind":"lean_declaration_pinned","label":"half_mem_mersenneAchievementSet_of_governedFrozenMarginProducer","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderFiniteShadow.lean#L1275","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/30","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"half_cylinder_finite_shadow"}}],"expect":"object"},"prior_art":{"family_id":"half_carry_compactness_consumers","status_summary":"Conditional compactness consumers from progressively weaker finite-word or carry supplies.","prior_art_posture":"No executable seam/closure antecedent is recorded beyond the global achievement-set geometry.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/5","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z13::finite_shadow_square_root_strip","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z13::half_membership_from_frozen_margin_sockets","prior_art_state":"not_assessed"},{"semantic_node_id":"Z13::seam_hole_scalar_normal_forms","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:halfcylindershadow","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1618","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:fixed_coefficient_rewind_contraction","assertion_kind":"registered_claim","source_id":"fixed_coefficient_rewind_contraction","reviewed":true,"granularity":"single_registered_claim","label":"Fixed-coefficient rewind contraction","statement":"For coefficient c, the canonical inverse parent is T_c(k) = ceil((k+c)/2). For a length-L coefficient list with accumulated offset O, the iterated rewind is R(k) = floor((k+O)/2^L), with phase theta = (1+O) mod 2^L. If 1 <= B <= 2^L, all targets in [1,B] have one ancestor exactly when theta + B - 1 < 2^L; otherwise the endpoint ancestors differ by one and every target maps to one of that adjacent pair. The width bound does not exclude the seam-pair alternative or prove half-membership.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"rewind_eq_closedForm","module":"Erdos249257/HalfCarryCeilParentContraction.lean","line":49,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarryCeilParentContraction.lean#L49","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryCeilParentContraction.lean#L49","kind":"theorem","signature":"theorem rewind_eq_closedForm (cs : List ℕ) (k : ℕ) : rewind cs k = (k + offset cs) / denom cs","docstring":"Exact iterate formula. Floor-of-floor composition is absorbed into a single division because every later translation is scaled by the appropriate power of two in `offset`.","generated_certificate":false},{"name":"rewind_endpoint_eq_iff_phase_fit","module":"Erdos249257/HalfCarryCeilParentContraction.lean","line":99,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarryCeilParentContraction.lean#L99","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryCeilParentContraction.lean#L99","kind":"theorem","signature":"theorem rewind_endpoint_eq_iff_phase_fit (cs : List ℕ) {B : ℕ} (hB : 1 ≤ B) (hwidth : B ≤ denom cs) : rewind cs B = rewind cs 1 ↔ phase cs + (B - 1) < denom cs","docstring":"Exact singleton criterion for the endpoints.","generated_certificate":false},{"name":"rewind_singleton_or_isSeamPair","module":"Erdos249257/HalfCarryCeilParentContraction.lean","line":180,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarryCeilParentContraction.lean#L180","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryCeilParentContraction.lean#L180","kind":"theorem","signature":"theorem rewind_singleton_or_isSeamPair (cs : List ℕ) {B : ℕ} (hB : 1 ≤ B) (hwidth : B ≤ denom cs) : (∀ k : ℕ, 1 ≤ k → k ≤ B → rewind cs k = rewind cs 1) ∨ IsSeamPair cs B","docstring":"Correct fixed-tail contraction dichotomy. Width alone proves exactly a singleton-or-seam-pair result, never an unconditional singleton.","generated_certificate":false}],"paper_label":"res:fixedcoeffrewind","paper_anchor":{"paper_label":"res:fixedcoeffrewind","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1635,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Fixed-coefficient rewind","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Finite-to-unbounded-depth criteria: windows, seams, and compactness","title_tex":"Finite-to-unbounded-depth criteria: windows, seams, and compactness","line":1609},{"level":"subsubsection","title":"Fixed-coefficient rewind","title_tex":"Fixed-coefficient rewind","line":1635}],"source_excerpt":{"text":"\\subsubsection{Fixed-coefficient rewind}\\label{res:fixedcoeffrewind}","media_type":"text/x-tex","source_char_start":84589,"source_char_end":84657,"start_line":1635,"end_line":1635,"truncated":false,"sha256":"sha256:19bacc66dd1ac43e23d33ac63cc294042fdf4d218c0855678564746a54a89239","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:selected_half_carry_window_compactness","assertion_kind":"registered_claim","source_id":"selected_half_carry_window_compactness","reviewed":true,"granularity":"single_registered_claim","label":"Selected half-carry windows","statement":"A selected half-carry window at depth N and radius R assigns every k in [1,R] an admissible Boolean word of depth N whose terminal carry is k. If such windows exist at every sufficiently large depth with R >= 1, their k = 1 representatives give cofinal finite admissibility; compactness then yields an infinite support A with erdosSupportSeries 2 A = 1/2. The unproved object-level input is a coherent one-row history: either all representatives agree on membership of every divisor of N+1, making their next coefficient common, or the adjacent outcome realizes the canonical one-hole seam. The module supplies no such cofinal history.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"SelectedHalfWindow.HistoryAlternative","module":"Erdos249257/HalfCarrySelectedWindow.lean","line":236,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarrySelectedWindow.lean#L236","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarrySelectedWindow.lean#L236","kind":"def","signature":"def SelectedHalfWindow.HistoryAlternative {N R : ℕ} (W : SelectedHalfWindow N R) : Prop","docstring":"Exact unresolved global history alternative. Repeated common inverse maps contract a carry interval to either one ancestor or two adjacent ancestors. The singleton case must establish `NextRowDivisorAgreement`; the adjacent-pair case must identify the actual transition with the already landed canonical one-hole seam. No false phase-free singleton claim is encoded here.","generated_certificate":false},{"name":"cofinalAdmissibility_of_selectedHalfWindows","module":"Erdos249257/HalfCarrySelectedWindow.lean","line":831,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarrySelectedWindow.lean#L831","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarrySelectedWindow.lean#L831","kind":"theorem","signature":"theorem cofinalAdmissibility_of_selectedHalfWindows (N₀ : ℕ) (R : ℕ → ℕ) (hR : ∀ N : ℕ, N₀ ≤ N → 1 ≤ R N) (hwindow : ∀ N : ℕ, N₀ ≤ N → SelectedHalfWindow N (R N)) : HalfCarryCofinalAdmissibility","docstring":"Protected selected windows at all sufficiently large levels imply the landed minimal producer interface. Only one representative (carry `1`) is needed at each chosen level.","generated_certificate":false},{"name":"exists_infinite_support_half_of_selectedHalfWindows","module":"Erdos249257/HalfCarrySelectedWindow.lean","line":864,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarrySelectedWindow.lean#L864","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarrySelectedWindow.lean#L864","kind":"theorem","signature":"theorem exists_infinite_support_half_of_selectedHalfWindows (N₀ : ℕ) (R : ℕ → ℕ) (hR : ∀ N : ℕ, N₀ ≤ N → 1 ≤ R N) (hwindow : ∀ N : ℕ, N₀ ≤ N → SelectedHalfWindow N (R N)) : ∃ A : Set ℕ, A.Infinite ∧ erdosSupportSeries 2 A = (1 : ℝ) / 2","docstring":"A supply of selected protected windows already yields an infinite support whose Mersenne achievement sum is exactly `1/2`.","generated_certificate":false}],"paper_label":"res:selectedhalfwindow","paper_anchor":{"paper_label":"res:selectedhalfwindow","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1652,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Selected half-carry windows","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Finite-to-unbounded-depth criteria: windows, seams, and compactness","title_tex":"Finite-to-unbounded-depth criteria: windows, seams, and compactness","line":1609},{"level":"subsubsection","title":"Selected half-carry windows","title_tex":"Selected half-carry windows","line":1652}],"source_excerpt":{"text":"\\subsubsection{Selected half-carry windows}\\label{res:selectedhalfwindow}","media_type":"text/x-tex","source_char_start":85658,"source_char_end":85731,"start_line":1652,"end_line":1652,"truncated":false,"sha256":"sha256:aabef7f852b4142ee60485bfb48f47835b45319630555d3217a09a84db4bf46f","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2},"edges":[{"semantic_node_id":"Z15::selected_window_cofinal_consumer","prior_art_state":"not_assessed"},{"semantic_node_id":"Z15::selected_window_history_alternative_interface","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:selectedhalfwindow","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1652","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. 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The equation `offset + suffix + 2 = denom * (ancestor + 1)` is the natural-number form obtained by unrolling `1 = denom * ancestor - coefficientNumeral - suffixNumeral` and using `offset = coefficientNumeral + denom - 1`.","generated_certificate":false},{"name":"rewind_interval_singleton_iff_suffixNumeral_covers","module":"Erdos249257/HalfCarryRewindPhase.lean","line":65,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarryRewindPhase.lean#L65","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryRewindPhase.lean#L65","kind":"theorem","signature":"theorem rewind_interval_singleton_iff_suffixNumeral_covers (cs : List ℕ) (bits : List Bool) (ancestor B : ℕ) (hlen : bits.length = cs.length) (hrealized : offset cs + suffixNumeral bits + 2 = denom cs * (ancestor + 1)) (hB : 1 ≤ B) (hwidth : B ≤ denom cs) : (∀ k : ℕ, 1 ≤ k → k ≤ B → rewind cs k = rewind cs 1) ↔ B - 1 ≤ suffixNumeral bits","docstring":"The exact fixed-coefficient singleton criterion in support-history coordinates: the terminal interval `[1,B]` rewinds to one ancestor exactly when the carry-one suffix numeral covers the other `B-1` targets.","generated_certificate":false},{"name":"rewind_supportCoeffHistory_singleton_iff_prefixCarry_covers","module":"Erdos249257/HalfCarryRewindPhase.lean","line":352,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarryRewindPhase.lean#L352","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryRewindPhase.lean#L352","kind":"theorem","signature":"theorem rewind_supportCoeffHistory_singleton_iff_prefixCarry_covers (A : Set ℕ) [DecidablePred (· ∈ A)] (M L ancestor B : ℕ) (hM : 1 ≤ M) (hhorizon : M + L ≤ 2 * M + 1) (hterminal : integerHalfCarry A (M + L - 1) = 1) (hancestor : integerHalfCarry A (M - 1) = (ancestor : ℤ)) (hB : 1 ≤ B) (hwidth : B ≤ denom (supportCoeffRewindHistory (A ∩ Set.Iic M) M L)) : (∀ k : ℕ, 1 ≤ k → k ≤ B → rewind (supportCoeffRewindHistory (A ∩ Set.Iic M) M L) k = rewind (supportCoeffRewindHistory (A ∩ Set.Iic M) M L) 1) ↔ (B : ℤ) ≤ integerHalfCarry (A ∩ Set.Iic M) (M + L - 1)","docstring":"Equivalent frozen-prefix form of the exact criterion. Under terminal carry one the prefix-zero-extension endpoint is `Z = 1 + X`, so singleton rewind is exactly `B ≤ Z`; no phase or suffix encoding remains downstream.","generated_certificate":false}],"paper_label":"res:rewindphase","paper_anchor":{"paper_label":"res:rewindphase","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1668,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"A realised rewind criterion","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Finite-to-unbounded-depth criteria: windows, seams, and compactness","title_tex":"Finite-to-unbounded-depth criteria: windows, seams, and compactness","line":1609},{"level":"subsubsection","title":"A realised rewind criterion","title_tex":"A realised rewind criterion","line":1668}],"source_excerpt":{"text":"\\subsubsection{A realised rewind criterion}\\label{res:rewindphase}","media_type":"text/x-tex","source_char_start":86733,"source_char_end":86799,"start_line":1668,"end_line":1668,"truncated":false,"sha256":"sha256:534f91eb509b5f1d2d7f285dad55ec0165cf7c65291c3fc4b3914ac7092f528d","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1668","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"phase_eq_denom_sub_one_sub_suffixNumeral","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryRewindPhase.lean#L39","cite_this":true},{"kind":"lean_declaration_pinned","label":"rewind_interval_singleton_iff_suffixNumeral_covers","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryRewindPhase.lean#L65","cite_this":true},{"kind":"lean_declaration_pinned","label":"rewind_supportCoeffHistory_singleton_iff_prefixCarry_covers","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryRewindPhase.lean#L352","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/33","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"realized_rewind_phase_identity"}}],"expect":"object"},"prior_art":{"family_id":"rewind_windows_and_protected_seams","status_summary":"Conditional rewind/window consumers plus one verified depth-18 base.","prior_art_posture":"No theorem-level match is recorded; any novelty posture remains provisional.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/7","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2},"edges":[{"semantic_node_id":"Z15::live_support_singleton_criterion","prior_art_state":"not_assessed"},{"semantic_node_id":"Z15::realized_history_phase_identity","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:rewindphase","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1668","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:localized_protected_seam_compactness","assertion_kind":"registered_claim","source_id":"localized_protected_seam_compactness","reviewed":true,"granularity":"single_registered_claim","label":"Localized protected seam compactness","statement":"At depth M, realized localized seam means only the forward implication: every seam-reachable carry k in [3,27] is the terminal carry of an admissible finite word. The seam's unique hole is H = 2 delta - c, so choosing k = 4 when H = 3 and k = 3 otherwise supplies one admissible word. If such realizations occur cofinally, then cofinal admissibility and an infinite support A with erdosSupportSeries 2 A = 1/2 follow. The cofinal realization supply is not proved, so Erdős #257 remains open.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"ProtectedEvenSeamRealizedAt","module":"Erdos249257/HalfCarryProtectedSeamConsumer.lean","line":18,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarryProtectedSeamConsumer.lean#L18","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryProtectedSeamConsumer.lean#L18","kind":"def","signature":"def ProtectedEvenSeamRealizedAt (N : ℕ) (δ c : ℤ) : Prop","docstring":"At one depth, every scalar carry in the protected window that survives the exact seam is realized by an admissible finite word. Only the forward realization direction is needed; neither full-strip equivalence nor seam escape is assumed.","generated_certificate":false},{"name":"cofinalAdmissibility_of_cofinalProtectedEvenSeamRealization","module":"Erdos249257/HalfCarryProtectedSeamConsumer.lean","line":31,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarryProtectedSeamConsumer.lean#L31","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryProtectedSeamConsumer.lean#L31","kind":"theorem","signature":"theorem cofinalAdmissibility_of_cofinalProtectedEvenSeamRealization (hsupply : HalfCarryCofinalProtectedEvenSeamRealization) : HalfCarryCofinalAdmissibility","docstring":"A one-hole seam cannot delete both protected carries `3` and `4`. Choosing the other one yields the sole finite word required by cofinal admissibility.","generated_certificate":false},{"name":"exists_infinite_support_half_of_cofinalProtectedEvenSeamRealization","module":"Erdos249257/HalfCarryProtectedSeamConsumer.lean","line":53,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarryProtectedSeamConsumer.lean#L53","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryProtectedSeamConsumer.lean#L53","kind":"theorem","signature":"theorem exists_infinite_support_half_of_cofinalProtectedEvenSeamRealization (hsupply : HalfCarryCofinalProtectedEvenSeamRealization) : ∃ A : Set ℕ, A.Infinite ∧ erdosSupportSeries 2 A = (1 : ℝ) / 2","docstring":"Localized seam realization is already a complete arithmetic producer for an infinite Mersenne support summing to `1/2`.","generated_certificate":false}],"paper_label":"res:protectedseam","paper_anchor":{"paper_label":"res:protectedseam","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1681,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Localised protected seams","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Finite-to-unbounded-depth criteria: windows, seams, and compactness","title_tex":"Finite-to-unbounded-depth criteria: windows, seams, and compactness","line":1609},{"level":"subsubsection","title":"Localised protected seams","title_tex":"Localised protected seams","line":1681}],"source_excerpt":{"text":"\\subsubsection{Localised protected seams}\\label{res:protectedseam}","media_type":"text/x-tex","source_char_start":87548,"source_char_end":87614,"start_line":1681,"end_line":1681,"truncated":false,"sha256":"sha256:46e643fbf68601e423d797436775797a770074288c46bb3e9506f4605ce0d315","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1681","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"ProtectedEvenSeamRealizedAt","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryProtectedSeamConsumer.lean#L18","cite_this":true},{"kind":"lean_declaration_pinned","label":"cofinalAdmissibility_of_cofinalProtectedEvenSeamRealization","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryProtectedSeamConsumer.lean#L31","cite_this":true},{"kind":"lean_declaration_pinned","label":"exists_infinite_support_half_of_cofinalProtectedEvenSeamRealization","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryProtectedSeamConsumer.lean#L53","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/34","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"localized_protected_seam_compactness"}}],"expect":"object"},"prior_art":{"family_id":"rewind_windows_and_protected_seams","status_summary":"Conditional rewind/window consumers plus one verified depth-18 base.","prior_art_posture":"No theorem-level match is recorded; any novelty posture remains provisional.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/7","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2},"edges":[{"semantic_node_id":"Z15::cofinal_protected_seam_consumer","prior_art_state":"not_assessed"},{"semantic_node_id":"Z15::protected_even_seam_realization_interface","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:protectedseam","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1681","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:mixed_cofinal_half_carry_fan_in","assertion_kind":"registered_claim","source_id":"mixed_cofinal_half_carry_fan_in","reviewed":true,"granularity":"single_registered_claim","label":"Mixed cofinal half-carry fan-in","statement":"For every N, suppose there is M >= N carrying either a selected protected half-carry window or a localized realized one-hole seam. The window branch chooses carry 1; the seam branch chooses 4 when its unique hole is 3 and 3 otherwise. Either branch supplies one admissible finite word, and the branches may alternate with N. Hence cofinal admissibility and, by compactness, an infinite support A with erdosSupportSeries 2 A = 1/2 follow. This mixed cofinal hypothesis is not proved, so Erdős #257 remains open.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"HalfCarryCofinalSelectedWindowOrProtectedSeam","module":"Erdos249257/HalfCarryCofinalWindowOrSeamConsumer.lean","line":25,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarryCofinalWindowOrSeamConsumer.lean#L25","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryCofinalWindowOrSeamConsumer.lean#L25","kind":"def","signature":"def HalfCarryCofinalSelectedWindowOrProtectedSeam : Prop","docstring":"Cofinal mixed producer: a full selected window or a localized realized seam may witness each requested scale.","generated_certificate":false},{"name":"cofinalAdmissibility_of_cofinalSelectedWindowOrProtectedSeam","module":"Erdos249257/HalfCarryCofinalWindowOrSeamConsumer.lean","line":32,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarryCofinalWindowOrSeamConsumer.lean#L32","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryCofinalWindowOrSeamConsumer.lean#L32","kind":"theorem","signature":"theorem cofinalAdmissibility_of_cofinalSelectedWindowOrProtectedSeam (hsupply : HalfCarryCofinalSelectedWindowOrProtectedSeam) : HalfCarryCofinalAdmissibility","docstring":"Every mixed cofinal receipt contains the one admissible word required by the landed inverse-system compactness theorem.","generated_certificate":false},{"name":"exists_infinite_support_half_of_cofinalSelectedWindowOrProtectedSeam","module":"Erdos249257/HalfCarryCofinalWindowOrSeamConsumer.lean","line":58,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarryCofinalWindowOrSeamConsumer.lean#L58","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryCofinalWindowOrSeamConsumer.lean#L58","kind":"theorem","signature":"theorem exists_infinite_support_half_of_cofinalSelectedWindowOrProtectedSeam (hsupply : HalfCarryCofinalSelectedWindowOrProtectedSeam) : ∃ A : Set ℕ, A.Infinite ∧ erdosSupportSeries 2 A = (1 : ℝ) / 2","docstring":"The mixed cofinal producer is already sufficient for an infinite support whose Mersenne achievement sum is exactly `1 / 2`.","generated_certificate":false}],"paper_label":"res:cofinalwindowseam","paper_anchor":{"paper_label":"res:cofinalwindowseam","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1693,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Combining the two unbounded-depth half-carry criteria","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Finite-to-unbounded-depth criteria: windows, seams, and compactness","title_tex":"Finite-to-unbounded-depth criteria: windows, seams, and compactness","line":1609},{"level":"subsubsection","title":"Combining the two unbounded-depth half-carry criteria","title_tex":"Combining the two unbounded-depth half-carry criteria","line":1693}],"source_excerpt":{"text":"\\subsubsection{Combining the two unbounded-depth half-carry criteria}\\label{res:cofinalwindowseam}","media_type":"text/x-tex","source_char_start":88380,"source_char_end":88478,"start_line":1693,"end_line":1693,"truncated":false,"sha256":"sha256:42703d66a1cc33ee8043f246dc57be6b4787eca0b2ceb17bae956d3da15581fc","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z13::fatal_gap_eventual_right_classification","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:fatalright","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1251","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:twenty_one_quotient_greedy_frontier","assertion_kind":"registered_claim","source_id":"twenty_one_quotient_greedy_frontier","reviewed":true,"granularity":"single_registered_claim","label":"Canonical quotient-greedy and scaled-remainder frontier at 1/21","statement":"Membership of 1/21 in the Mersenne achievement set is equivalent both to excluding one explicit fatal/cofinite/aligned branch and to cofinally bounded scaled actual greedy remainders. Every closed Boolean quotient row is the deterministic quotient-greedy row; any unbounded sequence of canonical rows with remainder at most 2^R gives membership. On the complementary fatal branch the remainder is eventually strictly supercapacity, the boundary rank is always selected, and support and remainder follow one exact affine recurrence. A final skipped exponent also forces a structured one-sided approximation to the Erdős–Borwein constant with exact denominator-order constraints. These results isolate permanent affine supercapacity as the surviving late regime; they do not contradict it or prove 1/21 membership.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-257-mersenne-support-subseries","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"twentyOneEvenQuotientGreedySupport","module":"Erdos249257/TwentyOneQuotientGreedy.lean","line":32,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L32","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L32","kind":"def","signature":"def twentyOneEvenQuotientGreedySupport (R : ℕ) : Finset ℕ","docstring":"The deterministic finite support obtained from the quotient-greedy word at even depth `2R`.","generated_certificate":false},{"name":"twentyOneEvenQuotientGreedyRemainder","module":"Erdos249257/TwentyOneQuotientGreedy.lean","line":39,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L39","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L39","kind":"def","signature":"def twentyOneEvenQuotientGreedyRemainder (R : ℕ) : ℕ","docstring":"The terminal scalar state of the same quotient-greedy row.","generated_certificate":false},{"name":"twentyOneGreedyDefect","module":"Erdos249257/BooleanMobiusCarry.lean","line":1721,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/BooleanMobiusCarry.lean#L1721","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/BooleanMobiusCarry.lean#L1721","kind":"def","signature":"noncomputable def twentyOneGreedyDefect (N : ℕ) : ℕ","docstring":"The genuinely dynamical part of the denominator-`21` carry, after removing the six-periodic residue of `2^N mod 21`. Unlike the raw carry, this coordinate starts at zero and is an ordinary natural number.","generated_certificate":false},{"name":"one_div_twenty_one_mem_iff_scaledGreedyRemainderCofinallyBounded","module":"Erdos249257/BooleanMobiusCarry.lean","line":2716,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/BooleanMobiusCarry.lean#L2716","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/BooleanMobiusCarry.lean#L2716","kind":"theorem","signature":"theorem one_div_twenty_one_mem_iff_scaledGreedyRemainderCofinallyBounded : (1 / 21 : ℝ) ∈ mersenneAchievementSet ↔ TwentyOneScaledGreedyRemainderCofinallyBounded","docstring":"**Exact cofinal scaled-remainder characterization.** Membership at `1/21` is equivalent to the scaled greedy remainder returning cofinally to one fixed bounded interval. On the surviving branch the scaled remainder is bounded by the scaled complete tail, which tends to `1`. On the fatal branch one fixed positive tail excess is conserved, so multiplication by `2^N` eventually exceeds every proposed ceiling.","generated_certificate":false},{"name":"one_div_twenty_one_mem_mersenneAchievementSet_of_defect_le_one_cofinally","module":"Erdos249257/BooleanMobiusCarry.lean","line":2843,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/BooleanMobiusCarry.lean#L2843","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/BooleanMobiusCarry.lean#L2843","kind":"theorem","signature":"theorem one_div_twenty_one_mem_mersenneAchievementSet_of_defect_le_one_cofinally (hreturn : ∀ N : ℕ, ∃ M : ℕ, N ≤ M ∧ twentyOneGreedyDefect M ≤ 1) : (1 / 21 : ℝ) ∈ mersenneAchievementSet","docstring":"The concrete band seen by exact computation is `Q_N ≤ 1`. Proving that it recurs cofinally is already enough; no global bound on the defect or on the gaps between returns is required.","generated_certificate":false},{"name":"one_div_twenty_one_mem_iff_not_fatalAlignedBranch","module":"Erdos249257/TwentyOneQuotientGreedy.lean","line":3507,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L3507","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L3507","kind":"theorem","signature":"theorem one_div_twenty_one_mem_iff_not_fatalAlignedBranch : (1 / 21 : ℝ) ∈ mersenneAchievementSet ↔ ¬ TwentyOneFatalAlignedBranch","docstring":"Consequently the prescribed point is represented exactly when the fatal/cofinite/aligned branch is impossible.","generated_certificate":false},{"name":"lastTwentyOneSkip_oddDoublingOrder_eq_lcm","module":"Erdos249257/TwentyOneQuotientGreedy.lean","line":3569,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L3569","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L3569","kind":"theorem","signature":"theorem lastTwentyOneSkip_oddDoublingOrder_eq_lcm {M : ℕ} (hlast : IsLastTwentyOneGreedySkip M) : oddDoublingOrder (finiteErdosSum (twentyOneSkippedPrefix M) 2).den (finiteErdosSum_den_odd (twentyOneSkippedPrefix M) (zero_not_mem_twentyOneSkippedPrefix M)) = (twentyOneSkippedPrefix M).lcm id","docstring":"The exact reduced denominator of the final skipped-prefix sum has binary order equal to the skipped-exponent LCM.","generated_certificate":false},{"name":"lastTwentyOneSkip_erdosBorwein_fatalInterval","module":"Erdos249257/TwentyOneQuotientGreedy.lean","line":3735,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L3735","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L3735","kind":"theorem","signature":"theorem lastTwentyOneSkip_erdosBorwein_fatalInterval {M : ℕ} (hlast : IsLastTwentyOneGreedySkip M) : 0 < (1 / 21 : ℝ) + positiveMersenneSupportValue (↑(twentyOneSkippedPrefix M) : Set ℕ) - erdosBorweinMersenneConstant ∧ (1 / 21 : ℝ) + positiveMersenneSupportValue (↑(twentyOneSkippedPrefix M) : Set ℕ) - erdosBorweinMersenneConstant < mersenneGap M","docstring":"A final denominator-`21` skip gives an exact one-sided approximation to the Erdős--Borwein constant, with error strictly smaller than the superincreasing gap at the final omitted exponent.","generated_certificate":false},{"name":"twentyOneCofinalEvenQuotientGreedyDecay_of_closedRows","module":"Erdos249257/TwentyOneQuotientGreedy.lean","line":5554,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L5554","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L5554","kind":"theorem","signature":"theorem twentyOneCofinalEvenQuotientGreedyDecay_of_closedRows {R : ℕ → ℕ} (hR : Tendsto R atTop atTop) (hrow : ∀ k : ℕ, 2 ≤ R k ∧ twentyOneEvenQuotientGreedyRemainder (R k) ≤ 2 ^ (R k)) : TwentyOneCofinalEvenQuotientGreedyDecay","docstring":"Closed canonical rows along any unbounded sequence already give the cap-free decay required by compactness. Strictness is irrelevant at this stage: the quotient normalization is quadratic in the binary scale.","generated_certificate":false},{"name":"twentyOneFatalAlignedBranch_eventually_affine_supercapacity","module":"Erdos249257/TwentyOneQuotientGreedy.lean","line":5658,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L5658","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L5658","kind":"theorem","signature":"theorem twentyOneFatalAlignedBranch_eventually_affine_supercapacity (hbranch : TwentyOneFatalAlignedBranch) : ∃ K : ℕ, ∀ R : ℕ, K ≤ R → twentyOneEvenQuotientGreedySupport (R + 1) = insert (R + 1) (twentyOneEvenQuotientGreedySupport R) ∧ twentyOneEvenQuotientGreedyRemainder (R + 1) = (4 * twentyOneEvenQuotientGreedyRemainder R + twentyOneTargetTwoStepPulse (2 * R) - localPrefixTwoStepPulse (twentyOneEvenQuotientGreedySupport R) (2 * R)) - (2 ^ (R + 1) + 1)","docstring":"**Affine normal form of the only remaining fatal regime.** Once the canonical remainder has escaped strictly above `2^R`, its exponential margin dominates the entire linear divisor pulse. Consequently the new boundary coin always fits: the support appends `R+1`, and the scalar follows one literal affine recurrence with no residual Boolean branch.","generated_certificate":false}],"paper_label":"res:one-over-twenty-one-frontier","paper_anchor":{"paper_label":"res:one-over-twenty-one-frontier","owner_source":"paper/257/erdos-257-mersenne-support-subseries.tex","source_line":1267,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":1266,"anchor_title":"integer-quotient tests for 1/21","anchor_title_tex":"integer-quotient tests for $1/21$","section_title":"Integer quotients for 1/21.","section_path":[{"level":"section","title":"Integer quotients, approximation and further questions","title_tex":"Integer quotients, approximation and further questions","line":1221},{"level":"paragraph","title":"Integer quotients for 1/21.","title_tex":"Integer quotients for $1/21$.","line":1226}],"source_excerpt":{"text":"\\begin{theorem}[integer-quotient tests for $1/21$]\n\\label{res:one-over-twenty-one-frontier}\nThe following statements hold.\n\\begin{enumerate}\n\\item $1/21\\in\\Ach$ if and only if $\\mathcal F_{21}$ does not hold.\n\\item If there is an unbounded sequence of ranks $R$ with $s_R\\le 2^R$,\n  then $1/21\\in\\Ach$.\n\\item On $\\mathcal F_{21}$, eventually $s_R>2^R$, the boundary rank $R+1$\n  belongs to $D_{R+1}$, and, for all sufficiently large $R$,\n\\[\n \\begin{aligned}\n D_{R+1}&=D_R\\cup\\{R+1\\},\\\\\n s_{R+1}&=4s_R+\n \\left\\lfloor\\frac{4(2^{2R}\\bmod21)}{21}\\right\\rfloor\n -2c_{D_R}(2R+1)-c_{D_R}(2R+2)\\\\\n &\\hspace{3em}{}-(2^{R+1}+1).\n \\end{aligned}\n\\]\nHere $c_{D_R}(m)=\\#\\{d\\in D_R:d\\mid m\\}$.\n\\end{enumerate}\n\\end{theorem}","media_type":"text/x-tex","source_char_start":59665,"source_char_end":60373,"start_line":1266,"end_line":1286,"truncated":false,"sha256":"sha256:c45560e732a799ae9896dca0fae21ddade98434a00d144472efc2c0dafbd94fb","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/257/erdos-257-mersenne-support-subseries.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/257/erdos-257-mersenne-support-subseries.tex#L1267","owner_problem_erdos_number":257,"owner_paper_id":"erdos-257-mersenne-support-subseries","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-257-mersenne-support-subseries.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-257-mersenne-support-subseries.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/257/erdos-257-mersenne-support-subseries.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.twenty_one_permanent_affine_supercapacity","statement":"Decide the exact 1/21 branch: equivalently, exclude the fatal/cofinite/aligned branch, contradict its eventual permanent affine-supercapacity recurrence, prove cofinally bounded scaled greedy remainders, or force an unbounded sequence of closed canonical quotient rows. A positive membership result would refute universal #257; no such producer is proved.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/5"},{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"twentyOneEvenQuotientGreedySupport","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L32","cite_this":true},{"kind":"lean_declaration_pinned","label":"twentyOneEvenQuotientGreedyRemainder","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L39","cite_this":true},{"kind":"lean_declaration_pinned","label":"twentyOneGreedyDefect","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/BooleanMobiusCarry.lean#L1721","cite_this":true},{"kind":"lean_declaration_pinned","label":"one_div_twenty_one_mem_iff_scaledGreedyRemainderCofinallyBounded","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/BooleanMobiusCarry.lean#L2716","cite_this":true},{"kind":"lean_declaration_pinned","label":"one_div_twenty_one_mem_mersenneAchievementSet_of_defect_le_one_cofinally","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/BooleanMobiusCarry.lean#L2843","cite_this":true},{"kind":"lean_declaration_pinned","label":"one_div_twenty_one_mem_iff_not_fatalAlignedBranch","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L3507","cite_this":true},{"kind":"lean_declaration_pinned","label":"lastTwentyOneSkip_oddDoublingOrder_eq_lcm","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L3569","cite_this":true},{"kind":"lean_declaration_pinned","label":"lastTwentyOneSkip_erdosBorwein_fatalInterval","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L3735","cite_this":true},{"kind":"lean_declaration_pinned","label":"twentyOneCofinalEvenQuotientGreedyDecay_of_closedRows","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L5554","cite_this":true},{"kind":"lean_declaration_pinned","label":"twentyOneFatalAlignedBranch_eventually_affine_supercapacity","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TwentyOneQuotientGreedy.lean#L5658","cite_this":true},{"kind":"paper_pdf","label":"erdos-257-mersenne-support-subseries","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-257-mersenne-support-subseries.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/39","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"twenty_one_quotient_greedy_frontier"}}],"expect":"object"},"prior_art":{"family_id":"achievement_geometry_and_half_target_interface","status_summary":"Classical geometry formalised, exact local bands, a conditional second channel, and scoped no-go boundaries.","prior_art_posture":"Kovač–Tao supply the strict-tail subsum geometry; the classical Lambert–Möbius identity is the baseline for the scoped sign-support no-go; Crandall and Campbell concern adjacent binary-digit questions and are not proof inputs. No priority claim is made for the sign separation.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/4","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex"},"source_lane":"registry_claim"},{"assertion_id":"claim:final_middle_cell_escape","assertion_kind":"registered_claim","source_id":"final_middle_cell_escape","reviewed":true,"granularity":"single_registered_claim","label":"Upper and first middle producers cannot feed an all-right tail","statement":"At a putative last false rank D ≥ 13, the upper producer is impossible unconditionally. On the middle branch, let u be the finite support below D and B_D = u union {n : n > D} the lazy support that skips D and selects every later rank. Its value is V(u) + T_D < 1/2, while the exact carry swap sends transition carry C_D to centred carry C_D + 3 at row 2D+1. Thus C_D = -3 gives centred carry zero; the next coefficient rows have at least one and two forced divisors and make the carry negative by row 2D+3, contradicting finite-level nonnegativity below 1/2. This removes the upper branch and one middle cell, not the right-tail branch or the remaining cells -2 and -1.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"upperProducer_not_last","module":"Erdos249257/HalfCylinderLastProducerContradiction.lean","line":299,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderLastProducerContradiction.lean#L299","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderLastProducerContradiction.lean#L299","kind":"theorem","signature":"theorem upperProducer_not_last (D : ℕ) (hD13 : 13 ≤ D) (hcarry : (seamAdjacentCut D (by omega)).successorCarries) (htrue : ∀ (s : ℕ) (hDs : D < s), SeamRowWord.terminal (by omega) (seamGreedyWord (s + 1)) = true) : False","generated_certificate":false},{"name":"middleProducer_neg_three_not_last","module":"Erdos249257/HalfCylinderLastProducerContradiction.lean","line":364,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderLastProducerContradiction.lean#L364","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderLastProducerContradiction.lean#L364","kind":"theorem","signature":"theorem middleProducer_neg_three_not_last (D : ℕ) (hD13 : 13 ≤ D) (hncarry : ¬ (seamAdjacentCut D (by omega)).successorCarries) (hmiddle : 4 * (seamAdjacentCut D (by omega)).remainder + (seamPerturbedFamily D (by omega)).gap - (seamAdjacentCut D (by omega)).belowPulse < (seamAdjacentCut D (by omega)).terminalWeight) (htrue : ∀ (s : ℕ) (hDs : D < s), SeamRowWord.terminal (by omega) (seamGreedyWord (s + 1)) = true) (hcell : 4 * ((seamAdjacentCut D (by omega)).remainder : ℤ) - ((seamAdjacentCut D (by omega)).belowPulse : ℤ) - 4 = -3) :","docstring":"The new lazy-tail carry bridge removes the first exceptional middle cell directly in the last-false-terminal coordinates used by this module. Keeping this adapter beside the global fan-in makes the downstream route discoverable without asking a later proof to reconstruct `hright`.","generated_certificate":false},{"name":"finalMiddleCell_neg_three_not_last","module":"Erdos249257/HalfCylinderFinalMiddleCellEscape.lean","line":587,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderFinalMiddleCellEscape.lean#L587","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderFinalMiddleCellEscape.lean#L587","kind":"theorem","signature":"theorem finalMiddleCell_neg_three_not_last (D : ℕ) (hD13 : 13 ≤ D) (hncarry : ¬ (seamAdjacentCut D (by omega)).successorCarries) (hmiddle : 4 * (seamAdjacentCut D (by omega)).remainder + (seamPerturbedFamily D (by omega)).gap - (seamAdjacentCut D (by omega)).belowPulse < (seamAdjacentCut D (by omega)).terminalWeight) (hright : ∀ s : ℕ, D + 1 ≤ s → seamGreedyWord (s + 1) = (seamGreedyWord s).extend true) (hcell : 4 * ((seamAdjacentCut D (by omega)).remainder : ℤ) - ((seamAdjacentCut D (by omega)).belowPulse : ℤ) - 4 = -3) : False","docstring":"The first of the three exceptional final-middle cells, coordinate `-3`, cannot precede an all-right tail. The finite producer cell becomes centred charge zero after the exact lazy-tail swap, while the same all-right word puts the resulting cofinite support strictly below one half.","generated_certificate":false}],"paper_label":"res:finalskipexclusions","paper_anchor":{"paper_label":"res:finalskipexclusions","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1266,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":1266,"anchor_title":"excluded final-skip branches","anchor_title_tex":"excluded final-skip branches","section_title":"The final-skip reduction","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"The final-skip reduction","title_tex":"The final-skip reduction","line":1180}],"source_excerpt":{"text":"\\begin{proposition}[excluded final-skip branches]\\label{res:finalskipexclusions}\nAt a final skipped exponent \\(D\\geq13\\), the upper successor is impossible.\nOn the middle branch, the only carry values outside the ranges\n\\(C_D\\leq-4\\) and \\(C_D\\geq0\\) are \\(C_D\\in\\{-3,-2,-1\\}\\).  At a final\nskip, the value \\(C_D=-3\\) is impossible; the values \\(-2\\) and \\(-1\\)\nremain unresolved.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":64450,"source_char_end":64848,"start_line":1266,"end_line":1272,"truncated":false,"sha256":"sha256:b54b40b88d7945773f4b643e8d7f92ca0a2cda80f1ced811957f3e22edfe8a0c","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1266","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.half_value_membership","statement":"Decide whether 1/2 lies in the Mersenne achievement set. Yes refutes universal #257; no closes only this route.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/4"}],"links":[{"kind":"lean_declaration_pinned","label":"upperProducer_not_last","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderLastProducerContradiction.lean#L299","cite_this":true},{"kind":"lean_declaration_pinned","label":"middleProducer_neg_three_not_last","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderLastProducerContradiction.lean#L364","cite_this":true},{"kind":"lean_declaration_pinned","label":"finalMiddleCell_neg_three_not_last","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderFinalMiddleCellEscape.lean#L587","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/40","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"final_middle_cell_escape"}}],"expect":"object"},"prior_art":{"family_id":"fatal_gap_and_last_producer_spine","status_summary":"Exact half-membership/nonmembership classifications, proved branch eliminations, a rank-26 phase sieve for the middle -2 cell, and explicit conditional tail consumers.","prior_art_posture":"No exact local seam or last-producer antecedent is recorded beyond strict-tail achievement-set background.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/8","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"routine_corollary_or_specialisation":2},"edges":[{"semantic_node_id":"Z14::final_middle_cell_neg_three_impossible","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z14::upper_last_producer_impossible","prior_art_state":"routine_corollary_or_specialisation"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:finalskipexclusions","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1266","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:final_middle_neg_two_phase_sieve","assertion_kind":"registered_claim","source_id":"final_middle_neg_two_phase_sieve","reviewed":true,"granularity":"single_registered_claim","label":"Rank-26 phase sieve for the final middle -2 cell","statement":"For every seam row s ≥ 27, the greedy decisions through rank 26 select exactly ranks 2, 3, 6, 7, 14, 20, 21, and 26. Consequently, in a hypothetical final middle -2 cell at rank D ≥ 27 followed by an all-right suffix, D mod 21 lies in {11, 14, 17, 20}, D mod 10 is not 8, the mod-10 class 7 cannot pair with the mod-21 class 11, D mod 13 is not 11, and the mod-13 class 10 cannot pair with the mod-21 class 11. Exactly 412 of the 2730 joint residue classes survive. This is a finite necessary condition, not an exclusion; even a later survivor density tending to zero would not suffice without finite extinction or an Archimedean contradiction. The -1 and right-tail branches remain open.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"seamGreedy_postTwentyOne_decisions_through_twentySix","module":"Erdos249257/HalfFinalMiddlePhaseSieve.lean","line":699,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfFinalMiddlePhaseSieve.lean#L699","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfFinalMiddlePhaseSieve.lean#L699","kind":"theorem","signature":"theorem seamGreedy_postTwentyOne_decisions_through_twentySix (s : ℕ) (hs : 27 ≤ s) : let R","docstring":"After rank `21`, rows `s ≥ 27` skip ranks `22` through `25` and take rank `26`. The exact reciprocal comparison at rank `25` has excess `4303731711383 / 614097164000829974850`, while adjoining rank `26` leaves deficit `100171945053757 / 12691341578466215591850`.","generated_certificate":false},{"name":"seamGreedyBits_head2367_14_20_21_26","module":"Erdos249257/HalfFinalMiddlePhaseSieve.lean","line":899,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfFinalMiddlePhaseSieve.lean#L899","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfFinalMiddlePhaseSieve.lean#L899","kind":"theorem","signature":"theorem seamGreedyBits_head2367_14_20_21_26 (s : ℕ) (hs : 27 ≤ s) : integerGreedyBits (seamWeights s) (seamSubsetTarget s) = true :: true :: false :: false :: true :: true :: false :: false :: false :: false :: false :: false :: true :: false :: false :: false :: false :: false :: true :: true :: false :: false :: false :: false :: true :: integerGreedyBits (seamWeightsFrom s 27) (seamSubsetTarget s - truncatedMersenneWeight s 2 - truncatedMersenneWeight s 3 - truncatedMersenneWeight s 6 - truncatedMersenneWeight s 7 - truncatedMersenneWeight s 14 - truncatedMersenneWeight s 20 - truncatedMersenneWeight s 21 - truncatedMersenneWeight s 26)","docstring":"List-level normal form of the forced seam decisions through rank `26`.","generated_certificate":false},{"name":"finalMiddleCell_neg_two_forces_phase_refinement_through_twentySix","module":"Erdos249257/HalfFinalMiddlePhaseSieve.lean","line":1622,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfFinalMiddlePhaseSieve.lean#L1622","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfFinalMiddlePhaseSieve.lean#L1622","kind":"theorem","signature":"theorem finalMiddleCell_neg_two_forces_phase_refinement_through_twentySix (D : ℕ) (hD27 : 27 ≤ D) (hncarry : ¬ (seamAdjacentCut D (by omega)).successorCarries) (hmiddle : 4 * (seamAdjacentCut D (by omega)).remainder + (seamPerturbedFamily D (by omega)).gap - (seamAdjacentCut D (by omega)).belowPulse < (seamAdjacentCut D (by omega)).terminalWeight) (hright : ∀ s : ℕ, D + 1 ≤ s → seamGreedyWord (s + 1) = (seamGreedyWord s).extend true) (hcell : 4 * ((seamAdjacentCut D (by omega)).remainder : ℤ) - ((seamAdjacentCut D (by omega)).belowPulse : ℤ) - 4 = -2) : D % 21 ∈ ({11, 14, 17, 20} : Finset ℕ) ∧","docstring":"The next forced rank, `26`, removes every surviving class with `D = 11 (mod 13)`. In the adjacent `D = 10 (mod 13)` phase it also removes the full `D = 11 (mod 21)` slice. Thus the hypothetical final `-2` middle cell is confined to 412 of the 2730 joint phases.","generated_certificate":false},{"name":"finalMiddleTwentySixPhaseSurvivors_card","module":"Erdos249257/HalfFinalMiddlePhaseSieve.lean","line":1759,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfFinalMiddlePhaseSieve.lean#L1759","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfFinalMiddlePhaseSieve.lean#L1759","kind":"theorem","signature":"theorem finalMiddleTwentySixPhaseSurvivors_card : finalMiddleTwentySixPhaseSurvivors.card = 412","docstring":"The rank-`26` refinement leaves exactly 412 of the 2730 joint phases.","generated_certificate":false}],"paper_label":"res:finalskipexclusions","paper_anchor":{"paper_label":"res:finalskipexclusions","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1266,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":1266,"anchor_title":"excluded final-skip branches","anchor_title_tex":"excluded final-skip branches","section_title":"The final-skip reduction","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"The final-skip reduction","title_tex":"The final-skip reduction","line":1180}],"source_excerpt":{"text":"\\begin{proposition}[excluded final-skip branches]\\label{res:finalskipexclusions}\nAt a final skipped exponent \\(D\\geq13\\), the upper successor is impossible.\nOn the middle branch, the only carry values outside the ranges\n\\(C_D\\leq-4\\) and \\(C_D\\geq0\\) are \\(C_D\\in\\{-3,-2,-1\\}\\).  At a final\nskip, the value \\(C_D=-3\\) is impossible; the values \\(-2\\) and \\(-1\\)\nremain unresolved.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":64450,"source_char_end":64848,"start_line":1266,"end_line":1272,"truncated":false,"sha256":"sha256:b54b40b88d7945773f4b643e8d7f92ca0a2cda80f1ced811957f3e22edfe8a0c","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Yes refutes universal #257; no closes only this route.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/4"}],"links":[{"kind":"lean_declaration_pinned","label":"seamGreedy_postTwentyOne_decisions_through_twentySix","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfFinalMiddlePhaseSieve.lean#L699","cite_this":true},{"kind":"lean_declaration_pinned","label":"seamGreedyBits_head2367_14_20_21_26","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfFinalMiddlePhaseSieve.lean#L899","cite_this":true},{"kind":"lean_declaration_pinned","label":"finalMiddleCell_neg_two_forces_phase_refinement_through_twentySix","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfFinalMiddlePhaseSieve.lean#L1622","cite_this":true},{"kind":"lean_declaration_pinned","label":"finalMiddleTwentySixPhaseSurvivors_card","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfFinalMiddlePhaseSieve.lean#L1759","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/41","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"final_middle_neg_two_phase_sieve"}}],"expect":"object"},"prior_art":{"family_id":"fatal_gap_and_last_producer_spine","status_summary":"Exact half-membership/nonmembership classifications, proved branch eliminations, a rank-26 phase sieve for the middle -2 cell, and explicit conditional tail consumers.","prior_art_posture":"No exact local seam or last-producer antecedent is recorded beyond strict-tail achievement-set background.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/8","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"exact_local_id","state_counts":{"candidate_new_statement":1},"edges":[{"semantic_node_id":"Z14::final_middle_neg_two_phase_sieve","prior_art_state":"candidate_new_statement"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:finalskipexclusions","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"requires_mathematical_adjudication","render_policy":"historical_only_with_recorded_difference","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1266","finding_axes":["negative_clause_loss"],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:last_producer_tail_escape_reduction","assertion_kind":"registered_claim","source_id":"last_producer_tail_escape_reduction","reviewed":true,"granularity":"single_registered_claim","label":"Last-producer tail-escape reduction","statement":"If every actual middle transition at D ≥ 13 other than the discharged cell C_D = -3 satisfies Theta_D < C_D, then an eventual right seam is impossible and 1/2 belongs to the Mersenne achievement set. Since Theta_D ≥ 0, this leaves two open inputs: exclude C_D = -2,-1 and prove Theta_D < C_D at every remaining transition with C_D ≥ 0. A square-root producer lower bound is a separate sufficient hypothesis. None of these arithmetic inputs is proved.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"half_mem_mersenneAchievementSet_of_middleProducerTailEscapeExceptNegThree","module":"Erdos249257/HalfCylinderLastProducerContradiction.lean","line":436,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderLastProducerContradiction.lean#L436","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderLastProducerContradiction.lean#L436","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_of_middleProducerTailEscapeExceptNegThree (hmiddleEscape : SeamMiddleProducerTailEscapeExceptNegThree) : (1 / 2 : ℝ) ∈ mersenneAchievementSet","docstring":"The reduced middle socket, with cell `-3` discharged locally, rules out an eventual right seam tail and proves the Erdős #257 endpoint.","generated_certificate":false},{"name":"half_mem_mersenneAchievementSet_of_middleProducerSqrtEscape","module":"Erdos249257/HalfCylinderLastProducerContradiction.lean","line":458,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderLastProducerContradiction.lean#L458","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderLastProducerContradiction.lean#L458","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_of_middleProducerSqrtEscape (hsqrt : SeamMiddleProducerSqrtEscape) : (1 / 2 : ℝ) ∈ mersenneAchievementSet","docstring":"It is enough to prove the square-root middle-producer lower bound.","generated_certificate":false}],"paper_label":"res:taildominance","paper_anchor":{"paper_label":"res:taildominance","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1283,"anchor_kind":"environment","anchor_unit":"corollary","anchor_unit_line":1283,"anchor_title":"tail-dominance membership criterion","anchor_title_tex":"tail-dominance membership criterion","section_title":"The final-skip reduction","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"The final-skip reduction","title_tex":"The final-skip reduction","line":1180}],"source_excerpt":{"text":"\\begin{corollary}[tail-dominance membership criterion]\\label{res:taildominance}\nIf every middle transition at a row \\(D\\geq13\\) with \\(C_D\\neq-3\\) satisfies\n\\[\n \\Theta_D<C_D, \\tag{4.2}\n\\]\nthen no final skipped exponent exists, and hence \\(1/2\\in\\mathcal A\\).\n\\end{corollary}","media_type":"text/x-tex","source_char_start":65469,"source_char_end":65743,"start_line":1283,"end_line":1289,"truncated":false,"sha256":"sha256:92b30e44470b4af50698a998a7bce4f4f109842795b85fe69ffbec0af0e0364b","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1283","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.half_value_membership","statement":"Decide whether 1/2 lies in the Mersenne achievement set. Yes refutes universal #257; no closes only this route.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/4"}],"links":[{"kind":"lean_declaration_pinned","label":"half_mem_mersenneAchievementSet_of_middleProducerTailEscapeExceptNegThree","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderLastProducerContradiction.lean#L436","cite_this":true},{"kind":"lean_declaration_pinned","label":"half_mem_mersenneAchievementSet_of_middleProducerSqrtEscape","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderLastProducerContradiction.lean#L458","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/42","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"last_producer_tail_escape_reduction"}}],"expect":"object"},"prior_art":{"family_id":"fatal_gap_and_last_producer_spine","status_summary":"Exact half-membership/nonmembership classifications, proved branch eliminations, a rank-26 phase sieve for the middle -2 cell, and explicit conditional tail consumers.","prior_art_posture":"No exact local seam or last-producer antecedent is recorded beyond strict-tail achievement-set background.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/8","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z14::half_membership_conditional_endpoints","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:taildominance","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1283","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:middle_producer_cardinality_reduction","assertion_kind":"registered_claim","source_id":"middle_producer_cardinality_reduction","reviewed":true,"granularity":"single_registered_claim","label":"Finite-prefix middle-producer cardinality reduction","statement":"Put g_D = #G_D. The augmented finite support A_D = G_D union {D} has g_D + 1 elements, so its complete future divisor-incidence tail satisfies Theta_D ≤ g_D + 1. Since C_D = 4 R_(D,D) - P_D - 4, the inequality g_D + P_D + 5 < 4 R_(D,D) is exactly g_D + 1 < C_D; required at every late middle transition, it excludes the negative cells and proves tail dominance, hence half-membership. The stronger row-scale hypothesis R_(D,D) ≥ D also suffices. Neither hypothesis is proved.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"half_mem_mersenneAchievementSet_of_middleProducerCardEscape","module":"Erdos249257/HalfCylinderMiddleCarryLowerBound.lean","line":2364,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderMiddleCarryLowerBound.lean#L2364","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderMiddleCarryLowerBound.lean#L2364","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_of_middleProducerCardEscape (hcard : SeamMiddleProducerCardEscape) : (1 / 2 : ℝ) ∈ mersenneAchievementSet","docstring":"The exact cardinality-scale middle bound rules out the final middle producer of an eventual right seam and therefore proves the half endpoint.","generated_certificate":false},{"name":"half_mem_mersenneAchievementSet_of_middleProducerRowEscape","module":"Erdos249257/HalfCylinderMiddleCarryLowerBound.lean","line":2371,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderMiddleCarryLowerBound.lean#L2371","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderMiddleCarryLowerBound.lean#L2371","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_of_middleProducerRowEscape (hrow : SeamMiddleProducerRowEscape) : (1 / 2 : ℝ) ∈ mersenneAchievementSet","docstring":"It is enough that every late middle remainder be at least its row index.","generated_certificate":false}],"paper_label":"res:taildominance","paper_anchor":{"paper_label":"res:taildominance","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1283,"anchor_kind":"environment","anchor_unit":"corollary","anchor_unit_line":1283,"anchor_title":"tail-dominance membership criterion","anchor_title_tex":"tail-dominance membership criterion","section_title":"The final-skip reduction","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"The final-skip reduction","title_tex":"The final-skip reduction","line":1180}],"source_excerpt":{"text":"\\begin{corollary}[tail-dominance membership criterion]\\label{res:taildominance}\nIf every middle transition at a row \\(D\\geq13\\) with \\(C_D\\neq-3\\) satisfies\n\\[\n \\Theta_D<C_D, \\tag{4.2}\n\\]\nthen no final skipped exponent exists, and hence \\(1/2\\in\\mathcal A\\).\n\\end{corollary}","media_type":"text/x-tex","source_char_start":65469,"source_char_end":65743,"start_line":1283,"end_line":1289,"truncated":false,"sha256":"sha256:92b30e44470b4af50698a998a7bce4f4f109842795b85fe69ffbec0af0e0364b","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1283","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"half_mem_mersenneAchievementSet_of_middleProducerCardEscape","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderMiddleCarryLowerBound.lean#L2364","cite_this":true},{"kind":"lean_declaration_pinned","label":"half_mem_mersenneAchievementSet_of_middleProducerRowEscape","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderMiddleCarryLowerBound.lean#L2371","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/43","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"middle_producer_cardinality_reduction"}}],"expect":"object"},"prior_art":{"family_id":"fatal_gap_and_last_producer_spine","status_summary":"Exact half-membership/nonmembership classifications, proved branch eliminations, a rank-26 phase sieve for the middle -2 cell, and explicit conditional tail consumers.","prior_art_posture":"No exact local seam or last-producer antecedent is recorded beyond strict-tail achievement-set background.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/8","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z14::half_membership_conditional_endpoints","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:taildominance","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1283","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:upper_reset_band_certificates_through_thirty","assertion_kind":"registered_claim","source_id":"upper_reset_band_certificates_through_thirty","reviewed":true,"granularity":"single_registered_claim","label":"Upper-reset dyadic-band certificates through row 30","statement":"At an upper transition G_(d+1) = H_d, put Q_d = 4 O_d + A_d from the upper overshoot and divisor pulse and B_(d,j) = 2^(d-j+1). For every 13 <= d <= 30 and j <= d, Lean proves B_(d,j) < Q_d or Q_d + 2(d+j) <= B_(d,j), equivalently avoidance of Q_d <= B_(d,j) < Q_d + 2(d+j). For fixed d, one exact successor remainder R_(d+1,d+1) certifies every j. This is bounded evidence and proves no global upper-reset band-escape hypothesis.","status":"verified finite instance","status_meaning":"Kernel-checked computation at a bounded range","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"gateway_section","warrant":"the claim's anchor sits inside the retired gateway section 'Erdős--Borwein-type irrationality (the #257 direction)', which is the author's own partition of that manuscript by problem"},"problem_edges":[{"problem_id":"erdos_257","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"seamUpperResetDyadicBandEscape_through_thirty","module":"Erdos249257/HalfCylinderUpperResetBandCertificates.lean","line":78,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderUpperResetBandCertificates.lean#L78","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderUpperResetBandCertificates.lean#L78","kind":"theorem","signature":"theorem seamUpperResetDyadicBandEscape_through_thirty (d : ℕ) (hd13 : 13 ≤ d) (hd30 : d ≤ 30) (hd5 : 5 ≤ d) (hcarry : (seamAdjacentCut d hd5).successorCarries) : ∀ j : ℕ, j ≤ d → 2 ^ (d - j + 1) < 4 * (seamAdjacentCut d hd5).overshoot + (seamAdjacentCut d hd5).abovePulse ∨ 4 * (seamAdjacentCut d hd5).overshoot + (seamAdjacentCut d hd5).abovePulse + 2 * (d + j) ≤ 2 ^ (d - j + 1)","docstring":"Every upper reset from row `13` through row `30` avoids every danger band relevant to a following right run. One exact successor-remainder certificate is shared across all band indices in each row, rather than recomputing the same greedy orbit separately for every `(d,j)` pair.","generated_certificate":false}],"paper_label":"res:finalskipexclusions","paper_anchor":{"paper_label":"res:finalskipexclusions","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1266,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":1266,"anchor_title":"excluded final-skip branches","anchor_title_tex":"excluded final-skip branches","section_title":"The final-skip reduction","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"The final-skip reduction","title_tex":"The final-skip reduction","line":1180}],"source_excerpt":{"text":"\\begin{proposition}[excluded final-skip branches]\\label{res:finalskipexclusions}\nAt a final skipped exponent \\(D\\geq13\\), the upper successor is impossible.\nOn the middle branch, the only carry values outside the ranges\n\\(C_D\\leq-4\\) and \\(C_D\\geq0\\) are \\(C_D\\in\\{-3,-2,-1\\}\\).  At a final\nskip, the value \\(C_D=-3\\) is impossible; the values \\(-2\\) and \\(-1\\)\nremain unresolved.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":64450,"source_char_end":64848,"start_line":1266,"end_line":1272,"truncated":false,"sha256":"sha256:b54b40b88d7945773f4b643e8d7f92ca0a2cda80f1ced811957f3e22edfe8a0c","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1266","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"seamUpperResetDyadicBandEscape_through_thirty","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderUpperResetBandCertificates.lean#L78","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/44","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"upper_reset_band_certificates_through_thirty"}}],"expect":"object"},"scope_limit":{"text":"Upper transitions at every row 13 <= d <= 30 and every band index j <= d; no row d > 30 is checked.","source":"claims_registry_bounded_domain"},"prior_art":{"family_id":"largest_skip_and_reset_deficit_frontier","status_summary":"Conditional largest-skip propagation, exact pulse geometry, and finite reset-band evidence.","prior_art_posture":"No close theorem-level antecedent is recorded.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/9","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z13::upper_reset_band_escape_rows_13_to_30","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:finalskipexclusions","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1266","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:half_membership_seam_classification","assertion_kind":"registered_claim","source_id":"half_membership_seam_classification","reviewed":true,"granularity":"single_registered_claim","label":"Positive half-membership classification in seam coordinates","statement":"Membership of 1/2 in the Mersenne achievement set is equivalent to false successor terminal bits beyond every bound, to a cofinal sequence of such bits, and to a cofinal integer-seam sequence whose skipped ranks tend to infinity. These equivalences do not establish that any positive producer exists.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":126,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L126","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L126","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse : (1 / 2 : ℝ) ∈ mersenneAchievementSet ↔ SeamGreedyUnboundedTerminalFalse","docstring":"Exact positive complement of the eventual-terminal classification: half membership is equivalent to false successor terminal bits beyond every bound.","generated_certificate":false},{"name":"half_mem_mersenneAchievementSet_iff_unboundedUpperOrMiddle","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":156,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L156","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L156","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_iff_unboundedUpperOrMiddle : (1 / 2 : ℝ) ∈ mersenneAchievementSet ↔ SeamGreedyUnboundedUpperOrMiddle","docstring":"Strongest current arithmetic socket: half membership is exactly cofinal occurrence of the explicit upper-or-middle successor disjunction.","generated_certificate":false},{"name":"unboundedTerminalFalse_iff_not_seamGreedyEventuallyRight","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":164,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L164","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L164","kind":"theorem","signature":"theorem unboundedTerminalFalse_iff_not_seamGreedyEventuallyRight : SeamGreedyUnboundedTerminalFalse ↔ ¬ SeamGreedyEventuallyRight","docstring":"The unbounded-terminal predicate is precisely failure of eventual right extension.","generated_certificate":false},{"name":"cofinalTerminalFalse_of_unboundedTerminalFalse","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":172,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L172","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L172","kind":"theorem","signature":"theorem cofinalTerminalFalse_of_unboundedTerminalFalse (hunbounded : SeamGreedyUnboundedTerminalFalse) : SeamGreedyCofinalTerminalFalse","docstring":"Choice turns the beyond-every-bound form into one cofinal false-terminal sequence.","generated_certificate":false},{"name":"unboundedTerminalFalse_of_cofinalTerminalFalse","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":186,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L186","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L186","kind":"theorem","signature":"theorem unboundedTerminalFalse_of_cofinalTerminalFalse (hcofinal : SeamGreedyCofinalTerminalFalse) : SeamGreedyUnboundedTerminalFalse","docstring":"A cofinal false-terminal sequence is, in particular, unbounded.","generated_certificate":false},{"name":"unboundedTerminalFalse_iff_cofinalTerminalFalse","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":196,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L196","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L196","kind":"theorem","signature":"theorem unboundedTerminalFalse_iff_cofinalTerminalFalse : SeamGreedyUnboundedTerminalFalse ↔ SeamGreedyCofinalTerminalFalse","docstring":"The quantifier and cofinal-sequence forms of terminal-false supply are exactly equivalent.","generated_certificate":false},{"name":"half_mem_mersenneAchievementSet_iff_cofinalTerminalFalse","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":203,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L203","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L203","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_iff_cofinalTerminalFalse : (1 / 2 : ℝ) ∈ mersenneAchievementSet ↔ SeamGreedyCofinalTerminalFalse","docstring":"Cofinal false successor terminals are an exact half-membership endpoint.","generated_certificate":false},{"name":"half_mem_mersenneAchievementSet_iff_exists_unboundedSkippedRanksAlong","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":213,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L213","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L213","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_iff_exists_unboundedSkippedRanksAlong : (1 / 2 : ℝ) ∈ mersenneAchievementSet ↔ ∃ rows : ℕ → ℕ, SeamGreedyUnboundedSkippedRanksAlong rows","docstring":"Half membership is also equivalent to existence of a cofinal seam sequence whose skipped ranks themselves tend to infinity. 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Yes refutes universal #257; no closes only this route.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/4"}],"links":[{"kind":"lean_declaration_pinned","label":"half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L126","cite_this":true},{"kind":"lean_declaration_pinned","label":"half_mem_mersenneAchievementSet_iff_unboundedUpperOrMiddle","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L156","cite_this":true},{"kind":"lean_declaration_pinned","label":"unboundedTerminalFalse_iff_not_seamGreedyEventuallyRight","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L164","cite_this":true},{"kind":"lean_declaration_pinned","label":"cofinalTerminalFalse_of_unboundedTerminalFalse","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L172","cite_this":true},{"kind":"lean_declaration_pinned","label":"unboundedTerminalFalse_of_cofinalTerminalFalse","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L186","cite_this":true},{"kind":"lean_declaration_pinned","label":"unboundedTerminalFalse_iff_cofinalTerminalFalse","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L196","cite_this":true},{"kind":"lean_declaration_pinned","label":"half_mem_mersenneAchievementSet_iff_cofinalTerminalFalse","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L203","cite_this":true},{"kind":"lean_declaration_pinned","label":"half_mem_mersenneAchievementSet_iff_exists_unboundedSkippedRanksAlong","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L213","cite_this":true},{"kind":"lean_declaration_pinned","label":"unboundedTerminalFalse_iff_greedyMersenneSkippedSupport_infinite","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L228","cite_this":true},{"kind":"lean_declaration_pinned","label":"half_mem_mersenneAchievementSet_iff_no_lastHalfGreedySkip","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean#L235","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/45","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"half_membership_seam_classification"}}],"expect":"object"},"prior_art":{"family_id":"fatal_gap_and_last_producer_spine","status_summary":"Exact half-membership/nonmembership classifications, proved branch eliminations, a rank-26 phase sieve for the middle -2 cell, and explicit conditional tail consumers.","prior_art_posture":"No exact local seam or last-producer antecedent is recorded beyond strict-tail achievement-set background.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/8","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. 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The global statement forcing an upper or middle step when 2(s+1) < 3d first fails is not proved.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"three_mul_largestSkipLowerWeight_add_exactLateGap_eq_upperWeight","module":"Erdos249257/HalfCylinderLargestSkipGap.lean","line":259,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderLargestSkipGap.lean#L259","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderLargestSkipGap.lean#L259","kind":"theorem","signature":"theorem three_mul_largestSkipLowerWeight_add_exactLateGap_eq_upperWeight {s d : ℕ} {u : Finset ℕ} (hd2 : 2 ≤ d) (hds : d < s) (hu : ∀ e ∈ u, 2 ≤ e ∧ e < d) (hlate : 2 * s < 3 * d) : 3 * wordWeightSum s (largestSkipLowerWord s d u).toNatWord + (3 * 2 ^ (s + 1) + 2 * 4 ^ (s - d) + 4) = 3 * wordWeightSum s (largestSkipUpperWord s d u).toNatWord","docstring":"Exact late-rank adjacent word gap, including an arbitrary common lower prefix.","generated_certificate":false},{"name":"seamGreedyWord_succ_isLargestFalseRank_terminal_of_upperOrMiddle","module":"Erdos249257/HalfCylinderLargestSkipGap.lean","line":328,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderLargestSkipGap.lean#L328","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderLargestSkipGap.lean#L328","kind":"theorem","signature":"theorem seamGreedyWord_succ_isLargestFalseRank_terminal_of_upperOrMiddle (s : ℕ) (hs : 5 ≤ s) (hUM : SeamGreedyUpperOrMiddleAt s hs) : IsLargestFalseRank (seamGreedyWord (s + 1)) s","docstring":"The combined arithmetic `U/M` predicate makes the successor terminal rank the largest false rank.","generated_certificate":false},{"name":"IsLargestFalseRank.seamGreedyWord_succ_of_rightBranch","module":"Erdos249257/HalfCylinderLargestSkipGap.lean","line":339,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderLargestSkipGap.lean#L339","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderLargestSkipGap.lean#L339","kind":"theorem","signature":"theorem IsLargestFalseRank.seamGreedyWord_succ_of_rightBranch {s d : ℕ} (hs : 5 ≤ s) (hmax : IsLargestFalseRank (seamGreedyWord s) d) (hcarry : ¬ (seamAdjacentCut s hs).successorCarries) (hright : (seamAdjacentCut s hs).terminalWeight ≤ 4 * (seamAdjacentCut s hs).remainder + (seamPerturbedFamily s (by omega)).gap - (seamAdjacentCut s hs).belowPulse) : IsLargestFalseRank (seamGreedyWord (s + 1)) d","docstring":"On the `R` branch, the old largest false rank is preserved.","generated_certificate":false}],"paper_label":"res:largestskip","paper_anchor":{"paper_label":"res:largestskip","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1840,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Largest skipped ranks","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Exact final-skip reduction and its exceptional carries","title_tex":"Exact final-skip reduction and its exceptional carries","line":1724},{"level":"subsubsection","title":"Largest skipped ranks","title_tex":"Largest skipped ranks","line":1840}],"source_excerpt":{"text":"\\subsubsection{Largest skipped ranks}\\label{res:largestskip}","media_type":"text/x-tex","source_char_start":97511,"source_char_end":97571,"start_line":1840,"end_line":1840,"truncated":false,"sha256":"sha256:a635b1f231131a235eb34c36a8e96376f481d1eb302caff007d6994620e0161a","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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What is recorded is that nobody has reconciled the two.","reader_rule":"This row's sentence may be quoted, with the recorded difference beside it. Do not present it as settled either way."},"render_policy":{"schema":"plectis_render_policy_v1","quotation_policy":"quote_with_the_recorded_difference","headline_eligible":true,"instruction":"This row's statement may be quoted, but the recorded difference must travel with it. 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This is the local one-rank form of the eventual-right obstruction.","generated_certificate":false},{"name":"no_lastHalfGreedySkip_iff_every_skip_survives","module":"Erdos249257/HalfCylinderFixedTailSocket.lean","line":57,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderFixedTailSocket.lean#L57","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderFixedTailSocket.lean#L57","kind":"theorem","signature":"theorem no_lastHalfGreedySkip_iff_every_skip_survives : (¬ ∃ M : ℕ, IsLastHalfGreedySkip M) ↔ ∀ M : ℕ, M ∈ greedyMersenneSkippedSupport (1 / 2 : ℝ) → greedyMersenneRemainder (1 / 2 : ℝ) M ≤ mersenneTail M","docstring":"Exact remaining arithmetic target: no final skipped rank iff every actual skip remains on the nonfatal side of the full Mersenne tail.","generated_certificate":false},{"name":"half_mem_iff_every_actual_skip_survives","module":"Erdos249257/HalfCylinderFixedTailSocket.lean","line":73,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderFixedTailSocket.lean#L73","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderFixedTailSocket.lean#L73","kind":"theorem","signature":"theorem half_mem_iff_every_actual_skip_survives : (1 / 2 : ℝ) ∈ mersenneAchievementSet ↔ ∀ M : ℕ, M ∈ greedyMersenneSkippedSupport (1 / 2 : ℝ) → greedyMersenneRemainder (1 / 2 : ℝ) M ≤ mersenneTail M","docstring":"Half membership is equivalently the rank-local statement that every actual skipped decision leaves a residual on the surviving side of the full remaining tail.","generated_certificate":false}],"paper_label":"res:fixedtailsocket","paper_anchor":{"paper_label":"res:fixedtailsocket","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1886,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"The fixed-tail survival criterion","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Exact final-skip reduction and its exceptional carries","title_tex":"Exact final-skip reduction and its exceptional carries","line":1724},{"level":"subsubsection","title":"The fixed-tail survival criterion","title_tex":"The fixed-tail survival criterion","line":1886}],"source_excerpt":{"text":"\\subsubsection{The fixed-tail survival criterion}\\label{res:fixedtailsocket}","media_type":"text/x-tex","source_char_start":100362,"source_char_end":100438,"start_line":1886,"end_line":1886,"truncated":false,"sha256":"sha256:2f274bcfdcdba9383e5c45073d1129ebca6cdd54652b251474608a805775f636","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z13::half_membership_iff_every_skip_survives","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:fixedtailsocket","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1886","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:producer_carry_future_tail_socket","assertion_kind":"registered_claim","source_id":"producer_carry_future_tail_socket","reviewed":true,"granularity":"single_registered_claim","label":"Exact producer-carry residual and sign criteria","statement":"For A contained in {2,...,d}, let f_A(n) count support divisors, let C_(A,d) be the affine integer state after 2d+1 updates from 1, and let Theta_(A,d) be the complete scaled coefficient tail after depth 2d+2. Then 1/2 - X_A = 2^(-(2d+2))(C_(A,d) - Theta_(A,d)). Hence C_(A,d) > Theta_(A,d) iff X_A < 1/2 and C_(A,d) < Theta_(A,d) iff X_A > 1/2; a negative carry is sufficient for the latter because the tail is nonnegative, and carry at most -8 gives a full Mersenne-gap margin. 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The support hypothesis matches an adjacent-above producer prefix, whose ranks are all at most `d`.","generated_certificate":false},{"name":"producerCarry_gt_tail_iff_supportSeries_lt_half","module":"Erdos249257/HalfCylinderProducerCarrySocket.lean","line":48,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderProducerCarrySocket.lean#L48","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderProducerCarrySocket.lean#L48","kind":"theorem","signature":"theorem producerCarry_gt_tail_iff_supportSeries_lt_half (A : Set ℕ) (d : ℕ) (hone : 1 ∉ A) (hA : A ⊆ Set.Iic d) : binaryCoeffTail (supportCoeff A) (2 * d + 2) < (producerCarry A d : ℝ) ↔ erdosSupportSeries 2 A < (1 : ℝ) / 2","docstring":"Exact middle-producer safety criterion: the adjacent-above prefix is below `1/2` iff its positive integer landing carry beats the complete future divisor-incidence tail.","generated_certificate":false},{"name":"producerCarry_lt_tail_iff_half_lt_supportSeries","module":"Erdos249257/HalfCylinderProducerCarrySocket.lean","line":78,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderProducerCarrySocket.lean#L78","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderProducerCarrySocket.lean#L78","kind":"theorem","signature":"theorem producerCarry_lt_tail_iff_half_lt_supportSeries (A : Set ℕ) (d : ℕ) (hone : 1 ∉ A) (hA : A ⊆ Set.Iic d) : (producerCarry A d : ℝ) < binaryCoeffTail (supportCoeff A) (2 * d + 2) ↔ (1 : ℝ) / 2 < erdosSupportSeries 2 A","docstring":"The complementary exact upper-producer criterion.","generated_certificate":false},{"name":"producerCarry_neg_forces_half_lt_supportSeries","module":"Erdos249257/HalfCylinderProducerCarrySocket.lean","line":145,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderProducerCarrySocket.lean#L145","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderProducerCarrySocket.lean#L145","kind":"theorem","signature":"theorem producerCarry_neg_forces_half_lt_supportSeries (A : Set ℕ) (d : ℕ) (hone : 1 ∉ A) (hA : A ⊆ Set.Iic d) (hcarry : producerCarry A d < 0) : (1 : ℝ) / 2 < erdosSupportSeries 2 A","docstring":"A negative upper-producer integer carry is automatically strictly below the nonnegative incidence tail.","generated_certificate":false},{"name":"producerCarry_le_neg_eight_forces_gap_margin","module":"Erdos249257/HalfCylinderProducerCarrySocket.lean","line":190,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderProducerCarrySocket.lean#L190","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderProducerCarrySocket.lean#L190","kind":"theorem","signature":"theorem producerCarry_le_neg_eight_forces_gap_margin (A : Set ℕ) (d : ℕ) (hone : 1 ∉ A) (hA : A ⊆ Set.Iic d) (hd : 2 ≤ d) (hcarry : producerCarry A d ≤ -8) : (1 : ℝ) / 2 + mersenneGap d < erdosSupportSeries 2 A","docstring":"An upper-producer carry of at most `-8` puts the adjacent-above prefix more than one full Mersenne gap above `1/2`.","generated_certificate":false}],"paper_label":"res:producercarry","paper_anchor":{"paper_label":"res:producercarry","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1898,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Residual identity for the middle-transition support","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Exact final-skip reduction and its exceptional carries","title_tex":"Exact final-skip reduction and its exceptional carries","line":1724},{"level":"subsubsection","title":"Residual identity for the middle-transition support","title_tex":"Residual identity for the middle-transition support","line":1898}],"source_excerpt":{"text":"\\subsubsection{Residual identity for the middle-transition support}\\label{res:producercarry}","media_type":"text/x-tex","source_char_start":101212,"source_char_end":101304,"start_line":1898,"end_line":1898,"truncated":false,"sha256":"sha256:9aea1925ebb5c20893738fe2241b0feb3435f62ff5ee57da582c6cd6f136fd9e","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1898","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"producerCarry_residual_identity","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderProducerCarrySocket.lean#L27","cite_this":true},{"kind":"lean_declaration_pinned","label":"producerCarry_gt_tail_iff_supportSeries_lt_half","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderProducerCarrySocket.lean#L48","cite_this":true},{"kind":"lean_declaration_pinned","label":"producerCarry_lt_tail_iff_half_lt_supportSeries","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderProducerCarrySocket.lean#L78","cite_this":true},{"kind":"lean_declaration_pinned","label":"producerCarry_neg_forces_half_lt_supportSeries","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderProducerCarrySocket.lean#L145","cite_this":true},{"kind":"lean_declaration_pinned","label":"producerCarry_le_neg_eight_forces_gap_margin","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderProducerCarrySocket.lean#L190","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/50","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"producer_carry_future_tail_socket"}}],"expect":"object"},"prior_art":{"family_id":"fatal_gap_and_last_producer_spine","status_summary":"Exact half-membership/nonmembership classifications, proved branch eliminations, a rank-26 phase sieve for the middle -2 cell, and explicit conditional tail consumers.","prior_art_posture":"No exact local seam or last-producer antecedent is recorded beyond strict-tail achievement-set background.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/8","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"formalisation_only":2},"edges":[{"semantic_node_id":"Z14::producer_carry_exact_half_criterion","prior_art_state":"formalisation_only"},{"semantic_node_id":"Z14::producer_carry_negative_forces_above_half","prior_art_state":"formalisation_only"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:producercarry","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1898","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:quarter_band_endpoint_cells","assertion_kind":"registered_claim","source_id":"quarter_band_endpoint_cells","reviewed":true,"granularity":"single_registered_claim","label":"Exact quarter-band failure cells for adjacent seam cuts","statement":"At row s put H = 2^(s-1), and let R, U, R', p^-, p^+ be the adjacent cut's remainder, overshoot, next remainder, below pulse, and above pulse. On the non-carry middle branch, failure R' >= 8H is equivalent to the unique cell R = H+k with p^- <= 4k < p^-+4. On the terminal-reset branch, failure is equivalent to R = 3H+k with p^-+4 <= 4k and 4k+16H < 4(R+U)+p^+. These are exact failure classifications, not an avoidance theorem for the concrete seam orbit.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"pulseCell_unique","module":"Erdos249257/HalfCylinderQuarterBandEndpoints.lean","line":34,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderQuarterBandEndpoints.lean#L34","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderQuarterBandEndpoints.lean#L34","kind":"theorem","signature":"theorem pulseCell_unique {pulse k l : ℕ} (hk : pulse ≤ 4 * k ∧ 4 * k < pulse + 4) (hl : pulse ≤ 4 * l ∧ 4 * l < pulse + 4) : k = l","docstring":"The integer cell in a four-unit pulse slab is unique.","generated_certificate":false},{"name":"seamAdjacentCut_middleNextUpperFailure_iff_pulseCell","module":"Erdos249257/HalfCylinderQuarterBandEndpoints.lean","line":153,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderQuarterBandEndpoints.lean#L153","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderQuarterBandEndpoints.lean#L153","kind":"theorem","signature":"theorem seamAdjacentCut_middleNextUpperFailure_iff_pulseCell {s : ℕ} (hs : 5 ≤ s) (hncarry : ¬ (seamAdjacentCut s hs).successorCarries) (hmiddle : 4 * (seamAdjacentCut s hs).remainder + (seamPerturbedFamily s (by omega)).gap - (seamAdjacentCut s hs).belowPulse < (seamAdjacentCut s hs).terminalWeight) : ¬ (seamAdjacentCut s hs).nextRemainder < 2 * (seamPerturbedFamily s (by omega)).gap ↔ ∃ k : ℕ, (seamAdjacentCut s hs).remainder = 2 ^ (s - 1) + k ∧ (seamAdjacentCut s hs).belowPulse ≤ 4 * k ∧ 4 * k < (seamAdjacentCut s hs).belowPulse + 4","generated_certificate":false},{"name":"seamAdjacentCut_rightNextUpperFailure_iff_pulseCell","module":"Erdos249257/HalfCylinderQuarterBandEndpoints.lean","line":175,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderQuarterBandEndpoints.lean#L175","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderQuarterBandEndpoints.lean#L175","kind":"theorem","signature":"theorem seamAdjacentCut_rightNextUpperFailure_iff_pulseCell {s : ℕ} (hs : 5 ≤ s) (hncarry : ¬ (seamAdjacentCut s hs).successorCarries) (hright : (seamAdjacentCut s hs).terminalWeight ≤ 4 * (seamAdjacentCut s hs).remainder + (seamPerturbedFamily s (by omega)).gap - (seamAdjacentCut s hs).belowPulse) : ¬ (seamAdjacentCut s hs).nextRemainder < 2 * (seamPerturbedFamily s (by omega)).gap ↔ ∃ k : ℕ, (seamAdjacentCut s hs).remainder = 3 * 2 ^ (s - 1) + k ∧ (seamAdjacentCut s hs).belowPulse + 4 ≤ 4 * k ∧ 4 * k + 16 * 2 ^ (s - 1) < 4 * ((seamAdjacentCut s hs).remainder +","generated_certificate":false}],"paper_label":"res:quarterband","paper_anchor":{"paper_label":"res:quarterband","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1917,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Quarter-band endpoint cases","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Exact final-skip reduction and its exceptional carries","title_tex":"Exact final-skip reduction and its exceptional carries","line":1724},{"level":"subsubsection","title":"Quarter-band endpoint cases","title_tex":"Quarter-band endpoint cases","line":1917}],"source_excerpt":{"text":"\\subsubsection{Quarter-band endpoint cases}\\label{res:quarterband}","media_type":"text/x-tex","source_char_start":102457,"source_char_end":102523,"start_line":1917,"end_line":1917,"truncated":false,"sha256":"sha256:b312604a60c1644e3b64ac162ed70c59847ef92d724cadc8e7bac59e42c7b1bc","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. 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Historical preservation does not make the manuscript authoritative."},"fidelity":{"schema":"plectis_claim_fidelity_v1","assertion_id":"claim:quarter_band_endpoint_cells","verdict":"not_mechanically_decidable","axes":["negative_clause_loss"],"findings":[{"axis":"negative_clause_loss","verdict":"not_mechanically_decidable","declaration":"seamAdjacentCut_middleNextUpperFailure_iff_pulseCell","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderQuarterBandEndpoints.lean#L153","signature_requires":"5 ≤ s","manuscript_also_states_the_bound":false,"why":"the theorem holds for s ≥ 5; the sentence states no range, so it reads as holding for every s. 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What is recorded is that nobody has reconciled the two.","reader_rule":"This row's sentence may be quoted, with the recorded difference beside it. Do not present it as settled either way."},"render_policy":{"schema":"plectis_render_policy_v1","quotation_policy":"quote_with_the_recorded_difference","headline_eligible":true,"instruction":"This row's statement may be quoted, but the recorded difference must travel with it. Do not present the sentence as unexamined.","quote_instead":{"manuscript_theorem":{"authority":"the author's own statement, quoted verbatim in this packet","paper_label":"res:quarterband","read_it_at":"this row's paper_anchor.source_excerpt.text","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1917"},"lean_proposition":{"authority":"proof authority; the statement the kernel accepted","declaration":"seamAdjacentCut_middleNextUpperFailure_iff_pulseCell","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderQuarterBandEndpoints.lean#L153"}},"why_not_a_corrected_sentence":"A replacement sentence written by this build would be an unreviewed mathematical claim. The sources above are authored and checked."},"source_lane":"registry_claim"},{"assertion_id":"claim:reset_deficit_escape_geometry","assertion_kind":"registered_claim","source_id":"reset_deficit_escape_geometry","reviewed":true,"granularity":"single_registered_claim","label":"Exact right-branch window and reset-deficit escape law","statement":"Write R_s for the final integer-greedy remainder and p_s^- for the below pulse. At a largest skip d with 2s < 3d, a right branch forces 2^(s+1)+4 <= 4R_s and 12R_s+3*2^(s+1) < 4*(3*2^(s+1)+2*4^(s-d)+4)+6(s-2), and it satisfies R_(s+1)+2^(s+1)+p_s^-+4 = 4R_s. If R_s <= 2^s and delta_s = 2^s-R_s, every further right step gives delta_(s+1) >= 4 delta_s while the reference scale doubles; hence 2^s <= 2^k delta_s forces an upper-or-middle row t in [s,s+k]. The global anti-concentration needed to exclude the pinned interval along the actual orbit is not proved.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"rightBranch_remainder_window","module":"Erdos249257/HalfCylinderResetDeficitEscape.lean","line":200,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderResetDeficitEscape.lean#L200","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderResetDeficitEscape.lean#L200","kind":"theorem","signature":"theorem rightBranch_remainder_window {s d : ℕ} (hs : 5 ≤ s) (hd : IsLargestFalseRank (seamGreedyWord s) d) (hlate : 2 * s < 3 * d) (hR : ¬ SeamGreedyUpperOrMiddleAt s hs) : 2 ^ (s + 1) + 4 ≤ 4 * seamIntegerGreedyRemainder s ∧ 12 * seamIntegerGreedyRemainder s + 3 * 2 ^ (s + 1) < 4 * (3 * 2 ^ (s + 1) + 2 * 4 ^ (s - d) + 4) + 3 * (2 * (s - 2))","docstring":"A right branch at a late row pins the greedy remainder into one explicit interval: above the quarter point, and strictly below the carry threshold. At the boundary cells `3*d ∈ {2*s+1, 2*s+2}` the emptiness of this interval for the actual orbit is exactly the remaining socket.","generated_certificate":false},{"name":"rightBranch_remainder_succ_eq","module":"Erdos249257/HalfCylinderResetDeficitEscape.lean","line":218,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderResetDeficitEscape.lean#L218","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderResetDeficitEscape.lean#L218","kind":"theorem","signature":"theorem rightBranch_remainder_succ_eq {s : ℕ} (hs : 5 ≤ s) (hR : ¬ SeamGreedyUpperOrMiddleAt s hs) : seamIntegerGreedyRemainder (s + 1) + 2 ^ (s + 1) + (seamAdjacentCut s hs).belowPulse + 4 = 4 * seamIntegerGreedyRemainder s","docstring":"Exact right-branch remainder recurrence.","generated_certificate":false},{"name":"upperOrMiddle_within_of_two_pow_deficit","module":"Erdos249257/HalfCylinderResetDeficitEscape.lean","line":261,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderResetDeficitEscape.lean#L261","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderResetDeficitEscape.lean#L261","kind":"theorem","signature":"theorem upperOrMiddle_within_of_two_pow_deficit : ∀ k s : ℕ, 5 ≤ s → seamIntegerGreedyRemainder s ≤ 2 ^ s → 2 ^ s ≤ 2 ^ k * (2 ^ s - seamIntegerGreedyRemainder s) → ∃ t : ℕ, s ≤ t ∧ t ≤ s + k ∧ ∃ h : 5 ≤ t, SeamGreedyUpperOrMiddleAt t h","docstring":"Deficit-side run law: once the doubled-scale deficit `2^s - rem(s)` reaches `2^(s-k)` (stated multiplicatively to avoid truncated subtraction in the exponent), an upper-or-middle branch fires within `k` rows. Contrapositive: an all-right run of length `k` from row `s` forces `rem(s)` within `2^(s-k)` of the half-point `2^s`.","generated_certificate":false}],"paper_label":"res:resetdeficit","paper_anchor":{"paper_label":"res:resetdeficit","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1936,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Right-branch deficit escape","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Exact final-skip reduction and its exceptional carries","title_tex":"Exact final-skip reduction and its exceptional carries","line":1724},{"level":"subsubsection","title":"Right-branch deficit escape","title_tex":"Right-branch deficit escape","line":1936}],"source_excerpt":{"text":"\\subsubsection{Right-branch deficit escape}\\label{res:resetdeficit}","media_type":"text/x-tex","source_char_start":103395,"source_char_end":103462,"start_line":1936,"end_line":1936,"truncated":false,"sha256":"sha256:30ec9a7cc426174124a912232fe50ef51e7ad9955e7b408fb64f749c64faa471","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z13::deficit_run_length_law","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z13::right_branch_remainder_recurrence","prior_art_state":"not_assessed"},{"semantic_node_id":"Z13::right_branch_remainder_window","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:resetdeficit","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"requires_mathematical_adjudication","render_policy":"historical_only_with_recorded_difference","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1936","finding_axes":["negative_clause_loss"],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:seam_producer_carry_alignment","assertion_kind":"registered_claim","source_id":"seam_producer_carry_alignment","reviewed":true,"granularity":"single_registered_claim","label":"Exact seam-hole to producer-carry alignment identity","statement":"Let P omit s, put A = P union {s}, and let delta be the half-carry of A at row 2s-2. Define the first-wrap hole H = 2 delta - f_P(2s) and paired incidence pulse Pi = 2 f_A(2s+1) + f_A(2s+2). Then the producer carry is exactly C(A,s) = 4(H-1) - Pi. This aligns the seam and producer coordinates but supplies no sign, size, or occurrence estimate.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"gateway_section","warrant":"the claim's anchor sits inside the retired gateway section 'Erdős--Borwein-type irrationality (the #257 direction)', which is the author's own partition of that manuscript by problem"},"problem_edges":[{"problem_id":"erdos_257","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"producerCarry_insert_eq_four_mul_shiftedHole_sub_pulse","module":"Erdos249257/HalfCylinderSeamProducerAlignment.lean","line":24,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderSeamProducerAlignment.lean#L24","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderSeamProducerAlignment.lean#L24","kind":"theorem","signature":"theorem producerCarry_insert_eq_four_mul_shiftedHole_sub_pulse (P : Set ℕ) (s : ℕ) (hs : 1 ≤ s) (hsP : s ∉ P) (δ : ℤ) (hδ : integerHalfCarry (insert s P) (2 * s - 2) = δ) : producerCarry (insert s P) s = 4 * (2 * δ - (supportCoeff P (2 * s) : ℤ) - 1) - (2 * (supportCoeff (insert s P) (2 * s + 1) : ℤ) + (supportCoeff (insert s P) (2 * s + 2) : ℤ))","docstring":"If `insert s P` is the left boundary support at the first feedback row, its producer-aligned carry is four times the shifted seam hole, minus the explicit odd/even paired incidence pulse at rows `2s+1` and `2s+2`. The seam hole is `2δ - supportCoeff P (2s)`, as in `actualFirstWrap_hole_eq_leftZeroCarry_add_one`.","generated_certificate":false}],"paper_label":"res:seamproducer","paper_anchor":{"paper_label":"res:seamproducer","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1956,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Alignment of seam and transition carries","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Exact final-skip reduction and its exceptional carries","title_tex":"Exact final-skip reduction and its exceptional carries","line":1724},{"level":"subsubsection","title":"Alignment of seam and transition carries","title_tex":"Alignment of seam and transition carries","line":1956}],"source_excerpt":{"text":"\\subsubsection{Alignment of seam and transition carries}\\label{res:seamproducer}","media_type":"text/x-tex","source_char_start":104414,"source_char_end":104494,"start_line":1956,"end_line":1956,"truncated":false,"sha256":"sha256:82e3beb51fbec7bd8ff6242a31e397b9f26c144ebfb05c00ac622cbf944d0a94","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1956","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"producerCarry_insert_eq_four_mul_shiftedHole_sub_pulse","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderSeamProducerAlignment.lean#L24","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/53","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"seam_producer_carry_alignment"}}],"expect":"object"},"prior_art":{"family_id":"fatal_gap_and_last_producer_spine","status_summary":"Exact half-membership/nonmembership classifications, proved branch eliminations, a rank-26 phase sieve for the middle -2 cell, and explicit conditional tail consumers.","prior_art_posture":"No exact local seam or last-producer antecedent is recorded beyond strict-tail achievement-set background.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/8","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z13::seam_hole_to_producer_carry_alignment","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:seamproducer","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1956","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:selected_window_depth18_base","assertion_kind":"registered_claim","source_id":"selected_window_depth18_base","reviewed":true,"granularity":"single_registered_claim","label":"Computed depth-18 selected half-carry window base","statement":"A kernel-reflected Boolean table supplies a selected half-carry window at depth 18 for every terminal carry in the half strip [1,12]. The representatives share a common depth-13 restriction, yielding the first next-row divisor-agreement certificate. This is a finite base and does not construct selected windows at unbounded depths.","status":"verified finite instance","status_meaning":"Kernel-checked computation at a bounded range","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"depth18TableCheck_eq_true","module":"Erdos249257/HalfCarrySelectedWindowBase.lean","line":79,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarrySelectedWindowBase.lean#L79","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarrySelectedWindowBase.lean#L79","kind":"theorem","signature":"theorem depth18TableCheck_eq_true : depth18TableCheck = true","docstring":"Kernel-reflected evaluation of the complete depth-18 table. The strip bound calls `Nat.sqrt`, whose well-founded recursion the elaborator will not unfold, so the goal is discharged by kernel reduction rather than `decide`.","generated_certificate":false},{"name":"depth18SelectedHalfWindow","module":"Erdos249257/HalfCarrySelectedWindowBase.lean","line":129,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarrySelectedWindowBase.lean#L129","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarrySelectedWindowBase.lean#L129","kind":"def","signature":"noncomputable def depth18SelectedHalfWindow : SelectedHalfWindow 18 (halfStripBound 18)","docstring":"The exact depth-18 selected full strip used as the global induction base.","generated_certificate":false},{"name":"depth18SelectedHalfWindow_nextRowDivisorAgreement","module":"Erdos249257/HalfCarrySelectedWindowBase.lean","line":172,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarrySelectedWindowBase.lean#L172","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarrySelectedWindowBase.lean#L172","kind":"theorem","signature":"theorem depth18SelectedHalfWindow_nextRowDivisorAgreement : depth18SelectedHalfWindow.NextRowDivisorAgreement","docstring":"Concrete initial coherent-history certificate: the depth-18 selected window agrees on every divisor inspected by row 19.","generated_certificate":false}],"paper_label":"res:selectedwindowbase","paper_anchor":{"paper_label":"res:selectedwindowbase","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1975,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"A verified initial selected window","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Finite evidence, coordinate changes, and eliminated special cases","title_tex":"Finite evidence, coordinate changes, and eliminated special cases","line":1967},{"level":"subsubsection","title":"A verified initial selected window","title_tex":"A verified initial selected window","line":1975}],"source_excerpt":{"text":"\\subsubsection{A verified initial selected window}\\label{res:selectedwindowbase}","media_type":"text/x-tex","source_char_start":105333,"source_char_end":105413,"start_line":1975,"end_line":1975,"truncated":false,"sha256":"sha256:375aaa27e6032416a0029c7544e48bed7e2f1f30c0ed2ec04f8eb102532fe949","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1975","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"depth18TableCheck_eq_true","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarrySelectedWindowBase.lean#L79","cite_this":true},{"kind":"lean_declaration_pinned","label":"depth18SelectedHalfWindow","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarrySelectedWindowBase.lean#L129","cite_this":true},{"kind":"lean_declaration_pinned","label":"depth18SelectedHalfWindow_nextRowDivisorAgreement","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarrySelectedWindowBase.lean#L172","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/54","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"selected_window_depth18_base"}}],"expect":"object"},"scope_limit":{"text":"The 12 terminal carries in the half strip at depth 18, together with their common depth-13 restriction and the next-row divisor-agreement certificate.","source":"claims_registry_bounded_domain"},"prior_art":{"family_id":"rewind_windows_and_protected_seams","status_summary":"Conditional rewind/window consumers plus one verified depth-18 base.","prior_art_posture":"No theorem-level match is recorded; any novelty posture remains provisional.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/7","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z15::depth18_selected_window_base","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:selectedwindowbase","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1975","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:rewind_seam_operational_bridge","assertion_kind":"registered_claim","source_id":"rewind_seam_operational_bridge","reviewed":true,"granularity":"single_registered_claim","label":"Operational bridge from rewind seams to protected next-row seams","statement":"Let W = (a^h)_(1 <= h <= R) be a selected depth-N window whose targets rewind to two adjacent base words at depth M. If D and phi are the rewind denominator and phase, delta = D-phi, M <= N, (N+1)/2 <= M, R <= D, and the adjacent base words have row-(N+1) coefficients c+1 and c, then f_(a^h)(N+1) equals c+1 for h <= delta and c for h > delta. If 27 <= S <= 2 floor(sqrt(N+1))+4 and S+c+1 <= 2R, the profile realizes a protected one-hole seam at row N+1. The rewind provenance, seam pair, coefficient drop, strip, and buffer are inputs. For a full half-strip window the divisor bound supplies the buffer, but the coefficient drop remains unproved.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"nextRow_profile_of_rewindSeam","module":"Erdos249257/RewindSeamOperationalBridge.lean","line":140,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/RewindSeamOperationalBridge.lean#L140","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/RewindSeamOperationalBridge.lean#L140","kind":"theorem","signature":"theorem nextRow_profile_of_rewindSeam {M N R₀ R c : ℕ} (W₀ : SelectedHalfWindow M R₀) (W : SelectedHalfWindow N R) (hMN : M ≤ N) (hhalf : (N + 1) / 2 ≤ M) (cs : List ℕ) (hwidth : R ≤ denom cs) (hprov : W.HasRewindRestrictionAt W₀ hMN cs) (hseam : IsSeamPair cs R) (hbaseProfile : RewindBaseUnitDropAt W₀ (N + 1) cs c) : ∀ h (hh : 1 ≤ h) (hhR : h ≤ R), supportCoeff (wordSupport (W.word h hh hhR)) (N + 1) = if h ≤ seamCut cs then c + 1 else c","docstring":"Exact contribution of rewind provenance in the seam branch. It turns a two-coefficient fact about the adjacent base ancestors into the piecewise coefficient profile of the live selected window.","generated_certificate":false},{"name":"protectedEvenSeamRealizedAt_succ_of_profile","module":"Erdos249257/RewindSeamOperationalBridge.lean","line":217,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/RewindSeamOperationalBridge.lean#L217","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/RewindSeamOperationalBridge.lean#L217","kind":"theorem","signature":"theorem protectedEvenSeamRealizedAt_succ_of_profile {N R S δ c : ℕ} (hN : 1 ≤ N) (W : SelectedHalfWindow N R) (hSstrip : S ≤ halfStripBound (N + 1)) (h27 : 27 ≤ S) (hbuffer : S + (c + 1) ≤ 2 * R) (hprofile : ∀ h (hh : 1 ≤ h) (hhR : h ≤ R), supportCoeff (wordSupport (W.word h hh hhR)) (N + 1) = if h ≤ δ then c + 1 else c) : ProtectedEvenSeamRealizedAt (N + 1) (δ : ℤ) (c : ℤ)","docstring":"A two-valued next-row coefficient profile realizes the exact scalar one-hole seam at the next depth. The buffer is the usual selected-window headroom, with `c+1` the larger of the two coefficients.","generated_certificate":false},{"name":"protectedEvenSeamRealizedAt_succ_of_rewindSeam","module":"Erdos249257/RewindSeamOperationalBridge.lean","line":339,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/RewindSeamOperationalBridge.lean#L339","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/RewindSeamOperationalBridge.lean#L339","kind":"theorem","signature":"theorem protectedEvenSeamRealizedAt_succ_of_rewindSeam {M N R₀ R S c : ℕ} (W₀ : SelectedHalfWindow M R₀) (W : SelectedHalfWindow N R) (hMN : M ≤ N) (hhalf : (N + 1) / 2 ≤ M) (cs : List ℕ) (hN : 1 ≤ N) (hwidth : R ≤ denom cs) (hprov : W.HasRewindRestrictionAt W₀ hMN cs) (hseam : IsSeamPair cs R) (hSstrip : S ≤ halfStripBound (N + 1)) (h27 : 27 ≤ S) (hbuffer : S + (c + 1) ≤ 2 * R) (hbaseProfile : RewindBaseUnitDropAt W₀ (N + 1) cs c) : ProtectedEvenSeamRealizedAt (N + 1) (seamCut cs : ℤ) (c : ℤ)","docstring":"Operational seam fan-in. Exact rewind provenance plus an actual scalar seam pair produces a localized realized seam one row later, provided the adjacent base ancestors have the required unit-drop next-row coefficient profile. No global `IsCanonicalEvenSeam` equivalence is needed.","generated_certificate":false},{"name":"protectedEvenSeamRealizedAt_succ_of_fullStrip_rewindSeam","module":"Erdos249257/RewindSeamOperationalBridge.lean","line":358,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/RewindSeamOperationalBridge.lean#L358","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/RewindSeamOperationalBridge.lean#L358","kind":"theorem","signature":"theorem protectedEvenSeamRealizedAt_succ_of_fullStrip_rewindSeam {M N R₀ c : ℕ} (W₀ : SelectedHalfWindow M R₀) (W : SelectedHalfWindow N (halfStripBound N)) (hMN : M ≤ N) (hhalf : (N + 1) / 2 ≤ M) (cs : List ℕ) (hN : 1 ≤ N) (hwidth : halfStripBound N ≤ denom cs) (hprov : W.HasRewindRestrictionAt W₀ hMN cs) (hseam : IsSeamPair cs (halfStripBound N)) (h27 : 27 ≤ halfStripBound (N + 1)) (hbaseProfile : RewindBaseUnitDropAt W₀ (N + 1) cs c) : ProtectedEvenSeamRealizedAt (N + 1) (seamCut cs : ℤ) (c : ℤ)","docstring":"Full-strip specialization: the standard divisor envelope discharges the buffer automatically. Thus the only semantic input not already present in `HasRewindRestrictionAt + IsSeamPair` is the unit coefficient drop between the two adjacent base ancestors.","generated_certificate":false}],"paper_label":"res:rewindseambridge","paper_anchor":{"paper_label":"res:rewindseambridge","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1986,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Rewind-to-seam equivalence","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Finite evidence, coordinate changes, and eliminated special cases","title_tex":"Finite evidence, coordinate changes, and eliminated special cases","line":1967},{"level":"subsubsection","title":"Rewind-to-seam equivalence","title_tex":"Rewind-to-seam equivalence","line":1986}],"source_excerpt":{"text":"\\subsubsection{Rewind-to-seam equivalence}\\label{res:rewindseambridge}","media_type":"text/x-tex","source_char_start":106022,"source_char_end":106092,"start_line":1986,"end_line":1986,"truncated":false,"sha256":"sha256:f69a84b961eb87cd5be853c67066ff091e87b450c34d0ea3fea05cec41297c96","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1986","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"nextRow_profile_of_rewindSeam","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/RewindSeamOperationalBridge.lean#L140","cite_this":true},{"kind":"lean_declaration_pinned","label":"protectedEvenSeamRealizedAt_succ_of_profile","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/RewindSeamOperationalBridge.lean#L217","cite_this":true},{"kind":"lean_declaration_pinned","label":"protectedEvenSeamRealizedAt_succ_of_rewindSeam","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/RewindSeamOperationalBridge.lean#L339","cite_this":true},{"kind":"lean_declaration_pinned","label":"protectedEvenSeamRealizedAt_succ_of_fullStrip_rewindSeam","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/RewindSeamOperationalBridge.lean#L358","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/55","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"rewind_seam_operational_bridge"}}],"expect":"object"},"prior_art":{"family_id":"rewind_windows_and_protected_seams","status_summary":"Conditional rewind/window consumers plus one verified depth-18 base.","prior_art_posture":"No theorem-level match is recorded; any novelty posture remains provisional.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/7","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":3},"edges":[{"semantic_node_id":"Z15::rewind_seam_plus_unit_drop_realizes_protected_seam","prior_art_state":"not_assessed"},{"semantic_node_id":"Z15::rewind_seam_transports_unit_drop_to_profile","prior_art_state":"not_assessed"},{"semantic_node_id":"Z15::threshold_profile_realizes_protected_seam","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:rewindseambridge","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"requires_mathematical_adjudication","render_policy":"historical_only_with_recorded_difference","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L1986","finding_axes":["negative_clause_loss"],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:selected_suffix_cylinder_propagation","assertion_kind":"registered_claim","source_id":"selected_suffix_cylinder_propagation","reviewed":true,"granularity":"single_registered_claim","label":"Conditional suffix-cylinder feedback and child-gap recurrence","statement":"For the depth-18 selected window, the common depth-13 prefix and suffix numeral satisfy ν + k = 17. Under a common next coefficient C the endpoint obeys E' = 2E - C. At a feedback row the full cylinder either advances or leaves one carry gap [g_-,g_+]. If the lower and upper families have coefficients C_- and C_+ and satisfy the explicit positivity, ordering, strip, and buffer inequalities, the next gap is exactly [2g_- - C_- - 1, 2g_+ - C_+]. Cutoff promotion appends forced bits 0 and 1 and preserves consecutive prefixes. A swallowed two-row update forces the sharp quarter-band bound B <= 4(g_+ - g_- + 1) + 2. Cofinal full stages would produce an infinite support of value 1/2, but no such cofinal family or uniform coefficient inequalities are proved.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"depth18SelectedHalfWindow_hasSuffixCylinderAt_thirteen","module":"Erdos249257/SelectedSuffixCylinder.lean","line":88,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SelectedSuffixCylinder.lean#L88","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SelectedSuffixCylinder.lean#L88","kind":"theorem","signature":"theorem depth18SelectedHalfWindow_hasSuffixCylinderAt_thirteen : HasSuffixCylinderAt depth18SelectedHalfWindow (by omega : 13 ≤ 18) 17","docstring":"Exact cylinder certificate for the existing certified induction base.","generated_certificate":false},{"name":"SelectedHalfWindow.hasSuffixCylinderAt_stepExplicit","module":"Erdos249257/SelectedSuffixCylinder.lean","line":136,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SelectedSuffixCylinder.lean#L136","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SelectedSuffixCylinder.lean#L136","kind":"theorem","signature":"theorem SelectedHalfWindow.hasSuffixCylinderAt_stepExplicit {M N R S C E : ℕ} (hN : 1 ≤ N) (W : SelectedHalfWindow N R) (hMN : M ≤ N) (hSstrip : S ≤ halfStripBound (N + 1)) (hp_common : ∀ k : ℕ, 1 ≤ k → k ≤ R → selectedRowCoeff W k = C) (hbuffer : S + C ≤ 2 * R) (hcyl : HasSuffixCylinderAt W hMN E) (hRE : R ≤ E) : HasSuffixCylinderAt (W.stepExplicit hN hSstrip hp_common hbuffer) (hMN.trans (Nat.le_succ N)) (2 * E - C) ∧ S ≤ 2 * E - C","docstring":"Exact cylinder propagation. The endpoint update is `E' = 2E-C`; the ordinary selected-window buffer also proves that the new endpoint still covers the entire new carry interval.","generated_certificate":false},{"name":"promotedSuffixCylinder_or_protectedEvenSeamRealizedAt","module":"Erdos249257/SuffixCylinderPromotion.lean","line":322,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderPromotion.lean#L322","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderPromotion.lean#L322","kind":"theorem","signature":"theorem promotedSuffixCylinder_or_protectedEvenSeamRealizedAt {K M N R₀ E₀ E : ℕ} (W₀ : SelectedHalfWindow M R₀) (W : SelectedHalfWindow N (halfStripBound N)) (hK1M : K + 1 ≤ M) (hMlt : M < 2 * (K + 1)) (hMN : M ≤ N) (hrow : N + 1 = 2 * (K + 1)) (cs : List ℕ) (hN : 1 ≤ N) (hwidth : halfStripBound N ≤ denom cs) (hcyl₀ : HasSuffixCylinderAt W₀ ((Nat.le_succ K).trans hK1M) E₀) (hcyl : HasSuffixCylinderAt W ((Nat.le_succ K).trans (hK1M.trans hMN)) E) (hprov : W.HasRewindRestrictionAt W₀ hMN cs) (h27 : 27 ≤ halfStripBound (N + 1)) :","docstring":"Full operational feedback-row dichotomy. A width-valid exact history either promotes the endpoint cylinder through the next half-divisor bit, or it realizes the localized one-hole seam at the produced even row.","generated_certificate":false},{"name":"CylinderStage.feedbackStep_or_protectedSeam","module":"Erdos249257/SuffixCylinderGlobalProducer.lean","line":174,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderGlobalProducer.lean#L174","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderGlobalProducer.lean#L174","kind":"theorem","signature":"theorem CylinderStage.feedbackStep_or_protectedSeam {K N : ℕ} (S : CylinderStage K N) (hN : 1 ≤ N) (hK1N : K + 1 ≤ N) (hrow : N + 1 = 2 * (K + 1)) (h27 : 27 ≤ halfStripBound (N + 1)) : Nonempty (CylinderStage (K + 1) (N + 1)) ∨ ∃ δ c : ℤ, ProtectedEvenSeamRealizedAt (N + 1) δ c","docstring":"**Total feedback step.** At an exact half-divisor horizon, every full suffix-cylinder stage either promotes through the new bit and advances one row, or emits a realized localized one-hole seam. There is no fourth object-level case.","generated_certificate":false},{"name":"SelectedHalfWindow.word_stepTailExplicit","module":"Erdos249257/SuffixCylinderBelow.lean","line":117,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderBelow.lean#L117","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderBelow.lean#L117","kind":"theorem","signature":"theorem SelectedHalfWindow.word_stepTailExplicit {N R S C δ : ℕ} (hN : 1 ≤ N) (W : SelectedHalfWindow N R) (hSstrip : S ≤ halfStripBound (N + 1)) (hbuffer : S + C ≤ 2 * R) (hparentTail : ∀ q : ℕ, 1 ≤ q → q ≤ S → δ < parent C q) (hcoeff : ∀ k (hk : 1 ≤ k) (hkR : k ≤ R), δ < k → supportCoeff (wordSupport (W.word k hk hkR)) (N + 1) = C) (q : ℕ) (hq : 1 ≤ q) (hqS : q ≤ S) : (Erdos249257.SuffixCylinderBelow.SelectedHalfWindow.stepTailExplicit hN W hSstrip hbuffer hparentTail hcoeff).word q hq hqS = extendHalfWord (W.word (parent C q) (explicitParent_pos hq) (explicitParent_le hqS hbuffer)) (decide (explicitParentBitZ C q = 1))","docstring":"The upper-sheet successor is definitionally the canonical fixed-`C` parent with its parity bit appended.","generated_certificate":false},{"name":"hasSuffixCylinderAt_take","module":"Erdos249257/SuffixCylinderTwoSheet.lean","line":26,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderTwoSheet.lean#L26","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderTwoSheet.lean#L26","kind":"theorem","signature":"theorem hasSuffixCylinderAt_take {M N R E : ℕ} (W : SelectedHalfWindow N R) (hMN : M ≤ N) (hcyl : HasSuffixCylinderAt W hMN E) (S : ℕ) (hSR : S ≤ R) : HasSuffixCylinderAt (Erdos249257.SuffixCylinderTwoSheet.SelectedHalfWindow.take W S hSR) hMN E","docstring":"Taking an initial carry interval preserves an endpoint cylinder.","generated_certificate":false},{"name":"CylinderStage.feedbackAdvance_or_protectedSeam_inStrip","module":"Erdos249257/SuffixCylinderFeedbackFanIn.lean","line":62,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderFeedbackFanIn.lean#L62","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderFeedbackFanIn.lean#L62","kind":"theorem","signature":"theorem CylinderStage.feedbackAdvance_or_protectedSeam_inStrip {K N : ℕ} (S : CylinderStage K N) (hN : 1 ≤ N) (hK1N : K + 1 ≤ N) (hrow : N + 1 = 2 * (K + 1)) (h27 : 27 ≤ halfStripBound (N + 1)) : Nonempty (CylinderStage (K + 1) (N + 1)) ∨ ∃ δ c : ℕ, ProtectedEvenSeamRealizedAt (N + 1) (δ : ℤ) (c : ℤ) ∧ (1 : ℤ) ≤ 2 * (δ : ℤ) - (c : ℤ) ∧ 2 * (δ : ℤ) - (c : ℤ) ≤ (halfStripBound (N + 1) : ℤ)","docstring":"**Complete feedback fan-in.** At an exact half-divisor horizon, low and high common-bit cases advance ordinarily. A crossing whose one-hole seam escapes below or above the next strip is recylindrified by the corresponding sheet constructor. Therefore the only non-advancing result is a realized seam whose exact integer hole lies inside the next full strip.","generated_certificate":false},{"name":"CylinderStage.feedbackAdvance_or_inStripTwoSheet","module":"Erdos249257/SuffixCylinderInStrip.lean","line":665,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderInStrip.lean#L665","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderInStrip.lean#L665","kind":"theorem","signature":"theorem CylinderStage.feedbackAdvance_or_inStripTwoSheet {K N : ℕ} (S : CylinderStage K N) (hN : 1 ≤ N) (hK1N : K + 1 ≤ N) (hrow : N + 1 = 2 * (K + 1)) : Nonempty (CylinderStage (K + 1) (N + 1)) ∨ Nonempty (InStripTwoSheetStage K (N + 1))","docstring":"**Operational one-depth feedback split.** At an exact half-divisor horizon, a full cylinder either advances as a full cylinder or produces the exact two-sheet state above. Unlike the protected-seam existential, the residual branch retains every admissible word off the hole, both cylinder endpoints, the complementary literal prefixes, and the canonical-parent failure coordinate needed by the finite-gap iterator. This theorem alone is not an iteration closure: later rows still need `HasCommonNextCoeff` receipts.","generated_certificate":false},{"name":"InStripGapStage.advanceOfCommonNextCoeff","module":"Erdos249257/SuffixCylinderFiniteGap.lean","line":256,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderFiniteGap.lean#L256","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderFiniteGap.lean#L256","kind":"def","signature":"noncomputable def InStripGapStage.advanceOfCommonNextCoeff {K N C : ℕ} (T : InStripGapStage K N) (hN : 1 ≤ N) (hcommon : T.HasCommonNextCoeff C) (hchildLoPos : 1 ≤ 2 * T.gapLo - C - 1) (hchildHiStrip : 2 * T.gapHi - C ≤ halfStripBound (N + 1)) (hbuffer : halfStripBound (N + 1) + C ≤ 2 * halfStripBound N) : InStripGapStage K (N + 1)","docstring":"**Structure-preserving finite-gap successor.** If both live sheets have one common next coefficient and the doubled child gap remains inside the next strip, propagate the complete two-sheet package by one row.","generated_certificate":false},{"name":"ProfiledGapStage.advanceOfNextCoeffProfile","module":"Erdos249257/SuffixCylinderProfiledGap.lean","line":124,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderProfiledGap.lean#L124","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderProfiledGap.lean#L124","kind":"def","signature":"noncomputable def ProfiledGapStage.advanceOfNextCoeffProfile {M N lowerCoeff upperCoeff : ℕ} (T : ProfiledGapStage M N) (hN : 1 ≤ N) (hprofile : T.HasNextCoeffProfile lowerCoeff upperCoeff) (hchildLoPos : 1 ≤ 2 * T.gapLo - lowerCoeff - 1) (hupperCoeff : upperCoeff ≤ 2 * T.gapHi) (hchildOrder : 2 * T.gapLo - lowerCoeff - 1 ≤ 2 * T.gapHi - upperCoeff) (hchildHiStrip : 2 * T.gapHi - upperCoeff ≤ halfStripBound (N + 1)) (hupperBuffer : halfStripBound (N + 1) + upperCoeff ≤ 2 * halfStripBound N) : ProfiledGapStage M (N + 1)","docstring":"Structure-preserving successor with independent coefficients on the two live sheets. The exact child gap is `[2*gapLo-lowerCoeff-1, 2*gapHi-upperCoeff]`; in particular, this constructor continues to apply after cutoff promotion has made the two coefficients different.","generated_certificate":false},{"name":"selectedHalfIntervalPromoteHigh","module":"Erdos249257/SuffixCylinderProfiledPromotion.lean","line":107,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderProfiledPromotion.lean#L107","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderProfiledPromotion.lean#L107","kind":"theorem","signature":"theorem selectedHalfIntervalPromoteHigh {M N L U E : ℕ} (W : SelectedHalfInterval N L U) (hM1N : M + 1 ≤ N) (pfx : HalfWord M) (hpfx : ∀ q (hqL : L ≤ q) (hqU : q ≤ U), restrictWord ((Nat.le_succ M).trans hM1N) (W.word q hqL hqU) = pfx) (hcyl : HasSuffixCylinderOnInterval W ((Nat.le_succ M).trans hM1N) E) (hmargin : 2 ^ (N - (M + 1)) + U ≤ E) : HasSuffixCylinderOnInterval W hM1N (E - 2 ^ (N - (M + 1))) ∧ (∀ q (hqL : L ≤ q) (hqU : q ≤ U), restrictWord hM1N (W.word q hqL hqU) = extendHalfWord pfx true) ∧ U ≤ E - 2 ^ (N - (M + 1))","docstring":"High-side promotion for an endpoint-indexed selected interval. The entire interval lies above the new binary head, so the literal prefix gains a true bit and the endpoint loses exactly that head weight.","generated_certificate":false},{"name":"ProfiledGapStage.lowerNextCoeff_le_upperNextCoeff_add_one","module":"Erdos249257/SuffixCylinderProfiledAdjacency.lean","line":232,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderProfiledAdjacency.lean#L232","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderProfiledAdjacency.lean#L232","kind":"theorem","signature":"theorem ProfiledGapStage.lowerNextCoeff_le_upperNextCoeff_add_one {M N : ℕ} (T : ProfiledGapStage M N) (hadjacent : T.HasAdjacentPrefixes) : supportCoeff (wordSupport T.lowerPrefix) (N + 1) ≤ supportCoeff (wordSupport T.upperPrefix) (N + 1) + 1","docstring":"Stage form of the binary-increment bound: adjacent profiled prefixes have lower next-row coefficient at most one above the upper coefficient.","generated_certificate":false},{"name":"ProfiledGapStage.promoteCutoff_hasAdjacentPrefixes","module":"Erdos249257/SuffixCylinderProfiledPromotionAdjacency.lean","line":65,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderProfiledPromotionAdjacency.lean#L65","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderProfiledPromotionAdjacency.lean#L65","kind":"theorem","signature":"theorem ProfiledGapStage.promoteCutoff_hasAdjacentPrefixes {M N : ℕ} (T : ProfiledGapStage M N) (hadjacent : T.HasAdjacentPrefixes) (hM : 4 ≤ M) (hboundary : N = 2 * M + 1) : (T.promoteCutoff hM hboundary).HasAdjacentPrefixes","docstring":"The low-false/high-true cutoff promotion preserves the binary boundary between the two profiled sheets.","generated_certificate":false},{"name":"ProfiledGapStage.promoteThenAdvance_hasAdjacentPrefixes","module":"Erdos249257/SuffixCylinderProfiledAutoStep.lean","line":110,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderProfiledAutoStep.lean#L110","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderProfiledAutoStep.lean#L110","kind":"theorem","signature":"theorem ProfiledGapStage.promoteThenAdvance_hasAdjacentPrefixes {M N : ℕ} (T : ProfiledGapStage M N) (hadjacent : T.HasAdjacentPrefixes) (hM : 4 ≤ M) (hboundary : N = 2 * M + 1) (hnext : (T.promoteCutoff hM hboundary).NextArithmetic) : (T.promoteThenAdvance hM hboundary hnext).HasAdjacentPrefixes","docstring":"The complete promote-then-advance operation preserves binary adjacency.","generated_certificate":false},{"name":"nextBound_le_four_gapWidth_add_two_of_macro_swallow","module":"Erdos249257/SuffixCylinderProfiledSwallow.lean","line":254,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderProfiledSwallow.lean#L254","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderProfiledSwallow.lean#L254","kind":"theorem","signature":"theorem nextBound_le_four_gapWidth_add_two_of_macro_swallow {a b nextBound lowerOddCoeff lowerEvenCoeff upperOddCoeff upperEvenCoeff : ℕ} (hab : a ≤ b) (hpulse : 2 * lowerOddCoeff + lowerEvenCoeff ≤ 2 * upperOddCoeff + upperEvenCoeff + 2) (hswallowLo : macroGapLoZ a lowerOddCoeff lowerEvenCoeff ≤ 1) (hswallowHi : (nextBound : ℤ) ≤ macroGapHiZ b upperOddCoeff upperEvenCoeff) : nextBound ≤ 4 * (b - a + 1) + 2","docstring":"A macro-step swallow together with a coefficient-pulse difference at most two forces the old gap to occupy at least one quarter of the next strip, up to the sharp additive two. This is the exact remaining quarter-band branch; it does not assert that the branch is impossible.","generated_certificate":false},{"name":"cofinalTerminalOnlyStrip_of_cofinalCylinderStages","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":264,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L264","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L264","kind":"theorem","signature":"theorem cofinalTerminalOnlyStrip_of_cofinalCylinderStages (hstages : ∀ N : ℕ, ∃ M K : ℕ, max N 1 ≤ M ∧ Nonempty (CylinderStage K M)) : HalfCarryCofinalTerminalOnlyStrip","docstring":"Cofinal full-cylinder stages imply the strictly weaker terminal-only cofinal producer expected by the closed achievement-set consumer.","generated_certificate":false},{"name":"exists_infinite_support_half_of_cofinalCylinderStages","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":277,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L277","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L277","kind":"theorem","signature":"theorem exists_infinite_support_half_of_cofinalCylinderStages (hstages : ∀ N : ℕ, ∃ M K : ℕ, max N 1 ≤ M ∧ Nonempty (CylinderStage K M)) : ∃ A : Set ℕ, A.Infinite ∧ erdosSupportSeries 2 A = (1 : ℝ) / 2","docstring":"Direct end-to-end consumer: cofinal suffix-cylinder stages produce an infinite support whose Mersenne support series is the rational value `1/2`.","generated_certificate":false},{"name":"exists_infinite_positive_support_half_of_cofinalCylinderStages","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":287,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L287","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L287","kind":"theorem","signature":"theorem exists_infinite_positive_support_half_of_cofinalCylinderStages (hstages : ∀ N : ℕ, ∃ M K : ℕ, max N 1 ≤ M ∧ Nonempty (CylinderStage K M)) : ∃ A : Set ℕ, 0 ∉ A ∧ A.Infinite ∧ erdosSupportSeries 2 A = (1 : ℝ) / 2","docstring":"Cofinal full-cylinder stages produce an infinite support of strictly positive exponents whose Mersenne support series is `1/2`.","generated_certificate":false}],"paper_label":"res:selectsuffix","paper_anchor":{"paper_label":"res:selectsuffix","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2051,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":2050,"anchor_title":"verified base and one-row cylinder update","anchor_title_tex":"verified base and one-row cylinder update","section_title":"Selected suffix cylinders","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Finite evidence, coordinate changes, and eliminated special cases","title_tex":"Finite evidence, coordinate changes, and eliminated special cases","line":1967},{"level":"subsubsection","title":"Selected suffix cylinders","title_tex":"Selected suffix cylinders","line":1998}],"source_excerpt":{"text":"\\begin{proposition}[verified base and one-row cylinder update]\n\\label{res:selectsuffix}\nThe depth-\\(18\\) selected window has \\(R=12\\) and a suffix cylinder at cutoff\n\\(M=13\\) with endpoint \\(E=17\\).\n\nMore generally, let a depth-\\(N\\), radius-\\(R\\) selected window\n\\((a^{(k)})_{1\\leq k\\leq R}\\) have a suffix cylinder at cutoff \\(M\\leq N\\)\nwith endpoint \\(E\\), where \\(1\\leq N\\) and \\(R\\leq E\\).  Let\n\\(1\\leq R'\\leq H_{N+1}\\).  If\n\\[\n f_{a^{(k)}}(N+1)=C\\qquad(1\\leq k\\leq R)\n\\]\nand\n\\[\n R'+C\\leq2R,\n\\]\nthen the radius-\\(R'\\) successor remains a suffix cylinder, with endpoint\n\\[\n E'=2E-C.\n\\]\n\\end{proposition}","media_type":"text/x-tex","source_char_start":108775,"source_char_end":109382,"start_line":2050,"end_line":2070,"truncated":false,"sha256":"sha256:fc466dc93e974d19611924382e03fca7e87b4f52eaa7de41ed8ed0692c26954e","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2051","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.half_value_membership","statement":"Decide whether 1/2 lies in the Mersenne achievement set. Yes refutes universal #257; no closes only this route.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/4"}],"links":[{"kind":"lean_declaration_pinned","label":"depth18SelectedHalfWindow_hasSuffixCylinderAt_thirteen","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SelectedSuffixCylinder.lean#L88","cite_this":true},{"kind":"lean_declaration_pinned","label":"SelectedHalfWindow.hasSuffixCylinderAt_stepExplicit","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SelectedSuffixCylinder.lean#L136","cite_this":true},{"kind":"lean_declaration_pinned","label":"promotedSuffixCylinder_or_protectedEvenSeamRealizedAt","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderPromotion.lean#L322","cite_this":true},{"kind":"lean_declaration_pinned","label":"CylinderStage.feedbackStep_or_protectedSeam","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderGlobalProducer.lean#L174","cite_this":true},{"kind":"lean_declaration_pinned","label":"SelectedHalfWindow.word_stepTailExplicit","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderBelow.lean#L117","cite_this":true},{"kind":"lean_declaration_pinned","label":"hasSuffixCylinderAt_take","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderTwoSheet.lean#L26","cite_this":true},{"kind":"lean_declaration_pinned","label":"CylinderStage.feedbackAdvance_or_protectedSeam_inStrip","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderFeedbackFanIn.lean#L62","cite_this":true},{"kind":"lean_declaration_pinned","label":"CylinderStage.feedbackAdvance_or_inStripTwoSheet","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderInStrip.lean#L665","cite_this":true},{"kind":"lean_declaration_pinned","label":"InStripGapStage.advanceOfCommonNextCoeff","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderFiniteGap.lean#L256","cite_this":true},{"kind":"lean_declaration_pinned","label":"ProfiledGapStage.advanceOfNextCoeffProfile","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderProfiledGap.lean#L124","cite_this":true},{"kind":"lean_declaration_pinned","label":"selectedHalfIntervalPromoteHigh","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderProfiledPromotion.lean#L107","cite_this":true},{"kind":"lean_declaration_pinned","label":"ProfiledGapStage.lowerNextCoeff_le_upperNextCoeff_add_one","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderProfiledAdjacency.lean#L232","cite_this":true},{"kind":"lean_declaration_pinned","label":"ProfiledGapStage.promoteCutoff_hasAdjacentPrefixes","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderProfiledPromotionAdjacency.lean#L65","cite_this":true},{"kind":"lean_declaration_pinned","label":"ProfiledGapStage.promoteThenAdvance_hasAdjacentPrefixes","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderProfiledAutoStep.lean#L110","cite_this":true},{"kind":"lean_declaration_pinned","label":"nextBound_le_four_gapWidth_add_two_of_macro_swallow","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderProfiledSwallow.lean#L254","cite_this":true},{"kind":"lean_declaration_pinned","label":"cofinalTerminalOnlyStrip_of_cofinalCylinderStages","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L264","cite_this":true},{"kind":"lean_declaration_pinned","label":"exists_infinite_support_half_of_cofinalCylinderStages","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L277","cite_this":true},{"kind":"lean_declaration_pinned","label":"exists_infinite_positive_support_half_of_cofinalCylinderStages","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L287","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/56","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"selected_suffix_cylinder_propagation"}}],"expect":"object"},"prior_art":{"family_id":"suffix_cylinders_and_feedback_boundary","status_summary":"An exact conditional child-gap recurrence with a cofinal half-value consequence, verified finite stages through depth 52, an exact unit drop, and a concrete scalar-seam obstruction.","prior_art_posture":"Kovač–Tao supply the surrounding strict-tail achievement-set geometry. No exact antecedent for this finite carry recurrence is recorded, and no novelty claim is made.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/10","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":12},"edges":[{"semantic_node_id":"Z16::adjacent_prefix_invariant","prior_art_state":"not_assessed"},{"semantic_node_id":"Z16::binary_increment_coefficient_bound","prior_art_state":"not_assessed"},{"semantic_node_id":"Z16::cofinal_stages_conditional_export","prior_art_state":"not_assessed"},{"semantic_node_id":"Z16::common_next_coeff_gap_advance","prior_art_state":"not_assessed"},{"semantic_node_id":"Z16::feedback_fan_in_hole_in_strip","prior_art_state":"not_assessed"},{"semantic_node_id":"Z16::feedback_trichotomy_advance_or_seam","prior_art_state":"not_assessed"},{"semantic_node_id":"Z16::in_strip_two_sheet_split","prior_art_state":"not_assessed"},{"semantic_node_id":"Z16::macro_swallow_quarter_band_bound","prior_art_state":"not_assessed"},{"semantic_node_id":"Z16::one_bit_cutoff_promotion","prior_art_state":"not_assessed"},{"semantic_node_id":"Z16::profiled_gap_advance","prior_art_state":"not_assessed"},{"semantic_node_id":"Z16::rewind_history_promotion_dichotomy","prior_art_state":"not_assessed"},{"semantic_node_id":"Z16::suffix_cylinder_endpoint_propagation","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:selectsuffix","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"requires_mathematical_adjudication","render_policy":"historical_only_with_recorded_difference","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2051","finding_axes":["negative_clause_loss"],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:selected_suffix_pre_feedback_run","assertion_kind":"registered_claim","source_id":"selected_suffix_pre_feedback_run","reviewed":true,"granularity":"single_registered_claim","label":"Verified finite suffix-cylinder stages through depth 52","statement":"A common depth-13 prefix supports full selected windows at every depth 18 through 27, and the threshold continuation reaches depth 29 while covering the protected strip. At depth 29, cutoffs 14 through 25 remain common and cutoff 26 does not. A separate exact recurrence reaches depth 51 with endpoint 51,327,745 beyond the next head threshold, and the promoted depth-52 successor supplies a terminal-strip witness. These are finite stages, not an unbounded induction.","status":"verified finite instance","status_meaning":"Kernel-checked computation at a bounded range","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"nonempty_stage_of_eighteen_le_of_le_twentySeven","module":"Erdos249257/SelectedSuffixCylinderPreFeedback.lean","line":118,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SelectedSuffixCylinderPreFeedback.lean#L118","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SelectedSuffixCylinderPreFeedback.lean#L118","kind":"theorem","signature":"theorem nonempty_stage_of_eighteen_le_of_le_twentySeven {N : ℕ} (h18 : 18 ≤ N) (h27 : N ≤ 27) : Nonempty (Stage N)","docstring":"Uniform finite-range form of the pre-feedback construction. This is the consumer-facing statement: every depth from the certified base through the last row before feedback carries a full selected window with the same depth-13 cylinder invariant.","generated_certificate":false},{"name":"depth27SelectedHalfWindow_hasSuffixCylinderAt_thirteen","module":"Erdos249257/SelectedSuffixCylinderPreFeedback.lean","line":141,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SelectedSuffixCylinderPreFeedback.lean#L141","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SelectedSuffixCylinderPreFeedback.lean#L141","kind":"theorem","signature":"theorem depth27SelectedHalfWindow_hasSuffixCylinderAt_thirteen : HasSuffixCylinderAt depth27SelectedHalfWindow (by omega : 13 ≤ 27) stage27.endpoint","docstring":"Every carry in the depth-27 protected strip shares the depth-13 prefix and has an endpoint-indexed suffix.","generated_certificate":false},{"name":"depth27SelectedHalfWindow_endpoint_covers_strip","module":"Erdos249257/SelectedSuffixCylinderPreFeedback.lean","line":147,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SelectedSuffixCylinderPreFeedback.lean#L147","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SelectedSuffixCylinderPreFeedback.lean#L147","kind":"theorem","signature":"theorem depth27SelectedHalfWindow_endpoint_covers_strip : halfStripBound 27 ≤ stage27.endpoint","docstring":"The propagated endpoint still covers the full depth-27 carry strip.","generated_certificate":false},{"name":"stage27_endpoint_eq","module":"Erdos249257/SuffixCylinderFirstFeedback.lean","line":144,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderFirstFeedback.lean#L144","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderFirstFeedback.lean#L144","kind":"theorem","signature":"theorem stage27_endpoint_eq : stage27.endpoint = 8389","generated_certificate":false},{"name":"depth29SelectedHalfWindow_endpoint_covers_strip","module":"Erdos249257/SuffixCylinderThreshold.lean","line":700,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderThreshold.lean#L700","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderThreshold.lean#L700","kind":"theorem","signature":"theorem depth29SelectedHalfWindow_endpoint_covers_strip : halfStripBound 29 ≤ 782","generated_certificate":false},{"name":"depth29SelectedHalfWindow_not_commonRestrictionAt_twentySix","module":"Erdos249257/SuffixCylinderThreshold.lean","line":806,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderThreshold.lean#L806","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderThreshold.lean#L806","kind":"theorem","signature":"theorem depth29SelectedHalfWindow_not_commonRestrictionAt_twentySix : ¬ depth29SelectedHalfWindow.CommonRestrictionAt (by omega : 26 ≤ 29)","docstring":"Cutoff 25 is maximal: the next suffix bit splits the protected interval exactly across weight eight.","generated_certificate":false},{"name":"fullCylinderStage51_feedback_headThreshold_margin","module":"Erdos249257/SuffixCylinderStage51.lean","line":536,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderStage51.lean#L536","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderStage51.lean#L536","kind":"theorem","signature":"theorem fullCylinderStage51_feedback_headThreshold_margin : 2 ^ (51 - (25 + 1)) + halfStripBound 51 ≤ fullCylinderStage51.endpoint","generated_certificate":false},{"name":"fullCylinderStage52_halfTerminalOnlyStripWitness","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":238,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L238","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L238","kind":"theorem","signature":"theorem fullCylinderStage52_halfTerminalOnlyStripWitness : HalfTerminalOnlyStripWitness 52","docstring":"The explicitly promoted second-feedback successor gives the next concrete terminal strip witness as well.","generated_certificate":false}],"paper_label":"res:prefeedback","paper_anchor":{"paper_label":"res:prefeedback","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2149,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Verified finite suffix-cylinder stages","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Finite evidence, coordinate changes, and eliminated special cases","title_tex":"Finite evidence, coordinate changes, and eliminated special cases","line":1967},{"level":"subsubsection","title":"Verified finite suffix-cylinder stages","title_tex":"Verified finite suffix-cylinder stages","line":2149}],"source_excerpt":{"text":"\\subsubsection{Verified finite suffix-cylinder stages}\\label{res:prefeedback}","media_type":"text/x-tex","source_char_start":113008,"source_char_end":113085,"start_line":2149,"end_line":2149,"truncated":false,"sha256":"sha256:27400e698ce58524ad50ff49043ea4b4f9d6b91a73bb8fae94c337bf72e4115f","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Yes refutes universal #257; no closes only this route.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/4"}],"links":[{"kind":"lean_declaration_pinned","label":"nonempty_stage_of_eighteen_le_of_le_twentySeven","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SelectedSuffixCylinderPreFeedback.lean#L118","cite_this":true},{"kind":"lean_declaration_pinned","label":"depth27SelectedHalfWindow_hasSuffixCylinderAt_thirteen","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SelectedSuffixCylinderPreFeedback.lean#L141","cite_this":true},{"kind":"lean_declaration_pinned","label":"depth27SelectedHalfWindow_endpoint_covers_strip","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SelectedSuffixCylinderPreFeedback.lean#L147","cite_this":true},{"kind":"lean_declaration_pinned","label":"stage27_endpoint_eq","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderFirstFeedback.lean#L144","cite_this":true},{"kind":"lean_declaration_pinned","label":"depth29SelectedHalfWindow_endpoint_covers_strip","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderThreshold.lean#L700","cite_this":true},{"kind":"lean_declaration_pinned","label":"depth29SelectedHalfWindow_not_commonRestrictionAt_twentySix","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderThreshold.lean#L806","cite_this":true},{"kind":"lean_declaration_pinned","label":"fullCylinderStage51_feedback_headThreshold_margin","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderStage51.lean#L536","cite_this":true},{"kind":"lean_declaration_pinned","label":"fullCylinderStage52_halfTerminalOnlyStripWitness","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L238","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/57","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"selected_suffix_pre_feedback_run"}}],"expect":"object"},"scope_limit":{"text":"Every depth from 18 through 29, plus explicit full stages at depths 51 and 52; depth 29 has common suffix cylinders at cutoffs 14 through 25 but not 26.","source":"claims_registry_bounded_domain"},"prior_art":{"family_id":"suffix_cylinders_and_feedback_boundary","status_summary":"An exact conditional child-gap recurrence with a cofinal half-value consequence, verified finite stages through depth 52, an exact unit drop, and a concrete scalar-seam obstruction.","prior_art_posture":"Kovač–Tao supply the surrounding strict-tail achievement-set geometry. No exact antecedent for this finite carry recurrence is recorded, and no novelty claim is made.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/10","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z16::certified_terminal_witnesses_51_52","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z16::depth29_cutoff26_promotion_impossible","prior_art_state":"not_assessed"},{"semantic_node_id":"Z16::pre_feedback_run_to_depth_27","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:prefeedback","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"requires_mathematical_adjudication","render_policy":"historical_only_with_recorded_difference","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2149","finding_axes":["negative_clause_loss"],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:half_divisor_unit_drop","assertion_kind":"registered_claim","source_id":"half_divisor_unit_drop","reviewed":true,"granularity":"single_registered_claim","label":"Half-divisor unit coefficient drop at the feedback row","statement":"At the doubled feedback row, switching only the terminal half-divisor bit from false to true changes the support coefficient from C to C+1 and, under K' = 2K-C, drops the successor carry by exactly one. The same identity holds for any explicitly identified adjacent boundary pair, and the rewind adapter transfers such a shared-cutoff pair to the next protected seam while retaining the unit-drop witness. This does not construct such pairs globally.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"gateway_section","warrant":"the claim's anchor sits inside the retired gateway section 'Erdős--Borwein-type irrationality (the #257 direction)', which is the author's own partition of that manuscript by problem"},"problem_edges":[{"problem_id":"erdos_257","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"supportCoeff_extend_true_eq_false_add_one_at_double","module":"Erdos249257/HalfDivisorUnitDrop.lean","line":20,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfDivisorUnitDrop.lean#L20","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfDivisorUnitDrop.lean#L20","kind":"theorem","signature":"theorem supportCoeff_extend_true_eq_false_add_one_at_double {N : ℕ} (a : HalfWord N) : supportCoeff (wordSupport (extendHalfWord a true)) (2 * (N + 1)) = supportCoeff (wordSupport (extendHalfWord a false)) (2 * (N + 1)) + 1","docstring":"At row `2(N+1)`, changing only exponent `N+1` from false to true adds exactly the half-divisor incidence.","generated_certificate":false},{"name":"supportCoeff_boundaryPair_unitDrop_at_double","module":"Erdos249257/HalfDivisorUnitDrop.lean","line":35,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfDivisorUnitDrop.lean#L35","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfDivisorUnitDrop.lean#L35","kind":"theorem","signature":"theorem supportCoeff_boundaryPair_unitDrop_at_double {N : ℕ} (a : HalfWord N) (left right : HalfWord (N + 1)) (hleft : left = extendHalfWord a true) (hright : right = extendHalfWord a false) : supportCoeff (wordSupport left) (2 * (N + 1)) = supportCoeff (wordSupport right) (2 * (N + 1)) + 1","docstring":"Proof-irrelevant word equalities may be supplied by any concrete adjacent-boundary construction.","generated_certificate":false},{"name":"rewindBaseUnitDropAt_of_halfDivisorBoundaryPair","module":"Erdos249257/RewindHalfDivisorAdapter.lean","line":86,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/RewindHalfDivisorAdapter.lean#L86","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/RewindHalfDivisorAdapter.lean#L86","kind":"theorem","signature":"theorem rewindBaseUnitDropAt_of_halfDivisorBoundaryPair {K M R₀ : ℕ} (W₀ : SelectedHalfWindow M R₀) (hKM : K + 1 ≤ M) (hMlt : M < 2 * (K + 1)) (cs : List ℕ) (a : HalfWord K) (hpair : RewindBaseHalfDivisorBoundaryPairAt W₀ hKM cs a) : RewindBaseUnitDropAt W₀ (2 * (K + 1)) cs (supportCoeff (wordSupport (extendHalfWord a false)) (2 * (K + 1)))","docstring":"A literal half-divisor boundary pair supplies exactly the operational unit-drop profile at the doubled row inspected by that half divisor.","generated_certificate":false},{"name":"protectedEvenSeamRealizedAt_succ_of_fullStrip_rewindHalfDivisorBoundaryPair","module":"Erdos249257/RewindHalfDivisorAdapter.lean","line":147,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/RewindHalfDivisorAdapter.lean#L147","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/RewindHalfDivisorAdapter.lean#L147","kind":"theorem","signature":"theorem protectedEvenSeamRealizedAt_succ_of_fullStrip_rewindHalfDivisorBoundaryPair {K M N R₀ : ℕ} (W₀ : SelectedHalfWindow M R₀) (W : SelectedHalfWindow N (halfStripBound N)) (hKM : K + 1 ≤ M) (hMlt : M < 2 * (K + 1)) (hMN : M ≤ N) (hhalf : (N + 1) / 2 ≤ M) (hrow : N + 1 = 2 * (K + 1)) (cs : List ℕ) (a : HalfWord K) (hN : 1 ≤ N) (hwidth : halfStripBound N ≤ denom cs) (hprov : W.HasRewindRestrictionAt W₀ hMN cs) (hseam : IsSeamPair cs (halfStripBound N)) (h27 : 27 ≤ halfStripBound (N + 1)) (hpair : RewindBaseHalfDivisorBoundaryPairAt W₀ hKM cs a) : ProtectedEvenSeamRealizedAt (N + 1) (seamCut cs : ℤ) (supportCoeff (wordSupport (extendHalfWord a false)) (N + 1) : ℤ)","docstring":"Full-strip specialization. The existing divisor envelope supplies the buffer once the literal half-divisor boundary pair is identified.","generated_certificate":false}],"paper_label":"res:halfdivisordrop","paper_anchor":{"paper_label":"res:halfdivisordrop","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2178,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Half-divisor unit drop","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Finite evidence, coordinate changes, and eliminated special cases","title_tex":"Finite evidence, coordinate changes, and eliminated special cases","line":1967},{"level":"subsubsection","title":"Half-divisor unit drop","title_tex":"Half-divisor unit drop","line":2178}],"source_excerpt":{"text":"\\subsubsection{Half-divisor unit drop}\\label{res:halfdivisordrop}","media_type":"text/x-tex","source_char_start":114767,"source_char_end":114832,"start_line":2178,"end_line":2178,"truncated":false,"sha256":"sha256:8089cb852abb9176636e7037332cb7ccdfdc43f2d424455bf2dcfd17e3be94a0","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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No exact antecedent for this finite carry recurrence is recorded, and no novelty claim is made.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/10","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z15::boundary_pair_realizes_protected_seam","prior_art_state":"not_assessed"},{"semantic_node_id":"Z15::boundary_pair_supplies_unit_drop","prior_art_state":"not_assessed"},{"semantic_node_id":"Z15::half_divisor_bit_unit_drop","prior_art_state":"routine_corollary_or_specialisation"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:halfdivisordrop","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2178","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:rewind_boundary_pair_obstruction","assertion_kind":"registered_claim","source_id":"rewind_boundary_pair_obstruction","reviewed":true,"granularity":"single_registered_claim","label":"Finite rewind boundary-pair obstruction","statement":"A concrete depth-26 selected two-word window has a depth-27 explicit rewind step with history [1], a width-two numerical seam, and the doubled-row identity, but no depth-13 half-divisor boundary-pair witness. Therefore scalar one-row seam data alone does not entail the structural boundary predicate required by the cofinal consumer.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"gateway_section","warrant":"the claim's anchor sits inside the retired gateway section 'Erdős--Borwein-type irrationality (the #257 direction)', which is the author's own partition of that manuscript by problem"},"problem_edges":[{"problem_id":"erdos_257","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"depth27_actualRewindSeam_not_boundaryPair","module":"Erdos249257/RewindBoundaryPairCounterexample.lean","line":169,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/RewindBoundaryPairCounterexample.lean#L169","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/RewindBoundaryPairCounterexample.lean#L169","kind":"theorem","signature":"theorem depth27_actualRewindSeam_not_boundaryPair : depth27FirstTwo.HasRewindRestrictionAt depth26FirstTwo (by omega : 26 ≤ 27) [1] ∧ IsSeamPair [1] 2 ∧ 2 ≤ denom [1] ∧ 27 + 1 = 2 * (13 + 1) ∧ ¬ ∃ a : HalfWord 13, RewindBaseHalfDivisorBoundaryPairAt depth26FirstTwo (by omega : 13 + 1 ≤ 26) [1] a","docstring":"Full concrete falsifier. This includes an actual explicit selected window step, exact rewind provenance, the width hypothesis, the scalar seam, and the doubled-row identity. Only the large protected-window buffer used by the cofinal consumer is intentionally absent.","generated_certificate":false}],"paper_label":"res:rewindboundarycounterexample","paper_anchor":{"paper_label":"res:rewindboundarycounterexample","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2195,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"A finite rewind-boundary obstruction","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Finite evidence, coordinate changes, and eliminated special cases","title_tex":"Finite evidence, coordinate changes, and eliminated special cases","line":1967},{"level":"subsubsection","title":"A finite rewind-boundary obstruction","title_tex":"A finite rewind-boundary obstruction","line":2195}],"source_excerpt":{"text":"\\subsubsection{A finite rewind-boundary obstruction}\\label{res:rewindboundarycounterexample}","media_type":"text/x-tex","source_char_start":115914,"source_char_end":116006,"start_line":2195,"end_line":2195,"truncated":false,"sha256":"sha256:4c654938af674a4a999b46dfbf2b47d4a27b91be1e3104c0c068082f74416e4e","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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No exact antecedent for this finite carry recurrence is recorded, and no novelty claim is made.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/10","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z15::boundary_pair_not_forced_by_scalar_seam","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:rewindboundarycounterexample","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2195","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:concrete_producer_carry_coordinates","assertion_kind":"registered_claim","source_id":"concrete_producer_carry_coordinates","reviewed":true,"granularity":"single_registered_claim","label":"Concrete adjacent-cut producer-carry coordinates","statement":"For every adjacent cut at rank s ≥ 5, the below and above words are the neighbouring subset sums bracketing the target; their remainder and overshoot are the unsigned distances, and their pulses are the paired divisor-incidence contributions of the next two rows. After adjoining terminal rank s, the below producer carry is 4 times the remainder minus the below pulse minus 4, while the above producer carry is the negative of 4 times the overshoot plus the above pulse plus 4. Hence the upper carry is negative and the middle sign depends on the remainder-pulse competition. These exact identities supply no uniform exponential lower bound, sign persistence at unbounded ranks, or universal escape argument.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"gateway_section","warrant":"the claim's anchor sits inside the retired gateway section 'Erdős--Borwein-type irrationality (the #257 direction)', which is the author's own partition of that manuscript by problem"},"problem_edges":[{"problem_id":"erdos_257","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"producerCarry_insert_seamWordSupport_eq","module":"Erdos249257/HalfCylinderProducerLowerBound.lean","line":105,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderProducerLowerBound.lean#L105","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderProducerLowerBound.lean#L105","kind":"theorem","signature":"theorem producerCarry_insert_seamWordSupport_eq {s : ℕ} (hs : 5 ≤ s) (b : SeamRowWord s) : producerCarry (insert s (↑(seamWordSupport b) : Set ℕ)) s = 4 * (wordDefect s b.toNatWord - 1) - (wordPulse s b.toNatWord : ℤ)","docstring":"Inserting the terminal rank into any concrete seam word turns its signed integer subset-sum defect into the producer-aligned affine carry.","generated_certificate":false},{"name":"producerCarry_insert_seamBelowSupport_eq_middleCoordinate","module":"Erdos249257/HalfCylinderProducerLowerBound.lean","line":150,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderProducerLowerBound.lean#L150","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderProducerLowerBound.lean#L150","kind":"theorem","signature":"theorem producerCarry_insert_seamBelowSupport_eq_middleCoordinate (s : ℕ) (hs : 5 ≤ s) : producerCarry (insert s (↑(seamWordSupport (seamAdjacentCut s hs).below) : Set ℕ)) s = 4 * ((seamAdjacentCut s hs).remainder : ℤ) - ((seamAdjacentCut s hs).belowPulse : ℤ) - 4","docstring":"The middle-producer coordinate is exactly the producer carry of the terminal-augmented below support.","generated_certificate":false},{"name":"producerCarry_insert_seamAboveSupport_eq_neg_upperCoordinate","module":"Erdos249257/HalfCylinderProducerLowerBound.lean","line":183,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCylinderProducerLowerBound.lean#L183","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderProducerLowerBound.lean#L183","kind":"theorem","signature":"theorem producerCarry_insert_seamAboveSupport_eq_neg_upperCoordinate (s : ℕ) (hs : 5 ≤ s) : producerCarry (insert s (↑(seamWordSupport (seamAdjacentCut s hs).above) : Set ℕ)) s = -((4 * (seamAdjacentCut s hs).overshoot + (seamAdjacentCut s hs).abovePulse + 4 : ℕ) : ℤ)","docstring":"The upper-producer coordinate, with the unavoidable four-unit terminal shift, is the negative producer carry of the terminal-augmented above support.","generated_certificate":false}],"paper_label":"res:producercarrycoordinates","paper_anchor":{"paper_label":"res:producercarrycoordinates","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2206,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Explicit carry coordinates at the final non-right transition","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Finite evidence, coordinate changes, and eliminated special cases","title_tex":"Finite evidence, coordinate changes, and eliminated special cases","line":1967},{"level":"subsubsection","title":"Explicit carry coordinates at the final non-right transition","title_tex":"Explicit carry coordinates at the final non-right transition","line":2205}],"source_excerpt":{"text":"\\subsubsection{Explicit carry coordinates at the final non-right transition}\n\\label{res:producercarrycoordinates}","media_type":"text/x-tex","source_char_start":116564,"source_char_end":116677,"start_line":2205,"end_line":2206,"truncated":false,"sha256":"sha256:cd4f3e41f90f3c97edf28a8c23774c6e800259b5703bb1c6e1d9263098a0d9f1","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2206","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"producerCarry_insert_seamWordSupport_eq","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderProducerLowerBound.lean#L105","cite_this":true},{"kind":"lean_declaration_pinned","label":"producerCarry_insert_seamBelowSupport_eq_middleCoordinate","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderProducerLowerBound.lean#L150","cite_this":true},{"kind":"lean_declaration_pinned","label":"producerCarry_insert_seamAboveSupport_eq_neg_upperCoordinate","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCylinderProducerLowerBound.lean#L183","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/60","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"concrete_producer_carry_coordinates"}}],"expect":"object"},"prior_art":{"family_id":"fatal_gap_and_last_producer_spine","status_summary":"Exact half-membership/nonmembership classifications, proved branch eliminations, a rank-26 phase sieve for the middle -2 cell, and explicit conditional tail consumers.","prior_art_posture":"No exact local seam or last-producer antecedent is recorded beyond strict-tail achievement-set background.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/8","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"formalisation_only":1},"edges":[{"semantic_node_id":"Z14::producer_carry_concrete_seam_coordinates","prior_art_state":"formalisation_only"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:producercarrycoordinates","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"requires_mathematical_adjudication","render_policy":"historical_only_with_recorded_difference","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2206","finding_axes":["negative_clause_loss"],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:mersenne_lambert_ladder","assertion_kind":"registered_claim","source_id":"mersenne_lambert_ladder","reviewed":true,"granularity":"single_registered_claim","label":"The Mersenne–Lambert ladder identities","statement":"L(μ) = 1/2, L(φ) = 2, L(φ*μ) = S, and the Möbius-square lens S = 1/2 + ∑ μ(d)/(2^d − 1)², placing the open constant one convolution from the machine-checked irrational E.","status":"formalised here","status_meaning":"Known mathematics represented and checked in Lean","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"programme_representative","warrant":"orientation.json lists this claim among the representative claims of a mathematical programme whose remaining open propositions all name one problem"},"problem_edges":[{"problem_id":"erdos_249","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":["lambert_obstruction_interfaces"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"tsum_totient_div_two_pow_sub_one_eq_two","module":"Erdos249257/CertificateKernel.lean","line":18429,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CertificateKernel.lean#L18429","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L18429","kind":"theorem","signature":"theorem tsum_totient_div_two_pow_sub_one_eq_two : ∑' d : ℕ+, (Nat.totient (d : ℕ) : ℝ) / ((2 : ℝ) ^ (d : ℕ) - 1) = 2","docstring":"**Rational rung `L(φ) = 2`**: `∑_{d≥1} φ(d)/(2^d-1) = 2`, exactly. The Lambert transform of the #249 coefficients at the #249 point is rational — the difficulty of #249 lives entirely on the power-series side.","generated_certificate":false},{"name":"totient_series_eq_half_add_moebius_mersenne_square","module":"Erdos249257/CertificateKernel.lean","line":18454,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CertificateKernel.lean#L18454","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L18454","kind":"theorem","signature":"theorem totient_series_eq_half_add_moebius_mersenne_square : (∑' n : ℕ, ((Nat.totient n : ℝ)) / (2 : ℝ) ^ n) = 1 / 2 + ∑' d : ℕ+, ((ArithmeticFunction.moebius (d : ℕ) : ℤ) : ℝ) / ((2 : ℝ) ^ (d : ℕ) - 1) ^ 2","docstring":"**Möbius-square lens**: `∑ φ(n)/2ⁿ = 1/2 + ∑_{d≥1} μ(d)/(2^d-1)²`. The #249 constant as a signed squared-Mersenne Lambert series.","generated_certificate":false}],"paper_label":"res:lens","paper_anchor":{"paper_label":"res:lens","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":3929,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":3929,"anchor_title":"Möbius-square identity for S","anchor_title_tex":"M\\\"obius-square identity for \\(S\\)","section_title":"Finite denominator-clearing boundaries.","section_path":[{"level":"section","title":"Exact remaining problems and further questions","title_tex":"Exact remaining problems and further questions","line":3819},{"level":"subsection","title":"Finite algebraic boundaries and squared-Lambert coordinates","title_tex":"Finite algebraic boundaries and squared-Lambert coordinates","line":3873},{"level":"paragraph","title":"Finite denominator-clearing boundaries.","title_tex":"Finite denominator-clearing boundaries.","line":3882}],"source_excerpt":{"text":"\\begin{proposition}[M\\\"obius-square identity for \\(S\\)]\\label{res:lens}\n\\[\nS \\;=\\; \\tfrac12 \\;+\\; \\sum_{d\\ge1}\\frac{\\mu(d)}{(2^d-1)^2}.\n\\]\n\\end{proposition}","media_type":"text/x-tex","source_char_start":206722,"source_char_end":206878,"start_line":3929,"end_line":3933,"truncated":false,"sha256":"sha256:16b31ed945262de604cbeddb92a2055509878f78815b01a7a9e92db57715fa49","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3929","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"tsum_totient_div_two_pow_sub_one_eq_two","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L18429","cite_this":true},{"kind":"lean_declaration_pinned","label":"totient_series_eq_half_add_moebius_mersenne_square","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L18454","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/61","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"mersenne_lambert_ladder"}}],"expect":"object"},"prior_art":{"family_id":"mersenne_lambert_classical_ladder","status_summary":"Classical identities formalised and one transcendental rung cited only.","prior_art_posture":"Apostol, Merca, and Merca–Schmidt own the transform background; Nesterenko owns the cited transcendence theorem.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/11","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"known_classical":1,"not_assessed":1,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z01::lambert_ladder_rational_rungs","prior_art_state":"known_classical"},{"semantic_node_id":"Z01::totient_series_moebius_square_lens","prior_art_state":"not_assessed"},{"semantic_node_id":"Z07::lambert_rung_totient_rational","prior_art_state":"routine_corollary_or_specialisation"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:lens","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3929","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:totient_coprimality_probability","assertion_kind":"registered_claim","source_id":"totient_coprimality_probability","reviewed":true,"granularity":"single_registered_claim","label":"Fair-coin coprimality form of the totient constant","statement":"For independent fair-coin geometric waiting times X and Y on the positive integers, S = 1/2 + P(gcd(X,Y)=1), where the probability is the visible coprime-pair series. That probability has no rational representation with denominator at most 39819823323350687661677887437915526. This is an exact reformulation and quantitative corollary; irrationality remains open.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":["probabilistic_gcd_geometry"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"totient_series_eq_half_add_visible_coprime_pairs","module":"Erdos249257/CertificateKernel.lean","line":18557,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CertificateKernel.lean#L18557","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L18557","kind":"theorem","signature":"theorem totient_series_eq_half_add_visible_coprime_pairs : (∑' n : ℕ, ((Nat.totient n : ℝ)) / (2 : ℝ) ^ n) = 1 / 2 + ∑' p : ℕ × ℕ, (if 0 < p.1 ∧ 0 < p.2 ∧ Nat.Coprime p.1 p.2 then ((1 : ℝ) / 2) ^ (p.1 + p.2) else 0)","docstring":"**Fair-coin geometric coprimality**: `∑ φ(n)/2ⁿ = 1/2 + P(gcd(X,Y) = 1)`, where the probability is spelled as its defining series `∑_{(a,b) coprime, a,b ≥ 1} 2^{-(a+b)}`. The boundary term `1/2` is `P(X = 1) = L(μ)` — the ladder's Möbius rung. Erdős #249 is equivalent to irrationality of this coprimality probability.","generated_certificate":false},{"name":"tsum_visible_coprime_pairs_ne_int_div_of_den_le_39819823323350687661677887437915526","module":"Erdos249257/CertificateKernel.lean","line":18572,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CertificateKernel.lean#L18572","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L18572","kind":"theorem","signature":"theorem tsum_visible_coprime_pairs_ne_int_div_of_den_le_39819823323350687661677887437915526 : ∀ (a : ℤ) (d : ℕ), 0 < d → d ≤ 39819823323350687661677887437915526 → (∑' p : ℕ × ℕ, if 0 < p.1 ∧ 0 < p.2 ∧ Nat.Coprime p.1 p.2 then ((1 : ℝ) / 2) ^ (p.1 + p.2) else 0) ≠ (a : ℝ) / (d : ℝ)","docstring":"**Record-bound transfer to the coprimality probability.** The probability that two independent fair-coin waiting times are coprime, `P(gcd(X,Y) = 1) = ∑_{(a,b)=1} 2^{-(a+b)}`, has no rational representation with denominator `≤ 39819823323350687661677887437915526` (half the wave-17.5 Farey record): `P = a/d` would force `S = (d+2a)/(2d)` with `2d` under the record bound.","generated_certificate":false}],"paper_label":"res:coprime","paper_anchor":{"paper_label":"res:coprime","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2409,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":2409,"anchor_title":"coprimality probability","anchor_title_tex":"coprimality probability","section_title":"The geometric picture: S as coprime-pair mass","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"The geometric picture: S as coprime-pair mass","title_tex":"The geometric picture: $S$ as coprime-pair mass","line":2398}],"source_excerpt":{"text":"\\begin{proposition}[coprimality probability]\\label{res:coprime}\n\\[\nS \\;=\\; \\tfrac12 \\;+\\; \\Pr\\!\\big(\\gcd(X,Y)=1\\big)\n\\;=\\; \\tfrac12 \\;+\\!\\!\\sum_{\\substack{a,b\\ge1\\\\ \\gcd(a,b)=1}}\\!\\! 2^{-(a+b)}.\n\\]\n\\end{proposition}","media_type":"text/x-tex","source_char_start":128238,"source_char_end":128453,"start_line":2409,"end_line":2414,"truncated":false,"sha256":"sha256:41b656f949fa3b223deb9ceec2c6df40bba459628cc23dc06e900921bf34001a","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2409","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_249_irrationality","statement":"Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/1"}],"links":[{"kind":"lean_declaration_pinned","label":"totient_series_eq_half_add_visible_coprime_pairs","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L18557","cite_this":true},{"kind":"lean_declaration_pinned","label":"tsum_visible_coprime_pairs_ne_int_div_of_den_le_39819823323350687661677887437915526","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L18572","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/62","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"totient_coprimality_probability"}}],"expect":"object"},"prior_art":{"family_id":"probabilistic_gcd_and_stern_brocot_geometry","status_summary":"Exact coprimality and gcd-moment identities, a Stern–Brocot cylinder law with uniform geometric convergence, and source-level Fibonacci/continuant run stability.","prior_art_posture":"Classical Lambert-series, convolution, Pillai gcd-sum, probability, and Stern–Brocot background; Postelmans–Van Assche own the cited irrationality theorem for the constant-weight squared-Lambert value. No priority claim is made for this assembled formal package.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/12","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1,"prior_art_found":1},"edges":[{"semantic_node_id":"Z01::totient_denominator_exclusion_ladder","prior_art_state":"prior_art_found"},{"semantic_node_id":"Z01::visible_lattice_coprimality","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:coprime","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2409","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:squared_lambert_gcd_moments","assertion_kind":"registered_claim","source_id":"squared_lambert_gcd_moments","reviewed":true,"granularity":"single_registered_claim","label":"Squared-Lambert identities for divisor count and the first gcd moment","statement":"The squared-Mersenne Lambert series with weight 1 equals the base-1/2 series of sigma(n)-tau(n), and the series with weight phi equals the base-1/2 series of P(n)-n, where P is Pillai's gcd-sum function. These are the exact divisor-count and first-moment identities for the fair-coin gcd law; the separately cited irrationality of the constant-weight value is not claimed as formalised here.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"programme_representative","warrant":"orientation.json lists this claim among the representative claims of a mathematical programme whose remaining open propositions all name one problem"},"problem_edges":[{"problem_id":"erdos_249","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":["probabilistic_gcd_geometry"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"tsum_one_div_mersenne_sq_eq_sigma_sub_tau_series","module":"Erdos249257/GcdMomentCalculus.lean","line":216,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GcdMomentCalculus.lean#L216","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GcdMomentCalculus.lean#L216","kind":"theorem","signature":"theorem tsum_one_div_mersenne_sq_eq_sigma_sub_tau_series : ∑' d : ℕ+, 1 / ((2 : ℝ) ^ (d : ℕ) - 1) ^ 2 = ∑' n : ℕ+, ((∑ e ∈ (n : ℕ).divisors, (((n : ℕ) / e : ℕ) : ℝ)) - ((n : ℕ).divisors.card : ℝ)) * ((1 : ℝ) / 2) ^ (n : ℕ)","docstring":"**The q-zeta anchor of the level-2 ladder**: `∑_{d≥1} 1/(2ᵈ-1)² = ∑_{n≥1} (σ(n) - τ(n))·(1/2)ⁿ`, exactly. Probability reading: `E[τ(gcd(X,Y))] = ∑_d P(d ∣ gcd)`. In q-zeta terms this value is `ζ_q(2) - ζ_q(1)` at `q = 1/2` (Jouhet–Mosaki convention), irrational by the Postelmans–Van Assche q-Padé theorem — cited, NOT formalised; what is machine-checked here is the identity.","generated_certificate":false},{"name":"tsum_totient_div_mersenne_sq_eq_gcd_moment_series","module":"Erdos249257/GcdMomentCalculus.lean","line":235,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GcdMomentCalculus.lean#L235","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GcdMomentCalculus.lean#L235","kind":"theorem","signature":"theorem tsum_totient_div_mersenne_sq_eq_gcd_moment_series : ∑' d : ℕ+, (Nat.totient (d : ℕ) : ℝ) / ((2 : ℝ) ^ (d : ℕ) - 1) ^ 2 = ∑' n : ℕ+, ((∑ e ∈ (n : ℕ).divisors, (Nat.totient e : ℝ) * (((n : ℕ) / e : ℕ) : ℝ)) - ((n : ℕ) : ℝ)) * ((1 : ℝ) / 2) ^ (n : ℕ)","docstring":"**The first gcd moment**: `∑_{d≥1} φ(d)/(2ᵈ-1)² = ∑_{n≥1} (P(n) - n)·(1/2)ⁿ` with `P` Pillai's gcd-sum function. Probability reading: `E[gcd(X,Y)] = ∑_d φ(d)·P(d ∣ gcd)` — the expected gcd of two independent fair-coin waiting times, as a totient-weighted squared-Mersenne series.","generated_certificate":false}],"paper_label":"res:gcdmoment","paper_anchor":{"paper_label":"res:gcdmoment","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":3950,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":3950,"anchor_title":"squared Lambert identity","anchor_title_tex":"squared Lambert identity","section_title":"Squared Lambert calculus.","section_path":[{"level":"section","title":"Exact remaining problems and further questions","title_tex":"Exact remaining problems and further questions","line":3819},{"level":"subsection","title":"Finite algebraic boundaries and squared-Lambert coordinates","title_tex":"Finite algebraic boundaries and squared-Lambert coordinates","line":3873},{"level":"paragraph","title":"Squared Lambert calculus.","title_tex":"Squared Lambert calculus.","line":3942}],"source_excerpt":{"text":"\\begin{proposition}[squared Lambert identity]\\label{res:gcdmoment}\nThe three weights \\(f=\\mu,\\mathbf 1,\\ph\\) give respectively\n\\[\n L^{(2)}(\\mu)=S-\\tfrac12,\\qquad\n L^{(2)}(\\mathbf 1)=\\sum_{n\\ge1}\\frac{\\sigma(n)-\\tau(n)}{2^n},\\qquad\n L^{(2)}(\\ph)=\\sum_{n\\ge1}\\frac{P(n)-n}{2^n},\n\\]\nwhere \\(P\\) is Pillai's gcd-sum function.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":207954,"source_char_end":208293,"start_line":3950,"end_line":3958,"truncated":false,"sha256":"sha256:2c5a5a73e8c0f17c626d9e1fb42d4bcedfcb8b0eee98d1c0e538a6a8d12dc044","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3950","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"tsum_one_div_mersenne_sq_eq_sigma_sub_tau_series","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GcdMomentCalculus.lean#L216","cite_this":true},{"kind":"lean_declaration_pinned","label":"tsum_totient_div_mersenne_sq_eq_gcd_moment_series","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GcdMomentCalculus.lean#L235","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/63","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"squared_lambert_gcd_moments"}}],"expect":"object"},"prior_art":{"family_id":"probabilistic_gcd_and_stern_brocot_geometry","status_summary":"Exact coprimality and gcd-moment identities, a Stern–Brocot cylinder law with uniform geometric convergence, and source-level Fibonacci/continuant run stability.","prior_art_posture":"Classical Lambert-series, convolution, Pillai gcd-sum, probability, and Stern–Brocot background; Postelmans–Van Assche own the cited irrationality theorem for the constant-weight squared-Lambert value. No priority claim is made for this assembled formal package.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/12","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"known_classical":1,"not_assessed":1,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z01::gcd_moment_squared_lambert","prior_art_state":"not_assessed"},{"semantic_node_id":"Z07::squared_kernel_totient_rung","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z07::squared_kernel_zeta_rung","prior_art_state":"known_classical"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:gcdmoment","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3950","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:stern_brocot_cylinder_law","assertion_kind":"registered_claim","source_id":"stern_brocot_cylinder_law","reviewed":true,"granularity":"single_registered_claim","label":"Reduced-direction normalization and Stern–Brocot cylinder law","statement":"The Mersenne weights of positive reduced slopes sum exactly to one. At every positive Stern–Brocot node (a,b), the cylinder mass 1/((2^a-1)(2^b-1)) splits into its stop mass and two child cylinders; the depth-d unfolding has error between zero and (2/3)^d times the cylinder mass and converges to the closed form.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249","erdos_257"],"problem_erdos_numbers":[249,257],"problem_binding":{"method":"core_programme_joint","warrant":"the claim's only paper anchor is owned by the retired combined #249/#257 exposition, which treats both problems together and publishes nothing that separates them. Bound to both rather than guessed apart.","separability":"not_separable_from_published_source"},"problem_edges":[{"problem_id":"erdos_249","relation":"shared_programme_context","basis":"problem_binding"},{"problem_id":"erdos_257","relation":"shared_programme_context","basis":"problem_binding"}],"programme_ids":["joint_249_257_gateway"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"tsum_pos_coprime_inv_mersenne_eq_one","module":"Erdos249257/GcdMomentCalculus.lean","line":349,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GcdMomentCalculus.lean#L349","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GcdMomentCalculus.lean#L349","kind":"theorem","signature":"theorem tsum_pos_coprime_inv_mersenne_eq_one : (∑' p : ℕ × ℕ, if 0 < p.1 ∧ 0 < p.2 ∧ Nat.Coprime p.1 p.2 then 1 / ((2 : ℝ) ^ (p.1 + p.2) - 1) else 0) = 1","docstring":"**The reduced-direction law**: `∑_{(a,b) coprime, a,b ≥ 1} 1/(2^{a+b}-1) = 1`. Every positive pair `(X,Y)` has a unique reduced slope, and `P(slope = a/b) = ∑_g 2^{-g(a+b)} = 1/(2^{a+b}-1)` for coprime `(a,b)`; the masses sum to one. Proved by expanding each Mersenne reciprocal as its geometric layer series and swapping with the wave-19 gcd-layer normalization. This is the root cylinder of the Stern–Brocot law: `M(1,1) = 1`.","generated_certificate":false},{"name":"cylinderMass_split","module":"Erdos249257/GcdMomentCalculus.lean","line":474,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GcdMomentCalculus.lean#L474","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GcdMomentCalculus.lean#L474","kind":"theorem","signature":"theorem cylinderMass_split (a b : ℕ+) : cylinderMass a b = 1 / ((2 : ℝ) ^ ((a : ℕ) + (b : ℕ)) - 1) + cylinderMass (a + b) b + cylinderMass a (a + b)","docstring":"**The exact telescoping recursion of the cylinder mass**: `M(a,b) = 1/(2^{a+b}-1) + M(a+b, b) + M(a, a+b)`. With `A = 2ᵃ`, `B = 2ᵇ` this is `1/((A-1)(B-1)) = 1/(AB-1) + 1/((AB-1)(B-1)) + 1/((A-1)(AB-1))` — the algebra that makes the reduced-direction distribution an exact rational Markov measure on the Stern–Brocot tree.","generated_certificate":false},{"name":"sternBrocotDepthMass_error","module":"Erdos249257/GcdMomentCalculus.lean","line":525,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GcdMomentCalculus.lean#L525","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GcdMomentCalculus.lean#L525","kind":"theorem","signature":"theorem sternBrocotDepthMass_error (dp : ℕ) : ∀ a b : ℕ+, 0 ≤ cylinderMass a b - sternBrocotDepthMass dp a b ∧ cylinderMass a b - sternBrocotDepthMass dp a b ≤ (2 / 3 : ℝ) ^ dp * cylinderMass a b","docstring":"**Finite-depth cylinder approximation with explicit geometric rate**: `0 ≤ M(a,b) - M_d(a,b) ≤ (2/3)ᵈ·M(a,b)`. The depth-`d` unfolding of the mediant recursion exhausts the closed form at rate `(2/3)ᵈ`, uniformly over all nodes — the error at depth `d` is exactly the frontier mass, and each generation retains at most `2/3` of its parent mass.","generated_certificate":false},{"name":"tendsto_sternBrocotDepthMass","module":"Erdos249257/GcdMomentCalculus.lean","line":560,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GcdMomentCalculus.lean#L560","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GcdMomentCalculus.lean#L560","kind":"theorem","signature":"theorem tendsto_sternBrocotDepthMass (a b : ℕ+) : Filter.Tendsto (fun dp : ℕ => sternBrocotDepthMass dp a b) Filter.atTop (nhds (cylinderMass a b))","docstring":"**The Stern–Brocot cylinder law, limit form**: the depth-`d` unfolding of the mediant recursion at `(a,b)` converges to the closed form `M(a,b) = 1/((2ᵃ-1)(2ᵇ-1))`, with explicit rate `(2/3)ᵈ` from `sternBrocotDepthMass_error`. At the root: `M(1,1) = 1`, matching the reduced-direction law `tsum_pos_coprime_inv_mersenne_eq_one`.","generated_certificate":false}],"paper_label":"res:sternbrocotcylinders","paper_anchor":{"paper_label":"res:sternbrocotcylinders","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":4005,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":4004,"anchor_title":"Stern--Brocot cylinder recursion and convergence","anchor_title_tex":"Stern--Brocot cylinder recursion and convergence","section_title":"Reduced directions and cylinder masses.","section_path":[{"level":"section","title":"Exact remaining problems and further questions","title_tex":"Exact remaining problems and further questions","line":3819},{"level":"subsection","title":"Probability-coordinate geometry: reduced slopes and cylinders","title_tex":"Probability-coordinate geometry: reduced slopes and cylinders","line":3973},{"level":"paragraph","title":"Reduced directions and cylinder masses.","title_tex":"Reduced directions and cylinder masses.","line":3981}],"source_excerpt":{"text":"\\begin{theorem}[Stern--Brocot cylinder recursion and convergence]\n\\label{res:sternbrocotcylinders}\nAt every positive node,\n\\[\n  M(a,b)=\\frac1{2^{a+b}-1}+M(a+b,b)+M(a,a+b),\n\\]\nand, uniformly in \\(a,b\\) and \\(d\\),\n\\[\n  0\\le M(a,b)-M_d(a,b)\n  \\le \\left(\\frac23\\right)^d M(a,b).\n\\]\nConsequently \\(M_d(a,b)\\to M(a,b)\\); at the root \\(M(1,1)=1\\).\n\\end{theorem}","media_type":"text/x-tex","source_char_start":210530,"source_char_end":210884,"start_line":4004,"end_line":4016,"truncated":false,"sha256":"sha256:10dbe000f3aa3b965d33fedb002d59921abb0bad53427883210877eb733cf9aa","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L4005","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"tsum_pos_coprime_inv_mersenne_eq_one","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GcdMomentCalculus.lean#L349","cite_this":true},{"kind":"lean_declaration_pinned","label":"cylinderMass_split","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GcdMomentCalculus.lean#L474","cite_this":true},{"kind":"lean_declaration_pinned","label":"sternBrocotDepthMass_error","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GcdMomentCalculus.lean#L525","cite_this":true},{"kind":"lean_declaration_pinned","label":"tendsto_sternBrocotDepthMass","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GcdMomentCalculus.lean#L560","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/64","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"stern_brocot_cylinder_law"}}],"expect":"object"},"prior_art":{"family_id":"probabilistic_gcd_and_stern_brocot_geometry","status_summary":"Exact coprimality and gcd-moment identities, a Stern–Brocot cylinder law with uniform geometric convergence, and source-level Fibonacci/continuant run stability.","prior_art_posture":"Classical Lambert-series, convolution, Pillai gcd-sum, probability, and Stern–Brocot background; Postelmans–Van Assche own the cited irrationality theorem for the constant-weight squared-Lambert value. No priority claim is made for this assembled formal package.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/12","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"exact_local_id","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z07::stern_brocot_cylinder_law","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:sternbrocotcylinders","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L4005","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:stern_brocot_run_fibonacci_stability","assertion_kind":"registered_claim","source_id":"stern_brocot_run_fibonacci_stability","reviewed":true,"granularity":"single_registered_claim","label":"Fibonacci and continuant stability of Stern–Brocot runs","statement":"A Stern–Brocot word with r nonempty alternating runs has height at least F_(r+3), with equality on the all-unit run spine. Writing each run length as 1+e_i gives an exact multiaffine continuant-defect expansion, the sharp global gain F_(r+1) sum e_i, and exact one-site coefficient F_(i+2)F_(r-i+1). Under the gcd-one weight 2^-(a+b), the first induced run layer has mass 1/2, while the natural denominator-bookkeeping exponent is exactly F_(r+3)-2. These are source-level combinatorial and arithmetic facts; no analytic denominator survival or irrationality conclusion is claimed.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_249","erdos_257"],"problem_erdos_numbers":[249,257],"problem_binding":{"method":"core_programme_joint","warrant":"the claim's only paper anchor is owned by the retired combined #249/#257 exposition, which treats both problems together and publishes nothing that separates them. Bound to both rather than guessed apart.","separability":"not_separable_from_published_source"},"problem_edges":[{"problem_id":"erdos_249","relation":"shared_programme_context","basis":"problem_binding"},{"problem_id":"erdos_257","relation":"shared_programme_context","basis":"problem_binding"}],"programme_ids":["joint_249_257_gateway"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"sbPair_runWord","module":"Erdos249257/SternBrocotRunGeometry.lean","line":125,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SternBrocotRunGeometry.lean#L125","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SternBrocotRunGeometry.lean#L125","kind":"theorem","signature":"theorem sbPair_runWord (newest : LR) (ns : List ℕ) : sbPair (runWord newest ns) = match newest with | .left => runBoundaryPair ns | .right => ((runBoundaryPair ns).2, (runBoundaryPair ns).1)","docstring":"The run continuant is not a new coordinate detached from the tree: it is exactly the pair reached by the corresponding alternating Stern--Brocot word, up to the expected coordinate swap for a newest right run.","generated_certificate":false},{"name":"runHeight_fib_lower","module":"Erdos249257/SternBrocotRunGeometry.lean","line":185,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SternBrocotRunGeometry.lean#L185","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SternBrocotRunGeometry.lean#L185","kind":"theorem","signature":"theorem runHeight_fib_lower (ns : List ℕ) (hpos : ∀ n ∈ ns, 0 < n) : Nat.fib (ns.length + 3) ≤ runHeight ns","docstring":"**Fibonacci minimum height.** Every alternating word with `r` nonempty runs has height at least `F_{r+3}`.","generated_certificate":false},{"name":"runHeight_defectRunLengths","module":"Erdos249257/SternBrocotRunGeometry.lean","line":263,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SternBrocotRunGeometry.lean#L263","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SternBrocotRunGeometry.lean#L263","kind":"theorem","signature":"theorem runHeight_defectRunLengths (e : List ℕ) : runHeight (defectRunLengths e) = Nat.fib (e.length + 3) + (exactDefectPair e).1 + (exactDefectPair e).2","docstring":"Exact height expansion. In particular, `exactDefectPair` is not an estimate: it owns every mixed coefficient of the continuant defect.","generated_certificate":false},{"name":"runHeight_defect_fib_sum_lower","module":"Erdos249257/SternBrocotRunGeometry.lean","line":343,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SternBrocotRunGeometry.lean#L343","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SternBrocotRunGeometry.lean#L343","kind":"theorem","signature":"theorem runHeight_defect_fib_sum_lower (e : List ℕ) : Nat.fib (e.length + 3) + Nat.fib (e.length + 1) * e.sum ≤ runHeight (defectRunLengths e)","docstring":"**Sharp global stability floor.** If the positive run lengths are `1 + eᵢ`, their height gains at least `F_{r+1}` for every unit of total defect. Boundary one-site rays attain this coefficient exactly.","generated_certificate":false},{"name":"runHeight_oneSiteDefect","module":"Erdos249257/SternBrocotRunGeometry.lean","line":413,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SternBrocotRunGeometry.lean#L413","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SternBrocotRunGeometry.lean#L413","kind":"theorem","signature":"theorem runHeight_oneSiteDefect {r i : ℕ} (hi : i < r) (e : ℕ) : runHeight (List.replicate i 1 ++ (e + 1) :: List.replicate (r - i - 1) 1) = Nat.fib (r + 3) + oneSiteWeight r i * e","docstring":"A single defect is exactly linear, with coefficient `F_{i+2} F_{r-i+1}`. This is the sharp coefficient formula and includes the arbitrary defect magnitude `e`, not only the derivative at `e = 0`.","generated_certificate":false},{"name":"oneRunMass_eq_half","module":"Erdos249257/SternBrocotRunGeometry.lean","line":467,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SternBrocotRunGeometry.lean#L467","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SternBrocotRunGeometry.lean#L467","kind":"theorem","signature":"theorem oneRunMass_eq_half : oneRunMass = 1 / 2","docstring":"Exact first induced layer: the two parabolic cusps together carry mass `1/2`.","generated_certificate":false},{"name":"naturalRunDenominatorExponent_add_two","module":"Erdos249257/SternBrocotRunGeometry.lean","line":491,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SternBrocotRunGeometry.lean#L491","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SternBrocotRunGeometry.lean#L491","kind":"theorem","signature":"theorem naturalRunDenominatorExponent_add_two (r : ℕ) : naturalRunDenominatorExponent r + 2 = Nat.fib (r + 3)","docstring":"The natural exponent is exactly two below the minimum run height scale: `F₂ + ... + F_{r+1} + 2 = F_{r+3}`.","generated_certificate":false}],"paper_label":"res:sternbrocotruns","paper_anchor":{"paper_label":"res:sternbrocotruns","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":4039,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":4038,"anchor_title":"Fibonacci height bound and exact run-length defects","anchor_title_tex":"Fibonacci height bound and exact run-length defects","section_title":"Run lengths and arithmetic height.","section_path":[{"level":"section","title":"Exact remaining problems and further questions","title_tex":"Exact remaining problems and further questions","line":3819},{"level":"subsection","title":"Probability-coordinate geometry: reduced slopes and cylinders","title_tex":"Probability-coordinate geometry: reduced slopes and cylinders","line":3973},{"level":"paragraph","title":"Run lengths and arithmetic height.","title_tex":"Run lengths and arithmetic height.","line":4025}],"source_excerpt":{"text":"\\begin{theorem}[Fibonacci height bound and exact run-length defects]\n\\label{res:sternbrocotruns}\nEvery \\(r\\)-run word satisfies\n\\[\n H(1+e_1,\\ldots,1+e_r)\n \\ge F_{r+3}+F_{r+1}\\sum_{i=1}^r e_i,\n\\]\nand \\(H(1,\\ldots,1)=F_{r+3}\\).  A defect at position\n\\(i\\in\\{0,\\ldots,r-1\\}\\) (the newest run is $i=0$) has the exact size\n\\[\n H(\\underbrace{1,\\ldots,1}_{i},1+e,\n   \\underbrace{1,\\ldots,1}_{r-i-1})\n =F_{r+3}+F_{i+2}F_{r-i+1}e.\n\\]\nMore generally, the excess over $F_{r+3}$ is a polynomial in the $e_i$, with\nnonnegative coefficients and degree at most one in each variable; the linked\nrecursive formula determines every mixed coefficient.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":211949,"source_char_end":212595,"start_line":4038,"end_line":4055,"truncated":false,"sha256":"sha256:c0e8d942275d37705c5adb7491e94887e6f5a308985d41d07d4724fbb3fa36d1","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L4039","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"sbPair_runWord","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SternBrocotRunGeometry.lean#L125","cite_this":true},{"kind":"lean_declaration_pinned","label":"runHeight_fib_lower","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SternBrocotRunGeometry.lean#L185","cite_this":true},{"kind":"lean_declaration_pinned","label":"runHeight_defectRunLengths","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SternBrocotRunGeometry.lean#L263","cite_this":true},{"kind":"lean_declaration_pinned","label":"runHeight_defect_fib_sum_lower","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SternBrocotRunGeometry.lean#L343","cite_this":true},{"kind":"lean_declaration_pinned","label":"runHeight_oneSiteDefect","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SternBrocotRunGeometry.lean#L413","cite_this":true},{"kind":"lean_declaration_pinned","label":"oneRunMass_eq_half","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SternBrocotRunGeometry.lean#L467","cite_this":true},{"kind":"lean_declaration_pinned","label":"naturalRunDenominatorExponent_add_two","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SternBrocotRunGeometry.lean#L491","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/65","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"stern_brocot_run_fibonacci_stability"}}],"expect":"object"},"prior_art":{"family_id":"probabilistic_gcd_and_stern_brocot_geometry","status_summary":"Exact coprimality and gcd-moment identities, a Stern–Brocot cylinder law with uniform geometric convergence, and source-level Fibonacci/continuant run stability.","prior_art_posture":"Classical Lambert-series, convolution, Pillai gcd-sum, probability, and Stern–Brocot background; Postelmans–Van Assche own the cited irrationality theorem for the constant-weight squared-Lambert value. No priority claim is made for this assembled formal package.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/12","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"known_classical":3,"not_assessed":1,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z07::continuant_defect_sensitivity","prior_art_state":"not_assessed"},{"semantic_node_id":"Z07::fibonacci_height_floor","prior_art_state":"known_classical"},{"semantic_node_id":"Z07::fibonacci_partial_sum_denominator_scale","prior_art_state":"known_classical"},{"semantic_node_id":"Z07::induced_run_layer_masses","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z07::run_coordinate_normalisation","prior_art_state":"known_classical"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:sternbrocotruns","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L4039","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:primitive_coordinate_index_obstruction","assertion_kind":"registered_claim","source_id":"primitive_coordinate_index_obstruction","reviewed":true,"granularity":"single_registered_claim","label":"Primitive Euler-coordinate finite-jet obstruction","statement":"For the primitive-conductor weight A = φ*μ, every positive index clearing the normalized coordinates A(n)/n through horizon N is divisible by the two-tier primorial: 4 at the prime 2 once N ≥ 4, p² for odd p with p² ≤ N, and p otherwise. Hence no fixed positive index clears all normalized coordinates. This is a finite Euler/Witt-coordinate obstruction only; it gives no irrationality criterion for S.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"programme_representative","warrant":"orientation.json lists this claim among the representative claims of a mathematical programme whose remaining open propositions all name one problem"},"problem_edges":[{"problem_id":"erdos_249","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":["lambert_obstruction_interfaces"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"twoTierPrimorial_dvd_of_integral_jet","module":"Erdos249257/PrimitiveDeterminantLift.lean","line":148,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PrimitiveDeterminantLift.lean#L148","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimitiveDeterminantLift.lean#L148","kind":"theorem","signature":"theorem twoTierPrimorial_dvd_of_integral_jet {N D : ℕ} (hN : 4 ≤ N) (hjet : IntegralPrimitiveEulerJet N D) : twoTierPrimorial N ∣ D","docstring":"The complete finite-jet obstruction: the two-tier primorial divides every common index clearing all primitive Euler coordinates through `N`.","generated_certificate":false},{"name":"no_fixed_integral_primitive_euler_index","module":"Erdos249257/PrimitiveDeterminantLift.lean","line":169,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PrimitiveDeterminantLift.lean#L169","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimitiveDeterminantLift.lean#L169","kind":"theorem","signature":"theorem no_fixed_integral_primitive_euler_index (D : ℕ) (hD : 0 < D) : ¬ ∀ n : ℕ, 0 < n → ∃ z : ℤ, (D : ℚ) * primEulerCoord n = (z : ℚ)","docstring":"No fixed positive integer clears all normalized primitive coordinates.","generated_certificate":false}],"paper_label":"res:primitiveindex","paper_anchor":{"paper_label":"res:primitiveindex","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":3886,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":3885,"anchor_title":"finite denominator obstruction","anchor_title_tex":"finite denominator obstruction","section_title":"Finite denominator-clearing boundaries.","section_path":[{"level":"section","title":"Exact remaining problems and further questions","title_tex":"Exact remaining problems and further questions","line":3819},{"level":"subsection","title":"Finite algebraic boundaries and squared-Lambert coordinates","title_tex":"Finite algebraic boundaries and squared-Lambert coordinates","line":3873},{"level":"paragraph","title":"Finite denominator-clearing boundaries.","title_tex":"Finite denominator-clearing boundaries.","line":3882}],"source_excerpt":{"text":"\\begin{proposition}[finite denominator obstruction]\n\\label{res:primitiveindex}\nFor $N\\ge4$, if a positive integer $D$ makes $Da_n$ integral for every\n$1\\le n\\le N$, then $\\mathcal P_N\\mid D$.  Hence no fixed positive $D$ clears every\n$a_n$.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":204989,"source_char_end":205247,"start_line":3885,"end_line":3890,"truncated":false,"sha256":"sha256:82b0485f83338c20871060a858639334f994b87e02f977ee7006494d2f632f60","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3886","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"twoTierPrimorial_dvd_of_integral_jet","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimitiveDeterminantLift.lean#L148","cite_this":true},{"kind":"lean_declaration_pinned","label":"no_fixed_integral_primitive_euler_index","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimitiveDeterminantLift.lean#L169","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/66","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"primitive_coordinate_index_obstruction"}}],"expect":"object"},"prior_art":{"family_id":"alternative_249_algebraic_boundaries","status_summary":"Exact finite obstructions and interfaces, a constructive dyadic basis, and an all-base arithmetic/spanning layer. The reusable rank wrapper is conditional; AllBaseTotientKernel.lean separately proves its independence premise and the complete rank unconditionally. The residue-series claims classify the rational totient residue series and prove the dilated series linearly independent.","prior_art_posture":"Classical Euler/Möbius, CRT, determinant, and Lambert tools; Coons already implies full-kernel infinite rank, and Martin supplies a broader external affine-independence antecedent. The Lean all-base proof uses CRT and Dirichlet independently. Erdős 1957 (Lemma 1) and Kaneko–Suzuki–Tachiya (Theorem 3) already give the irrationality of the totient residue series. No #249 irrationality endpoint or priority claim.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/17","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z07::primitive_euler_index_no_lift_barrier","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:primitiveindex","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3886","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:primitive_unit_gap_ceiling","assertion_kind":"registered_claim","source_id":"primitive_unit_gap_ceiling","reviewed":true,"granularity":"single_registered_claim","label":"Prime-power reduced-denominator unit-gap ceiling","statement":"For the reduced-denominator unit-gap predicate at an odd prime power p^e, the candidate interval is empty or has exactly one candidate divisible by p; in particular its count is at most one. The exact row (u,N,K) = (3,3,5) satisfies the unit-gap predicate while the corresponding ordinary empty-gap inequality fails. This is a finite certificate-level strengthening, not an unbounded supply theorem or an irrationality criterion for S.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"gateway_section","warrant":"the claim's anchor sits inside the retired gateway section 'The totient constant S (Erdős #249)', which is the author's own partition of that manuscript by problem"},"problem_edges":[{"problem_id":"erdos_249","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"reducedDenominatorUnitGapCert_prime_pow_iff","module":"Erdos249257/PrimitiveWeightCertificate.lean","line":54,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PrimitiveWeightCertificate.lean#L54","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimitiveWeightCertificate.lean#L54","kind":"theorem","signature":"theorem reducedDenominatorUnitGapCert_prime_pow_iff {p e N K : ℕ} (hp : Nat.Prime p) (he : 0 < e) : ReducedDenominatorUnitGapCert (p ^ e) N K ↔ let M","docstring":"At a prime-power reduced denominator, nonunits are exactly multiples of the underlying prime. Since two consecutive integers are coprime, the unit-gap extension can admit at most one candidate lattice point.","generated_certificate":false},{"name":"reducedDenominatorCandidateCount_prime_pow_le_one","module":"Erdos249257/PrimitiveWeightCertificate.lean","line":111,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PrimitiveWeightCertificate.lean#L111","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimitiveWeightCertificate.lean#L111","kind":"theorem","signature":"theorem reducedDenominatorCandidateCount_prime_pow_le_one {p e N K : ℕ} (hp : Nat.Prime p) (he : 0 < e) (hcert : ReducedDenominatorUnitGapCert (p ^ e) N K) : reducedDenominatorCandidateCount (p ^ e) N K ≤ 1","docstring":"Numerical form of the prime-power ceiling: a certified interval for an odd prime power contains zero or one candidate, never two.","generated_certificate":false},{"name":"reducedDenominatorUnitGapCert_three_window_3_5","module":"Erdos249257/PrimitiveWeightCertificate.lean","line":151,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PrimitiveWeightCertificate.lean#L151","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimitiveWeightCertificate.lean#L151","kind":"theorem","signature":"theorem reducedDenominatorUnitGapCert_three_window_3_5 : ReducedDenominatorUnitGapCert 3 3 5","generated_certificate":false},{"name":"not_ordinary_gap_certificate_three_window_3_5","module":"Erdos249257/PrimitiveWeightCertificate.lean","line":157,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PrimitiveWeightCertificate.lean#L157","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimitiveWeightCertificate.lean#L157","kind":"theorem","signature":"theorem not_ordinary_gap_certificate_three_window_3_5 : ¬ ((3 * primitiveCarry 3 5) % 2 ^ 5 + 3 * (3 + 5 + 2) < 2 ^ 5)","docstring":"The same row is not an ordinary empty-gap certificate: its scaled residue plus tail height is `38`, while the dyadic modulus is `32`.","generated_certificate":false}],"paper_label":"res:unitgap","paper_anchor":{"paper_label":"res:unitgap","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2770,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":2770,"anchor_title":"prime-power unit-gap ceiling","anchor_title_tex":"prime-power unit-gap ceiling","section_title":"A reduced-denominator finite predicate.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Fixed-scale mechanisms and conditional unbounded implications","title_tex":"Fixed-scale mechanisms and conditional unbounded implications","line":2720},{"level":"paragraph","title":"A reduced-denominator finite predicate.","title_tex":"A reduced-denominator finite predicate.","line":2761}],"source_excerpt":{"text":"\\begin{proposition}[prime-power unit-gap ceiling]\\label{res:unitgap}\nFor every prime power $p^e$ with $e>0$, a\n$\\mathsf{UnitGap}(p^e,N,K)$ has candidate count at most\none.  The instance $(u,N,K)=(3,3,5)$ satisfies this predicate, although its\ncorresponding ordinary empty-gap inequality fails.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":145555,"source_char_end":145866,"start_line":2770,"end_line":2775,"truncated":false,"sha256":"sha256:209f2781f15e7fdafd419a97a41b79d4a69933d8b7665492c2283fc5a666fd86","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2770","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"reducedDenominatorUnitGapCert_prime_pow_iff","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimitiveWeightCertificate.lean#L54","cite_this":true},{"kind":"lean_declaration_pinned","label":"reducedDenominatorCandidateCount_prime_pow_le_one","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimitiveWeightCertificate.lean#L111","cite_this":true},{"kind":"lean_declaration_pinned","label":"reducedDenominatorUnitGapCert_three_window_3_5","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimitiveWeightCertificate.lean#L151","cite_this":true},{"kind":"lean_declaration_pinned","label":"not_ordinary_gap_certificate_three_window_3_5","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimitiveWeightCertificate.lean#L157","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/67","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"primitive_unit_gap_ceiling"}}],"expect":"object"},"prior_art":{"family_id":"alternative_249_algebraic_boundaries","status_summary":"Exact finite obstructions and interfaces, a constructive dyadic basis, and an all-base arithmetic/spanning layer. The reusable rank wrapper is conditional; AllBaseTotientKernel.lean separately proves its independence premise and the complete rank unconditionally. The residue-series claims classify the rational totient residue series and prove the dilated series linearly independent.","prior_art_posture":"Classical Euler/Möbius, CRT, determinant, and Lambert tools; Coons already implies full-kernel infinite rank, and Martin supplies a broader external affine-independence antecedent. The Lean all-base proof uses CRT and Dirichlet independently. Erdős 1957 (Lemma 1) and Kaneko–Suzuki–Tachiya (Theorem 3) already give the irrationality of the totient residue series. No #249 irrationality endpoint or priority claim.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/17","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"formalisation_only":1,"not_assessed":1,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z06::unit_gap_prime_power_ceiling","prior_art_state":"not_assessed"},{"semantic_node_id":"Z06::unit_gap_prime_power_classification","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z06::unit_gap_strict_strengthening_instance","prior_art_state":"formalisation_only"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:unitgap","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2770","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:mersenne_tail_atom_boundary","assertion_kind":"registered_claim","source_id":"mersenne_tail_atom_boundary","reviewed":true,"granularity":"single_registered_claim","label":"Finite Mersenne tail-atom determinant boundary","statement":"A periodic Mersenne tail atom has the exact doubling-and-wrap carry law. For any finite atom matrix, clearing the product of its column Mersenne denominators yields the integral numerator determinant; a nonzero cleared weighted minor therefore has absolute value at least one. This is a finite-denominator boundary for an atom-minor route, not an irrationality criterion for S.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249","erdos_257"],"problem_erdos_numbers":[249,257],"problem_binding":{"method":"core_programme_joint","warrant":"the claim's only paper anchor is owned by the retired combined #249/#257 exposition, which treats both problems together and publishes nothing that separates them. Bound to both rather than guessed apart.","separability":"not_separable_from_published_source"},"problem_edges":[{"problem_id":"erdos_249","relation":"shared_programme_context","basis":"problem_binding"},{"problem_id":"erdos_257","relation":"shared_programme_context","basis":"problem_binding"}],"programme_ids":["joint_249_257_gateway"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"mersenneTailAtom_step","module":"Erdos249257/MersenneTailAtoms.lean","line":28,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/MersenneTailAtoms.lean#L28","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/MersenneTailAtoms.lean#L28","kind":"theorem","signature":"theorem mersenneTailAtom_step (d : ℕ+) (N : ℕ) : 2 * mersenneTailAtom d N - mersenneTailAtom d (N + 1) = if (d : ℕ) ∣ N + 1 then 1 else 0","docstring":"A periodic atom advances by doubling except at its unique wrap point, where the Mersenne numerator contributes exactly one carry.","generated_certificate":false},{"name":"mersenneAtom_denominatorProduct_mul_det","module":"Erdos249257/MersenneTailAtoms.lean","line":122,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/MersenneTailAtoms.lean#L122","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/MersenneTailAtoms.lean#L122","kind":"theorem","signature":"theorem mersenneAtom_denominatorProduct_mul_det {m : ℕ} (d : Fin m → ℕ+) (idx : Fin m → Fin m → ℕ) : (∏ j, ((2 : ℚ) ^ (d j : ℕ) - 1)) * Matrix.det (mersenneAtomMatrix d idx) = ((Matrix.det (mersenneAtomNumeratorMatrix d idx) : ℤ) : ℚ)","docstring":"Clearing the finite Mersenne denominator product recovers the integer numerator determinant exactly.","generated_certificate":false},{"name":"one_le_abs_weightedMersenneAtom_cleared_det","module":"Erdos249257/MersenneTailAtoms.lean","line":181,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/MersenneTailAtoms.lean#L181","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/MersenneTailAtoms.lean#L181","kind":"theorem","signature":"theorem one_le_abs_weightedMersenneAtom_cleared_det {m : ℕ} (d : Fin m → ℕ+) (idx : Fin m → Fin m → ℕ) (w : Fin m → ℤ) (hdet : Matrix.det (mersenneAtomNumeratorMatrix d idx) * ∏ j, w j ≠ 0) : (1 : ℚ) ≤ |(∏ j, ((2 : ℚ) ^ (d j : ℕ) - 1)) * Matrix.det (weightedMersenneAtomMatrix d idx w)|","docstring":"Weighted height-tight boundary in absolute-value form.","generated_certificate":false}],"paper_label":"res:mersennetailatoms","paper_anchor":{"paper_label":"res:mersennetailatoms","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":3905,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":3904,"anchor_title":"finite Mersenne-atom determinant identity","anchor_title_tex":"finite Mersenne-atom determinant identity","section_title":"Finite denominator-clearing boundaries.","section_path":[{"level":"section","title":"Exact remaining problems and further questions","title_tex":"Exact remaining problems and further questions","line":3819},{"level":"subsection","title":"Finite algebraic boundaries and squared-Lambert coordinates","title_tex":"Finite algebraic boundaries and squared-Lambert coordinates","line":3873},{"level":"paragraph","title":"Finite denominator-clearing boundaries.","title_tex":"Finite denominator-clearing boundaries.","line":3882}],"source_excerpt":{"text":"\\begin{proposition}[finite Mersenne-atom determinant identity]\n\\label{res:mersennetailatoms}\nFix \\(m\\ge1\\), conductors \\(d_j\\ge1\\), indices \\(n_{ij}\\ge0\\), and integer\ncolumn weights \\(w_j\\), and set\n\\[\n W_{ij}=w_j\\omega_{d_j}(n_{ij}),\n \\qquad U_{ij}=2^{\\,n_{ij}\\bmod d_j}.\n\\]\nThen\n\\[\n \\left(\\prod_{j=1}^m(2^{d_j}-1)\\right)\\det W\n =\\left(\\prod_{j=1}^m w_j\\right)\\det U\\in\\Z.\n\\]\nIf the integer on the right is nonzero, then the absolute value of the\ncleared determinant on the left is at least \\(1\\).\n\\end{proposition}","media_type":"text/x-tex","source_char_start":205635,"source_char_end":206152,"start_line":3904,"end_line":3919,"truncated":false,"sha256":"sha256:c758dab2b273a6ec40abecf750cefe88915c9fd878d2b78cddd737eefc9c9302","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3905","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"mersenneTailAtom_step","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/MersenneTailAtoms.lean#L28","cite_this":true},{"kind":"lean_declaration_pinned","label":"mersenneAtom_denominatorProduct_mul_det","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/MersenneTailAtoms.lean#L122","cite_this":true},{"kind":"lean_declaration_pinned","label":"one_le_abs_weightedMersenneAtom_cleared_det","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/MersenneTailAtoms.lean#L181","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/68","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"mersenne_tail_atom_boundary"}}],"expect":"object"},"prior_art":{"family_id":"alternative_249_algebraic_boundaries","status_summary":"Exact finite obstructions and interfaces, a constructive dyadic basis, and an all-base arithmetic/spanning layer. The reusable rank wrapper is conditional; AllBaseTotientKernel.lean separately proves its independence premise and the complete rank unconditionally. The residue-series claims classify the rational totient residue series and prove the dilated series linearly independent.","prior_art_posture":"Classical Euler/Möbius, CRT, determinant, and Lambert tools; Coons already implies full-kernel infinite rank, and Martin supplies a broader external affine-independence antecedent. The Lean all-base proof uses CRT and Dirichlet independently. Erdős 1957 (Lemma 1) and Kaneko–Suzuki–Tachiya (Theorem 3) already give the irrationality of the totient residue series. No #249 irrationality endpoint or priority claim.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/17","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"routine_corollary_or_specialisation":2},"edges":[{"semantic_node_id":"Z07::finite_atom_determinant_clearing_barrier","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z07::mersenne_tail_atom_carry","prior_art_state":"routine_corollary_or_specialisation"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:mersennetailatoms","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3905","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:mobius_sign_support_no_go","assertion_kind":"registered_claim","source_id":"mobius_sign_support_no_go","reviewed":true,"granularity":"single_registered_claim","label":"Möbius-sign support one-half no-go","statement":"The Boolean support selected by the negative Möbius values has sum 1/2 plus the positive Möbius tail and is therefore at least 1/2 + 1/63. This closes the sign-truncation route to the target value 1/2; it does not exclude another infinite Boolean support with that value.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"gateway_section","warrant":"the claim's anchor sits inside the retired gateway section 'Exact #257 coordinates: Boolean--Möbius carry certificates', which is the author's own partition of that manuscript by problem"},"problem_edges":[{"problem_id":"erdos_257","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"tsum_negativeMobius_eq_half_add_positiveMobiusTail","module":"Erdos249257/MobiusSignSupportNoGo.lean","line":111,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/MobiusSignSupportNoGo.lean#L111","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/MobiusSignSupportNoGo.lean#L111","kind":"theorem","signature":"theorem tsum_negativeMobius_eq_half_add_positiveMobiusTail : (∑' d : ℕ+, negativeMobiusTerm d) = 1 / 2 + ∑' d : ℕ+, positiveMobiusTailTerm d","docstring":"Exact sign-separation identity. The negative-Möbius Boolean support overshoots `1/2` by the complete positive-Möbius tail.","generated_certificate":false},{"name":"half_add_one_div_sixty_three_le_tsum_negativeMobius","module":"Erdos249257/MobiusSignSupportNoGo.lean","line":150,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/MobiusSignSupportNoGo.lean#L150","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/MobiusSignSupportNoGo.lean#L150","kind":"theorem","signature":"theorem half_add_one_div_sixty_three_le_tsum_negativeMobius : (1 : ℝ) / 2 + 1 / 63 ≤ ∑' d : ℕ+, negativeMobiusTerm d","docstring":"Quantitative no-go: sign truncation overshoots by at least `1/63`.","generated_certificate":false}],"paper_label":"res:mobiussignnogo","paper_anchor":{"paper_label":"res:mobiussignnogo","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":4747,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":4747,"anchor_title":"Möbius-sign support no-go","anchor_title_tex":"M\\\"obius-sign support no-go","section_title":"Exact #257 coordinates: Boolean--Möbius carry certificates","section_path":[{"level":"section","title":"Auxiliary rationality criteria from binary carry systems","title_tex":"Auxiliary rationality criteria from binary carry systems","line":4612},{"level":"subsection","title":"Exact #257 coordinates: Boolean--Möbius carry certificates","title_tex":"Exact \\#257 coordinates: Boolean--M\\\"obius carry certificates","line":4739}],"source_excerpt":{"text":"\\begin{proposition}[M\\\"obius-sign support no-go]\\label{res:mobiussignnogo}\nLet\n\\[\n A_-=\\{d\\ge2:\\mu(d)=-1\\},\n \\qquad A_+=\\{d\\ge2:\\mu(d)=1\\}.\n\\]\nThen\n\\[\n \\sum_{d\\in A_-}\\frac1{2^d-1}\n =\\frac12+\\sum_{d\\in A_+}\\frac1{2^d-1}\n \\ge\\frac12+\\frac1{63}>\\frac12.\n\\]\nIn particular, the signed identity \\(L(\\mu)=1/2\\) cannot be converted into a\nBoolean support of value \\(1/2\\) merely by selecting the negative M\\\"obius\nindices.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":249143,"source_char_end":249576,"start_line":4747,"end_line":4762,"truncated":false,"sha256":"sha256:9d53d581d2b260df9e58a7fd2a3ab24cf96ab6272b0333e4a616806058d3c95b","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L4747","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"tsum_negativeMobius_eq_half_add_positiveMobiusTail","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/MobiusSignSupportNoGo.lean#L111","cite_this":true},{"kind":"lean_declaration_pinned","label":"half_add_one_div_sixty_three_le_tsum_negativeMobius","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/MobiusSignSupportNoGo.lean#L150","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/69","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"mobius_sign_support_no_go"}}],"expect":"object"},"prior_art":{"family_id":"achievement_geometry_and_half_target_interface","status_summary":"Classical geometry formalised, exact local bands, a conditional second channel, and scoped no-go boundaries.","prior_art_posture":"Kovač–Tao supply the strict-tail subsum geometry; the classical Lambert–Möbius identity is the baseline for the scoped sign-support no-go; Crandall and Campbell concern adjacent binary-digit questions and are not proof inputs. No priority claim is made for the sign separation.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/4","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z10::mobius_sign_truncation_overshoots_one_half","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:mobiussignnogo","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L4747","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:totient_carry_kernel_anti_compression","assertion_kind":"registered_claim","source_id":"totient_carry_kernel_anti_compression","reviewed":true,"granularity":"single_registered_claim","label":"Totient carry anti-compression","statement":"If the binary totient coefficient series were rational, one positive-multiplier tempered integral carry would have uniformly eventually-periodic dyadic sections modulo its multiplier while retaining canonical carry-section span dimension at least 2^e − 1 for every e. No finite-rank bound for every rationality-supplied carry is proved, so this quotient-periodicity/anti-compression conjunction is a necessary consequence rather than an irrationality criterion for S.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"not_irrational_totientSeries_implies_mod_period_and_unbounded_rank","module":"Erdos249257/TotientTailCarryPeriod.lean","line":224,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientTailCarryPeriod.lean#L224","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientTailCarryPeriod.lean#L224","kind":"theorem","signature":"theorem not_irrational_totientSeries_implies_mod_period_and_unbounded_rank (hirr : ¬ Irrational (binaryCoeffSeries Nat.totient)) : ∃ v : ℕ, 0 < v ∧ ∃ u : ℕ → ℤ, IsTemperedBinaryOrbit Nat.totient v u ∧ (∀ e : ℕ, 2 ^ e - 1 ≤ Module.finrank ℚ (Submodule.span ℚ (Set.range (canonicalCarryKernelFamily u e)))) ∧ ∃ h : ℕ, 0 < h ∧ ∃ N₀ : ℕ, CarrySectionsEventuallyPeriodicMod v h N₀ u","docstring":"**Exact rationality frontier.** The same rationality-supplied carry has uniform eventual periodicity modulo `v` and unbounded torsion-free dyadic section rank. Hence quotient periodicity is the valid compression theorem; it cannot be promoted to finite `ℚ`-rank without extra totient arithmetic.","generated_certificate":false},{"name":"finrank_canonicalCarryKernel_ge","module":"Erdos249257/TotientCarryKernelRigidity.lean","line":263,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientCarryKernelRigidity.lean#L263","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientCarryKernelRigidity.lean#L263","kind":"theorem","signature":"theorem finrank_canonicalCarryKernel_ge {v : ℕ} {u : ℕ → ℤ} (hv : 0 < v) (hu : IsTemperedBinaryOrbit Nat.totient v u) (e : ℕ) : 2 ^ e - 1 ≤ finrank ℚ (Submodule.span ℚ (Set.range (canonicalCarryKernelFamily u e)))","docstring":"**Universal finite-level carry anti-compression.** Every positive-scale tempered totient orbit retains at least `2^e-1` dimensions among its canonical carry sections through level `e`. The bound is unconditional once the orbit is given: the CRT--Dirichlet minor has already discharged the coefficient-side independence hypothesis. Rationality is used only downstream to produce such an orbit.","generated_certificate":false},{"name":"carrySectionsEventuallyPeriodicMod_of_shift_dvd","module":"Erdos249257/TotientTailCarryPeriod.lean","line":180,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientTailCarryPeriod.lean#L180","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientTailCarryPeriod.lean#L180","kind":"theorem","signature":"theorem carrySectionsEventuallyPeriodicMod_of_shift_dvd {v h N₀ : ℕ} {u : ℕ → ℤ} (hint : ∀ N, N₀ ≤ N → (v : ℤ) ∣ u (N + h) - u N) : CarrySectionsEventuallyPeriodicMod v h N₀ u","docstring":"An eventual period of the base carry displacement gives the same eventual period modulo `v` on every dyadic section, with one common threshold for all levels and residues.","generated_certificate":false},{"name":"totient_forcing_vanishes_mod_of_dvd_multiplier","module":"Erdos249257/TotientTailCarryPeriod.lean","line":126,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientTailCarryPeriod.lean#L126","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientTailCarryPeriod.lean#L126","kind":"theorem","signature":"theorem totient_forcing_vanishes_mod_of_dvd_multiplier {p v : ℕ} {u : ℕ → ℤ} (hpv : p ∣ v) (hu : IsTemperedBinaryOrbit Nat.totient v u) (N : ℕ) : u (N + 1) ≡ 2 * u N [ZMOD (p : ℤ)]","docstring":"Reduction modulo any divisor of the orbit multiplier erases the totient forcing term. Thus a finite-field reduction at a prime dividing `v` sees only the homogeneous doubling recurrence, not the totient grid minor.","generated_certificate":false},{"name":"totient_carry_modEq_geometric_of_dvd_multiplier","module":"Erdos249257/TotientTailCarryPeriod.lean","line":140,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientTailCarryPeriod.lean#L140","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientTailCarryPeriod.lean#L140","kind":"theorem","signature":"theorem totient_carry_modEq_geometric_of_dvd_multiplier {p v : ℕ} {u : ℕ → ℤ} (hpv : p ∣ v) (hu : IsTemperedBinaryOrbit Nat.totient v u) : ∀ N : ℕ, u N ≡ (2 : ℤ) ^ N * u 0 [ZMOD (p : ℤ)]","docstring":"The preceding loss is global: modulo a divisor of `v`, the whole carry is the geometric orbit generated by its initial state.","generated_certificate":false},{"name":"finrank_canonicalCarryKernel_ge_of_linearIndependent","module":"Erdos249257/TotientCarryKernelRigidity.lean","line":211,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientCarryKernelRigidity.lean#L211","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientCarryKernelRigidity.lean#L211","kind":"theorem","signature":"theorem finrank_canonicalCarryKernel_ge_of_linearIndependent {v : ℕ} {u : ℕ → ℤ} (hv : 0 < v) (hu : IsTemperedBinaryOrbit Nat.totient v u) (e : ℕ) (hcanon : LinearIndependent ℚ (canonicalTotientKernelFamily e)) : 2 ^ e - 1 ≤ finrank ℚ (Submodule.span ℚ (Set.range (canonicalCarryKernelFamily u e)))","docstring":"**Finite-level carry anti-compression.** A nonzero-multiplier orbit whose canonical totient family is linearly independent must retain at least `2^e-1` dimensions among its carry sections through level `e`. Temperedness is not used beyond supplying the exact recurrence through the existing orbit interface.","generated_certificate":false},{"name":"not_irrational_totientSeries_implies_unbounded_carryRank_unconditional","module":"Erdos249257/TotientCarryKernelRigidity.lean","line":300,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientCarryKernelRigidity.lean#L300","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientCarryKernelRigidity.lean#L300","kind":"theorem","signature":"theorem not_irrational_totientSeries_implies_unbounded_carryRank_unconditional (hirr : ¬ Irrational (binaryCoeffSeries Nat.totient)) : ∃ v : ℕ, 0 < v ∧ ∃ u : ℕ → ℤ, IsTemperedBinaryOrbit Nat.totient v u ∧ ∀ e : ℕ, 2 ^ e - 1 ≤ finrank ℚ (Submodule.span ℚ (Set.range (canonicalCarryKernelFamily u e)))","docstring":"**Unconditional anti-compression consequence of the CRT--Dirichlet minor.** If the totient series were rational, its tempered integral carry would have unbounded dyadic section rank. This does not by itself prove irrationality: a separate theorem bounding the dyadic section rank of every rationality-supplied tempered carry is still required, and no such finite-rank compression theorem follows from the recurrence and temperedness alone.","generated_certificate":false}],"paper_label":"res:carryrank","paper_anchor":{"paper_label":"res:carryrank","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":3188,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":3188,"anchor_title":"carry-section rank forced by rationality","anchor_title_tex":"carry-section rank forced by rationality","section_title":"Relation to prior work.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Unconditional dyadic rank and the rationality-side carry bound","title_tex":"Unconditional dyadic rank and the rationality-side carry bound","line":3130},{"level":"paragraph","title":"Relation to prior work.","title_tex":"Relation to prior work.","line":3169}],"source_excerpt":{"text":"\\begin{proposition}[carry-section rank forced by rationality]\\label{res:carryrank}\nIf $S$ were rational, there would be an integer $v>0$ and integers $u(N)$ such\nthat $u(N+1)=2u(N)-v\\ph(N+1)$ and $u(N)/2^N\\to0$.  For every $e$, the\nsections $n\\mapsto u(2^jn+r)$ with\n$1\\leq j\\leq e$ would span a space of dimension at least $2^e-1$.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":168299,"source_char_end":168649,"start_line":3188,"end_line":3193,"truncated":false,"sha256":"sha256:e8c754d5a4cbe98fdec6753f8a11dcfd709c83959f2e9f2e05ed4f9dfb1e3aad","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3188","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_249_irrationality","statement":"Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/1"}],"links":[{"kind":"lean_declaration_pinned","label":"not_irrational_totientSeries_implies_mod_period_and_unbounded_rank","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientTailCarryPeriod.lean#L224","cite_this":true},{"kind":"lean_declaration_pinned","label":"finrank_canonicalCarryKernel_ge","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientCarryKernelRigidity.lean#L263","cite_this":true},{"kind":"lean_declaration_pinned","label":"carrySectionsEventuallyPeriodicMod_of_shift_dvd","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientTailCarryPeriod.lean#L180","cite_this":true},{"kind":"lean_declaration_pinned","label":"totient_forcing_vanishes_mod_of_dvd_multiplier","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientTailCarryPeriod.lean#L126","cite_this":true},{"kind":"lean_declaration_pinned","label":"totient_carry_modEq_geometric_of_dvd_multiplier","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientTailCarryPeriod.lean#L140","cite_this":true},{"kind":"lean_declaration_pinned","label":"finrank_canonicalCarryKernel_ge_of_linearIndependent","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientCarryKernelRigidity.lean#L211","cite_this":true},{"kind":"lean_declaration_pinned","label":"not_irrational_totientSeries_implies_unbounded_carryRank_unconditional","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientCarryKernelRigidity.lean#L300","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/70","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"totient_carry_kernel_anti_compression"}}],"expect":"object"},"prior_art":{"family_id":"alternative_249_algebraic_boundaries","status_summary":"Exact finite obstructions and interfaces, a constructive dyadic basis, and an all-base arithmetic/spanning layer. The reusable rank wrapper is conditional; AllBaseTotientKernel.lean separately proves its independence premise and the complete rank unconditionally. The residue-series claims classify the rational totient residue series and prove the dilated series linearly independent.","prior_art_posture":"Classical Euler/Möbius, CRT, determinant, and Lambert tools; Coons already implies full-kernel infinite rank, and Martin supplies a broader external affine-independence antecedent. The Lean all-base proof uses CRT and Dirichlet independently. Erdős 1957 (Lemma 1) and Kaneko–Suzuki–Tachiya (Theorem 3) already give the irrationality of the totient residue series. No #249 irrationality endpoint or priority claim.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/17","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z09::finite_level_carry_anti_compression","prior_art_state":"not_assessed"},{"semantic_node_id":"Z09::rationality_forces_unbounded_carry_rank","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z28::rationality_mod_period_and_unbounded_rank","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:carryrank","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3188","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:denominator_exclusion","assertion_kind":"registered_claim","source_id":"denominator_exclusion","reviewed":true,"granularity":"single_registered_claim","label":"Unconditional denominator exclusion for the totient constant","statement":"S = ∑ φ(n)/2ⁿ is not equal to p/q for any integer p and any 1 ≤ q ≤ 79 639 646 646 701 375 323 355 774 875 831 053 (≈ 7.96 × 10³⁴). A lower bound on any rational representation, not a proof of irrationality.","status":"verified finite instance","status_meaning":"Kernel-checked computation at a bounded range","readme_headline":true,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":["erdos249_certificate_story"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"tsum_totient_div_pow_two_ne_ratCast_of_den_le_79639646646701375323355774875831053","module":"Erdos249257/CertificateKernel.lean","line":18384,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CertificateKernel.lean#L18384","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L18384","kind":"theorem","signature":"theorem tsum_totient_div_pow_two_ne_ratCast_of_den_le_79639646646701375323355774875831053 : ∀ p : ℚ, p.den ≤ 79639646646701375323355774875831053 → (∑' n : ℕ, ((Nat.totient n : ℝ)) / (2 : ℝ) ^ n) ≠ (p : ℝ)","docstring":"**Erdős #249 denominator exclusion to `79639646646701375323355774875831053`, `ℚ` form.** If `∑_{n≥0} φ(n)/2^n` is rational, its reduced denominator exceeds `79639646646701375323355774875831053`.","generated_certificate":false}],"paper_label":"res:farey","paper_anchor":{"paper_label":"res:farey","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2331,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":2331,"anchor_title":"denominator exclusion","anchor_title_tex":"denominator exclusion","section_title":"Unconditional: no small denominator","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Unconditional: no small denominator","title_tex":"Unconditional: no small denominator","line":2329}],"source_excerpt":{"text":"\\begin{theorem}[denominator exclusion]\\label{res:farey}\nSet\n\\[\n\\Qzero:=79\\,639\\,646\\,646\\,701\\,375\\,323\\,355\\,774\\,875\\,831\\,053\n       \\approx7.96\\times10^{34}.\n\\]\nFor every integer $p$ and every $q$ with $1\\le q\\le\\Qzero$,\n\\[\nS \\;\\ne\\; \\frac{p}{q}.\n\\]\nEquivalently, if $S$ is rational then its reduced denominator exceeds that bound.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":123480,"source_char_end":123829,"start_line":2331,"end_line":2342,"truncated":false,"sha256":"sha256:1f1b0fad86fa9ef7849377e4799103995ed01a010999a9b4f95011be2c50bf6f","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2331","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_249_irrationality","statement":"Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/1"}],"links":[{"kind":"lean_declaration_pinned","label":"tsum_totient_div_pow_two_ne_ratCast_of_den_le_79639646646701375323355774875831053","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L18384","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/71","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"denominator_exclusion"}}],"expect":"object"},"scope_limit":{"text":"Integer p and 1 <= q <= 79 639 646 646 701 375 323 355 774 875 831 053; no larger denominator is excluded.","source":"claims_registry_bounded_domain"},"prior_art":{"family_id":"erdos249_quantitative_headline","status_summary":"One unconditional quantitative theorem and the open irrationality claim.","prior_art_posture":"Farey is historical language only; no exact antecedent for the explicit bound is identified, without asserting novelty.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/13","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"prior_art_found":1},"edges":[{"semantic_node_id":"Z01::totient_denominator_exclusion_ladder","prior_art_state":"prior_art_found"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:farey","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2331","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:certificate_reduction","assertion_kind":"registered_claim","source_id":"certificate_reduction","reviewed":true,"granularity":"single_registered_claim","label":"Exact reduction of #249 to finite decidable certificates","statement":"S is irrational if and only if certified non-integrality witnesses occur at unbounded parameters. The pointwise completeness theorem and the diagonal supply theorem together show that an unbounded lcm-diagonal supply is equivalent as well. Producing that supply remains open.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":true,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"claim_prose"}],"programme_ids":["erdos249_certificate_story"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"irrational_totient_series_of_certificate_supply","module":"Erdos249257/TotientTailPeriodKiller.lean","line":394,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientTailPeriodKiller.lean#L394","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientTailPeriodKiller.lean#L394","kind":"theorem","signature":"theorem irrational_totient_series_of_certificate_supply (hsupply : ∀ h : ℕ, 0 < h → ∀ N₀ : ℕ, ∃ N, N₀ ≤ N ∧ ∃ L, certifiedKill h N L) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"**The wave-21 reduction (the throat of #249).** A certified period-kill supply — for every period `h ≥ 1` and every threshold `N₀`, a certified kill at some `(N ≥ N₀, L)` — proves the totient series irrational. Each individual certificate is a finite, decidable residue computation.","generated_certificate":false},{"name":"irrational_totient_series_of_lcm_diagonal_certificate_supply","module":"Erdos249257/LcmDiagonalReduction.lean","line":112,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/LcmDiagonalReduction.lean#L112","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmDiagonalReduction.lean#L112","kind":"theorem","signature":"theorem irrational_totient_series_of_lcm_diagonal_certificate_supply (hsupply : ∀ t₀ : ℕ, ∃ t, t₀ ≤ t ∧ ∃ L, certifiedKill (periodLcm t) (periodLcm t) L) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"**The wave-23 diagonal collapse.** #249 follows from a SINGLE ℕ-indexed sequence of decidable statements: the endpoint certificate firing on the diagonal `(h, N) = (periodLcm t, periodLcm t)` for infinitely many `t`. Rationality would supply a primitive period `h₀` and pre-period `N₀`; any `t ≥ max(h₀, N₀)` gives `h₀ ∣ periodLcm t` and `periodLcm t ≥ t ≥ N₀` simultaneously, so a diagonal kill at such `t` is a contradiction. No second parameter remains.","generated_certificate":false},{"name":"rational_totient_series_forces_lcm_cone_flatness","module":"Erdos249257/LcmConeFlatness.lean","line":90,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/LcmConeFlatness.lean#L90","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmConeFlatness.lean#L90","kind":"theorem","signature":"theorem rational_totient_series_forces_lcm_cone_flatness (hrat : ¬ Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)) : ∃ t₁ : ℕ, ∀ t, t₁ ≤ t → ∀ q m : ℕ, 0 < q → totientTail (q * periodLcm t + m * periodLcm t) - totientTail (q * periodLcm t) ∈ Set.range ((↑) : ℤ → ℝ)","docstring":"**The lcm-cone flatness law.** If the totient series is rational, then from some scale on, the ENTIRE lcm cone is fractionally flat: every difference `R_{q·H_t + m·H_t} - R_{q·H_t}` is an integer, for every `t` past the threshold, every base multiplier `q ≥ 1`, and every step multiplier `m`. One constant fractional part governs every cone point — the maximal spend of the rationality hypothesis on the lcm lattice.","generated_certificate":false},{"name":"irrational_totient_series_iff_pointwise_certificates","module":"Erdos249257/LcmConeFlatness.lean","line":399,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/LcmConeFlatness.lean#L399","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmConeFlatness.lean#L399","kind":"theorem","signature":"theorem irrational_totient_series_iff_pointwise_certificates : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n) ↔ ∀ h : ℕ, 0 < h → ∀ N : ℕ, ∃ L, certifiedKill h N L","docstring":"Certificate completeness turns the pointwise non-integrality normal form into a pointwise finite-receipt normal form.","generated_certificate":false},{"name":"irrational_totient_series_iff_certificate_supply","module":"Erdos249257/LcmConeFlatness.lean","line":412,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/LcmConeFlatness.lean#L412","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmConeFlatness.lean#L412","kind":"theorem","signature":"theorem irrational_totient_series_iff_certificate_supply : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n) ↔ ∀ h : ℕ, 0 < h → ∀ N₀ : ℕ, ∃ N, N₀ ≤ N ∧ ∃ L, certifiedKill h N L","docstring":"The wave-21 cofinal certificate supply is not a weaker large-measure surrogate: together with completeness it is exactly equivalent to #249.","generated_certificate":false},{"name":"irrational_totient_series_iff_lcm_diagonal_certificate_supply","module":"Erdos249257/LcmConeFlatness.lean","line":426,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/LcmConeFlatness.lean#L426","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmConeFlatness.lean#L426","kind":"theorem","signature":"theorem irrational_totient_series_iff_lcm_diagonal_certificate_supply : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n) ↔ ∀ t₀ : ℕ, ∃ t, t₀ ≤ t ∧ ∃ L, certifiedKill (periodLcm t) (periodLcm t) L","docstring":"The one-parameter lcm-diagonal certificate supply is an exact normal form for #249, not only a sufficient criterion. In the forward direction, pointwise certificate completeness supplies the diagonal certificate already at `t = t₀`; the reverse direction is the diagonal collapse.","generated_certificate":false}],"paper_label":"res:diag","paper_anchor":{"paper_label":"res:diag","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2542,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":2542,"anchor_title":"exact diagonal characterisation","anchor_title_tex":"exact diagonal characterisation","section_title":"Why tail differences detect rationality.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Exact certificate characterisation","title_tex":"Exact certificate characterisation","line":2429},{"level":"paragraph","title":"Why tail differences detect rationality.","title_tex":"Why tail differences detect rationality.","line":2436}],"source_excerpt":{"text":"\\begin{theorem}[exact diagonal characterisation]\\label{res:diag}\nThe following are equivalent:\n\\[\n\\begin{aligned}\n &S\\notin\\Q;\\\\\n &\\forall\\,t_0\\in\\N\\ \\exists\\,t\\geq t_0\\ \\exists\\,L,\\qquad\n   \\mathsf{Sep}(M_t,M_t,L).\n\\end{aligned}\n\\]\n\\end{theorem}","media_type":"text/x-tex","source_char_start":134667,"source_char_end":134913,"start_line":2542,"end_line":2551,"truncated":false,"sha256":"sha256:fdb28bb05df5148eb7b270dd9858b0b55d74b87ef394f2664f976cc22a6156f5","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2542","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/3"}],"links":[{"kind":"lean_declaration_pinned","label":"irrational_totient_series_of_certificate_supply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientTailPeriodKiller.lean#L394","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totient_series_of_lcm_diagonal_certificate_supply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmDiagonalReduction.lean#L112","cite_this":true},{"kind":"lean_declaration_pinned","label":"rational_totient_series_forces_lcm_cone_flatness","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmConeFlatness.lean#L90","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totient_series_iff_pointwise_certificates","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmConeFlatness.lean#L399","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totient_series_iff_certificate_supply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmConeFlatness.lean#L412","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totient_series_iff_lcm_diagonal_certificate_supply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmConeFlatness.lean#L426","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/72","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"certificate_reduction"}}],"expect":"object"},"prior_art":{"family_id":"exact_certificate_equivalence_and_deposits","status_summary":"Exact equivalence and completeness with bounded verified deposits.","prior_art_posture":"Classical Lambert and Möbius identities underlie the construction; no exact certificate-equivalence antecedent is recorded, without a novelty conclusion.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/14","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":5},"edges":[{"semantic_node_id":"Z01::lcm_cone_flatness_law","prior_art_state":"not_assessed"},{"semantic_node_id":"Z06::certificate_supply_is_equivalent_to_249","prior_art_state":"not_assessed"},{"semantic_node_id":"Z06::lcm_cone_flatness_law","prior_art_state":"not_assessed"},{"semantic_node_id":"Z06::reduction_lcm_diagonal_supply","prior_art_state":"not_assessed"},{"semantic_node_id":"Z06::reduction_pointwise_certificate_supply","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:diag","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2542","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:certificate_completeness","assertion_kind":"registered_claim","source_id":"certificate_completeness","reviewed":true,"granularity":"single_registered_claim","label":"Certificate completeness","statement":"At fixed parameters, a certified kill exists exactly when the corresponding tail difference is non-integral. At every positive shift, integrality of one tail difference forces S to be rational; thus irrationality is equivalent to pointwise non-integrality, and to pointwise finite certificates.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":true,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"gateway_section","warrant":"the claim's anchor sits inside the retired gateway section 'The totient constant S (Erdős #249)', which is the author's own partition of that manuscript by problem"},"problem_edges":[{"problem_id":"erdos_249","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"exists_certifiedKill_iff_tail_diff_notMem_int","module":"Erdos249257/LcmConeFlatness.lean","line":316,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/LcmConeFlatness.lean#L316","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmConeFlatness.lean#L316","kind":"theorem","signature":"theorem exists_certifiedKill_iff_tail_diff_notMem_int (h N : ℕ) : (∃ L, certifiedKill h N L) ↔ totientTail (N + h) - totientTail N ∉ Set.range ((↑) : ℤ → ℝ)","docstring":"**Certificate completeness, iff form.** Some depth certifies `(h, N)` exactly when the tail difference is a non-integer. Finite certificates are receipts; non-integrality is the object.","generated_certificate":false},{"name":"tail_diff_mem_int_iff_scaled_series_mem_int","module":"Erdos249257/LcmConeFlatness.lean","line":327,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/LcmConeFlatness.lean#L327","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmConeFlatness.lean#L327","kind":"theorem","signature":"theorem tail_diff_mem_int_iff_scaled_series_mem_int (h N : ℕ) : (totientTail (N + h) - totientTail N ∈ Set.range ((↑) : ℤ → ℝ)) ↔ ((2 : ℝ) ^ N * ((2 : ℝ) ^ h - 1) * (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n) ∈ Set.range ((↑) : ℤ → ℝ))","docstring":"Integrality of one shifted tail difference is exactly integrality of the corresponding integer multiple `2^N (2^h - 1) S` of the #249 constant. This is the algebraic converse left implicit by the period-killer reduction.","generated_certificate":false},{"name":"irrational_totient_series_iff_all_tail_diffs_nonintegral","module":"Erdos249257/LcmConeFlatness.lean","line":386,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/LcmConeFlatness.lean#L386","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmConeFlatness.lean#L386","kind":"theorem","signature":"theorem irrational_totient_series_iff_all_tail_diffs_nonintegral : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n) ↔ ∀ h : ℕ, 0 < h → ∀ N : ℕ, totientTail (N + h) - totientTail N ∉ Set.range ((↑) : ℤ → ℝ)","docstring":"#249 is equivalent to pointwise non-integrality of every positive shifted tail difference.","generated_certificate":false},{"name":"irrational_totient_series_iff_pointwise_certificates","module":"Erdos249257/LcmConeFlatness.lean","line":399,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/LcmConeFlatness.lean#L399","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmConeFlatness.lean#L399","kind":"theorem","signature":"theorem irrational_totient_series_iff_pointwise_certificates : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n) ↔ ∀ h : ℕ, 0 < h → ∀ N : ℕ, ∃ L, certifiedKill h N L","docstring":"Certificate completeness turns the pointwise non-integrality normal form into a pointwise finite-receipt normal form.","generated_certificate":false}],"paper_label":"res:pointwisecomplete","paper_anchor":{"paper_label":"res:pointwisecomplete","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2507,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":2506,"anchor_title":"pointwise certificate characterisation","anchor_title_tex":"pointwise certificate characterisation","section_title":"Why tail differences detect rationality.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Exact certificate characterisation","title_tex":"Exact certificate characterisation","line":2429},{"level":"paragraph","title":"Why tail differences detect rationality.","title_tex":"Why tail differences detect rationality.","line":2436}],"source_excerpt":{"text":"\\begin{theorem}[pointwise certificate characterisation]\n\\label{res:pointwisecomplete}\nThe following are equivalent:\n\\[\n\\begin{aligned}\n &S\\notin\\Q,\\\\\n &\\forall\\,h\\ge1\\ \\forall\\,N\\ge0,\\qquad R_{N+h}-R_N\\notin\\Z,\\\\\n &\\forall\\,h\\ge1\\ \\forall\\,N\\ge0\\ \\exists\\,L,\\qquad\n      \\mathsf{Sep}(h,N,L).\n\\end{aligned}\n\\]\n\\end{theorem}","media_type":"text/x-tex","source_char_start":132928,"source_char_end":133250,"start_line":2506,"end_line":2517,"truncated":false,"sha256":"sha256:0bf37bfd782f86bdd2fa2bf4f6b50e25193d2090e031930da22f9cad36716c0a","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2507","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"exists_certifiedKill_iff_tail_diff_notMem_int","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmConeFlatness.lean#L316","cite_this":true},{"kind":"lean_declaration_pinned","label":"tail_diff_mem_int_iff_scaled_series_mem_int","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmConeFlatness.lean#L327","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totient_series_iff_all_tail_diffs_nonintegral","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmConeFlatness.lean#L386","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totient_series_iff_pointwise_certificates","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmConeFlatness.lean#L399","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/73","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"certificate_completeness"}}],"expect":"object"},"prior_art":{"family_id":"exact_certificate_equivalence_and_deposits","status_summary":"Exact equivalence and completeness with bounded verified deposits.","prior_art_posture":"Classical Lambert and Möbius identities underlie the construction; no exact certificate-equivalence antecedent is recorded, without a novelty conclusion.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/14","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2},"edges":[{"semantic_node_id":"Z06::certificate_supply_is_equivalent_to_249","prior_art_state":"not_assessed"},{"semantic_node_id":"Z06::endpoint_certificate_completeness","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:pointwisecomplete","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2507","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:certified_kill_instances","assertion_kind":"registered_claim","source_id":"certified_kill_instances","reviewed":true,"granularity":"single_registered_claim","label":"Initial certified-kill segment and generated certificates","statement":"Kernel-checked certified kills for the listed small periods, every lcm-diagonal scale t ≤ 82, and the finite certificate shards. The historical 28 explicit deposits through t = 64 form a strict subset of this contiguous finite band.","status":"verified finite instance","status_meaning":"Kernel-checked computation at a bounded range","readme_headline":true,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"certifiedKill_all_small","module":"Erdos249257/TotientTailPeriodKiller.lean","line":404,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientTailPeriodKiller.lean#L404","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientTailPeriodKiller.lean#L404","kind":"theorem","signature":"theorem certifiedKill_all_small : ∀ h ∈ Finset.Icc 1 8, certifiedKill h 12 16","docstring":"Machine-checked kills: for every period `h ∈ [1,8]` the residue certificate holds at `(N, L) = (12, 16)`. Kernel-decidable: 256 totient values below 37.","generated_certificate":false},{"name":"certifiedKill_periodLcm_diagonal_upto_six","module":"Erdos249257/LcmDiagonalReduction.lean","line":257,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/LcmDiagonalReduction.lean#L257","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmDiagonalReduction.lean#L257","kind":"theorem","signature":"theorem certifiedKill_periodLcm_diagonal_upto_six : ∀ t ∈ Finset.Icc 1 6, certifiedKill (periodLcm t) (periodLcm t) (diagonalKillDepth t)","docstring":"**Diagonal deposit.** The wave-23 predicate `P t` holds for every `1 ≤ t ≤ 6`: the endpoint certificate fires at `(H_t, H_t)` with the tabulated depth. Totient arguments stay ≤ 130 — a light kernel `decide`.","generated_certificate":false},{"name":"certifiedKill_periodLcm_diagonal_at_seven","module":"Erdos249257/LcmDiagonalReduction.lean","line":262,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/LcmDiagonalReduction.lean#L262","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmDiagonalReduction.lean#L262","kind":"theorem","signature":"theorem certifiedKill_periodLcm_diagonal_at_seven : certifiedKill (periodLcm 7) (periodLcm 7) 14","docstring":"`P 7`: the diagonal certificate at `lcm(1..7) = 420`, depth `14`.","generated_certificate":false},{"name":"certifiedKill_periodLcm_diagonal_at_eight","module":"Erdos249257/LcmDiagonalReduction.lean","line":266,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/LcmDiagonalReduction.lean#L266","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmDiagonalReduction.lean#L266","kind":"theorem","signature":"theorem certifiedKill_periodLcm_diagonal_at_eight : certifiedKill (periodLcm 8) (periodLcm 8) 15","docstring":"`P 8`: the diagonal certificate at `lcm(1..8) = 840`, depth `15`.","generated_certificate":false},{"name":"certifiedKill_diagonal_all_imported_through_t64","module":"Erdos249257/DiagonalPincerCertificatesT64.lean","line":1967,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DiagonalPincerCertificatesT64.lean#L1967","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalPincerCertificatesT64.lean#L1967","kind":"theorem","signature":"theorem certifiedKill_diagonal_all_imported_through_t64 : ∀ t ∈ diagonalPincerCertificateScalesThroughT64, certifiedKill (periodLcm t) (periodLcm t) (diagonalPincerKillDepthThroughT64 t)","docstring":"All 28 imported scales through t=64 fire.","generated_certificate":true},{"name":"exists_diagonalKill_le_82","module":"ErdosProblems/Skip/LadderT67.lean","line":71264,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Skip/LadderT67.lean#L71264","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Skip/LadderT67.lean#L71264","kind":"theorem","signature":"theorem exists_diagonalKill_le_82 (t : ℕ) (ht : t ≤ 82) : ∃ L, certifiedKill (periodLcm t) (periodLcm t) L","docstring":"**The certified band closes `t ≤ 82`, unconditionally.** Every scale from `1` to `82` carries a diagonal kill certificate, with no holes: `t ≤ 66` from the corpus, `67 … 70` from `Recon67`, `71, 72` from `CertT67`, `73 … 78`, `79, 80` and `81, 82` from the three certificates proved here together with their plateaus. The next jump is the prime `83`.","generated_certificate":false}],"paper_label":"res:deposits","paper_anchor":{"paper_label":"res:deposits","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2557,"anchor_kind":"environment","anchor_unit":"example","anchor_unit_line":2557,"anchor_title":"verified finite range","anchor_title_tex":"verified finite range","section_title":"Why tail differences detect rationality.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Exact certificate characterisation","title_tex":"Exact certificate characterisation","line":2429},{"level":"paragraph","title":"Why tail differences detect rationality.","title_tex":"Why tail differences detect rationality.","line":2436}],"source_excerpt":{"text":"\\begin{example}[verified finite range]\\label{res:deposits}\nLean checks:\n\\begin{itemize}[leftmargin=1.4em]\n\\item $\\mathsf{Sep}(h,12,16)$ for every $1\\le h\\le8$;\n\\item a fixed-window certificate for every $1\\le h\\le16$;\n\\item diagonal certificates $\\mathsf{Sep}(M_t,M_t,L)$ at $t=1$ and every\nlcm-height change endpoint $2\\le t\\le64$ (that is, $M_{t-1}<M_t$), namely\n\\[\n\\begin{aligned}\nt\\in\\{&1,2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,\\\\\n      &29,31,32,37,41,43,47,49,53,59,61,64\\};\n\\end{aligned}\n\\]\n\\item three finite lists of tails that cannot all share one fractional part.\n\\end{itemize}\n\\end{example}","media_type":"text/x-tex","source_char_start":135212,"source_char_end":135813,"start_line":2557,"end_line":2572,"truncated":false,"sha256":"sha256:6a010a51c9aee456da450f8d36322b239316594c2c01af8667be0bd6f2d87232","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2557","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/3"}],"links":[{"kind":"lean_declaration_pinned","label":"certifiedKill_all_small","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientTailPeriodKiller.lean#L404","cite_this":true},{"kind":"lean_declaration_pinned","label":"certifiedKill_periodLcm_diagonal_upto_six","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmDiagonalReduction.lean#L257","cite_this":true},{"kind":"lean_declaration_pinned","label":"certifiedKill_periodLcm_diagonal_at_seven","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmDiagonalReduction.lean#L262","cite_this":true},{"kind":"lean_declaration_pinned","label":"certifiedKill_periodLcm_diagonal_at_eight","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmDiagonalReduction.lean#L266","cite_this":true},{"kind":"lean_declaration_pinned","label":"certifiedKill_diagonal_all_imported_through_t64","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalPincerCertificatesT64.lean#L1967","cite_this":true},{"kind":"lean_declaration_pinned","label":"exists_diagonalKill_le_82","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Skip/LadderT67.lean#L71264","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/74","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"certified_kill_instances"}}],"expect":"object"},"scope_limit":{"text":"The explicitly listed small periods; every natural lcm-diagonal scale t with t ≤ 82; and the parameters encoded by the finite certificate shards. No certificate at t = 83 and no unbounded or cofinal supply is claimed.","source":"claims_registry_bounded_domain"},"prior_art":{"family_id":"exact_certificate_equivalence_and_deposits","status_summary":"Exact equivalence and completeness with bounded verified deposits.","prior_art_posture":"Classical Lambert and Möbius identities underlie the construction; no exact certificate-equivalence antecedent is recorded, without a novelty conclusion.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/14","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"formalisation_only":1,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z02::aggregate_imported_scales_fire","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z06::finite_kill_deposits","prior_art_state":"formalisation_only"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:deposits","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2557","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:diagonal_pincer_reduction","assertion_kind":"registered_claim","source_id":"diagonal_pincer_reduction","reviewed":true,"granularity":"single_registered_claim","label":"Exact full-target reduction on the lcm diagonal","statement":"At each lcm-diagonal scale, integrality holds if and only if a foreign-defect term lands in an explicit full-target interval. Globally, S is irrational if and only if the full target is avoided at arbitrarily large diagonal scales; that cofinal avoidance is not proved.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":true,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":["erdos249_diagonal_arithmetic"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"diagonal_int_iff_foreignDiagonalDefect_hits_fullTarget","module":"Erdos249257/DiagonalPincerDecomposition.lean","line":215,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DiagonalPincerDecomposition.lean#L215","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalPincerDecomposition.lean#L215","kind":"theorem","signature":"theorem diagonal_int_iff_foreignDiagonalDefect_hits_fullTarget (t : ℕ) : IsIntegralValue (diagonalTailDifference t) ↔ ForeignDiagonalDefectHitsFullTarget t","docstring":"The primary complement-free pincer. Integrality of the actual diagonal is exactly a hit by `foreignDiagonalDefect` alone on the full reduced target `-a_t / d_t` modulo `ℤ`.","generated_certificate":false},{"name":"irrational_totientSeries_of_full_target_avoidance_supply","module":"Erdos249257/DiagonalPincerDecomposition.lean","line":290,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DiagonalPincerDecomposition.lean#L290","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalPincerDecomposition.lean#L290","kind":"theorem","signature":"theorem irrational_totientSeries_of_full_target_avoidance_supply (hsupply : FullTargetAvoidanceSupply) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"Full-target avoidance at arbitrarily large LCM heights closes the existing diagonal non-integrality socket and hence conditionally proves #249.","generated_certificate":false}],"paper_label":"res:fulltargetavoidance","paper_anchor":{"paper_label":"res:fulltargetavoidance","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2702,"anchor_kind":"environment","anchor_unit":"corollary","anchor_unit_line":2701,"anchor_title":"exact unbounded target-avoidance characterisation","anchor_title_tex":"exact unbounded target-avoidance characterisation","section_title":"An exact target residue class.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Exact unbounded-depth normal forms and cone consequences","title_tex":"Exact unbounded-depth normal forms and cone consequences","line":2639},{"level":"paragraph","title":"An exact target residue class.","title_tex":"An exact target residue class.","line":2676}],"source_excerpt":{"text":"\\begin{corollary}[exact unbounded target-avoidance characterisation]\n\\label{res:fulltargetavoidance}\nThe following are equivalent:\n\\[\n\\begin{aligned}\n &S\\notin\\Q;\\\\\n &\\forall\\,t_0\\in\\N\\ \\exists\\,t\\geq t_0,\\qquad\n   \\neg\\mathsf{Hit}(t).\n\\end{aligned}\n\\]\n\\end{corollary}","media_type":"text/x-tex","source_char_start":142256,"source_char_end":142524,"start_line":2701,"end_line":2711,"truncated":false,"sha256":"sha256:62445688c66946102412911e3e87c6d01fd88f7171bc31bd89f62276b90943a5","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2702","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/3"}],"links":[{"kind":"lean_declaration_pinned","label":"diagonal_int_iff_foreignDiagonalDefect_hits_fullTarget","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalPincerDecomposition.lean#L215","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totientSeries_of_full_target_avoidance_supply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalPincerDecomposition.lean#L290","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/75","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"diagonal_pincer_reduction"}}],"expect":"object"},"prior_art":{"family_id":"diagonal_pincer_and_denominator_survival","status_summary":"Exact diagonal targets, sharp enclosures and denominators, and conditional avoidance projections.","prior_art_posture":"Classical Möbius/Lambert algebra only; no theorem-specific antecedent is identified.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/15","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z05::exact_full_target_pincer","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z05::supply_conditional_irrationality","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:fulltargetavoidance","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2702","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:diagonal_fresh_loss_reduction","assertion_kind":"registered_claim","source_id":"diagonal_fresh_loss_reduction","reviewed":true,"granularity":"single_registered_claim","label":"Fresh-loss projection and suffix-supply reduction","statement":"The fresh-loss residue certificate is exactly an integer projection inequality, and such a projection forces full-target avoidance. An unbounded projection supply—or the stronger adjacent-suffix gap supplies formalised here—implies irrationality; no unbounded supply is asserted.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":["erdos249_diagonal_arithmetic"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"diagonalFreshLossResidueCert_iff_projectionCert","module":"Erdos249257/DiagonalFreshLossBridge.lean","line":72,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DiagonalFreshLossBridge.lean#L72","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalFreshLossBridge.lean#L72","kind":"theorem","signature":"theorem diagonalFreshLossResidueCert_iff_projectionCert (t L : ℕ) : diagonalFreshLossResidueCert t L ↔ diagonalFreshLossProjectionCert t L","docstring":"On the two-vertex menu, the generic cone certificate is exactly the asymmetric residue-safe interval. This pins the operational shape used by the diagnostic probe to a kernel theorem rather than a comment-level analogy.","generated_certificate":false},{"name":"irrational_totientSeries_of_diagonalFreshLossProjectionSupply","module":"Erdos249257/DiagonalFreshLossBridge.lean","line":202,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DiagonalFreshLossBridge.lean#L202","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalFreshLossBridge.lean#L202","kind":"theorem","signature":"theorem irrational_totientSeries_of_diagonalFreshLossProjectionSupply (hsupply : DiagonalFreshLossProjectionSupply) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"Unbounded exact `[1, 2]` projection certificates feed the full-target avoidance consumer and therefore prove irrationality of the totient series.","generated_certificate":false}],"paper_label":"res:freshloss","paper_anchor":{"paper_label":"res:freshloss","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2916,"anchor_kind":"environment","anchor_unit":"corollary","anchor_unit_line":2916,"anchor_title":"finite residue projection","anchor_title_tex":"finite residue projection","section_title":"Exact finite divisor-index split and sufficient projection test.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Fixed-scale mechanisms and conditional unbounded implications","title_tex":"Fixed-scale mechanisms and conditional unbounded implications","line":2720},{"level":"paragraph","title":"Exact finite divisor-index split and sufficient projection test.","title_tex":"Exact finite divisor-index split and sufficient projection test.","line":2820}],"source_excerpt":{"text":"\\begin{corollary}[finite residue projection]\\label{res:freshloss}\nFor every \\(t,L\\in\\N\\), the diagonal projection certificate is equivalent to\n\\[\n M_t+L+2<\\eta_{t,L}<2^L-(2M_t+L+2),\n\\]\nand either condition forces full-target avoidance at height \\(t\\).\nConsequently,\n\\[\n \\forall t_0\\in\\N\\;\\exists t\\geq t_0\\;\\exists L\\in\\N,\\qquad\n M_t+L+2<\\eta_{t,L}<2^L-(2M_t+L+2)\n\\]\nimplies that \\(S\\) is irrational.\n\\end{corollary}","media_type":"text/x-tex","source_char_start":152686,"source_char_end":153102,"start_line":2916,"end_line":2928,"truncated":false,"sha256":"sha256:ee95336573f5dfc1e1343ceec1c2871cf1146b581dacd27e2f05d5fb36c0165a","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2916","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/3"}],"links":[{"kind":"lean_declaration_pinned","label":"diagonalFreshLossResidueCert_iff_projectionCert","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalFreshLossBridge.lean#L72","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totientSeries_of_diagonalFreshLossProjectionSupply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalFreshLossBridge.lean#L202","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/76","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"diagonal_fresh_loss_reduction"}}],"expect":"object"},"prior_art":{"family_id":"diagonal_pincer_and_denominator_survival","status_summary":"Exact diagonal targets, sharp enclosures and denominators, and conditional avoidance projections.","prior_art_posture":"Classical Möbius/Lambert algebra only; no theorem-specific antecedent is identified.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/15","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2},"edges":[{"semantic_node_id":"Z05::supply_conditional_irrationality","prior_art_state":"not_assessed"},{"semantic_node_id":"Z05::two_vertex_certificate_forces_full_target_miss","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:freshloss","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2916","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:diagonal_pincer_t64_primality_closure","assertion_kind":"registered_claim","source_id":"diagonal_pincer_t64_primality_closure","reviewed":true,"granularity":"single_registered_claim","label":"T64 diagonal-pincer primality closure","statement":"A generated proof-producing Lucas-certificate module supplies the primality dependencies for the t=47,49,53,59,61,64 diagonal-pincer certificates. This is finite certificate infrastructure; it neither supplies an unbounded certificate family nor proves irrationality.","status":"verified finite instance","status_meaning":"Kernel-checked computation at a bounded range","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"gateway_section","warrant":"the claim's anchor sits inside the retired gateway section 'The totient constant S (Erdős #249)', which is the author's own partition of that manuscript by problem"},"problem_edges":[{"problem_id":"erdos_249","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"prime_lucas_2364533768205644535022723273","module":"Erdos249257/DiagonalPincerPrimeCertificates/ClosureT64/Level1A.lean","line":2398,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DiagonalPincerPrimeCertificates/ClosureT64/Level1A.lean#L2398","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalPincerPrimeCertificates/ClosureT64/Level1A.lean#L2398","kind":"theorem","signature":"theorem prime_lucas_2364533768205644535022723273 : Nat.Prime 2364533768205644535022723273","generated_certificate":true}],"paper_label":"res:t64closure","paper_anchor":{"paper_label":"res:t64closure","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2941,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Primality certificates at changing-lcm scales through t=64.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Fixed-scale mechanisms and conditional unbounded implications","title_tex":"Fixed-scale mechanisms and conditional unbounded implications","line":2720},{"level":"paragraph","title":"Primality certificates at changing-lcm scales through t=64.","title_tex":"Primality certificates at changing-lcm scales through $t=64$.","line":2941}],"source_excerpt":{"text":"\\paragraph{Primality certificates at changing-lcm scales through $t=64$.}\\label{res:t64closure}\nA finite family of Lucas certificates proves the primality facts required by\nthe diagonal certificates at every $t\\leq64$ where $M_t$ increases;\nfor example,\n\\lword{DiagonalPincerPrimeCertificates/ClosureT64/Level1A.lean}{2398}{prime_lucas_2364533768205644535022723273}{representative Lucas primality certificate}.\nThis finite theorem family verifies the required primality inputs only through\n$t=64$.","media_type":"text/x-tex","source_char_start":153840,"source_char_end":154337,"start_line":2941,"end_line":2947,"truncated":false,"sha256":"sha256:eda7b48318df6d02945ffc003229efde6629459247fda58e5e6c0732f5d4f6d1","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2941","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"prime_lucas_2364533768205644535022723273","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalPincerPrimeCertificates/ClosureT64/Level1A.lean#L2398","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/77","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"diagonal_pincer_t64_primality_closure"}}],"expect":"object"},"scope_limit":{"text":"The generated Lucas primality dependencies required by the diagonal-pincer certificate family through t = 64.","source":"claims_registry_bounded_domain"},"prior_art":{"family_id":"exact_certificate_equivalence_and_deposits","status_summary":"Exact equivalence and completeness with bounded verified deposits.","prior_art_posture":"Classical Lambert and Möbius identities underlie the construction; no exact certificate-equivalence antecedent is recorded, without a novelty conclusion.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/14","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex"},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:t64closure","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2941","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:diagonal_pincer_t64_endpoint_certificate","assertion_kind":"registered_claim","source_id":"diagonal_pincer_t64_endpoint_certificate","reviewed":true,"granularity":"single_registered_claim","label":"T64 diagonal-pincer endpoint certificate","statement":"Lean checks the diagonal-pincer certified kill at t = 64 using the committed T64 Lucas-primality closure. This is one finite certificate and does not assert an unbounded certificate supply or irrationality.","status":"verified finite instance","status_meaning":"Kernel-checked computation at a bounded range","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"gateway_section","warrant":"the claim's anchor sits inside the retired gateway section 'The totient constant S (Erdős #249)', which is the author's own partition of that manuscript by problem"},"problem_edges":[{"problem_id":"erdos_249","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"certifiedKill_diagonal_t64","module":"Erdos249257/DiagonalPincerCertificatesT64.lean","line":1928,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DiagonalPincerCertificatesT64.lean#L1928","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalPincerCertificatesT64.lean#L1928","kind":"theorem","signature":"theorem certifiedKill_diagonal_t64 : certifiedKill (periodLcm 64) (periodLcm 64) 93","generated_certificate":true}],"paper_label":"res:t64endpoint","paper_anchor":{"paper_label":"res:t64endpoint","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2949,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"The diagonal certificate at t=64.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Fixed-scale mechanisms and conditional unbounded implications","title_tex":"Fixed-scale mechanisms and conditional unbounded implications","line":2720},{"level":"paragraph","title":"The diagonal certificate at t=64.","title_tex":"The diagonal certificate at $t=64$.","line":2949}],"source_excerpt":{"text":"\\paragraph{The diagonal certificate at $t=64$.}\\label{res:t64endpoint}\nUsing those primality facts, Lean checks the individual diagonal certificate\nat $t=64$,\n\\lword{DiagonalPincerCertificateT64Endpoint.lean}{1928}{certifiedKill_diagonal_t64}{scale-64 diagonal certificate}.\nThis endpoint is one of the 28 finite diagonal certificates in\nExample~\\ref{res:deposits}.","media_type":"text/x-tex","source_char_start":154339,"source_char_end":154704,"start_line":2949,"end_line":2954,"truncated":false,"sha256":"sha256:2755a853581b2256ba274591a6af5299bae4181954855f49f1b5dbfc6405074d","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2949","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"certifiedKill_diagonal_t64","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalPincerCertificatesT64.lean#L1928","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/78","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"diagonal_pincer_t64_endpoint_certificate"}}],"expect":"object"},"scope_limit":{"text":"The single lcm-diagonal endpoint t = 64.","source":"claims_registry_bounded_domain"},"prior_art":{"family_id":"exact_certificate_equivalence_and_deposits","status_summary":"Exact equivalence and completeness with bounded verified deposits.","prior_art_posture":"Classical Lambert and Möbius identities underlie the construction; no exact certificate-equivalence antecedent is recorded, without a novelty conclusion.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/14","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex"},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:t64endpoint","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2949","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:fresh_prime_deficit_decomposition","assertion_kind":"registered_claim","source_id":"fresh_prime_deficit_decomposition","reviewed":true,"granularity":"single_registered_claim","label":"Fresh-prime endpoint-deficit decomposition","statement":"At each positive height, every old-prime endpoint mass dominates the actual totient. The diagonal increment and literal foreign channel split exactly into the old Möbius increment and lower-minus-upper endpoint fresh deficits. The second and five-point curvature operators retain this split, and the nonnegative deficits bound the exact adverse weighted loss from the old-prime curvature margins. No cofinal deficit bound or irrationality conclusion is proved.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"endpointFreshDeficit_nonneg","module":"Erdos249257/FreshPrimeDeficitDecomposition.lean","line":91,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/FreshPrimeDeficitDecomposition.lean#L91","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FreshPrimeDeficitDecomposition.lean#L91","kind":"theorem","signature":"theorem endpointFreshDeficit_nonneg (H k s : ℕ) (hH : 0 < H) (hk : 0 < k) : 0 ≤ endpointFreshDeficit H k s","docstring":"Every old-prime endpoint mass dominates the actual totient.","generated_certificate":false},{"name":"finiteForeignChannelIncrement_eq_endpointFreshDeficit_sub","module":"Erdos249257/FreshPrimeDeficitDecomposition.lean","line":184,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/FreshPrimeDeficitDecomposition.lean#L184","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FreshPrimeDeficitDecomposition.lean#L184","kind":"theorem","signature":"theorem finiteForeignChannelIncrement_eq_endpointFreshDeficit_sub (H s : ℕ) (hH : 0 < H) : finiteForeignChannelIncrement H s = endpointFreshDeficit H 1 s - endpointFreshDeficit H 2 s","docstring":"The literal foreign Möbius channel is precisely lower endpoint deficit minus upper endpoint deficit.","generated_certificate":false},{"name":"oldFivePoint_sub_secondBranchAdverseDeficit_le_actual","module":"Erdos249257/FreshPrimeDeficitDecomposition.lean","line":269,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/FreshPrimeDeficitDecomposition.lean#L269","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FreshPrimeDeficitDecomposition.lean#L269","kind":"theorem","signature":"theorem oldFivePoint_sub_secondBranchAdverseDeficit_le_actual (H s : ℕ) (hH : 0 < H) : fourCurvaturePlusShift (oldMobiusIncrement H) s - secondBranchAdverseDeficit H s ≤ fourCurvaturePlusShift (diagonalHeightIncrement H) s","docstring":"Nonnegative endpoint deficits reduce the old five-point margin by at most the exact adverse weighted deficit used by the finite probe.","generated_certificate":false}],"paper_label":"res:freshprimedeficit","paper_anchor":{"paper_label":"res:freshprimedeficit","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2998,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":2998,"anchor_title":"fresh-prime deficit decomposition","anchor_title_tex":"fresh-prime deficit decomposition","section_title":"Exact fresh-prime deficit decomposition.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Candidate unbounded inputs and no-go boundaries","title_tex":"Candidate unbounded inputs and no-go boundaries","line":2956},{"level":"paragraph","title":"Exact fresh-prime deficit decomposition.","title_tex":"Exact fresh-prime deficit decomposition.","line":2988}],"source_excerpt":{"text":"\\begin{proposition}[fresh-prime deficit decomposition]\\label{res:freshprimedeficit}\nThe endpoint fresh deficits are nonnegative, give an exact decomposition of\nthe foreign-defect increment, and bound the loss from the old-prime curvature margins.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":157582,"source_char_end":157846,"start_line":2998,"end_line":3001,"truncated":false,"sha256":"sha256:44d8762f2174e6d78da8d15353995324db3c50639ba313c3a29a0c14f269363c","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2998","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/3"}],"links":[{"kind":"lean_declaration_pinned","label":"endpointFreshDeficit_nonneg","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FreshPrimeDeficitDecomposition.lean#L91","cite_this":true},{"kind":"lean_declaration_pinned","label":"finiteForeignChannelIncrement_eq_endpointFreshDeficit_sub","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FreshPrimeDeficitDecomposition.lean#L184","cite_this":true},{"kind":"lean_declaration_pinned","label":"oldFivePoint_sub_secondBranchAdverseDeficit_le_actual","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FreshPrimeDeficitDecomposition.lean#L269","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/79","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"fresh_prime_deficit_decomposition"}}],"expect":"object"},"prior_art":{"family_id":"fresh_loss_local_arithmetic","status_summary":"Exact local arithmetic, conditional band/slack reductions, and scoped parity ceilings.","prior_art_posture":"No close statement comparison is recorded; novelty remains unasserted.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/16","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z05::endpoint_fresh_deficit_decomposition","prior_art_state":"routine_corollary_or_specialisation"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:freshprimedeficit","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2998","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:flexible_odd_window_reduction","assertion_kind":"registered_claim","source_id":"flexible_odd_window_reduction","reviewed":true,"granularity":"single_registered_claim","label":"Flexible three-rank odd-window reduction","statement":"At an analytically admissible odd power-of-two rank, centrality in any one of the canonical odd rank or its next two base-four successors supplies the existing adjacent-suffix gap interface and hence irrationality. The remaining assertion is a cofinal three-rank band supply; no such supply is proved.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"diagonalAdjacentSuffixGapSupply_of_powerTwoOddGuardThreeRankBand","module":"Erdos249257/DiagonalFlexibleOddWindowSupply.lean","line":93,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DiagonalFlexibleOddWindowSupply.lean#L93","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalFlexibleOddWindowSupply.lean#L93","kind":"theorem","signature":"theorem diagonalAdjacentSuffixGapSupply_of_powerTwoOddGuardThreeRankBand (hsupply : PowerTwoOddGuardThreeRankBandSupply) : DiagonalAdjacentSuffixGapSupply","docstring":"The three-rank producer supplies the landed adjacent-gap interface.","generated_certificate":false},{"name":"irrational_totientSeries_of_powerTwoOddGuardThreeRankBandSupply","module":"Erdos249257/DiagonalFlexibleOddWindowSupply.lean","line":137,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DiagonalFlexibleOddWindowSupply.lean#L137","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalFlexibleOddWindowSupply.lean#L137","kind":"theorem","signature":"theorem irrational_totientSeries_of_powerTwoOddGuardThreeRankBandSupply (hsupply : PowerTwoOddGuardThreeRankBandSupply) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"The flexible three-rank band supply implies irrationality of the totient series. The supply itself remains unproved.","generated_certificate":false}],"paper_label":"res:flexibleoddwindow","paper_anchor":{"paper_label":"res:flexibleoddwindow","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2968,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Candidate unbounded input: a three-rank odd window.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Candidate unbounded inputs and no-go boundaries","title_tex":"Candidate unbounded inputs and no-go boundaries","line":2956},{"level":"paragraph","title":"Candidate unbounded input: a three-rank odd window.","title_tex":"Candidate unbounded input: a three-rank odd window.","line":2968}],"source_excerpt":{"text":"\\paragraph{Candidate unbounded input: a three-rank odd window.}\\label{res:flexibleoddwindow}\nFix $t=2^a$ and increase the canonical suffix depth by at most one to write it\nas $m=2q+1$.  Call a rank \\emph{central} when its half-word residue lies\nbetween $4^q/32$ and $31\\cdot4^q/32$.  Test $q,q+1,q+2$; centrality at any one\nyields the required adjacent-suffix gap condition,\n\\lword{DiagonalFlexibleOddWindowSupply.lean}{93}{diagonalAdjacentSuffixGapSupply_of_powerTwoOddGuardThreeRankBand}{three-rank suffix-gap implication}.\nThus success of this three-rank test for arbitrarily large $a$ implies\nirrationality,\n\\lword{DiagonalFlexibleOddWindowSupply.lean}{137}{irrational_totientSeries_of_powerTwoOddGuardThreeRankBandSupply}{unbounded three-rank criterion};\nthat supply is not proved.","media_type":"text/x-tex","source_char_start":155371,"source_char_end":156157,"start_line":2968,"end_line":2977,"truncated":false,"sha256":"sha256:73f1fac43f88a999e6fb98944ab80845be0b64861cc7b502dd71332a9af07952","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2968","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. 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Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2},"edges":[{"semantic_node_id":"Z05::power_two_supply_ladder_equivalences","prior_art_state":"not_assessed"},{"semantic_node_id":"Z05::supply_conditional_irrationality","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:flexibleoddwindow","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2968","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:flexible_odd_window_affine_frontier","assertion_kind":"registered_claim","source_id":"flexible_odd_window_affine_frontier","reviewed":true,"granularity":"single_registered_claim","label":"Affine three-scale frontier for the flexible odd window","statement":"Under the stated short-correction bounds, the three half-word central-band predicates are exactly a three-scale affine escape disjunction. A uniform correction envelope alone cannot force that escape, as a zero-start affine orbit remains inside every scaled edge tube. Thus arithmetic state-correction correlation remains required.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"zeroAffineOrbit_stays_in_scaled_edge","module":"Erdos249257/DiagonalFlexibleOddWindowAffine.lean","line":225,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DiagonalFlexibleOddWindowAffine.lean#L225","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalFlexibleOddWindowAffine.lean#L225","kind":"theorem","signature":"theorem zeroAffineOrbit_stays_in_scaled_edge {e : ℤ} (he : 0 < e) (C : ℕ → ℤ) (hC : ∀ n, |C n| < 2 * e) (j : ℕ) : |zeroAffineOrbit C j| < (4 : ℤ) ^ j * e","docstring":"The zero-start affine orbit remains inside every scaled edge tube. Thus no theorem using only a uniform `< 2e` correction envelope can force escape by increasing the number of inspected ranks.","generated_certificate":false},{"name":"threeRankBandAt_iff_threeScaleAffineEscapeAt_of_shortCorrections","module":"Erdos249257/DiagonalFlexibleOddWindowAffine.lean","line":236,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DiagonalFlexibleOddWindowAffine.lean#L236","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalFlexibleOddWindowAffine.lean#L236","kind":"theorem","signature":"theorem threeRankBandAt_iff_threeScaleAffineEscapeAt_of_shortCorrections {a q : ℕ} (ha : 2 ≤ a) (hq : 3 ≤ q) (hC : |actualOddHalfCorrection a q| < 2 * edge q) (hF : |actualOddHalfCorrection a (q + 1)| < 2 * edge q) : ThreeRankBandAt a q ↔ ThreeScaleAffineEscapeAt a q","docstring":"Exact replacement of the three half-word predicates by one affine three-scale escape predicate. Wrapping disappears conditionally on the preceding edge hits.","generated_certificate":false}],"paper_label":"res:flexibleoddaffine","paper_anchor":{"paper_label":"res:flexibleoddaffine","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2979,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Exact affine form and bounded zero-orbit obstruction.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Candidate unbounded inputs and no-go boundaries","title_tex":"Candidate unbounded inputs and no-go boundaries","line":2956},{"level":"paragraph","title":"Exact affine form and bounded zero-orbit obstruction.","title_tex":"Exact affine form and bounded zero-orbit obstruction.","line":2979}],"source_excerpt":{"text":"\\paragraph{Exact affine form and bounded zero-orbit obstruction.}\\label{res:flexibleoddaffine}\nLet $x_q$ be the centred half-word residue and put $e_q=4^q/32$.  Centrality is\n$|x_q|\\geq e_q$; under the stated bounds on the next two corrections, the three\ntests are equivalent to one affine escape disjunction,\n\\lword{DiagonalFlexibleOddWindowAffine.lean}{236}{threeRankBandAt_iff_threeScaleAffineEscapeAt_of_shortCorrections}{three-scale affine equivalence}.\nYet the generic recurrence $y_{j+1}=4y_j+c_j$, started at zero, can satisfy\n$|y_j|<4^j e$ for every $j$ even when every $|c_j|<2e$,\n\\lword{DiagonalFlexibleOddWindowAffine.lean}{225}{zeroAffineOrbit_stays_in_scaled_edge}{zero-orbit obstruction}.","media_type":"text/x-tex","source_char_start":156159,"source_char_end":156862,"start_line":2979,"end_line":2986,"truncated":false,"sha256":"sha256:5dd629e4d3e38900fb10c19f73b737dc24f3c135d016dab4091eca3fe6aec8e7","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2979","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/3"}],"links":[{"kind":"lean_declaration_pinned","label":"zeroAffineOrbit_stays_in_scaled_edge","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalFlexibleOddWindowAffine.lean#L225","cite_this":true},{"kind":"lean_declaration_pinned","label":"threeRankBandAt_iff_threeScaleAffineEscapeAt_of_shortCorrections","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalFlexibleOddWindowAffine.lean#L236","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/81","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"flexible_odd_window_affine_frontier"}}],"expect":"object"},"prior_art":{"family_id":"fresh_loss_local_arithmetic","status_summary":"Exact local arithmetic, conditional band/slack reductions, and scoped parity ceilings.","prior_art_posture":"No close statement comparison is recorded; novelty remains unasserted.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/16","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2},"edges":[{"semantic_node_id":"Z05::envelope_only_no_go","prior_art_state":"not_assessed"},{"semantic_node_id":"Z05::power_two_halfword_pointwise_equivalences","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:flexibleoddaffine","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2979","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:power_two_bit_lift","assertion_kind":"registered_claim","source_id":"power_two_bit_lift","reviewed":true,"granularity":"single_registered_claim","label":"Exact power-of-two one-bit lift","statement":"Given agreement modulo 2^k, agreement modulo 2^(k+1) is exactly evenness of the signed quotient after division by 2^k; failure gives the unique offset class. In particular, a signed difference 16*c agrees modulo 32 exactly when c is even. Under the stated correction and margin budget, the centered signed-margin socket is exactly failure of this next lift. This provides a finite fifth-bit interface and does not supply cofactor parity at unbounded scales.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"two_pow_succ_dvd_iff_quotient_even","module":"Erdos249257/PowerTwoBitLift.lean","line":34,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PowerTwoBitLift.lean#L34","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PowerTwoBitLift.lean#L34","kind":"theorem","signature":"theorem two_pow_succ_dvd_iff_quotient_even {z : ℤ} {k : ℕ} (hk : (2 : ℤ) ^ k ∣ z) : (2 : ℤ) ^ (k + 1) ∣ z ↔ Even (z / (2 : ℤ) ^ k)","docstring":"Exact divisibility lift from `2^k` to `2^(k+1)`: after the known power of two is divided out, only the parity of the integral quotient remains.","generated_certificate":false},{"name":"modEq_thirtyTwo_iff_even_difference_cofactor","module":"Erdos249257/PowerTwoBitLift.lean","line":111,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PowerTwoBitLift.lean#L111","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PowerTwoBitLift.lean#L111","kind":"theorem","signature":"theorem modEq_thirtyTwo_iff_even_difference_cofactor {P N c : ℤ} (hfactor : P - N = 16 * c) : Int.ModEq 32 P N ↔ Even c","docstring":"Factor-certificate form adapted to a signed difference `P-N`; changing the orientation of the difference does not change the parity test.","generated_certificate":false},{"name":"not_modEq_thirtyTwo_iff_offset_sixteen","module":"Erdos249257/PowerTwoBitLift.lean","line":129,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PowerTwoBitLift.lean#L129","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PowerTwoBitLift.lean#L129","kind":"theorem","signature":"theorem not_modEq_thirtyTwo_iff_offset_sixteen {x y : ℤ} (hxy : Int.ModEq 16 x y) : ¬ Int.ModEq 32 x y ↔ Int.ModEq 32 x (y + 16)","docstring":"The offset class is not merely a possible failure class: under fourth-bit agreement it is equivalent to failure of fifth-bit agreement.","generated_certificate":false},{"name":"actualPenultimateSignedMargin_centeredLift_iff_not_twice_dvd","module":"Erdos249257/PowerTwoCenteredBitLift.lean","line":121,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PowerTwoCenteredBitLift.lean#L121","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PowerTwoCenteredBitLift.lean#L121","kind":"theorem","signature":"theorem actualPenultimateSignedMargin_centeredLift_iff_not_twice_dvd {M z e c : ℤ} (hM : 0 < M) (hz : M ∣ z) (hc : |c| < e) (hbudget : e + |c| ≤ 4 * M) : ActualPenultimateSignedMargin (actualCenteredLift z (2 * M)) c e ↔ ¬ 2 * M ∣ z","docstring":"Divisibility version of the bridge: the signed margin is equivalent to failure of the next factor-of-two lift.","generated_certificate":false},{"name":"actualPenultimateSignedMargin_centeredThirtyTwo_iff_not_thirtyTwo_dvd","module":"Erdos249257/PowerTwoCenteredBitLift.lean","line":159,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PowerTwoCenteredBitLift.lean#L159","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PowerTwoCenteredBitLift.lean#L159","kind":"theorem","signature":"theorem actualPenultimateSignedMargin_centeredThirtyTwo_iff_not_thirtyTwo_dvd {z e c : ℤ} (hz : (16 : ℤ) ∣ z) (hc : |c| < e) (hbudget : e + |c| ≤ 64) : ActualPenultimateSignedMargin (actualCenteredLift z 32) c e ↔ ¬ (32 : ℤ) ∣ z","docstring":"Equivalently, at the concrete scale it is precisely failure of the fifth bit to vanish.","generated_certificate":false}],"paper_label":"res:powertwobitlift","paper_anchor":{"paper_label":"res:powertwobitlift","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":3018,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":3018,"anchor_title":"one-bit signed-margin lift","anchor_title_tex":"one-bit signed-margin lift","section_title":"Exact one-bit lifting.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Candidate unbounded inputs and no-go boundaries","title_tex":"Candidate unbounded inputs and no-go boundaries","line":2956},{"level":"paragraph","title":"Exact one-bit lifting.","title_tex":"Exact one-bit lifting.","line":3010}],"source_excerpt":{"text":"\\begin{proposition}[one-bit signed-margin lift]\\label{res:powertwobitlift}\nThe fifth-bit question after fourth-bit agreement is exactly one cofactor\nparity test, with the complementary offset class characterising failure.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":158898,"source_char_end":159137,"start_line":3018,"end_line":3021,"truncated":false,"sha256":"sha256:0a8d18fb60394e30ae62cccddcd8baabc2970d4219c3ab3243ceee54e384cccf","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3018","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/3"}],"links":[{"kind":"lean_declaration_pinned","label":"two_pow_succ_dvd_iff_quotient_even","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PowerTwoBitLift.lean#L34","cite_this":true},{"kind":"lean_declaration_pinned","label":"modEq_thirtyTwo_iff_even_difference_cofactor","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PowerTwoBitLift.lean#L111","cite_this":true},{"kind":"lean_declaration_pinned","label":"not_modEq_thirtyTwo_iff_offset_sixteen","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PowerTwoBitLift.lean#L129","cite_this":true},{"kind":"lean_declaration_pinned","label":"actualPenultimateSignedMargin_centeredLift_iff_not_twice_dvd","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PowerTwoCenteredBitLift.lean#L121","cite_this":true},{"kind":"lean_declaration_pinned","label":"actualPenultimateSignedMargin_centeredThirtyTwo_iff_not_thirtyTwo_dvd","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PowerTwoCenteredBitLift.lean#L159","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/82","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"power_two_bit_lift"}}],"expect":"object"},"prior_art":{"family_id":"fresh_loss_local_arithmetic","status_summary":"Exact local arithmetic, conditional band/slack reductions, and scoped parity ceilings.","prior_art_posture":"No close statement comparison is recorded; novelty remains unasserted.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/16","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"known_classical":1,"not_assessed":1},"edges":[{"semantic_node_id":"Z05::one_bit_lift_arithmetic","prior_art_state":"known_classical"},{"semantic_node_id":"Z05::signed_margin_socket_is_failure_of_next_lift","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:powertwobitlift","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3018","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:parity_socket_obstruction","assertion_kind":"registered_claim","source_id":"parity_socket_obstruction","reviewed":true,"granularity":"single_registered_claim","label":"Parity obstruction for the reduced margin socket","statement":"For every positive even modulus and every radius in the stated nontrivial range, each parity class contains both a reduced-margin-socket hit and a miss. The same holds for a positive power-of-two modulus. Thus quotient parity alone cannot decide the reduced signed-margin socket. No t = 827 numeral or signed-margin certificate is asserted.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"reducedMarginSocket_not_decided_by_parity","module":"Erdos249257/Totient827ParitySocketCeiling.lean","line":73,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/Totient827ParitySocketCeiling.lean#L73","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/Totient827ParitySocketCeiling.lean#L73","kind":"theorem","signature":"theorem reducedMarginSocket_not_decided_by_parity {m A : ℤ} (hmEven : Even m) (hmPos : 0 < m) (h16A : 16 < A) (hA : A < 8 * m - 16) : (∃ yHit yMiss : ℤ, Even yHit ∧ Even yMiss ∧ ReducedMarginSocket m A yHit ∧ ¬ ReducedMarginSocket m A yMiss) ∧ (∃ yHit yMiss : ℤ, Odd yHit ∧ Odd yMiss ∧ ReducedMarginSocket m A yHit ∧ ¬ ReducedMarginSocket m A yMiss)","docstring":"Knowing only the parity of a reduced quotient cannot decide its margin socket. In each parity class there is both a socket hit and a socket miss. The hits are `0` and `1`. Writing the even modulus as `2*r`, the misses are the adjacent central residues `r` and `r+1`; their parities are opposite, so they cover the two classes in one order or the other.","generated_certificate":false},{"name":"powerTwo_reducedMarginSocket_not_decided_by_parity","module":"Erdos249257/Totient827ParitySocketCeiling.lean","line":110,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/Totient827ParitySocketCeiling.lean#L110","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/Totient827ParitySocketCeiling.lean#L110","kind":"theorem","signature":"theorem powerTwo_reducedMarginSocket_not_decided_by_parity {k : ℕ} {A : ℤ} (hk : 0 < k) (h16A : 16 < A) (hA : A < 8 * (2 : ℤ) ^ k - 16) : (∃ yHit yMiss : ℤ, Even yHit ∧ Even yMiss ∧ ReducedMarginSocket ((2 : ℤ) ^ k) A yHit ∧ ¬ ReducedMarginSocket ((2 : ℤ) ^ k) A yMiss) ∧ (∃ yHit yMiss : ℤ, Odd yHit ∧ Odd yMiss ∧ ReducedMarginSocket ((2 : ℤ) ^ k) A yHit ∧ ¬ ReducedMarginSocket ((2 : ℤ) ^ k) A yMiss)","docstring":"Power-of-two specialization. This is the form directly applicable to the reduced modulus `2^1198` at the canonical `t = 827` row once its margin constant is exported by the arithmetic certificate module.","generated_certificate":false}],"paper_label":"res:paritysocket","paper_anchor":{"paper_label":"res:paritysocket","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":3046,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":3045,"anchor_title":"parity does not decide the reduced margin condition","anchor_title_tex":"parity does not decide the reduced margin condition","section_title":"Independent no-go: parity is not a margin certificate.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Candidate unbounded inputs and no-go boundaries","title_tex":"Candidate unbounded inputs and no-go boundaries","line":2956},{"level":"paragraph","title":"Independent no-go: parity is not a margin certificate.","title_tex":"Independent no-go: parity is not a margin certificate.","line":3036}],"source_excerpt":{"text":"\\begin{proposition}[parity does not decide the reduced margin condition]\n\\label{res:paritysocket}\nLet \\(m,a\\in\\Z\\), with \\(m>0\\) even and\n\\[\n 16<a<8m-16,\n\\]\nand put\n\\[\n \\mathcal M_{m,a}\n =\\{y\\in\\Z:\\exists k\\in\\Z,\\ 16|y-km|<a\\}.\n\\]\nEach parity class contains both a member and a nonmember of\n\\(\\mathcal M_{m,a}\\).  In particular, the conclusion holds for\n\\(m=2^j\\) whenever \\(j\\geq1\\).\n\\end{proposition}","media_type":"text/x-tex","source_char_start":160546,"source_char_end":160948,"start_line":3045,"end_line":3059,"truncated":false,"sha256":"sha256:396b7474e57c80d3c6239987c3f64708cc5d87c7c866a791875275648f197495","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3046","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. 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Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. 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Cofinal nonnegative slack at strict lcm jumps implies irrationality of the totient constant; that cofinal sign statement remains open.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"canonicalAdjacentSuffixCentral_iff_slack_nonneg","module":"Erdos249257/DiagonalFreshLossBridge.lean","line":2667,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DiagonalFreshLossBridge.lean#L2667","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalFreshLossBridge.lean#L2667","kind":"theorem","signature":"theorem canonicalAdjacentSuffixCentral_iff_slack_nonneg (t : ℕ) : (2 ^ (canonicalAdjacentSuffixDepth t - 5) ≤ diagonalAdjacentSuffixResidue t 0 (canonicalAdjacentSuffixDepth t) ∧ diagonalAdjacentSuffixResidue t 0 (canonicalAdjacentSuffixDepth t) ≤ 2 ^ canonicalAdjacentSuffixDepth t - 2 ^ (canonicalAdjacentSuffixDepth t - 5)) ↔ 0 ≤ canonicalAdjacentSuffixCentralSlack t","docstring":"The two central-band inequalities collapse to one exact integer sign.","generated_certificate":false},{"name":"canonicalAdjacentSuffixJumpCentralSupply_iff_slackSupply","module":"Erdos249257/DiagonalFreshLossBridge.lean","line":2757,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DiagonalFreshLossBridge.lean#L2757","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalFreshLossBridge.lean#L2757","kind":"theorem","signature":"theorem canonicalAdjacentSuffixJumpCentralSupply_iff_slackSupply : CanonicalAdjacentSuffixJumpCentralSupply ↔ CanonicalAdjacentSuffixJumpSlackSupply","generated_certificate":false},{"name":"irrational_totientSeries_of_canonicalAdjacentSuffixJumpSlackSupply","module":"Erdos249257/DiagonalFreshLossBridge.lean","line":2932,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/DiagonalFreshLossBridge.lean#L2932","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/DiagonalFreshLossBridge.lean#L2932","kind":"theorem","signature":"theorem irrational_totientSeries_of_canonicalAdjacentSuffixJumpSlackSupply (hsupply : CanonicalAdjacentSuffixJumpSlackSupply) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"Nonnegative canonical slack at arbitrarily large strict LCM jumps likewise implies irrationality of the totient series.","generated_certificate":false}],"paper_label":"res:slackcriterion","paper_anchor":{"paper_label":"res:slackcriterion","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":4131,"anchor_kind":"environment","anchor_unit":"corollary","anchor_unit_line":4130,"anchor_title":"central-band supply criterion for irrationality","anchor_title_tex":"central-band supply criterion for irrationality","section_title":"Two concrete scalar criteria","section_path":[{"level":"section","title":"Exact remaining problems and further questions","title_tex":"Exact remaining problems and further questions","line":3819},{"level":"subsection","title":"Two concrete scalar criteria","title_tex":"Two concrete scalar criteria","line":4094}],"source_excerpt":{"text":"\\begin{corollary}[central-band supply criterion for irrationality]\n\\label{res:slackcriterion}\nIf, for every $t_0$, there is a $t\\ge\\max\\{3,t_0\\}$ such that\n\\[\n M_t<M_{t+1}\\qquad\\text{and}\\qquad \\sigma_t\\ge0,\n\\]\nthen $S$ is irrational.\n\\end{corollary}","media_type":"text/x-tex","source_char_start":216165,"source_char_end":216415,"start_line":4130,"end_line":4137,"truncated":false,"sha256":"sha256:2852f26961649dadad842c049cda49b15b7c956550d0bace23be618e7030e62f","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Hence separation of\n$C_{H,D}$ from every integer by more than this bound forces\n$\\mathcal D_H\\notin\\Z$ and full-target avoidance.  No converse is asserted.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":144135,"source_char_end":144737,"start_line":2740,"end_line":2755,"truncated":false,"sha256":"sha256:0b88aa58346b82a14466f891a7135aaab0cb2fd2b84823b25eb5683103d62bfa","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:actual_foreign_residue_projection","assertion_kind":"registered_claim","source_id":"actual_foreign_residue_projection","reviewed":true,"granularity":"single_registered_claim","label":"Finite foreign-residue projection reduction","statement":"The finite Möbius residue diagonal splits exactly into projected foreign and divisor channels, and every finite stable tail window has a closed geometric bound. If an analytic kernel/complement bridge controls the actual foreign defect by that bound and the projection is separated from every integer, then the full target is missed. The analytic bridge and any unbounded separated-projection supply are not proved.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"finiteResidueDiagonal_eq_projectedForeign_add_divisor","module":"Erdos249257/ActualForeignResidueProjection.lean","line":308,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/ActualForeignResidueProjection.lean#L308","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ActualForeignResidueProjection.lean#L308","kind":"theorem","signature":"theorem finiteResidueDiagonal_eq_projectedForeign_add_divisor (H D : ℕ) : finiteResidueDiagonal H D = projectedForeignDefect H D + projectedDivisorChannels H D","docstring":"The finite residue state splits exactly into the retained foreign projection and the complementary divisor state.","generated_certificate":false},{"name":"abs_foreignTailWindow_le_foreignComplementBound","module":"Erdos249257/ActualForeignResidueProjection.lean","line":276,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/ActualForeignResidueProjection.lean#L276","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ActualForeignResidueProjection.lean#L276","kind":"theorem","signature":"theorem abs_foreignTailWindow_le_foreignComplementBound {H D L : ℕ} (hcutoff : 2 * H ≤ D) : |foreignTailWindow H D L| ≤ foreignComplementBound H D","docstring":"The finite foreign tail is bounded by the closed complement budget. The only remaining analytic obligation is to identify the actual complement with the limit of these windows.","generated_certificate":false},{"name":"scaleFullTarget_miss_of_projected_separation","module":"Erdos249257/ActualForeignResidueProjection.lean","line":414,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/ActualForeignResidueProjection.lean#L414","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ActualForeignResidueProjection.lean#L414","kind":"theorem","signature":"theorem scaleFullTarget_miss_of_projected_separation {H D : ℕ} (hcontrol : ControlledForeignProjection H D) (hseparation : ProjectedFullTargetSeparation H D) : ¬ScaleFullTargetHit H","docstring":"**Complement noncancellation consumer.** A controlled projection whose finite rational state is farther from every integer than the whole omitted complement can move it forces a genuine full-target miss.","generated_certificate":false}],"paper_label":"res:foreignresidue","paper_anchor":{"paper_label":"res:foreignresidue","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2834,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":2834,"anchor_title":"finite sufficient test for diagonal nonintegrality","anchor_title_tex":"finite sufficient test for diagonal nonintegrality","section_title":"Exact finite divisor-index split and sufficient projection test.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Fixed-scale mechanisms and conditional unbounded implications","title_tex":"Fixed-scale mechanisms and conditional unbounded implications","line":2720},{"level":"paragraph","title":"Exact finite divisor-index split and sufficient projection test.","title_tex":"Exact finite divisor-index split and sufficient projection test.","line":2820}],"source_excerpt":{"text":"\\begin{proposition}[finite sufficient test for diagonal nonintegrality]\\label{res:foreignresidue}\nLet \\(F_H^{(D)}\\) be the retained nondivisor projection at cutoff \\(D\\), let\n\\(F_H\\) be the actual foreign defect, and let \\(Q_H\\) be the scaled explicit\ndivisor shadow.  The finite residue diagonal is exactly the sum of\n\\(F_H^{(D)}\\) and its divisor-index complement.  For \\(2H\\leq D\\), put\n\\[\n B_{H,D}=c_H\\left(\\frac{2}{2^D}+\\frac{4}{3\\,4^D}\\right).\n\\]\nIf\n\\[\n |F_H-F_H^{(D)}|\\leq B_{H,D}\n \\quad\\text{and}\\quad\n |Q_H+F_H^{(D)}-z|>B_{H,D}\\quad\\text{for every }z\\in\\Z,\n\\]\nthen the full target at height \\(H\\) is not attained.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":149318,"source_char_end":149958,"start_line":2834,"end_line":2849,"truncated":false,"sha256":"sha256:8317e4787bdf0a5564d9fbc1cabe9d764c227ca9d3b8795472f6bb16681ecc16","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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A sufficient one-parameter form is to produce them along the lcm diagonal.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/3"}],"links":[{"kind":"lean_declaration_pinned","label":"finiteResidueDiagonal_eq_projectedForeign_add_divisor","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ActualForeignResidueProjection.lean#L308","cite_this":true},{"kind":"lean_declaration_pinned","label":"abs_foreignTailWindow_le_foreignComplementBound","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ActualForeignResidueProjection.lean#L276","cite_this":true},{"kind":"lean_declaration_pinned","label":"scaleFullTarget_miss_of_projected_separation","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ActualForeignResidueProjection.lean#L414","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/86","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"actual_foreign_residue_projection"}}],"expect":"object"},"prior_art":{"family_id":"diagonal_pincer_and_denominator_survival","status_summary":"Exact diagonal targets, sharp enclosures and denominators, and conditional avoidance projections.","prior_art_posture":"Classical Möbius/Lambert algebra only; no theorem-specific antecedent is identified.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/15","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2},"edges":[{"semantic_node_id":"Z05::controlled_projection_miss_is_conditional","prior_art_state":"not_assessed"},{"semantic_node_id":"Z05::foreign_residue_kernel_geometric_budget","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:foreignresidue","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2834","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:mersenne_shadow_denominator","assertion_kind":"registered_claim","source_id":"mersenne_shadow_denominator","reviewed":true,"granularity":"single_registered_claim","label":"Exact Mersenne-shadow denominator and survivor bounds","statement":"The reduced denominator of the lcm-height scaled Möbius shadow is computed exactly at every height, and for t ≥ 5 an explicit upper-half Mersenne product both divides it and supplies a quantitative lower bound.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"programme_representative","warrant":"orientation.json lists this claim among the representative claims of a mathematical programme whose remaining open propositions all name one problem"},"problem_edges":[{"problem_id":"erdos_249","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":["arithmetic_obstruction_interfaces"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"exists_upperHalf_mersenne_channel_dvd_den_with_bounds","module":"Erdos249257/MersenneShadowDenominatorGrowth.lean","line":99,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/MersenneShadowDenominatorGrowth.lean#L99","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/MersenneShadowDenominatorGrowth.lean#L99","kind":"theorem","signature":"theorem exists_upperHalf_mersenne_channel_dvd_den_with_bounds {t : ℕ} (ht : 5 ≤ t) : ∃ p ∈ upperHalfPrimes t, 2 ^ (t / 2) ≤ RadicalMobiusShadow.mersenne p ∧ RadicalMobiusShadow.mersenne p < 2 ^ t ∧ RadicalMobiusShadow.mersenne p ∣ ((lcmHeight t : ℚ) * RadicalMobiusShadow.numericMobiusShadow (lcmHeight t)).den","docstring":"A concrete upper-half channel survives in the reduced denominator, with both the lower size bound used for growth and the upper bit-scale bound needed by a later discrete channel-separation argument.","generated_certificate":false},{"name":"lcmHeight_scaledMobiusShadow_den_exact","module":"Erdos249257/MersenneShadowDenominatorGrowth.lean","line":147,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/MersenneShadowDenominatorGrowth.lean#L147","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/MersenneShadowDenominatorGrowth.lean#L147","kind":"theorem","signature":"theorem lcmHeight_scaledMobiusShadow_den_exact (t : ℕ) : ((lcmHeight t : ℚ) * RadicalMobiusShadow.numericMobiusShadow (lcmHeight t)).den = RadicalMobiusShadow.mersenne (lcmRadical t) / Nat.gcd (RadicalMobiusShadow.mersenne (lcmRadical t)) (lcmScale t * (CyclicTensorMobiusShadow.oddJordanScalar (lcmRadical t)).natAbs)","docstring":"The exact reduced denominator at every LCM height. The numerator is replaced completely by the scale times the odd Jordan scalar.","generated_certificate":false}],"paper_label":"res:shadowden","paper_anchor":{"paper_label":"res:shadowden","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":2875,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":2875,"anchor_title":"exact denominator of the M\"obius shadow","anchor_title_tex":"exact denominator of the M\"obius shadow","section_title":"Exact finite divisor-index split and sufficient projection test.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Fixed-scale mechanisms and conditional unbounded implications","title_tex":"Fixed-scale mechanisms and conditional unbounded implications","line":2720},{"level":"paragraph","title":"Exact finite divisor-index split and sufficient projection test.","title_tex":"Exact finite divisor-index split and sufficient projection test.","line":2820}],"source_excerpt":{"text":"\\begin{proposition}[exact denominator of the M\"obius shadow]\\label{res:shadowden}\nLet \\(r_t\\) be the square-free kernel of \\(M_t\\), let \\(s_t=M_t/r_t\\), and\nlet \\(J(r_t)\\) be the odd Jordan scalar.  The reduced denominator \\(d_t\\) of\nthe scaled M\\\"obius shadow is\n\\[\n d_t=\n \\frac{2^{r_t}-1}\n {\\gcd\\!\\bigl(2^{r_t}-1,\\,s_t\\lvert J(r_t)\\rvert\\bigr)}.\n\\]\nIf \\(t\\geq5\\), then\n\\[\n \\prod_{\\substack{p\\ \\mathrm{prime}\\\\ t/2<p\\leq t}}(2^p-1)\n \\ \\mid\\ d_t,\n \\qquad d_t\\geq2^{\\lfloor t/2\\rfloor}.\n\\]\n\\end{proposition}","media_type":"text/x-tex","source_char_start":151229,"source_char_end":151735,"start_line":2875,"end_line":2890,"truncated":false,"sha256":"sha256:35aad509979a485fff30876033e2d2794e27f339bbc04eb4f49f5efbef3e9812","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2875","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"exists_upperHalf_mersenne_channel_dvd_den_with_bounds","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/MersenneShadowDenominatorGrowth.lean#L99","cite_this":true},{"kind":"lean_declaration_pinned","label":"lcmHeight_scaledMobiusShadow_den_exact","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/MersenneShadowDenominatorGrowth.lean#L147","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/87","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"mersenne_shadow_denominator"}}],"expect":"object"},"prior_art":{"family_id":"diagonal_pincer_and_denominator_survival","status_summary":"Exact diagonal targets, sharp enclosures and denominators, and conditional avoidance projections.","prior_art_posture":"Classical Möbius/Lambert algebra only; no theorem-specific antecedent is identified.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/15","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z04::lcm_height_exact_denominator","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z04::single_channel_with_two_sided_bounds","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:shadowden","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2875","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:adelic_height_obstruction","assertion_kind":"registered_claim","source_id":"adelic_height_obstruction","reviewed":true,"granularity":"single_registered_claim","label":"Scalar-localisation height obstruction","statement":"For every rational x, integer c and natural number H, if H divides the reduced denominator of x and the reduced denominator of c·x divides H, then the complementary factor x.den/H divides |c|. Scalar localisation therefore moves that denominator information into the scalar; this local obstruction is not an irrationality criterion.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":true,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"gateway_section","warrant":"the claim's anchor sits inside the retired gateway section 'The totient constant S (Erdős #249)', which is the author's own partition of that manuscript by problem"},"problem_edges":[{"problem_id":"erdos_249","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"scalarLocalization_complement_dvd","module":"Erdos249257/AdelicHeightObstruction.lean","line":23,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/AdelicHeightObstruction.lean#L23","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/AdelicHeightObstruction.lean#L23","kind":"theorem","signature":"theorem scalarLocalization_complement_dvd (x : ℚ) (c : ℤ) {H : ℕ} (hH : H ∣ x.den) (hscaled : ((c : ℚ) * x).den ∣ H) : x.den / H ∣ c.natAbs","docstring":"If a scalar multiple of a reduced rational has denominator contained in `H`, then the complementary denominator `x.den / H` divides the scalar. This is the exact scalar-localisation tax: shrinking the displayed denominator to a surviving channel does not discard the complement; it moves that complement into the coefficient.","generated_certificate":false},{"name":"scalarLocalization_integer_eq_mul_num","module":"Erdos249257/AdelicHeightObstruction.lean","line":56,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/AdelicHeightObstruction.lean#L56","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/AdelicHeightObstruction.lean#L56","kind":"theorem","signature":"theorem scalarLocalization_integer_eq_mul_num (x : ℚ) (c : ℤ) {H : ℕ} (hH : H ∣ x.den) (hscaled : ((c : ℚ) * x).den ∣ H) : ∃ t : ℤ, (H : ℚ) * (c : ℚ) * x = (t : ℚ) * x.num","docstring":"After scalar localisation, the resulting channel-cleared rational is an integer multiple of the *original reduced numerator*. This is the equality form of height conservation: the omitted denominator has been transferred to the coefficient exactly, not erased.","generated_certificate":false}],"paper_label":"res:adelic","paper_anchor":{"paper_label":"res:adelic","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":3092,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":3092,"anchor_title":"independent no-go: scalar localisation","anchor_title_tex":"independent no-go: scalar localisation","section_title":"Independent no-go: prime adjunction.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Candidate unbounded inputs and no-go boundaries","title_tex":"Candidate unbounded inputs and no-go boundaries","line":2956},{"level":"paragraph","title":"Independent no-go: prime adjunction.","title_tex":"Independent no-go: prime adjunction.","line":3067}],"source_excerpt":{"text":"\\begin{proposition}[independent no-go: scalar localisation]\\label{res:adelic}\nLet $x\\in\\Q$ and $c\\in\\Z$.  If $H\\mid\\operatorname{den}(x)$ and the reduced denominator\nof $cx$ divides $H$, then\n\\[\n \\frac{\\operatorname{den}(x)}{H}\\mid |c|.\n\\]\n\\end{proposition}","media_type":"text/x-tex","source_char_start":162766,"source_char_end":163023,"start_line":3092,"end_line":3098,"truncated":false,"sha256":"sha256:3191bc62460eb232cbce52cc40d75c778aa868d6904214ad267a221ff7de2727","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z09::scalar_localisation_height_barrier","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:adelic","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3092","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:square_crt_correction_suppression","assertion_kind":"registered_claim","source_id":"square_crt_correction_suppression","reviewed":true,"granularity":"single_registered_claim","label":"Square-CRT correction suppression","statement":"For every finite family of distinct primes and prescribed anchors, a bounded simultaneous square-CRT base removes the prime-dilation correction at every admissible finite shift. Checked clean cubes can have vanishing coefficients and can also have a nonzero coefficient; this finite mechanism supplies neither an unbounded certificate family nor an irrationality criterion for the totient constant.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"gateway_section","warrant":"the claim's anchor sits inside the retired gateway section 'The totient constant S (Erdős #249)', which is the author's own partition of that manuscript by problem"},"problem_edges":[{"problem_id":"erdos_249","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"exists_squareCRT_clean_totient_family","module":"Erdos249257/SquareCRTCube.lean","line":297,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SquareCRTCube.lean#L297","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SquareCRTCube.lean#L297","kind":"theorem","signature":"theorem exists_squareCRT_clean_totient_family {ι : Type*} [DecidableEq ι] (E : Finset ι) (p A a : ι → ℕ) (hprime : ∀ i ∈ E, (p i).Prime) (hinj : Set.InjOn p (E : Set ι)) : ∃ n : ℕ, n < (∏ i ∈ E, p i ^ 2) * (E.sup (fun i => A i + p i * a i) + 2) ∧ ∀ i ∈ E, ∀ h : ℕ, ¬p i ∣ a i + h → ∃ t : ℕ, n = A i + p i * (a i + p i * t) ∧ Nat.totient (n + p i * h - A i) = (p i - 1) * Nat.totient (a i + p i * t + h)","docstring":"Simultaneous square-CRT correction suppression on a finite family. One common base works at every label and every shift whose chosen cofactor residue avoids zero modulo the corresponding prime.","generated_certificate":false},{"name":"exists_squareCRT_clean_horizon","module":"Erdos249257/SquareCRTCube.lean","line":327,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SquareCRTCube.lean#L327","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SquareCRTCube.lean#L327","kind":"theorem","signature":"theorem exists_squareCRT_clean_horizon {ι : Type*} [DecidableEq ι] (E : Finset ι) (p A : ι → ℕ) (J : ℕ) (hprime : ∀ i ∈ E, (p i).Prime) (hinj : Set.InjOn p (E : Set ι)) (horizon : ∀ i ∈ E, J < p i) : ∃ n : ℕ, n < (∏ i ∈ E, p i ^ 2) * (E.sup A + 2) ∧ ∀ i ∈ E, ∀ h : ℕ, 0 < h → h ≤ J → ∃ t : ℕ, n = A i + p i ^ 2 * t ∧ Nat.totient (n + p i * h - A i) = (p i - 1) * Nat.totient (p i * t + h)","docstring":"Canonical finite-horizon corollary. If every distinct construction prime exceeds `J`, the residue choice `a = 0` removes every totient dilation correction simultaneously for all `1 ≤ h ≤ J`.","generated_certificate":false},{"name":"squareCRT_clean_block_can_vanish","module":"Erdos249257/SquareCRTCube.lean","line":459,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SquareCRTCube.lean#L459","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SquareCRTCube.lean#L459","kind":"theorem","signature":"theorem squareCRT_clean_block_can_vanish : (13 : ℕ).Prime ∧ (5 : ℕ).Prime ∧ 52 ≡ 52 [MOD 13 ^ 2] ∧ 52 ≡ 2 [MOD 5 ^ 2] ∧ actualOneCubeCoeff 52 13 13 = 0 ∧ actualOneCubeCoeff 52 26 18 = 0","docstring":"A clean square-localized cube need not separate: the returned minimal `k = 1`, `J = 2` model has both finite coefficients equal to zero.","generated_certificate":false},{"name":"squareCRT_clean_block_can_be_nonzero","module":"Erdos249257/SquareCRTCube.lean","line":477,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SquareCRTCube.lean#L477","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SquareCRTCube.lean#L477","kind":"theorem","signature":"theorem squareCRT_clean_block_can_be_nonzero : (3 : ℕ).Prime ∧ (5 : ℕ).Prime ∧ 27 ≡ 0 [MOD 3 ^ 2] ∧ 27 ≡ 2 [MOD 5 ^ 2] ∧ actualOneCubeCoeff 27 3 3 = 0 ∧ actualOneCubeCoeff 27 6 8 = -4","docstring":"Correction suppression is not identically trivial: a nearby clean cube has a nonzero second coefficient.","generated_certificate":false}],"paper_label":"res:squarecrt","paper_anchor":{"paper_label":"res:squarecrt","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":3107,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":3107,"anchor_title":"independent no-go: square-CRT correction suppression","anchor_title_tex":"independent no-go: square-CRT correction suppression","section_title":"Independent no-go: prime adjunction.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Candidate unbounded inputs and no-go boundaries","title_tex":"Candidate unbounded inputs and no-go boundaries","line":2956},{"level":"paragraph","title":"Independent no-go: prime adjunction.","title_tex":"Independent no-go: prime adjunction.","line":3067}],"source_excerpt":{"text":"\\begin{proposition}[independent no-go: square-CRT correction suppression]\\label{res:squarecrt}\nLet $\\mathcal E$ be a finite family indexed by distinct primes $p_i$, with prescribed\nanchors $A_i$ and residues $a_i$.  There is a bounded common base $n$ such\nthat, for each $i\\in\\mathcal E$ and each shift $h$ with $p_i\\nmid a_i+h$, one has\n\\[\n n=A_i+p_i(a_i+p_i t)\\quad\\text{and}\\quad\n \\ph(n+p_i h-A_i)=(p_i-1)\\ph(a_i+p_i t+h)\n\\]\nfor a suitable $t\\in\\N$.  In particular, if every $p_i$ exceeds a fixed\nhorizon $J$, the choice $a_i=0$ works simultaneously for all $1\\le h\\le J$.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":163572,"source_char_end":164165,"start_line":3107,"end_line":3117,"truncated":false,"sha256":"sha256:177d590cf258627cfdbf6ef4032a0c70f4d286d08f82e8f1a9d5d293cad831ff","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"exact_local_id","state_counts":{"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z09::square_crt_correction_suppression","prior_art_state":"routine_corollary_or_specialisation"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:squarecrt","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3107","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. 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This is finite algebraic substrate only: it proves neither non-vanishing of the actual totient Hankel determinants nor an irrationality criterion.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"gateway_section","warrant":"the claim's anchor sits inside the retired gateway section 'The totient constant S (Erdős #249)', which is the author's own partition of that manuscript by problem"},"problem_edges":[{"problem_id":"erdos_249","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"det_mul_rectangular","module":"Erdos249257/SignedQMomentObstruction.lean","line":29,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SignedQMomentObstruction.lean#L29","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SignedQMomentObstruction.lean#L29","kind":"theorem","signature":"theorem det_mul_rectangular {R ι κ : Type*} [CommRing R] [Fintype ι] [DecidableEq ι] [Fintype κ] (M : Matrix ι κ R) (N : Matrix κ ι R) : Matrix.det (M * N) = ∑ p : ι → κ, Matrix.det (fun i j ↦ M i (p j)) * ∏ i, N (p i) i","generated_certificate":false},{"name":"scaled_dyadic_sum_odd","module":"Erdos249257/SignedQMomentObstruction.lean","line":78,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SignedQMomentObstruction.lean#L78","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SignedQMomentObstruction.lean#L78","kind":"theorem","signature":"theorem scaled_dyadic_sum_odd {α : Type*} (s : Finset α) (u : α → ℤ) (e : α → ℕ) (m : α) (hm : m ∈ s) (hu : ¬ Even (u m)) (hmax : ∀ i ∈ s, i ≠ m → e i < e m) : (∑ i ∈ s, u i * (2 : ℤ) ^ (e m - e i)) % 2 = 1","docstring":"If one dyadic denominator exponent is uniquely largest and its numerator is odd, the common numerator obtained by clearing that denominator is odd.","generated_certificate":false},{"name":"scaled_dyadic_sum_ne_zero","module":"Erdos249257/SignedQMomentObstruction.lean","line":96,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SignedQMomentObstruction.lean#L96","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SignedQMomentObstruction.lean#L96","kind":"theorem","signature":"theorem scaled_dyadic_sum_ne_zero {α : Type*} (s : Finset α) (u : α → ℤ) (e : α → ℕ) (m : α) (hm : m ∈ s) (hu : ¬ Even (u m)) (hmax : ∀ i ∈ s, i ≠ m → e i < e m) : (∑ i ∈ s, u i * (2 : ℤ) ^ (e m - e i)) ≠ 0","docstring":"The same unique-terminal hypothesis gives non-vanishing after common dyadic denominator clearing.","generated_certificate":false}],"paper_label":"res:signedmoment","paper_anchor":{"paper_label":"res:signedmoment","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":3222,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":3222,"anchor_title":"exact rectangular determinant expansion","anchor_title_tex":"exact rectangular determinant expansion","section_title":"Relation to prior work.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Unconditional dyadic rank and the rationality-side carry bound","title_tex":"Unconditional dyadic rank and the rationality-side carry bound","line":3130},{"level":"paragraph","title":"Relation to prior work.","title_tex":"Relation to prior work.","line":3169}],"source_excerpt":{"text":"\\begin{proposition}[exact rectangular determinant expansion]\\label{res:signedmoment}\nFor matrices $M:\\iota\\times\\kappa\\to R$ and\n$N:\\kappa\\times\\iota\\to R$ over a commutative ring, the determinant of\n$MN$ is the sum over all maps $p:\\iota\\to\\kappa$ of the determinant of the\ncorresponding selected-column matrix times the diagonal product\n$\\prod_i N_{p(i),i}$.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":170282,"source_char_end":170660,"start_line":3222,"end_line":3228,"truncated":false,"sha256":"sha256:46e17e16857954a38f7e7f19ceb155f45d878648d19a6d3670ffae36de16d06b","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3222","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"det_mul_rectangular","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SignedQMomentObstruction.lean#L29","cite_this":true},{"kind":"lean_declaration_pinned","label":"scaled_dyadic_sum_odd","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SignedQMomentObstruction.lean#L78","cite_this":true},{"kind":"lean_declaration_pinned","label":"scaled_dyadic_sum_ne_zero","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SignedQMomentObstruction.lean#L96","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/90","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"signed_dyadic_moment_substrate"}}],"expect":"object"},"prior_art":{"family_id":"alternative_249_algebraic_boundaries","status_summary":"Exact finite obstructions and interfaces, a constructive dyadic basis, and an all-base arithmetic/spanning layer. 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Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"known_classical":1,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z07::rectangular_leibniz_determinant_expansion","prior_art_state":"known_classical"},{"semantic_node_id":"Z07::unique_dyadic_terminal_parity","prior_art_state":"routine_corollary_or_specialisation"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:signedmoment","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3222","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. 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This is an independent constructive proof route; its infinite-dimensionality consequence is not an independent contribution because Coons's non-2-regularity theorem already implies it.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-249-binary-totient-series","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"linearIndependent_oddCoreTotientKernelFamily","module":"Erdos249257/TotientMahlerDefect.lean","line":1265,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientMahlerDefect.lean#L1265","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L1265","kind":"theorem","signature":"theorem linearIndependent_oddCoreTotientKernelFamily : LinearIndependent ℚ oddCoreTotientKernelFamily","docstring":"**The odd-core dyadic totient channels are linearly independent over `ℚ`.** Linear independence has finite character: every finite subfamily sits inside the canonical level-`e` family for `e` above the largest level occurring, and those are independent by the CRT--Dirichlet minor above.","generated_certificate":false},{"name":"span_range_fullTotientKernel_eq_span_range_oddCore","module":"Erdos249257/TotientMahlerDefect.lean","line":1380,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientMahlerDefect.lean#L1380","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L1380","kind":"theorem","signature":"theorem span_range_fullTotientKernel_eq_span_range_oddCore : Submodule.span ℚ (Set.range fullTotientKernelFamily) = Submodule.span ℚ (Set.range oddCoreTotientKernelFamily)","docstring":"**The odd-core family spans every dyadic section of `φ`.** With `linearIndependent_oddCoreTotientKernelFamily` this fixes the complete `ℚ`-linear relation structure of the dyadic sections of Euler's totient.","generated_certificate":false},{"name":"totientDyadicSectionBasis","module":"Erdos249257/TotientMahlerDefect.lean","line":1392,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientMahlerDefect.lean#L1392","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L1392","kind":"def","signature":"noncomputable def totientDyadicSectionBasis : Basis TotientOddCoreIndex ℚ (Submodule.span ℚ (Set.range fullTotientKernelFamily))","docstring":"**An explicit basis for the `ℚ`-span of all dyadic sections of Euler's totient**, indexed by the two zero-residue base channels together with one channel per odd residue per level.","generated_certificate":false}],"paper_label":"res:basis","paper_anchor":{"paper_label":"res:basis","owner_source":"paper/249/erdos-249-binary-totient-series.tex","source_line":468,"anchor_kind":"environment","anchor_unit":"corollary","anchor_unit_line":468,"anchor_title":"Dyadic basis","anchor_title_tex":"Dyadic basis","section_title":"Further consequences for the sections","section_path":[{"level":"section","title":"Further consequences for the sections","title_tex":"Further consequences for the sections","line":462}],"source_excerpt":{"text":"\\begin{corollary}[Dyadic basis]\\label{res:basis}\nThe family\n\\[\n \\{\\ph_{0,0},\\ph_{1,0}\\}\n \\cup\\{\\ph_{j,r}:j\\ge1,\\ 0<r<2^j,\\ r\\text{ odd}\\},\n \\qquad \\ph_{j,r}(n)=\\ph(2^jn+r),\n\\]\nis a basis for the rational span of the full dyadic kernel. Every rational\nrelation is generated by the scalar reductions. The complete level-zero\ntruncation has dimension one.\n\\end{corollary}","media_type":"text/x-tex","source_char_start":21247,"source_char_end":21615,"start_line":468,"end_line":478,"truncated":false,"sha256":"sha256:be9c99c6a0511eb3df014e9346bfd73f737e70f4362d59845eacfe9a85c857fd","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos-249-binary-totient-series.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/249/erdos-249-binary-totient-series.tex#L468","owner_problem_erdos_number":249,"owner_paper_id":"erdos-249-binary-totient-series","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-249-binary-totient-series.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-249-binary-totient-series.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/249/erdos-249-binary-totient-series.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_249_irrationality","statement":"Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/1"}],"links":[{"kind":"lean_declaration_pinned","label":"linearIndependent_oddCoreTotientKernelFamily","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L1265","cite_this":true},{"kind":"lean_declaration_pinned","label":"span_range_fullTotientKernel_eq_span_range_oddCore","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L1380","cite_this":true},{"kind":"lean_declaration_pinned","label":"totientDyadicSectionBasis","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L1392","cite_this":true},{"kind":"paper_pdf","label":"erdos-249-binary-totient-series","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-249-binary-totient-series.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/91","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"dyadic_totient_basis"}}],"expect":"object"},"prior_art":{"family_id":"alternative_249_algebraic_boundaries","status_summary":"Exact finite obstructions and interfaces, a constructive dyadic basis, and an all-base arithmetic/spanning layer. The reusable rank wrapper is conditional; AllBaseTotientKernel.lean separately proves its independence premise and the complete rank unconditionally. The residue-series claims classify the rational totient residue series and prove the dilated series linearly independent.","prior_art_posture":"Classical Euler/Möbius, CRT, determinant, and Lambert tools; Coons already implies full-kernel infinite rank, and Martin supplies a broader external affine-independence antecedent. The Lean all-base proof uses CRT and Dirichlet independently. Erdős 1957 (Lemma 1) and Kaneko–Suzuki–Tachiya (Theorem 3) already give the irrationality of the totient residue series. No #249 irrationality endpoint or priority claim.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/17","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z09::dyadic_section_basis_classification","prior_art_state":"routine_corollary_or_specialisation"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:totient_residue_series_classification","assertion_kind":"registered_claim","source_id":"totient_residue_series_classification","reviewed":true,"granularity":"single_registered_claim","label":"Irrational totient residue series and the rational dyadic residue observables","statement":"For every m ≥ 3 the series ∑_{n≥1} (φ(n) mod m)/2^n is irrational, and for k ≥ 1 and f: Z/2^kZ → Q the series ∑_{n≥1} f(φ(n) mod 2^k)/2^n is rational exactly when f is constant on the even residue classes, with value 3f(1)/4 + c/4 where c is that constant. The first assertion, in every integer base, already follows from Erdős's Lemma 1 (Indag. Math. 19 (1957), p. 213; the case b = 0 of his Lemma 4, pp. 215-218) because φ(n) mod m vanishes outside a set of density zero, and equally from Theorem 3 of Kaneko, Suzuki and Tachiya (arXiv:2601.20743v1, p. 5). Those criteria also give the classification when the values f(r) - f(0) on the even classes have one sign; with both signs they need a further spacing hypothesis, which the isolated-coefficient proof does not need. Wong posted the case in which the base equals the modulus in 2015. No novelty claim is made for the first assertion. The reduced series say nothing about the unreduced series ∑ φ(n)/2^n.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-249-binary-totient-series","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"short_note_residue_theorem","module":"ErdosProblems/Erdos249/PaperCompleteR7/RationalObservableClassification.lean","line":276,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/PaperCompleteR7/RationalObservableClassification.lean#L276","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/PaperCompleteR7/RationalObservableClassification.lean#L276","kind":"theorem","signature":"theorem short_note_residue_theorem : (∀ m : ℕ, 3 ≤ m → Irrational (totientResidueValue m)) ∧ (∀ k : ℕ, 1 ≤ k → ∀ f : ZMod (2 ^ k) → ℚ, ((∃ q : ℚ, (∑' n : ℕ, (f (Nat.totient (n + 1) : ZMod (2 ^ k)) : ℝ) / 2 ^ (n + 1)) = (q : ℝ)) ↔ ∀ r : ℕ, r < 2 ^ k → r % 2 = 0 → f (r : ZMod (2 ^ k)) = f 0)) ∧ (∀ k : ℕ, 1 ≤ k → ∀ f : ZMod (2 ^ k) → ℚ, ∀ c : ℚ, (∀ r : ℕ, r < 2 ^ k → r % 2 = 0 → f (r : ZMod (2 ^ k)) = c) → (∑' n : ℕ, (f (Nat.totient (n + 1) : ZMod (2 ^ k)) : ℝ) / 2 ^ (n + 1)) = ((3 * f 1 / 4 + c / 4 : ℚ) : ℝ))","docstring":"One end-to-end assembly for short-note res:residueseries.","generated_certificate":false}],"paper_label":"res:residueseries","paper_anchor":{"paper_label":"res:residueseries","owner_source":"paper/249/erdos-249-binary-totient-series.tex","source_line":366,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":366,"anchor_title":"Series of totient residues","anchor_title_tex":"Series of totient residues","section_title":"Bounded functions of totient residues","section_path":[{"level":"section","title":"Bounded functions of totient residues","title_tex":"Bounded functions of totient residues","line":358}],"source_excerpt":{"text":"\\begin{theorem}[Series of totient residues]\\label{res:residueseries}\nFor every $m\\ge3$,\n\\[\n \\sum_{n\\ge1}\\frac{\\ph(n)\\bmod m}{2^n}\\notin\\Q.\n\\]\nFor $k\\ge1$ and $f:\\Z/2^k\\Z\\to\\Q$, the series\n$\\sum_{n\\ge1}f(\\ph(n)\\bmod2^k)2^{-n}$ is rational exactly when $f$ is\nconstant on the even residue classes. If that constant is $c$, its value is\n$3f(1)/4+c/4$.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":16499,"source_char_end":16861,"start_line":366,"end_line":375,"truncated":false,"sha256":"sha256:e311b7f35a220b621e991c33da100b63f20831097386fcde6416cbb685c3f005","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos-249-binary-totient-series.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/249/erdos-249-binary-totient-series.tex#L366","owner_problem_erdos_number":249,"owner_paper_id":"erdos-249-binary-totient-series","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-249-binary-totient-series.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-249-binary-totient-series.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/249/erdos-249-binary-totient-series.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_249_irrationality","statement":"Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/1"}],"links":[{"kind":"lean_declaration_pinned","label":"short_note_residue_theorem","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/PaperCompleteR7/RationalObservableClassification.lean#L276","cite_this":true},{"kind":"paper_pdf","label":"erdos-249-binary-totient-series","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-249-binary-totient-series.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/92","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"totient_residue_series_classification"}}],"expect":"object"},"prior_art":{"family_id":"alternative_249_algebraic_boundaries","status_summary":"Exact finite obstructions and interfaces, a constructive dyadic basis, and an all-base arithmetic/spanning layer. The reusable rank wrapper is conditional; AllBaseTotientKernel.lean separately proves its independence premise and the complete rank unconditionally. The residue-series claims classify the rational totient residue series and prove the dilated series linearly independent.","prior_art_posture":"Classical Euler/Möbius, CRT, determinant, and Lambert tools; Coons already implies full-kernel infinite rank, and Martin supplies a broader external affine-independence antecedent. The Lean all-base proof uses CRT and Dirichlet independently. Erdős 1957 (Lemma 1) and Kaneko–Suzuki–Tachiya (Theorem 3) already give the irrationality of the totient residue series. No #249 irrationality endpoint or priority claim.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/17","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:totient_residue_series_integer_radix","assertion_kind":"registered_claim","source_id":"totient_residue_series_integer_radix","reviewed":true,"granularity":"single_registered_claim","label":"Totient residue series in every integer base","statement":"For every integer t ≥ 2 and m ≥ 3 the series ∑_{n≥1} (φ(n) mod m)/t^n is irrational, and for k ≥ 1 and f: Z/2^kZ → Q the series ∑_{n≥1} f(φ(n) mod 2^k)/t^n is rational exactly when f is constant on the even residue classes, with value (t+1)f(1)/t^2 + f(0)/(t^2(t-1)). The first assertion already follows from Erdős's Lemma 1 (1957) and from Theorem 3 of Kaneko, Suzuki and Tachiya, as recorded for the binary case; no novelty claim is made for it.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-totient-reasoning-surface","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"radix_residue_series_irrational","module":"ErdosProblems/Erdos249/PaperCompleteR7/RationalIntegerRadixClassification.lean","line":128,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/PaperCompleteR7/RationalIntegerRadixClassification.lean#L128","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/PaperCompleteR7/RationalIntegerRadixClassification.lean#L128","kind":"theorem","signature":"theorem radix_residue_series_irrational (B : ℕ) (hB : 2 ≤ B) {m : ℕ} (hm : 3 ≤ m) : Irrational (radixValue B (fun n => (Nat.totient n % m : ℤ)))","docstring":"The least-residue totient series is irrational for every integer radix at least two and every modulus at least three.","generated_certificate":false},{"name":"rational_zmod_radix_observable_iff","module":"ErdosProblems/Erdos249/PaperCompleteR7/RationalIntegerRadixClassification.lean","line":145,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/PaperCompleteR7/RationalIntegerRadixClassification.lean#L145","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/PaperCompleteR7/RationalIntegerRadixClassification.lean#L145","kind":"theorem","signature":"theorem rational_zmod_radix_observable_iff (B : ℕ) (hB : 2 ≤ B) {k : ℕ} (hk : 1 ≤ k) (f : ZMod (2 ^ k) → ℚ) : (∃ q : ℚ, (∑' n : ℕ, (f (Nat.totient (n + 1) : ZMod (2 ^ k)) : ℝ) / (B : ℝ) ^ (n + 1)) = (q : ℝ)) ↔ ∀ r : ℕ, r < 2 ^ k → r % 2 = 0 → f (r : ZMod (2 ^ k)) = f 0","docstring":"The same classification in the finite residue domain used by the public binary theorem.","generated_certificate":false},{"name":"positiveRadixValue_eq_of_even_constant","module":"ErdosProblems/Erdos249/PaperCompleteR7/IntegerRadixObservables.lean","line":303,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/PaperCompleteR7/IntegerRadixObservables.lean#L303","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/PaperCompleteR7/IntegerRadixObservables.lean#L303","kind":"theorem","signature":"theorem positiveRadixValue_eq_of_even_constant (B : ℕ) (hB : 2 ≤ B) {k : ℕ} (hk : 1 ≤ k) (f : ℕ → ℚ) (c : ℚ) (hc : ∀ r, r < 2 ^ k → r % 2 = 0 → f r = c) : positiveRadixValue B f (2 ^ k) = ((B : ℝ) + 1) / (B : ℝ) ^ 2 * (f 1 : ℝ) + (c : ℝ) / ((B : ℝ) ^ 2 * ((B : ℝ) - 1))","docstring":"Exact rational case of the all-integer-base observable classification.","generated_certificate":false}],"paper_label":"prop:radixresidue","paper_anchor":{"paper_label":"prop:radixresidue","owner_source":"paper/249/erdos249-totient-reasoning-surface.tex","source_line":544,"anchor_kind":"inline","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Reduced coefficients","section_path":[{"level":"section","title":"Totient sections and tail differences","title_tex":"Totient sections and tail differences","line":249},{"level":"subsection","title":"Reduced coefficients","title_tex":"Reduced coefficients","line":529}],"source_excerpt":{"text":"\\begin{prop}[Residue series in every integer base]\n\\label{prop:radixresidue}\nLet $t\\ge2$ be an integer.\n\\begin{enumerate}[label=(\\alph*),leftmargin=*]\n\\item For every $m\\ge3$, $\\sum_{n\\ge1}(\\varphi(n)\\bmod m)\\,t^{-n}$ is\n  irrational\n  (\\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/Erdos249/PaperCompleteR7/RationalIntegerRadixClassification.lean\\#L128}{\\texttt{radix\\_residue\\_series\\_irrational}}).\n\\item For $k\\ge1$ and $f:\\Z/2^k\\Z\\to\\Q$, the series\n  $\\sum_{n\\ge1}f(\\varphi(n)\\bmod2^k)\\,t^{-n}$ is rational exactly when $f$\n  is constant on the even residue classes, and its value is then\n  $(t+1)f(1)/t^2+f(0)/\\bigl(t^2(t-1)\\bigr)$\n  (\\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/Erdos249/PaperCompleteR7/RationalIntegerRadixClassification.lean\\#L145}{\\texttt{rational\\_zmod\\_radix\\_observable\\_iff}},\n  \\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/Erdos249/PaperCompleteR7/IntegerRadixObservables.lean\\#L303}{\\texttt{positiveRadixValue\\_eq\\_of\\_even\\_constant}}).\n\\end{enumerate}\n\\end{prop}","media_type":"text/x-tex","source_char_start":25744,"source_char_end":26941,"start_line":543,"end_line":557,"truncated":false,"sha256":"sha256:0a04c3a635774039d744d80f7d645e03ceab6e47587ed82d317468a913946406","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos249-totient-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/249/erdos249-totient-reasoning-surface.tex#L544","owner_problem_erdos_number":249,"owner_paper_id":"erdos249-totient-reasoning-surface","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos249-totient-reasoning-surface.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos249-totient-reasoning-surface.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/249/erdos249-totient-reasoning-surface.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_249_irrationality","statement":"Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/1"}],"links":[{"kind":"lean_declaration_pinned","label":"radix_residue_series_irrational","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/PaperCompleteR7/RationalIntegerRadixClassification.lean#L128","cite_this":true},{"kind":"lean_declaration_pinned","label":"rational_zmod_radix_observable_iff","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/PaperCompleteR7/RationalIntegerRadixClassification.lean#L145","cite_this":true},{"kind":"lean_declaration_pinned","label":"positiveRadixValue_eq_of_even_constant","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/PaperCompleteR7/IntegerRadixObservables.lean#L303","cite_this":true},{"kind":"paper_pdf","label":"erdos249-totient-reasoning-surface","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos249-totient-reasoning-surface.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/93","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"totient_residue_series_integer_radix"}}],"expect":"object"},"prior_art":{"family_id":"alternative_249_algebraic_boundaries","status_summary":"Exact finite obstructions and interfaces, a constructive dyadic basis, and an all-base arithmetic/spanning layer. The reusable rank wrapper is conditional; AllBaseTotientKernel.lean separately proves its independence premise and the complete rank unconditionally. The residue-series claims classify the rational totient residue series and prove the dilated series linearly independent.","prior_art_posture":"Classical Euler/Möbius, CRT, determinant, and Lambert tools; Coons already implies full-kernel infinite rank, and Martin supplies a broader external affine-independence antecedent. The Lean all-base proof uses CRT and Dirichlet independently. Erdős 1957 (Lemma 1) and Kaneko–Suzuki–Tachiya (Theorem 3) already give the irrationality of the totient residue series. No #249 irrationality endpoint or priority claim.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/17","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:totient_residue_series_dilations","assertion_kind":"registered_claim","source_id":"totient_residue_series_dilations","reviewed":true,"granularity":"single_registered_claim","label":"Linear independence of dilated totient residue series","statement":"For every integer t ≥ 2 and m ≥ 3, the numbers 1 and ∑_{n≥1} (φ(n) mod m)/t^{dn}, d = 1, 2, ..., are linearly independent over Q. For a finite set D of positive integers and residue functions f_d: Z/2^{k_d}Z → Q with k_d ≥ 1, any rational constant plus ∑_{d∈D} ∑_{n≥1} f_d(φ(n) mod 2^{k_d})/t^{dn} is rational exactly when every f_d is constant on the even residue classes. The proof isolates a nonzero coefficient at the least active dilation by the Chinese remainder theorem and Dirichlet's theorem. When all coefficients of a relation have one sign, Erdős's Lemma 1 (1957) already excludes it; the mixed-sign case is not covered by that criterion without a further spacing hypothesis.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-totient-reasoning-surface","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"linearIndependent_one_and_least_residue_values","module":"ErdosProblems/Erdos249/FiniteDilationLinearIndependent.lean","line":42,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/FiniteDilationLinearIndependent.lean#L42","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/FiniteDilationLinearIndependent.lean#L42","kind":"theorem","signature":"theorem linearIndependent_one_and_least_residue_values (m B : ℕ) (hm : 3 ≤ m) (hB : 2 ≤ B) : LinearIndependent ℚ (fun d : ℕ => if d = 0 then (1 : ℝ) else positiveRadixValue (B ^ d) (fun r : ℕ => (r : ℚ)) m)","docstring":"The exact countable formulation of the finite affine obstruction: index zero is the constant 1, and positive index d is the least-residue totient series in base B^d.","generated_certificate":false},{"name":"rational_mixed_moduli_with_constant_iff","module":"ErdosProblems/Erdos249/FiniteDilationMixedModuli.lean","line":95,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/FiniteDilationMixedModuli.lean#L95","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/FiniteDilationMixedModuli.lean#L95","kind":"theorem","signature":"theorem rational_mixed_moduli_with_constant_iff (D : Finset ℕ) (k : ℕ → ℕ) (f : (d : ℕ) → ZMod (2 ^ (k d)) → ℚ) (B : ℕ) (q₀ : ℚ) (hB : 2 ≤ B) (hpos : ∀ d ∈ D, 0 < d) (hk : ∀ d ∈ D, 0 < k d) : (∃ q : ℚ, (q₀ : ℝ) + (∑ d ∈ D, ∑' n : ℕ, (f d (Nat.totient (n + 1) : ZMod (2 ^ (k d))) : ℝ) / ((B : ℝ) ^ d) ^ (n + 1)) = (q : ℝ)) ↔ ∀ d ∈ D, ∀ r : ℕ, r < 2 ^ (k d) → Even r → f d (r : ZMod (2 ^ (k d))) = f d 0","docstring":"A rational constant cannot cancel a nonconstant even-residue component.","generated_certificate":false}],"paper_label":"prop:dilations","paper_anchor":{"paper_label":"prop:dilations","owner_source":"paper/249/erdos249-totient-reasoning-surface.tex","source_line":579,"anchor_kind":"inline","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Reduced coefficients","section_path":[{"level":"section","title":"Totient sections and tail differences","title_tex":"Totient sections and tail differences","line":249},{"level":"subsection","title":"Reduced coefficients","title_tex":"Reduced coefficients","line":529}],"source_excerpt":{"text":"\\begin{prop}[Finitely many dilations]\n\\label{prop:dilations}\nLet $t\\ge2$ be an integer.\n\\begin{enumerate}[label=(\\alph*),leftmargin=*]\n\\item For every $m\\ge3$, the numbers $1$ and\n  $\\sum_{n\\ge1}(\\varphi(n)\\bmod m)\\,t^{-dn}$, $d=1,2,\\ldots$, are linearly\n  independent over $\\Q$\n  (\\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/Erdos249/FiniteDilationLinearIndependent.lean\\#L42}{\\texttt{linearIndependent\\_one\\_and\\_least\\_residue\\_values}}).\n\\item Let $D$ be a finite set of positive integers, and for $d\\in D$ let\n  $k_d\\ge1$ and $f_d:\\Z/2^{k_d}\\Z\\to\\Q$. For every $c\\in\\Q$, the number\n  \\[\n   c+\\sum_{d\\in D}\\sum_{n\\ge1}f_d\\bigl(\\varphi(n)\\bmod2^{k_d}\\bigr)\\,t^{-dn}\n  \\]\n  is rational exactly when every $f_d$ is constant on the even residue\n  classes\n  (\\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/Erdos249/FiniteDilationMixedModuli.lean\\#L95}{\\texttt{rational\\_mixed\\_moduli\\_with\\_constant\\_iff}}).\n\\end{enumerate}\n\\end{prop}","media_type":"text/x-tex","source_char_start":27988,"source_char_end":29055,"start_line":578,"end_line":595,"truncated":false,"sha256":"sha256:7a532fda5f365f32e69ceef9fda0442a2e60d70c797313bbd3746892b583a554","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos249-totient-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/249/erdos249-totient-reasoning-surface.tex#L579","owner_problem_erdos_number":249,"owner_paper_id":"erdos249-totient-reasoning-surface","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos249-totient-reasoning-surface.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos249-totient-reasoning-surface.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/249/erdos249-totient-reasoning-surface.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_249_irrationality","statement":"Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/1"}],"links":[{"kind":"lean_declaration_pinned","label":"linearIndependent_one_and_least_residue_values","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/FiniteDilationLinearIndependent.lean#L42","cite_this":true},{"kind":"lean_declaration_pinned","label":"rational_mixed_moduli_with_constant_iff","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/FiniteDilationMixedModuli.lean#L95","cite_this":true},{"kind":"paper_pdf","label":"erdos249-totient-reasoning-surface","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos249-totient-reasoning-surface.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/94","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"totient_residue_series_dilations"}}],"expect":"object"},"prior_art":{"family_id":"alternative_249_algebraic_boundaries","status_summary":"Exact finite obstructions and interfaces, a constructive dyadic basis, and an all-base arithmetic/spanning layer. The reusable rank wrapper is conditional; AllBaseTotientKernel.lean separately proves its independence premise and the complete rank unconditionally. The residue-series claims classify the rational totient residue series and prove the dilated series linearly independent.","prior_art_posture":"Classical Euler/Möbius, CRT, determinant, and Lambert tools; Coons already implies full-kernel infinite rank, and Martin supplies a broader external affine-independence antecedent. The Lean all-base proof uses CRT and Dirichlet independently. Erdős 1957 (Lemma 1) and Kaneko–Suzuki–Tachiya (Theorem 3) already give the irrationality of the totient residue series. No #249 irrationality endpoint or priority claim.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/17","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:dyadic_totient_certificate_interface","assertion_kind":"registered_claim","source_id":"dyadic_totient_certificate_interface","reviewed":true,"granularity":"single_registered_claim","label":"Dyadic totient-kernel finite-level rank","statement":"For every e, a CRT–Dirichlet construction produces a separated nonzero evaluation minor for the canonical dyadic totient-kernel family, which is linearly independent with exact span dimension 2^e + 1. The same Lean development proves that the full dyadic kernel span is not finite-dimensional, but that corollary is not an independent contribution: it is equivalent, for this integer-valued sequence module, to Coons's non-2-regularity theorem. Even residues reduce to odd-core channels, and the compressed-adjoint boundary remains available. These coefficient-side theorems do not prove that S is irrational.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-249-binary-totient-series","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"exists_separatedMinorCertificate_totientAffineOddFamily","module":"Erdos249257/TotientMahlerDefect.lean","line":882,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientMahlerDefect.lean#L882","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L882","kind":"theorem","signature":"theorem exists_separatedMinorCertificate_totientAffineOddFamily (e : ℕ) : Nonempty (SeparatedMinorCertificate (totientAffineOddFamily e))","docstring":"The CRT--Dirichlet rows assemble into a nonzero evaluation minor for the normalized family on odd inputs.","generated_certificate":false},{"name":"linearIndependent_canonicalTotientKernelFamily","module":"Erdos249257/TotientMahlerDefect.lean","line":935,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientMahlerDefect.lean#L935","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L935","kind":"theorem","signature":"theorem linearIndependent_canonicalTotientKernelFamily (e : ℕ) : LinearIndependent ℚ (canonicalTotientKernelFamily e)","docstring":"The full canonical family is linearly independent. Odd inputs first separate the combined pair `φ(n), φ(2n)` from every odd residue channel; the single even input `n = 2` then separates those two remaining columns.","generated_certificate":false},{"name":"finrank_canonicalTotientKernel_eq","module":"Erdos249257/TotientMahlerDefect.lean","line":989,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientMahlerDefect.lean#L989","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L989","kind":"theorem","signature":"theorem finrank_canonicalTotientKernel_eq (e : ℕ) : finrank ℚ (Submodule.span ℚ (Set.range (canonicalTotientKernelFamily e))) = 2 ^ e + 1","docstring":"Exact unconditional finite-level rank of the canonical totient kernel.","generated_certificate":false},{"name":"finrank_totientKernelThroughLevelFamily_eq","module":"Erdos249257/TotientMahlerDefect.lean","line":1084,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientMahlerDefect.lean#L1084","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L1084","kind":"theorem","signature":"theorem finrank_totientKernelThroughLevelFamily_eq (e : ℕ) (he : 1 ≤ e) : finrank ℚ (Submodule.span ℚ (Set.range (totientKernelThroughLevelFamily e))) = 2 ^ e + 1","docstring":"**Exact rank of the actual finite truncation.** For every `e ≥ 1`, the span of all channels `n ↦ φ(2^j n + r)` with `j ≤ e` and `r < 2^j` has dimension exactly `2^e+1`.","generated_certificate":false},{"name":"not_finiteDimensional_span_fullTotientKernel","module":"Erdos249257/TotientMahlerDefect.lean","line":1145,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientMahlerDefect.lean#L1145","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L1145","kind":"theorem","signature":"theorem not_finiteDimensional_span_fullTotientKernel : ¬ FiniteDimensional ℚ (Submodule.span ℚ (Set.range fullTotientKernelFamily))","docstring":"**Unconditional infinite dyadic-kernel rank.** The span of all dyadic sections `n ↦ φ(2^j n + r)` is not finite-dimensional over `ℚ`. Indeed, at every depth `e` it contains the canonical independent family of exact rank `2^e+1`; choosing `e` equal to a hypothetical ambient finrank gives an immediate contradiction.","generated_certificate":false},{"name":"totientKernel_even_residue_reduce","module":"Erdos249257/TotientMahlerDefect.lean","line":169,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientMahlerDefect.lean#L169","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L169","kind":"theorem","signature":"theorem totientKernel_even_residue_reduce (j t s n : ℕ) (ht : t + 1 < j) (hs : Odd s) : totientKernelSeq j (2 ^ (t + 1) * s) n = (2 ^ t : ℚ) * totientKernelSeq (j - (t + 1)) s n","docstring":"Exact reduction of an even residue to its odd-core channel.","generated_certificate":false},{"name":"false_of_compressedAdjointCertificate","module":"Erdos249257/TotientMahlerDefect.lean","line":1183,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientMahlerDefect.lean#L1183","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L1183","kind":"theorem","signature":"theorem false_of_compressedAdjointCertificate {v : ℕ} (cert : CompressedAdjointCertificate v) : False","docstring":"A compressed-adjoint certificate is impossible: its boundary is both a nonzero multiple of `Q*v` and strictly smaller than `Q*v`.","generated_certificate":false}],"paper_label":"thm:kkernelrank","paper_anchor":{"paper_label":"thm:kkernelrank","owner_source":"paper/249/erdos-249-binary-totient-series.tex","source_line":87,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":86,"anchor_title":"A basis through each finite level","anchor_title_tex":"A basis through each finite level","section_title":"Introduction","section_path":[{"level":"section","title":"Introduction","title_tex":"Introduction","line":48}],"source_excerpt":{"text":"\\begin{theorem}[A basis through each finite level]\n\\label{thm:kkernelrank}\nLet $k\\ge2$ and $e\\ge1$ be integers, write $F^{(k)}_{j,r}(n)=\\ph(k^jn+r)$, and put\n\\[\n V_{k,e}=\\operatorname{span}_{\\Q}\n \\{\\,F^{(k)}_{j,r}:0\\le j\\le e,\\ 0\\le r<k^j\\,\\}.\n\\]\nThen $\\dim_{\\Q}V_{k,e}=k^e+1$, and\n\\[\n \\mathcal B_{k,e}\n =\\{F^{(k)}_{0,0},F^{(k)}_{1,0}\\}\n \\cup\\{\\,F^{(k)}_{j,r}:1\\le j\\le e,\\ 1\\le r<k^j,\\ k\\nmid r\\,\\}\n\\]\nis a basis.  Every omitted section reduces to a basis element by an explicit\nscalar: $F^{(k)}_{j,0}=k^{j-1}F^{(k)}_{1,0}$ for $j\\ge1$, and if $r=k^tu$ with\n$t=\\max\\{s:k^s\\mid r\\}\\ge1$ and $k\\nmid u$, then\n\\[\n F^{(k)}_{j,r}=C_k(t,u)\\,F^{(k)}_{j-t,u},\n \\qquad\n C_k(t,u)=k^t\\prod_{\\substack{p\\mid k\\\\ p\\nmid u}}\\Bigl(1-\\tfrac1p\\Bigr).\n\\]\n\\end{theorem}","media_type":"text/x-tex","source_char_start":3717,"source_char_end":4468,"start_line":86,"end_line":107,"truncated":false,"sha256":"sha256:40f371d1722f6d0e17a89a41e4a8a6a9e74a326897c192ffcd60bfb5cdae19f2","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos-249-binary-totient-series.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/249/erdos-249-binary-totient-series.tex#L87","owner_problem_erdos_number":249,"owner_paper_id":"erdos-249-binary-totient-series","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-249-binary-totient-series.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-249-binary-totient-series.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/249/erdos-249-binary-totient-series.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_249_irrationality","statement":"Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/1"}],"links":[{"kind":"lean_declaration_pinned","label":"exists_separatedMinorCertificate_totientAffineOddFamily","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L882","cite_this":true},{"kind":"lean_declaration_pinned","label":"linearIndependent_canonicalTotientKernelFamily","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L935","cite_this":true},{"kind":"lean_declaration_pinned","label":"finrank_canonicalTotientKernel_eq","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L989","cite_this":true},{"kind":"lean_declaration_pinned","label":"finrank_totientKernelThroughLevelFamily_eq","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L1084","cite_this":true},{"kind":"lean_declaration_pinned","label":"not_finiteDimensional_span_fullTotientKernel","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L1145","cite_this":true},{"kind":"lean_declaration_pinned","label":"totientKernel_even_residue_reduce","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L169","cite_this":true},{"kind":"lean_declaration_pinned","label":"false_of_compressedAdjointCertificate","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientMahlerDefect.lean#L1183","cite_this":true},{"kind":"paper_pdf","label":"erdos-249-binary-totient-series","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-249-binary-totient-series.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/95","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"dyadic_totient_certificate_interface"}}],"expect":"object"},"prior_art":{"family_id":"alternative_249_algebraic_boundaries","status_summary":"Exact finite obstructions and interfaces, a constructive dyadic basis, and an all-base arithmetic/spanning layer. The reusable rank wrapper is conditional; AllBaseTotientKernel.lean separately proves its independence premise and the complete rank unconditionally. The residue-series claims classify the rational totient residue series and prove the dilated series linearly independent.","prior_art_posture":"Classical Euler/Möbius, CRT, determinant, and Lambert tools; Coons already implies full-kernel infinite rank, and Martin supplies a broader external affine-independence antecedent. The Lean all-base proof uses CRT and Dirichlet independently. Erdős 1957 (Lemma 1) and Kaneko–Suzuki–Tachiya (Theorem 3) already give the irrationality of the totient residue series. No #249 irrationality endpoint or priority claim.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/17","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"known_classical":1,"not_assessed":1,"routine_corollary_or_specialisation":2},"edges":[{"semantic_node_id":"Z09::compressed_adjoint_contradiction_consumer","prior_art_state":"known_classical"},{"semantic_node_id":"Z09::dyadic_channel_odd_core_reduction","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z09::finite_level_dyadic_totient_independence","prior_art_state":"not_assessed"},{"semantic_node_id":"Z09::full_dyadic_kernel_infinite_dimensional","prior_art_state":"routine_corollary_or_specialisation"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:all_base_totient_kernel_index_reduction","assertion_kind":"registered_claim","source_id":"all_base_totient_kernel_index_reduction","reviewed":true,"granularity":"single_registered_claim","label":"All-base totient-kernel reduction, spanning, and conditional rank","statement":"For every integer base k at least 2, Lean checks the zero-residue scaling identity, the exact nonzero scalar used in composite-base residue reduction, unique quotient/nonzero-digit coordinates, and cardinality k^e + 1 of the canonical finite-level index. It proves unconditionally that this canonical family spans the complete kernel through level e, and proves exact rank k^e + 1 under the explicit hypothesis that the family is linearly independent. Lean does not prove that hypothesis here; the note's evaluation-matrix argument and AllBaseTotientKernel.lean at public checkpoint 25ef6245d15a discharge it by an independent finite-determinant construction, so the unconditional basis and rank theorem takes Martin's positive-density theorem as prior art rather than as an input.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":true,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-249-binary-totient-series","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"totient_pow_mul_eq","module":"Erdos249257/TotientKernelReduction.lean","line":60,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientKernelReduction.lean#L60","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelReduction.lean#L60","kind":"theorem","signature":"theorem totient_pow_mul_eq (k : ℕ) (hk : 0 < k) (n : ℕ) : ∀ j : ℕ, 1 ≤ j → Nat.totient (k ^ j * n) = k ^ (j - 1) * Nat.totient (k * n)","docstring":"The zero-residue relation of the totient `k`-kernel: for `j ≥ 1`, `φ (k^j * n) = k^(j-1) * φ (k * n)`.","generated_certificate":false},{"name":"totient_pow_mul_affine_gcd_cross_eq","module":"Erdos249257/TotientKernelReduction.lean","line":117,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientKernelReduction.lean#L117","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelReduction.lean#L117","kind":"theorem","signature":"theorem totient_pow_mul_affine_gcd_cross_eq (k : ℕ) (hk : 0 < k) (h n u t : ℕ) (hh : 1 ≤ h) (ht : 1 ≤ t) : Nat.totient (Nat.gcd k u) * Nat.totient (k ^ t * (k ^ h * n + u)) = k ^ (t - 1) * Nat.totient k * Nat.gcd k u * Nat.totient (k ^ h * n + u)","docstring":"Exact affine-section specialization of `totient_pow_mul_gcd_cross_eq`. For `m = k^h n + u` with `h ≥ 1`, the correction factor is controlled by `gcd(k,u)` and is therefore constant along the section. This is the arithmetic layer needed before the finite-dimensional all-base kernel basis argument; it does not assert the remaining linear independence theorem.","generated_certificate":false},{"name":"totientKernelSectionIndex_spec","module":"Erdos249257/TotientKernelIndex.lean","line":124,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientKernelIndex.lean#L124","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelIndex.lean#L124","kind":"theorem","signature":"theorem totientKernelSectionIndex_spec {k e : ℕ} (hk : 2 ≤ k) (i : TotientKernelSectionIndex k e) : 1 ≤ totientKernelSectionLevel i ∧ totientKernelSectionLevel i ≤ e ∧ 1 ≤ totientKernelSectionResidue i ∧ totientKernelSectionResidue i < k ^ totientKernelSectionLevel i ∧ ¬k ∣ totientKernelSectionResidue i","docstring":"Every section coordinate has the advertised level, residue, and nondivisibility properties.","generated_certificate":false},{"name":"totientKernelResidueAtLevel_injective","module":"Erdos249257/TotientKernelIndex.lean","line":190,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientKernelIndex.lean#L190","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelIndex.lean#L190","kind":"theorem","signature":"theorem totientKernelResidueAtLevel_injective (k j : ℕ) (hk : 2 ≤ k) : Function.Injective (totientKernelResidueAtLevel k : Fin (k ^ j) × Fin (k - 1) → ℕ)","docstring":"Quotient/nonzero-digit coordinates are injective at each fixed level.","generated_certificate":false},{"name":"existsUnique_totientKernelResidueAtLevel","module":"Erdos249257/TotientKernelIndex.lean","line":213,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientKernelIndex.lean#L213","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelIndex.lean#L213","kind":"theorem","signature":"theorem existsUnique_totientKernelResidueAtLevel (k j r : ℕ) (hk : 2 ≤ k) (_hrpos : 1 ≤ r) (hrlt : r < k ^ (j + 1)) (hnotdvd : ¬k ∣ r) : ∃! qd : Fin (k ^ j) × Fin (k - 1), totientKernelResidueAtLevel k qd = r","docstring":"Every positive nonmultiple below `k^(j+1)` has a unique fixed-level quotient/nonzero-digit representation. The representative is Euclidean: `q = r / k` and `d = r % k - 1`.","generated_certificate":false},{"name":"card_totientKernelSectionLevel","module":"Erdos249257/TotientKernelIndex.lean","line":241,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientKernelIndex.lean#L241","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelIndex.lean#L241","kind":"theorem","signature":"theorem card_totientKernelSectionLevel (k j : ℕ) : Fintype.card (Fin (k ^ j) × Fin (k - 1)) = k ^ (j + 1) - k ^ j","docstring":"At level `j + 1`, the quotient/nonzero-digit coordinates have cardinality `k^(j+1) - k^j`.","generated_certificate":false},{"name":"card_totientKernelSectionIndex","module":"Erdos249257/TotientKernelIndex.lean","line":249,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientKernelIndex.lean#L249","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelIndex.lean#L249","kind":"theorem","signature":"theorem card_totientKernelSectionIndex (k e : ℕ) : Fintype.card (TotientKernelSectionIndex k e) = ∑ j ∈ Finset.range e, (k ^ (j + 1) - k ^ j)","docstring":"The section-coordinate cardinality is the transparent telescoping sum `sum_{j<e} (k^(j+1) - k^j)`.","generated_certificate":false},{"name":"card_totientKernelIndex_eq_two_add_sum","module":"Erdos249257/TotientKernelIndex.lean","line":264,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientKernelIndex.lean#L264","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelIndex.lean#L264","kind":"theorem","signature":"theorem card_totientKernelIndex_eq_two_add_sum (k e : ℕ) : Fintype.card (TotientKernelIndex k e) = 2 + ∑ j ∈ Finset.range e, k ^ j * (k - 1)","docstring":"Before telescoping, the complete index cardinality is two plus one block of `k^j * (k - 1)` coordinates at every `j < e`.","generated_certificate":false},{"name":"card_totientKernelIndex","module":"Erdos249257/TotientKernelIndex.lean","line":277,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientKernelIndex.lean#L277","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelIndex.lean#L277","kind":"theorem","signature":"theorem card_totientKernelIndex (k e : ℕ) (hk : 2 ≤ k) : Fintype.card (TotientKernelIndex k e) = k ^ e + 1","docstring":"The all-base combinatorial index has cardinality `k^e + 1` for every `k ≥ 2`. The statement is valid also at `e = 0`; the intended finite-level application uses `e ≥ 1`.","generated_certificate":false},{"name":"totientKernelReductionScalar_ne_zero","module":"Erdos249257/TotientKernelConditional.lean","line":35,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientKernelConditional.lean#L35","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelConditional.lean#L35","kind":"theorem","signature":"theorem totientKernelReductionScalar_ne_zero (k r : ℕ) (hk : 0 < k) : totientKernelReductionScalar k r ≠ 0","docstring":"The reduction scalar is nonzero for every positive base.","generated_certificate":false},{"name":"allBaseTotientKernelSeq_mul_residue_step","module":"Erdos249257/TotientKernelConditional.lean","line":54,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientKernelConditional.lean#L54","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelConditional.lean#L54","kind":"theorem","signature":"theorem allBaseTotientKernelSeq_mul_residue_step (k j r : ℕ) (hk : 0 < k) (hj : 1 ≤ j) : allBaseTotientKernelSeq k (j + 1) (k * r) = totientKernelReductionScalar k r • allBaseTotientKernelSeq k j r","docstring":"**Composite residue scalar reduction.** Once the lower level is positive, multiplying its residue and modulus by `k` scales the entire totient section by a constant depending only on `k` and the residue `r`. The factor is `totient(k) * gcd(k,r) / totient(gcd(k,r))`; it is not in general `totient(k)` for composite `k`.","generated_certificate":false},{"name":"allBaseTotientKernelSeq_mem_span_canonical_of_le","module":"Erdos249257/TotientKernelConditional.lean","line":102,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientKernelConditional.lean#L102","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelConditional.lean#L102","kind":"theorem","signature":"theorem allBaseTotientKernelSeq_mem_span_canonical_of_le {k e j r : ℕ} (hk : 2 ≤ k) (hj : j ≤ e) (hr : r < k ^ j) : allBaseTotientKernelSeq k j r ∈ Submodule.span ℚ (Set.range (canonicalAllBaseTotientKernelFamily k e))","docstring":"Every channel through a positive depth belongs to the span of the filtration-compatible canonical all-base family. This spanning theorem is unconditional: composite residues are reduced recursively by `allBaseTotientKernelSeq_mul_residue_step`.","generated_certificate":false},{"name":"span_allBaseTotientKernelThroughLevelFamily_eq_canonical","module":"Erdos249257/TotientKernelConditional.lean","line":185,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientKernelConditional.lean#L185","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelConditional.lean#L185","kind":"theorem","signature":"theorem span_allBaseTotientKernelThroughLevelFamily_eq_canonical (k e : ℕ) (hk : 2 ≤ k) (he : 1 ≤ e) : Submodule.span ℚ (Set.range (allBaseTotientKernelThroughLevelFamily k e)) = Submodule.span ℚ (Set.range (canonicalAllBaseTotientKernelFamily k e))","docstring":"The complete all-base kernel through a positive level and the canonical filtration-compatible family span the same subspace.","generated_certificate":false},{"name":"finrank_canonicalAllBaseTotientKernel_eq_of_linearIndependent","module":"Erdos249257/TotientKernelConditional.lean","line":203,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientKernelConditional.lean#L203","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelConditional.lean#L203","kind":"theorem","signature":"theorem finrank_canonicalAllBaseTotientKernel_eq_of_linearIndependent (k e : ℕ) (hk : 2 ≤ k) (hcanon : LinearIndependent ℚ (canonicalAllBaseTotientKernelFamily k e)) : finrank ℚ (Submodule.span ℚ (Set.range (canonicalAllBaseTotientKernelFamily k e))) = k ^ e + 1","docstring":"Linear independence of the canonical all-base family implies the exact cardinality-sized rank. This is the explicit external-input boundary: the hypothesis is not proved in this module.","generated_certificate":false},{"name":"finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent","module":"Erdos249257/TotientKernelConditional.lean","line":215,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientKernelConditional.lean#L215","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelConditional.lean#L215","kind":"theorem","signature":"theorem finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent (k e : ℕ) (hk : 2 ≤ k) (he : 1 ≤ e) (hcanon : LinearIndependent ℚ (canonicalAllBaseTotientKernelFamily k e)) : finrank ℚ (Submodule.span ℚ (Set.range (allBaseTotientKernelThroughLevelFamily k e))) = k ^ e + 1","docstring":"**Conditional exact rank of the actual all-base truncation.** The arithmetic spanning and index-cardinality layers are formalised above. The only hypothesis is linear independence of the explicit canonical family, which is the external affine-ordering boundary and is not discharged here.","generated_certificate":false}],"paper_label":"thm:kkernelrank","paper_anchor":{"paper_label":"thm:kkernelrank","owner_source":"paper/249/erdos-249-binary-totient-series.tex","source_line":87,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":86,"anchor_title":"A basis through each finite level","anchor_title_tex":"A basis through each finite level","section_title":"Introduction","section_path":[{"level":"section","title":"Introduction","title_tex":"Introduction","line":48}],"source_excerpt":{"text":"\\begin{theorem}[A basis through each finite level]\n\\label{thm:kkernelrank}\nLet $k\\ge2$ and $e\\ge1$ be integers, write $F^{(k)}_{j,r}(n)=\\ph(k^jn+r)$, and put\n\\[\n V_{k,e}=\\operatorname{span}_{\\Q}\n \\{\\,F^{(k)}_{j,r}:0\\le j\\le e,\\ 0\\le r<k^j\\,\\}.\n\\]\nThen $\\dim_{\\Q}V_{k,e}=k^e+1$, and\n\\[\n \\mathcal B_{k,e}\n =\\{F^{(k)}_{0,0},F^{(k)}_{1,0}\\}\n \\cup\\{\\,F^{(k)}_{j,r}:1\\le j\\le e,\\ 1\\le r<k^j,\\ k\\nmid r\\,\\}\n\\]\nis a basis.  Every omitted section reduces to a basis element by an explicit\nscalar: $F^{(k)}_{j,0}=k^{j-1}F^{(k)}_{1,0}$ for $j\\ge1$, and if $r=k^tu$ with\n$t=\\max\\{s:k^s\\mid r\\}\\ge1$ and $k\\nmid u$, then\n\\[\n F^{(k)}_{j,r}=C_k(t,u)\\,F^{(k)}_{j-t,u},\n \\qquad\n C_k(t,u)=k^t\\prod_{\\substack{p\\mid k\\\\ p\\nmid u}}\\Bigl(1-\\tfrac1p\\Bigr).\n\\]\n\\end{theorem}","media_type":"text/x-tex","source_char_start":3717,"source_char_end":4468,"start_line":86,"end_line":107,"truncated":false,"sha256":"sha256:40f371d1722f6d0e17a89a41e4a8a6a9e74a326897c192ffcd60bfb5cdae19f2","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos-249-binary-totient-series.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/249/erdos-249-binary-totient-series.tex#L87","owner_problem_erdos_number":249,"owner_paper_id":"erdos-249-binary-totient-series","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-249-binary-totient-series.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-249-binary-totient-series.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/249/erdos-249-binary-totient-series.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_249_irrationality","statement":"Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/1"}],"links":[{"kind":"lean_declaration_pinned","label":"totient_pow_mul_eq","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelReduction.lean#L60","cite_this":true},{"kind":"lean_declaration_pinned","label":"totient_pow_mul_affine_gcd_cross_eq","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelReduction.lean#L117","cite_this":true},{"kind":"lean_declaration_pinned","label":"totientKernelSectionIndex_spec","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelIndex.lean#L124","cite_this":true},{"kind":"lean_declaration_pinned","label":"totientKernelResidueAtLevel_injective","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelIndex.lean#L190","cite_this":true},{"kind":"lean_declaration_pinned","label":"existsUnique_totientKernelResidueAtLevel","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelIndex.lean#L213","cite_this":true},{"kind":"lean_declaration_pinned","label":"card_totientKernelSectionLevel","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelIndex.lean#L241","cite_this":true},{"kind":"lean_declaration_pinned","label":"card_totientKernelSectionIndex","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelIndex.lean#L249","cite_this":true},{"kind":"lean_declaration_pinned","label":"card_totientKernelIndex_eq_two_add_sum","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelIndex.lean#L264","cite_this":true},{"kind":"lean_declaration_pinned","label":"card_totientKernelIndex","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelIndex.lean#L277","cite_this":true},{"kind":"lean_declaration_pinned","label":"totientKernelReductionScalar_ne_zero","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelConditional.lean#L35","cite_this":true},{"kind":"lean_declaration_pinned","label":"allBaseTotientKernelSeq_mul_residue_step","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelConditional.lean#L54","cite_this":true},{"kind":"lean_declaration_pinned","label":"allBaseTotientKernelSeq_mem_span_canonical_of_le","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelConditional.lean#L102","cite_this":true},{"kind":"lean_declaration_pinned","label":"span_allBaseTotientKernelThroughLevelFamily_eq_canonical","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelConditional.lean#L185","cite_this":true},{"kind":"lean_declaration_pinned","label":"finrank_canonicalAllBaseTotientKernel_eq_of_linearIndependent","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelConditional.lean#L203","cite_this":true},{"kind":"lean_declaration_pinned","label":"finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelConditional.lean#L215","cite_this":true},{"kind":"paper_pdf","label":"erdos-249-binary-totient-series","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-249-binary-totient-series.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/96","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"all_base_totient_kernel_index_reduction"}}],"expect":"object"},"prior_art":{"family_id":"alternative_249_algebraic_boundaries","status_summary":"Exact finite obstructions and interfaces, a constructive dyadic basis, and an all-base arithmetic/spanning layer. The reusable rank wrapper is conditional; AllBaseTotientKernel.lean separately proves its independence premise and the complete rank unconditionally. The residue-series claims classify the rational totient residue series and prove the dilated series linearly independent.","prior_art_posture":"Classical Euler/Möbius, CRT, determinant, and Lambert tools; Coons already implies full-kernel infinite rank, and Martin supplies a broader external affine-independence antecedent. The Lean all-base proof uses CRT and Dirichlet independently. Erdős 1957 (Lemma 1) and Kaneko–Suzuki–Tachiya (Theorem 3) already give the irrationality of the totient residue series. No #249 irrationality endpoint or priority claim.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/17","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"formalisation_only":6,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z09::all_base_totient_canonical_rank_of_linear_independence","prior_art_state":"formalisation_only"},{"semantic_node_id":"Z09::all_base_totient_canonical_span_equality","prior_art_state":"formalisation_only"},{"semantic_node_id":"Z09::all_base_totient_channel_mem_canonical_span","prior_art_state":"formalisation_only"},{"semantic_node_id":"Z09::all_base_totient_kernel_arithmetic_index","prior_art_state":"formalisation_only"},{"semantic_node_id":"Z09::all_base_totient_reduction_scalar_nonzero","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z09::all_base_totient_residue_scalar_step","prior_art_state":"formalisation_only"},{"semantic_node_id":"Z09::all_base_totient_truncation_rank_of_linear_independence","prior_art_state":"formalisation_only"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:residual_gauge_obstruction","assertion_kind":"registered_claim","source_id":"residual_gauge_obstruction","reviewed":true,"granularity":"single_registered_claim","label":"Residual-gauge obstruction for monomial minors","statement":"For a residual-weighted monomial matrix, common nonzero residual weights act by a right diagonal gauge and preserve determinant nonvanishing. Under the locked inverse-phase gauge, an exponent-one row is identically 1 even when the determinant remains nonzero; allowing row-dependent residuals reconstructs a consecutive-power matrix with zeroth row identically 1. Thus residual-blind rank, determinant, and conditioning criteria alone cannot exclude the target. This does not rule out a criterion with an additional arithmetic coupling identity, and it gives no irrationality criterion.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"gateway_section","warrant":"the claim's anchor sits inside the retired gateway section 'The totient constant S (Erdős #249)', which is the author's own partition of that manuscript by problem"},"problem_edges":[{"problem_id":"erdos_249","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"det_residualMonomialMatrix_ne_zero_iff","module":"Erdos249257/ResidualGaugeObstruction.lean","line":55,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/ResidualGaugeObstruction.lean#L55","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ResidualGaugeObstruction.lean#L55","kind":"theorem","signature":"theorem det_residualMonomialMatrix_ne_zero_iff {d : ℕ} (e : Fin d → ℕ) (W z : Fin d → ℂ) (hW : ∏ j, W j ≠ 0) : Matrix.det (residualMonomialMatrix e W z) ≠ 0 ↔ Matrix.det (phasePowerMatrix e z) ≠ 0","docstring":"Nonzero column gauges preserve determinant nonvanishing at every rank.","generated_certificate":false},{"name":"locked_reconstruction_preserves_nonzero_minor","module":"Erdos249257/ResidualGaugeObstruction.lean","line":94,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/ResidualGaugeObstruction.lean#L94","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ResidualGaugeObstruction.lean#L94","kind":"theorem","signature":"theorem locked_reconstruction_preserves_nonzero_minor {d : ℕ} (e : Fin d → ℕ) (z : Fin d → ℂ) (i : Fin d) (hi : e i = 1) (hz : ∀ j, z j ≠ 0) (hdet : Matrix.det (phasePowerMatrix e z) ≠ 0) : Matrix.det (residualMonomialMatrix e (lockedResidual z) z) ≠ 0 ∧ ∀ j, residualMonomialMatrix e (lockedResidual z) z i j = 1","docstring":"The locked bad row and a nonzero coefficient-side determinant coexist. This is the all-rank countermodel to a residual-blind minor certificate.","generated_certificate":false},{"name":"rowReconstruction_zero_row_one","module":"Erdos249257/ResidualGaugeObstruction.lean","line":145,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/ResidualGaugeObstruction.lean#L145","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ResidualGaugeObstruction.lean#L145","kind":"theorem","signature":"theorem rowReconstruction_zero_row_one {d : ℕ} (e : Fin d → ℕ) (z : Fin d → ℂ) (hz : ∀ j, z j ≠ 0) (i : Fin d) (hi : i.1 = 0) : ∀ j, rowResidualMonomialMatrix e (rowReconstructingResidual e z) z i j = 1","docstring":"The reconstructed zeroth row is identically `1`. Thus even allowing row-dependent unit residuals does not let a fixed-rank nonzero minor exclude the locked affine target without an additional arithmetic coupling law.","generated_certificate":false}],"paper_label":"res:residualgauge","paper_anchor":{"paper_label":"res:residualgauge","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":3269,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":3269,"anchor_title":"column-rescaling obstruction for monomial minors","anchor_title_tex":"column-rescaling obstruction for monomial minors","section_title":"Relation to prior work.","section_path":[{"level":"section","title":"The totient constant S (Erdős #249)","title_tex":"The totient constant $S$ (Erd\\H{o}s \\#249)","line":2304},{"level":"subsection","title":"Unconditional dyadic rank and the rationality-side carry bound","title_tex":"Unconditional dyadic rank and the rationality-side carry bound","line":3130},{"level":"paragraph","title":"Relation to prior work.","title_tex":"Relation to prior work.","line":3169}],"source_excerpt":{"text":"\\begin{proposition}[column-rescaling obstruction for monomial minors]\\label{res:residualgauge}\nLet $z_j,W_j\\ne0$ and $M_{ij}=W_jz_j^{e_i}$.  Then\n\\(det M=\\det(z_j^{e_i})\\prod_jW_j\\), so the column weights preserve\ndeterminant nonvanishing.  If $e_i=1$, choosing $W_j=z_j^{-1}$ makes row $i$\nidentically $1$.  Allowing row-dependent weights\n$W_{ij}=z_j^i/z_j^{e_i}$ reconstructs a consecutive-power matrix whose\nzeroth row is identically $1$.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":172868,"source_char_end":173327,"start_line":3269,"end_line":3276,"truncated":false,"sha256":"sha256:41bdfb7f8b5aeae4894ac9660091f1f7b2f135c6a6a45bebf902869b038c12c0","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3269","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"det_residualMonomialMatrix_ne_zero_iff","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ResidualGaugeObstruction.lean#L55","cite_this":true},{"kind":"lean_declaration_pinned","label":"locked_reconstruction_preserves_nonzero_minor","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ResidualGaugeObstruction.lean#L94","cite_this":true},{"kind":"lean_declaration_pinned","label":"rowReconstruction_zero_row_one","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ResidualGaugeObstruction.lean#L145","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/97","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"residual_gauge_obstruction"}}],"expect":"object"},"prior_art":{"family_id":"alternative_249_algebraic_boundaries","status_summary":"Exact finite obstructions and interfaces, a constructive dyadic basis, and an all-base arithmetic/spanning layer. The reusable rank wrapper is conditional; AllBaseTotientKernel.lean separately proves its independence premise and the complete rank unconditionally. The residue-series claims classify the rational totient residue series and prove the dilated series linearly independent.","prior_art_posture":"Classical Euler/Möbius, CRT, determinant, and Lambert tools; Coons already implies full-kernel infinite rank, and Martin supplies a broader external affine-independence antecedent. The Lean all-base proof uses CRT and Dirichlet independently. Erdős 1957 (Lemma 1) and Kaneko–Suzuki–Tachiya (Theorem 3) already give the irrationality of the totient residue series. No #249 irrationality endpoint or priority claim.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/17","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"known_classical":1,"not_assessed":2},"edges":[{"semantic_node_id":"Z07::locked_residual_countermodel","prior_art_state":"not_assessed"},{"semantic_node_id":"Z07::residual_gauge_determinant_factorisation","prior_art_state":"known_classical"},{"semantic_node_id":"Z07::row_dependent_residual_countermodel","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:residualgauge","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L3269","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:tail_orbit_rigidity","assertion_kind":"registered_claim","source_id":"tail_orbit_rigidity","reviewed":true,"granularity":"single_registered_claim","label":"Tempered tail-orbit rigidity criterion","statement":"For coefficients c(n) ≤ n, the binary series ∑ c(n)/2ⁿ is rational exactly when a tempered integer carry orbit exists, and every tempered orbit equals the scaled tail.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"programme_representative","warrant":"orientation.json lists this claim among the representative claims of a mathematical programme whose remaining open propositions all name one problem"},"problem_edges":[{"problem_id":"erdos_257","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":["boolean_mobius_constraints"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit","module":"Erdos249257/GenericTailOrbitRigidity.lean","line":426,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GenericTailOrbitRigidity.lean#L426","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GenericTailOrbitRigidity.lean#L426","kind":"theorem","signature":"theorem binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit (c : ℕ → ℕ) (hgrowth : ∀ n : ℕ, c n ≤ n) : HasRationalValue (binaryCoeffSeries c) ↔ ∃ v : ℕ, 0 < v ∧ ∃ u : ℕ → ℤ, IsTemperedBinaryOrbit c v u","docstring":"**Subexponential tail-orbit rigidity, equivalence form (T7).**","generated_certificate":false}],"paper_label":"res:rigidity","paper_anchor":{"paper_label":"res:rigidity","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":4712,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":4712,"anchor_title":"tail-orbit rigidity","anchor_title_tex":"tail-orbit rigidity","section_title":"Exact normal form: tail-orbit rigidity","section_path":[{"level":"section","title":"Auxiliary rationality criteria from binary carry systems","title_tex":"Auxiliary rationality criteria from binary carry systems","line":4612},{"level":"subsection","title":"Exact normal form: tail-orbit rigidity","title_tex":"Exact normal form: tail-orbit rigidity","line":4680}],"source_excerpt":{"text":"\\begin{theorem}[tail-orbit rigidity]\\label{res:rigidity}\nLet $\\gamma(n)\\le n$ for all $n$. Every tempered orbit is the scaled tail,\n$u(N)=v\\,T_\\gamma(N)$ for all $N$; and $X_\\gamma$ is rational if and only if a tempered\norbit exists for some multiplier $v\\ge1$.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":247138,"source_char_end":247413,"start_line":4712,"end_line":4716,"truncated":false,"sha256":"sha256:ddd494ea73791791911573c147c741846fc51e531602d8f43f9c5bd928ce1055","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L4712","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GenericTailOrbitRigidity.lean#L426","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/98","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"tail_orbit_rigidity"}}],"expect":"object"},"prior_art":{"family_id":"generic_binary_carry_rigidity","status_summary":"Exact rationality characterisations and proved necessary consequences.","prior_art_posture":"Han Wang supplies the closest weighted-binary carry recurrence; the abstraction, converse, and downstream constraints make no priority claim.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/3","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z11::rationality_iff_tempered_orbit","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:rigidity","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L4712","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:boolean_mobius_carry","assertion_kind":"registered_claim","source_id":"boolean_mobius_carry","reviewed":true,"granularity":"single_registered_claim","label":"Boolean–Möbius carry certificate equivalence","statement":"A nonempty normalised support, one omitting 0, with series value p/q exists exactly when a positive, square-root-bounded integer orbit with Boolean Möbius quotient exists; the support is reconstructed from the certificate.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"programme_representative","warrant":"orientation.json lists this claim among the representative claims of a mathematical programme whose remaining open propositions all name one problem"},"problem_edges":[{"problem_id":"erdos_257","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":["boolean_mobius_constraints"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"exists_normalized_support_fraction_iff_exists_booleanMobiusCarry","module":"Erdos249257/BooleanMobiusCarry.lean","line":949,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/BooleanMobiusCarry.lean#L949","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/BooleanMobiusCarry.lean#L949","kind":"theorem","signature":"theorem exists_normalized_support_fraction_iff_exists_booleanMobiusCarry (p : ℤ) (q : ℕ) (hq : 0 < q) : (∃ A : Set ℕ, 0 ∉ A ∧ (∃ a : ℕ, 0 < a ∧ a ∈ A) ∧ erdosSupportSeries 2 A = (p : ℝ) / (q : ℝ)) ↔ ∃ U : ℕ → ℤ, BooleanMobiusCarryCertificate p q U","docstring":"**Certificate T8 equivalence.** Positive normalized supports with value `p/q` are in exact correspondence at the existence level with quotient-only Boolean Möbius carry certificates. The converse support is not guessed: it is reconstructed as `booleanMobiusSupport` of the carry quotient.","generated_certificate":false}],"paper_label":"res:carrycert","paper_anchor":{"paper_label":"res:carrycert","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":4816,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":4816,"anchor_title":"support fraction--Boolean-carry equivalence","anchor_title_tex":"support fraction--Boolean-carry equivalence","section_title":"The exact coordinate chain.","section_path":[{"level":"section","title":"Auxiliary rationality criteria from binary carry systems","title_tex":"Auxiliary rationality criteria from binary carry systems","line":4612},{"level":"subsection","title":"Exact #257 coordinates: Boolean--Möbius carry certificates","title_tex":"Exact \\#257 coordinates: Boolean--M\\\"obius carry certificates","line":4739},{"level":"paragraph","title":"The exact coordinate chain.","title_tex":"The exact coordinate chain.","line":4771}],"source_excerpt":{"text":"\\begin{theorem}[support fraction--Boolean-carry equivalence]\\label{res:carrycert}\nFix $p\\in\\mathbb Z$ and an integer $q\\ge1$. There is a nonempty support $A$ of\npositive integers with $\\sum_{n\\in A}1/(2^n-1)=p/q$ if and only if there is an\ninteger sequence $U$, with $U(0)=p$, such that, for every $N$,\n\\[\n  0<U(N)\\le q\\,(2\\sqrt N+4)\n  \\qquad\\text{and}\\qquad\n  q\\mid 2U(N)-U(N+1),\n\\]\nand the M\\\"obius transform of the quotient sequence\n$N\\mapsto\\bigl(2U(N)-U(N+1)\\bigr)/q$ is Boolean. The support is not guessed:\nit is reconstructed from the certificate by Proposition~\\ref{res:boolmob}.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":253030,"source_char_end":253631,"start_line":4816,"end_line":4828,"truncated":false,"sha256":"sha256:83ccc7f3a3870344f9e35fc187399704ba5972bb0320a5b5ffc8367521bf2803","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z10::boolean_mobius_carry_certificate_equivalence","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:carrycert","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"contradicted_by_formal_source","render_policy":"historical_only_not_authoritative","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L4816","finding_axes":["negative_clause_loss"],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:reciprocal_mass_bound","assertion_kind":"registered_claim","source_id":"reciprocal_mass_bound","reviewed":true,"granularity":"single_registered_claim","label":"Rationality-forced reciprocal-mass lower bound","statement":"A rational support-series value with odd denominator part v forces reciprocal mass at least 1/ord_v(2), or divergence; an infinite support with dyadic-rational value has reciprocal mass divergent or greater than one.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"one_div_oddOrder_le_reciprocalMass_of_support_fraction","module":"Erdos249257/RationalSupportCarrySkeleton.lean","line":1480,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/RationalSupportCarrySkeleton.lean#L1480","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/RationalSupportCarrySkeleton.lean#L1480","kind":"theorem","signature":"theorem one_div_oddOrder_le_reciprocalMass_of_support_fraction (A : Set ℕ) (hA : ∃ a : ℕ, 0 < a ∧ a ∈ A) (hsum : Summable (reciprocalSupportTerm A)) (p : ℤ) (c : ℕ) {v : ℕ} (hv : 1 < v) (hvodd : Odd v) (hpv : p.toNat.Coprime v) (hvalue : erdosSupportSeries 2 A = (p : ℝ) / ((2 ^ c * v : ℕ) : ℝ)) : (1 : ℝ) / (oddDoublingOrder v hvodd : ℝ) ≤ reciprocalMass A","generated_certificate":false}],"paper_label":"res:orderwrap","paper_anchor":{"paper_label":"res:orderwrap","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":4975,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":4975,"anchor_title":"order-wrap lower bound, with exact excess mean","anchor_title_tex":"order-wrap lower bound, with exact excess mean","section_title":"Necessary conditions II: wraps, mass, and unbounded states","section_path":[{"level":"section","title":"Auxiliary rationality criteria from binary carry systems","title_tex":"Auxiliary rationality criteria from binary carry systems","line":4612},{"level":"subsection","title":"Necessary conditions II: wraps, mass, and unbounded states","title_tex":"Necessary conditions II: wraps, mass, and unbounded states","line":4898}],"source_excerpt":{"text":"\\begin{proposition}[order-wrap lower bound, with exact excess mean]\\label{res:orderwrap}\nLet \\(A\\subseteq\\Npos\\) be nonempty and suppose\n\\[\n X_A=\\frac{p}{2^cv},\n \\qquad p\\in\\Z,\\quad c\\in\\N,\\quad v>1\\ \\text{odd}.\n\\]\nLet \\(h=\\operatorname{ord}_v(2)\\), let \\(w\\) be the number of wraps in the\nleast-residue cycle \\(p2^N\\bmod v\\) for \\(0\\leq N<h\\), and set\n\\[\n u(N)=vT_{f_A}(c+N),\\qquad e_N=\\left\\lfloor\\frac{u(N)}v\\right\\rfloor.\n\\]\nThen $\\sum_{a\\in A}1/a$ diverges or $\\rho(A)\\ge w/h$; if moreover $p$ is\ncoprime to $v$, then $w\\ge1$, so\n$\\rho(A)\\ge1/\\operatorname{ord}_v(2)$. In the summable case the bound is the\nperiodic part of an exact identity: the Ces\\`aro mean of \\(e_N\\) converges,\nand\n\\[\n\\rho(A)\\;=\\;\\frac{w}{h}\\;+\\;\\lim_{N\\to\\infty}\\frac1N\\sum_{n<N}e_n .\n\\]\n\\end{proposition}","media_type":"text/x-tex","source_char_start":261217,"source_char_end":262000,"start_line":4975,"end_line":4994,"truncated":false,"sha256":"sha256:a9d1f33b231b5eacc2de2440fe9c4645913b38e0befe681fdbe5ffe337c6338b","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L4975","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"one_div_oddOrder_le_reciprocalMass_of_support_fraction","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/RationalSupportCarrySkeleton.lean#L1480","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/100","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"reciprocal_mass_bound"}}],"expect":"object"},"prior_art":{"family_id":"generic_binary_carry_rigidity","status_summary":"Exact rationality characterisations and proved necessary consequences.","prior_art_posture":"Han Wang supplies the closest weighted-binary carry recurrence; the abstraction, converse, and downstream constraints make no priority claim.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/3","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z10::rational_value_forces_order_wrap_ratio_bound","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:orderwrap","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L4975","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:unbounded_carry_states","assertion_kind":"registered_claim","source_id":"unbounded_carry_states","reviewed":true,"granularity":"single_registered_claim","label":"Carry states are unbounded over infinite rational-valued supports","statement":"An infinite support with rational series value has an unbounded positive carry state, closing finite-state readings of the carry system.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249","erdos_257"],"problem_erdos_numbers":[249,257],"problem_binding":{"method":"core_programme_joint","warrant":"the claim's only paper anchor is owned by the retired combined #249/#257 exposition, which treats both problems together and publishes nothing that separates them. Bound to both rather than guessed apart.","separability":"not_separable_from_published_source"},"problem_edges":[{"problem_id":"erdos_249","relation":"shared_programme_context","basis":"problem_binding"},{"problem_id":"erdos_257","relation":"shared_programme_context","basis":"problem_binding"}],"programme_ids":["joint_249_257_gateway"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"exists_unbounded_shifted_odd_tail_nat_state_of_support_fraction","module":"Erdos249257/RationalSupportCarrySkeleton.lean","line":2383,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/RationalSupportCarrySkeleton.lean#L2383","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/RationalSupportCarrySkeleton.lean#L2383","kind":"theorem","signature":"theorem exists_unbounded_shifted_odd_tail_nat_state_of_support_fraction (A : Set ℕ) (hAinf : A.Infinite) (p : ℤ) (c v : ℕ) (hv : 0 < v) (hvalue : erdosSupportSeries 2 A = (p : ℝ) / ((2 ^ c * v : ℕ) : ℝ)) : ∃ u : ℕ → ℕ, (∀ n : ℕ, (u n : ℝ) = (v : ℝ) * binaryCoeffTail (supportCoeff A) (c + n)) ∧ (∀ n : ℕ, 0 < u n) ∧ (∀ n : ℕ, u (n + 1) + v * supportCoeff A (c + n + 1) = 2 * u n) ∧ (∀ n : ℕ, u n ≡ p.toNat * 2 ^ n [MOD v]) ∧ (∀ B : ℕ, ∃ n : ℕ, B < u n)","docstring":"If an infinite support has the displayed rational value, its positive shifted integer tail state is unbounded.","generated_certificate":false}],"paper_label":"res:unbounded","paper_anchor":{"paper_label":"res:unbounded","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":5025,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":5025,"anchor_title":"carry states are unbounded over infinite supports","anchor_title_tex":"carry states are unbounded over infinite supports","section_title":"Necessary conditions II: wraps, mass, and unbounded states","section_path":[{"level":"section","title":"Auxiliary rationality criteria from binary carry systems","title_tex":"Auxiliary rationality criteria from binary carry systems","line":4612},{"level":"subsection","title":"Necessary conditions II: wraps, mass, and unbounded states","title_tex":"Necessary conditions II: wraps, mass, and unbounded states","line":4898}],"source_excerpt":{"text":"\\begin{theorem}[carry states are unbounded over infinite supports]\\label{res:unbounded}\nLet $A$ be infinite with value $p/(2^c v)$, $v\\ge1$. Then the positive integer\ncarry state $u(N)=v\\,T_{f_A}(c+N)$ is unbounded: for every $B$ there is an $N$\nwith $u(N)>B$.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":264081,"source_char_end":264355,"start_line":5025,"end_line":5029,"truncated":false,"sha256":"sha256:cea4deabac1b5688fc91bd5606a52a91bb4a00d31aa06411759e2042c53a202e","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L5025","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"exists_unbounded_shifted_odd_tail_nat_state_of_support_fraction","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/RationalSupportCarrySkeleton.lean#L2383","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/101","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"unbounded_carry_states"}}],"expect":"object"},"prior_art":{"family_id":"generic_binary_carry_rigidity","status_summary":"Exact rationality characterisations and proved necessary consequences.","prior_art_posture":"Han Wang supplies the closest weighted-binary carry recurrence; the abstraction, converse, and downstream constraints make no priority claim.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/3","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z10::carry_state_unbounded_at_common_multiples","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:unbounded","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L5025","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:sublog_zero_windows","assertion_kind":"registered_claim","source_id":"sublog_zero_windows","reviewed":true,"granularity":"single_registered_claim","label":"Sublogarithmic divisor-coverage zero windows","statement":"A rational support-series value forces every zero window in the divisor-count coefficient to have length at most ε·log₂(N+1) + B, where B ≥ 0 is built from ε, c and v but is bound after the support and numerator, so no single constant is claimed uniform across supports. This constrains coverage; it does not solve universal #257.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":true,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"supportCoeffZeroWindow_length_le_eps_logb_add","module":"Erdos249257/SublogDivisorCoverage.lean","line":392,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SublogDivisorCoverage.lean#L392","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SublogDivisorCoverage.lean#L392","kind":"theorem","signature":"theorem supportCoeffZeroWindow_length_le_eps_logb_add (A : Set ℕ) (hA : ∃ a : ℕ, 0 < a ∧ a ∈ A) (p : ℤ) (c v : ℕ) (hv : 0 < v) (hvalue : erdosSupportSeries 2 A = (p : ℝ) / ((2 ^ c * v : ℕ) : ℝ)) (ε : ℝ) (hε : 0 < ε) : ∃ B : ℝ, 0 ≤ B ∧ ∀ N h : ℕ, SupportCoeffZeroWindow A (c + N) h → (h : ℝ) ≤ ε * Real.logb 2 (N + 1 : ℝ) + B","docstring":"**Rationality-conditional T11.** For every positive `ε`, the displayed rational support-series hypothesis forces every support-coefficient zero window to have length at most `ε * log₂(N+1)` plus a constant. The proof constructs that constant from `ε`, `v`, and `c`; because `A` and `p` precede the existential in this theorem's signature, this declaration does not export one common constant quantified uniformly across all supports and numerators.","generated_certificate":false}],"paper_label":"res:sublog","paper_anchor":{"paper_label":"res:sublog","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":4859,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":4859,"anchor_title":"sublogarithmic zero windows","anchor_title_tex":"sublogarithmic zero windows","section_title":"Necessary condition I: sublogarithmic divisor coverage","section_path":[{"level":"section","title":"Auxiliary rationality criteria from binary carry systems","title_tex":"Auxiliary rationality criteria from binary carry systems","line":4612},{"level":"subsection","title":"Necessary condition I: sublogarithmic divisor coverage","title_tex":"Necessary condition I: sublogarithmic divisor coverage","line":4846}],"source_excerpt":{"text":"\\begin{theorem}[sublogarithmic zero windows]\\label{res:sublog}\nLet $A$ contain a positive integer and suppose\n\\[\n  \\sum_{a\\in A}\\frac1{2^a-1}=\\frac{p}{2^c v},\n  \\qquad p\\in\\Z,\\quad c\\in\\N,\\quad v\\ge1.\n\\]\nFor every $\\varepsilon>0$ there is a constant $B\\ge0$, depending only on\n$\\varepsilon,c,$ and $v$, such that every zero window after $c+N$ satisfies\n\\[\n  h\\le \\varepsilon\\log_2(N+1)+B,\n  \\qquad \\log_2 x:=\\frac{\\log x}{\\log 2}.\n\\]\nThe bound is uniform in $A,p,N,$ and $h$.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":255190,"source_char_end":255679,"start_line":4859,"end_line":4872,"truncated":false,"sha256":"sha256:c5c247a172e60d2f45c0dcec243f91a179abc9d2f0fd6b1678e4bf2905c3c5d5","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/archive/erdos249-257-main-paper.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L4859","owner_paper_id":"erdos249-257-main-paper","owner_status":"retired_but_public","offline_read_route":"embedded at paper_anchor.source_excerpt; reads offline","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/archive/erdos249-257-main-paper.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"supportCoeffZeroWindow_length_le_eps_logb_add","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SublogDivisorCoverage.lean#L392","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/102","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"sublog_zero_windows"}}],"expect":"object"},"prior_art":{"family_id":"generic_binary_carry_rigidity","status_summary":"Exact rationality characterisations and proved necessary consequences.","prior_art_posture":"Han Wang supplies the closest weighted-binary carry recurrence; the abstraction, converse, and downstream constraints make no priority claim.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/3","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z11::sublogarithmic_zero_window_bound","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:sublog","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"contradicted_by_formal_source","render_policy":"historical_only_not_authoritative","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L4859","finding_axes":["docstring_contradiction","quantifier_scope"],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:greedy_achievement_geometry","assertion_kind":"registered_claim","source_id":"greedy_achievement_geometry","reviewed":true,"granularity":"single_registered_claim","label":"Greedy geometry of the Mersenne achievement set","statement":"The Mersenne achievement set is compact, perfect, totally disconnected, nowhere dense, and has Lebesgue measure one. Membership is equivalent to greedy survival at every level; for the target 1/2 it is exactly equivalent to the canonical greedy orbit omitting infinitely many exponents. No finite normalized support has value 1/2, so positive membership would automatically produce an infinite rational-valued support and refute universal #257. Finite exact-rational death certificates soundly prove non-membership (3/4 is excluded at level one). The strict-tail Cantor geometry is recorded by Kovač–Tao; no novelty claim is made for the formalised refinements.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":true,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"claim_prose"}],"programme_ids":["erdos257_half_story"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"volume_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":996,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GreedyAchievementSet.lean#L996","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GreedyAchievementSet.lean#L996","kind":"theorem","signature":"theorem volume_mersenneAchievementSet : volume mersenneAchievementSet = 1","docstring":"**measure one.** Continuity from above transfers the exact cylinder measures to their decreasing intersection.","generated_certificate":false},{"name":"perfect_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":1656,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GreedyAchievementSet.lean#L1656","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GreedyAchievementSet.lean#L1656","kind":"theorem","signature":"theorem perfect_mersenneAchievementSet : Perfect mersenneAchievementSet","docstring":"**perfection.** The compact achievement set has no isolated points.","generated_certificate":false},{"name":"isTotallyDisconnected_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":1672,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GreedyAchievementSet.lean#L1672","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GreedyAchievementSet.lean#L1672","kind":"theorem","signature":"theorem isTotallyDisconnected_mersenneAchievementSet : IsTotallyDisconnected mersenneAchievementSet","generated_certificate":false},{"name":"isNowhereDense_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":1681,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GreedyAchievementSet.lean#L1681","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GreedyAchievementSet.lean#L1681","kind":"theorem","signature":"theorem isNowhereDense_mersenneAchievementSet : IsNowhereDense mersenneAchievementSet","docstring":"**nowhere density.** If the closed achievement set contained an open ball, total disconnectedness would put a complementary point between two points of that ball.","generated_certificate":false},{"name":"mem_mersenneAchievementSet_iff_greedy_survival","module":"Erdos249257/GreedyAchievementSet.lean","line":1458,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GreedyAchievementSet.lean#L1458","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GreedyAchievementSet.lean#L1458","kind":"theorem","signature":"theorem mem_mersenneAchievementSet_iff_greedy_survival (x : ℝ) : x ∈ mersenneAchievementSet ↔ 0 ≤ x ∧ ∀ n : ℕ, greedyMersenneRemainder x n ≤ mersenneTail n","docstring":"**Greedy membership criterion.** A real target belongs to the Mersenne achievement set exactly when it is nonnegative and every residual is at most the full remaining tail.","generated_certificate":false},{"name":"three_fourths_not_mem_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":1784,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GreedyAchievementSet.lean#L1784","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GreedyAchievementSet.lean#L1784","kind":"theorem","signature":"theorem three_fourths_not_mem_mersenneAchievementSet : (3 / 4 : ℝ) ∉ mersenneAchievementSet","docstring":"Consequently `3/4` is not in the Mersenne achievement set.","generated_certificate":false},{"name":"half_mem_mersenneAchievementSet_iff_greedySkippedSupport_infinite","module":"Erdos249257/GreedyAchievementSet.lean","line":2583,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GreedyAchievementSet.lean#L2583","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GreedyAchievementSet.lean#L2583","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_iff_greedySkippedSupport_infinite : (1 / 2 : ℝ) ∈ mersenneAchievementSet ↔ (greedyMersenneSkippedSupport (1 / 2 : ℝ)).Infinite","docstring":"**Greedy skip dichotomy for the half target.** Exact half-membership is equivalent to the canonical greedy orbit omitting infinitely many exponents. The forward direction uses Erdős–Borwein irrationality; the reverse direction uses absorbing fatal states.","generated_certificate":false},{"name":"finite_boolSupport_ne_half","module":"Erdos249257/HalfCarryReachability.lean","line":589,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfCarryReachability.lean#L589","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfCarryReachability.lean#L589","kind":"theorem","signature":"theorem finite_boolSupport_ne_half (A : Set ℕ) (hfinite : A.Finite) (hzero : 0 ∉ A) : erdosSupportSeries 2 A ≠ (1 : ℝ) / 2","docstring":"A finite positive-index support cannot have support-series value `1/2`, because its audited reduced denominator is odd.","generated_certificate":false}],"paper_label":"res:greedy","paper_anchor":{"paper_label":"res:greedy","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":805,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":805,"anchor_title":"strict-tail greedy coding","anchor_title_tex":"strict-tail greedy coding","section_title":"The base-2 achievement set","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"The base-2 achievement set","title_tex":"The base-$2$ achievement set","line":780}],"source_excerpt":{"text":"\\begin{theorem}[strict-tail greedy coding]\\label{res:greedy}\nFor every \\(n\\geq1\\),\n\\[\n T_n<w_n.\n\\]\nHence every \\(x\\in\\mathcal A\\)\nhas a unique support by positive exponents (the ``normalised'' support in the\nformal source).\nMoreover, a real number \\(x\\) lies in \\(\\mathcal A\\) if and only if \\(x\\geq0\\)\nand \\(r_n(x)\\leq T_n\\) for every \\(n\\geq0\\).  In that case\n\\(x=\\sum_{n\\geq1}\\varepsilon_n(x)w_n\\).\n\\end{theorem}","media_type":"text/x-tex","source_char_start":41921,"source_char_end":42336,"start_line":805,"end_line":816,"truncated":false,"sha256":"sha256:7c3a1b5d8d04dd5fd90a915fc9d0bb146df1fc5dd55c159e2e5a4f6dfab383a7","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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No priority claim is made for the sign separation.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/4","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"known_classical":2,"not_assessed":1,"routine_corollary_or_specialisation":3},"edges":[{"semantic_node_id":"Z12::finite_support_never_half","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z12::greedy_survival_characterisation","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z12::half_membership_iff_infinite_greedy_skips","prior_art_state":"not_assessed"},{"semantic_node_id":"Z12::mersenne_achievement_set_cantor_geometry","prior_art_state":"known_classical"},{"semantic_node_id":"Z12::mersenne_achievement_set_measure_one","prior_art_state":"known_classical"},{"semantic_node_id":"Z12::three_fourths_excluded","prior_art_state":"routine_corollary_or_specialisation"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:greedy","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L805","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:half_greedy_two_thirds_band","assertion_kind":"registered_claim","source_id":"half_greedy_two_thirds_band","reviewed":true,"granularity":"single_registered_claim","label":"Exact band localization of the final skip between two takes","statement":"Suppose a take at rank b is followed by the next take at c > b+1, put q = 2^b − 1 and m = 2^(c−1), and write the pre-take residual as 1/R with 0 < R < q. The final intervening skip is dyadically unsafe exactly when q(m−1)/(q+m−1) < R < qm/(q+m), an interval of exact width q^2/((q+m)(q+m−1)). For one intervening skip this specializes to the two-thirds band (q(2q+1)/(3q+1), 2q(q+1)/(3q+2)), whose width is below 1/9; integral reciprocals never lie there, and the corresponding positive odd numerator is at least 7. These are local constraints: no theorem here says that the actual greedy orbit for 1/2 avoids a band or identifies the surviving last-producer cells with unsafe skips.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":["erdos257_half_story"],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"postTakeUnsafeAt_iff_band","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":88,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfGreedyTwoThirdsBand.lean#L88","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfGreedyTwoThirdsBand.lean#L88","kind":"theorem","signature":"theorem postTakeUnsafeAt_iff_band {R q m : ℚ} (hR : 0 < R) (hRq : R < q) (hm : 1 ≤ m) : PostTakeUnsafeAt R q m ↔ Band R q m","docstring":"**Band localisation, general form.** Under a take at `b` (`0 < R < q`) and a last-skip threshold `m ≥ 1`, the last skip is dyadically unsafe if and only if the pre-take reciprocal lies in the band.","generated_certificate":false},{"name":"band_width_general","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":104,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfGreedyTwoThirdsBand.lean#L104","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfGreedyTwoThirdsBand.lean#L104","kind":"theorem","signature":"theorem band_width_general {q m : ℚ} (hq : 0 < q) (hm : 1 ≤ m) : q * m / (q + m) - q * (m - 1) / (q + m - 1) = q ^ 2 / ((q + m) * (q + m - 1))","docstring":"**The general band has width exactly `q²/((q+m)(q+m-1))`.**","generated_certificate":false},{"name":"singletonUnsafe_iff_twoThirdsBand","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":127,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfGreedyTwoThirdsBand.lean#L127","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfGreedyTwoThirdsBand.lean#L127","kind":"theorem","signature":"theorem singletonUnsafe_iff_twoThirdsBand {R q : ℚ} (hR : 0 < R) (hRq : R < q) : SingletonUnsafe R q ↔ TwoThirdsBand R q","docstring":"**Band localisation, single-skip run.** Under a take at `b` (`0 < R < q`), the single intervening skip is unsafe if and only if the pre-take reciprocal lies in the two-thirds band. This is `postTakeUnsafeAt_iff_band` at `m = 2q + 2`.","generated_certificate":false},{"name":"band_width","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":154,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfGreedyTwoThirdsBand.lean#L154","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfGreedyTwoThirdsBand.lean#L154","kind":"theorem","signature":"theorem band_width {q : ℚ} (hq : 0 < q) : 2 * q * (q + 1) / (3 * q + 2) - q * (2 * q + 1) / (3 * q + 1) = q ^ 2 / ((3 * q + 1) * (3 * q + 2))","docstring":"**The band has width exactly `q²/((3q+1)(3q+2))`.**","generated_certificate":false},{"name":"band_width_lt_ninth","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":163,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfGreedyTwoThirdsBand.lean#L163","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfGreedyTwoThirdsBand.lean#L163","kind":"theorem","signature":"theorem band_width_lt_ninth {q : ℚ} (hq : 0 < q) : q ^ 2 / ((3 * q + 1) * (3 * q + 2)) < 1 / 9","docstring":"The band width stays below `1/9`, uniformly in `q`.","generated_certificate":false},{"name":"not_twoThirdsBand_of_int","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":185,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfGreedyTwoThirdsBand.lean#L185","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfGreedyTwoThirdsBand.lean#L185","kind":"theorem","signature":"theorem not_twoThirdsBand_of_int {R q : ℚ} (hq : 0 < q) (m n : ℤ) (hRm : R = (m : ℚ)) (hqn : q = (n : ℚ)) : ¬ TwoThirdsBand R q","docstring":"**Integral reciprocals are safe.** If `R` and `q` are integers, the singleton skip cannot be unsafe. This covers a unit residual `rem = 1/(2D)`, for which `R = 2D`.","generated_certificate":false},{"name":"seven_le_of_intBand_odd","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":231,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/HalfGreedyTwoThirdsBand.lean#L231","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/HalfGreedyTwoThirdsBand.lean#L231","kind":"theorem","signature":"theorem seven_le_of_intBand_odd {p D q : ℤ} (hp : 0 < p) (hD : 0 < D) (hq : 0 < q) (hpodd : Odd p) (hDodd : Odd D) (hqodd : Odd q) (hband : IntBand p D q) : 7 ≤ p","docstring":"**The 2-adic bound.** For odd positive `p`, `D`, `q`, an unsafe singleton skip forces `p ≥ 7`; equivalently, `p ∈ {1, 3, 5}` is always safe. The bound is sharp for these hypotheses: odd triples with `p = 7` do lie in the band. Reachability by an actual greedy orbit is a separate question and is not addressed here.","generated_certificate":false}],"paper_label":"res:twothirdsband","paper_anchor":{"paper_label":"res:twothirdsband","owner_source":"paper/archive/erdos249-257-main-paper.tex","source_line":1361,"anchor_kind":"sectioning","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"Exact band for a final skip between two takes","section_path":[{"level":"section","title":"Erdős--Borwein-type irrationality (the #257 direction)","title_tex":"Erd\\H{o}s--Borwein-type irrationality (the \\#257 direction)","line":598},{"level":"subsection","title":"Exact band for a final skip between two takes","title_tex":"Exact band for a final skip between two takes","line":1360}],"source_excerpt":{"text":"\\subsection{Exact band for a final skip between two takes}\n\\label{res:twothirdsband}\n\\label{app:half-localizer}","media_type":"text/x-tex","source_char_start":69076,"source_char_end":69187,"start_line":1360,"end_line":1362,"truncated":false,"sha256":"sha256:b7b33bd4274fc5c075320b75403f3563cd4c56aae620070677fb29c8e759a30a","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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The infinite producer can therefore start at `n = 7`, where the observed margin is already large.","generated_certificate":false},{"name":"half_mem_mersenneAchievementSet_of_secondChannelSeparationRat_from_seven","module":"Erdos249257/GreedyAchievementSet.lean","line":3151,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GreedyAchievementSet.lean#L3151","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GreedyAchievementSet.lean#L3151","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_of_secondChannelSeparationRat_from_seven (hseparate : ∀ n : ℕ, 7 ≤ n → HalfSecondChannelSeparatedRat n) : (1 / 2 : ℝ) ∈ mersenneAchievementSet","docstring":"Proof-search form of the half-membership consumer: all levels through six are checked exactly, and the remaining hypothesis is a decidable statement about the rational greedy orbit only.","generated_certificate":false},{"name":"half_mem_mersenneAchievementSet_of_skipped_twoChannelCap","module":"Erdos249257/GreedyAchievementSet.lean","line":1425,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GreedyAchievementSet.lean#L1425","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GreedyAchievementSet.lean#L1425","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_of_skipped_twoChannelCap (hskip : ∀ n : ℕ, ¬ mersenneWeight (n + 1) ≤ greedyMersenneRemainder (1 / 2 : ℝ) n → greedyMersenneRemainder (1 / 2 : ℝ) n ≤ halfTwoChannelCap (n + 1)) : (1 / 2 : ℝ) ∈ mersenneAchievementSet","docstring":"It is enough to establish the two-channel cap only when the actual greedy orbit skips the next exponent. Selected branches and the final membership composition are discharged here.","generated_certificate":false},{"name":"half_mem_mersenneAchievementSet_iff_greedySkippedSupport_infinite","module":"Erdos249257/GreedyAchievementSet.lean","line":2583,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/GreedyAchievementSet.lean#L2583","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/GreedyAchievementSet.lean#L2583","kind":"theorem","signature":"theorem half_mem_mersenneAchievementSet_iff_greedySkippedSupport_infinite : (1 / 2 : ℝ) ∈ mersenneAchievementSet ↔ (greedyMersenneSkippedSupport (1 / 2 : ℝ)).Infinite","docstring":"**Greedy skip dichotomy for the half target.** Exact half-membership is equivalent to the canonical greedy orbit omitting infinitely many exponents. 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Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:campbell_shift_synchronization_boundary","assertion_kind":"registered_claim","source_id":"campbell_shift_synchronization_boundary","reviewed":true,"granularity":"single_registered_claim","label":"Campbell shift-synchronization strategy boundary","statement":"Campbell's application of the Alford-Granville-Pomerance estimate chooses delta = 1/4, so at prime-search height X the progression modulus d must satisfy d <= X^(1/4), equivalently d^4 <= X. In the proposed last-skip synchronization, P is the fatal-window prime cap, t the unavoidable phase period, and d the combined progression modulus. Taking X = P, the requirements 2 <= d, t <= d, P <= 4t, and d^4 <= P are inconsistent. The shifted-zero interface is guarded by a putative last skip and is exactly equivalent to infinitely many actual greedy skips and to membership of 1/2 in the Mersenne achievement set. It is an endpoint restatement, not an independent Campbell input, and proves no half-membership.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"gateway_section","warrant":"the claim's anchor sits inside the retired gateway section 'Erdős--Borwein-type irrationality (the #257 direction)', which is the author's own partition of that manuscript by problem"},"problem_edges":[{"problem_id":"erdos_257","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-257-main-paper","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"campbellQuarterExponent_periodFreeze_noSync","module":"Erdos249257/CampbellShiftSynchronization.lean","line":299,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CampbellShiftSynchronization.lean#L299","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CampbellShiftSynchronization.lean#L299","kind":"theorem","signature":"theorem campbellQuarterExponent_periodFreeze_noSync {d t P : ℕ} (hd : 2 ≤ d) (htd : t ≤ d) (hP : P ≤ 4 * t) (hAGP : d ^ 4 ≤ P) : False","docstring":"If `d` is the combined prime-progression modulus, `t` is the unavoidable phase factor, and `P` is the prime cap forced by the fatal window, Campbell's quarter-exponent applicability condition is impossible.","generated_certificate":false},{"name":"campbellPhaseFreeze_parameters_impossible","module":"Erdos249257/CampbellShiftSynchronization.lean","line":320,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CampbellShiftSynchronization.lean#L320","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CampbellShiftSynchronization.lean#L320","kind":"theorem","signature":"theorem campbellPhaseFreeze_parameters_impossible {M q0sq t d P : ℕ} (hq : 0 < q0sq) (htpos : 0 < t) (hd : 2 ≤ d) (ht : M ≤ t * q0sq) (htd : t ≤ d) (hwindow : P * q0sq ≤ 2 * M + 2) (hAGP : d ^ 4 ≤ P) : False","docstring":"Substrate-native parameter map. The hypotheses expose exactly where a future analytic construction must improve: the phase-period factor `t`, the fatal-window cap `P`, or the sublinear modulus exponent.","generated_certificate":false},{"name":"shiftWindowZero_iff_greedyMersenneSkippedSupport_infinite","module":"Erdos249257/CampbellShiftSynchronization.lean","line":471,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CampbellShiftSynchronization.lean#L471","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CampbellShiftSynchronization.lean#L471","kind":"theorem","signature":"theorem shiftWindowZero_iff_greedyMersenneSkippedSupport_infinite : GreedyShiftWindowZero ↔ (greedyMersenneSkippedSupport (1 / 2 : ℝ)).Infinite","docstring":"Exact endpoint classification of the Campbell-shift interface. 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Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2},"edges":[{"semantic_node_id":"Z12::campbell_agp_parameter_barrier","prior_art_state":"not_assessed"},{"semantic_node_id":"Z12::shift_window_zero_endpoint_restatement","prior_art_state":"not_assessed"}]}},"historical_anchor_fidelity":{"anchor_role":"selected_historical_owner","paper_id":"erdos249-257-main-paper","paper_label":"res:campbellboundary","owner_source":"paper/archive/erdos249-257-main-paper.tex","owner_status":"retired_but_public","relationship_to_formal_source":"historical_compatible_or_weaker","render_policy":"historical_only","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/archive/erdos249-257-main-paper.tex#L2233","finding_axes":[],"why":"Compared this preserved manuscript's exact source excerpt against the same Lean signature and docstring as the selected claim row. Historical preservation does not make the manuscript authoritative."},"source_lane":"registry_claim"},{"assertion_id":"claim:sigma_transcendence","assertion_kind":"registered_claim","source_id":"sigma_transcendence","reviewed":true,"granularity":"single_registered_claim","label":"Transcendence of ∑ σ(m)/2^m","statement":"Nesterenko (1996); the ladder rung L(Id). Cited for positioning, not formalised here.","status":"cited only","status_meaning":"External theorem used for positioning, not formalised","readme_headline":false,"problem_ids":["erdos_249","erdos_257"],"problem_erdos_numbers":[249,257],"problem_binding":{"method":"core_programme_joint","warrant":"this claim belongs to the reviewed Erdos249257 core library, which is the joint #249/#257 programme. No source surface -- not its remaining-open propositions, not its anchor's enclosing gateway section, not the orientation programme map -- assigns it to one rather than both. Both are listed because both are correct, not because the binding is unknown.","separability":"not_separable_from_published_source"},"problem_edges":[{"problem_id":"erdos_249","relation":"shared_programme_context","basis":"problem_binding"},{"problem_id":"erdos_257","relation":"shared_programme_context","basis":"problem_binding"}],"programme_ids":["joint_249_257_gateway"],"publication_edges":[],"evidence_authority":"registry_statement_only","evidence_resolution":"statement_only","declarations":[],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[],"links":[],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/107","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"sigma_transcendence"}}],"expect":"object"},"prior_art":{"family_id":"mersenne_lambert_classical_ladder","status_summary":"Classical identities formalised and one transcendental rung cited only.","prior_art_posture":"Apostol, Merca, and Merca–Schmidt own the transform background; Nesterenko owns the cited transcendence theorem.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/11","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex"},"source_lane":"registry_claim"},{"assertion_id":"claim:squarefree_support_irrationality","assertion_kind":"registered_claim","source_id":"squarefree_support_irrationality","reviewed":true,"granularity":"single_registered_claim","label":"Irrationality of squarefree-support series at power-of-two bases","statement":"Duverney–Tachiya Corollary 1.2 and Example 1.1 prove linear independence of 1 and the squarefree Lambert sums at bases 2^j (j≥1), including n=1; subtracting each rational n=1 term settles the support {n≥2:n squarefree} at every power-of-two base. Their condition |q|L≤s, with L=lcm(1,…,ℓ), here permits q=2 but does not cover integer bases that are not powers of two. This result is not formalised here.","status":"cited only","status_meaning":"External theorem used for positioning, not formalised","readme_headline":false,"problem_ids":["erdos_249","erdos_257"],"problem_erdos_numbers":[249,257],"problem_binding":{"method":"core_programme_joint","warrant":"this claim belongs to the reviewed Erdos249257 core library, which is the joint #249/#257 programme. No source surface -- not its remaining-open propositions, not its anchor's enclosing gateway section, not the orientation programme map -- assigns it to one rather than both. Both are listed because both are correct, not because the binding is unknown.","separability":"not_separable_from_published_source"},"problem_edges":[{"problem_id":"erdos_249","relation":"shared_programme_context","basis":"problem_binding"},{"problem_id":"erdos_257","relation":"direct_special_case","basis":"source_family"}],"programme_ids":["joint_249_257_gateway"],"publication_edges":[],"evidence_authority":"registry_statement_only","evidence_resolution":"statement_only","declarations":[],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[],"links":[],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/108","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"squarefree_support_irrationality"}}],"expect":"object"},"prior_art":{"family_id":"classical_full_support_and_named_257_families","status_summary":"Classical formalisation, cited special cases, and the open universal target.","prior_art_posture":"Erdős owns full support and the pairwise-coprime family; Duverney–Tachiya own squarefree-support irrationality at every power-of-two base; Tao–Teräväinen own the base-2 prime-support theorem. The cited squarefree theorem does not cover integer bases that are not powers of two, and the cited prime-power extension is remark-only. No priority claim is made for the formalised families.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/0","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. 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No source surface -- not its remaining-open propositions, not its anchor's enclosing gateway section, not the orientation programme map -- assigns it to one rather than both. Both are listed because both are correct, not because the binding is unknown.","separability":"not_separable_from_published_source"},"problem_edges":[{"problem_id":"erdos_249","relation":"shared_programme_context","basis":"problem_binding"},{"problem_id":"erdos_257","relation":"direct_special_case","basis":"source_family"}],"programme_ids":["joint_249_257_gateway"],"publication_edges":[],"evidence_authority":"registry_statement_only","evidence_resolution":"statement_only","declarations":[],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. 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The cited squarefree theorem does not cover integer bases that are not powers of two, and the cited prime-power extension is remark-only. No priority claim is made for the formalised families.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/0","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. 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Every supply hypothesis remains open.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"claim_prose"}],"programme_ids":["transport_curvature_programme"],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"irrational_totientSeries_of_sharpCurvatureSupply","module":"Erdos249257/CurvatureCarry.lean","line":206,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CurvatureCarry.lean#L206","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CurvatureCarry.lean#L206","kind":"theorem","signature":"theorem irrational_totientSeries_of_sharpCurvatureSupply (hsupply : SharpCurvatureSupply) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"The sharp curvature supply is a complete proof socket for #249.","generated_certificate":false},{"name":"irrational_totient_series_of_exponentOnlyThreeTransportSupply","module":"Erdos249257/ExponentOnlyTransport.lean","line":205,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/ExponentOnlyTransport.lean#L205","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ExponentOnlyTransport.lean#L205","kind":"theorem","signature":"theorem irrational_totient_series_of_exponentOnlyThreeTransportSupply (hsupply : ExponentOnlyThreeTransportSupply) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"The fixed-three transport supply contradicts rational LCM-cone flatness and therefore proves irrationality of the totient series.","generated_certificate":false},{"name":"irrational_totient_series_of_anchoredJoint35ConeSupply","module":"Erdos249257/JointExponentTransport.lean","line":217,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/JointExponentTransport.lean#L217","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/JointExponentTransport.lean#L217","kind":"theorem","signature":"theorem irrational_totient_series_of_anchoredJoint35ConeSupply (hsupply : AnchoredJoint35ConeSupply) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"An unbounded joint certificate supply contradicts rational LCM-cone flatness and proves irrationality of the totient series.","generated_certificate":false},{"name":"irrational_totient_series_of_primeJumpSharpKill_supply","module":"Erdos249257/PrimeJumpWindow.lean","line":193,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PrimeJumpWindow.lean#L193","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimeJumpWindow.lean#L193","kind":"theorem","signature":"theorem irrational_totient_series_of_primeJumpSharpKill_supply (hsupply : ∀ t₀ : ℕ, ∃ t, t₀ ≤ t ∧ ∃ p L : ℕ, 0 < p ∧ primeJumpSharpKill (periodLcm t) p L) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"An unbounded supply of sharp prime-jump window kills on the LCM cone would prove Erdős #249. This is the exact remaining consumer socket; the supply is not asserted here.","generated_certificate":false},{"name":"irrational_totientSeries_of_roughnessBoundaryLatticeSupply","module":"Erdos249257/ThreeTransportBoundary.lean","line":177,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/ThreeTransportBoundary.lean#L177","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ThreeTransportBoundary.lean#L177","kind":"theorem","signature":"theorem irrational_totientSeries_of_roughnessBoundaryLatticeSupply (hsupply : RoughnessBoundaryLatticeSupply) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"The boundary-certificate supply contradicts eventual LCM-cone flatness.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/3"}],"links":[{"kind":"lean_declaration_pinned","label":"irrational_totientSeries_of_sharpCurvatureSupply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CurvatureCarry.lean#L206","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totient_series_of_exponentOnlyThreeTransportSupply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ExponentOnlyTransport.lean#L205","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totient_series_of_anchoredJoint35ConeSupply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/JointExponentTransport.lean#L217","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totient_series_of_primeJumpSharpKill_supply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimeJumpWindow.lean#L193","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totientSeries_of_roughnessBoundaryLatticeSupply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ThreeTransportBoundary.lean#L177","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/110","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"transport_curvature_reductions"}}],"expect":"object"},"prior_art":{"family_id":"transport_curvature_finite_difference_architecture","status_summary":"Exact affine annihilation, conditional finite-difference consumers, and two verified windows.","prior_art_posture":"Finite differences and affine moments are general techniques; no matching combined theorem was identified, and that absence is not a novelty claim.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/18","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":5},"edges":[{"semantic_node_id":"Z09::curvature_supply_implies_249","prior_art_state":"not_assessed"},{"semantic_node_id":"Z09::exponent_only_three_transport_supply_implies_249","prior_art_state":"not_assessed"},{"semantic_node_id":"Z09::joint35_supply_implies_249","prior_art_state":"not_assessed"},{"semantic_node_id":"Z09::prime_jump_supply_implies_249","prior_art_state":"not_assessed"},{"semantic_node_id":"Z09::three_transport_supply_implies_249","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:affine_transport_annihilation","assertion_kind":"registered_claim","source_id":"affine_transport_annihilation","reviewed":true,"granularity":"single_registered_claim","label":"Affine residue state and old-channel annihilation","statement":"The exact Möbius residue kernel has a periodic affine slope-and-intercept state. First-order exponent transport and every coefficient system with vanishing zeroth and first moments annihilate all divisor channels already present at the base height; a prime jump additionally has an explicit migrated-channel correction.","status":"proved here","status_meaning":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"programme_representative","warrant":"orientation.json lists this claim among the representative claims of a mathematical programme whose remaining open propositions all name one problem"},"problem_edges":[{"problem_id":"erdos_249","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":["transport_curvature_programme"],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"transportResidueKernel_eq_affineState","module":"Erdos249257/ExponentOnlyTransport.lean","line":141,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/ExponentOnlyTransport.lean#L141","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ExponentOnlyTransport.lean#L141","kind":"theorem","signature":"theorem transportResidueKernel_eq_affineState {d N : ℕ} (hd : 0 < d) : transportResidueKernel d N = (N : ℝ) * residueSlope d N + residueIntercept d N","docstring":"The exact kernel is affine in the scale coordinate and periodic in the residue coordinate. This is the state omitted by a residue-permutation-only transport matrix.","generated_certificate":false},{"name":"oldChannel_affine_moment_annihilation","module":"Erdos249257/JointExponentTransport.lean","line":36,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/JointExponentTransport.lean#L36","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/JointExponentTransport.lean#L36","kind":"theorem","signature":"theorem oldChannel_affine_moment_annihilation {ι : Type*} [Fintype ι] (m : ι → ℕ) (c : ι → ℝ) (hzero : ∑ i, c i = 0) (hfirst : ∑ i, c i * (m i : ℝ) = 0) {d H : ℕ} (hd : 0 < d) (hdH : d ∣ H) : ∑ i, c i * transportResidueKernel d (m i * H) = 0","docstring":"Any finite real combination with zero constant and first moments kills an old affine residue channel. This is independent of the particular `(3,5)` operator.","generated_certificate":false},{"name":"residueHeckeDefect_eq_postJump_add_migration","module":"Erdos249257/PrimeJumpMigration.lean","line":38,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PrimeJumpMigration.lean#L38","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimeJumpMigration.lean#L38","kind":"theorem","signature":"theorem residueHeckeDefect_eq_postJump_add_migration (d H p : ℕ) : residueHeckeDefect d H p = postJumpResidueChannel d H p + preJumpMigrationChannel d H p","docstring":"The actual transition channel is post-jump contribution plus same-channel migration.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"transportResidueKernel_eq_affineState","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ExponentOnlyTransport.lean#L141","cite_this":true},{"kind":"lean_declaration_pinned","label":"oldChannel_affine_moment_annihilation","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/JointExponentTransport.lean#L36","cite_this":true},{"kind":"lean_declaration_pinned","label":"residueHeckeDefect_eq_postJump_add_migration","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimeJumpMigration.lean#L38","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/111","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"affine_transport_annihilation"}}],"expect":"object"},"prior_art":{"family_id":"transport_curvature_finite_difference_architecture","status_summary":"Exact affine annihilation, conditional finite-difference consumers, and two verified windows.","prior_art_posture":"Finite differences and affine moments are general techniques; no matching combined theorem was identified, and that absence is not a novelty claim.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/18","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1,"routine_corollary_or_specialisation":2},"edges":[{"semantic_node_id":"Z09::old_channel_moment_annihilation","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z09::prime_jump_channel_migration","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z09::residue_kernel_affine_state","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:transport_certificate_instances","assertion_kind":"registered_claim","source_id":"transport_certificate_instances","reviewed":true,"granularity":"single_registered_claim","label":"Verified transport-window certificates","statement":"Lean verifies the sharp prime-jump certificate at (H,p,L)=(12,5,15) and the balanced three-transport certificate at (H,L)=(60,12). These bounded instances do not establish any unbounded supply.","status":"verified finite instance","status_meaning":"Kernel-checked computation at a bounded range","readme_headline":false,"problem_ids":["erdos_249","erdos_257"],"problem_erdos_numbers":[249,257],"problem_binding":{"method":"core_programme_joint","warrant":"this claim belongs to the reviewed Erdos249257 core library, which is the joint #249/#257 programme. No source surface -- not its remaining-open propositions, not its anchor's enclosing gateway section, not the orientation programme map -- assigns it to one rather than both. Both are listed because both are correct, not because the binding is unknown.","separability":"not_separable_from_published_source"},"problem_edges":[{"problem_id":"erdos_249","relation":"shared_programme_context","basis":"problem_binding"},{"problem_id":"erdos_257","relation":"shared_programme_context","basis":"problem_binding"}],"programme_ids":["joint_249_257_gateway"],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"primeJumpSharpKill_twelve_five","module":"Erdos249257/PrimeJumpWindow.lean","line":186,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PrimeJumpWindow.lean#L186","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimeJumpWindow.lean#L186","kind":"theorem","signature":"theorem primeJumpSharpKill_twelve_five : primeJumpSharpKill 12 5 15","docstring":"A concrete new-prime-jump deposit.","generated_certificate":false},{"name":"sharpThreeTransportCert_60_12","module":"Erdos249257/ThreeTransportBoundary.lean","line":203,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/ThreeTransportBoundary.lean#L203","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ThreeTransportBoundary.lean#L203","kind":"theorem","signature":"theorem sharpThreeTransportCert_60_12 : sharpThreeTransportCert 60 12","docstring":"Kernel-checked finite lattice separation at `t = 6`, `H = 60`.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"primeJumpSharpKill_twelve_five","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PrimeJumpWindow.lean#L186","cite_this":true},{"kind":"lean_declaration_pinned","label":"sharpThreeTransportCert_60_12","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/ThreeTransportBoundary.lean#L203","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/112","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"transport_certificate_instances"}}],"expect":"object"},"scope_limit":{"text":"Exactly the prime-jump parameter triple (H,p,L)=(12,5,15) and the balanced three-transport pair (H,L)=(60,12).","source":"claims_registry_bounded_domain"},"prior_art":{"family_id":"transport_curvature_finite_difference_architecture","status_summary":"Exact affine annihilation, conditional finite-difference consumers, and two verified windows.","prior_art_posture":"Finite differences and affine moments are general techniques; no matching combined theorem was identified, and that absence is not a novelty claim.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/18","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"formalisation_only":2},"edges":[{"semantic_node_id":"Z09::prime_jump_finite_certificate_12_5_15","prior_art_state":"formalisation_only"},{"semantic_node_id":"Z09::three_transport_finite_certificate_60_12","prior_art_state":"formalisation_only"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:first_harmonic_certificate_interface","assertion_kind":"registered_claim","source_id":"first_harmonic_certificate_interface","reviewed":true,"granularity":"single_registered_claim","label":"First-harmonic gap implies a finite certificate","statement":"Under the explicit dyadic room inequality, a constant-saving first-harmonic bound on one block, or on any nonempty finite set of indices below its right endpoint, forces a certified non-integrality witness in that set. No cofinal harmonic estimate is asserted.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"exists_certifiedKill_of_first_harmonic_gap_subset","module":"Erdos249257/FirstHarmonicGap.lean","line":104,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/FirstHarmonicGap.lean#L104","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicGap.lean#L104","kind":"theorem","signature":"theorem exists_certifiedKill_of_first_harmonic_gap_subset {h X L : ℕ} (T : Finset ℕ) (hTlt : ∀ N ∈ T, N < 2 * X) (hTne : T.Nonempty) (hroom : 16 * (2 * X + h + L + 2) ≤ 2 ^ L) (hgap : (∑ N ∈ T, windowFirstCos h N L) ≤ (9 / 10 : ℝ) * T.card) : ∃ N ∈ T, certifiedKill h N L","docstring":"A first-harmonic saving on any nonempty finite subset of a dyadic block forces a certificate in that subset. This is the subset consumer needed by supplier-fibre arguments; no density or partition hypothesis is used.","generated_certificate":false},{"name":"exists_certifiedKill_of_first_harmonic_gap","module":"Erdos249257/FirstHarmonicGap.lean","line":128,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/FirstHarmonicGap.lean#L128","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicGap.lean#L128","kind":"theorem","signature":"theorem exists_certifiedKill_of_first_harmonic_gap {h X L : ℕ} (hX : 0 < X) (hroom : 16 * (2 * X + h + L + 2) ≤ 2 ^ L) (hgap : (∑ N ∈ Finset.Ico X (2 * X), windowFirstCos h N L) ≤ (9 / 10 : ℝ) * X) : ∃ N ∈ Finset.Ico X (2 * X), certifiedKill h N L","docstring":"A constant-saving first-harmonic gap on one dyadic block forces a finite certificate somewhere in that block. Establishing such gaps cofinally is the analytic input still required for #249.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/3"}],"links":[{"kind":"lean_declaration_pinned","label":"exists_certifiedKill_of_first_harmonic_gap_subset","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicGap.lean#L104","cite_this":true},{"kind":"lean_declaration_pinned","label":"exists_certifiedKill_of_first_harmonic_gap","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicGap.lean#L128","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/113","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"first_harmonic_certificate_interface"}}],"expect":"object"},"prior_art":{"family_id":"harmonic_and_phase_reductions","status_summary":"Conditional harmonic and phase consumers, including an exact four-term pivot decomposition and budget, an exact window-pair equivalence, and the two-branch tail-orbit phase-density and natural-prime strict-gap producers that feed the same certificate socket.","prior_art_posture":"No matching combined theorem is identified; all novelty language remains provisional.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/19","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z08::first_harmonic_saving_forces_certificate","prior_art_state":"routine_corollary_or_specialisation"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:first_harmonic_pivot_decomposition","assertion_kind":"registered_claim","source_id":"first_harmonic_pivot_decomposition","reviewed":true,"granularity":"single_registered_claim","label":"Complex first-harmonic pivot decomposition","statement":"For each positive cofactor m at most sqrt(X)/2, the canonical supplier fibre is exactly the injective image of the primes p in the shifted product interval under N = mp - (L-s+1). This exact bijection does not make a supplier prime globally isolated: at X=16, L=20, s=1, m=2, N=18 the pivot is 38=2*19, while the same prime 19 divides the distinct window argument N+1=19. Removing the pivot phase gives an exact decomposition of the block first-harmonic sum into a centred supplier-fibre correlation, a supplier-fibre mean term, a bad-supplier term, and a non-supplier term. At a positive scale satisfying the explicit dyadic-room inequality, if the real part of the centred term is at most 14X/25 and the other three norms are at most X/100, X/100, and 8X/25, then the real first-harmonic sum is at most 9X/10 and yields a finite certificate. A cofinal supply of these budgets implies irrationality of S. No global supplier-prime isolation, prime-distribution estimate, or decorrelation estimate supplying the budgets is proved.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"mem_pivotFiber_iff_exists_supplierPrime","module":"Erdos249257/FirstHarmonicPivot.lean","line":299,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/FirstHarmonicPivot.lean#L299","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L299","kind":"theorem","signature":"theorem mem_pivotFiber_iff_exists_supplierPrime {X L s m N : ℕ} (hm : 0 < m) (hmsmall : m ≤ Nat.sqrt X / 2) : N ∈ pivotFiber X L s m ↔ ∃ p ∈ pivotSupplierPrimes X L s m, pivotBaseOfPrime L s m p = N","docstring":"Exact membership form of the supplier-fibre parametrisation. For a positive cofactor `m ≤ sqrt(X)/2`, a prime `p` in the shifted product interval maps to `N = m*p - pivotOffset L s`. Conversely every member of the canonical largest-prime supplier fibre arises uniquely this way.","generated_certificate":false},{"name":"pivotBaseOfPrime_injective_on_supplierPrimes","module":"Erdos249257/FirstHarmonicPivot.lean","line":367,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/FirstHarmonicPivot.lean#L367","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L367","kind":"theorem","signature":"theorem pivotBaseOfPrime_injective_on_supplierPrimes {X L s m p q : ℕ} (hm : 0 < m) (hp : p ∈ pivotSupplierPrimes X L s m) (hq : q ∈ pivotSupplierPrimes X L s m) (heq : pivotBaseOfPrime L s m p = pivotBaseOfPrime L s m q) : p = q","docstring":"The prime-to-base map is injective on the exact supplier interval.","generated_certificate":false},{"name":"image_pivotSupplierPrimes_eq_pivotFiber","module":"Erdos249257/FirstHarmonicPivot.lean","line":384,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/FirstHarmonicPivot.lean#L384","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L384","kind":"theorem","signature":"theorem image_pivotSupplierPrimes_eq_pivotFiber {X L s m : ℕ} (hm : 0 < m) (hmsmall : m ≤ Nat.sqrt X / 2) : (pivotSupplierPrimes X L s m).image (pivotBaseOfPrime L s m) = pivotFiber X L s m","docstring":"Set-level form of the exact bijection between supplier primes and one canonical fibre. Together with `pivotBaseOfPrime_injective_on_supplierPrimes`, this is a genuine bijection, not merely a surjective parametrisation.","generated_certificate":false},{"name":"supplierPrime_not_globally_isolated_counterexample","module":"Erdos249257/FirstHarmonicPivot.lean","line":394,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/FirstHarmonicPivot.lean#L394","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L394","kind":"theorem","signature":"theorem supplierPrime_not_globally_isolated_counterexample : pivotOffset 20 1 = 20 ∧ 18 ∈ pivotFiber 16 20 1 2 ∧ pivotPrime 18 20 1 = 19 ∧ 19 ∣ pivotArgument 18 20 1 ∧ 19 ∣ 18 + 1","docstring":"Supplier primes are not automatically isolated from all other arguments in the same window. Here the pivot is `38 = 2*19`, while the same prime also divides the distinct window argument `18+1 = 19`.","generated_certificate":false},{"name":"windowFirstExp_sum_eq_pivot_decomposition","module":"Erdos249257/FirstHarmonicPivot.lean","line":514,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/FirstHarmonicPivot.lean#L514","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L514","kind":"theorem","signature":"theorem windowFirstExp_sum_eq_pivot_decomposition (h X L s : ℕ) (η : ℝ) : (∑ N ∈ Finset.Ico X (2 * X), windowFirstExp h N L) = pivotCenteredCorrelation h X L s η + pivotFiberMeanContribution h X L s η + pivotBadContribution h X L s η + pivotNonSupplierContribution h X L s","docstring":"Exact decomposition into the four actual, evaluator-visible terms.","generated_certificate":false},{"name":"PivotBudgetAt","module":"Erdos249257/FirstHarmonicPivot.lean","line":543,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/FirstHarmonicPivot.lean#L543","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L543","kind":"def","signature":"def PivotBudgetAt (h X L s : ℕ) (η : ℝ) : Prop","docstring":"Unlike the earlier `X/2` complex-norm proposal, the consumer only needs a one-sided real bound. With the same other budgets, `14/25` is sufficient.","generated_certificate":false},{"name":"first_harmonic_gap_of_pivotBudgetAt","module":"Erdos249257/FirstHarmonicPivot.lean","line":549,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/FirstHarmonicPivot.lean#L549","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L549","kind":"theorem","signature":"theorem first_harmonic_gap_of_pivotBudgetAt {h X L s : ℕ} {η : ℝ} (hbudget : PivotBudgetAt h X L s η) : (∑ N ∈ Finset.Ico X (2 * X), windowFirstCos h N L) ≤ (9 / 10 : ℝ) * X","generated_certificate":false},{"name":"DTWPivotResidualDecorrelation","module":"Erdos249257/FirstHarmonicPivot.lean","line":569,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/FirstHarmonicPivot.lean#L569","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L569","kind":"def","signature":"def DTWPivotResidualDecorrelation : Prop","docstring":"The genuinely narrower cofinal socket: all four terms are fixed finite sums over canonical largest-prime supplier fibres. The remaining estimate is only `Re centred ≤ 14X/25`, not `‖centred‖ ≤ X/2`.","generated_certificate":false},{"name":"irrational_totient_series_of_pivotResidualDecorrelation","module":"Erdos249257/FirstHarmonicPivot.lean","line":577,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/FirstHarmonicPivot.lean#L577","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L577","kind":"theorem","signature":"theorem irrational_totient_series_of_pivotResidualDecorrelation (hmix : DTWPivotResidualDecorrelation) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/3"}],"links":[{"kind":"lean_declaration_pinned","label":"mem_pivotFiber_iff_exists_supplierPrime","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L299","cite_this":true},{"kind":"lean_declaration_pinned","label":"pivotBaseOfPrime_injective_on_supplierPrimes","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L367","cite_this":true},{"kind":"lean_declaration_pinned","label":"image_pivotSupplierPrimes_eq_pivotFiber","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L384","cite_this":true},{"kind":"lean_declaration_pinned","label":"supplierPrime_not_globally_isolated_counterexample","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L394","cite_this":true},{"kind":"lean_declaration_pinned","label":"windowFirstExp_sum_eq_pivot_decomposition","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L514","cite_this":true},{"kind":"lean_declaration_pinned","label":"PivotBudgetAt","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L543","cite_this":true},{"kind":"lean_declaration_pinned","label":"first_harmonic_gap_of_pivotBudgetAt","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L549","cite_this":true},{"kind":"lean_declaration_pinned","label":"DTWPivotResidualDecorrelation","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L569","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totient_series_of_pivotResidualDecorrelation","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/FirstHarmonicPivot.lean#L577","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/114","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"first_harmonic_pivot_decomposition"}}],"expect":"object"},"prior_art":{"family_id":"harmonic_and_phase_reductions","status_summary":"Conditional harmonic and phase consumers, including an exact four-term pivot decomposition and budget, an exact window-pair equivalence, and the two-branch tail-orbit phase-density and natural-prime strict-gap producers that feed the same certificate socket.","prior_art_posture":"No matching combined theorem is identified; all novelty language remains provisional.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/19","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z08::first_harmonic_socket_implies_249","prior_art_state":"not_assessed"},{"semantic_node_id":"Z08::pivot_budget_forces_first_harmonic_gap","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z08::pivot_four_term_decomposition","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:first_harmonic_good_index_reduction","assertion_kind":"registered_claim","source_id":"first_harmonic_good_index_reduction","reviewed":true,"granularity":"single_registered_claim","label":"Irrationality of the totient series from one first-harmonic bound on the good indices","statement":"At the minimal admissible depth with s = 26 and η = 1/1000, if for every h ≥ 1 there are arbitrarily large X at which the real part of the first-harmonic sum over the good indices (the assigned indices N in [X, 2X) whose cofactor m satisfies φ(m) ≥ m/1000) is at most 603X/1000, then ∑ φ(n)/2^n is irrational. For all large X more than 67X/100 indices are good, by the unconditional unassigned count and the Chebyshev excluded-cofactor count, and 603/1000 = (9/10)(67/100); the underlying interface asks only for nonempty sets of indices beyond every threshold whose average first cosine is at most 9/10 at a depth with room. The bound is open, sufficient and not known to be necessary. It needs neither the prime number theorem nor a bound on the group means, and it supersedes the prime-number-theorem route at 11X/20. No computation in the record bears on it.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-totient-reasoning-surface","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"irrational_totient_series_of_support_gap","module":"ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean","line":44,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean#L44","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean#L44","kind":"theorem","signature":"theorem irrational_totient_series_of_support_gap (hgap : ∀ h : ℕ, 0 < h → ∀ A : ℕ, ∃ X L : ℕ, ∃ T : Finset ℕ, 16 * (2 * X + h + L + 2) ≤ 2 ^ L ∧ T.Nonempty ∧ (∀ N ∈ T, A ≤ N ∧ N < 2 * X) ∧ (∑ N ∈ T, windowFirstCos h N L) ≤ (9 / 10 : ℝ) * T.card) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"**The certificate interface of the first-harmonic family.** For every `h ≥ 1` and every threshold `A`, some nonempty set of bases in `[A, 2X)` whose average first cosine is at most `9/10`, at a depth `L` with room, gives the irrationality of `∑ φ(n)/2^n`.","generated_certificate":false},{"name":"eventually_card_pivotGoodBases_gt","module":"ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean","line":75,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean#L75","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean#L75","kind":"theorem","signature":"theorem eventually_card_pivotGoodBases_gt (h : ℕ) : ∀ᶠ X : ℕ in atTop, (67 / 100 : ℝ) * X < ((pivotGoodBases X (minimalDepth h 26 X) 26 (1 / 1000 : ℝ)).card : ℝ)","docstring":"At the minimal depth, with `s = 26` and `η = 1/1000`, more than `67X/100` of the bases in `[X, 2X)` are good, for all large `X`: Chebyshev's bound leaves fewer than `X/100` bad bases and `prop_dickman` fewer than `8X/25` non-supplier bases.","generated_certificate":false},{"name":"irrational_totient_series_of_goodBase_gap","module":"ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean","line":100,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean#L100","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean#L100","kind":"theorem","signature":"theorem irrational_totient_series_of_goodBase_gap (hgap : ∀ h : ℕ, 0 < h → ∀ A : ℕ, ∃ X : ℕ, max A 1 ≤ X ∧ (∑ N ∈ pivotGoodBases X (minimalDepth h 26 X) 26 (1 / 1000 : ℝ), windowFirstExp h N (minimalDepth h 26 X)).re ≤ (603 / 1000 : ℝ) * X) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"**The irrationality of `∑ φ(n) / 2 ^ n` from a first-harmonic gap on the good bases, with no further input**: for every `h ≥ 1`, arbitrarily large `X` at which the real part of the first harmonic summed over the good bases (minimal depth, `s = 26`, `η = 1/1000`) is at most `603X/1000`.","generated_certificate":false}],"paper_label":"thm:goodbasegap","paper_anchor":{"paper_label":"thm:goodbasegap","owner_source":"paper/249/erdos249-totient-reasoning-surface.tex","source_line":1364,"anchor_kind":"inline","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"One bound on the good indices","section_path":[{"level":"section","title":"Conditions that would imply irrationality","title_tex":"Conditions that would imply irrationality","line":1113},{"level":"subsection","title":"One bound on the good indices","title_tex":"One bound on the good indices","line":1354}],"source_excerpt":{"text":"\\begin{thm}[Irrationality from one bound on the good indices]\n\\label{thm:goodbasegap}\n\\begin{enumerate}[label=(\\alph*),leftmargin=*]\n\\item Suppose that for every $h\\ge1$ and every $A$ there are $X$, $L$ with\n  $16(2X+h+L+2)\\le2^L$ and a nonempty finite set $T$ of integers in\n  $[A,2X)$ with\n  \\[\n   \\sum_{N\\in T}\\operatorname{Re}E(h,N,L)\\le\\tfrac{9}{10}\\,\\#T.\n  \\]\n  Then $S\\notin\\Q$\n  (\\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean\\#L44}{\\texttt{irrational\\_totient\\_series\\_of\\_support\\_gap}}).\n\\item Take $s=26$, $\\eta=1/1000$ and $L=L(X)$. Suppose that for every\n  $h\\ge1$ there are arbitrarily large $X$ with\n  \\[\n   \\operatorname{Re}\\sum_{N\\in\\mathcal G}E\\bigl(h,N,L(X)\\bigr)\n   \\le\\tfrac{603}{1000}X.\n  \\]\n  Then $S\\notin\\Q$\n  (\\href{https://github.com/wcook04/plectis-erdos/blob/24edbddbe2bd68e920327a701aad0b9dd0d69675/lean/ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean\\#L100}{\\texttt{irrational\\_totient\\_series\\_of\\_goodBase\\_gap}}).\n\\end{enumerate}\n\\end{thm}","media_type":"text/x-tex","source_char_start":67415,"source_char_end":68508,"start_line":1363,"end_line":1383,"truncated":false,"sha256":"sha256:5c64216be444988daef9c845f8c3380f84a2830c045c593a0dc3115d138e69bb","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos249-totient-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/249/erdos249-totient-reasoning-surface.tex#L1364","owner_problem_erdos_number":249,"owner_paper_id":"erdos249-totient-reasoning-surface","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos249-totient-reasoning-surface.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos249-totient-reasoning-surface.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/249/erdos249-totient-reasoning-surface.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/3"}],"links":[{"kind":"lean_declaration_pinned","label":"irrational_totient_series_of_support_gap","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean#L44","cite_this":true},{"kind":"lean_declaration_pinned","label":"eventually_card_pivotGoodBases_gt","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean#L75","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totient_series_of_goodBase_gap","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean#L100","cite_this":true},{"kind":"paper_pdf","label":"erdos249-totient-reasoning-surface","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos249-totient-reasoning-surface.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/115","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"first_harmonic_good_index_reduction"}}],"expect":"object"},"prior_art":{"family_id":"harmonic_and_phase_reductions","status_summary":"Conditional harmonic and phase consumers, including an exact four-term pivot decomposition and budget, an exact window-pair equivalence, and the two-branch tail-orbit phase-density and natural-prime strict-gap producers that feed the same certificate socket.","prior_art_posture":"No matching combined theorem is identified; all novelty language remains provisional.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/19","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:adjacent_phase_separation_reduction","assertion_kind":"registered_claim","source_id":"adjacent_phase_separation_reduction","reviewed":true,"granularity":"single_registered_claim","label":"Adjacent-pair phase separation reduction","statement":"At scales satisfying the explicit dyadic room inequality, a squared chordal separation of at least 19/25 between two adjacent first-harmonic phases in a dyadic block forces a certified non-integrality witness at one of the two indices. A cofinal supply of such adjacent pairs implies irrationality of the totient constant; no such supply is proved.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"exists_certifiedKill_of_phasePairSeparation","module":"Erdos249257/AdjacentPhaseSeparation.lean","line":258,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/AdjacentPhaseSeparation.lean#L258","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/AdjacentPhaseSeparation.lean#L258","kind":"theorem","signature":"theorem exists_certifiedKill_of_phasePairSeparation {h X L N M : ℕ} (hNM : N ≠ M) (hNlt : N < 2 * X) (hMlt : M < 2 * X) (hroom : 16 * (2 * X + h + L + 2) ≤ 2 ^ L) (hsep : (19 / 25 : ℝ) ≤ ‖windowFirstExp h N L - windowFirstExp h M L‖ ^ 2) : ∃ K ∈ ({N, M} : Finset ℕ), certifiedKill h K L","docstring":"Separation of any two distinct phases in the standard dyadic block forces a certificate at one member of the pair.","generated_certificate":false},{"name":"irrational_totient_series_of_adjacentPhaseSeparation","module":"Erdos249257/AdjacentPhaseSeparation.lean","line":295,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/AdjacentPhaseSeparation.lean#L295","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/AdjacentPhaseSeparation.lean#L295","kind":"theorem","signature":"theorem irrational_totient_series_of_adjacentPhaseSeparation (hsep : DTWAdjacentPhaseSeparation) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"A cofinal adjacent chord-separation supply also proves irrationality, via the two-element first-harmonic consumer.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/3"}],"links":[{"kind":"lean_declaration_pinned","label":"exists_certifiedKill_of_phasePairSeparation","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/AdjacentPhaseSeparation.lean#L258","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totient_series_of_adjacentPhaseSeparation","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/AdjacentPhaseSeparation.lean#L295","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/116","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"adjacent_phase_separation_reduction"}}],"expect":"object"},"prior_art":{"family_id":"harmonic_and_phase_reductions","status_summary":"Conditional harmonic and phase consumers, including an exact four-term pivot decomposition and budget, an exact window-pair equivalence, and the two-branch tail-orbit phase-density and natural-prime strict-gap producers that feed the same certificate socket.","prior_art_posture":"No matching combined theorem is identified; all novelty language remains provisional.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/19","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z08::adjacent_socket_implies_249","prior_art_state":"not_assessed"},{"semantic_node_id":"Z08::two_point_chord_separation_forces_certificate","prior_art_state":"routine_corollary_or_specialisation"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:pivot_anti_reconstruction_reduction","assertion_kind":"registered_claim","source_id":"pivot_anti_reconstruction_reduction","reviewed":true,"granularity":"single_registered_claim","label":"Pivot anti-reconstruction and window-pair reduction","statement":"On a finite supplier fibre, sufficient counted phase-pair separation feeds a certified non-integrality witness through exact reconstruction-defect and variance identities. Cofinal pivot anti-reconstruction, reconstruction variance, or counted-pair producers imply irrationality of S. The fibre-free cofinal window-pair predicate is exactly equivalent to irrationality of S. No cofinal arithmetic producer is proved.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"exists_certifiedKill_of_pivotFiberSeparatedPairs","module":"Erdos249257/PivotAntiReconstruction.lean","line":1125,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PivotAntiReconstruction.lean#L1125","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PivotAntiReconstruction.lean#L1125","kind":"theorem","signature":"theorem exists_certifiedKill_of_pivotFiberSeparatedPairs {h X L s m : ℕ} (hne : (pivotFiber X L s m).Nonempty) (hroom : 16 * (2 * X + h + L + 2) ≤ 2 ^ L) (P : Finset (ℕ × ℕ)) (δ : ℝ) (hP : P ⊆ (pivotFiber X L s m).product (pivotFiber X L s m)) (hδ : 0 ≤ δ) (hsep : ∀ p ∈ P, δ ≤ ‖pivotReconstructionError h p.1 L s - pivotReconstructionError h p.2 L s‖) (hmass : 2 * ((pivotFiber X L s m).card : ℝ) ^ 2 / 5 ≤ (P.card : ℝ) * δ ^ 2) : ∃ N ∈ pivotFiber X L s m, certifiedKill h N L","docstring":"Let `P` be a subset of the ordered fibre square whose pairs are separated by at least `δ`. If `|P|δ² ≥ 2|F|²/5`, then the fibre contains a certified kill. This is a finite statement; it does not assert that such sets exist cofinally.","generated_certificate":false},{"name":"irrational_totient_series_of_pivotAntiReconstruction","module":"Erdos249257/PivotAntiReconstruction.lean","line":1523,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PivotAntiReconstruction.lean#L1523","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PivotAntiReconstruction.lean#L1523","kind":"theorem","signature":"theorem irrational_totient_series_of_pivotAntiReconstruction (hmix : DTWPivotAntiReconstruction) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"Cofinal pivot anti-reconstruction implies irrationality of the #249 totient series by producing certified kills at arbitrarily large indices.","generated_certificate":false},{"name":"dtwWindowSeparatedPairs_iff_irrational_totient_series","module":"Erdos249257/PivotAntiReconstruction.lean","line":1765,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/PivotAntiReconstruction.lean#L1765","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PivotAntiReconstruction.lean#L1765","kind":"theorem","signature":"theorem dtwWindowSeparatedPairs_iff_irrational_totient_series : DTWWindowSeparatedPairs ↔ Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"The fibre-free window-pair condition is exactly equivalent to the irrationality of the #249 totient series. Thus it is a geometric reformulation, not a logically weaker unresolved condition.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/3"}],"links":[{"kind":"lean_declaration_pinned","label":"exists_certifiedKill_of_pivotFiberSeparatedPairs","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PivotAntiReconstruction.lean#L1125","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totient_series_of_pivotAntiReconstruction","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PivotAntiReconstruction.lean#L1523","cite_this":true},{"kind":"lean_declaration_pinned","label":"dtwWindowSeparatedPairs_iff_irrational_totient_series","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/PivotAntiReconstruction.lean#L1765","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/117","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"pivot_anti_reconstruction_reduction"}}],"expect":"object"},"prior_art":{"family_id":"harmonic_and_phase_reductions","status_summary":"Conditional harmonic and phase consumers, including an exact four-term pivot decomposition and budget, an exact window-pair equivalence, and the two-branch tail-orbit phase-density and natural-prime strict-gap producers that feed the same certificate socket.","prior_art_posture":"No matching combined theorem is identified; all novelty language remains provisional.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/19","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2,"routine_corollary_or_specialisation":1},"edges":[{"semantic_node_id":"Z08::finite_fibre_intake_forces_certificate","prior_art_state":"routine_corollary_or_specialisation"},{"semantic_node_id":"Z08::pivot_fibre_socket_implies_249","prior_art_state":"not_assessed"},{"semantic_node_id":"Z08::window_separated_pairs_equivalent_to_249","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:fixed_precision_transport_no_go","assertion_kind":"registered_claim","source_id":"fixed_precision_transport_no_go","reviewed":true,"granularity":"single_registered_claim","label":"Fixed-precision valuation-unit no-go theorem","statement":"At every fixed positive precision, every finite word of odd valuation-unit symbols admits a compatible carry orbit from an arbitrary incoming integer state, with each successive state within the corresponding valuation-unit radius. Thus bounded local signatures alone cannot exclude all finite centred endpoints; growing precision or global information remains necessary for an endpoint argument.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_249","erdos_257"],"problem_erdos_numbers":[249,257],"problem_binding":{"method":"core_programme_joint","warrant":"this claim belongs to the reviewed Erdos249257 core library, which is the joint #249/#257 programme. No source surface -- not its remaining-open propositions, not its anchor's enclosing gateway section, not the orientation programme map -- assigns it to one rather than both. Both are listed because both are correct, not because the binding is unknown.","separability":"not_separable_from_published_source"},"problem_edges":[{"problem_id":"erdos_249","relation":"shared_programme_context","basis":"problem_binding"},{"problem_id":"erdos_257","relation":"shared_programme_context","basis":"problem_binding"}],"programme_ids":["joint_249_257_gateway"],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"fixedPrecisionTropicalNoGo","module":"Erdos249257/TropicalCurvatureCarry.lean","line":137,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TropicalCurvatureCarry.lean#L137","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TropicalCurvatureCarry.lean#L137","kind":"theorem","signature":"theorem fixedPrecisionTropicalNoGo (u : ℕ) (hu : 0 < u) (symbols : List VUSymbol) (hodd : ∀ σ ∈ symbols, Odd σ.unit) (e : ℤ) : ∃ states : List ℤ, VUOrbit u e symbols states ∧ List.Forall₂ (fun σ e' => |e'| ≤ vuRadius u σ) symbols states","docstring":"Formal fixed-precision tropical no-go: bounded local valuation--unit data alone never excludes all finite centred carry completions.","generated_certificate":false},{"name":"vu_step_has_centred_completion","module":"Erdos249257/TropicalCurvatureCarry.lean","line":69,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TropicalCurvatureCarry.lean#L69","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TropicalCurvatureCarry.lean#L69","kind":"theorem","signature":"theorem vu_step_has_centred_completion (u : ℕ) (σ : VUSymbol) (hu : 0 < u) (hodd : Odd σ.unit) (e : ℤ) : ∃ c e' : ℤ, VUCompatible u σ c ∧ e' = 2 * e + c ∧ |e'| ≤ vuRadius u σ","docstring":"Every one-step fixed-precision symbol has a compatible digit that recentres an arbitrary incoming carry state.","generated_certificate":false},{"name":"vu_word_has_prefix_locked_completion","module":"Erdos249257/TropicalCurvatureCarry.lean","line":115,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TropicalCurvatureCarry.lean#L115","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TropicalCurvatureCarry.lean#L115","kind":"theorem","signature":"theorem vu_word_has_prefix_locked_completion (u : ℕ) (hu : 0 < u) (symbols : List VUSymbol) (hodd : ∀ σ ∈ symbols, Odd σ.unit) (e : ℤ) : ∃ states : List ℤ, VUOrbit u e symbols states ∧ List.Forall₂ (fun σ e' => |e'| ≤ vuRadius u σ) symbols states","docstring":"Every finite word of odd valuation--unit symbols has a compatible carry orbit centred at every prefix. The initial state is arbitrary.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"fixedPrecisionTropicalNoGo","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TropicalCurvatureCarry.lean#L137","cite_this":true},{"kind":"lean_declaration_pinned","label":"vu_step_has_centred_completion","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TropicalCurvatureCarry.lean#L69","cite_this":true},{"kind":"lean_declaration_pinned","label":"vu_word_has_prefix_locked_completion","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TropicalCurvatureCarry.lean#L115","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/118","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"fixed_precision_transport_no_go"}}],"expect":"object"},"prior_art":{"family_id":"transport_strategy_no_go_boundaries","status_summary":"Two unconditional scoped countermodels to fixed local precision and finite shift algebra.","prior_art_posture":"No matching theorem is identified; the record explicitly refuses to infer novelty from that.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/20","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z09::fixed_precision_tropical_no_go","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:lcm_factor_ideal_anchor_pulse_no_go","assertion_kind":"registered_claim","source_id":"lcm_factor_ideal_anchor_pulse_no_go","reviewed":true,"granularity":"single_registered_claim","label":"LCM factor-ideal anchor-pulse no-go theorem","statement":"For every t ≥ 3, a nonzero synthetic affine carry has forcing letters in the principal ideal generated by φ(lcm(1,…,t)), matches every stated whole-ray anchor, and remains within the diagonal bounds while its state is a strict survivor. The same countermodel survives every finite integer shift polynomial. Thus factor ideals, whole-ray anchors, and finite-rank shift algebra alone do not force carry escape. The construction does not assert that its forcing letters are actual totient differences and does not rule out nonlinear use of fresh divisor channels.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"paper_owner","warrant":"the claim's paper_label 'res:factorideal' is defined only in the manuscript treating Erdős #249"},"problem_edges":[{"problem_id":"erdos_249","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-totient-reasoning-surface","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient","module":"Erdos249257/LcmFactorIdealPulseObstruction.lean","line":798,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/LcmFactorIdealPulseObstruction.lean#L798","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmFactorIdealPulseObstruction.lean#L798","kind":"theorem","signature":"theorem lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient (t : ℕ) (ht : 3 ≤ t) : ∃ c a : ℕ → ℤ, (∃ k, c k ≠ 0) ∧ (∀ i, a i = dyadicCoboundary c i) ∧ ∀ terms : List (ℕ × ℤ), ∃ d b : ℕ → ℤ, (∀ i, d i = shiftLinearCombination terms c i) ∧ (∀ i, b i = shiftLinearCombination terms a i) ∧ (∀ i, b i = dyadicCoboundary d i) ∧ (∀ n L, dyadicClearedPrefix b n L = (2 : ℤ) ^ L * d n - d (n + L)) ∧ (∀ i, (Nat.totient (periodLcm t) : ℤ) ∣ b i) ∧ (∀ j, j ∣ periodLcm t → ∀ i, (Nat.totient j : ℤ) ∣ b i) ∧","docstring":"**All-rank shift-algebra no-go.** For every `t ≥ 3`, one nonzero synthetic pulse simultaneously compensates every finite integer shift polynomial. Each transformed word remains a dyadic coboundary, every cleared prefix is cancelled exactly by its terminal transformed state, every lower factor ideal is inherited, and uniform coefficient-weight bounds remain available. The shifts need not be commensurate. Hence the theorem covers every finite family of rank-two/rank-three LCM cubes, arbitrary finite rank, and each entry of a matrix built from finitely many such linear observations. It makes no claim about nonlinear combinations such as determinants, and it does not assert that the base or transformed letters are actual totient differences.","generated_certificate":false},{"name":"lcm_factorIdeal_sparseAnchor_not_sufficient","module":"Erdos249257/LcmFactorIdealPulseObstruction.lean","line":866,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/LcmFactorIdealPulseObstruction.lean#L866","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmFactorIdealPulseObstruction.lean#L866","kind":"theorem","signature":"theorem lcm_factorIdeal_sparseAnchor_not_sufficient (t : ℕ) (ht : 3 ≤ t) : ∃ c a : ℕ → ℤ, LcmFactorIdealAnchorAdmissible t c a ∧ (∃ k, c k ≠ 0) ∧ ∀ K i, i ≤ K → |c i| < (2 * periodLcm t + i + 2 : ℕ)","docstring":"**LCM factor-ideal sparse-anchor no-go.** For every `t ≥ 3` there is a nonzero synthetic carry orbit satisfying the exact affine recurrence, all homogeneous lower factor ideals, the natural letter and strip bounds, and every prescribed whole-ray anchor. It survives every finite horizon. This does not assert arithmetic realisability of the compensation letters; it proves that the named factor-only hypotheses are not sufficient.","generated_certificate":false},{"name":"lcmAnchorPulse_t3_letters","module":"Erdos249257/LcmFactorIdealPulseObstruction.lean","line":895,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/LcmFactorIdealPulseObstruction.lean#L895","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmFactorIdealPulseObstruction.lean#L895","kind":"theorem","signature":"theorem lcmAnchorPulse_t3_letters : lcmAnchorPulseLetter 3 5 = 2 ∧ lcmAnchorPulseLetter 3 6 = -4","generated_certificate":false}],"paper_label":"res:factorideal","paper_anchor":{"paper_label":"res:factorideal","owner_source":"paper/249/erdos249-totient-reasoning-surface.tex","source_line":9803,"anchor_kind":"inline","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"LCM tails and rational comparisons","section_path":[{"level":"section","title":"Complementary proofs for the short paper","title_tex":"Complementary proofs for the short paper","line":9583},{"level":"subsection","title":"LCM tails and rational comparisons","title_tex":"LCM tails and rational comparisons","line":9761}],"source_excerpt":{"text":"\\item\\label{res:factorideal}\n\\textbf{Divisibility and finite linear combinations do not suffice.}\nA dyadic coboundary is a sequence of the form $2u_j-u_{j+1}$; its weighted\nsum telescopes. For every $t\\ge3$, with $H=\\lcm(1,\\ldots,t)$, the linked\nsource constructs a nonzero comparison sequence of this form. Its\ncoefficients reproduce the actual totient differences at $t-2$\nprescribed indices and are divisible by $\\varphi(H)$. Every finite integer\nlinear combination of forward shifts preserves the telescoping identity and\ndivisibility by $\\varphi(H)$ and the corresponding lower-level factors.\nThe bounds remain uniform in the index, but are multiplied by the sum\nof the absolute values of the combination's coefficients.\nA variant preserves sparse prescribed values along entire arithmetic\nprogressions. These properties alone do not force a contradiction: the\ncomparison need not consist of actual totient differences. The theorem\nconcerns linear combinations of shifts, not nonlinear operations, and\ndoes not supply certificates at arbitrarily large scales.\n{\\footnotesize\\emph{Source declarations:}\n\\mword{Erdos249257/LcmFactorIdealPulseObstruction.lean}{798}{lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient}{obstruction for finite integer combinations of shifts}.}\n\\end{enumerate}","media_type":"text/x-tex","source_char_start":488134,"source_char_end":489430,"start_line":9803,"end_line":9821,"truncated":false,"sha256":"sha256:9a5f3c262dd9d6141ce8d4ca08594e754e8678668b6f3687ea4846b427e9eff7","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos249-totient-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/249/erdos249-totient-reasoning-surface.tex#L9803","owner_problem_erdos_number":249,"owner_paper_id":"erdos249-totient-reasoning-surface","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos249-totient-reasoning-surface.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos249-totient-reasoning-surface.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/249/erdos249-totient-reasoning-surface.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmFactorIdealPulseObstruction.lean#L798","cite_this":true},{"kind":"lean_declaration_pinned","label":"lcm_factorIdeal_sparseAnchor_not_sufficient","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmFactorIdealPulseObstruction.lean#L866","cite_this":true},{"kind":"lean_declaration_pinned","label":"lcmAnchorPulse_t3_letters","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/LcmFactorIdealPulseObstruction.lean#L895","cite_this":true},{"kind":"paper_pdf","label":"erdos249-totient-reasoning-surface","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos249-totient-reasoning-surface.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/119","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"lcm_factor_ideal_anchor_pulse_no_go"}}],"expect":"object"},"prior_art":{"family_id":"transport_strategy_no_go_boundaries","status_summary":"Two unconditional scoped countermodels to fixed local precision and finite shift algebra.","prior_art_posture":"No matching theorem is identified; the record explicitly refuses to infer novelty from that.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/20","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z06::barrier_finite_rank_shift_algebra","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:actual_lcm_orbit_nonintegrality_frontier","assertion_kind":"registered_claim","source_id":"actual_lcm_orbit_nonintegrality_frontier","reviewed":true,"granularity":"single_registered_claim","label":"Actual LCM-orbit non-integrality frontier","statement":"Let H_a = lcm(1,…,2^a) and R_a = totientTail(2H_a) − totientTail(H_a). The totient series S is irrational if and only if, beyond every threshold a₀, some a ≥ a₀ has R_a ∉ ℤ. This is an exact cofinal non-integrality criterion for the actual LCM diagonal. That equivalence itself is not a quantitative separation estimate; the same source also provides a distinct sufficient route in which, cofinally in a₀, some a,q satisfy max 2 a₀ ≤ a, oddGuardedCanonicalAdjacentSuffixDepth (2^a) = 2q+1, and every integer z obeys 1/32 + actualLcmRawErrorRadius a q ≤ |actualLcmTailOrbit a − z|. This quantitative supply is not equivalent to the frontier criterion and remains unproved.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-totient-reasoning-surface","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"irrational_totientSeries_iff_actualLcmOrbitNonintegralitySupply","module":"Erdos249257/TotientActualLcmOrbitNonintegrality.lean","line":37,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientActualLcmOrbitNonintegrality.lean#L37","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientActualLcmOrbitNonintegrality.lean#L37","kind":"theorem","signature":"theorem irrational_totientSeries_iff_actualLcmOrbitNonintegralitySupply : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n) ↔ PowerTwoActualLcmOrbitNonintegralitySupply","docstring":"The exact power-two actual-LCM orbit frontier is equivalent to Erdős #249.","generated_certificate":false},{"name":"irrational_totientSeries_of_actualLcmOrbitNonintegralitySupply","module":"Erdos249257/TotientActualLcmOrbitNonintegrality.lean","line":53,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientActualLcmOrbitNonintegrality.lean#L53","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientActualLcmOrbitNonintegrality.lean#L53","kind":"theorem","signature":"theorem irrational_totientSeries_of_actualLcmOrbitNonintegralitySupply (hsupply : PowerTwoActualLcmOrbitNonintegralitySupply) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"Cofinal actual-LCM orbit non-integrality implies irrationality of the totient series directly, with no quantitative separation hypothesis. The non-integrality condition remains unproved.","generated_certificate":false},{"name":"irrational_totientSeries_of_actualLcmOrbitSeparationSupply","module":"Erdos249257/TotientActualLcmOrbitSeparation.lean","line":305,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientActualLcmOrbitSeparation.lean#L305","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientActualLcmOrbitSeparation.lean#L305","kind":"theorem","signature":"theorem irrational_totientSeries_of_actualLcmOrbitSeparationSupply (hsupply : PowerTwoActualLcmOrbitSeparationSupply) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"The orbit-separation supply implies irrationality of the totient series. The supply itself remains unproved.","generated_certificate":false}],"paper_label":"res:actualorbit","paper_anchor":{"paper_label":"res:actualorbit","owner_source":"paper/249/erdos249-totient-reasoning-surface.tex","source_line":9769,"anchor_kind":"inline","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"LCM tails and rational comparisons","section_path":[{"level":"section","title":"Complementary proofs for the short paper","title_tex":"Complementary proofs for the short paper","line":9583},{"level":"subsection","title":"LCM tails and rational comparisons","title_tex":"LCM tails and rational comparisons","line":9761}],"source_excerpt":{"text":"\\begin{enumerate}\n\\item\\label{res:actualorbit}\n\\textbf{Tail differences at least common multiples.}  Let\n$H_a=\\lcm(1,\\ldots,2^a)$. Then $S$ is irrational if and only if,\nbeyond every threshold $a_0$, some $a\\ge a_0$ satisfies\n$\\Delta_{H_a}(H_a)\\notin\\Z$.  This is the\n\\mword{Erdos249257/TotientActualLcmOrbitNonintegrality.lean}{37}{irrational_totientSeries_iff_actualLcmOrbitNonintegralitySupply}{exact cofinal nonintegrality criterion}\nfor the actual LCM diagonal, not a quantitative\nseparation estimate and not a proof that such indices occur; the\n\\mword{Erdos249257/TotientActualLcmOrbitNonintegrality.lean}{53}{irrational_totientSeries_of_actualLcmOrbitNonintegralitySupply}{forward implication}\nis recorded separately.","media_type":"text/x-tex","source_char_start":486311,"source_char_end":487035,"start_line":9768,"end_line":9778,"truncated":false,"sha256":"sha256:6d6f6ab6e52b6b5bf355e6e911d9caef630042f2c732e9021fdf0cdfb5c454fa","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos249-totient-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/249/erdos249-totient-reasoning-surface.tex#L9769","owner_problem_erdos_number":249,"owner_paper_id":"erdos249-totient-reasoning-surface","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos249-totient-reasoning-surface.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos249-totient-reasoning-surface.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/249/erdos249-totient-reasoning-surface.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/3"}],"links":[{"kind":"lean_declaration_pinned","label":"irrational_totientSeries_iff_actualLcmOrbitNonintegralitySupply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientActualLcmOrbitNonintegrality.lean#L37","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totientSeries_of_actualLcmOrbitNonintegralitySupply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientActualLcmOrbitNonintegrality.lean#L53","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_totientSeries_of_actualLcmOrbitSeparationSupply","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientActualLcmOrbitSeparation.lean#L305","cite_this":true},{"kind":"paper_pdf","label":"erdos249-totient-reasoning-surface","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos249-totient-reasoning-surface.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/120","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"actual_lcm_orbit_nonintegrality_frontier"}}],"expect":"object"},"prior_art":{"family_id":"actual_lcm_orbit_and_sign_frontier","status_summary":"An exact actual-LCM cofinal non-integrality reformulation and an unconditional positive-corridor/top-edge boundary.","prior_art_posture":"The actual-LCM tail equivalence and sign/carry analysis are source-specific; no novelty or priority claim is made.","primary_narrative_owner":"paper/249/erdos-249-binary-totient-series.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/21","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":2},"edges":[{"semantic_node_id":"Z28::actual_lcm_orbit_nonintegrality_exact_frontier","prior_art_state":"not_assessed"},{"semantic_node_id":"Z64::cofinal_actual_lcm_orbit_separation_conditionally_forces_totient_irrationality","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:actual_lcm_positive_corridor_top_edge","assertion_kind":"registered_claim","source_id":"actual_lcm_positive_corridor_top_edge","reviewed":true,"granularity":"single_registered_claim","label":"Actual LCM positive corridor and top-edge boundary","statement":"For a ≥ 8 and J + (a + 6) < 2·2^a, the true translated LCM-orbit difference is strictly positive. If it is represented by an integer and the dyadic modulus has room for the directed endpoint strip, the carry survivor is strictly negative and the discrepancy residue is exactly the top-edge representative 2^K − carry_K, strictly between the strip’s lower edge and 2^K. Thus the positive corridor eliminates the nonnegative true survivor but does not itself produce a central-band contradiction; an independent arithmetic exclusion of the top boundary is still required.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"paper_owner","warrant":"the claim's paper_label 'res:actualsign' is defined only in the manuscript treating Erdős #249"},"problem_edges":[{"problem_id":"erdos_249","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos249-totient-reasoning-surface","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"actualLcmTailDiff_shift_pos","module":"Erdos249257/TotientActualLcmOrbitSign.lean","line":39,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientActualLcmOrbitSign.lean#L39","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientActualLcmOrbitSign.lean#L39","kind":"theorem","signature":"theorem actualLcmTailDiff_shift_pos {a J : ℕ} (ha : 8 ≤ a) (hshort : J + (a + 6) < 2 * 2 ^ a) : 0 < totientTail (2 * periodLcm (2 ^ a) + J) - totientTail (periodLcm (2 ^ a) + J)","docstring":"**Unconditional positive-sign corridor.** At a power-of-two LCM height, the true translated tail difference stays positive as long as an `a+6`-term lookahead remains inside the short arithmetic window. The first lookahead letter already contributes more than enough: its quarter-height lower bound, amplified by `2^(a+5) = 32 * 2^a`, dominates four copies of the LCM height. All remaining lookahead letters are positive, and the directed lower bound controls the untranslated infinite remainder.","generated_certificate":false},{"name":"actualLcm_trueEndpointSurvivor_neg","module":"Erdos249257/TotientActualLcmOrbitSign.lean","line":172,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientActualLcmOrbitSign.lean#L172","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientActualLcmOrbitSign.lean#L172","kind":"theorem","signature":"theorem actualLcm_trueEndpointSurvivor_neg {a J K : ℕ} (ha : 8 ≤ a) (hshort : J + K + (a + 6) < 2 * 2 ^ a) {d : ℤ} (hd : (d : ℝ) = totientTail (2 * periodLcm (2 ^ a) + J) - totientTail (periodLcm (2 ^ a) + J)) : -carryOrbit (periodLcm (2 ^ a)) (periodLcm (2 ^ a) + J) d K < 0 ∧ endpointSurvivor (periodLcm (2 ^ a)) (periodLcm (2 ^ a) + J) K (-carryOrbit (periodLcm (2 ^ a)) (periodLcm (2 ^ a) + J) d K)","docstring":"**Survivor-sign consumer.** If a translated actual LCM orbit is integral at the start of the positive corridor, then at every later endpoint which still has an `a+6`-letter lookahead, the survivor supplied by the true carry is strictly negative. Thus the nonnegative true-survivor branch is eliminated; this does not assert that the finite survivor set has no nonnegative spurious elements.","generated_certificate":false},{"name":"actualLcm_integral_forces_topEdgeResidue","module":"Erdos249257/TotientActualLcmOrbitSign.lean","line":211,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientActualLcmOrbitSign.lean#L211","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientActualLcmOrbitSign.lean#L211","kind":"theorem","signature":"theorem actualLcm_integral_forces_topEdgeResidue {a J K : ℕ} (ha : 8 ≤ a) (hshort : J + K + (a + 6) < 2 * 2 ^ a) {d : ℤ} (hd : (d : ℝ) = totientTail (2 * periodLcm (2 ^ a) + J) - totientTail (periodLcm (2 ^ a) + J)) (hroom : ((2 * periodLcm (2 ^ a) + J + K + 2 : ℕ) : ℤ) < (2 : ℤ) ^ K) : let H","docstring":"**Exact obstruction to a sign-only modular kill.** Once the dyadic modulus is wider than the directed endpoint strip, integrality together with the positive-sign corridor does not put the discrepancy residue in the central arc. It forces the opposite: the residue is exactly the top-edge representative `2^K - carry_K`. Consequently the upper inequality required by `directedCertifiedKill` fails strictly. Any contradiction after the sign theorem therefore needs an independent arithmetic exclusion of this top boundary band; the carry reset alone merely re-encodes the persistent negative survivor.","generated_certificate":false}],"paper_label":"res:actualsign","paper_anchor":{"paper_label":"res:actualsign","owner_source":"paper/249/erdos249-totient-reasoning-surface.tex","source_line":9780,"anchor_kind":"inline","anchor_unit":null,"anchor_unit_line":null,"anchor_title":null,"anchor_title_tex":null,"section_title":"LCM tails and rational comparisons","section_path":[{"level":"section","title":"Complementary proofs for the short paper","title_tex":"Complementary proofs for the short paper","line":9583},{"level":"subsection","title":"LCM tails and rational comparisons","title_tex":"LCM tails and rational comparisons","line":9761}],"source_excerpt":{"text":"\\item\\label{res:actualsign}\n\\textbf{Positive tail differences need not be nonintegral.} For $a\\ge8$ and\n$J+(a+6)<2\\cdot2^a$, one has $\\Delta_{H_a}(H_a+J)>0$.\nSuppose this difference is an integer. If a truncation depth $K$ satisfies\n\\[\n J+K+(a+6)<2\\cdot2^a,\n \\qquad 2H_a+J+K+2<2^K,\n\\]\nthen\n\\[\n D_{H_a,H_a+J,K}\\bmod2^K\n   =2^K-\\Delta_{H_a}(H_a+J+K)\n\\]\nlies strictly between $2^K-(2H_a+J+K+2)$ and $2^K$.\nThe later tail difference is a positive integer; its \\emph{negative} is\nthe representative near zero. Thus positivity determines which endpoint\ncontains the residue, but does not exclude an integral tail difference.\nBoth displayed bounds on $K$ are required for this conclusion.\n{\\footnotesize\\emph{Source declarations:}\n\\mword{Erdos249257/TotientActualLcmOrbitSign.lean}{39}{actualLcmTailDiff_shift_pos}{positive tail differences},\n\\mword{Erdos249257/TotientActualLcmOrbitSign.lean}{172}{actualLcm_trueEndpointSurvivor_neg}{negative endpoint representative},\n\\mword{Erdos249257/TotientActualLcmOrbitSign.lean}{211}{actualLcm_integral_forces_topEdgeResidue}{residue near the upper endpoint}.}","media_type":"text/x-tex","source_char_start":487037,"source_char_end":488132,"start_line":9780,"end_line":9801,"truncated":false,"sha256":"sha256:0ae7922b2341e974e556277dcbd569a34513d7954572fcf2529aa7d03b304dc0","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/249/erdos249-totient-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/249/erdos249-totient-reasoning-surface.tex#L9780","owner_problem_erdos_number":249,"owner_paper_id":"erdos249-totient-reasoning-surface","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos249-totient-reasoning-surface.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos249-totient-reasoning-surface.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/249/erdos249-totient-reasoning-surface.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"actualLcmTailDiff_shift_pos","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientActualLcmOrbitSign.lean#L39","cite_this":true},{"kind":"lean_declaration_pinned","label":"actualLcm_trueEndpointSurvivor_neg","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientActualLcmOrbitSign.lean#L172","cite_this":true},{"kind":"lean_declaration_pinned","label":"actualLcm_integral_forces_topEdgeResidue","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientActualLcmOrbitSign.lean#L211","cite_this":true},{"kind":"paper_pdf","label":"erdos249-totient-reasoning-surface","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos249-totient-reasoning-surface.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/121","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"actual_lcm_positive_corridor_top_edge"}}],"expect":"object"},"prior_art":{"family_id":"actual_lcm_orbit_and_sign_frontier","status_summary":"An exact actual-LCM cofinal non-integrality reformulation and an unconditional positive-corridor/top-edge boundary.","prior_art_posture":"The actual-LCM tail equivalence and sign/carry analysis are source-specific; no novelty or priority claim is made.","primary_narrative_owner":"paper/249/erdos-249-binary-totient-series.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/21","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"formalisation_only":2,"not_assessed":1},"edges":[{"semantic_node_id":"Z51::actual_lcm_translated_tail_difference_is_positive_in_the_short_corridor","prior_art_state":"not_assessed"},{"semantic_node_id":"Z51::integral_actual_lcm_tail_difference_has_negative_true_endpoint_survivor","prior_art_state":"formalisation_only"},{"semantic_node_id":"Z51::integrality_places_the_discrepancy_residue_at_the_top_edge","prior_art_state":"formalisation_only"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:strict_prime_tail_orbit_gap","assertion_kind":"registered_claim","source_id":"strict_prime_tail_orbit_gap","reviewed":true,"granularity":"single_registered_claim","label":"Tail-orbit phase-density and natural-prime strict-gap reductions","statement":"A uniform 11/100 nonpositive-phase density on arbitrarily late dyadic blocks yields an 89/100 first-harmonic saving and hence a finite certificate; alternatively, cofinally many natural prime indices with tail-orbit real part below 9/10 yield adaptive pivot escape. The exact initial phase is the scaled-angle character of (2^h - 1) times the series, and the tail phase follows a power-of-two squaring orbit from it; phase 1 is equivalent to an integral actual tail difference, while an entered dyadic root obstructs the strict-gap producer.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"TotientTailOrbitBlockGap","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":25,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L25","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L25","kind":"def","signature":"noncomputable def TotientTailOrbitBlockGap : Prop","docstring":"On arbitrarily late dyadic blocks, the mean real part of the infinite tail-orbit phase is at most `89/100`. The hypotheses include every positive shift `h`; no such estimate is currently known for the totient orbit.","generated_certificate":false},{"name":"TotientTailOrbitNonpositiveBlockDensity","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":35,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L35","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L35","kind":"def","signature":"noncomputable def TotientTailOrbitNonpositiveBlockDensity : Prop","docstring":"On arbitrarily late dyadic blocks, at least an `11/100` proportion of the tail-orbit phases have nonpositive real part. This is a sufficient density hypothesis, not a statement proved here for the totient orbit.","generated_certificate":false},{"name":"tailOrbitBlockGap_of_nonpositiveBlockDensity","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":45,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L45","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L45","kind":"theorem","signature":"theorem tailOrbitBlockGap_of_nonpositiveBlockDensity (hdensity : TotientTailOrbitNonpositiveBlockDensity) : TotientTailOrbitBlockGap","docstring":"A density of nonpositive phases gives the `89/100` block gap. The other phases contribute at most `1` because every tail-orbit phase has norm one.","generated_certificate":false},{"name":"irrational_totient_series_of_tailOrbitBlockGap","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":96,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L96","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L96","kind":"theorem","signature":"theorem irrational_totient_series_of_tailOrbitBlockGap (hgap : TotientTailOrbitBlockGap) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"A uniform `89/100` block gap for the infinite orbit yields the finite `9/10` gap after choosing one truncation depth for the whole block.","generated_certificate":false},{"name":"DTWNaturalPrimeTailOrbitStrictGap","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":157,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L157","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L157","kind":"def","signature":"def DTWNaturalPrimeTailOrbitStrictGap : Prop","docstring":"Cofinal prime indices at which the infinite-orbit real part is strictly below `9/10`. The positive margin may depend on the chosen index.","generated_certificate":false},{"name":"naturalPivotPointEscape_of_naturalPrimeTailOrbitStrictGap","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":485,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L485","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L485","kind":"theorem","signature":"theorem naturalPivotPointEscape_of_naturalPrimeTailOrbitStrictGap (hgap : DTWNaturalPrimeTailOrbitStrictGap) : DTWNaturalPivotPointEscape","docstring":"A strict `9/10` gap in the infinite phase leaves a positive truncation budget and gives a finite `9/10` pointwise gap.","generated_certificate":false},{"name":"irrational_totient_series_of_naturalPrimeTailOrbitStrictGap","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":524,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L524","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L524","kind":"theorem","signature":"theorem irrational_totient_series_of_naturalPrimeTailOrbitStrictGap (hgap : DTWNaturalPrimeTailOrbitStrictGap) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"The pointwise prime condition implies irrationality. The condition itself remains unproved for the totient orbit.","generated_certificate":false},{"name":"tailOrbitFirstExp_succ","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":167,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L167","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L167","kind":"lemma","signature":"lemma tailOrbitFirstExp_succ (h M : ℕ) : tailOrbitFirstExp h (M + 1) = (tailOrbitFirstExp h M) ^ 2","docstring":"The first tail-orbit phase follows the exact doubling map. The integer carry increment disappears after exponentiation, so the producer questions above are questions about the orbit of one phase under repeated squaring.","generated_certificate":false},{"name":"tailOrbitFirstExp_add","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":191,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L191","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L191","kind":"lemma","signature":"lemma tailOrbitFirstExp_add (h M k : ℕ) : tailOrbitFirstExp h (M + k) = (tailOrbitFirstExp h M) ^ (2 ^ k)","docstring":"Every later first phase is obtained from an earlier one by an exact power-of-two iterate. Thus no carry information survives in the phase orbit beyond the single starting value.","generated_certificate":false},{"name":"tailOrbitFirstExp_eq_initial_pow_two","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":201,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L201","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L201","kind":"lemma","signature":"lemma tailOrbitFirstExp_eq_initial_pow_two (h N : ℕ) : tailOrbitFirstExp h N = (tailOrbitFirstExp h 0) ^ (2 ^ N)","docstring":"Initial-phase normal form for the whole tail orbit.","generated_certificate":false},{"name":"naturalPrimeTailOrbitStrictGap_iff_initial_phase","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":278,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L278","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L278","kind":"theorem","signature":"theorem naturalPrimeTailOrbitStrictGap_iff_initial_phase : DTWNaturalPrimeTailOrbitStrictGap ↔ ∀ h : ℕ, 0 < h → ∀ N₀ : ℕ, ∃ p : ℕ, max (N₀ + h + 1) (h + 5) ≤ p ∧ p.Prime ∧ ((tailOrbitFirstExp h 0) ^ (2 ^ (p - h - 1))).re < (9 / 10 : ℝ)","docstring":"The prime-index strict-gap hypothesis is exactly a statement about the power-of-two orbit of one initial phase for each shift.","generated_certificate":false},{"name":"tailOrbitFirstExp_eq_one_of_le","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":294,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L294","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L294","kind":"lemma","signature":"lemma tailOrbitFirstExp_eq_one_of_le {h M N : ℕ} (hMN : M ≤ N) (hM : tailOrbitFirstExp h M = 1) : tailOrbitFirstExp h N = 1","docstring":"The phase `1` is absorbing for the exact squaring dynamics.","generated_certificate":false},{"name":"not_naturalPrimeTailOrbitStrictGap_of_phase_one","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":308,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L308","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L308","kind":"theorem","signature":"theorem not_naturalPrimeTailOrbitStrictGap_of_phase_one {h M : ℕ} (hh : 0 < h) (hM : tailOrbitFirstExp h M = 1) : ¬ DTWNaturalPrimeTailOrbitStrictGap","docstring":"An absorbing phase at any time rigorously obstructs the cofinal strict-gap producer for that shift. This does not show that the actual totient orbit ever enters the absorbing phase.","generated_certificate":false},{"name":"not_naturalPrimeTailOrbitStrictGap_of_dyadic_root","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":322,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L322","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L322","kind":"theorem","signature":"theorem not_naturalPrimeTailOrbitStrictGap_of_dyadic_root {h M k : ℕ} (hh : 0 < h) (hroot : (tailOrbitFirstExp h M) ^ (2 ^ k) = 1) : ¬ DTWNaturalPrimeTailOrbitStrictGap","docstring":"More generally, entering any dyadic root of unity obstructs the strict-gap producer: finitely many squarings reach the absorbing phase `1`. The source does not prove that an actual totient phase is such a root.","generated_certificate":false},{"name":"tailOrbitFirstExp_zero_eq_scaled_angle","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":209,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L209","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L209","kind":"theorem","signature":"theorem tailOrbitFirstExp_zero_eq_scaled_angle (h : ℕ) : tailOrbitFirstExp h 0 = Complex.exp (((2 * Real.pi * (((2 : ℝ) ^ h - 1) * (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)) : ℝ) : ℂ) * Complex.I)","docstring":"Exact initial-angle formula for the actual totient phase. The discarded prefix is integral, so the starting phase is the additive character of `(2^h - 1)` times the original #249 series.","generated_certificate":false},{"name":"tailOrbitFirstExp_eq_one_iff_tail_diff_mem_int","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":221,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L221","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L221","kind":"lemma","signature":"lemma tailOrbitFirstExp_eq_one_iff_tail_diff_mem_int (h N : ℕ) : tailOrbitFirstExp h N = 1 ↔ totientTail (N + h) - totientTail N ∈ Set.range ((↑) : ℤ → ℝ)","docstring":"Landing at phase `1` is exactly integrality of the corresponding actual tail difference.","generated_certificate":false},{"name":"exists_tailOrbitFirstExp_zero_pow_two_eq_one_iff_dyadic","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":249,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L249","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L249","kind":"theorem","signature":"theorem exists_tailOrbitFirstExp_zero_pow_two_eq_one_iff_dyadic (h : ℕ) : (∃ k : ℕ, (tailOrbitFirstExp h 0) ^ (2 ^ k) = 1) ↔ IsDyadicReal (((2 : ℝ) ^ h - 1) * (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n))","docstring":"Exact classification of the exceptional arithmetic obstruction: the initial phase reaches `1` after finitely many squarings exactly when its real angle `(2^h - 1) S` is dyadic.","generated_certificate":false},{"name":"tailOrbitFirstExp_zero_pow_two_ne_one_upto_sixteen","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":333,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L333","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L333","kind":"theorem","signature":"theorem tailOrbitFirstExp_zero_pow_two_ne_one_upto_sixteen {h k : ℕ} (h1 : 1 ≤ h) (h16 : h ≤ 16) (hk : k ≤ 14) : (tailOrbitFirstExp h 0) ^ (2 ^ k) ≠ 1","docstring":"The existing exact certificate at time `14` excludes every dyadic-root landing of depth at most `14` for each shift `1 ≤ h ≤ 16`. 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sets\n$A_j\\subseteq\\{2,\\ldots,M_j\\}$ such that\n\\[\n \\frac{|K_{A_j}(M_j)|}{2^{M_j}}\\longrightarrow0.\n\\]\nThen $X_A(2)=1/2$ for some infinite set $A\\subseteq\\Npos$.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":63646,"source_char_end":63972,"start_line":1350,"end_line":1358,"truncated":false,"sha256":"sha256:794d00528fb7ca206bee491d982f0790d92397dc9048fcf79250b6428211eecc","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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This is a conditional countermodel frontier: the cofinal full-cylinder family is not constructed, so neither half-membership nor universal Erdős #257 is settled.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-257-mersenne-support-subseries","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"cofinalTerminalOnlyStrip_of_cofinalCylinderStages","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":264,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L264","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L264","kind":"theorem","signature":"theorem cofinalTerminalOnlyStrip_of_cofinalCylinderStages (hstages : ∀ N : ℕ, ∃ M K : ℕ, max N 1 ≤ M ∧ Nonempty (CylinderStage K M)) : HalfCarryCofinalTerminalOnlyStrip","docstring":"Cofinal full-cylinder stages imply the strictly weaker terminal-only cofinal producer expected by the closed achievement-set consumer.","generated_certificate":false},{"name":"exists_infinite_support_half_of_cofinalCylinderStages","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":277,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L277","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L277","kind":"theorem","signature":"theorem exists_infinite_support_half_of_cofinalCylinderStages (hstages : ∀ N : ℕ, ∃ M K : ℕ, max N 1 ≤ M ∧ Nonempty (CylinderStage K M)) : ∃ A : Set ℕ, A.Infinite ∧ erdosSupportSeries 2 A = (1 : ℝ) / 2","docstring":"Direct end-to-end consumer: cofinal suffix-cylinder stages produce an infinite support whose Mersenne support series is the rational value `1/2`.","generated_certificate":false},{"name":"exists_infinite_positive_support_half_of_cofinalCylinderStages","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":287,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L287","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L287","kind":"theorem","signature":"theorem exists_infinite_positive_support_half_of_cofinalCylinderStages (hstages : ∀ N : ℕ, ∃ M K : ℕ, max N 1 ≤ M ∧ Nonempty (CylinderStage K M)) : ∃ A : Set ℕ, 0 ∉ A ∧ A.Infinite ∧ erdosSupportSeries 2 A = (1 : ℝ) / 2","docstring":"Cofinal full-cylinder stages produce an infinite support of strictly positive exponents whose Mersenne support series is `1/2`.","generated_certificate":false}],"paper_label":"res:cylinderhalf","paper_anchor":{"paper_label":"res:cylinderhalf","owner_source":"paper/257/erdos-257-mersenne-support-subseries.tex","source_line":1415,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":1414,"anchor_title":"unbounded shared-prefix families represent one half","anchor_title_tex":"unbounded shared-prefix families represent one half","section_title":"Finite families with a shared prefix.","section_path":[{"level":"section","title":"Integer quotients, approximation and further questions","title_tex":"Integer quotients, approximation and further questions","line":1221},{"level":"paragraph","title":"Finite families with a shared prefix.","title_tex":"Finite families with a shared prefix.","line":1397}],"source_excerpt":{"text":"\\begin{theorem}[unbounded shared-prefix families represent one half]\n\\label{res:cylinderhalf}\nSuppose that for every $N$ there are $M,K$ with\n$\\max\\{N,1\\}\\le M$, $0\\le K\\le M$, and a family satisfying all the\nconditions in the preceding paragraph.  Then $X_A(2)=1/2$ for some\ninfinite set $A\\subseteq\\Npos$.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":66159,"source_char_end":66480,"start_line":1414,"end_line":1420,"truncated":false,"sha256":"sha256:69670cdc5a0d77e541b6bf7116c9406ee76e43fd56156da24d1d590ece1143e7","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Yes refutes universal #257; no closes only this route.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/4"},{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"cofinalTerminalOnlyStrip_of_cofinalCylinderStages","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L264","cite_this":true},{"kind":"lean_declaration_pinned","label":"exists_infinite_support_half_of_cofinalCylinderStages","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L277","cite_this":true},{"kind":"lean_declaration_pinned","label":"exists_infinite_positive_support_half_of_cofinalCylinderStages","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/SuffixCylinderTerminalOnlyBridge.lean#L287","cite_this":true},{"kind":"paper_pdf","label":"erdos-257-mersenne-support-subseries","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-257-mersenne-support-subseries.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/124","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"cofinal_cylinder_half_countermodel"}}],"expect":"object"},"prior_art":{"family_id":"conditional_half_countermodel_frontiers","status_summary":"Two conditional routes to an infinite rational half-valued Mersenne support: terminal scaled-vanishing and cofinal full-cylinder stages.","prior_art_posture":"These are source-specific conditional consumers; no existence, novelty, priority, or significance claim is made for either antecedent.","primary_narrative_owner":"paper/257/erdos-257-mersenne-support-subseries.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/25","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. 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The universal rational-base assertion remains open; 3/2 lies outside this region and is unresolved here.","status":"open","status_meaning":"Not proved by this release","readme_headline":false,"problem_ids":["erdos_1049"],"problem_erdos_numbers":[1049],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_1049","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_1049","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"registry_statement_only","evidence_resolution":"statement_only","declarations":[],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.erdos_1049_irrationality","statement":"Determine whether F(t) = ∑_{n≥1} 1/(tⁿ − 1) is irrational for every rational t > 1. Irrationality of F((31/4)^r) for every positive integer r is a Lean-checked Zudilin specialization; the 3/2 approximant with remainder control and the universal assertion remain open.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/9"}],"links":[],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/125","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"erdos_1049"}}],"expect":"object"},"prior_art":{"family_id":"rational_base_coordinatewise_boundaries","status_summary":"An open universal rational-base endpoint, a base-3/2 coordinatewise exclusion, exact cleared-tail dynamics and forcing bounds, and a finite q-Apéry diagonal non-transfer result.","prior_art_posture":"These are source-specific formal arithmetic results and an open target; no novelty, priority, significance, or irrationality claim is made for the new declarations.","primary_narrative_owner":"paper/1049/erdos-1049-rational-base-lambert.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/29","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. 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This excludes only that literal coordinatewise clearing architecture, not rationality or irrationality of F(3/2) or any other proof architecture.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_1049"],"problem_erdos_numbers":[1049],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_1049","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_1049","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos1049-rational-base-lambert-reasoning-surface","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"threeHalves_no_coordinatewiseCorridor","module":"ErdosProblems/Erdos1049/RationalBaseLambert.lean","line":155,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean#L155","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean#L155","kind":"theorem","signature":"theorem threeHalves_no_coordinatewiseCorridor {N K Q digit : ℕ} (hN : 1 ≤ N) (hK : 1 ≤ K) : ¬ CoordinatewiseCorridor 3 2 N K Q digit","docstring":"The literal coordinatewise Erdős corridor cannot occur at base `3/2` once both the shift and cleared window are nonempty.","generated_certificate":false}],"paper_label":"long1049:res:nocorridor","paper_anchor":{"paper_label":"long1049:res:nocorridor","owner_source":"paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","source_line":3948,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":3948,"anchor_title":"failure of the stated clearing conditions at 3/2","anchor_title_tex":"failure of the stated clearing conditions at $3/2$","section_title":"Direct clearing of a partial sum.","section_path":[{"level":"section","title":"Failure of the stated clearing conditions at 3/2","title_tex":"Failure of the stated clearing conditions at \\texorpdfstring{$3/2$}{3/2}","line":3811},{"level":"paragraph","title":"Direct clearing of a partial sum.","title_tex":"Direct clearing of a partial sum.","line":3830}],"source_excerpt":{"text":"\\begin{theorem}[failure of the stated clearing conditions at $3/2$]\\label{long1049:res:nocorridor}\nFor all $N\\ge1$ and $K\\ge1$ and all natural $Q,D$, the tuple\n$(3,2,N,K,Q,D)$ does not satisfy the clearing conditions.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":184079,"source_char_end":184310,"start_line":3948,"end_line":3951,"truncated":false,"sha256":"sha256:1b4552fb2e3dabc5b9f4562ccc06ab486d763f814a1a06c2a4feb023866200a0","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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This rules out that selected diagonal for that recurrence at n = 0; it proves no general recurrence, endpoint, or #1049 irrationality result.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_1049"],"problem_erdos_numbers":[1049],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_1049","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_1049","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"vanAsscheQAperyResidualAtZero_eq","module":"ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean","line":67,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean#L67","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean#L67","kind":"theorem","signature":"theorem vanAsscheQAperyResidualAtZero_eq (p : ℝ) : vanAsscheQAperyResidualAtZero p = -p * (p - 1) ^ 2 * (p + 1) * (p ^ 5 + 2 * p ^ 4 + 2 * p ^ 3 + 2 * p ^ 2 + 2)","docstring":"Exact factorisation of the `n = 0` residual.","generated_certificate":false},{"name":"vanAsscheQAperyResidualAtZero_ne_zero","module":"ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean","line":94,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean#L94","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean#L94","kind":"theorem","signature":"theorem vanAsscheQAperyResidualAtZero_ne_zero {p : ℝ} (hp : 1 < p) : vanAsscheQAperyResidualAtZero p ≠ 0","docstring":"Van Assche's diagonal fails the Amdeberhan--Zeilberger recurrence already at `n = 0`.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. 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Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:erdos_243","assertion_kind":"registered_claim","source_id":"erdos_243","reviewed":true,"granularity":"single_registered_claim","label":"Erdős #243: eventual Sylvester recurrence","statement":"Lean proves that if a is a strictly increasing sequence of positive integers with n^λ(a(n)²/a(n+1) − 1 − λ/n) → 0, then the reciprocal sum ∑ 1/a(n) is irrational, for every non-integer λ > 1, where Lean also derives convergence (nonintegral_regular_rate_irrational), and for λ = 3 whenever the sum converges (cubic_rate_irrationality_unconditional). Both theorems are zero-indexed; the paper transfers them to one-based indexing by an ordinary finite-prefix argument. Without such a rate the eventual Sylvester recurrence assertion remains open: the bounded-rise barrier and the conditional negative-part reductions leave the mixed-sign unbounded regime unresolved.","status":"open","status_meaning":"Not proved by this release","readme_headline":false,"problem_ids":["erdos_243"],"problem_erdos_numbers":[243],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_243","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_243","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"registry_statement_only","evidence_resolution":"statement_only","declarations":[],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.erdos_243_eventual_recurrence","statement":"Prove the Erdős #243 assertion: if 1 ≤ a₁ < a₂ < ··· has aₙ/aₙ₋₁² → 1 and rational reciprocal sum, then aₙ = aₙ₋₁² − aₙ₋₁ + 1 eventually; the checked barrier does not settle the mixed-sign unbounded regime.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/10"}],"links":[],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/130","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"erdos_243"}}],"expect":"object"},"prior_art":{"family_id":"reciprocal_tail_rigidity_frontier","status_summary":"An open eventual-Sylvester-recurrence endpoint plus Lean-checked irrationality theorems for the stated cubic rate and every nonintegral regular rate above one, unconditional bounded-rise and eventually-periodic negative-orbit obstructions, and a conditional cofinally bounded negative-magnitude exclusion.","prior_art_posture":"These are source-specific formal results; no novelty, priority, significance, or #243 solution claim is made, and normalised vanishing, mixed-sign, and unbounded-negative boundaries remain explicit.","primary_narrative_owner":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/27","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:bounded_rise_fresh_modulus_barrier","assertion_kind":"registered_claim","source_id":"bounded_rise_fresh_modulus_barrier","reviewed":true,"granularity":"single_registered_claim","label":"Bounded-rise fresh-modulus coprimality barrier","statement":"For natural sequences u and m, if u tends to infinity with upward increments bounded by a fixed positive B, the moduli m are eventually greater than 1 and pairwise coprime, and each earlier m(i) is coprime to every later u(t), then the hypotheses are inconsistent. This is a reusable state-level obstruction, not a proof of the full Erdős #243 assertion.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_243"],"problem_erdos_numbers":[243],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_243","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_243","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-243-reciprocal-tail-rigidity","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"no_boundedRise_of_tailAvoidance","module":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","line":903,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L903","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L903","kind":"theorem","signature":"theorem no_boundedRise_of_tailAvoidance (u m : ℕ → ℕ) (N B : ℕ) (hB : 0 < B) (hm : ∀ n, N ≤ n → 1 < m n) (hpair : ∀ {i j : ℕ}, N ≤ i → N ≤ j → i ≠ j → Nat.Coprime (m i) (m j)) (havoid : ∀ {i t : ℕ}, N ≤ i → i < t → Nat.Coprime (m i) (u t)) (hrise : ∀ n, N ≤ n → u (n + 1) ≤ u n + B) (huTop : Filter.Tendsto u Filter.atTop Filter.atTop) : False","docstring":"A sequence tending to infinity cannot have uniformly bounded upward increments while permanently avoiding infinitely many fresh pairwise-coprime moduli.","generated_certificate":false}],"paper_label":"res:barrier","paper_anchor":{"paper_label":"res:barrier","owner_source":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","source_line":595,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":595,"anchor_title":"Chinese remainder theorem and first crossing","anchor_title_tex":"Chinese remainder theorem and first crossing","section_title":"A stable gcd and a forbidden block","section_path":[{"level":"section","title":"Bounded increases","title_tex":"Bounded increases","line":471},{"level":"subsection","title":"A stable gcd and a forbidden block","title_tex":"A stable gcd and a forbidden block","line":561}],"source_excerpt":{"text":"\\begin{theorem}[Chinese remainder theorem and first crossing]\\label{res:barrier}\nLet $u:\\N\\to\\N$ tend to infinity and let $B\\ge1$ be an integer with\n$u_{n+1}\\le u_n+B$ for every $n$. There is no sequence of pairwise coprime\nintegers $m_i\\ge2$ for which $\\gcd(m_i,u_t)=1$ whenever $i<t$.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":25402,"source_char_end":25702,"start_line":595,"end_line":599,"truncated":false,"sha256":"sha256:db73d1081dc50706482861fe153dbaf598de086efab3554fbf8b5cb346ace379","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/243/erdos-243-reciprocal-tail-rigidity.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L595","owner_problem_erdos_number":243,"owner_paper_id":"erdos-243-reciprocal-tail-rigidity","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-243-reciprocal-tail-rigidity.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-243-reciprocal-tail-rigidity.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/243/erdos-243-reciprocal-tail-rigidity.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_243_eventual_recurrence","statement":"Prove the Erdős #243 assertion: if 1 ≤ a₁ < a₂ < ··· has aₙ/aₙ₋₁² → 1 and rational reciprocal sum, then aₙ = aₙ₋₁² − aₙ₋₁ + 1 eventually; the checked barrier does not settle the mixed-sign unbounded regime.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/10"}],"links":[{"kind":"lean_declaration_pinned","label":"no_boundedRise_of_tailAvoidance","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L903","cite_this":true},{"kind":"paper_pdf","label":"erdos-243-reciprocal-tail-rigidity","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-243-reciprocal-tail-rigidity.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/131","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"bounded_rise_fresh_modulus_barrier"}}],"expect":"object"},"prior_art":{"family_id":"reciprocal_tail_rigidity_frontier","status_summary":"An open eventual-Sylvester-recurrence endpoint plus Lean-checked irrationality theorems for the stated cubic rate and every nonintegral regular rate above one, unconditional bounded-rise and eventually-periodic negative-orbit obstructions, and a conditional cofinally bounded negative-magnitude exclusion.","prior_art_posture":"These are source-specific formal results; no novelty, priority, significance, or #243 solution claim is made, and normalised vanishing, mixed-sign, and unbounded-negative boundaries remain explicit.","primary_narrative_owner":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/27","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z17::n243_bounded_rise_barrier","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:eventually_periodic_negative_no_go","assertion_kind":"registered_claim","source_id":"eventually_periodic_negative_no_go","reviewed":true,"granularity":"single_registered_claim","label":"Eventually-periodic negative-orbit exclusion","statement":"For natural sequences a, D, C, and e with 2 ≤ a(n), positive e(N+n) < a(N+n), the exact recurrences D(n+1) = a(n)·D(n) and C(n+1) = C(n) + e(n), the shape equation D(n) + e(n) = (a(n)−1)·C(n), and a positive-period, positive-drift tail e(N+n+h) = e(N+n) and C(N+n+h) = C(N+n) + M with h > 0 and M > 0, the hypotheses are inconsistent. This is only the eventually-periodic negative-magnitude obstruction; mixed-sign and unbounded-negative behaviour remain outside the claim and the full Erdős #243 endpoint remains open.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_243"],"problem_erdos_numbers":[243],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_243","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_243","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"no_eventuallyPeriodicNegative_orbit","module":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","line":802,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L802","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L802","kind":"theorem","signature":"theorem no_eventuallyPeriodicNegative_orbit (a D C e : ℕ → ℕ) (N h M : ℕ) (hh : 0 < h) (hM : 0 < M) (ha : ∀ n, N ≤ n → 2 ≤ a n) (hepos : ∀ n, 0 < e (N + n)) (helt : ∀ n, e (N + n) < a (N + n)) (hD : ∀ n, N ≤ n → D (n + 1) = a n * D n) (hC : ∀ n, N ≤ n → C (n + 1) = C n + e n) (hshape : ∀ n, N ≤ n → D n + e n = (a n - 1) * C n) (hperiod : ∀ n, e (N + n + h) = e (N + n)) (hphase : ∀ n, C (N + n + h) = C (N + n) + M) : False","docstring":"An eventually periodic negative-magnitude tail is impossible in the same natural regime: shift to the first periodic index and apply `no_periodicNegative_orbit`. Every hypothesis is imposed only from the offset `N` onwards. The multiplier bound, the denominator recurrence, the numerator recurrence and the shape equation are consumed by the proof only at indices `N + n`, so restricting them to the tail costs nothing and matches the target object, whose product-cleared recurrence holds only from the clearing index onwards.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.erdos_243_eventual_recurrence","statement":"Prove the Erdős #243 assertion: if 1 ≤ a₁ < a₂ < ··· has aₙ/aₙ₋₁² → 1 and rational reciprocal sum, then aₙ = aₙ₋₁² − aₙ₋₁ + 1 eventually; the checked barrier does not settle the mixed-sign unbounded regime.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/10"}],"links":[{"kind":"lean_declaration_pinned","label":"no_eventuallyPeriodicNegative_orbit","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L802","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/132","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"eventually_periodic_negative_no_go"}}],"expect":"object"},"prior_art":{"family_id":"reciprocal_tail_rigidity_frontier","status_summary":"An open eventual-Sylvester-recurrence endpoint plus Lean-checked irrationality theorems for the stated cubic rate and every nonintegral regular rate above one, unconditional bounded-rise and eventually-periodic negative-orbit obstructions, and a conditional cofinally bounded negative-magnitude exclusion.","prior_art_posture":"These are source-specific formal results; no novelty, priority, significance, or #243 solution claim is made, and normalised vanishing, mixed-sign, and unbounded-negative boundaries remain explicit.","primary_narrative_owner":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/27","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z17::n243_periodic_negative_excluded","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:bounded_negative_exclusion","assertion_kind":"registered_claim","source_id":"bounded_negative_exclusion","reviewed":true,"granularity":"single_registered_claim","label":"Cofinally bounded negative-magnitude exclusion","statement":"For natural sequences a, C, D, and magnitude and B > 0 satisfying 1 < a(n), C(n) > 0, C(n+1) + D(n) = a(n)·C(n), D(n+1) = a(n)·D(n), bounded rise C(n+1) ≤ C(n) + B, magnitude(n) > 0, division-free normalised vanishing (for every K, eventually K·magnitude(n) < C(n)), and a cofinal negative branch with 0 < e ≤ B and D(t) + e = (a(t)−1)·C(t), the hypotheses are inconsistent. This conditional no-go does not decide the full Erdős #243 endpoint or the mixed-sign and unbounded-negative regimes.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_243"],"problem_erdos_numbers":[243],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_243","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_243","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"no_cofinallyBoundedNegative_of_normalizedVanishes","module":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","line":1754,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L1754","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L1754","kind":"theorem","signature":"theorem no_cofinallyBoundedNegative_of_normalizedVanishes (a C D magnitude : ℕ → ℕ) (B : ℕ) (hB : 0 < B) (ha : ∀ n, 1 < a n) (hCpos : ∀ n, 0 < C n) (hC : ∀ n, C (n + 1) + D n = a n * C n) (hD : ∀ n, D (n + 1) = a n * D n) (hRise : ∀ n, C (n + 1) ≤ C n + B) (hmagnitude : ∀ n, 0 < magnitude n) (hvanish : ∀ K, ∃ N, ∀ n, N ≤ n → K * magnitude n < C n) (hnegative : ∀ n, ∃ t e, n ≤ t ∧ 0 < e ∧ e ≤ B ∧ D t + e = (a t - 1) * C t) : False","docstring":"Division-free normalized vanishing closes the cofinally bounded negative branch by supplying the tail divergence needed above.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.erdos_243_eventual_recurrence","statement":"Prove the Erdős #243 assertion: if 1 ≤ a₁ < a₂ < ··· has aₙ/aₙ₋₁² → 1 and rational reciprocal sum, then aₙ = aₙ₋₁² − aₙ₋₁ + 1 eventually; the checked barrier does not settle the mixed-sign unbounded regime.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/10"}],"links":[{"kind":"lean_declaration_pinned","label":"no_cofinallyBoundedNegative_of_normalizedVanishes","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L1754","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/133","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"bounded_negative_exclusion"}}],"expect":"object"},"prior_art":{"family_id":"reciprocal_tail_rigidity_frontier","status_summary":"An open eventual-Sylvester-recurrence endpoint plus Lean-checked irrationality theorems for the stated cubic rate and every nonintegral regular rate above one, unconditional bounded-rise and eventually-periodic negative-orbit obstructions, and a conditional cofinally bounded negative-magnitude exclusion.","prior_art_posture":"These are source-specific formal results; no novelty, priority, significance, or #243 solution claim is made, and normalised vanishing, mixed-sign, and unbounded-negative boundaries remain explicit.","primary_narrative_owner":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/27","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z17::n243_no_cofinally_bounded_negative","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:cubic_survivor_phase_density","assertion_kind":"registered_claim","source_id":"cubic_survivor_phase_density","reviewed":true,"granularity":"single_registered_claim","label":"Phase-specific density obstruction for the two cubic survivor profiles","statement":"For either surviving cubic profile Q_{12,c}(n)=2n(n+1)(n+2)+c with c=1 or c=-1, the exact tail recurrences force a mismatch in every sufficiently late four-index window at the corresponding phase modulo 7. The mismatch set therefore has lower density at least 1/7, and hence at least the 1/28 stated in the long paper. No phase-free or arbitrary-profile density statement is claimed.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_243"],"problem_erdos_numbers":[243],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_243","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_243","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos243-reciprocal-tail-reasoning-surface","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"plus_phase_hit","module":"ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean","line":117,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean#L117","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean#L117","kind":"theorem","signature":"theorem plus_phase_hit (a u v : ℕ → ℤ) (T k : ℕ) (hT : T ≤ 7 * k) (hnum : ∀ j, T ≤ j → u (j + 1) + v j = a j * u j) (hden : ∀ j, T ≤ j → v (j + 1) = a j * v j) : ∃ i : ℕ, i < 4 ∧ u (7 * k + i) ≠ plusCubic (7 * k + i)","generated_certificate":false},{"name":"minus_phase_hit","module":"ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean","line":141,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean#L141","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean#L141","kind":"theorem","signature":"theorem minus_phase_hit (a u v : ℕ → ℤ) (T k : ℕ) (hT : T ≤ 1 + 7 * k) (hnum : ∀ j, T ≤ j → u (j + 1) + v j = a j * u j) (hden : ∀ j, T ≤ j → v (j + 1) = a j * v j) : ∃ i : ℕ, i < 4 ∧ u (1 + 7 * k + i) ≠ minusCubic (1 + 7 * k + i)","generated_certificate":false},{"name":"plus_profile_one_seventh","module":"ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean","line":165,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean#L165","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean#L165","kind":"theorem","signature":"theorem plus_profile_one_seventh (a u v : ℕ → ℤ) (T : ℕ) (hnum : ∀ j, T ≤ j → u (j + 1) + v j = a j * u j) (hden : ∀ j, T ≤ j → v (j + 1) = a j * v j) : ∀ X : ℕ, X ≤ 7 * exceptionCount {n | u n ≠ plusCubic n} X + (7 * T + 7)","docstring":"Correct one-seventh bound, for the PLUS terminal profile only.","generated_certificate":false},{"name":"minus_profile_one_seventh","module":"ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean","line":175,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean#L175","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean#L175","kind":"theorem","signature":"theorem minus_profile_one_seventh (a u v : ℕ → ℤ) (T : ℕ) (hnum : ∀ j, T ≤ j → u (j + 1) + v j = a j * u j) (hden : ∀ j, T ≤ j → v (j + 1) = a j * v j) : ∀ X : ℕ, X ≤ 7 * exceptionCount {n | u n ≠ minusCubic n} X + (7 * T + 8)","docstring":"Correct one-seventh bound, for the MINUS terminal profile only.","generated_certificate":false},{"name":"lowerDensityAtLeast_of_linear_bound","module":"ErdosProblems/Erdos243/PaperCompleteR11/WindowIncidence.lean","line":131,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/PaperCompleteR11/WindowIncidence.lean#L131","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/PaperCompleteR11/WindowIncidence.lean#L131","kind":"theorem","signature":"theorem lowerDensityAtLeast_of_linear_bound (E : Set ℕ) (a b c : ℝ) (hb : 0 < b) (h : ∀ X : ℕ, a * (X : ℝ) ≤ b * (exceptionCount E X : ℝ) + c) : LowerDensityAtLeast E (a / b)","docstring":"Convert an explicit real linear counting inequality into lower density.","generated_certificate":false}],"paper_label":"long243:res:modseven","paper_anchor":{"paper_label":"long243:res:modseven","owner_source":"paper/243/erdos243-reciprocal-tail-reasoning-surface.tex","source_line":563,"anchor_kind":"environment","anchor_unit":"lemma","anchor_unit_line":563,"anchor_title":"two incompatible four-term residue patterns","anchor_title_tex":"two incompatible four-term residue patterns","section_title":"The two remaining cubics fail modulo seven","section_path":[{"level":"section","title":"Irrationality at the cubic rate","title_tex":"Irrationality at the cubic rate","line":135},{"level":"subsection","title":"The two remaining cubics fail modulo seven","title_tex":"The two remaining cubics fail modulo seven","line":561}],"source_excerpt":{"text":"\\begin{lemma}[two incompatible four-term residue patterns]\\label{long243:res:modseven}\nFor $Q_{12,1}$, every sufficiently late four-index window beginning at\n$n\\equiv0\\pmod7$ contains an exceptional index. For $Q_{12,-1}$, the same\nholds for windows beginning at $n\\equiv1\\pmod7$. In either case,\n$\\underline d(S)\\ge1/7$.\nThe specified starting residue classes are part of the assertion; no\nclaim is made for every block of four consecutive indices.\n\\end{lemma}","media_type":"text/x-tex","source_char_start":25797,"source_char_end":26258,"start_line":563,"end_line":570,"truncated":false,"sha256":"sha256:9efabe7598665f7f532e0b652873dca5d608d57ec917a894669faaf12d333886","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex#L563","owner_problem_erdos_number":243,"owner_paper_id":"erdos243-reciprocal-tail-reasoning-surface","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos243-reciprocal-tail-reasoning-surface.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos243-reciprocal-tail-reasoning-surface.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/243/erdos243-reciprocal-tail-reasoning-surface.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_243_eventual_recurrence","statement":"Prove the Erdős #243 assertion: if 1 ≤ a₁ < a₂ < ··· has aₙ/aₙ₋₁² → 1 and rational reciprocal sum, then aₙ = aₙ₋₁² − aₙ₋₁ + 1 eventually; the checked barrier does not settle the mixed-sign unbounded regime.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/10"}],"links":[{"kind":"lean_declaration_pinned","label":"plus_phase_hit","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean#L117","cite_this":true},{"kind":"lean_declaration_pinned","label":"minus_phase_hit","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean#L141","cite_this":true},{"kind":"lean_declaration_pinned","label":"plus_profile_one_seventh","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean#L165","cite_this":true},{"kind":"lean_declaration_pinned","label":"minus_profile_one_seventh","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean#L175","cite_this":true},{"kind":"lean_declaration_pinned","label":"lowerDensityAtLeast_of_linear_bound","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/PaperCompleteR11/WindowIncidence.lean#L131","cite_this":true},{"kind":"paper_pdf","label":"erdos243-reciprocal-tail-reasoning-surface","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos243-reciprocal-tail-reasoning-surface.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/134","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"cubic_survivor_phase_density"}}],"expect":"object"},"prior_art":{"family_id":"reciprocal_tail_rigidity_frontier","status_summary":"An open eventual-Sylvester-recurrence endpoint plus Lean-checked irrationality theorems for the stated cubic rate and every nonintegral regular rate above one, unconditional bounded-rise and eventually-periodic negative-orbit obstructions, and a conditional cofinally bounded negative-magnitude exclusion.","prior_art_posture":"These are source-specific formal results; no novelty, priority, significance, or #243 solution claim is made, and normalised vanishing, mixed-sign, and unbounded-negative boundaries remain explicit.","primary_narrative_owner":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/27","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z17::n243_cubic_survivor_phase_density","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:cubic_rate_irrationality_unconditional","assertion_kind":"registered_claim","source_id":"cubic_rate_irrationality_unconditional","reviewed":true,"granularity":"single_registered_claim","label":"Cubic-rate reciprocal-sum irrationality under the stated asymptotic","statement":"For a zero-indexed strictly increasing sequence of positive natural numbers a, if n³(a(n)²/a(n+1) − (1 + 3/n)) tends to zero and the reciprocal series has sum S, then S is irrational. Lean proves the square-specialisation input through the Dedekind-zeta simple pole, so this theorem has no Chebotarev premise. The paper states the ordinary one-based bridge. The quantitative cubic-rate hypothesis is essential here; this does not settle the original eventual Sylvester recurrence assertion.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_243"],"problem_erdos_numbers":[243],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_243","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_243","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-243-reciprocal-tail-rigidity","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"cubic_rate_irrationality_unconditional","module":"ErdosProblems/Erdos243/PaperCompleteR21/SquareSpecialisationUnconditional.lean","line":70,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/PaperCompleteR21/SquareSpecialisationUnconditional.lean#L70","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/PaperCompleteR21/SquareSpecialisationUnconditional.lean#L70","kind":"theorem","signature":"theorem cubic_rate_irrationality_unconditional (a : ℕ → ℕ) (ha : StrictMono a) (hpos : ∀ n, 0 < a n) (hrate : Filter.Tendsto (fun n : ℕ => (n : ℝ) ^ 3 * ((a n : ℝ) ^ 2 / (a (n + 1) : ℝ) - (1 + 3 / (n : ℝ)))) Filter.atTop (nhds 0)) (Sv : ℝ) (hS : HasSum (fun n : ℕ => 1 / (a n : ℝ)) Sv) : Irrational Sv","docstring":"`res:cubicrate` / `long243:res:cubicrate`, unconditionally: a strictly increasing sequence of positive integers with `a n ^ 2 / a (n + 1) = 1 + 3 / n + o (n ^ (-3))` has irrational reciprocal sum.","generated_certificate":false}],"paper_label":"res:cubicrate","paper_anchor":{"paper_label":"res:cubicrate","owner_source":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","source_line":74,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":74,"anchor_title":"cubic-rate irrationality","anchor_title_tex":"cubic-rate irrationality","section_title":"Introduction","section_path":[{"level":"section","title":"Introduction","title_tex":"Introduction","line":48}],"source_excerpt":{"text":"\\begin{theorem}[cubic-rate irrationality]\\label{res:cubicrate}\nA strictly increasing sequence of positive integers with\n\\[\n a_n^2/a_{n+1}=1+\\frac3n+o(n^{-3})\n\\]\nhas irrational reciprocal sum.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":3158,"source_char_end":3363,"start_line":74,"end_line":80,"truncated":false,"sha256":"sha256:15b89b4f3d0c965c45e62dc66a5fa7e40bfbf22144d21b7c1f54bbf7ee4b15be","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/243/erdos-243-reciprocal-tail-rigidity.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L74","owner_problem_erdos_number":243,"owner_paper_id":"erdos-243-reciprocal-tail-rigidity","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-243-reciprocal-tail-rigidity.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-243-reciprocal-tail-rigidity.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/243/erdos-243-reciprocal-tail-rigidity.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_243_eventual_recurrence","statement":"Prove the Erdős #243 assertion: if 1 ≤ a₁ < a₂ < ··· has aₙ/aₙ₋₁² → 1 and rational reciprocal sum, then aₙ = aₙ₋₁² − aₙ₋₁ + 1 eventually; the checked barrier does not settle the mixed-sign unbounded regime.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/10"}],"links":[{"kind":"lean_declaration_pinned","label":"cubic_rate_irrationality_unconditional","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/PaperCompleteR21/SquareSpecialisationUnconditional.lean#L70","cite_this":true},{"kind":"paper_pdf","label":"erdos-243-reciprocal-tail-rigidity","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-243-reciprocal-tail-rigidity.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/135","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"cubic_rate_irrationality_unconditional"}}],"expect":"object"},"prior_art":{"family_id":"reciprocal_tail_rigidity_frontier","status_summary":"An open eventual-Sylvester-recurrence endpoint plus Lean-checked irrationality theorems for the stated cubic rate and every nonintegral regular rate above one, unconditional bounded-rise and eventually-periodic negative-orbit obstructions, and a conditional cofinally bounded negative-magnitude exclusion.","prior_art_posture":"These are source-specific formal results; no novelty, priority, significance, or #243 solution claim is made, and normalised vanishing, mixed-sign, and unbounded-negative boundaries remain explicit.","primary_narrative_owner":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/27","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:nonintegral_regular_rate_irrationality","assertion_kind":"registered_claim","source_id":"nonintegral_regular_rate_irrationality","reviewed":true,"granularity":"single_registered_claim","label":"Nonintegral regular-rate reciprocal-sum irrationality","statement":"For a zero-indexed strictly increasing sequence of positive natural numbers a and a real noninteger lambda greater than one, if n^lambda (a(n)^2/a(n+1) - (1 + lambda/n)) tends to zero, then the reciprocal series converges and its sum is irrational. The Lean theorem derives summability from the rate; the one-based paper form uses an ordinary finite-prefix bridge. The little-o remainder is essential to the integer-extraction argument. This does not settle the unrestricted Erdős #243 problem.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_243"],"problem_erdos_numbers":[243],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_243","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_243","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos243-reciprocal-tail-reasoning-surface","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"nonintegral_regular_rate_irrational","module":"ErdosProblems/Erdos243/PaperCompleteR21/NonintegralRegularRate.lean","line":196,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos243/PaperCompleteR21/NonintegralRegularRate.lean#L196","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/PaperCompleteR21/NonintegralRegularRate.lean#L196","kind":"theorem","signature":"theorem nonintegral_regular_rate_irrational (a : ℕ → ℕ) (ha : StrictMono a) (hpos : ∀ n, 0 < a n) (l : ℝ) (hl : 1 < l) (hnonint : ∀ d : ℕ, l ≠ (d : ℝ)) (hrate : Tendsto (fun n : ℕ => (n : ℝ) ^ l * ((a n : ℝ) ^ 2 / (a (n + 1) : ℝ) - (1 + l / (n : ℝ)))) atTop (nhds 0)) : Irrational (∑' n : ℕ, 1 / (a n : ℝ))","docstring":"Irrationality of the reciprocal sum at every nonintegral regular rate `l > 1`, under the sharp `o(n⁻ˡ)` ratio remainder. The summability of the reciprocal sequence is derived from the rate and strict increase.","generated_certificate":false}],"paper_label":"long243:res:nonintegralrate","paper_anchor":{"paper_label":"long243:res:nonintegralrate","owner_source":"paper/243/erdos243-reciprocal-tail-reasoning-surface.tex","source_line":154,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":154,"anchor_title":"nonintegral regular-rate irrationality","anchor_title_tex":"nonintegral regular-rate irrationality","section_title":"Irrationality at the cubic rate","section_path":[{"level":"section","title":"Irrationality at the cubic rate","title_tex":"Irrationality at the cubic rate","line":135}],"source_excerpt":{"text":"\\begin{theorem}[nonintegral regular-rate irrationality]\\label{long243:res:nonintegralrate}\nLet $\\lambda>1$ be a real number that is not an integer. If strictly\nincreasing positive integers satisfy\n$a_n^2/a_{n+1}=1+\\lambda/n+o(n^{-\\lambda})$, then\n$\\sum_n1/a_n$ is irrational.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":7055,"source_char_end":7344,"start_line":154,"end_line":159,"truncated":false,"sha256":"sha256:358ea1aa6744a5eb44a517af5d0656258ab325eaf4084f49aae16380f245a750","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex#L154","owner_problem_erdos_number":243,"owner_paper_id":"erdos243-reciprocal-tail-reasoning-surface","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos243-reciprocal-tail-reasoning-surface.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos243-reciprocal-tail-reasoning-surface.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/243/erdos243-reciprocal-tail-reasoning-surface.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_243_eventual_recurrence","statement":"Prove the Erdős #243 assertion: if 1 ≤ a₁ < a₂ < ··· has aₙ/aₙ₋₁² → 1 and rational reciprocal sum, then aₙ = aₙ₋₁² − aₙ₋₁ + 1 eventually; the checked barrier does not settle the mixed-sign unbounded regime.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/10"}],"links":[{"kind":"lean_declaration_pinned","label":"nonintegral_regular_rate_irrational","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos243/PaperCompleteR21/NonintegralRegularRate.lean#L196","cite_this":true},{"kind":"paper_pdf","label":"erdos243-reciprocal-tail-reasoning-surface","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos243-reciprocal-tail-reasoning-surface.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/136","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"nonintegral_regular_rate_irrationality"}}],"expect":"object"},"prior_art":{"family_id":"reciprocal_tail_rigidity_frontier","status_summary":"An open eventual-Sylvester-recurrence endpoint plus Lean-checked irrationality theorems for the stated cubic rate and every nonintegral regular rate above one, unconditional bounded-rise and eventually-periodic negative-orbit obstructions, and a conditional cofinally bounded negative-magnitude exclusion.","prior_art_posture":"These are source-specific formal results; no novelty, priority, significance, or #243 solution claim is made, and normalised vanishing, mixed-sign, and unbounded-negative boundaries remain explicit.","primary_narrative_owner":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/27","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:finite_period_noncollapse","assertion_kind":"registered_claim","source_id":"finite_period_noncollapse","reviewed":true,"granularity":"single_registered_claim","label":"Finite-support denominator period noncollapse","statement":"For every finite nonempty support F with 0 ∉ F and integer base b ≥ 2, the reduced denominator of the literal finite sum ∑_{n∈F} 1/(bⁿ−1) is coprime to b and the multiplicative order of b modulo that denominator equals F.lcm id. This finite-support theorem supplies no unbounded support certificate and does not settle the universal infinite-support Erdős #257 endpoint.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-257-mersenne-support-subseries","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"finite_period_noncollapse_rat_den","module":"Erdos249257/CertificateKernel.lean","line":5246,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/CertificateKernel.lean#L5246","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L5246","kind":"theorem","signature":"theorem finite_period_noncollapse_rat_den (F : Finset Nat) (b : Nat) (hF : F.Nonempty) (h0 : 0 ∉ F) (hb : 2 ≤ b) : orderOf (ZMod.unitOfCoprime b (coprime_base_den_finiteErdosSum F b h0 hb)) = F.lcm id","docstring":"The unconditional finite theorem stated over `Rat.den` of the literal rational partial sum: `ord_Q(b) = lcm F` where `Q` is the denominator in lowest terms of `∑_{n ∈ F} 1 / (b ^ n - 1)` as a rational number.","generated_certificate":false}],"paper_label":"res:period","paper_anchor":{"paper_label":"res:period","owner_source":"paper/257/erdos-257-mersenne-support-subseries.tex","source_line":995,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":995,"anchor_title":"the exact denominator period","anchor_title_tex":"the exact denominator period","section_title":"Finite-support denominator periods","section_path":[{"level":"section","title":"Finite-support denominator periods","title_tex":"Finite-support denominator periods","line":990}],"source_excerpt":{"text":"\\begin{theorem}[the exact denominator period]\\label{res:period}\nLet $F\\subseteq\\Npos$ be finite and nonempty, let $b\\ge2$ be an integer, and\nlet $D_F>0$ be the denominator of $X_F(b)$ in lowest terms.  Then $D_F$ is\ncoprime to $b$, and\n\\[\n  \\operatorname{ord}_{D_F}(b)=\\lcm\\{n:n\\in F\\}.\n\\]\nIf moreover $\\lcm(F)\\ge2$, then $\\lcm(F)<D_F$.  We use\n$\\operatorname{ord}_1(b)=1$, so the statement includes $F=\\{1\\}$\nat $b=2$.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":46619,"source_char_end":47052,"start_line":995,"end_line":1005,"truncated":false,"sha256":"sha256:a55a97c09113568eee0fde7f55bf2ce0ae55228c8701592e89c9f4f945f3aeb3","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/257/erdos-257-mersenne-support-subseries.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/257/erdos-257-mersenne-support-subseries.tex#L995","owner_problem_erdos_number":257,"owner_paper_id":"erdos-257-mersenne-support-subseries","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-257-mersenne-support-subseries.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-257-mersenne-support-subseries.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/257/erdos-257-mersenne-support-subseries.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"finite_period_noncollapse_rat_den","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/CertificateKernel.lean#L5246","cite_this":true},{"kind":"paper_pdf","label":"erdos-257-mersenne-support-subseries","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-257-mersenne-support-subseries.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/137","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"finite_period_noncollapse"}}],"expect":"object"},"prior_art":{"family_id":"finite_period_noncollapse_frontier","status_summary":"An unconditional finite-support denominator noncollapse theorem for Mersenne-type sums, with the infinite-support irrationality boundary still open.","prior_art_posture":"The finite-support order theorem is a source-specific formal result; no novelty, priority, significance, or universal #257 solution claim is made.","primary_narrative_owner":"paper/257/erdos-257-mersenne-support-subseries.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/28","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"exact_local_id","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z01::finite_period_noncollapse","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:all_base_totient_kernel_conditional_rank","assertion_kind":"registered_claim","source_id":"all_base_totient_kernel_conditional_rank","reviewed":true,"granularity":"single_registered_claim","label":"Conditional exact rank of the all-base totient kernel","statement":"For integers k and e with 2 ≤ k and 1 ≤ e, if the explicit canonical all-base family is linearly independent over ℚ, then the span of the actual all-base totient kernel through level e has dimension k^e + 1. This reusable interface retains the independence input; AllBaseTotientKernel.lean separately discharges it at public checkpoint 25ef6245d15a. As before, this coefficient-space result does not prove irrationality of the binary totient series.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent","module":"Erdos249257/TotientKernelConditional.lean","line":215,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/TotientKernelConditional.lean#L215","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelConditional.lean#L215","kind":"theorem","signature":"theorem finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent (k e : ℕ) (hk : 2 ≤ k) (he : 1 ≤ e) (hcanon : LinearIndependent ℚ (canonicalAllBaseTotientKernelFamily k e)) : finrank ℚ (Submodule.span ℚ (Set.range (allBaseTotientKernelThroughLevelFamily k e))) = k ^ e + 1","docstring":"**Conditional exact rank of the actual all-base truncation.** The arithmetic spanning and index-cardinality layers are formalised above. The only hypothesis is linear independence of the explicit canonical family, which is the external affine-ordering boundary and is not discharged here.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.erdos_249_irrationality","statement":"Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/1"}],"links":[{"kind":"lean_declaration_pinned","label":"finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/TotientKernelConditional.lean#L215","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/138","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"all_base_totient_kernel_conditional_rank"}}],"expect":"object"},"prior_art":{"family_id":"alternative_249_algebraic_boundaries","status_summary":"Exact finite obstructions and interfaces, a constructive dyadic basis, and an all-base arithmetic/spanning layer. The reusable rank wrapper is conditional; AllBaseTotientKernel.lean separately proves its independence premise and the complete rank unconditionally. The residue-series claims classify the rational totient residue series and prove the dilated series linearly independent.","prior_art_posture":"Classical Euler/Möbius, CRT, determinant, and Lambert tools; Coons already implies full-kernel infinite rank, and Martin supplies a broader external affine-independence antecedent. The Lean all-base proof uses CRT and Dirichlet independently. Erdős 1957 (Lemma 1) and Kaneko–Suzuki–Tachiya (Theorem 3) already give the irrationality of the totient residue series. No #249 irrationality endpoint or priority claim.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/17","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"formalisation_only":1},"edges":[{"semantic_node_id":"Z09::all_base_totient_truncation_rank_of_linear_independence","prior_art_state":"formalisation_only"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:erdos_249","assertion_kind":"registered_claim","source_id":"erdos_249","reviewed":true,"granularity":"single_registered_claim","label":"Erdős #249: irrationality of S = ∑ φ(n)/2ⁿ","statement":"Lean proves that S = ∑ φ(n)/2ⁿ is irrational if, for every h ≥ 1, there are arbitrarily large X at which the real part of the first-harmonic sum over the good indices is at most 603X/1000 (irrational_totient_series_of_goodBase_gap); the good indices are the assigned indices N in [X, 2X) at the minimal admissible depth with s = 26 whose cofactor m satisfies φ(m) ≥ m/1000. For all large X more than 67X/100 indices are good (eventually_card_pivotGoodBases_gt), and 603/1000 = (9/10)(67/100). Lean also proves that S is irrational if and only if certified non-integrality witnesses occur at unbounded parameters (irrational_totient_series_iff_certificate_supply). The 603X/1000 bound and that unbounded certificate supply are open, so S is not proved irrational.","status":"open","status_meaning":"Not proved by this release","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"registry_statement_only","evidence_resolution":"statement_only","declarations":[],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.erdos_249_irrationality","statement":"Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/1"},{"open_proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/3"}],"links":[],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/139","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"erdos_249"}}],"expect":"object"},"prior_art":{"family_id":"erdos249_quantitative_headline","status_summary":"One unconditional quantitative theorem and the open irrationality claim.","prior_art_posture":"Farey is historical language only; no exact antecedent for the explicit bound is identified, without asserting novelty.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/13","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex"},"source_lane":"registry_claim"},{"assertion_id":"claim:universal_257","assertion_kind":"registered_claim","source_id":"universal_257","reviewed":true,"granularity":"single_registered_claim","label":"Erdős #257, universal form","statement":"Lean proves that for every integer base b ≥ 2 and every finite nonempty set P of primes, with h(a) the largest divisor of a supported on P, every infinite set A of positive integers with ∑_{a∈A} h(a)/(a(b^{h(a)} − 1)) finite has ∑_{n∈A} 1/(bⁿ − 1) irrational (divisibilityWeightedClaim). Lean also proves irrationality at every such base for every infinite A with ∑_{a∈A} 1/a finite (irrational_erdosSupportSeries_of_summable_reciprocal), and exhibits sets with divergent reciprocal sum all of whose infinite subsets have irrational series at every such base (finite_monotone_witness_rule_realised). Irrationality of ∑_{n∈A} 1/(2ⁿ − 1) for every infinite A remains open.","status":"open","status_meaning":"Not proved by this release","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"registry_statement_only","evidence_resolution":"statement_only","declarations":[],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.half_value_membership","statement":"Decide whether 1/2 lies in the Mersenne achievement set. Yes refutes universal #257; no closes only this route.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/4"},{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/140","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"universal_257"}}],"expect":"object"},"prior_art":{"family_id":"classical_full_support_and_named_257_families","status_summary":"Classical formalisation, cited special cases, and the open universal target.","prior_art_posture":"Erdős owns full support and the pairwise-coprime family; Duverney–Tachiya own squarefree-support irrationality at every power-of-two base; Tao–Teräväinen own the base-2 prime-support theorem. The cited squarefree theorem does not cover integer bases that are not powers of two, and the cited prime-power extension is remark-only. No priority claim is made for the formalised families.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/0","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","primary_narrative_owner_material_ref":"/mathematical_assertions/prior_art_owner_material/paper~1erdos249-257-main-paper.tex"},"source_lane":"registry_claim"},{"assertion_id":"claim:weighted_phase_carry_finite_observer","assertion_kind":"registered_claim","source_id":"weighted_phase_carry_finite_observer","reviewed":true,"granularity":"single_registered_claim","label":"Weighted-phase carry split and finite realised span for #269","statement":"The exact weighted-phase carry recurrence splits into a finite residue digit, each of the two coordinates staying inside its declared interval, and an integral quotient coboundary that is not controlled. An explicit factorisation through a finite-dimensional function-faithful observer then forces the realised symbolic span to be finite. Nothing here proves a finite-state quotient or a literal infinite realised span, and no rationality or irrationality conclusion follows.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_269"],"problem_erdos_numbers":[269],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_269","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_269","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"carry_eq_residueDigit_add_coboundary","module":"ErdosProblems/Erdos269/WeightedPhaseCarry.lean","line":109,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean#L109","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean#L109","kind":"theorem","signature":"theorem carry_eq_residueDigit_add_coboundary (B : ℤ) (hB : 0 < B) (base carry digit : ℕ → ℤ) (hrec : ∀ n, carry (n + 1) = base n * carry n - B * digit n) : let residue","docstring":"Rationality's exact carry recurrence decomposes into a finite residue digit and an integral coboundary. No recurrence or finite-state property of the quotient coordinate is assumed or concluded.","generated_certificate":false},{"name":"carryResidue_mem_interval","module":"ErdosProblems/Erdos269/WeightedPhaseCarry.lean","line":150,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean#L150","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean#L150","kind":"theorem","signature":"theorem carryResidue_mem_interval (B c : ℤ) (hB : 0 < B) : 0 ≤ carryResidue B c ∧ carryResidue B c < B","docstring":"The residue coordinate is genuinely finite for a positive modulus.","generated_certificate":false},{"name":"residueDigit_mem_interval","module":"ErdosProblems/Erdos269/WeightedPhaseCarry.lean","line":157,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean#L157","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean#L157","kind":"theorem","signature":"theorem residueDigit_mem_interval (B base residue nextResidue : ℤ) (hB : 0 < B) (hbase : 0 < base) (hresidue : 0 ≤ residue ∧ residue < B) (hnext : nextResidue = (base * residue) % B) : 0 ≤ residueDigit B base residue nextResidue ∧ residueDigit B base residue nextResidue < base","docstring":"When the next residue is the canonical reduction of `base * residue`, the corresponding digit lies in the expected finite alphabet.","generated_certificate":false},{"name":"finite_realisedSpan_of_factorisation","module":"ErdosProblems/Erdos269/WeightedPhaseCarry.lean","line":334,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean#L334","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean#L334","kind":"theorem","signature":"theorem finite_realisedSpan_of_factorisation {𝕜 State Func E : Type*} [DivisionRing 𝕜] [AddCommGroup State] [Module 𝕜 State] [AddCommGroup Func] [Module 𝕜 Func] [AddCommGroup E] [Module 𝕜 E] [FiniteDimensional 𝕜 E] (realise : State →ₗ[𝕜] Func) (C : State →ₗ[𝕜] E) (readout : E →ₗ[𝕜] Func) (hfactor : realise = readout.comp C) : FiniteDimensional 𝕜 (LinearMap.range realise)","docstring":"An explicit factorization through a finite-dimensional observer forces the realised symbolic span to be finite-dimensional. A scalar value alone does not provide this factorization.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.erdos_269_two_dimensional_representation","statement":"Transcendence of D_{2,3,5}, which is irrational: represent it faithfully by a specified two-dimensional Mahler-type value theorem with every hypothesis verified, give a conditional theorem under explicit logarithmic nondegeneracy, or prove no such representation exists.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/13"}],"links":[{"kind":"lean_declaration_pinned","label":"carry_eq_residueDigit_add_coboundary","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean#L109","cite_this":true},{"kind":"lean_declaration_pinned","label":"carryResidue_mem_interval","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean#L150","cite_this":true},{"kind":"lean_declaration_pinned","label":"residueDigit_mem_interval","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean#L157","cite_this":true},{"kind":"lean_declaration_pinned","label":"finite_realisedSpan_of_factorisation","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean#L334","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/141","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"weighted_phase_carry_finite_observer"}}],"expect":"object"},"prior_art":{"family_id":"three_prime_residue_escape_consumers","status_summary":"An open three-prime #269 endpoint, a finite least-positive-residue contradiction, a conditional cofinal-window extinction consumer, and the weighted-phase carry split with its finite realised-span observer.","prior_art_posture":"The residue classifier and bounded-carry consumer are source-specific formal results; no novelty, priority, significance, or three-prime irrationality claim is made.","primary_narrative_owner":"paper/269/erdos-269-three-prime-running-lcm.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/23","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:totient_v8_conditional_peripheral_assembly","assertion_kind":"registered_claim","source_id":"totient_v8_conditional_peripheral_assembly","reviewed":true,"granularity":"single_registered_claim","label":"Conditional peripheral-and-centred assembly for the #249 first-harmonic route","statement":"Under eventual peripheral estimates and a cofinal centered estimate at one shared depth function, the exact finite decomposition supplies the first-harmonic gap consumer and hence the conditional irrationality route. Both analytic suppliers remain explicit hypotheses; no unconditional Erdős #249 conclusion follows.","status":"conditional reduction","status_meaning":"Exact implication whose remaining hypothesis is open","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"card_pivotFiber_eq_card_supplierPrimes","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralAssembly.lean","line":12,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralAssembly.lean#L12","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralAssembly.lean#L12","kind":"theorem","signature":"theorem card_pivotFiber_eq_card_supplierPrimes {X L s m : ℕ} (hm : 0 < m) (hmsmall : m ≤ Nat.sqrt X / 2) : (pivotFiber X L s m).card = (pivotSupplierPrimes X L s m).card","generated_certificate":false},{"name":"dtw_of_eventual_peripheral_and_cofinal_centered","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralAssembly.lean","line":33,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralAssembly.lean#L33","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralAssembly.lean#L33","kind":"theorem","signature":"theorem dtw_of_eventual_peripheral_and_cofinal_centered (depth : ℕ → ℕ) (hperipheral : ∀ h : ℕ, 0 < h → ∃ A : ℕ, ∀ X : ℕ, A ≤ X → PeripheralAt depth h X) (hcentered : ∀ h : ℕ, 0 < h → ∀ A : ℕ, ∃ X : ℕ, max A 1 ≤ X ∧ (pivotCenteredCorrelation h X (depth X) 26 (1 / 1000 : ℝ)).re ≤ (14 / 25 : ℝ) * X) : DTWPivotResidualDecorrelation","docstring":"Eventual peripheral estimates intersect a cofinal centred estimate at the same depth. This is NOT a proof of either analytic hypothesis.","generated_certificate":false},{"name":"irrational_of_eventual_peripheral_and_cofinal_centered","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralAssembly.lean","line":54,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralAssembly.lean#L54","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralAssembly.lean#L54","kind":"theorem","signature":"theorem irrational_of_eventual_peripheral_and_cofinal_centered (depth : ℕ → ℕ) (hperipheral : ∀ h : ℕ, 0 < h → ∃ A : ℕ, ∀ X : ℕ, A ≤ X → PeripheralAt depth h X) (hcentered : ∀ h : ℕ, 0 < h → ∀ A : ℕ, ∃ X : ℕ, max A 1 ≤ X ∧ (pivotCenteredCorrelation h X (depth X) 26 (1 / 1000 : ℝ)).re ≤ (14 / 25 : ℝ) * X) : Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","docstring":"The conditional arithmetic endpoint, with both missing inputs explicit.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.erdos_249_irrationality","statement":"Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/1"}],"links":[{"kind":"lean_declaration_pinned","label":"card_pivotFiber_eq_card_supplierPrimes","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralAssembly.lean#L12","cite_this":true},{"kind":"lean_declaration_pinned","label":"dtw_of_eventual_peripheral_and_cofinal_centered","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralAssembly.lean#L33","cite_this":true},{"kind":"lean_declaration_pinned","label":"irrational_of_eventual_peripheral_and_cofinal_centered","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralAssembly.lean#L54","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/142","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"totient_v8_conditional_peripheral_assembly"}}],"expect":"object"},"prior_art":{"family_id":"harmonic_and_phase_reductions","status_summary":"Conditional harmonic and phase consumers, including an exact four-term pivot decomposition and budget, an exact window-pair equivalence, and the two-branch tail-orbit phase-density and natural-prime strict-gap producers that feed the same certificate socket.","prior_art_posture":"No matching combined theorem is identified; all novelty language remains provisional.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/19","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z08::n249_conditional_peripheral_assembly","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:totient_v8_finite_arithmetic_threshold","assertion_kind":"registered_claim","source_id":"totient_v8_finite_arithmetic_threshold","reviewed":true,"granularity":"single_registered_claim","label":"Exact finite arithmetic and run-threshold checks for #249","statement":"The four displayed discrepancy values at (h,X,L)=(1,4,9), their finite room and certified kills, the Farey denominator bound, and the corrected finite run threshold hold exactly. These finite facts do not supply a cofinal certificate producer.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_249"],"problem_erdos_numbers":[249],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_249","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_249","relation":"direct_target","basis":"claim_prose"}],"programme_ids":[],"publication_edges":[],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"discrepancy_1_4_9","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":9,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean#L9","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean#L9","kind":"theorem","signature":"theorem discrepancy_1_4_9 : windowDiscrepancy 1 4 9 = -62","generated_certificate":false},{"name":"discrepancy_1_5_9","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":10,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean#L10","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean#L10","kind":"theorem","signature":"theorem discrepancy_1_5_9 : windowDiscrepancy 1 5 9 = 902","generated_certificate":false},{"name":"discrepancy_1_6_9","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":11,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean#L11","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean#L11","kind":"theorem","signature":"theorem discrepancy_1_6_9 : windowDiscrepancy 1 6 9 = -244","generated_certificate":false},{"name":"discrepancy_1_7_9","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":12,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean#L12","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean#L12","kind":"theorem","signature":"theorem discrepancy_1_7_9 : windowDiscrepancy 1 7 9 = 544","generated_certificate":false},{"name":"finite_room","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":13,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean#L13","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean#L13","kind":"theorem","signature":"theorem finite_room : 16 * (2 * 4 + 1 + 9 + 2) ≤ 2 ^ 9","generated_certificate":false},{"name":"four_certified_kills","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":14,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean#L14","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean#L14","kind":"theorem","signature":"theorem four_certified_kills : certifiedKill 1 4 9 ∧ certifiedKill 1 5 9 ∧ certifiedKill 1 6 9 ∧ certifiedKill 1 7 9","generated_certificate":false},{"name":"farey_denominator_bound","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":19,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean#L19","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean#L19","kind":"theorem","signature":"theorem farey_denominator_bound (a b c d r s : ℤ) (hb : 0 ≤ b) (hd : 0 ≤ d) (hdet : b * c - a * d = 1) (hl : 0 < r * b - a * s) (hu : 0 < c * s - r * d) : b + d ≤ s","docstring":"The finite denominator lemma underlying both retained and new Farey certificates. 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The rounded constant 9.1 is not an equivalence.","generated_certificate":false},{"name":"run_threshold_iff","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":39,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean#L39","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean#L39","kind":"theorem","signature":"theorem run_threshold_iff (X C : ℝ) (hC : 0 ≤ C) (hX : 0 ≤ X) : (X + 1) / (C + 1) ≤ 100 * (X + 1) / (11 * X + 100) ↔ 11 * X ≤ 100 * C","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. 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non-suppliers.","generated_certificate":false},{"name":"pivotGoodBases_subset_Ico","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean","line":56,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean#L56","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean#L56","kind":"theorem","signature":"theorem pivotGoodBases_subset_Ico (X L s : ℕ) (η : ℝ) : pivotGoodBases X L s η ⊆ Finset.Ico X (2 * X)","docstring":"Good bases are a subset of the ambient dyadic block.","generated_certificate":false},{"name":"card_pivotGoodBases_le","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean","line":64,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean#L64","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean#L64","kind":"theorem","signature":"theorem card_pivotGoodBases_le (X L s : ℕ) (η : ℝ) : (pivotGoodBases X L s η).card ≤ X","docstring":"Hence there are at most `X` good bases.","generated_certificate":false},{"name":"norm_pivotFiberMeanContribution_le_of_uniform","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean","line":76,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean#L76","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean#L76","kind":"theorem","signature":"theorem norm_pivotFiberMeanContribution_le_of_uniform (h X L s : ℕ) (η ε : ℝ) (hε : 0 ≤ ε) (hmean : ∀ N ∈ pivotGoodBases X L s η, ‖pivotFiberMean h X L s (pivotCofactor N L s)‖ ≤ ε) : ‖pivotFiberMeanContribution h X L s η‖ ≤ ε * X","docstring":"A uniform fibre-mean estimate immediately bounds the whole mean contribution, because every residual has unit norm.","generated_certificate":false},{"name":"pivotBudgetAt_of_peripheral_estimates","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean","line":105,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean#L105","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean#L105","kind":"theorem","signature":"theorem pivotBudgetAt_of_peripheral_estimates (h X L s : ℕ) (η : ℝ) (hcentered : (pivotCenteredCorrelation h X L s η).re ≤ (14 / 25 : ℝ) * X) (hmean : ∀ N ∈ pivotGoodBases X L s η, ‖pivotFiberMean h X L s (pivotCofactor N L s)‖ ≤ (1 / 100 : ℝ)) (hbad : ((pivotBadBases X L s η).card : ℝ) ≤ (1 / 100 : ℝ) * X) (hnon : ((pivotNonSupplierBases X L s).card : ℝ) ≤ (8 / 25 : ℝ) * X) : PivotBudgetAt h X L s η","docstring":"Finite assembly: cardinal estimates plus a uniform fibre-mean estimate supply the three peripheral clauses, leaving only the centred real part.","generated_certificate":false}],"paper_label":null,"paper_anchor":null,"paper_anchor_absent_reason":"the claims registry records no paper_label for this claim. No anchor is synthesised: a manufactured section reference is indistinguishable from a real one to the reader who follows it.","does_not_establish":[{"open_proposition_id":"remaining_open.erdos_249_irrationality","statement":"Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/1"}],"links":[{"kind":"lean_declaration_pinned","label":"norm_pivotPhaseAt","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean#L24","cite_this":true},{"kind":"lean_declaration_pinned","label":"norm_pivotResidualAt_v8","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean#L28","cite_this":true},{"kind":"lean_declaration_pinned","label":"norm_pivotBadContribution_le_card","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean#L34","cite_this":true},{"kind":"lean_declaration_pinned","label":"norm_pivotNonSupplierContribution_le_card","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean#L45","cite_this":true},{"kind":"lean_declaration_pinned","label":"pivotGoodBases_subset_Ico","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean#L56","cite_this":true},{"kind":"lean_declaration_pinned","label":"card_pivotGoodBases_le","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean#L64","cite_this":true},{"kind":"lean_declaration_pinned","label":"norm_pivotFiberMeanContribution_le_of_uniform","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean#L76","cite_this":true},{"kind":"lean_declaration_pinned","label":"pivotBudgetAt_of_peripheral_estimates","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean#L105","cite_this":true}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/146","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"totient_v8_peripheral_finite_budget_bridge"}}],"expect":"object"},"prior_art":{"family_id":"harmonic_and_phase_reductions","status_summary":"Conditional harmonic and phase consumers, including an exact four-term pivot decomposition and budget, an exact window-pair equivalence, and the two-branch tail-orbit phase-density and natural-prime strict-gap producers that feed the same certificate socket.","prior_art_posture":"No matching combined theorem is identified; all novelty language remains provisional.","primary_narrative_owner":"docs/claims.json::claims","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/19","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z08::n249_peripheral_finite_budget_bridge","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:zudilin_rational_base_region","assertion_kind":"registered_claim","source_id":"zudilin_rational_base_region","reviewed":true,"granularity":"single_registered_claim","label":"Zudilin rational-base irrationality and irrationality-exponent bound","statement":"For natural numbers 0 < b < a satisfying ZudilinContourRegion a b, the Lambert value F(a/b) is irrational and its irrationality exponent is at most rationalBaseMeasureBound a b. The proof constructs the polynomial forms and proves their clearing, nonvanishing and asymptotic estimates. In particular, F((31/4)^r) is irrational for every positive natural r. These are formalizations of the stated Zudilin region and its consequences, not universal rational-base irrationality; 3/2 is outside the region.","status":"formalised here","status_meaning":"Known mathematics represented and checked in Lean","readme_headline":false,"problem_ids":["erdos_1049"],"problem_erdos_numbers":[1049],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_1049","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_1049","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-1049-rational-base-lambert","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"rational_base_region","module":"ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean","line":113,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean#L113","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean#L113","kind":"theorem","signature":"theorem rational_base_region (a b : ℕ) (hb : 0 < b) (hab : b < a) (hr : ZudilinContourRegion a b) : Irrational (paperLambert ((a : ℝ) / b))","docstring":"The paper's region, now with the supplier applied rather than assumed.","generated_certificate":false},{"name":"thirtyone_four_powers","module":"ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean","line":117,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean#L117","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean#L117","kind":"theorem","signature":"theorem thirtyone_four_powers (r : ℕ) (hr : 0 < r) : Irrational (paperLambert (((31 : ℝ) / 4) ^ r))","generated_certificate":false},{"name":"rational_base_measure","module":"ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean","line":154,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean#L154","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean#L154","kind":"theorem","signature":"theorem rational_base_measure (a b : ℕ) (hb : 0 < b) (hab : b < a) (hr : ZudilinContourRegion a b) : irrationalityExponent (paperLambert ((a : ℝ) / b)) ≤ rationalBaseMeasureBound a b","docstring":"The actual supremum definition from R11, not merely a named rate quotient.","generated_certificate":false}],"paper_label":"res:rational-base-threshold","paper_anchor":{"paper_label":"res:rational-base-threshold","owner_source":"paper/1049/erdos-1049-rational-base-lambert.tex","source_line":676,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":675,"anchor_title":"rational-base region for Zudilin's forms","anchor_title_tex":"rational-base region for Zudilin's forms","section_title":"Rational bases","section_path":[{"level":"section","title":"Rational bases","title_tex":"Rational bases","line":628}],"source_excerpt":{"text":"\\begin{theorem}[rational-base region for Zudilin's forms]\n\\label{res:rational-base-threshold}\nLet $a>b\\ge1$ be coprime integers with\n\\[\n\\begin{gathered}\n b^{\\mu}<a,\\qquad\\text{equivalently}\\qquad\n \\frac{\\log b}{\\log a}<\\theta^*,\\\\\n \\theta^*=\\frac{C_0}{C_1}=0.4056830213840605\\ldots,\\\\\n \\mu=\\frac{C_1}{C_0}=2.4649786835749750\\ldots.\n\\end{gathered}\n\\]\nThen $F(a/b)$ is irrational.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":28356,"source_char_end":28748,"start_line":675,"end_line":687,"truncated":false,"sha256":"sha256:1976c8b96b507846d0bcefd734f79c107cbe20c817fc5b831da4d0a19d34cc32","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1049/erdos-1049-rational-base-lambert.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1049/erdos-1049-rational-base-lambert.tex#L676","owner_problem_erdos_number":1049,"owner_paper_id":"erdos-1049-rational-base-lambert","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-1049-rational-base-lambert.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-1049-rational-base-lambert.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/1049/erdos-1049-rational-base-lambert.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_1049_irrationality","statement":"Determine whether F(t) = ∑_{n≥1} 1/(tⁿ − 1) is irrational for every rational t > 1. Irrationality of F((31/4)^r) for every positive integer r is a Lean-checked Zudilin specialization; the 3/2 approximant with remainder control and the universal assertion remain open.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/9"}],"links":[{"kind":"lean_declaration_pinned","label":"rational_base_region","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean#L113","cite_this":true},{"kind":"lean_declaration_pinned","label":"thirtyone_four_powers","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean#L117","cite_this":true},{"kind":"lean_declaration_pinned","label":"rational_base_measure","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean#L154","cite_this":true},{"kind":"paper_pdf","label":"erdos-1049-rational-base-lambert","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-1049-rational-base-lambert.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/147","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"zudilin_rational_base_region"}}],"expect":"object"},"prior_art":{"family_id":"zudilin_rational_base_and_hankel_theorems","status_summary":"Formalized rational-base irrationality and measure in the contour region, all-rank normalized Hankel order and coefficient formulas, and the coefficient-pencil proposition through rank eight.","prior_art_posture":"Zudilin supplies the cited 2004 polynomial-form construction and 2016 Hankel sequence. The formalizations preserve their hypotheses and make no novelty, priority or independent human-review claim.","primary_narrative_owner":"paper/1049/erdos-1049-rational-base-lambert.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/30","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:zudilin_all_rank_hankel_order","assertion_kind":"registered_claim","source_id":"zudilin_all_rank_hankel_order","reviewed":true,"granularity":"single_registered_claim","label":"All-rank normalized Hankel order and leading coefficient","statement":"For every natural N, including N = 0, the normalized Zudilin Hankel determinant has q-order N(N-1)(2N-1)/6. Its leading coefficient satisfies the division-free identity 2^N times the coefficient = (N!)^2 (N+1)!, with the corresponding rational quotient formula. The all-row producer is proved rather than assumed. This algebraic theorem does not by itself supply the analytic estimates or irrationality at 3/2. Zudilin (2016) proves the q-order inequality; the equality is proved here.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_1049"],"problem_erdos_numbers":[1049],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_1049","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_1049","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-1049-rational-base-lambert","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"zudilinSharpHankelOrderAndCoeff_all","module":"ErdosProblems/Erdos1049/AllRow/Producer.lean","line":163,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/AllRow/Producer.lean#L163","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/AllRow/Producer.lean#L163","kind":"theorem","signature":"theorem zudilinSharpHankelOrderAndCoeff_all (N : ℕ) : PowerSeries.order (zudilinNormalizedHankelDet N) = ((∑ j ∈ Finset.range N, j ^ 2 : ℕ) : ℕ∞) ∧ PowerSeries.coeff (∑ j ∈ Finset.range N, j ^ 2) (zudilinNormalizedHankelDet N) = ∏ j ∈ Finset.range N, (zudilinTransformedRowCoeff j : ℤ)","docstring":"Same-turn composition into the supplied conditional determinant consumer. No parallel determinant transformation or noncancellation proof is introduced. Verified locally on 2026-09-09.","generated_certificate":false},{"name":"order_zudilinNormalizedHankelDet_all","module":"ErdosProblems/Erdos1049/AllRow/Producer.lean","line":173,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/AllRow/Producer.lean#L173","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/AllRow/Producer.lean#L173","kind":"theorem","signature":"theorem order_zudilinNormalizedHankelDet_all (N : ℕ) : PowerSeries.order (zudilinNormalizedHankelDet N) = ((N * (N - 1) * (2 * N - 1) / 6 : ℕ) : ℕ∞)","docstring":"Exact closed q-order, including the empty determinant.","generated_certificate":false},{"name":"coeff_zudilinNormalizedHankelDet_all_rat","module":"ErdosProblems/Erdos1049/AllRow/Producer.lean","line":199,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/AllRow/Producer.lean#L199","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/AllRow/Producer.lean#L199","kind":"theorem","signature":"theorem coeff_zudilinNormalizedHankelDet_all_rat (N : ℕ) : ((PowerSeries.coeff (N * (N - 1) * (2 * N - 1) / 6) (zudilinNormalizedHankelDet N) : ℤ) : ℚ) = (N.factorial : ℚ) ^ 2 * ((N + 1).factorial : ℚ) / (2 : ℚ) ^ N","docstring":"The factorial quotient literally, after the injective embedding of the integer coefficient into the rationals. Verified locally on 2026-09-09.","generated_certificate":false}],"paper_label":"res:zudilin-sharp-qorder","paper_anchor":{"paper_label":"res:zudilin-sharp-qorder","owner_source":"paper/1049/erdos-1049-rational-base-lambert.tex","source_line":580,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":579,"anchor_title":"the first nonzero term of the Hankel determinant","anchor_title_tex":"the first nonzero term of the Hankel determinant","section_title":"The first nonzero formal term","section_path":[{"level":"section","title":"The weights of Zudilin's moments","title_tex":"The weights of Zudilin's moments","line":313},{"level":"subsection","title":"The first nonzero formal term","title_tex":"The first nonzero formal term","line":569}],"source_excerpt":{"text":"\\begin{theorem}[the first nonzero term of the Hankel determinant]\n\\label{res:zudilin-sharp-qorder}\nFor every $N\\ge1$,\n\\[\n \\operatorname{ord}_q V_N^*=\\frac{N(N-1)(2N-1)}6,\n\\]\nand the coefficient of the first nonzero monomial is\n\\[\n [q^{N(N-1)(2N-1)/6}]V_N^*\n   =\\frac{(N!)^2(N+1)!}{2^N}.\n\\]\n\\end{theorem}","media_type":"text/x-tex","source_char_start":24254,"source_char_end":24557,"start_line":579,"end_line":590,"truncated":false,"sha256":"sha256:b3df128567ced90359d65c5fdba2e5bbacbd9ab1d6e56a75c061ad147c720619","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1049/erdos-1049-rational-base-lambert.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1049/erdos-1049-rational-base-lambert.tex#L580","owner_problem_erdos_number":1049,"owner_paper_id":"erdos-1049-rational-base-lambert","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-1049-rational-base-lambert.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-1049-rational-base-lambert.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/1049/erdos-1049-rational-base-lambert.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_1049_irrationality","statement":"Determine whether F(t) = ∑_{n≥1} 1/(tⁿ − 1) is irrational for every rational t > 1. Irrationality of F((31/4)^r) for every positive integer r is a Lean-checked Zudilin specialization; the 3/2 approximant with remainder control and the universal assertion remain open.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/9"}],"links":[{"kind":"lean_declaration_pinned","label":"zudilinSharpHankelOrderAndCoeff_all","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/AllRow/Producer.lean#L163","cite_this":true},{"kind":"lean_declaration_pinned","label":"order_zudilinNormalizedHankelDet_all","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/AllRow/Producer.lean#L173","cite_this":true},{"kind":"lean_declaration_pinned","label":"coeff_zudilinNormalizedHankelDet_all_rat","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/AllRow/Producer.lean#L199","cite_this":true},{"kind":"paper_pdf","label":"erdos-1049-rational-base-lambert","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-1049-rational-base-lambert.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/148","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"zudilin_all_rank_hankel_order"}}],"expect":"object"},"prior_art":{"family_id":"zudilin_rational_base_and_hankel_theorems","status_summary":"Formalized rational-base irrationality and measure in the contour region, all-rank normalized Hankel order and coefficient formulas, and the coefficient-pencil proposition through rank eight.","prior_art_posture":"Zudilin supplies the cited 2004 polynomial-form construction and 2016 Hankel sequence. The formalizations preserve their hypotheses and make no novelty, priority or independent human-review claim.","primary_narrative_owner":"paper/1049/erdos-1049-rational-base-lambert.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/30","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:coefficient_pencil_finite_rank","assertion_kind":"registered_claim","source_id":"coefficient_pencil_finite_rank","reviewed":true,"granularity":"single_registered_claim","label":"Coefficient-pencil positivity and root interlacing through rank eight","statement":"For each real p > 1 and each natural rank N ≤ 8, the coefficient Hankel matrix A_N is positive definite, the polynomial det(Y A_N − B_N) splits over the reals, and every root is strictly below F(p). For adjacent ranks N and N+1 ≤ 8, the ordered root lists interlace non-strictly, with multiplicities. Lean includes N = 0; the paper states the ranks 1 through 8. No all-rank coefficient-pencil theorem, simple-root or strict-interlacing assertion, limiting result, or irrationality of F(3/2) follows from this finite proposition.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_1049"],"problem_erdos_numbers":[1049],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: erdos_1049","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_1049","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos1049-rational-base-lambert-reasoning-surface","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"coefficientPencil_finitePencil","module":"ErdosProblems/Erdos1049/PaperR20/FinitePencilProposition.lean","line":43,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1049/PaperR20/FinitePencilProposition.lean#L43","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/PaperR20/FinitePencilProposition.lean#L43","kind":"theorem","signature":"theorem coefficientPencil_finitePencil {p : ℝ} (hp : 1 < p) : (∀ N : ℕ, N ≤ 8 → (coefficientAlphaMatrix p N).PosDef ∧ (coefficientPencilPoly p N).Splits ∧ ∀ x : ℝ, (coefficientPencilPoly p N).IsRoot x → x < PaperR16.lambert (1 / p)) ∧ (∀ N : ℕ, N + 1 ≤ 8 → ∃ (large : Fin (N + 1) → ℝ) (small : Fin N → ℝ), Antitone large ∧ Antitone small ∧ (coefficientPencilPoly p (N + 1)).roots = Multiset.map large Finset.univ.val ∧ (coefficientPencilPoly p N).roots = Multiset.map small Finset.univ.val ∧ ∀ i : Fin N, small i ≤ large i.castSucc ∧ large i.succ ≤ small i)","docstring":"The paper's proposition `long1049:res:finite-pencil`, for every real `p > 1`. 1. For every `N ≤ 8`: `A_N` is positive definite, `det(Y A_N − B_N)` splits over `ℝ`, and each of its roots is strictly less than `F(p)`. 2. For every `N` with `N + 1 ≤ 8`: the roots at ranks `N + 1` and `N`, listed in decreasing order with multiplicity, interlace non-strictly.","generated_certificate":false}],"paper_label":"long1049:res:finite-pencil","paper_anchor":{"paper_label":"long1049:res:finite-pencil","owner_source":"paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","source_line":2943,"anchor_kind":"environment","anchor_unit":"proposition","anchor_unit_line":2942,"anchor_title":"finite coefficient positivity and pencil roots","anchor_title_tex":"finite coefficient positivity and pencil roots","section_title":"A finite spectral consequence.","section_path":[{"level":"section","title":"Power comparisons and Hankel determinants","title_tex":"Power comparisons and Hankel determinants","line":1336},{"level":"subsection","title":"Coefficient moments and cyclotomic content","title_tex":"Coefficient moments and cyclotomic content","line":2770},{"level":"paragraph","title":"A finite spectral consequence.","title_tex":"A finite spectral consequence.","line":2932}],"source_excerpt":{"text":"\\begin{proposition}[finite coefficient positivity and pencil roots]\n\\label{long1049:res:finite-pencil}\nFor every real $p>1$ and $1\\le N\\le8$, $A_N$ is positive definite and\nall roots of $\\det(YA_N-B_N)$ are real and strictly less than $F(p)$.\nThe roots at consecutive ranks $N,N+1\\le8$ interlace non-strictly.\n\\end{proposition}","media_type":"text/x-tex","source_char_start":136596,"source_char_end":136923,"start_line":2942,"end_line":2947,"truncated":false,"sha256":"sha256:fd21a13cd31179cba4feb8ada7ce0be2adc6bf9a047989cf0f70df919f8b5db2","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex#L2943","owner_problem_erdos_number":1049,"owner_paper_id":"erdos1049-rational-base-lambert-reasoning-surface","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos1049-rational-base-lambert-reasoning-surface.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos1049-rational-base-lambert-reasoning-surface.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.erdos_1049_irrationality","statement":"Determine whether F(t) = ∑_{n≥1} 1/(tⁿ − 1) is irrational for every rational t > 1. Irrationality of F((31/4)^r) for every positive integer r is a Lean-checked Zudilin specialization; the 3/2 approximant with remainder control and the universal assertion remain open.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/9"}],"links":[{"kind":"lean_declaration_pinned","label":"coefficientPencil_finitePencil","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1049/PaperR20/FinitePencilProposition.lean#L43","cite_this":true},{"kind":"paper_pdf","label":"erdos1049-rational-base-lambert-reasoning-surface","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos1049-rational-base-lambert-reasoning-surface.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/149","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"coefficient_pencil_finite_rank"}}],"expect":"object"},"prior_art":{"family_id":"zudilin_rational_base_and_hankel_theorems","status_summary":"Formalized rational-base irrationality and measure in the contour region, all-rank normalized Hankel order and coefficient formulas, and the coefficient-pencil proposition through rank eight.","prior_art_posture":"Zudilin supplies the cited 2004 polynomial-form construction and 2016 Hankel sequence. The formalizations preserve their hypotheses and make no novelty, priority or independent human-review claim.","primary_narrative_owner":"paper/1049/erdos-1049-rational-base-lambert.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/30","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:finite_prime_weighted_support","assertion_kind":"registered_claim","source_id":"finite_prime_weighted_support","reviewed":true,"granularity":"single_registered_claim","label":"Finite-prime weighted support irrationality","statement":"For every integer base b >= 2, every infinite positive support A with finite weighted mass for some finite nonempty prime set P has irrational support series. If a host H of positive integers has finite base-two weighted mass, then every infinite subset A of H has irrational support series at every integer base b >= 2. This is a sufficient support criterion, not the universal statement.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":true,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-257-mersenne-support-subseries","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"weightedDyadicMeanTarget","module":"ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean","line":40,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean#L40","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean#L40","kind":"theorem","signature":"theorem weightedDyadicMeanTarget : WeightedDyadicMeanTarget","docstring":"The complete weighted producer, with prefix divisibility and one actual finite dyadic observation window.","generated_certificate":false},{"name":"divisibilityWeightedClaim","module":"ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean","line":120,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean#L120","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean#L120","kind":"theorem","signature":"theorem divisibilityWeightedClaim : DivisibilityWeightedClaim","docstring":"The irrationality clauses of long-record `thm:257-weighted`: fixed-base irrationality and the binary-cost all-base hereditary clause. No new analytic target is assumed.","generated_certificate":false}],"paper_label":"res:weighted-support","paper_anchor":{"paper_label":"res:weighted-support","owner_source":"paper/257/erdos-257-mersenne-support-subseries.tex","source_line":75,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":74,"anchor_title":"a weighted condition on the support","anchor_title_tex":"a weighted condition on the support","section_title":"Introduction","section_path":[{"level":"section","title":"Introduction","title_tex":"Introduction","line":47}],"source_excerpt":{"text":"\\begin{theorem}[a weighted condition on the support]\n\\label{res:weighted-support}\nLet $b\\ge2$ be an integer, let $A\\subseteq\\Npos$ be infinite, and let $P$\nbe a finite nonempty set of primes.  Set\n$h(a)=\\prod_{p\\in P}p^{v_p(a)}$.  If\n\\begin{equation}\\label{eq:weighted-fixed-base}\n W_{b,P}(A):=\\sum_{a\\in A}\n \\frac{h(a)}{a(b^{h(a)}-1)}<\\infty,\n\\end{equation}\nthen $X_A(b)$ is irrational.  In particular, the base-two condition\n\\begin{equation*}\\label{eq:weighted-return}\n \\sum_{a\\in A}\\frac{h(a)}{a(2^{h(a)}-1)}<\\infty\n \\tag{W}\n\\end{equation*}\nimplies that $X_A(b)$ is irrational for every integer $b\\ge2$.  Both\nconclusions are hereditary under passage to infinite subsets.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":3355,"source_char_end":4043,"start_line":74,"end_line":90,"truncated":false,"sha256":"sha256:916b7f642c391ea200f3ef101740c3f0d350619a16ae1b5fbd96663510f9d785","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/257/erdos-257-mersenne-support-subseries.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/257/erdos-257-mersenne-support-subseries.tex#L75","owner_problem_erdos_number":257,"owner_paper_id":"erdos-257-mersenne-support-subseries","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-257-mersenne-support-subseries.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-257-mersenne-support-subseries.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/257/erdos-257-mersenne-support-subseries.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"weightedDyadicMeanTarget","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean#L40","cite_this":true},{"kind":"lean_declaration_pinned","label":"divisibilityWeightedClaim","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean#L120","cite_this":true},{"kind":"paper_pdf","label":"erdos-257-mersenne-support-subseries","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-257-mersenne-support-subseries.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/150","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"finite_prime_weighted_support"}}],"expect":"object"},"prior_art":{"family_id":"weighted_and_cover_support_criteria","status_summary":"Unconditional hereditary support criteria checked in Lean; universal Erdős #257 remains open.","prior_art_posture":"The formal statements preserve their weighted-summability and positive-cover assumptions. No novelty, priority, or independent human-review claim is made.","primary_narrative_owner":"paper/257/erdos-257-mersenne-support-subseries.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/31","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'.","semantic_states":{"join_method":"declaration_overlap","state_counts":{"not_assessed":1},"edges":[{"semantic_node_id":"Z10::finite_prime_weighted_support_irrationality","prior_art_state":"not_assessed"}]}},"source_lane":"registry_claim"},{"assertion_id":"claim:strengthened_positive_cover_support","assertion_kind":"registered_claim","source_id":"strengthened_positive_cover_support","reviewed":true,"granularity":"single_registered_claim","label":"Strengthened positive-cover support irrationality","statement":"A strengthened positive divisor-cover datum with summable scaled cost makes every infinite subset of its host irrational at every integer base b >= 2. The statement retains the nonlinear cover majorant and summability hypotheses; it does not cover arbitrary supports.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-257-mersenne-support-subseries","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"strengthenedPositiveCoverClaim","module":"ErdosProblems/Erdos257/PaperCompleteR8/PositiveCoverReturn.lean","line":241,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos257/PaperCompleteR8/PositiveCoverReturn.lean#L241","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos257/PaperCompleteR8/PositiveCoverReturn.lean#L241","kind":"theorem","signature":"theorem strengthenedPositiveCoverClaim : StrengthenedPositiveCoverClaim","docstring":"FULL short-note `thm:variable-fractional-cover`, including all bases and all infinite thinnings.","generated_certificate":false}],"paper_label":"thm:variable-fractional-cover","paper_anchor":{"paper_label":"thm:variable-fractional-cover","owner_source":"paper/257/erdos-257-mersenne-support-subseries.tex","source_line":364,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":364,"anchor_title":"a summable divisor-cover criterion","anchor_title_tex":"a summable divisor-cover criterion","section_title":"A finite divisor set and the cover condition","section_path":[{"level":"section","title":"Positive divisor majorants","title_tex":"Positive divisor majorants","line":313},{"level":"subsection","title":"A finite divisor set and the cover condition","title_tex":"A finite divisor set and the cover condition","line":341}],"source_excerpt":{"text":"\\begin{theorem}[a summable divisor-cover criterion]\\label{thm:variable-fractional-cover}\nFor each $j\\ge1$, let $F_j\\subseteq\\Npos$ be finite, let\n$0<\\alpha_j\\le1$, and let $c_{j,d}\\ge0$ satisfy\n\\[\n f_{F_j}(n)^{\\alpha_j}\\le\\sum_{d\\mid n}c_{j,d}\\quad(n\\ge1).\n\\]\nSet $C_j=\\sum_{d\\ge1}c_{j,d}/d$.  If\n\\begin{equation*}\n \\sum_{j\\ge1}\\frac{C_j2^{j\\alpha_j}}{2^{\\alpha_j}-1}<\\infty,\n \\tag{V}\\label{eq:strengthened-cover}\n\\end{equation*}\nthen $X_A(b)$ is irrational for every infinite\n$A\\subseteq\\bigcup_jF_j$ and every integer $b\\ge2$.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":17126,"source_char_end":17668,"start_line":364,"end_line":377,"truncated":false,"sha256":"sha256:69758eeebe46544876377ae4c5f2d7fec0dcf5288d82abeb34c1fcf473a371c5","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/257/erdos-257-mersenne-support-subseries.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/257/erdos-257-mersenne-support-subseries.tex#L364","owner_problem_erdos_number":257,"owner_paper_id":"erdos-257-mersenne-support-subseries","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-257-mersenne-support-subseries.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-257-mersenne-support-subseries.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/257/erdos-257-mersenne-support-subseries.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"strengthenedPositiveCoverClaim","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos257/PaperCompleteR8/PositiveCoverReturn.lean#L241","cite_this":true},{"kind":"paper_pdf","label":"erdos-257-mersenne-support-subseries","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-257-mersenne-support-subseries.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/151","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"strengthened_positive_cover_support"}}],"expect":"object"},"prior_art":{"family_id":"weighted_and_cover_support_criteria","status_summary":"Unconditional hereditary support criteria checked in Lean; universal Erdős #257 remains open.","prior_art_posture":"The formal statements preserve their weighted-summability and positive-cover assumptions. No novelty, priority, or independent human-review claim is made.","primary_narrative_owner":"paper/257/erdos-257-mersenne-support-subseries.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/31","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:mixed_weighted_cover_support","assertion_kind":"registered_claim","source_id":"mixed_weighted_cover_support","reviewed":true,"granularity":"single_registered_claim","label":"Mixed weighted and positive-cover support irrationality","statement":"Let E have finite base-two finite-prime weighted mass and let V lie in a strengthened positive-cover host. Every infinite support A contained in E union V has irrational support series at every integer base b >= 2. The theorem uses a common observation scale; it does not assume separate irrationality conclusions for the two components.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-257-mersenne-support-subseries","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"mixedSupportClaim","module":"ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean","line":126,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean#L126","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean#L126","kind":"theorem","signature":"theorem mixedSupportClaim : MixedSupportClaim","docstring":"FULL short-note `res:mixed-supports`. Uses the nonlinear cover test on the weighted schedule; does not assume a first-moment bound for the cover component and does not add individual irrationality premises.","generated_certificate":false}],"paper_label":"res:mixed-supports","paper_anchor":{"paper_label":"res:mixed-supports","owner_source":"paper/257/erdos-257-mersenne-support-subseries.tex","source_line":537,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":537,"anchor_title":"mixed weighted and cover supports","anchor_title_tex":"mixed weighted and cover supports","section_title":"A common index for the two criteria","section_path":[{"level":"section","title":"A common index for the two criteria","title_tex":"A common index for the two criteria","line":530}],"source_excerpt":{"text":"\\begin{theorem}[mixed weighted and cover supports]\\label{res:mixed-supports}\nLet $E,V\\subseteq\\Npos$.  Suppose $E$ satisfies\n\\eqref{eq:weighted-return} for a finite nonempty prime set $P$, and\n$V\\subseteq\\bigcup_jF_j$ for finite sets and nonnegative majorants\nsatisfying the hypotheses of Theorem~\\ref{thm:variable-fractional-cover},\nwith either~\\eqref{eq:strengthened-cover} or its positive-weight variant.\nThen $X_A(b)$ is irrational for every\ninfinite $A\\subseteq E\\cup V$ and every integer $b\\ge2$.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":25003,"source_char_end":25519,"start_line":537,"end_line":545,"truncated":false,"sha256":"sha256:d8f99256211cd576ea1907c2ae1a591d4dd6c0cab11daed841d873881c27f235","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/257/erdos-257-mersenne-support-subseries.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/257/erdos-257-mersenne-support-subseries.tex#L537","owner_problem_erdos_number":257,"owner_paper_id":"erdos-257-mersenne-support-subseries","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-257-mersenne-support-subseries.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-257-mersenne-support-subseries.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/257/erdos-257-mersenne-support-subseries.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"mixedSupportClaim","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean#L126","cite_this":true},{"kind":"paper_pdf","label":"erdos-257-mersenne-support-subseries","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-257-mersenne-support-subseries.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/152","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"mixed_weighted_cover_support"}}],"expect":"object"},"prior_art":{"family_id":"weighted_and_cover_support_criteria","status_summary":"Unconditional hereditary support criteria checked in Lean; universal Erdős #257 remains open.","prior_art_posture":"The formal statements preserve their weighted-summability and positive-cover assumptions. No novelty, priority, or independent human-review claim is made.","primary_narrative_owner":"paper/257/erdos-257-mersenne-support-subseries.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/31","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:reciprocal_summable_support","assertion_kind":"registered_claim","source_id":"reciprocal_summable_support","reviewed":true,"granularity":"single_registered_claim","label":"Reciprocal-summable support irrationality at every integer base","statement":"For every integer base b >= 2 and every infinite support A, summability of the reciprocal-support mass implies that the support series is irrational. No pairwise-coprimality hypothesis is required. Erdős (1968) stated this extension without proof.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_257"],"problem_erdos_numbers":[257],"problem_binding":{"method":"open_proposition_target","warrant":"the claims registry lists this claim against remaining open propositions whose open_target_claim names the problem: universal_257","source_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/remaining_open_propositions"}],"expect":"array"}},"problem_edges":[{"problem_id":"erdos_257","relation":"direct_target","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-257-mersenne-support-subseries","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"irrational_erdosSupportSeries_of_summable_reciprocal","module":"Erdos249257/AllBaseReciprocalSupportIrrationality.lean","line":395,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/Erdos249257/AllBaseReciprocalSupportIrrationality.lean#L395","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/AllBaseReciprocalSupportIrrationality.lean#L395","kind":"theorem","signature":"theorem irrational_erdosSupportSeries_of_summable_reciprocal (b : ℕ) (A : Set ℕ) (hb : 2 ≤ b) (hA : A.Infinite) (hsum : Summable (reciprocalSupportTerm A)) : Irrational (erdosSupportSeries b A)","docstring":"Every infinite support with convergent reciprocal mass gives an irrational reciprocal-power subseries at every integer base b ≥ 2.","generated_certificate":false}],"paper_label":"res:reciprocal-support","paper_anchor":{"paper_label":"res:reciprocal-support","owner_source":"paper/257/erdos-257-mersenne-support-subseries.tex","source_line":919,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":919,"anchor_title":"reciprocal-summable supports","anchor_title_tex":"reciprocal-summable supports","section_title":"Reciprocal-summable supports","section_path":[{"level":"section","title":"Reciprocal-summable supports","title_tex":"Reciprocal-summable supports","line":912}],"source_excerpt":{"text":"\\begin{theorem}[reciprocal-summable supports]\\label{res:reciprocal-support}\nLet $A\\subseteq\\Npos$ be infinite.  If\n\\[\n  \\sum_{a\\in A}\\frac1a<\\infty,\n\\]\nthen \\(X_A(b)\\) is irrational for every integer \\(b\\ge2\\).\n\\end{theorem}","media_type":"text/x-tex","source_char_start":43331,"source_char_end":43555,"start_line":919,"end_line":925,"truncated":false,"sha256":"sha256:fe066009ada3e5cce55fcd2263cc77e8660a9ea0c1d038a081b3e795b73248f6","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/257/erdos-257-mersenne-support-subseries.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/257/erdos-257-mersenne-support-subseries.tex#L919","owner_problem_erdos_number":257,"owner_paper_id":"erdos-257-mersenne-support-subseries","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-257-mersenne-support-subseries.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-257-mersenne-support-subseries.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/257/erdos-257-mersenne-support-subseries.tex"]},"does_not_establish":[{"open_proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/remaining_open_propositions/6"}],"links":[{"kind":"lean_declaration_pinned","label":"irrational_erdosSupportSeries_of_summable_reciprocal","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/Erdos249257/AllBaseReciprocalSupportIrrationality.lean#L395","cite_this":true},{"kind":"paper_pdf","label":"erdos-257-mersenne-support-subseries","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-257-mersenne-support-subseries.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/153","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"reciprocal_summable_support"}}],"expect":"object"},"prior_art":{"family_id":"weighted_and_cover_support_criteria","status_summary":"Unconditional hereditary support criteria checked in Lean; universal Erdős #257 remains open.","prior_art_posture":"The formal statements preserve their weighted-summability and positive-cover assumptions. No novelty, priority, or independent human-review claim is made.","primary_narrative_owner":"paper/257/erdos-257-mersenne-support-subseries.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/31","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:ani_degree_seven_total_variation_counterexample","assertion_kind":"registered_claim","source_id":"ani_degree_seven_total_variation_counterexample","reviewed":true,"granularity":"single_registered_claim","label":"Degree-seven Hausdorff-length counterexample","statement":"For ani's explicit monic complex polynomial of degree seven, whose distinct roots lie in the open unit disc, every preconnected subset of the strict unit lemniscate containing two distinct roots has one-dimensional Hausdorff measure greater than 2. In particular, Lean proves the negation and answer(False) forms of the exact Formal Conjectures path-image-length statement. The total-variation bound also remains checked. This formalises one polynomial, not the reported small-parameter family; independent human review of correspondence with the 1958 wording is not recorded.","status":"formalised here","status_meaning":"Known mathematics represented and checked in Lean","readme_headline":false,"problem_ids":["erdos_1041"],"problem_erdos_numbers":[1041],"problem_binding":{"method":"paper_owner","warrant":"the claim's paper_label 'res:ani-degree-seven-counterexample' is defined only in the manuscript treating Erdős #1041"},"problem_edges":[{"problem_id":"erdos_1041","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos-1041-lemniscate-newton-flow","relation":"published_or_explained_in"}],"evidence_authority":"lean_declarations","evidence_resolution":"declaration_level","declarations":[{"name":"erdos1041_ani_degree_seven","module":"ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean","line":25,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean#L25","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean#L25","kind":"theorem","signature":"theorem erdos1041_ani_degree_seven : ∃ (p : ℂ[X]), p.Monic ∧ p.natDegree = 7 ∧ (∀ z, p.IsRoot z → ‖z‖ < 1) ∧ p.roots.Nodup ∧ ∀ z₁ z₂, p.IsRoot z₁ → p.IsRoot z₂ → z₁ ≠ z₂ → ∀ γ : ℝ → ℂ, ContinuousOn γ (Set.Icc 0 1) → γ 0 = z₁ → γ 1 = z₂ → (∀ τ ∈ Set.Icc (0 : ℝ) 1, ‖p.eval (γ τ)‖ < 1) → (2 : ℝ≥0∞) < eVariationOn γ (Set.Icc 0 1)","docstring":"Existential wrapper matching `FormalConjectures.ErdosProblems.erdos_1041.variants.ani_degree_seven`.","generated_certificate":false},{"name":"erdos1041_counterexample_hausdorff","module":"ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","line":283,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L283","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L283","kind":"theorem","signature":"theorem erdos1041_counterexample_hausdorff : ∀ z₁ z₂, f.IsRoot z₁ → f.IsRoot z₂ → z₁ ≠ z₂ → ∀ K : Set ℂ, IsPreconnected K → z₁ ∈ K → z₂ ∈ K → K ⊆ Omega f → (2 : ℝ≥0∞) < μH[1] K","docstring":"Every preconnected subset of the strict lemniscate of `f` that contains two distinct roots has one-dimensional Hausdorff measure greater than two.","generated_certificate":false},{"name":"erdos1041_hausdorff_negation","module":"ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","line":407,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L407","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L407","kind":"theorem","signature":"theorem erdos1041_hausdorff_negation : ¬ ∀ (n : ℕ) (f : ℂ[X]), n ≥ 2 → f.natDegree = n → f.Monic → f.rootSet ℂ ⊆ Metric.ball 0 1 → ∃ (z₁ z₂ : ℂ) (h : ({z₁, z₂} : Multiset ℂ) ≤ f.roots) (γ : Path z₁ z₂), Set.range γ ⊆ { z : ℂ | ‖f.eval z‖ < 1 } ∧ fcLength (Set.range γ) < 2","docstring":"Negation of Formal Conjectures' `Erdos1041.erdos_1041`. Its section variables `n`, `f`, `hn`, `hnum`, `h_monic`, `h` are quantified in their declared order, and path length is `length (Set.range γ)`, the one-dimensional Hausdorff measure of the image. The polynomial `f` of `Defs.lean` refutes it.","generated_certificate":false},{"name":"erdos1041_hausdorff_answer_false","module":"ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","line":449,"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L449","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L449","kind":"theorem","signature":"theorem erdos1041_hausdorff_answer_false : False ↔ ∀ (n : ℕ) (f : ℂ[X]), n ≥ 2 → f.natDegree = n → f.Monic → f.rootSet ℂ ⊆ Metric.ball 0 1 → ∃ (z₁ z₂ : ℂ) (h : ({z₁, z₂} : Multiset ℂ) ≤ f.roots) (γ : Path z₁ z₂), Set.range γ ⊆ { z : ℂ | ‖f.eval z‖ < 1 } ∧ fcLength (Set.range γ) < 2","docstring":"The `answer(False) ↔ …` form of the same statement, which is how Formal Conjectures records a question settled in the negative (`answer(False)` elaborates to `False`).","generated_certificate":false}],"paper_label":"res:ani-degree-seven-counterexample","paper_anchor":{"paper_label":"res:ani-degree-seven-counterexample","owner_source":"paper/1041/erdos-1041-lemniscate-newton-flow.tex","source_line":90,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":90,"anchor_title":"ani's degree-seven example","anchor_title_tex":"\\texttt{ani}'s degree-seven example","section_title":"Introduction","section_path":[{"level":"section","title":"Introduction","title_tex":"Introduction","line":61}],"source_excerpt":{"text":"\\begin{theorem}[\\texttt{ani}'s degree-seven example]\\label{res:ani-degree-seven-counterexample}\nThe monic polynomial $f$ in \\eqref{eq:ani-f} has seven distinct zeros in\nthe open unit disc. Every connected set $K\\subset\\Omega_f$ containing\ntwo of its zeros satisfies $\\mathcal H^1(K)>2$.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":3876,"source_char_end":4176,"start_line":90,"end_line":94,"truncated":false,"sha256":"sha256:7b27abb34b867d2c26cf13fdab6b1d91ebaa11a7e355884f5bbee94360e77b22","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1041/erdos-1041-lemniscate-newton-flow.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1041/erdos-1041-lemniscate-newton-flow.tex#L90","owner_problem_erdos_number":1041,"owner_paper_id":"erdos-1041-lemniscate-newton-flow","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos-1041-lemniscate-newton-flow.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-1041-lemniscate-newton-flow.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/1041/erdos-1041-lemniscate-newton-flow.tex"]},"does_not_establish":[],"links":[{"kind":"lean_declaration_pinned","label":"erdos1041_ani_degree_seven","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean#L25","cite_this":true},{"kind":"lean_declaration_pinned","label":"erdos1041_counterexample_hausdorff","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L283","cite_this":true},{"kind":"lean_declaration_pinned","label":"erdos1041_hausdorff_negation","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L407","cite_this":true},{"kind":"lean_declaration_pinned","label":"erdos1041_hausdorff_answer_false","url":"https://github.com/wcook04/plectis-erdos/blob/2a8c1667566020d123a081b6165e3ee94550566d/lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L449","cite_this":true},{"kind":"paper_pdf","label":"erdos-1041-lemniscate-newton-flow","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos-1041-lemniscate-newton-flow.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/154","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"ani_degree_seven_total_variation_counterexample"}}],"expect":"object"},"prior_art":{"family_id":"degree_seven_total_variation_counterexample","status_summary":"Ani's explicit degree-seven polynomial refutes the exact Formal Conjectures path-image-length statement by a Lean-checked one-dimensional Hausdorff-measure bound; the total-variation formulation is also refuted.","prior_art_posture":"Formalisation of ani’s existing construction; no novelty or independent human review of historical correspondence is claimed.","primary_narrative_owner":"paper/1041/erdos-1041-lemniscate-newton-flow.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/32","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:separated_cut_algebraic_lambert_transcendence","assertion_kind":"registered_claim","source_id":"separated_cut_algebraic_lambert_transcendence","reviewed":true,"granularity":"single_registered_claim","label":"Algebraic-base transcendence on separated divisibility supports","statement":"For an infinite positive-integer host with cuts L_j < M_j, L_j tending to infinity and M_j-L_j tending to infinity, every host exponent at most L_j dividing L_j and every later exponent divisible by M_j, every infinite thinning with bounded positive integer weights has transcendental Lambert sum at every real algebraic base t > 1. This includes all divisibility chains and an explicit host of unbounded divisibility width and divergent reciprocal sum. The full number-field Subspace Theorem argument has two AI proof reviews; it is an ordinary proof with no Lean transcendence theorem, independent human review or asserted historical priority.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_249","erdos_257"],"problem_erdos_numbers":[249,257],"problem_binding":{"method":"core_programme_joint","warrant":"this claim belongs to the reviewed Erdos249257 core library, which is the joint #249/#257 programme. No source surface -- not its remaining-open propositions, not its anchor's enclosing gateway section, not the orientation programme map -- assigns it to one rather than both. Both are listed because both are correct, not because the binding is unknown.","separability":"not_separable_from_published_source"},"problem_edges":[{"problem_id":"erdos_249","relation":"shared_programme_context","basis":"problem_binding"},{"problem_id":"erdos_257","relation":"shared_programme_context","basis":"problem_binding"}],"programme_ids":["joint_249_257_gateway"],"publication_edges":[{"paper_id":"optimal-sparse-perturbations","relation":"published_or_explained_in"}],"evidence_authority":"registry_statement_only","evidence_resolution":"paper_level_only","declarations":[],"paper_label":"thm:separated-cuts","paper_anchor":{"paper_label":"thm:separated-cuts","owner_source":"paper/synthesis/optimal-sparse-perturbations.tex","source_line":1322,"anchor_kind":"environment","anchor_unit":"theorem","anchor_unit_line":1321,"anchor_title":"Lambert sums across divisibility cuts","anchor_title_tex":"Lambert sums across divisibility cuts","section_title":"Divisibility cuts at every algebraic base","section_path":[{"level":"section","title":"Lambert subsums across bases","title_tex":"Lambert subsums across bases","line":843},{"level":"subsection","title":"Divisibility cuts at every algebraic base","title_tex":"Divisibility cuts at every algebraic base","line":1310}],"source_excerpt":{"text":"\\begin{theorem}[Lambert sums across divisibility cuts]\n\\label{thm:separated-cuts}\nLet $H$ be an infinite subset of the positive integers. Suppose that positive\nintegers $L_j,M_j$ satisfy\n\\[\n L_j\\longrightarrow\\infty,\\qquad M_j>L_j,\\qquad M_j-L_j\\longrightarrow\\infty,\n\\]\nand, for every $j$ and $n\\in H$,\n\\[\n n\\le L_j\\ \\Longrightarrow\\ n\\mid L_j,\n \\qquad n>L_j\\ \\Longrightarrow\\ M_j\\mid n.\n\\]\nFor every infinite $B\\subseteq H$, every bounded family of positive integer\nweights $(w_n)_{n\\in B}$, and every real algebraic $t>1$, the sum\n\\[\n \\sum_{n\\in B}\\frac{w_n}{t^n-1}\n\\]\nis transcendental.\n\\end{theorem}","media_type":"text/x-tex","source_char_start":65975,"source_char_end":66579,"start_line":1321,"end_line":1339,"truncated":false,"sha256":"sha256:a8471d00eff501bd7df40a8280936428da9c5ea711a0b38ae5ab6f842f40dd82","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. 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Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:independent_clock_channel_relations","assertion_kind":"registered_claim","source_id":"independent_clock_channel_relations","reviewed":true,"granularity":"single_registered_claim","label":"Complete rational relations for integer-radix independent-clock sums","statement":"For positive rationally independent clock rates with no simultaneous positive crossings, and integer radices b_i >= 2, all rational relations among 1 and the channel sums S_i are multiples of (-1,b_1-1,...,b_d-1). The proof uses genuine late swaps with a common infinite suffix in one orbit closure, a fixed rational lattice, and density of the clock section. The explicit rates 1,sqrt(2),sqrt(3),sqrt(6) and radices 2,3,5,7 give rational span dimension four. This is an ordinary proof reviewed by AI agents, without Lean or independent human verification of the infinite theorem.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_269"],"problem_erdos_numbers":[269],"problem_binding":{"method":"paper_owner","warrant":"the claim's paper_label 'long269:thm:clock-relations' is defined only in the manuscript treating Erdős #269"},"problem_edges":[{"problem_id":"erdos_269","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos269-running-lcm-reasoning-surface","relation":"published_or_explained_in"}],"evidence_authority":"registry_statement_only","evidence_resolution":"paper_level_only","declarations":[],"paper_label":"long269:thm:clock-relations","paper_anchor":{"paper_label":"long269:thm:clock-relations","owner_source":"paper/269/erdos269-running-lcm-reasoning-surface.tex","source_line":1575,"anchor_kind":"environment","anchor_unit":"remark","anchor_unit_line":1574,"anchor_title":"rational relations among the subseries","anchor_title_tex":"rational relations among the subseries","section_title":"Words obtained from a linear flow","section_path":[{"level":"section","title":"Words obtained from a linear flow","title_tex":"Words obtained from a linear flow","line":1447}],"source_excerpt":{"text":"\\begin{remark}[rational relations among the subseries]\n\\label{long269:thm:clock-relations}\nUnder the preceding hypotheses on the linear flow, choose integers $b_i\\ge2$, put\n$Q_n=\\prod_{k\\le n}b_{w_k}$ and\n\\[\n S_i=\\sum_{n:\\,w_n=i}\\frac1{Q_n}\\qquad(1\\le i\\le d).\n\\]\nFor every $c_i\\in\\Q$,\n\\[\n \\sum_{i=1}^dc_iS_i\\in\\Q\n \\quad\\Longleftrightarrow\\quad\n c_i=\\lambda(b_i-1)\\quad\\hbox{for all $i$ and some $\\lambda\\in\\Q$}.\n\\]\nThe rational value is $\\lambda$.  Equivalently, all rational relations\n$a_0+\\sum_i a_iS_i=0$ are multiples of\n$(-1,b_1-1,\\ldots,b_d-1)$.  Thus the span of\n$1,S_1,\\ldots,S_d$ has dimension $d$; each $S_i$ and each nonempty\nproper unweighted sub-sum is irrational.  The full unweighted sum is\nirrational exactly when the radices are not all equal.\n\\end{remark}","media_type":"text/x-tex","source_char_start":78156,"source_char_end":78930,"start_line":1574,"end_line":1593,"truncated":false,"sha256":"sha256:5c8f955670d76662fbc6d483d468ecfa391cf299744410228a16f79a21ee8bb2","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/269/erdos269-running-lcm-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/269/erdos269-running-lcm-reasoning-surface.tex#L1575","owner_problem_erdos_number":269,"owner_paper_id":"erdos269-running-lcm-reasoning-surface","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos269-running-lcm-reasoning-surface.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos269-running-lcm-reasoning-surface.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/269/erdos269-running-lcm-reasoning-surface.tex"]},"does_not_establish":[],"links":[{"kind":"paper_pdf","label":"erdos269-running-lcm-reasoning-surface","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos269-running-lcm-reasoning-surface.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/156","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"independent_clock_channel_relations"}}],"expect":"object"},"prior_art":{"family_id":"independent_clock_channel_relations_family","status_summary":"Reviewed ordinary proof; full infinite statement not formalised and independent human review not recorded.","prior_art_posture":"The integer-radix recurrence and torus-density argument are standard ingredients; the stated clock-channel combination has no asserted historical priority. Bugeaud-Laurent is a distinct two-letter comparison.","primary_narrative_owner":"paper/269/erdos269-running-lcm-reasoning-surface.tex","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/machine_readable_paper/publication_assembly/contribution_families/34","novelty_rule":"prior_art_posture is the corpus's own statement of who owns the antecedent. Absence of a named antecedent is not evidence of novelty -- it means no assessment is recorded. Never render an unrecorded antecedent as 'new'."},"source_lane":"registry_claim"},{"assertion_id":"claim:calibrated_rational_hankel_countermodel","assertion_kind":"registered_claim","source_id":"calibrated_rational_hankel_countermodel","reviewed":true,"granularity":"single_registered_claim","label":"Rational-value countermodel to a Hankel-signature inference","statement":"An explicit positive atomic moment family and integer formal series have all-rank Hankel order sum_{j<N} j^2 and leading coefficient (N!)^2(N+1)!/2^N, with the same fixed-base asymptotic shape including N^(-8 xi), while xi=G_{2/3}(2/3)=1. For n>=1 the exact reduced moment denominator is 3^(31n+62)(3^(n+1)-2^(n+1))^4; the determinant denominator has logarithmic size Theta(N^3), and a proposed rank-two row-product clearer leaves denominator625. These are ordinary proofs with independent exact finite checks and AI proof reviews, not a Lean proof of the infinite family or human peer review.","status":"unconditional progress","status_meaning":"Genuine theorem that does not settle the open problem","readme_headline":false,"problem_ids":["erdos_1049"],"problem_erdos_numbers":[1049],"problem_binding":{"method":"paper_owner","warrant":"the claim's paper_label 'long1049:thm:calibrated-model' is defined only in the manuscript treating Erdős #1049"},"problem_edges":[{"problem_id":"erdos_1049","relation":"supporting_or_contextual","basis":"problem_binding"}],"programme_ids":[],"publication_edges":[{"paper_id":"erdos1049-rational-base-lambert-reasoning-surface","relation":"published_or_explained_in"}],"evidence_authority":"registry_statement_only","evidence_resolution":"paper_level_only","declarations":[],"paper_label":"long1049:thm:calibrated-model","paper_anchor":{"paper_label":"long1049:thm:calibrated-model","owner_source":"paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","source_line":2539,"anchor_kind":"environment","anchor_unit":"remark","anchor_unit_line":2538,"anchor_title":"a rational special value","anchor_title_tex":"a rational special value","section_title":"Positive Hankel determinants and a rational special value","section_path":[{"level":"section","title":"Power comparisons and Hankel determinants","title_tex":"Power comparisons and Hankel determinants","line":1336},{"level":"subsection","title":"Positive Hankel determinants and a rational special value","title_tex":"Positive Hankel determinants and a rational special value","line":2483}],"source_excerpt":{"text":"\\begin{remark}[a rational special value]\n\\label{long1049:thm:calibrated-model}\nAll the weights in~\\eqref{long1049:eq:calibrated-weights} are positive\nfor $0<q\\le2/3$. The two moment sequences $(\\mu_m)$ and $(\\nu_m)$\nhave positive Hankel determinants of every size, $G_q$ is not a rational\nfunction, and\n$G_q(q)=1$.  Their formal Hankel determinants both begin with\n\\[\n C_Nq^{B_N},\\qquad B_N=\\sum_{j<N}j^2,\\quad\n C_N=\\frac{(N!)^2(N+1)!}{2^N}\\qquad(N\\ge1).\n\\]\nAt the fixed base $q_0=2/3$, with\n$P(q)=\\prod_{j\\ge1}(1-q^j)$ and\n$\\mathcal M(q)=\\prod_{j\\ge1}(1-q^j)^{-j}$,\nthere is an explicit $\\mathcal A>0$ such that\n\\[\n \\det(\\mu_{i+j}(q_0))_{0\\le i,j<N}\n \\sim\\mathcal A\\mathcal M(q_0)^3 C_Nq_0^{B_N}\n             P(q_0)^{2N}N^{-8\\xi},\\qquad \\xi=G_{q_0}(q_0)=1.\n\\]\n\\end{remark}","media_type":"text/x-tex","source_char_start":118717,"source_char_end":119490,"start_line":2538,"end_line":2558,"truncated":false,"sha256":"sha256:68450e04bced4a7c3884f132079d1899d282a08cd0c395e0f89bef97268eaee4","what_this_is":"the exact LaTeX around this anchor in the pinned manuscript, quoted so the statement can be read without a network. Verify it against permalink_url; nothing here is a paraphrase."},"current_url":"https://github.com/wcook04/plectis-erdos/blob/main/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","manuscript_source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex#L2539","owner_problem_erdos_number":1049,"owner_paper_id":"erdos1049-rational-base-lambert-reasoning-surface","owner_status":"registry_active","owner_pdf_hosted_path":"papers/erdos1049-rational-base-lambert-reasoning-surface.pdf","owner_pdf_url":"https://wcook04.github.io/practical-formal-verification/papers/erdos1049-rational-base-lambert-reasoning-surface.pdf","resolution_status":"resolved","resolution_rule":"T0_unique","owner_confidence":"decided","candidate_sources":["paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex"]},"does_not_establish":[],"links":[{"kind":"paper_pdf","label":"erdos1049-rational-base-lambert-reasoning-surface","url":"https://wcook04.github.io/practical-formal-verification/papers/erdos1049-rational-base-lambert-reasoning-surface.pdf"}],"packet_pointer":"/scholarly_corpus/formal_claim_control_plane/0/body/claims/157","packet_selector":{"resolver":"plectis_selector_v1","steps":[{"path":"/scholarly_corpus/formal_claim_control_plane","where":{"field":"record_id","equals":"lean_claims_registry"}},{"path":"/body/claims","where":{"field":"id","equals":"calibrated_rational_hankel_countermodel"}}],"expect":"object"},"prior_art":{"family_id":"calibrated_rational_hankel_countermodel_family","status_summary":"Reviewed ordinary proof; full infinite statement not formalised and independent human review not recorded.","prior_art_posture":"Reuses standard positive-moment/Christoffel machinery and the existing geometric-universality theorem. 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The full proof includes the allowance-dependent block bound and first-harmonic rational countermodels. 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Do not present them as the evidence chosen for this statement -- the published index does not make that mapping, and inventing one would be a more confident citation than the source supports.","problem_join":"/mathematical_assertions/problem_digest/erdos_1049","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/2/body/problems/7/what_is_not_checked/6","source_lane":"problem_index_negative"},{"assertion_id":"erdos_1049.open_obligation.three_halves_growing_rank_endpoint_jet_kernel","assertion_kind":"open_obligation","problem_id":"erdos_1049","problem_ids":["erdos_1049"],"erdos_number":1049,"problem_erdos_numbers":[1049],"problem_binding":{"method":"problem_index_membership","warrant":"this obligation is published inside the problem index row for erdos_1049"},"reviewed":false,"granularity":"problem_index_open_obligation","statement":"After rowwise primitive normalisation, exhibit at least 4R_n+2S_n coefficient pairs at positive quadratic depths from a non-collapsed deformation; preserve the two- and three-adic gain; then show that one checked {-1,0,1} collision has a nonzero polynomial pair outside the analytic remainder nullspace, or prove an exact rank obstruction.","status":"open","evidence_authority":"none_this_is_what_remains","evidence_resolution":"open_by_construction","problem_join":"/mathematical_assertions/problem_digest/erdos_1049","packet_pointer":"/scholarly_corpus/formal_claim_control_plane/2/body/problems/7/open_obligations/0","source_lane":"problem_index_open"},{"assertion_id":"paper:erdos_68#res:problem","assertion_kind":"paper_posed_problem","reviewed":false,"statement":"Erdős Problem #68","status":"posed in the manuscript","problem_ids":["erdos_68"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:problem","owner_paper_id":"erdos-68-factorial-denominator-irrationality","source_line":52,"anchor_unit":"problem","section_title":"Introduction","source_excerpt":{"text":"\\begin{problem}[Erd\\H{o}s Problem~\\#68]\\label{res:problem}\nIs\n\\[\n  \\Ser=\\sum_{n\\ge2}\\frac1{n!-1}\n\\]\nirrational?\n\\end{problem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/68/erdos-68-factorial-denominator-irrationality.tex#L52"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_problem","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/0","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_68#res:divisor-channel-coordinates","assertion_kind":"paper_argued_result","reviewed":false,"statement":"an integer basis with prescribed weighted sums","status":"proved in the manuscript","problem_ids":["erdos_68"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:divisor-channel-coordinates","owner_paper_id":"erdos-68-factorial-denominator-irrationality","source_line":157,"anchor_unit":"theorem","section_title":"An integral basis","source_excerpt":{"text":"\\begin{theorem}[an integer basis with prescribed weighted sums]\\label{res:divisor-channel-coordinates}\nSet\n\\[\nT_n=ne_{n-1}-e_n,\\qquad\nU_n=T_n-\\sum_{\\substack{d\\mid n\\\\2\\le d<n}}W_{d,n}U_d\n\\quad(n\\ge2).\n\\]\nThen\n\\[\nM(U_n)=0,\\qquad V_d(U_n)=(d!-1)\\mathbf1_{d=n}.\n\\]\nThe vectors $e_1,U_2,U_3,\\ldots$ form an integral basis.  Every finite vector\nhas the unique finite expansion\n\\begin{equation}\n\\lambda=M(\\lambda)e_1+\n\\sum_{d\\ge2}\\frac{V_d(\\lambda)-M(\\lambda)}{d!-1}U_d.\n\\label{eq:channel-basis-expansion}\n\\end{equation}\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/68/erdos-68-factorial-denominator-irrationality.tex#L157"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/0","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_68#res:finite-channel-moment-certificate","assertion_kind":"paper_argued_result","reviewed":false,"statement":"a finite formula for the gcd","status":"proved in the manuscript","problem_ids":["erdos_68"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:finite-channel-moment-certificate","owner_paper_id":"erdos-68-factorial-denominator-irrationality","source_line":293,"anchor_unit":"theorem","section_title":"A finite gcd calculation","source_excerpt":{"text":"\\begin{theorem}[a finite formula for the gcd]\n\\label{res:finite-channel-moment-certificate}\nChoose a prime $\\ell$ with $D/2<\\ell\\le D$ and put $H=D(2\\ell-1)$.\nThen\n\\[\ng_D=\\gcd(u_{D+1},\\ldots,u_H),\\qquad H<2D^2.\n\\]\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/68/erdos-68-factorial-denominator-irrationality.tex#L293"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/0","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:problem","assertion_kind":"paper_posed_problem","reviewed":false,"statement":"Erdős Problem 243","status":"posed in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:problem","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":59,"anchor_unit":"problem","section_title":"Introduction","source_excerpt":{"text":"\\begin{problem}[Erd\\H{o}s Problem 243]\\label{res:problem}\nLet $1\\le a_1<a_2<\\cdots$ be a sequence of integers with\n\\[\n \\lim_{n\\to\\infty}\\frac{a_n}{a_{n-1}^{2}}=1\n \\qquad\\text{and}\\qquad\n \\sum\\frac{1}{a_n}\\in\\Q .\n\\]\nThen $a_n=a_{n-1}^{2}-a_{n-1}+1$ for all sufficiently large $n$.\n\\end{problem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L59"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_problem","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:reduced","assertion_kind":"paper_argued_result","reviewed":false,"statement":"persistent coprimality","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:reduced","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":249,"anchor_unit":"proposition","section_title":"Reduction and the constant term","source_excerpt":{"text":"\\begin{proposition}[persistent coprimality]\\label{res:reduced}\nLet $a_n,u_n,v_n$ be integer sequences with $u_n>0$, $v_n\\ge0$ and\n$\\gcd(u_n,v_n)=1$ satisfying\n\\[\n u_{n+1}=a_nu_n-v_n,\\qquad v_{n+1}=a_nv_n.\n\\]\nThen $\\gcd(a_n,v_n)=1$. The $a_n$ are pairwise coprime, and\n$\\gcd(a_i,u_t)=1$ whenever $i<t$.\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L249"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:bounded","assertion_kind":"paper_argued_result","reviewed":false,"statement":"bounded negative part","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:bounded","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":483,"anchor_unit":"theorem","section_title":"Bounded increases","source_excerpt":{"text":"\\begin{theorem}[bounded negative part]\\label{res:bounded}\nLet $a,C,D:\\N\\to\\N$ satisfy $a_n>1$, $C_n>0$, and\n\\[\n C_{n+1}+D_n=a_nC_n,\\qquad D_{n+1}=a_nD_n\n \\quad(n\\ge0).\n\\]\nPut $E_n=D_n-(a_n-1)C_n$. If\n\\[\n \\frac{E_n}{C_n}\\longrightarrow0\n \\quad\\text{and}\\quad E_n\\ge-B\\quad\\text{for all sufficiently large }n\n\\]\nfor some integer $B\\ge0$, then $E_n=0$ for all sufficiently large $n$.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L483"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:update","assertion_kind":"paper_argued_result","reviewed":false,"statement":"error identities","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:update","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":506,"anchor_unit":"proposition","section_title":"Zero errors and descent","source_excerpt":{"text":"\\begin{proposition}[error identities]\\label{res:update}\\label{res:defect}\nFor integer sequences satisfying \\eqref{eq:recurrences},\n\\[\n C_{n+1}=C_n-E_n,\\qquad\n \\bigl(a_{n+1}-a_n^2+a_n-1\\bigr)C_{n+1}=a_n^2E_n-E_{n+1}.\n\\]\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L506"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:defect","assertion_kind":"paper_argued_result","reviewed":false,"statement":"error identities","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:defect","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":506,"anchor_unit":"proposition","section_title":"Zero errors and descent","source_excerpt":{"text":"\\begin{proposition}[error identities]\\label{res:update}\\label{res:defect}\nFor integer sequences satisfying \\eqref{eq:recurrences},\n\\[\n C_{n+1}=C_n-E_n,\\qquad\n \\bigl(a_{n+1}-a_n^2+a_n-1\\bigr)C_{n+1}=a_n^2E_n-E_{n+1}.\n\\]\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L506"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:absorb","assertion_kind":"paper_argued_result","reviewed":false,"statement":"absorption and descent","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:absorb","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":530,"anchor_unit":"theorem","section_title":"Zero errors and descent","source_excerpt":{"text":"\\begin{theorem}[absorption and descent]\\label{res:absorb}\\label{res:descent}\nLet $a,C,D:\\N\\to\\N$ satisfy $a_n>1$, $C_n>0$ and\n\\eqref{eq:recurrences}. If $|E_n|<C_n$ for every $n$, then $E_n=0$\nimplies $E_{n+1}=0$. For any positive integer sequence $C_n$ with\n$C_{n+1}=C_n-E_n$, eventual nonnegativity of $E_n$ implies its eventual\nvanishing.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L530"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:descent","assertion_kind":"paper_argued_result","reviewed":false,"statement":"absorption and descent","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:descent","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":530,"anchor_unit":"theorem","section_title":"Zero errors and descent","source_excerpt":{"text":"\\begin{theorem}[absorption and descent]\\label{res:absorb}\\label{res:descent}\nLet $a,C,D:\\N\\to\\N$ satisfy $a_n>1$, $C_n>0$ and\n\\eqref{eq:recurrences}. If $|E_n|<C_n$ for every $n$, then $E_n=0$\nimplies $E_{n+1}=0$. For any positive integer sequence $C_n$ with\n$C_{n+1}=C_n-E_n$, eventual nonnegativity of $E_n$ implies its eventual\nvanishing.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L530"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:step","assertion_kind":"paper_argued_result","reviewed":false,"statement":"two zero errors","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:step","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":545,"anchor_unit":"corollary","section_title":"Zero errors and descent","source_excerpt":{"text":"\\begin{corollary}[two zero errors]\\label{res:step}\\label{res:eventual}\nFor integer sequences satisfying \\eqref{eq:recurrences}, if\n$E_n=E_{n+1}=0$ and $C_{n+1}\\ne0$, then\n$a_{n+1}=a_n^2-a_n+1$. In particular, if $a,C,D:\\N\\to\\N$\nalso satisfy $a_n>1$ and $C_n>0$, eventual zero error implies\nthe eventual Sylvester recurrence.\n\\end{corollary}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L545"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:eventual","assertion_kind":"paper_argued_result","reviewed":false,"statement":"two zero errors","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:eventual","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":545,"anchor_unit":"corollary","section_title":"Zero errors and descent","source_excerpt":{"text":"\\begin{corollary}[two zero errors]\\label{res:step}\\label{res:eventual}\nFor integer sequences satisfying \\eqref{eq:recurrences}, if\n$E_n=E_{n+1}=0$ and $C_{n+1}\\ne0$, then\n$a_{n+1}=a_n^2-a_n+1$. In particular, if $a,C,D:\\N\\to\\N$\nalso satisfy $a_n>1$ and $C_n>0$, eventual zero error implies\nthe eventual Sylvester recurrence.\n\\end{corollary}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L545"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:gcdstab","assertion_kind":"paper_argued_result","reviewed":false,"statement":"gcd stabilisation","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:gcdstab","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":562,"anchor_unit":"proposition","section_title":"A stable gcd and a forbidden block","source_excerpt":{"text":"\\begin{proposition}[gcd stabilisation]\\label{res:gcdstab}\nLet $a,C,D:\\N\\to\\N$ satisfy $a_n>1$, $C_n>0$ and\n\\eqref{eq:recurrences}. If some fixed integer $B\\ge1$ satisfies\n$-B\\le E_n<0$ at infinitely many indices, then\n$G_n=\\gcd(C_n,D_n)$ is eventually constant. Dividing $C_n,D_n$ by\nits eventual value gives coprime sequences satisfying the same recurrences.\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L562"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:crt","assertion_kind":"paper_argued_result","reviewed":false,"statement":"consecutive multiples","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:crt","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":586,"anchor_unit":"lemma","section_title":"A stable gcd and a forbidden block","source_excerpt":{"text":"\\begin{lemma}[consecutive multiples]\\label{res:crt}\nFor pairwise coprime integers $m_0,\\ldots,m_{B-1}\\ge2$ and every lower\nbound, there is a larger $t$ such that $m_i\\mid t+i$ for each $i<B$.\n\\end{lemma}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L586"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:cor","assertion_kind":"paper_argued_result","reviewed":false,"statement":"res:cor","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:cor","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":631,"anchor_unit":"corollary","section_title":"A stable gcd and a forbidden block","source_excerpt":{"text":"\\begin{corollary}\\label{res:cor}\nUnder Theorem~\\ref{res:bounded}, the multipliers satisfy\n$a_{n+1}=a_n^2-a_n+1$ eventually.\n\\end{corollary}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L631"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"label_only"},{"assertion_id":"paper:erdos_243#res:originalbounded","assertion_kind":"paper_argued_result","reviewed":false,"statement":"bounded increments of P_n/a_n","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:originalbounded","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":650,"anchor_unit":"corollary","section_title":"The condition in the original sequence","source_excerpt":{"text":"\\begin{corollary}[bounded increments of $P_n/a_n$]\\label{res:originalbounded}\nLet $a_1<a_2<\\cdots$ be positive integers, $a_{n+1}/a_n^2\\to1$, and\n$\\sum_{n\\ge1}1/a_n\\in\\Q$.  Put $P_n=\\prod_{j<n}a_j$.  If\n\\[\n \\limsup_{n\\to\\infty}\\frac{P_n}{a_n}\n \\left(\\frac{a_n^2}{a_{n+1}}-1\\right)<+\\infty,\n\\]\nthen $a_{n+1}=a_n^2-a_n+1$ for all sufficiently large $n$.\n\\end{corollary}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L650"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:inclusiveone","assertion_kind":"paper_argued_result","reviewed":false,"statement":"an inclusive one-sided 1/n bound","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:inclusiveone","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":686,"anchor_unit":"corollary","section_title":"The condition in the original sequence","source_excerpt":{"text":"\\begin{corollary}[an inclusive one-sided $1/n$ bound]\\label{res:inclusiveone}\nLet $a_1<a_2<\\cdots$ be positive integers with $a_{n+1}/a_n^2\\to1$\nand $\\sum_n1/a_n\\in\\Q$. Suppose that for some $K\\ge0$ and $\\varepsilon>0$,\n\\[\n \\gamma_n:=a_n^2/a_{n+1}-1\\le \\frac1n+\\frac{K}{n^{1+\\varepsilon}}\n \\quad\\hbox{for all large }n.\n\\]\nThen the sequence is eventually Sylvester. In particular, the conclusion\nholds under the pointwise eventual bound $\\gamma_n\\le1/n$.\n\\end{corollary}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L686"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:massscalar","assertion_kind":"paper_argued_result","reviewed":false,"statement":"a convergent sum of relative increases","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:massscalar","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":733,"anchor_unit":"theorem","section_title":"Finite total relative increase","source_excerpt":{"text":"\\begin{theorem}[a convergent sum of relative increases]\\label{res:massscalar}\\label{res:mass}\nLet $C_n$ be positive integers and $E_n$ integers satisfying\n$C_{n+1}=C_n-E_n$. If\n\\[\n \\sum_n\\frac{(-E_n)_+}{C_n}<\\infty,\n\\]\nthen $E_n=0$ eventually. Neither denominator dynamics nor vanishing relative error is required.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L733"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:mass","assertion_kind":"paper_argued_result","reviewed":false,"statement":"a convergent sum of relative increases","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:mass","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":733,"anchor_unit":"theorem","section_title":"Finite total relative increase","source_excerpt":{"text":"\\begin{theorem}[a convergent sum of relative increases]\\label{res:massscalar}\\label{res:mass}\nLet $C_n$ be positive integers and $E_n$ integers satisfying\n$C_{n+1}=C_n-E_n$. If\n\\[\n \\sum_n\\frac{(-E_n)_+}{C_n}<\\infty,\n\\]\nthen $E_n=0$ eventually. Neither denominator dynamics nor vanishing relative error is required.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L733"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:weightedrecord","assertion_kind":"paper_argued_result","reviewed":false,"statement":"a convergent weighted sum over new maxima","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:weightedrecord","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":814,"anchor_unit":"theorem","section_title":"New maxima of an LCM numerator","source_excerpt":{"text":"\\begin{theorem}[a convergent weighted sum over new maxima]\\label{res:weightedrecord}\nAssume the growth and rationality hypotheses of Problem~\\ref{res:problem}.\nLet $f:[1,\\infty)\\to[0,\\infty)$ be finite and nonincreasing, with\n$\\int_1^\\infty f(t)\\,dt=\\infty$. Then the sequence is eventually Sylvester\nif and only if, for some integer $B\\ge0$,\n\\[\n \\sum_{n\\in\\mathcal R}(-V_n-B)_+f(U_n)<\\infty.\n\\]\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L814"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:lcmbounded","assertion_kind":"paper_argued_result","reviewed":false,"statement":"a bound using the least common multiple","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:lcmbounded","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":939,"anchor_unit":"corollary","section_title":"New maxima of an LCM numerator","source_excerpt":{"text":"\\begin{corollary}[a bound using the least common multiple]\\label{res:lcmbounded}\nAssume the hypotheses of Problem~\\ref{res:problem}. Write\n$A_n=\\operatorname{lcm}(a_1,\\ldots,a_{n-1})$ with $A_1=1$. If\n\\[\n \\limsup_{n\\to\\infty}\\frac{A_n}{a_n}\n \\left(\\frac{a_n^2}{a_{n+1}}-1\\right)<\\infty,\n\\]\nthen the sequence is eventually Sylvester.\n\\end{corollary}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L939"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:frontier","assertion_kind":"paper_argued_result","reviewed":false,"statement":"necessary profile","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:frontier","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":995,"anchor_unit":"proposition","section_title":"Further questions","source_excerpt":{"text":"\\begin{proposition}[necessary profile]\\label{res:frontier}\nThe integer tail of a sequence satisfying Problem~\\ref{res:problem}'s\nhypotheses but not its conclusion has $E_n\\ne0$ eventually,\n$|E_n|/C_n\\to0$, unbounded negative magnitudes along negative indices,\nand\n\\[\n \\sum_n\\frac{(-E_n)_+}{C_n}=\\infty.\n\\]\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L995"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:weights","assertion_kind":"paper_argued_result","reviewed":false,"statement":"weights and linear density","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:weights","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":1030,"anchor_unit":"lemma","section_title":"Further questions","source_excerpt":{"text":"\\begin{lemma}[weights and linear density]\\label{res:weights}\nLet $u_j$ be positive integers, let $w_j\\ge0$, and put\n$F(X)=\\sum_{u_j\\le X}w_j$, with the sum allowed a priori to be $+\\infty$. Then $\\liminf_{X\\to\\infty}F(X)/X=0$ if and only if\nthere is a finite nonincreasing $f:[1,\\infty)\\to[0,\\infty)$ with\n$\\int_1^\\infty f(t)\\,dt=\\infty$ and $\\sum_jw_jf(u_j)<\\infty$.\n\\end{lemma}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L1030"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:lcmheight","assertion_kind":"paper_posed_problem","reviewed":false,"statement":"growth of repeated denominator factors","status":"posed in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:lcmheight","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":1087,"anchor_unit":"problem","section_title":"Arithmetic compatibility","source_excerpt":{"text":"\\begin{problem}[growth of repeated denominator factors]\\label{res:lcmheight}\nFor every rational-tail orbit satisfying the hypotheses but not the conclusion of Problem~\\ref{res:problem},\nmust\n\\[\n \\limsup_{n\\to\\infty}\\frac{\\log M_n}{n}>0?\n\\]\nEquivalently, must there be a $K\\ge1$ for which\n\\[\n 2^n\\le M_n^K\n\\]\nat infinitely many indices?\n\\end{problem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L1087"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_problem","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_243#res:residue","assertion_kind":"paper_argued_result","reviewed":false,"statement":"factorial residue reduction","status":"proved in the manuscript","problem_ids":["erdos_243"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:residue","owner_paper_id":"erdos-243-reciprocal-tail-rigidity","source_line":1138,"anchor_unit":"theorem","section_title":"A factorial modulus for integral recursion","source_excerpt":{"text":"\\begin{theorem}[factorial residue reduction]\\label{res:residue}\nLet $h$ be a nonnegative integer and let $a,b$ be integers with $a\\equiv b\\pmod{(h+1)!}$.\nThe first $h$ steps of the recursion\n$a_{n+1}=\\fnum(n,a_n)/(n+2)$, starting at index zero, are integral\nfor $a_0=a$ if and only if they are integral for $a_0=b$.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/243/erdos-243-reciprocal-tail-rigidity.tex#L1138"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/2","statement_grain":"environment_title"},{"assertion_id":"paper:erdos-249-binary-totient-series#res:fulldepth","assertion_kind":"paper_argued_result","reviewed":false,"statement":"Propagation of one nonintegral tail difference","status":"proved in the manuscript","problem_ids":["erdos_249"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:fulldepth","owner_paper_id":"erdos-249-binary-totient-series","source_line":608,"anchor_unit":"theorem","section_title":"Equivalent residue conditions","source_excerpt":{"text":"\\begin{theorem}[Propagation of one nonintegral tail difference]\\label{res:fulldepth}\nFix $d\\ge1$ and $N\\ge0$. If $\\Delta_d(N)\\notin\\Z$, then every sufficiently\nlate pair $\\{t,t+1\\}$ contains an $m$ such that $K(md,N,md)$ holds.\nConsequently\n\\[\n \\exists t\\ge1:\\ K(td,N,td)\n \\quad\\Longleftrightarrow\\quad\\Delta_d(N)\\notin\\Z.\n\\]\nRequiring this for every $d\\ge1,N\\ge0$ is equivalent to $S\\notin\\Q$.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/249/erdos-249-binary-totient-series.tex#L608"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/4","statement_grain":"environment_title"},{"assertion_id":"paper:erdos-249-binary-totient-series#res:rankonefloor","assertion_kind":"paper_argued_result","reviewed":false,"statement":"A lower bound for the rank-one quotients","status":"proved in the manuscript","problem_ids":["erdos_249"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:rankonefloor","owner_paper_id":"erdos-249-binary-totient-series","source_line":699,"anchor_unit":"theorem","section_title":"Comparison sequences","source_excerpt":{"text":"\\begin{theorem}[A lower bound for the rank-one quotients]\\label{res:rankonefloor}\nFor $e\\ge1$ and $Y\\ge4$, the denominator of $Q(e,Y)$ is positive, and the\nunique minimiser is $(e,Y)=(1,5)$. Every admissible quotient and every\nnonempty finite positive weighted average of such quotients satisfies\n\\[\n Q-\\Theta_2>\\frac{21}{320}.\n\\]\nThe uniform bound $Q-\\Theta_2>1/15$ is false already at $(1,5)$.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/249/erdos-249-binary-totient-series.tex#L699"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/4","statement_grain":"environment_title"},{"assertion_id":"paper:erdos249-totient-reasoning-surface#res:fulldepth","assertion_kind":"paper_argued_result","reviewed":false,"statement":"Propagation of one nonintegral tail difference","status":"proved in the manuscript","problem_ids":["erdos_249"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:fulldepth","owner_paper_id":"erdos249-totient-reasoning-surface","source_line":9649,"anchor_unit":"theorem","section_title":"The finite test.","source_excerpt":{"text":"\\begin{theorem}[Propagation of one nonintegral tail difference]\\label{res:fulldepth}\nFix $d\\ge1$ and $N\\ge0$. If $\\Delta_d(N)\\notin\\Z$, then every sufficiently\nlate pair $\\{t,t+1\\}$ contains an $m$ such that $K(md,N,md)$ holds.\nConsequently\n\\[\n \\exists t\\ge1:\\ K(td,N,td)\n \\quad\\Longleftrightarrow\\quad\\Delta_d(N)\\notin\\Z.\n\\]\nRequiring this for every $d\\ge1,N\\ge0$ is equivalent to $S\\notin\\Q$.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/249/erdos249-totient-reasoning-surface.tex#L9649"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/5","statement_grain":"environment_title"},{"assertion_id":"paper:erdos249-totient-reasoning-surface#res:rankonefloor","assertion_kind":"paper_argued_result","reviewed":false,"statement":"A lower bound for the rank-one quotients","status":"proved in the manuscript","problem_ids":["erdos_249"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:rankonefloor","owner_paper_id":"erdos249-totient-reasoning-surface","source_line":9875,"anchor_unit":"theorem","section_title":"The sharp rank-one quotient bound","source_excerpt":{"text":"\\begin{theorem}[A lower bound for the rank-one quotients]\\label{res:rankonefloor}\nFor $e\\ge1$ and $Y\\ge4$, the denominator of $Q(e,Y)$ is positive, and the\nunique minimiser is $(e,Y)=(1,5)$. Every admissible quotient and every\nnonempty finite positive weighted average of such quotients satisfies\n\\[\n Q-\\Theta_2>\\frac{21}{320}.\n\\]\nThe uniform bound $Q-\\Theta_2>1/15$ is false already at $(1,5)$.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/249/erdos249-totient-reasoning-surface.tex#L9875"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/5","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_251#res:sparserationalisation","assertion_kind":"paper_argued_result","reviewed":false,"statement":"sparse changes preserving congruences","status":"proved in the manuscript","problem_ids":["erdos_251"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:sparserationalisation","owner_paper_id":"erdos-251-prime-gap-dyadic-series","source_line":104,"anchor_unit":"proposition","section_title":"Introduction","source_excerpt":{"text":"\\begin{proposition}[sparse changes preserving congruences]\\label{res:sparserationalisation}\nLet $a:\\N\\to\\N$ satisfy $A=\\sum_{n\\ge0}a_n2^{-(n+1)}<\\infty$,\nlet $K\\in\\N$, and let $f:\\N\\to\\R$ tend to $+\\infty$.\nThere exist a set $S\\subseteq[K,\\infty)$ of upper Banach density zero\nand a nondegenerate interval $I\\subset(A,\\infty)$ such that, for every\n$r\\in I$, there is an integer correction $e:\\N\\to\\N$ satisfying\n\\[\n \\operatorname{supp}e\\subseteq S,\\qquad e_n\\le f(n)\\ \\text{eventually},\n \\qquad \\sum_{n\\ge0}(a_n+e_n)2^{-(n+1)}=r.\n\\]\nFor each integer $q\\ge1$ there is a cutoff $N_q$, independent of $r$,\nsuch that $q\\mid e_n$ and $q\\mid\\sum_{i<n}e_i$ for all $n\\ge N_q$.\nFor every $\\varepsilon>0$, the choice $f(n)=(\\log(n+3))^\\varepsilon$\ncan be made with\n\\[\n |S\\cap[X,2X)|=O_\\varepsilon\\!\\left(\\frac{X}{\\log\\log X}\\right),\n \\qquad\n d_{\\rm TV}(\\mu_{a,X,m},\\mu_{a+e,X,m})\n \\le\\frac{m|S\\cap[X,2X+m)|}{X}.\n\\]\nIn particular the distance tends to zero for every integer-valued\n$m=m(X)\\ge1$ with $m(X)=o(\\log\\log X)$, uniformly over target values.\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/251/erdos-251-prime-gap-dyadic-series.tex#L104"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/6","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_251#res:jointcountermodel","assertion_kind":"paper_argued_result","reviewed":false,"statement":"a rational sum with the stated prime-gap statistics","status":"proved in the manuscript","problem_ids":["erdos_251"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:jointcountermodel","owner_paper_id":"erdos-251-prime-gap-dyadic-series","source_line":404,"anchor_unit":"corollary","section_title":"Application to prime gaps","source_excerpt":{"text":"\\begin{corollary}[a rational sum with the stated prime-gap statistics]\\label{res:jointcountermodel}\nLet $p_0=2,p_1=3,\\ldots$ be the primes and $g_n=p_{n+1}-p_n$.\nGiven $K\\in\\N$ and $0<\\varepsilon\\le1$, there is $b:\\N\\to\\N$ with\nrational dyadic sum such that $b_n=g_n$ for $n<K$, $b_n\\ge g_n$, and\n$b_n-g_n\\le(\\log(n+3))^\\varepsilon$ eventually.  For every fixed positive\nmodulus, both the coefficients and the cumulative positions eventually\nretain their corresponding residues.  The empirical distributions of\nunnormalised blocks have total variation distance tending to zero\nfor lengths $o(\\log\\log X)$, and for every fixed nonzero\n$F\\in\\Z[x_0,\\ldots,x_k]$,\n\\[\n \\bigl|\\{n<N:F(b_n,\\ldots,b_{n+k})=0\\}\\bigr|=o(N).\n\\]\nMoreover, with $P_n=2+\\sum_{i<n}b_i$,\n\\[\n 0\\le P_n-p_n=O_\\varepsilon\\!\\left(\n       \\frac{n(\\log(n+3))^\\varepsilon}{\\log\\log n}\\right),\n \\qquad P_n\\sim n\\log n.\n\\]\nThese positions are not asserted to be prime.\n\\end{corollary}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/251/erdos-251-prime-gap-dyadic-series.tex#L404"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/6","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_251#res:infinite","assertion_kind":"paper_argued_result","reviewed":false,"statement":"prime-to-gap identity","status":"proved in the manuscript","problem_ids":["erdos_251"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:infinite","owner_paper_id":"erdos-251-prime-gap-dyadic-series","source_line":489,"anchor_unit":"theorem","section_title":"The prime series and its actual tails","source_excerpt":{"text":"\\begin{theorem}[prime-to-gap identity]\\label{res:infinite}\nThe actual prime and gap series satisfy $\\Pi=2+G$.  Their complete tails\n\\[\n T_N=\\sum_{j\\ge1}g_{N+j}2^{-j}\n\\]\nsatisfy $T_0=2G-1$ and $T_{N+1}=2T_N-g_{N+1}$.\nThus $\\Pi$, $G$ and $T_0$ have the same rationality status.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/251/erdos-251-prime-gap-dyadic-series.tex#L489"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/6","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_251#res:true-tail","assertion_kind":"paper_argued_result","reviewed":false,"statement":"the boundary condition for the tail recurrence","status":"proved in the manuscript","problem_ids":["erdos_251"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:true-tail","owner_paper_id":"erdos-251-prime-gap-dyadic-series","source_line":523,"anchor_unit":"lemma","section_title":"The prime series and its actual tails","source_excerpt":{"text":"\\begin{lemma}[the boundary condition for the tail recurrence]\\label{res:true-tail}\nLet $\\sum_{j\\ge1}|a_j|2^{-j}<\\infty$ and $U_{N+1}=2U_N-a_{N+1}$.\nThen $U_N=\\sum_{j\\ge1}a_{N+j}2^{-j}$ for every $N$ if and only if\n$2^{-N}U_N\\to0$.\n\\end{lemma}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/251/erdos-251-prime-gap-dyadic-series.tex#L523"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/6","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_251#res:escape-irrational","assertion_kind":"paper_argued_result","reviewed":false,"statement":"exact rationality classification","status":"proved in the manuscript","problem_ids":["erdos_251"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:escape-irrational","owner_paper_id":"erdos-251-prime-gap-dyadic-series","source_line":544,"anchor_unit":"theorem","section_title":"Integral shifts","source_excerpt":{"text":"\\begin{theorem}[exact rationality classification]\\label{res:escape-irrational}\nFor a real integer-coefficient recurrence, the following are equivalent:\n$U_0\\in\\Q$; $D_h(N)\\in\\Z$ for some $h\\ge1,N\\ge0$; and, for some fixed\n$h\\ge1$, $D_h(N)\\in\\Z$ at every sufficiently large $N$.\nMore precisely, if $U_0=u/(2^sd)$ is in lowest terms, with $d$ odd, then\n\\[\n \\operatorname{den}(U_N)=2^{\\max(s-N,0)}d,\\qquad\n D_h(N)\\in\\Z\\ \\Longleftrightarrow\\ N\\ge s\\ \\text{and}\\ d\\mid2^h-1\n \\quad(h\\ge1).\n\\]\nConsequently $U_0$ is irrational exactly when every positive shift is\nnonintegral at every index, equivalently when for each fixed $h\\ge1$\nthere are arbitrarily late nonintegral shifts.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/251/erdos-251-prime-gap-dyadic-series.tex#L544"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/6","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_251#res:signedwindow","assertion_kind":"paper_argued_result","reviewed":false,"statement":"two consecutive differences of absolute value less than one","status":"proved in the manuscript","problem_ids":["erdos_251"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:signedwindow","owner_paper_id":"erdos-251-prime-gap-dyadic-series","source_line":590,"anchor_unit":"proposition","section_title":"Small tail differences and finite tests","source_excerpt":{"text":"\\begin{proposition}[two consecutive differences of absolute value less than one]\\label{res:signedwindow}\nLet $D,D'\\in\\R$, $\\delta\\in2\\Z$ and $D'=2D-\\delta$.\nThe conditions $|D|<1$, $|D'|<1$ and $\\delta\\ne0$ hold exactly when,\nfor some $s\\in\\{-1,1\\}$,\n\\[\n \\delta=2s,\\qquad \\tfrac12<sD<1.\n\\]\nIn that case $sD'\\in(-1,0)$ and both $D$ and $D'$ are nonintegral.\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/251/erdos-251-prime-gap-dyadic-series.tex#L590"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/6","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_251#res:truncation","assertion_kind":"paper_argued_result","reviewed":false,"statement":"finite separation criterion","status":"proved in the manuscript","problem_ids":["erdos_251"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:truncation","owner_paper_id":"erdos-251-prime-gap-dyadic-series","source_line":657,"anchor_unit":"proposition","section_title":"Finite separation from the integers","source_excerpt":{"text":"\\begin{proposition}[finite separation criterion]\\label{res:truncation}\nIf for every $h\\ge1$ and every cutoff $N_0$ there are $N\\ge N_0,L\\ge1$ with\n\\begin{equation}\\label{eq:truncation}\n \\operatorname{dist}(S_{h,N,L},\\Z)>R_{h,N,L}(M),\n\\end{equation}\nthen $\\Pi$ is irrational.\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/251/erdos-251-prime-gap-dyadic-series.tex#L657"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/6","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_257#res:problem","assertion_kind":"paper_posed_problem","reviewed":false,"statement":"Erdős #257","status":"posed in the manuscript","problem_ids":["erdos_257"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:problem","owner_paper_id":"erdos-257-mersenne-support-subseries","source_line":61,"anchor_unit":"problem","section_title":"Introduction","source_excerpt":{"text":"\\begin{problem}[Erd\\H{o}s \\#257]\\label{res:problem}\nIs $X_A(2)$ irrational for every infinite $A\\subseteq\\Npos$?\n\\end{problem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/257/erdos-257-mersenne-support-subseries.tex#L61"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_problem","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/8","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_257#res:weighted-cover-incomparability","assertion_kind":"paper_argued_result","reviewed":false,"statement":"incomparable support criteria","status":"proved in the manuscript","problem_ids":["erdos_257"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:weighted-cover-incomparability","owner_paper_id":"erdos-257-mersenne-support-subseries","source_line":614,"anchor_unit":"proposition","section_title":"Comparison of the support classes","source_excerpt":{"text":"\\begin{proposition}[incomparable support criteria]\n\\label{res:weighted-cover-incomparability}\nThere are infinite positive supports $E$ and $V$ such that\n\\[\n E\\in\\mathcal W_2,\\quad E\\notin\\mathcal C,\\qquad\n V\\in\\mathcal C,\\quad V\\notin\\mathcal W_b\\ (b\\ge2),\\qquad\n \\sum_{a\\in V}\\frac1a=\\infty.\n\\]\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/257/erdos-257-mersenne-support-subseries.tex#L614"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/8","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_257#res:strict-mixed-supports","assertion_kind":"paper_argued_result","reviewed":false,"statement":"a support requiring the mixed criterion","status":"proved in the manuscript","problem_ids":["erdos_257"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:strict-mixed-supports","owner_paper_id":"erdos-257-mersenne-support-subseries","source_line":764,"anchor_unit":"corollary","section_title":"A strict enlargement and finite unions","source_excerpt":{"text":"\\begin{corollary}[a support requiring the mixed criterion]\n\\label{res:strict-mixed-supports}\nThere is an infinite positive support $U$ with\n$U\\notin\\mathcal C$ and $U\\notin\\mathcal W_b$ for every integer $b\\ge2$,\nsuch that $X_A(b)$ is irrational for every infinite $A\\subseteq U$\nand every integer $b\\ge2$.\n\\end{corollary}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/257/erdos-257-mersenne-support-subseries.tex#L764"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/8","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_257#res:general-repair","assertion_kind":"paper_argued_result","reviewed":false,"statement":"membership and nonincreasing integer remainders","status":"proved in the manuscript","problem_ids":["erdos_257"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:general-repair","owner_paper_id":"erdos-257-mersenne-support-subseries","source_line":1148,"anchor_unit":"theorem","section_title":"The greedy recurrence","source_excerpt":{"text":"\\begin{theorem}[membership and nonincreasing integer remainders]\\label{res:general-repair}\nFor every real $x\\ge0$, the following are equivalent:\n\\[\n \\begin{gathered}\n x\\in\\Ach;\\\\\n \\forall K\\ge0\\ \\exists N\\ge K:\\quad Q_{N+1}\\le Q_N;\\\\\n \\forall K\\ge0\\ \\exists N\\in[K,K+2\\lfloor\\sqrt K\\rfloor+12):\n \\quad Q_{N+1}\\le Q_N.\n \\end{gathered}\n\\]\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/257/erdos-257-mersenne-support-subseries.tex#L1148"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/8","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_269#res:distinct-height-235","assertion_kind":"paper_argued_result","reviewed":false,"statement":"the sum over distinct running LCMs for {2,3,5}","status":"proved in the manuscript","problem_ids":["erdos_269"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:distinct-height-235","owner_paper_id":"erdos-269-three-prime-running-lcm","source_line":84,"anchor_unit":"theorem","section_title":"Two sums from running least common multiples","source_excerpt":{"text":"\\begin{theorem}[the sum over distinct running LCMs for $\\{2,3,5\\}$]\\label{res:distinct-height-235}\nThe number $\\mathcal D_{\\{2,3,5\\}}$ is irrational.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/269/erdos-269-three-prime-running-lcm.tex#L84"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/12","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_269#res:dyadic-alphabet","assertion_kind":"paper_argued_result","reviewed":false,"statement":"integer coefficients and four possible bases","status":"proved in the manuscript","problem_ids":["erdos_269"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:dyadic-alphabet","owner_paper_id":"erdos-269-three-prime-running-lcm","source_line":357,"anchor_unit":"lemma","section_title":"The repeated three-prime sum","source_excerpt":{"text":"\\begin{lemma}[integer coefficients and four possible bases]\n\\label{res:dyadic-alphabet}\nFor every $a\\ge0$, $m_a$ is a positive integer and\n$b_a\\in\\{2,6,10,30\\}$.  The word ``numerator'' does not impose the\npositional-digit restriction $m_a<b_a$; that restriction need not hold.\n\\end{lemma}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/269/erdos-269-three-prime-running-lcm.tex#L357"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/12","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_269#res:actual-orbit","assertion_kind":"paper_argued_result","reviewed":false,"statement":"the tail recurrence and a quadratic bound","status":"proved in the manuscript","problem_ids":["erdos_269"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:actual-orbit","owner_paper_id":"erdos-269-three-prime-running-lcm","source_line":371,"anchor_unit":"proposition","section_title":"The repeated three-prime sum","source_excerpt":{"text":"\\begin{proposition}[the tail recurrence and a quadratic bound]\n\\label{res:actual-orbit}\nThe series defining $S,T_a$ converge.  For every $a\\ge0$,\n\\[\n X_{a+1}=b_aX_a-m_a,\\qquad\n 0<X_a\\le\\frac{8640}{343}(a+1)^2<90(a+1)^2.\n\\]\nFor every integer $B\\ge1$, either $BX_a$ is integral at some index (and hence at every later index), or $\\operatorname{dist}(BX_a,\\mathbb Z)\\ge1/31$\nat arbitrarily large indices.\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/269/erdos-269-three-prime-running-lcm.tex#L371"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/12","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_269#res:denominator-reduction","assertion_kind":"paper_argued_result","reviewed":false,"statement":"rationality gives positive integer tails","status":"proved in the manuscript","problem_ids":["erdos_269"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:denominator-reduction","owner_paper_id":"erdos-269-three-prime-running-lcm","source_line":425,"anchor_unit":"theorem","section_title":"The repeated three-prime sum","source_excerpt":{"text":"\\begin{theorem}[rationality gives positive integer tails]\n\\label{res:denominator-reduction}\nIf $S=A/D$ in lowest terms, where\n$D=2^u3^v5^wB$ and $\\gcd(B,30)=1$, then for every\n$a\\ge a_0=u+1+2v+3w$,\n\\[\n d_a=BX_a\\in\\mathbb Z_{>0},\\qquad\n d_{a+1}=b_ad_a-Bm_a,\\qquad d_a\\le90B(a+1)^2.\n\\]\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/269/erdos-269-three-prime-running-lcm.tex#L425"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/12","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_269#res:exact-onset","assertion_kind":"paper_argued_result","reviewed":false,"statement":"the first index at which the denominator clears","status":"proved in the manuscript","problem_ids":["erdos_269"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:exact-onset","owner_paper_id":"erdos-269-three-prime-running-lcm","source_line":447,"anchor_unit":"corollary","section_title":"The repeated three-prime sum","source_excerpt":{"text":"\\begin{corollary}[the first index at which the denominator clears]\\label{res:exact-onset}\nUnder the same lowest-terms hypothesis, put $M=2^u3^v5^w$ and let\n$\\operatorname{den}$ denote the positive reduced denominator.  For $a\\ge1$,\n\\[\n \\operatorname{den}(BX_a)=\\frac{M}{\\gcd(M,h_a)}.\n\\]\nConsequently $BX_a$ is integral exactly when\n$2^a\\ge\\max(2^{u+1},3^v,5^w)$.  The first such $a$ can be found by\ninteger comparisons, without logarithmic rounding.\n\\end{corollary}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/269/erdos-269-three-prime-running-lcm.tex#L447"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/12","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_269#res:two-prime-transcendence","assertion_kind":"paper_argued_result","reviewed":false,"statement":"both two-prime sums","status":"proved in the manuscript","problem_ids":["erdos_269"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:two-prime-transcendence","owner_paper_id":"erdos-269-three-prime-running-lcm","source_line":633,"anchor_unit":"theorem","section_title":"Two-prime transcendence","source_excerpt":{"text":"\\begin{theorem}[both two-prime sums]\n\\label{res:two-prime-transcendence}\n\\label{res:two-prime-repeated-transcendence}\nLet $p<q$ be distinct primes.  Put $\\theta=\\log p/\\log q$ and\n$A=\\sum_{n\\ge0}p^{-n}q^{-\\lfloor n\\theta\\rfloor}$.  Let\n$\\mathcal R_{p,q}$ sum the reciprocal running LCM at every positive\n$\\{p,q\\}$-smooth integer, and let $\\mathcal D_{p,q}$ count each distinct\nrunning LCM once.  Then\n\\begin{equation}\\label{eq:two-prime-affine}\n \\mathcal D_{p,q}=\\frac{(q-p)A+p}{q-1},\\qquad\n \\mathcal R_{p,q}=\\frac{(p+q-1)A-(p-1)A^2}{q-1}.\n\\end{equation}\nBoth numbers are transcendental.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/269/erdos-269-three-prime-running-lcm.tex#L633"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/12","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_269#res:two-prime-repeated-transcendence","assertion_kind":"paper_argued_result","reviewed":false,"statement":"both two-prime sums","status":"proved in the manuscript","problem_ids":["erdos_269"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:two-prime-repeated-transcendence","owner_paper_id":"erdos-269-three-prime-running-lcm","source_line":634,"anchor_unit":"theorem","section_title":"Two-prime transcendence","source_excerpt":{"text":"\\begin{theorem}[both two-prime sums]\n\\label{res:two-prime-transcendence}\n\\label{res:two-prime-repeated-transcendence}\nLet $p<q$ be distinct primes.  Put $\\theta=\\log p/\\log q$ and\n$A=\\sum_{n\\ge0}p^{-n}q^{-\\lfloor n\\theta\\rfloor}$.  Let\n$\\mathcal R_{p,q}$ sum the reciprocal running LCM at every positive\n$\\{p,q\\}$-smooth integer, and let $\\mathcal D_{p,q}$ count each distinct\nrunning LCM once.  Then\n\\begin{equation}\\label{eq:two-prime-affine}\n \\mathcal D_{p,q}=\\frac{(q-p)A+p}{q-1},\\qquad\n \\mathcal R_{p,q}=\\frac{(p+q-1)A-(p-1)A^2}{q-1}.\n\\end{equation}\nBoth numbers are transcendental.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/269/erdos-269-three-prime-running-lcm.tex#L634"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/12","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_269#res:lcm","assertion_kind":"paper_argued_result","reviewed":false,"statement":"the running least common multiple","status":"proved in the manuscript","problem_ids":["erdos_269"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:lcm","owner_paper_id":"erdos-269-three-prime-running-lcm","source_line":723,"anchor_unit":"proposition","section_title":"Identities for the running LCM and multiplicities","source_excerpt":{"text":"\\begin{proposition}[the running least common multiple]\\label{res:lcm}\nLet $p,q,r$ be pairwise distinct primes and $x\\ge1$. Then the running\nleast common multiple is given by $\\Lprefix(x)=\\hgt(x)$.\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/269/erdos-269-three-prime-running-lcm.tex#L723"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/12","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_269#res:cell","assertion_kind":"paper_argued_result","reviewed":false,"statement":"cells and jumps","status":"proved in the manuscript","problem_ids":["erdos_269"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:cell","owner_paper_id":"erdos-269-three-prime-running-lcm","source_line":737,"anchor_unit":"proposition","section_title":"Identities for the running LCM and multiplicities","source_excerpt":{"text":"\\begin{proposition}[cells and jumps]\\label{res:cell}\nThe running LCM is constant when the three integer logarithms are constant.\nA jump in exactly one logarithm multiplies it by the corresponding prime.\nThe first $n$ positive powers of each prime, together with $1$, form\n$3n+1$ distinct points.\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/269/erdos-269-three-prime-running-lcm.tex#L737"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/12","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_269#res:fibre-prop","assertion_kind":"paper_argued_result","reviewed":false,"statement":"grouping terms with the same running LCM","status":"proved in the manuscript","problem_ids":["erdos_269"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:fibre-prop","owner_paper_id":"erdos-269-three-prime-running-lcm","source_line":749,"anchor_unit":"proposition","section_title":"Identities for the running LCM and multiplicities","source_excerpt":{"text":"\\begin{proposition}[grouping terms with the same running LCM]\\label{res:fibre-prop}\nFor a finite exponent box $\\mathcal B$, set\n$F(H)=\\{(i,j,k)\\in\\mathcal B:\\hgt(p^iq^jr^k)=H\\}$.  Then\n\\begin{equation}\\label{res:fibre}\n \\sum_{(i,j,k)\\in\\mathcal B}\\Kernel(i,j,k)=\\sum_H\\frac{\\#F(H)}H.\n\\end{equation}\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/269/erdos-269-three-prime-running-lcm.tex#L749"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/12","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_269#res:fibre","assertion_kind":"paper_argued_result","reviewed":false,"statement":"grouping terms with the same running LCM","status":"proved in the manuscript","problem_ids":["erdos_269"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:fibre","owner_paper_id":"erdos-269-three-prime-running-lcm","source_line":752,"anchor_unit":"proposition","section_title":"Identities for the running LCM and multiplicities","source_excerpt":{"text":"\\begin{proposition}[grouping terms with the same running LCM]\\label{res:fibre-prop}\nFor a finite exponent box $\\mathcal B$, set\n$F(H)=\\{(i,j,k)\\in\\mathcal B:\\hgt(p^iq^jr^k)=H\\}$.  Then\n\\begin{equation}\\label{res:fibre}\n \\sum_{(i,j,k)\\in\\mathcal B}\\Kernel(i,j,k)=\\sum_H\\frac{\\#F(H)}H.\n\\end{equation}\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/269/erdos-269-three-prime-running-lcm.tex#L752"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/12","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_269#res:infinite-rank","assertion_kind":"paper_argued_result","reviewed":false,"statement":"no finite separation of the kernel","status":"proved in the manuscript","problem_ids":["erdos_269"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:infinite-rank","owner_paper_id":"erdos-269-three-prime-running-lcm","source_line":780,"anchor_unit":"theorem","section_title":"The three-prime kernel","source_excerpt":{"text":"\\begin{theorem}[no finite separation of the kernel]\\label{res:infinite-rank}\nLet $p,q,r$ be primes with $p\\ne q$, $p\\ne r$ and $q\\ne r$.  For every\n$n\\ge0$ there are injective maps $I,J:\\{0,\\ldots,n-1\\}\\to\\N$ such that, for\nevery $k\\ge0$,\n\\[\n \\det\\bigl(\\Kernel(I(a),J(b),k)\\bigr)_{0\\le a,b<n}\\ne0.\n\\]\nConsequently, for no finite $d$ do there exist rational-valued functions\n$f_\\ell(i)$ and $G_\\ell(j,k)$, $0\\le\\ell<d$, satisfying\n\\[\n \\Kernel(i,j,k)=\\sum_{\\ell<d}f_\\ell(i)G_\\ell(j,k)\n \\qquad\\hbox{for all }i,j,k.\n\\]\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/269/erdos-269-three-prime-running-lcm.tex#L780"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/12","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_269#res:admissible-modular-minors","assertion_kind":"paper_argued_result","reviewed":false,"statement":"the same minors modulo integers coprime to 30","status":"proved in the manuscript","problem_ids":["erdos_269"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:admissible-modular-minors","owner_paper_id":"erdos-269-three-prime-running-lcm","source_line":869,"anchor_unit":"corollary","section_title":"The three-prime kernel","source_excerpt":{"text":"\\begin{corollary}[the same minors modulo integers coprime to $30$]\n\\label{res:admissible-modular-minors}\nFor $(p,q,r)=(2,3,5)$ and every $n\\ge1$, there are injective maps\n$I,J:\\{0,\\ldots,n-1\\}\\to\\N$, chosen independently of $B$ and $k$, such that\nfor every $B\\ge2$ coprime to $30$ and every $k\\ge0$, the selected $n\\times n$\nkernel matrix has unit determinant over $\\mathbb Z/B\\mathbb Z$, with each\nreciprocal prime power interpreted by its modular inverse.\n\\end{corollary}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/269/erdos-269-three-prime-running-lcm.tex#L869"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/12","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_269#res:finite-cut-rank","assertion_kind":"paper_argued_result","reviewed":false,"statement":"rank of a matrix of threshold columns","status":"proved in the manuscript","problem_ids":["erdos_269"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:finite-cut-rank","owner_paper_id":"erdos-269-three-prime-running-lcm","source_line":902,"anchor_unit":"proposition","section_title":"Leading minors.","source_excerpt":{"text":"\\begin{proposition}[rank of a matrix of threshold columns]\\label{res:finite-cut-rank}\nLet $m\\ge1$ and let $c$ lie in a field with $c\\ne 0,1$.\nFor $0\\le k\\le m$ write $v_k$ for the length-$m$ column with a $1$ in each of\nthe first $k$ coordinates and $c$ thereafter.\nA matrix whose distinct columns are $v_k$ for $k$ in a nonempty set $E$ has\nrank $|E|-\\mathbf 1_{\\{0,m\\}\\subseteq E}$.\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/269/erdos-269-three-prime-running-lcm.tex#L902"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/12","statement_grain":"environment_title"},{"assertion_id":"paper:erdos-1041-lemniscate-newton-flow#res:problem","assertion_kind":"paper_posed_problem","reviewed":false,"statement":"Erdős #1041","status":"posed in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:problem","owner_paper_id":"erdos-1041-lemniscate-newton-flow","source_line":75,"anchor_unit":"problem","section_title":"Introduction","source_excerpt":{"text":"\\begin{problem}[Erd\\H{o}s \\#1041]\\label{res:problem}\nLet $f(z)=\\prod_{i=1}^{n}(z-z_i)$ be monic with $n\\ge2$ and all\n$z_i$ in the open unit disc.  Must two root occurrences be joined by a curve\nof length less than $2$ inside $\\{|f|<1\\}$?\n\\end{problem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos-1041-lemniscate-newton-flow.tex#L75"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_problem","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/13","statement_grain":"environment_title"},{"assertion_id":"paper:erdos-1041-lemniscate-newton-flow#res:trinomial-all-degree","assertion_kind":"paper_argued_result","reviewed":false,"statement":"all-degree monic trinomials","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:trinomial-all-degree","owner_paper_id":"erdos-1041-lemniscate-newton-flow","source_line":414,"anchor_unit":"theorem","section_title":"Trinomials and radial segments","source_excerpt":{"text":"\\begin{theorem}[all-degree monic trinomials]\n\\label{res:trinomial-all-degree}\nLet $1\\le m<n$ and\n\\[\n f(z)=z^n+az^m+b,\n\\]\nwith every zero in the open unit disc.  For any zero $\\zeta$, the entire\nsegment $[0,\\zeta]$ lies in $\\{|f|<1\\}$.  Distinct zeros $\\zeta_1,\\zeta_2$\nare therefore joined by the broken line $\\zeta_1\\to0\\to\\zeta_2$ of length\n$\\|\\zeta_1\\|+\\|\\zeta_2\\|<2$ inside the open unit lemniscate.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos-1041-lemniscate-newton-flow.tex#L414"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/13","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:sextic-spoke","assertion_kind":"paper_argued_result","reviewed":false,"statement":"failure of a prescribed radial segment","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:sextic-spoke","owner_paper_id":"erdos-1041-lemniscate-newton-flow","source_line":454,"anchor_unit":"proposition","section_title":"Trinomials and radial segments","source_excerpt":{"text":"\\begin{proposition}[failure of a prescribed radial segment]\n\\label{res:sextic-spoke}\nThere exist $r\\in(0,1)$ for which every zero of\n\\[\n f_r(z)=z^6+\\tfrac15 r^2 z^4-\\tfrac15 r^4 z^2-r^6\n\\]\nlies in the open unit disc, yet the radial segment from the origin to the zero\n$r$ leaves $\\{|f_r|<1\\}$.\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos-1041-lemniscate-newton-flow.tex#L454"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/13","statement_grain":"environment_title"},{"assertion_id":"paper:erdos1041-lemniscate-reasoning-surface#res:problem","assertion_kind":"paper_posed_problem","reviewed":false,"statement":"Erdős #1041","status":"posed in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:problem","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":85,"anchor_unit":"problem","section_title":"The question and the degree-seven counterexample","source_excerpt":{"text":"\\begin{problem}[Erd\\H{o}s \\#1041]\\label{res:problem}\nLet $f(z)=\\prod_{i=1}^{n}(z-z_i)$ be monic of degree $n\\ge2$, with\n$z_i\\in\\dbar$, the open unit disc.  Must two of the roots be joined by a curve of length less than\n$2$ lying in the open lemniscate $\\{z\\in\\mathbb{C}:|f(z)|<1\\}$?\n\\end{problem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L85"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_problem","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:ani-degree-seven-counterexample-long","assertion_kind":"paper_argued_result","reviewed":false,"statement":"ani's counterexample","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:ani-degree-seven-counterexample-long","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":106,"anchor_unit":"theorem","section_title":"The question and the degree-seven counterexample","source_excerpt":{"text":"\\begin{theorem}[\\texttt{ani}'s counterexample]\\label{res:ani-degree-seven-counterexample-long}\nThe monic polynomial in \\eqref{eq:ani-f-long} has seven distinct roots\nin the open unit disc. Every connected subset of its strict unit\nlemniscate containing two roots has one-dimensional Hausdorff measure\ngreater than $2$.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L106"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos1041-lemniscate-reasoning-surface#res:trinomial-all-degree","assertion_kind":"paper_argued_result","reviewed":false,"statement":"trinomial root connections","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:trinomial-all-degree","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":546,"anchor_unit":"theorem","section_title":"Trinomials","source_excerpt":{"text":"\\begin{theorem}[trinomial root connections]\n\\label{res:trinomial-all-degree}\nLet $n,m$ be integers with $1\\le m<n$, and let $f(z)=z^n+az^m+b$ have every\nzero in $\\dbar$.  For every zero $\\zeta$, the segment $[0,\\zeta]$ lies in\n$E_f$.  Consequently any two zeros $\\zeta_1,\\zeta_2$ are joined in $E_f$ by\nthe broken line $\\zeta_1\\to0\\to\\zeta_2$, of length $|\\zeta_1|+|\\zeta_2|<2$.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L546"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:circle-slice-packing","assertion_kind":"paper_argued_result","reviewed":false,"statement":"circle-slice packing","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:circle-slice-packing","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":805,"anchor_unit":"lemma","section_title":"A lower bound for the number of roots.","source_excerpt":{"text":"\\begin{lemma}[circle-slice packing]\\label{res:circle-slice-packing}\nUnder~\\eqref{eq:lc-separation}, for every $r>0$,\n\\[\n \\sum_{j=1}^{k}w(d_j,r)\\le\\pi,\\qquad\n w(d,r)=\\arccos\\Bigl(\\operatorname{clamp}\n   \\frac{\\cosh d\\cosh r-\\cosh(D/2)}{\\sinh d\\sinh r}\\Bigr),\n\\]\nwhere $\\operatorname{clamp}$ truncates its argument to $[-1,1]$.\n\\end{lemma}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L805"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:one-root-gamma-false","assertion_kind":"paper_argued_result","reviewed":false,"statement":"a counterexample to the proposed one-root perimeter bound","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:one-root-gamma-false","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":1537,"anchor_unit":"proposition","section_title":"A path estimate from area and boundary length","source_excerpt":{"text":"\\begin{proposition}[a counterexample to the proposed one-root perimeter bound]\n\\label{res:one-root-gamma-false}\nLet $p(z)=z^8-(3/2)z$ and let $C$ be the component of $\\{|p|\\le1\\}$\ncontaining the origin.  Then $C$ contains exactly one zero and a\nneighbourhood of the closed disc of radius $5/8$, so\n$\\mathcal H^1(\\partial C)>5\\pi/4$.  The constant\n$\\Gamma(1/4)^2/(2\\sqrt{\\pi})$ is at most $(\\pi/2)(1+\\sqrt2)<5\\pi/4$.\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L1537"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:degree-three","assertion_kind":"paper_argued_result","reviewed":false,"statement":"the cubic case","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:degree-three","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":1707,"anchor_unit":"theorem","section_title":"Degree three","source_excerpt":{"text":"\\begin{theorem}[the cubic case]\n\\label{res:degree-three}\nLet $f(z)=\\prod_{j=1}^{3}(z-z_j)$ with $|z_j|<1$, the roots listed with\nmultiplicity.  Then two listed root occurrences are joined inside\n$\\{|f|<1\\}$ by a polygonal path of length strictly below $2$.  If $f$ is\nsquarefree the two are distinct.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L1707"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:sep-or-false","assertion_kind":"paper_argued_result","reviewed":false,"statement":"failure of two critical-value criteria to cover all polynomials","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:sep-or-false","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":2156,"anchor_unit":"proposition","section_title":"Examples separating the geometric hypotheses","source_excerpt":{"text":"\\begin{proposition}[failure of two critical-value criteria to cover all polynomials]\n\\label{res:sep-or-false}\nLet $f(z)=z^3+(3/100)z-3/4$.  Every root lies in the open unit disc, both\ncritical points are simple, the critical values lie on distinct positive\nrays, $\\mu>13/25$, and\n\\[\n \\bigl|1-f(c_-)/f(c_+)\\bigr|<2/375<2.\n\\]\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L2156"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:arity-not-capacity","assertion_kind":"paper_argued_result","reviewed":false,"statement":"root count does not force a capacity gap","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:arity-not-capacity","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":2182,"anchor_unit":"proposition","section_title":"Examples separating the geometric hypotheses","source_excerpt":{"text":"\\begin{proposition}[root count does not force a capacity gap]\n\\label{res:arity-not-capacity}\nLet $g(z)=z^3-(3/400)z-3/32$. All roots lie in the open unit disc and\n$\\mu=187/2000\\le1/2$. The first merger joins two root components, so\n$k_0=2$, but the component at level $2\\mu$ containing that pair has\nnormalised capacity $1$.\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L2182"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:complementary-binomial-chords","assertion_kind":"paper_argued_result","reviewed":false,"statement":"complementary binomial chords","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:complementary-binomial-chords","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":2769,"anchor_unit":"theorem","section_title":"The exact binomial chord calculation","source_excerpt":{"text":"\\begin{theorem}[complementary binomial chords]\\label{res:complementary-binomial-chords}\nTwo adjacent zeros of $z^n-a$ can be joined by an explicit polygonal path\ninside $\\{|z^n-a|<1\\}$ of length strictly below $2$.  For $r<r_*$ the\nadjacent-root chord itself works.  For $r\\ge r_*$, two radial legs and an\ninner adjacent crossing chord work after an arbitrarily small radial\ncontraction.  These two constructions meet at $r=r_*$, where the outer chord attains\n$|f|=1$ at its midpoint and therefore lies only in the closed lemniscate.\nOpen containment at and above the switch uses the inner chord\nafter a radial contraction.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L2769"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:critical-value-budget","assertion_kind":"paper_argued_result","reviewed":false,"statement":"a mean bound for critical values","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:critical-value-budget","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":3193,"anchor_unit":"theorem","section_title":"A mean bound for critical values in every degree","source_excerpt":{"text":"\\begin{theorem}[a mean bound for critical values]\n\\label{res:critical-value-budget}\nLet $f$ be monic of degree $n\\ge2$, with roots in a closed disc of radius\n$R\\ge0$.  If $c_1,\\ldots,c_{n-1}$ are its critical points counted with\nmultiplicity, then\n\\begin{equation}\\label{eq:critical-value-quadratic-budget}\n \\sum_{j=1}^{n-1}|f(c_j)|^{2/(n-1)}\n \\le(n-1)R^{2n/(n-1)}.\n\\end{equation}\nConsequently the lower exponents used elsewhere in the record satisfy\n\\begin{equation}\\label{eq:critical-value-power-budget}\n \\sum_{j=1}^{n-1}|f(c_j)|^{1/(n-1)}\n \\le(n-1)R^{n/(n-1)}\n\\end{equation}\nand\n\\[\n \\sum_{j=1}^{n-1}|f(c_j)|^{1/n}\\le(n-1)R.\n\\]\nThe constant is attained by $f(z)=(z-\\tau)^n-\\lambda$ with enclosing\ndisk centred at $\\tau$ and radius $R=|\\lambda|^{1/n}$.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L3193"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:reflected-critical-value","assertion_kind":"paper_argued_result","reviewed":false,"statement":"reflected-derivative bound","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:reflected-critical-value","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":3224,"anchor_unit":"lemma","section_title":"A mean bound for critical values in every degree","source_excerpt":{"text":"\\begin{lemma}[reflected-derivative bound]\\label{res:reflected-critical-value}\nIf $f$ is monic of degree $n\\ge2$ with roots in the closed unit disk,\nand $c_1,\\ldots,c_{n-1}$ list its critical points with multiplicity, then\n\\begin{equation}\\label{eq:critical-reflected-product}\n |f(c_j)|\\le\\prod_{k=1}^{n-1}|1-\\overline{c_j}c_k|.\n\\end{equation}\n\\end{lemma}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L3224"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:fp-weighted-all-degree","assertion_kind":"paper_argued_result","reviewed":false,"statement":"a weighted inequality for points in a disc","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:fp-weighted-all-degree","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":3305,"anchor_unit":"theorem","section_title":"The weighted Poisson inequality and its equality case.","source_excerpt":{"text":"\\begin{theorem}[a weighted inequality for points in a disc]\n\\label{res:fp-weighted-all-degree}\nLet $c_1,\\ldots,c_m\\in\\overline{\\mathbb D}$ and let $w_j>0$ satisfy\n$\\sum_j w_j=1$.  Set\n\\[\n G(z)=\\prod_k |1-\\overline{c_k}z|^{w_k}.\n\\]\nThen\n\\[\n \\sum_j w_j G(c_j)^2\\le 1,\n\\]\nwith equality if and only if every $c_j=0$.  Equal weights therefore give\n$\\sum_j(\\prod_k|1-\\overline{c_k}c_j|)^{1/m}\\le m$ for every $m$.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L3305"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:critical-proximity","assertion_kind":"paper_argued_result","reviewed":false,"statement":"a geometric-mean bound for distances to a critical point","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:critical-proximity","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":3634,"anchor_unit":"theorem","section_title":"A sufficient critical-value threshold in degree five","source_excerpt":{"text":"\\begin{theorem}[a geometric-mean bound for distances to a critical point]\n\\label{res:critical-proximity}\nLet $n\\ge2$, let $z_1,\\ldots,z_n,c\\in\\mathbb C$ with $c\\ne z_k$ for every\n$k$, and suppose\n\\[\n  \\sum_{k=1}^n\\frac1{c-z_k}=0.\n\\]\nIf $r>0$ is determined by\n\\[\n  r^n=\\prod_{k=1}^n|c-z_k|,\n\\]\nthen there are distinct indices $i,j$ such that\n\\[\n  |c-z_i|+|c-z_j|\\le2r.\n\\]\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L3634"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:two-nearest-roots","assertion_kind":"paper_argued_result","reviewed":false,"statement":"two nearest roots","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:two-nearest-roots","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":3670,"anchor_unit":"corollary","section_title":"The distance bound for the two nearest roots.","source_excerpt":{"text":"\\begin{corollary}[two nearest roots]\\label{res:two-nearest-roots}\nIf the roots lie in the open unit disc and $c$ is a non-root critical point,\nthe two nearest roots to $c$ have total distance strictly below $2$.\n\\end{corollary}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L3670"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:straight-no-go","assertion_kind":"paper_argued_result","reviewed":false,"statement":"two counterexamples to straight-path assertions","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:straight-no-go","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":3716,"anchor_unit":"proposition","section_title":"Two failures of straight segments.","source_excerpt":{"text":"\\begin{proposition}[two counterexamples to straight-path assertions]\n\\label{res:straight-no-go}\nThere is a monic quintic with all roots in the open unit disc and a non-root\ncritical point $c$ whose unique nearest root has a point on the straight segment\nto $c$ outside $\\{|f|<1\\}$.  There is also a monic cubic with all roots in the\nopen unit disc such that the midpoint of every pair of distinct roots lies\noutside $\\{|f|<1\\}$.\n\\end{proposition}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L3716"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:orlicz-currency","assertion_kind":"paper_argued_result","reviewed":false,"statement":"the relation between two merger-scale integrals","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:orlicz-currency","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":4179,"anchor_unit":"theorem","section_title":"An integral for successive merger scales","source_excerpt":{"text":"\\begin{theorem}[the relation between two merger-scale integrals]\n\\label{res:orlicz-currency}\nDefine\n\\[\n \\Phi(x)=\\int_0^x\\frac{dt}{\\log(\\coth t)}\\qquad(x\\ge0).\n\\]\nAt $t=0$ the integrand is understood by its continuous limiting value $0$\n(equivalently, the integral is improper at that endpoint).\nFor every integer $k\\ge1$ and $0<r\\le1$, with\n$x=k^{-1}\\log(1/r)$,\n\\[\n I_k(r)=k\\Phi(x).                                      \\tag{O1}\n\\]\nThe function $\\Phi$ is increasing and strictly convex on $(0,\\infty)$, and\n\\[\n \\frac{\\Phi(x)}x\\longrightarrow0\\qquad(x\\downarrow0). \\tag{O2}\n\\]\nConsequently, for every fixed $k\\ge1$ and every $c>0$, some $0<r<1$\nsatisfies\n\\[\n I_k(r)<c\\,\\frac1k\\log\\frac1r.                        \\tag{O3}\n\\]\nIn particular, no positive universal constant bounds $I_k(r)$ below\nby that constant times $k^{-1}\\log(1/r)$.\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L4179"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:value","assertion_kind":"paper_argued_result","reviewed":false,"statement":"value equation","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:value","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":4259,"anchor_unit":"theorem","section_title":"The Newton value equation","source_excerpt":{"text":"\\begin{theorem}[value equation]\\label{res:value}\nFor a polynomial $f$ and a differentiable curve $z:I\\to\\mathbb C$ on an\ninterval $I$, assume $f'(z(t))\\ne0$ and\n$z'(t)=-f(z(t))/f'(z(t))$ throughout $I$.  Then $w=f\\circ z$ satisfies\n$w'=-w$, and\n\\[\n f(z(t))=e^{-(t-t_0)}f(z(t_0))\\qquad(t,t_0\\in I).\n\\]\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L4259"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1041#res:locus","assertion_kind":"paper_argued_result","reviewed":false,"statement":"ray-collision locus","status":"proved in the manuscript","problem_ids":["erdos_1041"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:locus","owner_paper_id":"erdos1041-lemniscate-reasoning-surface","source_line":4351,"anchor_unit":"theorem","section_title":"Separating the rays of critical values","source_excerpt":{"text":"\\begin{theorem}[ray-collision locus]\\label{res:locus}\nLet $a\\ne b$ be complex.  Every common translation $\\beta$ for which $a+\\beta$\nand $b+\\beta$ lie on the same positive ray has the form\n\\[\n  \\beta=\\frac{ra-b}{1-r},\n  \\qquad r\\in\\mathbb{R}_{>0},\\ r\\ne1 .\n\\]\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1041/erdos1041-lemniscate-reasoning-surface.tex#L4351"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/14","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1049#res:sharp-fixed-base","assertion_kind":"paper_argued_result","reviewed":false,"statement":"the size of V_N^* at a fixed base","status":"proved in the manuscript","problem_ids":["erdos_1049"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:sharp-fixed-base","owner_paper_id":"erdos-1049-rational-base-lambert","source_line":65,"anchor_unit":"theorem","section_title":"Introduction","source_excerpt":{"text":"\\begin{theorem}[the size of $V_N^*$ at a fixed base]\\label{res:sharp-fixed-base}\nFix $0<q<1$ and write $P=(q;q)_\\infty$, $B_N=N(N-1)(2N-1)/6$ and\n$C_N=(N!)^2(N+1)!/2^N$. There is a $K(q)>0$ with\n\\[\n V_N^*(q)\\sim K(q)\\,C_Nq^{B_N}P^{2N}N^{-8F(1/q)}\n \\qquad(N\\to\\infty).\n\\]\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1049/erdos-1049-rational-base-lambert.tex#L65"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/15","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1049#res:thirtyone-four","assertion_kind":"paper_argued_result","reviewed":false,"statement":"res:thirtyone-four","status":"proved in the manuscript","problem_ids":["erdos_1049"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:thirtyone-four","owner_paper_id":"erdos-1049-rational-base-lambert","source_line":928,"anchor_unit":"corollary","section_title":"Rational bases","source_excerpt":{"text":"\\begin{corollary}\\label{res:thirtyone-four}\n$F\\bigl((31/4)^r\\bigr)$ is irrational for every integer $r\\ge1$.\n\\end{corollary}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1049/erdos-1049-rational-base-lambert.tex#L928"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/15","statement_grain":"label_only"},{"assertion_id":"paper:erdos_1049#res:bottomjet","assertion_kind":"paper_argued_result","reviewed":false,"statement":"a residue and divisibility by 3^R","status":"proved in the manuscript","problem_ids":["erdos_1049"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:bottomjet","owner_paper_id":"erdos1049-rational-base-lambert-reasoning-surface","source_line":3490,"anchor_unit":"lemma","section_title":"Congruences after evaluation at 3/2","source_excerpt":{"text":"\\begin{lemma}[a residue and divisibility by $3^R$]\\label{res:bottomjet}\nVanishing of the\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{191}{bottomJet3_eq_zero_iff_dvd}{residue modulo $3^R$}\nis exactly divisibility by the corresponding power of three:\n$J_{3,R}(P)=0$ if and only if $3^R\\mid H_W(P)$.\n\\end{lemma}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex#L3490"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/16","statement_grain":"environment_title"},{"assertion_id":"paper:erdos_1049#res:boundedfibre","assertion_kind":"paper_argued_result","reviewed":false,"statement":"equal residues with close, distinct values","status":"proved in the manuscript","problem_ids":["erdos_1049"],"problem_binding":{"method":"manuscript_subject","warrant":"the paper id names the problem this manuscript is about"},"evidence_authority":"manuscript_argument","paper_anchor":{"paper_label":"res:boundedfibre","owner_paper_id":"erdos1049-rational-base-lambert-reasoning-surface","source_line":3633,"anchor_unit":"theorem","section_title":"Congruences after evaluation at 3/2","source_excerpt":{"text":"\\begin{theorem}[equal residues with close, distinct values]\\label{res:boundedfibre}\nLet $A,B,J$ be finite sets and let $f:A\\to B$, $g:A\\to\\mathbb R$ and\n$\\iota:A\\to J$.  Suppose that each simultaneous fibre of $(f,g)$ has at\nmost $k$ elements and that, for some $\\delta>0$,\n\\[\n \\iota(x)=\\iota(y)\\quad\\Longrightarrow\\quad |g(x)-g(y)|<\\delta.\n\\]\nIf $|B||J|k<|A|$, then some distinct $x,y\\in A$ satisfy\n\\[\n f(x)=f(y),\\qquad 0<|g(x)-g(y)|<\\delta.\n\\]\n\\end{theorem}"},"permalink_url":"https://github.com/wcook04/plectis-lean-erdos249-257/blob/2a8c1667566020d123a081b6165e3ee94550566d/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex#L3633"},"lean_coverage_status":"not_declared_for_this_result","source_lane":"manuscript_result","evidence_resolution":"paper_level_only","packet_pointer":"/scholarly_corpus/papers/16","statement_grain":"environment_title"}],"claim_fidelity":{"schema":"plectis_claim_fidelity_v1","role":"does_the_public_sentence_match_the_theorem_it_cites","why_this_exists":"Every other gate checks that a citation lands. This one checks that the sentence at the near end of the link says what the theorem at the far end proves. The founding case: a claim said a bound held 'uniformly in the support' while the declaration it cited bound the support before the existential, and the declaration's own docstring said outright that it exports no such uniform constant. Coordinates all resolved. The mathematics did not agree.","surfaces_compared":["registry statement (assertions[].statement)","Lean proposition (assertions[].declarations[].signature)","Lean docstring (assertions[].declarations[].docstring)","manuscript excerpt and \\stag (assertions[].paper_anchor.source_excerpt)"],"authority_order":"checked Lean source is proof authority; the manuscript owns exposition; the registry sentence is a projection of both. When they disagree the projection is what is wrong.","mechanical_only":"Findings are produced by parsing binder order, dependency subscripts, denial sentences and status tags. Differences that need a mathematician are reported as 'not_mechanically_decidable' rather than decided here.","disclosure_rule":"A claim carrying a finding must also carry the finding on its own row. The build stops otherwise. The correction lives upstream in the published claims registry, so this packet cannot fix the sentence -- it can only guarantee no reader meets it alone.","verdict_legend":{"overstrong":"a surface the claim cites contradicts its own sentence. Upstream correction required; carried in upstream_defects.","not_mechanically_decidable":"the surfaces differ in a way this check cannot adjudicate. Recorded on the row so nobody relies on the sentence unexamined, and not asserted as an error."},"counts":{"registered_claims_inspected":160,"claims_with_a_lean_signature_to_compare":144,"claims_with_a_finding":3,"material":0,"needing_a_mathematician":3,"axes_triggered":["negative_clause_loss"]},"affected_assertion_ids":[],"advisory_assertion_ids":["claim:concrete_seam_adapter","claim:largest_skip_gap_propagation","claim:quarter_band_endpoint_cells"]},"paper_results":{"schema":"plectis_paper_results_v1","role":"results_the_manuscripts_prove_that_no_registered_claim_carries","why_this_exists":"The row layer was built from the Lean claims registry and the per-problem index, so it indexed what is formalised rather than what is proved. A paper-only theorem had nowhere to be a row. The #269 two-prime transcendence theorems -- unconditional, complete, and the strongest finished mathematics in the corpus -- were invisible to every query route while the headline said they were recorded in their problem row.","selection_rule":"Every label the manuscripts prefix 'res:', which is the authors' own marker for a result, tagged with the environment they wrote it in. This build ranks nothing and omits nothing it can read; the selection was made in the manuscript.","what_a_row_here_means":"This statement is proved in this manuscript at this line. It does not mean a Lean declaration covers it, and it does not mean the claims registry reviewed it.","row_shape":{"statement":"the title the author gave the environment, which names the result rather than stating it. The proved statement is the LaTeX in paper_anchor.source_excerpt.text -- quoted exactly, so it reads without a network","evidence_authority":"'manuscript_argument' on every row here: proved in the manuscript by a written argument. Whether a Lean declaration also covers it is not determined by this index, and is not implied","reviewed":"false on every row here. The claims registry carries reviewed claims for #249 and #257 only, so a manuscript result outside it is authored mathematics that no registry entry reviews. This is a statement about the registry's scope, not about the mathematics","problem_ids":"the manuscript's subject, read from the paper id. The joint #249/#257 gateway is bound to neither, because binding it to one would be a false narrowing","assertion_id":"paper:<problem>#<label> where unambiguous; labels reused by companion manuscripts use paper:<exact-paper-id>#<label>. Problem binding remains explicit in problem_ids.","paper_status_tag":"present only where the author wrote a \\stag marker; one manuscript uses them","lean_coverage":"lean_coverage_status is present on every paper-only row. A manuscript status tag can establish declared formalisation; otherwise not_declared_for_this_result is an explicit unknown, not a claim that no Lean theorem bears on the result. A source-authored per-result manifest is the upgrade contract.","excerpt_media_type":"text/x-tex"},"counts":{"rows_added":79,"by_kind":{"paper_posed_problem":6,"paper_argued_result":73},"papers_scanned":22},"per_paper":[{"paper_id":"erdos-68-factorial-denominator-irrationality","problem_id":"erdos_68","labelled_results":5,"already_a_registered_claim":0,"new_rows":3},{"paper_id":"erdos68-factorial-reasoning-surface","problem_id":"erdos_68","labelled_results":0,"already_a_registered_claim":0,"new_rows":0},{"paper_id":"erdos-243-reciprocal-tail-rigidity","problem_id":"erdos_243","labelled_results":24,"already_a_registered_claim":2,"new_rows":22},{"paper_id":"erdos243-reciprocal-tail-reasoning-surface","problem_id":"erdos_243","labelled_results":0,"already_a_registered_claim":0,"new_rows":0},{"paper_id":"erdos-249-binary-totient-series","problem_id":"erdos_249","labelled_results":5,"already_a_registered_claim":2,"new_rows":2},{"paper_id":"erdos249-totient-reasoning-surface","problem_id":"erdos_249","labelled_results":6,"already_a_registered_claim":3,"new_rows":2},{"paper_id":"erdos-251-prime-gap-dyadic-series","problem_id":"erdos_251","labelled_results":10,"already_a_registered_claim":2,"new_rows":7},{"paper_id":"erdos251-prime-gap-reasoning-surface","problem_id":"erdos_251","labelled_results":2,"already_a_registered_claim":2,"new_rows":0},{"paper_id":"erdos-257-mersenne-support-subseries","problem_id":"erdos_257","labelled_results":11,"already_a_registered_claim":7,"new_rows":4},{"paper_id":"erdos257-mersenne-reasoning-surface","problem_id":"erdos_257","labelled_results":0,"already_a_registered_claim":0,"new_rows":0},{"paper_id":"erdos257-expository-sums-never-fractions","problem_id":"erdos_257","labelled_results":0,"already_a_registered_claim":0,"new_rows":0},{"paper_id":"erdos269-running-lcm-reasoning-surface","problem_id":"erdos_269","labelled_results":0,"already_a_registered_claim":0,"new_rows":0},{"paper_id":"erdos-269-three-prime-running-lcm","problem_id":"erdos_269","labelled_results":18,"already_a_registered_claim":2,"new_rows":14},{"paper_id":"erdos-1041-lemniscate-newton-flow","problem_id":"erdos_1041","labelled_results":27,"already_a_registered_claim":2,"new_rows":3},{"paper_id":"erdos1041-lemniscate-reasoning-surface","problem_id":"erdos_1041","labelled_results":31,"already_a_registered_claim":1,"new_rows":18},{"paper_id":"erdos-1049-rational-base-lambert","problem_id":"erdos_1049","labelled_results":4,"already_a_registered_claim":2,"new_rows":2},{"paper_id":"erdos1049-rational-base-lambert-reasoning-surface","problem_id":"erdos_1049","labelled_results":2,"already_a_registered_claim":0,"new_rows":2},{"paper_id":"optimal-sparse-perturbations","problem_id":null,"labelled_results":0,"already_a_registered_claim":0,"new_rows":0},{"paper_id":"writing-a-good-mathematical-paper","problem_id":null,"labelled_results":0,"already_a_registered_claim":0,"new_rows":0},{"paper_id":"writing-mathematics-from-reviewed-revisions","problem_id":null,"labelled_results":0,"already_a_registered_claim":0,"new_rows":0},{"paper_id":"claim-faithful-publication-systems","problem_id":null,"labelled_results":0,"already_a_registered_claim":0,"new_rows":0},{"paper_id":"plectis-public-system","problem_id":null,"labelled_results":0,"already_a_registered_claim":0,"new_rows":0}],"coverage_before":"the paper anchor index scanned only manuscripts a registered claim cited: 3 of 13","lean_coverage_manifest":{"available":false,"upgrade_contract":"Publish a source-authored per-result manifest mapping paper id and result label to checked Lean declaration ids. Until then, absence of a manuscript status tag stays explicitly unknown."}},"source_lane_legend":{"what_this_is":"the body of source a row's authority comes from, on every row. It is derived from assertion_kind, which maps one-to-one onto it -- so it adds no evidence, and agreement between the two fields corroborates nothing. It is here so that any sentence about where rows come from can be generated from the axis it is about, instead of from 'reviewed', which answers a different question.","registry_claim":"docs/claims.json -- a curated claim the release reviews and publishes with a status. reviewed=true","registry_boundary":"docs/claims.json -- a statement the registry explicitly does not claim","registry_open":"docs/claims.json -- a proposition the registry records as still open","problem_index_checked":"docs/problems.json -- kernel-checked Lean declarations in a problem's own expansion library, which the registry deliberately does not promote into reviewed public claims. reviewed=false","problem_index_negative":"docs/problems.json -- what that problem's library explicitly does not check","problem_index_open":"docs/problems.json -- an obligation that problem's library leaves open","manuscript_result":"an embedded manuscript -- a result proved by written argument at a quoted line. reviewed=false, and no Lean declaration is asserted to cover it unless the author's own status tag says so","manuscript_problem":"an embedded manuscript -- a question the paper poses rather than settles","why_two_unreviewed_lanes_matter":"reviewed=false now covers both a kernel-checked Lean declaration and a paper argument with no formalisation asserted. Collapsing them loses the difference between 'the machine checked it, the registry did not review it' and 'a person proved it in prose'."},"lane_binding":{"schema":"plectis_assertion_lane_binding_v1","role":"consumers_of_the_row_collection_reconciled_after_the_manuscript_lane_landed","why_this_exists":"The manuscript rows were appended to a collection that four finished surfaces already described, and each kept describing the collection it was written against. The published count said 256 while the array held 347; the single-row contract promised two fields ninety-one rows did not carry; every per-problem frontier said an empty result meant the evidence was in the expansion lane, including for the problem whose strongest finished theorem had just been indexed. None of it raised: an empty list is a valid answer and a stale integer is still an integer.","rows_total":414,"declared_count_before_binding":335,"manuscript_rows_bound":79,"problems_given_a_manuscript_lane":["erdos_1041","erdos_1049","erdos_243","erdos_249","erdos_251","erdos_257","erdos_269","erdos_68"],"label_only_statements":["paper:erdos_1049#res:thirtyone-four","paper:erdos_243#res:cor"],"label_only_meaning":"the manuscript gave the environment no title, so the row's statement field falls back to the label. statement_grain says so on the row; quote paper_anchor.source_excerpt.text instead.","standing_rule":"A new assertion kind is not landed until the count, the row contract, the per-problem frontier, the lane map, the recap and the route tests have all been told it exists. plectis_query_contracts sweeps the whole return domain of every route for exactly this reason."}},"scholarly_maps":{"schema":"plectis_public_scholarly_corpus_v0","role":"scholarly_corpus_maps_without_duplicating_paper_bodies","source":"review_packet.scholarly_corpus","counts":{"paper_count":22,"papers_embedded":22,"control_plane_record_count":8,"control_plane_records_embedded":8,"repository_map_count":2,"repository_maps_available":2},"licences":{"CC-BY-4.0":"manuscript layer: every active paper source and its local dependencies","Apache-2.0":"Lean repository documentation and machine-readable claim records"},"reading_graph":{"schema":"plectis_public_reading_graph_v1","pointer_format":"RFC 6901 JSON Pointer from the complete packet root","boundary":"Reading order and source relationships, not proof dependencies or instructions to execute.","edge_columns":["from","to","relation"],"nodes":[{"id":"paper:erdos-68-factorial-denominator-irrationality","kind":"paper","label":"Integer Linear Forms for a Factorial Reciprocal Series","purpose":"Which integer coefficient vectors cancel initial weighted sums in factorial-denominator linear forms?","form":"Problem note","pointer":"/scholarly_corpus/papers/0","body_fields":["body","dependencies"],"sha256":"sha256:b4eba8fb48d81d851b0679a5d77bdc3b20e60913d48f403184365009d4f30edd"},{"id":"paper:erdos68-factorial-reasoning-surface","kind":"paper","label":"Factorial Linear Forms and Denominators: Detailed Proofs and Rationality Criteria","purpose":"What growth, valuation, and finite-denominator facts are proved for ∑_{n≥2}(n!−1)^{-1}, and which tail inequalities remain?","form":"Reasoning surface","pointer":"/scholarly_corpus/papers/1","body_fields":["body","dependencies"],"sha256":"sha256:0af0d85ec1beab46b63ef9a54e0326abb872d3c38403af575b2ca63ae9dea60d"},{"id":"paper:erdos-243-reciprocal-tail-rigidity","kind":"paper","label":"Reciprocal Sums and the Sylvester Recurrence","purpose":"Which growth rates force irrational reciprocal sums, and which additional tail conditions force the Sylvester recurrence?","form":"Problem note","pointer":"/scholarly_corpus/papers/2","body_fields":["body","dependencies"],"sha256":"sha256:b393a9190d51137fb1ba6d3ba15bb1915219b468de6fe7279cbfaec46abd9b9c"},{"id":"paper:erdos243-reciprocal-tail-reasoning-surface","kind":"paper","label":"Reciprocal Sums and the Sylvester Recurrence: Further Results and Proofs","purpose":"Which growth hypotheses make a reciprocal sum irrational, and which sufficient conditions force the Sylvester recurrence without settling Erdős #243?","form":"Reasoning surface","pointer":"/scholarly_corpus/papers/3","body_fields":["body","dependencies"],"sha256":"sha256:92512a9aa35a79c5e4d2fb5690c56912a7a7dff9a4a244e356b1c2e886b37bac"},{"id":"paper:erdos-249-binary-totient-series","kind":"paper","label":"Integral Relations among Totient Sections","purpose":"What explicit basis and integral relations does the totient k-kernel have at every integer base, and why does that not decide Erdős #249?","form":"Problem 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Complete arguments are ordinary proofs, with the sharp dyadic input and the p-adic Subspace Theorem cited. Lean checks specified ingredients, not the complete theorems. No Erdos claim status changes.","anchor_label_allowlist":["sec:map","sec:dyadic-shifts","thm:restricted-dyadic-shifts","sec:polynomial-shifts","cor:polynomial-shifts","capacity:sec:criterion","capacity:eq:capacity","capacity:thm:capacity","capacity:eq:class","capacity:eq:congruence","capacity:thm:main","capacity:eq:gaps","capacity:eq:both","capacity:cor:density","capacity:sec:capacity-proof","capacity:eq:overlap","capacity:sec:negative","capacity:eq:tailbound","capacity:lem:feedback","capacity:eq:step","capacity:eq:selector","capacity:prop:unrestricted","capacity:sec:carry","capacity:thm:gcd","sec:results","thm:dichotomy","thm:bases","thm:chains","thm:chain-transcendence","thm:forced","sec:dichotomy","sec:bases","sec:base2","sec:lemmas","lem:gaps","lem:translate","prob:membership","sec:second","sec:inventory","sec:misread","sec:eliminated","sec:known","capacity:sec:sources","sec:questions","capacity:thm:jets","capacity:sec:jets-intro","capacity:sec:jets-proof","capacity:thm:rational-jets","capacity:sec:rational-jets"],"formal_source_ref":"8d6596fbde2c4aacf946adec4b0226ba1a97545d"}]},"publication_assembly":{"purpose":"Partition every public claim into one mathematical contribution family and bind its status, prior-art posture, narrative owner, source route, consumer or open obligation, and rendered-view decision.","coverage_rule":"Every docs/claims.json claim id appears in exactly one contribution family. The release checker enforces completeness and disjointness.","publication_architecture":{"canonical_gateway":{"source":"paper/archive/erdos249-257-main-paper.tex","decision":"archive_only_not_current_entrance","thesis":"Archival joint #249/#257 manuscript. Current mathematical entrances are the eight short problem notes."},"retained_companions":[{"id":"erdos_249_reasoning_record","source":"paper/249/erdos249-totient-reasoning-surface.tex","decision":"retain_as_full_problem_reasoning_record","thesis":"Complete working record for Erdős #249: checked premises, finite certificates, closed routes, evidence bands, and surviving obligations."},{"id":"erdos_257_reasoning_record","source":"paper/257/erdos257-mersenne-reasoning-surface.tex","decision":"retain_as_full_problem_reasoning_record","thesis":"Ordinary proofs and Lean-checked finite-prime weighted, strengthened positive-cover, common-witness mixed, and reciprocal-summable support criteria. 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The reported counterexample family is not formalised; correspondence with the 1958 wording awaits independent human review."},{"id":"erdos_1049_reasoning_record","source":"paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","decision":"retain_as_full_problem_reasoning_record","thesis":"Rational-base irrationality and irrationality-exponent bounds in the Zudilin contour region, including powers of 31/4, plus all-rank normalized Hankel order and coefficient formulas. The coefficient pencil is Lean checked only through rank eight. Irrationality at 3/2 and the universal rational-base assertion remain open."},{"id":"erdos_68_reasoning_record","source":"paper/68/erdos68-factorial-reasoning-surface.tex","decision":"retain_as_full_problem_reasoning_record","thesis":"Complete reasoning record for Erdős #68. Lean Core carry equivalences are distinct from ordinary LCM-liminf and finite prime-cancellation examples. The problem remains open."},{"id":"erdos_243_reasoning_record","source":"paper/243/erdos243-reciprocal-tail-reasoning-surface.tex","decision":"retain_as_full_problem_reasoning_record","thesis":"Complete reasoning record for Erdős #243. The zero-indexed positive StrictMono cubic-rate irrationality theorem and its square-specialisation chain have unconditional Lean declarations under their exact hypotheses; the paper's one-based transfer uses an ordinary finite-prefix bridge; the ordinary proof uses Chebotarev, while the formal proof uses a Dedekind-zeta pole comparison. The integer-state bounded-negative criteria remain separate. The unrestricted problem and mixed-sign unbounded regime remain open."},{"id":"erdos_251_reasoning_record","source":"paper/251/erdos251-prime-gap-reasoning-surface.tex","decision":"retain_as_full_problem_reasoning_record","thesis":"Complete reasoning record for Erdős #251: countermodels and finite counting reductions, the already-known prime-to-gap identity without the prime-number theorem, and an ordinary PNT-based compression argument for hypothetical rational actual-prime tails. Equal tails do not supply the still-open nonintegral tail witness."},{"id":"erdos_269_reasoning_record","source":"paper/269/erdos269-running-lcm-reasoning-surface.tex","decision":"retain_as_full_problem_reasoning_record","thesis":"Complete reasoning record for Erdős #269. Two-prime credit is Steve Fan's; three-prime Lean rank/window versus unproved cofinal escape. The problem remains open."}],"problem_series":[{"id":"erdos_68_note","source":"paper/68/erdos-68-factorial-denominator-irrationality.tex","decision":"retain_as_problem_owned_note","thesis":"The factorial-series carry equivalences, two incomparable finite denominator exclusions, and the still-open cofinal-carry producer. FactorialGapPlateauCore owns the carry definitions; FactorialZeroPlateau retains compatibility imports."},{"id":"erdos_243_note","source":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","decision":"retain_as_problem_owned_note","thesis":"A Lean-checked zero-indexed positive StrictMono cubic-rate irrationality theorem for a_n²/a_(n+1)=1+3/n+o(n^-3), transferred to the one-based paper form by an ordinary finite-prefix argument, alongside the bounded-increment criterion for eventual Sylvester recurrence under rationality. The latter requires an additional increment bound not derived from the original hypotheses. 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One hypothetical rational carry has both unbounded section ranks and periodic congruence sections, with no contradiction or cofinal producer proved."},{"id":"erdos_251_note","source":"paper/251/erdos-251-prime-gap-dyadic-series.tex","decision":"retain_as_problem_owned_note","thesis":"A coefficient-only countermodel and sparse rationalising perturbation as the supported lead; the already-known prime-to-gap identity checked without the prime-number theorem; cofinal small-mismatch tail witness open."},{"id":"erdos_257_note","source":"paper/257/erdos-257-mersenne-support-subseries.tex","decision":"retain_as_problem_owned_note","thesis":"Ordinary proofs and Lean-checked finite-prime weighted, strengthened positive-cover, common-witness mixed, and reciprocal-summable support criteria. Named separating examples retain ordinary proofs. 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The family is not formalised; historical correspondence and ordinary analytic assemblies remain separate from Lean verification."},{"id":"erdos_1049_note","source":"paper/1049/erdos-1049-rational-base-lambert.tex","decision":"retain_as_problem_owned_note","thesis":"Rational-base irrationality and irrationality-exponent bounds in the Zudilin contour region, including powers of 31/4, plus all-rank normalized Hankel order and coefficient formulas. The finite coefficient pencil is Lean checked only through rank eight. Irrationality at 3/2 and the universal rational-base assertion remain open."}],"problem_series_boundary":"A problem note expounds declarations and ordinary mathematics. Formal status belongs to exact registered statements and their evidence, not to every assertion in a note. Using ani's degree-seven polynomial, Lean refutes the exact Formal Conjectures path-image-length statement by a one-dimensional Hausdorff-measure bound; the total-variation result remains checked. Historical correspondence remains unreviewed, and the other seven targets remain unresolved here.","retirement_notices":[{"source":"paper/erdos249-transport-curvature-companion-note.tex","decision":"retired_one_sided_companion","thesis":"The former one-sided Erdős Problem 249 specialist note is not an active mathematical paper; its archived predecessor is historical evidence only, and the registered Lean claims remain queryable from the claim registry."},{"source":"paper/synthesis/erdos-synthesis-subsums-across-bases.tex","decision":"consolidated_into_one_cross_problem_paper","thesis":"Content retained in paper/synthesis/optimal-sparse-perturbations.tex; prior versions remain in Git history."},{"source":"paper/synthesis/erdos-synthesis-reading-together-record.tex","decision":"consolidated_into_one_cross_problem_paper","thesis":"Content retained in paper/synthesis/optimal-sparse-perturbations.tex; prior versions remain in Git history."}],"architecture_guides":[{"id":"repository_architecture_guide","source":"paper/systems/claim-faithful-publication-systems-paper.tex","decision":"retain_as_primary_repository_architecture_guide","thesis":"The research architecture: each open problem kept as a persistent, checkable record (short paper, long record, Lean source, reviewed claims), the path from a proof to a public claim through Lean, Comparator and reviewed wording, and the contribution cycle published as agent skills, with the exact boundary between human judgement and automatic checks."}],"agent_native_guides":[{"id":"agent_native_navigation_guide","source":"paper/systems/cold-clone-to-proof-receipt.tex","decision":"retain_as_agent_navigation_and_receipt_guide","thesis":"A cold-clone-to-proof-receipt architecture: exhaustive and selective navigation layers, bounded intent routing, a session notary whose verdicts come from Lean, focused changed-cone builds, exact dependency-index cache receipts, one replayed dogfood session, and explicit transfer limits."}],"qualified_future_companion":{"id":"erdos257_half_value_carry_companion","decision":"qualified_but_not_created_in_this_wave","thesis":"Unique greedy coding, fatal-gap and half-membership equivalences, last-producer residuals, rewind/cylinder propagation, and counterexamples to naive local induction form an independent #257 specialist arc.","creation_gate":"Create only after a citable abstract, selected theorem spine, and low-duplication transfer of the detailed seam/cylinder material from the gateway are authored together."},"rejected_companions":[{"id":"additional_erdos249_companion","decision":"do_not_create","reason":"The certificate and diagonal-pincer material is the mathematical centre of the gateway; the retained transport paper already owns the distinct specialist #249 thesis."},{"id":"erdos_synthesis_note","decision":"consolidated_into_one_cross_problem_paper","reason":"The results and research record now belong to optimal-sparse-perturbations; do not publish a separate companion."},{"id":"erdos_synthesis_record","decision":"consolidated_into_one_cross_problem_paper","reason":"The results and research record now belong to optimal-sparse-perturbations; do not publish a separate companion."}]},"editorial_state":{"schema":"erdos249257-editorial-state/1","scope":"continuous_whole_main_paper_exposition","current_priority":{"id":"main_paper_source_link_semantic_label_audit","surface":"paper/archive/erdos249-257-main-paper.tex","defect_class":"A source link may resolve successfully yet still be a poor reader interface when its visible phrase is broader, narrower, or more internal than the declaration it opens. Link text should state the mathematical role a reader will find at the target without exposing module-local vocabulary unnecessarily.","statement":"Audit visible Lean-link labels for semantic agreement with the surrounding theorem and linked declaration. Replace misleading or internal labels with literal mathematical descriptions, retain immutable targets and exact declaration names behind the links, rerun link and release checks, and preserve 53 pages."},"last_verified_wave":{"id":"main_paper_redundant_boundary_sentence_audit","surface":"paper/archive/erdos249-257-main-paper.tex","result":"Audited repeated finite-only, does-not-prove, supplies-no, and remains-open fences by theorem cluster. Repetition at the abstract, first-reading map, local theorem, appendix entry, and final open-problem surfaces remains intentional because each is an independent reader entry point. Sharpened four generic fences by naming the unresolved actual values of C_D and comparisons with Theta_D, the after-every-cutoff Sep certificates, and the exact unbounded-supply equation. Consolidated the adjacent changing-lcm primality and t=64 certificate warnings into one verified range statement plus a reference to the 28-certificate finite record. Removed the two residual appendix labels Reader map. No theorem, formula, citation, source target, or problem status changed; the rendered boundary and 53-page target pass."},"active_inconsistencies":[],"blocked_decisions":[]},"contribution_families":[{"id":"classical_full_support_and_named_257_families","claim_ids":["eb_full_support","eb_constant","support_families","squarefree_support_irrationality","prime_support_irrationality","universal_257"],"status_summary":"Classical formalisation, cited special cases, and the open universal target.","prior_art_posture":"Erdős owns full support and the pairwise-coprime family; Duverney–Tachiya own squarefree-support irrationality at every power-of-two base; Tao–Teräväinen own the base-2 prime-support theorem. The cited squarefree theorem does not cover integer bases that are not powers of two, and the cited prime-power extension is remark-only. No priority claim is made for the formalised families.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim eb_full_support","consumer_or_open_obligation":"Establishes the classical benchmark and leaves remaining_open.universal_257_all_infinite_supports.","view_decision":"gateway_headline"},{"id":"periodic_weighted_lambert_theorems","claim_ids":["periodic_nonnegative_weight_irrationality","periodic_signed_weight_dichotomy","period_four_signed_residue_obstruction"],"status_summary":"One classical/broader antecedent, one exact signed dichotomy, and one proved residue obstruction.","prior_art_posture":"Luca–Tachiya prove the broader eventually-periodic rational-coefficient theorem by another method; the signed endpoint formalised here is only irrational-or-b-power-denominator.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim periodic_nonnegative_weight_irrationality","consumer_or_open_obligation":"Supports structured #257 families; unconditional signed irrationality would require excluding base termination.","view_decision":"gateway_summary_reference_detail"},{"id":"orthogonal_petal_and_divisor_layers","claim_ids":["orthogonal_petal_sunflower_reduction","composite_dilation_defect_identity","mixed_prime_power_layer_identity"],"status_summary":"One conditional support-family reduction and two exact local divisor identities.","prior_art_posture":"Classical divisor algebra is background; no combined selector theorem is identified, and absence of a match is not a novelty claim.","primary_narrative_owner":"docs/claims.json::claims","source_route":"python3 scripts/query_corpus.py --claim orthogonal_petal_sunflower_reduction","consumer_or_open_obligation":"The uniform tail-selection predicate for orthogonal-petal bouquets remains open.","view_decision":"living_reference_only"},{"id":"generic_binary_carry_rigidity","claim_ids":["tail_orbit_rigidity","boolean_mobius_carry","reciprocal_mass_bound","unbounded_carry_states","sublog_zero_windows"],"status_summary":"Exact rationality characterisations and proved necessary consequences.","prior_art_posture":"Han Wang supplies the closest weighted-binary carry recurrence; the abstraction, converse, and downstream constraints make no priority claim.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim tail_orbit_rigidity","consumer_or_open_obligation":"Constrains any rational #257 counterexample without settling universal irrationality.","view_decision":"gateway_appendix_reference_detail"},{"id":"achievement_geometry_and_half_target_interface","claim_ids":["campbell_shift_synchronization_boundary","greedy_achievement_geometry","half_greedy_two_thirds_band","half_second_channel_reduction","mobius_sign_support_no_go","twenty_one_quotient_greedy_frontier"],"status_summary":"Classical geometry formalised, exact local bands, a conditional second channel, and scoped no-go boundaries.","prior_art_posture":"Kovač–Tao supply the strict-tail subsum geometry; the classical Lambert–Möbius identity is the baseline for the scoped sign-support no-go; Crandall and Campbell concern adjacent binary-digit questions and are not proof inputs. No priority claim is made for the sign separation.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim greedy_achievement_geometry","consumer_or_open_obligation":"Decide whether 1/2 belongs to the achievement set; positive membership refutes universal #257.","view_decision":"gateway_headline"},{"id":"half_carry_compactness_consumers","claim_ids":["half_trapping_return_carry_interface","half_carry_reachability_compactness","cofinal_strip_return_closure","terminal_only_cofinal_closure","terminal_scaled_vanishing_closure","half_cylinder_finite_shadow"],"status_summary":"Conditional compactness consumers from progressively weaker finite-word or carry supplies.","prior_art_posture":"No executable seam/closure antecedent is recorded beyond the global achievement-set geometry.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim half_carry_reachability_compactness","consumer_or_open_obligation":"Supply canonical seams, strip returns, terminal witnesses, or scaled-vanishing words cofinally.","view_decision":"gateway_appendix_future_257_companion"},{"id":"dyadic_prefix_block_safety_interfaces","claim_ids":["dyadic_prefix_block_safety_interfaces"],"status_summary":"Exact reduced-denominator transport, block-safety and primitive-overshoot equivalences, conditional half-membership consumers, and explicit failures of naive induction.","prior_art_posture":"Exact rational and greedy-block formalisation; no priority claim is made for the assembled source-level package.","primary_narrative_owner":"docs/SOURCE_MAP.md","source_route":"python3 scripts/query_corpus.py --claim dyadic_prefix_block_safety_interfaces","consumer_or_open_obligation":"Supply actual block-safety, raw-margin, or primitive-overshoot witnesses cofinally; no such unbounded producer is proved. Until the qualified #257 companion has a citable low-duplication theorem spine, this package remains source-map material rather than another gateway inventory.","view_decision":"source_map_now_qualified_future_257_companion_source_only"},{"id":"rewind_windows_and_protected_seams","claim_ids":["fixed_coefficient_rewind_contraction","selected_half_carry_window_compactness","realized_rewind_phase_identity","localized_protected_seam_compactness","mixed_cofinal_half_carry_fan_in","rewind_seam_operational_bridge","selected_window_depth18_base"],"status_summary":"Conditional rewind/window consumers plus one verified depth-18 base.","prior_art_posture":"No theorem-level match is recorded; any novelty posture remains provisional.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim selected_half_carry_window_compactness","consumer_or_open_obligation":"Produce repeated object-level coefficient agreement or protected seams cofinally.","view_decision":"gateway_appendix_future_257_companion"},{"id":"fatal_gap_and_last_producer_spine","claim_ids":["fatal_gap_right_tail_classification","final_middle_cell_escape","final_middle_neg_two_phase_sieve","half_membership_seam_classification","seam_producer_carry_alignment","concrete_producer_carry_coordinates","last_producer_tail_escape_reduction","middle_producer_cardinality_reduction","fixed_tail_survival_socket","producer_carry_future_tail_socket","integer_greedy_first_wrap_reduction","concrete_seam_adapter"],"status_summary":"Exact half-membership/nonmembership classifications, proved branch eliminations, a rank-26 phase sieve for the middle -2 cell, and explicit conditional tail consumers.","prior_art_posture":"No exact local seam or last-producer antecedent is recorded beyond strict-tail achievement-set background.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim half_membership_seam_classification","consumer_or_open_obligation":"Eliminate the 412 surviving classes modulo 2730 in the middle -2 cell, then handle the middle -1 and right-tail branches or prove future-tail escape directly.","view_decision":"gateway_spine_future_257_companion_detail"},{"id":"largest_skip_and_reset_deficit_frontier","claim_ids":["largest_skip_gap_propagation","boundary_pulse_normalization","largest_skip_induction_reduction","quarter_band_endpoint_cells","reset_deficit_escape_geometry","upper_reset_band_certificates_through_thirty"],"status_summary":"Conditional largest-skip propagation, exact pulse geometry, and finite reset-band evidence.","prior_art_posture":"No close theorem-level antecedent is recorded.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim largest_skip_gap_propagation","consumer_or_open_obligation":"Supply a global first-crossing or anti-concentration theorem preventing persistent right-branch pinning.","view_decision":"gateway_appendix_future_257_companion"},{"id":"suffix_cylinders_and_feedback_boundary","claim_ids":["selected_suffix_cylinder_propagation","selected_suffix_pre_feedback_run","half_divisor_unit_drop","rewind_boundary_pair_obstruction"],"status_summary":"An exact conditional child-gap recurrence with a cofinal half-value consequence, verified finite stages through depth 52, an exact unit drop, and a concrete scalar-seam obstruction.","prior_art_posture":"Kovač–Tao supply the surrounding strict-tail achievement-set geometry. No exact antecedent for this finite carry recurrence is recorded, and no novelty claim is made.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim selected_suffix_cylinder_propagation","consumer_or_open_obligation":"Establish the child-gap inequalities at arbitrarily large depths, or otherwise construct cofinal full suffix-cylinder stages.","view_decision":"future_257_companion_reference_stages"},{"id":"mersenne_lambert_classical_ladder","claim_ids":["mersenne_lambert_ladder","sigma_transcendence"],"status_summary":"Classical identities formalised and one transcendental rung cited only.","prior_art_posture":"Apostol, Merca, and Merca–Schmidt own the transform background; Nesterenko owns the cited transcendence theorem.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim mersenne_lambert_ladder","consumer_or_open_obligation":"Provides the conceptual bridge between the classical #257 benchmark and the open #249 constant.","view_decision":"gateway_spine"},{"id":"probabilistic_gcd_and_stern_brocot_geometry","claim_ids":["totient_coprimality_probability","squared_lambert_gcd_moments","stern_brocot_cylinder_law","stern_brocot_run_fibonacci_stability"],"status_summary":"Exact coprimality and gcd-moment identities, a Stern–Brocot cylinder law with uniform geometric convergence, and source-level Fibonacci/continuant run stability.","prior_art_posture":"Classical Lambert-series, convolution, Pillai gcd-sum, probability, and Stern–Brocot background; Postelmans–Van Assche own the cited irrationality theorem for the constant-weight squared-Lambert value. No priority claim is made for this assembled formal package.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --route probabilistic_gcd_geometry","consumer_or_open_obligation":"Recasts #249 and exposes independent gcd-distribution and continued-fraction structure; the run package has no endpoint consumer, and irrationality remains open.","view_decision":"gateway_supporting_theorems_and_living_reference_detail"},{"id":"erdos249_quantitative_headline","claim_ids":["denominator_exclusion","erdos_249"],"status_summary":"One unconditional quantitative theorem and the open irrationality claim.","prior_art_posture":"Farey is historical language only; no exact antecedent for the explicit bound is identified, without asserting novelty.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim denominator_exclusion","consumer_or_open_obligation":"Leaves remaining_open.erdos_249_irrationality.","view_decision":"gateway_headline"},{"id":"exact_certificate_equivalence_and_deposits","claim_ids":["certificate_reduction","certificate_completeness","certified_kill_instances","diagonal_pincer_t64_primality_closure","diagonal_pincer_t64_endpoint_certificate"],"status_summary":"Exact equivalence and completeness with bounded verified deposits.","prior_art_posture":"Classical Lambert and Möbius identities underlie the construction; no exact certificate-equivalence antecedent is recorded, without a novelty conclusion.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim certificate_completeness","consumer_or_open_obligation":"Produce certificates at unbounded parameters.","view_decision":"gateway_headline_finite_matrix"},{"id":"diagonal_pincer_and_denominator_survival","claim_ids":["diagonal_pincer_reduction","diagonal_fresh_loss_reduction","actual_foreign_residue_projection","squared_mersenne_enclosure","mersenne_shadow_denominator"],"status_summary":"Exact diagonal targets, sharp enclosures and denominators, and conditional avoidance projections.","prior_art_posture":"Classical Möbius/Lambert algebra only; no theorem-specific antecedent is identified.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim diagonal_pincer_reduction","consumer_or_open_obligation":"Supply unbounded full-target avoidance, fresh-loss separation, or the missing analytic complement bridge.","view_decision":"gateway_supporting_spine"},{"id":"fresh_loss_local_arithmetic","claim_ids":["fresh_prime_deficit_decomposition","power_two_bit_lift","parity_socket_obstruction","flexible_odd_window_reduction","flexible_odd_window_affine_frontier","canonical_jump_slack_reduction"],"status_summary":"Exact local arithmetic, conditional band/slack reductions, and scoped parity ceilings.","prior_art_posture":"No close statement comparison is recorded; novelty remains unasserted.","primary_narrative_owner":"paper/archive/erdos249-257-main-paper.tex","source_route":"python3 scripts/query_corpus.py --claim canonical_jump_slack_reduction","consumer_or_open_obligation":"Prove cofinal nonnegative slack or a correlated multi-rank band supply.","view_decision":"gateway_appendix_reference_detail"},{"id":"alternative_249_algebraic_boundaries","claim_ids":["primitive_coordinate_index_obstruction","primitive_unit_gap_ceiling","mersenne_tail_atom_boundary","adelic_height_obstruction","square_crt_correction_suppression","signed_dyadic_moment_substrate","dyadic_totient_basis","dyadic_totient_certificate_interface","all_base_totient_kernel_index_reduction","all_base_totient_kernel_conditional_rank","residual_gauge_obstruction","totient_carry_kernel_anti_compression","totient_residue_series_classification","totient_residue_series_integer_radix","totient_residue_series_dilations"],"status_summary":"Exact finite obstructions and interfaces, a constructive dyadic basis, and an all-base arithmetic/spanning layer. The reusable rank wrapper is conditional; AllBaseTotientKernel.lean separately proves its independence premise and the complete rank unconditionally. The residue-series claims classify the rational totient residue series and prove the dilated series linearly independent.","prior_art_posture":"Classical Euler/Möbius, CRT, determinant, and Lambert tools; Coons already implies full-kernel infinite rank, and Martin supplies a broader external affine-independence antecedent. The Lean all-base proof uses CRT and Dirichlet independently. Erdős 1957 (Lemma 1) and Kaneko–Suzuki–Tachiya (Theorem 3) already give the irrationality of the totient residue series. No #249 irrationality endpoint or priority claim.","primary_narrative_owner":"docs/claims.json::claims","source_route":"python3 scripts/query_corpus.py --route lambert_obstruction_interfaces","consumer_or_open_obligation":"No rationality-to-finite-rank bridge or #249 irrationality endpoint is proved. The all-base independence premise is discharged in AllBaseTotientKernel.lean; the reusable conditional interface remains separately recorded.","view_decision":"gateway_selected_boundaries_and_living_reference_detail"},{"id":"transport_curvature_finite_difference_architecture","claim_ids":["transport_curvature_reductions","affine_transport_annihilation","transport_certificate_instances"],"status_summary":"Exact affine annihilation, conditional finite-difference consumers, and two verified windows.","prior_art_posture":"Finite differences and affine moments are general techniques; no matching combined theorem was identified, and that absence is not a novelty claim.","primary_narrative_owner":"docs/claims.json::claims","source_route":"python3 scripts/query_corpus.py --claim affine_transport_annihilation","consumer_or_open_obligation":"Produce an unbounded supply of a named transport certificate.","view_decision":"registry_only_after_one_sided_companion_retirement"},{"id":"harmonic_and_phase_reductions","claim_ids":["first_harmonic_certificate_interface","first_harmonic_pivot_decomposition","first_harmonic_good_index_reduction","adjacent_phase_separation_reduction","pivot_anti_reconstruction_reduction","strict_prime_tail_orbit_gap","totient_v8_conditional_peripheral_assembly","totient_v8_finite_arithmetic_threshold","totient_v8_generic_centering_countermodel","totient_v8_one_finite_first_harmonic_instance","totient_v8_peripheral_finite_budget_bridge"],"status_summary":"Conditional harmonic and phase consumers, including an exact four-term pivot decomposition and budget, an exact window-pair equivalence, and the two-branch tail-orbit phase-density and natural-prime strict-gap producers that feed the same certificate socket.","prior_art_posture":"No matching combined theorem is identified; all novelty language remains provisional.","primary_narrative_owner":"docs/claims.json::claims","source_route":"python3 scripts/query_corpus.py --claim pivot_anti_reconstruction_reduction","consumer_or_open_obligation":"Supply a cofinal first-harmonic pivot budget, a cofinal 603X/1000 first-harmonic bound on the good indices, adjacent-pair separator, anti-reconstruction producer, or one of the two tail-orbit strict-gap antecedents (late-block nonpositive phase density, or cofinally many natural primes with tail-orbit real part below 9/10).","view_decision":"registry_only_after_one_sided_companion_retirement"},{"id":"transport_strategy_no_go_boundaries","claim_ids":["fixed_precision_transport_no_go","lcm_factor_ideal_anchor_pulse_no_go"],"status_summary":"Two unconditional scoped countermodels to fixed local precision and finite shift algebra.","prior_art_posture":"No matching theorem is identified; the record explicitly refuses to infer novelty from that.","primary_narrative_owner":"docs/claims.json::claims","source_route":"python3 scripts/query_corpus.py --claim fixed_precision_transport_no_go","consumer_or_open_obligation":"Future arguments must use growing precision, global correlation, nonlinear arithmetic, or actual fresh-channel structure.","view_decision":"registry_only_after_one_sided_companion_retirement"},{"id":"actual_lcm_orbit_and_sign_frontier","claim_ids":["actual_lcm_orbit_nonintegrality_frontier","actual_lcm_positive_corridor_top_edge"],"status_summary":"An exact actual-LCM cofinal non-integrality reformulation and an unconditional positive-corridor/top-edge boundary.","prior_art_posture":"The actual-LCM tail equivalence and sign/carry analysis are source-specific; no novelty or priority claim is made.","primary_narrative_owner":"paper/249/erdos-249-binary-totient-series.tex","source_route":"python3 scripts/query_corpus.py --claim actual_lcm_orbit_nonintegrality_frontier","consumer_or_open_obligation":"Produce a cofinal actual-LCM non-integrality supply or independently exclude the top-edge boundary.","view_decision":"registry_only_after_one_sided_companion_retirement"},{"id":"factorial_quotient_band_breakpoint","claim_ids":["erdos_68","channel_quotient_band_breakpoint"],"status_summary":"An open factorial-denominator endpoint plus an unconditional finite quotient-band factorisation and breakpoint under channel cancellation.","prior_art_posture":"The quotient-band identity and its breakpoint consequence are source-specific formal results; no novelty, priority, or significance claim is made.","primary_narrative_owner":"paper/68/erdos-68-factorial-denominator-irrationality.tex","source_route":"python3 scripts/query_corpus.py --claim channel_quotient_band_breakpoint","consumer_or_open_obligation":"The finite-family result supplies no cancelling family, simultaneous-channel control, residual estimate, or cofinal non-unit carry for the still-open #68 endpoint.","view_decision":"problem_note_reference_detail"},{"id":"three_prime_residue_escape_consumers","claim_ids":["erdos_269","three_prime_finite_residue_contradiction","three_prime_cofinal_window_carry_extinction","weighted_phase_carry_finite_observer","three_prime_exact_modulus_pinning","three_prime_octic_window_equivalence","three_prime_finite_prefix_completion_boundary","three_prime_bounded_certificate_safety"],"status_summary":"An open three-prime #269 endpoint, a finite least-positive-residue contradiction, a conditional cofinal-window extinction consumer, and the weighted-phase carry split with its finite realised-span observer.","prior_art_posture":"The residue classifier and bounded-carry consumer are source-specific formal results; no novelty, priority, significance, or three-prime irrationality claim is made.","primary_narrative_owner":"paper/269/erdos-269-three-prime-running-lcm.tex","source_route":"python3 scripts/query_corpus.py --claim three_prime_finite_residue_contradiction","consumer_or_open_obligation":"The actual rationality-to-positive-reduced-carry bridge is established. The denominator-dependent cofinal local-window escape remains open, so the consumers do not settle the three-prime problem; the observer supplies no two-dimensional representation and excludes none.","view_decision":"problem_note_reference_detail"},{"id":"prime_gap_tail_shift_frontier","claim_ids":["erdos_251","prime_gap_unboundedness_and_nonperiodicity","prime_gap_irrationality_equivalence","prime_gap_identity_tao"],"status_summary":"An open prime-index endpoint, exact coefficient constraints, a Lean-formalised prime/gap irrationality equivalence under an elementary polynomial prime bound, and Tao's already-known identity recorded as cited only.","prior_art_posture":"The prime-to-gap identity is Tao's (erdosproblems.com/251, 7 October 2025). This checkout formalises a version that discharges summability without the prime-number theorem. No novelty, priority, significance, or irrationality claim is made.","primary_narrative_owner":"paper/251/erdos-251-prime-gap-dyadic-series.tex","source_route":"python3 scripts/query_corpus.py --claim prime_gap_unboundedness_and_nonperiodicity","consumer_or_open_obligation":"The exact prime-gap coefficient facts do not supply the cofinal non-integrality or small-shift escape required for Erdős #251.","view_decision":"problem_note_reference_detail"},{"id":"conditional_half_countermodel_frontiers","claim_ids":["terminal_scaled_vanishing_half_countermodel","cofinal_cylinder_half_countermodel"],"status_summary":"Two conditional routes to an infinite rational half-valued Mersenne support: terminal scaled-vanishing and cofinal full-cylinder stages.","prior_art_posture":"These are source-specific conditional consumers; no existence, novelty, priority, or significance claim is made for either antecedent.","primary_narrative_owner":"paper/257/erdos-257-mersenne-support-subseries.tex","source_route":"python3 scripts/query_corpus.py --claim terminal_scaled_vanishing_half_countermodel","consumer_or_open_obligation":"Construct or refute the corresponding terminal-scaled or cofinal-cylinder antecedent; neither route settles half-membership or universal Erdős #257 without that antecedent.","view_decision":"future_257_companion_reference_stages"},{"id":"newton_ray_separation_frontier","claim_ids":["erdos_1041","newton_ray_separation_consumer"],"status_summary":"Finite Newton-flow ray-separation and perturbative root-retention inputs, alongside the historical short-connection question whose correspondence with the separately checked total-variation counterexample awaits independent review.","prior_art_posture":"The finite ray and perturbation statements are source-specific formal results. Their checked hypotheses do not imply unrestricted connection; no novelty, priority, significance or independent human-review claim is made.","primary_narrative_owner":"paper/1041/erdos-1041-lemniscate-newton-flow.tex","source_route":"python3 scripts/query_corpus.py --claim newton_ray_separation_consumer","consumer_or_open_obligation":"These finite conditional consumers do not construct a universal short connector. The exact Formal Conjectures path-image-length statement and the total-variation formulation are refuted by the separate degree-seven result; historical correspondence and valid positive-family hypotheses remain distinct questions.","view_decision":"problem_note_reference_detail"},{"id":"reciprocal_tail_rigidity_frontier","claim_ids":["erdos_243","bounded_rise_fresh_modulus_barrier","eventually_periodic_negative_no_go","bounded_negative_exclusion","cubic_survivor_phase_density","cubic_rate_irrationality_unconditional","nonintegral_regular_rate_irrationality"],"status_summary":"An open eventual-Sylvester-recurrence endpoint plus Lean-checked irrationality theorems for the stated cubic rate and every nonintegral regular rate above one, unconditional bounded-rise and eventually-periodic negative-orbit obstructions, and a conditional cofinally bounded negative-magnitude exclusion.","prior_art_posture":"These are source-specific formal results; no novelty, priority, significance, or #243 solution claim is made, and normalised vanishing, mixed-sign, and unbounded-negative boundaries remain explicit.","primary_narrative_owner":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","source_route":"python3 scripts/query_corpus.py --claim bounded_rise_fresh_modulus_barrier","consumer_or_open_obligation":"The cubic and nonintegral-rate theorems apply only under their respective quantitative asymptotics. The three no-go consumers remove fresh-modulus avoidance, eventually-periodic negative tails, and the cofinally bounded negative branch under normalised vanishing; the original eventual-recurrence assertion still requires control of the mixed-sign unbounded orbit regime.","view_decision":"problem_note_reference_detail"},{"id":"finite_period_noncollapse_frontier","claim_ids":["finite_period_noncollapse"],"status_summary":"An unconditional finite-support denominator noncollapse theorem for Mersenne-type sums, with the infinite-support irrationality boundary still open.","prior_art_posture":"The finite-support order theorem is a source-specific formal result; no novelty, priority, significance, or universal #257 solution claim is made.","primary_narrative_owner":"paper/257/erdos-257-mersenne-support-subseries.tex","source_route":"python3 scripts/query_corpus.py --claim finite_period_noncollapse","consumer_or_open_obligation":"The finite order identity supplies denominator growth for finite partial sums, but no unbounded support certificate is supplied and the universal infinite-support #257 endpoint remains open.","view_decision":"problem_note_reference_detail"},{"id":"rational_base_coordinatewise_boundaries","claim_ids":["erdos_1049","three_halves_coordinatewise_corridor_no_go","rational_base_cleared_tail_recurrence","rational_base_forcing_growth","van_assche_qapery_diagonal_no_go"],"status_summary":"An open universal rational-base endpoint, a base-3/2 coordinatewise exclusion, exact cleared-tail dynamics and forcing bounds, and a finite q-Apéry diagonal non-transfer result.","prior_art_posture":"These are source-specific formal arithmetic results and an open target; no novelty, priority, significance, or irrationality claim is made for the new declarations.","primary_narrative_owner":"paper/1049/erdos-1049-rational-base-lambert.tex","source_route":"python3 scripts/query_corpus.py --claim three_halves_coordinatewise_corridor_no_go","consumer_or_open_obligation":"The corridor exclusion covers only its stated six-condition architecture, while the recurrence and forcing bound supply no rationality bridge or bounded-state contradiction; irrationality at 3/2 and the universal #1049 assertion remain open.","view_decision":"problem_note_reference_detail"},{"id":"zudilin_rational_base_and_hankel_theorems","claim_ids":["zudilin_rational_base_region","zudilin_all_rank_hankel_order","coefficient_pencil_finite_rank"],"status_summary":"Formalized rational-base irrationality and measure in the contour region, all-rank normalized Hankel order and coefficient formulas, and the coefficient-pencil proposition through rank eight.","prior_art_posture":"Zudilin supplies the cited 2004 polynomial-form construction and 2016 Hankel sequence. The formalizations preserve their hypotheses and make no novelty, priority or independent human-review claim.","primary_narrative_owner":"paper/1049/erdos-1049-rational-base-lambert.tex","source_route":"python3 scripts/query_corpus.py --claim zudilin_rational_base_region","consumer_or_open_obligation":"The contour-region theorems include all positive powers of 31/4. The coefficient-pencil result is finite-rank only; its all-rank extension, irrationality at base 3/2, and universal rational-base irrationality remain open.","view_decision":"problem_note_reference_detail"},{"id":"weighted_and_cover_support_criteria","claim_ids":["finite_prime_weighted_support","strengthened_positive_cover_support","mixed_weighted_cover_support","reciprocal_summable_support"],"status_summary":"Unconditional hereditary support criteria checked in Lean; universal Erdős #257 remains open.","prior_art_posture":"The formal statements preserve their weighted-summability and positive-cover assumptions. No novelty, priority, or independent human-review claim is made.","primary_narrative_owner":"paper/257/erdos-257-mersenne-support-subseries.tex","source_route":"python3 scripts/query_corpus.py --claim finite_prime_weighted_support","consumer_or_open_obligation":"remaining_open.universal_257_all_infinite_supports","view_decision":"problem_note_reference_detail"},{"id":"degree_seven_total_variation_counterexample","claim_ids":["ani_degree_seven_total_variation_counterexample"],"status_summary":"Ani's explicit degree-seven polynomial refutes the exact Formal Conjectures path-image-length statement by a Lean-checked one-dimensional Hausdorff-measure bound; the total-variation formulation is also refuted.","prior_art_posture":"Formalisation of ani’s existing construction; no novelty or independent human review of historical correspondence is claimed.","primary_narrative_owner":"paper/1041/erdos-1041-lemniscate-newton-flow.tex","source_route":"python3 scripts/query_corpus.py --claim ani_degree_seven_total_variation_counterexample","consumer_or_open_obligation":"Independent human review of correspondence with the 1958 curve-length wording remains open; the exact Formal Conjectures Hausdorff-length statement is checked and refuted.","view_decision":"problem_note_reference_detail"},{"id":"separated_cut_algebraic_lambert_transcendence_family","claim_ids":["separated_cut_algebraic_lambert_transcendence"],"status_summary":"Reviewed ordinary proof; full infinite statement not formalised and independent human review not recorded.","prior_art_posture":"Uses Evertse-Ferretti normalization of the number-field Subspace Theorem and the Corvaja-Zannier approximation method; no historical priority claim.","primary_narrative_owner":"paper/synthesis/optimal-sparse-perturbations.tex","source_route":"python3 scripts/query_corpus.py --claim separated_cut_algebraic_lambert_transcendence","consumer_or_open_obligation":"The full support at 3/2 and arbitrary infinite supports at base two remain unresolved. Finite cut arithmetic is not a proof of transcendence.","view_decision":"retain_in_companion"},{"id":"independent_clock_channel_relations_family","claim_ids":["independent_clock_channel_relations"],"status_summary":"Reviewed ordinary proof; full infinite statement not formalised and independent human review not recorded.","prior_art_posture":"The integer-radix recurrence and torus-density argument are standard ingredients; the stated clock-channel combination has no asserted historical priority. Bugeaud-Laurent is a distinct two-letter comparison.","primary_narrative_owner":"paper/269/erdos269-running-lcm-reasoning-surface.tex","source_route":"python3 scripts/query_corpus.py --claim independent_clock_channel_relations","consumer_or_open_obligation":"No rational independence of the original reciprocal prime logarithms is assumed or proved; the repeated running-LCM endpoint remains open.","view_decision":"retain_in_companion"},{"id":"calibrated_rational_hankel_countermodel_family","claim_ids":["calibrated_rational_hankel_countermodel"],"status_summary":"Reviewed ordinary proof; full infinite statement not formalised and independent human review not recorded.","prior_art_posture":"Reuses standard positive-moment/Christoffel machinery and the existing geometric-universality theorem. Exact construction and arithmetic controls carry no historical priority claim.","primary_narrative_owner":"paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex","source_route":"python3 scripts/query_corpus.py --claim calibrated_rational_hankel_countermodel","consumer_or_open_obligation":"These are not Zudilin's actual moments. They refute an inference from the shared signature alone; they do not settle F(3/2) or refute the attributed divisibility-based criterion.","view_decision":"retain_in_companion"},{"id":"selected_staircase_integral_height_boundary_family","claim_ids":["selected_staircase_integral_height_boundary"],"status_summary":"Reviewed ordinary proof; full infinite statement not formalised and independent human review not recorded.","prior_art_posture":"Smith form and the Schur-complement identity are standard; Stanley is the attributed Smith-form reference. This is subordinate arithmetic for the existing selected kernel.","primary_narrative_owner":"paper/269/erdos269-running-lcm-reasoning-surface.tex","source_route":"python3 scripts/query_corpus.py --claim selected_staircase_integral_height_boundary","consumer_or_open_obligation":"The optimality concerns diagonal clearing and one rational target. It does not rule out multi-target, nonlinear, recurrence, or new approximation methods and does not prove running-LCM irrationality.","view_decision":"retain_in_companion"},{"id":"signed_totient_entire_defect_interpolation_family","claim_ids":["signed_totient_entire_defect_interpolation"],"status_summary":"Reviewed ordinary proof; full infinite statement not formalised and independent human review not recorded.","prior_art_posture":"Finite-difference moment cancellation is standard. Comparisons cite Crmaric-Kovac, Coons with version-specific numbering, and Habiro; no historical priority is asserted. The arbitrary divergent allowance strengthens the returned polylogarithmic/power cases.","primary_narrative_owner":"paper/249/erdos249-totient-reasoning-surface.tex","source_route":"python3 scripts/query_corpus.py --claim signed_totient_entire_defect_interpolation","consumer_or_open_obligation":"The correction is signed, so it does not subsume the one-sided prime-gap theorem. It preserves singularities and prescribed finite jets, not full analytic functions or all Taylor coefficients. The unperturbed totient and prime series remain unresolved.","view_decision":"retain_in_companion"}]},"projections":[{"id":"bounded_machine_orientation","path":"docs/orientation.json","role":"Generated first-read capsule of release scale, the proved/open boundary, principal claim routes, and typed drilldowns.","authority":"navigation_projection_not_proof_authority","build":"python3 scripts/build_corpus_descriptor.py","check":"python3 scripts/build_corpus_descriptor.py --check"},{"id":"bounded_human_orientation","path":"docs/ORIENTATION.md","role":"Generated human projection of the same bounded first-read capsule.","authority":"navigation_projection_not_proof_authority","build":"python3 scripts/build_corpus_descriptor.py","check":"python3 scripts/build_corpus_descriptor.py --check"},{"id":"external_corpus_descriptor","path":"docs/corpus_descriptor.json","role":"Generated registration envelope for external mathematical-corpus systems, with bounded inline handles and digest-bound 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it neither proves that statement nor settles arbitrary infinite supports."}},{"from":"transport_curvature_reductions","relation":"builds_on","to":"certificate_reduction"},{"from":"transport_curvature_reductions","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"re_expresses","statement":"Re-expresses diagonal non-integrality through several exact transport and curvature observables while leaving every unbounded supply open."}},{"from":"affine_transport_annihilation","relation":"builds_on","to":"transport_curvature_reductions"},{"from":"transport_certificate_instances","relation":"instantiates","to":"transport_curvature_reductions"},{"from":"first_harmonic_certificate_interface","relation":"builds_on","to":"certificate_reduction"},{"from":"first_harmonic_certificate_interface","relation":"builds_on","to":"certificate_completeness"},{"from":"first_harmonic_certificate_interface","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"re_expresses","statement":"Turns a cofinal first-harmonic estimate into the existing finite certificate supply; no such estimate is proved."}},{"from":"first_harmonic_pivot_decomposition","relation":"builds_on","to":"first_harmonic_certificate_interface"},{"from":"first_harmonic_pivot_decomposition","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"re_expresses","statement":"Gives an exact four-term first-harmonic decomposition and a sufficient 9X/10 budget; no prime-distribution or decorrelation estimate supplies those budgets cofinally."}},{"from":"adjacent_phase_separation_reduction","relation":"builds_on","to":"first_harmonic_certificate_interface"},{"from":"adjacent_phase_separation_reduction","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"re_expresses","statement":"Re-expresses the missing supply as cofinal adjacent-pair phase separation; no such supply is proved."}},{"from":"pivot_anti_reconstruction_reduction","relation":"builds_on","to":"adjacent_phase_separation_reduction"},{"from":"pivot_anti_reconstruction_reduction","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"re_expresses","statement":"Re-expresses the missing supply as cofinal pivot anti-reconstruction or window-pair separation; no cofinal arithmetic producer is proved."}},{"from":"fixed_precision_transport_no_go","relation":"builds_on","to":"transport_curvature_reductions"},{"from":"lcm_factor_ideal_anchor_pulse_no_go","relation":"builds_on","to":"transport_curvature_reductions"},{"from":"actual_lcm_orbit_nonintegrality_frontier","relation":"re_expresses_open_branch","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"re_expresses","statement":"Re-expresses the open #249 endpoint as cofinal non-integrality of the actual power-two LCM orbit; the required supply remains unproved."}},{"from":"actual_lcm_positive_corridor_top_edge","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"narrows","statement":"Eliminates the nonnegative true-survivor branch in the positive corridor but leaves top-edge exclusion and cofinal actual-LCM non-integrality open."}},{"from":"terminal_scaled_vanishing_half_countermodel","relation":"advances_open_target","to":"universal_257","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.half_value_membership","operation":"re_expresses","statement":"Re-expresses half-membership as the existence of a terminal scaled-vanishing sequence; the sequence is not constructed, and the resulting conditional countermodel does not settle universal #257."}},{"from":"cofinal_cylinder_half_countermodel","relation":"advances_open_target","to":"universal_257","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.half_value_membership","operation":"re_expresses","statement":"Re-expresses half-membership as the existence of cofinal full-cylinder stages; the cofinal family is not constructed, and the resulting conditional countermodel does not settle universal #257."}},{"from":"sigma_transcendence","relation":"positions","to":"mersenne_lambert_ladder"},{"from":"totient_coprimality_probability","relation":"builds_on","to":"mersenne_lambert_ladder"},{"from":"squared_lambert_gcd_moments","relation":"builds_on","to":"totient_coprimality_probability"},{"from":"stern_brocot_cylinder_law","relation":"builds_on","to":"squared_lambert_gcd_moments"},{"from":"stern_brocot_run_fibonacci_stability","relation":"positions","to":"stern_brocot_cylinder_law"},{"from":"prime_support_irrationality","relation":"positions","to":"universal_257"},{"from":"three_prime_finite_residue_contradiction","relation":"advances_open_target","to":"erdos_269","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.erdos_269_reduced_tail_nonintegrality","operation":"constrains","statement":"The exact residue classifier excludes a bounded positive carry state whose canonical least-positive representative exceeds the bound. It constrains the reduced tail without excluding the integral branch, which is what pointwise nonintegrality of every reduced tail would do."}},{"from":"three_prime_cofinal_window_carry_extinction","relation":"re_expresses_open_branch","to":"erdos_269","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.erdos_269_cofinal_local_window_escape","operation":"re_expresses","statement":"The checked consumer takes the denominator-dependent cofinal local-window residue escape as its hypothesis, so this proposition is exactly the input that would discharge it for every multiplier coprime to 30."}},{"from":"weighted_phase_carry_finite_observer","relation":"advances_open_target","to":"erdos_269","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.erdos_269_two_dimensional_representation","operation":"constrains","statement":"The finite-observer formalisation isolates the faithfulness requirement a two-dimensional representation must meet, and shows that a genuine finite-dimensional factorisation forces a finite realised span. It supplies no representation and excludes none."}},{"from":"bounded_rise_fresh_modulus_barrier","relation":"advances_open_target","to":"erdos_243","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.erdos_243_overlap_height_growth","operation":"constrains","statement":"The bounded-rise barrier is a state-level obstruction on the canonical orbit and is one of the inputs the overlap-height growth question would feed. It proves no lower bound on the growth of the overlap height."}},{"from":"first_harmonic_pivot_decomposition","relation":"re_expresses_open_branch","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.erdos_249_first_harmonic_anti_concentration","operation":"re_expresses","statement":"The exact supplier-fibre bijection and its counterexample to global supplier isolation set up the four-term decomposition this proposition asks for, and leave the constant saving on the first additive character unproved."}},{"from":"twenty_one_quotient_greedy_frontier","relation":"re_expresses_open_branch","to":"universal_257","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.twenty_one_scaled_remainder_cofinal_return","operation":"re_expresses","statement":"Membership of 1/21 is proved equivalent to cofinally bounded scaled actual greedy remainders, so this proposition is that criterion stated as the remaining obligation."}},{"from":"twenty_one_quotient_greedy_frontier","relation":"advances_open_target","to":"universal_257","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.twenty_one_actual_orbit_invariant","operation":"constrains","statement":"The exact six-step defect recurrence and the slope-aware repair-load equivalence say which quantities a bounded-memory invariant would have to track. No invariant is constructed and none is excluded."}},{"from":"twenty_one_quotient_greedy_frontier","relation":"advances_open_target","to":"universal_257","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.twenty_one_final_skip_exclusion","operation":"narrows","statement":"On the complementary fatal branch the last-skip approximant is pinned inside an exact one-sided interval around the Erdos-Borwein constant. Turning that interval into a contradiction still needs the final-skip signatures this proposition asks for."}}]}},"release":{"version":"0.10.0","tag":"v0.10.0","date":"2026-09-11","lean_toolchain":"leanprover/lean4:v4.29.1","repository":"https://github.com/wcook04/plectis-erdos","formal_source":{"ref":"2a8c1667566020d123a081b6165e3ee94550566d","ref_kind":"commit","publication_state":"committed_checkpoint_pending_remote_publication","relationship_to_last_tag":"post_tag_checkpoint","public_tag":null},"public_projection":{"posture":"self_contained_public_projection_from_a_larger_ongoing_research_workflow","meaning":"This checkout is a deliberately curated, independently verifiable public output of a larger ongoing formal-mathematics workflow. No public claim depends on private or unreleased work.","public_evidence":["pinned formal-source identity","claim and open-proposition registry","machine-readable argument and module graphs","generated declaration atlas","bounded cold-clone and adversarial checks"],"boundary":"The workflow provenance explains the release discipline; it does not imply hidden proof authority or equivalence with private work."}},"status_taxonomy":{"proved here":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","formalised here":"Known mathematics represented and checked in Lean","unconditional progress":"Genuine theorem that does not settle the open problem","conditional reduction":"Exact implication whose remaining hypothesis is open","verified finite instance":"Kernel-checked computation at a bounded range","cited only":"External theorem used for positioning, not formalised","open":"Not proved by this release","formal statement refuted":"Lean refutes the exact Formal Conjectures statement; correspondence with the original wording awaits independent review"},"external_verification_packet":{"schema":"erdos-external-verification-packet/1","scope":"all_eight_public_problem_programmes","purpose":"Disclose the checked frontier for each of the eight public problem programmes, and isolate a small Comparator-executable review unit with a separately declared statement and axiom budget.","boundary":"Using the degree-seven polynomial constructed by the erdosproblems.com contributor ani, Lean proves that every preconnected strict-lemniscate set containing two distinct roots has one-dimensional Hausdorff measure greater than two. This refutes the exact Formal Conjectures path-image-length statement; the separate total-variation bound is also checked. The other seven targets remain open. Independent human review of correspondence with the 1958 wording has not been recorded. Comparator checks only selected exact statements, axioms and kernel acceptance; it does not assess novelty or historical correspondence.","review_status":"self-assessed and agent-checked; no human mathematical peer review is claimed","automation":{"posture":"Substantial language-model and agent assistance was used in formalisation, proof engineering, checking, and exposition.","human_responsibility":"Will Cook selected the public scope, accepts responsibility for the claims and correspondence, and does not present CI as mathematical peer review."},"formalization_schema":{"version":"v0.4","repository":"https://github.com/mathlib-initiative/formalization.yaml","commit":"99c678e569c7c4c0772db297c5ddd5e4c9b6322e"},"problem_index_owner":"docs/problem_index_source.json","problem_index_projection":"docs/problems.json","problem_ids":["erdos_68","erdos_243","erdos_249","erdos_251","erdos_257","erdos_269","erdos_1041","erdos_1049"],"comparator":{"config":"verification/comparator.json","challenge_module":"ExternalVerification.Challenge","solution_module":"ExternalVerification.Solution","challenge_trust_boundary":"ExternalVerification/Statements.lean and its Mathlib-only import closure, plus the pinned lakefile.toml, lean-toolchain, and lake-manifest.json build inputs required by Comparator's threat model","permitted_axioms":["propext","Quot.sound","Classical.choice"],"enable_nanoda":false,"pins":{"comparator":"789279735fe44c1c05dc54bb9f46ba4d9b8c7611","lean4export":"6f4e21dd70c3c11d7fbd07d39e3192792c657448","landrun":"811cfff51ceaf3d9843708aa6d22e9b84ccac8b4"}},"claim_status_contract":{"registered_claim":"A selected interface with claim_id supports an existing reviewed claim family; Comparator selection does not widen that family.","unregistered_interface":"A selected interface without claim_id is an executable review interface, not a canonical reviewed claim.","local_proof_provenance":"Every selected solution is proved in this repository and accepted by the pinned Lean kernel.","novelty":"Local proof provenance is not priority evidence. Novelty is unassessed unless a separate source-comparison review says otherwise."},"source_fidelity":{"prime_gap_unboundedness":{"source_statement":"The gaps between consecutive primes are unbounded; the elementary factorial construction gives arbitrarily long prime-free intervals.","mapping":"The local theorem uses the zero-based consecutive-prime gap sequence and proves that for every bound B some indexed gap exceeds B. This is the same unboundedness statement after the repository's indexing convention is unfolded.","references":["docs/PRIOR_ART.md#expansion-note-principal-sources-68-243-251-269-1041-1049","paper/251/erdos-251-prime-gap-dyadic-series.tex"],"historical_attribution":"Classical elementary factorial prime-free interval argument; modern large-gap theorems in the paper are contextual and strictly stronger.","logical_dependency":"The Lean proof reconstructs an elementary argument locally; no external prime-gap theorem is imported as a proof dependency."},"eventually_periodic_lambert":{"source_statement":"For an eventually periodic sequence of nonnegative rational coefficients that is not eventually zero, the associated Lambert series at an integer base b at least 2 is irrational.","mapping":"The local theorem formalises the nonnegative rational eventually-periodic coefficient case. Its coefficient, base, sign, and non-eventual-zero hypotheses are kept explicit; it does not claim the broader mixed-sign theorem.","references":["https://doi.org/10.1142/S1793042113501121","docs/PRIOR_ART.md"],"historical_attribution":"Luca and Tachiya, Irrationality of Lambert series associated with a periodic sequence (2014), with the later RIMS exposition used for statement comparison.","logical_dependency":"The local Lean proof uses a separate periodic-divisor certificate route; the cited paper establishes attribution and source fidelity, not a kernel dependency."},"known_irrational_supports":{"source_statement":"Several structured support families for the Mersenne reciprocal subseries are known to give irrational sums, including full support and other classical or later special families.","mapping":"The review family is an aggregate disclosure. Each local theorem retains its own support predicate; the Comparator selects full support as the classical representative and pairwise-coprime support as a dedicated exact non-dilation representative.","references":["https://users.renyi.hu/~p_erdos/1948-04.pdf","https://users.renyi.hu/~p_erdos/1969-09.pdf","https://doi.org/10.1142/S1793042113501121","https://arxiv.org/abs/2512.01739v2","docs/PRIOR_ART.md"],"historical_attribution":"Erdos (1948) for full support; Erdos (1968), Luca--Tachiya, and Tao--Teravainen for the specific additional families identified in the paper and prior-art ledger.","logical_dependency":"The full-support and pairwise-coprime Comparator theorems are proved locally in Lean. Other family rows have their stated local or cited proof boundaries; historical attribution is not an imported Lean dependency."},"pairwise_coprime_support":{"source_statement":"For every integer base b at least 2 and every infinite pairwise-coprime support A with summable reciprocal mass, the support-restricted Mersenne reciprocal series is irrational.","mapping":"The local theorem keeps the base, infinitude, pairwise-coprimality, and summable-reciprocal hypotheses explicit and concludes irrationality of erdosSupportSeries b A. At base 2 this is exactly the pairwise-coprime support subfamily of Erdős #257; it is a separate non-dilation representative from the full-support row.","references":["https://users.renyi.hu/~p_erdos/1969-09.pdf","docs/PRIOR_ART.md","paper/257/erdos-257-mersenne-support-subseries.tex"],"historical_attribution":"Erdos (1968), as recorded in the repository's prior-art ledger and the #257 problem note.","logical_dependency":"The proof-bearing source supplies the adaptive CRT and weighted coefficient certificate argument locally; the cited source establishes historical attribution and statement fidelity, not an imported Lean dependency."},"achievement_set_geometry":{"source_statement":"Kakeya's strict-tail criterion places rapidly decreasing subsum sets in the Cantor-set regime; for the fixed-base Mersenne sequence, Kovač--Tao verify the strict-tail inequality and distinct subsums.","mapping":"At base 2 the local development formalises compactness, perfection, total disconnectedness, nowhere density, and measure one. The strict-tail Cantor geometry aligns with the cited antecedents; the measure-one clause is locally derived and carries no novelty assertion.","references":["https://doi.org/10.11429/ptmps1907.7.14_250","https://doi.org/10.1007/s10474-025-01528-0","docs/PRIOR_ART.md"],"historical_attribution":"Kakeya (1914) for the strict-tail subsum-set criterion and Kovač--Tao (2025), Remark 4.1, for the fixed-base Mersenne antecedent.","logical_dependency":"The topology and measure theorem are proved locally in Lean; the cited sources fix historical/source alignment and are not imported proof objects."},"erdos_borwein_full_support":{"source_statement":"For every integer b at least 2, the full-support Lambert--Mersenne series sum over n at least 1 of 1/(b^n-1) is irrational.","mapping":"The local theorem uses Nat indexing with exponent k+1, which is exactly the source sum after reindexing n=k+1; the base assumption is the same integer condition b at least 2.","references":["https://users.renyi.hu/~p_erdos/1948-04.pdf","docs/PRIOR_ART.md","paper/257/erdos-257-mersenne-support-subseries.tex"],"historical_attribution":"Erdos, On arithmetical properties of Lambert series (1948), theorem on p. 63 as recorded in the repository's prior-art ledger.","logical_dependency":"The theorem is independently represented and proved in Lean in this repository; the historical source is attribution and fidelity evidence, not an imported kernel dependency."},"totient_kernel_rank":{"source_statement":"Coons proved that Euler's totient sequence is not k-regular for every integer k at least 2, while Martin's affine-totient ordering theorem implies linear independence of every finite family of pairwise nonproportional positive-slope affine totient sections.","mapping":"The dyadic Lean development gives an independent constructive finite-level basis and rank proof. Its full-kernel infinite-dimensionality corollary is kernel-checked here but is not an independent contribution: for integer-valued sequence modules it is equivalent to Coons's non-k-regularity result. The all-base Lean modules check zero-channel and composite-base reductions, residue coordinates, index cardinality k^e+1, canonical spanning and the reusable conditional-rank interface. Separately, AllBaseTotientKernel.lean at public checkpoint 25ef6245d15a proves canonical independence and the complete rank unconditionally by CRT and Dirichlet. This does not enlarge the Comparator target represented below or formalise Martin's broader theorem.","references":["paper/249/erdos-249-binary-totient-series.tex","docs/PRIOR_ART.md","https://doi.org/10.1017/S0305004108001665","https://doi.org/10.1007/s11139-004-4633-4"],"historical_attribution":"Coons for non-k-regularity of Euler's totient sequence; Martin for the positive-density affine ordering theorem from which the paper's finite-family independence step follows.","logical_dependency":"Arithmetic reduction and canonical spanning are unconditional. The recorded reusable rank wrapper takes independence as an input. AllBaseTotientKernel.lean discharges that input independently; Martin's broader positive-density theorem remains cited external prior work, not an axiom of the Lean proof."},"prime_gap_irrationality_equivalence":{"source_statement":"The prime-value dyadic series equals 2 plus the prime-gap dyadic series by summation by parts, so one series is irrational exactly when the other is.","mapping":"The local named iff still takes a Summable hypothesis. This checkout discharges that hypothesis by the elementary polynomial bound p_n ≤ 1250(n+1)^4 and the unconditional identity, not by the prime-number theorem. Neither side is proved irrational.","references":["https://www.erdosproblems.com/forum/thread/251","paper/251/erdos-251-prime-gap-dyadic-series.tex","docs/PRIOR_ART.md#expansion-note-principal-sources-68-243-251-269-1041-1049"],"historical_attribution":"Tao, erdosproblems.com/251, 7 October 2025. Recorded as cited only on prime_gap_identity_tao. This is not a novelty or priority claim.","logical_dependency":"The Lean proof reconstructs summation by parts and an elementary prime bound locally; the forum post is attribution, not a kernel import."}},"main_results":[{"id":"factorial_strict_successor_equivalence","problem":68,"review_family":"factorial_carry_characterisation","contribution_class":"locally proved result; novelty unassessed","original_declaration":"ErdosProblems.Erdos68.irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses","original_source":"ErdosProblems/Erdos68/FactorialGapPlateauCore.lean","wrapper_declaration":"Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses","statement":"Irrationality is equivalent to cofinally many strict-successor divisibility failures.","boundary":"The equivalence does not prove that the cofinal failures occur."},{"id":"factorial_nonunit_carry_equivalence","problem":68,"review_family":"factorial_carry_characterisation","contribution_class":"locally proved result; novelty unassessed","original_declaration":"ErdosProblems.Erdos68.irrational_factorialGapSeries_iff_cofinal_nonunit_carries","original_source":"ErdosProblems/Erdos68/FactorialGapPlateauCore.lean","wrapper_declaration":"Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_nonunit_carries","statement":"Irrationality is equivalent to cofinally many non-unit carries.","boundary":"The equivalence does not supply a cofinal carry producer."},{"id":"bounded_negative_exclusion","problem":243,"review_family":"bounded_negative_exclusion","contribution_class":"conditional reduction and no-go result","original_declaration":"ErdosProblems.Erdos243.no_cofinallyBoundedNegative_of_normalizedVanishes","original_source":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","wrapper_declaration":"Erdos249257.ExternalVerification.no_cofinallyBoundedNegative_of_normalizedVanishes","statement":"Normalised vanishing and bounded rise exclude a cofinally bounded negative part.","boundary":"All dynamics, bounded-rise, positivity, and vanishing hypotheses remain explicit."},{"id":"centered_state_dynamics","problem":243,"review_family":"centered_state_dynamics","contribution_class":"locally proved signed recovery/rigidity theorem; novelty unassessed","original_declaration":"ErdosProblems.Erdos243.boundedNegativePart_eventually_zero","original_source":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","wrapper_declaration":"Erdos249257.ExternalVerification.boundedNegativePart_eventually_zero","statement":"Under exact natural C/D dynamics with a > 1, C > 0, E = D - (a - 1)C, strict |E| < C, a uniform lower bound on E, and division-free normalized vanishing, boundedNegativePart_eventually_zero forces E = 0 eventually; centered-zero plus an eventually nonzero next tail recovers eventual Sylvester recurrence.","boundary":"The signed theorem has a C D : ℕ → ℕ, E : ℕ → ℤ, and B : ℕ with all exact hypotheses explicit: a > 1, C > 0, C(n + 1) + D(n) = a(n) · C(n), D(n + 1) = a(n) · D(n), E is the centered state D − (a − 1)C, |E| < C, a uniform lower bound on E, and division-free normalized vanishing. It excludes persistent bounded negative behaviour and forces eventual centered defect zero; centered-zero plus an eventually nonzero next tail recovers Sylvester, but no reciprocal-tail irrationality follows. The unbounded negative mixed-sign branch and the prime-specific producer remain open. This is distinct from the periodic negative-orbit and cofinally bounded-negative no-go families; no novelty, priority, significance, or external-review claim is made, and no unrestricted #243 solution follows."},{"id":"negative_mass_recovery","problem":243,"review_family":"negative_mass_recovery","contribution_class":"conditional analytic recovery criterion; novelty unassessed","original_declaration":"ErdosProblems.Erdos243.sylvesterNext_eventually_of_summable_negativeRelativeMass","original_source":"ErdosProblems/Erdos243/SparseResetRecovery.lean","wrapper_declaration":"Erdos249257.ExternalVerification.sylvesterNext_eventually_of_summable_negativeRelativeMass","statement":"Under exact next-denominator and next-tail C/D dynamics with a, D : ℕ → ℤ, C : ℕ → ℕ, C > 0, the strict centered step equation, division-free normalized vanishing, and Summable normalized negative centered mass, the centered state vanishes eventually and a follows the Sylvester recurrence eventually.","boundary":"This is a conditional recovery criterion: the source theorem assumes every displayed dynamics, positivity, strict centered-step, normalized-vanishing, and summability hypothesis, and does not prove them for the original Erdős #243 orbit. Any surviving canonical orbit must have divergent normalized negative mass. It supplies no prime-specific producer, unconditional #243 endpoint, or reciprocal-tail irrationality, and makes no novelty, priority, significance, or external-review claim. It is distinct from centered_state_dynamics, which uses a uniform lower-bound hypothesis rather than summability of normalized negative mass."},{"id":"eventually_periodic_negative_no_go","problem":243,"review_family":"negative_orbit_no_go","contribution_class":"no-go result","original_declaration":"ErdosProblems.Erdos243.no_eventuallyPeriodicNegative_orbit","original_source":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","wrapper_declaration":"Erdos249257.ExternalVerification.no_eventuallyPeriodicNegative_orbit","statement":"An eventually periodic negative-error orbit with positive phase growth is impossible.","boundary":"The theorem does not cover arbitrary unbounded negative behaviour."},{"id":"totient_kernel_finite_rank","problem":249,"review_family":"totient_kernel_rank","claim_id":"dyadic_totient_certificate_interface","contribution_class":"locally proved result; novelty unassessed","original_declaration":"Erdos249257.finrank_totientKernelThroughLevelFamily_eq","original_source":"Erdos249257/TotientMahlerDefect.lean","wrapper_declaration":"Erdos249257.ExternalVerification.finrank_totientKernelThroughLevelFamily_eq","statement":"For every e >= 1, the complete dyadic totient kernel through level e has rational span dimension 2^e + 1.","boundary":"This does not prove irrationality of the totient series."},{"id":"totient_kernel_infinite_rank","problem":249,"review_family":"totient_kernel_rank","claim_id":"dyadic_totient_certificate_interface","contribution_class":"Lean formalisation of an existing consequence of Coons","original_declaration":"Erdos249257.not_finiteDimensional_span_fullTotientKernel","original_source":"Erdos249257/TotientMahlerDefect.lean","wrapper_declaration":"Erdos249257.ExternalVerification.not_finiteDimensional_span_fullTotientKernel","statement":"The rational span of the full dyadic totient kernel is not finite-dimensional.","boundary":"The declaration is kernel-checked locally, but full-kernel infinite-dimensionality is not an independent contribution of this release; no rationality-to-finite-rank bridge is proved."},{"id":"totient_kernel_basis","problem":249,"review_family":"totient_kernel_basis","claim_id":"dyadic_totient_basis","contribution_class":"locally proved result; novelty unassessed","original_declaration":"Erdos249257.totientDyadicSectionBasis","original_source":"Erdos249257/TotientMahlerDefect.lean","wrapper_declaration":"Erdos249257.ExternalVerification.exists_totientDyadicSectionBasis","statement":"The full dyadic totient kernel span admits the explicit odd-core-indexed basis.","boundary":"The basis theorem does not connect rationality to finite kernel rank."},{"id":"totient_kernel_all_base_rank","problem":249,"review_family":"totient_kernel_all_base_index","contribution_class":"formalised unconditional spanning and conditional rank with external independence input","original_declaration":"Erdos249257.finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent","original_source":"Erdos249257/TotientKernelConditional.lean","wrapper_declaration":"Erdos249257.ExternalVerification.finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent","statement":"Under 2 ≤ k, 1 ≤ e, and linear independence of the explicit canonical all-base family, the actual all-base totient kernel through level e has rational span dimension k^e + 1.","boundary":"The linear-independence hypothesis is the external affine-ordering boundary and is not proved here; this does not prove irrationality of the binary totient series."},{"id":"totient_visible_coprime_lattice_mass","problem":249,"review_family":"totient_visible_coprime_lattice_mass","contribution_class":"formalised exact visible-lattice normalization identity","original_declaration":"GcdMomentCalculus.tsum_pos_coprime_inv_mersenne_eq_one","original_source":"Erdos249257/GcdMomentCalculus.lean","wrapper_declaration":"Erdos249257.ExternalVerification.tsum_pos_coprime_inv_mersenne_eq_one","statement":"The positive coprime Mersenne weights over the visible lattice sum exactly to one, giving the reduced-direction mass behind the fair-coin coprimality form of the totient constant.","boundary":"This is a rational normalization identity and does not prove irrationality of the binary totient series or settle Erdős #249."},{"id":"totient_squared_lambert_gcd_moment","problem":249,"review_family":"squared_lambert_gcd_moments","contribution_class":"formalised exact first-gcd-moment squared-Lambert transfer identity","original_declaration":"GcdMomentCalculus.tsum_totient_div_mersenne_sq_eq_gcd_moment_series","original_source":"Erdos249257/GcdMomentCalculus.lean","wrapper_declaration":"Erdos249257.ExternalVerification.tsum_totient_div_mersenne_sq_eq_gcd_moment_series","statement":"At base one half, the squared-Mersenne Lambert series weighted by Euler's totient equals the dyadic series of Pillai's gcd-sum function minus n, the exact first gcd-moment transfer.","boundary":"This exact identity does not prove irrationality of the binary totient series; the constant-weight squared-Lambert irrationality is cited external mathematics, and no Erdős #249 endpoint follows."},{"id":"stern_brocot_run_fibonacci_stability","problem":249,"review_family":"stern_brocot_run_fibonacci_stability","contribution_class":"formalised exact Fibonacci/continuant run stability floor","original_declaration":"SternBrocotRunGeometry.runHeight_defect_fib_sum_lower","original_source":"Erdos249257/SternBrocotRunGeometry.lean","wrapper_declaration":"Erdos249257.ExternalVerification.runHeight_defect_fib_sum_lower","statement":"For every finite defect list e, the height of the corresponding positive Stern-Brocot run word is at least F_(r+3) + F_(r+1) times the total defect, where r is the number of runs.","boundary":"This exact combinatorial stability floor does not establish analytic denominator survival, irrationality of the binary totient series, or novelty; no Erdős #249 endpoint follows."},{"id":"stern_brocot_cylinder_mass_split","problem":249,"review_family":"stern_brocot_cylinder_law","contribution_class":"formalised exact Stern-Brocot cylinder telescoping identity","original_declaration":"GcdMomentCalculus.cylinderMass_split","original_source":"Erdos249257/GcdMomentCalculus.lean","wrapper_declaration":"Erdos249257.ExternalVerification.cylinderMass_split","statement":"At every positive Stern-Brocot node (a,b), the closed cylinder mass splits exactly into the node's stop mass and the two child cylinder masses.","boundary":"This exact telescoping recursion and its geometric interpretation do not prove irrationality of the binary totient series or novelty; the #249 endpoint remains open."},{"id":"lcm_factor_ideal_shift_algebra_no_go","problem":249,"review_family":"lcm_factor_ideal_shift_algebra_no_go","claim_id":"lcm_factor_ideal_anchor_pulse_no_go","contribution_class":"unconditional progress","original_declaration":"Erdos249257.TotientTailPeriodKiller.lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient","original_source":"Erdos249257/LcmFactorIdealPulseObstruction.lean","wrapper_declaration":"Erdos249257.ExternalVerification.lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient","statement":"For every t >= 3, a nonzero synthetic dyadic coboundary survives every finite integer shift polynomial while retaining the period-LCM totient factor ideals, exact cleared-prefix cancellation, and uniform state and letter bounds.","boundary":"The witness is synthetic: it does not assert that its forcing letters are actual totient differences, does not address nonlinear combinations, and leaves the unbounded certificate supply for Erdős #249 open."},{"id":"fixed_precision_transport_no_go","problem":249,"review_family":"fixed_precision_transport_no_go","claim_id":"fixed_precision_transport_no_go","contribution_class":"unconditional synthetic no-go; novelty unassessed","original_declaration":"Erdos249257.TotientTailPeriodKiller.fixedPrecisionTropicalNoGo","original_source":"Erdos249257/TropicalCurvatureCarry.lean","wrapper_declaration":"Erdos249257.ExternalVerification.fixedPrecisionTropicalNoGo","statement":"For every positive fixed precision, every finite word of odd valuation-unit symbols and every incoming integer carry state admit a compatible orbit whose successive states stay within the corresponding valuation-unit radii; fixed-precision local signatures therefore do not by themselves exclude finite centred completions.","boundary":"This is a synthetic finite-word construction with unrestricted incoming/high quotient states. It does not impose the actual totient-difference relation or provide growing-precision/global correlation, and it gives no Erdős #249 endpoint or irrationality result. No novelty or priority claim is made. The one-step and finite-word completion theorems are subordinate mechanisms, not separate families."},{"id":"actual_lcm_orbit_separation","problem":249,"review_family":"actual_lcm_orbit_separation","claim_id":"actual_lcm_orbit_nonintegrality_frontier","contribution_class":"conditional reduction","original_declaration":"Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_actualLcmOrbitSeparationSupply","original_source":"Erdos249257/TotientActualLcmOrbitSeparation.lean","wrapper_declaration":"Erdos249257.ExternalVerification.irrational_totientSeries_of_actualLcmOrbitSeparationSupply","statement":"A cofinal quantitative separation supply for the actual power-two LCM tail orbit implies irrationality of the binary totient series.","boundary":"The cofinal separation supply is an open premise, stronger than mere non-integrality; this sufficient route is not an equivalence and leaves the punctured/top-edge boundary and the Erdős #249 endpoint open."},{"id":"actual_foreign_residue_projection","problem":249,"review_family":"actual_foreign_residue_projection","claim_id":"actual_foreign_residue_projection","contribution_class":"conditional reduction","original_declaration":"Erdos249257.ActualForeignResidueProjection.scaleFullTarget_miss_of_projected_separation","original_source":"Erdos249257/ActualForeignResidueProjection.lean","wrapper_declaration":"Erdos249257.ExternalVerification.scaleFullTarget_miss_of_projected_separation","statement":"The finite Möbius residue diagonal splits into projected foreign and divisor channels, every finite stable foreign-tail window obeys a closed geometric bound, and a controlled projection separated from every integer forces the full target to be missed.","boundary":"This is a conditional finite partition/geometric-majorant reduction with a generic complement-noncancellation consumer. The ControlledForeignProjection analytic bridge and any unbounded separated-projection supply remain unproved; no Erdős #249 endpoint, novelty, significance, priority, or external-review claim is made."},{"id":"first_harmonic_pivot_decomposition","problem":249,"review_family":"first_harmonic_pivot_decomposition","claim_id":"first_harmonic_pivot_decomposition","contribution_class":"conditional reduction","original_declaration":"Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_pivotResidualDecorrelation","original_source":"Erdos249257/FirstHarmonicPivot.lean","wrapper_declaration":"Erdos249257.ExternalVerification.irrational_totient_series_of_pivotResidualDecorrelation","statement":"The exact first-harmonic pivot decomposition and its 9X/10 gap yield a finite certificate whenever the stated PivotBudgetAt bounds hold, and the cofinal pivot-residual-decorrelation socket implies irrationality of the binary totient series.","boundary":"The four-term decomposition and 9X/10 gap remain conditional on the explicit dyadic-room and PivotBudgetAt hypotheses; the cofinal decorrelation producer is unproved. The fibre/image lemmas do not give global supplier-prime isolation or a prime-distribution estimate, and no Erdős #249 irrationality conclusion is claimed."},{"id":"strict_prime_tail_orbit_gap","problem":249,"review_family":"strict_prime_tail_orbit_gap","claim_id":"strict_prime_tail_orbit_gap","contribution_class":"conditional reduction","original_declaration":"ErdosProblems.Erdos249.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap","original_source":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","wrapper_declaration":"Erdos249257.ExternalVerification.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap","statement":"An 11/100 nonpositive-phase density on arbitrarily late dyadic blocks yields an 89/100 first-harmonic saving and a finite 9/10 gap; alternatively, a cofinal natural-prime tail-orbit strict gap below 9/10 yields adaptive pivot escape and conditional irrationality. The exact initial phase is the scaled-angle character of (2^h - 1) times the series, and the strict-gap route is an orbit of repeated squaring from that phase.","boundary":"The phase-density, cofinal natural-prime strict-gap, associated uniform-margin, and cofinal nonpositive-prime-shift producers remain unproved for the actual totient orbit. Exact phase/integrality equivalence, finite dyadic-root exclusion for 1 ≤ h ≤ 16 and k ≤ 14, and non-dyadicity-to-cofinal-nonpositive-phase implications are classification or finite evidence only; they do not supply density or prime alignment, and the source does not prove actual root entry. This is a conditional reduction only: no unconditional Erdős #249 endpoint, novelty, priority, significance, or external-review claim is made. It is distinct from the finite first_harmonic_pivot_decomposition and actual_lcm_orbit_separation families."},{"id":"totient_carry_anti_compression","problem":249,"review_family":"totient_carry_anti_compression","claim_id":"totient_carry_kernel_anti_compression","contribution_class":"locally proved necessary consequence; novelty unassessed","original_declaration":"Erdos249257.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank","original_source":"Erdos249257/TotientTailCarryPeriod.lean","wrapper_declaration":"Erdos249257.ExternalVerification.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank","statement":"If the binary totient coefficient series were rational, one positive-multiplier tempered integral carry has dyadic sections uniformly eventually periodic modulo its multiplier and canonical carry-section span dimension at least 2^e − 1 for every e.","boundary":"This is a necessary consequence of rationality only: it gives no contradiction, no finite-rank upper bound for a rationality-supplied carry, and no irrationality proof for Erdős #249. Modulo a divisor of the multiplier, the forcing disappears and the carry becomes geometric, so quotient periodicity alone supplies no finite-ℚ-rank upper bound. The coefficient-side kernel-rank and certificate-equivalence families are not duplicated, and #249 remains open."},{"id":"prime_gap_unboundedness","problem":251,"review_family":"coefficient_only_no_go","contribution_class":"formalisation of existing result","original_declaration":"ErdosProblems.Erdos251.exists_primeGap0_gt","original_source":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","wrapper_declaration":"Erdos249257.ExternalVerification.exists_primeGap0_gt","statement":"The consecutive prime gaps are unbounded.","boundary":"Unbounded coefficients alone do not force irrationality."},{"id":"coefficient_only_no_go","problem":251,"review_family":"coefficient_only_no_go","contribution_class":"formalisation of exact finite identity and nonperiodicity contrast; novelty and significance unassessed","original_declaration":"ErdosProblems.Erdos251.carryPartialSum_natCast_eq","original_source":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","wrapper_declaration":"Erdos249257.ExternalVerification.coefficientOnlyNoGo","statement":"The coefficient-only wrapper records the exact finite carry partial-sum identity together with non-eventual-periodicity of the synthetic coefficient stream and the actual prime-gap stream, exposing the limits of coefficient-only reasoning.","boundary":"The wrapper proves the finite identity and the two non-eventual-periodicity statements only. It does not assert the infinite rational-sum limit of the synthetic countermodel, identify the synthetic coefficient stream with actual prime gaps, establish a prime-gap tail bridge, prove #251 irrationality, or make novelty, priority, significance, or external-review claims."},{"id":"prime_gap_irrationality_equivalence","problem":251,"review_family":"prime_gap_reformulation","contribution_class":"formalisation of existing result; novelty unassessed","original_declaration":"ErdosProblems.Erdos251.irrational_tsum_primeDyadicTerm_iff_primeGap","original_source":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","wrapper_declaration":"Erdos249257.ExternalVerification.irrational_tsum_primeDyadicTerm_iff_primeGap","statement":"Lean checks that the prime-value dyadic series is irrational exactly when the prime-gap dyadic series is, discharging summability by an elementary polynomial prime bound rather than the prime-number theorem. Neither side is proved irrational. The already-known identity is Tao's (7 October 2025) and is recorded separately as cited only.","boundary":"The equivalence does not prove irrationality of either series."},{"id":"dyadic_tail_integrality_classification","problem":251,"review_family":"dyadic_tail_integrality_classification","contribution_class":"abstract Lean-checked classification and normal form; novelty unassessed","original_declaration":"ErdosProblems.Erdos251.irrational_initial_iff_all_positive_tailShifts_nonintegral","original_source":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","wrapper_declaration":"Erdos249257.ExternalVerification.irrational_initial_iff_all_positive_tailShifts_nonintegral","statement":"For any integer-digit real dyadic tail recurrence T(N + 1) = 2*T(N) - g(N + 1), the initial value T(0) is irrational exactly when every positive-length tail shift is nonintegral.","boundary":"This exact normal form covers arbitrary integer-digit dyadic tail recurrences with the declared real state type; it does not supply a cofinal nonintegral shift for the consecutive-prime-gap recurrence. The prime-specific producer remains open, so no prime-gap irrationality, novelty, significance, or external-review claim is made and Erdős #251 remains open."},{"id":"small_mismatch_criterion","problem":251,"review_family":"small_mismatch_criterion","contribution_class":"conditional actual-prime-gap endpoint reduction; novelty and significance unassessed","original_declaration":"ErdosProblems.Erdos251.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch","original_source":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","wrapper_declaration":"Erdos249257.ExternalVerification.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch","statement":"For the actual prime-gap dyadic tail, cofinally many adjacent pairs of strictly small h-shifts with unequal corresponding prime gaps force failure of eventual RatIntegral h-shifts.","boundary":"The representative assumes T : ℕ → ℚ, the DyadicTailRecurrence for the actual prime-gap digits, and for every N₀ an N ≥ N₀ whose adjacent h-shifts both lie strictly in (-1, 1) while the corresponding prime gaps differ. Under that cofinal small-mismatch supply it excludes eventual RatIntegral h-shifts; it does not prove the supply or actual smallness. rationalPrimeGapTail_has_positive_shift_not_eventually_small records the contrary rational-state obstruction for at least one positive shift. This is a conditional reduction only: no #251 irrationality, infinite rational-sum limit, universal producer, novelty, priority, significance, or external-review claim is made; #251 remains open. The actual-prime-gap specialization is distinct from coefficient_only_no_go and does not duplicate prime_gap_reformulation."},{"id":"mersenne_achievement_measure_one","problem":257,"review_family":"achievement_set_geometry","claim_id":"greedy_achievement_geometry","contribution_class":"mixed prior-art geometry and locally proved metric result; novelty unassessed","original_declaration":"Erdos249257.volume_mersenneAchievementSet","original_source":"Erdos249257/GreedyAchievementSet.lean","wrapper_declaration":"Erdos249257.ExternalVerification.volume_mersenneAchievementSet","statement":"The base-2 Mersenne achievement set has Lebesgue measure exactly one.","boundary":"This does not decide whether every infinite-support sum is irrational."},{"id":"erdos_borwein_full_support","problem":257,"review_family":"known_irrational_supports","claim_id":"eb_full_support","contribution_class":"formalisation of an existing theorem","original_declaration":"Erdos249257.irrational_erdosSum_full_support","original_source":"Erdos249257/CertificateKernel.lean","wrapper_declaration":"Erdos249257.ExternalVerification.irrational_erdosSum_full_support","statement":"For every integer base b >= 2, the full-support Erdos-Borwein series is irrational.","boundary":"This is the classical full-support theorem, not universal Erdos #257."},{"id":"finite_prime_weighted_support","problem":257,"review_family":"finite_prime_weighted_support","claim_id":"finite_prime_weighted_support","contribution_class":"locally proved result; novelty unassessed","original_declaration":"ErdosProblems.Erdos257.PaperCompleteR8.divisibilityWeightedClaim","original_source":"ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean","wrapper_declaration":"Erdos249257.ExternalVerification.divisibilityWeightedClaim","statement":"For every base b >= 2, an infinite positive support A with finite prime-weighted mass in base b has irrational support series; if a positive host H has finite prime-weighted mass in base 2, every infinite subset of H has irrational support series at every base b >= 2.","boundary":"The finite-prime weighted-summability premise is explicit in both clauses. This criterion does not cover every infinite support or settle universal Erdős #257; no novelty, priority, significance, or external human-review claim is made."},{"id":"finite_prime_weighted_fixed_base_hereditary","problem":257,"review_family":"finite_prime_weighted_support","contribution_class":"locally proved result; novelty unassessed","original_declaration":"ErdosProblems.Erdos257.PaperCompleteR8.finitePrimeWeighted_fixedBase_hereditary","original_source":"ErdosProblems/Erdos257/PaperCompleteR8/WeightedHereditaryClaim.lean","wrapper_declaration":"Erdos249257.ExternalVerification.finitePrimeWeighted_fixedBase_hereditary","statement":"For every base b >= 2, if a positive host H has finite prime-weighted mass in base b, every infinite subset A of H has irrational support series in the same base b.","boundary":"This hereditary result retains the same-base weighted-summability hypothesis; it does not infer all-base irrationality from weighted mass at an arbitrary base, or settle universal Erdős #257."},{"id":"erdos_support_pairwise_coprime","problem":257,"review_family":"pairwise_coprime_support","claim_id":"support_families","contribution_class":"formalisation of an existing theorem","original_declaration":"Erdos249257.irrational_erdosSupportSeries_pairwise_coprime","original_source":"Erdos249257/CertificateKernel.lean","wrapper_declaration":"Erdos249257.ExternalVerification.irrational_erdosSupportSeries_pairwise_coprime","statement":"For every integer base b >= 2 and every infinite pairwise-coprime support A with summable reciprocal mass, the support-restricted Mersenne series is irrational.","boundary":"This is the pairwise-coprime support theorem, not universal Erdos #257; the infinitude, pairwise-coprimality, and summable-reciprocal hypotheses remain explicit."},{"id":"periodic_nonnegative_weight_irrationality","problem":257,"review_family":"periodic_nonnegative_weight_irrationality","claim_id":"periodic_nonnegative_weight_irrationality","contribution_class":"formalised here; novelty unassessed","original_declaration":"Erdos249257.irrational_ratWeightSeries_eventuallyPeriodic","original_source":"Erdos249257/CertificateKernel.lean","wrapper_declaration":"Erdos249257.ExternalVerification.irrational_ratWeightSeries_eventuallyPeriodic","statement":"For every integer base b >= 2 and every eventually-periodic nonnegative rational coefficient sequence γ with a positive value at a positive index in its periodic region, the series sum_a γ(a)/(b^a - 1) is irrational.","boundary":"This is the nonnegative rational eventually-periodic coefficient theorem; its nonnegativity, eventual periodicity, and positive-tail hypotheses remain explicit, and the broader mixed-sign periodic theorem is not claimed."},{"id":"orthogonal_petal_sunflower_reduction","problem":257,"review_family":"orthogonal_petal_sunflower_reduction","claim_id":"orthogonal_petal_sunflower_reduction","contribution_class":"conditional reduction; novelty unassessed","original_declaration":"Erdos249257.SupportSunflowerDichotomy.irrational_erdosSupportSeries_of_orthogonalPetalBouquet","original_source":"Erdos249257/SupportSunflowerDichotomy.lean","wrapper_declaration":"Erdos249257.ExternalVerification.irrational_erdosSupportSeries_of_orthogonalPetalBouquet","statement":"For every support A carrying the finite-core OrthogonalPetalBouquet structure and satisfying the exact uniform SunflowerForcedSlotTailSelection predicate, the base-2 support-restricted Mersenne series erdosSupportSeries 2 A is irrational.","boundary":"This is a conditional reduction: the bouquet structure and uniform tail-selection predicate remain hypotheses; neither is constructed for arbitrary supports, so this does not add an unconditional Erdos #257 support family or solve the universal problem."},{"id":"finite_period_noncollapse","problem":257,"review_family":"finite_period_noncollapse","contribution_class":"locally proved result; novelty unassessed","original_declaration":"Erdos249257.finite_period_noncollapse_rat_den","original_source":"Erdos249257/CertificateKernel.lean","wrapper_declaration":"Erdos249257.ExternalVerification.finite_period_noncollapse_rat_den_interface","statement":"For a reduced finite Mersenne sum, the base order modulo the denominator equals the support lcm.","boundary":"The finite-support theorem does not settle any infinite-support sum."},{"id":"supported_achievement_measure_dichotomy","problem":257,"review_family":"restricted_achievement_sets","contribution_class":"locally proved result; novelty unassessed","original_declaration":"ErdosProblems.Erdos257.volume_supportedMersenneAchievementSet_dichotomy","original_source":"ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean","wrapper_declaration":"Erdos249257.ExternalVerification.volume_supportedMersenneAchievementSet_dichotomy","statement":"A support-restricted Mersenne achievement set has exact finite-complement measure or measure zero for infinite complement.","boundary":"The measure dichotomy does not classify rational points."},{"id":"shifted_odd_tail_nat_state","problem":257,"review_family":"shifted_odd_tail_state","contribution_class":"conditional reduction","original_declaration":"Erdos249257.exists_shifted_odd_tail_nat_states_of_support_fraction","original_source":"Erdos249257/RationalSupportCarrySkeleton.lean","wrapper_declaration":"Erdos249257.ExternalVerification.exists_shifted_odd_tail_nat_states_of_support_fraction","statement":"A rational support value with a positive support seed and denominator 2^c*v produces a positive natural shifted tail state with an exact binary recurrence and a modular doubling orbit modulo v.","boundary":"This state-extraction theorem does not exclude rational infinite supports or settle universal Erdős #257; the support and rational-value hypotheses remain explicit."},{"id":"boolean_mobius_carry","problem":257,"review_family":"boolean_mobius_carry","claim_id":"boolean_mobius_carry","contribution_class":"locally proved exact equivalence plus conditional consequences; novelty unassessed","original_declaration":"Erdos249257.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry","original_source":"Erdos249257/BooleanMobiusCarry.lean","wrapper_declaration":"Erdos249257.ExternalVerification.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry","statement":"For p : ℤ and q > 0, a nonempty positive-index support with erdosSupportSeries 2 A = p/q exists exactly when a quotient-only Boolean–Möbius carry certificate exists, with positive square-root-bounded integer orbit, divisible carry differences, and support reconstructed from its Boolean Möbius transform.","boundary":"The equivalence is stated at the existence level with q > 0, 0 ∉ A, and a positive support witness; it gives no contradiction, no universal Erdős #257 irrationality result, and no producer of cofinally nonintegral shifts for the actual recurrence. The generic tempered-orbit theorem and the reciprocal-mass, unbounded-tail, and zero-window results are subordinate consequences, not separate families; the zero-window constant depends on the support and numerator and is not uniform. This does not settle every infinite support, and no novelty, priority, significance, external-review, or unrestricted #257 endpoint claim is made."},{"id":"boolean_mobius_exact_row_dynamics","problem":257,"review_family":"boolean_mobius_exact_row_dynamics","contribution_class":"conditional reduction; novelty unassessed","original_declaration":"Erdos249257.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips","original_source":"Erdos249257/BooleanMobiusSkipRowCofinal.lean","wrapper_declaration":"Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips","statement":"If positive rational-greedy skips occur cofinally in the declared sense, the skipped-core construction supplies cofinally many exact local Mersenne half rows and hence (1 / 2 : ℝ) belongs to the Mersenne achievement set.","boundary":"This route is conditional on the unproved CofinalPositiveHalfGreedySkips premise: for every N there is c ≥ max N 4 with a positive greedy remainder below mersenneWeightRat c. A finite exact row or any finite collection of exact rows does not establish that cofinal supply. The result is a conditional 1/2-membership/counterexample route only, not an unconditional solution of Erdős #257; the exact-row, upper-half Boolean-word, rank-two, and endpoint-defect declarations are subordinate mechanism evidence, and no novelty, priority, significance, or external-review claim is made."},{"id":"half_membership_seam_classification","problem":257,"review_family":"half_membership_seam_classification","claim_id":"half_membership_seam_classification","contribution_class":"locally proved exact endpoint classification; novelty unassessed","original_declaration":"Erdos249257.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse","original_source":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","wrapper_declaration":"Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse","statement":"Membership of 1/2 in the Mersenne achievement set is equivalent to unbounded false successor terminal bits, equivalently to an unbounded upper-or-middle successor branch, cofinal terminal-false positions, unbounded skipped ranks along the integer seam, infinite greedy skipped support, and no last half-greedy skip.","boundary":"This exact endpoint-socket classification does not supply any unbounded terminal-false or upper-or-middle occurrences, cofinal supply, skipped-support producer, or half-membership witness; finite exact rows do not establish the required global supply. It is not an unconditional or universal Erdős #257 result, and no novelty, priority, significance, or external-review claim is made."},{"id":"composite_dilation_defect_split","problem":257,"review_family":"composite_dilation_defect_identity","claim_id":"composite_dilation_defect_identity","contribution_class":"unconditional progress","original_declaration":"Erdos249257.CompositeDilationDefect.supportCoeff_mul_eq_add_defect","original_source":"Erdos249257/CompositeDilationDefect.lean","wrapper_declaration":"Erdos249257.ExternalVerification.supportCoeff_mul_eq_add_defect","statement":"For a positive support element a and positive x, the divisor-count coefficient at a*x splits into the coefficient at x, the distinguished new divisor when a does not divide x, and an exact finite composite-dilation defect.","boundary":"The exact defect is not bounded for arbitrary supports and supplies no tail selector or irrationality theorem; the separate bouquet budget remains conditional on its structural hypotheses."},{"id":"half_two_adic_band_bound","problem":257,"review_family":"half_and_twenty_one_frontiers","claim_id":"half_greedy_two_thirds_band","contribution_class":"no-go result","original_declaration":"Erdos249257.HalfGreedyTwoThirdsBand.seven_le_of_intBand_odd","original_source":"Erdos249257/HalfGreedyTwoThirdsBand.lean","wrapper_declaration":"Erdos249257.ExternalVerification.seven_le_of_intBand_odd","statement":"For positive odd integers p, D, and q, the integer form of the two-thirds band forces p ≥ 7; the small odd numerator classes 1, 3, and 5 are therefore excluded.","boundary":"This is local 2-adic band arithmetic: p = 7 is sharp for the stated hypotheses, and the theorem does not prove that an actual half-greedy orbit reaches or avoids the band, nor does it prove half-membership."},{"id":"three_prime_kernel_minor","problem":269,"review_family":"rank_two_kernel_no_go","contribution_class":"no-go result","original_declaration":"ErdosProblems.Erdos269.kernel_235_minor_eq_neg_one_fifteen","original_source":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","wrapper_declaration":"Erdos249257.ExternalVerification.kernel_235_minor_eq_neg_one_fifteen","statement":"The smallest displayed 2,3,5 kernel minor is exactly -1/15.","boundary":"Failure of rank one does not imply irrationality."},{"id":"three_prime_running_lcm","problem":269,"review_family":"three_prime_lcm_cells","contribution_class":"locally proved result; novelty unassessed","original_declaration":"ErdosProblems.Erdos269.smoothPrefixLcm_eq_threePrimeHeight","original_source":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","wrapper_declaration":"Erdos249257.ExternalVerification.smoothPrefixLcm_eq_threePrimeHeight","statement":"The running lcm of the three-prime smooth prefix equals the exact coordinatewise height.","boundary":"The identity does not supply the missing irrationality bridge."},{"id":"conditional_carry_escape","problem":269,"review_family":"conditional_carry_escape","contribution_class":"conditional no-go consumer; novelty and significance unassessed","original_declaration":"ErdosProblems.Erdos269.no_positive_reducedCarry_of_cofinalLocalWindowEscape","original_source":"ErdosProblems/Erdos269/RestrictedFloorSum.lean","wrapper_declaration":"Erdos249257.ExternalVerification.no_positive_reducedCarry_of_cofinalLocalWindowEscape","statement":"Under the denominator-dependent cofinal local-window residue-escape predicate, no positive reduced carry can satisfy the exact multiplier recurrence together with its short bound.","boundary":"The representative assumes b,m : ℕ → ℕ, shortBound : ℕ → ℕ → ℕ, CofinalLocalWindowEscape b m shortBound, and for each positive B coprime to 30 a positive integer-valued d with d(n+1) = (b n : ℤ) d(n) − (B : ℤ) (m n : ℤ) and Int.natAbs (d n) ≤ shortBound B n; it then gives False. The representative remains a conditional consumer for arbitrary words. For the actual three-prime word, the rationality-to-reduced-carry bridge is established separately; the source-specific cofinal escape producer is not proved. This is not a #269 endpoint or irrationality proof, and no actual-series identification, novelty, priority, significance, or external-review claim is made. It is one conditional window-carry family, distinct from the finite residue, rank-two, and weighted-phase observer families."},{"id":"newton_small_translation","problem":1041,"review_family":"translation_avoidance","contribution_class":"locally proved result; novelty unassessed","original_declaration":"ErdosProblems.Erdos1041.exists_small_translation_separating_arguments","original_source":"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","wrapper_declaration":"Erdos249257.ExternalVerification.exists_small_translation_separating_arguments","statement":"An arbitrarily small common translation separates a finite injective family from zero and from shared positive rays.","boundary":"Finite-family avoidance does not provide the global topological or metric gluing."},{"id":"newton_root_retention","problem":1041,"review_family":"root_retention","contribution_class":"locally proved result; novelty unassessed","original_declaration":"ErdosProblems.Erdos1041.constant_perturbation_roots_in_unitDisk","original_source":"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","wrapper_declaration":"Erdos249257.ExternalVerification.constant_perturbation_roots_in_unitDisk","statement":"A quantified small constant perturbation keeps every polynomial root inside the open unit disc.","boundary":"Root retention is only one input to an incomplete global route."},{"id":"three_halves_corridor_no_go","problem":1049,"review_family":"coordinatewise_corridor_no_go","contribution_class":"no-go result","original_declaration":"ErdosProblems.Erdos1049.threeHalves_no_coordinatewiseCorridor","original_source":"ErdosProblems/Erdos1049/RationalBaseLambert.lean","wrapper_declaration":"Erdos249257.ExternalVerification.threeHalves_no_coordinatewiseCorridor","statement":"The coordinatewise divisibility corridor is impossible at base 3/2.","boundary":"This excludes one proof architecture and proves no irrationality statement."},{"id":"rational_base_tail_recurrence","problem":1049,"review_family":"rational_base_tail_recurrence","contribution_class":"locally proved result; novelty unassessed","original_declaration":"ErdosProblems.Erdos1049.rationalBaseClearedTailQ_succ","original_source":"ErdosProblems/Erdos1049/RationalBaseLambert.lean","wrapper_declaration":"Erdos249257.ExternalVerification.rationalBaseClearedTailQ_succ","statement":"The rational-base cleared tail obeys its exact first-order recurrence.","boundary":"The recurrence is not derived from a rationality contradiction."},{"id":"rectangular_hp_threshold","problem":1049,"review_family":"height_and_pade_arithmetic","contribution_class":"locally proved sharp model-specific no-go; novelty unassessed","original_declaration":"ErdosProblems.Erdos1049.rectangular_hp_threshold_eq_classical_iff","original_source":"ErdosProblems/Erdos1049/HermitePadeNoGo.lean","wrapper_declaration":"Erdos249257.ExternalVerification.rectangular_hp_threshold_eq_classical_iff","statement":"For the explicit rectangular two-function exponent model, with real rho and sigma satisfying 0 <= rho and 1 + rho <= sigma, the threshold equals 1/2 - 1/pi^2 exactly when rho = 0 and sigma = 1.","boundary":"This is a sharp no-go only for the explicit rectangular two-function model under the stated real hypotheses; it constructs no approximating polynomials or remainders and is not a universal Pade or Hermite-Pade no-go. It proves no irrationality at 3/2 or any general rational-base endpoint, and the analytic remainder/nonvanishing input remains untreated. No novelty, priority, significance, external-review, or endpoint claim is made; Erdős #1049 remains open."}],"review_matrix":[{"problem":68,"paper":"paper/68/erdos-68-factorial-denominator-irrationality.tex","families":[{"id":"factorial_carry_characterisation","contribution_class":"locally proved result; novelty unassessed","summary":"Exact equivalence between irrationality, cofinally many non-unit carries, and cofinal strict-successor divisibility failures.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","declarations":["ErdosProblems.Erdos68.irrational_factorialGapSeries_iff_cofinal_nonunit_carries","ErdosProblems.Erdos68.irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses"],"boundary":"An exact reformulation does not supply the required cofinal failures."},{"id":"factorial_channel_and_projection_rigidity","contribution_class":"locally proved result; novelty unassessed","summary":"Finite quotient-band channel factorisation and breakpoint, channel congruences, a two-term prime-channel corrector, and endpoint-weighted projection rigidity.","evidence_mode":"Lean kernel","declarations":["Erdos68.factorialMoment_eq_factorial_pow_mul_channelNumerator_band","Erdos68.exists_index_ge_two_mul_of_factorialMoment_ne_zero_of_channel_eq_zero","Erdos68.channelNumerator_mod_factorialMoment","Erdos68.primeTranslator_channelResidual_eq_one","ErdosProblems.Erdos68.projection_disagreement_excludes_bounded_endpoint"],"comparator_disposition":"represented_by_isolated_headline","boundary":"These finite and structural results do not produce a cofinal obstruction."},{"id":"factorial_conditional_producers","contribution_class":"conditional reduction","summary":"Cofinal zero-branch or large private-factor hypotheses imply the required irrationality conclusion.","evidence_mode":"Lean kernel","comparator_disposition":"not_selected_deep_project_predicate_stack","boundary":"The producer hypotheses are unproved."},{"id":"factorial_finite_certificates","contribution_class":"finite computation","summary":"Lean checks small exact misses; a hash-bound GMP scan reaches index 300000 and yields the corresponding finite denominator exclusion.","evidence_mode":"Lean finite theorem plus external exact certificate","comparator_disposition":"not_applicable_to_external_execution","boundary":"Finite computation does not change the cofinal quantifier."},{"id":"factorial_lcm_growth","contribution_class":"ordinary paper argument","summary":"An elementary terminal-block argument gives liminf log L_N / (N^{3/2} log N) ≥ 2√2/3 for the lcm of factorial-gap denominators. The displayed bound needs no external multiplicity theorem; an older, weaker multiplicity-based route is historical context only.","evidence_mode":"Lean kernel plus paper argument","comparator_disposition":"not_selected_separate_release_comparator_entry","boundary":"The LCM asymptotic has an ordinary paper proof and is checked in Lean as common_denominator_growth_liminf in PaperCompleteLiminf.lean. The finite-block inequality has a separate Lean source. The Comparator entry for this bound belongs to the separate public release wcook04/plectis-erdos-lean. The bound does not decide irrationality."}]},{"problem":243,"paper":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","families":[{"id":"centered_state_dynamics","contribution_class":"locally proved signed recovery/rigidity theorem; novelty unassessed","summary":"Under exact natural C/D dynamics with a > 1, C > 0, E = D - (a - 1)C, strict |E| < C, a uniform lower bound on E, and normalized vanishing, boundedNegativePart_eventually_zero forces E = 0 eventually. centeredState_eventually_zero supplies the nonnegative stabilization mechanism, and sylvesterNext_eventually_of_centered_zero recovers eventual Sylvester recurrence when the next tail is nonzero.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","declarations":["ErdosProblems.Erdos243.boundedNegativePart_eventually_zero","ErdosProblems.Erdos243.centeredState_eventually_zero","ErdosProblems.Erdos243.sylvesterNext_eventually_of_centered_zero"],"boundary":"The signed theorem has a C D : ℕ → ℕ, E : ℕ → ℤ, and B : ℕ with all exact hypotheses explicit: a > 1, C > 0, C(n + 1) + D(n) = a(n) · C(n), D(n + 1) = a(n) · D(n), E is the centered state D − (a − 1)C, |E| < C, a uniform lower bound on E, and division-free normalized vanishing. It excludes persistent bounded negative behaviour and forces eventual centered defect zero; centered-zero plus an eventually nonzero next tail recovers Sylvester, but no reciprocal-tail irrationality follows. The unbounded negative mixed-sign branch and the prime-specific producer remain open. This is distinct from the periodic negative-orbit and cofinally bounded-negative no-go families; no novelty, priority, significance, or external-review claim is made, and no unrestricted #243 solution follows."},{"id":"negative_orbit_no_go","contribution_class":"no-go result","summary":"Constant, eventually constant, periodic, and eventually periodic negative-error orbits are excluded in their stated regimes.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted_representative_periodic_theorem","boundary":"The exclusions do not cover arbitrary unbounded negative behaviour."},{"id":"bounded_rise_coprimality","contribution_class":"locally proved result; novelty unassessed","summary":"A bounded-rise sequence cannot remain coprime to fresh pairwise-coprime moduli; reduced tails inherit this obstruction.","evidence_mode":"Lean kernel","declarations":["ErdosProblems.Erdos243.no_eventuallyBoundedRise_reducedTail","ErdosProblems.Erdos243.reducedTail_pairwiseCoprime"],"comparator_disposition":"represented_by_stronger_selected_boundary_theorem","boundary":"The required bounded-rise input is not automatic for the original sequence."},{"id":"bounded_negative_exclusion","contribution_class":"conditional reduction and no-go result","summary":"Normalised vanishing excludes a cofinally bounded negative part and forces eventual Sylvester behaviour.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"Every positivity, dynamics, bounded-rise, and vanishing hypothesis remains explicit."},{"id":"negative_mass_recovery","contribution_class":"conditional analytic recovery criterion; novelty unassessed","summary":"Under exact reciprocal-tail C/D dynamics, positive tail, the strict centered-step equation, and division-free normalized vanishing, summability of the normalized negative centered mass forces centeredState to vanish eventually and recovers the Sylvester recurrence. The load-bearing theorem is supported by the exact tail-growth bound and the finite-negative-mass stabilization theorem.","evidence_mode":"Lean kernel","comparator_disposition":"targeted","declarations":["ErdosProblems.Erdos243.tail_growth_le_one_add_negativeRelativeMass","ErdosProblems.Erdos243.eventually_zero_of_summable_negativeRelativeMass","ErdosProblems.Erdos243.sylvesterNext_eventually_of_summable_negativeRelativeMass"],"boundary":"The exact criterion assumes a, D : ℕ → ℤ and C : ℕ → ℕ with the declared next-denominator and next-tail dynamics, C > 0, the strict centered step equation, division-free normalized vanishing, and Summable (negativeRelativeMass C (fun n ↦ centeredState (a n) (D n) (C n))). It does not prove these hypotheses for the original Erdős #243 orbit, supply a prime-specific producer, or establish an unconditional #243 endpoint or reciprocal-tail irrationality. Any surviving canonical orbit must have divergent normalized negative mass. This is distinct from centered_state_dynamics, whose recovery criterion uses a uniform lower-bound hypothesis; no novelty, priority, significance, or external-review claim is made."}]},{"problem":249,"paper":"paper/249/erdos-249-binary-totient-series.tex","families":[{"id":"totient_kernel_basis","contribution_class":"locally proved result; novelty unassessed","summary":"An explicit odd-core basis and relation normal form for the full dyadic totient kernel.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"The basis theorem does not connect rationality of the series to finite kernel rank."},{"id":"totient_kernel_rank","contribution_class":"mixed local finite-level theorem and formalised prior consequence","summary":"An independent constructive proof of exact finite-level dyadic rank 2^e+1, together with a kernel-checked full-kernel infinite-dimensionality corollary already implied by Coons's non-2-regularity theorem.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"Only the finite-level rank is presented as an additional result; full-kernel infinite-dimensionality is prior-art context, and no rationality-to-finite-rank bridge is proved."},{"id":"totient_kernel_all_base_index","contribution_class":"formalised arithmetic, spanning and reusable conditional rank interface","summary":"For every base k at least 2, Lean checks the arithmetic reduction, exact nonmultiple-residue coordinates, unconditional canonical spanning, and exact rank k^e+1 conditional on canonical-family linear independence.","evidence_mode":"Lean kernel for this arithmetic, finite indexing, spanning and conditional-rank interface; the independent all-base proof is a separate public source result","comparator_disposition":"targeted","boundary":"This Comparator target reviews the reusable conditional interface. AllBaseTotientKernel.lean separately proves canonical all-base independence and exact rank unconditionally at 25ef6245d15a; it does not formalise Martin's broader theorem or solve #249."},{"id":"totient_visible_coprime_lattice_mass","contribution_class":"formalised exact visible-lattice normalization identity","summary":"The positive coprime Mersenne weights on the visible lattice sum to one, exposing the reduced-direction mass behind the fair-coin coprimality form of #249.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"This rational normalization identity does not prove irrationality of the binary totient series or settle #249."},{"id":"squared_lambert_gcd_moments","contribution_class":"formalised exact squared-Lambert first-moment transfer","summary":"At base one half, the totient-weighted squared-Mersenne Lambert series is the dyadic Pillai gcd-sum-minus-n series, exposing the first fair-coin gcd moment.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"This exact transfer identity does not prove irrationality of the binary totient series; the cited constant-weight q-Padé result remains external and no #249 endpoint follows."},{"id":"stern_brocot_run_fibonacci_stability","contribution_class":"formalised exact Fibonacci/continuant run stability floor","summary":"For a positive Stern-Brocot run word, the global height gain from run defects is bounded below by the sharp Fibonacci coefficient F_(r+1) times the total defect.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"This exact combinatorial stability floor does not establish analytic denominator survival, irrationality of the binary totient series, or novelty; no #249 endpoint follows."},{"id":"stern_brocot_cylinder_law","contribution_class":"formalised exact Stern-Brocot cylinder telescoping identity","summary":"At every positive Stern-Brocot node, the closed cylinder mass splits into stop mass and the two child cylinders, giving an exact Markov recursion.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"This exact telescoping recursion and its geometric interpretation do not prove irrationality of the binary totient series or novelty; the #249 endpoint remains open."},{"id":"lcm_factor_ideal_shift_algebra_no_go","contribution_class":"unconditional progress","summary":"For every t at least 3, a nonzero synthetic dyadic coboundary survives every finite integer shift polynomial while retaining the period-LCM totient factor ideals, exact cleared-prefix cancellation, and uniform state and letter bounds.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"The witness is synthetic: it does not assert that its forcing letters are actual totient differences, does not address nonlinear combinations, and leaves the unbounded certificate supply for Erdős #249 open."},{"id":"fixed_precision_transport_no_go","contribution_class":"unconditional synthetic no-go; novelty unassessed","summary":"At every fixed positive precision, every finite word of odd valuation-unit symbols admits a compatible centred carry completion from any incoming integer state; bounded local signatures therefore cannot exclude all finite centred endpoints.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","declarations":["Erdos249257.TotientTailPeriodKiller.fixedPrecisionTropicalNoGo","Erdos249257.TotientTailPeriodKiller.vu_step_has_centred_completion","Erdos249257.TotientTailPeriodKiller.vu_word_has_prefix_locked_completion"],"boundary":"This is a synthetic finite-word construction with unrestricted incoming/high quotient states. It does not impose the actual totient-difference relation or provide growing-precision/global correlation, and it gives no Erdős #249 endpoint or irrationality result. No novelty or priority claim is made. The one-step and finite-word completion theorems are subordinate mechanisms, not separate families."},{"id":"actual_lcm_orbit_separation","contribution_class":"conditional reduction","summary":"A cofinal quantitative 1/32 separation supply at canonically guarded odd ranks transfers through the explicit raw-block approximation and error radius to the existing half-word/signed-margin irrationality consumer. The landed actual-LCM sign/top-edge analysis is supporting and contrary-boundary evidence: positivity leaves the true survivor negative, and integrality forces the exact top-edge residue.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","declarations":["Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_iff_actualLcmOrbitNonintegralitySupply","Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_actualLcmOrbitNonintegralitySupply","Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_actualLcmOrbitSeparationSupply","Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcm_trueEndpointSurvivor_neg","Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcm_integral_forces_topEdgeResidue","Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcmTailDiff_shift_pos"],"boundary":"The quantitative 1/32 separation supply is sufficient but stronger than mere non-integrality, is not an equivalence, and remains open. The sign positivity alone does not exclude the negative true survivor; the top-edge forcing does not exclude that residue or supply cofinal non-integrality. These source declarations are supporting/contrary-boundary evidence, not a second actual-LCM family, and the punctured/top-edge boundary and #249 endpoint remain open."},{"id":"actual_foreign_residue_projection","contribution_class":"conditional reduction","summary":"The finite Möbius residue diagonal splits exactly into projected foreign and divisor channels, the omitted finite foreign tail has a closed geometric majorant, and the generic complement-noncancellation consumer turns a controlled projection separated from every integer into a full-target miss.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","declarations":["Erdos249257.ActualForeignResidueProjection.scaleFullTarget_miss_of_projected_separation","Erdos249257.ActualForeignResidueProjection.finiteResidueDiagonal_eq_projectedForeign_add_divisor","Erdos249257.ActualForeignResidueProjection.abs_foreignTailWindow_le_foreignComplementBound"],"boundary":"This is only a finite partition/geometric-majorant reduction feeding a generic complement-noncancellation consumer. The ControlledForeignProjection analytic bridge and any unbounded supply of separated projections remain unproved; no Erdős #249 endpoint, novelty, significance, priority, or external-review claim follows."},{"id":"first_harmonic_pivot_decomposition","contribution_class":"conditional reduction","summary":"The supplier fibre is an exact prime-image/fibre bijection, while a checked finite counterexample shows that supplier-prime global isolation fails. The load-bearing result is an exact four-term first-harmonic decomposition; under the explicit PivotBudgetAt budgets, the centred real-part bound and three contribution bounds give the 9X/10 gap and a finite certificate, and the conditional pivot-residual-decorrelation theorem transfers a cofinal supply to irrationality.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","declarations":["Erdos249257.TotientTailPeriodKiller.mem_pivotFiber_iff_exists_supplierPrime","Erdos249257.TotientTailPeriodKiller.pivotBaseOfPrime_injective_on_supplierPrimes","Erdos249257.TotientTailPeriodKiller.image_pivotSupplierPrimes_eq_pivotFiber","Erdos249257.TotientTailPeriodKiller.supplierPrime_not_globally_isolated_counterexample","Erdos249257.TotientTailPeriodKiller.windowFirstExp_sum_eq_pivot_decomposition","Erdos249257.TotientTailPeriodKiller.PivotBudgetAt","Erdos249257.TotientTailPeriodKiller.first_harmonic_gap_of_pivotBudgetAt","Erdos249257.TotientTailPeriodKiller.DTWPivotResidualDecorrelation","Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_pivotResidualDecorrelation"],"boundary":"The fibre/image lemmas and the explicit counterexample establish neither global supplier-prime isolation nor a prime-distribution estimate. The 9X/10 gap is conditional on PivotBudgetAt and the stated dyadic-room inequality; irrationality is conditional on DTWPivotResidualDecorrelation, whose exact cofinal producer remains open. No decorrelation estimate supplying those budgets is proved, and the #249 endpoint remains open."},{"id":"strict_prime_tail_orbit_gap","contribution_class":"conditional reduction","summary":"A single two-branch source mechanism gives an 89/100 first-harmonic block saving from an 11/100 nonpositive-phase density, or an adaptive pivot escape from a cofinal natural-prime tail-orbit strict gap below 9/10. The exact squaring law reduces the latter to a power-of-two orbit of one initial phase and shows that an entered dyadic root obstructs that producer, without proving such an entry. The load-bearing representative is the natural-prime irrationality reduction; block-gap, density, and phase-classification declarations remain subordinate.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","declarations":["ErdosProblems.Erdos249.TotientTailOrbitBlockGap","ErdosProblems.Erdos249.TotientTailOrbitNonpositiveBlockDensity","ErdosProblems.Erdos249.tailOrbitBlockGap_of_nonpositiveBlockDensity","ErdosProblems.Erdos249.irrational_totient_series_of_tailOrbitBlockGap","ErdosProblems.Erdos249.DTWNaturalPrimeTailOrbitStrictGap","ErdosProblems.Erdos249.naturalPivotPointEscape_of_naturalPrimeTailOrbitStrictGap","ErdosProblems.Erdos249.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap","ErdosProblems.Erdos249.tailOrbitFirstExp_succ","ErdosProblems.Erdos249.tailOrbitFirstExp_add","ErdosProblems.Erdos249.tailOrbitFirstExp_eq_initial_pow_two","ErdosProblems.Erdos249.naturalPrimeTailOrbitStrictGap_iff_initial_phase","ErdosProblems.Erdos249.tailOrbitFirstExp_eq_one_of_le","ErdosProblems.Erdos249.not_naturalPrimeTailOrbitStrictGap_of_phase_one","ErdosProblems.Erdos249.not_naturalPrimeTailOrbitStrictGap_of_dyadic_root","ErdosProblems.Erdos249.tailOrbitFirstExp_zero_eq_scaled_angle","ErdosProblems.Erdos249.tailOrbitFirstExp_eq_one_iff_tail_diff_mem_int","ErdosProblems.Erdos249.exists_tailOrbitFirstExp_zero_pow_two_eq_one_iff_dyadic","ErdosProblems.Erdos249.tailOrbitFirstExp_zero_pow_two_ne_one_upto_sixteen","ErdosProblems.Erdos249.cofinally_tailOrbitFirstExp_re_nonpos_of_not_dyadic","ErdosProblems.Erdos249.naturalPrimeTailOrbitStrictGap_of_cofinal_nonpositive_prime_shift"],"boundary":"The 11/100 phase-density, cofinal natural-prime strict-gap, associated uniform-margin, and cofinal nonpositive-prime-shift producers remain unproved for the actual totient orbit. The scaled-angle and phase/integrality equivalences, finite 1 ≤ h ≤ 16 and k ≤ 14 root exclusion, and non-dyadicity-to-cofinal-nonpositive-phase theorem provide classification or finite evidence only; they do not prove a density/equidistribution theorem, prime alignment, or actual dyadic-root entry. Both branches are conditional certificate/irrationality reductions only: they do not close Erdős #249, duplicate first_harmonic_pivot_decomposition or actual_lcm_orbit_separation, or justify novelty, priority, significance, or external review."},{"id":"totient_carry_anti_compression","contribution_class":"locally proved necessary consequence; novelty unassessed","summary":"If the binary totient coefficient series were rational, one positive-multiplier tempered integral carry would have dyadic sections uniformly eventually periodic modulo its multiplier while its canonical carry-section span has dimension at least 2^e − 1 for every e. This is the orbit-level quotient-periodicity/anti-compression conjunction, distinct from the coefficient-side kernel-rank family.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","declarations":["Erdos249257.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank","Erdos249257.finrank_canonicalCarryKernel_ge","Erdos249257.carrySectionsEventuallyPeriodicMod_of_shift_dvd","Erdos249257.totient_forcing_vanishes_mod_of_dvd_multiplier","Erdos249257.totient_carry_modEq_geometric_of_dvd_multiplier"],"boundary":"This is a necessary consequence of rationality only: it gives no contradiction, no finite-rank upper bound for a rationality-supplied carry, and no irrationality proof for Erdős #249. Modulo a divisor of the multiplier, the forcing disappears and the carry becomes geometric, so quotient periodicity alone supplies no finite-ℚ-rank upper bound. The coefficient-side kernel-rank and certificate-equivalence families are not duplicated, and #249 remains open."},{"id":"totient_finite_denominator_exclusion","contribution_class":"finite computation","summary":"A Farey denominator exclusion through 7.96e34, sharp for its selected window, and diagonal certificates through t=82.","evidence_mode":"Lean kernel and generated finite certificates","comparator_disposition":"not_selected_finite_family_has_separate_receipt","boundary":"The selected finite window and certificate ceiling do not imply an unbounded producer."},{"id":"totient_carry_rank","contribution_class":"conditional reduction","summary":"Rationality forces a tempered integral tail orbit with unbounded finite-level carry rank.","evidence_mode":"Lean kernel","comparator_disposition":"not_selected_deep_orbit_vocabulary","boundary":"This is a necessary consequence, not a contradiction."},{"id":"totient_lambert_coefficients","contribution_class":"locally proved result; novelty unassessed","summary":"The totient series is rewritten as a Lambert series with explicit nonnegative, unbounded prime-power coefficients.","evidence_mode":"Lean kernel","declarations":["MersenneLambertLadder.tsum_primWeight_div_two_pow_sub_one","Erdos249257.tsum_primWeight_div_two_pow_sub_one_eq_totient_series"],"comparator_disposition":"represented_by_basis_and_rank_targets","boundary":"The coefficient identities do not prove irrationality."},{"id":"eventually_periodic_lambert","contribution_class":"Lean formalisation of existing result","summary":"Eventually periodic nonnegative rational weights give an irrational Lambert series.","evidence_mode":"Lean kernel","declarations":["Erdos249257.irrational_ratWeightSeries_eventuallyPeriodic"],"comparator_disposition":"not_selected_prior_result","boundary":"The totient-derived weights are not eventually periodic."},{"id":"totient_certificate_equivalences","contribution_class":"exact equivalence and conditional reduction","summary":"Pointwise, cofinal, lcm-diagonal, and separated-window certificate supplies are exact or sufficient reformulations.","evidence_mode":"Lean kernel","declarations":["Erdos249257.TotientTailPeriodKiller.irrational_totient_series_iff_lcm_diagonal_certificate_supply","ErdosProblems.Erdos249.PeriodMultipleEscape.periodMultipleKillSupply_iff_irrational","Erdos249257.TotientTailPeriodKiller.dtwWindowSeparatedPairs_iff_irrational_totient_series"],"comparator_disposition":"not_selected_deep_certificate_vocabulary","boundary":"Equivalent producer statements are as hard as the unresolved target."},{"id":"signed_totient_entire_defect_interpolation","contribution_class":"ordinary paper argument","summary":"For any integer beta>=2 and positive allowance f(n) tending to infinity, an interval around the totient value is filled by integer sequences b=phi+e with 0<=b(n)<=n and |e(n)|<=f(n). One target-independent even support has upper Banach density zero, leaving any fixed prefix and odd coefficients unchanged. The construction preserves every fixed eventual coefficient and cumulative congruence, gives eventually constant progression moment defects outside common hulls of density zero, entire progression Dirichlet defects, and smooth root-of-unity radial germs with prescribed finite zero jets. The full proof includes the allowance-dependent block bound and first-harmonic rational countermodels. It is an ordinary proof with two AI proof reviews, not a Lean proof of the infinite theorem or independent human review.","evidence_mode":"ordinary proof; two AI reviews, no independent human review","comparator_disposition":"not_applicable_without_exact_Lean_statement","declarations":[],"boundary":"The correction is signed, so it does not subsume the one-sided prime-gap theorem. It preserves singularities and prescribed finite jets, not full analytic functions or all Taylor coefficients. The unperturbed totient and prime series remain unresolved."}]},{"problem":251,"paper":"paper/251/erdos-251-prime-gap-dyadic-series.tex","families":[{"id":"prime_gap_reformulation","contribution_class":"locally proved result; novelty unassessed","summary":"Exact finite summation by parts, the unconditional infinite prime-gap identity, and irrationality equivalence.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"The equivalence does not prove irrationality of either series."},{"id":"integral_shift_classification","contribution_class":"locally proved result; novelty unassessed","summary":"Block identities and exact denominator criteria classify rationality through integral positive tail shifts.","evidence_mode":"Lean kernel","declarations":["ErdosProblems.Erdos251.irrational_initial_iff_all_positive_tailShifts_nonintegral","ErdosProblems.Erdos251.not_irrational_initial_iff_exists_eventually_integral_positive_tailShift","ErdosProblems.Erdos251.tailShift_integral_iff_den_dvd_mersenne","ErdosProblems.Erdos251.tailShift_integral_iff_two_pow_modEq_one"],"comparator_disposition":"not_selected_abstract_family_represented_in_manifest","boundary":"The concrete prime-gap producer remains missing."},{"id":"dyadic_tail_integrality_classification","contribution_class":"abstract Lean-checked classification and normal form; novelty unassessed","summary":"For any integer-digit dyadic tail recurrence T(N+1) = 2*T(N) - g(N+1), the rational h-shift is integral exactly when the reduced denominator of T(N) divides 2^h - 1, equivalently when 2^h is congruent to 1 modulo that denominator. For the real recurrence, the load-bearing normal form says that irrationality of T(0) is equivalent to every positive tail shift being nonintegral; the eventual-integrality equivalence and denominator/order laws are supporting mechanism evidence.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","declarations":["ErdosProblems.Erdos251.tailShift_integral_iff_den_dvd_mersenne","ErdosProblems.Erdos251.tailShift_integral_iff_two_pow_modEq_one","ErdosProblems.Erdos251.not_irrational_initial_iff_exists_eventually_integral_positive_tailShift","ErdosProblems.Erdos251.irrational_initial_iff_all_positive_tailShifts_nonintegral"],"boundary":"These exact statements cover arbitrary integer-digit dyadic tail recurrences with the declared rational and real state types; they classify integrality and give an irrationality normal form but do not supply a cofinal nonintegral shift for the consecutive-prime-gap recurrence. The generic hypotheses therefore do not close the prime-specific producer gap: no prime-gap irrationality, novelty, significance, or external review is claimed, and Erdős #251 remains open. This is one abstract classification family, distinct from prime_gap_reformulation, not a duplicate frontier claim."},{"id":"totient_shift_propagation","contribution_class":"locally proved result; novelty unassessed","summary":"Odd rational denominators give totient-length integral shifts, and integrality propagates through the recurrence.","evidence_mode":"Lean kernel","declarations":["ErdosProblems.Erdos251.tailShift_integral_totient_of_odd_den","ErdosProblems.Erdos251.tailShift_integral_succ","ErdosProblems.Erdos251.tailShift_integral_add"],"comparator_disposition":"represented_by_integral_shift_family","boundary":"This describes rational states and supplies no contradiction for actual prime gaps."},{"id":"small_mismatch_criterion","contribution_class":"conditional actual-prime-gap endpoint reduction; novelty and significance unassessed","summary":"For the actual prime-gap dyadic tail, cofinally many adjacent pairs of strictly small h-shifts with unequal corresponding prime gaps force failure of eventual RatIntegral h-shifts. The adjacent-pair zero obstruction is the load-bearing mechanism; exact non-eventual periodicity of consecutive prime gaps supplies the actual-prime-gap contrast.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","declarations":["ErdosProblems.Erdos251.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch","ErdosProblems.Erdos251.not_eventuallyIntegralTailShift_of_cofinal_small_mismatch","ErdosProblems.Erdos251.tailShift_not_both_integral_of_small_pair_of_digit_ne","ErdosProblems.Erdos251.primeGap0_not_eventually_periodic","ErdosProblems.Erdos251.rationalPrimeGapTail_has_positive_shift_not_eventually_small"],"boundary":"The representative assumes T : ℕ → ℚ, the DyadicTailRecurrence for the actual prime-gap digits, and for every N₀ an N ≥ N₀ whose adjacent h-shifts both lie strictly in (-1, 1) while the corresponding prime gaps differ. Under that cofinal small-mismatch supply it excludes eventual RatIntegral h-shifts; it does not prove the supply or actual smallness. rationalPrimeGapTail_has_positive_shift_not_eventually_small records the contrary rational-state obstruction for at least one positive shift. This is a conditional reduction only: no #251 irrationality, infinite rational-sum limit, universal producer, novelty, priority, significance, or external-review claim is made; #251 remains open. The actual-prime-gap specialization is distinct from coefficient_only_no_go and does not duplicate prime_gap_reformulation."},{"id":"coefficient_only_no_go","contribution_class":"coefficient-only barrier; novelty and significance unassessed","summary":"The exact finite carry identity and the two non-eventual-periodicity statements expose the coefficient-only barrier: irregularity and growth of a coefficient stream do not by themselves force irrationality, while the synthetic carry stream remains distinct from the actual prime-gap stream.","evidence_mode":"Lean kernel plus authored synthesis","comparator_disposition":"targeted","declarations":["ErdosProblems.Erdos251.carryPartialSum_natCast_eq","ErdosProblems.Erdos251.carryCoeff_natCast_not_eventually_periodic","ErdosProblems.Erdos251.primeGap0_not_eventually_periodic"],"boundary":"CoefficientOnlyNoGo checks the exact finite partial-sum identity and the two non-eventual-periodicity statements. It does not assert the infinite limit of the synthetic countermodel, identify its coefficient stream with actual prime gaps, establish a prime-gap tail bridge, prove #251 irrationality, or make novelty, priority, significance, or external-review claims."}]},{"problem":257,"paper":"paper/257/erdos-257-mersenne-support-subseries.tex","families":[{"id":"finite_period_noncollapse","contribution_class":"locally proved result; novelty unassessed","summary":"The multiplicative order of the base modulo a reduced finite-sum denominator is the support lcm, forcing denominator growth.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"A finite-support denominator theorem does not settle an infinite-support sum."},{"id":"boolean_mobius_carry","contribution_class":"locally proved exact equivalence plus conditional consequences; novelty unassessed","summary":"For p : ℤ and q > 0, a nonempty normalized positive-index support with erdosSupportSeries 2 A = p/q exists exactly when a quotient-only Boolean–Möbius carry certificate U exists. The certificate has U 0 = p, positive U, the square-root bound U(N) ≤ q(2√N + 4), q-divisible carry differences, and a Boolean Möbius transform; the converse reconstructs the support from that transform. This quotient-level equivalence is distinct from the shifted natural-tail state family.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","declarations":["Erdos249257.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry","Erdos249257.BooleanMobiusCarryCertificate","Erdos249257.binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit","Erdos249257.one_div_oddOrder_le_reciprocalMass_of_support_fraction","Erdos249257.exists_unbounded_shifted_odd_tail_nat_state_of_support_fraction","Erdos249257.supportCoeffZeroWindow_length_le_eps_logb_add"],"boundary":"The equivalence is stated at the existence level with q > 0, 0 ∉ A, and a positive support witness; it gives no contradiction, no universal Erdős #257 irrationality result, and no producer of cofinally nonintegral shifts for the actual recurrence. The generic tempered-orbit theorem and the reciprocal-mass, unbounded-tail, and zero-window results are subordinate consequences, not separate families; the zero-window constant depends on the support and numerator and is not uniform. This does not settle every infinite support, and no novelty, priority, significance, external-review, or unrestricted #257 endpoint claim is made."},{"id":"boolean_mobius_exact_row_dynamics","contribution_class":"conditional reduction; novelty unassessed","summary":"If CofinalPositiveHalfGreedySkips holds—cofinally many positive rational-greedy remainders below the next Mersenne weight—the skipped-core and upper-half Boolean-word construction supplies cofinally many exact local half rows, yielding (1 / 2 : ℝ) ∈ mersenneAchievementSet. The load-bearing representative is half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips; exact-row construction, endpoint-defect transition, forced rank-two membership, and Boolean-word filling are subordinate mechanisms.","evidence_mode":"Lean kernel","comparator_disposition":"targeted","declarations":["Erdos249257.CofinalPositiveHalfGreedySkips","Erdos249257.exactLocalMersenneHalfRow_of_positiveHalfGreedySkip","Erdos249257.cofinalExactLocalMersenneHalfRows_of_positiveHalfGreedySkips","Erdos249257.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips","Erdos249257.two_mem_of_exact_localMersenneQuotient","Erdos249257.localEndpointDefect_succ","Erdos249257.exists_upperHalfBooleanWord_of_skippedCore"],"boundary":"CofinalPositiveHalfGreedySkips is an unproved global supply premise: for every N there must be c ≥ max N 4 with a positive greedy remainder below mersenneWeightRat c. Finite exact rows, including one produced by a skipped core, do not establish cofinality. This is a conditional 1/2-membership/counterexample route, not an unconditional Erdős #257 solution; the rank-two, endpoint-defect, and upper-half Boolean-word declarations remain subordinate evidence, and no novelty, priority, significance, or external-review claim is made."},{"id":"half_membership_seam_classification","contribution_class":"locally proved exact endpoint classification; novelty unassessed","summary":"Membership of 1/2 in the Mersenne achievement set is exactly equivalent to unbounded false successor terminal bits, the unbounded upper-or-middle seam branch, cofinal terminal-false positions, unbounded skipped ranks along the integer seam, infinite greedy skipped support, and the absence of a last half-greedy skip. The load-bearing representative is half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse; the remaining equivalences are subordinate endpoint-socket evidence.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","declarations":["Erdos249257.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse","Erdos249257.half_mem_mersenneAchievementSet_iff_unboundedUpperOrMiddle","Erdos249257.unboundedTerminalFalse_iff_not_seamGreedyEventuallyRight","Erdos249257.cofinalTerminalFalse_of_unboundedTerminalFalse","Erdos249257.unboundedTerminalFalse_of_cofinalTerminalFalse","Erdos249257.unboundedTerminalFalse_iff_cofinalTerminalFalse","Erdos249257.half_mem_mersenneAchievementSet_iff_cofinalTerminalFalse","Erdos249257.half_mem_mersenneAchievementSet_iff_exists_unboundedSkippedRanksAlong","Erdos249257.unboundedTerminalFalse_iff_greedyMersenneSkippedSupport_infinite","Erdos249257.half_mem_mersenneAchievementSet_iff_no_lastHalfGreedySkip"],"boundary":"This exact endpoint-socket classification does not supply any unbounded terminal-false or upper-or-middle occurrences, cofinal supply, skipped-support producer, or half-membership witness; finite exact rows do not establish the required global supply. It is not an unconditional or universal Erdős #257 result, and no novelty, priority, significance, or external-review claim is made."},{"id":"shifted_odd_tail_state","contribution_class":"conditional reduction","summary":"A rational support fraction produces a positive natural shifted tail state satisfying the exact binary recurrence and modular doubling orbit modulo v.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"The state-extraction theorem does not exclude rational infinite supports or settle universal Erdős #257; the support and rational-value hypotheses remain explicit."},{"id":"composite_dilation_defect_identity","contribution_class":"unconditional progress","summary":"The composite support-dilation identity splits the new divisor count into the old coefficient, the distinguished divisor, and an exact finite defect; its bouquet specialization supplies a separate structural budget.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"The exact defect is not bounded for arbitrary supports and supplies no tail selector or irrationality theorem; the bouquet budget remains conditional on its structural hypotheses."},{"id":"known_irrational_supports","contribution_class":"Lean formalisation of existing results","summary":"Full, factorial, power-of-two, multiple, eventually periodic, residue-class, and odd supports are formalised; pairwise-coprime support is split out as a dedicated non-dilation representative.","evidence_mode":"Lean kernel plus Comparator for full and pairwise-coprime support","comparator_disposition":"targeted_full_support_representative","boundary":"Structured known families do not cover arbitrary infinite supports."},{"id":"finite_prime_weighted_support","contribution_class":"locally proved result; novelty unassessed","summary":"Finite prime-weighted mass gives fixed-base irrationality for infinite positive supports; binary weighted mass gives all-base irrationality for every infinite subset of a positive host, while weighted mass at another fixed base gives same-base hereditary irrationality.","evidence_mode":"Lean kernel; exact Comparator transport configured, external replay pending","comparator_disposition":"targeted_exact_weighted_and_hereditary_interfaces_pending_replay","boundary":"Each clause retains its exact finite-prime weighted-summability and positive-support hypotheses. Binary mass is required for the all-base hereditary clause; arbitrary infinite supports remain open."},{"id":"pairwise_coprime_support","contribution_class":"formalisation of an existing theorem","summary":"Infinite pairwise-coprime supports with summable reciprocal mass have irrational Mersenne support series by an adaptive CRT and weighted coefficient certificate argument.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted_pairwise_coprime_support","boundary":"This structured support theorem does not settle arbitrary infinite supports; its infinitude, pairwise-coprimality, and summable-reciprocal hypotheses remain explicit."},{"id":"periodic_nonnegative_weight_irrationality","contribution_class":"formalised here; novelty unassessed","summary":"An eventually-periodic nonnegative rational coefficient sequence with a positive value in its periodic region has an irrational Lambert–Mersenne series; denominator clearing reduces the rational sequence to a periodic Nat-weight certificate.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted_strongest_nonnegative_rational_weight","boundary":"This does not claim the broader mixed-sign periodic theorem; all coefficient, positivity, base, period, and threshold hypotheses remain explicit."},{"id":"orthogonal_petal_sunflower_reduction","contribution_class":"conditional reduction","summary":"A finite-core orthogonal-petal bouquet together with the exact uniform tail-selection predicate composes through the forced-carry certificate consumer to irrationality; the explicit alternating-core, squared-prime-petal witness shows the structural class is non-vacuous, but the selector remains open.","evidence_mode":"Lean kernel plus Comparator plus paper argument","comparator_disposition":"targeted_conditional_sunflower_reduction","boundary":"The bouquet and selector hypotheses remain explicit; the selector is not proved, so this is not an unconditional #257 support family."},{"id":"squarefree_support","contribution_class":"Lean structure plus paper and external theorem","summary":"Lean checks squarefree incidence and no-go statements; the irrationality conclusion uses an external analytic theorem in the paper.","evidence_mode":"Lean kernel plus paper argument plus cited theorem","comparator_disposition":"not_applicable_to_external_irrationality_input","boundary":"Comparator must not badge the paper-plus-external conclusion."},{"id":"achievement_set_geometry","contribution_class":"mixed prior-art geometry and locally proved metric result; novelty unassessed","summary":"Compactness, perfectness, total disconnectedness, nowhere density, and measure one for the unrestricted achievement set.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted_measure_theorem","boundary":"Geometry of the full set does not decide irrationality of every coded point."},{"id":"restricted_achievement_sets","contribution_class":"locally proved result; novelty unassessed","summary":"Support-restricted coding is injective and the associated achievement set has exact finite-complement measure or zero for infinite complement.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted_measure_dichotomy","boundary":"The measure dichotomy does not classify rational points."},{"id":"half_and_twenty_one_frontiers","contribution_class":"conditional reduction and no-go result","summary":"Exact greedy characterisations, finite-support exclusions, the odd 2-adic two-thirds-band bound, and two conditional rational-half countermodel frontiers isolate local constraints around the remaining 1/2 and 1/21 alternatives.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"Neither membership question is decided; the 2-adic bound is sharp locally, and the two countermodel routes require unconstructed terminal-scaled or cofinal-cylinder antecedents."},{"id":"separated_cut_algebraic_lambert_transcendence","contribution_class":"ordinary paper argument","summary":"For an infinite positive-integer host with cuts L_j < M_j, L_j tending to infinity and M_j-L_j tending to infinity, every host exponent at most L_j dividing L_j and every later exponent divisible by M_j, every infinite thinning with bounded positive integer weights has transcendental Lambert sum at every real algebraic base t > 1. This includes all divisibility chains and an explicit host of unbounded divisibility width and divergent reciprocal sum. The full number-field Subspace Theorem argument has two AI proof reviews; it is an ordinary proof with no Lean transcendence theorem, independent human review or asserted historical priority.","evidence_mode":"ordinary proof; two AI reviews, no independent human review","comparator_disposition":"not_applicable_without_exact_Lean_statement","declarations":[],"boundary":"The full support at 3/2 and arbitrary infinite supports at base two remain unresolved. Finite cut arithmetic is not a proof of transcendence."}]},{"problem":269,"paper":"paper/269/erdos-269-three-prime-running-lcm.tex","families":[{"id":"three_prime_lcm_cells","contribution_class":"locally proved result; novelty unassessed","summary":"Exact running-LCM product, logarithmic-cell constancy, coordinate jump ratios, and jump count.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted_running_lcm_identity","boundary":"Cell structure alone does not prove irrationality."},{"id":"two_prime_transcendence","contribution_class":"paper plus external theorem","summary":"Steve Fan posted the two-prime factorisation, reduction and transcendence first (erdosproblems.com/269, 26 June 2026, post-7218). This checkout records that ordinary deduction from Bugeaud–Laurent and does not claim priority or a Lean theorem.","evidence_mode":"paper argument plus cited theorem","comparator_disposition":"not_applicable_not_a_lean_declaration","boundary":"Comparator cannot certify the external analytic input."},{"id":"height_fibre_and_shell","contribution_class":"locally proved result; novelty unassessed","summary":"Finite height-fibre normal form and a quadratic smooth-shell multiplicity bound.","evidence_mode":"Lean kernel","declarations":["ErdosProblems.Erdos269.finiteSmoothKernelSum_groupedByHeight","ErdosProblems.Erdos269.smoothExponentShell_card_quadratic"],"comparator_disposition":"represented_by_three_prime_structure_target","boundary":"These fibre bounds are counting input, not an escape producer. The separate actual-series divisibility bridge is established."},{"id":"rank_two_kernel_no_go","contribution_class":"no-go result","summary":"The 2,3,5 kernel is not rank one and its smallest displayed minor equals -1/15.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"Failure of rank one does not itself imply irrationality."},{"id":"dyadic_block_alphabet","contribution_class":"locally proved result; novelty unassessed","summary":"The exact dyadic block alphabet is 2, 6, 10, and 30.","evidence_mode":"Lean kernel","declarations":["ErdosProblems.Erdos269.dyadicBlockBase235_cases"],"comparator_disposition":"represented_by_three_prime_structure_target","boundary":"The finite alphabet does not supply the needed carry escape."},{"id":"conditional_carry_escape","contribution_class":"conditional no-go consumer; novelty and significance unassessed","summary":"Under the denominator-dependent cofinal local-window residue-escape predicate, no positive reduced carry can satisfy the exact multiplier recurrence together with its short bound. The load-bearing consumer is no_positive_reducedCarry_of_cofinalLocalWindowEscape; an absorbed nonzero common-factor carry reduces exactly to that consumer. CofinalLocalWindowEscape, windowBase, windowForcing, leastPositiveResidue, and the absorbed-carry bridge are subordinate finite-window mechanism evidence.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","declarations":["ErdosProblems.Erdos269.CofinalLocalWindowEscape","ErdosProblems.Erdos269.no_positive_reducedCarry_of_cofinalLocalWindowEscape","ErdosProblems.Erdos269.windowBase","ErdosProblems.Erdos269.windowForcing","ErdosProblems.Erdos269.leastPositiveResidue","ErdosProblems.Erdos269.no_positive_absorbedCarry_of_cofinalLocalWindowEscape"],"boundary":"The representative assumes b,m : ℕ → ℕ, shortBound : ℕ → ℕ → ℕ, CofinalLocalWindowEscape b m shortBound, and for each positive B coprime to 30 a positive integer-valued d with d(n+1) = (b n : ℤ) d(n) − (B : ℤ) (m n : ℤ) and Int.natAbs (d n) ≤ shortBound B n; it then gives False. The absorbed-carry bridge additionally assumes smoothFactor ≠ 0, c n = smoothFactor * d n, and the corresponding exact absorbed recurrence, then cancels that factor to the same reduced consumer. The representative remains a conditional consumer for arbitrary words. For the actual three-prime word, the rationality-to-reduced-carry bridge is established separately; the source-specific cofinal escape producer is not proved. This is not a #269 endpoint or irrationality proof, and no actual-series identification, novelty, priority, significance, or external-review claim is made. It is one conditional window-carry family, distinct from the finite residue, rank-two, and weighted-phase observer families."},{"id":"weighted_phase_carry_observer","contribution_class":"locally proved result; novelty unassessed","summary":"An exact weighted-phase carry recurrence splits into a finite residue digit and an uncontrolled integral quotient coboundary; an explicit function-faithful finite-dimensional observer then forces finite realised span.","evidence_mode":"Lean kernel","comparator_disposition":"targeted","declarations":["ErdosProblems.Erdos269.carry_eq_residueDigit_add_coboundary","ErdosProblems.Erdos269.carryResidue_mem_interval","ErdosProblems.Erdos269.residueDigit_mem_interval","ErdosProblems.Erdos269.finite_realisedSpan_of_factorisation"],"boundary":"The recurrence supplies a finite residue coordinate but leaves an uncontrolled integral quotient coboundary; it proves neither a finite-state quotient nor a literal infinite realised span. Finite realised span requires an explicit factorisation through a finite-dimensional function-faithful observer, and scalar evaluation alone is insufficient. No irrationality conclusion follows from this observer alone. The actual rationality-to-positive-reduced-carry bridge is established separately; cofinal local-window escape remains open."},{"id":"three_prime_finite_search","contribution_class":"finite computation","summary":"A local-window checker searches 106666 denominator and start pairs and records small certificates.","evidence_mode":"external exact computation","comparator_disposition":"not_applicable_not_a_lean_theorem","boundary":"Finite search is not a cofinal statement."},{"id":"independent_clock_channel_relations","contribution_class":"ordinary paper argument","summary":"For positive rationally independent clock rates with no simultaneous positive crossings, and integer radices b_i >= 2, all rational relations among 1 and the channel sums S_i are multiples of (-1,b_1-1,...,b_d-1). The proof uses genuine late swaps with a common infinite suffix in one orbit closure, a fixed rational lattice, and density of the clock section. The explicit rates 1,sqrt(2),sqrt(3),sqrt(6) and radices 2,3,5,7 give rational span dimension four. This is an ordinary proof reviewed by AI agents, without Lean or independent human verification of the infinite theorem.","evidence_mode":"ordinary proof; two AI reviews, no independent human review","comparator_disposition":"not_applicable_without_exact_Lean_statement","declarations":[],"boundary":"No rational independence of the original reciprocal prime logarithms is assumed or proved; the repeated running-LCM endpoint remains open."},{"id":"selected_staircase_integral_height_boundary","contribution_class":"ordinary paper argument","summary":"The selected staircase matrix with ones on and below the diagonal and integer r>=2 above it has Smith factors 1,r-1,...,r-1. For the selected {2,3,5} kernel, any entrywise-integral diagonal clearing has determinant magnitude at least 4^(N-1), whereas the reduced rational determinant has numerator one up to sign. A rational single-target border has primitive polynomial plus or minus (tX-s), where s/t is the reduced Schur complement. These ordinary arithmetic arguments were reviewed by AI agents; they are not the existing Lean nonvanishing theorem and have no independent human review.","evidence_mode":"ordinary proof; two AI reviews, no independent human review","comparator_disposition":"not_applicable_without_exact_Lean_statement","declarations":[],"boundary":"The optimality concerns diagonal clearing and one rational target. It does not rule out multi-target, nonlinear, recurrence, or new approximation methods and does not prove running-LCM irrationality."}]},{"problem":1041,"paper":"paper/1041/erdos-1041-lemniscate-newton-flow.tex","families":[{"id":"newton_value_decay","contribution_class":"locally proved result and formalised calculus; novelty unassessed","summary":"Lean checks the local Newton value equation and the zero-derivative form of its exponentially scaled value; constancy along a real-time interval is an ordinary calculus consequence in the paper.","evidence_mode":"Lean kernel plus authored synthesis","declarations":["ErdosProblems.Erdos1041.newtonFlow_value_hasDerivAt","ErdosProblems.Erdos1041.newtonFlow_scaledValue_hasDerivAt_zero"],"comparator_disposition":"not_selected_compact_calculus_family","boundary":"The cited declarations are pointwise derivative statements. The real-time first integral requires the paper’s interval integration step, and value decay alone does not construct a global trajectory or give a short connecting curve."},{"id":"ray_separation","contribution_class":"locally proved result; novelty unassessed","summary":"Lean checks that distinct positive rays contradict an assumed exponential endpoint relation; the paper derives that relation for a finite real-time Newton trajectory by ordinary integration and endpoint continuity.","evidence_mode":"Lean kernel plus authored synthesis","declarations":["ErdosProblems.Erdos1041.samePositiveRay_of_real_exp_decay","ErdosProblems.Erdos1041.no_newtonConnection_of_not_samePositiveRay"],"comparator_disposition":"represented_by_translation_avoidance_targets","boundary":"The finite-trajectory conclusion retains continuity on the closed interval, differentiability and noncritical Newton evolution on its interior, and the ordinary endpoint passage. It supplies neither global trajectory existence nor geometric length control."},{"id":"translation_avoidance","contribution_class":"locally proved result; novelty unassessed","summary":"Collision translations are parameterised by real affine lines and an arbitrarily small common translation avoids finitely many of them.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"The finite-family avoidance theorem does not perform the global topology or metric gluing."},{"id":"root_retention","contribution_class":"locally proved result; novelty unassessed","summary":"A quantified small constant perturbation keeps every polynomial root inside the open unit disc.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"Root retention is one input to a still-incomplete route."},{"id":"published_proof_gap","contribution_class":"paper diagnosis","summary":"The paper identifies an invalid local saddle block in a claimed unrestricted proof.","evidence_mode":"authored source analysis","comparator_disposition":"not_applicable_not_a_lean_proposition","boundary":"The diagnosis does not prove that no repair exists."},{"id":"lemniscate_finite_search","contribution_class":"finite computation","summary":"A bounded search records finite geometric evidence.","evidence_mode":"external computation","comparator_disposition":"not_applicable_not_a_lean_theorem","boundary":"Finite evidence does not settle the universal problem."}]},{"problem":1049,"paper":"paper/1049/erdos-1049-rational-base-lambert.tex","families":[{"id":"scalar_content_no_go","contribution_class":"no-go result","summary":"Integer scalar content changes analytic error and exterior determinant by matching factors, yielding no margin by itself.","evidence_mode":"Lean kernel","comparator_disposition":"not_selected_scalar_content_scaling_family","boundary":"The conclusion is scoped to one normalisation strategy."},{"id":"endpoint_residues","contribution_class":"no-go result","summary":"Endpoint residues at 2 and 3 exclude a common multiplier under unit-endpoint hypotheses.","evidence_mode":"Lean kernel","comparator_disposition":"not_selected_endpoint_residue_and_common_multiplier_family","boundary":"Other local or cross-row arithmetic remains possible."},{"id":"four_jet_collision","contribution_class":"locally proved result and no-go result; novelty unassessed","summary":"Binary selectors collide in the four-jet signature at the stated rank and depth.","evidence_mode":"Lean kernel","comparator_disposition":"not_selected_finite_combinatorial_family","boundary":"A signed relation is not nonzero analytic data."},{"id":"coordinatewise_corridor_no_go","contribution_class":"no-go result","summary":"The coordinatewise corridor forces a power-versus-linear inequality and cannot occur at base 3/2.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"This excludes one proof architecture and proves no irrationality statement."},{"id":"rational_base_tail_recurrence","contribution_class":"locally proved result; novelty unassessed","summary":"The exact rational-base cleared-tail recurrence exposes exponential denominator-base forcing absent at integer bases.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","boundary":"The recurrence is not derived from a rationality contradiction."},{"id":"seven_halves_irrationality","contribution_class":"paper plus external theorem","summary":"Irrationality at 7/2 is obtained from an external analytic criterion whose elementary height hypothesis is checked in Lean.","evidence_mode":"paper argument plus cited theorem and Lean arithmetic","comparator_disposition":"not_applicable_to_external_irrationality_conclusion","boundary":"Comparator may check the height inequality, not the cited theorem application."},{"id":"height_and_pade_arithmetic","contribution_class":"locally proved sharp model-specific no-go; novelty unassessed","summary":"For the explicit rectangular two-function exponent model, the cleared gap is nonpositive throughout rho >= 0 and 1 + rho <= sigma, and the threshold is at most 1/2 - 1/pi^2 with equality exactly at rho = 0, sigma = 1. The gap sign and unique-zero mechanism support this sharp threshold no-go within the wider height-and-Padé arithmetic family.","evidence_mode":"Lean kernel plus Comparator","comparator_disposition":"targeted","declarations":["ErdosProblems.Erdos1049.hpClearedGap_nonpos","ErdosProblems.Erdos1049.hpClearedGap_eq_zero_iff","ErdosProblems.Erdos1049.rectangular_hp_threshold_le_classical","ErdosProblems.Erdos1049.rectangular_hp_threshold_eq_classical_iff"],"boundary":"This is a sharp no-go only for the explicit rectangular two-function exponent model under the stated real hypotheses; it constructs no approximating polynomials or remainders and is not a universal Padé or Hermite-Padé no-go. It proves no irrationality at 3/2 or any general rational-base endpoint, and the analytic remainder/nonvanishing input remains untreated. No novelty, priority, significance, external-review, or endpoint claim is made; Erdős #1049 remains open."},{"id":"calibrated_rational_hankel_countermodel","contribution_class":"ordinary paper argument","summary":"An explicit positive atomic moment family and integer formal series have all-rank Hankel order sum_{j<N} j^2 and leading coefficient (N!)^2(N+1)!/2^N, with the same fixed-base asymptotic shape including N^(-8 xi), while xi=G_{2/3}(2/3)=1. For n>=1 the exact reduced moment denominator is 3^(31n+62)(3^(n+1)-2^(n+1))^4; the determinant denominator has logarithmic size Theta(N^3), and a proposed rank-two row-product clearer leaves denominator625. These are ordinary proofs with independent exact finite checks and AI proof reviews, not a Lean proof of the infinite family or human peer review.","evidence_mode":"ordinary proof; two AI reviews, no independent human review","comparator_disposition":"not_applicable_without_exact_Lean_statement","declarations":[],"boundary":"These are not Zudilin's actual moments. They refute an inference from the shared signature alone; they do not settle F(3/2) or refute the attributed divisibility-based criterion."}]}],"receipt_contract":{"runtime_output":"artifacts/external-verification-receipt.json","binds":["repository commit","Lean and mathlib pins","Comparator, lean4export, and landrun revisions","observed Comparator, lean4export, and landrun binary digests","config digest","challenge import-closure digest","named theorem set and permitted axioms","Comparator exit status","exact negative statement-mismatch diagnostic","CI run identity and timestamp"],"public_wording":"Comparator-checked against a separately declared statement and axiom budget","forbidden_wording":"independently verified"},"projections":["formalization.yaml","verification/comparator.json","docs/external_verification_packet.json","docs/EXTERNAL_VERIFICATION.md","docs/reference/OUTREACH_EVIDENCE_CAPSULES.md"]},"remaining_open_propositions":[{"id":"remaining_open.erdos_68_irrationality","status":"open","open_target_claim":"erdos_68","statement":"Prove that S = ∑_{n≥2} 1/(n! − 1) is irrational.","paper_anchor":{"source":"paper/68/erdos-68-factorial-denominator-irrationality.tex","environment":"problem","title":"Erdős Problem~#68","line":52}},{"id":"remaining_open.erdos_249_irrationality","status":"open","open_target_claim":"erdos_249","statement":"Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator.","paper_anchor":{"source":"paper/archive/erdos249-257-main-paper.tex","environment":"problem","title":"Erd\\H{o}s \\#249","line":4431}},{"id":"remaining_open.erdos_251_irrationality","status":"open","open_target_claim":"erdos_251","statement":"Prove that Π = ∑_{n≥1} p_n/2ⁿ is irrational; the exact tail-shift equivalence and prime-gap constraints do not supply the required cofinal escape.","paper_anchor":{"source":"paper/251/erdos-251-prime-gap-dyadic-series.tex","environment":"section","title":"Introduction","line":47}},{"id":"remaining_open.unbounded_certificate_supply","status":"open","open_target_claim":"erdos_249","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","paper_anchor":{"source":"paper/archive/erdos249-257-main-paper.tex","environment":"problem","title":"unbounded certificate supply","line":4442}},{"id":"remaining_open.half_value_membership","status":"open","open_target_claim":"universal_257","statement":"Decide whether 1/2 lies in the Mersenne achievement set. Yes refutes universal #257; no closes only this route.","paper_anchor":{"source":"paper/archive/erdos249-257-main-paper.tex","environment":"problem","title":"the half-value counterexample route","line":4466}},{"id":"remaining_open.twenty_one_permanent_affine_supercapacity","status":"open","open_target_claim":"universal_257","statement":"Decide the exact 1/21 branch: equivalently, exclude the fatal/cofinite/aligned branch, contradict its eventual permanent affine-supercapacity recurrence, prove cofinally bounded scaled greedy remainders, or force an unbounded sequence of closed canonical quotient rows. A positive membership result would refute universal #257; no such producer is proved.","paper_anchor":{"source":"paper/257/erdos-257-mersenne-support-subseries.tex","environment":"problem","title":"membership of 1/21 in the Mersenne achievement set","line":1433}},{"id":"remaining_open.universal_257_all_infinite_supports","status":"open","open_target_claim":"universal_257","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","paper_anchor":{"source":"paper/archive/erdos249-257-main-paper.tex","environment":"problem","title":"universal Erd\\H{o}s \\#257","line":4457}},{"id":"remaining_open.erdos_269_three_prime_irrationality","status":"open","open_target_claim":"erdos_269","statement":"Determine irrationality of the running-LCM reciprocal sum for finite prime sets of size at least three. For {2,3,5}, the rationality-to-carry bridge is established and cofinal local-window escape remains open.","paper_anchor":{"source":"paper/269/erdos-269-three-prime-running-lcm.tex","environment":"section","title":"Two sums from running least common multiples","line":58}},{"id":"remaining_open.erdos_1041_lemniscate_connection","status":"open","open_target_claim":"erdos_1041","statement":"Independently review whether the 1958 curve-length wording matches the exact Formal Conjectures path-image-length statement, whose negation is now Lean-checked using ani's explicit degree-seven polynomial. The checked one-dimensional Hausdorff-measure bound settles that formal statement negatively; the reported small-parameter family is not formalised.","paper_anchor":{"source":"paper/1041/erdos-1041-lemniscate-newton-flow.tex","environment":"problem","title":"Erdős #1041","line":75}},{"id":"remaining_open.erdos_1049_irrationality","status":"open","open_target_claim":"erdos_1049","statement":"Determine whether F(t) = ∑_{n≥1} 1/(tⁿ − 1) is irrational for every rational t > 1. Irrationality of F((31/4)^r) for every positive integer r is a Lean-checked Zudilin specialization; the 3/2 approximant with remainder control and the universal assertion remain open.","paper_anchor":{"source":"paper/1049/erdos-1049-rational-base-lambert.tex","environment":"section","title":"Introduction","line":50}},{"id":"remaining_open.erdos_243_eventual_recurrence","status":"open","open_target_claim":"erdos_243","statement":"Prove the Erdős #243 assertion: if 1 ≤ a₁ < a₂ < ··· has aₙ/aₙ₋₁² → 1 and rational reciprocal sum, then aₙ = aₙ₋₁² − aₙ₋₁ + 1 eventually; the checked barrier does not settle the mixed-sign unbounded regime.","paper_anchor":{"source":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","environment":"problem","title":"Erdős Problem 243","line":59}},{"id":"remaining_open.erdos_269_reduced_tail_nonintegrality","status":"open","open_target_claim":"erdos_269","statement":"Prove that B·T_a ∉ ℤ for every B ≥ 1 coprime to 30 and every a ≥ 1.","paper_anchor":{"source":"paper/269/erdos-269-three-prime-running-lcm.tex","environment":"problem","title":"no reduced tail is an integer","line":609}},{"id":"remaining_open.erdos_269_cofinal_local_window_escape","status":"open","open_target_claim":"erdos_269","statement":"Prove that for every B ≥ 1 coprime to 30 and every a₀ ≥ 1 there are ℓ ≥ a₀ and h ≥ 1 with lpr_{W(ℓ,h)}(−B·F(ℓ,h)) > K(B, ℓ+h), where K is the defined long-paper cap.","paper_anchor":{"source":"paper/269/erdos-269-three-prime-running-lcm.tex","environment":"problem","title":"residue inequalities after every starting index","line":615}},{"id":"remaining_open.erdos_269_two_dimensional_representation","status":"open","open_target_claim":"erdos_269","statement":"Transcendence of D_{2,3,5}, which is irrational: represent it faithfully by a specified two-dimensional Mahler-type value theorem with every hypothesis verified, give a conditional theorem under explicit logarithmic nondegeneracy, or prove no such representation exists.","paper_anchor":{"source":"paper/269/erdos269-running-lcm-reasoning-surface.tex","environment":"problem","title":"an exact function representing the distinct-height sum","line":3580}},{"id":"remaining_open.erdos_243_overlap_height_growth","status":"open","open_target_claim":"erdos_243","statement":"Decide whether every nonterminal canonical orbit of the reduced-orbit problem has limsup_n (log M_n)/n > 0, equivalently 2^n ≤ M_n^K infinitely often for some K ≥ 1.","paper_anchor":{"source":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","environment":"problem","title":"growth of repeated denominator factors","line":1087}},{"id":"remaining_open.erdos_249_first_harmonic_anti_concentration","status":"open","open_target_claim":"erdos_249","statement":"Decide whether, for each h ≥ 1, some s ≥ 1 and 0 < η < 1 give, for every X₀, admissible X and L meeting the displayed size conditions for which the four first-harmonic bounds hold.","paper_anchor":{"source":"paper/249/erdos-249-binary-totient-series.tex","environment":"problem","title":"Four estimates for the first-harmonic sum","line":802}},{"id":"remaining_open.twenty_one_scaled_remainder_cofinal_return","status":"open","open_target_claim":"universal_257","statement":"Decide whether the scaled actual greedy remainder returns cofinally to one bounded interval: whether some B < ∞ has 2^N r_N ≤ B at arbitrarily large N.","paper_anchor":{"source":"paper/257/erdos-257-mersenne-support-subseries.tex","environment":"problem","title":"bounded returns of the scaled remainder","line":1449}},{"id":"remaining_open.twenty_one_actual_orbit_invariant","status":"open","open_target_claim":"universal_257","statement":"Decide whether a bounded-memory 2-adic or discrepancy invariant of the actual support forces a closed return or forbids permanent supercapacity, or prove no such invariant exists.","paper_anchor":{"source":"paper/257/erdos-257-mersenne-support-subseries.tex","environment":"problem","title":"arithmetic tests for the greedy sequence","line":1458}},{"id":"remaining_open.twenty_one_final_skip_exclusion","status":"open","open_target_claim":"universal_257","statement":"Decide whether the final-skip signatures prove |E − a_M| ≥ gap_M, contradicting 0 < a_M − E < gap_M, for E = ∑_{n≥1}(2^n − 1)^{−1} and the last-skip approximant a_M.","paper_anchor":{"source":"paper/257/erdos-257-mersenne-support-subseries.tex","environment":"problem","title":"final-skip Diophantine exclusion","line":1469}}],"claims":[{"id":"erdos_68","label":"Erdős #68: irrationality of S = ∑_{n≥2} 1/(n! − 1)","status":"open","statement":"Lean proves that S = ∑_{n≥2} 1/(n! − 1) is irrational if and only if, for infinitely many m, m does not divide ⌊m!·S_m⌋ + 1, where S_m is the prefix sum through n = m (irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses); equivalently, the factorial-gap carry differs from 1 for infinitely many m (irrational_factorialGapSeries_iff_cofinal_nonunit_carries). Lean also proves liminf log L_N/(N^{3/2} log N) ≥ 2√2/3 for the common denominator L_N = lcm_{2≤n≤N}(n! − 1) before cancellation (common_denominator_growth_liminf). The irrationality of S remains open: the checked reductions and finite obstructions do not supply the required cofinal non-unit carries.","remaining_open_proposition_ids":["remaining_open.erdos_68_irrationality"],"paper_label":null,"readme_headline":false,"declarations":[]},{"id":"erdos_269","label":"Erdős #269: irrationality of running-LCM reciprocal sums","status":"open","statement":"For every finite set P of at least two primes, the distinct-height sum D_P (each running LCM value of the P-smooth integers counted once) is irrational (res:distinct-height-all; ordinary proof checked by a second AI agent, no human review), as Erdős asserted without an argument in 1973; for |P| = 2 it follows from Hecke-Mahler transcendence, and D_{2,3,5} is Lean-checked (res:distinct-height-235; Comparator not yet run). Each single-prime sub-sum E_p of the catalogue sum is irrational (res:single-prime-subsums; same evidence class as D_P). The catalogue sum for three or more primes is unresolved in this release: for {2,3,5} the rationality-to-positive-reduced-carry bridge is checked, and the remaining endpoint is cofinal local-window escape, equivalently exclusion of the integral branch.","remaining_open_proposition_ids":["remaining_open.erdos_269_three_prime_irrationality","remaining_open.erdos_269_reduced_tail_nonintegrality","remaining_open.erdos_269_cofinal_local_window_escape","remaining_open.erdos_269_two_dimensional_representation"],"paper_label":null,"readme_headline":false,"declarations":[]},{"id":"erdos_251","label":"Erdős #251: irrationality of Π = ∑_{n≥1} p_n/2ⁿ","status":"open","statement":"Lean proves that Π = ∑_{n≥1} p_n/2ⁿ equals 2 plus the prime-gap dyadic series and that Π is irrational if and only if that gap series is (tsum_primeDyadicTerm_eq_two_add_primeGap_unconditional, irrational_tsum_primeDyadicTerm_iff_primeGap), with summability from the elementary bound p_n ≤ 1250(n+1)^4; the identity is the known summation by parts. Lean also proves that the consecutive prime gaps are unbounded and not eventually periodic (exists_primeGap0_gt, primeGap0_not_eventually_periodic). The irrationality of Π remains open: these facts do not supply the required cofinal escape for the actual prime gaps.","remaining_open_proposition_ids":["remaining_open.erdos_251_irrationality"],"paper_label":null,"readme_headline":false,"declarations":[]},{"id":"prime_gap_unboundedness_and_nonperiodicity","label":"Unbounded and non-eventually-periodic consecutive prime gaps","status":"unconditional progress","statement":"The consecutive-prime-gap sequence is unbounded and is not eventually periodic with any positive period. These coefficient facts do not by themselves imply irrationality of the prime-gap or prime-index dyadic series.","remaining_open_proposition_ids":["remaining_open.erdos_251_irrationality"],"paper_label":"res:gap-nonperiodic","readme_headline":false,"declarations":[{"name":"exists_primeGap0_gt","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","line":57},{"name":"primeGap0_not_eventually_periodic","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","line":1023}]},{"id":"prime_gap_irrationality_equivalence","label":"Exact prime-gap dyadic irrationality equivalence","status":"formalised here","statement":"Lean checks that the prime-value dyadic series is irrational exactly when the prime-gap dyadic series is. The named iff still takes a Summable hypothesis; this checkout discharges that hypothesis by the elementary bound p_n ≤ 1250(n+1)^4 (not the prime-number theorem) and the unconditional identity. Neither side is proved irrational. This formalises known summation-by-parts; the public attribution of that identity is the cited-only row prime_gap_identity_tao.","remaining_open_proposition_ids":["remaining_open.erdos_251_irrationality"],"paper_label":"res:irr-equivalence","readme_headline":false,"declarations":[{"name":"summable_primeDyadicTerm","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","line":379},{"name":"tsum_primeDyadicTerm_eq_two_add_primeGap_unconditional","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","line":427},{"name":"irrational_tsum_primeDyadicTerm_iff_primeGap","module":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","line":435}]},{"id":"prime_gap_identity_tao","label":"Tao's already-known prime-to-gap identity","status":"cited only","statement":"Tao posted the prime-to-gap identity by summation by parts (erdosproblems.com/251, 7 October 2025). That identity is already-known mathematics, not a theorem of this checkout, and is not the public lead; the supported lead is the coefficient-only countermodel and sparse rationalising perturbation.","declarations":[]},{"id":"erdos_1041","label":"Erdős #1041: short connections inside polynomial lemniscates","status":"formal statement refuted","statement":"Using ani's explicit monic degree-seven polynomial, Lean proves that every preconnected subset of its strict unit lemniscate containing two distinct roots has one-dimensional Hausdorff measure greater than 2. This refutes the exact Formal Conjectures path-image-length statement and its total-variation counterpart. The reported family is not formalised, and independent human review of correspondence with the 1958 wording is not recorded.","remaining_open_proposition_ids":["remaining_open.erdos_1041_lemniscate_connection"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"erdos1041_counterexample_hausdorff","module":"ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","line":283},{"name":"erdos1041_hausdorff_negation","module":"ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","line":407},{"name":"erdos1041_hausdorff_answer_false","module":"ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","line":449},{"name":"erdos1041_ani_degree_seven","module":"ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean","line":25}]},{"id":"newton_ray_separation_consumer","label":"Finite Newton-flow ray separation and perturbation consumer","status":"unconditional progress","statement":"An explicit exponential-decay Newton connection can occur only when its endpoint values lie on the same positive ray; conversely, every finite injective complex family admits an arbitrarily small common translation making all values nonzero and all positive-ray arguments distinct. These are finite perturbative and conditional connection statements.","remaining_open_proposition_ids":["remaining_open.erdos_1041_lemniscate_connection"],"paper_label":"res:ray","readme_headline":false,"declarations":[{"name":"exists_small_translation_separating_arguments","module":"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","line":197},{"name":"samePositiveRay_of_real_exp_decay","module":"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","line":325},{"name":"no_newtonConnection_of_not_samePositiveRay","module":"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","line":334}]},{"id":"eb_full_support","label":"Full-support Erdős–Borwein irrationality, every integer base b ≥ 2","status":"formalised here","statement":"For every integer b ≥ 2, the series ∑_{n≥1} 1/(bⁿ − 1) is irrational (Erdős, 1948).","paper_label":"res:full","readme_headline":true,"declarations":[{"name":"irrational_erdosSum_full_support","module":"Erdos249257/CertificateKernel.lean","line":8328}]},{"id":"eb_constant","label":"The Erdős–Borwein constant E is irrational","status":"formalised here","statement":"E = ∑_{n≥1} 1/(2ⁿ − 1) is irrational, as the base-2 corollary of the full-support theorem.","paper_label":"res:full","readme_headline":true,"declarations":[{"name":"irrational_erdosBorwein_series","module":"Erdos249257/CertificateKernel.lean","line":8335}]},{"id":"support_families","label":"Named infinite-support irrationality families toward #257","status":"formalised here","statement":"For every base b ≥ 2, ∑_{n∈A} 1/(bⁿ − 1) is irrational for factorial support, power-of-two support, multiples, pairwise-coprime supports with summable reciprocals, eventually-periodic supports, residue classes, and the odd numbers. Proved families, not the universal statement.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:support","readme_headline":true,"declarations":[{"name":"erdos257_family_factorial_instance","module":"Erdos249257/CertificateKernel.lean","line":6082},{"name":"erdos257_family_two_pow_instance","module":"Erdos249257/CertificateKernel.lean","line":6090},{"name":"irrational_erdosSupportSeries_pairwise_coprime","module":"Erdos249257/CertificateKernel.lean","line":10776}]},{"id":"channel_quotient_band_breakpoint","label":"Quotient-band channel breakpoint for factorial moments","status":"unconditional progress","statement":"For a finite integer coefficient family whose indices all lie in one quotient band k·d ≤ i_j < (k+1)·d, the factorial moment is exactly (d!)^k times the d-channel numerator. In the first band d ≤ i_j < 2d, channel cancellation forces the factorial moment to vanish; with all indices at least d and a nonzero moment, some index is therefore at least 2d.","remaining_open_proposition_ids":["remaining_open.erdos_68_irrationality"],"paper_label":"long68:res:bandbreakpoint","readme_headline":false,"declarations":[{"name":"factorialMoment_eq_factorial_pow_mul_channelNumerator_band","module":"ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean","line":71},{"name":"factorialMoment_eq_zero_of_channelNumerator_eq_zero_band","module":"ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean","line":91},{"name":"factorialMoment_eq_factorial_mul_channelNumerator_firstBand","module":"ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean","line":101},{"name":"exists_index_ge_two_mul_of_factorialMoment_ne_zero_of_channel_eq_zero","module":"ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean","line":130}]},{"id":"three_prime_finite_residue_contradiction","label":"Finite least-positive-residue contradiction for #269","status":"unconditional progress","statement":"For a positive modulus C, a positive integer state c bounded by K cannot be congruent to x modulo C when the canonical least-positive representative of x is greater than K; the zero residue class is represented by C. The integer carry form and exact residue classifier are checked in Lean.","remaining_open_proposition_ids":["remaining_open.erdos_269_three_prime_irrationality"],"paper_label":"res:consumer","readme_headline":false,"declarations":[{"name":"no_bounded_positive_state_of_residue_escape","module":"ErdosProblems/Erdos269/ResidueEscape.lean","line":76},{"name":"no_bounded_positive_int_state_of_leastPositiveResidue","module":"ErdosProblems/Erdos269/ResidueEscape.lean","line":110},{"name":"leastPositiveResidue_eq_natAbs_of_pos_le_modEq","module":"ErdosProblems/Erdos269/ResidueEscape.lean","line":138}]},{"id":"three_prime_cofinal_window_carry_extinction","label":"Conditional cofinal-window extinction of bounded carries for #269","status":"conditional reduction","statement":"If the denominator-dependent cofinal local-window residue escape holds, then for every positive multiplier B coprime to 30 there is no positive integral sequence satisfying the exact cleared recurrence d_{n+1}=b_n d_n−B m_n together with the corresponding short bound. The same contradiction applies to an explicitly absorbed nonzero common-factor carry after exact cancellation. For the actual {2,3,5} series, the rationality-to-positive-reduced-carry bridge is established separately; the cofinal escape producer remains open.","remaining_open_proposition_ids":["remaining_open.erdos_269_three_prime_irrationality"],"paper_label":"res:windowconsumer","readme_headline":false,"declarations":[{"name":"no_positive_reducedCarry_of_cofinalLocalWindowEscape","module":"ErdosProblems/Erdos269/RestrictedFloorSum.lean","line":645},{"name":"no_positive_absorbedCarry_of_cofinalLocalWindowEscape","module":"ErdosProblems/Erdos269/RestrictedFloorSum.lean","line":689}]},{"id":"three_prime_exact_modulus_pinning","label":"Exact-modulus pinning for an actual #269 window","status":"conditional reduction","statement":"If an actual finite {2,3,5} window has a least-positive forcing residue no larger than the exact carry cap, then the scaled normalized launch lies within that cap divided by the actual window modulus of an integer. Irrationality plus a separately supplied exact-modulus decay estimate therefore yields cofinal local-window escape. The decay estimate and irrationality input are not proved by this claim.","remaining_open_proposition_ids":["remaining_open.erdos_269_cofinal_local_window_escape"],"paper_label":"res:windowconsumer","readme_headline":false,"declarations":[{"name":"exact_modulus_pinning","module":"ErdosProblems/Erdos269/R12/ExactModulus.lean","line":45},{"name":"actual_exact_modulus_pinning","module":"ErdosProblems/Erdos269/R12/ExactModulus.lean","line":88}]},{"id":"three_prime_octic_window_equivalence","label":"Conditional o(8^a) equivalence band for #269","status":"conditional reduction","statement":"For a cap G which dominates the long-paper cap for every positive denominator and satisfies G(B,n)/8^n -> 0 for each positive B, cofinal local-window escape is equivalent to irrationality of the three-prime paper series. Both the dominance and little-o hypotheses are explicit; the theorem proves neither side unconditionally.","remaining_open_proposition_ids":["remaining_open.erdos_269_cofinal_local_window_escape","remaining_open.erdos_269_three_prime_irrationality"],"paper_label":"long269:res:actual-escape-endpoint","readme_headline":false,"declarations":[{"name":"octic_cap_is_exactly_beaten","module":"ErdosProblems/Erdos269/R12/OcticWindowBand.lean","line":19},{"name":"octic_window_band","module":"ErdosProblems/Erdos269/R12/OcticWindowBand.lean","line":77}]},{"id":"three_prime_finite_prefix_completion_boundary","label":"Finite #269 radix prefixes admit an altered rational completion","status":"unconditional progress","statement":"Every finite positive integral radix prefix with nonnegative integral digits extends to a positive rational orbit by changing the future to radix 2 and digit 1. In particular, the actual word through any finite horizon has such an abstract completion. This is a countermodel to finite-prefix-only inference and is not a rational model of the actual infinite smooth-number series.","remaining_open_proposition_ids":["remaining_open.erdos_269_three_prime_irrationality"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"finite_prefix_has_positive_rational_completion","module":"ErdosProblems/Erdos269/R12/PrefixExtension.lean","line":75},{"name":"actual_prefix_has_abstract_rational_completion","module":"ErdosProblems/Erdos269/R12/PrefixExtension.lean","line":105}]},{"id":"three_prime_bounded_certificate_safety","label":"Size-authenticated bounded certificate checks for #269","status":"unconditional progress","statement":"The bounded width and cumulative-array checkers expose their size hypotheses, are sound on every accessed entry, compose across adjacent ranges, and carry explicit regressions showing that default-zero array access, half-height cancellation, and finite coverage cannot certify an infinite/cofinal conclusion. No paper numerical table is executed or validated by these safety theorems alone.","remaining_open_proposition_ids":["remaining_open.erdos_269_cofinal_local_window_escape"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"widthCheckBoundedR12_sound","module":"ErdosProblems/Erdos269/CertificateSafetyR12.lean","line":14},{"name":"widthCheckBoundedR12_actual","module":"ErdosProblems/Erdos269/CertificateSafetyR12.lean","line":25},{"name":"widthCheckBoundedR12_append","module":"ErdosProblems/Erdos269/CertificateSafetyR12.lean","line":34},{"name":"cumulativeCheckBoundedR12_sound","module":"ErdosProblems/Erdos269/CertificateSafetyR12.lean","line":54},{"name":"width_shape_regressions_R12","module":"ErdosProblems/Erdos269/CertificateSafetyR12.lean","line":64},{"name":"factor_two_cannot_be_cancelled_R12","module":"ErdosProblems/Erdos269/CertificateSafetyR12.lean","line":73},{"name":"finite_coverage_not_cofinal_R12","module":"ErdosProblems/Erdos269/CertificateSafetyR12.lean","line":84}]},{"id":"periodic_nonnegative_weight_irrationality","label":"Eventually-periodic nonnegative weighted Lambert irrationality","status":"formalised here","statement":"For every integer base b ≥ 2, an eventually-periodic nonnegative rational coefficient sequence with a positive value at a positive index in its periodic region has irrational series ∑ γ(a)/(b^a − 1). This includes eventually-periodic natural-number weights and indicator supports. Luca–Tachiya prove a broader periodic rational-coefficient theorem by different methods; no priority claim is made here.","paper_label":"res:periodicnonnegative","readme_headline":false,"declarations":[{"name":"irrational_weightedErdosSeries_periodic","module":"Erdos249257/CertificateKernel.lean","line":12698},{"name":"irrational_weightedErdosSeries_eventuallyPeriodic","module":"Erdos249257/CertificateKernel.lean","line":12725},{"name":"irrational_ratWeightSeries_eventuallyPeriodic","module":"Erdos249257/CertificateKernel.lean","line":12811}]},{"id":"periodic_signed_weight_dichotomy","label":"Periodic signed-weight irrational-or-terminating dichotomy","status":"proved here","statement":"For every integer base b ≥ 2 and every periodic integer weight w, the signed series ∑ w(a)/(b^a − 1) is irrational or some power of b times its value is an integer. Excluding that terminating alternative implies irrationality. The terminating alternative is not excluded in general, so the full mixed-sign periodic-coefficient theorem is not claimed.","paper_label":"res:periodicsigned","readme_headline":false,"declarations":[{"name":"irrational_or_bpow_mul_eq_intCast_intWeightedErdosSeries_periodic","module":"Erdos249257/CertificateKernel.lean","line":14175},{"name":"irrational_intWeightedErdosSeries_periodic_of_bpow_mul_ne_intCast","module":"Erdos249257/CertificateKernel.lean","line":14200}]},{"id":"period_four_signed_residue_obstruction","label":"Period-four signed residue-selection obstruction","status":"proved here","statement":"For the period-four signed weight 1,0,−1,0, the associated signed divisor coefficient vanishes for every n congruent to 3 modulo 4. A whole arithmetic progression can therefore carry zero coefficient, so residue-blind selection cannot force a protected nonzero residue for every periodic signed weight.","paper_label":"res:periodfourobstruction","readme_headline":false,"declarations":[{"name":"intWeightedCoeff_periodFourSignWeight_eq_zero_of_mod_four_eq_three","module":"Erdos249257/CertificateKernel.lean","line":14226}]},{"id":"orthogonal_petal_sunflower_reduction","label":"Conditional orthogonal-petal sunflower reduction","status":"conditional reduction","statement":"For a finite-core orthogonal-petal bouquet at base 2, the exact uniform tail-selection predicate implies a forced-carry certificate supply and hence irrationality of the associated support series. Lean formalises the carry conversion and an explicit infinite alternating-core, squared-prime-petal bouquet whose elements have three prime factors with multiplicity. The tail-selection predicate itself is not proved, so this does not add an unconditional #257 support family.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:sunflower","readme_headline":false,"declarations":[{"name":"sunflowerForcedCarrySupply_of_orthogonalPetalBouquet","module":"Erdos249257/SupportSunflowerDichotomy.lean","line":531},{"name":"irrational_erdosSupportSeries_of_orthogonalPetalBouquet","module":"Erdos249257/SupportSunflowerDichotomy.lean","line":540},{"name":"alternatingPrimeSquareSupport_cardFactors_eq_three","module":"Erdos249257/SupportSunflowerDichotomy.lean","line":343}]},{"id":"composite_dilation_defect_identity","label":"Composite dilation defect identity","status":"proved here","statement":"For every positive support element a and positive x, the divisor-count coefficient at a·x is the coefficient at x, plus the distinguished new divisor when a does not divide x, plus an exact finite composite-dilation defect. For a finite-core orthogonal-petal bouquet, the ray defect is bounded by the finite exceptional-frame cardinality plus the petal-support divisor count. This local identity supplies no correlation estimate, tail selector, or irrationality theorem.","paper_label":"res:compositedefect","readme_headline":false,"declarations":[{"name":"supportCoeff_mul_eq_add_defect","module":"Erdos249257/CompositeDilationDefect.lean","line":30},{"name":"compositeDilationDefect_le_exceptional_add_petalCoeff","module":"Erdos249257/CompositeDilationDefect.lean","line":151},{"name":"compositeDilationDefect_two_six_fixture","module":"Erdos249257/CompositeDilationDefect.lean","line":218}]},{"id":"mixed_prime_power_layer_identity","label":"Mixed prime-power support-layer identity","status":"proved here","statement":"Exact p-adic and q-adic coefficient-layer operators commute. For distinct primes and a residual argument coprime to their product, the two-layer operator on the support coefficient equals the iterated exact-prime-power pullback. The checked singleton {12} fixture has mixed (2²,3) layer value one at residual argument one. This is finite coefficient algebra only, with no decimation transport, bounded-Ω endpoint, or irrationality theorem.","paper_label":"res:mixedlayer","readme_headline":false,"declarations":[{"name":"primePowerLayer_comm","module":"Erdos249257/MaximalOmegaLayer.lean","line":29},{"name":"mixedPrimePowerLayerTwo_supportCoeffInt","module":"Erdos249257/MaximalOmegaLayer.lean","line":39},{"name":"mixedPrimePowerLayerTwo_twelve_fixture","module":"Erdos249257/MaximalOmegaLayer.lean","line":65}]},{"id":"dyadic_prefix_block_safety_interfaces","label":"Dyadic-prefix, block-safety, and primitive-overshoot interfaces","status":"proved here","statement":"Inside a hypothetical dyadic-rational support value, a reduced finite positive prefix transports exactly to a residual rational support value and wrap-ratio certificate. For the half-greedy orbit, reduced and raw margins, three-depth take-run excess, primitive-lattice cap violation, support-coefficient tail gaps, and gcd-scaled predecessor overshoot are connected by exact identities and equivalences; cofinal supplies of the named witnesses imply that 1/2 belongs to the Mersenne achievement set. Checked fixtures show that safe brackets with numerator monotonicity and a factor-11 predecessor rule are not inductive. No cofinal witness supply is proved, so neither half-membership nor universal Erdős #257 follows.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"readme_headline":false,"declarations":[{"name":"finiteFragment_dyadicPrefixCompression","module":"Erdos249257/DyadicPrefixCompression.lean","line":1636},{"name":"halfGreedyThreeDepthTakeRunExcessBound_iff_skippedExcessBound","module":"Erdos249257/DyadicPrefixCompression.lean","line":2176},{"name":"greedyHalf_badSkipCapViolation_iff_primitiveLattice","module":"Erdos249257/DyadicPrefixCompression.lean","line":2188},{"name":"halfGreedyPrefixSupportTailGap_eq_excess_div_denominator","module":"Erdos249257/DyadicPrefixCompression.lean","line":2281},{"name":"greedyHalf_badSkipCapViolation_iff_supportCoeffTailGap","module":"Erdos249257/DyadicPrefixCompression.lean","line":2319},{"name":"rawSafe_but_not_gcdScaled_fixture","module":"Erdos249257/DyadicPrefixCompression.lean","line":3450},{"name":"factorEleven_predecessorOvershoot_not_inductive_fixture","module":"Erdos249257/DyadicPrefixCompression.lean","line":3472},{"name":"unsafeSkipGcdOvershootSupply_iff_primitive","module":"Erdos249257/DyadicPrefixCompression.lean","line":4521},{"name":"half_mem_mersenneAchievementSet_of_unsafeSkipPrimitiveOvershootSupply","module":"Erdos249257/DyadicPrefixCompression.lean","line":4562}]},{"id":"half_trapping_return_carry_interface","label":"Half-trapping reverse-carry interface","status":"conditional reduction","statement":"Scalar-relation-invariant rational-linear channels have no nontrivial square determinant at any rank. Separately, a common coefficient word with a 1/0 seam and a common following overlap forces the terminal reverse-carry difference to be a power of two times an odd integer; a common nonnegative real carry bound yields the corresponding dyadic spacing inequality. The module neither supplies reverse-carry presentations nor a predecessor overlap, so it is not a greedy separation or irrationality theorem.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:halftrapping","readme_headline":false,"declarations":[{"name":"relationInvariantLinearChannels_det_eq_zero","module":"Erdos249257/HalfTrappingReturnCarry.lean","line":42},{"name":"overlappingReverseCarryWords_carryDifference_eq_twoPow_mul_odd","module":"Erdos249257/HalfTrappingReturnCarry.lean","line":124},{"name":"overlappingReverseCarryWords_twoPow_le_realBound","module":"Erdos249257/HalfTrappingReturnCarry.lean","line":191}]},{"id":"half_carry_reachability_compactness","label":"Half-carry reachability and compactness","status":"conditional reduction","statement":"At an unresolved even seam, the scalar transition reaches exactly every integer terminal carry other than the one value 2*δ-c. If canonical seam coordinates are supplied at every sufficiently large even depth, then admissible finite half-carry words occur cofinally; compactness yields an infinite support A with erdosSupportSeries 2 A = 1/2. The canonical seam-supply hypothesis is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:halfcarryreachability","readme_headline":false,"declarations":[{"name":"evenSeamReachable_iff","module":"Erdos249257/HalfCarryReachability.lean","line":235},{"name":"cofinalAdmissibility_of_canonicalEvenSeamSupply","module":"Erdos249257/HalfCarryReachability.lean","line":1028},{"name":"exists_infinite_support_half_of_canonicalEvenSeamSupply","module":"Erdos249257/HalfCarryReachability.lean","line":1056}]},{"id":"cofinal_strip_return_closure","label":"Cofinal square-root-strip return closure","status":"conditional reduction","statement":"The binary-scaled uncentred greedy half carry is antitone, and the square-root strip envelope divided by 2^N tends to zero. Therefore cofinal returns of the canonical greedy carry to that strip force the scaled carry to tend to zero and yield membership of 1/2 in the Mersenne achievement set. The cofinal return hypothesis is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:cofinalstripreturn","readme_headline":false,"declarations":[{"name":"greedy_integerHalfCarry_scaled_tendsto_zero_of_cofinalStripReturn","module":"Erdos249257/CofinalStripReturn.lean","line":84},{"name":"greedy_half_infinite_of_cofinalStripReturn","module":"Erdos249257/CofinalStripReturn.lean","line":130},{"name":"half_mem_mersenneAchievementSet_of_cofinalStripReturn","module":"Erdos249257/CofinalStripReturn.lean","line":157}]},{"id":"terminal_only_cofinal_closure","label":"Terminal-only cofinal square-root-strip closure","status":"conditional reduction","statement":"A normalized finite word with terminal half carry in the square-root strip has support-series value within (4M+12)/2^M of 1/2. Thus cofinally many such terminal-only witnesses, with neither coherent histories nor earlier-row bounds, force 1/2 into the Mersenne achievement set and yield an infinite support with support-series value 1/2. The cofinal terminal-witness supply is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:terminalonlycofinal","readme_headline":false,"declarations":[{"name":"dist_half_erdosSupportSeries_wordSupport_le_of_terminalOnlyWitness","module":"Erdos249257/TerminalOnlyCofinal.lean","line":68},{"name":"half_mem_mersenneAchievementSet_of_cofinalTerminalOnlyStrip","module":"Erdos249257/TerminalOnlyCofinal.lean","line":134},{"name":"exists_infinite_support_half_of_cofinalTerminalOnlyStrip","module":"Erdos249257/TerminalOnlyCofinal.lean","line":192}]},{"id":"terminal_scaled_vanishing_closure","label":"Scaled terminal-vanishing closure","status":"conditional reduction","statement":"A normalized sequence of finite terminal words whose depths tend to infinity and whose terminal carries divided by the corresponding powers of 2 tend to zero has support-series values converging to 1/2. Closedness of the Mersenne achievement set then gives membership of 1/2 and an infinite support with that value. The prior cofinal square-root-strip condition maps into this weaker socket, but no scaled-vanishing sequence is constructed.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:terminalscaledvanishing","readme_headline":false,"declarations":[{"name":"dist_half_erdosSupportSeries_wordSupport_le_carry_add_tail","module":"Erdos249257/TerminalOnlyScaledVanishing.lean","line":24},{"name":"half_mem_mersenneAchievementSet_of_terminalScaledVanishing","module":"Erdos249257/TerminalOnlyScaledVanishing.lean","line":165},{"name":"exists_infinite_support_half_of_terminalScaledVanishing","module":"Erdos249257/TerminalOnlyScaledVanishing.lean","line":221}]},{"id":"half_cylinder_finite_shadow","label":"Finite shadows of the half cylinder","status":"conditional reduction","statement":"For an admissible depth-N half-carry word with support A, terminal carry K_N, and future tail T_N, the residual is exactly 1/2 - X_A = (K_N - T_N)/2^N. At a first even seam the sole missing carry is H = 2 delta - c, so the integer strip [1,B] is fully reachable exactly when H lies outside it. The rank-three fixture A = {2,3} has residual 1/42 < 1/32 but its depth-six strip [1,8] contains H = 3, so raw dyadic safety does not force seam escape. The sole conditional membership input is the unproved governed frozen-margin producer, which requires future coefficient rows to cover the centred carry within J <= k+1 after every skipped rank k+1.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:halfcylindershadow","readme_headline":false,"declarations":[{"name":"halfStripAdmissible_residual_eq","module":"Erdos249257/HalfCylinderFiniteShadow.lean","line":55},{"name":"evenSeam_fullIntegerStrip_iff_escape","module":"Erdos249257/HalfCylinderFiniteShadow.lean","line":824},{"name":"half_mem_mersenneAchievementSet_of_governedFrozenMarginProducer","module":"Erdos249257/HalfCylinderFiniteShadow.lean","line":1275}]},{"id":"fixed_coefficient_rewind_contraction","label":"Fixed-coefficient rewind contraction","status":"conditional reduction","statement":"For coefficient c, the canonical inverse parent is T_c(k) = ceil((k+c)/2). For a length-L coefficient list with accumulated offset O, the iterated rewind is R(k) = floor((k+O)/2^L), with phase theta = (1+O) mod 2^L. If 1 <= B <= 2^L, all targets in [1,B] have one ancestor exactly when theta + B - 1 < 2^L; otherwise the endpoint ancestors differ by one and every target maps to one of that adjacent pair. The width bound does not exclude the seam-pair alternative or prove half-membership.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:fixedcoeffrewind","readme_headline":false,"declarations":[{"name":"rewind_eq_closedForm","module":"Erdos249257/HalfCarryCeilParentContraction.lean","line":49},{"name":"rewind_endpoint_eq_iff_phase_fit","module":"Erdos249257/HalfCarryCeilParentContraction.lean","line":99},{"name":"rewind_singleton_or_isSeamPair","module":"Erdos249257/HalfCarryCeilParentContraction.lean","line":180}]},{"id":"selected_half_carry_window_compactness","label":"Selected half-carry windows","status":"conditional reduction","statement":"A selected half-carry window at depth N and radius R assigns every k in [1,R] an admissible Boolean word of depth N whose terminal carry is k. If such windows exist at every sufficiently large depth with R >= 1, their k = 1 representatives give cofinal finite admissibility; compactness then yields an infinite support A with erdosSupportSeries 2 A = 1/2. The unproved object-level input is a coherent one-row history: either all representatives agree on membership of every divisor of N+1, making their next coefficient common, or the adjacent outcome realizes the canonical one-hole seam. The module supplies no such cofinal history.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:selectedhalfwindow","readme_headline":false,"declarations":[{"name":"SelectedHalfWindow.HistoryAlternative","module":"Erdos249257/HalfCarrySelectedWindow.lean","line":236},{"name":"cofinalAdmissibility_of_selectedHalfWindows","module":"Erdos249257/HalfCarrySelectedWindow.lean","line":831},{"name":"exists_infinite_support_half_of_selectedHalfWindows","module":"Erdos249257/HalfCarrySelectedWindow.lean","line":864}]},{"id":"realized_rewind_phase_identity","label":"Realized rewind phase identity","status":"conditional reduction","statement":"For a realized carry-one fixed-coefficient history, list the suffix bits newest row first and let X be their binary numeral, so the newest bit has weight 2^0. If D = 2^L, the rewind phase is exactly theta = D - 1 - X. The interval [1,B] rewinds to one ancestor exactly when B - 1 <= X; equivalently, for the frozen-prefix carry Z = 1 + X, exactly when B <= Z. No theorem proves either coverage inequality at all sufficiently large depths.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:rewindphase","readme_headline":false,"declarations":[{"name":"phase_eq_denom_sub_one_sub_suffixNumeral","module":"Erdos249257/HalfCarryRewindPhase.lean","line":39},{"name":"rewind_interval_singleton_iff_suffixNumeral_covers","module":"Erdos249257/HalfCarryRewindPhase.lean","line":65},{"name":"rewind_supportCoeffHistory_singleton_iff_prefixCarry_covers","module":"Erdos249257/HalfCarryRewindPhase.lean","line":352}]},{"id":"localized_protected_seam_compactness","label":"Localized protected seam compactness","status":"conditional reduction","statement":"At depth M, realized localized seam means only the forward implication: every seam-reachable carry k in [3,27] is the terminal carry of an admissible finite word. The seam's unique hole is H = 2 delta - c, so choosing k = 4 when H = 3 and k = 3 otherwise supplies one admissible word. If such realizations occur cofinally, then cofinal admissibility and an infinite support A with erdosSupportSeries 2 A = 1/2 follow. The cofinal realization supply is not proved, so Erdős #257 remains open.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:protectedseam","readme_headline":false,"declarations":[{"name":"ProtectedEvenSeamRealizedAt","module":"Erdos249257/HalfCarryProtectedSeamConsumer.lean","line":18},{"name":"cofinalAdmissibility_of_cofinalProtectedEvenSeamRealization","module":"Erdos249257/HalfCarryProtectedSeamConsumer.lean","line":31},{"name":"exists_infinite_support_half_of_cofinalProtectedEvenSeamRealization","module":"Erdos249257/HalfCarryProtectedSeamConsumer.lean","line":53}]},{"id":"mixed_cofinal_half_carry_fan_in","label":"Mixed cofinal half-carry fan-in","status":"conditional reduction","statement":"For every N, suppose there is M >= N carrying either a selected protected half-carry window or a localized realized one-hole seam. The window branch chooses carry 1; the seam branch chooses 4 when its unique hole is 3 and 3 otherwise. Either branch supplies one admissible finite word, and the branches may alternate with N. Hence cofinal admissibility and, by compactness, an infinite support A with erdosSupportSeries 2 A = 1/2 follow. This mixed cofinal hypothesis is not proved, so Erdős #257 remains open.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:cofinalwindowseam","readme_headline":false,"declarations":[{"name":"HalfCarryCofinalSelectedWindowOrProtectedSeam","module":"Erdos249257/HalfCarryCofinalWindowOrSeamConsumer.lean","line":25},{"name":"cofinalAdmissibility_of_cofinalSelectedWindowOrProtectedSeam","module":"Erdos249257/HalfCarryCofinalWindowOrSeamConsumer.lean","line":32},{"name":"exists_infinite_support_half_of_cofinalSelectedWindowOrProtectedSeam","module":"Erdos249257/HalfCarryCofinalWindowOrSeamConsumer.lean","line":58}]},{"id":"integer_greedy_first_wrap_reduction","label":"Integer-greedy first-wrap reduction","status":"conditional reduction","statement":"For every s >= 3, let C_s = 2^(2s-1) - 2^s, let B_s = 2 floor(sqrt(2s)) + 4, and let r_s be the remainder obtained by greedily subtracting each descending truncated seam weight from C_s whenever possible. Each weight exceeds its complete tail by at least 2^(s+1) > B_s. Hence a positive subset-sum defect at most B_s exists exactly when 1 <= r_s <= B_s. The downstream skipped-seam escape hypothesis requires B_s < r_s at every genuine skipped rank. That hypothesis is not proved, so Erdős #257 remains open.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:integergreedy","readme_headline":false,"declarations":[{"name":"seamWeights_gapDominates","module":"Erdos249257/HalfCylinderIntegerGreedy.lean","line":420},{"name":"seam_smallPositiveDefect_iff_integerGreedyRemainder","module":"Erdos249257/HalfCylinderIntegerGreedy.lean","line":654},{"name":"HalfGreedySkippedSeamEscape","module":"Erdos249257/HalfCylinderFullShellSeamBridge.lean","line":687},{"name":"quarterBandEscape_iff_bound_lt_nextRemainder","module":"Erdos249257/HalfCylinderIntegerGreedy.lean","line":1528}]},{"id":"concrete_seam_adapter","label":"Concrete finite seam adapter","status":"conditional reduction","statement":"Concrete finite seam words realize the abstract adjacent-cut recurrence, and the next seam word agrees with the integer-greedy choice. This implements the finite recurrence but does not establish the unbounded remainder estimate.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:concreteseam","readme_headline":false,"declarations":[{"name":"seamAdjacentCut_newCapacity","module":"Erdos249257/HalfCylinderConcreteSeamAdapter.lean","line":451},{"name":"seamCutNextWord_eq_greedyWord_succ","module":"Erdos249257/HalfCylinderConcreteSeamAdapter.lean","line":667}]},{"id":"fatal_gap_right_tail_classification","label":"Fatal-gap and eventual-right seam classification","status":"proved here","statement":"The concrete seam is eventually right when some S >= 5 makes every later seam word append true. A finite prefix u through rank d is a fatal half gap when V(u) + T_(d+1) < 1/2 < V(u) + w_(d+1), which makes every continuation miss 1/2. Eventual rightness is equivalent to the existence of such a gap, to nonmembership of 1/2 in the Mersenne achievement set, and to a final skipped exponent in the real greedy orbit. This exact classification does not decide which branch holds.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:fatalright","readme_headline":false,"declarations":[{"name":"seamGreedyEventuallyRight_iff_existsFatalHalfGap","module":"Erdos249257/HalfCylinderFatalGapRightTail.lean","line":781},{"name":"seamGreedyEventuallyRight_iff_half_not_mem","module":"Erdos249257/HalfCylinderFatalGapRightTail.lean","line":787},{"name":"seamGreedyEventuallyRight_iff_exists_isLastHalfGreedySkip","module":"Erdos249257/HalfCylinderFatalGapRightTail.lean","line":802}]},{"id":"twenty_one_quotient_greedy_frontier","label":"Canonical quotient-greedy and scaled-remainder frontier at 1/21","status":"unconditional progress","statement":"Membership of 1/21 in the Mersenne achievement set is equivalent both to excluding one explicit fatal/cofinite/aligned branch and to cofinally bounded scaled actual greedy remainders. Every closed Boolean quotient row is the deterministic quotient-greedy row; any unbounded sequence of canonical rows with remainder at most 2^R gives membership. On the complementary fatal branch the remainder is eventually strictly supercapacity, the boundary rank is always selected, and support and remainder follow one exact affine recurrence. A final skipped exponent also forces a structured one-sided approximation to the Erdős–Borwein constant with exact denominator-order constraints. These results isolate permanent affine supercapacity as the surviving late regime; they do not contradict it or prove 1/21 membership.","remaining_open_proposition_ids":["remaining_open.twenty_one_permanent_affine_supercapacity","remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:one-over-twenty-one-frontier","readme_headline":false,"declarations":[{"name":"twentyOneEvenQuotientGreedySupport","module":"Erdos249257/TwentyOneQuotientGreedy.lean","line":32},{"name":"twentyOneEvenQuotientGreedyRemainder","module":"Erdos249257/TwentyOneQuotientGreedy.lean","line":39},{"name":"twentyOneGreedyDefect","module":"Erdos249257/BooleanMobiusCarry.lean","line":1721},{"name":"one_div_twenty_one_mem_iff_scaledGreedyRemainderCofinallyBounded","module":"Erdos249257/BooleanMobiusCarry.lean","line":2716},{"name":"one_div_twenty_one_mem_mersenneAchievementSet_of_defect_le_one_cofinally","module":"Erdos249257/BooleanMobiusCarry.lean","line":2843},{"name":"one_div_twenty_one_mem_iff_not_fatalAlignedBranch","module":"Erdos249257/TwentyOneQuotientGreedy.lean","line":3507},{"name":"lastTwentyOneSkip_oddDoublingOrder_eq_lcm","module":"Erdos249257/TwentyOneQuotientGreedy.lean","line":3569},{"name":"lastTwentyOneSkip_erdosBorwein_fatalInterval","module":"Erdos249257/TwentyOneQuotientGreedy.lean","line":3735},{"name":"twentyOneCofinalEvenQuotientGreedyDecay_of_closedRows","module":"Erdos249257/TwentyOneQuotientGreedy.lean","line":5554},{"name":"twentyOneFatalAlignedBranch_eventually_affine_supercapacity","module":"Erdos249257/TwentyOneQuotientGreedy.lean","line":5658}]},{"id":"final_middle_cell_escape","label":"Upper and first middle producers cannot feed an all-right tail","status":"proved here","statement":"At a putative last false rank D ≥ 13, the upper producer is impossible unconditionally. On the middle branch, let u be the finite support below D and B_D = u union {n : n > D} the lazy support that skips D and selects every later rank. Its value is V(u) + T_D < 1/2, while the exact carry swap sends transition carry C_D to centred carry C_D + 3 at row 2D+1. Thus C_D = -3 gives centred carry zero; the next coefficient rows have at least one and two forced divisors and make the carry negative by row 2D+3, contradicting finite-level nonnegativity below 1/2. This removes the upper branch and one middle cell, not the right-tail branch or the remaining cells -2 and -1.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:finalskipexclusions","readme_headline":false,"declarations":[{"name":"upperProducer_not_last","module":"Erdos249257/HalfCylinderLastProducerContradiction.lean","line":299},{"name":"middleProducer_neg_three_not_last","module":"Erdos249257/HalfCylinderLastProducerContradiction.lean","line":364},{"name":"finalMiddleCell_neg_three_not_last","module":"Erdos249257/HalfCylinderFinalMiddleCellEscape.lean","line":587}]},{"id":"final_middle_neg_two_phase_sieve","label":"Rank-26 phase sieve for the final middle -2 cell","status":"conditional reduction","statement":"For every seam row s ≥ 27, the greedy decisions through rank 26 select exactly ranks 2, 3, 6, 7, 14, 20, 21, and 26. Consequently, in a hypothetical final middle -2 cell at rank D ≥ 27 followed by an all-right suffix, D mod 21 lies in {11, 14, 17, 20}, D mod 10 is not 8, the mod-10 class 7 cannot pair with the mod-21 class 11, D mod 13 is not 11, and the mod-13 class 10 cannot pair with the mod-21 class 11. Exactly 412 of the 2730 joint residue classes survive. This is a finite necessary condition, not an exclusion; even a later survivor density tending to zero would not suffice without finite extinction or an Archimedean contradiction. The -1 and right-tail branches remain open.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:finalskipexclusions","readme_headline":false,"declarations":[{"name":"seamGreedy_postTwentyOne_decisions_through_twentySix","module":"Erdos249257/HalfFinalMiddlePhaseSieve.lean","line":699},{"name":"seamGreedyBits_head2367_14_20_21_26","module":"Erdos249257/HalfFinalMiddlePhaseSieve.lean","line":899},{"name":"finalMiddleCell_neg_two_forces_phase_refinement_through_twentySix","module":"Erdos249257/HalfFinalMiddlePhaseSieve.lean","line":1622},{"name":"finalMiddleTwentySixPhaseSurvivors_card","module":"Erdos249257/HalfFinalMiddlePhaseSieve.lean","line":1759}]},{"id":"last_producer_tail_escape_reduction","label":"Last-producer tail-escape reduction","status":"conditional reduction","statement":"If every actual middle transition at D ≥ 13 other than the discharged cell C_D = -3 satisfies Theta_D < C_D, then an eventual right seam is impossible and 1/2 belongs to the Mersenne achievement set. Since Theta_D ≥ 0, this leaves two open inputs: exclude C_D = -2,-1 and prove Theta_D < C_D at every remaining transition with C_D ≥ 0. A square-root producer lower bound is a separate sufficient hypothesis. None of these arithmetic inputs is proved.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:taildominance","readme_headline":false,"declarations":[{"name":"half_mem_mersenneAchievementSet_of_middleProducerTailEscapeExceptNegThree","module":"Erdos249257/HalfCylinderLastProducerContradiction.lean","line":436},{"name":"half_mem_mersenneAchievementSet_of_middleProducerSqrtEscape","module":"Erdos249257/HalfCylinderLastProducerContradiction.lean","line":458}]},{"id":"middle_producer_cardinality_reduction","label":"Finite-prefix middle-producer cardinality reduction","status":"conditional reduction","statement":"Put g_D = #G_D. The augmented finite support A_D = G_D union {D} has g_D + 1 elements, so its complete future divisor-incidence tail satisfies Theta_D ≤ g_D + 1. Since C_D = 4 R_(D,D) - P_D - 4, the inequality g_D + P_D + 5 < 4 R_(D,D) is exactly g_D + 1 < C_D; required at every late middle transition, it excludes the negative cells and proves tail dominance, hence half-membership. The stronger row-scale hypothesis R_(D,D) ≥ D also suffices. Neither hypothesis is proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:taildominance","readme_headline":false,"declarations":[{"name":"half_mem_mersenneAchievementSet_of_middleProducerCardEscape","module":"Erdos249257/HalfCylinderMiddleCarryLowerBound.lean","line":2364},{"name":"half_mem_mersenneAchievementSet_of_middleProducerRowEscape","module":"Erdos249257/HalfCylinderMiddleCarryLowerBound.lean","line":2371}]},{"id":"upper_reset_band_certificates_through_thirty","label":"Upper-reset dyadic-band certificates through row 30","status":"verified finite instance","bounded_domain":"Upper transitions at every row 13 <= d <= 30 and every band index j <= d; no row d > 30 is checked.","statement":"At an upper transition G_(d+1) = H_d, put Q_d = 4 O_d + A_d from the upper overshoot and divisor pulse and B_(d,j) = 2^(d-j+1). For every 13 <= d <= 30 and j <= d, Lean proves B_(d,j) < Q_d or Q_d + 2(d+j) <= B_(d,j), equivalently avoidance of Q_d <= B_(d,j) < Q_d + 2(d+j). For fixed d, one exact successor remainder R_(d+1,d+1) certifies every j. This is bounded evidence and proves no global upper-reset band-escape hypothesis.","paper_label":"res:finalskipexclusions","readme_headline":false,"declarations":[{"name":"seamUpperResetDyadicBandEscape_through_thirty","module":"Erdos249257/HalfCylinderUpperResetBandCertificates.lean","line":78}]},{"id":"half_membership_seam_classification","label":"Positive half-membership classification in seam coordinates","status":"proved here","statement":"Membership of 1/2 in the Mersenne achievement set is equivalent to false successor terminal bits beyond every bound, to a cofinal sequence of such bits, and to a cofinal integer-seam sequence whose skipped ranks tend to infinity. These equivalences do not establish that any positive producer exists.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:halfmembership","readme_headline":false,"declarations":[{"name":"half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":126},{"name":"half_mem_mersenneAchievementSet_iff_unboundedUpperOrMiddle","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":156},{"name":"unboundedTerminalFalse_iff_not_seamGreedyEventuallyRight","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":164},{"name":"cofinalTerminalFalse_of_unboundedTerminalFalse","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":172},{"name":"unboundedTerminalFalse_of_cofinalTerminalFalse","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":186},{"name":"unboundedTerminalFalse_iff_cofinalTerminalFalse","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":196},{"name":"half_mem_mersenneAchievementSet_iff_cofinalTerminalFalse","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":203},{"name":"half_mem_mersenneAchievementSet_iff_exists_unboundedSkippedRanksAlong","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":213},{"name":"unboundedTerminalFalse_iff_greedyMersenneSkippedSupport_infinite","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":228},{"name":"half_mem_mersenneAchievementSet_iff_no_lastHalfGreedySkip","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":235}]},{"id":"largest_skip_gap_propagation","label":"Largest skipped-rank gap and successor propagation","status":"conditional reduction","statement":"At row s, let d be the largest skipped rank, u the selected prefix below d, L the integer weight of u together with the filled suffix above d, and U the weight obtained by replacing that suffix with d. In the late range d > 2s/3, equivalently 2s < 3d, Lean proves 3L + (3*2^(s+1) + 2*4^(s-d) + 4) = 3U. Upper or middle successors make the new terminal rank s the largest skip, while right successors preserve d. The global statement forcing an upper or middle step when 2(s+1) < 3d first fails is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:largestskip","readme_headline":false,"declarations":[{"name":"three_mul_largestSkipLowerWeight_add_exactLateGap_eq_upperWeight","module":"Erdos249257/HalfCylinderLargestSkipGap.lean","line":259},{"name":"seamGreedyWord_succ_isLargestFalseRank_terminal_of_upperOrMiddle","module":"Erdos249257/HalfCylinderLargestSkipGap.lean","line":328},{"name":"IsLargestFalseRank.seamGreedyWord_succ_of_rightBranch","module":"Erdos249257/HalfCylinderLargestSkipGap.lean","line":339}]},{"id":"boundary_pulse_normalization","label":"Boundary-pulse normalization at the largest-skip crossing","status":"conditional reduction","statement":"The row-s pulse of a selected rank e contributes two when e divides 2s+1 and one when e divides 2s+2. For a largest skip d with 2s < 3d, every filled suffix rank d < e < s divides neither index, so the word pulse is exactly the selected-prefix pulse below d. At the first crossing 2s < 3d <= 2(s+1), either 3d = 2s+1 with pulse two or 3d = 2s+2 with pulse one. The transition-carry estimate that would use these exact counts is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:boundarypulse","readme_headline":false,"declarations":[{"name":"exists_lowerPrefix_wordPulse_eq_of_largestFalse_late","module":"Erdos249257/HalfCylinderBoundaryPulse.lean","line":85},{"name":"rowPulse_boundary_of_late_firstCrossing","module":"Erdos249257/HalfCylinderBoundaryPulse.lean","line":150}]},{"id":"largest_skip_induction_reduction","label":"Largest-skip induction from a quantified first-crossing hypothesis","status":"conditional reduction","statement":"Assume that for every s >= 14 and largest skip d with 2s < 3d, either 2(s+1) < 3d or the transition to row s+1 is upper or middle. Lean checks the base s = 14, d = 13 with 28 < 39. Under the hypothesis, induction gives a largest skip beyond two thirds at every row s >= 14; these skipped ranks tend to infinity, so 1/2 belongs to the Mersenne achievement set. The universal next-row hypothesis is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:largestskipinduction","readme_headline":false,"declarations":[{"name":"largestSkipLateAt_fourteen","module":"Erdos249257/HalfCylinderLargestSkipInduction.lean","line":73},{"name":"LargestSkipLateStepSocket.largestSkipLateAt_succ","module":"Erdos249257/HalfCylinderLargestSkipInduction.lean","line":90},{"name":"largestSkipLateAt_of_stepSocket","module":"Erdos249257/HalfCylinderLargestSkipInduction.lean","line":120},{"name":"seamGreedyUnboundedSkippedRanksAlong_of_largestSkipLateStepSocket","module":"Erdos249257/HalfCylinderLargestSkipInduction.lean","line":130},{"name":"half_mem_mersenneAchievementSet_of_largestSkipLateStepSocket","module":"Erdos249257/HalfCylinderLargestSkipInduction.lean","line":167}]},{"id":"fixed_tail_survival_socket","label":"Fixed-tail survival equivalence at every actual skip","status":"conditional reduction","statement":"Writing r_M(1/2) for the residual after ranks through M and T_M for the complete tail after M, an actual skip M is final exactly when T_M < r_M(1/2). Negating the existence of a final skip yields r_M(1/2) <= T_M at every actual skip; together with the no-final-skip classification, this is equivalent to membership of 1/2. The required inequalities along the actual greedy orbit are not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:fixedtailsocket","readme_headline":false,"declarations":[{"name":"isLastHalfGreedySkip_iff_skip_and_fatal","module":"Erdos249257/HalfCylinderFixedTailSocket.lean","line":22},{"name":"no_lastHalfGreedySkip_iff_every_skip_survives","module":"Erdos249257/HalfCylinderFixedTailSocket.lean","line":57},{"name":"half_mem_iff_every_actual_skip_survives","module":"Erdos249257/HalfCylinderFixedTailSocket.lean","line":73}]},{"id":"producer_carry_future_tail_socket","label":"Exact producer-carry residual and sign criteria","status":"conditional reduction","statement":"For A contained in {2,...,d}, let f_A(n) count support divisors, let C_(A,d) be the affine integer state after 2d+1 updates from 1, and let Theta_(A,d) be the complete scaled coefficient tail after depth 2d+2. Then 1/2 - X_A = 2^(-(2d+2))(C_(A,d) - Theta_(A,d)). Hence C_(A,d) > Theta_(A,d) iff X_A < 1/2 and C_(A,d) < Theta_(A,d) iff X_A > 1/2; a negative carry is sufficient for the latter because the tail is nonnegative, and carry at most -8 gives a full Mersenne-gap margin. No occurrence theorem for these carries is proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:producercarry","readme_headline":false,"declarations":[{"name":"producerCarry_residual_identity","module":"Erdos249257/HalfCylinderProducerCarrySocket.lean","line":27},{"name":"producerCarry_gt_tail_iff_supportSeries_lt_half","module":"Erdos249257/HalfCylinderProducerCarrySocket.lean","line":48},{"name":"producerCarry_lt_tail_iff_half_lt_supportSeries","module":"Erdos249257/HalfCylinderProducerCarrySocket.lean","line":78},{"name":"producerCarry_neg_forces_half_lt_supportSeries","module":"Erdos249257/HalfCylinderProducerCarrySocket.lean","line":145},{"name":"producerCarry_le_neg_eight_forces_gap_margin","module":"Erdos249257/HalfCylinderProducerCarrySocket.lean","line":190}]},{"id":"quarter_band_endpoint_cells","label":"Exact quarter-band failure cells for adjacent seam cuts","status":"conditional reduction","statement":"At row s put H = 2^(s-1), and let R, U, R', p^-, p^+ be the adjacent cut's remainder, overshoot, next remainder, below pulse, and above pulse. On the non-carry middle branch, failure R' >= 8H is equivalent to the unique cell R = H+k with p^- <= 4k < p^-+4. On the terminal-reset branch, failure is equivalent to R = 3H+k with p^-+4 <= 4k and 4k+16H < 4(R+U)+p^+. These are exact failure classifications, not an avoidance theorem for the concrete seam orbit.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:quarterband","readme_headline":false,"declarations":[{"name":"pulseCell_unique","module":"Erdos249257/HalfCylinderQuarterBandEndpoints.lean","line":34},{"name":"seamAdjacentCut_middleNextUpperFailure_iff_pulseCell","module":"Erdos249257/HalfCylinderQuarterBandEndpoints.lean","line":153},{"name":"seamAdjacentCut_rightNextUpperFailure_iff_pulseCell","module":"Erdos249257/HalfCylinderQuarterBandEndpoints.lean","line":175}]},{"id":"reset_deficit_escape_geometry","label":"Exact right-branch window and reset-deficit escape law","status":"conditional reduction","statement":"Write R_s for the final integer-greedy remainder and p_s^- for the below pulse. At a largest skip d with 2s < 3d, a right branch forces 2^(s+1)+4 <= 4R_s and 12R_s+3*2^(s+1) < 4*(3*2^(s+1)+2*4^(s-d)+4)+6(s-2), and it satisfies R_(s+1)+2^(s+1)+p_s^-+4 = 4R_s. If R_s <= 2^s and delta_s = 2^s-R_s, every further right step gives delta_(s+1) >= 4 delta_s while the reference scale doubles; hence 2^s <= 2^k delta_s forces an upper-or-middle row t in [s,s+k]. The global anti-concentration needed to exclude the pinned interval along the actual orbit is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:resetdeficit","readme_headline":false,"declarations":[{"name":"rightBranch_remainder_window","module":"Erdos249257/HalfCylinderResetDeficitEscape.lean","line":200},{"name":"rightBranch_remainder_succ_eq","module":"Erdos249257/HalfCylinderResetDeficitEscape.lean","line":218},{"name":"upperOrMiddle_within_of_two_pow_deficit","module":"Erdos249257/HalfCylinderResetDeficitEscape.lean","line":261}]},{"id":"seam_producer_carry_alignment","label":"Exact seam-hole to producer-carry alignment identity","status":"proved here","statement":"Let P omit s, put A = P union {s}, and let delta be the half-carry of A at row 2s-2. Define the first-wrap hole H = 2 delta - f_P(2s) and paired incidence pulse Pi = 2 f_A(2s+1) + f_A(2s+2). Then the producer carry is exactly C(A,s) = 4(H-1) - Pi. This aligns the seam and producer coordinates but supplies no sign, size, or occurrence estimate.","paper_label":"res:seamproducer","readme_headline":false,"declarations":[{"name":"producerCarry_insert_eq_four_mul_shiftedHole_sub_pulse","module":"Erdos249257/HalfCylinderSeamProducerAlignment.lean","line":24}]},{"id":"selected_window_depth18_base","label":"Computed depth-18 selected half-carry window base","status":"verified finite instance","bounded_domain":"The 12 terminal carries in the half strip at depth 18, together with their common depth-13 restriction and the next-row divisor-agreement certificate.","statement":"A kernel-reflected Boolean table supplies a selected half-carry window at depth 18 for every terminal carry in the half strip [1,12]. The representatives share a common depth-13 restriction, yielding the first next-row divisor-agreement certificate. This is a finite base and does not construct selected windows at unbounded depths.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:selectedwindowbase","readme_headline":false,"declarations":[{"name":"depth18TableCheck_eq_true","module":"Erdos249257/HalfCarrySelectedWindowBase.lean","line":79},{"name":"depth18SelectedHalfWindow","module":"Erdos249257/HalfCarrySelectedWindowBase.lean","line":129},{"name":"depth18SelectedHalfWindow_nextRowDivisorAgreement","module":"Erdos249257/HalfCarrySelectedWindowBase.lean","line":172}]},{"id":"rewind_seam_operational_bridge","label":"Operational bridge from rewind seams to protected next-row seams","status":"conditional reduction","statement":"Let W = (a^h)_(1 <= h <= R) be a selected depth-N window whose targets rewind to two adjacent base words at depth M. If D and phi are the rewind denominator and phase, delta = D-phi, M <= N, (N+1)/2 <= M, R <= D, and the adjacent base words have row-(N+1) coefficients c+1 and c, then f_(a^h)(N+1) equals c+1 for h <= delta and c for h > delta. If 27 <= S <= 2 floor(sqrt(N+1))+4 and S+c+1 <= 2R, the profile realizes a protected one-hole seam at row N+1. The rewind provenance, seam pair, coefficient drop, strip, and buffer are inputs. For a full half-strip window the divisor bound supplies the buffer, but the coefficient drop remains unproved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:rewindseambridge","readme_headline":false,"declarations":[{"name":"nextRow_profile_of_rewindSeam","module":"Erdos249257/RewindSeamOperationalBridge.lean","line":140},{"name":"protectedEvenSeamRealizedAt_succ_of_profile","module":"Erdos249257/RewindSeamOperationalBridge.lean","line":217},{"name":"protectedEvenSeamRealizedAt_succ_of_rewindSeam","module":"Erdos249257/RewindSeamOperationalBridge.lean","line":339},{"name":"protectedEvenSeamRealizedAt_succ_of_fullStrip_rewindSeam","module":"Erdos249257/RewindSeamOperationalBridge.lean","line":358}]},{"id":"selected_suffix_cylinder_propagation","label":"Conditional suffix-cylinder feedback and child-gap recurrence","status":"conditional reduction","statement":"For the depth-18 selected window, the common depth-13 prefix and suffix numeral satisfy ν + k = 17. Under a common next coefficient C the endpoint obeys E' = 2E - C. At a feedback row the full cylinder either advances or leaves one carry gap [g_-,g_+]. If the lower and upper families have coefficients C_- and C_+ and satisfy the explicit positivity, ordering, strip, and buffer inequalities, the next gap is exactly [2g_- - C_- - 1, 2g_+ - C_+]. Cutoff promotion appends forced bits 0 and 1 and preserves consecutive prefixes. A swallowed two-row update forces the sharp quarter-band bound B <= 4(g_+ - g_- + 1) + 2. Cofinal full stages would produce an infinite support of value 1/2, but no such cofinal family or uniform coefficient inequalities are proved.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:selectsuffix","readme_headline":false,"declarations":[{"name":"depth18SelectedHalfWindow_hasSuffixCylinderAt_thirteen","module":"Erdos249257/SelectedSuffixCylinder.lean","line":88},{"name":"SelectedHalfWindow.hasSuffixCylinderAt_stepExplicit","module":"Erdos249257/SelectedSuffixCylinder.lean","line":136},{"name":"promotedSuffixCylinder_or_protectedEvenSeamRealizedAt","module":"Erdos249257/SuffixCylinderPromotion.lean","line":322},{"name":"CylinderStage.feedbackStep_or_protectedSeam","module":"Erdos249257/SuffixCylinderGlobalProducer.lean","line":174},{"name":"SelectedHalfWindow.word_stepTailExplicit","module":"Erdos249257/SuffixCylinderBelow.lean","line":117},{"name":"hasSuffixCylinderAt_take","module":"Erdos249257/SuffixCylinderTwoSheet.lean","line":26},{"name":"CylinderStage.feedbackAdvance_or_protectedSeam_inStrip","module":"Erdos249257/SuffixCylinderFeedbackFanIn.lean","line":62},{"name":"CylinderStage.feedbackAdvance_or_inStripTwoSheet","module":"Erdos249257/SuffixCylinderInStrip.lean","line":665},{"name":"InStripGapStage.advanceOfCommonNextCoeff","module":"Erdos249257/SuffixCylinderFiniteGap.lean","line":256},{"name":"ProfiledGapStage.advanceOfNextCoeffProfile","module":"Erdos249257/SuffixCylinderProfiledGap.lean","line":124},{"name":"selectedHalfIntervalPromoteHigh","module":"Erdos249257/SuffixCylinderProfiledPromotion.lean","line":107},{"name":"ProfiledGapStage.lowerNextCoeff_le_upperNextCoeff_add_one","module":"Erdos249257/SuffixCylinderProfiledAdjacency.lean","line":232},{"name":"ProfiledGapStage.promoteCutoff_hasAdjacentPrefixes","module":"Erdos249257/SuffixCylinderProfiledPromotionAdjacency.lean","line":65},{"name":"ProfiledGapStage.promoteThenAdvance_hasAdjacentPrefixes","module":"Erdos249257/SuffixCylinderProfiledAutoStep.lean","line":110},{"name":"nextBound_le_four_gapWidth_add_two_of_macro_swallow","module":"Erdos249257/SuffixCylinderProfiledSwallow.lean","line":254},{"name":"cofinalTerminalOnlyStrip_of_cofinalCylinderStages","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":264},{"name":"exists_infinite_support_half_of_cofinalCylinderStages","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":277},{"name":"exists_infinite_positive_support_half_of_cofinalCylinderStages","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":287}]},{"id":"selected_suffix_pre_feedback_run","label":"Verified finite suffix-cylinder stages through depth 52","status":"verified finite instance","bounded_domain":"Every depth from 18 through 29, plus explicit full stages at depths 51 and 52; depth 29 has common suffix cylinders at cutoffs 14 through 25 but not 26.","statement":"A common depth-13 prefix supports full selected windows at every depth 18 through 27, and the threshold continuation reaches depth 29 while covering the protected strip. At depth 29, cutoffs 14 through 25 remain common and cutoff 26 does not. A separate exact recurrence reaches depth 51 with endpoint 51,327,745 beyond the next head threshold, and the promoted depth-52 successor supplies a terminal-strip witness. These are finite stages, not an unbounded induction.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:prefeedback","readme_headline":false,"declarations":[{"name":"nonempty_stage_of_eighteen_le_of_le_twentySeven","module":"Erdos249257/SelectedSuffixCylinderPreFeedback.lean","line":118},{"name":"depth27SelectedHalfWindow_hasSuffixCylinderAt_thirteen","module":"Erdos249257/SelectedSuffixCylinderPreFeedback.lean","line":141},{"name":"depth27SelectedHalfWindow_endpoint_covers_strip","module":"Erdos249257/SelectedSuffixCylinderPreFeedback.lean","line":147},{"name":"stage27_endpoint_eq","module":"Erdos249257/SuffixCylinderFirstFeedback.lean","line":144},{"name":"depth29SelectedHalfWindow_endpoint_covers_strip","module":"Erdos249257/SuffixCylinderThreshold.lean","line":700},{"name":"depth29SelectedHalfWindow_not_commonRestrictionAt_twentySix","module":"Erdos249257/SuffixCylinderThreshold.lean","line":806},{"name":"fullCylinderStage51_feedback_headThreshold_margin","module":"Erdos249257/SuffixCylinderStage51.lean","line":536},{"name":"fullCylinderStage52_halfTerminalOnlyStripWitness","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":238}]},{"id":"half_divisor_unit_drop","label":"Half-divisor unit coefficient drop at the feedback row","status":"proved here","statement":"At the doubled feedback row, switching only the terminal half-divisor bit from false to true changes the support coefficient from C to C+1 and, under K' = 2K-C, drops the successor carry by exactly one. The same identity holds for any explicitly identified adjacent boundary pair, and the rewind adapter transfers such a shared-cutoff pair to the next protected seam while retaining the unit-drop witness. This does not construct such pairs globally.","paper_label":"res:halfdivisordrop","readme_headline":false,"declarations":[{"name":"supportCoeff_extend_true_eq_false_add_one_at_double","module":"Erdos249257/HalfDivisorUnitDrop.lean","line":20},{"name":"supportCoeff_boundaryPair_unitDrop_at_double","module":"Erdos249257/HalfDivisorUnitDrop.lean","line":35},{"name":"rewindBaseUnitDropAt_of_halfDivisorBoundaryPair","module":"Erdos249257/RewindHalfDivisorAdapter.lean","line":86},{"name":"protectedEvenSeamRealizedAt_succ_of_fullStrip_rewindHalfDivisorBoundaryPair","module":"Erdos249257/RewindHalfDivisorAdapter.lean","line":147}]},{"id":"rewind_boundary_pair_obstruction","label":"Finite rewind boundary-pair obstruction","status":"proved here","statement":"A concrete depth-26 selected two-word window has a depth-27 explicit rewind step with history [1], a width-two numerical seam, and the doubled-row identity, but no depth-13 half-divisor boundary-pair witness. Therefore scalar one-row seam data alone does not entail the structural boundary predicate required by the cofinal consumer.","paper_label":"res:rewindboundarycounterexample","readme_headline":false,"declarations":[{"name":"depth27_actualRewindSeam_not_boundaryPair","module":"Erdos249257/RewindBoundaryPairCounterexample.lean","line":169}]},{"id":"concrete_producer_carry_coordinates","label":"Concrete adjacent-cut producer-carry coordinates","status":"proved here","statement":"For every adjacent cut at rank s ≥ 5, the below and above words are the neighbouring subset sums bracketing the target; their remainder and overshoot are the unsigned distances, and their pulses are the paired divisor-incidence contributions of the next two rows. After adjoining terminal rank s, the below producer carry is 4 times the remainder minus the below pulse minus 4, while the above producer carry is the negative of 4 times the overshoot plus the above pulse plus 4. Hence the upper carry is negative and the middle sign depends on the remainder-pulse competition. These exact identities supply no uniform exponential lower bound, sign persistence at unbounded ranks, or universal escape argument.","paper_label":"res:producercarrycoordinates","readme_headline":false,"declarations":[{"name":"producerCarry_insert_seamWordSupport_eq","module":"Erdos249257/HalfCylinderProducerLowerBound.lean","line":105},{"name":"producerCarry_insert_seamBelowSupport_eq_middleCoordinate","module":"Erdos249257/HalfCylinderProducerLowerBound.lean","line":150},{"name":"producerCarry_insert_seamAboveSupport_eq_neg_upperCoordinate","module":"Erdos249257/HalfCylinderProducerLowerBound.lean","line":183}]},{"id":"mersenne_lambert_ladder","label":"The Mersenne–Lambert ladder identities","status":"formalised here","statement":"L(μ) = 1/2, L(φ) = 2, L(φ*μ) = S, and the Möbius-square lens S = 1/2 + ∑ μ(d)/(2^d − 1)², placing the open constant one convolution from the machine-checked irrational E.","paper_label":"res:lens","readme_headline":false,"declarations":[{"name":"tsum_totient_div_two_pow_sub_one_eq_two","module":"Erdos249257/CertificateKernel.lean","line":18429},{"name":"totient_series_eq_half_add_moebius_mersenne_square","module":"Erdos249257/CertificateKernel.lean","line":18454}]},{"id":"totient_coprimality_probability","label":"Fair-coin coprimality form of the totient constant","status":"proved here","statement":"For independent fair-coin geometric waiting times X and Y on the positive integers, S = 1/2 + P(gcd(X,Y)=1), where the probability is the visible coprime-pair series. That probability has no rational representation with denominator at most 39819823323350687661677887437915526. This is an exact reformulation and quantitative corollary; irrationality remains open.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":"res:coprime","readme_headline":false,"declarations":[{"name":"totient_series_eq_half_add_visible_coprime_pairs","module":"Erdos249257/CertificateKernel.lean","line":18557},{"name":"tsum_visible_coprime_pairs_ne_int_div_of_den_le_39819823323350687661677887437915526","module":"Erdos249257/CertificateKernel.lean","line":18572}]},{"id":"squared_lambert_gcd_moments","label":"Squared-Lambert identities for divisor count and the first gcd moment","status":"proved here","statement":"The squared-Mersenne Lambert series with weight 1 equals the base-1/2 series of sigma(n)-tau(n), and the series with weight phi equals the base-1/2 series of P(n)-n, where P is Pillai's gcd-sum function. These are the exact divisor-count and first-moment identities for the fair-coin gcd law; the separately cited irrationality of the constant-weight value is not claimed as formalised here.","paper_label":"res:gcdmoment","readme_headline":false,"declarations":[{"name":"tsum_one_div_mersenne_sq_eq_sigma_sub_tau_series","module":"Erdos249257/GcdMomentCalculus.lean","line":216},{"name":"tsum_totient_div_mersenne_sq_eq_gcd_moment_series","module":"Erdos249257/GcdMomentCalculus.lean","line":235}]},{"id":"stern_brocot_cylinder_law","label":"Reduced-direction normalization and Stern–Brocot cylinder law","status":"proved here","statement":"The Mersenne weights of positive reduced slopes sum exactly to one. At every positive Stern–Brocot node (a,b), the cylinder mass 1/((2^a-1)(2^b-1)) splits into its stop mass and two child cylinders; the depth-d unfolding has error between zero and (2/3)^d times the cylinder mass and converges to the closed form.","paper_label":"res:sternbrocotcylinders","readme_headline":false,"declarations":[{"name":"tsum_pos_coprime_inv_mersenne_eq_one","module":"Erdos249257/GcdMomentCalculus.lean","line":349},{"name":"cylinderMass_split","module":"Erdos249257/GcdMomentCalculus.lean","line":474},{"name":"sternBrocotDepthMass_error","module":"Erdos249257/GcdMomentCalculus.lean","line":525},{"name":"tendsto_sternBrocotDepthMass","module":"Erdos249257/GcdMomentCalculus.lean","line":560}]},{"id":"stern_brocot_run_fibonacci_stability","label":"Fibonacci and continuant stability of Stern–Brocot runs","status":"unconditional progress","statement":"A Stern–Brocot word with r nonempty alternating runs has height at least F_(r+3), with equality on the all-unit run spine. Writing each run length as 1+e_i gives an exact multiaffine continuant-defect expansion, the sharp global gain F_(r+1) sum e_i, and exact one-site coefficient F_(i+2)F_(r-i+1). Under the gcd-one weight 2^-(a+b), the first induced run layer has mass 1/2, while the natural denominator-bookkeeping exponent is exactly F_(r+3)-2. These are source-level combinatorial and arithmetic facts; no analytic denominator survival or irrationality conclusion is claimed.","paper_label":"res:sternbrocotruns","readme_headline":false,"declarations":[{"name":"sbPair_runWord","module":"Erdos249257/SternBrocotRunGeometry.lean","line":125},{"name":"runHeight_fib_lower","module":"Erdos249257/SternBrocotRunGeometry.lean","line":185},{"name":"runHeight_defectRunLengths","module":"Erdos249257/SternBrocotRunGeometry.lean","line":263},{"name":"runHeight_defect_fib_sum_lower","module":"Erdos249257/SternBrocotRunGeometry.lean","line":343},{"name":"runHeight_oneSiteDefect","module":"Erdos249257/SternBrocotRunGeometry.lean","line":413},{"name":"oneRunMass_eq_half","module":"Erdos249257/SternBrocotRunGeometry.lean","line":467},{"name":"naturalRunDenominatorExponent_add_two","module":"Erdos249257/SternBrocotRunGeometry.lean","line":491}]},{"id":"primitive_coordinate_index_obstruction","label":"Primitive Euler-coordinate finite-jet obstruction","status":"proved here","statement":"For the primitive-conductor weight A = φ*μ, every positive index clearing the normalized coordinates A(n)/n through horizon N is divisible by the two-tier primorial: 4 at the prime 2 once N ≥ 4, p² for odd p with p² ≤ N, and p otherwise. Hence no fixed positive index clears all normalized coordinates. This is a finite Euler/Witt-coordinate obstruction only; it gives no irrationality criterion for S.","paper_label":"res:primitiveindex","readme_headline":false,"declarations":[{"name":"twoTierPrimorial_dvd_of_integral_jet","module":"Erdos249257/PrimitiveDeterminantLift.lean","line":148},{"name":"no_fixed_integral_primitive_euler_index","module":"Erdos249257/PrimitiveDeterminantLift.lean","line":169}]},{"id":"primitive_unit_gap_ceiling","label":"Prime-power reduced-denominator unit-gap ceiling","status":"proved here","statement":"For the reduced-denominator unit-gap predicate at an odd prime power p^e, the candidate interval is empty or has exactly one candidate divisible by p; in particular its count is at most one. The exact row (u,N,K) = (3,3,5) satisfies the unit-gap predicate while the corresponding ordinary empty-gap inequality fails. This is a finite certificate-level strengthening, not an unbounded supply theorem or an irrationality criterion for S.","paper_label":"res:unitgap","readme_headline":false,"declarations":[{"name":"reducedDenominatorUnitGapCert_prime_pow_iff","module":"Erdos249257/PrimitiveWeightCertificate.lean","line":54},{"name":"reducedDenominatorCandidateCount_prime_pow_le_one","module":"Erdos249257/PrimitiveWeightCertificate.lean","line":111},{"name":"reducedDenominatorUnitGapCert_three_window_3_5","module":"Erdos249257/PrimitiveWeightCertificate.lean","line":151},{"name":"not_ordinary_gap_certificate_three_window_3_5","module":"Erdos249257/PrimitiveWeightCertificate.lean","line":157}]},{"id":"mersenne_tail_atom_boundary","label":"Finite Mersenne tail-atom determinant boundary","status":"proved here","statement":"A periodic Mersenne tail atom has the exact doubling-and-wrap carry law. For any finite atom matrix, clearing the product of its column Mersenne denominators yields the integral numerator determinant; a nonzero cleared weighted minor therefore has absolute value at least one. This is a finite-denominator boundary for an atom-minor route, not an irrationality criterion for S.","paper_label":"res:mersennetailatoms","readme_headline":false,"declarations":[{"name":"mersenneTailAtom_step","module":"Erdos249257/MersenneTailAtoms.lean","line":28},{"name":"mersenneAtom_denominatorProduct_mul_det","module":"Erdos249257/MersenneTailAtoms.lean","line":122},{"name":"one_le_abs_weightedMersenneAtom_cleared_det","module":"Erdos249257/MersenneTailAtoms.lean","line":181}]},{"id":"mobius_sign_support_no_go","label":"Möbius-sign support one-half no-go","status":"proved here","statement":"The Boolean support selected by the negative Möbius values has sum 1/2 plus the positive Möbius tail and is therefore at least 1/2 + 1/63. This closes the sign-truncation route to the target value 1/2; it does not exclude another infinite Boolean support with that value.","paper_label":"res:mobiussignnogo","readme_headline":false,"declarations":[{"name":"tsum_negativeMobius_eq_half_add_positiveMobiusTail","module":"Erdos249257/MobiusSignSupportNoGo.lean","line":111},{"name":"half_add_one_div_sixty_three_le_tsum_negativeMobius","module":"Erdos249257/MobiusSignSupportNoGo.lean","line":150}]},{"id":"totient_carry_kernel_anti_compression","label":"Totient carry anti-compression","status":"unconditional progress","statement":"If the binary totient coefficient series were rational, one positive-multiplier tempered integral carry would have uniformly eventually-periodic dyadic sections modulo its multiplier while retaining canonical carry-section span dimension at least 2^e − 1 for every e. No finite-rank bound for every rationality-supplied carry is proved, so this quotient-periodicity/anti-compression conjunction is a necessary consequence rather than an irrationality criterion for S.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":"res:carryrank","readme_headline":false,"declarations":[{"name":"not_irrational_totientSeries_implies_mod_period_and_unbounded_rank","module":"Erdos249257/TotientTailCarryPeriod.lean","line":224},{"name":"finrank_canonicalCarryKernel_ge","module":"Erdos249257/TotientCarryKernelRigidity.lean","line":263},{"name":"carrySectionsEventuallyPeriodicMod_of_shift_dvd","module":"Erdos249257/TotientTailCarryPeriod.lean","line":180},{"name":"totient_forcing_vanishes_mod_of_dvd_multiplier","module":"Erdos249257/TotientTailCarryPeriod.lean","line":126},{"name":"totient_carry_modEq_geometric_of_dvd_multiplier","module":"Erdos249257/TotientTailCarryPeriod.lean","line":140},{"name":"finrank_canonicalCarryKernel_ge_of_linearIndependent","module":"Erdos249257/TotientCarryKernelRigidity.lean","line":211},{"name":"not_irrational_totientSeries_implies_unbounded_carryRank_unconditional","module":"Erdos249257/TotientCarryKernelRigidity.lean","line":300}]},{"id":"denominator_exclusion","label":"Unconditional denominator exclusion for the totient constant","status":"verified finite instance","bounded_domain":"Integer p and 1 <= q <= 79 639 646 646 701 375 323 355 774 875 831 053; no larger denominator is excluded.","statement":"S = ∑ φ(n)/2ⁿ is not equal to p/q for any integer p and any 1 ≤ q ≤ 79 639 646 646 701 375 323 355 774 875 831 053 (≈ 7.96 × 10³⁴). A lower bound on any rational representation, not a proof of irrationality.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":"res:farey","readme_headline":true,"declarations":[{"name":"tsum_totient_div_pow_two_ne_ratCast_of_den_le_79639646646701375323355774875831053","module":"Erdos249257/CertificateKernel.lean","line":18384}]},{"id":"certificate_reduction","label":"Exact reduction of #249 to finite decidable certificates","status":"conditional reduction","statement":"S is irrational if and only if certified non-integrality witnesses occur at unbounded parameters. The pointwise completeness theorem and the diagonal supply theorem together show that an unbounded lcm-diagonal supply is equivalent as well. Producing that supply remains open.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:diag","readme_headline":true,"declarations":[{"name":"irrational_totient_series_of_certificate_supply","module":"Erdos249257/TotientTailPeriodKiller.lean","line":394},{"name":"irrational_totient_series_of_lcm_diagonal_certificate_supply","module":"Erdos249257/LcmDiagonalReduction.lean","line":112},{"name":"rational_totient_series_forces_lcm_cone_flatness","module":"Erdos249257/LcmConeFlatness.lean","line":90},{"name":"irrational_totient_series_iff_pointwise_certificates","module":"Erdos249257/LcmConeFlatness.lean","line":399},{"name":"irrational_totient_series_iff_certificate_supply","module":"Erdos249257/LcmConeFlatness.lean","line":412},{"name":"irrational_totient_series_iff_lcm_diagonal_certificate_supply","module":"Erdos249257/LcmConeFlatness.lean","line":426}]},{"id":"certificate_completeness","label":"Certificate completeness","status":"proved here","statement":"At fixed parameters, a certified kill exists exactly when the corresponding tail difference is non-integral. At every positive shift, integrality of one tail difference forces S to be rational; thus irrationality is equivalent to pointwise non-integrality, and to pointwise finite certificates.","paper_label":"res:pointwisecomplete","readme_headline":true,"declarations":[{"name":"exists_certifiedKill_iff_tail_diff_notMem_int","module":"Erdos249257/LcmConeFlatness.lean","line":316},{"name":"tail_diff_mem_int_iff_scaled_series_mem_int","module":"Erdos249257/LcmConeFlatness.lean","line":327},{"name":"irrational_totient_series_iff_all_tail_diffs_nonintegral","module":"Erdos249257/LcmConeFlatness.lean","line":386},{"name":"irrational_totient_series_iff_pointwise_certificates","module":"Erdos249257/LcmConeFlatness.lean","line":399}]},{"id":"certified_kill_instances","label":"Initial certified-kill segment and generated certificates","status":"verified finite instance","statement":"Kernel-checked certified kills for the listed small periods, every lcm-diagonal scale t ≤ 82, and the finite certificate shards. The historical 28 explicit deposits through t = 64 form a strict subset of this contiguous finite band.","bounded_domain":"The explicitly listed small periods; every natural lcm-diagonal scale t with t ≤ 82; and the parameters encoded by the finite certificate shards. No certificate at t = 83 and no unbounded or cofinal supply is claimed.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:deposits","readme_headline":true,"declarations":[{"name":"certifiedKill_all_small","module":"Erdos249257/TotientTailPeriodKiller.lean","line":404},{"name":"certifiedKill_periodLcm_diagonal_upto_six","module":"Erdos249257/LcmDiagonalReduction.lean","line":257},{"name":"certifiedKill_periodLcm_diagonal_at_seven","module":"Erdos249257/LcmDiagonalReduction.lean","line":262},{"name":"certifiedKill_periodLcm_diagonal_at_eight","module":"Erdos249257/LcmDiagonalReduction.lean","line":266},{"name":"certifiedKill_diagonal_all_imported_through_t64","module":"Erdos249257/DiagonalPincerCertificatesT64.lean","line":1967},{"name":"exists_diagonalKill_le_82","module":"ErdosProblems/Skip/LadderT67.lean","line":71264}]},{"id":"diagonal_pincer_reduction","label":"Exact full-target reduction on the lcm diagonal","status":"conditional reduction","statement":"At each lcm-diagonal scale, integrality holds if and only if a foreign-defect term lands in an explicit full-target interval. Globally, S is irrational if and only if the full target is avoided at arbitrarily large diagonal scales; that cofinal avoidance is not proved.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:fulltargetavoidance","readme_headline":true,"declarations":[{"name":"diagonal_int_iff_foreignDiagonalDefect_hits_fullTarget","module":"Erdos249257/DiagonalPincerDecomposition.lean","line":215},{"name":"irrational_totientSeries_of_full_target_avoidance_supply","module":"Erdos249257/DiagonalPincerDecomposition.lean","line":290}]},{"id":"diagonal_fresh_loss_reduction","label":"Fresh-loss projection and suffix-supply reduction","status":"conditional reduction","statement":"The fresh-loss residue certificate is exactly an integer projection inequality, and such a projection forces full-target avoidance. An unbounded projection supply—or the stronger adjacent-suffix gap supplies formalised here—implies irrationality; no unbounded supply is asserted.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:freshloss","readme_headline":false,"declarations":[{"name":"diagonalFreshLossResidueCert_iff_projectionCert","module":"Erdos249257/DiagonalFreshLossBridge.lean","line":72},{"name":"irrational_totientSeries_of_diagonalFreshLossProjectionSupply","module":"Erdos249257/DiagonalFreshLossBridge.lean","line":202}]},{"id":"diagonal_pincer_t64_primality_closure","label":"T64 diagonal-pincer primality closure","status":"verified finite instance","bounded_domain":"The generated Lucas primality dependencies required by the diagonal-pincer certificate family through t = 64.","statement":"A generated proof-producing Lucas-certificate module supplies the primality dependencies for the t=47,49,53,59,61,64 diagonal-pincer certificates. This is finite certificate infrastructure; it neither supplies an unbounded certificate family nor proves irrationality.","paper_label":"res:t64closure","readme_headline":false,"declarations":[{"name":"prime_lucas_2364533768205644535022723273","module":"Erdos249257/DiagonalPincerPrimeCertificates/ClosureT64/Level1A.lean","line":2398}]},{"id":"diagonal_pincer_t64_endpoint_certificate","label":"T64 diagonal-pincer endpoint certificate","status":"verified finite instance","bounded_domain":"The single lcm-diagonal endpoint t = 64.","statement":"Lean checks the diagonal-pincer certified kill at t = 64 using the committed T64 Lucas-primality closure. This is one finite certificate and does not assert an unbounded certificate supply or irrationality.","paper_label":"res:t64endpoint","readme_headline":false,"declarations":[{"name":"certifiedKill_diagonal_t64","module":"Erdos249257/DiagonalPincerCertificatesT64.lean","line":1928}]},{"id":"fresh_prime_deficit_decomposition","label":"Fresh-prime endpoint-deficit decomposition","status":"proved here","statement":"At each positive height, every old-prime endpoint mass dominates the actual totient. The diagonal increment and literal foreign channel split exactly into the old Möbius increment and lower-minus-upper endpoint fresh deficits. The second and five-point curvature operators retain this split, and the nonnegative deficits bound the exact adverse weighted loss from the old-prime curvature margins. No cofinal deficit bound or irrationality conclusion is proved.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:freshprimedeficit","readme_headline":false,"declarations":[{"name":"endpointFreshDeficit_nonneg","module":"Erdos249257/FreshPrimeDeficitDecomposition.lean","line":91},{"name":"finiteForeignChannelIncrement_eq_endpointFreshDeficit_sub","module":"Erdos249257/FreshPrimeDeficitDecomposition.lean","line":184},{"name":"oldFivePoint_sub_secondBranchAdverseDeficit_le_actual","module":"Erdos249257/FreshPrimeDeficitDecomposition.lean","line":269}]},{"id":"flexible_odd_window_reduction","label":"Flexible three-rank odd-window reduction","status":"conditional reduction","statement":"At an analytically admissible odd power-of-two rank, centrality in any one of the canonical odd rank or its next two base-four successors supplies the existing adjacent-suffix gap interface and hence irrationality. The remaining assertion is a cofinal three-rank band supply; no such supply is proved.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:flexibleoddwindow","readme_headline":false,"declarations":[{"name":"diagonalAdjacentSuffixGapSupply_of_powerTwoOddGuardThreeRankBand","module":"Erdos249257/DiagonalFlexibleOddWindowSupply.lean","line":93},{"name":"irrational_totientSeries_of_powerTwoOddGuardThreeRankBandSupply","module":"Erdos249257/DiagonalFlexibleOddWindowSupply.lean","line":137}]},{"id":"flexible_odd_window_affine_frontier","label":"Affine three-scale frontier for the flexible odd window","status":"conditional reduction","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"statement":"Under the stated short-correction bounds, the three half-word central-band predicates are exactly a three-scale affine escape disjunction. A uniform correction envelope alone cannot force that escape, as a zero-start affine orbit remains inside every scaled edge tube. Thus arithmetic state-correction correlation remains required.","paper_label":"res:flexibleoddaffine","readme_headline":false,"declarations":[{"name":"zeroAffineOrbit_stays_in_scaled_edge","module":"Erdos249257/DiagonalFlexibleOddWindowAffine.lean","line":225},{"name":"threeRankBandAt_iff_threeScaleAffineEscapeAt_of_shortCorrections","module":"Erdos249257/DiagonalFlexibleOddWindowAffine.lean","line":236}]},{"id":"power_two_bit_lift","label":"Exact power-of-two one-bit lift","status":"proved here","statement":"Given agreement modulo 2^k, agreement modulo 2^(k+1) is exactly evenness of the signed quotient after division by 2^k; failure gives the unique offset class. In particular, a signed difference 16*c agrees modulo 32 exactly when c is even. Under the stated correction and margin budget, the centered signed-margin socket is exactly failure of this next lift. This provides a finite fifth-bit interface and does not supply cofactor parity at unbounded scales.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:powertwobitlift","readme_headline":false,"declarations":[{"name":"two_pow_succ_dvd_iff_quotient_even","module":"Erdos249257/PowerTwoBitLift.lean","line":34},{"name":"modEq_thirtyTwo_iff_even_difference_cofactor","module":"Erdos249257/PowerTwoBitLift.lean","line":111},{"name":"not_modEq_thirtyTwo_iff_offset_sixteen","module":"Erdos249257/PowerTwoBitLift.lean","line":129},{"name":"actualPenultimateSignedMargin_centeredLift_iff_not_twice_dvd","module":"Erdos249257/PowerTwoCenteredBitLift.lean","line":121},{"name":"actualPenultimateSignedMargin_centeredThirtyTwo_iff_not_thirtyTwo_dvd","module":"Erdos249257/PowerTwoCenteredBitLift.lean","line":159}]},{"id":"parity_socket_obstruction","label":"Parity obstruction for the reduced margin socket","status":"proved here","statement":"For every positive even modulus and every radius in the stated nontrivial range, each parity class contains both a reduced-margin-socket hit and a miss. The same holds for a positive power-of-two modulus. Thus quotient parity alone cannot decide the reduced signed-margin socket. No t = 827 numeral or signed-margin certificate is asserted.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:paritysocket","readme_headline":false,"declarations":[{"name":"reducedMarginSocket_not_decided_by_parity","module":"Erdos249257/Totient827ParitySocketCeiling.lean","line":73},{"name":"powerTwo_reducedMarginSocket_not_decided_by_parity","module":"Erdos249257/Totient827ParitySocketCeiling.lean","line":110}]},{"id":"canonical_jump_slack_reduction","label":"Canonical strict-jump slack reduction","status":"conditional reduction","statement":"The two canonical adjacent-suffix central-band inequalities are equivalent to nonnegativity of one explicit integer slack. Cofinal nonnegative slack at strict lcm jumps implies irrationality of the totient constant; that cofinal sign statement remains open.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:slackcriterion","readme_headline":false,"declarations":[{"name":"canonicalAdjacentSuffixCentral_iff_slack_nonneg","module":"Erdos249257/DiagonalFreshLossBridge.lean","line":2667},{"name":"canonicalAdjacentSuffixJumpCentralSupply_iff_slackSupply","module":"Erdos249257/DiagonalFreshLossBridge.lean","line":2757},{"name":"irrational_totientSeries_of_canonicalAdjacentSuffixJumpSlackSupply","module":"Erdos249257/DiagonalFreshLossBridge.lean","line":2932}]},{"id":"squared_mersenne_enclosure","label":"Sharp squared-Mersenne diagonal enclosure","status":"proved here","statement":"The scaled diagonal tail difference is written as an explicit Lambert-projected centre plus a signed squared-Mersenne tail, with a sharp symmetric bound and a checked separation criterion for missing the full target.","paper_label":"res:squareenclosure","readme_headline":false,"declarations":[{"name":"scaleDiagonalTailDifference_sub_lambertProjectedDiagonal","module":"Erdos249257/SquaredMersenneDiagonalEnclosure.lean","line":138},{"name":"scaleFullTarget_miss_of_lambert_projected_separation","module":"Erdos249257/SquaredMersenneDiagonalEnclosure.lean","line":426}]},{"id":"actual_foreign_residue_projection","label":"Finite foreign-residue projection reduction","status":"conditional reduction","statement":"The finite Möbius residue diagonal splits exactly into projected foreign and divisor channels, and every finite stable tail window has a closed geometric bound. If an analytic kernel/complement bridge controls the actual foreign defect by that bound and the projection is separated from every integer, then the full target is missed. The analytic bridge and any unbounded separated-projection supply are not proved.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:foreignresidue","readme_headline":false,"declarations":[{"name":"finiteResidueDiagonal_eq_projectedForeign_add_divisor","module":"Erdos249257/ActualForeignResidueProjection.lean","line":308},{"name":"abs_foreignTailWindow_le_foreignComplementBound","module":"Erdos249257/ActualForeignResidueProjection.lean","line":276},{"name":"scaleFullTarget_miss_of_projected_separation","module":"Erdos249257/ActualForeignResidueProjection.lean","line":414}]},{"id":"mersenne_shadow_denominator","label":"Exact Mersenne-shadow denominator and survivor bounds","status":"proved here","statement":"The reduced denominator of the lcm-height scaled Möbius shadow is computed exactly at every height, and for t ≥ 5 an explicit upper-half Mersenne product both divides it and supplies a quantitative lower bound.","paper_label":"res:shadowden","readme_headline":false,"declarations":[{"name":"exists_upperHalf_mersenne_channel_dvd_den_with_bounds","module":"Erdos249257/MersenneShadowDenominatorGrowth.lean","line":99},{"name":"lcmHeight_scaledMobiusShadow_den_exact","module":"Erdos249257/MersenneShadowDenominatorGrowth.lean","line":147}]},{"id":"adelic_height_obstruction","label":"Scalar-localisation height obstruction","status":"proved here","statement":"For every rational x, integer c and natural number H, if H divides the reduced denominator of x and the reduced denominator of c·x divides H, then the complementary factor x.den/H divides |c|. Scalar localisation therefore moves that denominator information into the scalar; this local obstruction is not an irrationality criterion.","paper_label":"res:adelic","readme_headline":true,"declarations":[{"name":"scalarLocalization_complement_dvd","module":"Erdos249257/AdelicHeightObstruction.lean","line":23},{"name":"scalarLocalization_integer_eq_mul_num","module":"Erdos249257/AdelicHeightObstruction.lean","line":56}]},{"id":"square_crt_correction_suppression","label":"Square-CRT correction suppression","status":"proved here","statement":"For every finite family of distinct primes and prescribed anchors, a bounded simultaneous square-CRT base removes the prime-dilation correction at every admissible finite shift. Checked clean cubes can have vanishing coefficients and can also have a nonzero coefficient; this finite mechanism supplies neither an unbounded certificate family nor an irrationality criterion for the totient constant.","paper_label":"res:squarecrt","readme_headline":false,"declarations":[{"name":"exists_squareCRT_clean_totient_family","module":"Erdos249257/SquareCRTCube.lean","line":297},{"name":"exists_squareCRT_clean_horizon","module":"Erdos249257/SquareCRTCube.lean","line":327},{"name":"squareCRT_clean_block_can_vanish","module":"Erdos249257/SquareCRTCube.lean","line":459},{"name":"squareCRT_clean_block_can_be_nonzero","module":"Erdos249257/SquareCRTCube.lean","line":477}]},{"id":"signed_dyadic_moment_substrate","label":"Signed dyadic-moment algebra","status":"proved here","statement":"A rectangular determinant expansion over all maps and a unique-largest-dyadic-exponent parity lemma are formalised. The latter makes a cleared finite dyadic sum odd, hence nonzero, when its selected numerator is odd and its exponent is strictly largest. This is finite algebraic substrate only: it proves neither non-vanishing of the actual totient Hankel determinants nor an irrationality criterion.","paper_label":"res:signedmoment","readme_headline":false,"declarations":[{"name":"det_mul_rectangular","module":"Erdos249257/SignedQMomentObstruction.lean","line":29},{"name":"scaled_dyadic_sum_odd","module":"Erdos249257/SignedQMomentObstruction.lean","line":78},{"name":"scaled_dyadic_sum_ne_zero","module":"Erdos249257/SignedQMomentObstruction.lean","line":96}]},{"id":"dyadic_totient_basis","label":"Explicit basis of the dyadic totient kernel span","status":"unconditional progress","statement":"Lean constructs an explicit odd-core-indexed basis of the rational span of the full dyadic totient kernel, with exact span equality and linear independence. This is an independent constructive proof route; its infinite-dimensionality consequence is not an independent contribution because Coons's non-2-regularity theorem already implies it.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":"res:basis","readme_headline":false,"declarations":[{"name":"linearIndependent_oddCoreTotientKernelFamily","module":"Erdos249257/TotientMahlerDefect.lean","line":1265},{"name":"span_range_fullTotientKernel_eq_span_range_oddCore","module":"Erdos249257/TotientMahlerDefect.lean","line":1380},{"name":"totientDyadicSectionBasis","module":"Erdos249257/TotientMahlerDefect.lean","line":1392}]},{"id":"totient_residue_series_classification","label":"Irrational totient residue series and the rational dyadic residue observables","status":"unconditional progress","statement":"For every m ≥ 3 the series ∑_{n≥1} (φ(n) mod m)/2^n is irrational, and for k ≥ 1 and f: Z/2^kZ → Q the series ∑_{n≥1} f(φ(n) mod 2^k)/2^n is rational exactly when f is constant on the even residue classes, with value 3f(1)/4 + c/4 where c is that constant. The first assertion, in every integer base, already follows from Erdős's Lemma 1 (Indag. Math. 19 (1957), p. 213; the case b = 0 of his Lemma 4, pp. 215-218) because φ(n) mod m vanishes outside a set of density zero, and equally from Theorem 3 of Kaneko, Suzuki and Tachiya (arXiv:2601.20743v1, p. 5). Those criteria also give the classification when the values f(r) - f(0) on the even classes have one sign; with both signs they need a further spacing hypothesis, which the isolated-coefficient proof does not need. Wong posted the case in which the base equals the modulus in 2015. No novelty claim is made for the first assertion. The reduced series say nothing about the unreduced series ∑ φ(n)/2^n.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":"res:residueseries","readme_headline":false,"declarations":[{"name":"short_note_residue_theorem","module":"ErdosProblems/Erdos249/PaperCompleteR7/RationalObservableClassification.lean","line":276}]},{"id":"totient_residue_series_integer_radix","label":"Totient residue series in every integer base","status":"unconditional progress","statement":"For every integer t ≥ 2 and m ≥ 3 the series ∑_{n≥1} (φ(n) mod m)/t^n is irrational, and for k ≥ 1 and f: Z/2^kZ → Q the series ∑_{n≥1} f(φ(n) mod 2^k)/t^n is rational exactly when f is constant on the even residue classes, with value (t+1)f(1)/t^2 + f(0)/(t^2(t-1)). The first assertion already follows from Erdős's Lemma 1 (1957) and from Theorem 3 of Kaneko, Suzuki and Tachiya, as recorded for the binary case; no novelty claim is made for it.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":"prop:radixresidue","readme_headline":false,"declarations":[{"name":"radix_residue_series_irrational","module":"ErdosProblems/Erdos249/PaperCompleteR7/RationalIntegerRadixClassification.lean","line":128},{"name":"rational_zmod_radix_observable_iff","module":"ErdosProblems/Erdos249/PaperCompleteR7/RationalIntegerRadixClassification.lean","line":145},{"name":"positiveRadixValue_eq_of_even_constant","module":"ErdosProblems/Erdos249/PaperCompleteR7/IntegerRadixObservables.lean","line":303}]},{"id":"totient_residue_series_dilations","label":"Linear independence of dilated totient residue series","status":"unconditional progress","statement":"For every integer t ≥ 2 and m ≥ 3, the numbers 1 and ∑_{n≥1} (φ(n) mod m)/t^{dn}, d = 1, 2, ..., are linearly independent over Q. For a finite set D of positive integers and residue functions f_d: Z/2^{k_d}Z → Q with k_d ≥ 1, any rational constant plus ∑_{d∈D} ∑_{n≥1} f_d(φ(n) mod 2^{k_d})/t^{dn} is rational exactly when every f_d is constant on the even residue classes. The proof isolates a nonzero coefficient at the least active dilation by the Chinese remainder theorem and Dirichlet's theorem. When all coefficients of a relation have one sign, Erdős's Lemma 1 (1957) already excludes it; the mixed-sign case is not covered by that criterion without a further spacing hypothesis.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":"prop:dilations","readme_headline":false,"declarations":[{"name":"linearIndependent_one_and_least_residue_values","module":"ErdosProblems/Erdos249/FiniteDilationLinearIndependent.lean","line":42},{"name":"rational_mixed_moduli_with_constant_iff","module":"ErdosProblems/Erdos249/FiniteDilationMixedModuli.lean","line":95}]},{"id":"dyadic_totient_certificate_interface","label":"Dyadic totient-kernel finite-level rank","status":"unconditional progress","statement":"For every e, a CRT–Dirichlet construction produces a separated nonzero evaluation minor for the canonical dyadic totient-kernel family, which is linearly independent with exact span dimension 2^e + 1. The same Lean development proves that the full dyadic kernel span is not finite-dimensional, but that corollary is not an independent contribution: it is equivalent, for this integer-valued sequence module, to Coons's non-2-regularity theorem. Even residues reduce to odd-core channels, and the compressed-adjoint boundary remains available. These coefficient-side theorems do not prove that S is irrational.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":"thm:kkernelrank","readme_headline":false,"declarations":[{"name":"exists_separatedMinorCertificate_totientAffineOddFamily","module":"Erdos249257/TotientMahlerDefect.lean","line":882},{"name":"linearIndependent_canonicalTotientKernelFamily","module":"Erdos249257/TotientMahlerDefect.lean","line":935},{"name":"finrank_canonicalTotientKernel_eq","module":"Erdos249257/TotientMahlerDefect.lean","line":989},{"name":"finrank_totientKernelThroughLevelFamily_eq","module":"Erdos249257/TotientMahlerDefect.lean","line":1084},{"name":"not_finiteDimensional_span_fullTotientKernel","module":"Erdos249257/TotientMahlerDefect.lean","line":1145},{"name":"totientKernel_even_residue_reduce","module":"Erdos249257/TotientMahlerDefect.lean","line":169},{"name":"false_of_compressedAdjointCertificate","module":"Erdos249257/TotientMahlerDefect.lean","line":1183}]},{"id":"all_base_totient_kernel_index_reduction","label":"All-base totient-kernel reduction, spanning, and conditional rank","status":"proved here","statement":"For every integer base k at least 2, Lean checks the zero-residue scaling identity, the exact nonzero scalar used in composite-base residue reduction, unique quotient/nonzero-digit coordinates, and cardinality k^e + 1 of the canonical finite-level index. It proves unconditionally that this canonical family spans the complete kernel through level e, and proves exact rank k^e + 1 under the explicit hypothesis that the family is linearly independent. Lean does not prove that hypothesis here; the note's evaluation-matrix argument and AllBaseTotientKernel.lean at public checkpoint 25ef6245d15a discharge it by an independent finite-determinant construction, so the unconditional basis and rank theorem takes Martin's positive-density theorem as prior art rather than as an input.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":"thm:kkernelrank","readme_headline":true,"declarations":[{"name":"totient_pow_mul_eq","module":"Erdos249257/TotientKernelReduction.lean","line":60},{"name":"totient_pow_mul_affine_gcd_cross_eq","module":"Erdos249257/TotientKernelReduction.lean","line":117},{"name":"totientKernelSectionIndex_spec","module":"Erdos249257/TotientKernelIndex.lean","line":124},{"name":"totientKernelResidueAtLevel_injective","module":"Erdos249257/TotientKernelIndex.lean","line":190},{"name":"existsUnique_totientKernelResidueAtLevel","module":"Erdos249257/TotientKernelIndex.lean","line":213},{"name":"card_totientKernelSectionLevel","module":"Erdos249257/TotientKernelIndex.lean","line":241},{"name":"card_totientKernelSectionIndex","module":"Erdos249257/TotientKernelIndex.lean","line":249},{"name":"card_totientKernelIndex_eq_two_add_sum","module":"Erdos249257/TotientKernelIndex.lean","line":264},{"name":"card_totientKernelIndex","module":"Erdos249257/TotientKernelIndex.lean","line":277},{"name":"totientKernelReductionScalar_ne_zero","module":"Erdos249257/TotientKernelConditional.lean","line":35},{"name":"allBaseTotientKernelSeq_mul_residue_step","module":"Erdos249257/TotientKernelConditional.lean","line":54},{"name":"allBaseTotientKernelSeq_mem_span_canonical_of_le","module":"Erdos249257/TotientKernelConditional.lean","line":102},{"name":"span_allBaseTotientKernelThroughLevelFamily_eq_canonical","module":"Erdos249257/TotientKernelConditional.lean","line":185},{"name":"finrank_canonicalAllBaseTotientKernel_eq_of_linearIndependent","module":"Erdos249257/TotientKernelConditional.lean","line":203},{"name":"finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent","module":"Erdos249257/TotientKernelConditional.lean","line":215}]},{"id":"residual_gauge_obstruction","label":"Residual-gauge obstruction for monomial minors","status":"proved here","statement":"For a residual-weighted monomial matrix, common nonzero residual weights act by a right diagonal gauge and preserve determinant nonvanishing. Under the locked inverse-phase gauge, an exponent-one row is identically 1 even when the determinant remains nonzero; allowing row-dependent residuals reconstructs a consecutive-power matrix with zeroth row identically 1. Thus residual-blind rank, determinant, and conditioning criteria alone cannot exclude the target. This does not rule out a criterion with an additional arithmetic coupling identity, and it gives no irrationality criterion.","paper_label":"res:residualgauge","readme_headline":false,"declarations":[{"name":"det_residualMonomialMatrix_ne_zero_iff","module":"Erdos249257/ResidualGaugeObstruction.lean","line":55},{"name":"locked_reconstruction_preserves_nonzero_minor","module":"Erdos249257/ResidualGaugeObstruction.lean","line":94},{"name":"rowReconstruction_zero_row_one","module":"Erdos249257/ResidualGaugeObstruction.lean","line":145}]},{"id":"tail_orbit_rigidity","label":"Tempered tail-orbit rigidity criterion","status":"proved here","statement":"For coefficients c(n) ≤ n, the binary series ∑ c(n)/2ⁿ is rational exactly when a tempered integer carry orbit exists, and every tempered orbit equals the scaled tail.","paper_label":"res:rigidity","readme_headline":false,"declarations":[{"name":"binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit","module":"Erdos249257/GenericTailOrbitRigidity.lean","line":426}]},{"id":"boolean_mobius_carry","label":"Boolean–Möbius carry certificate equivalence","status":"proved here","statement":"A nonempty normalised support, one omitting 0, with series value p/q exists exactly when a positive, square-root-bounded integer orbit with Boolean Möbius quotient exists; the support is reconstructed from the certificate.","paper_label":"res:carrycert","readme_headline":false,"declarations":[{"name":"exists_normalized_support_fraction_iff_exists_booleanMobiusCarry","module":"Erdos249257/BooleanMobiusCarry.lean","line":949}]},{"id":"reciprocal_mass_bound","label":"Rationality-forced reciprocal-mass lower bound","status":"proved here","statement":"A rational support-series value with odd denominator part v forces reciprocal mass at least 1/ord_v(2), or divergence; an infinite support with dyadic-rational value has reciprocal mass divergent or greater than one.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:orderwrap","readme_headline":false,"declarations":[{"name":"one_div_oddOrder_le_reciprocalMass_of_support_fraction","module":"Erdos249257/RationalSupportCarrySkeleton.lean","line":1480}]},{"id":"unbounded_carry_states","label":"Carry states are unbounded over infinite rational-valued supports","status":"proved here","statement":"An infinite support with rational series value has an unbounded positive carry state, closing finite-state readings of the carry system.","paper_label":"res:unbounded","readme_headline":false,"declarations":[{"name":"exists_unbounded_shifted_odd_tail_nat_state_of_support_fraction","module":"Erdos249257/RationalSupportCarrySkeleton.lean","line":2383}]},{"id":"sublog_zero_windows","label":"Sublogarithmic divisor-coverage zero windows","status":"proved here","statement":"A rational support-series value forces every zero window in the divisor-count coefficient to have length at most ε·log₂(N+1) + B, where B ≥ 0 is built from ε, c and v but is bound after the support and numerator, so no single constant is claimed uniform across supports. This constrains coverage; it does not solve universal #257.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:sublog","readme_headline":true,"declarations":[{"name":"supportCoeffZeroWindow_length_le_eps_logb_add","module":"Erdos249257/SublogDivisorCoverage.lean","line":392}]},{"id":"greedy_achievement_geometry","label":"Greedy geometry of the Mersenne achievement set","status":"unconditional progress","statement":"The Mersenne achievement set is compact, perfect, totally disconnected, nowhere dense, and has Lebesgue measure one. Membership is equivalent to greedy survival at every level; for the target 1/2 it is exactly equivalent to the canonical greedy orbit omitting infinitely many exponents. No finite normalized support has value 1/2, so positive membership would automatically produce an infinite rational-valued support and refute universal #257. Finite exact-rational death certificates soundly prove non-membership (3/4 is excluded at level one). The strict-tail Cantor geometry is recorded by Kovač–Tao; no novelty claim is made for the formalised refinements.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:greedy","readme_headline":true,"declarations":[{"name":"volume_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":996},{"name":"perfect_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":1656},{"name":"isTotallyDisconnected_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":1672},{"name":"isNowhereDense_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":1681},{"name":"mem_mersenneAchievementSet_iff_greedy_survival","module":"Erdos249257/GreedyAchievementSet.lean","line":1458},{"name":"three_fourths_not_mem_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":1784},{"name":"half_mem_mersenneAchievementSet_iff_greedySkippedSupport_infinite","module":"Erdos249257/GreedyAchievementSet.lean","line":2583},{"name":"finite_boolSupport_ne_half","module":"Erdos249257/HalfCarryReachability.lean","line":589}]},{"id":"half_greedy_two_thirds_band","label":"Exact band localization of the final skip between two takes","status":"proved here","statement":"Suppose a take at rank b is followed by the next take at c > b+1, put q = 2^b − 1 and m = 2^(c−1), and write the pre-take residual as 1/R with 0 < R < q. The final intervening skip is dyadically unsafe exactly when q(m−1)/(q+m−1) < R < qm/(q+m), an interval of exact width q^2/((q+m)(q+m−1)). For one intervening skip this specializes to the two-thirds band (q(2q+1)/(3q+1), 2q(q+1)/(3q+2)), whose width is below 1/9; integral reciprocals never lie there, and the corresponding positive odd numerator is at least 7. These are local constraints: no theorem here says that the actual greedy orbit for 1/2 avoids a band or identifies the surviving last-producer cells with unsafe skips.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:twothirdsband","readme_headline":false,"declarations":[{"name":"postTakeUnsafeAt_iff_band","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":88},{"name":"band_width_general","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":104},{"name":"singletonUnsafe_iff_twoThirdsBand","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":127},{"name":"band_width","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":154},{"name":"band_width_lt_ninth","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":163},{"name":"not_twoThirdsBand_of_int","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":185},{"name":"seven_le_of_intBand_odd","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":231}]},{"id":"half_second_channel_reduction","label":"Exact rational second-channel reduction for the target one half","status":"conditional reduction","statement":"A decidable rational separation predicate controls the greedy Mersenne orbit for the target 1/2. Lean verifies levels 1 through 6 exactly; proving the predicate for every n >= 7 would place 1/2 in the Mersenne achievement set, equivalently would force infinitely many canonical greedy skips. The universal tail statement remains open.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:secondcriterion","readme_headline":false,"declarations":[{"name":"HalfSecondChannelSeparatedRat","module":"Erdos249257/GreedyAchievementSet.lean","line":2913},{"name":"halfSecondChannelSeparatedRat_of_pos_le_six","module":"Erdos249257/GreedyAchievementSet.lean","line":2973},{"name":"half_mem_mersenneAchievementSet_of_secondChannelSeparationRat_from_seven","module":"Erdos249257/GreedyAchievementSet.lean","line":3151},{"name":"half_mem_mersenneAchievementSet_of_skipped_twoChannelCap","module":"Erdos249257/GreedyAchievementSet.lean","line":1425},{"name":"half_mem_mersenneAchievementSet_iff_greedySkippedSupport_infinite","module":"Erdos249257/GreedyAchievementSet.lean","line":2583}]},{"id":"campbell_shift_synchronization_boundary","label":"Campbell shift-synchronization strategy boundary","status":"unconditional progress","statement":"Campbell's application of the Alford-Granville-Pomerance estimate chooses delta = 1/4, so at prime-search height X the progression modulus d must satisfy d <= X^(1/4), equivalently d^4 <= X. In the proposed last-skip synchronization, P is the fatal-window prime cap, t the unavoidable phase period, and d the combined progression modulus. Taking X = P, the requirements 2 <= d, t <= d, P <= 4t, and d^4 <= P are inconsistent. The shifted-zero interface is guarded by a putative last skip and is exactly equivalent to infinitely many actual greedy skips and to membership of 1/2 in the Mersenne achievement set. It is an endpoint restatement, not an independent Campbell input, and proves no half-membership.","paper_label":"res:campbellboundary","readme_headline":false,"declarations":[{"name":"campbellQuarterExponent_periodFreeze_noSync","module":"Erdos249257/CampbellShiftSynchronization.lean","line":299},{"name":"campbellPhaseFreeze_parameters_impossible","module":"Erdos249257/CampbellShiftSynchronization.lean","line":320},{"name":"shiftWindowZero_iff_greedyMersenneSkippedSupport_infinite","module":"Erdos249257/CampbellShiftSynchronization.lean","line":471},{"name":"shiftWindowZero_iff_half_mem_mersenneAchievementSet","module":"Erdos249257/CampbellShiftSynchronization.lean","line":488}]},{"id":"sigma_transcendence","label":"Transcendence of ∑ σ(m)/2^m","status":"cited only","statement":"Nesterenko (1996); the ladder rung L(Id). Cited for positioning, not formalised here.","declarations":[]},{"id":"squarefree_support_irrationality","label":"Irrationality of squarefree-support series at power-of-two bases","status":"cited only","statement":"Duverney–Tachiya Corollary 1.2 and Example 1.1 prove linear independence of 1 and the squarefree Lambert sums at bases 2^j (j≥1), including n=1; subtracting each rational n=1 term settles the support {n≥2:n squarefree} at every power-of-two base. Their condition |q|L≤s, with L=lcm(1,…,ℓ), here permits q=2 but does not cover integer bases that are not powers of two. This result is not formalised here.","declarations":[]},{"id":"prime_support_irrationality","label":"Irrationality of the prime-support series ∑_p 1/(2^p − 1)","status":"cited only","statement":"Tao–Teräväinen; settles the prime-support special case of #257. Only the identity bridge to ∑ ω(n)/2ⁿ is formalised, not the irrationality.","declarations":[]},{"id":"transport_curvature_reductions","label":"Transport and curvature reductions for Erdős #249","status":"conditional reduction","statement":"Curvature, fixed-three exponent transport, anchored (3,5) transport, prime-jump, and balanced three-transport observables each have exact finite non-integrality consumers whose unbounded supply would imply irrationality of the totient constant. Every supply hypothesis remains open.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"irrational_totientSeries_of_sharpCurvatureSupply","module":"Erdos249257/CurvatureCarry.lean","line":206},{"name":"irrational_totient_series_of_exponentOnlyThreeTransportSupply","module":"Erdos249257/ExponentOnlyTransport.lean","line":205},{"name":"irrational_totient_series_of_anchoredJoint35ConeSupply","module":"Erdos249257/JointExponentTransport.lean","line":217},{"name":"irrational_totient_series_of_primeJumpSharpKill_supply","module":"Erdos249257/PrimeJumpWindow.lean","line":193},{"name":"irrational_totientSeries_of_roughnessBoundaryLatticeSupply","module":"Erdos249257/ThreeTransportBoundary.lean","line":177}]},{"id":"affine_transport_annihilation","label":"Affine residue state and old-channel annihilation","status":"proved here","statement":"The exact Möbius residue kernel has a periodic affine slope-and-intercept state. First-order exponent transport and every coefficient system with vanishing zeroth and first moments annihilate all divisor channels already present at the base height; a prime jump additionally has an explicit migrated-channel correction.","paper_label":null,"readme_headline":false,"declarations":[{"name":"transportResidueKernel_eq_affineState","module":"Erdos249257/ExponentOnlyTransport.lean","line":141},{"name":"oldChannel_affine_moment_annihilation","module":"Erdos249257/JointExponentTransport.lean","line":36},{"name":"residueHeckeDefect_eq_postJump_add_migration","module":"Erdos249257/PrimeJumpMigration.lean","line":38}]},{"id":"transport_certificate_instances","label":"Verified transport-window certificates","status":"verified finite instance","bounded_domain":"Exactly the prime-jump parameter triple (H,p,L)=(12,5,15) and the balanced three-transport pair (H,L)=(60,12).","statement":"Lean verifies the sharp prime-jump certificate at (H,p,L)=(12,5,15) and the balanced three-transport certificate at (H,L)=(60,12). These bounded instances do not establish any unbounded supply.","paper_label":null,"readme_headline":false,"declarations":[{"name":"primeJumpSharpKill_twelve_five","module":"Erdos249257/PrimeJumpWindow.lean","line":186},{"name":"sharpThreeTransportCert_60_12","module":"Erdos249257/ThreeTransportBoundary.lean","line":203}]},{"id":"first_harmonic_certificate_interface","label":"First-harmonic gap implies a finite certificate","status":"conditional reduction","statement":"Under the explicit dyadic room inequality, a constant-saving first-harmonic bound on one block, or on any nonempty finite set of indices below its right endpoint, forces a certified non-integrality witness in that set. No cofinal harmonic estimate is asserted.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"exists_certifiedKill_of_first_harmonic_gap_subset","module":"Erdos249257/FirstHarmonicGap.lean","line":104},{"name":"exists_certifiedKill_of_first_harmonic_gap","module":"Erdos249257/FirstHarmonicGap.lean","line":128}]},{"id":"first_harmonic_pivot_decomposition","label":"Complex first-harmonic pivot decomposition","status":"conditional reduction","statement":"For each positive cofactor m at most sqrt(X)/2, the canonical supplier fibre is exactly the injective image of the primes p in the shifted product interval under N = mp - (L-s+1). This exact bijection does not make a supplier prime globally isolated: at X=16, L=20, s=1, m=2, N=18 the pivot is 38=2*19, while the same prime 19 divides the distinct window argument N+1=19. Removing the pivot phase gives an exact decomposition of the block first-harmonic sum into a centred supplier-fibre correlation, a supplier-fibre mean term, a bad-supplier term, and a non-supplier term. At a positive scale satisfying the explicit dyadic-room inequality, if the real part of the centred term is at most 14X/25 and the other three norms are at most X/100, X/100, and 8X/25, then the real first-harmonic sum is at most 9X/10 and yields a finite certificate. A cofinal supply of these budgets implies irrationality of S. No global supplier-prime isolation, prime-distribution estimate, or decorrelation estimate supplying the budgets is proved.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"mem_pivotFiber_iff_exists_supplierPrime","module":"Erdos249257/FirstHarmonicPivot.lean","line":299},{"name":"pivotBaseOfPrime_injective_on_supplierPrimes","module":"Erdos249257/FirstHarmonicPivot.lean","line":367},{"name":"image_pivotSupplierPrimes_eq_pivotFiber","module":"Erdos249257/FirstHarmonicPivot.lean","line":384},{"name":"supplierPrime_not_globally_isolated_counterexample","module":"Erdos249257/FirstHarmonicPivot.lean","line":394},{"name":"windowFirstExp_sum_eq_pivot_decomposition","module":"Erdos249257/FirstHarmonicPivot.lean","line":514},{"name":"PivotBudgetAt","module":"Erdos249257/FirstHarmonicPivot.lean","line":543},{"name":"first_harmonic_gap_of_pivotBudgetAt","module":"Erdos249257/FirstHarmonicPivot.lean","line":549},{"name":"DTWPivotResidualDecorrelation","module":"Erdos249257/FirstHarmonicPivot.lean","line":569},{"name":"irrational_totient_series_of_pivotResidualDecorrelation","module":"Erdos249257/FirstHarmonicPivot.lean","line":577}]},{"id":"first_harmonic_good_index_reduction","label":"Irrationality of the totient series from one first-harmonic bound on the good indices","status":"conditional reduction","statement":"At the minimal admissible depth with s = 26 and η = 1/1000, if for every h ≥ 1 there are arbitrarily large X at which the real part of the first-harmonic sum over the good indices (the assigned indices N in [X, 2X) whose cofactor m satisfies φ(m) ≥ m/1000) is at most 603X/1000, then ∑ φ(n)/2^n is irrational. For all large X more than 67X/100 indices are good, by the unconditional unassigned count and the Chebyshev excluded-cofactor count, and 603/1000 = (9/10)(67/100); the underlying interface asks only for nonempty sets of indices beyond every threshold whose average first cosine is at most 9/10 at a depth with room. The bound is open, sufficient and not known to be necessary. It needs neither the prime number theorem nor a bound on the group means, and it supersedes the prime-number-theorem route at 11X/20. No computation in the record bears on it.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"thm:goodbasegap","readme_headline":false,"declarations":[{"name":"irrational_totient_series_of_support_gap","module":"ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean","line":44},{"name":"eventually_card_pivotGoodBases_gt","module":"ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean","line":75},{"name":"irrational_totient_series_of_goodBase_gap","module":"ErdosProblems/ArgumentGraph/Results/Erdos249Endpoint.lean","line":100}]},{"id":"adjacent_phase_separation_reduction","label":"Adjacent-pair phase separation reduction","status":"conditional reduction","statement":"At scales satisfying the explicit dyadic room inequality, a squared chordal separation of at least 19/25 between two adjacent first-harmonic phases in a dyadic block forces a certified non-integrality witness at one of the two indices. A cofinal supply of such adjacent pairs implies irrationality of the totient constant; no such supply is proved.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"exists_certifiedKill_of_phasePairSeparation","module":"Erdos249257/AdjacentPhaseSeparation.lean","line":258},{"name":"irrational_totient_series_of_adjacentPhaseSeparation","module":"Erdos249257/AdjacentPhaseSeparation.lean","line":295}]},{"id":"pivot_anti_reconstruction_reduction","label":"Pivot anti-reconstruction and window-pair reduction","status":"conditional reduction","statement":"On a finite supplier fibre, sufficient counted phase-pair separation feeds a certified non-integrality witness through exact reconstruction-defect and variance identities. Cofinal pivot anti-reconstruction, reconstruction variance, or counted-pair producers imply irrationality of S. The fibre-free cofinal window-pair predicate is exactly equivalent to irrationality of S. No cofinal arithmetic producer is proved.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"exists_certifiedKill_of_pivotFiberSeparatedPairs","module":"Erdos249257/PivotAntiReconstruction.lean","line":1125},{"name":"irrational_totient_series_of_pivotAntiReconstruction","module":"Erdos249257/PivotAntiReconstruction.lean","line":1523},{"name":"dtwWindowSeparatedPairs_iff_irrational_totient_series","module":"Erdos249257/PivotAntiReconstruction.lean","line":1765}]},{"id":"fixed_precision_transport_no_go","label":"Fixed-precision valuation-unit no-go theorem","status":"unconditional progress","statement":"At every fixed positive precision, every finite word of odd valuation-unit symbols admits a compatible carry orbit from an arbitrary incoming integer state, with each successive state within the corresponding valuation-unit radius. Thus bounded local signatures alone cannot exclude all finite centred endpoints; growing precision or global information remains necessary for an endpoint argument.","paper_label":null,"readme_headline":false,"declarations":[{"name":"fixedPrecisionTropicalNoGo","module":"Erdos249257/TropicalCurvatureCarry.lean","line":137},{"name":"vu_step_has_centred_completion","module":"Erdos249257/TropicalCurvatureCarry.lean","line":69},{"name":"vu_word_has_prefix_locked_completion","module":"Erdos249257/TropicalCurvatureCarry.lean","line":115}]},{"id":"lcm_factor_ideal_anchor_pulse_no_go","label":"LCM factor-ideal anchor-pulse no-go theorem","status":"unconditional progress","statement":"For every t ≥ 3, a nonzero synthetic affine carry has forcing letters in the principal ideal generated by φ(lcm(1,…,t)), matches every stated whole-ray anchor, and remains within the diagonal bounds while its state is a strict survivor. The same countermodel survives every finite integer shift polynomial. Thus factor ideals, whole-ray anchors, and finite-rank shift algebra alone do not force carry escape. The construction does not assert that its forcing letters are actual totient differences and does not rule out nonlinear use of fresh divisor channels.","paper_label":"res:factorideal","readme_headline":false,"declarations":[{"name":"lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient","module":"Erdos249257/LcmFactorIdealPulseObstruction.lean","line":798},{"name":"lcm_factorIdeal_sparseAnchor_not_sufficient","module":"Erdos249257/LcmFactorIdealPulseObstruction.lean","line":866},{"name":"lcmAnchorPulse_t3_letters","module":"Erdos249257/LcmFactorIdealPulseObstruction.lean","line":895}]},{"id":"actual_lcm_orbit_nonintegrality_frontier","label":"Actual LCM-orbit non-integrality frontier","status":"conditional reduction","statement":"Let H_a = lcm(1,…,2^a) and R_a = totientTail(2H_a) − totientTail(H_a). The totient series S is irrational if and only if, beyond every threshold a₀, some a ≥ a₀ has R_a ∉ ℤ. This is an exact cofinal non-integrality criterion for the actual LCM diagonal. That equivalence itself is not a quantitative separation estimate; the same source also provides a distinct sufficient route in which, cofinally in a₀, some a,q satisfy max 2 a₀ ≤ a, oddGuardedCanonicalAdjacentSuffixDepth (2^a) = 2q+1, and every integer z obeys 1/32 + actualLcmRawErrorRadius a q ≤ |actualLcmTailOrbit a − z|. This quantitative supply is not equivalent to the frontier criterion and remains unproved.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:actualorbit","readme_headline":false,"declarations":[{"name":"irrational_totientSeries_iff_actualLcmOrbitNonintegralitySupply","module":"Erdos249257/TotientActualLcmOrbitNonintegrality.lean","line":37},{"name":"irrational_totientSeries_of_actualLcmOrbitNonintegralitySupply","module":"Erdos249257/TotientActualLcmOrbitNonintegrality.lean","line":53},{"name":"irrational_totientSeries_of_actualLcmOrbitSeparationSupply","module":"Erdos249257/TotientActualLcmOrbitSeparation.lean","line":305}]},{"id":"actual_lcm_positive_corridor_top_edge","label":"Actual LCM positive corridor and top-edge boundary","status":"unconditional progress","statement":"For a ≥ 8 and J + (a + 6) < 2·2^a, the true translated LCM-orbit difference is strictly positive. If it is represented by an integer and the dyadic modulus has room for the directed endpoint strip, the carry survivor is strictly negative and the discrepancy residue is exactly the top-edge representative 2^K − carry_K, strictly between the strip’s lower edge and 2^K. Thus the positive corridor eliminates the nonnegative true survivor but does not itself produce a central-band contradiction; an independent arithmetic exclusion of the top boundary is still required.","paper_label":"res:actualsign","readme_headline":false,"declarations":[{"name":"actualLcmTailDiff_shift_pos","module":"Erdos249257/TotientActualLcmOrbitSign.lean","line":39},{"name":"actualLcm_trueEndpointSurvivor_neg","module":"Erdos249257/TotientActualLcmOrbitSign.lean","line":172},{"name":"actualLcm_integral_forces_topEdgeResidue","module":"Erdos249257/TotientActualLcmOrbitSign.lean","line":211}]},{"id":"strict_prime_tail_orbit_gap","label":"Tail-orbit phase-density and natural-prime strict-gap reductions","status":"conditional reduction","statement":"A uniform 11/100 nonpositive-phase density on arbitrarily late dyadic blocks yields an 89/100 first-harmonic saving and hence a finite certificate; alternatively, cofinally many natural prime indices with tail-orbit real part below 9/10 yield adaptive pivot escape. The exact initial phase is the scaled-angle character of (2^h - 1) times the series, and the tail phase follows a power-of-two squaring orbit from it; phase 1 is equivalent to an integral actual tail difference, while an entered dyadic root obstructs the strict-gap producer.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"TotientTailOrbitBlockGap","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":25},{"name":"TotientTailOrbitNonpositiveBlockDensity","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":35},{"name":"tailOrbitBlockGap_of_nonpositiveBlockDensity","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":45},{"name":"irrational_totient_series_of_tailOrbitBlockGap","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":96},{"name":"DTWNaturalPrimeTailOrbitStrictGap","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":157},{"name":"naturalPivotPointEscape_of_naturalPrimeTailOrbitStrictGap","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":485},{"name":"irrational_totient_series_of_naturalPrimeTailOrbitStrictGap","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":524},{"name":"tailOrbitFirstExp_succ","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":167},{"name":"tailOrbitFirstExp_add","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":191},{"name":"tailOrbitFirstExp_eq_initial_pow_two","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":201},{"name":"naturalPrimeTailOrbitStrictGap_iff_initial_phase","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":278},{"name":"tailOrbitFirstExp_eq_one_of_le","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":294},{"name":"not_naturalPrimeTailOrbitStrictGap_of_phase_one","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":308},{"name":"not_naturalPrimeTailOrbitStrictGap_of_dyadic_root","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":322},{"name":"tailOrbitFirstExp_zero_eq_scaled_angle","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":209},{"name":"tailOrbitFirstExp_eq_one_iff_tail_diff_mem_int","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":221},{"name":"exists_tailOrbitFirstExp_zero_pow_two_eq_one_iff_dyadic","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":249},{"name":"tailOrbitFirstExp_zero_pow_two_ne_one_upto_sixteen","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":333},{"name":"cofinally_tailOrbitFirstExp_re_nonpos_of_not_dyadic","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":360},{"name":"naturalPrimeTailOrbitStrictGap_of_cofinal_nonpositive_prime_shift","module":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","line":465}]},{"id":"terminal_scaled_vanishing_half_countermodel","label":"Conditional terminal-scaled-vanishing half countermodel","status":"conditional reduction","statement":"Any HalfTerminalOnlyScaledVanishingSequence—finite words avoiding ranks 0 and 1, depths tending to infinity, and terminal carry divided by 2^M tending to zero—produces an infinite support A with ∑_{a∈A} 1/(2^a − 1) = 1/2. Hence that antecedent would refute universal Erdős #257, but it is not constructed here and does not itself settle half-membership.","remaining_open_proposition_ids":["remaining_open.half_value_membership","remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:terminalhalf","readme_headline":false,"declarations":[{"name":"half_mem_mersenneAchievementSet_of_terminalScaledVanishing","module":"Erdos249257/TerminalOnlyScaledVanishing.lean","line":165},{"name":"exists_infinite_support_half_of_terminalScaledVanishing","module":"Erdos249257/TerminalOnlyScaledVanishing.lean","line":221},{"name":"exists_rational_half_counterexample_of_terminalScaledVanishing","module":"ErdosProblems/Erdos257/HalfCounterexampleFrontier.lean","line":31},{"name":"not_universal_of_terminalScaledVanishing","module":"ErdosProblems/Erdos257/HalfCounterexampleFrontier.lean","line":39}]},{"id":"cofinal_cylinder_half_countermodel","label":"Conditional cofinal-cylinder half countermodel","status":"conditional reduction","statement":"If for every N there are M,K with max(N,1) ≤ M and a nonempty CylinderStage K M, then an infinite positive support A satisfies ∑_{a∈A} 1/(2^a − 1) = 1/2. This is a conditional countermodel frontier: the cofinal full-cylinder family is not constructed, so neither half-membership nor universal Erdős #257 is settled.","remaining_open_proposition_ids":["remaining_open.half_value_membership","remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:cylinderhalf","readme_headline":false,"declarations":[{"name":"cofinalTerminalOnlyStrip_of_cofinalCylinderStages","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":264},{"name":"exists_infinite_support_half_of_cofinalCylinderStages","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":277},{"name":"exists_infinite_positive_support_half_of_cofinalCylinderStages","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":287}]},{"id":"erdos_1049","label":"Erdős #1049: irrationality of F(t) at rational bases","status":"open","statement":"For natural numbers 0 < b < a in the Zudilin contour region, Lean proves that F(a/b) is irrational with irrationality exponent at most rationalBaseMeasureBound a b (rational_base_region, rational_base_measure), from the constructed polynomial forms and analytic estimates; in particular F((31/4)^r) is irrational for every positive integer r (thirtyone_four_powers). The universal rational-base assertion remains open; 3/2 lies outside this region and is unresolved here.","remaining_open_proposition_ids":["remaining_open.erdos_1049_irrationality"],"paper_label":null,"readme_headline":false,"declarations":[]},{"id":"three_halves_coordinatewise_corridor_no_go","label":"Base 3/2 coordinatewise corridor exclusion","status":"unconditional progress","statement":"With N ≥ 1 and K ≥ 1, the six-condition CoordinatewiseCorridor at base 3/2 is impossible for every natural Q and digit. This excludes only that literal coordinatewise clearing architecture, not rationality or irrationality of F(3/2) or any other proof architecture.","remaining_open_proposition_ids":["remaining_open.erdos_1049_irrationality"],"paper_label":"long1049:res:nocorridor","readme_headline":false,"declarations":[{"name":"threeHalves_no_coordinatewiseCorridor","module":"ErdosProblems/Erdos1049/RationalBaseLambert.lean","line":155}]},{"id":"rational_base_cleared_tail_recurrence","label":"Exact rational-base cleared-tail recurrence","status":"unconditional progress","statement":"For rational r, s, B, and F, a coefficient sequence coeff : ℕ → ℚ, and r ≠ 0, the denominator-cleared tail state satisfies rationalBaseClearedTailQ (r, s, B, F, coeff) (N + 1) = r · rationalBaseClearedTailQ (r, s, B, F, coeff) N − B · coeff (N + 1) · s^(N + 1) for every N. The identity is not derived from rationality and is not itself a rationality contradiction.","remaining_open_proposition_ids":["remaining_open.erdos_1049_irrationality"],"paper_label":"long1049:res:tailrec","readme_headline":false,"declarations":[{"name":"rationalBaseClearedTailQ_succ","module":"ErdosProblems/Erdos1049/RationalBaseLambert.lean","line":187}]},{"id":"rational_base_forcing_growth","label":"Rational-base forcing-term growth boundary","status":"unconditional progress","statement":"For natural s, B, coeff, and N, if s ≥ 2, B ≥ 1, and coeff(N + 1) ≥ 1, then rationalBaseForcingNat s B coeff N ≥ 2^(N + 1); at s = 1 the same quantity is exactly B · coeff(N + 1). This is an elementary forcing-term bound, not a bounded-state contradiction or endpoint result.","remaining_open_proposition_ids":["remaining_open.erdos_1049_irrationality"],"paper_label":"long1049:res:forcing","readme_headline":false,"declarations":[{"name":"twoPow_le_rationalBaseForcingNat","module":"ErdosProblems/Erdos1049/RationalBaseLambert.lean","line":204},{"name":"rationalBaseForcingNat_one","module":"ErdosProblems/Erdos1049/RationalBaseLambert.lean","line":218}]},{"id":"van_assche_qapery_diagonal_no_go","label":"Van Assche q-Apéry diagonal recurrence non-transfer","status":"unconditional progress","statement":"For every real p > 1, the n = 0 residual obtained by applying the Amdeberhan–Zeilberger q-Apéry operator to Van Assche's moving diagonal is the displayed strictly negative factorisation, hence is nonzero. This rules out that selected diagonal for that recurrence at n = 0; it proves no general recurrence, endpoint, or #1049 irrationality result.","remaining_open_proposition_ids":["remaining_open.erdos_1049_irrationality"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"vanAsscheQAperyResidualAtZero_eq","module":"ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean","line":67},{"name":"vanAsscheQAperyResidualAtZero_ne_zero","module":"ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean","line":94}]},{"id":"erdos_243","label":"Erdős #243: eventual Sylvester recurrence","status":"open","statement":"Lean proves that if a is a strictly increasing sequence of positive integers with n^λ(a(n)²/a(n+1) − 1 − λ/n) → 0, then the reciprocal sum ∑ 1/a(n) is irrational, for every non-integer λ > 1, where Lean also derives convergence (nonintegral_regular_rate_irrational), and for λ = 3 whenever the sum converges (cubic_rate_irrationality_unconditional). Both theorems are zero-indexed; the paper transfers them to one-based indexing by an ordinary finite-prefix argument. Without such a rate the eventual Sylvester recurrence assertion remains open: the bounded-rise barrier and the conditional negative-part reductions leave the mixed-sign unbounded regime unresolved.","remaining_open_proposition_ids":["remaining_open.erdos_243_eventual_recurrence"],"paper_label":null,"readme_headline":false,"declarations":[]},{"id":"bounded_rise_fresh_modulus_barrier","label":"Bounded-rise fresh-modulus coprimality barrier","status":"unconditional progress","statement":"For natural sequences u and m, if u tends to infinity with upward increments bounded by a fixed positive B, the moduli m are eventually greater than 1 and pairwise coprime, and each earlier m(i) is coprime to every later u(t), then the hypotheses are inconsistent. This is a reusable state-level obstruction, not a proof of the full Erdős #243 assertion.","remaining_open_proposition_ids":["remaining_open.erdos_243_eventual_recurrence"],"paper_label":"res:barrier","readme_headline":false,"declarations":[{"name":"no_boundedRise_of_tailAvoidance","module":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","line":903}]},{"id":"eventually_periodic_negative_no_go","label":"Eventually-periodic negative-orbit exclusion","status":"unconditional progress","statement":"For natural sequences a, D, C, and e with 2 ≤ a(n), positive e(N+n) < a(N+n), the exact recurrences D(n+1) = a(n)·D(n) and C(n+1) = C(n) + e(n), the shape equation D(n) + e(n) = (a(n)−1)·C(n), and a positive-period, positive-drift tail e(N+n+h) = e(N+n) and C(N+n+h) = C(N+n) + M with h > 0 and M > 0, the hypotheses are inconsistent. This is only the eventually-periodic negative-magnitude obstruction; mixed-sign and unbounded-negative behaviour remain outside the claim and the full Erdős #243 endpoint remains open.","remaining_open_proposition_ids":["remaining_open.erdos_243_eventual_recurrence"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"no_eventuallyPeriodicNegative_orbit","module":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","line":802}]},{"id":"bounded_negative_exclusion","label":"Cofinally bounded negative-magnitude exclusion","status":"conditional reduction","statement":"For natural sequences a, C, D, and magnitude and B > 0 satisfying 1 < a(n), C(n) > 0, C(n+1) + D(n) = a(n)·C(n), D(n+1) = a(n)·D(n), bounded rise C(n+1) ≤ C(n) + B, magnitude(n) > 0, division-free normalised vanishing (for every K, eventually K·magnitude(n) < C(n)), and a cofinal negative branch with 0 < e ≤ B and D(t) + e = (a(t)−1)·C(t), the hypotheses are inconsistent. This conditional no-go does not decide the full Erdős #243 endpoint or the mixed-sign and unbounded-negative regimes.","remaining_open_proposition_ids":["remaining_open.erdos_243_eventual_recurrence"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"no_cofinallyBoundedNegative_of_normalizedVanishes","module":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","line":1754}]},{"id":"cubic_survivor_phase_density","label":"Phase-specific density obstruction for the two cubic survivor profiles","status":"unconditional progress","statement":"For either surviving cubic profile Q_{12,c}(n)=2n(n+1)(n+2)+c with c=1 or c=-1, the exact tail recurrences force a mismatch in every sufficiently late four-index window at the corresponding phase modulo 7. The mismatch set therefore has lower density at least 1/7, and hence at least the 1/28 stated in the long paper. No phase-free or arbitrary-profile density statement is claimed.","remaining_open_proposition_ids":["remaining_open.erdos_243_eventual_recurrence"],"paper_label":"long243:res:modseven","readme_headline":false,"declarations":[{"name":"plus_phase_hit","module":"ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean","line":117},{"name":"minus_phase_hit","module":"ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean","line":141},{"name":"plus_profile_one_seventh","module":"ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean","line":165},{"name":"minus_profile_one_seventh","module":"ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean","line":175},{"name":"lowerDensityAtLeast_of_linear_bound","module":"ErdosProblems/Erdos243/PaperCompleteR11/WindowIncidence.lean","line":131}]},{"id":"cubic_rate_irrationality_unconditional","label":"Cubic-rate reciprocal-sum irrationality under the stated asymptotic","status":"unconditional progress","statement":"For a zero-indexed strictly increasing sequence of positive natural numbers a, if n³(a(n)²/a(n+1) − (1 + 3/n)) tends to zero and the reciprocal series has sum S, then S is irrational. Lean proves the square-specialisation input through the Dedekind-zeta simple pole, so this theorem has no Chebotarev premise. The paper states the ordinary one-based bridge. The quantitative cubic-rate hypothesis is essential here; this does not settle the original eventual Sylvester recurrence assertion.","remaining_open_proposition_ids":["remaining_open.erdos_243_eventual_recurrence"],"paper_label":"res:cubicrate","readme_headline":false,"declarations":[{"name":"cubic_rate_irrationality_unconditional","module":"ErdosProblems/Erdos243/PaperCompleteR21/SquareSpecialisationUnconditional.lean","line":70}]},{"id":"nonintegral_regular_rate_irrationality","label":"Nonintegral regular-rate reciprocal-sum irrationality","status":"unconditional progress","statement":"For a zero-indexed strictly increasing sequence of positive natural numbers a and a real noninteger lambda greater than one, if n^lambda (a(n)^2/a(n+1) - (1 + lambda/n)) tends to zero, then the reciprocal series converges and its sum is irrational. The Lean theorem derives summability from the rate; the one-based paper form uses an ordinary finite-prefix bridge. The little-o remainder is essential to the integer-extraction argument. This does not settle the unrestricted Erdős #243 problem.","remaining_open_proposition_ids":["remaining_open.erdos_243_eventual_recurrence"],"paper_label":"long243:res:nonintegralrate","readme_headline":false,"declarations":[{"name":"nonintegral_regular_rate_irrational","module":"ErdosProblems/Erdos243/PaperCompleteR21/NonintegralRegularRate.lean","line":196}]},{"id":"finite_period_noncollapse","label":"Finite-support denominator period noncollapse","status":"unconditional progress","statement":"For every finite nonempty support F with 0 ∉ F and integer base b ≥ 2, the reduced denominator of the literal finite sum ∑_{n∈F} 1/(bⁿ−1) is coprime to b and the multiplicative order of b modulo that denominator equals F.lcm id. This finite-support theorem supplies no unbounded support certificate and does not settle the universal infinite-support Erdős #257 endpoint.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:period","readme_headline":false,"declarations":[{"name":"finite_period_noncollapse_rat_den","module":"Erdos249257/CertificateKernel.lean","line":5246}]},{"id":"all_base_totient_kernel_conditional_rank","label":"Conditional exact rank of the all-base totient kernel","status":"conditional reduction","statement":"For integers k and e with 2 ≤ k and 1 ≤ e, if the explicit canonical all-base family is linearly independent over ℚ, then the span of the actual all-base totient kernel through level e has dimension k^e + 1. This reusable interface retains the independence input; AllBaseTotientKernel.lean separately discharges it at public checkpoint 25ef6245d15a. As before, this coefficient-space result does not prove irrationality of the binary totient series.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent","module":"Erdos249257/TotientKernelConditional.lean","line":215}]},{"id":"erdos_249","label":"Erdős #249: irrationality of S = ∑ φ(n)/2ⁿ","status":"open","statement":"Lean proves that S = ∑ φ(n)/2ⁿ is irrational if, for every h ≥ 1, there are arbitrarily large X at which the real part of the first-harmonic sum over the good indices is at most 603X/1000 (irrational_totient_series_of_goodBase_gap); the good indices are the assigned indices N in [X, 2X) at the minimal admissible depth with s = 26 whose cofactor m satisfies φ(m) ≥ m/1000. For all large X more than 67X/100 indices are good (eventually_card_pivotGoodBases_gt), and 603/1000 = (9/10)(67/100). Lean also proves that S is irrational if and only if certified non-integrality witnesses occur at unbounded parameters (irrational_totient_series_iff_certificate_supply). The 603X/1000 bound and that unbounded certificate supply are open, so S is not proved irrational.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality","remaining_open.unbounded_certificate_supply"],"declarations":[]},{"id":"universal_257","label":"Erdős #257, universal form","status":"open","statement":"Lean proves that for every integer base b ≥ 2 and every finite nonempty set P of primes, with h(a) the largest divisor of a supported on P, every infinite set A of positive integers with ∑_{a∈A} h(a)/(a(b^{h(a)} − 1)) finite has ∑_{n∈A} 1/(bⁿ − 1) irrational (divisibilityWeightedClaim). Lean also proves irrationality at every such base for every infinite A with ∑_{a∈A} 1/a finite (irrational_erdosSupportSeries_of_summable_reciprocal), and exhibits sets with divergent reciprocal sum all of whose infinite subsets have irrational series at every such base (finite_monotone_witness_rule_realised). Irrationality of ∑_{n∈A} 1/(2ⁿ − 1) for every infinite A remains open.","remaining_open_proposition_ids":["remaining_open.half_value_membership","remaining_open.universal_257_all_infinite_supports"],"declarations":[]},{"id":"weighted_phase_carry_finite_observer","label":"Weighted-phase carry split and finite realised span for #269","status":"unconditional progress","statement":"The exact weighted-phase carry recurrence splits into a finite residue digit, each of the two coordinates staying inside its declared interval, and an integral quotient coboundary that is not controlled. An explicit factorisation through a finite-dimensional function-faithful observer then forces the realised symbolic span to be finite. Nothing here proves a finite-state quotient or a literal infinite realised span, and no rationality or irrationality conclusion follows.","remaining_open_proposition_ids":["remaining_open.erdos_269_two_dimensional_representation"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"carry_eq_residueDigit_add_coboundary","module":"ErdosProblems/Erdos269/WeightedPhaseCarry.lean","line":109},{"name":"carryResidue_mem_interval","module":"ErdosProblems/Erdos269/WeightedPhaseCarry.lean","line":150},{"name":"residueDigit_mem_interval","module":"ErdosProblems/Erdos269/WeightedPhaseCarry.lean","line":157},{"name":"finite_realisedSpan_of_factorisation","module":"ErdosProblems/Erdos269/WeightedPhaseCarry.lean","line":334}]},{"id":"totient_v8_conditional_peripheral_assembly","label":"Conditional peripheral-and-centred assembly for the #249 first-harmonic route","status":"conditional reduction","statement":"Under eventual peripheral estimates and a cofinal centered estimate at one shared depth function, the exact finite decomposition supplies the first-harmonic gap consumer and hence the conditional irrationality route. Both analytic suppliers remain explicit hypotheses; no unconditional Erdős #249 conclusion follows.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"card_pivotFiber_eq_card_supplierPrimes","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralAssembly.lean","line":12},{"name":"dtw_of_eventual_peripheral_and_cofinal_centered","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralAssembly.lean","line":33},{"name":"irrational_of_eventual_peripheral_and_cofinal_centered","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralAssembly.lean","line":54}]},{"id":"totient_v8_finite_arithmetic_threshold","label":"Exact finite arithmetic and run-threshold checks for #249","status":"unconditional progress","statement":"The four displayed discrepancy values at (h,X,L)=(1,4,9), their finite room and certified kills, the Farey denominator bound, and the corrected finite run threshold hold exactly. These finite facts do not supply a cofinal certificate producer.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"discrepancy_1_4_9","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":9},{"name":"discrepancy_1_5_9","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":10},{"name":"discrepancy_1_6_9","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":11},{"name":"discrepancy_1_7_9","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":12},{"name":"finite_room","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":13},{"name":"four_certified_kills","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":14},{"name":"farey_denominator_bound","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":19},{"name":"run_length_counterexample","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":35},{"name":"run_threshold_iff","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean","line":39}]},{"id":"totient_v8_generic_centering_countermodel","label":"Generic centering countermodel for the #249 decomposition","status":"unconditional progress","statement":"A two-point complex model has vanishing peripheral mean contributions while its centered contribution remains nonzero. This refutes the generic inference from peripheral cancellation to a centered budget; it is not a counterexample to the arithmetic totient-centered predicate.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"twoPhase_unit","module":"ErdosProblems/Erdos249/TypeBReturnV8/CenteringCounterexample.lean","line":14},{"name":"twoMean_zero","module":"ErdosProblems/Erdos249/TypeBReturnV8/CenteringCounterexample.lean","line":16},{"name":"twoCentered_eq_two","module":"ErdosProblems/Erdos249/TypeBReturnV8/CenteringCounterexample.lean","line":18},{"name":"twoMeanContribution_zero","module":"ErdosProblems/Erdos249/TypeBReturnV8/CenteringCounterexample.lean","line":20},{"name":"three_peripheral_zero_do_not_force_centered_budget","module":"ErdosProblems/Erdos249/TypeBReturnV8/CenteringCounterexample.lean","line":22}]},{"id":"totient_v8_one_finite_first_harmonic_instance","label":"One unconditional finite first-harmonic gap instance","status":"unconditional progress","statement":"At (h,X,L)=(1,4,9), the exact finite first-harmonic expression has the certified norm gap bounded by 21X/25. This is one finite instance and does not establish DTWFirstHarmonicNormGap or any cofinal statement.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"finite_first_harmonic_norm_gap","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteFirstHarmonic.lean","line":41},{"name":"one_unconditional_norm_gap_instance","module":"ErdosProblems/Erdos249/TypeBReturnV8/FiniteFirstHarmonic.lean","line":66}]},{"id":"totient_v8_peripheral_finite_budget_bridge","label":"Finite peripheral estimates assemble into PivotBudgetAt","status":"conditional reduction","statement":"Exact phase, residual, cardinality, and uniform fibre-mean estimates compose into PivotBudgetAt. The bridge proves neither the prime-distribution estimates nor the centered estimate required by the endpoint route.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":null,"readme_headline":false,"declarations":[{"name":"norm_pivotPhaseAt","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean","line":24},{"name":"norm_pivotResidualAt_v8","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean","line":28},{"name":"norm_pivotBadContribution_le_card","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean","line":34},{"name":"norm_pivotNonSupplierContribution_le_card","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean","line":45},{"name":"pivotGoodBases_subset_Ico","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean","line":56},{"name":"card_pivotGoodBases_le","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean","line":64},{"name":"norm_pivotFiberMeanContribution_le_of_uniform","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean","line":76},{"name":"pivotBudgetAt_of_peripheral_estimates","module":"ErdosProblems/Erdos249/TypeBReturnV8/PeripheralFiniteBridge.lean","line":105}]},{"id":"zudilin_rational_base_region","label":"Zudilin rational-base irrationality and irrationality-exponent bound","status":"formalised here","statement":"For natural numbers 0 < b < a satisfying ZudilinContourRegion a b, the Lambert value F(a/b) is irrational and its irrationality exponent is at most rationalBaseMeasureBound a b. The proof constructs the polynomial forms and proves their clearing, nonvanishing and asymptotic estimates. In particular, F((31/4)^r) is irrational for every positive natural r. These are formalizations of the stated Zudilin region and its consequences, not universal rational-base irrationality; 3/2 is outside the region.","remaining_open_proposition_ids":["remaining_open.erdos_1049_irrationality"],"paper_label":"res:rational-base-threshold","readme_headline":false,"declarations":[{"name":"rational_base_region","module":"ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean","line":113},{"name":"thirtyone_four_powers","module":"ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean","line":117},{"name":"rational_base_measure","module":"ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean","line":154}]},{"id":"zudilin_all_rank_hankel_order","label":"All-rank normalized Hankel order and leading coefficient","status":"unconditional progress","statement":"For every natural N, including N = 0, the normalized Zudilin Hankel determinant has q-order N(N-1)(2N-1)/6. Its leading coefficient satisfies the division-free identity 2^N times the coefficient = (N!)^2 (N+1)!, with the corresponding rational quotient formula. The all-row producer is proved rather than assumed. This algebraic theorem does not by itself supply the analytic estimates or irrationality at 3/2. Zudilin (2016) proves the q-order inequality; the equality is proved here.","remaining_open_proposition_ids":["remaining_open.erdos_1049_irrationality"],"paper_label":"res:zudilin-sharp-qorder","readme_headline":false,"declarations":[{"name":"zudilinSharpHankelOrderAndCoeff_all","module":"ErdosProblems/Erdos1049/AllRow/Producer.lean","line":163},{"name":"order_zudilinNormalizedHankelDet_all","module":"ErdosProblems/Erdos1049/AllRow/Producer.lean","line":173},{"name":"coeff_zudilinNormalizedHankelDet_all_rat","module":"ErdosProblems/Erdos1049/AllRow/Producer.lean","line":199}]},{"id":"coefficient_pencil_finite_rank","label":"Coefficient-pencil positivity and root interlacing through rank eight","status":"unconditional progress","statement":"For each real p > 1 and each natural rank N ≤ 8, the coefficient Hankel matrix A_N is positive definite, the polynomial det(Y A_N − B_N) splits over the reals, and every root is strictly below F(p). For adjacent ranks N and N+1 ≤ 8, the ordered root lists interlace non-strictly, with multiplicities. Lean includes N = 0; the paper states the ranks 1 through 8. No all-rank coefficient-pencil theorem, simple-root or strict-interlacing assertion, limiting result, or irrationality of F(3/2) follows from this finite proposition.","remaining_open_proposition_ids":["remaining_open.erdos_1049_irrationality"],"paper_label":"long1049:res:finite-pencil","readme_headline":false,"declarations":[{"name":"coefficientPencil_finitePencil","module":"ErdosProblems/Erdos1049/PaperR20/FinitePencilProposition.lean","line":43}]},{"declarations":[{"line":40,"module":"ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean","name":"weightedDyadicMeanTarget"},{"line":120,"module":"ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean","name":"divisibilityWeightedClaim"}],"id":"finite_prime_weighted_support","label":"Finite-prime weighted support irrationality","paper_label":"res:weighted-support","readme_headline":true,"remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"statement":"For every integer base b >= 2, every infinite positive support A with finite weighted mass for some finite nonempty prime set P has irrational support series. If a host H of positive integers has finite base-two weighted mass, then every infinite subset A of H has irrational support series at every integer base b >= 2. This is a sufficient support criterion, not the universal statement.","status":"unconditional progress"},{"declarations":[{"line":241,"module":"ErdosProblems/Erdos257/PaperCompleteR8/PositiveCoverReturn.lean","name":"strengthenedPositiveCoverClaim"}],"id":"strengthened_positive_cover_support","label":"Strengthened positive-cover support irrationality","paper_label":"thm:variable-fractional-cover","readme_headline":false,"remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"statement":"A strengthened positive divisor-cover datum with summable scaled cost makes every infinite subset of its host irrational at every integer base b >= 2. The statement retains the nonlinear cover majorant and summability hypotheses; it does not cover arbitrary supports.","status":"unconditional progress"},{"declarations":[{"line":126,"module":"ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean","name":"mixedSupportClaim"}],"id":"mixed_weighted_cover_support","label":"Mixed weighted and positive-cover support irrationality","paper_label":"res:mixed-supports","readme_headline":false,"remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"statement":"Let E have finite base-two finite-prime weighted mass and let V lie in a strengthened positive-cover host. Every infinite support A contained in E union V has irrational support series at every integer base b >= 2. The theorem uses a common observation scale; it does not assume separate irrationality conclusions for the two components.","status":"unconditional progress"},{"declarations":[{"line":395,"module":"Erdos249257/AllBaseReciprocalSupportIrrationality.lean","name":"irrational_erdosSupportSeries_of_summable_reciprocal"}],"id":"reciprocal_summable_support","label":"Reciprocal-summable support irrationality at every integer base","paper_label":"res:reciprocal-support","readme_headline":false,"remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"statement":"For every integer base b >= 2 and every infinite support A, summability of the reciprocal-support mass implies that the support series is irrational. No pairwise-coprimality hypothesis is required. Erdős (1968) stated this extension without proof.","status":"unconditional progress"},{"id":"ani_degree_seven_total_variation_counterexample","label":"Degree-seven Hausdorff-length counterexample","status":"formalised here","statement":"For ani's explicit monic complex polynomial of degree seven, whose distinct roots lie in the open unit disc, every preconnected subset of the strict unit lemniscate containing two distinct roots has one-dimensional Hausdorff measure greater than 2. In particular, Lean proves the negation and answer(False) forms of the exact Formal Conjectures path-image-length statement. The total-variation bound also remains checked. This formalises one polynomial, not the reported small-parameter family; independent human review of correspondence with the 1958 wording is not recorded.","paper_label":"res:ani-degree-seven-counterexample","readme_headline":false,"declarations":[{"name":"erdos1041_ani_degree_seven","module":"ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean","line":25},{"name":"erdos1041_counterexample_hausdorff","module":"ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","line":283},{"name":"erdos1041_hausdorff_negation","module":"ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","line":407},{"name":"erdos1041_hausdorff_answer_false","module":"ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","line":449}]},{"id":"separated_cut_algebraic_lambert_transcendence","label":"Algebraic-base transcendence on separated divisibility supports","statement":"For an infinite positive-integer host with cuts L_j < M_j, L_j tending to infinity and M_j-L_j tending to infinity, every host exponent at most L_j dividing L_j and every later exponent divisible by M_j, every infinite thinning with bounded positive integer weights has transcendental Lambert sum at every real algebraic base t > 1. This includes all divisibility chains and an explicit host of unbounded divisibility width and divergent reciprocal sum. The full number-field Subspace Theorem argument has two AI proof reviews; it is an ordinary proof with no Lean transcendence theorem, independent human review or asserted historical priority.","paper_label":"thm:separated-cuts","status":"unconditional progress","declarations":[],"readme_headline":false},{"id":"independent_clock_channel_relations","label":"Complete rational relations for integer-radix independent-clock sums","statement":"For positive rationally independent clock rates with no simultaneous positive crossings, and integer radices b_i >= 2, all rational relations among 1 and the channel sums S_i are multiples of (-1,b_1-1,...,b_d-1). The proof uses genuine late swaps with a common infinite suffix in one orbit closure, a fixed rational lattice, and density of the clock section. The explicit rates 1,sqrt(2),sqrt(3),sqrt(6) and radices 2,3,5,7 give rational span dimension four. This is an ordinary proof reviewed by AI agents, without Lean or independent human verification of the infinite theorem.","paper_label":"long269:thm:clock-relations","status":"unconditional progress","declarations":[],"readme_headline":false},{"id":"calibrated_rational_hankel_countermodel","label":"Rational-value countermodel to a Hankel-signature inference","statement":"An explicit positive atomic moment family and integer formal series have all-rank Hankel order sum_{j<N} j^2 and leading coefficient (N!)^2(N+1)!/2^N, with the same fixed-base asymptotic shape including N^(-8 xi), while xi=G_{2/3}(2/3)=1. For n>=1 the exact reduced moment denominator is 3^(31n+62)(3^(n+1)-2^(n+1))^4; the determinant denominator has logarithmic size Theta(N^3), and a proposed rank-two row-product clearer leaves denominator625. These are ordinary proofs with independent exact finite checks and AI proof reviews, not a Lean proof of the infinite family or human peer review.","paper_label":"long1049:thm:calibrated-model","status":"unconditional progress","declarations":[],"readme_headline":false},{"id":"selected_staircase_integral_height_boundary","label":"Smith index and primitive one-target border boundary","statement":"The selected staircase matrix with ones on and below the diagonal and integer r>=2 above it has Smith factors 1,r-1,...,r-1. For the selected {2,3,5} kernel, any entrywise-integral diagonal clearing has determinant magnitude at least 4^(N-1), whereas the reduced rational determinant has numerator one up to sign. A rational single-target border has primitive polynomial plus or minus (tX-s), where s/t is the reduced Schur complement. These ordinary arithmetic arguments were reviewed by AI agents; they are not the existing Lean nonvanishing theorem and have no independent human review.","paper_label":"long269:lem:staircase-smith","status":"unconditional progress","declarations":[],"readme_headline":false},{"id":"signed_totient_entire_defect_interpolation","label":"Arbitrarily slow signed interpolation preserving Dirichlet singularities","statement":"For any integer beta>=2 and positive allowance f(n) tending to infinity, an interval around the totient value is filled by integer sequences b=phi+e with 0<=b(n)<=n and |e(n)|<=f(n). One target-independent even support has upper Banach density zero, leaving any fixed prefix and odd coefficients unchanged. The construction preserves every fixed eventual coefficient and cumulative congruence, gives eventually constant progression moment defects outside common hulls of density zero, entire progression Dirichlet defects, and smooth root-of-unity radial germs with prescribed finite zero jets. The full proof includes the allowance-dependent block bound and first-harmonic rational countermodels. It is an ordinary proof with two AI proof reviews, not a Lean proof of the infinite theorem or independent human review.","paper_label":"long249:thm:signed-interpolation","status":"unconditional progress","declarations":[],"readme_headline":false}],"non_claims":[{"id":"not_erdos_257_solution","meaning":"This release does not settle the universal Erdős #257 problem"},{"id":"not_erdos_249_solution","meaning":"This release does not settle Erdős #249"},{"id":"not_publication_authority","meaning":"This release does not itself constitute a peer-reviewed publication record"},{"id":"not_private_root_equivalence","meaning":"This release is not asserted to be equivalent to any private development"},{"id":"not_provider_proof_authority","meaning":"Provider or model output is not proof authority; only the Lean kernel check is"},{"id":"not_hidden_proof_body_authority","meaning":"No unpublished proof sketch or hidden artefact carries proof authority for this release"}]},"body_format":"json_parsed","byte_parity_note":"body is the parsed object; re-serialisation differs from source bytes. Verify against sha256 by hashing the upstream file.","fidelity_note":{"why_this_is_here":"This body is quoted verbatim from the published registry, including sentences this build found to overstate the theorem they cite. It is not corrected here, because correcting a quotation would destroy the thing that makes it worth carrying.","statements_with_a_finding":["claim:concrete_seam_adapter","claim:largest_skip_gap_propagation","claim:quarter_band_endpoint_cells"],"read_the_findings_at":"/mathematical_assertions/claim_fidelity","read_the_corrections_at":"/upstream_defects"}},{"record_type":"formal_claim_control_plane_record","record_id":"lean_orientation","source_repository":"plectis-erdos","source_path":"docs/orientation.json","media_type":"application/json","licence":"Apache-2.0","copyright":"2026 Will Cook","why_an_offline_reader_needs_this":"Release tag, toolchain, and the pinned formal-source checkpoint the claims are stated against.","embedded":true,"availability":"embedded","sha256":"sha256:76b89af811a3c2898ddef0e9524992d5303b60df369c9a85c4fe93e7247e136c","source_byte_count":33423,"body":{"schema":"erdos249257-orientation/1","artifact_role":"bounded_first_read_navigation_projection","authority_posture":"navigation_projection_not_proof_authority","proof_authority":"Lean source checked by the pinned Lean kernel","release_provenance":{"posture":"self_contained_public_projection_from_a_larger_ongoing_research_workflow","meaning":"This checkout is a deliberately curated, independently verifiable public output of a larger ongoing formal-mathematics workflow. No public claim depends on private or unreleased work.","public_evidence":["pinned formal-source identity","claim and open-proposition registry","machine-readable argument and module graphs","generated declaration atlas","bounded cold-clone and adversarial checks"],"boundary":"The workflow provenance explains the release discipline; it does not imply hidden proof authority or equivalence with private work."},"mathematical_signal_first":[{"rank":1,"family_id":"finite_prime_weighted_support","selection_status":"selected","interface":"Erdos249257.ExternalVerification.divisibilityWeightedClaim","source_declaration":"ErdosProblems.Erdos257.PaperCompleteR8.divisibilityWeightedClaim","boundary":"The finite-prime weighted-summability premise is explicit in both clauses. This criterion does not cover every infinite support or settle universal Erdős #257; no novelty, priority, significance, or external human-review claim is made."},{"rank":2,"family_id":"known_irrational_supports","selection_status":"represented","interface":"Erdos249257.ExternalVerification.irrational_erdosSum_full_support","source_declaration":"Erdos249257.irrational_erdosSum_full_support","boundary":"This is the classical full-support theorem, not universal Erdos #257."},{"rank":3,"family_id":"pairwise_coprime_support","selection_status":"represented","interface":"Erdos249257.ExternalVerification.irrational_erdosSupportSeries_pairwise_coprime","source_declaration":"Erdos249257.irrational_erdosSupportSeries_pairwise_coprime","boundary":"This is the pairwise-coprime support theorem, not universal Erdos #257; the infinitude, pairwise-coprimality, and summable-reciprocal hypotheses remain explicit."},{"rank":4,"family_id":"orthogonal_petal_sunflower_reduction","selection_status":"represented","interface":"Erdos249257.ExternalVerification.irrational_erdosSupportSeries_of_orthogonalPetalBouquet","source_declaration":"Erdos249257.SupportSunflowerDichotomy.irrational_erdosSupportSeries_of_orthogonalPetalBouquet","boundary":"This is a conditional reduction: the bouquet structure and uniform tail-selection predicate remain hypotheses; neither is constructed for arbitrary supports, so this does not add an unconditional Erdos #257 support family or solve the universal problem."},{"rank":5,"family_id":"periodic_nonnegative_weight_irrationality","selection_status":"represented","interface":"Erdos249257.ExternalVerification.irrational_ratWeightSeries_eventuallyPeriodic","source_declaration":"Erdos249257.irrational_ratWeightSeries_eventuallyPeriodic","boundary":"This is the nonnegative rational eventually-periodic coefficient theorem; its nonnegativity, eventual periodicity, and positive-tail hypotheses remain explicit, and the broader mixed-sign periodic theorem is not claimed."},{"rank":6,"family_id":"actual_lcm_orbit_separation","selection_status":"represented","interface":"Erdos249257.ExternalVerification.irrational_totientSeries_of_actualLcmOrbitSeparationSupply","source_declaration":"Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_actualLcmOrbitSeparationSupply","boundary":"The cofinal separation supply is an open premise, stronger than mere non-integrality; this sufficient route is not an equivalence and leaves the punctured/top-edge boundary and the Erdős #249 endpoint open."},{"rank":7,"family_id":"first_harmonic_pivot_decomposition","selection_status":"represented","interface":"Erdos249257.ExternalVerification.irrational_totient_series_of_pivotResidualDecorrelation","source_declaration":"Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_pivotResidualDecorrelation","boundary":"The four-term decomposition and 9X/10 gap remain conditional on the explicit dyadic-room and PivotBudgetAt hypotheses; the cofinal decorrelation producer is unproved. The fibre/image lemmas do not give global supplier-prime isolation or a prime-distribution estimate, and no Erdős #249 irrationality conclusion is claimed."},{"rank":8,"family_id":"strict_prime_tail_orbit_gap","selection_status":"represented","interface":"Erdos249257.ExternalVerification.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap","source_declaration":"ErdosProblems.Erdos249.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap","boundary":"The phase-density, cofinal natural-prime strict-gap, associated uniform-margin, and cofinal nonpositive-prime-shift producers remain unproved for the actual totient orbit. Exact phase/integrality equivalence, finite dyadic-root exclusion for 1 ≤ h ≤ 16 and k ≤ 14, and non-dyadicity-to-cofinal-nonpositive-phase implications are classification or finite evidence only; they do not supply density or prime alignment, and the source does not prove actual root entry. This is a conditional reduction only: no unconditional Erdős #249 endpoint, novelty, priority, significance, or external-review claim is made. It is distinct from the finite first_harmonic_pivot_decomposition and actual_lcm_orbit_separation families."},{"rank":9,"family_id":"factorial_carry_characterisation","selection_status":"subordinate","interface":"Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_nonunit_carries","source_declaration":"ErdosProblems.Erdos68.irrational_factorialGapSeries_iff_cofinal_nonunit_carries","boundary":"The equivalence does not supply a cofinal carry producer."},{"rank":10,"family_id":"prime_gap_reformulation","selection_status":"subordinate","interface":"Erdos249257.ExternalVerification.irrational_tsum_primeDyadicTerm_iff_primeGap","source_declaration":"ErdosProblems.Erdos251.irrational_tsum_primeDyadicTerm_iff_primeGap","boundary":"The equivalence does not prove irrationality of either series."},{"rank":11,"family_id":"totient_carry_anti_compression","selection_status":"represented","interface":"Erdos249257.ExternalVerification.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank","source_declaration":"Erdos249257.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank","boundary":"This is a necessary consequence of rationality only: it gives no contradiction, no finite-rank upper bound for a rationality-supplied carry, and no irrationality proof for Erdős #249. Modulo a divisor of the multiplier, the forcing disappears and the carry becomes geometric, so quotient periodicity alone supplies no finite-ℚ-rank upper bound. The coefficient-side kernel-rank and certificate-equivalence families are not duplicated, and #249 remains open."},{"rank":12,"family_id":"half_membership_seam_classification","selection_status":"represented","interface":"Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse","source_declaration":"Erdos249257.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse","boundary":"This exact endpoint-socket classification does not supply any unbounded terminal-false or upper-or-middle occurrences, cofinal supply, skipped-support producer, or half-membership witness; finite exact rows do not establish the required global supply. It is not an unconditional or universal Erdős #257 result, and no novelty, priority, significance, or external-review claim is made."},{"rank":13,"family_id":"negative_mass_recovery","selection_status":"represented","interface":"Erdos249257.ExternalVerification.sylvesterNext_eventually_of_summable_negativeRelativeMass","source_declaration":"ErdosProblems.Erdos243.sylvesterNext_eventually_of_summable_negativeRelativeMass","boundary":"This is a conditional recovery criterion: the source theorem assumes every displayed dynamics, positivity, strict centered-step, normalized-vanishing, and summability hypothesis, and does not prove them for the original Erdős #243 orbit. Any surviving canonical orbit must have divergent normalized negative mass. It supplies no prime-specific producer, unconditional #243 endpoint, or reciprocal-tail irrationality, and makes no novelty, priority, significance, or external-review claim. It is distinct from centered_state_dynamics, which uses a uniform lower-bound hypothesis rather than summability of normalized negative mass."}],"mathematical_signal_presentation":{"authority":"docs/PALOMAR_RESULT_SHOWCASE.json::selection_contract","tiers":[{"order":1,"tier_id":"source_ranked_frontier","label":"source-ranked frontier","reader_role":"The explicitly ranked exceptional spine: completed direct results first, then the strongest conditional endpoint routes and deep structural results."},{"order":2,"tier_id":"conditional_endpoint_leverage","label":"conditional endpoint leverage","reader_role":"Exact target-facing consumers and reductions whose named producer or bridge remains open; their leverage is shown before medium machinery, with every premise adjacent."},{"order":3,"tier_id":"deep_mechanism_and_classification","label":"deep mechanism and classification","reader_role":"Reusable exact normal forms, classifications, geometric mechanisms, and structural obstructions that sharpen the problem without themselves being endpoint theorems."},{"order":4,"tier_id":"natural_friction_and_no_go","label":"natural friction and no-go","reader_role":"Countermodels and exact route obstructions that explain why tempting coefficient, finite, local, or model-specific arguments do not reach an endpoint."},{"order":5,"tier_id":"supporting_and_long_tail","label":"supporting and long tail","reader_role":"Exact supporting identities, finite interfaces, budgets, technical inputs, and pending rows retained for drilldown rather than first-contact prominence."}],"relational_placements":[{"family_id":"small_mismatch_criterion","tier_id":"conditional_endpoint_leverage","within_tier_order":1,"relative_judgement":"Below the ranked prime-gap equivalence but above the #269 carry-escape consumer, abstract dyadic-tail classification, and coefficient-only no-go: it is specialized to the actual prime-gap recurrence and identifies the precise cofinal adjacent-smallness producer, while that producer remains unproved."},{"family_id":"conditional_carry_escape","tier_id":"conditional_endpoint_leverage","within_tier_order":2,"relative_judgement":"Below the #251 small-mismatch route because the source-specific cofinal escape remains unproved, but above medium finite #269 machinery because the exact residue-window contradiction is the endpoint consumer that such machinery must feed. The actual-series reduction is given."},{"family_id":"dyadic_tail_integrality_classification","tier_id":"deep_mechanism_and_classification","within_tier_order":1,"relative_judgement":"Below the source-specific conditional endpoint consumers but above negative-method evidence: its exact denominator and all-positive-tail-shift normal forms are deeper and more reusable than a coefficient-only obstruction, while supplying no actual-prime producer."},{"family_id":"coefficient_only_no_go","tier_id":"natural_friction_and_no_go","within_tier_order":1,"relative_judgement":"Below the #251 small-mismatch route and abstract integrality classification: it is valuable natural friction against coefficient-only reasoning, but its synthetic carry stream has no infinite-limit or actual-prime-gap identification bridge."}]},"release":{"version":"0.10.0","tag":"v0.10.0","lean_toolchain":"leanprover/lean4:v4.29.1","formal_source_ref":"2a8c1667566020d123a081b6165e3ee94550566d","formal_source_public_tag":null,"formal_source_publication_state":"committed_checkpoint_pending_remote_publication"},"scale":{"module_count":1903,"declaration_count":164639,"theorem_like_count":160373,"generated_certificate_declaration_count":8171,"principal_claim_link_count":478,"kind_counts":{"abbrev":58,"def":4102,"inductive":12,"instance":14,"lemma":1448,"structure":80,"theorem":158925}},"status_taxonomy":{"proved here":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","formalised here":"Known mathematics represented and checked in Lean","unconditional progress":"Genuine theorem that does not settle the open problem","conditional reduction":"Exact implication whose remaining hypothesis is open","verified finite instance":"Kernel-checked computation at a bounded range","cited only":"External theorem used for positioning, not formalised","open":"Not proved by this release","formal statement refuted":"Lean refutes the exact Formal Conjectures statement; correspondence with the original wording awaits independent review"},"remaining_open_propositions":[{"id":"remaining_open.erdos_68_irrationality","open_target_claim":"erdos_68","statement":"Prove that S = ∑_{n≥2} 1/(n! − 1) is irrational."},{"id":"remaining_open.erdos_249_irrationality","open_target_claim":"erdos_249","statement":"Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator."},{"id":"remaining_open.erdos_251_irrationality","open_target_claim":"erdos_251","statement":"Prove that Π = ∑_{n≥1} p_n/2ⁿ is irrational; the exact tail-shift equivalence and prime-gap constraints do not supply the required cofinal escape."},{"id":"remaining_open.unbounded_certificate_supply","open_target_claim":"erdos_249","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal."},{"id":"remaining_open.half_value_membership","open_target_claim":"universal_257","statement":"Decide whether 1/2 lies in the Mersenne achievement set. Yes refutes universal #257; no closes only this route."},{"id":"remaining_open.twenty_one_permanent_affine_supercapacity","open_target_claim":"universal_257","statement":"Decide the exact 1/21 branch: equivalently, exclude the fatal/cofinite/aligned branch, contradict its eventual permanent affine-supercapacity recurrence, prove cofinally bounded scaled greedy remainders, or force an unbounded sequence of closed canonical quotient rows. A positive membership result would refute universal #257; no such producer is proved."},{"id":"remaining_open.universal_257_all_infinite_supports","open_target_claim":"universal_257","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here."},{"id":"remaining_open.erdos_269_three_prime_irrationality","open_target_claim":"erdos_269","statement":"Determine irrationality of the running-LCM reciprocal sum for finite prime sets of size at least three. For {2,3,5}, the rationality-to-carry bridge is established and cofinal local-window escape remains open."},{"id":"remaining_open.erdos_1041_lemniscate_connection","open_target_claim":"erdos_1041","statement":"Independently review whether the 1958 curve-length wording matches the exact Formal Conjectures path-image-length statement, whose negation is now Lean-checked using ani's explicit degree-seven polynomial. The checked one-dimensional Hausdorff-measure bound settles that formal statement negatively; the reported small-parameter family is not formalised."},{"id":"remaining_open.erdos_1049_irrationality","open_target_claim":"erdos_1049","statement":"Determine whether F(t) = ∑_{n≥1} 1/(tⁿ − 1) is irrational for every rational t > 1. Irrationality of F((31/4)^r) for every positive integer r is a Lean-checked Zudilin specialization; the 3/2 approximant with remainder control and the universal assertion remain open."},{"id":"remaining_open.erdos_243_eventual_recurrence","open_target_claim":"erdos_243","statement":"Prove the Erdős #243 assertion: if 1 ≤ a₁ < a₂ < ··· has aₙ/aₙ₋₁² → 1 and rational reciprocal sum, then aₙ = aₙ₋₁² − aₙ₋₁ + 1 eventually; the checked barrier does not settle the mixed-sign unbounded regime."},{"id":"remaining_open.erdos_269_reduced_tail_nonintegrality","open_target_claim":"erdos_269","statement":"Prove that B·T_a ∉ ℤ for every B ≥ 1 coprime to 30 and every a ≥ 1."},{"id":"remaining_open.erdos_269_cofinal_local_window_escape","open_target_claim":"erdos_269","statement":"Prove that for every B ≥ 1 coprime to 30 and every a₀ ≥ 1 there are ℓ ≥ a₀ and h ≥ 1 with lpr_{W(ℓ,h)}(−B·F(ℓ,h)) > K(B, ℓ+h), where K is the defined long-paper cap."},{"id":"remaining_open.erdos_269_two_dimensional_representation","open_target_claim":"erdos_269","statement":"Transcendence of D_{2,3,5}, which is irrational: represent it faithfully by a specified two-dimensional Mahler-type value theorem with every hypothesis verified, give a conditional theorem under explicit logarithmic nondegeneracy, or prove no such representation exists."},{"id":"remaining_open.erdos_243_overlap_height_growth","open_target_claim":"erdos_243","statement":"Decide whether every nonterminal canonical orbit of the reduced-orbit problem has limsup_n (log M_n)/n > 0, equivalently 2^n ≤ M_n^K infinitely often for some K ≥ 1."},{"id":"remaining_open.erdos_249_first_harmonic_anti_concentration","open_target_claim":"erdos_249","statement":"Decide whether, for each h ≥ 1, some s ≥ 1 and 0 < η < 1 give, for every X₀, admissible X and L meeting the displayed size conditions for which the four first-harmonic bounds hold."},{"id":"remaining_open.twenty_one_scaled_remainder_cofinal_return","open_target_claim":"universal_257","statement":"Decide whether the scaled actual greedy remainder returns cofinally to one bounded interval: whether some B < ∞ has 2^N r_N ≤ B at arbitrarily large N."},{"id":"remaining_open.twenty_one_actual_orbit_invariant","open_target_claim":"universal_257","statement":"Decide whether a bounded-memory 2-adic or discrepancy invariant of the actual support forces a closed return or forbids permanent supercapacity, or prove no such invariant exists."},{"id":"remaining_open.twenty_one_final_skip_exclusion","open_target_claim":"universal_257","statement":"Decide whether the final-skip signatures prove |E − a_M| ≥ gap_M, contradicting 0 < a_M − E < gap_M, for E = ∑_{n≥1}(2^n − 1)^{−1} and the last-skip approximant a_M."}],"non_claims":[{"id":"not_erdos_257_solution","meaning":"This release does not settle the universal Erdős #257 problem"},{"id":"not_erdos_249_solution","meaning":"This release does not settle Erdős #249"},{"id":"not_publication_authority","meaning":"This release does not itself constitute a peer-reviewed publication record"},{"id":"not_private_root_equivalence","meaning":"This release is not asserted to be equivalent to any private development"},{"id":"not_provider_proof_authority","meaning":"Provider or model output is not proof authority; 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Lean also proves liminf log L_N/(N^{3/2} log N) ≥ 2√2/3 for the common denominator L_N = lcm_{2≤n≤N}(n! − 1) before cancellation (common_denominator_growth_liminf). The irrationality of S remains open: the checked reductions and finite obstructions do not supply the required cofinal non-unit carries."},"directory":"lean/ErdosProblems/Erdos68","modules":[{"module":"ErdosProblems.Erdos68.FactorialGapPlateauCore","path":"lean/ErdosProblems/Erdos68/FactorialGapPlateauCore.lean","present":true,"line_count":1008,"declaration_count":36,"theorem_count":31,"content_digest":"sha256:8572bdb53144be1aa35b33395db25e3708e717c4dc2e1fd4aa774a12061bcf2a"},{"module":"ErdosProblems.Erdos68.CanonicalFactorialDigits","path":"lean/ErdosProblems/Erdos68/CanonicalFactorialDigits.lean","present":true,"line_count":278,"declaration_count":18,"theorem_count":15,"content_digest":"sha256:a0b711c869ded5d04b3f22b27ff57ea574cb28550c27535f3c8b4047f8f14827"},{"module":"ErdosProblems.Erdos68.ChannelBreakpointRigidity","path":"lean/ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean","present":true,"line_count":144,"declaration_count":10,"theorem_count":8,"content_digest":"sha256:0f62e658b9964a37ad2bf0310d99ec4fa35f385720932d66e91abac28b65a903"},{"module":"ErdosProblems.Erdos68.ChannelIntegralCongruence","path":"lean/ErdosProblems/Erdos68/ChannelIntegralCongruence.lean","present":true,"line_count":1198,"declaration_count":58,"theorem_count":49,"content_digest":"sha256:7fbb70bcd1087d98b0a31cf36d82eb71e787bf00f41d25158848923bd9d09203"},{"module":"ErdosProblems.Erdos68.DivisorFactorialCentre","path":"lean/ErdosProblems/Erdos68/DivisorFactorialCentre.lean","present":true,"line_count":303,"declaration_count":18,"theorem_count":13,"content_digest":"sha256:9fea8662f744c1d4bd948065e7a024315620bec3b3714c82fedea7f1b3de656b"},{"module":"ErdosProblems.Erdos68.EndpointWeightedPrivateSupport","path":"lean/ErdosProblems/Erdos68/EndpointWeightedPrivateSupport.lean","present":true,"line_count":6675,"declaration_count":223,"theorem_count":183,"content_digest":"sha256:5f999c8f50f693e12ecd059bcee5e6b47044f4f678e31b824f398931ae5224be"},{"module":"ErdosProblems.Erdos68.FactorialCarry","path":"lean/ErdosProblems/Erdos68/FactorialCarry.lean","present":true,"line_count":101,"declaration_count":8,"theorem_count":5,"content_digest":"sha256:45438410c87c1e0e21e1157cb75e067adf467f89ec0d836d0ab2c569b64da730"},{"module":"ErdosProblems.Erdos68.FactorialChannelCertificate","path":"lean/ErdosProblems/Erdos68/FactorialChannelCertificate.lean","present":true,"line_count":172,"declaration_count":17,"theorem_count":11,"content_digest":"sha256:170b6eadada6558fb37892ff30e4fbdb1e1c2474db33241da38867dcbd4213a6"},{"module":"ErdosProblems.Erdos68.FactorialZeroPlateau","path":"lean/ErdosProblems/Erdos68/FactorialZeroPlateau.lean","present":true,"line_count":37,"declaration_count":0,"theorem_count":0,"content_digest":"sha256:6c91bdef6b22856b59952eed00003fb2617d3cc6dff9a14eac475c9b05db0c62"},{"module":"ErdosProblems.Erdos68.FiniteDefectAutomaton","path":"lean/ErdosProblems/Erdos68/FiniteDefectAutomaton.lean","present":true,"line_count":120,"declaration_count":10,"theorem_count":6,"content_digest":"sha256:c072b7c22b1f86b69868db89aa9e2d2626d8637bbe5a08a2ca860c30c28fe6c3"},{"module":"ErdosProblems.Erdos68.PrimeUnitTranslator","path":"lean/ErdosProblems/Erdos68/PrimeUnitTranslator.lean","present":true,"line_count":1730,"declaration_count":91,"theorem_count":70,"content_digest":"sha256:ceec4294ac834abb7850fe8732a5eea455328ade40bf9a5eb66c6f37cbef2c9a"},{"module":"ErdosProblems.Erdos68.PrimeZeroBranch","path":"lean/ErdosProblems/Erdos68/PrimeZeroBranch.lean","present":true,"line_count":7293,"declaration_count":187,"theorem_count":155,"content_digest":"sha256:de5508cb858fdaaa5d9c4b40bcb65a6960f8c7ebb6209887bd57e800c6674900"},{"module":"ErdosProblems.Erdos68.StrictSuccessorArithmetic","path":"lean/ErdosProblems/Erdos68/StrictSuccessorArithmetic.lean","present":true,"line_count":204,"declaration_count":4,"theorem_count":4,"content_digest":"sha256:e03320c57d2ae0d42f52564023f4d62d92e805696bf010d6b48c66ecb053064b"}],"note":{"artifact_id":"erdos_68_note","title":"Integer Linear Forms for a\\\\Factorial Reciprocal Series","source_path":"paper/68/erdos-68-factorial-denominator-irrationality.tex","rendered_path":"paper/68/erdos-68-factorial-denominator-irrationality.pdf","authority_posture":"authored_mathematical_exposition_not_Lean_proof_authority; public_status_is_owned_per_claim_by_docs/claims.json"},"what_is_checked":["The exact integral normal form: the series is irrational if and only if the strict factorial successor Z_m misses divisibility by m cofinally.","The equivalent carry form: rationality is equivalent to eventual unit carry, so cofinally many non-unit carries imply irrationality.","Canonical factorial digits, exact remainder recurrences, zero-tail propagation, and the divisor-factorial centre recurrence.","Integral channel congruences, the lcm tax, the prime unit translator, and factorial-grid Cramer residual identities.","Weighted projection and factor-split rigidity consumers with their exact modular and Archimedean hypotheses.","The zero-plateau and first-exit carry dichotomy, prime-power prefix obstructions, and exact doubled-prime branch characterisations.","A Lean-checked implication from one exact non-unit carry to a rational-denominator lower bound; a fresh exact GMP certificate at m = 300000 supplies the finite input used by the papers; the event-trace and unit carries match the previously retained receipt."],"what_is_not_checked":["Irrationality of the factorial-denominator series.","A theorem producing cofinally many divisibility misses or non-unit carries.","Cofinal strict nonintegrality of the factorial-grid Cramer residual.","The weighted collision-product and complementary-residue estimates needed by the factor-split projection consumer.","The GMP computation is an exact finite external certificate, not a Lean evaluation and not an unbounded argument."],"open_obligations":[{"id":"weighted_collision_and_complementary_residue","statement":"On an unbounded family of tailored prime blocks, prove both the weighted collision-core product bound and the independent complementary-residue inequality consumed by the exact unit-factor scale split."},{"id":"cofinal_prime_power_amplification","statement":"At infinitely many genuinely nonterminal indices, prove quantitative repeated-support valuation amplification strong enough to meet the checked endpoint inequality."},{"id":"cofinal_lower_endpoint_escape","statement":"For infinitely many primes, force the reduced predecessor gap outside the exact lower unit-carry cylinder."},{"id":"cofinal_doubled_prime_branch_failure","statement":"For infinitely many odd primes, rule out both exact p^2-divisibility branches at index 2p by coupling the carry value to the predecessor residue."},{"id":"cramer_residual_nonintegrality","statement":"Construct an unbounded factorial-grid family whose exact Cramer residual is nonintegral, using a determinant, valuation, cancellation, or gcd-of-minors certificate that controls the finite sign-changing block."}],"finite_search":"A retained exact GMP receipt through m = 300000 records a non-unit carry at that index, so every positive rational denominator satisfies q ∤ 299999!. A byte-identical regeneration of that receipt with the current driver is outstanding in this revision; a separate exact replay through 4000 reproduces the unit-carry prefix. The Lean consumer turns one non-unit carry at m into q >= m; it does not prove a cofinal statement. An independent continued-fraction enclosure gives q ≥ 2^39990. At the prime 12487 and gap 12, exact polynomial arithmetic exhibits two critical roots 4626 and 7848, so the small-prime pattern of at most one critical root is not uniform; those roots have factorial values 442 and 6300, not 1.","paper":{"paper_id":"erdos-68-factorial-denominator-irrationality","title":"Integer Linear Forms for a Factorial Reciprocal Series","source":"paper/68/erdos-68-factorial-denominator-irrationality.tex","pdf":"paper/68/erdos-68-factorial-denominator-irrationality.pdf","full_text":"docs/papers/full-text/erdos-68-factorial-denominator-irrationality.md","resolution":"resolved"},"external_check":{"state":"reviewed","family_count":5,"by_class":{"not_applicable":1,"not_selected":2,"represented_by":1,"targeted":1},"dispositions":{"factorial_carry_characterisation":"targeted","factorial_channel_and_projection_rigidity":"represented_by_isolated_headline","factorial_conditional_producers":"not_selected_deep_project_predicate_stack","factorial_finite_certificates":"not_applicable_to_external_execution","factorial_lcm_growth":"not_selected_separate_release_comparator_entry"}}},{"problem_id":"erdos_243","erdos_number":243,"short_title":"Reciprocal-tail rigidity near the Sylvester recurrence","status":"open","question":"Under a rapid-growth hypothesis on an integer sequence, does rationality of its reciprocal sum force the sequence to satisfy the Sylvester recurrence eventually?","library_root":"ErdosProblems.lean","claim_registry_status":"registered_declarations","claim_registration":{"state":"registered_declarations","source":"docs/claims.json","programme_claim_id":"erdos_243","registered_claim_ids":["bounded_negative_exclusion","bounded_rise_fresh_modulus_barrier","eventually_periodic_negative_no_go"],"boundary":"Only the listed claim rows carry their recorded public status; other declarations are not promoted by module membership or a kernel build.","programme_status":"open","programme_statement":"Lean proves that if a is a strictly increasing sequence of positive integers with n^λ(a(n)²/a(n+1) − 1 − λ/n) → 0, then the reciprocal sum ∑ 1/a(n) is irrational, for every non-integer λ > 1, where Lean also derives convergence (nonintegral_regular_rate_irrational), and for λ = 3 whenever the sum converges (cubic_rate_irrationality_unconditional). Both theorems are zero-indexed; the paper transfers them to one-based indexing by an ordinary finite-prefix argument. Without such a rate the eventual Sylvester recurrence assertion remains open: the bounded-rise barrier and the conditional negative-part reductions leave the mixed-sign unbounded regime unresolved."},"directory":"lean/ErdosProblems/Erdos243","modules":[{"module":"ErdosProblems.Erdos243.ReciprocalTailRigidity","path":"lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","present":true,"line_count":2441,"declaration_count":78,"theorem_count":68,"content_digest":"sha256:516544a9e64edd668a83d9b0ea0179d32c2dbeea7a3bea773d498d6897695f18"},{"module":"ErdosProblems.Erdos243.FiniteHorizonResidue","path":"lean/ErdosProblems/Erdos243/FiniteHorizonResidue.lean","present":true,"line_count":141,"declaration_count":10,"theorem_count":7,"content_digest":"sha256:61d3b9deeb41875b031a0b50a27b336d3ebbbfe7111eda14450227c142d13006"},{"module":"ErdosProblems.Erdos243.SparseResetRecovery","path":"lean/ErdosProblems/Erdos243/SparseResetRecovery.lean","present":true,"line_count":720,"declaration_count":23,"theorem_count":18,"content_digest":"sha256:f207295595576c95ab9e7717f219eec3b1d1e33555b0a9a1a1e95c690f623177"},{"module":"ErdosProblems.Erdos243.DynamicCancellation","path":"lean/ErdosProblems/Erdos243/DynamicCancellation.lean","present":true,"line_count":624,"declaration_count":21,"theorem_count":21,"content_digest":"sha256:0da6cf5dd09100a815d79fce850ad458b4a492ccc1206573741f93f7752fdf76"},{"module":"ErdosProblems.Erdos243.FeedbackRealizability","path":"lean/ErdosProblems/Erdos243/FeedbackRealizability.lean","present":true,"line_count":165,"declaration_count":7,"theorem_count":7,"content_digest":"sha256:b5244899b15e5554ca45d0079023b9ec50622569c607b8ebaaf13ba2eba7c8e3"},{"module":"ErdosProblems.Erdos243.GlobalLcmHeight","path":"lean/ErdosProblems/Erdos243/GlobalLcmHeight.lean","present":true,"line_count":152,"declaration_count":13,"theorem_count":9,"content_digest":"sha256:9b9d070b01a39b6685f9aee7b7669e22589b2a2a5dbf57a9c299879429711d02"},{"module":"ErdosProblems.Erdos243.LcmCriticalBoundary","path":"lean/ErdosProblems/Erdos243/LcmCriticalBoundary.lean","present":true,"line_count":388,"declaration_count":15,"theorem_count":14,"content_digest":"sha256:da5323838ee6412f15f2a9a169188edd46d105ba5a9b36a4161a1ca3cff33fbe"},{"module":"ErdosProblems.Erdos243.PrimitivePrefixRigidity","path":"lean/ErdosProblems/Erdos243/PrimitivePrefixRigidity.lean","present":true,"line_count":136,"declaration_count":6,"theorem_count":4,"content_digest":"sha256:5bb23f80d66cb863e4504775476f0633d6814a749fe26a586596d390a0385327"},{"module":"ErdosProblems.Erdos243.WeightedCRTRepair","path":"lean/ErdosProblems/Erdos243/WeightedCRTRepair.lean","present":true,"line_count":333,"declaration_count":17,"theorem_count":11,"content_digest":"sha256:7bd0b18c719d15929bcee8039eb56cc7692853e917853f25e5f74065f603a763"}],"note":{"artifact_id":"erdos_243_note","title":"Reciprocal Sums and the Sylvester Recurrence","source_path":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","rendered_path":"paper/243/erdos-243-reciprocal-tail-rigidity.pdf","authority_posture":"authored_mathematical_exposition_not_Lean_proof_authority; public_status_is_owned_per_claim_by_docs/claims.json"},"what_is_checked":["Lean proves cubic_rate_irrationality_unconditional for positive StrictMono zero-indexed a_n with a_n²/a_(n+1) = 1 + 3/n + o(n^-3). The one-based paper uses a finite-prefix bridge; general #243 remains open.","Lean proves nonintegral_regular_rate_irrational for positive StrictMono zero-indexed a_n whenever the real coefficient lambda is greater than one and nonintegral, with a_n²/a_(n+1) = 1 + lambda/n + o(n^-lambda). Summability is derived. The one-based paper uses the same finite-prefix bridge; the unrestricted problem remains open.","The exact product-cleared tail dynamics and the defect identity, as identities in the integer state variables.","That a vanishing centred state is absorbing under strict centring, so the state is either eventually zero or nowhere zero.","That eventual vanishing of the centred state, with the next tail state nonvanishing, forces the Sylvester recurrence eventually.","That a nonnegative centred state vanishes eventually, by descent.","That the centred state cannot be eventually constant and negative, at any magnitude and any common scale.","That the negative magnitude cannot be eventually periodic with positive drift, in the regime where each magnitude is below its multiplier.","A coprimality barrier: a sequence tending to infinity with uniformly bounded upward increments cannot permanently avoid infinitely many fresh pairwise coprime moduli.","That a reduced exact tail has pairwise coprime multipliers which its numerator permanently avoids, so no such tail has a divergent numerator with bounded rise.","That the tail gcd stabilises under cofinally bounded negative magnitudes.","Conditionally: exact dynamics, eventual strict centring, an eventually bounded negative part, and normalised vanishing together force the centred state to vanish eventually.","That survival through a fixed number of forced updates depends only on the initial state modulo a factorial of the horizon.","From a strictly increasing positive integer sequence, a rational reciprocal sum and a_(n+1)/a_n^2 tending to one, canonical_integer_tail_normalized derives eventual strict centring and normalised vanishing of the exact integer tail."],"what_is_not_checked":["The unrestricted problem.","A uniform lower bound on the centred error for every sequence in the original problem; the canonical tail transfer does not supply this additional hypothesis.","Any exclusion of negative excursions whose magnitudes are unbounded along a cofinal set.","The Erdős–Straus weighted criterion and Duverney's conditional characterisation, which are cited rather than formalised here."],"open_obligations":[{"id":"unbounded_negative_excursions","statement":"Exclude or construct exact orbits with infinitely many unbounded negative centred excursions. Checked exclusions assume bounded negative part, a fixed period, or fixed prime divisors."},{"id":"derive_negative_part_bound","statement":"Derive a lower bound on centred error from the original growth and rational-sum hypotheses, control unbounded negative excursions another way, or construct a counterexample."},{"id":"formalise_the_published_criteria","statement":"Kernel-check the Erdos-Straus weighted criterion and Duverney's conditional characterisation under explicit analytic hypotheses."}],"finite_search":"An exact search over initial states below five thousand produced forced prefixes of length seventeen and no longer. It is superseded: the constant-negative template is now excluded outright, for every seed and at every scale.","paper":{"paper_id":"erdos-243-reciprocal-tail-rigidity","title":"Reciprocal Sums and the Sylvester Recurrence","source":"paper/243/erdos-243-reciprocal-tail-rigidity.tex","pdf":"paper/243/erdos-243-reciprocal-tail-rigidity.pdf","full_text":"docs/papers/full-text/erdos-243-reciprocal-tail-rigidity.md","resolution":"resolved"},"external_check":{"state":"reviewed","family_count":5,"by_class":{"represented_by":1,"targeted":4},"dispositions":{"centered_state_dynamics":"targeted","negative_orbit_no_go":"targeted_representative_periodic_theorem","bounded_rise_coprimality":"represented_by_stronger_selected_boundary_theorem","bounded_negative_exclusion":"targeted","negative_mass_recovery":"targeted"}}},{"problem_id":"erdos_249","erdos_number":249,"short_title":"The binary totient series","status":"open","question":"Is the binary Lambert series sum phi(n)/2^n irrational?","library_root":"ErdosProblems.lean","claim_registry_status":"registered_declarations","claim_registration":{"state":"registered_declarations","source":"docs/claims.json","programme_claim_id":"erdos_249","registered_claim_ids":["strict_prime_tail_orbit_gap"],"boundary":"Only the listed claim rows carry their recorded public status; other declarations are not promoted by module membership or a kernel build.","programme_status":"open","programme_statement":"Lean proves that S = ∑ φ(n)/2ⁿ is irrational if, for every h ≥ 1, there are arbitrarily large X at which the real part of the first-harmonic sum over the good indices is at most 603X/1000 (irrational_totient_series_of_goodBase_gap); the good indices are the assigned indices N in [X, 2X) at the minimal admissible depth with s = 26 whose cofactor m satisfies φ(m) ≥ m/1000. For all large X more than 67X/100 indices are good (eventually_card_pivotGoodBases_gt), and 603/1000 = (9/10)(67/100). Lean also proves that S is irrational if and only if certified non-integrality witnesses occur at unbounded parameters (irrational_totient_series_iff_certificate_supply). The 603X/1000 bound and that unbounded certificate supply are open, so S is not proved irrational."},"directory":"lean/ErdosProblems/Erdos249","modules":[{"module":"ErdosProblems.Erdos249.TotientStrictPrimeEscape","path":"lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","present":true,"line_count":530,"declaration_count":25,"theorem_count":21,"content_digest":"sha256:d1c192d9ea692136eb58382e87fba2fdc61372de70dd7213c6fc083efbf2c09b"},{"module":"ErdosProblems.Erdos249.FiniteEulerSieve","path":"lean/ErdosProblems/Erdos249/FiniteEulerSieve.lean","present":true,"line_count":69,"declaration_count":6,"theorem_count":4,"content_digest":"sha256:a0172184d831cb4d6004781027464922460d762c4a93b17b04fefc950dd62e93"},{"module":"ErdosProblems.Erdos249.PrimeRayCyclotomicCurvature","path":"lean/ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean","present":true,"line_count":268,"declaration_count":17,"theorem_count":11,"content_digest":"sha256:7c2833745d718e8efb4d9d824770fa7ce12a2feead633de32cd33cf3f32170e8"}],"note":{"artifact_id":"erdos_249_note","title":"Integral Relations among Totient Sections","source_path":"paper/249/erdos-249-binary-totient-series.tex","rendered_path":"paper/249/erdos-249-binary-totient-series.pdf","authority_posture":"authored_mathematical_exposition_not_Lean_proof_authority; public_status_is_owned_per_claim_by_docs/claims.json"},"what_is_checked":["For every e >= 1, the complete dyadic totient kernel through level e has rational span dimension 2^e+1.","The rational span of the full dyadic totient kernel is not finite-dimensional; for integer-valued sequence modules this is equivalent to Coons's non-k-regularity and is not an independent contribution.","Lean diagonal certificates through t = 82; neither t = 83 nor an unbounded family is supplied.","Conditionally: a cofinal strict 9/10 gap for the exact natural prime-tail orbit supplies the existing finite pivot-point escape and hence the reviewed irrationality endpoint.","The local s=1 and s=2 squared Möbius Euler-factor identities and the first two prime-power divisor-sum differences.","That the two-by-two checkerboard kills separable backgrounds and is unique up to scale under the four row and column cancellations.","An abstract four-point divisor-layer identity, stated from its four explicit factorisation hypotheses.","An affine completion no-go: every residue class modulo an even modulus has a digit that recentres one carry step.","Exact producer and consumer interfaces for prime-ray layer supply, bounded-degree order witnesses, and finite-prime-support escape."],"what_is_not_checked":["The strict prime-tail orbit gap, which is the unproved producer consumed by the conditional endpoint.","The packet's sharp parabolic Stern-Brocot cusp theorem and its passage from a geometric x-parameter to the arithmetic totient series.","Eventual Hankel nonvanishing for the Möbius-Mersenne power ladder, or its denominator obstruction.","The natural-boundary and quasimodular-transcendence steps of the infinite Euler-sieve tower.","Polynomial resultant realisability, exact-order finite-field supply, Frobenius freezing, or the Archimedean growth needed by the prime-ray strategy.","Unconditional irrationality of the binary totient series."],"open_obligations":[{"id":"strict_prime_tail_orbit_gap","statement":"Prove the cofinal strict 9/10 real-part gap for the exact natural prime-tail orbit, or replace it with another producer for the reviewed pivot-escape endpoint."},{"id":"hankel_denominator_fan_in","statement":"Prove eventual nonvanishing or a usable denominator obstruction for the Möbius-Mersenne power ladder and connect it to the target value."},{"id":"euler_sieve_limit_theorem","statement":"Control the infinite Euler-sieve limit strongly enough that finite-stage quasimodularity yields a valid arithmetic obstruction."},{"id":"prime_ray_resultant_supply","statement":"Realise the layer system by cyclic resultants and prove prime-support escape with the growth estimates needed to contradict a rational carry orbit."},{"id":"stern_brocot_arithmetic_bridge","statement":"Prove the sharp cusp asymptotic and then supply a non-formal bridge from the geometric parameter regime to the exact totient value."}],"finite_search":"The returned strict-orbit packet reports an exact finite deposit at parameters (147,23), verified through harmonic height 5022. It is research evidence, not a Lean-checked unbounded certificate.","paper":{"paper_id":"erdos-249-binary-totient-series","title":"Integral Relations among Totient Sections","source":"paper/249/erdos-249-binary-totient-series.tex","pdf":"paper/249/erdos-249-binary-totient-series.pdf","full_text":"docs/papers/full-text/erdos-249-binary-totient-series.md","resolution":"resolved"},"external_check":{"state":"reviewed","family_count":20,"by_class":{"not_applicable":1,"not_selected":4,"represented_by":1,"targeted":14},"dispositions":{"totient_kernel_basis":"targeted","totient_kernel_rank":"targeted","totient_kernel_all_base_index":"targeted","totient_visible_coprime_lattice_mass":"targeted","squared_lambert_gcd_moments":"targeted","stern_brocot_run_fibonacci_stability":"targeted","stern_brocot_cylinder_law":"targeted","lcm_factor_ideal_shift_algebra_no_go":"targeted","fixed_precision_transport_no_go":"targeted","actual_lcm_orbit_separation":"targeted","actual_foreign_residue_projection":"targeted","first_harmonic_pivot_decomposition":"targeted","strict_prime_tail_orbit_gap":"targeted","totient_carry_anti_compression":"targeted","totient_finite_denominator_exclusion":"not_selected_finite_family_has_separate_receipt","totient_carry_rank":"not_selected_deep_orbit_vocabulary","totient_lambert_coefficients":"represented_by_basis_and_rank_targets","eventually_periodic_lambert":"not_selected_prior_result","totient_certificate_equivalences":"not_selected_deep_certificate_vocabulary","signed_totient_entire_defect_interpolation":"not_applicable_without_exact_Lean_statement"}}},{"problem_id":"erdos_251","erdos_number":251,"short_title":"The prime-gap dyadic series","status":"open","question":"Is the dyadic series of consecutive primes irrational? Equivalently, is the corresponding consecutive-prime-gap dyadic series irrational?","library_root":"ErdosProblems.lean","claim_registry_status":"registered_declarations","claim_registration":{"state":"registered_declarations","source":"docs/claims.json","programme_claim_id":"erdos_251","registered_claim_ids":["prime_gap_irrationality_equivalence","prime_gap_unboundedness_and_nonperiodicity"],"boundary":"Only the listed claim rows carry their recorded public status; other declarations are not promoted by module membership or a kernel build.","programme_status":"open","programme_statement":"Lean proves that Π = ∑_{n≥1} p_n/2ⁿ equals 2 plus the prime-gap dyadic series and that Π is irrational if and only if that gap series is (tsum_primeDyadicTerm_eq_two_add_primeGap_unconditional, irrational_tsum_primeDyadicTerm_iff_primeGap), with summability from the elementary bound p_n ≤ 1250(n+1)^4; the identity is the known summation by parts. Lean also proves that the consecutive prime gaps are unbounded and not eventually periodic (exists_primeGap0_gt, primeGap0_not_eventually_periodic). The irrationality of Π remains open: these facts do not supply the required cofinal escape for the actual prime gaps."},"directory":"lean/ErdosProblems/Erdos251","modules":[{"module":"ErdosProblems.Erdos251.PrimeGapDyadicTail","path":"lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","present":true,"line_count":1665,"declaration_count":116,"theorem_count":92,"content_digest":"sha256:d284ca394ba94d6e43b2819757ce27eb86562949b525f8bbf81546d613abfcd1"}],"note":{"artifact_id":"erdos_251_note","title":"Sparse Congruence-Preserving Perturbations\\\\of Dyadic Series","source_path":"paper/251/erdos-251-prime-gap-dyadic-series.tex","rendered_path":"paper/251/erdos-251-prime-gap-dyadic-series.pdf","authority_posture":"authored_mathematical_exposition_not_Lean_proof_authority; public_status_is_owned_per_claim_by_docs/claims.json"},"what_is_checked":["The finite summation-by-parts identity relating the prime dyadic partial sums to the prime-gap dyadic partial sums, with an exact endpoint term and no convergence premise.","The factor-of-two normalisation between the two displayed indexings, at every finite horizon.","The unconditional prime-to-gap identity, using an elementary polynomial bound on primes to prove summability; irrationality of the prime series is equivalent to irrationality of the prime-gap series.","For an abstract integer-digit dyadic tail recurrence: the exact block identity, and that a shift of fixed length is integral exactly when an explicit scaled state is.","That one integral shift propagates to every later index.","That repeated doubling removes the power-of-two denominator, so every rational-valued dyadic recurrence has one fixed positive shift integral at all sufficiently late indices.","For a real-valued dyadic recurrence, cofinal non-integrality of every fixed positive shift implies irrationality of the initial state.","That the actual consecutive-prime-gap sequence is unbounded, by the factorial prime-free-interval construction, and therefore is not eventually periodic with any positive period.","For any integer-digit dyadic tail recurrence, two adjacent fixed-length shifts lying strictly between -1 and 1 cannot both be integral when the corresponding digits differ.","For the actual prime gaps, a cofinal supply of adjacent small-shift pairs with a mismatching gap excludes eventual integrality of that fixed shift.","The generic distance-to-integers core of the finite-truncation criterion: an approximation farther than its error bound from every integer certifies non-integrality."],"what_is_not_checked":["The target irrationality.","A cofinal nonintegrality theorem for the concrete prime-gap tails; defining the series and proving its recurrence do not supply this.","A theorem supplying cofinally many adjacent actual-prime-gap tail shifts in the open unit interval with the required gap mismatch.","Any theorem forcing periodicity or automaticity of the prime gaps from rationality; the free-carry identity disconfirms that route."],"open_obligations":[{"id":"prime_gap_cofinal_shift_escape","statement":"For every fixed h >= 1 and every N0, find N >= N0 for which sum_{j>=1} (g_{N+h+j}-g_{N+j})/2^j is not an integer."},{"id":"cofinal_adjacent_small_mismatch","statement":"For every fixed h >= 1 and every N0, find N >= N0 such that both adjacent full tail shifts have absolute value less than 1 and g_{N+h+1} != g_{N+1}. The checked local consumer then excludes eventual integrality of the h-shift."}],"finite_search":null,"paper":{"paper_id":"erdos-251-prime-gap-dyadic-series","title":"Sparse Congruence-Preserving Perturbations of Dyadic Series","source":"paper/251/erdos-251-prime-gap-dyadic-series.tex","pdf":"paper/251/erdos-251-prime-gap-dyadic-series.pdf","full_text":"docs/papers/full-text/erdos-251-prime-gap-dyadic-series.md","resolution":"resolved"},"external_check":{"state":"reviewed","family_count":6,"by_class":{"not_selected":1,"represented_by":1,"targeted":4},"dispositions":{"prime_gap_reformulation":"targeted","integral_shift_classification":"not_selected_abstract_family_represented_in_manifest","dyadic_tail_integrality_classification":"targeted","totient_shift_propagation":"represented_by_integral_shift_family","small_mismatch_criterion":"targeted","coefficient_only_no_go":"targeted"}}},{"problem_id":"erdos_257","erdos_number":257,"short_title":"Reciprocal sums over infinite exponent supports","status":"open","question":"Is the sum of 1/(2^n-1) over every infinite set of positive exponents irrational?","library_root":"ErdosProblems.lean","claim_registry_status":"programme_only","claim_registration":{"state":"programme_only","source":"docs/claims.json","programme_claim_id":"universal_257","registered_claim_ids":[],"boundary":"Only the listed claim rows carry their recorded public status; other declarations are not promoted by module membership or a kernel build.","programme_status":"open","programme_statement":"Lean proves that for every integer base b ≥ 2 and every finite nonempty set P of primes, with h(a) the largest divisor of a supported on P, every infinite set A of positive integers with ∑_{a∈A} h(a)/(a(b^{h(a)} − 1)) finite has ∑_{n∈A} 1/(bⁿ − 1) irrational (divisibilityWeightedClaim). Lean also proves irrationality at every such base for every infinite A with ∑_{a∈A} 1/a finite (irrational_erdosSupportSeries_of_summable_reciprocal), and exhibits sets with divergent reciprocal sum all of whose infinite subsets have irrational series at every such base (finite_monotone_witness_rule_realised). Irrationality of ∑_{n∈A} 1/(2ⁿ − 1) for every infinite A remains open."},"directory":"lean/ErdosProblems/Erdos257","modules":[{"module":"ErdosProblems.Erdos257.MersenneSubseriesRigidity","path":"lean/ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean","present":true,"line_count":421,"declaration_count":28,"theorem_count":23,"content_digest":"sha256:7dd147a2bef89f7f23a168c02a8dfb4c7fee887314a7a5367d49336330eb1419"}],"note":{"artifact_id":"erdos_257_note","title":"Irrationality criteria for Lambert subseries","source_path":"paper/257/erdos-257-mersenne-support-subseries.tex","rendered_path":"paper/257/erdos-257-mersenne-support-subseries.pdf","authority_posture":"authored_mathematical_exposition_not_Lean_proof_authority; public_status_is_owned_per_claim_by_docs/claims.json"},"what_is_checked":["Irrationality at every integer base for the classical full support, reciprocal-summable supports, and the finite-prime weighted supports in divisibilityWeightedClaim.","Every infinite subset of the union of a finite base-two weighted-mass host and a strengthened positive-cover host has irrational series at every integer base (mixedSupportClaim).","Unique Mersenne coding on every support; compactness and nowhere density; perfection for infinite support; finite-complement measure 2^(-|F|), otherwise zero."],"what_is_not_checked":["Universal irrationality over all infinite supports, or the proposed rational values 1/2 and 1/21.","Fair-bit laws, finer singularity results, and periodic-stride dimensions.","Any bridge from geometric uniqueness or measure rigidity to arithmetic irrationality."],"open_obligations":[{"id":"arithmetic_rigidity_for_thin_supports","statement":"Convert hereditary unique coding into an arithmetic obstruction to rational values for every infinite support, including zero-measure Cantor supports."},{"id":"formalise_measure_and_stride_geometry","statement":"Extend the measure dichotomy to Bernoulli laws and periodic-stride dimensions without implying arithmetic consequences."}],"finite_search":null,"paper":{"paper_id":"erdos-257-mersenne-support-subseries","title":"Irrationality criteria for Lambert subseries","source":"paper/257/erdos-257-mersenne-support-subseries.tex","pdf":"paper/257/erdos-257-mersenne-support-subseries.pdf","full_text":"docs/papers/full-text/erdos-257-mersenne-support-subseries.md","resolution":"resolved"},"external_check":{"state":"reviewed","family_count":16,"by_class":{"not_applicable":2,"targeted":14},"dispositions":{"finite_period_noncollapse":"targeted","boolean_mobius_carry":"targeted","boolean_mobius_exact_row_dynamics":"targeted","half_membership_seam_classification":"targeted","shifted_odd_tail_state":"targeted","composite_dilation_defect_identity":"targeted","known_irrational_supports":"targeted_full_support_representative","finite_prime_weighted_support":"targeted_exact_weighted_and_hereditary_interfaces_pending_replay","pairwise_coprime_support":"targeted_pairwise_coprime_support","periodic_nonnegative_weight_irrationality":"targeted_strongest_nonnegative_rational_weight","orthogonal_petal_sunflower_reduction":"targeted_conditional_sunflower_reduction","squarefree_support":"not_applicable_to_external_irrationality_input","achievement_set_geometry":"targeted_measure_theorem","restricted_achievement_sets":"targeted_measure_dichotomy","half_and_twenty_one_frontiers":"targeted","separated_cut_algebraic_lambert_transcendence":"not_applicable_without_exact_Lean_statement"}}},{"problem_id":"erdos_269","erdos_number":269,"short_title":"Three-prime running least common multiples","status":"open","question":"For a finite set of at least two primes, is the sum of reciprocals of the running least common multiples of the smooth numbers irrational? This library treats the three-prime case.","library_root":"ErdosProblems.lean","claim_registry_status":"registered_declarations","claim_registration":{"state":"registered_declarations","source":"docs/claims.json","programme_claim_id":"erdos_269","registered_claim_ids":["three_prime_cofinal_window_carry_extinction","three_prime_finite_residue_contradiction","weighted_phase_carry_finite_observer"],"boundary":"Only the listed claim rows carry their recorded public status; other declarations are not promoted by module membership or a kernel build.","programme_status":"open","programme_statement":"For every finite set P of at least two primes, the distinct-height sum D_P (each running LCM value of the P-smooth integers counted once) is irrational (res:distinct-height-all; ordinary proof checked by a second AI agent, no human review), as Erdős asserted without an argument in 1973; for |P| = 2 it follows from Hecke-Mahler transcendence, and D_{2,3,5} is Lean-checked (res:distinct-height-235; Comparator not yet run). Each single-prime sub-sum E_p of the catalogue sum is irrational (res:single-prime-subsums; same evidence class as D_P). The catalogue sum for three or more primes is unresolved in this release: for {2,3,5} the rationality-to-positive-reduced-carry bridge is checked, and the remaining endpoint is cofinal local-window escape, equivalently exclusion of the integral branch."},"directory":"lean/ErdosProblems/Erdos269","modules":[{"module":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","path":"lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","present":true,"line_count":912,"declaration_count":55,"theorem_count":39,"content_digest":"sha256:769d80481465f4fd8655d5d62c5eb3e372c80c39b0722e80659bf4e150c6c036"},{"module":"ErdosProblems.Erdos269.ResidueEscape","path":"lean/ErdosProblems/Erdos269/ResidueEscape.lean","present":true,"line_count":164,"declaration_count":8,"theorem_count":6,"content_digest":"sha256:826a3dd901c86f1876c7e03011447dafef8d3cb139a673227e921bef3a9dfe87"},{"module":"ErdosProblems.Erdos269.RestrictedFloorSum","path":"lean/ErdosProblems/Erdos269/RestrictedFloorSum.lean","present":true,"line_count":709,"declaration_count":43,"theorem_count":30,"content_digest":"sha256:7369d105e5cb014188e31fa44fdccb03b2c40bdfe4719f6e227555c1c30b0bd3"},{"module":"ErdosProblems.Erdos269.BoundedRadixTailEscape","path":"lean/ErdosProblems/Erdos269/BoundedRadixTailEscape.lean","present":true,"line_count":304,"declaration_count":10,"theorem_count":8,"content_digest":"sha256:741d610f12f3774500b993266f187c846be33cb8be466fa38a0b3a7b1503d8e5"},{"module":"ErdosProblems.Erdos269.CarryLiftExtinction","path":"lean/ErdosProblems/Erdos269/CarryLiftExtinction.lean","present":true,"line_count":351,"declaration_count":14,"theorem_count":12,"content_digest":"sha256:ace522da1f0a6833a200efbe18d5961cf3562e1e6756055427b85140d25aae4b"},{"module":"ErdosProblems.Erdos269.ThreeChannelBlockRigidity","path":"lean/ErdosProblems/Erdos269/ThreeChannelBlockRigidity.lean","present":true,"line_count":128,"declaration_count":6,"theorem_count":4,"content_digest":"sha256:5dbd4a448520c6c4de924480e381ee5408a4724a3081837d04bfb4ac1186ae5f"},{"module":"ErdosProblems.Erdos269.WeightedPhaseCarry","path":"lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean","present":true,"line_count":371,"declaration_count":23,"theorem_count":14,"content_digest":"sha256:ef3e57a4728534e108bb4e668ceef2cc102c2cf001673d59d9c15cff7f0beb9e"}],"note":{"artifact_id":"erdos_269_note","title":"Distinct running least common multiples","source_path":"paper/269/erdos-269-three-prime-running-lcm.tex","rendered_path":"paper/269/erdos-269-three-prime-running-lcm.pdf","authority_posture":"authored_mathematical_exposition_not_Lean_proof_authority; public_status_is_owned_per_claim_by_docs/claims.json"},"what_is_checked":["That for three pairwise distinct primes the least common multiple of the smooth prefix equals the product of the three maximal pure prime powers below the cutoff.","That equal logarithmic coordinates give an equal prefix least common multiple, so the running value is constant on logarithmic cells.","That advancing exactly one logarithmic coordinate multiplies the running value by the corresponding prime.","That the positive power channels of distinct primes are disjoint, so the first powers in three channels give an exact jump count.","That a finite smooth lattice sum groups exactly by running height, with the height fibre cardinality as coefficient.","A quadratic bound on the cardinality of the actual filtered exponent shell.","That each dyadic block contains at most one internal pure power from each non-dyadic channel, so the compressed {2,3,5} block radix belongs to the exact alphabet {2,6,10,30} and in particular lies between 2 and 30.","That the smallest {2,3,5} kernel rectangle has determinant exactly -1/15, so its two-by-two matrix has rank two over the rationals and no rank-one separation exists.","The canonical least-positive-residue interval and congruence facts, including the zero-residue representative.","The exact local-window identity for an integral carry.","For the actual {2,3,5} series, rationality produces a positive reduced carry; cofinal local-window escape rules out such carries under the recorded bound.","A bounded-radix dichotomy: under a bounded positive digit alphabet, a nonzero scaled tail must be separated from the integers cofinally.","Exact three-channel block-nullity as a coboundary condition, and extinction of a bounded carry lift under two independent anchors.","The exact residue-digit plus coboundary form of an integral carry, together with a function-faithful observer criterion and a finite realised-span consequence.","Irrationality of the distinct-height sum for {2,3,5}, which counts each running value once."],"what_is_not_checked":["Irrationality of the repeated sum, or transcendence, in any three-prime case.","CofinalLocalWindowEscape for those actual words; it is an explicit hypothesis of the checked consumer, not a theorem.","A proof that the actual three-channel word satisfies the block-null, anchor, or factorisation hypotheses of the migrated abstract consumers.","The infinite limit of the jump expansion; only its finite core is checked.","Any unbounded denominator exclusion. The finite certificates exclude denominators below an explicit bound and no further."],"open_obligations":[{"id":"actual_local_window_residue_escape","statement":"For every positive B coprime to 30 and every starting index, produce a later actual {2,3,5} window whose least positive forcing residue exceeds the exact denominator-dependent carry bound."},{"id":"nonstationary_analytic_theorem","statement":"Prove a genuinely higher-dimensional analytic theorem for the phase cocycle or the associated contraction."},{"id":"unbounded_height_certificate","statement":"Replace one finite denominator exclusion by a family whose exclusion bound tends to infinity."}],"finite_search":"The integer-only dyadic-window checker reproduces three exact packet certificates and, for all 106,666 pairs with denominator B <= 1000 coprime to 30 and dyadic start 100 <= a <= 500, finds an escaping window of length at most 18; the largest first successful length is 14. This is finite proof-consumer evidence only and proves neither unbounded-denominator nor cofinal escape.","paper":{"paper_id":"erdos-269-three-prime-running-lcm","title":"Distinct running least common multiples","source":"paper/269/erdos-269-three-prime-running-lcm.tex","pdf":"paper/269/erdos-269-three-prime-running-lcm.pdf","full_text":"docs/papers/full-text/erdos-269-three-prime-running-lcm.md","resolution":"resolved"},"external_check":{"state":"reviewed","family_count":10,"by_class":{"not_applicable":4,"represented_by":2,"targeted":4},"dispositions":{"three_prime_lcm_cells":"targeted_running_lcm_identity","two_prime_transcendence":"not_applicable_not_a_lean_declaration","height_fibre_and_shell":"represented_by_three_prime_structure_target","rank_two_kernel_no_go":"targeted","dyadic_block_alphabet":"represented_by_three_prime_structure_target","conditional_carry_escape":"targeted","weighted_phase_carry_observer":"targeted","three_prime_finite_search":"not_applicable_not_a_lean_theorem","independent_clock_channel_relations":"not_applicable_without_exact_Lean_statement","selected_staircase_integral_height_boundary":"not_applicable_without_exact_Lean_statement"}}},{"problem_id":"erdos_1041","erdos_number":1041,"short_title":"Short connections inside polynomial lemniscates","status":"formal statement refuted","question":"Must two roots of a monic polynomial in the open unit disc be joined by a sub-two-length curve inside its unit lemniscate? Ani’s degree-seven example refutes the exact Formal Conjectures statement; correspondence with the 1958 wording awaits human review.","library_root":"ErdosProblems.lean","claim_registry_status":"registered_declarations","claim_registration":{"state":"registered_declarations","source":"docs/claims.json","programme_claim_id":"erdos_1041","registered_claim_ids":["ani_degree_seven_total_variation_counterexample","erdos_1041","newton_ray_separation_consumer"],"boundary":"Only the listed claim rows carry their recorded public status; other declarations are not promoted by module membership or a kernel build.","programme_status":"formal statement refuted","programme_statement":"Using ani's explicit monic degree-seven polynomial, Lean proves that every preconnected subset of its strict unit lemniscate containing two distinct roots has one-dimensional Hausdorff measure greater than 2. This refutes the exact Formal Conjectures path-image-length statement and its total-variation counterpart. The reported family is not formalised, and independent human review of correspondence with the 1958 wording is not recorded."},"directory":"lean/ErdosProblems/Erdos1041","modules":[{"module":"ErdosProblems.Erdos1041.Counterexample.HausdorffLength","path":"lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean","present":true,"line_count":460,"declaration_count":10,"theorem_count":9,"content_digest":"sha256:069ea76e27304bd34e020cf8aab0fcc9200c495ff892127612e8afe571e3fcb4"},{"module":"ErdosProblems.Erdos1041.Counterexample.CatalogueAdapter","path":"lean/ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean","present":true,"line_count":90,"declaration_count":2,"theorem_count":2,"content_digest":"sha256:1f807ef3e453c12f382d4ce7a4f422397e1e2dee33ba87d68323273a1860a600"},{"module":"ErdosProblems.Erdos1041.NewtonFlowRaySeparation","path":"lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","present":true,"line_count":435,"declaration_count":28,"theorem_count":25,"content_digest":"sha256:9a2bc4dcafece55cd76b0bfe1a6b1dc8e1ebabf570c7ea92f9e53fe7629464fa"}],"note":{"artifact_id":"erdos_1041_note","title":"Paths in Polynomial Lemniscates:\\\\A Degree-Seven Counterexample\\\\and Radial Connections","source_path":"paper/1041/erdos-1041-lemniscate-newton-flow.tex","rendered_path":"paper/1041/erdos-1041-lemniscate-newton-flow.pdf","authority_posture":"authored_mathematical_exposition_not_Lean_proof_authority; public_status_is_owned_per_claim_by_docs/claims.json"},"what_is_checked":["For ani’s degree-seven polynomial, every preconnected strict-lemniscate set containing two distinct roots has one-dimensional Hausdorff measure above two. Lean refutes the exact Formal Conjectures path-image claim and checks a separate total-variation bound.","Along a trajectory tangent to the complex Newton vector field -f/f', the polynomial value satisfies w' = -w.","The exponentially rescaled polynomial value is a first integral, so the value of a Newton trajectory remains on one oriented ray.","A finite Newton-flow connection can only join endpoints whose polynomial values lie on the same positive ray; distinct critical-value arguments therefore exclude such a connection.","For two translated critical values, the ray-collision locus is a real affine line with an exact one-parameter description.","A finite family of ray-collision lines can be avoided by an arbitrarily small common translation.","An explicit root-retention estimate keeps roots of a sufficiently small constant perturbation inside the unit disc."],"what_is_not_checked":["The full small-parameter family reported by ani; only one explicit polynomial is formalised.","Independent human review adjudicating whether the exact Formal Conjectures path-image-length statement represents the 1958 curve-length wording.","The spanning-tree decomposition claimed in the March 2026 manuscript; its printed Proposition 12 proof uses an invalid three-ended local saddle block, and a Cassini example refutes the printed global tree budget.","A repair of Proposition 12 that retains the printed 1/(2π) spanning-tree coefficient; any later argument must change the metric inequality.","A perturbation theorem that first makes critical values injective while preserving the relevant lemniscate component, collars, and strict length slack.","A corrected restricted-family or alternative-metric gluing theorem; universal sub-two total-variation gluing is ruled out by the degree-seven counterexample."],"open_obligations":[{"id":"historical_curve_length_correspondence","statement":"Independently review whether the 1958 curve-length wording matches the refuted Formal Conjectures path-image Hausdorff statement."},{"id":"repair_or_refute_saddle_block","statement":"Repair or replace the false Proposition 12 spanning-tree estimate; account for positive attachment costs or use another metric inequality."},{"id":"compact_ray_cut_strip_decomposition","statement":"Verify a compact ray-cut strip or cell decomposition under explicit local-valence hypotheses for restricted families; it gives no universal sub-two connector."},{"id":"corrected_metric_gluing_for_restricted_families","statement":"For a restricted family or different historical length notion, prove a metric gluing inequality that pays every positive attachment cost; universal sub-two total-variation gluing is refuted."},{"id":"two_stage_stable_perturbation","statement":"Given a selected connector with strict slack, quantify a two-stage perturbation preserving roots, component, collars, ray separation, and slack; no universal existence claim follows."},{"id":"relative_global_newton_flow_theorem","statement":"Under explicit component, saddle, and corrected metric hypotheses, derive a restricted root-to-root Newton-flow connector without a universal sub-two claim."}],"finite_search":"Raster tests through degree ten suggest candidate bounds only; they are separate from the exact degree-seven Lean counterexample.","paper":{"paper_id":"erdos-1041-lemniscate-newton-flow","title":"Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections","source":"paper/1041/erdos-1041-lemniscate-newton-flow.tex","pdf":"paper/1041/erdos-1041-lemniscate-newton-flow.pdf","full_text":"docs/papers/full-text/erdos-1041-lemniscate-newton-flow.md","resolution":"resolved"},"external_check":{"state":"reviewed","family_count":6,"by_class":{"not_applicable":2,"not_selected":1,"represented_by":1,"targeted":2},"dispositions":{"newton_value_decay":"not_selected_compact_calculus_family","ray_separation":"represented_by_translation_avoidance_targets","translation_avoidance":"targeted","root_retention":"targeted","published_proof_gap":"not_applicable_not_a_lean_proposition","lemniscate_finite_search":"not_applicable_not_a_lean_theorem"}},"research_corpus":{"directory":"research_corpus/Erdos1041","authority_posture":"public_safe_research_evidence_not_reviewed_claim_registry_or_Lean_proof_authority","reading_rule":"Read FRONTIER.md for dated refutations and open gaps, then STRONGEST_RESULTS.json for source-pinned activation. Premises and falsifiers retain local hypotheses; research rows do not settle historical correspondence or become Comparator claims.","files":{"frontier":{"path":"research_corpus/Erdos1041/FRONTIER.md","content_digest":"sha256:8aef5071a9a2a35de6b2af79bf0a486d97306aa0b6d4b997144e16ba1d112a9a"},"strongest_results":{"path":"research_corpus/Erdos1041/STRONGEST_RESULTS.json","content_digest":"sha256:a5f888560836a710024bd6079538cbd62b39c1544a19514903ac57f9789d7e24"},"manifest":{"path":"research_corpus/Erdos1041/CORPUS_MANIFEST.json","content_digest":"sha256:6f456a92e30ed2318d6c342f2e80dbbc77be39a14e051828ef215ffb8b001bbd"},"checkpoint":{"path":"research_corpus/Erdos1041/PUBLIC_CORPUS_CHECKPOINT.json","content_digest":"sha256:3ec0d7b9f9ab8a3425678b52830ee3e6701382b9b7485787c3dac43f66ac7fb3"}},"strongest_result_summary":{"source_checkpoint":"fb5c8803c1498cad9755d1615473cb33b11af6ee","status":"open","result_count":294},"checkpoint_summary":{"source_subtree_tree":null,"exported_source_file_count":952,"status":"source_current_problem_snapshot"}}},{"problem_id":"erdos_1049","erdos_number":1049,"short_title":"Lambert-type series at rational bases","status":"open","question":"For which rational bases is the corresponding series irrational? The first resistant explicit base is three halves.","library_root":"ErdosProblems.lean","claim_registry_status":"registered_declarations","claim_registration":{"state":"registered_declarations","source":"docs/claims.json","programme_claim_id":"erdos_1049","registered_claim_ids":["rational_base_cleared_tail_recurrence","rational_base_forcing_growth","three_halves_coordinatewise_corridor_no_go","van_assche_qapery_diagonal_no_go"],"boundary":"Only the listed claim rows carry their recorded public status; other declarations are not promoted by module membership or a kernel build.","programme_status":"open","programme_statement":"For natural numbers 0 < b < a in the Zudilin contour region, Lean proves that F(a/b) is irrational with irrationality exponent at most rationalBaseMeasureBound a b (rational_base_region, rational_base_measure), from the constructed polynomial forms and analytic estimates; in particular F((31/4)^r) is irrational for every positive integer r (thirtyone_four_powers). The universal rational-base assertion remains open; 3/2 lies outside this region and is unresolved here."},"directory":"lean/ErdosProblems/Erdos1049","modules":[{"module":"ErdosProblems.Erdos1049.RationalBaseLambert","path":"lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean","present":true,"line_count":223,"declaration_count":16,"theorem_count":11,"content_digest":"sha256:593c3b3e776820de229bad00c7f374bb39416099de60250b375d52cfa5f6e56a"},{"module":"ErdosProblems.Erdos1049.RationalPadeArithmetic","path":"lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean","present":true,"line_count":222,"declaration_count":19,"theorem_count":14,"content_digest":"sha256:c7b54971556577d73bbc563dcf14eaf62aef8d31e1450c4e7337028e46013690"},{"module":"ErdosProblems.Erdos1049.ZudilinConeArithmetic","path":"lean/ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean","present":true,"line_count":450,"declaration_count":36,"theorem_count":25,"content_digest":"sha256:9001917590300c4f10a049af0dc56192f9db95f2700b9cfac5ac88b31505b7d0"},{"module":"ErdosProblems.Erdos1049.HermitePadeNoGo","path":"lean/ErdosProblems/Erdos1049/HermitePadeNoGo.lean","present":true,"line_count":153,"declaration_count":10,"theorem_count":5,"content_digest":"sha256:812f2ca2ea8b18f8ce8e0a68ae14897aa851fba2e050a07881d839168182f2cc"},{"module":"ErdosProblems.Erdos1049.ZudilinHeightRegion","path":"lean/ErdosProblems/Erdos1049/ZudilinHeightRegion.lean","present":true,"line_count":143,"declaration_count":13,"theorem_count":12,"content_digest":"sha256:bdb9ffaeab12363ed91a6385a6df8ef8e931003ad7862096ed28fc09aa3669fa"},{"module":"ErdosProblems.Erdos1049.QAperyDiagonalNonEquivalence","path":"lean/ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean","present":true,"line_count":102,"declaration_count":11,"theorem_count":4,"content_digest":"sha256:8094ead680a8a3be3e7ad2c6c2fbc89cd51c9f84a360bfb46f91f6fc4b90cda1"}],"note":{"artifact_id":"erdos_1049_note","title":"Hankel Determinants of Geometric Moments\\\\and Rational Lambert Values","source_path":"paper/1049/erdos-1049-rational-base-lambert.tex","rendered_path":"paper/1049/erdos-1049-rational-base-lambert.pdf","authority_posture":"authored_mathematical_exposition_not_Lean_proof_authority; public_status_is_owned_per_claim_by_docs/claims.json"},"what_is_checked":["Lean checks irrationality and the stated irrationality-exponent bound for rational bases in the exact Zudilin contour region, including every positive integral power of 31/4. The contour excludes 3/2.","For p>1 and N≤8, Lean checks positive definite coefficient matrices, real finite-pencil roots below F(p), and non-strict interlacing at adjacent ranks. Neither all-rank positivity nor irrationality at 3/2 follows.","That a literal coordinatewise transfer of the integer-base clearing argument forces a power-versus-linear inequality, and that this inequality is impossible at base three halves.","The exact denominator-cleared rational-base tail recurrence, including its forcing term.","That for every noninteger base with denominator at least two the forcing magnitude grows at least exponentially, while at denominator one it collapses to a bounded expression.","The complete elementary Archimedean height inequality used by the published Bundschuh-Väänänen criterion at base seven halves; the analytic irrationality theorem itself remains external.","Polynomial bounds on the doubled denominator exponents in the homogenised Padé construction, together with an exact gap identity.","The exact exterior-determinant elimination identity, and that rowwise contents scale the Padé errors, exterior determinant, and absolute determinant height by the same factors.","The exact Heine-Zudilin cone endpoint exponent arithmetic used to identify the bottom and top coefficient endpoints.","That homogeneous evaluation at (3,2) sees only the constant endpoint modulo three and the top endpoint modulo two, so unit endpoints obstruct common local factors.","That under unit top and constant endpoint hypotheses, a common multiplier of the two specialised coefficient channels is divisible by neither two nor three; the missing local gain must therefore survive primitive normalisation.","That vanishing of the higher bottom and top endpoint jets is exactly divisibility by the corresponding powers of three and two.","That the source integer-base ratio inequality implies a strictly negative scalar product-formula margin at three halves.","That the four-jet target has exact cardinality (3^R)^2(2^S)^2 and, for positive R, any family with n at least 4R+2S forces two distinct binary selectors to collide; their {-1,0,1} difference cancels all four endpoint jets.","That the rectangular Hermite-Pade threshold never improves on the classical one in the checked exponent model, with equality characterised exactly.","The exact 81/200 height region: 31/4 and every positive power lie inside it and outside the earlier Bundschuh-Vaananen region, while 3/2 lies outside both.","At n = 0, the Amdeberhan-Zeilberger q-Apery operator applied to Van Assche's moving diagonal has residual -p(p-1)^2(p+1)(p^5+2p^4+2p^3+2p^2+2), which is strictly negative and nonzero for every real p > 1."],"what_is_not_checked":["The universal rational-base conjecture, including the case 3/2, remains open. The checked 31/4 power family settles one region and leaves that conjecture untouched.","The published analytic height theorem, which is cited and used as an external result and is not formalised here.","The all-scale Heine-Zudilin coefficient construction at rational base three halves; the primary source assumes an integer base.","A sufficiently large primitive-normalised, non-collapsed coefficient family meeting the checked n >= 4R+2S threshold at the required quadratic depths.","That one of the resulting bounded-coefficient collision vectors has a nonzero polynomial pair, lies outside the analytic remainder nullspace, and has sufficient local gain.","The elementary Padé pilot checks exponent arithmetic only; it lacks remainder positivity or decay and does not qualify the separate Zudilin contour result.","The n=0 q-Apéry residual gives no general recurrence, lattice, valuation, denominator, or irrationality result; it only blocks the named scalar transfer."],"open_obligations":[{"id":"three_halves_growing_rank_endpoint_jet_kernel","statement":"After rowwise primitive normalisation, exhibit at least 4R_n+2S_n coefficient pairs at positive quadratic depths from a non-collapsed deformation; preserve the two- and three-adic gain; then show that one checked {-1,0,1} collision has a nonzero polynomial pair outside the analytic remainder nullspace, or prove an exact rank obstruction."}],"finite_search":null,"paper":{"paper_id":"erdos-1049-rational-base-lambert","title":"Hankel Determinants of Geometric Moments and Rational Lambert Values","source":"paper/1049/erdos-1049-rational-base-lambert.tex","pdf":"paper/1049/erdos-1049-rational-base-lambert.pdf","full_text":"docs/papers/full-text/erdos-1049-rational-base-lambert.md","resolution":"resolved"},"external_check":{"state":"reviewed","family_count":8,"by_class":{"not_applicable":2,"not_selected":3,"targeted":3},"dispositions":{"scalar_content_no_go":"not_selected_scalar_content_scaling_family","endpoint_residues":"not_selected_endpoint_residue_and_common_multiplier_family","four_jet_collision":"not_selected_finite_combinatorial_family","coordinatewise_corridor_no_go":"targeted","rational_base_tail_recurrence":"targeted","seven_halves_irrationality":"not_applicable_to_external_irrationality_conclusion","height_and_pade_arithmetic":"targeted","calibrated_rational_hankel_countermodel":"not_applicable_without_exact_Lean_statement"}}}],"reviewed_problems":[{"problem_id":"erdos_249","erdos_number":249,"short_title":"The totient constant","status":"open","library_root":"Erdos249257.lean","archived_provenance":"paper/archive/erdos249-257-main-paper.tex","problem_note":"paper/249/erdos-249-binary-totient-series.tex","route":"python3 scripts/query_corpus.py --route erdos249_diagonal_arithmetic","note":"The individual problem paper and reviewed claim registry own the active reading route and status. The former combined manuscript is retained only as provenance.","paper":{"paper_id":"erdos-249-binary-totient-series","title":"Integral Relations among Totient Sections","source":"paper/249/erdos-249-binary-totient-series.tex","pdf":"paper/249/erdos-249-binary-totient-series.pdf","full_text":"docs/papers/full-text/erdos-249-binary-totient-series.md","resolution":"resolved"}},{"problem_id":"erdos_257","erdos_number":257,"short_title":"Reciprocal sums over infinite exponent supports","status":"open","library_root":"Erdos249257.lean","archived_provenance":"paper/archive/erdos249-257-main-paper.tex","problem_note":"paper/257/erdos-257-mersenne-support-subseries.tex","route":"python3 scripts/query_corpus.py --route erdos257_half_story","note":"The individual problem paper and reviewed claim registry own the active reading route and status. The former combined manuscript is retained only as provenance.","paper":{"paper_id":"erdos-257-mersenne-support-subseries","title":"Irrationality criteria for Lambert subseries","source":"paper/257/erdos-257-mersenne-support-subseries.tex","pdf":"paper/257/erdos-257-mersenne-support-subseries.pdf","full_text":"docs/papers/full-text/erdos-257-mersenne-support-subseries.md","resolution":"resolved"}}],"validation_commands":["python3 scripts/build_problem_index.py --check","python3 scripts/check_problem_note_sources.py --coverage","python3 scripts/check_publication_contract.py"]},"body_format":"json_parsed","byte_parity_note":"body is the parsed object; re-serialisation differs from source bytes. Verify against sha256 by hashing the upstream file."},{"record_type":"formal_claim_control_plane_record","record_id":"lean_methodology","source_repository":"plectis-erdos","source_path":"docs/methodology.json","media_type":"application/json","licence":"Apache-2.0","copyright":"2026 Will Cook","why_an_offline_reader_needs_this":"Distinguishes Lean kernel checking from repository checking and from human mathematical review; owns the standing rules for public claims.","embedded":true,"availability":"embedded","sha256":"sha256:cef19d9092c49ba9c9b269a4b04c2c6030164bba6b80561117577713f60c3750","source_byte_count":51034,"body":{"schema":"erdos249257-methodology/2","title":"Formal proofs, mathematical claims, and what remains open","artifact_role":"rules_for_mathematical_reasoning_and_public_claim_changes_not_proof_authority","scope":"This file defines how this repository distinguishes formal proof, intended mathematical meaning, public claims, and remaining open propositions. It does not prove any mathematical statement and does not replace human mathematical review.","human_preamble":["This page records the rules used to relate the Lean development to the mathematical claims made in this repository, and to keep partial results from being overstated. It is generated from `docs/methodology.json`; run `python3 scripts/build_methodology.py` after changing that source. Lean source checked by the pinned Lean kernel remains proof authority.","The machine source calls the six standing rules `method_axiom` records. They are repository rules, not Lean `axiom` declarations."],"imports_contracts":["docs/claims.json::machine_readable_paper.authority_order","docs/claims.json::status_taxonomy","docs/claims.json::remaining_open_propositions","docs/claims.json::non_claims"],"evidence_classes":{"lean_check":{"human_name":"Lean check","decides":"Whether the declaration elaborates and its proof term is accepted in the pinned environment. 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The packet builder exports a registered historical cut without Git history and verifies that the target declaration is absent. It does not isolate the participant from other files, the network, or prior knowledge. Packet preparation is not evaluation evidence: current explanatory prose needs a hint review, controls need capsule and workload matching, and a scored run needs fresh participants and independent scoring.","does_not_decide":"Whether the recovered statement is novel, significant, or correctly proved; whether the mechanism credited for the recovery is the one the original proof used; or whether a recovery generalises beyond the held-out case."}},"human_capsule":["The exact boundary between proved and open must remain explicit.","Lean checks the stated proposition; it does not check that the proposition expresses the intended mathematics.","A public claim is transferred between representations only through a proved implication or equivalence.","A reduction must state the hypothesis or quantifier that remains open.","A finite result remains finite unless a theorem supplies the unbounded or universal step.","Progress toward an open problem must state which outstanding obligation it proves, 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projections.","evidence_classes":["repository_check","human_mathematical_review"],"applies_to":["change_type:lean_proposition_changed","change_type:claim_status_changed","change_type:authored_exposition_changed","change_type:projection_changed"],"checked_by":["release.methodology_schema_and_references","release.methodology_routes_exist"],"scope_limit":"A proof-body-only change does not automatically authorize stronger prose."},{"id":"principle.public_checkout_explains_claims_and_open_boundary","title":"A public checkout must explain the claims and open boundary","group":"publishing","statement":"A reader using only the public checkout must be able to find the main argument, proof routes, public claim statuses, and remaining open propositions.","evidence_classes":["repository_check","human_mathematical_review"],"applies_to":["object:projection","change_type:projection_changed"],"checked_by":["release.methodology_schema_and_references","release.methodology_projection_matches_source","release.methodology_routes_exist"],"scope_limit":"Path and projection checks establish availability, not reader comprehension."},{"id":"principle.claim_assurance_is_two_sided","title":"Claim assurance checks underclaiming as well as overclaiming","group":"publishing","statement":"Public claim review checks both ceilings and coverage: prose must not exceed the checked theorem chain, and a source-backed high-value mechanism, exact reformulation, quantitative reduction, or distinct endpoint must be mechanically discoverable when omitting it would materially understate the corpus.","evidence_classes":["lean_check","repository_check","human_mathematical_review"],"applies_to":["object:lean_declaration","object:public_claim","object:projection","change_type:claim_support_linkage_changed","change_type:authored_exposition_changed","change_type:projection_changed"],"checked_by":["release.methodology_schema_and_references","release.methodology_projection_matches_source","release.methodology_routes_exist"],"scope_limit":"Coverage does not turn support lemmas into separate headline claims: group plumbing under the smallest faithful contribution and require human review for significance."}],"anti_principles":[{"id":"anti_principle.do_not_equate_lean_acceptance_with_intended_meaning","title":"Do not equate Lean acceptance with intended meaning","statement":"Do not use a successful Lean check to bypass review of whether the proposition expresses the intended mathematics.","evidence_classes":["human_mathematical_review"],"applies_to":["object:lean_declaration","object:public_claim","change_type:lean_proposition_changed"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"The repository can require review but cannot automate the judgement."},{"id":"anti_principle.do_not_report_reduction_as_solution","title":"Do not report a reduction as a solution","statement":"Do not narrate a conditional implication, equivalence, or criterion as closure of the original open problem.","evidence_classes":["repository_check","human_mathematical_review"],"applies_to":["claim_status:conditional reduction","object:proved_reduction"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"An exact and useful reduction remains a genuine result when its remaining hypothesis is explicit."},{"id":"anti_principle.do_not_generalize_finite_result_without_theorem","title":"Do not generalize a finite result without a theorem","statement":"Do not treat a large checked range as evidence for an unbounded or universal assertion.","evidence_classes":["repository_check","human_mathematical_review"],"applies_to":["claim_status:verified finite instance","object:finite_certificate","change_type:finite_certificate_range_changed"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"Finite results may be valuable without changing the universal claim."},{"id":"anti_principle.do_not_identify_representations_without_transport","title":"Do not identify two representations without proved transport","statement":"Do not speak of a transformed representation as the original claim unless a checked implication or equivalence supplies that relation.","evidence_classes":["lean_check","human_mathematical_review"],"applies_to":["object:representation","change_type:representation_changed"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"This rule concerns claim transfer, not the informal use of notation."},{"id":"anti_principle.do_not_treat_graph_model_or_private_work_as_proof","title":"Do not treat a graph, model output, or private work as proof authority","statement":"A dependency graph, generated explanation, language-model output, or private proof sketch is not public proof authority.","evidence_classes":["repository_check"],"applies_to":["object:projection","change_type:projection_changed"],"checked_by":["release.methodology_schema_and_references","release.methodology_routes_exist","release.proof_trust"],"scope_limit":"These artefacts may help navigation or discovery without becoming proof."},{"id":"anti_principle.do_not_use_output_volume_as_mathematical_progress","title":"Do not use output volume as mathematical progress","statement":"Counts of declarations, proof lines, certificates, or agent activity do not by themselves establish progress on an open problem.","evidence_classes":["human_mathematical_review"],"applies_to":["argument_relation:advances_open_target","change_type:open_problem_relation_changed"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"Scale metrics may describe the artefact; they do not change a mathematical claim."},{"id":"anti_principle.do_not_edit_projection_as_source","title":"Do not edit a projection as though it were the source","statement":"README, generated Markdown, the corpus descriptor, and declaration maps must remain below their owning source files.","evidence_classes":["repository_check"],"applies_to":["object:projection","change_type:projection_changed"],"checked_by":["release.methodology_schema_and_references","release.methodology_projection_matches_source","release.methodology_routes_exist"],"scope_limit":"The paper is authored exposition and is checked for bounded agreement rather than generated from this file."},{"id":"anti_principle.do_not_keep_rule_without_owner_check_and_scope","title":"Do not keep a rule without an owner, a check or review route, and a scope limit","statement":"A methodology rule is removed if it cannot name the objects or changes it applies to, the evidence it requires, the route that evaluates it, and what it does not establish.","evidence_classes":["repository_check"],"applies_to":["object:projection","change_type:methodology_changed"],"checked_by":["release.methodology_schema_and_references"],"scope_limit":"A human-review route is valid when the judgement is not machine-decidable; it must still be explicit."}],"worked_examples":[{"id":"example.exact_reduction_without_closure","title":"An exact reduction that does not close #249","claim_ids":["certificate_reduction","certificate_completeness"],"remaining_open_ids":["remaining_open.unbounded_certificate_supply"],"route":["a hypothetical rational value of S = ∑ φ(n)/2ⁿ","an integrality constraint on totient tail differences","a finite non-integrality certificate at fixed parameters (certifiedKill)","an exact conditional reduction (irrational_totient_series_of_lcm_diagonal_certificate_supply)","the unbounded supply of such certificates, which remains open"],"narrative":"If S were rational, its totient tails would satisfy an integrality constraint. The certifiedKill predicate refutes that constraint at fixed parameters, and the completeness theorem shows the finite computation loses no information at those parameters. The reduction theorem then proves: if certificates occur at unbounded parameters, S is irrational. Producing that unbounded supply is the remaining open proposition; nothing in this repository claims it."},{"id":"example.finite_verification_stays_finite","title":"Finite verification that stays finite","claim_ids":["certified_kill_instances"],"remaining_open_ids":["remaining_open.unbounded_certificate_supply"],"narrative":"certifiedKill_all_small and the lcm-diagonal deposits through t = 8 (certifiedKill_periodLcm_diagonal_upto_six and the t = 7 and t = 8 endpoints) are kernel-checked with decide at fixed bounds. Each names a specific bounded region the checker has cleared; together with the generated certificate shards they show the reduction is not vacuous. The claim keeps the status verified finite instance and states its bounded domain. Raising the checked bound changes the stated bound; it does not change the quantifier of any public claim."},{"id":"example.necessary_structure_without_contradiction","title":"Necessary structure that does not yet contradict","claim_ids":["reciprocal_mass_bound","unbounded_carry_states","sublog_zero_windows"],"remaining_open_ids":["remaining_open.universal_257_all_infinite_supports"],"narrative":"The carry and zero-window theorems state what a rational series value would force: reciprocal-mass lower bounds, unbounded carry states, and sublogarithmic zero windows in divisor coverage. These are necessary consequences of rationality. They constrain a hypothetical rational value; none of them rules it out, and the universal #257 statement remains open."}],"human_review_triggers":[{"id":"claim_text_changed","description":"The wording of a public claim changes."},{"id":"claim_status_changed","description":"The status of a public claim changes."},{"id":"intended_meaning_changed","description":"The reviewed correspondence between a proposition and its intended statement changes."},{"id":"paper_statement_changed","description":"A paper-facing mathematical statement changes."},{"id":"significance_claim_changed","description":"What the repository says a result means for an open problem changes."},{"id":"rule_semantics_changed","description":"The meaning of a methodology rule changes, not only its wording or projection."}],"forbidden_effect_vocabulary":[{"id":"change_open_target_status","description":"The change must not alter the status of an open target claim."},{"id":"infer_unbounded_result","description":"The change must not derive an unbounded or universal statement from bounded evidence."},{"id":"strengthen_public_claim_wording","description":"The change must not strengthen the wording of any public claim."},{"id":"identify_representations_without_transport","description":"The change must not present a transformed representation as the original claim without a checked transport theorem."},{"id":"bypass_projection_owner","description":"The change must not edit a generated projection except by regenerating it from its owner."}],"change_types":["proof_body_changed","lean_proposition_changed","assumptions_changed","representation_changed","finite_certificate_range_changed","claim_support_linkage_changed","open_problem_relation_changed","claim_status_changed","authored_exposition_changed","projection_changed","methodology_changed"],"change_classes":[{"id":"proof_body_changed","summary":"A proof body changes; the stated proposition does not","minimum_evidence":["lean_check","repository_check"],"human_review":"on_trigger","review_triggers":["intended_meaning_changed","claim_text_changed"],"forbidden_effects":["strengthen_public_claim_wording","change_open_target_status"],"public_consequence":"No claim or exposition change follows automatically. Compare the axiom audit before and after."},{"id":"lean_proposition_changed","summary":"A stated proposition or its hypotheses change","minimum_evidence":["lean_check","repository_check","human_mathematical_review"],"human_review":"required","forbidden_effects":["change_open_target_status"],"public_consequence":"Reconcile the claim wording, the remaining-open boundary, and the paper statement."},{"id":"assumptions_changed","summary":"The transitive assumptions of a declaration change","minimum_evidence":["lean_check","repository_check","human_mathematical_review"],"human_review":"required","forbidden_effects":["strengthen_public_claim_wording"],"public_consequence":"Reassess the proof posture even when the proposition is unchanged."},{"id":"representation_changed","summary":"A claim moves between coordinates, encodings, or models","minimum_evidence":["lean_check","human_mathematical_review"],"human_review":"required","forbidden_effects":["identify_representations_without_transport"],"public_consequence":"Record the transport theorem, its direction, and anything not preserved."},{"id":"finite_certificate_range_changed","summary":"A checked bounded range or certificate family is extended","minimum_evidence":["lean_check","repository_check"],"human_review":"on_trigger","review_triggers":["claim_text_changed","claim_status_changed","intended_meaning_changed","paper_statement_changed"],"forbidden_effects":["change_open_target_status","infer_unbounded_result"],"public_consequence":"Keep the finite status; update only the stated bound."},{"id":"claim_support_linkage_changed","summary":"A claim's declaration coordinates or links are corrected; the claim is otherwise unchanged","minimum_evidence":["lean_check","repository_check"],"human_review":"on_trigger","review_triggers":["claim_text_changed","claim_status_changed","intended_meaning_changed"],"forbidden_effects":["strengthen_public_claim_wording","change_open_target_status"],"public_consequence":"Link or correct the supporting declarations of an unchanged claim, then regenerate dependent projections. The mathematical claim does not change; a Lean check confirms the linked declaration exists and elaborates."},{"id":"open_problem_relation_changed","summary":"An advances-open-target relation is added or changed","minimum_evidence":["repository_check","human_mathematical_review"],"human_review":"required","forbidden_effects":["change_open_target_status"],"public_consequence":"Record the outstanding obligation affected and the exact relation."},{"id":"claim_status_changed","summary":"The status of a public claim changes","minimum_evidence":["repository_check","human_mathematical_review"],"human_review":"required","forbidden_effects":["infer_unbounded_result"],"public_consequence":"State the theorem or review that licenses the new status."},{"id":"authored_exposition_changed","summary":"Authored paper or documentation prose changes mathematically","minimum_evidence":["repository_check","human_mathematical_review"],"human_review":"required","forbidden_effects":["change_open_target_status"],"public_consequence":"Prose must not exceed the status recorded in the claim registry."},{"id":"projection_changed","summary":"A generated projection is rebuilt","minimum_evidence":["repository_check"],"human_review":"not_required","forbidden_effects":["bypass_projection_owner","strengthen_public_claim_wording"],"public_consequence":"Regenerate below the existing owners; no mathematical effect."},{"id":"methodology_changed","summary":"docs/methodology.json changes","minimum_evidence":["repository_check"],"human_review":"on_trigger","review_triggers":["rule_semantics_changed"],"forbidden_effects":["change_open_target_status"],"public_consequence":"Regenerate docs/METHODOLOGY.md and the corpus descriptor. Rules route review; they do not decide mathematics."}],"transition_contracts":[{"id":"transition.lean_declaration_to_public_claim","title":"From a Lean declaration to a public claim","trigger":"A new declaration or a changed proposition is proposed as support for a public claim.","required_evidence_classes":["lean_check","repository_check","human_mathematical_review"],"requirements":["The declaration exists in the pinned public source and passes the Lean check.","The public claim links to the declaration and uses a declared claim status.","Human mathematical review confirms that the proposition expresses the intended paper-facing statement.","Any remaining open proposition is stated explicitly."],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"]},{"id":"transition.representation_change_to_public_claim","title":"From a changed representation to a public claim","trigger":"A claim is moved into different coordinates, an encoding, a recurrence, or another model.","required_evidence_classes":["lean_check","human_mathematical_review"],"requirements":["A checked implication or equivalence identifies the direction of transfer.","The claim states anything not preserved by the transfer.","Human mathematical review confirms that the transferred statement is the intended one."],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"]},{"id":"transition.conditional_reduction_to_public_claim","title":"From a conditional reduction to a public claim","trigger":"A checked implication or equivalence is published with status conditional reduction.","required_evidence_classes":["lean_check","repository_check","human_mathematical_review"],"requirements":["The supporting declaration is linked.","The claim names at least one remaining open proposition from docs/claims.json.","The public statement says what the reduction does not prove."],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"]},{"id":"transition.finite_certificate_to_public_claim","title":"From a finite certificate to a public claim","trigger":"A checked finite computation or bounded family is added as a new public claim, or its interpretation changes.","required_evidence_classes":["lean_check","repository_check","human_mathematical_review"],"requirements":["The claim status remains verified finite instance.","The checked finite domain is explicit.","No open target changes status without a separate unbounded or universal theorem.","A purely mechanical extension of an already reviewed bounded family follows the finite_certificate_range_changed change class instead: Lean and repository checks, with review re-entered only when claim text, claim status, intended meaning, or the paper statement changes."],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"]},{"id":"transition.public_claim_to_open_target_edge","title":"From a public claim to an advances-open-target edge","trigger":"The argument graph says that a public claim advances an open target.","required_evidence_classes":["repository_check","human_mathematical_review"],"requirements":["The target claim has status open.","The edge names a remaining open proposition owned by that target.","The edge states whether the result proves, narrows, decomposes, re-expresses, or constrains that proposition."],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"]},{"id":"transition.methodology_source_to_public_projections","title":"From methodology source to public projections","trigger":"docs/methodology.json changes.","required_evidence_classes":["repository_check"],"requirements":["docs/METHODOLOGY.md is regenerated exactly from docs/methodology.json.","The corpus descriptor records the methodology path, schema, digest, and compact capsule.","docs/agents/AGENT_GUIDE.md and README.md route to the methodology without duplicating its full content.","README.md carries no methodology machine identifiers beyond the source link."],"checked_by":["release.methodology_schema_and_references","release.methodology_projection_matches_source","release.methodology_routes_exist"]}],"validation_guards":[{"id":"release.methodology_schema_and_references","description":"Checks the methodology schema, literal object and change references, evidence classes, rule fields, local-instance references, change-class evidence and review requirements, guard references, and reciprocal guard-to-rule bindings.","serves":["method_axiom.proof_is_exact_checked_proposition","method_axiom.formal_proof_does_not_verify_intended_meaning","method_axiom.representation_changes_require_proved_transport","method_axiom.reduction_states_remaining_open_hypothesis","method_axiom.finite_results_remain_finite_without_uniform_theorem","method_axiom.open_problem_progress_names_changed_open_proposition","principle.partial_result_states_remaining_open_proposition","principle.derive_exact_consequences_from_hypotheses","principle.exact_finite_checker_needs_soundness_and_completeness","principle.record_what_representation_changes_preserve","principle.record_failed_routes_and_counterexamples","principle.classify_change_before_updating_public_claims","principle.separate_lean_human_and_agent_authority","principle.update_authoritative_sources_before_projections","principle.public_checkout_explains_claims_and_open_boundary","principle.claim_assurance_is_two_sided","anti_principle.do_not_equate_lean_acceptance_with_intended_meaning","anti_principle.do_not_report_reduction_as_solution","anti_principle.do_not_generalize_finite_result_without_theorem","anti_principle.do_not_identify_representations_without_transport","anti_principle.do_not_treat_graph_model_or_private_work_as_proof","anti_principle.do_not_use_output_volume_as_mathematical_progress","anti_principle.do_not_edit_projection_as_source","anti_principle.do_not_keep_rule_without_owner_check_and_scope","transition.lean_declaration_to_public_claim","transition.representation_change_to_public_claim","transition.conditional_reduction_to_public_claim","transition.finite_certificate_to_public_claim","transition.public_claim_to_open_target_edge","transition.methodology_source_to_public_projections"]},{"id":"release.claim_transition_requirements","description":"Checks typed remaining-open propositions, conditional reductions, finite claim domains, and advances-open-target edge effects.","serves":["method_axiom.formal_proof_does_not_verify_intended_meaning","method_axiom.representation_changes_require_proved_transport","method_axiom.reduction_states_remaining_open_hypothesis","method_axiom.finite_results_remain_finite_without_uniform_theorem","method_axiom.open_problem_progress_names_changed_open_proposition","principle.partial_result_states_remaining_open_proposition","principle.exact_finite_checker_needs_soundness_and_completeness","principle.record_what_representation_changes_preserve","anti_principle.do_not_equate_lean_acceptance_with_intended_meaning","anti_principle.do_not_report_reduction_as_solution","anti_principle.do_not_generalize_finite_result_without_theorem","anti_principle.do_not_identify_representations_without_transport","anti_principle.do_not_use_output_volume_as_mathematical_progress","transition.lean_declaration_to_public_claim","transition.representation_change_to_public_claim","transition.conditional_reduction_to_public_claim","transition.finite_certificate_to_public_claim","transition.public_claim_to_open_target_edge"]},{"id":"release.methodology_projection_matches_source","description":"Checks that docs/METHODOLOGY.md is the exact deterministic projection of docs/methodology.json.","serves":["principle.public_checkout_explains_claims_and_open_boundary","principle.claim_assurance_is_two_sided","anti_principle.do_not_edit_projection_as_source","transition.methodology_source_to_public_projections"]},{"id":"release.methodology_routes_exist","description":"Checks that the claim-map entrypoint, corpus descriptor, docs/agents/AGENT_GUIDE.md, and README.md route to the methodology source and human projection, and that README carries no methodology machine identifiers.","serves":["principle.classify_change_before_updating_public_claims","principle.separate_lean_human_and_agent_authority","principle.update_authoritative_sources_before_projections","principle.public_checkout_explains_claims_and_open_boundary","principle.claim_assurance_is_two_sided","anti_principle.do_not_treat_graph_model_or_private_work_as_proof","anti_principle.do_not_edit_projection_as_source","transition.methodology_source_to_public_projections"]},{"id":"release.proof_trust","description":"Checks that public Lean source contains no sorry, admit, native_decide, or added axiom declarations.","serves":["method_axiom.proof_is_exact_checked_proposition","principle.separate_lean_human_and_agent_authority","anti_principle.do_not_treat_graph_model_or_private_work_as_proof"]}]},"body_format":"json_parsed","byte_parity_note":"body is the parsed object; re-serialisation differs from source bytes. Verify against sha256 by hashing the upstream file."},{"record_type":"formal_claim_control_plane_record","record_id":"lean_scope","source_repository":"plectis-erdos","source_path":"docs/SCOPE.md","media_type":"text/markdown","licence":"Apache-2.0","copyright":"2026 Will Cook","why_an_offline_reader_needs_this":"Names the exact propositions that remain open, and states that private work and generated prose are not public proof authority.","embedded":true,"availability":"embedded","sha256":"sha256:a49cfdf72b96920f35dcb3f05304988706f75a62589185d5ebd8db98b68b0659","source_byte_count":5804,"body":"<!-- SPDX-FileCopyrightText: 2026 Will Cook -->\n<!-- SPDX-License-Identifier: Apache-2.0 -->\n\n# Scope of this release\n\nUsing the degree-seven polynomial constructed by the erdosproblems.com\ncontributor ani, Lean proves that every preconnected strict-lemniscate set\ncontaining two distinct roots has one-dimensional Hausdorff measure greater\nthan two. This refutes the exact Formal Conjectures path-image-length\nstatement; the separate total-variation bound is also checked. The other\nseven targets remain open. Independent human review of correspondence with\nthe 1958 wording has not been recorded. Comparator checks only selected exact\nstatements, axioms and kernel acceptance; it does not assess novelty or\nhistorical correspondence.\n\nThe reported small-parameter family is not formalised. This release does not\nprove Erdős #68, #243, #249, #251, #257, #269, or #1049. Claims use only the\nimmutable formal-source checkpoint in [`docs/claims.json`](claims.json); the\nlast release tag is its citation identity,\nnot a substitute for an exact proof-source identity.\n\nLean source checked by the pinned kernel is proof authority. Unreleased work, private repositories, generated/provider output, and hidden sketches are not public proof artefact. Committed `research_corpus/Erdos1041` is source-only and extends neither the checkpoint nor `docs/claims.json`; it makes no peer-review, priority, novelty, or significance claim. Later work is out of scope even if locally checked; a later tag is a separate citation action.\n\n#1041: `python3 scripts/query_corpus.py --route erdos_1041` verifies four indexed digests and reports the 35-result/open envelope; resume with `python3 scripts/query_route_memory.py --problem 1041`. These navigation/evidence routes do not promote research rows into reviewed claims or Comparator or close #1041.\n\nThe [complete eight-problem return matrix](SOURCE_MAP.md#complete-eight-problem-return-matrix) binds each problem to its strongest source/paper route and frontier. The [Palomar showcase](PALOMAR_RESULT_SHOWCASE.json) maps source-reviewed family dispositions without adding claims. The generated [external-verification dossier](EXTERNAL_VERIFICATION.md) gives the exact spine and boundaries; neither replaces `docs/claims.json` or Lean proof authority.\n\n## Exact open propositions\n\nMachine-resolvable open propositions:\n\n- `remaining_open.erdos_1041_lemniscate_connection`: python3 scripts/query_corpus.py --open remaining_open.erdos_1041_lemniscate_connection\n- `remaining_open.erdos_1049_irrationality`: python3 scripts/query_corpus.py --open remaining_open.erdos_1049_irrationality\n- `remaining_open.erdos_243_eventual_recurrence`: python3 scripts/query_corpus.py --open remaining_open.erdos_243_eventual_recurrence\n- `remaining_open.erdos_243_overlap_height_growth`: python3 scripts/query_corpus.py --open remaining_open.erdos_243_overlap_height_growth\n- `remaining_open.erdos_249_first_harmonic_anti_concentration`: python3 scripts/query_corpus.py --open remaining_open.erdos_249_first_harmonic_anti_concentration\n- `remaining_open.erdos_249_irrationality`: python3 scripts/query_corpus.py --open remaining_open.erdos_249_irrationality\n- `remaining_open.erdos_251_irrationality`: python3 scripts/query_corpus.py --open remaining_open.erdos_251_irrationality\n- `remaining_open.erdos_269_cofinal_local_window_escape`: python3 scripts/query_corpus.py --open remaining_open.erdos_269_cofinal_local_window_escape\n- `remaining_open.erdos_269_reduced_tail_nonintegrality`: python3 scripts/query_corpus.py --open remaining_open.erdos_269_reduced_tail_nonintegrality\n- `remaining_open.erdos_269_three_prime_irrationality`: python3 scripts/query_corpus.py --open remaining_open.erdos_269_three_prime_irrationality\n- `remaining_open.erdos_269_two_dimensional_representation`: python3 scripts/query_corpus.py --open remaining_open.erdos_269_two_dimensional_representation\n- `remaining_open.erdos_68_irrationality`: python3 scripts/query_corpus.py --open remaining_open.erdos_68_irrationality\n- `remaining_open.half_value_membership`: python3 scripts/query_corpus.py --open remaining_open.half_value_membership\n- `remaining_open.twenty_one_actual_orbit_invariant`: python3 scripts/query_corpus.py --open remaining_open.twenty_one_actual_orbit_invariant\n- `remaining_open.twenty_one_final_skip_exclusion`: python3 scripts/query_corpus.py --open remaining_open.twenty_one_final_skip_exclusion\n- `remaining_open.twenty_one_permanent_affine_supercapacity`: python3 scripts/query_corpus.py --open remaining_open.twenty_one_permanent_affine_supercapacity\n- `remaining_open.twenty_one_scaled_remainder_cofinal_return`: python3 scripts/query_corpus.py --open remaining_open.twenty_one_scaled_remainder_cofinal_return\n- `remaining_open.unbounded_certificate_supply`: python3 scripts/query_corpus.py --open remaining_open.unbounded_certificate_supply\n- `remaining_open.universal_257_all_infinite_supports`: python3 scripts/query_corpus.py --open remaining_open.universal_257_all_infinite_supports\n\nFinite instances, conditional reductions, cited neighbours, and named infinite-support families do not discharge these propositions. Only a future claim transition against an exact identifier with evidence required by `docs/methodology.json` can change this boundary.\n\n## Machine identifiers\n\nMachine IDs in [`docs/claims.json`](claims.json), cross-checked by `scripts/check_release.py`:\n\n- `not_erdos_249_solution` / `not_erdos_257_solution`: #249 and universal #257 remain open.\n- `not_publication_authority`: not a peer-reviewed publication record.\n- `not_private_root_equivalence`: no equivalence to private work is asserted.\n- `not_provider_proof_authority`: provider/model output is not proof authority.\n- `not_hidden_proof_body_authority`: unpublished sketches are not proof authority.\n","body_format":"markdown"},{"record_type":"formal_claim_control_plane_record","record_id":"lean_source_map","source_repository":"plectis-erdos","source_path":"docs/SOURCE_MAP.md","media_type":"text/markdown","licence":"Apache-2.0","copyright":"2026 Will Cook","why_an_offline_reader_needs_this":"Bounded reading order over the Lean modules. Navigation, explicitly not proof authority.","embedded":true,"availability":"embedded","sha256":"sha256:07266bba272223f29e50612866983ff8c72b42b2f06781be69529d808315d9b5","source_byte_count":28839,"body":"<!--\nSPDX-FileCopyrightText: 2026 Will Cook\nSPDX-License-Identifier: Apache-2.0\n-->\n\n# Source map\n\n## Bounded reader route\n\nStart from one mathematical question rather than decoding the complete import\ngraph:\n\n1. Read `docs/orientation.json`, or its human projection\n   `docs/ORIENTATION.md`, and select one programme, claim, or exact remaining\n   open proposition.\n2. Run one bounded lookup:\n   `python3 scripts/query_corpus.py --route <programme_id>`,\n   `python3 scripts/query_corpus.py --claim <claim_id>`, or\n   `python3 scripts/query_corpus.py --open <remaining_open.id>`.\n3. Follow the returned paper label into the authored exposition and its\n   declaration or source handles into the modules grouped below.\n4. Read `Erdos249257.lean` or `ErdosProblems.lean` only when package topology\n   itself is the question. Those root files now live under `lean/`. Read `docs/reference/WAVE_INDEX.md` for chronology, not as the\n   recommended mathematical reading order.\n\nLean source checked by the pinned Lean kernel is proof authority. This map is\nauthored navigation: it does not strengthen a claim, replace\n`docs/claims.json`, or close an exact remaining-open proposition. Erdős #249\nand the universal form of #257 remain open.\n\n## Source result and Comparator spines\n\nUse the repository-local [result selection record](PALOMAR_RESULT_SHOWCASE.json)\nfor two related reading orders. It records a possible Palomar submission, not\nan external service decision, claim count, or proof authority. Its\n`source_result_spine` leads with the checked finite-prime weighted #257\nirrationality criterion, then the formalised Zudilin-region #1049\nirrationality theorem (including powers of 31/4), then ani's degree-seven\n#1041 counterexample to the exact Formal Conjectures path-image statement.\nUniversal arbitrary-support #257 and rational-base #1049 remain open;\ncorrespondence between the #1041 formal statement and the 1958 wording awaits\nindependent human review. The separate `candidate_ranking` orders\nComparator-qualified declarations. The generated [verification dossier](EXTERNAL_VERIFICATION.md#mathematical-signal-spine)\nexpands that Comparator spine with exact declarations, mechanisms, and\nboundaries. These selections are navigation and appraisal, not proof authority\nor novelty judgements; `docs/claims.json` and the named Lean sources own the\nexact claims. Support lemmas and conditional producers stay subordinate.\n\n## #1041 / source-current frontier\n\nThe current #1041 paper covers the exact Formal Conjectures path-image\nHausdorff refutation using ani's degree-seven polynomial, a separate\ntotal-variation counterexample, and Newton-flow results.\n[HausdorffLength](../lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean)\nis the Lean source for the path-image statement;\n[CatalogueAdapter](../lean/ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean)\nis the source entry for `ani_degree_seven_total_variation_counterexample`.\n`docs/claims.json` records the exact statements and attribution. Independent\nreview of correspondence with the 1958 wording remains outstanding. For the\nseparate source-only research corpus, start with the dated [frontier delta](../research_corpus/Erdos1041/FRONTIER.md),\nthen use the [strongest-result map](../research_corpus/Erdos1041/STRONGEST_RESULTS.json)\nand the [corpus manifest](../research_corpus/Erdos1041/CORPUS_MANIFEST.json).\nThe frontier delta is the authority for rows refuted after the activation map\nwas generated; do not rebuild those rows merely because they remain in the\nmachine map.\n\nThe high-signal reading routes are:\n\n- [hub selection and generic closure](../research_corpus/Erdos1041/AttachmentAwareReeb.md)\n  → [GenericSufficiencyClosure](../research_corpus/Erdos1041/GenericSufficiencyClosure.md):\n  the surviving parent carrier is `min_c L(c) <= 2` on the ray-separated locus,\n  followed by lower-semicontinuity closure.\n- [quantitative Fekete stability and radial/angular reduction](../research_corpus/Erdos1041/NearFeketeRadialAngularSplit.md)\n  → [degree-five sharpened cuts](../research_corpus/Erdos1041/Degree5AssemblyAndSharpenedCuts.md):\n  the new tools localise the residual, but do not prove the degree-five or\n  unrestricted theorem.\n- [partial-cluster perimeter identity](../research_corpus/Erdos1041/PartialClusterPreimagePerimeterIdentity.md)\n  → [tied-fibre moment hierarchy](../research_corpus/Erdos1041/TiedNewtonFaceFibreProduct.md):\n  these are exact geometric/algebraic premises for future selection work, not\n  a completed compatible-root selector.\n- [new exterior-root product covering](../research_corpus/Erdos1041/ExteriorRootProductCovering.md)\n  and [separatrix aggregate reduction](../research_corpus/Erdos1041/SeparatrixAggregateReduction.md):\n  read their explicit hypotheses and refutations before treating either as a\n  parent route.\n\nThe [current paper](papers/full-text/erdos-1041-lemniscate-newton-flow.md)\nlinks its statements to their Lean declarations. The separate research corpus is\nnot part of `docs/claims.json`'s reviewed claim authority: its files preserve\nsource-level premises, computations, no-go results, and open boundaries.\nThe exact Formal Conjectures path-image statement is refuted; the historical\nwording has not been independently adjudicated. The separate research corpus\nstill records open geometric continuations in `FRONTIER.md`, which must be read\nbefore new proof search.\n\n## Complete eight-problem return matrix\n\nThe per-problem [machine index](problems.json) lists module entry points,\npapers and open obligations. The [problem library](problem_library.json)\nexpands each programme into its papers and transitive Lean source map. Expand each row's\n`what_is_checked` and `what_is_not_checked` fields before treating one source\nentry as the whole result inventory: they are the evidence ceiling,\nnot additional claims. This single matrix makes the same route bidirectional:\nstart at a public problem id, reach source entry points and the paper\nrecord, then use the exact index handles to resume the open boundary; a paper\nor source reader can return with the `--route` command in the second column.\nUse `claim_registration.registered_claim_ids` to reach each registered\nstatement's status and declarations in `docs/claims.json`. The module or\nlibrary containing a statement does not determine its review status. Unlisted\ndeclarations acquire no public claim status from their neighbours. For #249\nand #257, follow the reviewed core routes as well as the problem-owned index.\n\n| Problem | Problem return / source back-route | Source entry points | Paper and source record |\n|---|---|---|---|\n| #68 | `python3 scripts/query_corpus.py --route erdos_68` | [`FactorialGapPlateauCore.lean`](../lean/ErdosProblems/Erdos68/FactorialGapPlateauCore.lean) · [`FactorialZeroPlateau.lean`](../lean/ErdosProblems/Erdos68/FactorialZeroPlateau.lean) | [paper note](papers/full-text/erdos-68-factorial-denominator-irrationality.md) · [TeX source](../paper/68/erdos-68-factorial-denominator-irrationality.tex) |\n| #243 | `python3 scripts/query_corpus.py --route erdos_243` | [`ReciprocalTailRigidity.lean`](../lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean) · [`SquareSpecialisationUnconditional.lean`](../lean/ErdosProblems/Erdos243/PaperCompleteR21/SquareSpecialisationUnconditional.lean#L70) | [paper note](papers/full-text/erdos-243-reciprocal-tail-rigidity.md) · [TeX source](../paper/243/erdos-243-reciprocal-tail-rigidity.tex) |\n| #249 | `python3 scripts/query_corpus.py --route erdos_249` · reviewed return: `python3 scripts/query_corpus.py --route erdos249_certificate_story` · family return: `python3 scripts/query_corpus.py --route probabilistic_gcd_geometry` | [`TotientStrictPrimeEscape.lean`](../lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean) · [`GcdMomentCalculus.lean`](../lean/Erdos249257/GcdMomentCalculus.lean#L235) · [`SternBrocotRunGeometry.lean`](../lean/Erdos249257/SternBrocotRunGeometry.lean#L343) | [paper note](papers/full-text/erdos-249-binary-totient-series.md) · [TeX source](../paper/249/erdos-249-binary-totient-series.tex) · [probability/cylinder explanation](papers/full-text/erdos-249-binary-totient-series.md#exact-identities-and-representations.) |\n| #251 | `python3 scripts/query_corpus.py --route erdos_251`<br>Palomar family spine (run `python3 scripts/query_semantic.py family-relations <family_id>`): `prime_gap_reformulation` → `small_mismatch_criterion` → `dyadic_tail_integrality_classification` → `integral_shift_classification` → `totient_shift_propagation` → `coefficient_only_no_go` | [`PrimeGapDyadicTail.lean`](../lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean) | [paper note](papers/full-text/erdos-251-prime-gap-dyadic-series.md) · [TeX source](../paper/251/erdos-251-prime-gap-dyadic-series.tex) |\n| #257 | `python3 scripts/query_corpus.py --route erdos_257` · reviewed return: `python3 scripts/query_corpus.py --route erdos257_half_story` | [`WeightedReturn.lean`](../lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean) · [`MersenneSubseriesRigidity.lean`](../lean/ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean) | [paper note](papers/full-text/erdos-257-mersenne-support-subseries.md) · [TeX source](../paper/257/erdos-257-mersenne-support-subseries.tex) |\n| #269 | `python3 scripts/query_corpus.py --route erdos_269`<br>Palomar family spine (run `python3 scripts/query_semantic.py family-relations <family_id>`): `conditional_carry_escape` → `weighted_phase_carry_observer` → `rank_two_kernel_no_go` → `height_fibre_and_shell` → `dyadic_block_alphabet` → `three_prime_lcm_cells` | [`RestrictedFloorSum.lean`](../lean/ErdosProblems/Erdos269/RestrictedFloorSum.lean) · [`WeightedPhaseCarry.lean`](../lean/ErdosProblems/Erdos269/WeightedPhaseCarry.lean) · [`ThreePrimeRunningLcm.lean`](../lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean) | [paper note](papers/full-text/erdos-269-three-prime-running-lcm.md) · [TeX source](../paper/269/erdos-269-three-prime-running-lcm.tex) |\n| #1041 | `python3 scripts/query_corpus.py --route erdos_1041` | [`HausdorffLength.lean`](../lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean) · [`CatalogueAdapter.lean`](../lean/ErdosProblems/Erdos1041/Counterexample/CatalogueAdapter.lean) · [`NewtonFlowRaySeparation.lean`](../lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean) | [dated frontier](../research_corpus/Erdos1041/FRONTIER.md) → [paper note](papers/full-text/erdos-1041-lemniscate-newton-flow.md) · [TeX source](../paper/1041/erdos-1041-lemniscate-newton-flow.tex) |\n| #1049 | `python3 scripts/query_corpus.py --route erdos_1049` | [`RationalBaseThreshold.lean`](../lean/ErdosProblems/Erdos1049/PaperCompleteR21/RationalBaseThreshold.lean) · [`RationalBaseLambert.lean`](../lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean) · [`FinitePencilProposition.lean`](../lean/ErdosProblems/Erdos1049/PaperR20/FinitePencilProposition.lean#L43) | [paper note](papers/full-text/erdos-1049-rational-base-lambert.md) · [TeX source](../paper/1049/erdos-1049-rational-base-lambert.tex) |\n\nThe exact open boundary each row stops at, as machine handles. A five-column\ntable put these sentences 354px past the rendered column, so they read here:\n\n- **#68** — `weighted_collision_and_complementary_residue`; `cofinal_prime_power_amplification`; `cofinal_lower_endpoint_escape`; `cofinal_doubled_prime_branch_failure`; `cramer_residual_nonintegrality`\n- **#243** — `unbounded_negative_excursions`; `derive_negative_part_bound`; `formalise_the_published_criteria`\n- **#249** — Index: `strict_prime_tail_orbit_gap`; `hankel_denominator_fan_in`; `euler_sieve_limit_theorem`; `prime_ray_resultant_supply`; `stern_brocot_arithmetic_bridge`. Family: `totient_coprimality_probability`; `squared_lambert_gcd_moments`; `stern_brocot_cylinder_law`; `stern_brocot_run_fibonacci_stability`. Reviewed: `remaining_open.erdos_249_irrationality`; `remaining_open.unbounded_certificate_supply`\n- **#251** — `prime_gap_cofinal_shift_escape`; `cofinal_adjacent_small_mismatch`. Under a rationality witness, the actual infinite-tail identification is proved in [`PrimeGapDyadicTail.lean`](../lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L494); the cofinal escape and small-shift supplies remain open.\n- **#257** — Index: `arithmetic_rigidity_for_thin_supports`; `formalise_measure_and_stride_geometry`. Reviewed: `remaining_open.half_value_membership`; `remaining_open.twenty_one_permanent_affine_supercapacity`; `remaining_open.universal_257_all_infinite_supports`\n- **#269** — `actual_local_window_residue_escape`; `nonstationary_analytic_theorem`; `unbounded_height_certificate`. The actual rationality-to-reduced-carry bridge is proved in [`RationalityCarryBridge.lean`](../lean/ErdosProblems/Erdos269/RationalityCarryBridge.lean#L324); the actual cofinal escape producer remains open.\n- **#1041** — `historical_curve_length_correspondence`; `repair_or_refute_saddle_block`; `compact_ray_cut_strip_decomposition`; `corrected_metric_gluing_for_restricted_families`; `two_stage_stable_perturbation`; `relative_global_newton_flow_theorem`\n- **#1049** — `three_halves_growing_rank_endpoint_jet_kernel`\n\nFrom a reviewed route, use the registered claim or paper-anchor query to\ntravel back to exact Lean declarations; use the open query for the registered\nunresolved boundary rather than treating a checked special case as a\nsolution:\n\n```sh\npython3 scripts/query_corpus.py --claim <claim_id>\npython3 scripts/query_corpus.py --paper-anchor <TeX_label_or_source_ref>\npython3 scripts/query_corpus.py --open <remaining_open.id>\n```\n\n## Build entrypoint\n\n- `Erdos249257.lean`\n- `ErdosProblems.lean`\n- Lake package: `erdos249257`\n- Lean libraries / stable root imports: `Erdos249257`, `ErdosProblems`\n\n`Erdos249257` is the reviewed #249/#257 corpus. `ErdosProblems` is the\nproblem-owned expansion surface; unregistered declarations there remain exact\nLean propositions rather than reviewed public mathematical claims.\n\n## #257 / Erdős–Borwein families\n\n- `Erdos249257.CertificateKernel`\n- Relevant declarations include: `irrational_erdosSum_full_support`,\n  `irrational_erdosBorwein_series`, `irrational_erdosSum_factorial_support`,\n  `irrational_erdosSum_two_pow_support`, and\n  `irrational_erdosSupportSeries_eventuallyPeriodic`.\n\n## Periodic weighted Lambert series\n\n- `Erdos249257.CertificateKernel`\n- Nonnegative weights:\n  `irrational_weightedErdosSeries_periodic`,\n  `irrational_weightedErdosSeries_eventuallyPeriodic`, and\n  `irrational_ratWeightSeries_eventuallyPeriodic`.\n- Signed weights:\n  `irrational_or_bpow_mul_eq_intCast_intWeightedErdosSeries_periodic` gives\n  the irrational-or-base-terminating dichotomy, while\n  `irrational_intWeightedErdosSeries_periodic_of_bpow_mul_ne_intCast`\n  consumes an explicit nontermination hypothesis.\n- `intWeightedCoeff_periodFourSignWeight_eq_zero_of_mod_four_eq_three` is the\n  residue-selection obstruction explaining why the signed endpoint is not\n  unconditional irrationality.\n\n## #249 / totient denominator exclusions\n\n- `Erdos249257.GapFareyBound`\n- `Erdos249257.CertificateKernel`\n- **Visible-coprime lattice mass** (`totient_visible_coprime_lattice_mass`) —\n  [`tsum_pos_coprime_inv_mersenne_eq_one`](../lean/Erdos249257/GcdMomentCalculus.lean#L349)\n  proves that the positive coprime Mersenne weights on the visible lattice sum\n  exactly to one. This is the rational normalization behind the fair-coin\n  coprimality form of #249, not an irrationality result. Follow the\n  [individual-paper explanation](papers/full-text/erdos-249-binary-totient-series.md#exact-identities-and-representations.)\n  or the reviewed claim route\n  `python3 scripts/query_corpus.py --claim stern_brocot_cylinder_law`; both\n  expose the explanation and reverse source links. The exact boundary is\n  unchanged: this identity does not settle #249.\n\n- **All-base totient-kernel reduction, spanning, and conditional rank**\n  (`all_base_totient_kernel_index_reduction`) —\n  [`TotientKernelConditional.lean`](../lean/Erdos249257/TotientKernelConditional.lean#L102)\n  gives the unconditional canonical spanning theorem, while\n  [`finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent`](../lean/Erdos249257/TotientKernelConditional.lean#L215)\n  gives rank `k^e + 1` only under the explicit linear-independence hypothesis.\n  That independence, and Martin's external positive-density theorem, are not\n  formalised here; this algebraic boundary does not prove irrationality of\n  #249. Retrieve the full claim/source record with\n  `python3 scripts/query_corpus.py --claim all_base_totient_kernel_index_reduction`,\n  then return to the open problem with\n  `python3 scripts/query_corpus.py --route erdos_249`.\n\n- **Prime-ray cyclotomic support escape** (`prime_ray_resultant_supply`) —\n  [`PrimeRayCyclotomicCurvature.lean`](../lean/ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean#L95)\n  turns bounded-degree order divisibility and eventual nontrivial clean layers\n  into finite-support escape and cofinal unbounded prime divisors. Its bounded\n  order is an existence of an exponent witness, not exact order. The actual\n  totient layer/resultant bridge and the route from unbounded prime support to\n  certificate or carry escape remain open; the Mersenne-only instantiation in\n  `CyclotomicAnchoredKill.lean:33-105` does not close #249 or justify novelty.\n\n- **Tail-orbit phase-density and natural-prime strict-gap reductions**\n  (`strict_prime_tail_orbit_gap`) —\n  [`TotientStrictPrimeEscape.lean`](../lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L25)\n  makes the two producer interfaces explicit: an `11/100` nonpositive-phase\n  density on arbitrarily late dyadic blocks implies an `89/100` first-harmonic\n  block saving, while the natural-prime strict-gap predicate gives a pointwise\n  `9/10` pivot escape through the wrapper\n  `Erdos249257.ExternalVerification.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap`.\n  The same module records the exact phase update\n  [`tailOrbitFirstExp_succ`](../lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean#L167):\n  after the integer carry disappears under exponentiation, the first phase\n  evolves by squaring. This is orbit-mechanism evidence only; it supplies none\n  of the density, cofinal prime strict-gap, or uniform-margin hypotheses.\n  The density, cofinal prime strict-gap, and positive truncation-margin\n  producers are unproved for the actual totient orbit. This is conditional\n  certificate/irrationality reduction only: it does not close #249, duplicate\n  the finite `first_harmonic_pivot_decomposition` or `actual_lcm_orbit_separation`\n  families, or justify novelty, priority, significance, or external review.\n\n## #249 / certificate reductions\n\n- `Erdos249257.TotientTailPeriodKiller`\n- `Erdos249257.CarrySurvivorExtinction`\n- `Erdos249257.LcmDiagonalReduction`\n- `Erdos249257.LcmConeFlatness`\n- `Erdos249257.LcmConeNonflat`\n\nSupporting ladder modules:\n\n- `Erdos249257.MersenneLambertLadder`\n- `Erdos249257.GeometricCoprimality`\n- `Erdos249257.GcdMomentCalculus`\n\n## #249 / diagonal pincer frontier\n\nRead this band after the certificate reductions above.  It sharpens the same\nopen obligation; it does not prove the required unbounded supply.\n\n- `Erdos249257.DiagonalPincerDecomposition` — exact full-target interval\n  characterisation and the target-avoidance-to-irrationality reduction.\n- `Erdos249257.SquaredMersenneDiagonalEnclosure` — Lambert-projected centre,\n  exact signed square tail, and a sharp separation criterion.\n- `Erdos249257.DiagonalFreshLossBridge` — exact fresh-loss projection and\n  stronger adjacent-suffix supply interfaces.\n- `Erdos249257.DiagonalPincerCertificates` and\n  `Erdos249257.DiagonalPincerCertificatesT*`, followed by\n  `ErdosProblems.Lift` and `ErdosProblems.Skip` — the finite diagonal segment\n  supplies a checked working depth for every\n  natural `t ≤ 82`; it supplies nothing at `t = 83`. This is finite source\n  extraction, not a cofinal or asymptotic result.\n- `Erdos249257.DiagonalPincerPrimeCertificates.*` — finite prime certificate\n  shards used by that assembled segment; read the aggregators first.\n\nAlgebraic support for the same band:\n\n- `MersenneShadowCyclotomicNoncollapse → MersenneShadowDenominatorGrowth`\n- `RadicalMobiusShadow → RepunitMobiusNumerator → CyclicTensorMobiusShadow`\n- `CyclotomicProjectionOfShadow → PrimePowerJumpDynamics`\n- `RationalDenominatorSurvival`, `LambertDiagonalEnclosure`,\n  and `FullTargetPrimeAdjunctionNoGo`\n\nExploratory exact interfaces with explicit conditional boundaries live in\n`PrimitiveRationalGapSupply`, `PrimitiveSupportBridge`,\n`ActualForeignResidueProjection`, `SternBrocotRunGeometry`, and\n`SupportDilationDifferences`.\n\n### Transport and curvature companion band\n\nRead `CurvatureCarry → ExponentOnlyTransport → JointExponentTransport` for the\nmain proof spine, then `PrimeJumpMigration → PrimeJumpWindow` and\n`ThreeTransportBoundary` for the new-prime and balanced finite consumers.\nFor the harmonic route, read\n`FirstHarmonicGap → FirstHarmonicPivot → AdjacentPhaseSeparation`.\n`FirstHarmonicGap` turns a constant-saving cosine bound into a direct\ncertificate. `FirstHarmonicPivot` decomposes the complex harmonic sum exactly\ninto centred supplier-fibre, fibre-mean, bad-supplier, and non-supplier terms;\nthe one-sided budgets `14X/25`, `X/100`, `X/100`, and `8X/25` give the required\n`9X/10` saving. `AdjacentPhaseSeparation` gives a pairwise alternative. No\ncofinal harmonic, prime-distribution, or decorrelation estimate is proved.\n`TropicalCurvatureCarry` is a proved negative result showing that bounded\nvaluation–unit precision alone cannot exclude all centred completions.\n\n- `CurvatureCarry` — exact curvature window, sharp radius, recurrence, and\n  conditional unbounded-supply endpoints.\n- `ExponentOnlyTransport` — squarefree support stability and the exact\n  periodic affine slope/intercept state.\n- `JointExponentTransport` — universal two-moment old-channel annihilation and\n  the anchored `(3,5)` four-vertex consumer.\n- `PrimeJumpMigration` / `PrimeJumpWindow` — migrated-channel correction,\n  sharp direct radius, and the finite `(12,5,15)` certificate.\n- `ThreeTransportBoundary` — balanced `(1,2,3,6)` consumer and the finite\n  `(60,12)` certificate.\n- `FirstHarmonicGap` — deterministic conversion of a blockwise harmonic gap\n  into a direct certificate.\n- `FirstHarmonicPivot` — exact four-term complex pivot decomposition and a\n  conditional cofinal budget-to-irrationality socket.\n- `TropicalCurvatureCarry` — fixed-precision local-signature no-go theorem.\n\nThe former one-sided transport companion is retired.  Its exact Lean results\nremain available through the claim registry and source routes; no separate\nactive manuscript or rendered companion is part of this release.\n\n### Paper-curation boundary\n\nThe paper treats `GcdMomentCalculus` as supporting mathematical exposition: its\nsquared Lambert transform, gcd moments, reduced-direction law, and cylinder\nrecursion explain the representation containing the #249 Möbius-square atom.\n`FullTargetPrimeAdjunctionNoGo` is included as a proved negative result because\nit closes a recurring prime-adjunction strategy without changing the open\nclaim. `SternBrocotRunGeometry` is registered as a source-level supporting\nclaim: its continuant/Fibonacci stability theorems are stated in the Lambert\ncomplements, but no endpoint argument currently consumes them. It therefore\nremains outside the paper's principal result spine; the cylinder law is owned\nby `GcdMomentCalculus`, while `SternBrocotRunGeometry` owns the induced-run\nstability package.\n\n## Binary-carry trunk (post-wave carry layer)\n\nBuilds on `Erdos249257.CertificateKernel`; not part of the wave chronology.\nReading order:\n`GenericTailOrbitRigidity → GreedyAchievementSet → CampbellShiftSynchronization → BooleanMobiusCarry → RationalSupportCarrySkeleton → DyadicPrefixCompression → SublogDivisorCoverage`.\n\n- `Erdos249257.GenericTailOrbitRigidity`\n- Relevant declarations include: `temperedBinaryOrbit_eq_scaledTail`, `binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit`, `balancedPulse_fanout_unbounded`, `balancedPulse_no_autonomous_decoder`\n\n- `Erdos249257.GreedyAchievementSet`\n- Relevant declarations include: `volume_mersenneAchievementSet`, `perfect_mersenneAchievementSet`, `isNowhereDense_mersenneAchievementSet`, `mem_mersenneAchievementSet_iff_greedy_survival`, `certifiedGreedyMersenneDeath_not_mem`, `three_fourths_not_mem_mersenneAchievementSet`, `rational_member_support_bit_iff`, `half_mem_mersenneAchievementSet_of_prefixForcingChain` (conditional interface only)\n\n- `Erdos249257.CampbellShiftSynchronization`\n- Relevant declarations include:\n  `campbellQuarterExponent_periodFreeze_noSync`,\n  `campbellPhaseFreeze_parameters_impossible`,\n  `shiftWindowZero_iff_greedyMersenneSkippedSupport_infinite`, and\n  `shiftWindowZero_iff_half_mem_mersenneAchievementSet`. These are a\n  synchronization no-go and an exact endpoint re-expression, not an\n  independent half-membership producer.\n\n- `Erdos249257.BooleanMobiusCarry`\n- Relevant declarations include: `erdosSupportSeries_rational_iff_exists_temperedCarry`, `support_fraction_iff_exists_temperedCarry`, `exists_normalized_support_fraction_iff_exists_booleanMobiusCarry`, `mobius_carryQuotient_recovers_support`, `support_half_iff_affineBinaryOrbit_tempered`\n\n- `Erdos249257.RationalSupportCarrySkeleton`\n- Relevant declarations include: `sum_doublingResidue_eq_mul_wrapCount`, `one_div_oddOrder_le_reciprocalMass_of_support_fraction`, `reciprocalMass_eq_residueMean_add_excessMean`, `dyadic_support_fraction_reciprocalMass_diverges_or_gt_one`, `exists_unbounded_shifted_odd_tail_nat_state_of_support_fraction`\n\n- `Erdos249257.DyadicPrefixCompression`\n- This is #257 support, not part of the #249 diagonal band. It gives exact\n  finite-fragment dyadic compression inside a hypothetical dyadic-rational\n  support value, reduced/raw block-safety coordinates, primitive-lattice and\n  support-tail-gap equivalences, conditional half-membership consumers, and\n  checked failures of naive induction.\n  Start with `finiteFragment_dyadicPrefixCompression`,\n  `halfGreedyThreeDepthTakeRunExcessBound_iff_skippedExcessBound`,\n  `greedyHalf_badSkipCapViolation_iff_primitiveLattice`,\n  `halfGreedyPrefixSupportTailGap_eq_excess_div_denominator`, and\n  `unsafeSkipGcdOvershootSupply_iff_primitive`. No cofinal witness supply is\n  proved.\n\n- `Erdos249257.SublogDivisorCoverage`\n- Relevant declarations include: `supportCoeffZeroWindow_length_le_eps_logb_add`\n\n## Half-value finite-prefix and boundary analysis\n\nThis band studies the test value `1/2` in ordinary finite-prefix language.\nA greedy prefix is the first `n` include/skip decisions. A seam is a pair of\nfinite words whose values bracket `1/2`. A cylinder is the set of infinite\nextensions of one fixed prefix. A producer is an unproved hypothesis that\nsupplies the required seams or return configurations arbitrarily far out.\n\nRead the band in four layers:\n\n1. `GreedyAchievementSet → HalfCutLocator` defines the achievement set, greedy\n   coding, and the finite cut around `1/2`.\n2. `HalfCarryReachability → HalfCylinderFiniteShadow` translates finite\n   prefixes into integer carry and finite-shadow coordinates.\n3. `HalfCylinderIntegerGreedy → HalfCylinderConcreteSeamAdapter →\n   HalfCylinderFullShellSeamBridge → HalfCylinderSkippedEndpointClassifier →\n   HalfCylinderFloorErrorReset → HalfCylinderSeamLimit` develops the exact\n   boundary arithmetic and limiting seam interface.\n4. `HalfCylinderFatalGapRightTail → HalfCylinderHalfMembershipClassification →\n   HalfCylinderLastProducerContradiction → HalfCylinderMiddleCarryLowerBound`\n   gives the endpoint classification, eliminates the upper and `-3` middle\n   cases, and isolates the remaining carry-tail inequality for `-2` and `-1`.\n\nThe floor-error and shell modules are supporting arithmetic layers, not\nseparate endpoint theorems. The last layer remains conditional: no producer\nsupplying the required configurations cofinally is proved.\n\n## Claim registry\n\n- `docs/claims.json` — the single owner of release identity, claim statuses, and declaration coordinates (file and line), cross-checked by `scripts/check_release.py`\n\n## Generated finite certificates\n\n- `Erdos249257.GeneratedCertificates`\n- `Erdos249257.GeneratedCertificates.*`\n\nThe complete import graph and every declaration are generated in\n`docs/claims.json::machine_readable_paper.module_graph` and\n`docs/declaration_atlas.json`; use those surfaces when a grouped route above is\nnot specific enough. Current corpus counts live in the generated\n`docs/ORIENTATION.md`, not in this authored route map.\n","body_format":"markdown"},{"record_type":"formal_claim_control_plane_record","record_id":"lean_results_map","source_repository":"plectis-erdos","source_path":"docs/RESULTS.md","media_type":"text/markdown","licence":"Apache-2.0","copyright":"2026 Will Cook","why_an_offline_reader_needs_this":"Per-problem human map: the strongest checked statement for each Erdős problem, its exact status, and where to verify it. The surface a mathematician reads before anything else.","embedded":true,"availability":"embedded","sha256":"sha256:4b2947142ea31b0a1aeb69a516296320d7d579cfd6ac739da4adc9a319bc934f","source_byte_count":141962,"body":"<!-- SPDX-FileCopyrightText: 2026 Will Cook -->\n<!-- SPDX-License-Identifier: Apache-2.0 -->\n\n# Results and limits\n\nThis repository formalises theorems, reductions, equivalences, and\nobstructions related to eight Erdős problem programmes —\n[#68](https://www.erdosproblems.com/68),\n[#243](https://www.erdosproblems.com/243),\n[#249](https://www.erdosproblems.com/249),\n[#251](https://www.erdosproblems.com/251),\n[#257](https://www.erdosproblems.com/257),\n[#269](https://www.erdosproblems.com/269),\n[#1041](https://www.erdosproblems.com/1041), and\n[#1049](https://www.erdosproblems.com/1049).\nEach problem has a short paper and a longer reasoning record; the retired\ncombined #249/#257 manuscript is archive/provenance only, not a current\ngateway. Using the degree-seven polynomial constructed by the erdosproblems.com\ncontributor ani, Lean proves that every preconnected strict-lemniscate set\ncontaining two distinct roots has one-dimensional Hausdorff measure greater\nthan two. This refutes the exact Formal Conjectures path-image-length\nstatement; the separate total-variation bound is also checked. The other\nseven targets remain open. Independent human review of correspondence with\nthe 1958 wording has not been recorded. Comparator checks only selected exact\nstatements, axioms and kernel acceptance; it does not assess novelty or\nhistorical correspondence.\n\nLean source checked by the pinned Lean kernel is proof authority. The audit log\nreports the headline declarations below with kernel assumptions\n`[propext, Classical.choice, Quot.sound]`. A theorem with hypotheses proves only\nthe displayed implication; it does not prove that its hypotheses occur.\n\n## The short version\n\nThe [front-page #257 example](../README.md) gives a complete irrationality\ntheorem for restricted supports. The other programmes have direct results,\nexact reductions, conditional routes and counterexamples to tempting methods.\nThe table gives one entry point per problem; the\n[guide below](#problem-by-problem-guide) keeps each result beside its missing\nstep. Mathematical importance requires separate judgement.\n\n| Problem | A result to start with | Where the result stops |\n|---|---|---|\n| [#68](#result-68) | Two exact finite denominator exclusions and a Lean-checked `3/2` lower growth exponent; the paper also gives a finite calculation of attainable factorial moments after prescribed cancellations. | Lean proves irrationality equivalent to cofinally many non-unit factorial carries; no cofinal supply of such carries has been produced, so irrationality is still unproved. |\n| [#243](#result-243) | Lean checks irrationality under the precise cubic rate and every nonintegral regular rate above one; the paper transfers these zero-indexed statements to one-based indexing. Signed-error criteria also force eventual Sylvester behaviour under their stated premises. | The unrestricted Sylvester-tail hypotheses remain unproved. |\n| [#249](#result-249) | The short paper and Lean classify the fixed-base-two totient-residue series for every positive modulus and every rational-valued observable at dyadic moduli. They also give the exact all-base totient-kernel rank `k^e+1`. | The original series with unreduced totients remains open; the conditional routes still need their cofinal arithmetic inputs. |\n| [#251](#result-251) | Lean checks a rich synthetic prime-gap countermodel, and the paper gives a separate sparse-perturbation obstruction. | These are not actual prime gaps; the prime-specific producer for irrationality remains open. |\n| [#257](#result-257) | Lean checks the finite-prime weighted-support theorem at every integer base, including supports with divergent reciprocal sum. | It does not cover every infinite support. [Palomar registered version 1](https://palomar-registry.org/entry?id=PALOMAR-2026-09-25-000009&version=1) of the exact five-declaration `E257_01` selection. |\n| [#269](#result-269) | Both two-prime running-LCM sums are transcendental by the paper's argument using a cited Hecke–Mahler theorem; Lean checks the formulas and the conditional transfer. | The cited transcendence input is not formalised, and the three-prime irrationality question remains open. |\n| [#1041](#result-1041) | Ani's degree-seven polynomial refutes the exact Formal Conjectures path-image-length statement in Lean; Lean also checks positive trinomial and sharp collinear families. | Independent review of correspondence with the 1958 wording is pending; other geometric results have their own hypotheses. |\n| [#1049](#result-1049) | Lean checks irrationality in Zudilin's rational-base contour region and exact Hankel orders. | `3/2` and the all-rational-base claim remain open. |\n\nTo read further, the [paper catalogue](../paper/README.md#problem-papers)\nhas each short paper and longer reasoning record; the\n[synthesis paper](../paper/synthesis/optimal-sparse-perturbations.pdf) reads\nacross the eight. To inspect a claim, use the [source map](SOURCE_MAP.md),\n[claim registry](claims.json), [prior art](PRIOR_ART.md) and\n[verification dossier](EXTERNAL_VERIFICATION.md); Comparator covers selected\nformal statements, not every theorem or its novelty. The\n[reproduction guide](REPRODUCIBILITY.md) starts with one claim without Lean.\nFor the system's design and its open-source collaboration model, read\n[the systems paper](../paper/systems/claim-faithful-publication-systems-paper.pdf).\nThe [systems paper index](../paper/systems/README.md) identifies the earlier\naccounts retained for historical detail.\nTo improve a proof, explanation or the architecture itself, use the\n[contribution route](../CONTRIBUTING.md),\n[architecture guide](research-commons/ARCHITECTURE_CONTRIBUTIONS.md) and\n[credit policy](research-commons/CREDIT_POLICY.md).\n\n### Problem-by-problem guide\n\n<a id=\"result-68\"></a>\n\n**[#68](https://www.erdosproblems.com/68).** Put\n`S = ∑_{n≥2} 1/(n!−1)` and `L_N = lcm_{2≤n≤N}(n!−1)`. Any rational\n`S = a/q` with `q>0` satisfies the two incomparable exclusions\n`q ∤ 299999!` and `q ≥ 2^{39990} > 10^{12038}`. The first is a factorial\ndivisibility constraint from a fresh exact GMP carry census through\n`m = 300000` together with a Lean consumer; that replay matches the previously\nretained certificate, while an\nindependent replay through `4000` reproduces the unit-carry prefix. The second\nis an independent continued-fraction size bound. Neither implication yields\nthe other, and neither proves irrationality. The paper and Lean proof give\n`liminf log L_N/(N^(3/2) log N) ≥ 2√2/3`. This is a common-denominator theorem:\nit shows why clearing every summand separately cannot make the positive tail\nsmall, but says nothing by itself about the denominator after cancellation.\nLean separately checks that irrationality of `S` is equivalent to\ncofinally many non-unit factorial carries, and checks a finite quotient-band\nchannel obstruction. The required cofinal carries are not produced. A local\nexact calculation at the prime `12487` exhibits two critical roots for gap\n`12`, so the earlier small-prime pattern of at most one critical root is not a\nuniform theorem; those roots have factorial values `442` and `6300`, not `1`.\nLean checks the `3/2` theorem as `common_denominator_growth_liminf` in\n`PaperCompleteLiminf.lean`. The separate public Lean release\n[`wcook04/plectis-erdos-lean`](https://github.com/wcook04/plectis-erdos-lean/blob/52f29ad173b04e3bac941b3663f2b9aebe5de0bb/PalomarCorpus/E68/Challenge.lean#L249)\nstates the `3/2` bound at commit `52f29ad1` as `common_denominator_growth` in\nits entry `PalomarCorpus/E68` and proves it; that repository's Linux replay of\nPalomar's Comparator stage accepted the entry with both kernels\n(run 34782407633).\nFurther conditional zero-branch and private-factor carry producers are in the\n[formal evidence](EXTERNAL_VERIFICATION.md#programme-68); neither supplies the\ncofinal property for the actual factorial-gap orbit. Read the\n[short paper](../paper/68/erdos-68-factorial-denominator-irrationality.pdf) or\n[long record](../paper/68/erdos68-factorial-reasoning-surface.pdf).\nThe separately prepared\n[`E68_05` Comparator selection](https://github.com/wcook04/plectis-erdos-lean/blob/dc779af057dbab4224ac4f0cc101384f340fabc2/PalomarCorpus/E68_05/comparator.json)\ncontains six structural carry, channel and criterion declarations. It does\nnot select the denominator exclusions or the `3/2` growth theorem above.\nIts [caller-side preflight](https://github.com/wcook04/plectis-erdos-lean/actions/runs/36019608937)\npassed at that exact source commit; this is not a Palomar registration.\n\nThe channel results have a direct finite use. For a finitely supported integer\nvector `λ` on indices `n≥2`, set `M(λ)=∑ n!λₙ` and let `V_d(λ)` be its\nfactorial channel at `d`, as defined in [Section 2 of the short\npaper](../paper/68/erdos-68-factorial-denominator-irrationality.pdf).\nGiven a depth `D≥2`, the paper classifies the moments possible when\n`V₂(λ)=⋯=V_D(λ)=0`. Its tail gcd needs only coefficients through\n`H=D(2p−1)<2D²` for a prime `D/2<p≤D`; this makes the minimum positive\nmoment a finite calculation. At `D=4`, the minimum is **1380**, attained by\n`λ=(-15318, 8176, -710, 1518, -253, -88, 11)` on indices `2,…,8`.\nDirect substitution gives `M(λ)=1380` and `V₂(λ)=V₃(λ)=V₄(λ)=0`; the\ncoordinate and finite-gcd theorems prove minimality. This certificate answers\nwhich moments survive a chosen finite cancellation. It does not establish\nnonintegrality of the remaining tail or irrationality of `S`.\nRun `python3 scripts/check_erdos68_channel_moment.py` for the\n[exact integer replay](../scripts/check_erdos68_channel_moment.py) of the\nrecurrence, finite gcd, moment and three vanishing channels. The paper's\nfinite tail-gcd theorem extends the computed gcd to all later indices.\n\n<a id=\"result-243\"></a>\n\n**[#243](https://www.erdosproblems.com/243).** Under the exact cubic rate\n`a_n²/a_(n+1)=1+3/n+o(n⁻³)`, every strictly increasing positive integer\nsequence has an irrational reciprocal sum. Lean checks the zero-indexed\ntheorem through a square-specialisation argument using Mathlib's\nDedekind-zeta simple pole; the short paper transfers it to one-based indexing\nby an ordinary finite-prefix argument. Its printed proof uses classical\nChebotarev instead. This rate does not cover the unrestricted Sylvester-tail\nquestion.\n\nMore generally, the [nonintegral regular-rate theorem](../lean/ErdosProblems/Erdos243/PaperCompleteR21/NonintegralRegularRate.lean)\nchecks irrationality when `λ > 1` is nonintegral and\n`a_n²/a_(n+1)=1+λ/n+o(n⁻λ)`. Its Lean statement derives convergence\nfor a positive, strictly increasing sequence indexed from zero. The long\npaper gives the ordinary finite-prefix transfer to one-based indexing.\nThe integer-extraction lemma rules out nonintegral coefficients; the\nseparate cubic argument rules out coefficient three. No rational examples\nare asserted for the remaining integer coefficients.\n\nThe rounded sequence `a_1 = 8`,\n`a_(n+1) = ⌈n a_n²/(n+3)⌉` has precisely this rate and product-ratio\nincrements of order `n²`. Thus the separate bounded-increment recurrence\ncriterion below gives no conclusion for it, while the cubic-rate theorem\nproves its reciprocal sum irrational. Duverney's Corollary 3.2 supplies an\nearlier signed recurrence criterion under its printed one-sided growth-defect\ncondition. In the all-positive, absolutely summable specialisation, the\nproduct-ratio increments tend to zero. For the rounded example,\n`a_(n+1)/a_n²−1 ∼ −3/n`, so its signed defect series diverges and that\nabsolutely summable specialisation does not apply. The exact `o(n⁻³)`\nremainder matters to the polynomial-exclusion proof. These comparisons locate\nthe contribution without deciding its novelty or priority; the\n[short paper's argument and example](papers/full-text/erdos-243-reciprocal-tail-rigidity.md#sec:secondaryrate)\ngive the proof and source locators.\n\nThe short paper also proves an ordinary original-sequence corollary. Let\n`a_1<a_2<⋯` be positive integers,\n`a_(n+1)/a_n²→1`, `∑ 1/a_n` rational, and `P_n=∏_{j<n} a_j`. If\n`limsup (P_n/a_n)(a_n²/a_(n+1)−1)<+∞`, then\n`a_(n+1)=a_n²−a_n+1` eventually. The needed limsup bound, or the paper's\nalternative weighted record budget, is not proved for the unrestricted problem.\nTake a positive denominator `q` of the full reciprocal sum and clear the nth\ntail by `qP_n`. The resulting integer error is minus `q` times the scaled\ndefect, up to a term tending to zero. This\ncancellation turns the upper bound in the original sequence into the lower\nbound on the integer error used below.\nThe proof passes through a signed bounded-negative theorem. Lean checks\nthat theorem and the canonical tail transfer described below. Let `a,C,D : ℕ → ℕ` and\n`E : ℕ → ℤ`. Assume\n`a(n)>1`, `C(n)>0`, the exact recurrences\n`C(n+1)+D(n)=a(n)C(n)` and `D(n+1)=a(n)D(n)`, and that `E(n)` is the exact\ncentered state with `|E(n)|<C(n)`. If one constant `B` satisfies `E(n)≥−B`\nfor every `n`, and for every `K` one eventually has `K|E(n)|<C(n)`, then\n`E(n)=0` eventually (`boundedNegativePart_eventually_zero`). The checked\ncentered-zero theorem then gives the Sylvester recurrence once the exact\nproduct-cleared orbit has an eventually nonzero next-tail state. Lean carries\nthat chain to the original denominators in\n`boundedNegativePart_sylvesterNext_eventually`, and checks the LCM-weighted\nbounded-defect corollary from the rational reciprocal sum in\n`original_coordinate_lcm_bounded_defect`\n(`ErdosProblems/Erdos243/PaperCompleteR7/LcmDefect.lean:49`); the paper derives\nthe displayed `P_n` form from that corollary by `A_n | P_n`. For the\nstate-system endpoint, no uniform lower bound on the centered error is proved.\nUnbounded negative excursions and the full Erdős endpoint remain open. The\nsigned recovery develops Koizumi's canonical-tail framework under an additional\nbounded-negative premise. Read the\n[short paper](../paper/243/erdos-243-reciprocal-tail-rigidity.pdf),\n[long record](../paper/243/erdos243-reciprocal-tail-reasoning-surface.pdf), or\n[selected formal checks](EXTERNAL_VERIFICATION.md#programme-243).\n\nThe long record also proves an exact maximal-gap theorem for a separate\nstatic sieve. Let `m₀<m₁<⋯` be pairwise coprime integers at least `2`, with\n`ℓ(m_j)=j+O(1)` for `ℓ(x)=log₂ log₂ max(4,x)`. If\n`σ=∏_j(1−1/m_j)>0` and `u_n` lists the positive integers divisible by none\nof the `m_j`, then\n`limsup (u_(n+1)−u_n)/ℓ(u_n)=1/σ`. Lean checks the general theorem\n`maximal_gap_limsup_eq_inv_sigma` with the convergence to `σ` stated\nexplicitly. The paper's finite-product calculation gives `σ=1/2` and thus\ncoefficient `2` for the Fermat moduli `m_j=2^(2^j)+1`. This concerns\n[avoidance of whole moduli](papers/full-text/erdos243-reciprocal-tail-reasoning-surface.md#long243:res:gapconstant),\nnot coprimality to composite moduli or construction of a reciprocal-tail\norbit; the unrestricted #243 question remains open.\n\n<a id=\"result-249\"></a>\n\n**[#249](https://www.erdosproblems.com/249).** The short paper and Lean\nclassify the fixed-base-two series with coefficients `φ(n) mod m`: it is `0`\nfor `m=1`, `3/4` for `m=2`, and irrational for every `m≥3`. More generally,\nfor `k≥1` and any rational-valued function `f` on the residues modulo `2^k`,\nthe series with coefficients `f(φ(n) mod 2^k)` is rational exactly when `f`\nis constant on the even residue classes. If that constant is `c`, the value\nis `3f(1)/4+c/4`. For example, at modulus four, the table\n`f=(5,7,5,-2)` in residue order `0,1,2,3` gives `13/2`, while\n`f=(0,0,1,0)` gives an irrational sum. The public Lean development checks the\n[irrationality theorem](https://github.com/wcook04/plectis-erdos/blob/a14777b3219873bc8343205cca0bb3bb6530e8fa/lean/ErdosProblems/Erdos249/ResidueClassTotientSeries.lean#L566-L580)\nand [dyadic classification](https://github.com/wcook04/plectis-erdos/blob/a14777b3219873bc8343205cca0bb3bb6530e8fa/lean/ErdosProblems/Erdos249/PaperCompleteR7/RationalObservableClassification.lean#L221-L233).\nThe series with unreduced totients remains open.\n\nThe short paper also proves the all-base finite-level totient-kernel rank\n`k^e+1` for every `k≥2` and `e≥1`, with canonical integral coordinates and\na basis of all integral relations; at prime base the rank is exponential in\nthe depth `e`. A rational `5/4` control that agrees with totient on odd\narguments still has tempered carry rank at least `2^e−1` at every depth, so\na generic rationality-driven carry-rank ceiling is false. The strongest\nchecked structural result on the hypothetical rational totient branch is\ncarry anti-compression:\none carry would have uniformly eventually-periodic dyadic sections modulo its\nmultiplier while retaining canonical section rank at least `2^e − 1` at every\nlevel. No finite-rank upper bound is proved, so this is a necessary\nconsequence rather than a contradiction. Finite denominator exclusions and\nconditional actual-LCM, first-harmonic and strict natural-prime tail-gap\nroutes remain useful, but none supplies its missing cofinal producer. Coons\nnon-regularity, Martin affine\nindependence, and Yazdani–Shallit CRT–Dirichlet separation are credited\nantecedents, not new claims of this release. Read the\n[short paper](../paper/249/erdos-249-binary-totient-series.pdf),\n[long record](../paper/249/erdos249-totient-reasoning-surface.pdf), and\n[selected formal checks](EXTERNAL_VERIFICATION.md#programme-249). For a\nconcrete use of the basis, the short paper's\n[base-six example](papers/full-text/erdos-249-binary-totient-series.md#sec:base-six-test)\nreduces all\n43 sections through depth two to 37 coordinates. The\n[exact normal-form tool](../scripts/totient_kernel_normal_form.py) prints\nthose coordinates, the six relation coefficients, and an integer\ncounterexample at one of the first 37 inputs whenever a proposed identity\nis false; its determinant check certifies that finite test.\nFor any integer base, the\n[sparse normal-form command](../scripts/totient_kernel_sparse_normal_form.py)\nreduces only the supplied sections to exact integral coordinates and decides\ntheir identity by the all-base basis theorem. For example,\n`--base 12 --term 8:29859840:1 --term 2:10:-1990656` returns an identity\nwithout constructing the ambient depth-eight matrix. Its optional\n`--witness-budget` searches for a concrete unequal input; exhaustion does not\nchange an exact nonidentity decision. The Python command is an implementation\nof the paper's reduction, not a Lean-verified executable artifact.\n\n<a id=\"result-251\"></a>\n\n**[#251](https://www.erdosproblems.com/251).** A\n[public Lean declaration](../lean/ErdosProblems/Erdos251/AllResidueLogarithmicR9.lean)\nconstructs a synthetic sequence of positive even digits bounded by\n`4 log(n+1)+24`. Its dyadic sum is exactly `6`, every complete scaled tail and\nevery tail shift is integral, the values `2` and `4` recur arbitrarily late in\nevery residue class, and its increasing odd cumulative positions satisfy\n`P_n/(n log n) → 1`. The\n[reasoning paper states the combined theorem explicitly](papers/full-text/erdos251-prime-gap-reasoning-surface.md#long251:res:all-residue-log-countermodel).\nThis rules out a much richer package of plausible soft inputs than\nunboundedness or nonperiodicity alone, but these digits are not the actual\nprime gaps. The declaration is source-backed and does not yet have a curated\nclaim row.\n\nIndependently, a sparse perturbation of the prime gaps can have a rational\ndyadic sum while retaining the prime growth scale, every fixed eventual\ncongruence, and asymptotically the same short-block statistics. The later\npositions are not asserted to be prime. This is complementary to Land's\nconditional result, not a refutation. Lean checks the sparse rational-target\nconstruction and the polylogarithmic schedule with growing-block transfer in\n`SparsePaperR11.lean`. The printed construction additionally gives congruence\ncutoffs uniform in the target; the linked Lean statement quantifies those\ncutoffs after the target. The prime-growth corollary uses cited analytic results. This obstruction shows that those coarse\nstatistics alone do not force irrationality. Lean also checks a precise\nconditional route: for one fixed shift, cofinally many adjacent tail shifts\nstrictly between `-1` and `1`, with unequal actual prime gaps, would prevent\neventual integrality of that shift. Those prime-specific mismatches are not\nproved. The exact prime-gap summation-by-parts equivalence still proves\nneither the prime-gap series nor the original series irrational. Read the\n[short paper](../paper/251/erdos-251-prime-gap-dyadic-series.pdf),\n[long record](../paper/251/erdos251-prime-gap-reasoning-surface.pdf), and\n[selected formal checks](EXTERNAL_VERIFICATION.md#programme-251).\n\n<a id=\"result-257\"></a>\n\n**[#257](https://www.erdosproblems.com/257).** Fix a finite nonempty set of\nprimes `P` and let `h(a)` be the largest divisor of `a` supported on `P`.\nFor an infinite support `A` and an integer base `b ≥ 2`, finiteness of\n`∑_{a∈A} h(a)/(a(b^{h(a)}−1))` implies irrationality of\n`∑_{a∈A} 1/(b^a−1)`. The\n[long paper explains the proof](../paper/257/erdos257-mersenne-reasoning-surface.pdf):\nif this sum were rational with denominator `v`, every positive displacement\nfrom an integer would be at least `1/v`. Choose a finite part of `A` and make\nits exponents divide an observation modulus `Q`, so their displacements vanish.\nFor the remaining exponents, a complete residue orbit gives the weighted main\nterm. Averaging over a finite block of dyadic observation lengths charges the\nincomplete orbits by weighted reciprocal mass, uniformly over finite\nsubfamilies; this uniform bound permits the passage to the infinite support.\nThe [short paper](../paper/257/erdos-257-mersenne-support-subseries.pdf)\nthen chooses the modulus and block length so that its error charged to the\nwhole weighted mass tends to zero. The Lean proof first makes the remaining\ntail weight small and uses a different block-length schedule. Both produce a\npositive displacement below `1/v`; their parameter schedules should be read\nwith their respective error bounds.\nLean proves this as `divisibilityWeightedClaim` in\n[`WeightedReturn.lean`](../lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean).\nThe short paper gives a shorter proof and an\nexplicit support with divergent reciprocal sum satisfying the criterion.\nThe comparison is with Erdős's earlier reciprocal-summable condition, which\nhe stated for every integer base after proving the pairwise-coprime case.\nFor `P = {2}`, write `a = 2^k m` with `m` odd: the weighted summand is\n`1/[m(b^{2^k}−1)]`. In the short paper's example\n`A★ = {2^k m : k ≥ 1, m odd, m ≤ 2^{2^k}}`, dyadic blocks show that the\nodd reciprocals in the `k`th layer sum to between `2^k/4` and `2^k`.\nThe layer therefore contributes at least `1/4` to `∑_{a∈A★} 1/a`, but at\nmost `2^{1−k}` to the base-two weighted sum. The ordinary reciprocal sum\ndiverges while the weighted sum converges. The theorem gives irrationality\nat every integer base for every infinite subset of `A★`. It does not cover\nfull support, all odd exponents, or the full prime support; Tao–Teräväinen\nprove the latter at base two by another method.\nThe [#257 reader exercise](research-commons/PROVE2ME_WEIGHTED_257_PACKET.md#try-changing-a-hypothesis)\ntests how changing a layer cutoff alters this certificate, and why a failed\nweighted test is not a rationality result.\nA [second test](research-commons/PROVE2ME_WEIGHTED_257_PACKET.md#try-a-changing-prime-set)\nshows why a prime set fitted separately to each finite prefix cannot certify\nthe infinite hypothesis.\nThe [weighted-support transfer exercise](../research/experiments/weighted_support_transfer/README.md#a-host-that-needs-every-chosen-prime)\ngives the complementary fixed-host phenomenon: for any prescribed finite\nnonempty prime set, its binary weighted certificate can require every prime\nin that set. This is a calculation about the sufficient criterion, not a\nclaim that each infinite subset needs the same witness or that the example\nhas been formalised separately in Lean.\nThe comparison identifies the added class and proof mechanism, but does not\nsettle independent novelty or priority assessment. See the\n[short paper's theorem, example and sources](papers/full-text/erdos-257-mersenne-support-subseries.md#an-example-beyond-reciprocal-summability).\n[One host realises any finite monotone witness rule](papers/full-text/erdos257-mersenne-reasoning-surface.md#prop%3A257-finite-witness-rule),\nchecked in Lean as `finite_monotone_witness_rule_realised`. Given a finite\nset $E$ of primes and a nonconstant upward-closed rule on its subsets, a\nfinite union of prime-cofactor blocks has divergent reciprocal sum, its\nweighted test converges at every base exactly when the witness primes in $E$\nsatisfy the rule, and every infinite subset has irrational subseries at every\ninteger base. For three primes, any two suffice and none is individually\nmandatory. The [worked variants](../research/experiments/weighted_support_transfer/README.md#a-host-with-several-minimal-witnesses)\ncompare this host with a different required-prime construction. The\nconstruction is AI-assisted and its novelty has not been independently\nassessed.\nAt base two, the weighted condition can also be combined with the positive\ndivisor-cover criterion: the long paper proves that a common finite averaging\nwindow makes both displacements small. Every infinite subset of their union\nthen has irrational subseries at every integer base. Lean proves this as\n`mixedSupportClaim` in the same module. Neither theorem establishes the\nseparately proposed hosts separating the two classes.\nEvery infinite reciprocal-summable support satisfies it, yielding the\ncoprimality-free extension stated by Erdős. Full-support\nirrationality at every integer base is classical (Erdős 1948) and\nLean-checked here, as are pairwise-coprime summable-reciprocal support and\nLebesgue measure one for the base-2 achievement set. Irrationality for\nevery infinite support and the `1/2` and `1/21` branches remain open.\nThe weighted theorem is Lean-checked. The exact five-declaration\n`PalomarCorpus/E257_01` entry, which includes `divisibilityWeightedClaim`,\npassed [Palomar mechanical verification](https://github.com/PalomarRegistry/PalomarSubmission/actions/runs/36009433226)\nfor source commit `b85ed30805188eb4390a686b111294b24363418e`.\nPalomar [registered version 1](https://palomar-registry.org/entry?id=PALOMAR-2026-09-25-000009&version=1)\nas `PALOMAR-2026-09-25-000009` on 25 September 2026. The\n[immutable record](https://data.palomar-registry.org/entries/PALOMAR-2026-09-25-000009-v1.json)\nbinds that version to submission `gid0ym5uu910`, the pinned source, and the\nfive selected declarations. Registration and mechanical verification cover\nthe selected interface, not every claim in the\npaper or the unrestricted Erdős problem. Read the\n[short paper](../paper/257/erdos-257-mersenne-support-subseries.pdf),\n[long record](../paper/257/erdos257-mersenne-reasoning-surface.pdf), and\n[separate Comparator verification status](EXTERNAL_VERIFICATION.md#programme-257).\n\n<a id=\"result-269\"></a>\n\n**[#269](https://www.erdosproblems.com/269).** For three pairwise distinct\nprimes, the threshold-column argument produces nonsingular selected kernel\nminors of every order and excludes every finite rational separated kernel\nrepresentation. The arbitrary-order theorem is present in the compiled\npublic Lean source, but the Wave-A receipt's named axiom audit does not audit\nthat rank declaration; Comparator checks only the displayed rank-two minor\n`-1/15`. For the actual `{2,3,5}` series, Lean checks the\nrationality-to-positive-reduced-carry bridge. Cofinal local-window escape then\nextinguishes such carries; under the explicit cap hypotheses recorded in the\nclaim registry, that escape statement is equivalent to irrationality. This is\nan exact interface, not an easier theorem whose premise has been proved. The\nsource-specific escape and three-prime irrationality remain open.\n\nFor every pair of distinct primes, both the repeated running-LCM reciprocal\nsum and the version retaining one term at each distinct LCM value are\ntranscendental. The\n[ordinary proof](../paper/269/erdos-269-three-prime-running-lcm.tex) expresses\nthem as nonconstant quadratic and affine polynomials in one transcendental\nHecke–Mahler value. Steve Fan posted the repeated-sum two-prime factorisation,\nreduction and transcendence conclusion first, on the problem's forum on\n26 June 2026; the note credits his priority, gives a proof found\nindependently, derives the de-duplicated formula, and uses the cited\nHecke–Mahler transcendence theorem. Lean also checks the two-prime affine and\nquadratic formulas and derives transcendence from the cited Hecke–Mahler\nresult as an explicit named hypothesis; that external theorem itself is not\nformalised here. Read the\n[short paper](../paper/269/erdos-269-three-prime-running-lcm.pdf),\n[long record](../paper/269/erdos269-running-lcm-reasoning-surface.pdf), and\n[selected formal checks](EXTERNAL_VERIFICATION.md#programme-269).\n\n<a id=\"result-1041\"></a>\n\n**[#1041](https://www.erdosproblems.com/1041).** A source-backed\n[ordinary theorem](../research_corpus/Erdos1041/ConcyclicAlternation.md) covers\nmonic degree-`n` polynomials, `n≥3`, whose distinct zeros lie on a circle of\nradius `ρ` and satisfy `2ρ^n≤1`: two adjacent zeros have chord length at most\n`2ρ sin(π/n)<2`, and their straight chord lies in `{|f|<1}`. A repeated zero is\nan immediate short connection. The\n[reasoning record](papers/full-text/erdos1041-lemniscate-reasoning-surface.md#unformalised-remark-a-bounded-radius-concyclic-class)\nexplains the alternation and chord argument. The proof is outside Lean; an\n[exact finite checker](../research_corpus/Erdos1041/scripts/check_erdos1041_concyclic_exact_witness.py)\ntests load-bearing identities and configurations, and the theorem does not yet\nhave a curated claim row. At each fixed degree, the sharp arc constant `2`\nleaves radii sufficiently close to `1` outside this method.\n\nEvery monic trinomial with roots in the open unit disc also has radial\nroot-to-origin segments inside `{|f|<1}`, so any two roots join through the\norigin with length less than `2`; Lean checks the complete displayed\ntrinomial statement. For collinear roots, Lean checks a sharp Chebyshev\nsegment bound and the resulting `<2` connector when the roots lie in the\nopen unit disc. These are solved families, not a general connector theorem.\nFor a squarefree monic polynomial, write `μ = min_{f'(c)=0} |f(c)|`. An\nordinary theorem gives a connector of length less than `2` in the open unit\nlemniscate whenever `μ ≤ 13/25`, with no root-location hypothesis; scaling\ngives a connector of length less than `(5/2) μ^{1/n}` in `{|f| < (25/13)μ}`.\nThe sharp critical-value mean on the closed unit disc is Lean-checked with\n[source-bound audit evidence](../verification/erdos1041-returned-r18-v5-full-audit-evidence.json),\nand the paper also gives an ordinary proof. The mean controls critical values,\nnot connectors.\n\nA further ordinary theorem gives a connector of length less than `2` in every\ndegree `n≥3` when a simple nonzero critical value admits the stated separated\ncritical-value disk at radius `4/3`; it asserts no existence of such a disk.\nOn the generic stratum where simple nonzero critical values have pairwise\ndistinct arguments and moduli, an ordinary slit-sheet theorem identifies the\ninverse-ray root-connection tree but gives no uniform length bound. Lean checks\nNewton-flow decay, ray-separating translations, and perturbative root\nretention. Using one degree-seven polynomial constructed by the\nerdosproblems.com contributor\n[`ani`](https://www.erdosproblems.com/forum/thread/1041#post-8861), Lean proves\nthat every preconnected strict-lemniscate set joining two distinct roots has\none-dimensional Hausdorff measure greater than two. This refutes the exact\nFormal Conjectures path-image-length statement and the separate total-variation\nformulation; correspondence with the 1958 wording remains unreviewed.\nThe short paper also gives a cubic showing why a two-root first merger alone\ndoes not force a capacity gap; its Lean capacity clause assumes the classical\ntransfinite-diameter formula. Read the\n[short paper](../paper/1041/erdos-1041-lemniscate-newton-flow.pdf),\n[long record](../paper/1041/erdos1041-lemniscate-reasoning-surface.pdf), and\n[selected formal checks](EXTERNAL_VERIFICATION.md#programme-1041).\n\n<a id=\"result-1049\"></a>\n\n**[#1049](https://www.erdosproblems.com/1049).** For positive integers\n`0<b<a` in the exact Zudilin contour region, Lean checks irrationality and\nthe stated irrationality-exponent bound for `F(a/b)`, including every positive\nintegral power of `31/4`. The short paper also gives an ordinary proof for\ncoprime `a,b` under `log b/log a < θ*`, where\n`θ* ≈ 0.4056830214`, using Zudilin's 2004 forms and estimates. Zudilin's\nprior work is credited; formalisation does not establish novelty. Lean also\nchecks the normalized Hankel determinant's order and leading coefficient at\nevery rank. Separately, for each real `p>1` and `1≤N≤8`, the Lean-checked\nfinite coefficient pencil has a positive definite first matrix, real roots\nstrictly below `F(p)`, and non-strict interlacing at adjacent ranks. The eight\nunshifted determinant certificates are kernel checked; the shifted eight and\n76 cyclotomic residue witnesses remain finite computations. These results\ngive no all-rank coefficient positivity or irrationality at `3/2`, which lies\noutside the contour region. The universal rational-base question remains open.\nRead the [short paper](../paper/1049/erdos-1049-rational-base-lambert.pdf),\n[long record](../paper/1049/erdos1049-rational-base-lambert-reasoning-surface.pdf),\nand [selected formal checks](EXTERNAL_VERIFICATION.md#programme-1049).\nThe prepared\n[`E1049_01` Comparator selection](https://github.com/wcook04/plectis-erdos-lean/blob/6bc4913c4ca42ac48829ad2d89985f8516361cb5/PalomarCorpus/E1049_01/comparator.json)\ncontains 25 rational-base contour and measure declarations, including\n`rational_base_threshold`. It does not select the later Hankel determinant\nor finite coefficient-pencil results in the same paper. The\n[caller-side preflight](https://github.com/wcook04/plectis-erdos-lean/actions/runs/36023637887)\npassed at that exact source commit; no registry intake is claimed.\n\nThis guide is not a new result ranking.\n[`PALOMAR_RESULT_SHOWCASE.json`](PALOMAR_RESULT_SHOWCASE.json) records a scoped\ncandidate ranking for a possible Palomar submission; it does not rank\nmathematical importance across the whole corpus. Exact public statuses and\nopen propositions live in [`claims.json`](claims.json).\n\nFor a narrative account, open the relevant item in the\n[eight-paper portfolio](../README.md#problem-papers). For exact\ndeclarations and assumptions, use the [source map](SOURCE_MAP.md). The sections\nbelow retain the complete technical inventory, including result-family\ndispositions, semantic census data, finite measurements, and retractions;\noperational procedures live in the [agent workbench](agents/AGENT_WORKBENCH.md).\n\n<a id=\"ten-minute-verdict\"></a>\n\n<details>\n<summary><strong>Open the complete technical results inventory</strong></summary>\n\n## Technical verdict\n\nUsing the degree-seven polynomial constructed by erdosproblems.com contributor\nani, Lean proves that every preconnected strict-lemniscate set joining two\ndistinct roots has one-dimensional Hausdorff measure greater than two. This\nrefutes the exact Formal Conjectures path-image-length statement; the separate\ntotal-variation bound is also checked. The other seven targets remain open.\nIndependent human review of correspondence with the 1958 wording has not been\nrecorded. Comparator checks only selected exact statements, axioms and kernel\nacceptance; it does not assess novelty or historical correspondence. The\ndevelopment is neither an empty collection of restatements nor\na claim to settle the seven unresolved targets or the unadjudicated historical\n#1041 formulation. For a first mathematical pass, the high-signal spine is:\n\n- **#257, finite-prime weighted supports and a completed full-support\n  case.** The checked `divisibilityWeightedClaim` proves irrationality at every\n  integer base under the stated base-two weighted mass condition for a finite\n  nonempty prime set. Some qualifying supports have divergent reciprocal mass;\n  full support does not satisfy this criterion, and arbitrary support remains\n  open. Separately, the checked divisor-block proof combines a bounded\n  Bertrand/CRT first block, middle-window divisor-pair averaging, weighted tail control, and\n  parameter closure (`irrational_erdosSum_full_support`). It proves the\n  canonical full-support series irrational in every integer base `b ≥ 2`, but\n  not universal #257. A distinct adaptive-CRT certificate proves irrationality\n  for every infinite pairwise-coprime support with summable reciprocal mass;\n  those structured-support hypotheses likewise leave arbitrary support open.\n  A deeper, genuinely distinct conditional route proves the base-2 support\n  series under a finite-core orthogonal-petal bouquet plus the uniform\n  `SunflowerForcedSlotTailSelection` predicate\n  (`irrational_erdosSupportSeries_of_orthogonalPetalBouquet`,\n  `Erdos249257/SupportSunflowerDichotomy.lean:540`); the selector and bouquet\n  are not constructed for arbitrary supports. A separate direct result covers\n  nonnegative rational eventually periodic coefficient sequences with a\n  positive periodic-tail value: one common denominator reduces the claim to\n  the periodic divisor-orbit sieve\n  (`irrational_ratWeightSeries_eventuallyPeriodic`,\n  `Erdos249257/CertificateKernel.lean:12811`); mixed-sign coefficients and\n  arbitrary supports remain outside its hypotheses. Separately, the full base-2\n  Mersenne achievement set has Lebesgue measure exactly one, while the\n  supported-set theorem gives exact finite-complement measure or zero for an\n  infinite complement (`volume_mersenneAchievementSet`,\n  `Erdos249257/GreedyAchievementSet.lean:996`;\n  `volume_supportedMersenneAchievementSet_dichotomy`,\n  `ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean:397`). This is\n  consequential global geometry, not a classification of rational points or\n  a universal irrationality theorem.\n- **#1049, irrationality throughout a rational-base region.** The Lean-checked\n  `rational_base_region` theorem constructs the required polynomial forms and\n  analytic estimates for bases in the stated Zudilin contour region. It proves\n  irrationality of `F(a/b)` there and an irrationality-exponent bound; in\n  particular, `F((31/4)^r)` is irrational for every positive integer `r`.\n  This formalises a stated region, not the universal rational-base claim.\n  The base `3/2` lies outside that region and remains unresolved here.\n- **#1041, ani's degree-seven counterexample.** For ani's explicit polynomial,\n  the checked `erdos1041_counterexample_hausdorff` theorem shows that every\n  preconnected strict-lemniscate set joining distinct roots has\n  one-dimensional Hausdorff measure greater than two. It refutes the exact\n  Formal Conjectures path-image-length statement; a separate total-variation\n  bound is checked. Ani supplied the polynomial. The reported family is not\n  formalised, and correspondence with the 1958 wording awaits independent\n  human review.\n- A second-layer Boolean–Möbius normal form is now an executable rationality\n  equivalence: any `p/q` support value (`q>0`) is represented by a positive,\n  square-root-bounded integer carry orbit whose quotient reconstructs the\n  support. This is a change of coordinates for hypothetical rational support,\n  not a contradiction or a universal #257 proof; arbitrary infinite support\n  and the `1/2` membership boundary remain open.\n- **Degree-restricted #1041 positive routes remain open.** The source\n  frontier's parent carrier is `min_c L(c) ≤ 2` over admissible hubs on the\n  ray-separated locus. Its Corollary G, with Theorem L and Corollary S,\n  ordinary proofs in the research corpus with no human review, shows that the\n  carrier in degree `n` gives the degree-`n` case of\n  #1041, so ani's example makes the carrier fail in degree seven. Its\n  degree-five `SPOKE-5` instance would settle degree five. This is source-only\n  research evidence with no Lean counterpart: hub choice, path containment,\n  and the `SPOKE-5` supply remain open, and the earlier minimum-critical and\n  aggregate shortcuts are explicitly refuted\n  (`research_corpus/Erdos1041/FRONTIER.md`).\n- **#249, one open bound.** Lean proves that `S = ∑ φ(n)/2ⁿ` is irrational if,\n  for every shift `h ≥ 1`, there are arbitrarily large `X` at which the real\n  part of the first-harmonic sum over the good indices is at most `603X/1000`,\n  at `s = 26`, `η = 1/1000` and minimal depth\n  (`irrational_totient_series_of_goodBase_gap`,\n  `ArgumentGraph/Results/Erdos249Endpoint.lean`). For all large `X` more than\n  `67X/100` of the indices in `[X, 2X)` are good\n  (`eventually_card_pivotGoodBases_gt`), so the bound asks for the average\n  first-harmonic cosine over them to stay below `9/10`. At these parameters the\n  four pivot budgets imply it, and it asks less than the prime-number-theorem\n  route's `11X/20`. The bound is open. The actual-LCM orbit route is a second\n  sufficient route: an explicit approximation of the orbit by an odd-rank raw block,\n  with an error radius, turns cofinal distance at least `1/32` plus that radius\n  from every integer into the existing signed-margin producer. The exact\n  source calls this `PowerTwoActualLcmOrbitSeparationSupply` and then derives\n  irrationality from it, but this is a sufficient route, not an equivalence to\n  the separate non-integrality criterion. The supply itself is not proved;\n  neither divisibility nor irrationality alone supplies the required\n  anti-concentration. Separately, the dyadic totient-kernel rank `2^e + 1`\n  and its odd-core basis are an unconditional finite-level structural result,\n  and so are the all-base rank `k^e + 1` and its explicit basis for every\n  `k ≥ 2` and `e ≥ 1`: `AllBaseTotientKernel.lean` discharges the\n  affine-independence premise of the reusable all-base wrapper. The actual-LCM, diagonal, and phase/prime\n  producers remain open, so neither structural spine solves the binary\n  totient endpoint.\n- A distinct orbit-level carry anti-compression result is now checked: under\n  hypothetical rationality, one positive multiplier supplies a tempered\n  integral carry whose dyadic sections share an eventual modular period, while\n  the canonical carry-kernel rank is at least `2^e-1` for every `e`. The hard\n  friction is that modulo a divisor of the multiplier the totient forcing\n  vanishes and the carry becomes geometric, so this gives no finite-rank\n  contradiction or irrationality proof (`ExternalVerification/Challenge.lean:137`;\n  `Erdos249257/TotientTailCarryPeriod.lean:224`).\n- **#68, an exact endpoint reduction.** Irrationality is equivalent to\n  cofinally many strict factorial-top divisibility misses, equivalently to\n  cofinally many non-unit carries. The equivalence transfers the hard\n  producer question exactly; it does not supply that producer.\n- **#243 and #251, boundary results that diagnose the missing input.** Fresh\n  pairwise-coprime moduli obstruct bounded rises, while periodic/vanishing\n  negative-error regimes are excluded; for #251, unbounded prime gaps and an\n  exact prime-gap irrationality equivalence still leave the concrete tail\n  bridge open. A public Comparator-backed abstract denominator classifier\n  reduces a dyadic tail's rationality to eventual integrality of one positive\n  shift. Its companion normal form says, for a `RealDyadicTailRecurrence`,\n  irrationality is equivalent to every positive tail shift being nonintegral;\n  neither classification supplies the prime-specific producer.\n- **#269 structural checks and further #1049 diagnostics.** Running-LCM height and\n  rank-two obstructions, plus a distinct weighted-phase observer that separates\n  a finite residue digit from an uncontrolled integral coboundary, isolate the\n  #269 mechanism without claiming its endpoint. For #1049, the growing-rank\n  four-jet/Padé kernel says that when `0 < R` and `4R + 2S ≤ n`, fewer finite\n  endpoint signatures than binary selectors force a collision; row-content\n  divisors remain as a second barrier. A separate explicit rectangular\n  two-function Hermite–Padé model has a sharp no-go under `0 ≤ ρ` and\n  `1 + ρ ≤ σ`, but neither route constructs approximants/remainders or proves\n  irrationality at `3/2`.\n\nThe finite #249 certificate band, Stern–Brocot and gcd-moment identities,\n#257 measure/noncollapse geometry, and the cited-only ladders remain useful\nsupporting routes. They are deliberately secondary to the mechanisms above:\nfinite ranges do not become cofinal theorems, and cited analytic inputs do\nnot become Lean results by appearing in the same paper.\n\n### Signal-audit dispositions\n\nThe [problem-by-problem guide](#problem-by-problem-guide) above gives the\ncurrent result selection, including stronger ordinary-paper results. This\ntable records an earlier scoped review of selected mechanisms and their\nsupporting roles; it does not rank the full current corpus. The detailed\nsections retain the exact statements and evidence boundaries.\n\nThat review compared consequence, endpoint proximity, mechanism depth,\ndistinctness, usefulness, evidence certainty, and the remaining burden.\nIts dispositions are editorial judgments, not novelty, priority, or\nproof-status claims.\n\n| Disposition | Reader-facing treatment | Mathematical reason |\n|---|---|---|\n| **promote** | #257's full-support irrationality theorem, pairwise-coprime extension, orthogonal-petal reduction, and nonnegative rational eventually-periodic coefficient theorem, together with #249's actual-LCM separation reduction, led that scoped review. | The full-support row is an unconditional endpoint case for each `b ≥ 2`; the pairwise CRT and periodic denominator-clearing rows are distinct direct mechanisms; the bouquet and actual-LCM rows add unusually deep endpoint-facing conditional structure, with their unproved supplies kept adjacent rather than hidden. |\n| **retain** | #257's achievement-set measure geometry, Boolean–Möbius exact-row conditional route, and sublogarithmic zero-window constraint, #68's exact carry equivalence, #249's diagonal full-target and foreign-residue reductions, finite kernel rank/basis, orbit-level carry anti-compression, and fixed-precision valuation/unit no-go, #243's bounded-rise, negative-orbit, and signed centered-recovery boundaries, #251's equivalence and countermodel, #269's height/rank obstruction and weighted-phase observer, #1041's admissible-hub/SPOKE-5 source frontier and Lean ray/root inputs, and #1049's recurrence, rectangular-threshold, and four-jet/no-go boundaries remain useful supporting mechanisms and open routes. | The measure rows give consequential global geometry without classifying rational points; the Boolean–Möbius exact-row route gives a concrete local-to-cofinal half-membership mechanism under an unproved positive-skip supply, so it is useful endpoint-facing structure but not a contradiction; the sublog row gives a rationality-conditional coverage bound with a support/numerator-dependent constant, not a contradiction; the #68 and #249 rows are exact endpoint-adjacent or infinite-dimensionality results without their missing producers, while the #249 diagonal row makes integrality a full-target foreign-defect hit, the foreign-residue row adds an exact finite foreign/divisor split and closed geometric tail bound, and the orbit conjunction exposes modular periodicity alongside rank growth and its forcing-erasure friction; the fixed-precision row gives a prefix-locked centred completion for every finite compatible odd-unit word at fixed positive precision, so bounded local signatures cannot exclude every finite endpoint; its unrestricted high quotient is synthetic rather than actual totient arithmetic, and no #249 endpoint follows; the foreign row still needs the unproved `ControlledForeignProjection` bridge and unbounded separated-projection supply; the signed #243 recovery forces a centered defect to zero under exact displayed dynamics and bounded-negative hypotheses but leaves the unbounded mixed-sign branch open; the #269 observer exposes a finite residue/uncontrolled-coboundary split and a conditional finite-span criterion, but no finite-state quotient or endpoint; the source-only #1041 selector remains the direct degree-five/unrestricted socket while its local Lean inputs do not supply the global topology or metric selection; the #1049 rectangular threshold is sharp only for its explicit two-function model and the four-jet collision remains finite, so neither supplies an approximant/remainder bridge; the other rows rule out named regimes or supply exact identities without the missing universal, cofinal, topological, or irrationality bridge. |\n| **consolidate** | The adjacent #249 gcd-moment, Stern–Brocot, finite-denominator, scalar-height, and synthetic LCM-shift results form one supporting arithmetic route; #257's finite-period, state, and defect results form one supporting universal-boundary route. | These results reuse finite-level or finite-support interfaces and share the same endpoint gap; the scalar-height lemma transfers denominator information but is not an irrationality criterion. Grouping these mechanisms preserves their reuse value without making repeated local identities look like separate endpoint advances. |\n| **demote** | Citation-only #249 context and other classical formalised variants follow the local checked mechanisms; the canonical #257 full-support theorem stays promoted. | Provenance and known formalised cases establish context or a restricted theorem, not a new #249 conclusion; the completed full-support mechanism is the explicit exception because it is a direct checked endpoint case. |\n| **replace** | That review replaced a citation-first/problem-order presentation with a comparison of selected mechanisms; the current problem guide above now supplies the result selection. | Consequence and load-bearing mechanism are more informative to a first reader than paper order, while the per-problem sections preserve complete coverage and exact source routes. |\n| **long-tail-only** | Finite searches, measurement logs, routine certificate shards, and unselected conditional predicate stacks remain discoverable through the source maps, but do not lead the reader. | A finite range, routine shard, or unconstructed antecedent cannot establish a cofinal family or universal endpoint, so it is evidence for follow-up rather than first-contact signal. |\n\nThe project also audits its conditional routes for endpoint equivalence. Some\nnamed hypotheses are exact restatements of an open endpoint rather than smaller\nsubproblems; that diagnosis prevents a reformulation from being mistaken for\nprogress.\n\n<!-- BEGIN semantic_public_census -->\nOnly after those theorem-level facts comes the corpus census. The current semantic graph provides three diagnostic views across every indexed Erdős problem:\n\n| View | #68 | #243 | #249 | #251 | #257 | #269 | #1041 | #1049 | both | shared | total |\n|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|\n| mechanically nonrecurring candidates | 0 | 3 | 90 | 0 | 169 | 0 | 0 | 5 | 0 | 20 | 287 |\n| classical/prior-art formalisations | 0 | 1 | 36 | 0 | 23 | 0 | 0 | 1 | 0 | 40 | 101 |\n| bare open-problem equivalences | 0 | 0 | 15 | 0 | 15 | 0 | 0 | 0 | 0 | 2 | 32 |\n\nThe nonrecurring view contains 185 unconditional object theorems, 56 scoped barriers, and 46 reductions or transports after aliases, open antecedents, bare equivalences, finite/generated instances, infrastructure, classical results, and routine corollaries are removed.\n\nThe internal adjudicated frontier shortlist contains 11 nodes; it is distinct from the 7-node public prior-art review queue. 232 nonrecurring candidates remain unassessed for prior art. The live authored open-antecedent surface has 52 clusters, of which 10 are marked endpoint-equivalent. None of these populations is a novelty census.\n\nAgents use the live owners rather than a frozen restatement sample:\n\n- `python3 scripts/query_corpus.py --overview --format card` for reviewed claim status and exact registered Lean interfaces;\n- `python3 scripts/query_corpus.py --route <route_id> --format card` for one problem programme and its open boundary;\n- `python3 scripts/query_semantic.py node <node_id>` and `family-relations <family_id>` for cross-paper statement and relation navigation;\n- `docs/PALOMAR_RESULT_SHOWCASE.json` for the current candidate universe and screening dispositions; and\n- `scripts/residual_evaluator.py` for a kernel-backed comparison of one proposed reduction with its target.\n\nThe claim registry covers selected registered claims, not every sentence in every paper. Semantic relations and Palomar screening are navigation and review surfaces; neither is proof, novelty, significance or peer review.\n<!-- END semantic_public_census -->\n\nThe large #249 denominator exclusion is a direct, kernel-checked instantiation\nof the classical neighbouring-fraction/mediant method. That method supplies the\ndisplayed bound exactly (numerical delta `0`). The local addition is the checked\narithmetic instance, its transfers, and its sharp first failure.\n\n## Strongest checked results by problem\n\nThis is a high-signal orientation by problem, not a one-result roster: it gives\nthe strongest checked statements, their exact status, and where to verify them.\nEach item carries the independent labels used across this repository: what Lean\nchecked, what the statement changes, and its literature relationship.\n`formalised here` renders known mathematics;\n`proved here` makes no novelty claim — that judgement is a human number\ntheorist's, and no query here supplies it. Reduction targets are named with\ntheir own status in the same sentence. The claim registry and generated\nverification projection are supporting navigation; they do not replace the\nsource declaration or enlarge its claim ceiling. The per-problem sections\nbelow are the human reading map, with evidence, attribution, and surviving\nboundaries kept together.\n\n**#68 — is `∑_{n≥2} 1/(n!−1)` irrational?**\n\n- Irrationality is equivalent, with no hypotheses, to one integer\n  divisibility test on exact rational prefixes failing infinitely often —\n  `irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses`,\n  `ErdosProblems/Erdos68/FactorialGapPlateauCore.lean:986`. Checked theorem;\n  exact equivalence, transporting the difficulty without reducing it;\n  registry: none.\n- A separate finite quotient-band obstruction is now explicit. For a finite\n  coefficient family whose indices all satisfy `k d ≤ i_j < (k+1)d`, the\n  factorial moment factors as `M = (d!)^k C_d`; in the first band, channel\n  cancellation `C_d = 0` therefore forces `M = 0`. More generally, if all\n  indices are at least `d`, channel cancellation with `M ≠ 0` forces some\n  index to be at least `2d` (`ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean`;\n  paper discussion in `erdos-68-factorial-denominator-irrationality.pdf`).\n  Registry: `channel_quotient_band_breakpoint` (`res:bandbreakpoint`); its\n  source and evidence boundary are the authority.\n  This is a finite-family breakpoint only: it supplies no cancelling family,\n  simultaneous-channel control, residual estimate, or cofinal non-unit carry.\n- The kernel-internal denominator bound is `q ≥ 67`\n  (`ErdosProblems/Erdos68/FactorialZeroPlateauCertificates.lean:136`); the `300000`\n  exclusion is a checked implication\n  (`ErdosProblems/Erdos68/FactorialGapPlateauCore.lean:812`) whose evaluation is an external\n  computation, not a kernel result.\n- Open: produce infinitely many non-unit carries.\n- A parallel carry form makes the same boundary explicit: irrationality is\n  equivalent to cofinally many non-unit carries\n  (`irrational_factorialGapSeries_iff_cofinal_nonunit_carries`,\n  `ErdosProblems/Erdos68/FactorialGapPlateauCore.lean:945`). The equivalence supplies\n  no cofinal carry producer, so #68 remains open.\n\n**#243 — does rationality force eventual Sylvester recurrence?**\n\n- A divergent natural sequence with upward steps bounded by `B` cannot stay\n  coprime to an infinite family of fresh pairwise-coprime moduli —\n  `no_boundedRise_of_tailAvoidance`,\n  `ErdosProblems/Erdos243/ReciprocalTailRigidity.lean:897`. Checked theorem;\n  reusable obstruction; claims registry: `bounded_rise_fresh_modulus_barrier`\n  (`res:barrier`).\n- Normalised vanishing for the canonical orbit is derived in\n  `PaperCompleteR7/CanonicalState.lean` from the original growth and rationality\n  hypotheses. The paper credits Koizumi (Integers 26 (2026), A28). With this\n  vanishing, a bounded negative part or\n  finite normalised negative mass forces the error to vanish and the\n  Sylvester recurrence to begin\n  (`ErdosProblems/Erdos243/ReciprocalTailRigidity.lean:2177`,\n  `ErdosProblems/Erdos243/ReciprocalTailRigidity.lean:2264`;\n  `ErdosProblems/Erdos243/SparseResetRecovery.lean:155`,\n  `ErdosProblems/Erdos243/SparseResetRecovery.lean:175`). Checked conditional theorems; the\n  note concedes its absorption and descent lemmas as Koizumi prior art.\n- A separate no-go rules out an eventually periodic negative-error orbit when\n  the phase growth is positive\n  (`no_eventuallyPeriodicNegative_orbit`,\n  `ErdosProblems/Erdos243/ReciprocalTailRigidity.lean:796`; claims registry:\n  `eventually_periodic_negative_no_go`). Its exact hypotheses include\n  `2 ≤ a(n)`, `0 < e(N+n) < a(N+n)`, the denominator, tail, and shape\n  recurrences, and positive-period/positive-drift equations. It does not cover\n  unbounded upward excursions of the unreduced numerator, so the open boundary\n  remains.\n- A separately declared Comparator interface\n  `no_cofinallyBoundedNegative_of_normalizedVanishes` restates another\n  conditional no-go under a fixed axiom budget: with `a(n) > 1`, positive `C`\n  and `magnitude`, the coupled recurrences `C(n+1) + D(n) = a(n)·C(n)` and\n  `D(n+1) = a(n)·D(n)`, a bounded rise `C(n+1) ≤ C(n) + B`, normalized\n  vanishing `K·magnitude(n) < C(n)` eventually for every `K`, and a cofinally\n  bounded negative part, the hypotheses are inconsistent\n  (`ErdosProblems/Erdos243/ReciprocalTailRigidity.lean`). Every dynamical,\n  positivity, bounded-rise, and vanishing assumption remains explicit; this\n  does not close unbounded upward excursions of the unreduced numerator\n  (claims registry: `bounded_negative_exclusion`; source coordinate:\n  `ErdosProblems/Erdos243/ReciprocalTailRigidity.lean:1748`). The interface is\n  not a fresh passing receipt for this release commit.\n- A distinct signed recovery family makes the centered mechanism executable:\n  `boundedNegativePart_eventually_zero` assumes `a(n)>1`, `C(n)>0`, the exact\n  natural dynamics `C(n+1)+D(n)=a(n)C(n)` and `D(n+1)=a(n)D(n)`,\n  `E(n)=D(n)−(a(n)−1)C(n)`, strict `|E(n)|<C(n)`, a uniform lower bound\n  `E(n)≥−B`, and division-free normalized vanishing; it forces `E=0`\n  eventually. With centered zero and an eventually nonzero next tail,\n  `sylvesterNext_eventually_of_centered_zero` recovers the Sylvester recurrence.\n  The latter two declarations are mechanism evidence for this one family, not\n  separate rows. Lean carries the chain to the original denominators in\n  `boundedNegativePart_sylvesterNext_eventually`\n  (`ErdosProblems/Erdos243/ReciprocalTailRigidity.lean:2412`), and checks the\n  LCM-weighted bounded-defect corollary from the rational reciprocal sum in\n  `original_coordinate_lcm_bounded_defect`\n  (`ErdosProblems/Erdos243/PaperCompleteR7/LcmDefect.lean:49`). The frontier\n  profile of a counterexample is checked in `canonical_frontier`\n  (`ErdosProblems/Erdos243/PaperCompleteR7/Frontier.lean:151`). This is\n  not reciprocal-tail irrationality: unbounded upward excursions of the\n  unreduced numerator remain open. The selected statement has a separately\n  declared Comparator interface; that is not a fresh passing receipt for this\n  release commit (Comparator wrapper:\n  `ExternalVerification/Challenge.lean:96-109`; source\n  `ErdosProblems/Erdos243/ReciprocalTailRigidity.lean:2265`, with supporting\n  declarations at `:1837` and `:1799`).\n- Open: unbounded upward excursions of the unreduced numerator.\n\n**#249 — is `∑ φ(n)/2ⁿ` irrational? (reviewed core)**\n\n- Cited-only source ceiling: Nesterenko's 1996 Corollary 2 gives algebraic\n  independence of the Ramanujan values at algebraic `q` with `0 < |q| < 1`.\n  At `q = 1/2`, its exact identity\n  `P(1/2) = 1 − 24 ∑_{n≥1} σ(n)/2ⁿ` makes the divisor-sum series\n  `∑_{n≥1} σ(n)/2ⁿ` transcendental by an elementary affine rewrite\n  ([source closure](primary-sources/totient-kernel/nesterenko-1996-source-closure.md)).\n  This is a distinct `σ`-coefficient ladder rung, not the Euler-totient\n  constant `S`; Nesterenko's analytic proof is not formalised here, and no\n  #249 irrationality or novelty claim follows.\n- Cited-only q-zeta antecedent: Postelmans and Van Assche prove that\n  `1`, `ζ_q(1)`, and `ζ_q(2)` are linearly independent for `q = 1/p` with\n  integer `p > 1` (Theorem 1.3). At `q = 1/2`, the release's separately\n  formalised identity `L₂(1) = ζ_{1/2}(2) − ζ_{1/2}(1)` therefore gives an\n  irrational constant-weight squared-Lambert value\n  ([source closure](primary-sources/reciprocal-tail/postelmans-van-assche-2007-source-closure.md)).\n  This does not transfer to the Möbius-weighted totient constant `S`, and the\n  source supplies no #249 or #257 solution.\n- A nearby cited-only sparse criterion of Kaneko, Suzuki, and Tachiya gives\n  irrationality of `∑ f(n)/t^{σ(n)}` and `∑ f(n)/t^{φ(n)}` under explicit\n  nonnegative infinite-support and growth hypotheses ([source closure](primary-sources/reciprocal-tail/kaneko-suzuki-tachiya-2026-source-closure.md)).\n  Here `φ(n)` is a denominator exponent, not the coefficient of `2⁻ⁿ`; this\n  result does not transfer to `S`, arbitrary-support #257, or the release's\n  gcd-moment identities.\n- Unconditional: the dyadic sections `n ↦ φ(2ʲn+r)` of Euler's totient have\n  an explicit rational basis. The canonical family `φ(n)`, `φ(2n)` and\n  `φ(2ʲn+r)` with `1 ≤ j ≤ e` and `r` odd spans a space of dimension exactly\n  `2ᵉ + 1` — `finrank_canonicalTotientKernel_eq`,\n  `Erdos249257/TotientMahlerDefect.lean:989`; basis object at\n  `Erdos249257/TotientMahlerDefect.lean:1392`. For every `e ≥ 1` this is the\n  span of all sections through level `e`; at level zero that span is `φ(n)`\n  alone, of dimension one. A\n  theorem about the coefficient sequence, not about irrationality of `S`;\n  it quantifies the known qualitative fact that `φ` is not 2-regular\n  (Coons 2008, arXiv:0810.3709, via ζ zero-density — no exact rank appears\n  there).\n  Claim `dyadic_totient_certificate_interface` (the basis declarations at\n  `Erdos249257/TotientMahlerDefect.lean:1265`,\n  `Erdos249257/TotientMahlerDefect.lean:1380`,\n  `Erdos249257/TotientMahlerDefect.lean:1392` — registry:\n  dyadic_totient_certificate_interface).\n- The Comparator interfaces make the rank frontier precise: for every\n  `e ≥ 1`, `finrank_totientKernelThroughLevelFamily_eq` gives dimension\n  `2ᵉ + 1` for the actual finite truncation, while\n  `exists_totientDyadicSectionBasis` supplies the explicit odd-core-indexed\n  basis. The full-kernel interface\n  `not_finiteDimensional_span_fullTotientKernel` proves that the rational span\n  is not finite-dimensional. These are coefficient-space facts, not a\n  rationality-to-finite-rank bridge or an irrationality proof.\n- For every base `k ≥ 2` and level `e ≥ 1`, an all-base extension has an\n  explicit canonical spanning family. Its reusable conditional theorem gives\n  dimension `kᵉ + 1` under a linear-independence hypothesis\n  (`finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent`,\n  `Erdos249257/TotientKernelConditional.lean:215`; claims registry:\n  `all_base_totient_kernel_conditional_rank`). The separate\n  `AllBaseTotientKernel.lean` proof discharges that hypothesis and gives the\n  unconditional rank and basis stated in the table below. This remains a\n  coefficient-space result, not an irrationality proof for `S`.\n- A scalar-localisation height lemma transfers, rather than discards, denominator\n  information: under its displayed divisibility hypotheses,\n  `scalarLocalization_complement_dvd` bounds the complementary factor of a\n  reduced denominator by the multiplying integer, with the integer-coordinate\n  identity supplied by `scalarLocalization_integer_eq_mul_num`\n  (`Erdos249257/AdelicHeightObstruction.lean:23,56`; claims registry:\n  `adelic_height_obstruction`). This is supporting arithmetic, not an\n  irrationality criterion.\n- A distinct actual-LCM orbit reduction makes the next analytic obligation\n  explicit. With `H_a = lcm(1, …, 2^a)` and\n  `R_a = totientTail(2H_a) − totientTail(H_a)`, the source defines\n  `PowerTwoActualLcmOrbitSeparationSupply` as a cofinal supply of canonical\n  odd ranks whose `R_a` stays at least `1/32 + actualLcmRawErrorRadius a q`\n  from every integer. The checked raw-block approximation and its explicit\n  error bound then force the existing half-word/signed-margin producer, and\n  `irrational_totientSeries_of_actualLcmOrbitSeparationSupply` derives\n  irrationality from that hypothesis. This is a new sufficient, non-equivalent\n  quantitative bridge, not a proof of its supply: the module explicitly leaves\n  that anti-concentration question open, and the earlier sign/top-edge result\n  does not establish it. See\n  `Erdos249257/TotientActualLcmOrbitSeparation.lean:254-310` and the transported\n  consumer `ExternalVerification/Challenge.lean:159`.\n- The public support transport keeps the boundary from collapsing into a sign\n  argument. `actualLcmTailDiff_shift_pos` gives the positive corridor under\n  `8 ≤ a` and its shortness inequality;\n  `actualLcm_trueEndpointSurvivor_neg` shows that the true translated survivor\n  can still be negative under the corresponding integer-difference hypothesis;\n  and `actualLcm_integral_forces_topEdgeResidue` puts an integral orbit in the\n  exact top-edge residue strip under its additional room inequality\n  (`ExternalVerification/Challenge.lean:198-239`). These three declarations\n  are support for the single `actual_lcm_orbit_separation` family, not three\n  independent routes: positivity does not exclude the survivor, and neither\n  row supplies cofinal separation or top-edge exclusion.\n- A complementary diagonal pincer exposes the exact full-target coordinate:\n  `diagonal_int_iff_foreignDiagonalDefect_hits_fullTarget` says that diagonal\n  integrality is equivalent to the foreign defect hitting the complete reduced\n  target (`Erdos249257/DiagonalPincerDecomposition.lean:215`). Its companion\n  `irrational_totientSeries_of_full_target_avoidance_supply` would turn cofinal\n  misses into irrationality (`:290`), but that full-target avoidance supply is\n  open. This is one exact reduction subordinate to the actual-LCM frontier,\n  not a new endpoint theorem or a second producer.\n- A separate public first-harmonic pivot route is represented in the Comparator\n  roster through its transported decomposition, budget, and conditional\n  endpoint in `ExternalVerification/Challenge.lean`. The exact four-term\n  decomposition of one block is\n  `windowFirstExp_sum_eq_pivot_decomposition`; under the explicit room\n  inequality, `PivotBudgetAt` bounds the centred real part by `14X/25` and the\n  other three norms by `X/100`, `X/100`, and `8X/25`. The exact\n  `first_harmonic_gap_of_pivotBudgetAt` consequence records the resulting real\n  first-harmonic gap of `9X/10` and hence a finite certificate\n  (`Erdos249257/FirstHarmonicPivot.lean:514-552`). The cofinal socket\n  `DTWPivotResidualDecorrelation` would imply irrationality, but its supplier\n  decorrelation budgets are unproved. The supporting\n  `image_pivotSupplierPrimes_eq_pivotFiber` identity and\n  `supplierPrime_not_globally_isolated_counterexample` counterexample expose\n  the exact support boundary: at `X=16, L=20, s=1, m=2, N=18`, a supplier\n  prime is not globally isolated (`:394`, `:569-588`). Comparator exposes this\n  conditional route, but the supplier decorrelation budgets are unproved, so\n  the unbounded certificate supply remains open.\n- A separate source-current `strict_prime_tail_orbit_gap` family is retained\n  below the actual-LCM and first-harmonic frontiers. Its Comparator wrapper\n  `irrational_totient_series_of_naturalPrimeTailOrbitStrictGap` has one coherent\n  two-branch interface. The subordinate `tailOrbitFirstExp_succ` identity records\n  the exact squaring evolution of each tail-orbit phase; it clarifies the phase\n  mechanism but supplies neither the `(11/100)` density nor the cofinal-prime\n  producer. The later `tailOrbitFirstExp_add` and\n  `tailOrbitFirstExp_eq_initial_pow_two` identities reduce every future phase to\n  a power-of-two iterate of its initial phase;\n  `not_naturalPrimeTailOrbitStrictGap_of_dyadic_root` shows that entering a\n  dyadic root blocks the strict-gap producer. These are contrary-mechanism\n  constraints, not density or cofinal-prime supply. In the block branch, for\n  every `h > 0` and cutoff\n  `X₀`, `TotientTailOrbitNonpositiveBlockDensity` assumes a late block with at\n  least `(11/100) * X` nonpositive values of\n  `Re(tailOrbitFirstExp h N)`; unit norm gives an `89/100` block saving, and a\n  single `1/100` truncation budget gives the finite `9/10` certificate gap. In\n  the prime branch, for every `h > 0` and `N₀`,\n  `DTWNaturalPrimeTailOrbitStrictGap` assumes a prime\n  `p ≥ max (N₀ + h + 1) (h + 5)` with\n  `Re(tailOrbitFirstExp h (p − h − 1)) < 9/10`; its positive margin feeds exact\n  natural pivot-fibre escape. The phase-density, cofinal-prime, and\n  uniform-margin producers are all open for the actual totient orbit, and\n  nonintegrality alone gives no fixed distance from the integers. This is\n  producer-side conditional evidence, distinct from the actual-LCM and\n  first-harmonic routes and not an #249 proof, novelty, or priority claim\n  (`ExternalVerification/Challenge.lean`;\n  `ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean:25-233`).\n- Irrationality of `S` has six checked exact characterisations, from\n  tail-difference certificates through the lcm-diagonal form to\n  window-separated pairs — `Erdos249257/LcmConeFlatness.lean:316–426`,\n  `Erdos249257/PivotAntiReconstruction.lean:1775`; claims\n  `certificate_completeness`, `certificate_reduction`.\n- Finite: the classical Farey/mediant window at `K = 240` excludes reduced\n  denominators through `7.96 × 10³⁴` — improvement over the classical bound:\n  zero (claim `denominator_exclusion`) — and diagonal certificates exist at\n  every `t ≤ 82`, with the plateau broken at `t = 83` and a proved depth\n  floor of `125` there\n  (`ErdosProblems/Skip/LadderT67.lean:71264`,\n  `ErdosProblems/Skip/LadderT67.lean:71285`,\n  `ErdosProblems/Skip/LadderT67.lean:71294`; claim `certified_kill_instances`;\n  no certificate at `t = 83` is\n  claimed).\n- Open: the unbounded certificate supply, which is equivalent to #249 itself;\n  a checked splice construction shows no finite inspection can settle it.\n- A separate Comparator-transported fixed-precision no-go,\n  `fixedPrecisionTropicalNoGo`, gives every finite compatible valuation/unit\n  word at fixed `u > 0` a bounded prefix-locked centred completion by varying\n  an unrestricted high quotient (`ExternalVerification/Challenge.lean:150-158`;\n  `Erdos249257/TropicalCurvatureCarry.lean:69-142`). This is synthetic finite-\n  word evidence: it blocks bounded local signatures alone, but does not encode\n  actual totient differences or supply growing precision, global correlation,\n  a #249 endpoint, or irrationality.\n- A checked synthetic no-go shows that, for every `t ≥ 3`, a nonzero dyadic\n  coboundary can survive every finite integer shift polynomial while retaining\n  the period-LCM totient factor ideals, exact cleared-prefix cancellation, and\n  uniform state and letter bounds\n  (`Erdos249257.TotientTailPeriodKiller.lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient`,\n  `Erdos249257/LcmFactorIdealPulseObstruction.lean`). This obstructs that\n  finite shift-algebra route only: the forcing letters are synthetic rather\n  than actual totient differences, nonlinear combinations are not covered,\n  and no unbounded certificate supply follows.\n- Two additional checked #249 families are directly inspectable: an exact\n  [squared-Lambert gcd-moment transfer](../lean/Erdos249257/GcdMomentCalculus.lean)\n  at base `1/2` — `tsum_totient_div_mersenne_sq_eq_gcd_moment_series` identifies\n  the totient-weighted squared-Mersenne Lambert series with the dyadic series\n  of Pillai's gcd-sum function minus `n` — and the Comparator-transported\n  `runHeight_defect_fib_sum_lower`\n  (`Erdos249257/SternBrocotRunGeometry.lean`). For every finite defect list\n  `e`, this theorem bounds the height of its positive Stern–Brocot run word\n  from below by `(F_{r+3} + F_{r+1}) · ∑ eᵢ`, where `r` is the number of runs.\n  These finite identities/inequalities do not establish analytic denominator\n  survival or the unbounded family, so #249 remains open.\n- At every positive Stern–Brocot node `(a,b)`,\n  `cylinderMass_split` exactly decomposes the closed cylinder mass into the\n  node's stop mass and the two child-cylinder masses\n  (`Erdos249257/GcdMomentCalculus.lean`). This telescoping recursion and its\n  geometric interpretation do not establish denominator survival,\n  irrationality, or novelty; the #249 endpoint remains open.\n- The same gcd-moment source gives the exact visible-coprime lattice\n  normalization `∑ 1/(2ⁿ−1) = 1` over positive coprime pairs\n  (`tsum_pos_coprime_inv_mersenne_eq_one`). It supplies the reduced-direction\n  mass behind the fair-coin form of the totient constant, but is a rational\n  normalization identity, not an irrationality result.\n- A newly executable #249 complement-noncancellation reduction is deliberately\n  subordinate to the actual-LCM frontier. The exact finite Möbius-residue\n  split `finiteResidueDiagonal_eq_projectedForeign_add_divisor` separates\n  foreign and divisor channels, while\n  `abs_foreignTailWindow_le_foreignComplementBound` gives each omitted stable\n  window a closed geometric budget. The consumer\n  `scaleFullTarget_miss_of_projected_separation` then rules out a full-target\n  hit when `ControlledForeignProjection` holds and the projected state is\n  farther from every integer than that budget\n  (`ExternalVerification/Challenge.lean:280-295`;\n  `Erdos249257/ActualForeignResidueProjection.lean:276-417`). The analytic\n  bridge identifying the actual complement with this bound, and any unbounded\n  separated-projection supply, remain unproved: this sharpens the reduction,\n  not #249 irrationality.\n\n**#251 — is `∑ pₙ/2ⁿ` irrational?**\n\n- Unconditional: `pₙ ≤ 1250(n+1)⁴` by an elementary argument, so the series\n  is summable and equals `2 +` the prime-gap dyadic series with no external\n  input (`ErdosProblems/Erdos251/PrimeGapDyadicTail.lean:360`,\n  `ErdosProblems/Erdos251/PrimeGapDyadicTail.lean:379`,\n  `ErdosProblems/Erdos251/PrimeGapDyadicTail.lean:427`); and for any sequence obeying the checked integer-digit dyadic\n  tail recurrence, irrationality is exactly equivalent to cofinal\n  non-integral tail shifts\n  (`irrational_initial_iff_cofinalNonintegralTailShifts`,\n  `ErdosProblems/Erdos251/PrimeGapDyadicTail.lean:1572`) — the\n  identification of the concrete prime tail with that recurrence is\n  paper-level. Claims registry: `prime_gap_unboundedness_and_nonperiodicity`\n  (`unconditional progress`), `prime_gap_irrationality_equivalence`\n  (`formalised here`: Lean iff plus elementary polynomial summability, not\n  PNT), and `prime_gap_identity_tao` (`cited only`: Tao, 7 October 2025).\n  The actual prime-tail bridge remains open.\n- The Comparator-facing boundary is also explicit: `exists_primeGap0_gt`\n  proves that consecutive prime gaps are unbounded, while\n  `irrational_tsum_primeDyadicTerm_iff_primeGap` is the named iff under a\n  `Summable` hypothesis; `summable_primeDyadicTerm` discharges that\n  hypothesis by `p_n ≤ 1250(n+1)^4`. Unbounded coefficients and this\n  equivalence alone prove irrationality of neither series.\n- A separate public Comparator transport makes the arithmetic boundary\n  explicit. For a `DyadicTailRecurrence`,\n  `tailShift_integral_iff_den_dvd_mersenne` and\n  `tailShift_integral_iff_two_pow_modEq_one` classify an integral shift by the\n  current reduced denominator and the congruence `2^h ≡ 1`; the source-level\n  `tailShift_integral_iff_scaledTail` gives the equivalent integrality of\n  `(2^h - 1) * T_N`. The transported\n  `not_irrational_initial_iff_exists_eventually_integral_positive_tailShift`\n  says that an integer-digit dyadic tail is rational exactly when one positive\n  shift is eventually integral, while\n  `irrational_initial_iff_all_positive_tailShifts_nonintegral` gives the exact\n  companion normal form: under the same real recurrence, every positive\n  shift at every index is nonintegral\n  (`ExternalVerification/Challenge.lean:245-269`; source\n  `ErdosProblems/Erdos251/PrimeGapDyadicTail.lean:802-807,1279-1295,1525-1556`).\n  The classifications are abstract even though their normal form is now in the\n  Comparator screen; they supply no prime-specific eventual-small or\n  cofinal-mismatch producer.\n- The exact finite carry identity and non-eventual-periodicity theorem close the\n  coefficient-only periodicity implication at the Lean level\n  (`carryPartialSum_natCast_eq`, `carryCoeff_natCast_not_eventually_periodic`,\n  `ErdosProblems/Erdos251/PrimeGapDyadicTail.lean:1154-1173`). The accompanying\n  rational-sum countermodel is an authored paper-level inference from the\n  endpoint decay using the standard exponential-dominance limit; the\n  Comparator interface does not assert that infinite limit. Its coefficients\n  are synthetic, not actual prime gaps, so this is a methodological no-go and\n  not an unconditional #251 refutation.\n- A sparse congruence-preserving perturbation of the actual prime gaps can\n  make the dyadic sum rational while retaining the prime growth scale,\n  eventual fixed-modulus congruences, and short-block statistics at the\n  stated scale. Lean checks the finite core, the sparse rational-target\n  construction, and the growing-block transfer (`SparseAmbientR9.lean`,\n  `SparsePaperR11.lean`, `GrowingBlocksR11.lean`). The support, interval and\n  support-count bound precede the target; the formal congruence cutoffs\n  follow it. The paper proves common congruence cutoffs for its different\n  triple construction. Keep that quantifier distinction when citing\n  Proposition 1.1; the source links do not verify every printed strengthening.\n  The proposition has no Comparator entry, and there is no separately\n  published `SparseRationalisation.md`. The later positions are\n  not asserted to be prime, and the construction does not refute Land.\n- Open: cofinally many adjacent index pairs with small tail shifts and\n  differing gaps; each smallness condition constrains a complete infinite\n  tail.\n\n**#257 — is `∑_{n∈A} 1/(2ⁿ−1)` irrational for every infinite `A`? (reviewed\ncore)**\n\n- Native unconditional: for every finite nonempty support `F` of positive\n  exponents and every base `b ≥ 2`, the reduced denominator of `∑_{n∈F} 1/(bⁿ−1)` is coprime to `b`\n  and has multiplicative order exactly `lcm F` — `finite_period_noncollapse`\n  (Comparator interface `finite_period_noncollapse_rat_den_interface`),\n  `Erdos249257/CertificateKernel.lean:5246`, with no witness-supply\n  hypothesis. Checked theorem; claims registry: `finite_period_noncollapse`\n  (`res:period`). It is a finite-support denominator result, not an\n  irrationality theorem for arbitrary infinite supports.\n- Formalised known results: the full-support series is irrational in every\n  base `b ≥ 2` (Erdős 1948; claim `eb_full_support`), as are pairwise-coprime\n  supports with summable reciprocals (Erdős 1968) and eventually periodic\n  supports.\n- The committed Comparator interface\n  `irrational_erdosSupportSeries_pairwise_coprime` makes the pairwise-coprime\n  row exact: for every base `b ≥ 2`, infinite support `A`, pairwise\n  `Nat.Coprime` elements of `A`, and summable reciprocal indicator\n  `Set.indicator A (fun a : ℕ => 1 / a)`, the support series\n  `erdosSupportSeries b A` is irrational\n  (`Erdos249257/CertificateKernel.lean`; independent statement wrapper in\n  `ExternalVerification/Challenge.lean`). At base `2` this is a dedicated\n  non-dilation family inside #257, not the universal infinite-support claim.\n- Historical source boundary: Chowla's 1947 paper defines the ordinary\n  Lambert series `f(x) = ∑ xⁿ/(1−xⁿ)` and a sine-twisted companion, conjectures\n  irrationality for both at nonzero rational arguments, but expressly says he\n  cannot prove the ordinary `f(x)` case. It proves only the sine-twisted\n  `g(1/t)` case for integer `t ≥ 5`, via long zero blocks in sums of two\n  squares ([source closure](primary-sources/reciprocal-tail/chowla-1947-source-closure.md)).\n  Thus Chowla is historical context for the full-support `x = 1/2` Lambert\n  value, not evidence for arbitrary-support #257 or a new release theorem.\n- A distinct periodic-support antecedent is due to Luca and Tachiya: for a\n  nonzero purely periodic integer coefficient sequence, their Theorem A gives\n  irrationality of the associated divisor-convolution Lambert value at every\n  integer base with absolute value greater than one; their stronger Theorem 1\n  gives linear independence for nonnegative periodic coefficients, including\n  full- and odd-support examples ([source closure](primary-sources/reciprocal-tail/luca-tachiya-2014-source-closure.md)).\n  The hypotheses are periodic and coefficient-specific, so this is not a\n  theorem for arbitrary infinite supports and remains distinct from the\n  release's periodic-divisor certificate and mixed-sign boundary.\n- The base-2 Mersenne achievement set — all subsums of `∑ 1/(2ⁿ−1)` — is\n  compact, perfect, totally disconnected, nowhere dense, and of Lebesgue\n  measure one (strict-tail Cantor structure after Kovač–Tao, Remark 4.1; the\n  measure clause is proved here; positive measure was asserted informally by\n  Kovač on the erdosproblems.com #257 discussion page in August 2025, and no\n  published source computes the exact value) — the Comparator theorem\n  `volume_mersenneAchievementSet`,\n  `Erdos249257/GreedyAchievementSet.lean:996`,\n  `Erdos249257/GreedyAchievementSet.lean:1633–1658`; claim\n  `greedy_achievement_geometry`. Membership of `1/2` is equivalent to\n  infinitely many greedy skips\n  (`Erdos249257/GreedyAchievementSet.lean:2527`) and to the absence of\n  a fatal half-gap (`Erdos249257/HalfCutLocator.lean:654`); the dyadic\n  membership question was posed publicly by Kovač in the same discussion,\n  and a positive answer would refute universal #257.\n\n- Historical topological source boundary: Kakeya's 1914 paper proves\n  perfectness for subsum sets of absolutely convergent real or complex\n  series, gives the real interval enclosure, and states the exact term-versus-\n  remaining-tail criterion for filling that interval. It also states\n  nowhere density when the criterion fails, but explicitly leaves the\n  “infinitely many failures” necessary-and-sufficient refinement unproved\n  ([source closure](primary-sources/reciprocal-tail/kakeya-1914-source-closure.md)).\n  This is classical subsum-set context only: it does not prove an arithmetic\n  statement about Mersenne denominators, rational points, or universal #257.\n\n- The Comparator theorem `seven_le_of_intBand_odd` adds a local two-adic\n  band exclusion: for positive odd integers `p`, `D`, and `q` satisfying the\n  stated `IntBand` inequalities, `7 ≤ p`\n  (`Erdos249257/HalfGreedyTwoThirdsBand.lean`). Thus numerator classes `1`,\n  `3`, and `5` are excluded in that band, while `p = 7` is sharp. The theorem\n  does not show that an actual half-greedy orbit reaches or avoids the band,\n  and does not decide half-membership.\n- A newly transported conditional reduction makes the rational-support boundary\n  concrete: if the support has a positive seed and its base-2 value is\n  `p/(2^c·v)` with `v > 0`, then a positive natural shifted-tail state exists\n  with the exact binary recurrence and modular orbit\n  (`exists_shifted_odd_tail_nat_states_of_support_fraction`,\n  `Erdos249257/RationalSupportCarrySkeleton.lean`). This exposes structure\n  under a rationality hypothesis; it does not exclude rational infinite\n  supports or settle universal #257.\n- For a support restriction `J`, the transported measure dichotomy\n  `volume_supportedMersenneAchievementSet_dichotomy`\n  (`ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean`) gives exact\n  measure `((2 : ℝ≥0∞) ^ F.card)⁻¹` when `J = (↑F)ᶜ` for a finite `F`, or\n  measure zero when `Jᶜ` is infinite. This classifies the restricted\n  achievement-set measure, not its rational points, so it does not settle\n  universal #257.\n- A separate unconditional composite-dilation identity now exposes the\n  lower-rank defect rather than hiding it in a prime-only adapter:\n  `supportCoeff_mul_eq_add_defect`\n  (`Erdos249257/CompositeDilationDefect.lean`). For `a ∈ A`, `0 < a`, and\n  `0 < x`, it states\n  `supportCoeff A (a·x) = supportCoeff A x + (if a ∣ x then 0 else 1) + compositeDilationDefect A a x`.\n  The exact defect is not bounded for arbitrary supports and supplies no tail\n  selector or irrationality theorem; the separate bouquet budget remains\n  conditional on its structural hypotheses.\n- Methodological: a formalised squarefree-support no-go theorem is shown to\n  be an artefact of normalisation — adjoining `1` to the support shifts the\n  value by a rational and flips the divisor-incidence parity, and the\n  finite-change principle is itself checked\n  (`ErdosProblems/Erdos257/SquarefreeSupportIncidence.lean:94`,\n  `ErdosProblems/Erdos257/SquarefreeSupportIncidence.lean:111`,\n  `ErdosProblems/Erdos257/SquarefreeSupportIncidence.lean:140`;\n  `Erdos249257/CertificateKernel.lean:9467`,\n  `Erdos249257/CertificateKernel.lean:9476`). The squarefree value\n  itself is settled at power-of-two bases in the literature (Duverney–Tachiya\n  2019; cited, not formalised). Registry: none.\n- Open: universal #257, and the `1/2` and `1/21` memberships.\n\n**#269 — are reciprocal sums of running lcms irrational?**\n\n- For every pair of distinct primes, both the de-duplicated and the repeated\n  running-lcm reciprocal sums are transcendental by a paper argument using\n  Loxton–van der Poorten 1977, quoted in the modern Bugeaud–Laurent form\n  (Theorem 1.1). Lean checks the exact affine and quadratic formulas and\n  derives transcendence under the explicit `BugeaudLaurentTranscendence`\n  hypothesis; the cited external theorem itself is not formalised\n  ([`TwoPrimeSums.lean`](../lean/ErdosProblems/Erdos269/PaperCompleteR21/TwoPrimeSums.lean#L841)).\n  Independently found and not first: the\n  same reduction and conclusion were posted by Steve Fan on the\n  erdosproblems.com #269 discussion page on 26 June 2026, before the note was\n  finalised, with follow-ups there extending it to arbitrary coprime pairs;\n  the note acknowledges this.\n- Lean-checked: the exact Comparator interface\n  `smoothPrefixLcm_eq_threePrimeHeight` states that, for primes `p`, `q`, and\n  `r` with `p ≠ q`, `p ≠ r`, `q ≠ r`, and `x ≠ 0`, the running lcm of the\n  three-prime smooth prefix is exactly `threePrimeHeight p q r x`\n  (`ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean:123`; formalisation of a\n  classical identity). The same source also gives the four-letter block\n  alphabet `{2, 6, 10, 30}`\n  (`ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean:699`), and a rank-two\n  kernel obstruction\n  (`ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean:479`,\n  `ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean:721`): the Comparator\n  theorem `kernel_235_minor_eq_neg_one_fifteen` identifies the smallest\n  displayed `2,3,5` kernel minor as exactly `-1/15`. This rules out the\n  displayed rank-one route only; it does not imply irrationality. The stronger\n  arbitrary-order minor and finite-nonseparation theorems are proved in\n  [`KernelCarryRank.lean`](../lean/ErdosProblems/Erdos269/KernelCarryRank.lean#L313),\n  whose exact source bytes belong to the successfully compiled public closure\n  in the [Wave-A validation record](../verification/erdos269-wavea-validation.json).\n  That record's named axiom audit covers the Wave-A declarations; it is not a\n  separate rank-declaration audit. Comparator covers the finite minor, not\n  the arbitrary-order theorem. The paper gives the threshold-column proof.\n- Open: any three-prime case. The actual-series reduction is given; the\n  source-specific cofinal escape remains unproved.\n- The finite residue contradiction and conditional cofinal-window carry\n  extinction are indexed as `three_prime_finite_residue_contradiction` and\n  `three_prime_cofinal_window_carry_extinction` in `docs/claims.json`, with\n  paper routes `res:consumer` and `res:windowconsumer`.\n- A separate Comparator-transported family in `ErdosProblems/Erdos269/WeightedPhaseCarry.lean`\n  keeps one coherent observer mechanism visible. For `B : ℤ` with `0 < B` and\n  the displayed carry recurrence, `carry_eq_residueDigit_add_coboundary`\n  gives the exact residue-digit plus integral-quotient-coboundary identity\n  (`WeightedPhaseCarry.lean:109`); `carryResidue_mem_interval` and\n  `residueDigit_mem_interval` give the finite intervals under their explicit\n  positivity and canonical-reduction hypotheses (`:150`, `:157`). The separate\n  `finite_realisedSpan_of_factorisation` theorem gives finite-dimensional\n  realised span only for `DivisionRing` modules with finite-dimensional\n  observer space and an explicit linear factorisation\n  `realise = readout.comp C` (`:334`). The quotient coboundary remains\n  uncontrolled: no finite-state quotient, literal infinite realised span,\n  scalar-evaluation implication, or #269 rationality/irrationality conclusion\n  follows. This is now Comparator-transported evidence, with the same boundary, and is\n  distinct from the\n  three-prime cell, rank-two no-go, and conditional carry-escape families.\n- A separate typed consumer makes the conditional carry-escape boundary concrete:\n  `no_positive_reducedCarry_of_cofinalLocalWindowEscape` derives `False` from\n  `CofinalLocalWindowEscape`, `B > 0`, `Nat.Coprime B 30`, positivity of an\n  integer-valued carry, the exact multiplier recurrence, and its\n  denominator-dependent short bound (`ExternalVerification/Challenge.lean:498-509`;\n  `ErdosProblems/Erdos269/RestrictedFloorSum.lean:629-654`). Its hard step is\n  the cofinal window expansion against that bound. The actual-series reduction\n  is given; the source-specific cofinal escape remains unproved, so this is no\n  #269 endpoint or irrationality proof; it is distinct from the finite residue\n  and weighted-phase observers.\n**#1041 — short connections inside polynomial lemniscates?**\n\n- Lean-checked all-degree theorem: every monic trinomial `z^n + a z^m + b` with\n  roots in the open unit disc has each root-to-origin segment inside `{|f|<1}`,\n  so any two roots join through the origin with length less than `2`. This is\n  coefficient-restricted, not a solution of the unrestricted path problem\n  ([`PaperTrinomialWholeR21.lean`](../lean/ErdosProblems/Erdos1041/PaperTrinomialWholeR21.lean#L23)).\n- Lean checks a sharp Chebyshev bound on a segment between collinear roots of\n  a monic polynomial and the `<2` connector corollary for roots in the open\n  unit disc ([`CollinearDiameterWhole.lean`](../lean/ErdosProblems/Erdos1041/PaperCompleteR21/CollinearDiameterWhole.lean#L967)).\n  A separate cubic shows that a first merger joining two roots need not give\n  a normalised capacity gap; the Lean capacity clause assumes the named\n  classical transfinite-diameter formula\n  ([`ArityNotCapacity.lean`](../lean/ErdosProblems/Erdos1041/PaperCompleteR21/ArityNotCapacity.lean#L677)).\n- Lean-checked sharp critical-value mean: for monic degree-`n ≥ 2` polynomials with\n  zeros in the closed unit disc,\n  `∑_{j=1}^{n-1} |f(c_j)|^{2/(n-1)} ≤ n-1`, with equality for `z^n − λ` when\n  `|λ| = 1`, counting critical points with multiplicity. The complete theorem\n  also gives `∑ |f(c_j)|^{1/n} ≤ (n−1)R` for roots in any closed disc of\n  radius `R ≥ 0`. See\n  [`paper_critical_value_mean`](../lean/ErdosProblems/Erdos1041/PaperCriticalValueMeanR10.lean#L101)\n  and the [successful source-bound audit](../verification/erdos1041-returned-r18-v5-full-audit-evidence.json),\n  whose directly audited adapter is `ReturnV5.critical_value_mean_exact_target`.\n  The paper supplies an ordinary proof too. These moment bounds alone do not\n  construct a connecting curve or bound its length.\n- Ordinary separated-value connector theorem: let `f` be monic of degree\n  `n ≥ 3`, let `c` be a simple critical point with `0 < |v| < 1`, where\n  `v=f(c)`, and suppose some real `w₀ ∈ [0,1]` satisfies\n  `|f(d)/v − w₀| ≥ 4/3` for every other critical point `d`. Then two roots\n  have a connector of length `<2` inside `{|f|<1}`. The long paper gives the\n  two-sheeted uniformization, Bergman length estimate and component-capacity\n  argument; the numerical coefficient bound has a separate Lean kernel.\n  No existence of such a separated critical-value disk is asserted.\n- Ordinary generic topology theorem: in a unit-sublevel component, simple\n  critical points with nonzero critical values of pairwise distinct arguments\n  and moduli give a slit-sheet tree of inverse-ray root connections.\n  This identifies the topology on that stratum; it provides no uniform\n  length bound and does not resolve simultaneous critical levels.\n- Lean checks supporting Newton-flow inputs: quantitative root retention under\n  constant perturbation (`constant_perturbation_roots_in_unitDisk`,\n  `ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean:287`) and arbitrarily\n  small translations separating all pairwise ray arguments. More precisely,\n  `exists_small_translation_separating_arguments` gives every finite injective\n  complex family, for each `ε > 0`, a common shift of norm `< ε` making every\n  value nonzero and every positive-ray argument distinct (`:197`). The\n  exponential-decay connection equation is a hypothesis of the local\n  ray-separation consumer, not a theorem (`:315`).\n- A Cassini example refutes the printed global tree budget of a March 2026\n  manuscript's Proposition 12; that obstruction is not a disproof of\n  Erdős #1041. The source-only frontier records hub selection on the\n  ray-separated locus as a parent carrier; by its Corollary G the carrier\n  fails in degree seven, so only degree-restricted versions stay open\n  (`research_corpus/Erdos1041/FRONTIER.md`).\n- Checked counterexample: using the explicit degree-seven polynomial of\n  erdosproblems.com contributor\n  [`ani`](https://www.erdosproblems.com/forum/thread/1041#post-8861),\n  Lean proves that every preconnected strict-lemniscate set containing two\n  distinct roots has one-dimensional Hausdorff measure greater than two. The\n  path-image case refutes the exact Formal Conjectures statement, and the\n  separate total-variation bound remains checked. The formal proof covers one\n  polynomial, not ani's reported small-parameter family. Independent human\n  review of correspondence with the 1958 curve-length wording remains open.\n\n- Current-source boundary: the committed [`research_corpus/Erdos1041/FRONTIER.md`](../research_corpus/Erdos1041/FRONTIER.md)\n  is the dated route for later source-only research evidence. Read it before\n  the generated `STRONGEST_RESULTS.json`: it records certified refutations of\n  several proposed strategies, surviving carriers, and the gaps open at that\n  snapshot. It predates the checked total-variation counterexample and is not\n  the owner of the current #1041 status. These rows are not reviewed claim\n  entries, peer review, priority, novelty, or significance findings; the\n  frontier itself flags priority/novelty as unchecked for named new rows.\n\n**#1049 — Lambert series at rational bases, beginning with `3/2`?**\n\n- Checked exclusions at `3/2`, each of one named mechanism only: the\n  coordinatewise clearing scheme is impossible for all parameters\n  (`threeHalves_no_coordinatewiseCorridor`,\n  `ErdosProblems/Erdos1049/RationalBaseLambert.lean:155`), scalar content is\n  neutral for the local-to-Archimedean balance, and unit endpoints exclude\n  both `2` and `3` from a common divisor\n  (`ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean:293`). Claims registry:\n  `three_halves_coordinatewise_corridor_no_go` (`res:nocorridor`) and\n  `rational_base_cleared_tail_recurrence` (`res:tailrec`); both are exact\n  formal boundaries. Separately, Lean constructs the polynomial forms and\n  analytic estimates for the stated Zudilin contour region, then proves\n  irrationality and an irrationality-exponent bound there, including\n  `F((31/4)^r)` for every positive integer `r`\n  (`ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean`). This formalises\n  a known region; `3/2` lies outside it.\n- The elementary height inequality used by Bundschuh–Väänänen's external\n  irrationality criterion at `7/2` is checked\n  (`ErdosProblems/Erdos1049/RationalBaseLambert.lean:83`);\n  the analytic theorem itself remains external. Formalisation of known\n  inputs.\n- A separate exact rational-base state recurrence is checked under `r ≠ 0`:\n  `rationalBaseClearedTailQ_succ` gives\n  `Q(N+1) = r·Q(N) − B·coeff(N+1)·s^(N+1)` for the denominator-cleared tail\n  state (`ErdosProblems/Erdos1049/RationalBaseLambert.lean`). The\n  `s^(N+1)` forcing term records the denominator-base tax; this recurrence is\n  not itself a rationality contradiction, so it yields no #1049 endpoint.\n  The companion forcing bound is indexed as `rational_base_forcing_growth`\n  (`res:forcing`): for `s ≥ 2` and positive data it gives a lower bound\n  `2^(N+1)`, while `s = 1` collapses exactly to `B·coeff(N+1)`.\n- A distinct source-level four-jet/Padé obstruction is deferred outside the\n  Comparator screen. When `0 < R` and `4R + 2S ≤ n`, the four-jet target has\n  cardinality `(3^R)^2 * (2^S)^2 < 2^n`, forcing two distinct binary selectors\n  to share a four-jet signature\n  (`ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean:129-167`). The\n  accompanying Padé exterior determinants retain their row-content divisors\n  (`ErdosProblems/Erdos1049/RationalPadeArithmetic.lean:80-220`), so the\n  package exposes finite collision and content barriers without supplying an\n  analytic remainder, nonvanishing, or irrationality theorem. It is distinct\n  source evidence, not a Comparator result or an endpoint closure.\n- A source-faithful rectangular Hermite–Padé threshold family is now exposed\n  through the external consumer. In the explicit two-function exponent model,\n  `0 ≤ ρ` and `1 + ρ ≤ σ` imply `hpClearedGap_nonpos` and\n  `hpClearedGap_eq_zero_iff`; equivalently,\n  `rectangular_hp_threshold_le_classical` and\n  `rectangular_hp_threshold_eq_classical_iff` give\n  `hpThreshold ≤ 1/2 − 1/π²`, with equality only at `(ρ,σ) = (0,1)`\n  (`ErdosProblems/Erdos1049/HermitePadeNoGo.lean:58-129`;\n  `ExternalVerification/Challenge.lean:399-420`). This is a sharp no-go for\n  the displayed model, not a universal Padé result: no approximating\n  polynomials or analytic remainders are constructed, and no #1049\n  irrationality conclusion follows.\n- The Van Assche moving diagonal fails the Amdeberhan–Zeilberger scalar\n  recurrence already at `n=0`: its exact residual is\n  `-p(p-1)^2(p+1)(p^5+2p^4+2p^3+2p^2+2)`, strictly negative for every real\n  `p>1`\n  (`ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean:67,94`). This finite obstruction\n  proves only non-transfer of that recurrence; it supplies no general\n  recurrence, endpoint, lattice, valuation, denominator, or irrationality\n  theorem. Claims registry: `van_assche_qapery_diagonal_no_go`; its source-bound\n  ceiling is recurrence non-transfer at `n=0`, not an endpoint or irrationality\n  result.\n- Open beyond the proved contour region: irrationality at `3/2`; the primitive\n  noncollapsed construction; and analytic remainder control for that route.\n  The short paper proves irrationality in Zudilin's rational-base region, and\n  the pinned Lean source checks `rational_base_region`,\n  `thirtyone_four_powers`, and `rational_base_measure`. Those theorems do not\n  cover `3/2`.\n\nThe semantic corpus marks exactly eight statements `candidate_new_statement` —\nthe only machine novelty marks in this release, against 5,505 of 6,089\nstatement nodes still `not_assessed`. They concentrate in the #249/#257 core\n(the totient Möbius-numerator normal form, a rank-one overshoot barrier, and\nthe `1/2` tail-escape and middle-producer family). Reproduce the list from\n`docs/semantic_corpus.json.gz` (`statement_nodes[].prior_art_state`); candidate\nstatus is a search outcome recorded for triage, not a novelty claim.\n\n## What is proved\n\n### Settled support families for Problem 257\n\nThe universal problem is open, but several infinite supports are settled.\nThe formal source proves irrationality in every integer base `b≥2` for the\nfull support, factorials, powers of two, multiples of a fixed positive integer,\nresidue classes, the odd numbers, eventually periodic supports, and pairwise\ncoprime supports with summable reciprocals. The relevant declarations include:\n\n- `irrational_erdosSum_full_support`;\n- `erdos257_family_factorial_instance` and\n  `erdos257_family_two_pow_instance`;\n- `irrational_erdosSupportSeries_multiples`,\n  `irrational_erdosSupportSeries_residueClass`, and\n  `irrational_erdosSupportSeries_odd`;\n- `irrational_erdosSupportSeries_eventuallyPeriodic`; and\n- `irrational_erdosSupportSeries_pairwise_coprime`.\n\nThese declarations are in `Erdos249257/CertificateKernel.lean`. The later\nreciprocal-summable, finite-prime weighted and mixed-support theorems remove\nsome of these restrictions; their exact hypotheses appear in the\n[programme summary](#problem-by-problem-guide). Cited prime and prime-power\nresults retain their literature attribution.\n\nA distinct second-layer rationality normal form is now a typed public\nconsumer: `exists_normalized_support_fraction_iff_exists_booleanMobiusCarry`\n(`ExternalVerification/Challenge.lean:363`, source\n`Erdos249257/BooleanMobiusCarry.lean:949`). For `p : ℤ` and `q > 0`, it is an\nexistence-level equivalence between a support omitting `0`, containing a\npositive index, and having value `p/q`, and a map `U : ℕ → ℤ` with\n`U 0 = p`, positivity, the square-root bound `U N ≤ q(2√N+4)`, divisible carry\ndifferences, and a Boolean–Möbius quotient; the quotient reconstructs the\nsupport. This is genuinely distinct from the shifted-natural-tail state: it\nis quotient-level and reconstructs literal support. It is only a\ncharacterisation, not a contradiction or a universal irrationality theorem;\nit does not produce cofinally nonintegral shifts for the actual recurrence,\nsettle arbitrary infinite support, or settle the `1/2` membership boundary.\n\nA separate Comparator-backed conditional row route makes the half-membership\nmechanism executable: under `CofinalPositiveHalfGreedySkips`, a positive\nrational-greedy remainder below the next Mersenne weight yields an exact local\nhalf row; cofinally many such skips give\n`(1 / 2 : ℝ) ∈ mersenneAchievementSet`\n(`half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips`,\n`ExternalVerification/Challenge.lean:400`, source\n`Erdos249257/BooleanMobiusSkipRowCofinal.lean:22,73`). The cofinal supply is\nunproved, and finite exact rows do not establish it, so this is a conditional\ncounterexample route rather than an unconditional #257 result; it does not\nsettle the universal problem.\n\nA separate necessary-condition family controls coverage under a hypothetical\nrational support value. `supportCoeffZeroWindow_length_le_eps_logb_add` forces\nevery zero window in the divisor-count coefficient to have length at most\n`ε·log₂(N+1) + B`; the additive constant is built after the support and\nnumerator, so it is not uniform across supports (`SublogDivisorCoverage.lean:392`;\nclaims registry: `sublog_zero_windows`). This is a genuine restriction on a\npossible rational support, not a contradiction or a proof of universal #257.\n\nThe squarefree-support value is already known to be irrational at every\npower-of-two base `2^j` by Duverney–Tachiya (2019, cited here, not\nformalised); their cited result does not decide the other integer bases.\nIts divisor incidence is proved to be `2^ω(n)-1`, hence odd at every `n≥2`.\nThe certificate engines in this development begin by selecting an\neven-incidence block, so they cannot act on that support at any block\nposition. The checked declarations\n`card_squarefreeDivisors`, `squarefreeIncidence_eq`, and\n`odd_squarefreeIncidence` are in\n`ErdosProblems/Erdos257/SquarefreeSupportIncidence.lean`. This is a limitation\nof the named engines, not a statement about the value's irrationality or the\nuniversal #257 question.\n\n### The reduction chain for the one-half test case of Problem 257\n\nThe Mersenne achievement set consists of subsums of\n`1/(2^n-1)`. If `1/2` belongs to it, its support gives an infinite set with a\nrational subseries, which would refute the universal statement in Problem 257.\nThe repository does not prove this membership.\n\nThe chain includes the declarations below. Public CI does not compile the\nfiles cited here under `ErdosProblems/Skip`, `Bit`, `Decl` or `Hlow`; the\ntwo `Erdos249257` files are in the default build.\n\n- `half_mem_mersenneAchievementSet_iff_greedySkippedSupport_infinite` in\n  `Erdos249257/GreedyAchievementSet.lean`: membership of `1/2` is equivalent to\n  infinitely many skips in its greedy expansion.\n- `half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse` in\n  `Erdos249257/HalfCylinderHalfMembershipClassification.lean`: the same\n  membership is equivalent to terminal false bits beyond every bound.\n- `half_mem_iff_unboundedLargestSkipLate` in\n  `ErdosProblems/Skip/D2.lean`: membership is equivalent to late largest skips\n  beyond every bound.\n- `exists_largestSkipLateAt_of_remainder_le_two_pow` in\n  `ErdosProblems/Bit/R2.lean`: a row with remainder at most `2^N` produces a\n  late largest skip in the explicit interval `[N+1,2N+1]`.\n- `half_mem_of_remainderReachesHalfPointCofinally` in\n  `ErdosProblems/Bit/R2.lean`: remainders at most `2^s` cofinally are sufficient\n  for membership.\n\nThe final two bullets are one-way sufficient criteria. They are not\nequivalences and are not known to hold cofinally.\n\nThe proposed universal `hlow` side condition is false:\n`not_hlow_seven_five` in `ErdosProblems/Decl/D4.lean` and\n`not_hlow_of_hhigh` in `ErdosProblems/Hlow/H1.lean` give the counterexample\n`(s,d) = (7,5)`. Its failure is useful rather than obstructive:\n`seamRemainder_lt_two_pow_of_not_hlow` and\n`seven_is_a_hit_from_failure` in `ErdosProblems/Hlow/H2.lean` convert that\nfailure into a hit certificate.\n\n### Exact ranks for dyadic sections of Euler's totient\n\nFor `j ≥ 0` and `0 ≤ r < 2^j`, let\n`φ_(j,r)(n) = φ(2^j n + r)`. The family consisting of `φ_(0,0)`,\n`φ_(1,0)`, and all `φ_(j,r)` with odd `r` is a rational basis for the span of\nall dyadic sections. At level `e`, the span has dimension exactly `2^e + 1`.\n\nLean names and sources:\n\n- `linearIndependent_oddCoreTotientKernelFamily`,\n  `span_range_fullTotientKernel_eq_span_range_oddCore`, and\n  `totientDyadicSectionBasis` in\n  `Erdos249257/TotientMahlerDefect.lean`;\n- `finrank_canonicalTotientKernel_eq` in the same file.\n\nThese are theorems about the coefficient sequence `φ`. They do not prove that\n`∑ φ(n)/2^n` is irrational.\n\n### A finite diagonal certificate band for Problem 249\n\nLet `periodLcm t = lcm(1,…,t)`. The predicate\n`certifiedKill h N L` is a finite modular-arithmetic certificate showing that a\nspecific binary totient-tail difference at shift `h` and position `N` is not an\ninteger.\n\n`exists_diagonalKill_le_82` proves\n\n```text\nfor every t ≤ 82, there is an L such that\ncertifiedKill (periodLcm t) (periodLcm t) L.\n```\n\nLean name and source:\n\n- `exists_diagonalKill_le_82` —\n  `ErdosProblems/Skip/LadderT67.lean`\n\nThe exact least depths at the new boundary scales are:\n\n| Scale `t` | Least depth `L` | Lean name | File |\n|---:|---:|---|---|\n| 67 | 100 | `t67_minimal_depth` | `ErdosProblems/Lift/Recon67.lean` |\n| 71 | 105 | `t71_minimal_depth` | `ErdosProblems/Lift/CertT67.lean` |\n| 73 | 113 | `t73_minimal_depth` | `ErdosProblems/Skip/LadderT67.lean` |\n| 79 | 120 | `t79_minimal_depth` | `ErdosProblems/Skip/LadderT67.lean` |\n| 81 | 120 | `t81_minimal_depth` | `ErdosProblems/Skip/LadderT67.lean` |\n\nThe same file proves\n\n```text\nperiodLcm 79 = 32433859254793982911622772305630400,\n```\n\na 35-digit modulus, as `periodLcm_79_value`.\n\nThe source-backed\n[`checked_diagonal_depth_roster.json`](measurements/checked_diagonal_depth_roster.json)\ncovers all 33 distinct `periodLcm` scales whose plateaux fill the positive band\n`t≤82`. Every recorded working depth lies `0…4` above the arithmetic\nadmissibility floor (median `1`, mean `40/33`). Only the five boundary scales\n`t=67,71,73,79,81` have separate theorems proving their recorded depths least;\nfor the other 28 scales, the excess is only an upper bound on the least-depth\nexcess. Its reproducibility route is maintained with the other operational\nchecks, not part of the mathematical claim.\n\nThis is a bounded result. The reduction for Problem 249 requires certificates\nat arbitrarily large scales. The theorem says nothing at `t = 83`.\n\n### Finite off-diagonal certificates\n\nThe source-bound roster records 125 verified rows at positions not constrained\nto equal `periodLcm t`; \"verified\" is the `lean_verified` flag of the source log\n`lean/ErdosProblems/FreePosition/data.jsonl`. Two pairs of rows repeat the same\n`(h,N,L)` triple, so the roster contains 123 distinct certificates matched to\n123 public Lean theorems across 122 files. Public CI does not compile the\n`ErdosProblems/FreePosition` files. The source-bound reproduction plan in\n`scripts/run_source_bound_reproduction.py` includes a focused recompilation of\n`FreeKill64OneHundredFifteenDI.lean`; no receipt of a run is committed. Its\nlargest recorded position is:\n\n```text\nfreeKill_64OneHundredFifteenDI :\n  certifiedKill 1 135960691671824579163806583999 101\n```\n\nSource:\n`ErdosProblems/FreePosition/FreeKill64OneHundredFifteenDI.lean`.\n\nThe generated\n[`off_diagonal_certificate_roster.json`](measurements/off_diagonal_certificate_roster.json)\nretains all selected source rows and their hashes, identifies both duplicate\ngroups, and binds each distinct triple to its exact theorem, line, and source\nhash. It is a navigation and evidence projection, not a replacement for the\ntheorem statements.\n\nThe source log also contains 54 later verified certificate triples above the\nhistorical position cutoff. The roster exposes that excluded census without\nsilently changing the historical result.\n\nThe relevant sufficient theorem for Problem 249 is\n`irrational_totient_series_of_multiple_window_kill_supply` in\n`Erdos249257/CertificateKernel.lean`. Its hypothesis requires, for every\npositive base shift and every lower bound on the position, a later certificate\nat some positive multiple of that shift. A finite list of 123 instances does\nnot establish that quantified supply.\n\n### Direct Farey comparison\n\nThe denominator exclusion is exactly the standard Farey/mediant bound for the\ntwo committed finite windows. The arithmetic below was recomputed from\n\n```text\nV_K = (∑_{r=1}^K φ(r+1) 2^(K-r)) mod 2^K\n```\n\nand from the explicit neighbouring fractions in\n`Erdos249257/GapFareyBound.lean`.\n\n| `K` | Interval width | Delta |\n|---:|---:|---:|\n| 120 | `123/2^120` | `0` |\n| 240 | `243/2^240` | `0` |\n\nThe exact denominators of each window are:\n\n```text\nK = 120\n  Farey neighbour denominators b, d   41326538792622579\n                                      207345787569745331\n  b+d-1                               248672326362367909\n  formal exclusion bound              248672326362367909\n  first failing denominator           248672326362367910\n\nK = 240\n  Farey neighbour denominators b, d   66013429300517684370587633721421187\n                                      13626217346183690952768141154409867\n  b+d-1                               79639646646701375323355774875831053\n  formal exclusion bound              79639646646701375323355774875831053\n  first failing denominator           79639646646701375323355774875831054\n```\n\nFor each window, the recomputation gives determinant `1` for the two neighbours,\nverifies both interval brackets, verifies the displayed bound, and verifies\nfailure at the next denominator. The corresponding proved declarations are\n`farey_gap`, `gap_check_window_1_120_le_248672326362367909`,\n`gap_check_window_1_120_first_failure`,\n`gap_check_window_1_240_le_79639646646701375323355774875831053`, and\n`gap_check_window_1_240_first_failure`.\n\nThe division of credit is exact: Farey supplies the bound directly;\nformalisation supplies a checked arithmetic instance and proves it is sharp\nfor that window. Neither finite window proves that the exclusion bounds are\nunbounded, so neither proves Erdős #249.\n\nThe magnitude is also a tunable window output, not an invariant measure of\nprogress. `scripts/check_farey_denominator_scaling.py` independently reproduces\nthe two formalised constants and cross-checks the producer by brute force on\nsix small windows. On this run, choosing `K=16000` produced a 2,406-digit bound\nin `0.08` seconds for the scaling row (`0.18` seconds for the complete script).\nEvery fixed `K` still gives only a finite exclusion, and increasing `K` does\nnot supply the unbounded theorem required by #249.\n\n### Scoped no-go theorems\n\nThese theorems rule out particular proof mechanisms. They do not rule out other\narguments. Public CI compiles none of the `Lift`, `Half`, `Rem`, `Three` and\n`Decl` files cited in this list.\n\n- `no_lift_from_lower_totient_data`, `ErdosProblems/Lift/AngleA5.lean` — no\n  universal lift of the displayed form can follow only from the stated lower\n  totient data, pointwise bound, and a lower certificate.\n- `no_window_local_lift`, `ErdosProblems/Lift/InduceLaw.lean` — for prime\n  multipliers `p ≥ 3`, agreement on the lower diagonal window does not force\n  any certificate in the lifted window.\n- `exists_certifiedKill_antitone`, `ErdosProblems/Lift/AngleA4.lean` — a\n  certificate at a larger position gives one at a smaller position; the\n  implication runs opposite to a forward lift.\n- `consecutiveFail_iff`, `ErdosProblems/Half/FreeWalk2.lean` — under its\n  explicit depth inequality, a run of `K` failures at depth `L` is equivalent\n  to failure at depth `L+K` at the base position.\n- `no_absolute_bound_on_consecutive_failures`,\n  `ErdosProblems/Half/FreeWalk1.lean` — the abstract admissible-run model\n  permits arbitrarily long failure runs under its stated budget.\n- `exists_persistent_surplus_model`, `ErdosProblems/Rem/C1.lean` — the listed\n  row recurrence and ceiling inequalities admit a sequence that stays above the\n  target forever.\n- `rowLaw_blind_at_16`, `ErdosProblems/Rem/C1.lean` — at one concrete row,\n  every listed row-level trigger is silent although the next row has a late\n  skip.\n- `IntervalCeiling.four_pow_le`, `ErdosProblems/Three/T1.lean` — every ceiling\n  satisfying the stated branch-blind rank-recursion obligations must grow on\n  the `4^s` scale.\n- `two_pow_mul_y`, `ErdosProblems/Three/T2.lean` — the normalised state used in\n  the doubling route is dyadic at every finite row.\n- `exists_persistent_odd_surplus_model`, `ErdosProblems/Three/T3.lean` — adding\n  parity to the listed row-level constraints still admits a persistent surplus\n  model.\n- `PairCeiling.terminal_ge_of_splits`, `ErdosProblems/Decl/D1.lean` — in the\n  stated two-sided abstraction, a failed branch prediction forces the terminal\n  upper bound above the target scale.\n\n### Other standalone exact obstructions\n\nThese five registry claims are not restatements of either endpoint. Each has a\nseparate limitation, and none settles an Erdős problem.\n\n- **Orbit-level carry anti-compression** — the checked consumer\n  `not_irrational_totientSeries_implies_mod_period_and_unbounded_rank`\n  exposes one positive multiplier `v` and tempered integral carry `u` under\n  hypothetical rationality: every dyadic section is eventually periodic modulo\n  `v`, while its canonical carry-kernel rank is at least `2^e-1` for every `e`\n  (`ExternalVerification/Challenge.lean:137`;\n  `Erdos249257/TotientTailCarryPeriod.lean:224`). This is one orbit-level\n  conjunction, not the coefficient-side rank-only family. Natural friction:\n  modulo a divisor of `v`, totient forcing disappears and `u` is geometric, so\n  quotient periodicity cannot yield a finite-`ℚ`-rank upper bound. Boundary:\n  necessary consequence of rationality only; no contradiction or #249\n  irrationality proof follows. Claims registry:\n  `totient_carry_kernel_anti_compression`.\n- `primitive_coordinate_index_obstruction` — clearing `(φ*μ)(n)/n` through\n  level `N` forces a two-tier primorial divisor; no fixed positive index clears\n  every level.\n  Boundary: finite Euler-coordinate obstruction, not an irrationality\n  criterion.\n- `lcm_factor_ideal_anchor_pulse_no_go` — for every `t≥3`, a synthetic survivor\n  satisfies the factor-ideal and whole-ray-anchor constraints and survives\n  every finite integer shift polynomial.\n  Boundary: its letters need not be actual totient differences; nonlinear\n  fresh-divisor arguments remain possible.\n- `reciprocal_mass_bound` — a rational support value with odd denominator part\n  `v` forces reciprocal mass at least `1/ord_v(2)`, or divergence; a\n  dyadic-rational infinite support forces divergence or mass greater than one.\n  Boundary: does not exclude every rational value.\n- `unbounded_carry_states` — every infinite support with rational series value\n  has an unbounded positive carry state.\n  Boundary: rules out finite-state interpretations, not rationality itself.\n\nTheir registry identifiers lead to the declarations and current source\ncoordinates; operational lookup details live in the\n[agent workbench](agents/AGENT_WORKBENCH.md).\n\n### Current relation and residual routes\n\nCross-paper conjectures and reductions are not assigned a score from a frozen\n#249/#257 sample. Use `python3 scripts/query_semantic.py node <node_id>` for one\ncurrent statement, `python3 scripts/query_semantic.py family-relations\n<family_id>` for its registered family relationships, and\n`scripts/residual_evaluator.py` when a proposed reduction needs a kernel-backed\ncomparison with its target. Public claim status and exact registered Lean\ninterfaces remain owned by `docs/claims.json` and\n`python3 scripts/query_corpus.py --claim <claim_id>`.\n\n## Finite evidence policy\n\nA finite measurement belongs on this page only when the release preserves its\nsample identities, executable owner and rerunnable validation route. Historical\nscans without that chain are omitted; a prose summary is not a substitute for\nthe missing evidence. Current registered finite-evidence claims are reached\nthrough `python3 scripts/query_corpus.py --overview --format card` and their\nemitted source and receipt handles.\n\n## Historical corrections retained in source\n\nThe current Lean module headers and declarations retain the operative\nboundaries. Public CI compiles none of the files listed below. The principal\ncorrections were:\n\n- `Lift/AngleB2.lean` no longer says that the survivor set is empty. Its theorem\n  is conditional on a false `hcof` instance at each surviving class and on an\n  unobserved `seamExcess = 2` hypothesis.\n- `Lift/AngleB3.lean` no longer presents\n  `ratchetOrbit_above_universal_barrier` as a live universal barrier. Its\n  intended hypothesis fails at the initial row, and the proved nonrecurrence\n  result covers only offsets `b ≥ Q+2`, not offsets `1,…,9`. The file now uses\n  an actual Lean bridge to `Recon257`.\n- `Skip/D1.lean` and `Skip/Wire1-3.lean` no longer claim to exclude residue\n  classes. Their existential threshold exceeds every case admitted by their\n  own hypotheses, and the same unobserved seam-excess premise remains.\n- `Rem/C6.lean` no longer says the residual invariant is empirically true on\n  `s = 6,…,3000`. It is false at `(13,7)`.\n- `Rem/C3.lean` now distinguishes the branch it proved from the branch it only\n  measured at the time, and points to `Three/T2.lean`, where the latter branch\n  was subsequently proved.\n- `Bit/R2.lean` labels `badRun_pins_remainder` as a conditional statement with\n  no observed instance: it needs at least 12 consecutive bad rows, while the\n  measured maximum was 3.\n- `Three/T4.lean` and `Bit/Corr1.lean` no longer call `hlow` open. It is false\n  at `(7,5)`, and that failure supplies a hit certificate.\n\nNo theorem was weakened or removed to make these corrections. The claims were\nnarrowed to the theorem types and the recorded counterexamples.\n\n## What is open\n\n### Exact questions for a human expert\n\nThe validated handoffs form a bounded set of exact questions. Each requested\ninput is `OPEN`; the consumer named under each question is a checked theorem.\nThe classification matters: the first question is the endpoint in exact normal\nform, the next two are sufficient analytic producers for #249, and the last two\nwould produce a counterexample to universal #257 rather than prove its positive\nstatement. Their complete packets are indexed from the agent workbench.\n\nNo checked strictly weaker handoff currently implies the universal positive\nstatement of #257 for every infinite support. Named support families,\nhalf-value analysis, reciprocal-mass bounds, and sublogarithmic zero-window\nconstraints are partial; treating any one as a universal proof route would\noverstate what its checked consumer establishes.\n\nThe packet does not make an expert start cold. Every ask also carries a\nlow-confidence current hypothesis, enumerated alternatives, current evidence,\nand the observation or theorem that would distinguish them; both are recorded\nwith each question below.\n\n#### `XQ249-lcm-diagonal-supply` — diagonal supply\n\nExact `OPEN` input: for every `t₀`, find `t ≥ t₀` and `L` with a diagonal\ncertificate at `(lcm(1,…,t), lcm(1,…,t))`.\n\nPayoff and boundary: equivalent to #249. A proof closes #249; until then this\nis a restatement, not intermediate progress.\n\nCurrent working guess: certificates occur beyond every scale, equivalently #249\nis true.\n\nEvidence that would change the guess: a rational representation or an\neventual-failure theorem decides the opposite branch. The exact rectangle has\nfirst-depth excess `0…11`; across all 33 distinct checked diagonal scales\ncovering `t≤82`, working-depth excess is `0…4` (median `1`, mean `40/33`). Only\nfive diagonal depths are proved least, and finite low upper bounds neither\nprove bounded excess nor decide the cofinal quantifier.\n\n`PROVED` consumer:\n\n```text\nirrational_totient_series_iff_lcm_diagonal_certificate_supply\n```\n\n#### `XQ249-pivot-decorrelation` — pivot decorrelation\n\nExact `OPEN` input: cofinally supply the exact four pivot budgets: centred real\ncorrelation `≤14X/25`, supplier mean and bad-supplier norms `≤X/100` each, and\nnon-supplier norm `≤8X/25`, with overlap `h≤L-s` and room\n`16(2X+h+L+2)≤2^L`.\n\nPayoff and boundary: sufficient for #249. The missing input is a\nprime-distribution or residual-decorrelation estimate, not another certificate\nreformulation.\n\nCurrent working guess: for every shift, fixed `s,η` permit cofinally many\nblocks satisfying overlap, room, and all four budgets; the centred real term is\nlikely the hard budget.\n\nEvidence that would change the guess: exact failure requires one shift for\nwhich every `s,η` eventually loses at least one clause. Infinitely many bad\nblocks do not suffice. An exact four-budget producer proves #249.\n\n`PROVED` consumer:\n\n```text\nirrational_totient_series_of_pivotResidualDecorrelation\n```\n\n#### `XQ249-adjacent-phase-separation` — adjacent phases\n\nExact `OPEN` input: for every `h>0` and `X₀`, find `X≥max(X₀,1)`, a depth `L`,\nand adjacent `N,N+1∈[X,2X)` such that `16(2X+h+L+2)≤2^L` and their\nfirst-harmonic phases have squared chordal separation at least `19/25`.\n\nPayoff and boundary: sufficient for #249. Finite scans or average separation do\nnot meet the cofinal quantifiers.\n\nCurrent working guess: qualifying adjacent separation occurs cofinally for\nevery shift.\n\nEvidence that would change the guess: one shift and a cutoff beyond which no\nadmissible block, depth and adjacent pair reaches `19/25` kill this route;\ninfinitely many bad blocks alone do not. A cofinal adjacent-discrepancy theorem\nat that threshold proves #249.\n\n`PROVED` consumer:\n\n```text\nirrational_totient_series_of_adjacentPhaseSeparation\n```\n\n#### `XQ257-second-channel-separation` — second channel\n\nExact `OPEN` input: prove `HalfSecondChannelSeparatedRat n` for every `n≥7`,\ni.e. that the exact rational second-channel phase stays at least\n`1/6+(37/56)2^{-n}` from `1/3`; Lean checks `1,…,6`, and an independent exact\nprobe finds no failure through `n=1000`.\n\nPayoff and boundary: proves `1/2` attainable. The finite-support exclusion then\nforces the witness to be infinite, giving a rational counterexample to\nuniversal #257. The finite probe does not move the quantifier; failure of this\nroute would not prove universal #257.\n\nCurrent working guess: the separation continues for all `n≥7`; confidence\nremains low despite an empty exact failure set on `1≤n≤1000`.\n\nEvidence that would change the guess: the least measured margin on the open\nrange is `1033253069/8193024` at `n=7`, but `n=1001` onward is unmeasured. One\nfailed level closes only this socket; a global reachability invariant or\nuniform sign estimate proves membership and, with finite-support exclusion, a\ncounterexample.\n\n`PROVED` consumers:\n\n```text\nhalf_mem_mersenneAchievementSet_of_secondChannelSeparationRat_from_seven\npositiveMersenneSupportValue_coe_finset_ne_half\npositiveMersenneSupportValue_eq_erdosSupportSeries\n```\n\nThe source-bound second-channel measurement re-evaluates the rational greedy\nrecurrence and the complete failure set\nfor `1≤n≤1000` from the definitions in\n`Erdos249257/GreedyAchievementSet.lean`. The generated artifact is\n[`docs/measurements/second_channel_separation_probe.json`](measurements/second_channel_separation_probe.json).\nIts status is `MEASURED`, not `PROVED`; in particular, the empty finite failure\nset does not establish the all-`n` hypothesis consumed by the Lean theorem.\n\n#### `XQ257-middle-producer-tail-escape` — middle producer\n\nExact `OPEN` input: at every actual middle transition from rank `13`, write `C`\nfor the checked producer carry and `Θ` for its nonnegative coefficient tail.\nProve `C=-3` or `(1≤C and Θ<C)`.\n\nPayoff and boundary: also proves `1/2` attainable; at the putative last-false\nrow the checked `-3` theorem removes the first disjunct, and the\nfinite-support exclusion then forces an infinite witness. The socket must\nexclude every actual `C≤0`, `C≠-3`, and control `Θ<C` for every positive cell.\n\nCurrent working guess: every actual middle row lies in the `C=-3` branch or has\n`1≤C` and `Θ<C`.\n\nEvidence that would change the guess: one actual nonpositive cell other than\n`-3`, or one positive cell with `Θ≥C`, refutes this socket. A proof of the\ndisplayed disjunction at every actual row proves membership; the final `-2`\nsieve alone does not.\n\n`PROVED` consumers:\n\n```text\nhalf_mem_mersenneAchievementSet_of_middleProducerTailEscapeExceptNegThree\npositiveMersenneSupportValue_coe_finset_ne_half\npositiveMersenneSupportValue_eq_erdosSupportSeries\n```\n\n### Cold-reader review boundary\n\nThe repository cannot certify its own human legibility. It therefore exposes a\nsecond `OPEN` question: can an independent reader, starting from a fresh clone,\nrecover the current eight-problem scope, authority and #1041 credit boundary in\nat most ten minutes? The request is exact, including the response schema, its\nconsumer, payoff, and boundary.\n\nThe reader replays the current overview, #1041 and semantic-relation routes;\nreports that ani's degree-seven counterexample refutes the exact Formal\nConjectures path-image-length statement and the total-variation formulation,\nwhile the historical curve-length correspondence remains\nunreviewed; identifies the claim registry as selected rather than exhaustive;\nand summarises checked results from at least three distinct problems. This\nreplaces the retired protocol built around a frozen #249/#257 binder sample.\n\nA structurally accepted response from an identified independent reader returns\nan explicit rubric for its result summaries, evidence paths and verdict. A\nsecond identified evaluator binds a final review to the response digest. The\nvalidator cannot authenticate either person or the elapsed time, and does not\npretend to judge free prose by itself. Neither receipt validates the mathematics\nor establishes priority or novelty. The same query without a domain filter\nreturns this systems question beside the five theorem-connected mathematical\nquestions; respondent-facing output omits the evaluator's scalar answer key.\n\n### Erdős Problem 249\n\nThe irrationality of\n\n```text\n∑_{n≥1} φ(n)/2^n\n```\n\nis not proved in this release; Lean proves it from one open bound, a real part\nat most `603X/1000` for the first-harmonic sum over the good indices at\narbitrarily large `X` (`irrational_totient_series_of_goodBase_gap`). The short\npaper proves all-base finite-level rank `k^e+1` for `k≥2, e≥1`, with canonical integral coordinates and all\nintegral relations; at prime base the rank is exponential in the level `e`.\nA rational `5/4` control agreeing with totient on odd arguments has tempered\ncarry rank at least `2^e−1` at every depth, so a generic rationality-driven\ncarry-rank ceiling is false, and that control is distinct from the bounded\n`3/2` parity-and-aperiodicity analysis. The finite band through `t=82` and\nthe 123 historical off-diagonal certificates do not supply a cofinal\ndiagonal or an all-ray seed. Coons non-regularity, Martin affine\nindependence, and Yazdani–Shallit CRT–Dirichlet separation are credited\nantecedents, not new claims of this release.\n\n### Erdős Problem 257\n\nThe statement that\n\n```text\n∑_{n∈A} 1/(2^n-1)\n```\n\nis irrational for every infinite `A` is open. In particular, membership of\n`1/2` in the Mersenne achievement set is not proved.\n\nAt level `d`, the rational compared with the remaining tail changes with `d`.\nA naive common-denominator construction gives an upper height bound of the\nform `2^O(d^2)`, but the repository proves neither a matching lower bound nor a\n`Theta(d^2)` law for the reduced denominator along the actual greedy support.\nThe displayed separation boundary has an explicit correction proportional to\n`2^{-d}`. No registered theorem controls the required sign uniformly at this\nmoving boundary, and there is no fixed rational approximation target to which\na fixed-number theorem applies directly.\n\nIrrationality alone is insufficient because the open branch asks for the sign\nof a nonzero tail difference, not merely for non-equality.\n\n## Verification and continuation\n\nThe exact build, query, replay, contribution, and source-bound reproduction\nroutes are maintained in the [agent workbench](agents/AGENT_WORKBENCH.md), where they\ncan stay current without interrupting the mathematical account here. The\nhistorical isolated-tree timings were a dated manual record, not a\nmachine-revalidated receipt for the present source tree. Lean checks certify\nthe displayed formal statements; they do not certify novelty, priority,\nsignificance, or the still-open Erdős endpoints.\n\n</details>\n\n## Problem statements, verification and status\n\n<!-- BEGIN problem_programme_card -->\n_Questions generated from `docs/problem_index_source.json`._\n\n| Problem | Question |\n|---|---|\n| [#68](../paper/68/erdos-68-factorial-denominator-irrationality.pdf) | Is the series sum_{n >= 2} 1/(n! - 1) irrational? |\n| [#243](../paper/243/erdos-243-reciprocal-tail-rigidity.pdf) | Under a rapid-growth hypothesis on an integer sequence, does rationality of its reciprocal sum force the sequence to satisfy the Sylvester recurrence eventually? |\n| [#249](../paper/249/erdos-249-binary-totient-series.pdf) | Is the binary Lambert series sum phi(n)/2^n irrational? |\n| [#251](../paper/251/erdos-251-prime-gap-dyadic-series.pdf) | Is the dyadic series of consecutive primes irrational? Equivalently, is the corresponding consecutive-prime-gap dyadic series irrational? |\n| [#257](../paper/257/erdos-257-mersenne-support-subseries.pdf) | Is the sum of 1/(2^n-1) over every infinite set of positive exponents irrational? |\n| [#269](../paper/269/erdos-269-three-prime-running-lcm.pdf) | For a finite set of at least two primes, is the sum of reciprocals of the running least common multiples of the smooth numbers irrational? This library treats the three-prime case. |\n| [#1041](../paper/1041/erdos-1041-lemniscate-newton-flow.pdf) | Must two roots of a monic polynomial in the open unit disc be joined by a sub-two-length curve inside its unit lemniscate? Ani’s degree-seven example refutes the exact Formal Conjectures statement; correspondence with the 1958 wording awaits human review. |\n| [#1049](../paper/1049/erdos-1049-rational-base-lambert.pdf) | For which rational bases is the corresponding series irrational? The first resistant explicit base is three halves. |\n<!-- END problem_programme_card -->\n\nThe questions above are readable here: no query is required, and this page\ndoes not require a private repository. The [problem papers](../paper/README.md#problem-papers) explain the eight questions.\n[Scope](SCOPE.md) records the public claim boundaries and status labels;\n[external verification](EXTERNAL_VERIFICATION.md) lists the selected Comparator\ninterfaces, axioms and replay limits. The [claim registry](claims.json) owns\nsource identity and current status. These are generated from their owners.\n\nThe claim registry pins the immutable formal-source checkpoint; this is not a\nnew tagged release. Its release tag is the latest tagged release and citation\nanchor, kept separate from the source currently under validation.\n\n### What the formal source establishes\n\nFarey's method supplies the number directly; it is not an improvement on the\nclassical bound.\n\nThe labels describe evidence. **Formalised here** means known mathematics\nrendered and kernel-checked in Lean; it makes no priority claim. **Proved\nhere** means the argument is this project's.\n**Unconditional progress** marks a theorem that does not settle the open\nproblem and asserts neither novelty nor prior knowledge. **Verified finite instance** means\nLean checked only the listed inputs; **conditional reduction** means the\nconclusion depends on a named open condition.\n\n| Status | Result |\n|---|---|\n| **formalised here** | For every integer `b ≥ 2`, the full-support series `∑ 1/(bⁿ - 1)` is irrational; Erdős (1948) is a checked rendering rather than a new result. Several named infinite-support families are also formalised; this does not cover every infinite support. |\n| **formalised here** | The base-2 Mersenne achievement set is compact, perfect, totally disconnected, nowhere dense, and has Lebesgue measure one. Membership is equivalent to greedy survival at every level. |\n| **proved here** | For the #257 test value `1/2`, achievement-set membership is equivalent to infinitely many greedy skips and would produce an infinite support of rational sum, refuting universal #257. Under the last-skip schema's hypotheses (a rank floor, a carry condition, and a strict middle-cell inequality), the upper branch and the middle coordinate `-3` are impossible. |\n| **conditional reduction** | Within that same last-skip contradiction schema, the two still-unexcluded middle coordinates, `-2` and `-1`, would also be ruled out if one current contribution were larger than the sum of all later possible contributions. That inequality is not proved. |\n| **formalised here** | The dyadic sections of Euler's totient have an explicit rational basis; for `e ≥ 1` the level-`e` span has dimension exactly `2ᵉ + 1`. The Lean proof is an independent constructive route to an independence consequence of Martin's stronger theorem; this is a theorem about the coefficient sequence, not the irrationality of `S`. |\n| **unconditional progress** | For every integer `k ≥ 2`, the sections through level `e ≥ 1` have rank `kᵉ + 1`, with an explicit basis and complete scalar relation normal form. [AllBaseTotientKernel.lean](https://github.com/wcook04/plectis-erdos/blob/25ef6245d15a47548c6926369ae8f1a0f0a14a80/Erdos249257/AllBaseTotientKernel.lean#L1118) proves canonical-family independence by CRT and Dirichlet, and [the complete rank theorem](https://github.com/wcook04/plectis-erdos/blob/25ef6245d15a47548c6926369ae8f1a0f0a14a80/Erdos249257/AllBaseTotientKernel.lean#L1239) discharges the reusable conditional wrapper's independence premise. Martin's broader affine-independence theorem remains an external antecedent, not an axiom of that Lean proof. |\n| **formalised here** | Applying the classical Farey/mediant bound directly to the committed `K=240` interval excludes rational denominators through `79 639 646 646 701 375 323 355 774 875 831 053` (about `7.96 × 10³⁴`). This is exactly the Farey bound, not an improvement on it; Lean also checks that the next denominator fails this finite window. |\n| **proved here** | `S` is irrational exactly when every positive binary tail difference is non-integral, equivalently when every fixed pair has a finite certificate. Finishing the argument would require certificates at arbitrarily large stages; that step is not proved. |\n| **verified finite instance** | Lean proves a diagonal certificate at every `t ≤ 82`. Historical free-position audit: 125 verified log rows represent 123 distinct off-diagonal `(h,N,L)` certificates in 122 Lean files. This finite evidence does not prove successful cases beyond every fixed cutoff. |\n\n#### Other exact mathematics in the corpus\n\n| Package | Exact checked content | Boundary |\n|---|---|---|\n| Fair-coin coprimality | `S = 1/2 + P(gcd(X,Y)=1)` for independent `P(X=n)=2⁻ⁿ`. | Irrationality remains open. |\n| Squared-Lambert gcd moments | Two exact divisor-sum identities for squared Lambert denominators. | No transfer to the open Möbius row. |\n| Stern–Brocot cylinder law | Exact stop/child splitting; depth error at most `(2/3)^d`. | Probability law, not irrationality. |\n| Fibonacci/continuant run stability | Height at least `F_{r+3}` with exact defect expansion. | No analytic denominator-clearing theorem. |\n| Tempered binary tail rigidity | Exact rationality/carry-orbit classification for `c(n) ≤ n`. | Needs problem-specific orbit control. |\n| Exact Möbius-shadow denominator | Exact reduced denominator and an explicit divisor lower bound. | No unbounded avoidance supply. |\n| Scalar-localisation height obstruction | If `H ∣ x.den` and `(c·x).den ∣ H`, then `x.den/H ∣ |c|`. | Local obstruction only. |\n\nTyped routes expose sources: `probabilistic_gcd_geometry` for the first four\nrows, `boolean_mobius_constraints` for tail rigidity, and\n`arithmetic_obstruction_interfaces` for the last two. Orientation also lists\neventually-periodic nonnegative weighted irrationality, a signed\nirrational-or-base-terminating dichotomy, five binary-carry\ncriteria/consequences, and two scoped #249 no-go countermodels.\nFive further obstructions are stated with their limits in\n[RESULTS](#other-standalone-exact-obstructions).\n\nAn exact final-skip band formula does not show that the actual orbit avoids\nan unsafe band.\n\n[Orientation](ORIENTATION.md) routes claims; the retained\n[mathematics paper](../paper/archive/erdos249-257-main-paper.pdf) preserves the joint\n#249/#257 exposition.\n\n\n## What remains open\n\nEach problem's section in the [problem-by-problem guide](#problem-by-problem-guide)\nstates its own open questions. For the two problems with working records:\n\n- Prove that `S = ∑ φ(n)/2ⁿ` is irrational without placing a bound on a possible\n  rational denominator.\n- Prove the single #249 bound: for every shift, a real part at most `603X/1000`\n  for the first-harmonic sum over the good indices at arbitrarily large `X`,\n  which Lean turns into irrationality.\n- Produce the unbounded certificate supply required by the exact #249\n  reduction.\n- Prove irrationality of `∑_{n∈A} 1/(2ⁿ - 1)` for every infinite\n  `A ⊆ ℕ`, rather than only the named support families formalised here.\n\nThe #249 and #257 working records each close with a section titled \"The wall\": every\nattempted argument class is stopped by a stated bound, recorded with what it\ndoes not rule out.\n\n[`docs/SCOPE.md`](SCOPE.md) is the short boundary statement. The exact expert\nhandoffs state what input is requested, current guess, alternatives,\ndiscriminating evidence, checked consumer, and endpoint-or-counterexample\nboundary. See\n[RESULTS](#exact-questions-for-a-human-expert) and\n`python3 scripts/query_expert_handoffs.py`. A refuted route is withdrawn in\nthe next edition and the refutation credited.\n","body_format":"markdown"},{"record_type":"formal_claim_control_plane_record","record_id":"lean_prior_art","source_repository":"plectis-erdos","source_path":"docs/PRIOR_ART.md","media_type":"text/markdown","licence":"Apache-2.0","copyright":"2026 Will Cook","why_an_offline_reader_needs_this":"The repository's own record of what is antecedent, what is independent, and what remains unassessed for novelty. Required before describing any result here as new.","embedded":true,"availability":"embedded","sha256":"sha256:da3391cb4b3ebd0308569046774412c88593a2e444ddf32a67aac3af8b64fd33","source_byte_count":51403,"body":"<!--\nSPDX-FileCopyrightText: 2026 Will Cook\nSPDX-License-Identifier: Apache-2.0\n-->\n\n# Prior art and attribution map\n\nThis is a reading map for the mathematical sources cited in the exposition. It\nrecords relationship, not proof authority: only the pinned Lean source\nestablishes what this release proves. A citation never widens a release claim,\nand a source listed as *context* is not an imported argument or a claim of\nmathematical priority. Formalisation/software dependencies are recorded in\n[CITATION.cff](../CITATION.cff); the Erdős Problems catalogue supplies\nnumbering and status context rather than mathematical priority.\n\nEach current [problem paper and research record](../paper/README.md) carries\nits own bibliography and credits sources at the point of use. The\n[joint #249/#257 exposition](../paper/archive/erdos249-257-main-paper.tex) is\narchival. [CITATION.cff](../CITATION.cff) supplies the current repository citation\nand selected mathematical and software references; it is not a duplicate of\nall the paper bibliographies. Its linked source records explain the local use\nand limits of the selected references. Record the commit used; for a historical\nrelease, use the citation file from that tag. This map explains why the\nprincipal sources are credited.\n\n## Browse all source credits\n\nThe [source attribution index](research-commons/SOURCE_ATTRIBUTIONS.md) connects\nnamed literature, public Erdős Problems contributions, and software sources to\nexact paper and Lean locations. It includes anonymous credit for implemented\nprivate advice until the correspondent confirms public naming. The\n[machine-readable view](research-commons/source-attribution-index.json) records\ncitation coverage, source locators, content digests, and unresolved matches;\nthe [authored registry](research-commons/source-attributions.json) owns the\ncredit decisions. These records complement the principal-source comparisons\nbelow and the separate accepted-contribution receipts in Research Commons.\n\n## Bounded comparison route\n\nStart with\n`python3 scripts/query_corpus.py --route trace_prior_art`, then select the\nexact local claim with `python3 scripts/query_corpus.py --claim <claim_id>`.\nRead its registry status, statement, assumptions, remaining-open links, and\nLean declarations before interpreting any row below. A status such as\n`proved here` or `formalised here` is an evidence statement, not a novelty\nstatement.\n\nFor a theorem-level comparison, record the source statement, assumptions,\nconclusion, specialisation map, proof mechanism, and residual difference\nagainst that selected local claim. Failure to identify a matching source is\nnot evidence of novelty. Lean source checked by the pinned Lean kernel remains\nproof authority; this bibliography and its search history are authored\nattribution evidence only.\n\n## Erdős #249: attribution update, 16 September 2026\n\nThe finite-level result is an explicit basis, rank \\(k^e+1\\) for \\(k\\ge2,e\\ge1\\), and an integral basis of all relations among the totient sections. Coons’s non-regularity theorem and Martin’s affine-independence theorem are antecedents, not new claims of this release. For composite bases the retained condition is \\(k\\nmid r\\).\n\nYazdani’s Theorem 2 and proof (2001, pp. 652–653) use CRT–Dirichlet separation and explicitly credit Shallit; this is the method antecedent of the evaluation-matrix proof. Yazdani’s Corollary 4 (p. 654) already gives non-automaticity of totient residues modulo every m≥3 in every base. Allouche–Shallit–Yassawi’s survey (published 2022; arXiv v1 2021, Theorem 3, Example 4 and Remark 7) explains that result. Wong’s earlier base-equals-modulus theorem remains credited.\n\n**Residue series, adjudicated 28 September 2026.** The first assertion of the short paper's residue theorem (`res:residueseries`: \\(\\sum_{n\\ge1}(\\varphi(n)\\bmod m)2^{-n}\\notin\\mathbb Q\\) for every \\(m\\ge3\\)) is implied, in every integer base \\(t\\ge2\\), by Erdős 1957 (Indag. Math. 19, 212–219). His Lemma 1 (printed p. 213) makes \\(\\sum_k a_kt^{-k}\\) irrational when the integers \\(a_k\\ge0\\) have bounded averages and infinitely many, but density-zero, nonzero terms; Erdős states it as known and notes that it is the case \\(b_k=0\\) of his Lemma 4 (stated pp. 215–216, proved pp. 216–218). The least residue \\(\\varphi(n)\\bmod m\\) is nonzero only when \\(n\\) has no prime factor \\(p\\equiv1\\pmod m\\), a set of density zero since \\(\\sum_{p\\equiv1\\,(m)}1/p\\) diverges, and equals \\(m-2\\) at every prime \\(p\\equiv-1\\pmod m\\). Lemma 4′ (p. 218, proof omitted by Erdős) and Theorem 3 of Kaneko–Suzuki–Tachiya (arXiv:2601.20743v1, PDF p. 5; they prove Lemma 4′ as their Theorem A, pp. 2 and 18) give the same. The same criteria cover the dyadic classification (the theorem's second assertion), its integer-base form (long record `prop:radixresidue`) and the finite-dilation independence (`prop:dilations`) only when the relevant values \\(f(r)-f(0)\\), or relation coefficients, have one sign; with both signs Lemmas 4 and 4′ and KST Theorem 3 need a spacing condition on the negative terms, which was not checked. The isolated-coefficient proof in the papers needs none. KST Theorem 3 with \\(b=0\\) also gives the irrational direction of the long record's slowly-growing-modulus dichotomy (`prop:slowmoduli`, ordinary proof), subject to a density bound for \\(n\\) with small \\(v_2(\\varphi(n))\\) that the record does not write out. No novelty claim is made for the residue-series irrationality; the papers cite Erdős 1957 and KST at the result. Page references are to the Rényi scan of Erdős 1957 and to the KST PDF recorded in its [source closure](primary-sources/reciprocal-tail/kaneko-suzuki-tachiya-2026-source-closure.md).\n\nBell–Smertnig (2026 preprint, Theorem 1.3) proves that a characteristic-zero Mahler series with multiplicative coefficients is regular, and explicitly excludes the totient series from every Mahler base. Bell–Bruin–Coons (2012, Theorems 1.5–1.6) supplies the earlier algebraic and D-finite classification context. These are generating-function statements, not proofs about the value at 1/2. Adamczewski–Drmota–Müllner (2022; arXiv v2 2021, Theorem 1.4 and Remark 1.5) provides the automatic-prime density context, with local conditions rather than unconditional positive density for every state.\n\nPrimary sources: [Yazdani](https://www.numdam.org/item/JTNB_2001__13_2_651_0/); [Allouche–Shallit–Yassawi](https://arxiv.org/abs/2104.13072v1); [Bell–Smertnig](https://arxiv.org/abs/2603.23456v1); [Bell–Bruin–Coons](https://arxiv.org/abs/1003.2221v2); [Adamczewski–Drmota–Müllner](https://arxiv.org/abs/2009.14773v2); [Martin](https://arxiv.org/abs/math/0603053v1). No priority claim for the local normal form is certified by this comparison.\n\n## Principal sources\n\nEach entry gives the source, then its relationship to this release, then the\nboundary of that relationship.\n\n- P. Erdős and R. L. Graham, [*Old and New Problems and Results in Combinatorial Number Theory* (1980)](https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf) ([read source closure](primary-sources/reciprocal-tail/erdos-graham-1980-source-closure.md)); P. Erdős, [*On the irrationality of certain series: problems and results* (1988)](https://doi.org/10.1017/CBO9780511897184.009) ([Renyi scan](https://users.renyi.hu/~p_erdos/1988-22.pdf); [read source closure](primary-sources/reciprocal-tail/erdos-1988-source-closure.md))\n\n  The 1980 monograph records the full-support divisor Lambert value \\(\\sum_{n\\ge1}d(n)/2^n\\) as irrational (printed p. 61), states that the analogous binary totient series could not then be proved irrational (p. 61), and identifies the full-support Mersenne identity \\(\\sum_{n\\ge1}(2^n-1)^{-1}=\\sum_{n\\ge1}d(n)/2^n\\) (p. 62). It also records the squarefree reciprocal question as open (p. 63). Erdős's 1988 article, printed p. 105, explicitly asks whether every increasing support gives an irrational \\(\\sum_k(2^{n_k}-1)^{-1}\\); its density/LCM irrationality theorem is a separate result on pp. 106--108.\n\n  **Boundary.** The 1980 full-support identity and the 1988 p. 105 subseries statement do not prove universal #257; the separate LCM theorem does not automatically apply to Mersenne denominators. The 1980 source explicitly leaves the binary totient irrationality question unresolved, so neither source proves #249, the release's Lean, Comparator, Palomar, geometry, finite-certificate, or totient-kernel claims.\n\n- S. Chowla, [*On Series of the Lambert Type which assume Irrational Values for Rational Values of the Argument* (1947)](https://insa.nic.in/UI/Archivesection.aspx?JID=MA%3D%3D&JYrs=MTk0Nw%3D%3D) ([read source closure](primary-sources/reciprocal-tail/chowla-1947-source-closure.md))\n\n  Historical conjectural predecessor: its rational-argument Lambert question contains the full-support specialisation \\(x=1/b\\).\n\n  **Boundary.** Chowla's source is a conjecture in this relation, not the proof source; the theorem formalised here is credited to Erdős (1948).\n\n- P. Erdős, [*On arithmetical properties of Lambert series* (1948)](https://users.renyi.hu/~p_erdos/1948-04.pdf) ([read source closure](primary-sources/totient-kernel/erdos-1948-lambert-source-closure.md))\n\n  **Full-support divisor row.** Printed p. 63 states irrationality of `f(1/t) = Σ_{r≥1}d(r)/t^r` for every integer `|t|>1`, the Erdős--Borwein divisor-weighted Lambert theorem used for the full-support row.\n\n  **Boundary.** Printed p. 66 explicitly says the analogous Euler-totient series `Σφ(n)/t^n` “seem[s] to present difficulties.” This source therefore does not support irrationality of the #249 totient series, any totient-kernel rank/basis, or the release’s Lean/Comparator claims; no novelty or priority claim is made.\n\n- P. Erdős, [*On the irrationality of certain series* (Math. Student 36, 1968)](https://users.renyi.hu/~p_erdos/1969-09.pdf) ([read source closure](primary-sources/reciprocal-tail/erdos-1968-source-closure.md))\n\n  **Pairwise-coprime support theorem.** Printed p. 222 states that if `n_1 < n_2 < ...` are pairwise coprime and `Σ 1/n_i < ∞`, then `Σ_i 1/(t^(n_i)-1)` is irrational for every integer `t ≥ 2`; the proof occupies printed pp. 223–225.\n\n  **Boundary.** Printed p. 222 says the pairwise-coprimality condition can be removed only by more complicated arguments and gives no details, and that `Σ 1/n_i < ∞` could be replaced by an unstated weaker but more complicated condition; printed p. 226 suggests `Σ_{n_i<x} 1/n_i = o(log log x)` via Brun's method for pairwise coprime supports and says the all-primes case is not handled. The source therefore states the coprimality-free reciprocal-summable extension without proof; the #257 note credits that stated extension to Erdős and gives a complete proof of it. The source does not settle universal #257 or supply the release's Lean/Comparator proofs, and the note does not identify Erdős's unstated weaker condition with its weighted criterion. No novelty or priority claim is made.\n\n- D. Duverney and Y. Tachiya, [*Refinement of the Chowla–Erdős method and linear independence of certain Lambert series* (2019)](https://doi.org/10.1515/forum-2018-0299) ([author preprint](https://danielduverney.fr/documents/theorie-des-nombres/DuverneyTachiya190522.pdf); [read source closure](primary-sources/reciprocal-tail/duverney-tachiya-2019-source-closure.md))\n\n  **Squarefree-support antecedent.** Corollary 1.2 (PDF p. 4; proof pp. 10–11), specialised to `E` equal to the primes, `s=2`, `ell=1`, and `q=2`, proves linear independence of `1` and the squarefree Lambert values at bases `2^j`; Example 1.1 displays the family using `|mu(n)|`.\n\n  **Averaging antecedent.** The proof of Theorem 1.1 (Section 2, (2.3)–(2.9), PDF pp. 5–6) selects its index by averaging a local coefficient mass over an arithmetic progression and taking a term attaining the minimum. The #257 note cites this after its weighted-support theorem as the antecedent of the selection step in its displacement averaging.\n\n  **Boundary.** The condition `|q| L <= s`, a product with `L = lcm(1,...,ell)`, is essential to this specialisation, so the result does not give squarefree irrationality at every integer base and does not settle universal #257. With `s = infinity` it covers the perfect `i`th-power supports at every integer base. It also does not prove the release's Lean, Comparator, Palomar, geometry, or totient-kernel claims; no novelty or priority claim is made.\n\n- P. B. Borwein, [*On the irrationality of ∑ 1/(q^n+r)* (1991)](https://doi.org/10.1016/S0022-314X(05)80041-1) ([publisher record](https://www.sciencedirect.com/science/article/pii/S0022314X05800411); [read source closure](primary-sources/reciprocal-tail/borwein-1991-qn-r-source-closure.md))\n\n  **Fixed-base rational-offset antecedent.** Theorem 4 (printed pp. 257--258) proves that ∑ 1/(q^n+r) is irrational and not a Liouville number when q is an integer greater than one and r is a nonzero rational avoiding the denominator zeros r = -q^n for n ≥ 1. The paper’s Padé/Stieltjes construction is the direct source for this full-support family.\n\n  **Boundary.** The result is for every positive index with an integer base; it does not settle arbitrary infinite supports, the release's rational noninteger-base claim at `3/2`, universal #257, or the release's Lean results. Its recurrence and Padé estimates are not the Amdeberhan--Zeilberger q-WZ operator, and no novelty or priority claim is made.\n\n- P. B. Borwein, [*On the irrationality of certain series* (1992)](https://doi.org/10.1017/S030500410007081X) ([author PDF](https://www.cecm.sfu.ca/~pborwein/PAPERS/P59.pdf); [read source closure](primary-sources/reciprocal-tail/borwein-1992-source-closure.md))\n\n  **Full-support antecedent.** Theorems 1 and 2 (printed pp. 142 and 145; proof pp. 142--146) prove irrationality, and non-Liouville behavior, for the full-support series \\(\\sum_{n\\ge1}(q^n+c)^{-1}\\) and its alternating variant under their stated integer/rational hypotheses. Theorem 1 with \\(q=2\\), \\(c=-1\\) contains the full-support Mersenne--Lambert specialization.\n\n  **Boundary.** The paper sums over every positive index, not an arbitrary infinite support, so it does not settle universal #257, prime-support or squarefree subseries, or the release's #249 reductions. Its contour-integral proof is not formalised here; no novelty or priority claim is made.\n\n- H. Kaneko, Y. Suzuki, and Y. Tachiya, [*Refinements of Erdős's irrationality criterion for certain sparse infinite series* (2026)](https://arxiv.org/abs/2601.20743) ([read source closure](primary-sources/reciprocal-tail/kaneko-suzuki-tachiya-2026-source-closure.md))\n\n  Nearby current work proving irrationality for sparse series such as \\(\\sum d(n)^k/t^{\\varphi(n)}\\).\n\n  **Averaged-tail antecedent for #257.** The quantity \\(R_c(q,x,z)\\) of (1.7) (PDF p. 3) and Lemmas 1–2 (PDF pp. 6–8) derive irrationality from scaled tails that are small on average. The #257 note cites them beside its weighted-support theorem. There the divisor-incidence coefficients are positive on every multiple of \\(\\min A\\), so the scaled tail is at least the series value and the note averages the displacement instead.\n\n  **Residue-series antecedent for #249.** Theorem 3 (PDF p. 5), with \\(b=0\\), implies the irrationality of \\(\\sum_{n\\ge1}(\\varphi(n)\\bmod m)t^{-n}\\) for every \\(m\\ge3\\) and integer \\(t\\ge2\\), because these coefficients are bounded and nonzero only on a set of density zero; Theorem A (p. 2, proof p. 18) is Erdős's 1957 Lemma 4′. The #249 papers cite this beside the residue theorem; see the adjudication under the #249 attribution update above.\n\n  **Boundary.** Here \\(\\varphi(n)\\) occurs in the exponent. It does not treat the coefficient-weighted constant \\(\\sum\\varphi(n)/2^n\\), its denominator bound, or its tail-certificate equivalence. The counting hypotheses of Theorems 1 and 3 fail for divisor-incidence coefficients, so these criteria do not apply to Mersenne–Lambert subseries.\n\n- P. Erdős, [*On the irrationality of certain series* (Indag. Math. 19 (1957), 212–219)](https://users.renyi.hu/~p_erdos/1957-07.pdf)\n\n  **Residue-series antecedent for #249.** Lemma 1 (printed p. 213), the case \\(b_k=0\\) of Lemma 4 (pp. 215–218), gives irrationality of \\(\\sum a_kt^{-k}\\) for nonnegative integers with bounded averages and infinitely many but density-zero nonzero terms, and hence of \\(\\sum(\\varphi(n)\\bmod m)t^{-n}\\) for every \\(m\\ge3\\) and integer \\(t\\ge2\\). Lemma 4′ (p. 218) is stated without proof.\n\n  **Boundary.** The paper's Theorem 1 concerns \\(\\sum 1/t^{\\varphi(n)}\\) and \\(\\sum1/t^{\\sigma(n)}\\), with the arithmetic function in the exponent; its introduction (printed p. 212) says the author cannot prove irrationality of \\(\\sum\\varphi(n)/t^n\\). It does not prove #249, the dyadic classification with mixed-sign values, the finite-dilation independence with mixed-sign relations, or any Lean or Comparator claim. No source closure file or local copy is recorded in this pass.\n\n- T. M. Apostol, [*Introduction to Analytic Number Theory* (1976)](https://doi.org/10.1007/978-1-4757-5579-4); M. Merca, [*The Lambert series factorization theorem* (2017)](https://doi.org/10.1007/s11139-016-9856-3); M. Merca and M. D. Schmidt, [*Generating Special Arithmetic Functions by Lambert Series Factorizations* (2019)](https://doi.org/10.55016/ojs/cdm.v14i1.62425) ([official journal PDF](https://cdm.ucalgary.ca/article/download/62425/53773); [read source closure](primary-sources/totient-kernel/merca-schmidt-2017-lambert-factorizations-source-closure.md))\n\n  Apostol supplies the classical Dirichlet/Möbius background; Merca gives the modern factorisation theorem and Merca--Schmidt its synthesis/unification treatment. Merca--Schmidt's equation (2) records the totient Lambert identity used as transform background, while Theorem 3.2 and Corollary 3.3 give their partition/inverse-matrix mechanism. Apostol/Merca 2017 remain cited background without a source closure in this pass.\n\n  **Boundary.** The release uses classical transforms and claims a Lean-checked composition, not new Möbius inversion, Lambert identities, or the cited factorisation theorems.\n\n- Y. V. Nesterenko, [*Modular functions and transcendence questions* (1996)](https://doi.org/10.1070/SM1996v187n09ABEH000158) ([read source closure](primary-sources/totient-kernel/nesterenko-1996-source-closure.md))\n\n  Direct source for the cited-only transcendence of the divisor-sum ladder rung \\(L(\\mathrm{Id})=\\sum_{m\\geq1}\\sigma(m)/2^m\\).\n\n  **Boundary.** This positions one ladder row only; no part of its modular-transcendence proof is formalised here.\n\n- K. Postelmans and W. Van Assche, [*Irrationality of \\(\\zeta_q(1)\\) and \\(\\zeta_q(2)\\)* (2007)](https://doi.org/10.1016/j.jnt.2006.11.011) ([read source closure](primary-sources/reciprocal-tail/postelmans-van-assche-2007-source-closure.md))\n\n  Primary source for the cited-only linear independence of \\(1,\\zeta_q(1),\\zeta_q(2)\\) when \\(q^{-1}\\) is an integer of absolute value greater than one. At \\(q=1/2\\), this implies the irrationality of the squared-ladder value \\(L_2(\\mathbf1)=\\zeta_{1/2}(2)-\\zeta_{1/2}(1)\\).\n\n  **Boundary.** The release formalises the identity placing \\(L_2(\\mathbf1)\\) in this q-zeta family, not the Hermite--Padé proof. Linear independence of the constant-weight row gives no irrationality transfer to the Möbius row \\(L_2(\\mu)=S-1/2\\), which remains open.\n\n- S. Kakeya, [*On the Set of Partial Sums of an Infinite Series* (1914)](https://doi.org/10.11429/ptmps1907.7.14_250) ([read source closure](primary-sources/reciprocal-tail/kakeya-1914-source-closure.md))\n\n  Classical strict-tail criterion for subsum sets: domination of every remaining tail by its term gives the Cantor-set regime.\n\n  **Boundary.** The release uses this as historical geometry, not as a claim that Kakeya supplied the local executable machinery.\n\n- V. Kovač and T. Tao, [*On several irrationality problems for Ahmes series* (2025)](https://doi.org/10.1007/s10474-025-01528-0) ([read source closure](primary-sources/reciprocal-tail/kovac-tao-2025-source-closure.md))\n\n  **Fixed-base strict-tail antecedent.** Their Remark 4.1 (PDF p. 13) verifies \\(\\sum_{\\ell>n}(t^\\ell-1)^{-1}<(t^n-1)^{-1}\\) for every integer \\(t\\ge2\\), and deduces distinct Lambert subsums and a Cantor set. Their Theorem 2.3 (PDF p. 5; proof pp. 13–14) separately constructs rational *merged* sums from several bases under a mass hypothesis.\n\n  **Boundary.** At \\(t=2\\) the strict-tail remark supplies the known Mersenne geometry, but it does not assert a measure formula. The multi-base theorem allows repeated terms and is not a fixed-base counterexample. The release now formalises compactness, perfection, total disconnectedness, nowhere density, and measure one as well as the two-scale gap, greedy interface, and finite death certificates. It makes no novelty claim for those refinements, and Kovač–Tao explicitly leave the universal fixed-base subseries question open; none of these results settles #257.\n\n- M. Coons, [*(Non)Automaticity of number theoretic functions* (Journal de Théorie des Nombres de Bordeaux 22 (2010), no. 2, 339--352)](https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.718/) ([official PDF](https://jtnb.centre-mersenne.org/item/10.5802/jtnb.718.pdf); [read source closure](primary-sources/totient-kernel/coons-2010-source-closure.md))\n\n  **Global totient-kernel boundary.** Coons's Theorem 3.2, stated on printed p. 348 and proved on p. 349, shows that Euler's totient function is not \\(k\\)-regular for any \\(k\\ge2\\), using the Dirichlet-series identity \\(\\sum_{n\\ge1}\\varphi(n)n^{-s}=\\zeta(s-1)/\\zeta(s)\\) and the pole-counting criterion of Corollary 3.1. This is the prior-art source for the release's global non-finite-generation context; it does not state the finite-level rank, explicit basis, or relation normal form proved or discussed locally.\n\n  **Boundary.** For integer-valued sequences, finite generation of the \\(\\mathbb Z\\)-module generated by the kernel is equivalent to finite dimension of its rational span: if the span has dimension \\(d\\), some \\(d\\) evaluations embed it in \\(\\mathbb Q^d\\) and the module in \\(\\mathbb Z^d\\). The #249 short note states this in §1, so the infinite-dimensionality of the totient-kernel span is Coons's theorem in another form, and the exact ranks recover it. Coons does not prove the release's Lean declarations, dyadic rank \\(2^e+1\\), CRT--Dirichlet--determinant construction, or Erdős Problem #249. No novelty or priority claim is made for either the cited theorem or the local finite-level results.\n\n- M. Coons, [*Regular sequences and the joint spectral radius* (International Journal of Foundations of Computer Science 28 (2017), no. 2, 135--140)](https://doi.org/10.1142/S0129054117500095) ([arXiv:1511.07535](https://arxiv.org/abs/1511.07535); [read source closure](primary-sources/totient-kernel/coons-2017-joint-spectral-radius-source-closure.md))\n\n  **Basis versus spanning family for regular sequences.** For a \\(k\\)-regular sequence, Theorem 1 identifies the growth exponent with the base-\\(k\\) logarithm of the joint spectral radius of matrices associated to a basis of the kernel span; Proposition 4 gives the upper bound for any spanning set, Corollary 7 compares the two, and the example after it shows that a redundant spanning set can overestimate. Appendix A shows that the evaluation vectors of a basis of sequences span the coefficient space. The #249 short note cites these results for the basis-versus-spanning-family distinction and for the evaluation argument in the equivalence above.\n\n  **Boundary.** The source concerns regular sequences and does not mention Euler's totient, which is not regular, so its growth theorem is not applied to the totient. It does not state the release's finite-level basis, ranks, section-map description, relation module, or Lean declarations. Statement numbers follow arXiv:1511.07535v1. No novelty or priority claim is made.\n\n- J. Farey, [*On a Curious Property of Vulgar Fractions* (1816)](https://doi.org/10.1080/14786441608628487)\n\n  Historical source for the Farey language. The standard neighbouring-fraction/mediant lemma is the method that directly produces the finite #249 denominator bound: for the committed interval it gives `b+d-1`, exactly the formal bound.\n\n  **Boundary.** The mathematical method and denominator estimate are classical. The local contribution is a kernel-checked arithmetic instance and proof that `b+d` is the first failure for each selected window; the improvement over the classical Farey bound is zero.\n\n- G. Martin, [*Simultaneous inequalities among values of the Euler phi-function* (arXiv:math/0603053, 2006)](https://arxiv.org/abs/math/0603053) ([read source closure](primary-sources/totient-kernel/martin-2006-source-closure.md))\n\n  **Subsuming source for affine-totient ratio comparisons.** His Theorem 1 assumes only that the slopes \\(a_i\\) are positive integers and that \\(a_ib_j\\neq a_jb_i\\), and proves that for every \\(C>0\\) the simultaneous ratio gaps \\(\\varphi(a_1n+b_1)/\\varphi(a_2n+b_2)>C,\\ldots\\) hold on a set of positive lower density; the symmetry discussion supplies strict ordering patterns. For the local all-base argument it is an earlier source that already implies the affine-totient independence, which the release proves separately; the finite dyadic Lean theorem remains separately formalised, not a theorem directly stated by Martin. His Corollary 4 transfers Theorem 1 and Corollaries 2–3 to \\(\\sigma\\).\n\n  **Boundary.** The release's dyadic Lean independence result is separately formalised from this broader comparison input by a finite CRT–Dirichlet–determinant argument; Martin does not present the release's Lean statement or proof. The all-base paper theorem proves its affine independence by a determinant that is diagonal modulo an auxiliary prime, built with the Chinese remainder theorem and Dirichlet's theorem, and cites Martin's Theorem 1 as an earlier source that already implies that independence. Lean proves the all-base independence, the exact rank `k^e + 1` for `k ≥ 2` and `e ≥ 1`, and the explicit basis unconditionally in [`AllBaseTotientKernel.lean`](../lean/Erdos249257/AllBaseTotientKernel.lean); it does not formalise Martin's positive-density theorem, which is not an input of that proof. Martin needs neither odd slopes, nor primitivity, nor a residue bound, and concludes strictly more. A \\(\\sigma\\) analogue would likewise not be new. Exact-title, DOI and venue searches did not locate a separate journal publication, so this is cited as a public preprint.\n\n- F. Luca and Y. Tachiya, [*Irrationality of Lambert series associated with a periodic sequence* (2014)](https://doi.org/10.1142/S1793042113501121) ([read source closure](primary-sources/reciprocal-tail/luca-tachiya-2014-source-closure.md))\n\n  Direct antecedent for the eventually-periodic coefficient family. Their rational-coefficient theorem contains the indicator-support case and the nonnegative rational weights formalised here.\n\n  **Boundary.** Their theorem is broader, and its Chowla--Erdős/large-modulus-prime proof is separate. This release uses a periodic-divisor certificate route. For signed integer weights it proves only an irrational-or-base-terminating dichotomy; it does not exclude the terminating alternative in general or claim the full mixed-sign theorem.\n\n- T. Tao and J. Teräväinen, [*Quantitative correlations and some problems on prime factors of consecutive integers* (arXiv v2, 2026)](https://arxiv.org/abs/2512.01739v2) ([read source closure](primary-sources/reciprocal-tail/tao-teravainen-2026-source-closure.md))\n\n  Direct source for the cited-only irrationality of the prime-support series: Theorem 1.3, equation (1.7), on PDF p. 4, with proof on pp. 44--56. It gives the \\(\\sum_n\\omega(n)/2^n = \\sum_p(2^p-1)^{-1}\\) formulation and this release's separately formalised identity bridge.\n\n  **Boundary.** It settles the prime-support family at base 2, not universal #257. The source only says that other integer bases and the prime-power variant can be treated by modifications and leaves those details to the reader; its quantitative correlation proof is not formalised here.\n\n- H. Wang, [*Sparse Polynomial-Weighted Expansions* (arXiv:2606.24972v4, 2026)](https://arxiv.org/abs/2606.24972v4) ([read source closure](primary-sources/dyadic-carry/wang-2026-source-closure.md))\n\n  Direct antecedent for the rationality-forced integral carry recurrence in the weighted-binary special case (c(n)=n d_n).\n\n  **Boundary.** The public generic criterion is a formalisation/abstraction; its converse and rigidity make no priority claim. The source's positive-density theorem and Erdős #260 corollary are neither used nor formalised here.\n\n- R. Crandall, [*The googol-th bit of the Erdős--Borwein constant* (2012)](https://doi.org/10.1515/integers-2012-0007); J. M. Campbell, [*On the binary digits of the Erdős--Borwein constant* (arXiv, 2026)](https://arxiv.org/abs/2605.24160)\n\n  Adjacent work on binary digits of the full-support constant; Campbell resolves Crandall's question on infinitely many occurrences of the block \\(11\\).\n\n  **Boundary.** Neither result is formalised or used by this release, and neither changes its irrationality or open-problem claims.\n\n## Expansion-note principal sources (#68, #243, #251, #269, #1041, #1049)\n\nThe same contract as above applies: relationship, not proof authority, and no\nrow widens a release claim. Each source below is load-bearing for exactly the\nstated use in its problem note, where the full statement-level comparison is\nrecorded.\n\nEntries carry the same three parts as above — source, relationship, boundary —\ngrouped by the problem whose note they serve.\n\n### #68\n\n- J. Louwsma and J. Martino, *Rational numbers with odd greedy expansion of fixed length*, [arXiv:2309.07280v1](https://arxiv.org/abs/2309.07280v1)\n\n  Lemma 4.1, p. 10, is the reciprocal-sum valuation formula. The notes' maximal prime-power survival test is its top valuation layer, written out for the denominators `n!−1`; the 139 and 2593 cancellations are the notes' own exact computations.\n\n  **Boundary.** The lemma concerns finite sums of reciprocals and gives no statement about the infinite series.\n\n- J. Hančl and R. Tijdeman, *On the irrationality of factorial series*, Acta Arith. 118 (2005), [doi:10.4064/aa118-4-5](https://doi.org/10.4064/aa118-4-5); W. Koepf and D. Schmersau, *Irrationality of certain infinite series II*, Analysis 31 (2011), [doi:10.1524/anly.2011.1094](https://doi.org/10.1524/anly.2011.1094)\n\n  Hančl–Tijdeman Lemma 2.1 and the following remark (p. 385) give the factorial-tail integrality argument behind the finite-denominator implication. Koepf–Schmersau Example 3.2 (p. 121) states the irrationality direction of the factorial-digit criterion, whose analogue for the companion `C = S − e + 2` the notes prove. The long record compares the Hančl–Tijdeman polynomial-coefficient classification (Theorem 3.1) and the Oppenheim criterion they reproduce (Lemma 2.2), naming the unmet hypothesis of each.\n\n  **Boundary.** The carry characterisation is proved directly in the notes; no source theorem supplies the needed cofinal event.\n\n- F. Luca and I. E. Shparlinski, Bull. London Math. Soc. 37 (2005), [doi:10.1112/S0024609305004923](https://doi.org/10.1112/S0024609305004923); L. Lai, *On the largest prime divisor of n!+1*, Bull. Aust. Math. Soc. 113 (2026), [doi:10.1017/S0004972725100543](https://doi.org/10.1017/S0004972725100543)\n\n  The factorial-gap gcd bound is the case `P = −1` of the subtraction that Luca and Shparlinski use in the proof of their Lemma 5 and Lai uses at display (2.5); Lai's Lemma 2.1 supplies the non-vanishing cutoff for polynomial shifts.\n\n  **Boundary.** The terminal-block optimisation and the resulting lcm growth theorem are the notes' own argument.\n\n- P. Erdős, *Some problems and results on the irrationality of the sum of infinite series* (1975), Theorem 1; D. Duverney (2001), [journal PDF](https://www.ms.u-tokyo.ac.jp/journal/pdf/jms080206.pdf); Barreto–Kang–Kim–Kovač–Zhang, [arXiv:2601.21442v3](https://arxiv.org/abs/2601.21442v3)\n\n  Rapid-growth and fast-series irrationality criteria, each checked in the long record against `n!−1` with the unmet hypothesis named.\n\n  **Boundary.** Non-applicability results; no criterion is claimed to transfer.\n\n- M. Z. Garaev, F. Luca and I. E. Shparlinski (Trans. AMS 356), doi:10.1090/S0002-9947-04-03612-8; C. L. Stewart (Publ. Math. Debrecen 65), doi:10.5486/PMD.2004.3190\n\n  Multiplicity (Theorem 12) and Wilson-reflection and least-prime-factor inputs (pp. 462–464) for the collision-core layer and the superseded lcm deduction.\n\n  **Boundary.** Not kernel-checked; load-bearing for nothing beyond the stated uses.\n\n### #243\n\n- P. Erdős and E. G. Straus, [*On the irrationality of certain Ahmes series*](https://users.renyi.hu/~p_erdos/1964-19.pdf) ([read source closure](primary-sources/reciprocal-tail/erdos-straus-1964-source-closure.md))\n\n  Original source for the conditional Ahmes-series rationality criterion cited at Theorem 3, p. 132. Its printed theorem uses the prefix quotient `N_k/n_(k+1)` and the next-index factor `n_(k+1)^2/n_(k+2)-1`; this is the source boundary for the note's correction of the shifted catalogue/survey display.\n\n  **Boundary.** The criterion is conditional and only gives eventual Sylvester recurrence. It does not settle unrestricted #243, the release's Lean results, or any novelty/priority claim.\n\n- D. Duverney, [*Irrationality of Fast Converging Series of Rational Numbers*](https://www.ms.u-tokyo.ac.jp/journal/pdf/jms080206.pdf) ([read source closure](primary-sources/reciprocal-tail/duverney-2001-source-closure.md))\n\n  Corollary 3.2, p. 287, gives a conditional signed form of the #243 criterion: with `a_n` in `{-1,1}`, rationality of `sum a_n/u_n` is equivalent to the eventual signed recurrence under Duverney's one-sided condition `sum (u_(n+1)/u_n^2-1) < infinity`.\n\n  **Boundary.** The source condition is not written with absolute values; absolute convergence is only a stronger sufficient specialization. The result remains conditional and does not settle unrestricted #243, the release's Lean results, or any novelty/priority claim.\n\n- C. Badea, [*A theorem on irrationality of infinite series and applications*](https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/63/4/107785/a-theorem-on-irrationality-of-infinite-series-and-applications) ([publisher PDF](https://matwbn.icm.edu.pl/ksiazki/aa/aa63/aa6342.pdf); [read source closure](primary-sources/reciprocal-tail/badea-1993-source-closure.md))\n\n  Adjacent positive-term prior art: Theorem A (p. 313), Theorem 2.10 and Corollary 2.2 (pp. 315–316) force eventual equality in Badea's explicit term-inequality regime, generalizing the positive Sylvester boundary.\n\n  **Boundary.** The hypotheses are positive rational terms plus an eventual inequality; they do not cover unrestricted #243, mixed-sign state orbits, the release's Lean results, or any novelty/priority claim.\n\n- J. Koizumi, [*Irrationality of the reciprocal sum of doubly exponential sequences* (Integers 26 (2026), A28)](https://math.colgate.edu/~integers/aa28/aa28.pdf); [arXiv:2504.05933](https://arxiv.org/abs/2504.05933) ([read source closure](primary-sources/reciprocal-tail/koizumi-2026-source-closure.md))\n\n  Koizumi's canonical-tail and normalised-vanishing arguments are prior art.\n  The note credits his published Lemma 3 and Proposition 1(2) (preprint-v1\n  Lemma 13 and Proposition 19(2)) for absorption and descent.\n\n  **Boundary.** Lean checks `canonical_integer_tail_normalized`,\n  `canonical_tail_ratio_quantitative` and\n  `cubic_rate_irrationality_unconditional`. The paper's one-based finite-prefix\n  bridge remains prose. Unrestricted #243 is open; no priority or independence\n  is adjudicated.\n\n### #251\n\n- V. Kovač, counterexample to Erdős's variable-denominator expectation (2026, Theorem 1)\n\n  Refutes the 1988 expectation that `∑ p_n/(g_1⋯g_n)` is irrational whenever `g_n ≥ 2`, `g_n = o(p_n)`; the note records that the live catalogue still repeats the refuted expectation.\n\n  **Boundary.** Concerns the variable-denominator variant, not the fixed dyadic series studied here.\n\n- J.-C. Schlage-Puchta, *The irrationality of some number theoretical series*, [arXiv:1105.1451](https://arxiv.org/abs/1105.1451)\n\n  Lemma 4, a Selberg-sieve density-zero statement for polynomial relations among consecutive prime gaps, is the input the reasoning record uses. Theorem 2 treats base-`b` digit strings that concatenate the representations of a slowly growing integer sequence, and Theorem 3 gives the `ℚ`-linear independence of `1, S_0, S_1, …` with `S_k = ∑ p_n^k / n!`.\n\n  **Boundary.** No antecedent for the tail criteria recorded here has been located. Theorem 2 is the nearest located neighbour and treats a different object; no declaration in this release uses Lemma 4.\n\n- Y. Zhang (2014); J. Maynard (2015), doi:10.4007/annals.2015.181.1.7; Ford–Green–Konyagin–Maynard–Tao (2018), doi:10.1090/jams/876\n\n  Bounded-gap and large-gap prime results, cited and each shown insufficient for the required joint dyadic distribution of a logarithmic block of gap differences. Maynard's Theorem 1.1 bounds `liminf (p_(n+m) - p_n)` for every fixed `m`; the explicit bound 600 for consecutive gaps is cited by page. The 2018 large-gap Theorem 1 gives an effective lower bound for the largest single consecutive-prime gap below a threshold.\n\n  **Boundary.** Context; none is used as a proof input.\n\n- J. Land, conditional proof of the irrationality of `∑_(n≥1) p_n 2^(-n)` (research draft, 5 September 2026), with an accompanying Lean formalisation of the conditional argument\n\n  Assumes Kuperberg's uniform Hardy-Littlewood prime-tuples conjecture and constructs weighted prime-gap tails; announced on the problem's discussion thread. The #251 note and reasoning record both cite it.\n\n  **Boundary.** Every #251 declaration in this release is unconditional and was proved independently of it. A proof conditional on a prime-tuples conjecture leaves Erdős #251 open unconditionally.\n\n- S. Ringer, *Local gap statistics, telescoping, and normality* (preprint, 11 September 2026, [github.com/StefanRinger/erdos-251](https://github.com/StefanRinger/erdos-251)), with a Lean formalisation whose conditional endpoint theorems take Kuperberg's Conjecture 1.3 as an explicit hypothesis\n\n  Corollary 1.2 deduces from Kuperberg's uniform Hardy-Littlewood prime-tuples conjecture that `∑_(n≥1) p_n B^(-n)` is normal to base `B` for each integer `B ≥ 2`, which gives irrationality at `B = 2`. Appendix C proves unconditionally that `∑_(n≥1) p_n B^(-S_n)` with `S_n = ∑_(j≤n) ⌈log_B log(j+3)⌉` is normal to base `B`. The preprint was submitted as a partial proof claim on the problem page on 13 September 2026. The #251 note and reasoning record both cite it.\n\n  **Boundary.** Every #251 declaration in this release is unconditional and was proved independently of it. A normality proof conditional on a prime-tuples conjecture leaves Erdős #251 open unconditionally.\n\n- W. van Doorn and V. Kovač, [*Lacunary sequences whose reciprocal sums represent all rational numbers in an interval* (Acta Arith. 223 (2026), 275–295)](https://doi.org/10.4064/aa251001-13-1) ([arXiv:2509.24971v3](https://arxiv.org/abs/2509.24971v3); [read source closure](primary-sources/reciprocal-tail/van-doorn-kovac-2026-source-closure.md))\n\n  Proposition 8 proves that the finite reciprocal sums take every rational value in an interval, combining the dense filling of Lemma 7 with a divisibility chain among distinguished terms that eventually absorbs every positive integer. The #251 note and reasoning record cite both statements beside the interval-filling and divisibility-chain steps of the sparse rationalisation construction.\n\n  **Boundary.** The source concerns finite sums of unit fractions and has no prime numerators. The #251 construction uses its own factorial moduli and does not invoke Proposition 8.\n\n### #269\n\n- J. H. Loxton and A. J. van der Poorten, *Arithmetic properties of certain functions in several variables III*, Bull. Austral. Math. Soc. 16 (1977), [doi:10.1017/S0004972700022978](https://doi.org/10.1017/S0004972700022978); Y. Bugeaud and M. Laurent, *Transcendence and continued fraction expansion of values of Hecke–Mahler series*, Acta Arith. 209 (2023), [doi:10.4064/aa220323-18-1](https://doi.org/10.4064/aa220323-18-1)\n\n  The Hecke–Mahler value theorem (Loxton–van der Poorten Theorem 8, p. 40, in the modern form of Bugeaud–Laurent Theorem 1.1) is the transcendence input for the note's two-prime theorems: both the de-duplicated and repeated running-lcm reciprocal sums are transcendental for every pair of distinct primes, by a paper argument.\n\n  **Boundary.** Lean proves the two-prime transcendence transfers conditionally on the named input `BugeaudLaurentTranscendence` (`two_prime_sums_transcendental` in [`TwoPrimeSums.lean`](../lean/ErdosProblems/Erdos269/PaperCompleteR21/TwoPrimeSums.lean)); the Hecke–Mahler value theorem itself is not formalised. Nothing follows for three or more primes.\n\n- S. Fan, [comment on Erdős Problem #269](https://www.erdosproblems.com/forum/thread/269#post-7218), 26 June 2026\n\n  Public forum post giving the same two-channel factorisation, Hecke–Mahler reduction and transcendence conclusion, and the running-LCM identity for every prime set; later comments extend it to coprime pairs. The note first appeared on 22 July 2026, 26 days later.\n\n  **Boundary.** Priority for the two-prime result is Fan's. The note's argument was developed independently and is not a Lean theorem; a broader search would be needed to identify the earliest public proof.\n\n- P. Erdős and E. G. Straus, *On the irrationality of certain series*, Pacific J. Math. 55 (1974), [doi:10.2140/pjm.1974.55.85](https://doi.org/10.2140/pjm.1974.55.85); J. Hančl and R. Tijdeman, *On the irrationality of Cantor and Ahmes series*, Publ. Math. Debrecen 65 (2004), [doi:10.5486/PMD.2004.3254](https://doi.org/10.5486/PMD.2004.3254)\n\n  The literal `{2,3,5}` value is a Cantor series whose normalised tails obey an integer carry recurrence, the classical setting of Erdős–Straus Theorem 2.1 and the Hančl–Tijdeman tail criteria (§§2–3, Theorem 3.1).\n\n  **Boundary.** Both criteria assume a small-numerator hypothesis that fails for this series, so the recurrence, the bounds and the denominator clearing are proved directly in the notes.\n\n- P. Erdős and S. J. Taylor, Proc. London Math. Soc. (3) 7 (1957), Theorem 1; S. Fan, *Strongly complete sets and a conjecture of Erdős*, [arXiv:2607.14071v1](https://arxiv.org/abs/2607.14071v1), Lemma 3.1\n\n  For an unbounded integer sequence with bounded successive ratios, the reals whose multiples by that sequence approach integers form a countable set; the note uses this to identify the integral branch of the bounded-radix alternative.\n\n  **Boundary.** Used only to identify that branch; it gives no irrationality statement.\n\n- P. Erdős, letter of 1 January 1973 (printed p. 335 as cited in the #269 note)\n\n  Asserted the de-duplicated irrationality for given primes without a printed proof; the note's route is an independent modern argument.\n\n  **Boundary.** Historical formulation and priority context only.\n\n### #1041\n\n- P. Erdős, F. Herzog, and G. Piranian, *Metric properties of polynomials* (J. Analyse Math. 6 (1958), Problem 5, p. 139), doi:10.1007/BF02790232; see the [read source closure](primary-sources/bergman-geodesic/erdos-herzog-piranian-1958-source-closure.md)\n\n  Original problem source, including the known input that one lemniscate component contains at least two zeros.\n\n  **Boundary.** Formulation, not proof; the note's saddle-block diagnosis of a recent manuscript is prose, not Lean.\n\n- Current evidence boundary: the committed [`research_corpus/Erdos1041/FRONTIER.md`](../research_corpus/Erdos1041/FRONTIER.md)\n  is the dated route for the current source-only research state, not an\n  antecedent source. It records certified strategy refutations, surviving\n  carriers, and open gaps that are not part of this historical bibliography.\n  Read it before generated `STRONGEST_RESULTS.json`, which can lag the dated\n  notes. The corpus is not thereby prior art or a reviewed claim: its rows make\n  no peer-review, priority, novelty, or significance assertion. The current\n  status of #1041 is owned by [`docs/claims.json`](claims.json): Lean refutes\n  the exact Formal Conjectures statement using the degree-seven polynomial\n  posted by `ani`, Formal Conjectures records the answer `False`, and\n  independent human review of correspondence with the 1958 wording has not\n  been recorded.\n\n### #1049\n\n- W. Zudilin, [*Heine's basic transform and a permutation group for $q$-harmonic series* (2004)](https://doi.org/10.4064/aa111-2-4) ([read source closure](primary-sources/reciprocal-tail/zudilin-2004-heine-permutation-group-source-closure.md)); W. Zudilin, [*On the irrationality of generalized $q$-logarithm* (2016)](https://doi.org/10.1007/s40993-016-0042-x) ([read source closure](primary-sources/reciprocal-tail/zudilin-2016-generalized-q-logarithm-source-closure.md))\n\n  The 2004 paper supplies the linear forms, Lemma 7 with display (23) on p. 161, the direction `(14, 12, 14; 27)`, the thirteen intervals, and the constant `C_1/C_0 = 2.46497868...` printed on p. 162 as an integer-base exponent bound, which defines the rational-base region of the #1049 notes. The 2016 paper proves the `q`-order inequality for the normalised Hankel determinant and remarks in its Section 2, on p. 4, that its results can be given at non-integer rational bases under a computable height restriction.\n\n  **Boundary.** Both papers state their theorems for integer bases, and the 2016 remark computes no constant. The rational specialisation, its denominator accounting, the explicit region and the equality case of the Hankel order are argued in the #1049 notes, and no priority claim is attached to them.\n\n- P. Bundschuh and K. Väänänen, *Compositio Math.* 91 (1994), Theorem 2, [official Numdam PDF](https://numdam.org/item/CM_1994__91_2_175_0.pdf) ([read source closure](primary-sources/totient-kernel/bundschuh-vaanenen-1994-source-closure.md))\n\n  External irrationality criterion at base `7/2`; the release checks only its elementary Archimedean height inequality.\n\n  **Boundary.** The analytic argument remains external, and nothing follows at `3/2`.\n\n- T. Amdeberhan and D. Zeilberger, [*$q$-Apéry irrationality proofs by $q$-WZ pairs* (1998)](https://doi.org/10.1006/aama.1997.0565) ([read source closure](primary-sources/reciprocal-tail/amdeberhan-zeilberger-1998-q-apery-source-closure.md)); W. Van Assche, [*Little $q$-Legendre polynomials and irrationality of certain Lambert series* (2001)](https://doi.org/10.1023/A:1012930828917) ([read source closure](primary-sources/reciprocal-tail/van-assche-2001-little-q-legendre-source-closure.md)); P. B. Borwein, [*On the irrationality of certain series* (1992)](https://doi.org/10.1017/S030500410007081X)\n\n  Amdeberhan--Zeilberger and Van Assche target the same Lambert value with the same little-$q$-Legendre kernel on different moving diagonals; Van Assche cites Borwein's Lemma 2 for the neighbouring evaluation used in the comparison.\n\n  **Boundary.** The exact nonzero residual in the paper disproves transfer of the Amdeberhan--Zeilberger scalar recurrence to Van Assche's diagonal. No endpoint, lattice, valuation, or denominator gain is transferred merely from the shared kernel. Borwein 1992, not the adjacent 1991 paper, is the cited evaluation source.\n\n## Provisional theorem-specific comparisons\n\nThese local rows come after the direct antecedents on purpose. They record\ndated searches and residual comparisons; failure to identify a closer theorem\nis not a novelty result.\n\nEach entry names the local package, the comparison this release recorded for\nit, and the boundary of that comparison.\n\n- **The complete tail-certificate normal form.** The comparison search covered the original problem sources, Erdős's and Borwein's Lambert-series theorems, periodic-coefficient results, and current arithmetic-function series literature. No theorem with the same pointwise certificate equivalence was identified as of 16 July 2026. The finite denominator number is deliberately excluded because it is obtained directly by the classical Farey/mediant method.\n\n  **Boundary.** This is a dated search result, not a priority or novelty theorem. A closer antecedent should replace it if one is found.\n\n- **Finite-difference and affine-moment transport.** General algebraic techniques are applied here to exact Möbius residue kernels and totient-tail windows. The primary-source search found no cited theorem matching the combined curvature, prime-jump, anchored `(3,5)`, or fixed-precision no-go statements.\n\n  **Boundary.** Absence of an identified match is not a novelty claim. The release claims only the pinned Lean theorems and keeps the open supply hypotheses explicit.\n\n- **First-harmonic and large-prime pivot reductions.** The additive-character step and \\(\\varphi(mp)=\\varphi(m)(p-1)\\) use standard ingredients; Apostol supplies the arithmetic background. The local statement is the four-term supplier-fibre decomposition, its one-sided \\(14/25,1/100,1/100,8/25\\) budget, and its composition with certificate completeness.\n\n  **Boundary.** No matching combined theorem is recorded, but no prime-distribution or residual-decorrelation estimate populating the four budgets is proved.\n\n- **Reduced-direction, Stern--Brocot cylinder, and continuant-run package.** The direct sources above supply the classical divisor/Möbius, mediant, and adjacent squared-Lambert settings. The release proves an exact Mersenne-weight probability normalization, a telescoping cylinder law with uniform geometric error, and sharp Fibonacci/continuant run estimates.\n\n  **Boundary.** No priority claim is made. The run estimates have no endpoint consumer, and matching Fibonacci and denominator scales does not prove denominator survival or irrationality of \\(S\\).\n\n- **Exact \\(1/2\\)-membership, fatal-gap, and last-producer package.** The strict-tail source above gives the global subsum geometry. The maintained search found no source stating the exact equivalence between \\(1/2\\)-membership, infinitely many greedy skips, cofinal false seam terminals, and absence of a last skip, nor the surviving middle-producer future-tail condition.\n\n  **Boundary.** This is provisional positioning, not a novelty assertion. Positive membership would refute universal #257; nonmembership would decide only this test value.\n\n## Audit rule for new sources\n\nAdd a public citation only when the source can be tied to a named statement,\nmethod, or historical formulation in the release. Before describing a source as\nan antecedent of a local theorem family, record the exact statement comparison:\nassumptions, conclusion, specialisation map, proof mechanism, and the remaining\ndelta. Otherwise, retain it as private research context rather than turning a\nbibliographic lead into a public priority claim.\n","body_format":"markdown"}],"verification_boundary":{"runtime_verification_complete":false,"proof_recheck_performed":false,"what_a_reader_can_do":["read the exact theorem statement and its paper exposition","follow a public claim to its Lean declaration, module, and line","separate formalised claims from the remaining open propositions","detect disagreement between the paper, the claims registry, and scope"],"what_a_reader_cannot_do":["assert the Lean kernel rechecked these proofs -- no kernel ran during this build, and reading Lean source is not verifying it","treat the embedded source as a runnable clone of either repository"],"authority_order":"checked Lean source is proof authority; the claims registry owns public status and coordinates; methodology owns transition rules; the paper owns 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A reader meets records wherever a page links out to something inspectable rather than describing it in prose.","example":"The doctrine card \"Generated-result record source inversion\" concerns exactly this: a generated record, such as a result file emitted by a tool, must stay subordinate to the source that produced it rather than being treated as the authority itself.","example_route":{"href":"doctrine.html#dcard-ap-13","label":"Generated-result record source inversion","gloss":"open a rule about how generated records relate to their source"},"non_example":"Recording an assertion preserves it for inspection; the act of recording does not make the assertion true.","visible_text_byte_count":1174,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::record","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"can_carry","target":"evidence_class","gloss":"A record can name the support type behind it."},{"verb":"bounded_by","target":"scope_limit","gloss":"A record's claim boundary must travel with it."},{"verb":"below","target":"source_authority","gloss":"A generated record is not stronger than its source authority."}],"source_refs":[{"ref":"sites/microcosm/content-graph.json","role":"generated_public_record_index","gloss":"Generated public index of components, paper modules, glossary terms, and evidence profiles."},{"ref":"tools/meta/dissemination/build_microcosm_public_site.py","role":"public_projection_builder","gloss":"Builds public records from source rows and generated projections."}],"evidence_commands":[],"used_in":{"total_occurrences":1942,"page_count":155,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":288},{"page":"doctrine.html","count":161},{"page":"paper-modules.html","count":60},{"page":"paper-module-pattern-assimilation.html","count":43},{"page":"paper-module-mcp-tool-authority-replay.html","count":30},{"page":"paper-module-navigation-hologram-route-plane.html","count":26}]}},{"schema":"plectis_public_glossary_term_v0","position":6,"id":"substrate","label":"substrate","preferred_label":"substrate","alternative_labels":["source substrate","workflow substrate"],"hidden_search_labels":["live substrate","runtime substrate"],"term_kind":"substrate","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"Substrate means the live files, records, or runtime state on which this system operates. It is a local systems metaphor, not a biological, chemical, or philosophical substrate claim.","meaning_contrasts":[],"status":"active","disposition":"deep_public_technical_term","anchor":"glossary-substrate","relative_href":"docs/glossary.html#glossary-substrate","public_phrase":"underlying working base","short_definition":"Here, substrate means the underlying files, stored state, and tools on which the system operates.","reader_preview":"Here, substrate means the underlying files, stored state, and tools on which the system operates.","reader_preview_basis":"definition","card_definition":"It includes the concrete material an agent can inspect or change, beneath the pages and summaries that describe it. Plectis uses the term when distinguishing direct access to working material from access to a copied explanation.","reader_rule":"The source's role determines what it can authoritatively establish.","deep_definition":"A webpage can accurately describe a file without granting access to that file or permission to change it. Type A and Type B describe this access distinction in the project's actor vocabulary.","scope_note":"The term covers the source layer of the whole system: files, standards, ledgers, recorded state, and tooling. A reader meets it wherever the pages distinguish the generated public view from the source it was derived from.","example":"The Source and license page is the entry point into this substrate, opening onto the source files and records that the rest of the site is generated from.","example_route":{"href":"source.html","label":"Source & license","gloss":"open the source layer the pages are generated from"},"non_example":"A substrate is not a vague word for the whole project, and not a mystical layer behind it. It is the specific, inspectable set of files and records that the pages are generated from.","visible_text_byte_count":1227,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::substrate","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"contains","target":"source_authority","gloss":"Source authority rests in substrate-owned records and tools."},{"verb":"projected_by","target":"plectis","gloss":"Plectis is a public map over source substrate."},{"verb":"contrasts_with","target":"record","gloss":"A record is one inspectable artifact inside or generated from substrate."}],"source_refs":[{"ref":"codex/doctrine/paper_modules/microcosm_substrate.md","role":"paper_module","gloss":"Doctrine roof for the source-open substrate concept."},{"ref":"AGENTS.override.md","role":"agent_seed","gloss":"Type A operating seed that distinguishes live substrate from chat."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":7,"id":"source_authority","label":"source authority","preferred_label":"source authority","alternative_labels":["source of record","governing source"],"hidden_search_labels":["source owner","authority posture"],"term_kind":"governance","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"This local source-of-record relation concerns a specific claim or artifact. It confers no legal, moral, academic, or general institutional authority.","meaning_contrasts":[],"status":"active","disposition":"public_authority_boundary_label","anchor":"glossary-source-authority","relative_href":"docs/glossary.html#glossary-source-authority","public_phrase":"where truth is owned","short_definition":"Source authority identifies the record or implementation controlling a particular question in this system.","reader_preview":"Source authority identifies the record or implementation controlling a particular question in this system.","reader_preview_basis":"authored_override","card_definition":"A registry can own identity, code can define implemented behaviour, and an execution record can document a run. Generated pages select and explain information from these sources; each claim must trace to the appropriate owner.","reader_rule":"Choose the controlling source by the question. A schema-valid record does not override implementation behaviour, and implementation code alone does not settle what happened in a particular run.","deep_definition":"When a source and projection disagree, inspect their derivation and versions before deciding what needs correction. This local rule reconciles system records; it does not make one file or institution authoritative about every subject.","scope_note":"Authority is assigned for a particular fact and traced to its governing source.","example":"The term standard governs required fields; the registry supplies authored wording; the glossary presents that wording.","example_route":{"href":"doctrine.html#dcard-ap-13","label":"Generated-result record source inversion","gloss":"open the anti-pattern that fires when a generated record is mistaken for the source that owns the fact"},"non_example":"A generated summary does not become the controlling source merely by presenting a fact; the governing contract must identify the owner for that particular question.","visible_text_byte_count":1010,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::source_authority","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"governs","target":"record","gloss":"Records point to the authority that owns their claim."},{"verb":"bounds","target":"plectis","gloss":"Plectis public pages stay below source authority."},{"verb":"requires","target":"scope_limit","gloss":"Authority and scope must be named together for public claims."}],"source_refs":[{"ref":"codex/standards/std_system_term.json","role":"standard","gloss":"Defines system vocabulary rows as projections below cited source authority."},{"ref":"tools/meta/dissemination/build_microcosm_public_site.py","role":"public_projection_builder","gloss":"Emits source refs and authority boundaries into public packets."},{"ref":"codex/standards/std_validation_artifact_ref.json","role":"standard","gloss":"Native standards control meaning when schemas only check shape."}],"evidence_commands":[],"used_in":{"total_occurrences":294,"page_count":85,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":142},{"page":"components.html","count":16},{"page":"paper-modules.html","count":13},{"page":"area-import-drift.html","count":8},{"page":"paper-module-routing-anti-patterns-registry.html","count":8},{"page":"area-architecture.html","count":4}]}},{"schema":"plectis_public_glossary_term_v0","position":8,"id":"organ","label":"organ (source identifier)","preferred_label":"organ (source identifier)","alternative_labels":["source organ","organ id"],"hidden_search_labels":["runtime organ","organ registry"],"term_kind":"compatibility_identifier","meaning_scope_class":"compatibility_identifier","meaning_scope_label":"Compatibility identifier","definition_profile":"governed_system_term","meaning_boundary":"Organ is a retained source/storage compatibility identifier. It is not the preferred public object name, a biological claim, or proof that a storage rename has completed.","meaning_contrasts":[],"status":"active_compatibility","disposition":"source_identifier_exposed_only_when_needed","anchor":"glossary-organ","relative_href":"docs/glossary.html#glossary-organ","public_phrase":"source-level runtime unit","short_definition":"Organ is the source-code term for a bounded unit that the public site usually calls a component.","reader_preview":"Organ is the source-code term for a bounded unit that the public site usually calls a component.","reader_preview_basis":"definition","card_definition":"It may appear in registry names and paths retained by the implementation. Each unit has a particular job, interfaces, and evidence describing its behaviour.","reader_rule":"The biological word is internal terminology; it does not add a separate guarantee or capability.","deep_definition":"When following a component card into source code, an organ identifier can refer to that same unit. The registry and linked source establish the identity and scope more precisely than the choice of label.","scope_note":"The term covers every bounded unit the system is assembled from. A reader meets it in source paths and identifiers such as organ_registry.json, and meets the same units as components throughout the public pages, where each can be opened on its own.","example":"A concrete organ is the Source Projection Import Protocol, a unit that brings source material across into the public slice; it is listed and openable among the components.","example_route":{"href":"components.html#component-macro_projection_import_protocol","label":"Source Projection Import Protocol","gloss":"browse the units, each openable on its own"},"non_example":"An organ is not a biological metaphor and not a marketing feature. It is a unit of running code with a stated boundary.","visible_text_byte_count":1136,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::organ","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"translated_publicly_as","target":"component","gloss":"Component is the preferred reader-facing word."},{"verb":"belongs_to","target":"source_slice","gloss":"Organ ids may appear in the public source slice."},{"verb":"bounded_by","target":"scope_limit","gloss":"Source-level organ records still need claim boundaries."}],"source_refs":[{"ref":"microcosm-substrate/core/organ_registry.json","role":"source_registry","gloss":"Source-level registry that still uses organ ids."},{"ref":"microcosm-substrate/core/organ_atlas.json","role":"source_atlas","gloss":"Source atlas consumed by public component projections."}],"evidence_commands":[{"command":"jq 'keys[0:5]' core/organ_registry.json","reason":"Inspect source-level organ registry ids."}],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":9,"id":"witness","label":"check result","preferred_label":"check result","alternative_labels":["verification record","witness"],"hidden_search_labels":["public witness","proof witness","witness route","witness surface"],"term_kind":"evidence","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"The governed internal row retains the software verification-record meaning. Public text also explains the ordinary mathematical existential witness, which is an object with a proved property rather than a synonym for a log.","meaning_contrasts":[],"status":"active_compatibility","disposition":"prefer_check_result_publicly","anchor":"glossary-witness","relative_href":"docs/glossary.html#glossary-witness","public_phrase":"source-linked check result","short_definition":"A check result records the outcome and evidence from a named software check.","reader_preview":"A check result records the outcome and evidence from a named software check.","reader_preview_basis":"authored_override","card_definition":"Local software uses witness for a verification record tied to a particular execution. Public pages call this a check result: it identifies what was checked, the inputs or source version, the method, and the reported outcome. The record can report failure or incomplete evidence as well as success.","reader_rule":"Interpret a check result within its recorded method and scope. In mathematics, witness has a different meaning: an object satisfying an existence claim.","deep_definition":"For example, 3 witnesses that some positive integer has square 9, with justification that 3 squared equals 9. That object and its proved property differ from a software log. A check record supports only what its producer and method establish; recording success alone does not prove a mathematical claim.","scope_note":"Mathematical witnesses and software check records support different kinds of claim.","example":"On the evidence page, the record of what a passing check examined, such as a source-row check or a validator run, is a witness: it names the command and the rows it inspected and the one claim that result supports.","example_route":{"href":"evidence.html","label":"What the pass proves (evidence)","gloss":"see the records that stand behind a passing check"},"non_example":"A log reporting success is not by itself a proof of a mathematical existence statement or of the whole system's correctness.","visible_text_byte_count":1206,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::witness","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"publicly_prefer","target":"record","gloss":"Verification record is the safer reader-facing object."},{"verb":"uses","target":"evidence_class","gloss":"A check result can carry an evidence class."},{"verb":"bounded_by","target":"scope_limit","gloss":"A witness/check result must not exceed its claim boundary."}],"source_refs":[{"ref":"microcosm-substrate/core/axiom_organ_routing.json","role":"source_routing","gloss":"Axiom support routes use witness-like evidence vocabulary."},{"ref":"tools/meta/dissemination/build_microcosm_public_site.py","role":"public_projection_builder","gloss":"Scrubs public copy toward check results and verification records."},{"ref":"codex/standards/std_operator_action_receipt.json","role":"standard","gloss":"Outcome claims and evidence completeness are independent receipt fields."}],"evidence_commands":[{"command":"jq '.support_cases[0]?' receipts/first_wave/axiom_support_cover/axiom_support_cover_result.json","reason":"Inspect a generated support record when present."}],"used_in":{"total_occurrences":307,"page_count":51,"top_page":"doctrine.html","top_pages":[{"page":"doctrine.html","count":57},{"page":"rules-and-ideas.html","count":36},{"page":"paper-module-formal-math-lean-proof-witness.html","count":25},{"page":"paper-module-bounded-autonomy-campaign-packet.html","count":20},{"page":"paper-modules.html","count":18},{"page":"paper-module-microcosm-axiom-substrate.html","count":15}]}},{"schema":"plectis_public_glossary_term_v0","position":10,"id":"navigable","label":"navigable","preferred_label":"navigable","alternative_labels":["routable","discoverable"],"hidden_search_labels":["routeable","navigability"],"term_kind":"navigation","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"Navigable means a cold reader can discover, select, open, and verify an artifact through supported routes; it is stronger than merely being searchable or linked somewhere.","meaning_contrasts":[],"status":"active","disposition":"public_route_property_label","anchor":"glossary-navigable","relative_href":"docs/glossary.html#glossary-navigable","public_phrase":"has a route","short_definition":"Material is navigable when a reader has usable routes from its summary to the relevant detail or next action.","reader_preview":"Material is navigable when a reader has usable routes from its summary to the relevant detail or next action.","reader_preview_basis":"definition","card_definition":"In Plectis, these routes can be links, anchors, source references, or commands. They help a person or agent move from a term or result to the object being described.","reader_rule":"A route must resolve to the intended material to be useful.","deep_definition":"The presence of a link alone does not ensure good navigation. Its label, destination, access requirements, and level of detail should fit the reader's question.","scope_note":"The property applies to glossary rows, components, records, and pages throughout the site, and is most visible on the architecture map, where each part connects through to the source material it stands for.","example":"The architecture map is a navigable surface: each area and component on it links through to the source files, standards, and checks it represents, so you can move from the overview down to the exact material behind any box.","example_route":{"href":"architecture.html","label":"How it fits together (architecture map)","gloss":"follow a map whose every part routes to its source"},"non_example":"Navigable does not mean important, prominent, or well written. A heading can look significant and still be non-navigable if it offers no anchor, owner, or source reference to follow.","visible_text_byte_count":1184,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::navigable","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"requires","target":"routing","gloss":"Navigable objects need routes."},{"verb":"protects","target":"cold_agent","gloss":"Navigability prevents cold readers from guessing."},{"verb":"uses","target":"record","gloss":"Records make route targets inspectable."}],"source_refs":[{"ref":"codex/doctrine/paper_modules/navigation_hologram_theory.md","role":"paper_module","gloss":"Navigation theory behind routeable, inspectable surfaces."},{"ref":"sites/microcosm/content-graph.json","role":"generated_public_graph","gloss":"Generated public graph carrying routes and anchors."}],"evidence_commands":[],"used_in":{"total_occurrences":6,"page_count":6,"top_page":"coding-agents.html","top_pages":[{"page":"coding-agents.html","count":1},{"page":"doctrine.html","count":1},{"page":"paper-module-first-screen-composition-root.html","count":1},{"page":"paper-module-navigation-hologram-route-plane.html","count":1},{"page":"paper-module-public-reveal-walkthrough.html","count":1},{"page":"rules-and-ideas.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":11,"id":"two_kernel","label":"2-kernel","preferred_label":"2-kernel","alternative_labels":["two-kernel","dyadic kernel"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The 2-kernel is a family of subsequences. It is distinct from a null space, a software core, or Lean’s proof-checking kernel.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-two-kernel","relative_href":"docs/glossary.html#glossary-two-kernel","public_phrase":"2-kernel","short_definition":"The 2-kernel of a sequence is the family of all subsequences \\(n\\mapsto a(2^{e}n+r)\\), with \\(e\\ge 0\\) and \\(0\\le r<2^{e}\\).","reader_preview":"The 2-kernel of a sequence is the family of all subsequences \\(n\\mapsto a(2^{e}n+r)\\), with \\(e\\ge 0\\) and \\(0\\le r<2^{e}\\).","reader_preview_basis":"definition","card_definition":"At \\(e=1\\) it contains the even-indexed and odd-indexed subsequences. Higher levels split by residue classes modulo 4,8,…, and \\(e=0\\) includes the original sequence.","reader_rule":"A finite 2-kernel characterizes a 2-automatic sequence. For integer-valued sequences, 2-regularity instead asks that the integer module generated by this family be finitely generated.","deep_definition":"The #249 paper gives exact bases and ranks after stopping at a finite depth, then describes the full rational span. Unbounded finite-level ranks show that the totient sequence is not 2-regular. This kernel is a family of sequences, distinct from a linear map’s null space or Lean’s proof checker.","scope_note":"A finite 2-kernel characterizes a 2-automatic sequence. For integer-valued sequences, 2-regularity instead asks that the integer module generated by this family be finitely generated.","example":"At level e, the original sequence is split by residue classes modulo \\(2^e\\).","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":890,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::two_kernel","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/249/erdos249-totient-reasoning-surface.tex::line 175","role":"mathematical_source","gloss":"full dyadic totient kernel"},{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/249/erdos-249-binary-totient-series.tex","role":"mathematical_source","gloss":"Primary manuscript context and mathematical application."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":12,"id":"two_regular_sequence","label":"2-regular sequence","preferred_label":"2-regular sequence","alternative_labels":["2-regular","two-regular sequence","k-regular sequence"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the cited paper specifies the parameters and hypotheses for its application.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-two-regular-sequence","relative_href":"docs/glossary.html#glossary-two-regular-sequence","public_phrase":"2-regular sequence","short_definition":"An integer-valued sequence is 2-regular when its dyadic subsequences generate a finitely generated module over the integers.","reader_preview":"An integer-valued sequence is 2-regular when its dyadic subsequences generate a finitely generated module over the integers.","reader_preview_basis":"definition","card_definition":"Take all subsequences \\(n\\mapsto a(2^{e}n+r)\\). Finite generation means finitely many sequences suffice to express every such subsequence as an integer linear combination. The analogous definition uses base \\(k\\ge 2\\).","reader_rule":"This is weaker than having only finitely many distinct dyadic subsequences. A theorem at each finite level does not establish finite generation across all levels.","deep_definition":"The #249 paper computes the ranks of the totient sequence’s finite-level modules and shows that these ranks grow without bound. That proves nonregularity while also describing the exact finite-level structure; it does not settle the irrationality of the totient series value.","scope_note":"This is weaker than having only finitely many distinct dyadic subsequences. A theorem at each finite level does not establish finite generation across all levels.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/249/erdos-249-binary-totient-series.tex.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":978,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::two_regular_sequence","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/249/erdos-249-binary-totient-series.tex","role":"mathematical_source","gloss":"Primary manuscript context and mathematical application."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":13,"id":"microcosm_instance","label":"a microcosm","preferred_label":"a microcosm","alternative_labels":["microcosm of","a microcosm of"],"hidden_search_labels":["microcosmic"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"This row is the ordinary representative-case sense. The capital-M compatibility name of the old public identity is a different row, Microcosm.","meaning_contrasts":["microcosm","guinea_pig","plectis"],"status":"active","disposition":"active","anchor":"glossary-microcosm-instance","relative_href":"docs/glossary.html#glossary-microcosm-instance","public_phrase":"a microcosm","short_definition":"A microcosm is a small example that reflects features of a larger system or situation.","reader_preview":"A microcosm is a small example that reflects features of a larger system or situation.","reader_preview_basis":"definition","card_definition":"Calling a local case a microcosm suggests that it makes a wider pattern easier to see. The comparison is useful only to the extent that the relevant relationships are actually shared.","reader_rule":"A representative example does not automatically establish every claim about the larger whole.","deep_definition":"This ordinary sense is distinct from Microcosm, the project's earlier name. When a component illustrates a general mechanism, its evidence still applies first to the particular case that was built and checked.","scope_note":"Ordinary representative-case usage on the why paragraph. Not the legacy product name.","example":"A single public problem page with failed routes and remaining obligations can be a microcosm of how the whole project wants to be read.","example_route":null,"non_example":"A `.microcosm/` folder or `microcosm-substrate` package id is the compatibility name, not this sense.","visible_text_byte_count":832,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::microcosm_instance","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":4,"page_count":4,"top_page":"paper-module-microcosm-axiom-substrate.html","top_pages":[{"page":"paper-module-microcosm-axiom-substrate.html","count":1},{"page":"paper-module-navigation-hologram-route-plane.html","count":1},{"page":"paper-modules.html","count":1},{"page":"rules-and-ideas.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":14,"id":"above_and_beyond","label":"above and beyond","preferred_label":"above and beyond","alternative_labels":["go above and beyond","going above and beyond"],"hidden_search_labels":["above-and-beyond","beyond the ask","extra mile","go further","overdeliver"],"term_kind":"judgment_trait","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"Inside ai_workflow this means bounded durable follow-through, not overwork, speculative expansion, or permission to exceed the user's scope.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-above-and-beyond","relative_href":"docs/glossary.html#glossary-above-and-beyond","public_phrase":"close the durable loop","short_definition":"In Plectis, “above and beyond” means doing the useful follow-through that makes a completed task easy to find, check, and continue.","reader_preview":"In Plectis, “above and beyond” means doing the useful follow-through that makes a completed task easy to find, check, and continue.","reader_preview_basis":"definition","card_definition":"Finishing a change can leave a broken link, an outdated instruction, or a lesson that exists only in the conversation. This term asks the agent to fix the directly related loose ends while the context is still available.","reader_rule":"The extra work needs a clear stopping point and must fit the user's authorised scope.","deep_definition":"For example, a new feature may need its existing guide updated and its check recorded so the next person can use it.","scope_note":"The term covers how a unit of work is judged complete across the system, and it appears wherever a pass reports what it finished and what it deliberately left for later, most directly on the surfaces that audit whether a pass closed its loop honestly.","example":"The Agent Completion Faithfulness Audit is a concrete instance: it is the check that compares what a pass claims it finished against what it actually closed, which is the boundary that above and beyond stays inside.","example_route":{"href":"paper-modules.html#paper-module-agent-closeout-faithfulness-audit","label":"Agent Completion Faithfulness Audit","gloss":"open the audit that checks whether a pass honestly closed its loop"},"non_example":"Above and beyond is not speculative scope expansion or doing more than was asked for its own sake. It is bounded follow-through that makes one result durable and then stops at a named boundary.","visible_text_byte_count":1343,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::above_and_beyond","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"specializes","target":"type_a_judgment","gloss":"Above-and-beyond is one routeable Type A judgment trait."},{"verb":"balances","target":"passion","gloss":"Vision can justify follow-through only when it stays source-backed and bounded."},{"verb":"balances","target":"care","gloss":"Care turns extra effort into upkeep for future agents and humans."},{"verb":"operates_through","target":"common_sense","gloss":"Common sense chooses the smallest truthful improvement instead of broadening scope reflexively."},{"verb":"uses","target":"up_propagate","gloss":"Reusable above-and-beyond work moves into the owning durable plane."},{"verb":"bounded_by","target":"standard","gloss":"Validation, authority, and projection standards cap the extra work."}],"source_refs":[{"ref":"codex/doctrine/skills/doctrine/local_to_general_propagation.md","role":"skill","gloss":"Plane-home workflow that turns extra effort into bounded durable refinement."},{"ref":"codex/doctrine/skills/kernel/navigation_seed.md","role":"skill","gloss":"Opening ladder that should receive new cold-start rungs when they ship."},{"ref":"codex/doctrine/paper_modules/agent_entry_surfaces.md","role":"paper_module","gloss":"Compressed entry-surface rule that keeps cold-start affordances pointer-shaped."},{"ref":"codex/doctrine/system_vocabulary/term_registry.json::type_a_operating_register","role":"vocabulary","gloss":"Parent operating register for care, passion, common sense, and useful-finding routing."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":15,"id":"absolute_convergence","label":"absolute convergence","preferred_label":"absolute convergence","alternative_labels":["absolutely convergent","converges absolutely"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-absolute-convergence","relative_href":"docs/glossary.html#glossary-absolute-convergence","public_phrase":"absolute convergence","short_definition":"A series converges absolutely if the sum of the absolute values of its terms is finite.","reader_preview":"A series converges absolutely if the sum of the absolute values of its terms is finite.","reader_preview_basis":"definition","card_definition":"For a real or complex series \\(\\sum a_{n}\\), this means \\(\\sum |a_{n}|\\) converges. It implies convergence of the original series and permits rearranging its terms without changing the sum.","reader_rule":"Ordinary convergence does not imply absolute convergence: the alternating harmonic series is a standard counterexample.","deep_definition":"For nonnegative terms, the two conditions coincide. Absolute convergence is useful in the papers when expanding denominators, regrouping contributions, or bounding an omitted tail by the sum of its magnitudes.","scope_note":"Ordinary convergence does not imply absolute convergence: the alternating harmonic series is a standard counterexample.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/68/erdos-68-factorial-denominator-irrationality.tex::line 107.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":943,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::absolute_convergence","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/68/erdos-68-factorial-denominator-irrationality.tex::line 107","role":"mathematical_source","gloss":"Published manuscript context for this mathematical term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":16,"id":"absolute_value","label":"absolute value","preferred_label":"absolute value","alternative_labels":["absolute values","complex modulus"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Real and complex numbers. For vectors and functions the analogous object is usually called a norm.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-absolute-value","relative_href":"docs/glossary.html#glossary-absolute-value","public_phrase":"absolute value","short_definition":"A number's nonnegative size, ignoring sign; for complex numbers it is distance from zero.","reader_preview":"A number's nonnegative size, ignoring sign; for complex numbers it is distance from zero.","reader_preview_basis":"definition","card_definition":"Absolute value measures size without sign. Thus \\(|-3|=3\\), while for the complex number \\(z=3+4i\\), \\(|z|=5\\) because the point \\((3,4)\\) lies five units from the origin. It is zero only at zero.","reader_rule":"The distance between real or complex numbers x and y is \\(|x-y|\\). Products satisfy \\(|xy|=|x||y|\\), and sums satisfy the triangle inequality \\(|x+y|\\le |x|+|y|\\); cancellation can make the left side much smaller.","deep_definition":"In these papers absolute values measure approximation errors, bound nonzero integer expressions from below, and define polynomial regions such as \\(\\{|f(z)|<1\\}\\). For vectors or functions, the related size concept usually needs a separately specified norm.","scope_note":"The distance between real or complex numbers x and y is \\(|x-y|\\). Products satisfy \\(|xy|=|x||y|\\), and sums satisfy the triangle inequality \\(|x+y|\\le |x|+|y|\\); cancellation can make the left side much smaller.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex::line 258.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":951,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::absolute_value","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex::line 258","role":"mathematical_source","gloss":"Published manuscript context for this term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":17,"id":"accessibility","label":"accessibility","preferred_label":"accessibility","alternative_labels":["accessible design","digital accessibility"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Design and implementation that let people with disabilities perceive, operate, and understand a system; general availability or speed alone is insufficient.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-accessibility","relative_href":"docs/glossary.html#glossary-accessibility","public_phrase":"accessibility","short_definition":"Accessibility is the extent to which people, including disabled people, can perceive, understand, navigate, and use information or a system.","reader_preview":"Accessibility is the extent to which people, including disabled people, can perceive, understand, navigate, and use information or a system.","reader_preview_basis":"definition","card_definition":"Accessible design removes barriers such as controls that require a mouse, text with insufficient contrast, or images without useful alternatives. Different people need different ways to access the same content; a clear visual layout alone does not make a page accessible.","reader_rule":"Assess accessibility against specific readers, tasks, technologies, and recognised criteria rather than treating it as a general claim of ease.","deep_definition":"Accessibility concerns whether disabled people can perceive, understand, navigate, and operate a product or document. A feature can help one group while creating a barrier for another, so testing should cover concrete tasks and assistive technologies. Plectis uses keyboard controls, focus management, semantic HTML, and preserved mathematical accessibility data. Accessibility does not imply that specialist subject matter has become simple.","scope_note":"Assess accessibility against specific readers, tasks, technologies, and recognised criteria rather than treating it as a general claim of ease.","example":"A glossary tooltip can be opened by keyboard and dismissed with Escape while retaining focus.","example_route":null,"non_example":"Calling a page accessible because it loads quickly does not establish disability access.","visible_text_byte_count":995,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::accessibility","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/lean/experience/papers/open-source-mathematics-strategy.html","role":"reading_context","gloss":"The accessibility claim is role-bounded. A non-specialist compute contributor is not asked to decide theorem correctness"}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":18,"id":"accountability_layer","label":"accountability layer","preferred_label":"accountability layer","alternative_labels":["evidence accountability layer","claim accountability layer"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"Here, accountability layer names the records and routes around a mechanism that make one claim inspectable. It is not the only accepted use of the phrase, and it does not imply that the wider system is public or independently verified.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-accountability-layer","relative_href":"docs/glossary.html#glossary-accountability-layer","public_phrase":"records that bound claims","short_definition":"The accountability layer is the evidence and source links that let someone check what the system says it did.","reader_preview":"The accountability layer is the evidence and source links that let someone check what the system says it did.","reader_preview_basis":"definition","card_definition":"A visible result should lead to the run that produced it, the inputs used, and the claim that was tested. Plectis groups those connections under this name so a reader can inspect the work behind a summary.","reader_rule":"An evidence record supports the particular action or property it captures.","deep_definition":"Keeping the record and its limits together helps expose a claim that has grown broader than the work supporting it.","scope_note":"The term covers the public records and routes that make one mechanism or claim traceable. 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It includes the empty selection, finite selections, and infinite selections.","reader_rule":"Different selections can give the same value unless additional conditions ensure uniqueness. Its possible shapes include intervals and Cantor sets.","deep_definition":"For the #257 Mersenne weights \\(w_{n}=1/(2^{n}-1)\\), every weight exceeds the sum of its successors. At the first differing selection digit, later terms cannot cancel the difference, so each achieved value determines a unique selection. This does not itself decide which rational targets are achieved by infinite selections.","scope_note":"Different selections can give the same value unless additional conditions ensure uniqueness. Its possible shapes include intervals and Cantor sets.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":932,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::achievement_set","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex","role":"mathematical_source","gloss":"Primary manuscript context and mathematical application."}],"evidence_commands":[],"used_in":{"total_occurrences":1,"page_count":1,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":20,"id":"actor_axes","label":"actor axes","preferred_label":"actor axes","alternative_labels":["actor_axes","Type A/B actor axes"],"hidden_search_labels":["con_016::actor_axes","substrate authority axes","type a type b concept"],"term_kind":"actor","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"These are local actor-classification axes. They are not personality types, intelligence rankings, capability tiers, or a universal taxonomy of agents.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-actor-axes","relative_href":"docs/glossary.html#glossary-actor-axes","public_phrase":"actor classification axes","short_definition":"Actor axes are separate descriptions of an agent's access, abilities, tools, and role in a task.","reader_preview":"Actor axes are separate descriptions of an agent's access, abilities, tools, and role in a task.","reader_preview_basis":"definition","card_definition":"Plectis uses several axes because one ranking would mix different things. Type A and Type B describe access to the working system; capability describes what the actor can do; its role describes how it participates in the work.","reader_rule":"A model's reasoning ability does not determine its file permissions or access to evidence.","deep_definition":"The same model can have different authority in two sessions, depending on the tools and material available to it.","scope_note":"Actor axes appear wherever the system labels who or what is acting, including handoffs between a Type B service and a Type A agent, prompt and route descriptions, and the documentation for working through a coding agent.","example":"A coding agent running with live access to the repository is an actor that sits at the Type A end of the substrate-authority axis, with a high capability tier and a delegated topology role, all distinguished at once by the axes.","example_route":{"href":"coding-agents.html","label":"With a coding agent","gloss":"see a Type A actor, a coding agent with live repository access, in use"},"non_example":"Actor axes are not a single ranking of actors from least to most powerful, and substrate authority is not the same as intelligence or cost. 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It is a local necessary condition for infinitely many simultaneous prime translates, not a sufficiency theorem.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-admissible-prime-tuple","relative_href":"docs/glossary.html#glossary-admissible-prime-tuple","public_phrase":"admissible prime tuple","short_definition":"An admissible prime tuple is a finite set of distinct integer shifts that leaves at least one residue class unoccupied modulo every prime.","reader_preview":"An admissible prime tuple is a finite set of distinct integer shifts that leaves at least one residue class unoccupied modulo every prime.","reader_preview_basis":"definition","card_definition":"For \\(H=\\{h_1,\\ldots,h_k\\}\\), admissibility means \\(\\nu_H(p)<p\\) for every prime \\(p\\), where \\(\\nu_H(p)\\) counts occupied residue classes. 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It differs from an explicit hypothesis of a conditional theorem and from permitted foundational axioms.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-admitted-proof-step","relative_href":"docs/glossary.html#glossary-admitted-proof-step","public_phrase":"admitted proof step","short_definition":"An admitted proof step fills a gap with an unproved assertion so unfinished formal work can continue.","reader_preview":"An admitted proof step fills a gap with an unproved assertion so unfinished formal work can continue.","reader_preview_basis":"authored_override","card_definition":"Lean's sorry placeholder marks a step whose proof has not been supplied. Surrounding development can proceed with a warning, but declarations depending on that step inherit the gap.","reader_rule":"Audit the target declaration's dependencies, not just visible placeholders in its source. A gap can arrive through an imported lemma.","deep_definition":"A stated hypothesis describes a conditional theorem; an admission supplies missing justification inside its development. 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Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-algorithm","relative_href":"docs/glossary.html#glossary-algorithm","public_phrase":"algorithm","short_definition":"An algorithm is an explicit procedure for turning permitted inputs into outputs or decisions.","reader_preview":"An algorithm is an explicit procedure for turning permitted inputs into outputs or decisions.","reader_preview_basis":"definition","card_definition":"Algorithms range from sorting recipes to search and learning procedures. A description should identify valid inputs, steps, stopping conditions, and outputs. Correctness concerns whether the procedure meets its specification; efficiency concerns resources such as time and memory. An algorithm is the method, while one execution is a particular run on particular inputs.","reader_rule":"A named algorithm does not establish that its implementation is correct, efficient, unbiased, or suitable for the present data.","deep_definition":"An algorithm is an abstract procedure, independent of any particular programming language or machine. To assess one, specify its admissible inputs, output relation, sequence of steps, stopping behavior, and resource costs. Correctness asks whether every permitted execution meets the specification; complexity studies how time or space grows with input size. Randomized algorithms also expose a probability model. An implementation can mistranslate a sound algorithm, and a sound method can still be unsuitable for biased data, hostile inputs, or operational constraints.","scope_note":"An algorithm covers the declared input domain, steps, and output contract.","example":"A resolver can use a longest-match algorithm to choose one glossary phrase without overlapping links.","example_route":null,"non_example":"A vague strategy such as “try likely options” is not an algorithm until its steps and stopping condition are specified.","visible_text_byte_count":1012,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::algorithm","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":1,"page_count":1,"top_page":"credit-ledger.html","top_pages":[{"page":"credit-ledger.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":34,"id":"almost_surely","label":"almost surely","preferred_label":"almost surely","alternative_labels":["almost-sure","with probability one","probability one"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The probability-one meaning is intended, equivalent to almost everywhere for a probability measure. It differs from high probability, eventual occurrence on every trajectory, and a universal assertion.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-almost-surely","relative_href":"docs/glossary.html#glossary-almost-surely","public_phrase":"almost surely","short_definition":"A property holds almost surely when the set of outcomes where it fails has probability zero.","reader_preview":"A property holds almost surely when the set of outcomes where it fails has probability zero.","reader_preview_basis":"authored_override","card_definition":"The phrase allows exceptional outcomes; it does not mean the property holds everywhere. On a continuous probability space, even a nonempty or infinite set of outcomes can have probability zero.","reader_rule":"Specify the measure and the property, including any quantifiers. Probability one for each separate event is not automatically a simultaneous assertion over an uncountable family.","deep_definition":"The #257 note's limiting profinite observable is infinite almost surely, although its values at positive integer inputs are finite. Those exceptional inputs do not contradict the measure-theoretic statement or justify a finite integral.","scope_note":"Specify the measure and the property, including any quantifiers. Probability one for each separate event is not automatically a simultaneous assertion over an uncountable family.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/257/erdos257-mersenne-reasoning-surface.tex.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1026,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::almost_surely","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/257/erdos257-mersenne-reasoning-surface.tex","role":"local_manuscript","gloss":"Finite residue averages, Haar probability on profinite integers, independent prime divisibility events, almost-sure divergence, countable covers, and disjoint sets."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":35,"id":"analytic_continuation","label":"analytic continuation","preferred_label":"analytic continuation","alternative_labels":["holomorphic continuation"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Continuation preserves complex analyticity. Different paths can produce different branches, even when continuation is locally unique.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-analytic-continuation","relative_href":"docs/glossary.html#glossary-analytic-continuation","public_phrase":"analytic continuation","short_definition":"Analytic continuation extends a holomorphic function beyond its initial domain while agreeing with the original function wherever both are defined.","reader_preview":"Analytic continuation extends a holomorphic function beyond its initial domain while agreeing with the original function wherever both are defined.","reader_preview_basis":"definition","card_definition":"The geometric series \\(1+z+z^{2}+\\ldots\\) defines \\(1/(1-z)\\) only for \\(|z|<1\\), but the rational function supplies an analytic continuation to every complex z except 1. Agreement on an overlap makes the extension unique on a connected domain.","reader_rule":"Continuation may proceed along paths and can fail at singularities. Different paths can lead to different branches unless a monodromy condition removes the ambiguity. It is more restrictive than any smooth extension because it must remain complex analytic.","deep_definition":"Here an inverse branch is continued across a larger disc. The proof must rule out critical or branch points and escape to infinity; merely writing the same formula outside the starting neighbourhood would not justify continuation.","scope_note":"Continuation may proceed along paths and can fail at singularities. Different paths can lead to different branches unless a monodromy condition removes the ambiguity. It is more restrictive than any smooth extension because it must remain complex analytic.","example":"The geometric series \\(1+z+z^{2}+\\ldots\\) defines \\(1/(1-z)\\) only for \\(|z|<1\\), but the rational function supplies an analytic continuation to every complex z except 1.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1187,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::analytic_continuation","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 1080","role":"mathematical_source","gloss":"The analytic continuation, monodromy, area formula, change of variables and"}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":36,"id":"analytic_function","label":"analytic function","preferred_label":"analytic function","alternative_labels":["real-analytic function","complex-analytic function"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The adjective analytic can also mean reasoning-based rather than computational; this entry is only the mathematical regularity property.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-analytic-function","relative_href":"docs/glossary.html#glossary-analytic-function","public_phrase":"analytic function","short_definition":"A function that near every point is represented by a convergent power series; over the complex numbers this is equivalent to being holomorphic.","reader_preview":"A function that near every point is represented by a convergent power series; over the complex numbers this is equivalent to being holomorphic.","reader_preview_basis":"definition","card_definition":"An analytic function agrees locally with the power series obtained from its derivatives. The exponential is an example. In real analysis, having derivatives of every order is weaker: a smooth function can have a Taylor series that fails to recover the function.","reader_rule":"Specify whether the variables are real or complex: complex analytic means holomorphic, while real analytic is stronger than merely smooth.","deep_definition":"A function is analytic on an open set if every point has a neighbourhood on which the function equals a convergent power series centred at that point. For functions of one complex variable, complex differentiability throughout an open set implies this power-series property, so analytic and holomorphic are equivalent there. For real functions, possessing derivatives of every order is insufficient: smooth functions can have Taylor series that fail to recover the function. References to analytic continuation or analytic inverse branches usually use the complex sense and require a connected domain and compatible local continuations.","scope_note":"Specify whether the variables are real or complex: complex analytic means holomorphic, while real analytic is stronger than merely smooth.","example":"The exponential function is analytic because its power series converges everywhere and equals the function.","example_route":null,"non_example":"The real function \\(\\exp (-1/x^2)\\) for x not equal to zero, extended by zero at \\(x=0\\), is smooth but not analytic at zero.","visible_text_byte_count":1031,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::analytic_function","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":37,"id":"annex","label":"annex","preferred_label":"annex","alternative_labels":["annex note","annex substrate"],"hidden_search_labels":["curated external","adapted substrate","annex family","pattern fuel","annex prior art","annex prior-art pressure","prior art pressure field"],"term_kind":"substrate","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"Here annex means curated external prior art adapted into the local doctrine graph, not merely an appendix attached to a document.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-annex","relative_href":"docs/glossary.html#glossary-annex","public_phrase":"curated external substrate","short_definition":"An annex is Plectis's source-linked account of ideas or techniques drawn from an outside project.","reader_preview":"An annex is Plectis's source-linked account of ideas or techniques drawn from an outside project.","reader_preview_basis":"definition","card_definition":"It records what an external framework, repository, or document offers and how a useful pattern might apply locally. The aim is to make prior work available before an agent invents another implementation.","reader_rule":"The original source remains the authority on what the outside project actually does.","deep_definition":"Adapting a technique requires a separate local decision and check; an annex does not establish that the technique has already been integrated or works here.","scope_note":"Annex covers the layer of curated external references the system draws on for design patterns and prior art. A reader meets it in the import and drift-control area and wherever a component records that its shape was informed by outside work.","example":"The Pattern Binding Contract is a concrete instance: a component that governs how a borrowed external pattern is bound into the system's own form before it is allowed to take effect.","example_route":{"href":"components.html#component-pattern_binding_contract","label":"Pattern Binding Contract","gloss":"open the component that binds a borrowed pattern into local form"},"non_example":"An annex is not a dependency, a fork, or imported source code that the system runs. It is a pattern-level note about outside work, kept separate from the system's own authority.","visible_text_byte_count":1256,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::annex","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"adapted_into","target":"standard","gloss":"Annex patterns can crystallize into local standards."},{"verb":"adapted_into","target":"skill","gloss":"Annex patterns can refine local skills."},{"verb":"contrasts_with","target":"raw_seed","gloss":"Annex is external pattern evidence; raw seed is operator voice."},{"verb":"pressures","target":"failure_class_propagation","gloss":"Annex patterns can help classify whether a local failure implies a broader repair class."}],"source_refs":[{"ref":"codex/doctrine/paper_modules/codex_annex_substrate.md","role":"paper_module","gloss":"Annex substrate subsystem."},{"ref":"codex/doctrine/skills/annex/annex_pattern_transfer.md","role":"skill","gloss":"Translation discipline (adapt-not-adopt)."},{"ref":"codex/standards/annex/std_annex_notes.json","role":"standard","gloss":"Annex-note authority contract."}],"evidence_commands":[],"used_in":{"total_occurrences":2,"page_count":1,"top_page":"component-annex-knowledge-routing.html","top_pages":[{"page":"component-annex-knowledge-routing.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":38,"id":"annihilator_subspace","label":"annihilator of a subspace","preferred_label":"annihilator of a subspace","alternative_labels":["subspace annihilator"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"This is the dual-space linear-algebra annihilator, not an ideal-theoretic annihilator; rationality of a relation requires a separate rational-structure argument.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-annihilator-subspace","relative_href":"docs/glossary.html#glossary-annihilator-subspace","public_phrase":"annihilator of a subspace","short_definition":"The linear functionals that vanish on every vector of a specified subspace.","reader_preview":"The linear functionals that vanish on every vector of a specified subspace.","reader_preview_basis":"authored_override","card_definition":"For a linear subspace U of V, its annihilator \\(U^{0}\\) is \\(\\{\\ell\\in V^*: \\ell(u)=0\\text{ for every }u\\in U\\}\\). It belongs to the dual space of linear functionals. In finite dimensions, \\(\\dim U^{0}=\\dim V-\\dim U\\).","reader_rule":"No inner product is needed. An orthogonal complement lives in the original space after choosing an inner product; identifying it with the annihilator requires that additional identification. Both the ambient space and scalar field must be specified.","deep_definition":"The #269 torus proof studies the rational annihilator of the velocity-containing subspace. If two coordinate forms were dependent there, a rationally defined relation supported on those coordinates would yield an integer relation and contradict the irrational ratio of the corresponding logarithms.","scope_note":"No inner product is needed. An orthogonal complement lives in the original space after choosing an inner product; identifying it with the annihilator requires that additional identification. Both the ambient space and scalar field must be specified.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/269/erdos269-running-lcm-reasoning-surface.tex::line 809.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1074,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::annihilator_subspace","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/269/erdos269-running-lcm-reasoning-surface.tex::line 809","role":"mathematical_source","gloss":"Published manuscript use of annihilator of a subspace with the local hypotheses and conclusion stated in the referenced passage."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":39,"id":"annulus","label":"annulus","preferred_label":"annulus","alternative_labels":["annular region"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Open and closed annuli differ in whether their boundary circles are included. A topological annulus need not have circular boundaries.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-annulus","relative_href":"docs/glossary.html#glossary-annulus","public_phrase":"annulus","short_definition":"An annulus is the region between two concentric circles, usually written \\(\\{z: r<|z-a|<R\\}\\) with \\(0\\le r<R\\).","reader_preview":"An annulus is the region between two concentric circles, usually written \\(\\{z: r<|z-a|<R\\}\\) with \\(0\\le r<R\\).","reader_preview_basis":"definition","card_definition":"The set \\(1<|z|<2\\) is an open annulus. Unlike a disc, it has a hole: a loop going once around the origin cannot be shrunk to a point while remaining in the annulus.","reader_rule":"Open or closed boundary conventions must be stated. In complex analysis, “annulus” often means the open region \\(r<|z-a|<R\\); a topological annulus may mean any space homeomorphic to such a region.","deep_definition":"The displayed inequality defines an open circular annulus; including one or both boundary circles gives a closed or partly closed version. When \\(r=0\\), the centre is still excluded, producing a punctured disc. The word is also used for regions with the same topology. In the paper, cutting an annulus along an arc joining its boundaries changes its topology and can remove the closed loops that carry winding information.","scope_note":"Open or closed boundary conventions must be stated. In complex analysis, “annulus” often means the open region \\(r<|z-a|<R\\); a topological annulus may mean any space homeomorphic to such a region.","example":"The set \\(1<|z|<2\\) is an open annulus.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":850,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::annulus","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::line 752","role":"mathematical_source","gloss":"cut an adjoining regular annulus along a separatrix or regular flow arc before"}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":40,"id":"anti_principle","label":"anti-principle","preferred_label":"anti-principle","alternative_labels":["anti-principles"],"hidden_search_labels":["named shortcut to refuse"],"term_kind":"doctrine","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"An anti-principle is a named shortcut this system refuses. It is not a moral prohibition, a legal ban, or a claim that the named pattern is always wrong outside this lattice.","meaning_contrasts":["principle","axiom"],"status":"active","disposition":"active","anchor":"glossary-anti-principle","relative_href":"docs/glossary.html#glossary-anti-principle","public_phrase":"named refuse shortcut","short_definition":"An anti-principle names a recurring shortcut or failure pattern that the system is meant to avoid.","reader_preview":"An anti-principle names a recurring shortcut or failure pattern that the system is meant to avoid.","reader_preview_basis":"definition","card_definition":"Where a principle describes a desired way of working, an anti-principle makes a specific bad practice recognisable. It connects the refusal to the requirement it protects, so the rule can be applied to real cases.","reader_rule":"The useful question is which concrete behaviour would violate the rule.","deep_definition":"For example, treating a plausible summary as evidence of a completed action would defeat a requirement to check the actual execution record.","scope_note":"Anti-principles appear as their own kind on the doctrine page and as numbered AP cards in the doctrine reference.","example":"AP-9 names the shortcut of writing a live value into durable prose as if it were fixed; AX-10 is the axiom that requires the value, as-of time, basis, and refresh route instead.","example_route":{"href":"doctrine.html#what-doctrine-refuses","label":"What doctrine refuses","gloss":"open the anti-principle cards on the doctrine page"},"non_example":"A vague warning such as \"be careful\" is not an anti-principle. The shortcut and the skipped obligation have to be named.","visible_text_byte_count":1013,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::anti_principle","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"belongs_to","target":"doctrine","gloss":"Anti-principles are a doctrine kind."},{"verb":"guards","target":"principle","gloss":"Each anti-principle names a shortcut a principle would refuse."},{"verb":"guards","target":"axiom","gloss":"Some anti-principles name shortcuts that would skip an axiom obligation."}],"source_refs":[{"ref":"tools/meta/dissemination/microcosm_public_narratives.json","role":"public_projection","gloss":"Doctrine page taxonomy names anti-principles as a governing kind."},{"ref":"codex/doctrine/paper_modules/system_constitution_seed.md","role":"paper_module","gloss":"Root ontology places anti-principles in the doctrine lattice."}],"evidence_commands":[],"used_in":{"total_occurrences":45,"page_count":4,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":22},{"page":"doctrine.html","count":20},{"page":"paper-modules.html","count":2},{"page":"reference.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":41,"id":"apache_2_0","label":"Apache-2.0","preferred_label":"Apache-2.0","alternative_labels":["Apache License 2.0"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"externally_anchored_term","meaning_scope_label":"Externally anchored","definition_profile":"public_contextual_gloss","meaning_boundary":"This registry only explains how the external term appears here. The external owner, specification, or primary source controls when exact meaning matters.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-apache-2-0","relative_href":"docs/glossary.html#glossary-apache-2-0","public_phrase":"Apache-2.0","short_definition":"Apache-2.0 is an open-source licence allowing use, modification, and redistribution, including commercial use, subject to its conditions.","reader_preview":"Apache-2.0 is an open-source licence allowing use, modification, and redistribution, including commercial use, subject to its conditions.","reader_preview_basis":"definition","card_definition":"Redistribution requires a copy of the licence, retention of applicable notices, and prominent notices of changed files. An included NOTICE file can add attribution requirements. The licence also contains a patent grant and its conditions.","reader_rule":"The licence attached to the particular repository or file determines what is covered.","deep_definition":"Apache-2.0 does not grant general trademark rights and provides the work without warranties. The linked licence text controls the exact terms.","scope_note":"Apache-2.0 is published by the Apache Software Foundation; its text is the authority, not this gloss.","example":"The public repositories are released under Apache-2.0.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":888,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::apache_2_0","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":42,"id":"api","label":"API","preferred_label":"API","alternative_labels":["application programming interface","APIs","application-programming interface"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The stated technical meaning; distinct disciplinary senses are identified where the site uses them.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-api","relative_href":"docs/glossary.html#glossary-api","public_phrase":"API","short_definition":"An API is a defined interface through which software requests operations or data from another software component.","reader_preview":"An API is a defined interface through which software requests operations or data from another software component.","reader_preview_basis":"definition","card_definition":"An API specifies available operations, input forms, returned values, and errors so separately built programs can cooperate. It may be a local library interface, an operating-system call, or a remote service.","reader_rule":"The API contract determines how callers interact with an implementation; transport over a network is common but not required.","deep_definition":"An API can expose functions inside one process, communication between local processes, hardware services, or requests to a remote server. Its documentation describes the public contract rather than the provider’s internal implementation. Versions may change accepted inputs or response fields. Authorization, quotas, latency, availability, pricing, and data retention are separate operational conditions. A successful API response proves that the interface returned that result under one request; interpreting or trusting the result still depends on provenance, semantics, validation, and error handling.","scope_note":"The API contract determines how callers interact with an implementation; transport over a network is common but not required.","example":"Representative source: sites/microcosm/docs/components.html.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":853,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::api","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/docs/components.html","role":"reading_context","gloss":"Public context in which readers encounter this term."},{"ref":"sites/microcosm/docs/paper-module-batch12-prediction-market-board-capsule.html","role":"reading_context","gloss":"Public context in which readers encounter this term."}],"evidence_commands":[],"used_in":{"total_occurrences":20,"page_count":14,"top_page":"paper-module-world-model-projection-drift-control-room.html","top_pages":[{"page":"paper-module-world-model-projection-drift-control-room.html","count":3},{"page":"paper-module-batch12-prediction-market-board-capsule.html","count":2},{"page":"paper-module-batch8-audio-level-rms-port.html","count":2},{"page":"paper-module-provider-context-recipe-budget.html","count":2},{"page":"paper-modules.html","count":2},{"page":"area-agent-reliability.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":43,"id":"appeal_lifecycle","label":"appeal lifecycle","preferred_label":"appeal lifecycle","alternative_labels":["appeal_lifecycle","operator court appeal"],"hidden_search_labels":["appeal row","appeal review"],"term_kind":"governance","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"An appeal lifecycle is the system’s internal path for contesting and repairing a recorded judgment; it is not a legal appeal process or an informal objection left in chat.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-appeal-lifecycle","relative_href":"docs/glossary.html#glossary-appeal-lifecycle","public_phrase":"judgment challenge lifecycle","short_definition":"An appeal lifecycle is the process for challenging a system decision and recording how the challenge was resolved.","reader_preview":"An appeal lifecycle is the process for challenging a system decision and recording how the challenge was resolved.","reader_preview_basis":"definition","card_definition":"A disputed label or result is examined, its cause is identified, and responsibility for any repair is assigned. 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A reader meets it wherever a label or decision can be formally disputed and tracked to resolution.","example":"The operator-court appeal ledger is a concrete appeal lifecycle in use: entries record judgments that operators have challenged, the classified cause, and the rationale that closes each one, fed by the Cognitive Operator Registry that holds the operator contract.","example_route":{"href":"components.html#component-cognitive_operator_registry","label":"Cognitive Operator Registry","gloss":"open the component that records and validates operator-raised challenges"},"non_example":"An appeal lifecycle is not a legal court process and not a record of a row that errored and was discarded. 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It is one defined level of a depth ladder, with a contract for what that level must contain.","visible_text_byte_count":1324,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::band","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"declared_by","target":"compression_profile","gloss":"A compression profile declares its legal bands and band contracts."},{"verb":"projects","target":"affordance_row","gloss":"Bands are how an affordance row shows itself at different costs."},{"verb":"used_by","target":"hologram","gloss":"Hologram nodes render rows through a requested band."}],"source_refs":[{"ref":"codex/doctrine/paper_modules/navigation_hologram_theory.md","role":"paper_module","gloss":"Bands-as-schemas refinement."},{"ref":"codex/standards/std_system_term.json","role":"standard","gloss":"System-term band contract."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":65,"id":"barrier","label":"barrier","preferred_label":"barrier","alternative_labels":["obstruction"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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In these papers, such results help identify which attempted routes have failed and what a future approach would have to change.","reader_rule":"Check the hypotheses and the precise class of approaches ruled out.","deep_definition":"A method failing does not show that the original problem is impossible. A barrier can be useful progress by preventing repetition and revealing which assumption caused the obstruction.","scope_note":"A barrier establishes that one route fails. 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In the irrationality papers, expressions involving a rational base can be studied as powers or series directly; conclusions about integer divisibility cannot be transferred without checking the assumptions.","scope_note":"Base covers the explicitly named numeral or exponential base in one series, expansion, or theorem.","example":"In base two, \\(0.11\\) represents \\(1/2 + 1/4\\); at a rational base such as \\(3/2\\), power-based series need the separate assumptions stated in the paper.","example_route":null,"non_example":"Base here does not mean a database, code superclass, or physical support.","visible_text_byte_count":975,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::base","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":5,"page_count":3,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":3},{"page":"introduction.html","count":1},{"page":"updates.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":67,"id":"basis","label":"basis","preferred_label":"basis","alternative_labels":["rational basis","linear basis","linear bases","vector-space basis"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Covers both the linear-algebra sense and the ordinary evidential sense. In linear algebra, a basis is a linearly independent spanning family; in an argument, a basis is the assumptions or evidence on which it rests. named_basis is the separate Plectis artifact that records selected premises for a bounded task.","meaning_contrasts":["named_basis"],"status":"active","disposition":"active","anchor":"glossary-basis","relative_href":"docs/glossary.html#glossary-basis","public_phrase":"basis","short_definition":"In linear algebra, a basis gives unique vector coordinates; in an argument, its basis is the assumptions or evidence supporting it.","reader_preview":"In linear algebra, a basis gives unique vector coordinates; in an argument, its basis is the assumptions or evidence supporting it.","reader_preview_basis":"definition","card_definition":"A vector-space basis both spans the space and is linearly independent: every vector is a unique finite sum of basis vectors multiplied by scalars. One horizontal and one vertical unit vector form a basis for the ordinary plane.","reader_rule":"The scalars and vector space are part of the mathematical definition; evidence supporting an argument does not have to form a vector space.","deep_definition":"Plectis uses “named basis” for the stated sources or assumptions behind a claim. Naming those sources makes them inspectable; it does not by itself show that they support the conclusion.","scope_note":"Covers the vector-space meaning and distinguishes the evidential use of the same word.","example":"The vectors \\((1,0)\\) and (0,1) form a basis of the plane.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1053,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::basis","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":201,"page_count":19,"top_page":"doctrine.html","top_pages":[{"page":"doctrine.html","count":154},{"page":"rules-and-ideas.html","count":18},{"page":"papers.html","count":8},{"page":"paper-module-microcosm-axiom-substrate.html","count":3},{"page":"paper-module-batch12-release-claim-language-gate.html","count":2},{"page":"paper-module-doctrine-fact-claim-audit.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":68,"id":"benchmark","label":"benchmark","preferred_label":"benchmark","alternative_labels":["evaluation benchmark","benchmark run"],"hidden_search_labels":["benchmark result"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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Its result describes performance under those conditions; it does not automatically rank systems for workloads the benchmark did not test.","reader_rule":"A comparison must identify the workload, scoring rule, environment, and any deliberately varied conditions; conclusions remain limited to what those choices represent.","deep_definition":"Benchmarks measure properties such as speed, accuracy, reliability, resource use, or cost on a defined workload. Fair comparison fixes the relevant conditions or explicitly identifies what is being varied, including hardware, scoring, workloads, and treatment of failures. A single score can hide trade-offs, distribution shifts, or poor tail behaviour. Repeated optimization against a public test may also make the score less representative because systems adapt to its examples. In finance, benchmark can instead mean a reference index or rate; the shared idea is comparison with a declared standard, not a universal measure of quality.","scope_note":"A benchmark covers the declared tasks, data, metric, model or system version, and execution conditions.","example":"A premise-retrieval benchmark can compare which relevant lemmas different search methods return for the same queries.","example_route":null,"non_example":"A single impressive demo or hand-picked example is not a benchmark.","visible_text_byte_count":900,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::benchmark","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":476,"page_count":79,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":64},{"page":"paper-modules.html","count":60},{"page":"components.html","count":34},{"page":"paper-module-agent-benchmark-integrity-anti-gaming-replay.html","count":26},{"page":"paper-module-engine-room-navigation-fitness-benchmark.html","count":15},{"page":"area-agent-reliability.html","count":14}]}},{"schema":"plectis_public_glossary_term_v0","position":69,"id":"bergman_kernel","label":"Bergman kernel","preferred_label":"Bergman kernel","alternative_labels":["Bergman reproducing kernel"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The paper uses the unweighted unit-disc Bergman kernel with ordinary area normalization; positivity of arbitrary off-diagonal values is not claimed.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-bergman-kernel","relative_href":"docs/glossary.html#glossary-bergman-kernel","public_phrase":"Bergman kernel","short_definition":"The kernel that reproduces square-integrable holomorphic functions by integration over their domain.","reader_preview":"The kernel that reproduces square-integrable holomorphic functions by integration over their domain.","reader_preview_basis":"authored_override","card_definition":"For the unit disc with ordinary area measure, \\(K(z,w)=1/[\\pi (1-z\\cdot \\overline{w})^{2}]\\). If f is holomorphic and square-integrable there, \\(f(z)=\\int K(z,w)f(w)dA(w)\\). The kernel is holomorphic in z and antiholomorphic in w.","reader_rule":"The normalization depends on the measure. Positive-definite kernel does not mean every kernel value is a positive real number. On the real diameter of the disc, this particular kernel is positive, which permits a special comparison.","deep_definition":"In #1041 the coefficient expansion of a Riemann map and Cauchy–Schwarz turn component area into a bound on the length of the image of a real interval. The positive disc kernel makes the resulting double integral explicitly computable.","scope_note":"The normalization depends on the measure. Positive-definite kernel does not mean every kernel value is a positive real number. On the real diameter of the disc, this particular kernel is positive, which permits a special comparison.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 719.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1066,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::bergman_kernel","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 719","role":"mathematical_source","gloss":"Published manuscript use of Bergman kernel with the local hypotheses and conclusion stated in the referenced passage."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":70,"id":"bernstein_basis","label":"Bernstein basis","preferred_label":"Bernstein basis","alternative_labels":["Bernstein polynomial basis","Bernstein basis polynomials"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The positivity implication requires the Bernstein representation on the specified interval; ordinary power coefficients have no equivalent implication.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-bernstein-basis","relative_href":"docs/glossary.html#glossary-bernstein-basis","public_phrase":"Bernstein basis","short_definition":"The degree-n polynomial basis \\(\\binom{n}{k}t^{k}(1-t)^{n-k}\\) for k from 0 to n.","reader_preview":"The degree-n polynomial basis \\(\\binom{n}{k}t^{k}(1-t)^{n-k}\\) for k from 0 to n.","reader_preview_basis":"authored_override","card_definition":"Each basis function is nonnegative on \\([0,1]\\), and the basis functions sum to one. Consequently a polynomial expressed in this basis lies between its smallest and largest Bernstein coefficients on that interval.","reader_rule":"Nonnegative coefficients are a sufficient positivity certificate, not a necessary condition in a fixed degree. Basis positivity holds on the stated interval; it cannot be used unchanged outside \\([0,1]\\). Bernstein approximation polynomials are a related construction, not this basis itself.","deep_definition":"The #1041 counterexample verifies inequalities by rewriting degree-fourteen polynomials along line segments in Bernstein form. Exact rational coefficients certify the whole interval, rather than only values at finitely many sampled points.","scope_note":"Nonnegative coefficients are a sufficient positivity certificate, not a necessary condition in a fixed degree. Basis positivity holds on the stated interval; it cannot be used unchanged outside \\([0,1]\\). Bernstein approximation polynomials are a related construction, not this basis itself.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 421.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1122,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::bernstein_basis","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 421","role":"mathematical_source","gloss":"Published manuscript use of Bernstein basis with the local hypotheses and conclusion stated in the referenced passage."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":71,"id":"bertrand_postulate","label":"Bertrand postulate","preferred_label":"Bertrand postulate","alternative_labels":["Bertrand's postulate","Bertrand’s postulate","Bertrand theorem"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the reader bands identify the ambient field, index set, convergence requirement, or theorem hypotheses relevant to the cited manuscript.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-bertrand-postulate","relative_href":"docs/glossary.html#glossary-bertrand-postulate","public_phrase":"Bertrand postulate","short_definition":"Bertrand’s postulate states that for every integer \\(n>1\\) there is a prime strictly between n and 2n.","reader_preview":"Bertrand’s postulate states that for every integer \\(n>1\\) there is a prime strictly between n and 2n.","reader_preview_basis":"definition","card_definition":"It guarantees a prime in a fixed multiplicative interval, without identifying which prime. For instance, at \\(n=3\\) the prime 5 lies between 3 and 6.","reader_rule":"Endpoint conventions and the small cases matter when reformulating the theorem. It gives no claim that every short interval of an arbitrary fixed length contains a prime.","deep_definition":"The #68 finite-gcd argument needs a prime ℓ with \\(D/2<\\ell \\le D\\). Bertrand supplies it for \\(D\\ge 3\\) after treating the integer endpoints; \\(D=2\\) is handled directly. This interval ensures a unique largest factorial channel contributes the relevant prime valuation, ruling out a nontrivial common divisor.","scope_note":"Endpoint conventions and the small cases matter when reformulating the theorem. 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Any parameters on which C may depend must be stated, often by a subscript such as \\(O_{\\varepsilon}\\).","deep_definition":"Big-O lets the papers separate a main arithmetic contribution from controlled errors or bound the number of exceptional indices. It hides constants and a finite initial range, so it cannot by itself certify a numerical inequality at a particular input.","scope_note":"The statement \\(f=O(g)\\) near a specified limit means \\(|f|\\le C|g|\\) there for some fixed C after a threshold. 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A claim covering every possible input requires a different argument or a suitable proof.","scope_note":"A claim establishes nothing by itself. 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This gives reviewers a way to check that the explanation says what the mathematics actually establishes.","reader_rule":"Registration records how a claim is presented and supported; it is not peer review.","deep_definition":"When the wording or proof changes, their connection must be checked again. 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Its accepted logical assumptions should remain explicit.","deep_definition":"The local Lean sources use classical blocks to make classical instances available. This permits standard mathematical reasoning while axiom audits report the resulting dependencies; the source keyword alone does not list every assumption of a theorem.","scope_note":"A classical proof can establish existence without yielding an executable procedure for the object. 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The same program can be a client in one exchange and a server in another, and a client may present a command line, library API, or graphical interface.","reader_rule":"Identify the service and protocol when calling something a client; the label alone does not state its permissions or trust.","deep_definition":"A client constructs requests, sends them through a protocol, interprets responses, and handles failures such as timeouts or rate limits. It may cache data or retry, subject to the service contract. Plectis component prose distinguishes a provider client from a theorem prover or local runner: the client communicates with a provider but does not supply the provider’s guarantees. Client input and server output both need validation at their boundaries.","scope_note":"Identify the service and protocol when calling something a client; the label alone does not state its permissions or trust.","example":"A browser acts as an HTTP client when requesting a page from a web server.","example_route":null,"non_example":"A standalone program doing local arithmetic is not acting as a client unless it requests another service.","visible_text_byte_count":781,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::client","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/docs/paper-module-formal-math-verifier-trace-repair-loop.html","role":"reading_context","gloss":"not a theorem prover, a Lean or Lake runner, a provider client, or a private import"}],"evidence_commands":[],"used_in":{"total_occurrences":1,"page_count":1,"top_page":"paper-module-formal-math-verifier-trace-repair-loop.html","top_pages":[{"page":"paper-module-formal-math-verifier-trace-repair-loop.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":109,"id":"clone","label":"clone","preferred_label":"clone","alternative_labels":["clones","cloned"],"hidden_search_labels":["cloning"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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The record should make clear what would have to change, and whether the closure rests on proof, exact calculation, or weaker evidence.","scope_note":"Covers failed approaches recorded in the papers and working records.","example":"For #249, the positive rank-one Möbius–Mersenne mechanism is a closed route: its cone sits a uniform gap above the target value.","example_route":null,"non_example":"An approach abandoned for lack of time is not a closed route, because nothing was shown.","visible_text_byte_count":844,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::closed_route","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":5,"page_count":3,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":3},{"page":"contact.html","count":1},{"page":"introduction.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":111,"id":"closed_set","label":"closed set","preferred_label":"closed set","alternative_labels":["closed subset","closed region"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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The result covers the configured jobs and their environment.","reader_rule":"A green CI result means the configured checks passed for that revision; it is not proof of correctness, security, deployability, or review.","deep_definition":"Continuous integration combines small repository changes frequently and runs declared automation against each selected revision. Typical jobs compile code, execute tests, check formatting, scan dependencies, or build documentation in a controlled environment. The practice detects integration faults early and leaves reviewable logs, but its authority ends at configured checks, fixtures, platforms, and permissions. A green result establishes only what the completed checks justify; a skipped job or an untested requirement is not covered by the colour of the overall status. Reproducibility also depends on pinned dependencies and a faithfully described runner environment.","scope_note":"Describes the practice; the specific service used is a separate question.","example":"A check in the CI floor runs on every commit to the public repository.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":953,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::continuous_integration","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":157,"id":"contradiction","label":"contradiction","preferred_label":"contradiction","alternative_labels":["proof by contradiction"],"hidden_search_labels":["contradict","contradicts"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-contradiction","relative_href":"docs/glossary.html#glossary-contradiction","public_phrase":"contradiction","short_definition":"A contradiction is an impossibility obtained from assumptions, such as a statement and its negation both being true.","reader_preview":"A contradiction is an impossibility obtained from assumptions, such as a statement and its negation both being true.","reader_preview_basis":"definition","card_definition":"In proof by contradiction, one assumes the opposite of the desired conclusion and derives such an impossibility. Classical reasoning then rules out that opposite assumption.","reader_rule":"Each inference must follow from the assumptions actually available at that point.","deep_definition":"For example, an irrationality argument may assume a number equals a fraction in lowest terms, then show both numerator and denominator must share a factor. That conflicts with the lowest-terms assumption.","scope_note":"A contradiction rejects the assumptions used to derive it within the logic and hypotheses of that proof.","example":"Assuming a sum is rational may force an integer strictly between zero and one, yielding a contradiction.","example_route":null,"non_example":"Two surprising or tension-filled statements are not a contradiction unless they cannot both be true.","visible_text_byte_count":867,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::contradiction","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":34,"page_count":10,"top_page":"paper-module-metabolism-queue-reconciliation.html","top_pages":[{"page":"paper-module-metabolism-queue-reconciliation.html","count":12},{"page":"paper-module-engine-room-metabolism-runtime.html","count":7},{"page":"paper-module-prediction-oracle-reconciliation.html","count":4},{"page":"paper-modules.html","count":3},{"page":"rules-and-ideas.html","count":3},{"page":"component-metabolism-queue-reconciliation.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":158,"id":"contrapositive","label":"contrapositive","preferred_label":"contrapositive","alternative_labels":["proof by contrapositive","contrapositive form"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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Either version expresses the same implication through a different route.","reader_rule":"The converse, “if Q, then P”, is a different statement and need not hold.","deep_definition":"The papers often prove that rationality would force a property, then establish failure of that property to deduce irrationality. The reasoning must negate the exact consequence, with its quantifiers intact.","scope_note":"The converse, “if Q, then P”, is a different statement and need not hold.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/251/erdos-251-prime-gap-dyadic-series.tex::line 790.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":890,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::contrapositive","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/251/erdos-251-prime-gap-dyadic-series.tex::line 790","role":"mathematical_source","gloss":"Published manuscript context for this mathematical term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":159,"id":"contribute","label":"contribute","preferred_label":"contribute","alternative_labels":["contribution","contributions"],"hidden_search_labels":["contributes","contributed","contributing","how to contribute"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Contribute keeps its ordinary meaning: to give work that can be accepted. Credit for accepted work is a separate record; see contributor credit.","meaning_contrasts":["contributor_credit","mining"],"status":"active","disposition":"active","anchor":"glossary-contribute","relative_href":"docs/glossary.html#glossary-contribute","public_phrase":"contribute","short_definition":"To contribute is to supply work that helps a shared project advance or correct itself.","reader_preview":"To contribute is to supply work that helps a shared project advance or correct itself.","reader_preview_basis":"definition","card_definition":"Here useful contributions include proofs, counterexamples, clearer explanations, software fixes, and reproducible reports of failed approaches. The public repositories provide concrete material against which a contribution can be inspected.","reader_rule":"Acceptance and attribution should follow the work and evidence, not the confidence of its presentation.","deep_definition":"A mathematical result remains attributable to the people who established it. Using Plectis as a tool or starting point does not transfer priority for their result to the project.","scope_note":"Names the act of giving work. 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Describing the contribution makes the credit more informative than simply listing a name.","reader_rule":"Human authorship and the use of AI or other tools should be recorded distinctly.","deep_definition":"The project's contribution record is not a substitute for scholarly priority or publication conventions. A result's authors retain credit for establishing it, including when this system helped them work.","scope_note":"Covers acknowledgement for accepted contributions to the two public repositories, including downstream results they lead to.","example":"A correction to a stated hypothesis, sent as an issue, is credited to its author in the contribution record.","example_route":null,"non_example":"A polite reply from a reader is not a review and earns no credit line.","visible_text_byte_count":909,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::contributor_credit","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":41,"page_count":6,"top_page":"credit-ledger.html","top_pages":[{"page":"credit-ledger.html","count":24},{"page":"paper-module-agent-memory-temporal-conflict-replay.html","count":5},{"page":"contact.html","count":4},{"page":"introduction.html","count":3},{"page":"paper-module-engine-room-lean-proof-search-lab.html","count":3},{"page":"papers.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":161,"id":"convergence","label":"convergence","preferred_label":"convergence","alternative_labels":["series convergence"],"hidden_search_labels":["convergent","converges"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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For a series, apply that condition to the sums of the first n terms.","reader_rule":"Terms tending to zero are necessary for a series to converge, but are not sufficient.","deep_definition":"For example, \\(1 + 1/2 + 1/3 + \\cdots\\) diverges. Proving convergence also leaves separate questions about whether the sum is rational, algebraic, or transcendental.","scope_note":"Convergence applies to one sequence or series under the stated ordering and notion of distance.","example":"A reciprocal series may converge because its denominators grow fast enough, while its sum's irrationality remains open.","example_route":null,"non_example":"Terms tending to zero are necessary but not sufficient for a series to converge.","visible_text_byte_count":861,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::convergence","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":3,"page_count":2,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":2},{"page":"paper-module-work-landing-control-spine.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":162,"id":"convex_combination","label":"convex combination","preferred_label":"convex combination","alternative_labels":["weighted average"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Convex geometry and analysis. 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This entry uses the lower-envelope convention: the greatest convex function lying below the given function. It is distinct from the convex hull of a set.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-convex-envelope","relative_href":"docs/glossary.html#glossary-convex-envelope","public_phrase":"convex envelope","short_definition":"The convex envelope of a function is the greatest convex function that lies everywhere at or below the original function.","reader_preview":"The convex envelope of a function is the greatest convex function that lies everywhere at or below the original function.","reader_preview_basis":"definition","card_definition":"On the interval \\([-1,1]\\), the function \\(f(x)=-x^{2}\\) bends downward and is not convex. Its convex envelope is the constant function \\(-1\\): that constant stays below f, and convexity prevents any convex function below f from rising above the chord joining the two endpoint values.","reader_rule":"State the convention and domain. Here “convex envelope” means the lower convex envelope of a function. It is not the convex hull of a set, although the envelope can be constructed from convex combinations and is closely related to the convex hull of the function’s epigraph.","deep_definition":"On a convex domain, take the pointwise supremum of all convex functions g satisfying \\(g\\le f\\). Equivalently, minimize weighted averages of f over finite convex combinations representing each point. If no finite convex minorant exists, the envelope may take \\(-\\infty\\) under an extended-value convention. A closed convex envelope additionally requires lower semicontinuity; that condition can change boundary values. The function construction is related to convex hulls through the epigraph, the set of points on or above a graph.","scope_note":"This is a construction on functions; the convex hull is the corresponding construction on sets.","example":"For \\(f(x)=-x^{2}\\) on \\([-1,1]\\), the lower convex envelope is the constant function \\(-1\\).","example_route":null,"non_example":"The filled triangle containing three points is their convex hull, not a convex envelope of a function.","visible_text_byte_count":993,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::convex_envelope","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"contrasts_with","target":"convex_hull","gloss":"An envelope is a function; a hull is the smallest convex set containing a set."},{"verb":"uses","target":"convexity","gloss":"The envelope is required to be convex."}],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 89","role":"mathematical_context","gloss":"The manuscript discusses an extremal envelope in the neighboring collinear-root argument; it does not use convex envelope as a synonym for convex hull."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":164,"id":"convex_hull","label":"convex hull","preferred_label":"convex hull","alternative_labels":[],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The hull includes the filled-in region, not just its boundary. 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The convex hull contains points, not merely the polygonal boundary; repeated or interior generating points do not change it.","deep_definition":"In the polynomial paper, a point is located relative to the convex hull of several partial sums. That geometric containment encodes every possible convex combination of those sums and gives stronger information than membership in their union.","scope_note":"A set is convex when the segment between any two of its points remains inside. 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This permits an indexed sum over the cover without asserting that the covered support itself is finite or that an optimal cover has been computed.","scope_note":"Countability bounds cardinality, not density, size of individual values, or existence of an efficient enumeration algorithm.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/257/erdos257-mersenne-reasoning-surface.tex.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":942,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::countable_set","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/257/erdos257-mersenne-reasoning-surface.tex","role":"local_manuscript","gloss":"Finite residue averages, Haar probability on profinite integers, independent prime divisibility events, almost-sure divergence, countable covers, and disjoint sets."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":169,"id":"counterexample","label":"counterexample","preferred_label":"counterexample","alternative_labels":["counterexamples"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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Naming the exact statement they refute preserves that distinction and explains what a revised approach must avoid.","scope_note":"A counterexample refutes the exact universal statement whose hypotheses it satisfies; it does not establish a broader negative claim or settle nearby formulations with different hypotheses.","example":"The prime 2 refutes the statement that every prime is odd; it does not refute the statement that every prime greater than 2 is odd.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":942,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::counterexample","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":9,"page_count":2,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":8},{"page":"introduction.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":170,"id":"coverage","label":"coverage","preferred_label":"coverage","alternative_labels":["covered set","coverage receipt"],"hidden_search_labels":["coverage count","coverage fraction"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Coverage measures a specified part of a stated inventory or space. The system records inventories and matching methods; test coverage, corpus coverage, and proof-obligation coverage remain distinct measurements.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-coverage","relative_href":"docs/glossary.html#glossary-coverage","public_phrase":"coverage","short_definition":"Coverage is the measured portion of a declared population, requirement set, behavior space, or source surface that has been examined or represented.","reader_preview":"Coverage is the measured portion of a declared population, requirement set, behavior space, or source surface that has been examined or represented.","reader_preview_basis":"definition","card_definition":"The denominator and matching method determine what the number means. Test coverage can report executed statements or branches; corpus coverage can report linked records; proof coverage can report which stated obligations have formal evidence. None of these automatically establishes correctness of covered items.","reader_rule":"Ask what total set was declared, how membership and matches were determined, when measurement occurred, and which uncovered cases remain.","deep_definition":"Coverage converts a bounded inventory into a numerator, denominator, and method. High test coverage can coexist with weak assertions or missing behaviors. Complete proof coverage may mean every declared obligation has a proof while the specification itself omits an important requirement. Source or documentation coverage can be exact within a known catalogue yet silent about objects outside it. Comparisons are meaningful only when inventories and methods align. A useful receipt therefore preserves the authority for the denominator, exclusions, measurement version, and concrete uncovered set.","scope_note":"Coverage is always relative to one declared inventory or behavior space, method, and measurement time.","example":"A report states that 92 of 100 declared routes were exercised and names the eight uncovered routes.","example_route":null,"non_example":"A high percentage does not prove covered items correct or show completeness beyond the declared denominator.","visible_text_byte_count":1096,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::coverage","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"codex/standards/std_compliance_coverage.json","role":"source_authority","gloss":"Grounds this Plectis-specific definition and its scope."},{"ref":"microcosm-substrate/src/microcosm_core/organs/self_ignorance_coverage_ledger.py","role":"source_authority","gloss":"Grounds this Plectis-specific definition and its scope."}],"evidence_commands":[],"used_in":{"total_occurrences":303,"page_count":56,"top_page":"doctrine.html","top_pages":[{"page":"doctrine.html","count":44},{"page":"rules-and-ideas.html","count":37},{"page":"paper-module-self-ignorance-coverage-ledger.html","count":34},{"page":"paper-modules.html","count":18},{"page":"paper-module-standards-meta-diagnostics.html","count":12},{"page":"component-self-ignorance-coverage-ledger.html","count":10}]}},{"schema":"plectis_public_glossary_term_v0","position":171,"id":"covering_map","label":"covering map","preferred_label":"covering map","alternative_labels":["finite cover","covering projection"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"A covering is a global surjective map with evenly covered neighbourhoods; branch points must be excluded in the polynomial use.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-covering-map","relative_href":"docs/glossary.html#glossary-covering-map","public_phrase":"covering map","short_definition":"A map whose target has neighbourhoods that lift to separate copies, each mapped homeomorphically onto that neighbourhood.","reader_preview":"A map whose target has neighbourhoods that lift to separate copies, each mapped homeomorphically onto that neighbourhood.","reader_preview_basis":"authored_override","card_definition":"A covering map is a continuous surjection with this evenly covered neighbourhood property. 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Local invertibility alone does not establish a covering: the neighbourhood must work simultaneously for every preimage.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::res:component-local-covering.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1013,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::covering_map","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::res:component-local-covering","role":"mathematical_source","gloss":"Published paper context for the definition and its stated boundary."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":172,"id":"credential","label":"credential","preferred_label":"credential","alternative_labels":["credentials","login credential","professional credential"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Evidence used to establish identity, authority, or qualification; a public account name alone is not a secret authentication credential.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-credential","relative_href":"docs/glossary.html#glossary-credential","public_phrase":"credential","short_definition":"A credential is evidence used to support a claim about identity, qualification, role, or authority.","reader_preview":"A credential is evidence used to support a claim about identity, qualification, role, or authority.","reader_preview_basis":"definition","card_definition":"In computing, credentials include passwords, keys, tokens, and certificates used during authentication. 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A reader meets it in the doctrine and agent-reliability material, and sees its results in the recorded rules and anti-patterns that stabilized criticism produces.","example":"The Routing Anti Patterns Registry is a concrete result of critic posture: a recorded set of failure modes, each one a criticism that held up under testing and was written down as a durable rule rather than left as a passing objection.","example_route":{"href":"components.html#component-routing_anti_patterns_registry","label":"Routing Anti Patterns Registry","gloss":"open the recorded failure modes that stabilized criticism produces"},"non_example":"Critic posture is not a separate reviewer role that must be spawned, and it is not a judgment that the whole system is flawed. 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In several real variables, all first partial derivatives must vanish together.","reader_rule":"A critical point is an input where the derivative vanishes; its image is a critical value. Neither need be a root, and distinct critical points can share one critical value.","deep_definition":"Some elementary conventions also include points of nondifferentiability. The polynomial setting here is differentiable everywhere, and critical points matter because the Newton expression \\(-p/p'\\) is undefined there.","scope_note":"Here the term covers critical points of complex polynomials and differentiable real-valued functions. Some broader conventions also count points where a derivative does not exist; that convention is not needed for the polynomial setting.","example":"For \\(p(z)=z^{2}-1\\), \\(p'(z)=2z\\), so 0 is the unique critical point even though the roots are 1 and \\(-1\\).","example_route":null,"non_example":"A root of a function need not be a critical point: \\(p(1)=0\\) for \\(p(z)=z^{2}-1\\), but \\(p'(1)=2\\).","visible_text_byte_count":1127,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::critical_point","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::line 455","role":"mathematical_source","gloss":"Published manuscript context for this mathematical term."}],"evidence_commands":[],"used_in":{"total_occurrences":1,"page_count":1,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":175,"id":"critical_value","label":"critical value","preferred_label":"critical value","alternative_labels":[],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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Distinct critical points can share a critical value, and different values can have the same magnitude.","scope_note":"The modulus \\(|p(c)|\\) is the magnitude of a critical value, not the value itself.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::line 151.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":818,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::critical_value","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::line 151","role":"mathematical_source","gloss":"Published manuscript context for this mathematical term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":176,"id":"css","label":"CSS","preferred_label":"CSS","alternative_labels":["Cascading Style Sheets","CSS stylesheet","CSS stylesheets"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The style language that controls presentation of structured documents, distinct from HTML structure and JavaScript behavior.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-css","relative_href":"docs/glossary.html#glossary-css","public_phrase":"CSS","short_definition":"CSS is a language for describing how structured documents are presented.","reader_preview":"CSS is a language for describing how structured documents are presented.","reader_preview_basis":"definition","card_definition":"CSS rules select parts of an HTML document and assign visual or layout properties such as colour, spacing, type, position, and responsive behaviour. The document remains HTML; CSS changes its presentation.","reader_rule":"CSS controls presentation, including layout and interaction cues such as visible keyboard focus.","deep_definition":"A browser combines the document tree with matching CSS rules to calculate its presentation. When rules compete, the cascade resolves their origin, importance, layer, specificity, and order; some properties are inherited from a parent. 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Density depends on the surrounding topology; changing the space can change the verdict.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/269/erdos-269-three-prime-running-lcm.tex::line 110.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":905,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::dense_subset","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/269/erdos-269-three-prime-running-lcm.tex::line 110","role":"mathematical_source","gloss":"Published manuscript context for this term."}],"evidence_commands":[],"used_in":{"total_occurrences":3,"page_count":2,"top_page":"paper-module-macro-projection-import-protocol.html","top_pages":[{"page":"paper-module-macro-projection-import-protocol.html","count":2},{"page":"paper-module-engine-room-annex-knowledge-router.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":191,"id":"dependency","label":"dependency","preferred_label":"dependency","alternative_labels":["dependencies","dependencies declared"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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A final answer alone may hide the assumption or transformation on which it depends.","scope_note":"The term covers proof calculations, formula transformations, and source-to-projection computation chains.","example":"A generated page has a derivation from registry data through the deterministic site builder.","example_route":null,"non_example":"A result asserted without intermediate rules or a reproducible procedure is not an inspectable derivation.","visible_text_byte_count":867,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::derivation","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":108,"page_count":58,"top_page":"doctrine.html","top_pages":[{"page":"doctrine.html","count":31},{"page":"rules-and-ideas.html","count":15},{"page":"paper-module-batch5-authority-systems-capsule.html","count":3},{"page":"paper-module-target-shape-tactic-routing.html","count":2},{"page":"paper-module-world-model-projection-drift-control-room.html","count":2},{"page":"paper-modules.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":195,"id":"derivation_certificate","label":"derivation certificate","preferred_label":"derivation certificate","alternative_labels":["derivation certificates"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"public_contextual_gloss","meaning_boundary":"A derivation certificate is the finite checkable artifact that lets an independent checker recompute a claim from its named basis. 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It also differs from reproducibility, which asks whether another execution environment can reconstruct the conditions closely enough to obtain the recorded result.","scope_note":"Deterministic applies to the named process under its declared inputs, versions, and environment.","example":"Generating the glossary twice from the same registry and builder revision should produce byte-identical output.","example_route":null,"non_example":"A process is not deterministic merely because two observed runs happened to agree.","visible_text_byte_count":885,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::deterministic","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":158,"page_count":53,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":24},{"page":"paper-modules.html","count":16},{"page":"paper-module-batch7-oracle-sibling-capsule.html","count":8},{"page":"paper-module-batch7-macro-engines-capsule.html","count":7},{"page":"paper-module-batch8-policy-engines-capsule.html","count":7},{"page":"paper-module-batch8-audio-level-rms-port.html","count":5}]}},{"schema":"plectis_public_glossary_term_v0","position":200,"id":"diebold_mariano_test","label":"Diebold-Mariano test","preferred_label":"Diebold-Mariano test","alternative_labels":["diebold-mariano","diebold mariano"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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Statistical significance neither identifies the cause of a difference nor guarantees that the better-performing method will remain better on future data.","scope_note":"Named for Francis Diebold and Roberto Mariano; the test is standard in forecast evaluation and is not a contribution of this work.","example":"Two forecasts of the same series can be compared by the size of their squared errors.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":948,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::diebold_mariano_test","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":12,"page_count":4,"top_page":"paper-module-finance-forecast-evaluation-spine.html","top_pages":[{"page":"paper-module-finance-forecast-evaluation-spine.html","count":7},{"page":"paper-modules.html","count":2},{"page":"rules-and-ideas.html","count":2},{"page":"component-finance-forecast-evaluation-spine.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":201,"id":"differentiable_function","label":"differentiable function","preferred_label":"differentiable function","alternative_labels":["differentiable","differentiability"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"This entry concerns mathematical differentiability, not whether two data records can be distinguished.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-differentiable-function","relative_href":"docs/glossary.html#glossary-differentiable-function","public_phrase":"differentiable function","short_definition":"A function with a linear first-order approximation at each point under discussion.","reader_preview":"A function with a linear first-order approximation at each point under discussion.","reader_preview_basis":"definition","card_definition":"The derivative gives a linear approximation whose error becomes negligible compared with the change in input. Squaring is differentiable everywhere on the reals; absolute value is continuous at zero but has no derivative there because its two one-sided slopes disagree.","reader_rule":"In several variables use the full linear approximation, not merely the existence of all partial derivatives; in complex analysis complex differentiability is substantially stronger than real differentiability.","deep_definition":"For a real function of one variable, differentiability at x means that the difference quotient approaches a finite limit. In several real variables, differentiability means there is a linear map that approximates the increment with an error small compared with the increment's size. For a curve \\(z(t)\\), its derivative is its instantaneous velocity. For complex functions, the same quotient must have one limit independent of the direction of approach; differentiability on an open set then leads to holomorphicity. 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Plectis uses hashes to connect generated material and result records to particular file contents.","reader_rule":"A mismatch proves the contents differ; a matching secure hash provides strong practical evidence of equality.","deep_definition":"Different inputs can theoretically share a digest, so it is not a mathematical guarantee of identical bytes. 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The identity element is 1 at \\(n=1\\) and 0 elsewhere.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex::line 110.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":976,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::dirichlet_convolution","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex::line 110","role":"mathematical_source","gloss":"Published manuscript context for this term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":206,"id":"dirichlet_energy","label":"Dirichlet energy","preferred_label":"Dirichlet energy","alternative_labels":[],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Finite Dirichlet energy bounds an integral of squared derivatives. 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Each output uses a finite sum, so the coefficient operation itself needs no convergence assumption.","reader_rule":"For sequences indexed by all integers, the corresponding sum can be infinite and needs convergence or finite-support hypotheses. Dirichlet convolution instead combines multiplicative divisor indices.","deep_definition":"The #1049 moment weights involve two-fold and three-fold convolutions of the sequence \\(1/(q;q)_n\\). The finite sums expose positivity and let the proof compare the leading constant-sequence contribution with its correction. They arise because multiplying generating functions convolves their coefficients.","scope_note":"For sequences indexed by all integers, the corresponding sum can be infinite and needs convergence or finite-support hypotheses. Dirichlet convolution instead combines multiplicative divisor indices.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex::line 416.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1062,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::discrete_convolution","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex::line 416","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":209,"id":"discrete_mathematics","label":"discrete mathematics","preferred_label":"discrete mathematics","alternative_labels":["discrete math"],"hidden_search_labels":["discrete structures"],"term_kind":"subject","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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The discriminant of an order or chosen basis can differ from the field discriminant.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-discriminant","relative_href":"docs/glossary.html#glossary-discriminant","public_phrase":"discriminant","short_definition":"A discriminant detects repeated roots or arithmetic degeneracy: a polynomial’s discriminant vanishes exactly when it has a repeated root, while a number field’s discriminant records how its integral arithmetic behaves modulo primes.","reader_preview":"A discriminant detects repeated roots or arithmetic degeneracy: a polynomial’s discriminant vanishes exactly when it has a repeated root, while a number field’s discriminant records how its integral arithmetic behaves modulo primes.","reader_preview_basis":"definition","card_definition":"For a quadratic \\(ax^{2}+bx+c\\) with \\(a\\ne 0\\), the discriminant is \\(b^{2}-4ac\\). For a number field, it is the determinant of the trace-pairing matrix of an integral basis. A rational prime ramifies in the number field exactly when it divides the field discriminant.","reader_rule":"Polynomial and number-field discriminants are related but are not identical objects. Their values depend on conventions and, for an order or basis discriminant, on the chosen integral data. Always identify which discriminant is meant.","deep_definition":"For a degree-n polynomial with leading coefficient a and roots \\(r_{1},\\ldots ,r_{n}\\), its discriminant is \\(a^{2n-2}\\) times the product of \\((r_{i}-r_{j})^{2}\\) over \\(i<j\\). A basis or order discriminant need not equal the field discriminant; an index factor can introduce additional prime divisors. The cited argument excludes the relevant discriminant primes to ensure the reduction and Frobenius steps meet their hypotheses.","scope_note":"Polynomial and number-field discriminants are related but are not identical objects. Their values depend on conventions and, for an order or basis discriminant, on the chosen integral data. Always identify which discriminant is meant.","example":"For \\(ax^{2}+bx+c\\), the discriminant is \\(b^{2}-4ac\\).","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1183,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::discriminant","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex::line 285","role":"mathematical_source","gloss":"primes dividing denominators or the discriminant"}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":211,"id":"disjoint_sets","label":"disjoint sets","preferred_label":"disjoint sets","alternative_labels":["disjoint set","pairwise disjoint sets","disjoint subsets"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The set-theoretic meaning is intended, including pairwise disjoint families. It does not mean probabilistic independence, noninteraction between systems, or merely distinct set names.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-disjoint-sets","relative_href":"docs/glossary.html#glossary-disjoint-sets","public_phrase":"disjoint sets","short_definition":"Two sets are disjoint when they have no common element, equivalently when their intersection is empty.","reader_preview":"Two sets are disjoint when they have no common element, equivalently when their intersection is empty.","reader_preview_basis":"authored_override","card_definition":"A family is pairwise disjoint if every two distinct members are disjoint. This is stronger than saying that the intersection of all members is empty: several sets can overlap in pairs while sharing no single element collectively.","reader_rule":"Distinguish disjointness from independence. Probability masses add for disjoint events, while independence concerns products of probabilities for intersections.","deep_definition":"The #257 constructions use disjoint sets of prime coordinates and disjoint events within a fixed parent group. Each kind of separation supports a different counting or probability step, so the set family being compared must be identified.","scope_note":"Distinguish disjointness from independence. Probability masses add for disjoint events, while independence concerns products of probabilities for intersections.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/257/erdos257-mersenne-reasoning-surface.tex.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1040,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::disjoint_sets","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/257/erdos257-mersenne-reasoning-surface.tex","role":"local_manuscript","gloss":"Finite residue averages, Haar probability on profinite integers, independent prime divisibility events, almost-sure divergence, countable covers, and disjoint sets."}],"evidence_commands":[],"used_in":{"total_occurrences":1,"page_count":1,"top_page":"paper-module-batch5-authority-systems-capsule.html","top_pages":[{"page":"paper-module-batch5-authority-systems-capsule.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":212,"id":"distance","label":"distance","preferred_label":"distance","alternative_labels":["metric distance"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Metric geometry. 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A problem may use a different metric, such as hyperbolic distance, so always identify which distance is intended.","deep_definition":"The lemniscate paper compares Euclidean separation, path length, and hyperbolic distance when selecting root connections. Two metrics can rank or scale pairs differently even on the same set; numerical bounds cannot be transferred without a proved comparison.","scope_note":"Distances obey \\(d(x,z)\\le d(x,y)+d(y,z)\\): traveling through y cannot beat the direct distance. A problem may use a different metric, such as hyperbolic distance, so always identify which distance is intended.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::line 192.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":942,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::distance","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::line 192","role":"mathematical_source","gloss":"Published manuscript context for this term."}],"evidence_commands":[],"used_in":{"total_occurrences":6,"page_count":3,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":3},{"page":"paper-module-batch8-validator-checker-capsule.html","count":2},{"page":"paper-module-prediction-oracle-reconciliation.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":213,"id":"divergence","label":"divergence","preferred_label":"divergence","alternative_labels":["divergent behavior"],"hidden_search_labels":["diverges","divergent","divergent series"],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-divergence","relative_href":"docs/glossary.html#glossary-divergence","public_phrase":"divergence","short_definition":"A sequence or series diverges when it does not converge in the sense being considered.","reader_preview":"A sequence or series diverges when it does not converge in the sense being considered.","reader_preview_basis":"definition","card_definition":"For real sequences, failure to approach a finite limit can mean growing without bound or oscillating indefinitely. A series diverges when its sequence of partial sums fails to converge.","reader_rule":"Individual terms tending to zero do not ensure that their sum converges.","deep_definition":"The harmonic series is a standard example: its terms shrink to zero while its partial sums grow without bound. Papers should specify whether “diverges” means unbounded growth or another failure of convergence.","scope_note":"Divergence covers the exact sequence, series, parameter, and convergence notion named by the statement.","example":"If partial sums grow without bound, the series diverges in the ordinary sense.","example_route":null,"non_example":"A slowly converging sequence is not divergent merely because a finite computation has not reached its limit.","visible_text_byte_count":833,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::divergence","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":4,"page_count":4,"top_page":"paper-module-batch8-tools-tail-primitives-capsule.html","top_pages":[{"page":"paper-module-batch8-tools-tail-primitives-capsule.html","count":1},{"page":"paper-module-lean-proof-search-lab-runtime.html","count":1},{"page":"paper-module-self-ignorance-coverage-ledger.html","count":1},{"page":"papers.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":214,"id":"divisibility","label":"divisibility","preferred_label":"divisibility","alternative_labels":["divides","divisor relation"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-divisibility","relative_href":"docs/glossary.html#glossary-divisibility","public_phrase":"divisibility","short_definition":"Divisibility is the relation in which one integer is an exact integer multiple of another.","reader_preview":"Divisibility is the relation in which one integer is an exact integer multiple of another.","reader_preview_basis":"definition","card_definition":"The notation \\(a \\mid b\\) means that \\(b = ak\\) for some integer k. For example, 3 divides 12 because \\(12 = 3\\cdot 4\\), while 5 does not divide 12.","reader_rule":"The direction matters: \\(a \\mid b\\) does not generally imply \\(b\\mid a\\).","deep_definition":"With this algebraic definition, every integer divides zero and zero divides only zero. The papers often impose positivity to exclude such cases, then use divisibility to track denominator cancellation, congruences, and prime factors.","scope_note":"The direction matters: \\(a \\mid b\\) does not generally imply \\(b\\mid a\\).","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/68/erdos-68-factorial-denominator-irrationality.tex::line 306.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":825,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::divisibility","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/68/erdos-68-factorial-denominator-irrationality.tex::line 306","role":"mathematical_source","gloss":"Published manuscript context for this mathematical term."}],"evidence_commands":[],"used_in":{"total_occurrences":6,"page_count":4,"top_page":"paper-module-finite-erdos-denominator-certificate-strike.html","top_pages":[{"page":"paper-module-finite-erdos-denominator-certificate-strike.html","count":3},{"page":"doctrine.html","count":1},{"page":"paper-module-batch7-secondary-runtime-capsule.html","count":1},{"page":"papers.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":215,"id":"divisor","label":"divisor","preferred_label":"divisor","alternative_labels":["factor of an integer"],"hidden_search_labels":["divisors"],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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With the stated algebraic definition, every integer divides zero, and zero divides only zero; additional positivity hypotheses often exclude these edge cases.","scope_note":"Divisor covers the integer relation and conventions stated in the surrounding theorem or definition.","example":"3 is a divisor of 12 because \\(12 = 3 \\times 4\\).","example_route":null,"non_example":"5 is not a divisor of 12 because no integer multiplied by 5 equals 12.","visible_text_byte_count":739,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::divisor","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":3,"page_count":2,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":2},{"page":"paper-module-batch8-audio-level-rms-port.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":216,"id":"divisor_transform","label":"divisor transform","preferred_label":"divisor transform","alternative_labels":["divisor-sum transform"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The convention used here sums an arithmetic function over the positive divisors of n. Other normalizations need their own formula.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-divisor-transform","relative_href":"docs/glossary.html#glossary-divisor-transform","public_phrase":"divisor transform","short_definition":"The divisor transform sends an arithmetic function a to \\(b(n)=\\sum _{d\\mid n}a(d)\\), summing its values over every positive divisor of n.","reader_preview":"The divisor transform sends an arithmetic function a to \\(b(n)=\\sum _{d\\mid n}a(d)\\), summing its values over every positive divisor of n.","reader_preview_basis":"definition","card_definition":"If a is the indicator of a selected set, then \\(b(n)\\) counts selected divisors of n. Möbius inversion recovers a from b by \\(a(n)=\\sum _{d\\mid n}\\mu (d)b(n/d)\\).","reader_rule":"Some authors use “divisor transform” for closely related normalizations, so the displayed formula should accompany the name. The sum is finite for every positive integer n and concerns divisors, not a Fourier transform.","deep_definition":"The paper transforms one Boolean selector into its coefficient sequence. This lets inclusion choices be encoded by divisor arithmetic and later recovered through Möbius inversion.","scope_note":"Some authors use “divisor transform” for closely related normalizations, so the displayed formula should accompany the name. The sum is finite for every positive integer n and concerns divisors, not a Fourier transform.","example":"If a is the indicator of a selected set, then \\(b(n)\\) counts selected divisors of n.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":965,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::divisor_transform","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex::line 110","role":"mathematical_source","gloss":"is the divisor transform"}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":217,"id":"divisor_function","label":"divisor-counting function","preferred_label":"divisor-counting function","alternative_labels":["divisor function","number-of-divisors function","divisor-counting function τ"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the reader bands identify the ambient field, index set, convergence requirement, or theorem hypotheses relevant to the cited manuscript.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-divisor-function","relative_href":"docs/glossary.html#glossary-divisor-function","public_phrase":"divisor-counting function","short_definition":"The divisor-counting function \\(\\tau (n)\\), also written \\(d(n)\\), counts the positive divisors of a positive integer n.","reader_preview":"The divisor-counting function \\(\\tau (n)\\), also written \\(d(n)\\), counts the positive divisors of a positive integer n.","reader_preview_basis":"definition","card_definition":"For example, \\(\\tau (12)=6\\) because 1,2,3,4,6,12 divide twelve. If \\(n=\\prod p_i^{a_i}\\), then \\(\\tau (n)=\\prod (a_{i}+1)\\).","reader_rule":"This row uses the counting function, not the sum-of-divisors function \\(\\sigma(n)\\). The positive-integer domain avoids the infinitely many divisors of zero.","deep_definition":"The #257 paper counts selected divisors with \\(c_A(n)\\). Since these form a subset of all positive divisors, \\(0\\le c_A(n)\\le \\tau (n)\\). This bound controls the coefficients after expanding the Mersenne-denominator series; it does not assert that every divisor belongs to the chosen support A.","scope_note":"This row uses the counting function, not the sum-of-divisors function \\(\\sigma(n)\\). The positive-integer domain avoids the infinitely many divisors of zero.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex::eq:incidence.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":993,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::divisor_function","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex::eq:incidence","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":218,"id":"doctrine","label":"doctrine","preferred_label":"doctrine","alternative_labels":["doctrine layer","principles concepts mechanisms","curated meaning"],"hidden_search_labels":[],"term_kind":"doctrine","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"Doctrine names this repository's curated principle-concept-mechanism layer. 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The local singular shape and a justified analytic remainder are needed for coefficient estimates; identifying one pole is not enough when competing singularities have not been excluded.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex::line 2230.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1238,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::dominant_singularity","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex::line 2230","role":"mathematical_source","gloss":"Published manuscript use of dominant singularity with the local hypotheses and conclusion stated in the referenced passage."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":223,"id":"dominated_convergence_theorem","label":"dominated convergence theorem","preferred_label":"dominated convergence theorem","alternative_labels":["dominated convergence","Lebesgue dominated convergence theorem"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage. 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No generally accepted external category, whole-system coverage, or equivalence with an unpublished system is claimed.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-executable-cross-section","relative_href":"docs/glossary.html#glossary-executable-cross-section","public_phrase":"runnable bounded public slice","short_definition":"An executable cross-section is a selected part of a larger system that can be run and inspected on its own.","reader_preview":"An executable cross-section is a selected part of a larger system that can be run and inspected on its own.","reader_preview_basis":"definition","card_definition":"Plectis publishes components with the source, inputs, checks, and records needed to examine their behaviour. 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It may finish successfully, stop partway, remain active, or complete with unwanted side effects. Logs and receipts can show which instructions ran, while outputs and state checks show what changed. A successful process exit establishes only the success condition defined by that process; it does not automatically prove semantic correctness, publication, deployment, or acceptance by another system.","scope_note":"The term covers command, program, workflow, test, and proof-checking runs.","example":"Executing the builder reads source data and writes generated public files.","example_route":null,"non_example":"Displaying a command in a code block does not execute it.","visible_text_byte_count":736,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::execution","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":461,"page_count":104,"top_page":"paper-modules.html","top_pages":[{"page":"paper-modules.html","count":64},{"page":"rules-and-ideas.html","count":61},{"page":"paper-module-verifier-lab-execution-spine.html","count":29},{"page":"paper-module-agent-sandbox-policy-escape-replay.html","count":19},{"page":"components.html","count":14},{"page":"paper-module-target-shape-tactic-routing.html","count":14}]}},{"schema":"plectis_public_glossary_term_v0","position":254,"id":"existential_quantifier","label":"existential quantifier","preferred_label":"existential quantifier","alternative_labels":["existential quantification"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The mathematical sense stated here; related software or ordinary-language uses are distinguished where relevant.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-existential-quantifier","relative_href":"docs/glossary.html#glossary-existential-quantifier","public_phrase":"existential quantifier","short_definition":"An existential quantifier states that at least one element of a specified domain satisfies a property.","reader_preview":"An existential quantifier states that at least one element of a specified domain satisfies a property.","reader_preview_basis":"definition","card_definition":"The symbol \\(\\exists\\) means “there exists.” To establish \\(\\exists n \\in \\mathbb{N}, n^{2} = 4\\), the number 2 is a witness. “At least one” does not mean exactly one: a separate uniqueness claim is needed to exclude other witnesses.","reader_rule":"A witness must satisfy every required condition; changing the domain can change whether a witness exists.","deep_definition":"In \\(\\forall x \\exists y\\), the choice of y may depend on x. 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Lean statements preserve this distinction, which can make a conditional or uniform claim substantially stronger.","scope_note":"A witness must satisfy every required condition; changing the domain can change whether a witness exists.","example":"Representative source: governed vocabulary term_registry.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":858,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::existential_quantifier","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"codex/doctrine/system_vocabulary/term_registry.json::terms[id=witness]","role":"reading_context","gloss":"Prerequisite vocabulary used by this public definition."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":255,"id":"exit_code","label":"exit code","preferred_label":"exit code","alternative_labels":["exit status"],"hidden_search_labels":["exit 1"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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Its direction differs from the rising factorial, so the notation and convention matter.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/synthesis/optimal-sparse-perturbations.tex::capacity:sec:rational-jets.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":945,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::falling_factorial","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/synthesis/optimal-sparse-perturbations.tex::capacity:sec:rational-jets","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":272,"id":"falsifiable","label":"falsifiable","preferred_label":"falsifiable","alternative_labels":["falsifiability","falsify"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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It does not mean the claim is false, easily tested, or scientifically important, and a failed test may also expose faulty assumptions or measurement.","reader_rule":"Falsifiable does not mean false; it means the claim states what evidence would conflict with it rather than accommodating every possible outcome.","deep_definition":"A falsifiable claim exposes at least one possible observation, experiment, counterexample, or result that would count against it under stated assumptions. This does not mean the claim is false, likely false, easy to test, or scientifically valuable. A negative result may challenge an auxiliary assumption, measurement method, or experimental setup instead of the central proposition, so interpretation still requires care. Conversely, repeatedly accommodating every outcome without changing the claim removes empirical content. Good statements identify both the conflicting evidence and the background conditions needed for that conflict to be meaningful.","scope_note":"Ordinary usage from philosophy of science, applied here to software claims.","example":"A component's claim is falsifiable because you can run its check and compare.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":895,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::falsifiable","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":24,"page_count":9,"top_page":"paper-modules.html","top_pages":[{"page":"paper-modules.html","count":5},{"page":"rules-and-ideas.html","count":5},{"page":"paper-module-agent-monitor-redteam-falsification-replay.html","count":4},{"page":"paper-module-annex-knowledge-routing.html","count":4},{"page":"paper-module-standards-meta-diagnostics.html","count":2},{"page":"paper-module-bridge-campaign-dag-validation.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":273,"id":"farey_sequence","label":"Farey sequence","preferred_label":"Farey sequence","alternative_labels":["Farey window"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-farey-sequence","relative_href":"docs/glossary.html#glossary-farey-sequence","public_phrase":"Farey sequence","short_definition":"The Farey sequence of order N lists all reduced fractions between 0 and 1 whose positive denominators are at most N.","reader_preview":"The Farey sequence of order N lists all reduced fractions between 0 and 1 whose positive denominators are at most N.","reader_preview_basis":"definition","card_definition":"The fractions appear in increasing order, including \\(0/1\\) and \\(1/1\\). For example, order 3 gives \\(0/1\\), \\(1/3\\), \\(1/2\\), \\(2/3\\), \\(1/1\\).","reader_rule":"“Reduced” means that numerator and denominator have no common factor greater than 1.","deep_definition":"The denominator bound makes the sequence finite. Its neighbouring fractions have useful arithmetic relationships, making Farey sequences a way to organise questions about rational approximation and the gaps between fractions.","scope_note":"The order N fixes the positive-denominator bound and the interval is \\([0,1]\\). Finite exclusion arguments based on these fractions need their own stated connection to the target real number.","example":"The Farey sequence of order three is \\(0/1\\), \\(1/3\\), \\(1/2\\), \\(2/3\\), \\(1/1\\).","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":903,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::farey_sequence","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":274,"id":"feedback_loop","label":"feedback loop","preferred_label":"feedback loop","alternative_labels":["feedback loops"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":["circuit_break","endogenous","exogenous"],"status":"active","disposition":"active","anchor":"glossary-feedback-loop","relative_href":"docs/glossary.html#glossary-feedback-loop","public_phrase":"feedback loop","short_definition":"A feedback loop occurs when a process's output influences its later behaviour.","reader_preview":"A feedback loop occurs when a process's output influences its later behaviour.","reader_preview_basis":"definition","card_definition":"Feedback can reinforce a change, counteract it, or create more complicated dynamics. In software, test results can guide revisions; in research, an interpretation can shape which evidence is sought next.","reader_rule":"Feedback is not inherently beneficial or self-confirming; its effect depends on what is measured and how it is used.","deep_definition":"Plectis records evidence and invites external criticism to help distinguish correction from a loop that merely repeats the assumptions of its own earlier outputs.","scope_note":"Names a self-feeding process. Endogenous says the feed is from inside; circuit-break names the interruption.","example":"Drafting with a model, then asking the same model whether the draft is important, is a feedback loop.","example_route":null,"non_example":"A stranger reading the public record and reporting an error is not another turn of that loop.","visible_text_byte_count":822,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::feedback_loop","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":3,"page_count":1,"top_page":"paper-module-formal-math-verifier-trace-repair-loop.html","top_pages":[{"page":"paper-module-formal-math-verifier-trace-repair-loop.html","count":3}]}},{"schema":"plectis_public_glossary_term_v0","position":275,"id":"field_extension","label":"field extension","preferred_label":"field extension","alternative_labels":["extension field"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The notation L/K names a field containing another field, not a quotient obtained by division. 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Properties such as finite, algebraic, normal, separable, and Galois are always relative to the chosen base K.","deep_definition":"The paper chooses one finite extension of Q containing several algebraic quantities so that all of their symmetries and reductions can be handled together.","scope_note":"The notation \\(L/K\\) names containment, not a quotient field. Properties such as finite, algebraic, normal, separable, and Galois are always relative to the chosen base K.","example":"The complex numbers form an extension of the real numbers, and \\(Q(\\sqrt 2)\\) is the smallest field containing Q and \\(\\sqrt 2\\).","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1019,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::field_extension","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex::line 280","role":"mathematical_source","gloss":"extension $L/\\Q$"}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":276,"id":"field_of_fractions","label":"field of fractions","preferred_label":"field of fractions","alternative_labels":["fraction field"],"hidden_search_labels":["Frac"],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"This construction applies to an integral domain. 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For a prime ideal p, \\(R/p\\) is an integral domain, so \\(\\operatorname{Frac}(R/p)\\) exists even when \\(R/p\\) is not already a field.","example":"The field of fractions of Z is Q.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":970,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::field_of_fractions","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex::line 287","role":"prerequisite_dependency","gloss":"For a prime ideal p, κ(p)=Frac(R/p)."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":277,"id":"field_trace","label":"field trace","preferred_label":"field trace","alternative_labels":["trace of a field element","field traces"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the reader bands identify the ambient field, index set, convergence requirement, or theorem hypotheses relevant to the cited manuscript.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-field-trace","relative_href":"docs/glossary.html#glossary-field-trace","public_phrase":"field trace","short_definition":"The field trace of an element in a finite field extension is the trace of the linear map that multiplies by that element.","reader_preview":"The field trace of an element in a finite field extension is the trace of the linear map that multiplies by that element.","reader_preview_basis":"definition","card_definition":"For \\(L/K\\), multiplication by \\(a\\in L\\) is a K-linear map \\(L\\to L\\). Its matrix trace is \\(\\operatorname{Tr}_{L/K}(a)\\), independent of the chosen basis. In a separable extension it also equals the sum over all K-embeddings into an algebraic closure.","reader_rule":"The ambient extension must be specified. Summing distinct roots of an element’s minimal polynomial can omit multiplicities when the element generates a proper subfield.","deep_definition":"The #243 proof computes \\(\\operatorname{Tr}_{K/\\mathbb{Q}}\\) of \\(\\alpha\\), z, and \\(z^{-1}\\) in a degree-three number field. Newton identities turn these traces into equations for the coefficients of z’s minimal polynomial. The trace here is an algebraic scalar, separate from an execution trace in system documentation.","scope_note":"The ambient extension must be specified. Summing distinct roots of an element’s minimal polynomial can omit multiplicities when the element generates a proper subfield.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::sec:cubic-trace.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1072,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::field_trace","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::sec:cubic-trace","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":278,"id":"finite","label":"finite","preferred_label":"finite","alternative_labels":["finitely many"],"hidden_search_labels":["finite set","finite sequence"],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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Extending that result to all instances requires a general argument, unless a theorem has already reduced the problem to those particular cases.","scope_note":"Finite covers the collection, process, or bound named in the statement.","example":"A computation checking all denominators up to a fixed bound is finite.","example_route":null,"non_example":"A billion-element set is still finite despite being too large for convenient manual inspection.","visible_text_byte_count":740,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::finite","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":121,"page_count":19,"top_page":"doctrine.html","top_pages":[{"page":"doctrine.html","count":24},{"page":"paper-module-finite-erdos-denominator-certificate-strike.html","count":20},{"page":"papers.html","count":18},{"page":"rules-and-ideas.html","count":14},{"page":"component-finite-erdos-denominator-certificate-strike.html","count":11},{"page":"paper-modules.html","count":8}]}},{"schema":"plectis_public_glossary_term_v0","position":279,"id":"finite_blaschke_product","label":"finite Blaschke product","preferred_label":"finite Blaschke product","alternative_labels":["finite Blaschke products"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Only finite products with zeros strictly inside the disc are defined here; degree need not equal the full polynomial degree.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-finite-blaschke-product","relative_href":"docs/glossary.html#glossary-finite-blaschke-product","public_phrase":"finite Blaschke product","short_definition":"A finite product of disc automorphism factors, multiplied by a complex constant of modulus one.","reader_preview":"A finite product of disc automorphism factors, multiplied by a complex constant of modulus one.","reader_preview_basis":"authored_override","card_definition":"It has the form \\(B(z)=e^{i\\theta }\\prod (z-a_j)/(1-\\overline{a_j}z)\\), with \\(|a_j|<1\\). The zeros \\(a_j\\) are counted with multiplicity. On the unit circle its modulus is one, and its degree is the number of factors.","reader_rule":"For positive degree it gives a proper map from the disc onto itself and can have critical points. It is not generally one-to-one. Infinite Blaschke products require an additional convergence condition and lie outside this finite row.","deep_definition":"After a Riemann map, a regular polynomial lemniscate component becomes a finite Blaschke product whose degree is the number of roots in that component. An exterior version records the excluded roots and supplies the boundary-fibre identity in the capacity estimate.","scope_note":"For positive degree it gives a proper map from the disc onto itself and can have critical points. It is not generally one-to-one. Infinite Blaschke products require an additional convergence condition and lie outside this finite row.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 700.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1042,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::finite_blaschke_product","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 700","role":"mathematical_source","gloss":"Published manuscript use of finite Blaschke product with the local hypotheses and conclusion stated in the referenced passage."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":280,"id":"finite_computation","label":"finite computation","preferred_label":"finite computation","alternative_labels":["finite calculation","computed finite case"],"hidden_search_labels":["finite check"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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The algorithm and arithmetic determine which kind of result it gives.","reader_rule":"The number of cases checked does not by itself extend the result beyond those cases.","deep_definition":"A finite computation can support an infinite theorem when combined with a proof of coverage or a suitable error bound. The connection must be stated rather than inferred from the amount of computation.","scope_note":"A finite computation covers its declared inputs, algorithm, numeric exactness, and execution record.","example":"The corpus checks thousands of finite denominator-order cases while the universal irrationality problem remains open.","example_route":null,"non_example":"Computing many cases is not a proof that every remaining case behaves the same way.","visible_text_byte_count":871,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::finite_computation","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":1,"page_count":1,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":281,"id":"finite_difference","label":"finite difference","preferred_label":"finite difference","alternative_labels":["forward difference","finite differences"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the reader bands identify the ambient field, index set, convergence requirement, or theorem hypotheses relevant to the cited manuscript.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-finite-difference","relative_href":"docs/glossary.html#glossary-finite-difference","public_phrase":"finite difference","short_definition":"A finite difference measures a function’s change over a fixed step; for a sequence, its forward difference is \\(\\Delta a_{n}=a_{n+1}-a_{n}\\).","reader_preview":"A finite difference measures a function’s change over a fixed step; for a sequence, its forward difference is \\(\\Delta a_{n}=a_{n+1}-a_{n}\\).","reader_preview_basis":"definition","card_definition":"Repeated differences apply the same operation again. A degree-d polynomial sequence has zero \\((d+1)\\text{st}\\) differences, just as its \\((d+1)\\text{st}\\) derivative vanishes.","reader_rule":"A small error in the values need not have the required small differences. Eventual vanishing of a fixed-order difference is stronger than an approximate polynomial growth estimate.","deep_definition":"In #243, the recurrence controls the first difference of the error. Three further differences show \\(\\Delta ^{4}C_{n}\\to 0\\). Since these differences are integers, they eventually equal zero, and Newton interpolation gives an eventual cubic. The paper explicitly shows why its little-o precision cannot simply be weakened to big-O.","scope_note":"A small error in the values need not have the required small differences. Eventual vanishing of a fixed-order difference is stronger than an approximate polynomial growth estimate.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::sec:cubic-extraction.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1044,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::finite_difference","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::sec:cubic-extraction","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":1,"page_count":1,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":282,"id":"finite_field","label":"finite field","preferred_label":"finite field","alternative_labels":["finite fields"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the reader bands identify the ambient field, index set, convergence requirement, or theorem hypotheses relevant to the cited manuscript.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-finite-field","relative_href":"docs/glossary.html#glossary-finite-field","public_phrase":"finite field","short_definition":"A finite field is a field with finitely many elements.","reader_preview":"A finite field is a field with finitely many elements.","reader_preview_basis":"definition","card_definition":"Its size is a prime power \\(p^{r}\\). 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A field with \\(p^{r}\\) elements is a different construction; its characteristic is p.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::eq:all-root-square.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":882,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::finite_field","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::eq:all-root-square","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":283,"id":"finite_truncation","label":"finite truncation","preferred_label":"finite truncation","alternative_labels":["truncation","cutoff approximation"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-finite-truncation","relative_href":"docs/glossary.html#glossary-finite-truncation","public_phrase":"finite truncation","short_definition":"A finite truncation keeps a specified finite part of an otherwise longer or infinite object.","reader_preview":"A finite truncation keeps a specified finite part of an otherwise longer or infinite object.","reader_preview_basis":"definition","card_definition":"For a series, truncating after N terms gives its Nth partial sum and leaves a tail. For a power series, it produces a polynomial retaining a chosen range of powers.","reader_rule":"A finite expression can be calculated exactly while still differing from the full object.","deep_definition":"The papers connect calculations to infinite claims through bounds or identities controlling what was omitted. The truncation point and error estimate determine whether the finite calculation is adequate for the stated conclusion.","scope_note":"A finite expression can be calculated exactly while still differing from the full object.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/251/erdos-251-prime-gap-dyadic-series.tex::line 1186.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":867,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::finite_truncation","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/251/erdos-251-prime-gap-dyadic-series.tex::line 1186","role":"mathematical_source","gloss":"Published manuscript context for this mathematical term."}],"evidence_commands":[],"used_in":{"total_occurrences":1,"page_count":1,"top_page":"paper-module-provider-context-recipe-budget.html","top_pages":[{"page":"paper-module-provider-context-recipe-budget.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":284,"id":"fixture","label":"fixture","preferred_label":"fixture","alternative_labels":["input fixture","synthetic fixture"],"hidden_search_labels":["input fixtures"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-fixture","relative_href":"docs/glossary.html#glossary-fixture","public_phrase":"fixture","short_definition":"A test fixture supplies the inputs, state, or environment used for a test.","reader_preview":"A test fixture supplies the inputs, state, or environment used for a test.","reader_preview_basis":"definition","card_definition":"In Plectis's public examples, fixtures are often small stored datasets or files chosen to exercise particular behaviours. A negative fixture deliberately represents a case the mechanism should reject or report.","reader_rule":"The fixture's role is to make the tested conditions explicit and repeatable.","deep_definition":"Using the same fixture helps comparison, but repeatability also depends on the program and environment. A fixture can test one failure mode well without representing the full range of real-world inputs.","scope_note":"Fixtures appear wherever a component states an input. A passing run against a fixture establishes the fixture case, never whole-system correctness.","example":"A component that classifies evidence ships a fixture file of sample rows; running the check reads that file, writes a result record, and proves the classifier behaves as claimed on those rows.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1014,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::fixture","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":2143,"page_count":193,"top_page":"paper-modules.html","top_pages":[{"page":"paper-modules.html","count":158},{"page":"rules-and-ideas.html","count":96},{"page":"paper-module-bridge-phase-continuity-runtime.html","count":39},{"page":"paper-module-engine-room-demo.html","count":36},{"page":"paper-module-mechanistic-interpretability-circuit-attribution-replay.html","count":33},{"page":"paper-module-executable-doctrine-grammar.html","count":32}]}},{"schema":"plectis_public_glossary_term_v0","position":285,"id":"flat_torus","label":"flat torus","preferred_label":"flat torus","alternative_labels":["flat tori","translation torus"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The paper uses a real translation torus with its flat metric; an abstract torus need not carry that chosen metric.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-flat-torus","relative_href":"docs/glossary.html#glossary-flat-torus","public_phrase":"flat torus","short_definition":"A periodic Euclidean space formed by identifying points that differ by lattice translations.","reader_preview":"A periodic Euclidean space formed by identifying points that differ by lattice translations.","reader_preview_basis":"authored_override","card_definition":"The standard two-dimensional example is a square with opposite edges identified: crossing one edge returns at the opposite edge. More generally \\(\\mathbb{R}^d/\\Lambda\\) uses a full-rank discrete translation lattice \\(\\Lambda\\). Its metric is inherited locally from Euclidean space, so its curvature is zero.","reader_rule":"The familiar doughnut drawn in three-dimensional space has the same topology but a different metric, with varying curvature. Flat refers to intrinsic geometry, not to the visual embedding.","deep_definition":"In #269 each coordinate is a real number modulo \\(\\log q\\). The product is a flat torus whose straight flow records prime-power crossings. A two-dimensional subtorus is used for an incidence count. The quotient keeps phase information while discarding whole periods.","scope_note":"The familiar doughnut drawn in three-dimensional space has the same topology but a different metric, with varying curvature. Flat refers to intrinsic geometry, not to the visual embedding.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/269/erdos269-running-lcm-reasoning-surface.tex::long269:sec:flow.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1057,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::flat_torus","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/269/erdos269-running-lcm-reasoning-surface.tex::long269:sec:flow","role":"mathematical_source","gloss":"Published paper context for this term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":286,"id":"floor_function","label":"floor function","preferred_label":"floor function","alternative_labels":["floor","greatest-integer function"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Floor rounds toward negative infinity, so it differs from truncation toward zero on negative nonintegers.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-floor-function","relative_href":"docs/glossary.html#glossary-floor-function","public_phrase":"floor function","short_definition":"The floor of a real number x, written ⌊x⌋, is the greatest integer less than or equal to x.","reader_preview":"The floor of a real number x, written ⌊x⌋, is the greatest integer less than or equal to x.","reader_preview_basis":"definition","card_definition":"Thus \\(\\lfloor 3.8\\rfloor =3\\) and \\(\\lfloor -1.2\\rfloor =-2\\). The second example matters: floor rounds downward, not toward zero. It is constant between consecutive integers and jumps at each integer.","reader_rule":"The defining inequalities are \\(\\lfloor x\\rfloor \\le x<\\lfloor x\\rfloor +1\\). At an integer x the floor equals x. Floor differs from ceiling, nearest-integer rounding, and truncation of decimal digits.","deep_definition":"The paper partitions a parameter range according to which floor function changes. Its intervals are controlled by the jump points where an underlying real expression crosses an integer.","scope_note":"The defining inequalities are \\(\\lfloor x\\rfloor \\le x<\\lfloor x\\rfloor +1\\). At an integer x the floor equals x. 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Visible focus styling shows the current target.","reader_rule":"Move focus only for a clear interaction reason, preserve a sensible return target, and avoid reopening dismissed content from the restoration event.","deep_definition":"Browsers maintain one active element in a document. Scripts sometimes need to move focus when opening a dialog or return it to the control that opened a dismissed popup. Plectis glossary behaviour keeps focus on the mathematical expression after Escape and suppresses the focus event that would immediately reopen the card. Focus management complements semantic HTML; it cannot repair controls with missing names or incorrect roles.","scope_note":"Move focus only for a clear interaction reason, preserve a sensible return target, and avoid reopening dismissed content from the restoration event.","example":"Escape closes a notation popup and restores focus to the expression that opened it.","example_route":null,"non_example":"Moving focus on every mouse movement is not useful focus management.","visible_text_byte_count":888,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::focus_management","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/index.html","role":"prerequisite_context","gloss":"Interactive glossary and navigation controls support keyboard use."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":288,"id":"forecasting","label":"forecasting","preferred_label":"forecasting","alternative_labels":["forecast skill"],"hidden_search_labels":["research and forecasting"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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Scenarios and conditional forecasts answer different questions from unconditional probability estimates.","reader_rule":"A forecast is evidence of a reasoned expectation, not knowledge of the future; evaluate it over comparable resolved cases rather than one memorable outcome.","deep_definition":"Forecasting states expectations about outcomes not yet known, ideally with a precise event definition, resolution rule, deadline, information cutoff, and probability or range. Evaluation across comparable resolved cases can measure calibration, discrimination, and scoring performance. Conditional forecasts describe what is expected if a premise holds; scenarios explore possibilities without necessarily assigning probabilities. One correct or incorrect prediction is weak evidence about skill. Forecast quality also depends on question selection, base rates, missing outcomes, revisions, and whether forecasters could observe information that later evaluators treat as unavailable.","scope_note":"Nothing here is investment advice or a claim about tradeable performance.","example":"Comparing a forecast against a benchmark and reporting whether the difference is larger than chance.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":955,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::forecasting","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":41,"page_count":14,"top_page":"area-research.html","top_pages":[{"page":"area-research.html","count":10},{"page":"components.html","count":9},{"page":"component-prediction-oracle-reconciliation.html","count":4},{"page":"paper-module-prediction-oracle-reconciliation.html","count":4},{"page":"paper-modules.html","count":3},{"page":"paper-module-finance-forecast-evaluation-spine.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":289,"id":"formal","label":"formal","preferred_label":"formal","alternative_labels":["formally specified","formal method","formal methods"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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A formal proof follows the system's rules to derive a conclusion from assumptions.","reader_rule":"Precision of form does not ensure that the statement expresses the intended question.","deep_definition":"The papers connect formal statements with ordinary mathematical explanations so readers can examine both the validity of the derivation and whether the encoded result matches the claim being made.","scope_note":"Here the term covers rule-governed mathematical and software descriptions, not ceremonial or stylistic formality.","example":"A Lean theorem declaration is formal because its meaning is fixed by Lean's language and kernel.","example_route":null,"non_example":"Carefully worded prose is not formal merely because it sounds rigorous.","visible_text_byte_count":837,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::formal","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":337,"page_count":54,"top_page":"doctrine.html","top_pages":[{"page":"doctrine.html","count":52},{"page":"paper-modules.html","count":34},{"page":"rules-and-ideas.html","count":28},{"page":"area-formal-math.html","count":26},{"page":"components.html","count":26},{"page":"paper-module-formal-math-readiness-gate.html","count":18}]}},{"schema":"plectis_public_glossary_term_v0","position":290,"id":"formal_mathematics","label":"formal mathematics","preferred_label":"formal mathematics","alternative_labels":["formal math"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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Authors may use tactics and automation to construct proofs, while the accepted proof must satisfy the system's checking rules.","reader_rule":"The exact statement and assumptions determine what has been established.","deep_definition":"Formal checking reduces dependence on unchecked inference steps. It leaves separate work in choosing the right formulation, explaining the ideas, assessing significance, and ensuring that the formal statement corresponds to the intended mathematics.","scope_note":"Covers mathematical definitions, statements, and proofs expressed under explicit formal rules; formalisation names the process of producing them.","example":"A Lean library can contain formal definitions, theorem statements, and checked proofs alongside explanatory prose.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":957,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::formal_mathematics","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":241,"page_count":44,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":97},{"page":"reference.html","count":16},{"page":"paper-modules.html","count":14},{"page":"architecture.html","count":12},{"page":"paper-module-formal-math-readiness-gate.html","count":9},{"page":"paper-module-formal-math-lean-proof-witness.html","count":8}]}},{"schema":"plectis_public_glossary_term_v0","position":291,"id":"formal_power_series","label":"formal power series","preferred_label":"formal power series","alternative_labels":["formal series"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the reader bands identify the ambient field, index set, convergence requirement, or theorem hypotheses relevant to the cited manuscript.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-formal-power-series","relative_href":"docs/glossary.html#glossary-formal-power-series","public_phrase":"formal power series","short_definition":"A formal power series is an infinite coefficient sequence written as powers of a symbol, with arithmetic defined coefficient by coefficient.","reader_preview":"A formal power series is an infinite coefficient sequence written as powers of a symbol, with arithmetic defined coefficient by coefficient.","reader_preview_basis":"definition","card_definition":"Over a coefficient ring, \\(\\sum a_{n}z^{n}\\) is an algebraic object. Addition is termwise, and each coefficient of a product is a finite sum \\(\\sum _{k=0}^{n}a_{k}b_{n-k}\\).","reader_rule":"Formal series do not require numerical convergence. Substitution into an infinite series needs its own algebraic condition, such as a substituted series with zero constant term.","deep_definition":"The #1049 moment sums define formal series because only finitely many summands contribute to each coefficient. Its determinant proof can truncate modulo a fixed power before taking limits. The manuscript separately proves convergence when q is a real number between zero and one.","scope_note":"Formal series do not require numerical convergence. Substitution into an infinite series needs its own algebraic condition, such as a substituted series with zero constant term.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex::line 61.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1001,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::formal_power_series","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex::line 61","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":292,"id":"formal_proof","label":"formal proof","preferred_label":"formal proof","alternative_labels":["machine-checked proof","formalised proof"],"hidden_search_labels":["formalized proof"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The ordinary logical meaning applies, with Lean as the local implementation. Mechanical acceptance is a further claim about a particular proof, environment, and run.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-formal-proof","relative_href":"docs/glossary.html#glossary-formal-proof","public_phrase":"formal proof","short_definition":"A formal proof is a derivation expressed in a logical system whose rules make each step precisely checkable.","reader_preview":"A formal proof is a derivation expressed in a logical system whose rules make each step precisely checkable.","reader_preview_basis":"authored_override","card_definition":"In Lean, a proposition is represented by a type and its proof by a term of that type. A term can be written directly or constructed by tactics; its derivation depends on definitions, hypotheses, and axioms.","reader_rule":"Distinguish a formal proof from a formalised statement with an unfinished proof. Also distinguish being formally expressible from having passed a recorded checker run.","deep_definition":"A checked conditional theorem establishes its conclusion under its hypotheses. Applying it requires those hypotheses and a formal statement matching the intended claim. Paper-to-source links expose that comparison but do not settle it automatically.","scope_note":"Read the exact proposition, hypotheses, axiom dependencies, and recorded checking environment.","example":"A Lean proof term checked against a theorem declaration is a machine-checked formal proof of that declaration.","example_route":null,"non_example":"A detailed paper proof or successful numerical experiment is not automatically a formal proof.","visible_text_byte_count":895,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::formal_proof","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":27,"page_count":13,"top_page":"paper-modules.html","top_pages":[{"page":"paper-modules.html","count":6},{"page":"paper-module-proof-diagnostic-evidence-spine.html","count":5},{"page":"paper-module-formal-math-readiness-gate.html","count":3},{"page":"paper-module-finite-erdos-denominator-certificate-strike.html","count":2},{"page":"rules-and-ideas.html","count":2},{"page":"tour.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":293,"id":"formal_verification","label":"formal verification","preferred_label":"formal verification","alternative_labels":["formally verified","formal check"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Formal verification has established software, hardware, and protocol applications. Local Lean checks verify encoded mathematical claims; neither use guarantees its specification fully captures the intended external problem.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-formal-verification","relative_href":"docs/glossary.html#glossary-formal-verification","public_phrase":"formal verification","short_definition":"Formal verification uses logic and mathematical models to establish a stated property against a precise specification.","reader_preview":"Formal verification uses logic and mathematical models to establish a stated property against a precise specification.","reader_preview_basis":"authored_override","card_definition":"In software or hardware, the property might concern an implementation model or all reachable states. In the mathematical work here, Lean checks whether a proof term establishes its formal theorem. These are different applications of mathematical checking.","reader_rule":"Identify the specification, model, assumptions, and trusted checking method before interpreting the guarantee. Passing sample tests supports a different kind of conclusion.","deep_definition":"Exact verification can concern an incomplete model or a specification missing the intended question. Reviewing that correspondence remains necessary. Locally, the linked declaration fixes what Lean checked, while the paper explains why that statement addresses the research problem.","scope_note":"Covers the specification and mathematical model verified by the stated method; the local Lean examples concern named formal declarations.","example":"Each problem page lists the statements Lean has checked, with a link to the exact source, separately from the prose that explains them.","example_route":null,"non_example":"Testing selected examples is not by itself formal verification of a universal specification, even when every tested example passes.","visible_text_byte_count":1106,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::formal_verification","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":3,"page_count":2,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":2},{"page":"doctrine.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":294,"id":"formalisation","label":"formalisation","preferred_label":"formalisation","alternative_labels":["formalization","formalised","formalized"],"hidden_search_labels":["formalise","formalize","formalising","formalizing","formal source"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The disciplinary process includes encoding statements and specifications as well as completed arguments. Lean is the main local example; a formalised statement alone does not imply a completed proof.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-formalisation","relative_href":"docs/glossary.html#glossary-formalisation","public_phrase":"formalisation","short_definition":"Formalisation translates definitions, statements, or arguments into a language with explicit logical rules.","reader_preview":"Formalisation translates definitions, statements, or arguments into a language with explicit logical rules.","reader_preview_basis":"authored_override","card_definition":"Mathematical formalisation chooses representations, states hypotheses and quantifiers, and expresses conclusions precisely. The Lean developments here also connect their declarations to statements discussed in accompanying papers.","reader_rule":"Report the stage reached: formal definitions, an encoded statement, a finished proof, and an inspected correspondence with the informal claim are distinct achievements.","deep_definition":"Representation choices can uncover hidden assumptions or change a theorem's formulation. Software specifications can likewise be formalised before anyone proves that an implementation satisfies them. Pinned locators let readers inspect this translation at a fixed source revision.","scope_note":"Encoding the intended statement and proving that statement are separate parts of the work.","example":"A paper names a Lean declaration and pinned source revision so readers can compare the informal and encoded statements.","example_route":null,"non_example":"Careful prose or familiar mathematical notation alone does not specify a formal language, and a formalised theorem statement is not yet a proof.","visible_text_byte_count":1047,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::formalisation","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":9,"page_count":5,"top_page":"contact.html","top_pages":[{"page":"contact.html","count":4},{"page":"papers.html","count":2},{"page":"area-formal-math.html","count":1},{"page":"credit-ledger.html","count":1},{"page":"introduction.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":295,"id":"forward_invariant_set","label":"forward-invariant set","preferred_label":"forward-invariant set","alternative_labels":["forward invariant","positively invariant"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Dynamical systems. 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It depends on the chosen dynamics and makes no promise about backward motion or continuation through singular points.","deep_definition":"Along regular Newton trajectories, the polynomial modulus decreases, so a polynomial sublevel set is preserved. This alone does not justify extending a trajectory through a critical point where the vector field -f/f′ is undefined.","scope_note":"Forward invariance means a trajectory stays in the set for every later time for which its evolution is defined. It depends on the chosen dynamics and makes no promise about backward motion or continuation through singular points.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::line 635.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1027,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::forward_invariant_set","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::line 635","role":"mathematical_source","gloss":"Published manuscript context for this term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":296,"id":"fraction","label":"fraction","preferred_label":"fraction","alternative_labels":["reduced fraction"],"hidden_search_labels":["numerator"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-fraction","relative_href":"docs/glossary.html#glossary-fraction","public_phrase":"fraction","short_definition":"A fraction writes a quotient as \\(a/b\\); for a rational number, a and b are integers and b is nonzero.","reader_preview":"A fraction writes a quotient as \\(a/b\\); for a rational number, a and b are integers and b is nonzero.","reader_preview_basis":"definition","card_definition":"The numerator a is divided by the denominator b. Different fractions can represent the same number: \\(2/4\\) and \\(1/2\\) are equal because the common factor 2 can be cancelled.","reader_rule":"“In lowest terms” means that numerator and denominator are coprime.","deep_definition":"An expression written as a quotient is not necessarily a rational number if its numerator or denominator is non-integer. For example, \\(\\sqrt 2/1\\) is still irrational.","scope_note":"Fraction covers the stated numerator-denominator representation, including whether it is reduced.","example":"A finite reciprocal sum can be computed exactly as a reduced fraction `P/Q`.","example_route":null,"non_example":"A decimal approximation is not the exact fraction unless the equality is separately established.","visible_text_byte_count":787,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::fraction","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":22,"page_count":6,"top_page":"paper-module-finite-erdos-denominator-certificate-strike.html","top_pages":[{"page":"paper-module-finite-erdos-denominator-certificate-strike.html","count":12},{"page":"papers.html","count":4},{"page":"component-finite-erdos-denominator-certificate-strike.html","count":2},{"page":"paper-module-formal-math-premise-retrieval.html","count":2},{"page":"paper-modules.html","count":1},{"page":"rules-and-ideas.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":297,"id":"fractional_part","label":"fractional part","preferred_label":"fractional part","alternative_labels":["fractional parts"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The fractional part lies between zero and one, even for negative inputs; it is not generally the signed decimal tail.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-fractional-part","relative_href":"docs/glossary.html#glossary-fractional-part","public_phrase":"fractional part","short_definition":"For a real number x, its fractional part is x minus the greatest integer at most x, so it lies in \\([0,1)\\).","reader_preview":"For a real number x, its fractional part is x minus the greatest integer at most x, so it lies in \\([0,1)\\).","reader_preview_basis":"definition","card_definition":"For example, 3.7 has fractional part 0.7, while -1.2 has fractional part 0.8 because the floor is -2. Fractional parts record position modulo one and are central when a sequence is studied around the unit circle.","reader_rule":"The fractional part \\(\\{x\\}=x-\\lfloor x\\rfloor\\) is zero exactly when x is an integer. Negative inputs still give a value in \\([0,1)\\), unlike a signed decimal tail produced by truncation.","deep_definition":"Here the paper studies fractional parts of factorially scaled numbers. Saying they vanish eventually means every sufficiently late scaled value is an integer. Bounds on fractional parts can force arithmetic conclusions even without proving that those parts are evenly spread.","scope_note":"The fractional part \\(\\{x\\}=x-\\lfloor x\\rfloor\\) is zero exactly when x is an integer. Negative inputs still give a value in \\([0,1)\\), unlike a signed decimal tail produced by truncation.","example":"For example, 3.7 has fractional part 0.7, while -1.2 has fractional part 0.8 because the floor is -2.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":943,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::fractional_part","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/68/erdos68-factorial-reasoning-surface.tex::line 1648","role":"mathematical_source","gloss":"For rational $x$ the fractional parts vanish eventually; if all large"}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":298,"id":"freshness","label":"freshness","preferred_label":"freshness","alternative_labels":["freshness contract","refresh rule"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"public_contextual_gloss","meaning_boundary":"Freshness here is whether a dated live reading is still current under its declared basis and refresh rule. It is not newness, recency of publication, or a quality judgement.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-freshness","relative_href":"docs/glossary.html#glossary-freshness","public_phrase":"freshness","short_definition":"Freshness describes whether stored information still reflects the source or state it is meant to represent.","reader_preview":"Freshness describes whether stored information still reflects the source or state it is meant to represent.","reader_preview_basis":"definition","card_definition":"A generated page, cached result, or copied file may have been accurate when created and become outdated after its source changes. Plectis uses source versions and refresh rules to assess that relationship.","reader_rule":"A recent timestamp alone does not show that the right source was read.","deep_definition":"A useful freshness record identifies the source, the version examined, and the condition that requires another check. Different kinds of information can have very different refresh needs.","scope_note":"Applies to dated live readings on doctrine, evidence, and generated pages, not to the age of a theorem or a paper.","example":"A count of remaining obligations is a cached read: it is only current while it carries its as-of time, its basis, and a refresh rule.","example_route":{"href":"doctrine.html#the-operating-loop","label":"The operating loop","gloss":"open the freshness question on the doctrine page"},"non_example":"A Lean-checked theorem does not go stale because a week has passed. 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The p-power Frobenius is always an injective field endomorphism and is an automorphism exactly for a perfect field; finite fields are perfect. This is distinct from a Frobenius element attached to a prime in a number-field extension.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-frobenius-automorphism","relative_href":"docs/glossary.html#glossary-frobenius-automorphism","public_phrase":"Frobenius automorphism","short_definition":"In characteristic p, Frobenius sends x to \\(x^p\\); it is an automorphism when this power map is onto, as it always is for a finite field.","reader_preview":"In characteristic p, Frobenius sends x to \\(x^p\\); it is an automorphism when this power map is onto, as it always is for a finite field.","reader_preview_basis":"definition","card_definition":"In \\(F_{4}\\), the map \\(x\\mapsto x^{2}\\) fixes 0 and 1 and swaps the two other elements, so it is a nontrivial automorphism. More generally, if \\(q=p^{n}\\), the p-power Frobenius generates the cyclic Galois group \\(Gal(F_q/F_p)\\) and has order n.","reader_rule":"On every field of characteristic p the p-power map is injective, but it need not be surjective on an infinite imperfect field. On \\(F_q\\) itself, \\(x\\mapsto x^q\\) is the identity. On \\(F_{q^m}\\) over \\(F_q\\), the q-power map is an automorphism of order m and generates \\(Gal(F_{q^m}/F_q)\\).","deep_definition":"The binomial identity \\((x+y)^p=x^p+y^p\\) in characteristic p makes Frobenius a field endomorphism. In characteristic p, a field is perfect precisely when this map is surjective and hence an automorphism; all finite fields are perfect because an injective self-map of a finite set is surjective. In number-field reduction, a Frobenius element at P above p acts on the upstairs residue field \\(\\kappa (P)\\) by \\(x\\mapsto x^q\\), where \\(q=|\\kappa (p)|\\) is the base residue-field size. That action is a field Frobenius automorphism, but the Frobenius element also records how it lifts into the number-field Galois group.","scope_note":"This is the power-map automorphism in characteristic p, distinct from a Frobenius element or conjugacy class attached to a prime.","example":"For \\(F_{8}/F_{2}\\), \\(x\\mapsto x^{2}\\) is an automorphism of order 3, and applying it three times gives \\(x\\mapsto x^{8}=x\\).","example_route":null,"non_example":"On \\(F_q\\), the q-power map is the identity; calling it a nontrivial generator of \\(Gal(F_q/F_p)\\) would confuse q-power and p-power Frobenius.","visible_text_byte_count":1189,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::frobenius_automorphism","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"contrasts_with","target":"frobenius_element","gloss":"The automorphism is a characteristic-p power map; the element is a lift attached to an unramified prime."},{"verb":"acts_on","target":"residue_field","gloss":"Prime-attached Frobenius elements induce Frobenius automorphisms on residue fields."}],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex::line 283","role":"prerequisite_dependency","gloss":"The manuscript's prime-attached Frobenius class acts through a power automorphism on the corresponding residue field."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":300,"id":"frobenius_element","label":"Frobenius element","preferred_label":"Frobenius element","alternative_labels":["Frobenius conjugacy class"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"At an unramified prime, choosing a prime above it specifies an element; without that choice the base prime specifies a conjugacy class.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-frobenius-element","relative_href":"docs/glossary.html#glossary-frobenius-element","public_phrase":"Frobenius element","short_definition":"A Frobenius element is a field symmetry attached to an unramified prime in a finite Galois extension of number fields; it describes a power map after reduction modulo a chosen prime above it.","reader_preview":"A Frobenius element is a field symmetry attached to an unramified prime in a finite Galois extension of number fields; it describes a power map after reduction modulo a chosen prime above it.","reader_preview_basis":"definition","card_definition":"In a number-field extension, reducing algebraic integers modulo a prime gives a finite field. 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Some geometry texts call the inverse element geometric Frobenius; the convention must be stated.","scope_note":"The exponent q is the cardinality of the base residue field \\(\\kappa (p)\\), not the usually larger upstairs field \\(\\kappa (P)\\). 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A shared name does not establish that correspondence.","scope_note":"Distinguishes mathematical assignments from software callables; the local domain or programming contract fixes the details.","example":"Euler's totient function assigns to n the number of positive integers up to n that are coprime to n.","example_route":null,"non_example":"A mathematical relation assigning two different outputs to the same input is not a function into that output space; returning a set as one output is a different construction.","visible_text_byte_count":1020,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::function","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":200,"page_count":90,"top_page":"paper-module-microcosm-axiom-substrate.html","top_pages":[{"page":"paper-module-microcosm-axiom-substrate.html","count":14},{"page":"paper-module-batch9-macro-engines-capsule.html","count":11},{"page":"paper-module-engine-room-command-run-singleflight.html","count":10},{"page":"paper-module-batch8-validator-checker-capsule.html","count":9},{"page":"doctrine.html","count":8},{"page":"paper-module-batch11-saturation-engines-capsule.html","count":6}]}},{"schema":"plectis_public_glossary_term_v0","position":304,"id":"function_composition","label":"function composition","preferred_label":"function composition","alternative_labels":["composition of functions","composite function"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The mathematical sense stated here; related software or ordinary-language uses are distinguished where relevant.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-function-composition","relative_href":"docs/glossary.html#glossary-function-composition","public_phrase":"function composition","short_definition":"Function composition applies one function to the output of another.","reader_preview":"Function composition applies one function to the output of another.","reader_preview_basis":"definition","card_definition":"The notation \\(g \\circ f\\) means “first f, then g,” so \\((g \\circ f)(x) = g(f(x))\\). 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The paper places several algebraic quantities inside one finite Galois extension so a chosen automorphism can later be realized as Frobenius behavior at suitable primes.","scope_note":"Galois is relative to a named base field K. In characteristic zero, algebraic extensions are separable, but normality still must be checked. 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Exponential generating functions instead use \\(a_{n}z^{n}/n!\\).","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex::eq:weight-generating-function.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1101,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::generating_function","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex::eq:weight-generating-function","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":313,"id":"geodesic","label":"geodesic","preferred_label":"geodesic","alternative_labels":["geodesics","closed geodesic","closed geodesics"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Local straightness depends on the metric; the paper uses the flat quotient metric.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-geodesic","relative_href":"docs/glossary.html#glossary-geodesic","public_phrase":"geodesic","short_definition":"A curve that is locally straight according to the geometry of its surrounding space.","reader_preview":"A curve that is locally straight according to the geometry of its surrounding space.","reader_preview_basis":"authored_override","card_definition":"On a flat torus, geodesics lift to straight lines in Euclidean space. 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The starting index matters because a ratio bound may fail for early terms even when it holds for all later ones.","scope_note":"The same ratio and initial bound must work throughout the claimed tail.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/68/erdos68-factorial-reasoning-surface.tex::line 333.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":893,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::geometric_majorant","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/68/erdos68-factorial-reasoning-surface.tex::line 333","role":"mathematical_source","gloss":"Published manuscript context for this mathematical term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":315,"id":"geometric_series","label":"geometric series","preferred_label":"geometric series","alternative_labels":["geometric progression series"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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Fixed-tuple and quantified uniform versions have distinct logical strength.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-hardy-littlewood-prime-tuple-conjecture","relative_href":"docs/glossary.html#glossary-hardy-littlewood-prime-tuple-conjecture","public_phrase":"Hardy–Littlewood prime-tuple conjecture","short_definition":"The Hardy–Littlewood prime-tuple conjecture predicts an asymptotic count of simultaneous prime translates of each fixed admissible set of shifts.","reader_preview":"The Hardy–Littlewood prime-tuple conjecture predicts an asymptotic count of simultaneous prime translates of each fixed admissible set of shifts.","reader_preview_basis":"definition","card_definition":"For a fixed nonempty admissible set \\(H=\\{h_1,\\ldots,h_k\\}\\) of distinct integer shifts, it predicts \\(\\#\\{1\\le n\\le x:n+h_i\\text{ is prime for every }i\\}\\sim\\mathfrak S(H)\\int_2^x(\\log t)^{-k}\\,dt\\) as \\(x\\to\\infty\\). The singular series supplies the local congruence correction.","reader_rule":"The usual fixed-tuple asymptotic gives no uniform rate for tuples whose sizes or shifts grow with the parameter. Such uniform versions are stronger hypotheses.","deep_definition":"#251 discusses a version with \\(k\\le(\\log\\log x)^3\\), shifts in \\([0,(\\log x)^2]\\), and error at most \\(Cx^{1-\\delta}\\). Constants are chosen before the tuple. The paper assumes none of these hypotheses; the cited irrationality and normality results remain conditional.","scope_note":"The usual fixed-tuple asymptotic gives no uniform rate for tuples whose sizes or shifts grow with the parameter. Such uniform versions are stronger hypotheses.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/251/erdos251-prime-gap-reasoning-surface.tex::lines 487–497.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1277,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::hardy_littlewood_prime_tuple_conjecture","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/251/erdos251-prime-gap-reasoning-surface.tex::lines 487–497","role":"mathematical_source","gloss":"Exact published manuscript occurrence and local claim boundary."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":333,"id":"harmonic_extension","label":"harmonic extension","preferred_label":"harmonic extension","alternative_labels":["Poisson extension"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The boundary data and allowed function class determine the extension problem. Harmonicity alone does not guarantee existence or uniqueness.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-harmonic-extension","relative_href":"docs/glossary.html#glossary-harmonic-extension","public_phrase":"harmonic extension","short_definition":"A harmonic extension fills boundary data into a domain by a harmonic function that takes those boundary values, when the relevant Dirichlet problem is solvable.","reader_preview":"A harmonic extension fills boundary data into a domain by a harmonic function that takes those boundary values, when the relevant Dirichlet problem is solvable.","reader_preview_basis":"definition","card_definition":"On the unit disc, continuous boundary data can be extended using the Poisson integral. Constant boundary data extend constantly; oscillatory Fourier modes decay toward the center according to their frequency.","reader_rule":"Existence and uniqueness depend on the domain and the function class. “Extension” means agreement in the appropriate boundary sense, which may be continuous or almost everywhere. Harmonic does not mean holomorphic, though real and imaginary parts of holomorphic functions are harmonic.","deep_definition":"The paper extends boundary information harmonically so interior estimates can be derived from boundary averages and maximum principles. The extension acts as a controlled interpolant rather than introducing new independent analytic data.","scope_note":"Existence and uniqueness depend on the domain and the function class. “Extension” means agreement in the appropriate boundary sense, which may be continuous or almost everywhere. Harmonic does not mean holomorphic, though real and imaginary parts of holomorphic functions are harmonic.","example":"On the unit disc, continuous boundary data can be extended using the Poisson integral.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1106,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::harmonic_extension","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::line 425","role":"mathematical_source","gloss":"Its harmonic extension controls"}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":334,"id":"harmonic_function","label":"harmonic function","preferred_label":"harmonic function","alternative_labels":["harmonicity"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Harmonic analysis is a larger subject; not every object described as a harmonic mode is itself a harmonic function in this sense.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-harmonic-function","relative_href":"docs/glossary.html#glossary-harmonic-function","public_phrase":"harmonic function","short_definition":"A harmonic function has continuous second derivatives and a zero Laplacian, the sum of its pure second partial derivatives.","reader_preview":"A harmonic function has continuous second derivatives and a zero Laplacian, the sum of its pure second partial derivatives.","reader_preview_basis":"definition","card_definition":"A harmonic function is locally balanced: its value at the centre of a sufficiently small ball equals its average over that ball or its boundary. For example, \\(x^{2}-y^{2}\\) is harmonic on the plane because its pure second derivatives, 2 and \\(-2\\), cancel.","reader_rule":"Check the domain and regularity; in the plane, real and imaginary parts of holomorphic functions are harmonic, but a general harmonic function need not be globally the real part of one on a domain with holes.","deep_definition":"A harmonic function u on an open region satisfies \\(\\Delta u=0\\), where the Laplacian adds the second derivative in every coordinate direction. Harmonic functions obey the mean-value property: the value at the centre of a sufficiently small ball equals the average on its boundary or interior. Consequently, a nonconstant harmonic function cannot attain a maximum or minimum strictly inside a connected domain. In two dimensions, the real and imaginary parts of holomorphic functions are harmonic. A harmonic conjugate exists locally, and globally on simply connected domains, but topology can obstruct a single global conjugate.","scope_note":"Check the domain and regularity; in the plane, real and imaginary parts of holomorphic functions are harmonic, but a general harmonic function need not be globally the real part of one on a domain with holes.","example":"On the plane, \\(u(x,y)=x^2-y^2\\) is harmonic because its two pure second derivatives cancel.","example_route":null,"non_example":"The function x squared plus y squared is not harmonic because its Laplacian is 4.","visible_text_byte_count":974,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::harmonic_function","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":335,"id":"harmonic_measure","label":"harmonic measure","preferred_label":"harmonic measure","alternative_labels":["harmonic measures","boundary harmonic measure"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"This row concerns planar boundary harmonic measure; continuous boundary-value formulas require the stated regularity, and uniform arclength is a special case.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-harmonic-measure","relative_href":"docs/glossary.html#glossary-harmonic-measure","public_phrase":"harmonic measure","short_definition":"A measure describing how strongly different parts of a domain boundary influence a harmonic value at a chosen interior point.","reader_preview":"A measure describing how strongly different parts of a domain boundary influence a harmonic value at a chosen interior point.","reader_preview_basis":"authored_override","card_definition":"For a sufficiently regular bounded domain D, continuous boundary data h determine a harmonic extension \\(u(z)=\\int h(\\xi )d\\omega _D(z,\\xi )\\). The measure \\(\\omega _D(z,\\cdot )\\) depends on the base point z. At the centre of the disc it is uniform angular measure; elsewhere it is weighted by the Poisson kernel.","reader_rule":"Harmonic measure is generally different from uniform boundary length. Conformal maps transport it, but an arc-length density needs boundary regularity and the appropriate derivative factor. The domain and base point must be specified.","deep_definition":"In #1041, pushing the uniform circle measure through a Blaschke product produces harmonic measure seen from \\(B(0)\\). Comparing its density with the inverse-branch boundary weights gives a bound on the capacity of the selected polynomial component.","scope_note":"Harmonic measure is generally different from uniform boundary length. Conformal maps transport it, but an arc-length density needs boundary regularity and the appropriate derivative factor. The domain and base point must be specified.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 1962.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1208,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::harmonic_measure","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 1962","role":"mathematical_source","gloss":"Published manuscript use of harmonic measure with the local hypotheses and conclusion stated in the referenced passage."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":336,"id":"harness","label":"harness","preferred_label":"harness","alternative_labels":["test harness"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-harness","relative_href":"docs/glossary.html#glossary-harness","public_phrase":"harness","short_definition":"A test harness is the code and setup that runs a test and captures its result.","reader_preview":"A test harness is the code and setup that runs a test and captures its result.","reader_preview_basis":"definition","card_definition":"It supplies inputs, prepares the environment, invokes the thing being tested, and records or compares outputs. Plectis uses harnesses to make a component's advertised check executable.","reader_rule":"The harness is separate from both the component and the claim being tested.","deep_definition":"Its own behaviour matters: a harness that runs the wrong command or ignores a failure can produce a misleading pass. Negative cases help check that relevant failures are actually detected.","scope_note":"A harness establishes results for the scenarios it runs.","example":"A replay harness feeds a recorded scenario through a component and asserts the outcome.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":762,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::harness","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":45,"page_count":24,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":5},{"page":"paper-module-pattern-binding-contract.html","count":4},{"page":"paper-module-provider-context-recipe-budget.html","count":4},{"page":"components.html","count":3},{"page":"area-agent-reliability.html","count":2},{"page":"component-ring2-premise-retrieval-precision-recall-harness.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":337,"id":"hausdorff_measure","label":"hausdorff measure","preferred_label":"hausdorff measure","alternative_labels":["Hausdorff length","one-dimensional Hausdorff measure"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Geometric measure theory. Not Hausdorff distance and not the Hausdorff separation axiom.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-hausdorff-measure","relative_href":"docs/glossary.html#glossary-hausdorff-measure","public_phrase":"hausdorff measure","short_definition":"Hausdorff measure measures a set by covering it with arbitrarily small pieces and summing powers of their diameters.","reader_preview":"Hausdorff measure measures a set by covering it with arbitrarily small pieces and summing powers of their diameters.","reader_preview_basis":"definition","card_definition":"The one-dimensional version measures length: it agrees with arc length for a simple rectifiable curve, and also applies to connected sets with no chosen parametrisation.","reader_rule":"Specify the dimension and normalization. The measure of a curve’s image counts each location once; the length of a parametrised path counts retraced portions again.","deep_definition":"For dimension s, first minimize \\(c_{s}\\sum _{i}\\operatorname{diam}(U_{i})^{s}\\) over countable covers with diameters at most \\(\\delta\\), then let \\(\\delta\\) decrease to zero; \\(c_{s}\\) fixes the standard normalization. The #1041 counterexample bounds one-dimensional Hausdorff measure of connecting sets, a statement distinct from a bound on a particular parametrised path.","scope_note":"Specify the dimension and normalization. The measure of a curve’s image counts each location once; the length of a parametrised path counts retraced portions again.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 367.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":925,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::hausdorff_measure","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 367","role":"mathematical_source","gloss":"Published manuscript context for this term."},{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex","role":"mathematical_source","gloss":"Primary manuscript context and mathematical application."}],"evidence_commands":[],"used_in":{"total_occurrences":2,"page_count":1,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":338,"id":"heine_determinant_identity","label":"Heine determinant identity","preferred_label":"Heine determinant identity","alternative_labels":["Heine moment identity","Heine expansion of a moment determinant"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The moment-determinant identity is intended; convergence and positivity depend on the stated moment measure or discrete weights.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-heine-determinant-identity","relative_href":"docs/glossary.html#glossary-heine-determinant-identity","public_phrase":"Heine determinant identity","short_definition":"An identity expressing a moment determinant as an integral or sum of a squared Vandermonde product.","reader_preview":"An identity expressing a moment determinant as an integral or sum of a squared Vandermonde product.","reader_preview_basis":"authored_override","card_definition":"For \\(n\\ge 1\\) and a finite positive measure \\(\\mu\\) on the real line with finite moments through degree \\(2n-2\\), \\(\\det (\\mu _{i+j})_{i,j<n}=1/n! \\int \\ldots \\int \\prod _{i<j}(x_i-x_j)^{2} d\\mu (x_{1})\\ldots d\\mu (x_{n})\\), where \\(\\mu _k=\\int x^{k}d\\mu\\). For discrete weights, strictly increasing node indices remove the \\(1/n!\\) factor; pairwise distinct ordered tuples retain it.","reader_rule":"The moments and multiple integral must exist, with hypotheses justifying interchange. For a positive measure every summand is nonnegative, which is useful for limits and lower bounds. Signed weights remove that positivity advantage.","deep_definition":"In #1049 the nodes are powers of q and the weights are positive. The Heine expansion tracks displaced node tuples and shows why the initial tuple alone misses a nontrivial asymptotic constant. The identity is not Heine’s transformation of a basic hypergeometric series.","scope_note":"The moments and multiple integral must exist, with hypotheses justifying interchange. For a positive measure every summand is nonnegative, which is useful for limits and lower bounds. Signed weights remove that positivity advantage.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex::line 199.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1253,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::heine_determinant_identity","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex::line 199","role":"mathematical_source","gloss":"Published manuscript use of Heine determinant identity with the local hypotheses and conclusion stated in the referenced passage."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":339,"id":"hereditary_property","label":"hereditary property","preferred_label":"hereditary property","alternative_labels":["hereditary criterion","hereditary support condition"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the cited paper specifies the parameters and hypotheses for its application.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-hereditary-property","relative_href":"docs/glossary.html#glossary-hereditary-property","public_phrase":"hereditary property","short_definition":"A property is hereditary under a specified restriction if every allowed restriction of an object retains that property.","reader_preview":"A property is hereditary under a specified restriction if every allowed restriction of an object retains that property.","reader_preview_basis":"definition","card_definition":"Here, a support condition is hereditary when it passes from a set of exponents to each infinite subset. Removing nonnegative terms cannot increase a finite weighted sum.","reader_rule":"State which restrictions are allowed. An irrational sum alone does not imply that every infinite subseries has an irrational sum.","deep_definition":"The #257 criteria establish conditions that survive passage to infinite subsets. Applying the theorem anew to each such subset proves irrationality throughout that class; the mechanism is preservation of the hypothesis, not a general rule about irrational numbers.","scope_note":"State which restrictions are allowed. An irrational sum alone does not imply that every infinite subseries has an irrational sum.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":898,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::hereditary_property","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex","role":"mathematical_source","gloss":"Primary manuscript context and mathematical application."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":340,"id":"hessian_matrix","label":"hessian matrix","preferred_label":"hessian matrix","alternative_labels":["Hessian"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"A Hessian-scale model or Hessian bound refers to second-order behaviour even when the full matrix is not displayed.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-hessian-matrix","relative_href":"docs/glossary.html#glossary-hessian-matrix","public_phrase":"hessian matrix","short_definition":"The square matrix of second partial derivatives of a scalar-valued function.","reader_preview":"The square matrix of second partial derivatives of a scalar-valued function.","reader_preview_basis":"definition","card_definition":"The Hessian records second-order change in a scalar-valued function. For \\(f(x,y) = x^{2}-y^{2}\\) it has diagonal entries 2 and \\(-2\\), showing upward curvature along one axis and downward curvature along the other at the origin.","reader_rule":"At a critical point of a twice continuously differentiable real function, positive or negative definiteness gives a strict local minimum or maximum, and indefiniteness gives a saddle.","deep_definition":"For n variables, the Hessian is the \\(n\\times n\\) matrix of second partial derivatives. Under continuous second differentiability, equality of mixed partials makes it symmetric. Its quadratic form, multiplied by one half, gives the second-order Taylor term. A semidefinite Hessian with zero directions needs further analysis: higher-order terms can change the classification. A singular Hessian can still be indefinite and establish a saddle, so singularity alone is not the classification rule.","scope_note":"At a critical point of a twice continuously differentiable real function, positive or negative definiteness gives a strict local minimum or maximum, and indefiniteness gives a saddle.","example":"For \\(f(x,y)=x^2-y^2\\), the Hessian is diagonal with entries 2 and -2, showing a saddle at the origin.","example_route":null,"non_example":"The Jacobian of a vector-valued map contains first derivatives and is not generally a Hessian.","visible_text_byte_count":873,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::hessian_matrix","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":341,"id":"heuristic","label":"heuristic","preferred_label":"heuristic","alternative_labels":["heuristics","heuristic method"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"A practical rule that guides search or choice without guaranteeing correctness or completeness.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-heuristic","relative_href":"docs/glossary.html#glossary-heuristic","public_phrase":"heuristic","short_definition":"A heuristic is a practical rule or method that often finds a useful answer without guaranteeing the best or correct answer.","reader_preview":"A heuristic is a practical rule or method that often finds a useful answer without guaranteeing the best or correct answer.","reader_preview_basis":"definition","card_definition":"Heuristics reduce search or decision cost by using patterns, estimates, or experience. They are valuable when exact calculation is too expensive, but their limits should be tested against representative cases.","reader_rule":"Treat a heuristic result as a candidate or ranking signal until a stronger check establishes the required claim.","deep_definition":"A heuristic may rank search results, choose a promising proof step, or estimate where to spend effort. It can be fast and useful while missing good answers on some inputs. A guaranteed algorithm also has assumptions; the distinction is whether the claimed guarantee has been established for the case at hand. Plectis uses heuristic retrieval and planning to choose work, then checks the resulting evidence separately.","scope_note":"Treat a heuristic result as a candidate or ranking signal until a stronger check establishes the required claim.","example":"Ranking files by shared keywords is a heuristic for finding relevant source.","example_route":null,"non_example":"A Lean kernel accepting a proof is a formal check, not a heuristic guess.","visible_text_byte_count":772,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::heuristic","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/docs/paper-module-verifier-lab-kernel.html","role":"reading_context","gloss":"keeps trusted checked results apart from heuristics and automation"}],"evidence_commands":[],"used_in":{"total_occurrences":1,"page_count":1,"top_page":"paper-module-verifier-lab-kernel.html","top_pages":[{"page":"paper-module-verifier-lab-kernel.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":342,"id":"hologram","label":"hologram","preferred_label":"hologram","alternative_labels":["portable hologram"],"hidden_search_labels":["Russian doll","holographic telescope"],"term_kind":"navigation","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"Hologram is a local navigation metaphor for a source-derived multi-resolution graph. 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Each is a consistent single-valued local inverse of squaring. Choosing a branch specifies which preimage is followed as w changes.","reader_rule":"For a holomorphic map, a nonzero derivative gives a local holomorphic inverse branch. Its domain and starting preimage matter, and extending the branch requires avoiding the relevant obstructions.","deep_definition":"If f is locally one-to-one near \\(z_0\\) and \\(w_0=f(z_0)\\), an inverse branch is a function g on a neighbourhood of \\(w_0\\) satisfying \\(f(g(w))=w\\) and \\(g(w_0)=z_0\\). For a holomorphic f with \\(f'(z_0)\\) nonzero, the inverse function theorem gives a holomorphic branch. A polynomial of degree greater than one has several possible preimages, so there is generally no single global inverse. Continuing a chosen branch can encounter a critical value or can return to another branch after travelling around a branch point.","scope_note":"For a holomorphic map, a nonzero derivative gives a local holomorphic inverse branch. Its domain and starting preimage matter, and extending the branch requires avoiding the relevant obstructions.","example":"Near \\(w=1\\), the square-root branch with value 1 is a local inverse branch of z maps to z squared.","example_route":null,"non_example":"The two-valued expression plus-or-minus square root of w is not a single-valued inverse branch until one choice and domain are fixed.","visible_text_byte_count":973,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::inverse_branch","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":376,"id":"inverse_function","label":"inverse function","preferred_label":"inverse function","alternative_labels":["inverse map"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The mathematical sense stated here; related software or ordinary-language uses are distinguished where relevant.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-inverse-function","relative_href":"docs/glossary.html#glossary-inverse-function","public_phrase":"inverse function","short_definition":"An inverse function undoes a function, returning each output to the unique input that produced it.","reader_preview":"An inverse function undoes a function, returning each output to the unique input that produced it.","reader_preview_basis":"definition","card_definition":"The map \\(f(x) = 2x\\) on the reals has inverse \\(y \\mapsto y/2\\). Squaring on all real numbers has no inverse function because 2 and \\(-2\\) have the same square; restricting the domain to nonnegative reals removes that ambiguity.","reader_rule":"A function has a two-sided inverse between its domain and codomain exactly when it is bijective.","deep_definition":"The notation \\(f^{-1}(B)\\) can instead mean the preimage of a set B, which is defined even without an inverse function. The papers' inverse branches make an inverse choice on a suitable restricted region.","scope_note":"A function has a two-sided inverse between its domain and codomain exactly when it is bijective.","example":"Representative source: governed vocabulary term_registry.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":806,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::inverse_function","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"codex/doctrine/system_vocabulary/term_registry.json::terms[id=preimage]","role":"reading_context","gloss":"Prerequisite vocabulary used by this public definition."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":377,"id":"irrational","label":"irrational","preferred_label":"irrational","alternative_labels":["irrationality"],"hidden_search_labels":["irrational numbers"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-irrational","relative_href":"docs/glossary.html#glossary-irrational","public_phrase":"irrational","short_definition":"A real number is irrational if it cannot equal \\(a/b\\) for any integers a and nonzero b.","reader_preview":"A real number is irrational if it cannot equal \\(a/b\\) for any integers a and nonzero b.","reader_preview_basis":"definition","card_definition":"For example, \\(\\sqrt 2\\) is irrational. In any fixed integer base greater than one, an irrational number's digit expansion never becomes permanently periodic.","reader_rule":"An infinite expansion alone does not imply irrationality: \\(1/3\\) has an infinite repeating decimal expansion.","deep_definition":"An irrationality proof must rule out every possible rational representation. Computing many digits or checking many candidate denominators can provide evidence, but it does not on its own exclude all fractions.","scope_note":"An irrationality result covers the specific number or family it names.","example":"Erdős #249 asks whether the sum of the totient function divided by powers of two is irrational.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":795,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::irrational","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":45,"page_count":4,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":41},{"page":"credit-ledger.html","count":2},{"page":"introduction.html","count":1},{"page":"updates.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":378,"id":"irrationality_criterion","label":"irrationality criterion","preferred_label":"irrationality criterion","alternative_labels":["criterion for irrationality","irrationality test"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-irrationality-criterion","relative_href":"docs/glossary.html#glossary-irrationality-criterion","public_phrase":"irrationality criterion","short_definition":"An irrationality criterion is a proved condition that can be used to establish that a number is irrational.","reader_preview":"An irrationality criterion is a proved condition that can be used to establish that a number is irrational.","reader_preview_basis":"definition","card_definition":"It may require a pattern of rational approximations, denominators, or scaled tails that a rational number cannot have. The criterion gives a route to irrationality when its requirements are verified.","reader_rule":"Some criteria are sufficient conditions; stronger characterisations give both necessary and sufficient conditions.","deep_definition":"The papers state these logical roles separately. Proving a criterion is useful, but applying it to an open problem still requires establishing the condition for the particular number under study.","scope_note":"Some criteria are sufficient conditions; stronger characterisations give both necessary and sufficient conditions.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/68/erdos-68-factorial-denominator-irrationality.tex::line 500.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":978,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::irrationality_criterion","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/68/erdos-68-factorial-denominator-irrationality.tex::line 500","role":"mathematical_source","gloss":"Published manuscript context for this mathematical term."}],"evidence_commands":[],"used_in":{"total_occurrences":2,"page_count":1,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":379,"id":"irrationality_exponent","label":"irrationality exponent","preferred_label":"irrationality exponent","alternative_labels":["irrationality measure exponent"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Diophantine approximation. Distinct from merely proving irrationality.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-irrationality-exponent","relative_href":"docs/glossary.html#glossary-irrationality-exponent","public_phrase":"irrationality exponent","short_definition":"The irrationality exponent is the supremum of the approximation powers achieved infinitely often by rational approximations to an irrational number.","reader_preview":"The irrationality exponent is the supremum of the approximation powers achieved infinitely often by rational approximations to an irrational number.","reader_preview_basis":"definition","card_definition":"For an irrational \\(\\xi\\), consider exponents \\(\\nu\\) for which \\(|\\xi -p/q| < 1/q^\\nu\\) holds for infinitely many coprime integers \\(p,q\\) with \\(q > 0\\). The supremum of those \\(\\nu\\) is at least 2 and can be infinite. Larger values allow unusually close rational approximations.","reader_rule":"A supremum need not itself satisfy the defining inequality infinitely often, and a finite approximation sample cannot establish the exponent.","deep_definition":"Lambert-series approximation estimates yield explicit upper bounds on how well the target values can be approximated.","scope_note":"A supremum need not itself satisfy the defining inequality infinitely often, and a finite approximation sample cannot establish the exponent.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex::line 291.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1010,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::irrationality_exponent","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex::line 291","role":"mathematical_source","gloss":"Published manuscript context for this term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":380,"id":"irreducible_polynomial","label":"irreducible polynomial","preferred_label":"irreducible polynomial","alternative_labels":["irreducible polynomials","polynomial irreducibility"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the reader bands identify the ambient field, index set, convergence requirement, or theorem hypotheses relevant to the cited manuscript.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-irreducible-polynomial","relative_href":"docs/glossary.html#glossary-irreducible-polynomial","public_phrase":"irreducible polynomial","short_definition":"An irreducible polynomial over a field is a nonconstant polynomial that cannot be factored into two polynomials of positive degree over that field.","reader_preview":"An irreducible polynomial over a field is a nonconstant polynomial that cannot be factored into two polynomials of positive degree over that field.","reader_preview_basis":"definition","card_definition":"The polynomial \\(T^{2}-2\\) is irreducible over \\(\\mathbb{Q}\\) but factors over \\(\\mathbb{R}\\). Irreducibility therefore always refers to a specified coefficient field.","reader_rule":"For degrees two and three over a field, irreducibility is equivalent to having no root in that field. This root test is insufficient in higher degrees.","deep_definition":"The #243 argument excludes a rational root of its cubic by reducing that root modulo suitable persistent primes and contradicting coprimality. The cubic root test then supplies irreducibility, allowing the proof to pass to a degree-three number field.","scope_note":"For degrees two and three over a field, irreducibility is equivalent to having no root in that field. This root test is insufficient in higher degrees.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::sec:reduction.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1025,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::irreducible_polynomial","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::sec:reduction","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":381,"id":"jacobi_matrix","label":"Jacobi matrix","preferred_label":"Jacobi matrix","alternative_labels":["symmetric Jacobi matrix","Jacobi operator matrix"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The symmetric orthonormal-polynomial convention is intended; generic tridiagonal matrices and Jacobian matrices are different objects.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-jacobi-matrix","relative_href":"docs/glossary.html#glossary-jacobi-matrix","public_phrase":"Jacobi matrix","short_definition":"The tridiagonal matrix representing multiplication by the variable in an orthonormal polynomial basis.","reader_preview":"The tridiagonal matrix representing multiplication by the variable in an orthonormal polynomial basis.","reader_preview_basis":"authored_override","card_definition":"For orthonormal polynomials on the real line, \\(xp_n=a_{n+1}p_{n+1}+b_n p_n+a_n p_{n-1}\\). The associated real symmetric matrix has \\(b_n\\) on the diagonal and positive \\(a_n\\) on neighbouring diagonals. Its finite truncations give quadrature nodes as eigenvalues.","reader_rule":"This row uses the symmetric orthonormal convention. A differently normalized polynomial basis can yield a nonsymmetric tridiagonal recurrence matrix. The Jacobi matrix is also distinct from a Jacobian matrix of partial derivatives.","deep_definition":"The #1049 finite moment construction becomes a Jacobi matrix after choosing an orthonormal polynomial basis. The paper explicitly distinguishes this positive-node quadrature construction from a different coefficient matrix pencil whose roots can include negative values.","scope_note":"This row uses the symmetric orthonormal convention. A differently normalized polynomial basis can yield a nonsymmetric tridiagonal recurrence matrix. The Jacobi matrix is also distinct from a Jacobian matrix of partial derivatives.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex::line 2926.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1117,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::jacobi_matrix","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex::line 2926","role":"mathematical_source","gloss":"Published manuscript use of Jacobi matrix with the local hypotheses and conclusion stated in the referenced passage."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":382,"id":"jacobian_matrix","label":"jacobian matrix","preferred_label":"jacobian matrix","alternative_labels":["derivative matrix"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Some prose says Jacobian when it means the Jacobian determinant; preserve the distinction unless the context clearly uses the scalar determinant.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-jacobian-matrix","relative_href":"docs/glossary.html#glossary-jacobian-matrix","public_phrase":"jacobian matrix","short_definition":"The matrix of all first partial derivatives of a map between finite-dimensional coordinate spaces.","reader_preview":"The matrix of all first partial derivatives of a map between finite-dimensional coordinate spaces.","reader_preview_basis":"definition","card_definition":"For \\(F(x,y) = (x+y,xy)\\), the Jacobian has rows \\((1,1)\\) and \\((y,x)\\). When F is differentiable, this matrix represents the linear map describing its first-order local change; the existence of individual partial derivatives alone is not enough.","reader_rule":"For n inputs and m outputs, the Jacobian is \\(m\\times n\\). A square Jacobian with nonzero determinant gives a local differentiable inverse when the map is continuously differentiable nearby.","deep_definition":"Each entry records how one output changes with one input coordinate. The inverse function theorem uses both local regularity and a nonzero determinant. A zero determinant does not exclude every kind of local inverse: \\(x \\mapsto x^{3}\\) is invertible near zero, but its inverse is not differentiable there. Some prose uses “Jacobian” for the determinant; the matrix and that scalar must be distinguished. Hessians instead collect second derivatives of a scalar-valued function.","scope_note":"For n inputs and m outputs, the Jacobian is \\(m\\times n\\). A square Jacobian with nonzero determinant gives a local differentiable inverse when the map is continuously differentiable nearby.","example":"For \\(F(x,y)=(x+y,xy)\\), the Jacobian has rows \\((1,1)\\) and \\((y,x)\\).","example_route":null,"non_example":"A list of partial derivatives that does not give the full linear approximation need not establish differentiability.","visible_text_byte_count":953,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::jacobian_matrix","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":383,"id":"javascript","label":"JavaScript","preferred_label":"JavaScript","alternative_labels":["JS"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The programming language used for browser and server scripts, distinct from HTML document structure and CSS presentation.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-javascript","relative_href":"docs/glossary.html#glossary-javascript","public_phrase":"JavaScript","short_definition":"JavaScript is a programming language used in web browsers and many server and tooling environments.","reader_preview":"JavaScript is a programming language used in web browsers and many server and tooling environments.","reader_preview_basis":"definition","card_definition":"Browsers use JavaScript to respond to actions such as clicks, update a page, and communicate through web APIs. Other environments use the same language for servers, scripts, and development tools. Its core language is standardized as ECMAScript.","reader_rule":"JavaScript being enabled permits code to run; it does not establish that the code is safe, correct, or necessary.","deep_definition":"The host environment supplies facilities beyond the core language: a browser provides documents, events, and timers, while a server runtime can provide file or network access. Plectis uses JavaScript to add glossary previews, search, maps, and navigation controls to generated HTML. Direct links and readable content remain useful when those enhancements are unavailable. JavaScript and Java are separate languages.","scope_note":"JavaScript being enabled permits code to run; it does not establish that the code is safe, correct, or necessary.","example":"A script listens for Escape and closes a glossary tooltip without reloading the page.","example_route":null,"non_example":"HTML that displays a paragraph is document markup, not JavaScript.","visible_text_byte_count":790,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::javascript","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/index.html","role":"reading_context","gloss":"Direct files, if a button fails or JavaScript is off"}],"evidence_commands":[],"used_in":{"total_occurrences":2,"page_count":2,"top_page":"lineage.html","top_pages":[{"page":"lineage.html","count":1},{"page":"paper-module-batch7-macro-engines-capsule.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":384,"id":"jitter","label":"jitter","preferred_label":"jitter","alternative_labels":["retry jitter","randomised retry delay","randomized retry delay"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Jitter can be unwanted timing variation in a network or deliberate variation in a retry schedule. Random changes to message content are neither timing sense.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-jitter","relative_href":"docs/glossary.html#glossary-jitter","public_phrase":"jitter","short_definition":"Jitter is variation in timing. In a retry schedule, deliberate random variation spreads requests out so clients do not all retry together.","reader_preview":"Jitter is variation in timing.","reader_preview_basis":"first_sentence","card_definition":"In networking, jitter often means unwanted variation in packet arrival times. Retry jitter deliberately varies waiting times: clients that failed together choose different delays, reducing another simultaneous traffic spike.","reader_rule":"Distinguish unwanted timing variation from the deliberate randomness used to spread retry attempts.","deep_definition":"Retry jitter is often combined with exponential backoff. One scheme draws a delay anywhere between zero and the current backoff limit; another varies around a planned delay. The variation need not be small. Plectis provider-routing discussions use this scheduling sense, where several clients may otherwise retry together after a shared rate limit. The delay changes timing; it does not make an unsafe operation safe to repeat.","scope_note":"Distinguish unwanted timing variation from the deliberate randomness used to spread retry attempts.","example":"One hundred workers each choose a different random delay within the next eight seconds.","example_route":null,"non_example":"Adding random data to a request payload is not retry jitter.","visible_text_byte_count":874,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::jitter","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/docs/paper-module-engine-room-annex-knowledge-router.html","role":"prerequisite_context","gloss":"provider rate-limit handling uses a back-off route across services"}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":385,"id":"jordan_domain","label":"jordan domain","preferred_label":"jordan domain","alternative_labels":["Jordan region"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Planar topology and complex analysis. Named for Camille Jordan; unrelated to Jordan normal form.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-jordan-domain","relative_href":"docs/glossary.html#glossary-jordan-domain","public_phrase":"jordan domain","short_definition":"A Jordan domain is the bounded open region enclosed by a simple closed curve in the plane.","reader_preview":"A Jordan domain is the bounded open region enclosed by a simple closed curve in the plane.","reader_preview_basis":"definition","card_definition":"The inside of a circle or any non-self-intersecting closed loop is a Jordan domain. A figure-eight boundary is excluded because it crosses itself and encloses more than one lobe rather than one simple inside region.","reader_rule":"A Jordan domain is the bounded open component enclosed by a simple closed plane curve. 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The public explanation also distinguishes Lean's externally defined proof-checking kernel; operating-system, statistical, and sequence kernels have other meanings.","meaning_contrasts":["lean_kernel"],"status":"active","disposition":"active","anchor":"glossary-kernel","relative_href":"docs/glossary.html#glossary-kernel","public_phrase":"repo control query spine","short_definition":"Here, kernel can mean Lean's proof checker or the repository's kernel.py command router.","reader_preview":"Here, kernel can mean Lean's proof checker or the repository's kernel.py command router.","reader_preview_basis":"authored_override","card_definition":"In Lean, the kernel checks terms against a small core type theory. In ai_workflow, kernel.py is a command entry point selecting files, records, tools, and state views. 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In particular, if no nonzero integer relation among the frequencies exists, the flow is dense in the whole torus.","reader_rule":"For a discrete orbit \\(n\\alpha\\), relations \\(k\\cdot \\alpha\\) that are integers must also be respected. Full density then requires rational independence of 1 and the coordinates of \\(\\alpha\\). The continuous and discrete forms must not be interchanged.","deep_definition":"The #269 proof uses the continuous form to identify its orbit closure as a subtorus determined by the rational relations among reciprocal logarithms. It then searches inside that subtorus. Qualitative approximation gives arbitrarily close returns; a numerical return-time bound would require additional estimates.","scope_note":"For a discrete orbit \\(n\\alpha\\), relations \\(k\\cdot \\alpha\\) that are integers must also be respected. Full density then requires rational independence of 1 and the coordinates of \\(\\alpha\\). 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A reader meets it most directly on the architecture map, where the parts are shown joined together, and implicitly whenever one page links across to a different kind of artifact.","example":"The architecture map on this site is a view of the lattice: it lays out the system's planes and units and shows the edges that connect a standard to the skill that applies it, or a route to the component it reaches.","example_route":{"href":"architecture.html","label":"How it fits together (architecture map)","gloss":"see the typed planes and units joined into one map"},"non_example":"A lattice is not a single diagram or a rigid hierarchy of layers stacked one above another. It is the typed web of relationships among the system's parts, where any kind of artifact can link to any other.","visible_text_byte_count":1408,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::lattice","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"contains","target":"doctrine","gloss":"Doctrine is one plane inside the larger system lattice."},{"verb":"supports","target":"lattice_transposition","gloss":"Transposition moves lessons through lattice coordinates."},{"verb":"projected_by","target":"hologram","gloss":"Navigation holograms expose lattice rows under bounded context."},{"verb":"uses","target":"routing","gloss":"Routing chooses paths through the lattice."}],"source_refs":[{"ref":"codex/doctrine/paper_modules/navigation_hologram_theory.md","role":"paper_module","gloss":"Whole-system lattice and affordance-machine theory roof."},{"ref":"codex/doctrine/skills/doctrine/local_to_general_propagation.md","role":"skill","gloss":"Lattice transposition and cross-plane propagation workflow."}],"evidence_commands":[],"used_in":{"total_occurrences":127,"page_count":108,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":8},{"page":"doctrine.html","count":6},{"page":"paper-module-formal-math-readiness-gate.html","count":3},{"page":"paper-module-bounded-autonomy-campaign-packet.html","count":2},{"page":"paper-module-executable-doctrine-grammar.html","count":2},{"page":"paper-module-first-screen-composition-root.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":396,"id":"lattice_transposition","label":"lattice transposition","preferred_label":"lattice transposition","alternative_labels":["lattice-transposition","generalized up-propagation","generalised up-propagation"],"hidden_search_labels":["portable invariant routing","cross-plane up-propagation"],"term_kind":"propagation","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"Lattice transposition re-expresses one governed meaning in another context while preserving its authority and relationships; it is not matrix transposition or unconstrained rewriting.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-lattice-transposition","relative_href":"docs/glossary.html#glossary-lattice-transposition","public_phrase":"portable invariant routing","short_definition":"Lattice transposition moves a reusable lesson into the kind of system record that can apply it to future work.","reader_preview":"Lattice transposition moves a reusable lesson into the kind of system record that can apply it to future work.","reader_preview_basis":"definition","card_definition":"A local failure may reveal a missing instruction, validation rule, or concept. Plectis retains the original case and places the general refinement in the narrowest existing skill, standard, or other owner that can carry it.","reader_rule":"The lesson's scope must survive the move.","deep_definition":"For example, a faulty glossary preview may justify changing the shared glossary rule. It does not automatically justify changing how every kind of document is written.","scope_note":"The term covers how the system turns a one-off lesson into a durable, reusable rule routed to the right governing plane. A reader meets it in the doctrine and routing material, where rules are shown alongside the cases that produced them and the boundaries that limit them.","example":"The doctrine card on provenance propagation and non-interference (AX-8) is a concrete instance: it is the rule that a lesson may move between planes only while carrying its origin and without contaminating the plane it enters, which is exactly the discipline lattice transposition follows.","example_route":{"href":"doctrine.html#dcard-ax-8","label":"Provenance propagation and non-interference","gloss":"open the doctrine rule that governs how a lesson moves between planes"},"non_example":"Lattice transposition is not copying a snippet of code or text from one place to another, and it is not a vague claim that everything in the system is connected. It is the specific, accountable act of lifting a lesson to its underlying rule and routing that rule to the plane that owns it, with the source case kept on record.","visible_text_byte_count":1614,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::lattice_transposition","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"specializes","target":"up_propagate","gloss":"Adds cross-plane invariant transfer to the upward routing move."},{"verb":"aliases_conceptually","target":"generalise_up_propagate","gloss":"Names the generalized up-propagation move."},{"verb":"moves_through","target":"lattice","gloss":"Transposition traverses typed edges of the system lattice."},{"verb":"may_deposit_to","target":"standard","gloss":"Schema-grade transpositions belong in standards."},{"verb":"may_deposit_to","target":"peer_propagate","gloss":"Immature cross-plane lessons can stage as PEER rows."}],"source_refs":[{"ref":"codex/doctrine/skills/doctrine/local_to_general_propagation.md","role":"skill","gloss":"Closeout skill defines lattice transposition and its owning-plane discipline."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":397,"id":"launch_scope_decision","label":"launch-scope decision","preferred_label":"launch-scope decision","alternative_labels":["launch-scope"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"public_contextual_gloss","meaning_boundary":"A launch-scope decision is the separate operator or release-owner judgment to host, ship, or share something; no component check or passing fixture possesses that authority.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-launch-scope-decision","relative_href":"docs/glossary.html#glossary-launch-scope-decision","public_phrase":"launch-scope decision","short_definition":"A launch-scope decision determines what is ready and authorised to be published or made available.","reader_preview":"A launch-scope decision determines what is ready and authorised to be published or made available.","reader_preview_basis":"definition","card_definition":"It considers the intended audience, included material, evidence, and unresolved limitations. In Plectis, component checks provide information for this decision but do not independently grant publication authority.","reader_rule":"A passing technical check answers its own conditions, not every question about release readiness.","deep_definition":"The person responsible for the release must make the decision within the authorised scope. A published selection should then match the material and evidence on which that decision was based.","scope_note":"Launch-scope decisions are outside every component's authority on this site.","example":"A step claiming a component is ready to host is refused with the release-overclaim finding code, even when the underlying check genuinely passes.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":934,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::launch_scope_decision","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"codex/standards/std_microcosm.json","role":"source_authority","gloss":"Grounds this Plectis-specific definition and its scope."},{"ref":"microcosm-substrate/src/microcosm_core/organs/batch12_release_claim_language_gate.py","role":"source_authority","gloss":"Grounds this Plectis-specific definition and its scope."}],"evidence_commands":[],"used_in":{"total_occurrences":514,"page_count":136,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":139},{"page":"paper-modules.html","count":114},{"page":"components.html","count":22},{"page":"area-import-drift.html","count":7},{"page":"area-agent-reliability.html","count":6},{"page":"paper-module-batch8-validator-checker-capsule.html","count":6}]}},{"schema":"plectis_public_glossary_term_v0","position":398,"id":"lay_person","label":"lay person","preferred_label":"lay person","alternative_labels":["lay people"],"hidden_search_labels":["non-specialist","layperson"],"term_kind":"public_projection","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"public_contextual_gloss","meaning_boundary":"Here a lay person is an intended collaborator on the public maths work, not a reader the site talks down to. The phrase does not certify lack of skill in every field.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-lay-person","relative_href":"docs/glossary.html#glossary-lay-person","public_phrase":"lay person","short_definition":"A lay person is someone who is not a specialist in the subject being discussed.","reader_preview":"A lay person is someone who is not a specialist in the subject being discussed.","reader_preview_basis":"definition","card_definition":"The term concerns subject knowledge, not intelligence or general ability. A mathematician can be a lay reader of software architecture, just as an experienced programmer can be new to number theory.","reader_rule":"The glossary aims to explain unfamiliar terms without requiring specialist training.","deep_definition":"Readers can still contribute useful questions, corrections, or work with tools. 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Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-lean","relative_href":"docs/glossary.html#glossary-lean","public_phrase":"Lean","short_definition":"Lean is a programming language and proof assistant used to state mathematics and check formal proofs.","reader_preview":"Lean is a programming language and proof assistant used to state mathematics and check formal proofs.","reader_preview_basis":"definition","card_definition":"Authors supply definitions, theorem statements, and proofs, often with tactics that help construct the proof object. Lean then checks that the object establishes the encoded proposition under its assumptions.","reader_rule":"Successful checking applies to that precise formal statement.","deep_definition":"Admissions, added axioms, or an incorrectly translated problem can limit what acceptance establishes. The papers therefore need readable statements and evidence of the relevant checks alongside the Lean source.","scope_note":"Lean establishes that a proof is valid for the statement as written, under the declared axioms.","example":"The Erdős problem work here is written in Lean, so each theorem's range is exactly what the source states.","example_route":null,"non_example":"A carefully written paper argument is not a Lean result until the assistant has accepted a precise form of it.","visible_text_byte_count":839,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::lean","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":927,"page_count":77,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":88},{"page":"paper-modules.html","count":71},{"page":"paper-module-lean-proof-search-lab-runtime.html","count":68},{"page":"area-formal-math.html","count":59},{"page":"components.html","count":59},{"page":"paper-module-formal-math-lean-proof-witness.html","count":48}]}},{"schema":"plectis_public_glossary_term_v0","position":400,"id":"lean_kernel","label":"Lean kernel","preferred_label":"Lean kernel","alternative_labels":["kernel acceptance","pinned kernel"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"externally_anchored_term","meaning_scope_label":"Externally anchored","definition_profile":"public_contextual_gloss","meaning_boundary":"Lean's implementation and reference control this term. 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An admitted statement can type-check because its gap is represented by an axiom.","deep_definition":"This separation lets complex automation propose proofs without requiring the kernel to reproduce the search. Readers still assess whether the encoded statement matches the intended mathematics. The repository's kernel.py supplies navigation and has no such proof-checking role.","scope_note":"A checker for Lean's core language, distinct from tactics, compilation, and repository navigation.","example":"A proof the Lean kernel accepts is machine-checked; its accompanying prose is not.","example_route":null,"non_example":"The command-line program kernel.py on the software pages is not the Lean kernel.","visible_text_byte_count":816,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::lean_kernel","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":11,"page_count":3,"top_page":"paper-module-batch4-proof-authority-runtime.html","top_pages":[{"page":"paper-module-batch4-proof-authority-runtime.html","count":6},{"page":"paper-module-engine-room-lean-proof-search-lab.html","count":4},{"page":"paper-module-certificate-kernel-execution-lab.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":401,"id":"lean_standard_axioms","label":"Lean's standard axioms","preferred_label":"Lean's standard axioms","alternative_labels":["propext, Quot.sound, Classical.choice"],"hidden_search_labels":["standard axioms","added axioms"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"This row explains the three permitted mathematical foundations in the local proof audit. 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Different parametrizations of the same traced curve have the same length under standard conditions.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-curve-length","relative_href":"docs/glossary.html#glossary-curve-length","public_phrase":"length of a curve","short_definition":"A curve’s length is the supremum of the lengths of polygonal paths through successive points on the curve.","reader_preview":"A curve’s length is the supremum of the lengths of polygonal paths through successive points on the curve.","reader_preview_basis":"definition","card_definition":"A segment from (0,0) to \\((3,4)\\) has length 5. For a parametrised curve, each ordered partition gives a polygonal approximation. Curve length is the supremum over all these lengths; suitable successive partitions approach that supremum, whereas an arbitrary choice of samples need not.","reader_rule":"A continuous curve may have infinite length; finite-length curves are called rectifiable. 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Retracing a segment counts its length again, even though no new points are added.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::line 122.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1041,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::curve_length","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::line 122","role":"mathematical_source","gloss":"Published manuscript context for this term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":407,"id":"level_curve","label":"level curve","preferred_label":"level curve","alternative_labels":["level curves","contour line"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The mathematical meaning stated here, with the ambient space and regularity assumptions made explicit.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-level-curve","relative_href":"docs/glossary.html#glossary-level-curve","public_phrase":"level curve","short_definition":"A level curve is a curve along which a real-valued function has a constant value.","reader_preview":"A level curve is a curve along which a real-valued function has a constant value.","reader_preview_basis":"definition","card_definition":"Contour lines on a height map connect points at the same height. For \\(g(x,y)=x^{2}+y^{2}\\), the level at 1 is the unit circle. For \\(g(x,y)=x^{2}-y^{2}\\), the level at 0 instead crosses at the origin, so a level set need not form one smooth curve.","reader_rule":"For a continuously differentiable planar function, a nonzero gradient at a point gives a continuously differentiable local level curve through that point.","deep_definition":"The papers study curves where a polynomial’s magnitude has a fixed value. Away from critical points, local regularity gives a well-defined tangent, perpendicular to the gradient of the real-valued magnitude function when that gradient is defined. Critical levels can have crossings or changes in the number of components. 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For a continuously differentiable planar function, a nonzero gradient gives a smooth local curve through that point.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::res:locus.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":896,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::level_set","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::res:locus","role":"mathematical_source","gloss":"Published paper context for the definition and its stated boundary."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":409,"id":"limit","label":"limit","preferred_label":"limit","alternative_labels":["evidence limit"],"hidden_search_labels":["evidence limits"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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No term needs equal the result, and a one-time early minimum is irrelevant.","deep_definition":"A finite liminf for consecutive prime gaps says gaps at most slightly above that level recur arbitrarily far out. It does not bound every sufficiently late gap, so large gaps may coexist with the bounded recurring subsequence.","scope_note":"Form the infimum of each tail \\(\\{a_{k}:k\\ge n\\}\\); these tail infima increase, and their limit is liminf \\(a_{n}\\), possibly infinite. 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A mechanism can exist before it is fully understood or documented.","scope_note":"The term covers the \"how it works\" layer that connects concepts, principles, skills, standards, and code. A reader meets it across the doctrine pages, where each rule is expected to be backed by an explanation of the process that carries it out.","example":"The Executable Doctrine Grammar component is a mechanism: it is the surface that requires a piece of doctrine to be run as actual steps before it is treated as authoritative, so the rule is bound to a working process rather than to wording alone.","example_route":{"href":"components.html#component-executable_doctrine_grammar","label":"Executable Doctrine Grammar","gloss":"open the component that turns a rule into runnable steps"},"non_example":"Saying “it detects drift” names a behaviour but leaves the mechanism unexplained.","visible_text_byte_count":1118,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::mechanism","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"belongs_to","target":"doctrine","gloss":"Mechanisms are doctrine nodes."},{"verb":"operates_on","target":"concept","gloss":"Mechanisms describe movement around concepts."}],"source_refs":[{"ref":"codex/standards/principles/std_mechanism.json","role":"standard","gloss":"Mechanism schema standard."},{"ref":"codex/doctrine/skills/doctrine/concept_mechanism_curation.md","role":"skill","gloss":"Concept/mechanism curation workflow."}],"evidence_commands":[],"used_in":{"total_occurrences":801,"page_count":121,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":246},{"page":"reference.html","count":109},{"page":"paper-module-batch10-governance-compilers-capsule.html","count":30},{"page":"paper-module-batch11-saturation-engines-capsule.html","count":26},{"page":"paper-module-batch4-proof-authority-runtime.html","count":24},{"page":"doctrine.html","count":18}]}},{"schema":"plectis_public_glossary_term_v0","position":442,"id":"mechanism_contract","label":"mechanism contract","preferred_label":"mechanism contract","alternative_labels":["mechanism contracts"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"public_contextual_gloss","meaning_boundary":"On the tour, a mechanism contract is the public component card's declared shape. 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A quotient may have removable points after common factors cancel, so its unreduced denominator alone does not locate genuine poles.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex::line 2550.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1086,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::meromorphic_function","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex::line 2550","role":"mathematical_source","gloss":"Published manuscript use of meromorphic function with the local hypotheses and conclusion stated in the referenced passage."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":444,"id":"mersenne_number","label":"Mersenne number","preferred_label":"Mersenne number","alternative_labels":["Mersenne"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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A reader meets it whenever a page notes that a surface is generated, populated, or kept in sync rather than authored directly.","example":"The Engine Room Metabolism Runtime is a concrete instance: the working machinery that runs these upkeep loops over the underlying material and keeps derived surfaces current.","example_route":{"href":"paper-modules.html#paper-module-engine-room-metabolism-runtime","label":"Engine Room Metabolism Runtime","gloss":"open the runtime that runs the upkeep loops"},"non_example":"Metabolism here is not biological metabolism and not a vague word for activity in general. 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These are power moments, not necessarily moments about the mean of a probability distribution.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":917,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::moment_sequence","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex","role":"mathematical_source","gloss":"Primary manuscript context and mathematical application."}],"evidence_commands":[],"used_in":{"total_occurrences":8,"page_count":1,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":8}]}},{"schema":"plectis_public_glossary_term_v0","position":458,"id":"monic","label":"monic","preferred_label":"monic","alternative_labels":["monic polynomial"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-monic","relative_href":"docs/glossary.html#glossary-monic","public_phrase":"monic","short_definition":"A polynomial is monic if its leading coefficient is 1.","reader_preview":"A polynomial is monic if its leading coefficient is 1.","reader_preview_basis":"definition","card_definition":"The leading coefficient multiplies its highest-degree nonzero term. For example, \\(z^{3} + 2z - 7\\) is monic, while \\(3z^{3} + 2z - 7\\) is not.","reader_rule":"This normalisation concerns the polynomial's scale, not the positions of its roots alone.","deep_definition":"In lemniscate questions, multiplying a polynomial by a constant changes sets defined by \\(|p(z)| < 1\\) or \\(|p(z)| = 1\\). A monic hypothesis therefore matters to geometric statements about those sets.","scope_note":"Monic in the leading-coefficient-1 sense, over the complex numbers unless a page says otherwise.","example":"\\(z^{2} - 1\\) is monic; \\(2z^{2} - 2\\) has the same roots and is not.","example_route":null,"non_example":"A monic person or a monic inscription is not this term.","visible_text_byte_count":642,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::monic","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":1,"page_count":1,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":459,"id":"monodromy","label":"monodromy","preferred_label":"monodromy","alternative_labels":[],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Monodromy records the effect of following a loop. The monodromy theorem is a separate result giving conditions for path independence.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-monodromy","relative_href":"docs/glossary.html#glossary-monodromy","public_phrase":"monodromy","short_definition":"Monodromy describes how a locally defined branch changes when analytically continued around loops; the monodromy theorem gives path independence under suitable simply connected hypotheses.","reader_preview":"Monodromy describes how a locally defined branch changes when analytically continued around loops; the monodromy theorem gives path independence under suitable simply connected hypotheses.","reader_preview_basis":"definition","card_definition":"A square root chosen near 1 returns with the opposite sign after continuation once around 0, showing nontrivial monodromy on the punctured plane. On a simply connected region avoiding 0, a consistent square-root branch can be chosen.","reader_rule":"Local uniqueness of analytic continuation need not give one branch around every loop. The monodromy theorem gives conditions for path independence; monodromy describes what happens when path independence fails as well.","deep_definition":"The paper invokes monodromy to turn a local inverse into one branch on a disc after excluding finite branch points and escape. The term names the global compatibility step, not merely the act of extending a function.","scope_note":"Local uniqueness of analytic continuation need not give one branch around every loop. The monodromy theorem gives conditions for path independence; monodromy describes what happens when path independence fails as well.","example":"A square root chosen near 1 returns with the opposite sign after continuation once around 0, showing nontrivial monodromy on the punctured plane.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1105,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::monodromy","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::line 513","role":"mathematical_source","gloss":"the monodromy theorem continues"}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":460,"id":"monodromy_theorem","label":"monodromy theorem","preferred_label":"monodromy theorem","alternative_labels":["analytic monodromy theorem"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Classical one-complex-variable analytic continuation. The theorem is a path-independence result under explicit continuation and simple-connectivity hypotheses; monodromy itself is the broader phenomenon it controls.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-monodromy-theorem","relative_href":"docs/glossary.html#glossary-monodromy-theorem","public_phrase":"monodromy theorem","short_definition":"The monodromy theorem says that a holomorphic function defined near one starting point extends to a single-valued function on a simply connected domain if it can be continued along every path there.","reader_preview":"The monodromy theorem says that a holomorphic function defined near one starting point extends to a single-valued function on a simply connected domain if it can be continued along every path there.","reader_preview_basis":"definition","card_definition":"A logarithm chosen near 1 can be continued consistently throughout the slit plane \\(\\mathbb{C}\\setminus(-\\infty,0]\\), which is simply connected and avoids 0. On the punctured plane \\(\\mathbb{C}\\setminus\\{0\\}\\), going once around 0 changes a continued logarithm by \\(2\\pi i\\), so the conclusion fails because the domain has a hole.","reader_rule":"Fix a base point and an initial holomorphic germ. One must know that analytic continuation exists along every path from that base point inside the domain. Simple connectivity then makes continuations along paths with the same endpoint agree. Local invertibility at the starting point alone supplies neither global continuation nor path independence.","deep_definition":"A germ means a function specified near a point, with two such descriptions considered equivalent when they agree on some smaller neighbourhood. Continuations along paths deformable into one another with endpoints fixed give the same final germ, under the continuation hypotheses. In a simply connected domain, paths with the same endpoints are related this way, so the local functions agree and assemble into one holomorphic function. Singularities, escape of inverse branches, or failure of continuation along even one path can prevent the theorem from applying.","scope_note":"Monodromy may be nontrivial; the monodromy theorem gives conditions under which analytic continuation is independent of the chosen path.","example":"A local logarithm at 1 becomes a single-valued holomorphic logarithm on the slit plane after continuation along all paths there.","example_route":null,"non_example":"Continuing a square root once around 0 in the punctured plane can reverse its sign, so no single-valued conclusion follows on that domain.","visible_text_byte_count":1244,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::monodromy_theorem","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"controls","target":"monodromy","gloss":"Under its hypotheses, the theorem forces continuation to be path independent."},{"verb":"supports","target":"analytic_continuation","gloss":"Compatible continued germs assemble into one analytic function."},{"verb":"requires","target":"simply_connected","gloss":"Simple connectivity removes path classes that could carry different branches."}],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::line 513","role":"mathematical_source","gloss":"The manuscript invokes the monodromy theorem only after excluding finite branch points and escape to infinity over the bounded value domain."},{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 2124","role":"mathematical_source","gloss":"The reasoning record explicitly identifies absence of a global monodromy theorem as a gap in a rejected route."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":461,"id":"monotone_sequence","label":"monotone sequence","preferred_label":"monotone sequence","alternative_labels":["monotonic sequence"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordered sequences. 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A proof using only monotonicity cannot silently assume this stronger condition.","scope_note":"State whether equality is allowed and whether the direction is increasing or decreasing.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/257/erdos257-mersenne-reasoning-surface.tex::line 1642.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":719,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::monotone_sequence","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/257/erdos257-mersenne-reasoning-surface.tex::line 1642","role":"mathematical_source","gloss":"Published manuscript context for this term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":462,"id":"morse_chart","label":"Morse chart","preferred_label":"Morse chart","alternative_labels":["Morse coordinate chart"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The quadratic normal form applies at a nondegenerate critical point. A degenerate critical point need not admit such coordinates.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-morse-chart","relative_href":"docs/glossary.html#glossary-morse-chart","public_phrase":"Morse chart","short_definition":"A Morse chart is a local coordinate system near a nondegenerate critical point in which a smooth function becomes a quadratic form with plus and minus squares.","reader_preview":"A Morse chart is a local coordinate system near a nondegenerate critical point in which a smooth function becomes a quadratic form with plus and minus squares.","reader_preview_basis":"definition","card_definition":"Near a saddle on a surface, suitable coordinates make the function look like \\(c+x^{2}-y^{2}\\). Its level set at c crosses at the critical point, while nearby levels form branches whose connections change across that value.","reader_rule":"The Morse lemma requires a nondegenerate Hessian. The number of negative squares is the Morse index and does not depend on the chosen chart. A degenerate critical point need not admit this normal form.","deep_definition":"The paper uses a Morse chart around an index-one critical point to model the local saddle. The audit then notes that the local quadratic picture alone cannot determine how branches reconnect through a surrounding annulus.","scope_note":"The Morse lemma requires a nondegenerate Hessian. The number of negative squares is the Morse index and does not depend on the chosen chart. A degenerate critical point need not admit this normal form.","example":"Near a saddle on a surface, suitable coordinates make the function look like \\(c+x^{2}-y^{2}\\).","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1022,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::morse_chart","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos-1041-lemniscate-newton-flow.tex::line 738","role":"mathematical_source","gloss":"the proof invokes a Morse chart"}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":463,"id":"morse_saddle","label":"Morse saddle","preferred_label":"Morse saddle","alternative_labels":["interior morse saddle"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-morse-saddle","relative_href":"docs/glossary.html#glossary-morse-saddle","public_phrase":"Morse saddle","short_definition":"A Morse saddle is a critical point of a smooth real-valued function whose second-order form is nondegenerate and has both positive and negative directions.","reader_preview":"A Morse saddle is a critical point of a smooth real-valued function whose second-order form is nondegenerate and has both positive and negative directions.","reader_preview_basis":"definition","card_definition":"In two dimensions, the local behaviour resembles \\(x^{2} - y^{2}\\) after a suitable change of coordinates: the value rises in some directions and falls in others. Nondegeneracy rules out flat second-order directions.","reader_rule":"At a planar saddle, the level set through the point has four local half-branches.","deep_definition":"The lemniscate arguments use this local geometry. It must be distinguished from the behaviour of a singular flow or a degenerate critical point, where additional analysis is required.","scope_note":"Applies to a critical point of a smooth real function, in the sense of Morse theory.","example":"The origin is a Morse saddle for x squared minus y squared.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":890,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::morse_saddle","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":464,"id":"multinomial_coefficient","label":"multinomial coefficient","preferred_label":"multinomial coefficient","alternative_labels":["multinomial coefficients"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the reader bands identify the ambient field, index set, convergence requirement, or theorem hypotheses relevant to the cited manuscript.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-multinomial-coefficient","relative_href":"docs/glossary.html#glossary-multinomial-coefficient","public_phrase":"multinomial coefficient","short_definition":"A multinomial coefficient counts ways to divide a finite set into labelled groups of prescribed sizes.","reader_preview":"A multinomial coefficient counts ways to divide a finite set into labelled groups of prescribed sizes.","reader_preview_basis":"definition","card_definition":"For nonnegative integers \\(k_{1},\\ldots ,k_{r}\\) with total n, it equals \\(n!/(k_{1}!\\cdots k_{r}!)\\). For two groups this is the ordinary binomial coefficient.","reader_rule":"The group labels matter, but the order within each group does not. The factorial quotient is an integer because it counts these allocations.","deep_definition":"In #68, writing \\(n=kd+r\\) identifies \\(n!/(d!)^{k}\\) as \\(r!\\) times the multinomial coefficient with k groups of size d and one of size r. This directly proves the channel weight is integral, an essential input before small-tail estimates can force an integer contradiction.","scope_note":"The group labels matter, but the order within each group does not. The factorial quotient is an integer because it counts these allocations.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/68/erdos-68-factorial-denominator-irrationality.tex::sec:channels.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":947,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::multinomial_coefficient","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/68/erdos-68-factorial-denominator-irrationality.tex::sec:channels","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":465,"id":"multiple","label":"multiple","preferred_label":"multiple","alternative_labels":["integer multiple"],"hidden_search_labels":["multiples"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Plectis uses both ordinary senses of multiple: several items, and the mathematical divisibility relation. The surrounding grammar determines which one applies.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-multiple","relative_href":"docs/glossary.html#glossary-multiple","public_phrase":"multiple","short_definition":"In arithmetic, a multiple of an integer d is a number d·k for some integer k.","reader_preview":"In arithmetic, a multiple of an integer d is a number d·k for some integer k.","reader_preview_basis":"definition","card_definition":"For example, 12 is a multiple of 3 because \\(12 = 3\\cdot 4\\). A common multiple of several positive integers is divisible by every one of them.","reader_rule":"The adjective “multiple” also means “more than one” in ordinary prose.","deep_definition":"In phrases such as “multiple root”, it has a more specific algebraic role: a root occurs with multiplicity greater than one. The surrounding noun therefore matters when reading this word in a paper.","scope_note":"This entry disambiguates the ordinary quantity sense from the mathematical integer-multiple relation.","example":"Multiple LLM providers means several providers, while 12 is a multiple of 3 because \\(12 = 3 \\times 4\\).","example_route":null,"non_example":"The phrase multiple providers is not a divisibility statement.","visible_text_byte_count":728,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::multiple","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":3,"page_count":3,"top_page":"paper-module-annex-knowledge-routing.html","top_pages":[{"page":"paper-module-annex-knowledge-routing.html","count":1},{"page":"paper-module-engine-room-annex-knowledge-router.html","count":1},{"page":"papers.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":466,"id":"multiple_root","label":"multiple root","preferred_label":"multiple root","alternative_labels":["repeated root"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-multiple-root","relative_href":"docs/glossary.html#glossary-multiple-root","public_phrase":"multiple root","short_definition":"A multiple root of a nonzero polynomial is a root appearing with multiplicity at least two.","reader_preview":"A multiple root of a nonzero polynomial is a root appearing with multiplicity at least two.","reader_preview_basis":"definition","card_definition":"For example, \\(z^{2} - 2z + 1 = (z - 1)^{2}\\) has 1 as a double root. In general, multiplicity k means \\((z - a)^{k}\\) divides the polynomial but the next power does not.","reader_rule":"Over the real or complex numbers, every multiple root is also a root of the derivative.","deep_definition":"This changes local geometry and root counting. A polynomial's degree counts roots over the complex numbers with multiplicity, while the number of distinct root locations can be smaller.","scope_note":"Over the real or complex numbers, every multiple root is also a root of the derivative.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 95.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":848,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::multiple_root","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 95","role":"mathematical_source","gloss":"Published manuscript context for this mathematical term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":467,"id":"multiplicative_function","label":"multiplicative function","preferred_label":"multiplicative function","alternative_labels":["multiplicative arithmetic function"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Number theory. A property of a function, not merely multiplication appearing in its formula.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-multiplicative-function","relative_href":"docs/glossary.html#glossary-multiplicative-function","public_phrase":"multiplicative function","short_definition":"An arithmetic function f is multiplicative if \\(f(1) = 1\\) and \\(f(mn) = f(m)f(n)\\) whenever m and n are coprime.","reader_preview":"An arithmetic function f is multiplicative if \\(f(1) = 1\\) and \\(f(mn) = f(m)f(n)\\) whenever m and n are coprime.","reader_preview_basis":"definition","card_definition":"Euler’s totient is multiplicative: since 3 and 5 are coprime, \\(\\varphi (15) = \\varphi (3)\\varphi (5) = 2\\cdot 4 = 8\\). Sharing a prime changes matters: \\(\\varphi (9) = 6\\), whereas \\(\\varphi (3)\\varphi (3) = 4\\).","reader_rule":"An arithmetic function f is multiplicative when \\(f(1)=1\\) and \\(f(mn)=f(m)f(n)\\) for coprime m,n. Completely multiplicative functions obey the product rule without the coprimality restriction.","deep_definition":"The totient papers split values into independent local prime factors by arranging coprime inputs. The distinction between multiplicative and completely multiplicative prevents applying a valid coprime factorization across overlapping prime powers.","scope_note":"An arithmetic function f is multiplicative when \\(f(1)=1\\) and \\(f(mn)=f(m)f(n)\\) for coprime m,n. Completely multiplicative functions obey the product rule without the coprimality restriction.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/249/erdos249-totient-reasoning-surface.tex::line 2000.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":991,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::multiplicative_function","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/249/erdos249-totient-reasoning-surface.tex::line 2000","role":"mathematical_source","gloss":"Published manuscript context for this term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":468,"id":"multiplicative_order","label":"multiplicative order","preferred_label":"multiplicative order","alternative_labels":["modular order","order modulo"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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Not gradient flow in general.","example":"If a regular Newton-flow trajectory is defined for times from 0 to t and p′ does not vanish there, then \\(p(z(t)) = p(z(0))e^{-t}\\).","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":901,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::newton_flow","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":479,"id":"newton_identities","label":"Newton identities","preferred_label":"Newton identities","alternative_labels":["Newton's identities","Newton’s identities"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the reader bands identify the ambient field, index set, convergence requirement, or theorem hypotheses relevant to the cited manuscript.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-newton-identities","relative_href":"docs/glossary.html#glossary-newton-identities","public_phrase":"Newton identities","short_definition":"Newton identities relate the sums of powers of a polynomial’s roots to its coefficients, through the elementary symmetric sums.","reader_preview":"Newton identities relate the sums of powers of a polynomial’s roots to its coefficients, through the elementary symmetric sums.","reader_preview_basis":"definition","card_definition":"If a monic polynomial begins \\(T^{d}+c_{1}T^{d-1}+c_{2}T^{d-2}\\), and \\(s_{j}\\) is the sum of jth powers of its roots, then \\(s_{1}+c_{1}=0\\) and \\(s_{2}+c_{1}s_{1}+2c_{2}=0\\).","reader_rule":"The identities themselves hold over commutative rings. Solving them by division requires the relevant integers to be invertible; characteristic-zero fields permit this.","deep_definition":"In #243, the first three identities compute traces of z and \\(z^{-1}\\) from their cubic minimal polynomials. Substituting \\(2\\alpha =z+z^{-1}\\) connects those coefficient equations to the known traces of \\(\\alpha\\). These identities are distinct from Newton interpolation, which reconstructs a polynomial from values or differences.","scope_note":"The identities themselves hold over commutative rings. Solving them by division requires the relevant integers to be invertible; characteristic-zero fields permit this.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::sec:cubic-trace.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1017,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::newton_identities","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::sec:cubic-trace","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":480,"id":"newton_vector_field","label":"Newton vector field","preferred_label":"Newton vector field","alternative_labels":["complex Newton field"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-newton-vector-field","relative_href":"docs/glossary.html#glossary-newton-vector-field","public_phrase":"Newton vector field","short_definition":"The Newton vector field assigns the velocity \\(-p(z)/p'(z)\\) to points where the polynomial derivative p′ is nonzero.","reader_preview":"The Newton vector field assigns the velocity \\(-p(z)/p'(z)\\) to points where the polynomial derivative p′ is nonzero.","reader_preview_basis":"definition","card_definition":"A vector field prescribes a direction and speed at each allowed point. Its integral curves solve the Newton-flow equation; the field and the resulting trajectories are related but distinct objects.","reader_rule":"The quotient is undefined where p′ vanishes, even if a special case allows a separately defined extension.","deep_definition":"Along a regular trajectory, the polynomial value satisfies \\(w' = -w\\). This explains its exponential decay, while the existence, endpoints, and length of a global trajectory require additional arguments in the lemniscate papers.","scope_note":"The quotient is undefined where p′ vanishes, even if a special case allows a separately defined extension.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 254.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":944,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::newton_vector_field","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 254","role":"mathematical_source","gloss":"Published manuscript context for this mathematical term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":481,"id":"nonsingular_matrix","label":"nonsingular matrix","preferred_label":"nonsingular matrix","alternative_labels":["invertible matrix","non-singular matrix"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Linear algebra. Here singularity is algebraic, not a blow-up of a function.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-nonsingular-matrix","relative_href":"docs/glossary.html#glossary-nonsingular-matrix","public_phrase":"nonsingular matrix","short_definition":"A square matrix over a field is nonsingular when it has an inverse, equivalently when its determinant is nonzero.","reader_preview":"A square matrix over a field is nonsingular when it has an inverse, equivalently when its determinant is nonzero.","reader_preview_basis":"definition","card_definition":"The identity matrix is nonsingular. A square matrix with two equal columns is singular because different input vectors can collapse to the same output. Over the chosen field, nonsingularity means a two-sided inverse exists.","reader_rule":"For a square matrix over a field, the following are equivalent: nonzero determinant, full rank, independent columns, independent rows, and invertibility. The field matters because a determinant may be nonzero in one characteristic and zero in another.","deep_definition":"Nonsingular minors of every order certify arbitrarily large independent arithmetic families in the running-LCM paper. The term concerns algebraic invertibility; it is different from a singularity where a function blows up or ceases to be analytic.","scope_note":"For a square matrix over a field, the following are equivalent: nonzero determinant, full rank, independent columns, independent rows, and invertibility. The field matters because a determinant may be nonzero in one characteristic and zero in another.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/269/erdos-269-three-prime-running-lcm.tex::line 46.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1030,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::nonsingular_matrix","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/269/erdos-269-three-prime-running-lcm.tex::line 46","role":"mathematical_source","gloss":"Published manuscript context for this term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":482,"id":"norm","label":"norm","preferred_label":"norm","alternative_labels":["vector norm","function norm","matrix norm"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"For a complex number, norm often means its absolute value; in algebraic number theory norm can instead mean a field norm, a different construction.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-norm","relative_href":"docs/glossary.html#glossary-norm","public_phrase":"norm","short_definition":"A norm measures a vector’s size, is zero only at zero, scales with absolute scalar size, and satisfies the triangle inequality.","reader_preview":"A norm measures a vector’s size, is zero only at zero, scales with absolute scalar size, and satisfies the triangle inequality.","reader_preview_basis":"definition","card_definition":"For the vector \\((3,4)\\), the Euclidean norm is 5 while the supremum norm is 4, the largest absolute coordinate. Functions and matrices can also be vectors in a vector space and have norms that measure different aspects of their size.","reader_rule":"The norm must be specified. Some names coincide in a setting: the Frobenius and Hilbert–Schmidt norms agree for finite matrices.","deep_definition":"A norm satisfies \\(||ax|| = |a|\\cdot ||x||\\) and \\(||x+y|| \\le ||x||+||y||\\), with \\(||x|| = 0\\) exactly when \\(x = 0\\). It gives the distance \\(||x-y||\\). In a fixed finite-dimensional real or complex vector space, all norms give the same convergent sequences, although numerical bounds differ. That equivalence can fail in infinite-dimensional spaces. A field norm in algebraic number theory is a different construction and need not be nonnegative.","scope_note":"The norm must be specified. Some names coincide in a setting: the Frobenius and Hilbert–Schmidt norms agree for finite matrices.","example":"The Euclidean norm of \\((3,4)\\) is 5, while its supremum norm is 4.","example_route":null,"non_example":"The squared Euclidean length is not itself a norm because scaling by a doubles the exponent, producing \\(|a|\\) squared.","visible_text_byte_count":910,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::norm","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":1,"page_count":1,"top_page":"open-questions.html","top_pages":[{"page":"open-questions.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":483,"id":"normal_extension","label":"normal extension","preferred_label":"normal extension","alternative_labels":["normal field extension"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Normality is relative to the base field. A normal extension need not have all its intermediate fields normal over that base.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-normal-extension","relative_href":"docs/glossary.html#glossary-normal-extension","public_phrase":"normal extension","short_definition":"An algebraic extension \\(L/K\\) is normal when every irreducible polynomial over K having one root in L splits completely into linear factors over L.","reader_preview":"An algebraic extension \\(L/K\\) is normal when every irreducible polynomial over K having one root in L splits completely into linear factors over L.","reader_preview_basis":"definition","card_definition":"\\(Q(\\sqrt 2)/Q\\) is normal because the conjugate roots \\(\\pm \\sqrt 2\\) both lie in the field. \\(Q(\\sqrt[3] 2)/Q\\) is not normal because its two nonreal conjugate roots are missing.","reader_rule":"Normality is relative to K and requires an algebraic extension. It says that L contains every K-conjugate of each of its elements, not that every subfield or intermediate extension is automatically normal.","deep_definition":"Normality ensures that every embedding of L into an algebraic closure which fixes K carries L back onto itself. Equivalently, once an irreducible polynomial over K has a root in L, all of its roots lie there. Combined with separability, normality makes an algebraic extension Galois; in the finite case the number of K-fixing automorphisms equals the extension degree.","scope_note":"Normality is relative to K and requires an algebraic extension. It says that L contains every K-conjugate of each of its elements, not that every subfield or intermediate extension is automatically normal.","example":"\\(Q(\\sqrt 2)/Q\\) is normal because the conjugate roots \\(\\pm \\sqrt 2\\) both lie in the field.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":975,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::normal_extension","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex::line 280","role":"prerequisite_dependency","gloss":"The Galois-extension entry defines Galois as normal and separable."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":484,"id":"normal_form","label":"normal form","preferred_label":"normal form","alternative_labels":["normal forms"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-normal-form","relative_href":"docs/glossary.html#glossary-normal-form","public_phrase":"normal form","short_definition":"A normal form is a specified standard representation of an object, expression, statement, or record.","reader_preview":"A normal form is a specified standard representation of an object, expression, statement, or record.","reader_preview_basis":"definition","card_definition":"A rational number can be written in lowest terms with a positive denominator. Software can similarly put equivalent data into a standard shape for comparison. Some normal forms are unique; others permit several representations.","reader_rule":"The transformation must preserve the properties it claims to preserve, and uniqueness needs its own justification.","deep_definition":"The papers use normal forms to express exact remaining obligations. Data normalisation can aid comparison or validation. In either case, arranging the representation does not by itself establish the claim made about the object.","scope_note":"Covers mathematical, logical, and data representations under their stated transformation rules.","example":"Writing a fraction in lowest terms is a normal form for rational numbers.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":860,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::normal_form","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":485,"id":"normal_number","label":"normal number","preferred_label":"normal number","alternative_labels":["normality in a base","base-b normality"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the reader bands identify the ambient field, index set, convergence requirement, or theorem hypotheses relevant to the cited manuscript.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-normal-number","relative_href":"docs/glossary.html#glossary-normal-number","public_phrase":"normal number","short_definition":"A real number is normal in an integer base \\(b\\ge 2\\) when every finite digit block appears in its base-b expansion with its expected limiting frequency.","reader_preview":"A real number is normal in an integer base \\(b\\ge 2\\) when every finite digit block appears in its base-b expansion with its expected limiting frequency.","reader_preview_basis":"definition","card_definition":"A length-k block must have frequency \\(b^{-k}\\). Equal frequencies of individual digits alone establish only simple normality, not the full condition for every block length.","reader_rule":"Normality is relative to a specified base. Absolute normality requires normality in every integer base; irrationality by itself implies neither property.","deep_definition":"The #251 reasoning surface reports conditional normality for a prime-based series under explicit prime-pattern hypotheses. Changing b changes the value of that series as well as its expansion. The cited conditional result should not be read as unconditional normality of the dyadic number or absolute normality of one fixed number.","scope_note":"Normality is relative to a specified base. Absolute normality requires normality in every integer base; irrationality by itself implies neither property.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/251/erdos251-prime-gap-reasoning-surface.tex::line 475.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1005,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::normal_number","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/251/erdos251-prime-gap-reasoning-surface.tex::line 475","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":486,"id":"normalized_vanishing","label":"normalised vanishing","preferred_label":"normalised vanishing","alternative_labels":["normalized vanishing","vanishing after scaling"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-normalized-vanishing","relative_href":"docs/glossary.html#glossary-normalized-vanishing","public_phrase":"normalised vanishing","short_definition":"Normalised vanishing means that a quantity tends to zero after division by a specified nonzero scale.","reader_preview":"Normalised vanishing means that a quantity tends to zero after division by a specified nonzero scale.","reader_preview_basis":"definition","card_definition":"For example, n divided by \\(n^{2}\\) tends to zero although n itself grows. The property describes relative size, often written \\(a_{n}/b_{n} \\to 0\\), with the denominator's allowed values and sign specified.","reader_rule":"Changing the normalising scale changes the statement.","deep_definition":"In the reciprocal-tail work, a centred error is compared with an arithmetic scale. This relative smallness can combine with discrete constraints or lower bounds, but it does not alone imply that the unscaled error eventually equals zero.","scope_note":"Changing the normalising scale changes the statement.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::line 96.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":901,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::normalized_vanishing","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::line 96","role":"mathematical_source","gloss":"Published manuscript context for this mathematical term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":487,"id":"nowhere_dense","label":"nowhere dense","preferred_label":"nowhere dense","alternative_labels":["nowhere-dense"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-nowhere-dense","relative_href":"docs/glossary.html#glossary-nowhere-dense","public_phrase":"nowhere dense","short_definition":"A set is nowhere dense if its closure has empty interior.","reader_preview":"A set is nowhere dense if its closure has empty interior.","reader_preview_basis":"definition","card_definition":"On the real line, this means that even after adding all limit points, the set contains no nonempty open interval. Every open interval therefore contains a smaller open interval avoiding the closure.","reader_rule":"Having empty interior alone is weaker: the rational numbers contain no interval, but their closure is the whole real line.","deep_definition":"The papers distinguish this topological property from measure. A nowhere-dense set can have positive Lebesgue measure, so the word does not simply mean “small in length”.","scope_note":"Applies to sets of real numbers here, not to sequences or to sets of indices.","example":"The Cantor set is nowhere dense.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":734,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::nowhere_dense","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":488,"id":"number_field","label":"number field","preferred_label":"number field","alternative_labels":["number fields"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the reader bands identify the ambient field, index set, convergence requirement, or theorem hypotheses relevant to the cited manuscript.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-number-field","relative_href":"docs/glossary.html#glossary-number-field","public_phrase":"number field","short_definition":"A number field is a field containing the rational numbers with finite dimension as a vector space over them.","reader_preview":"A number field is a field containing the rational numbers with finite dimension as a vector space over them.","reader_preview_basis":"definition","card_definition":"The dimension \\([K:\\mathbb{Q}]\\) is its degree. For example, \\(\\mathbb{Q}(\\sqrt 2)\\) has basis 1,\\(\\sqrt 2\\) and degree two. Its elements can be added, multiplied, and divided by any nonzero element.","reader_rule":"A number field need not be contained in the real numbers. Its degree concerns dimension over \\(\\mathbb{Q}\\), rather than the number of its elements.","deep_definition":"In #243, irreducibility of the cubic \\(f(T)=T^{3}-T+\\eta\\) makes \\(K=\\mathbb{Q}(\\alpha )\\) a number field of degree three. The proof studies a possible square root inside K and computes traces back to \\(\\mathbb{Q}\\); the degree controls both calculations.","scope_note":"A number field need not be contained in the real numbers. Its degree concerns dimension over \\(\\mathbb{Q}\\), rather than the number of its elements.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::sec:cubic-field.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":961,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::number_field","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::sec:cubic-field","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":489,"id":"number_theory","label":"number theory","preferred_label":"number theory","alternative_labels":["elementary number theory"],"hidden_search_labels":[],"term_kind":"subject","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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Problems about integer patterns can require tools from analysis, algebra, geometry, or combinatorics.","reader_rule":"The precise assumptions determine which arithmetic setting a theorem concerns.","deep_definition":"The papers here use number-theory ideas to examine reciprocal sums, prime factors, digit expansions, and related questions. 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A reader meets it wherever the public pages present a list of components, paper modules, or doctrine cards that can be opened one at a time.","example":"The Navigation Hologram Route Plane is the routing layer in which option surfaces operate: the place where artifact rows are listed by kind and a stable id is selected before the authoritative source is opened.","example_route":{"href":"paper-modules.html#paper-module-navigation-hologram-route-plane","label":"Navigation Hologram Route Plane","gloss":"open the routing plane where rows are listed and selected by id"},"non_example":"An option surface is not a keyword search box and not the source files themselves. It is a fixed catalogue of rows you select from by id, sitting one layer above the authority it links to.","visible_text_byte_count":1283,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::option_surface","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"belongs_to","target":"artifact_kind","gloss":"Each option surface is scoped by artifact kind."},{"verb":"implements","target":"rowability","gloss":"A supported option surface is the rowability browse layer."},{"verb":"projects_through","target":"band","gloss":"Option surfaces expose adapter-supported bands."},{"verb":"uses","target":"omission_receipt","gloss":"Cards record omitted source context and drilldown."},{"verb":"serves","target":"navigation","gloss":"Option surfaces make navigation selectable before retrieval."}],"source_refs":[{"ref":"system/lib/standard_option_surface.py","role":"code","gloss":"Public option-surface adapter implementation."},{"ref":"codex/standards/std_navigation_contract.json","role":"standard","gloss":"Navigation contract for row shape and drilldown behavior."},{"ref":"codex/standards/std_kind_atlas.json","role":"standard","gloss":"Kind Atlas contract for supported option-surface status."},{"ref":"codex/doctrine/system_vocabulary/term_registry.json","role":"registry","gloss":"Vocabulary rows now expose option-surface terminology."}],"evidence_commands":[],"used_in":{"total_occurrences":4,"page_count":3,"top_page":"paper-module-materials-chemistry-closed-loop-lab-safety-replay.html","top_pages":[{"page":"paper-module-materials-chemistry-closed-loop-lab-safety-replay.html","count":2},{"page":"paper-module-batch7-macro-engines-capsule.html","count":1},{"page":"paper-module-durable-agent-work-landing-replay.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":503,"id":"orbit","label":"orbit","preferred_label":"orbit","alternative_labels":["orbits","infinite orbit"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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In modular arithmetic the state space is finite, but further hypotheses determine whether repetition starts immediately or only after an initial segment.","scope_note":"Discrete-dynamics sense: iterating one map. 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That statement does not say its closure is the whole surrounding space, and does not give a rate or frequency of visits.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/269/erdos269-running-lcm-reasoning-surface.tex::long269:sec:flow.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":965,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::orbit_closure","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/269/erdos269-running-lcm-reasoning-surface.tex::long269:sec:flow","role":"mathematical_source","gloss":"Published paper context for this term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":505,"id":"orientation","label":"orientation","preferred_label":"orientation","alternative_labels":["orientation guide","concise guide"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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In the ring of integers of a number field, every nonzero prime ideal is maximal.","deep_definition":"Prime ideals play the role of generalized prime locations. Localizing at p focuses algebra near that location, and the residue field \\(\\kappa (p)=\\operatorname{Frac}(R/p)\\) records the scalars visible exactly at it.","scope_note":"A prime ideal must be proper. It need not be maximal: for example, the zero ideal in Z is prime but not maximal. 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The inequality convention matters at a prime input and must be checked.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/251/erdos251-prime-gap-reasoning-surface.tex::line 2143.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":928,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::prime_counting_function","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/251/erdos251-prime-gap-reasoning-surface.tex::line 2143","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":553,"id":"prime_tuple_singular_series","label":"prime-tuple singular series","preferred_label":"prime-tuple singular series","alternative_labels":["singular series of a prime tuple","Hardy–Littlewood singular series","Hardy-Littlewood singular series"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"This is the Hardy–Littlewood singular series for distinct integer prime shifts. 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Ramification indices give the exact multiplicity condition behind that exclusion.","scope_note":"The index belongs to a chosen upstairs prime \\(P_i\\) over a chosen base prime p. “Unramified at p” means all these indices equal one; it does not mean that p has only one prime above it.","example":"An index \\(e_i>1\\) means the upstairs prime occurs repeatedly in the ideal factorization.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1006,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::ramification_index","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/243/erdos243-reciprocal-tail-reasoning-surface.tex::line 283","role":"prerequisite_dependency","gloss":"Unramified means every prime above the base prime has ramification index one."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":593,"id":"ramification_point","label":"ramification point","preferred_label":"ramification point","alternative_labels":["ramified point"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"A ramification point belongs to the source of a map; its image is usually called a branch value. 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Away from the origin the map is locally one-to-one.","reader_rule":"A ramification point lies in the source, while a branch point or branch value often names its image in the target; terminology varies, so the page should state its convention. Multiplicity determines the local degree.","deep_definition":"The paper studies a projection from an algebraic curve. Excluding ramification over a disc makes the projection an unramified finite cover, which is what permits separate analytic inverse branches.","scope_note":"A ramification point lies in the source, while a branch point or branch value often names its image in the target; terminology varies, so the page should state its convention. Multiplicity determines the local degree.","example":"The map \\(z\\mapsto z^{2}\\) ramifies at 0: nearby nonzero values have two preimages, but both sheets meet at the origin.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1143,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::ramification_point","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 962","role":"mathematical_source","gloss":"A finite ramification point of its projection"}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":594,"id":"random_variable","label":"random variable","preferred_label":"random variable","alternative_labels":["random variables"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The measurable-function meaning is intended, with real-valued variables as the main example. A random draw is one realized value, and a distribution is the law induced by the function.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-random-variable","relative_href":"docs/glossary.html#glossary-random-variable","public_phrase":"random variable","short_definition":"A random variable is a measurable function from a probability space to numerical or other measurable values.","reader_preview":"A random variable is a measurable function from a probability space to numerical or other measurable values.","reader_preview_basis":"authored_override","card_definition":"It assigns a value to each outcome. Probabilities for its values come from the underlying measure: the event \\(X \\le t\\) consists of outcomes where that inequality holds. The function is deterministic once the outcome is fixed.","reader_rule":"Distinguish the random variable from its probability distribution. Different functions, even on different spaces, can have the same distribution.","deep_definition":"In the #257 finite-residue averaging argument, a cutoff divisor sum is a random variable when its residue input is chosen uniformly. Its expected value is the period average; taking an infinite cutoff requires additional convergence or integrability reasoning.","scope_note":"Distinguish the random variable from its probability distribution. 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The rank cannot exceed either matrix dimension.","deep_definition":"Evaluation matrices turn arithmetic functions into finite linear-algebra objects. Proving large rank shows many channels are independent; it does not by itself establish irrationality of an infinite sum, which requires a separate analytic or denominator bridge.","scope_note":"Row rank equals column rank. A nonzero \\(k\\times k\\) minor proves rank at least k, while all minors of order \\(k+1\\) vanishing proves rank at most k. The rank cannot exceed either matrix dimension.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/249/erdos-249-binary-totient-series.tex::line 103.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":894,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::rank_of_matrix","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/249/erdos-249-binary-totient-series.tex::line 103","role":"mathematical_source","gloss":"Published manuscript context for this term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":596,"id":"rate_limit","label":"rate limit","preferred_label":"rate limit","alternative_labels":["rate limits","request limit"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"A rate limit constrains requests or resource use over a specified interval or quota window. 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Clients often use bounded exponential backoff with jitter, while preserving deadlines and error visibility. Plectis provider routing treats rate limits as an operational condition across model services. A rate limit is not evidence that a request or result is invalid.","scope_note":"When a rate limit is reached, respect the server’s retry guidance and avoid synchronized or unlimited retries.","example":"An API permits 100 requests per minute per credential and returns a retry time when exceeded.","example_route":null,"non_example":"A mathematical convergence limit is unrelated to request rate limiting.","visible_text_byte_count":760,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::rate_limit","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/docs/paper-module-engine-room-annex-knowledge-router.html","role":"reading_context","gloss":"rate limit and back off across multiple LLM providers"}],"evidence_commands":[],"used_in":{"total_occurrences":4,"page_count":2,"top_page":"paper-module-annex-knowledge-routing.html","top_pages":[{"page":"paper-module-annex-knowledge-routing.html","count":2},{"page":"paper-module-engine-room-annex-knowledge-router.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":597,"id":"rational","label":"rational","preferred_label":"rational","alternative_labels":["rational number","rationality"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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The type of each term alone does not determine the type of the limit.","scope_note":"Covers numbers expressible as ratios of integers and rationality questions about values such as infinite sums.","example":"Rationality of the series forces an exact recurrence in the long division, which is where several of the constraints here come from.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":894,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::rational","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":27,"page_count":3,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":23},{"page":"paper-module-finite-erdos-denominator-certificate-strike.html","count":3},{"page":"credit-ledger.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":598,"id":"rational_approximation","label":"rational approximation","preferred_label":"rational approximation","alternative_labels":["rational approximant","approximation by rationals"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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A useful estimate states both the error and how it depends on q in \\(p/q\\).","reader_rule":"A small decimal error alone does not establish irrationality.","deep_definition":"Arguments in these papers combine structured approximations with denominator restrictions or tail bounds. The theorem must explain why their combined behaviour cannot occur if the target number is rational.","scope_note":"A small decimal error alone does not establish irrationality.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex::line 1.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":949,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::rational_approximation","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex::line 1","role":"mathematical_source","gloss":"Published manuscript context for this mathematical term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":599,"id":"rational_function","label":"rational function","preferred_label":"rational function","alternative_labels":["rational functions"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the reader bands identify the ambient field, index set, convergence requirement, or theorem hypotheses relevant to the cited manuscript.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-rational-function","relative_href":"docs/glossary.html#glossary-rational-function","public_phrase":"rational function","short_definition":"A rational function over a field is a quotient of two polynomials over that field, with a nonzero denominator polynomial.","reader_preview":"A rational function over a field is a quotient of two polynomials over that field, with a nonzero denominator polynomial.","reader_preview_basis":"definition","card_definition":"For example, \\((1+z)/(1-z)\\) is a rational function. Cancelling a common polynomial factor gives the same element of the field of rational functions.","reader_rule":"This describes the whole function, rather than whether one of its values is rational. Numerical evaluation also requires a denominator that is nonzero after cancellation.","deep_definition":"The #249 discussion of integer-coefficient Mahler series distinguishes rational functions from transcendental functions under its stated growth hypotheses. The #1049 reasoning surface warns that a nonrational generating function may still take a rational value at an individual rational input. Functional nonrationality alone does not resolve that specialization.","scope_note":"This describes the whole function, rather than whether one of its values is rational. Numerical evaluation also requires a denominator that is nonzero after cancellation.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex::line 4812.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":971,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::rational_function","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex::line 4812","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":600,"id":"rational_independence","label":"rational independence","preferred_label":"rational independence","alternative_labels":["linearly independent over the rationals","Q-linear independence"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Specify the exact family: pairwise irrational ratios, full rational independence, and independence after adjoining 1 are different conditions.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-rational-independence","relative_href":"docs/glossary.html#glossary-rational-independence","public_phrase":"rational independence","short_definition":"Numbers are rationally independent if no rational linear combination equals zero except the one with all coefficients zero.","reader_preview":"Numbers are rationally independent if no rational linear combination equals zero except the one with all coefficients zero.","reader_preview_basis":"authored_override","card_definition":"Clearing denominators gives the equivalent test using integer coefficients. The numbers 1 and \\(\\sqrt 2\\) pass it. Three numbers can fail even if every pair has an irrational ratio: 1, \\(\\sqrt 2\\) and \\(1+\\sqrt 2\\) have the relation \\(1+\\sqrt 2-(1+\\sqrt 2)=0\\).","reader_rule":"For a continuous linear flow on a torus, the frequencies must be rationally independent for a dense orbit in the whole torus. For the sampled translation \\(n\\alpha\\), the required family includes 1 as well as all coordinates of \\(\\alpha\\).","deep_definition":"The running-LCM paper works in the actual orbit closure of its logarithmic flow. It proves the pairwise independence needed for the two-wall argument without assuming that all frequencies are independent. A proof about two coordinates therefore need not establish density in the entire higher-dimensional torus.","scope_note":"For a continuous linear flow on a torus, the frequencies must be rationally independent for a dense orbit in the whole torus. For the sampled translation \\(n\\alpha\\), the required family includes 1 as well as all coordinates of \\(\\alpha\\).","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/269/erdos269-running-lcm-reasoning-surface.tex::long269:sec:flow.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1177,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::rational_independence","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/269/erdos269-running-lcm-reasoning-surface.tex::long269:sec:flow","role":"mathematical_source","gloss":"Published paper context for the definition and its stated boundary."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":601,"id":"rational_subspace","label":"rational subspace","preferred_label":"rational subspace","alternative_labels":["rational linear subspace","rational plane"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Rational means defined relative to the standard rational coordinate structure, not that every point of the real subspace is rational.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-rational-subspace","relative_href":"docs/glossary.html#glossary-rational-subspace","public_phrase":"rational subspace","short_definition":"A real linear subspace that has a basis consisting of vectors with rational coordinates.","reader_preview":"A real linear subspace that has a basis consisting of vectors with rational coordinates.","reader_preview_basis":"authored_override","card_definition":"In \\(\\mathbb{R}^{d}\\), clearing basis denominators makes this equivalent to being spanned by integer vectors. For example the line spanned by \\((1,2)\\) is rational, whereas the line spanned by \\((1,\\sqrt 2)\\) is not. Real multiples in a rational subspace need not themselves have rational coordinates.","reader_rule":"The ambient coordinate lattice is part of the meaning. A rational subspace projects to a closed connected subtorus of \\(\\mathbb{R}^{d}/\\mathbb{Z}^{d}\\); an irrationally directed line can instead have a larger dense closure.","deep_definition":"In #269 the smallest rational subspace containing the logarithmic-flow velocity determines its orbit closure. A rational two-dimensional plane inside it is chosen so that every pair of relevant coordinate forms remains independent, enabling the finite wall-arrangement count.","scope_note":"The ambient coordinate lattice is part of the meaning. A rational subspace projects to a closed connected subtorus of \\(\\mathbb{R}^{d}/\\mathbb{Z}^{d}\\); an irrationally directed line can instead have a larger dense closure.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/269/erdos269-running-lcm-reasoning-surface.tex::line 805.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1119,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::rational_subspace","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/269/erdos269-running-lcm-reasoning-surface.tex::line 805","role":"mathematical_source","gloss":"Published manuscript use of rational subspace with the local hypotheses and conclusion stated in the referenced passage."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":602,"id":"raw_seed","label":"raw seed","preferred_label":"raw seed","alternative_labels":["raw_seed","operator voice","Will voice"],"hidden_search_labels":["voice authority"],"term_kind":"substrate","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"Raw seed means the preserved operator-voice substrate in this repository. 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A reader meets it on the pages that show how recorded voice becomes structured doctrine, and in references back to source provenance.","example":"The Voice To Doctrine Self Improvement Loop is one place the raw seed is visible in use: it takes recorded operator voice as its input and turns it into structured doctrine, so the seed is the raw material entering one end of that loop.","example_route":{"href":"components.html#component-voice_to_doctrine_self_improvement_loop","label":"Voice To Doctrine Self Improvement Loop","gloss":"open the component that turns recorded operator voice into doctrine"},"non_example":"A raw seed is not a polished mission statement, a marketing narrative, or the system's final doctrine. 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The surrounding explanation should still be understandable enough that a glossary is optional support rather than a prerequisite for every sentence.","scope_note":"Reader covers the audience named by a page, packet, or route contract.","example":"The glossary gives a cold reader a one-sentence definition before deeper context.","example_route":null,"non_example":"The author of a page is not automatically its intended reader.","visible_text_byte_count":712,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::reader","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":365,"page_count":100,"top_page":"paper-module-first-screen-composition-root.html","top_pages":[{"page":"paper-module-first-screen-composition-root.html","count":38},{"page":"doctrine.html","count":33},{"page":"rules-and-ideas.html","count":24},{"page":"paper-module-cold-reader-route-map.html","count":20},{"page":"paper-modules.html","count":11},{"page":"paper-module-public-reveal-walkthrough.html","count":8}]}},{"schema":"plectis_public_glossary_term_v0","position":604,"id":"reader_packet","label":"reader packet","preferred_label":"reader packet","alternative_labels":["reader digest","review packet"],"hidden_search_labels":["AI handoff","copy packet","orientation packet"],"term_kind":"artifact","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"A reader packet is a generated portable guide to public routes, summaries, sources, and limits; it is not the source authority, a complete export, or publication permission.","meaning_contrasts":[],"status":"active","disposition":"public_handoff_artifact_label","anchor":"glossary-reader-packet","relative_href":"docs/glossary.html#glossary-reader-packet","public_phrase":"portable inspection guide","short_definition":"A reader packet gathers selected explanations, links, source references, and scope information for someone examining the project.","reader_preview":"A reader packet gathers selected explanations, links, source references, and scope information for someone examining the project.","reader_preview_basis":"definition","card_definition":"It can be downloaded or supplied to an assistant as a portable starting point. The selection helps the reader orient and identify which original material deserves closer attention.","reader_rule":"The packet is a generated guide with declared coverage, not a replacement for all its sources.","deep_definition":"Its version matters when the project changes. A useful packet preserves routes to current source and evidence while identifying which summaries or copied contents belong to the packet's own snapshot.","scope_note":"The term covers the generated digest and review artifacts that orient a reviewer at the public source slice, such as the AI reader digest and the review packet. A reader encounters it as the entry-and-orientation surface that hands off into components, routes, and source files.","example":"The Cold-Reader Route Map is a reader packet: a single artifact that lays out the first routes a newcomer should follow into the public slice and links each one back to its source.","example_route":{"href":"components.html#component-cold_reader_route_map","label":"Cold Reader Route Map","gloss":"open the packet that routes a first-time reader into the source"},"non_example":"A reader packet is not source authority and not proof that a reviewer read or endorsed the project. It is a generated map whose summaries hold only as far as the source records they link to.","visible_text_byte_count":1294,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::reader_packet","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"routes_to","target":"record","gloss":"Reader packets should route to concrete records."},{"verb":"summarizes","target":"source_slice","gloss":"Reader packets summarize the public source slice."},{"verb":"below","target":"source_authority","gloss":"Reader packets remain projections below source authority."}],"source_refs":[{"ref":"sites/microcosm/microcosm-ai-reader-digest.json","role":"generated_reader_digest","gloss":"Generated public reader digest."},{"ref":"sites/microcosm/microcosm-review-packet.json","role":"generated_review_packet","gloss":"Generated public review packet."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":605,"id":"readiness","label":"readiness","preferred_label":"readiness","alternative_labels":["ready","readiness check"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The stated technical meaning; distinct disciplinary senses are identified where the site uses them.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-readiness","relative_href":"docs/glossary.html#glossary-readiness","public_phrase":"readiness","short_definition":"Readiness is the condition of meeting the stated requirements for a particular next action or operating stage.","reader_preview":"Readiness is the condition of meeting the stated requirements for a particular next action or operating stage.","reader_preview_basis":"definition","card_definition":"A readiness decision compares current state with a declared gate for testing, review, publication, deployment, or operation. Evidence supports the decision, but the evidence itself is not readiness and one stage does not imply another.","reader_rule":"Readiness is relative to a named action and its required conditions; it holds when those conditions are satisfied, and it ends when a requirement fails or the target conditions change.","deep_definition":"Readiness describes whether an artifact, service, or process currently satisfies the conditions for a specified next step. A test-ready change may need required files and a working build; publication readiness may add source checks and review; production readiness may require operational capacity, rollback, security, and monitoring. Test reports, approvals, freshness records, and health signals are evidence used to judge those conditions, rather than readiness itself. Because gates differ, passing an earlier one cannot silently confer a stronger status. A readiness result also becomes stale when dependencies, requirements, or the evaluated artifact change.","scope_note":"Readiness is relative to a named action and its required conditions; it holds when those conditions are satisfied, and it ends when a requirement fails or the target conditions change.","example":"Representative source: sites/microcosm/docs/reference.html.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":897,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::readiness","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/docs/reference.html","role":"reading_context","gloss":"Public context in which readers encounter this term."},{"ref":"sites/microcosm/docs/paper-modules.html","role":"reading_context","gloss":"Public context in which readers encounter this term."}],"evidence_commands":[],"used_in":{"total_occurrences":345,"page_count":44,"top_page":"paper-module-formal-math-readiness-gate.html","top_pages":[{"page":"paper-module-formal-math-readiness-gate.html","count":65},{"page":"rules-and-ideas.html","count":59},{"page":"paper-module-corpus-readiness-mathlib-absence-gate.html","count":37},{"page":"doctrine.html","count":35},{"page":"paper-module-pattern-binding-contract.html","count":33},{"page":"paper-modules.html","count":27}]}},{"schema":"plectis_public_glossary_term_v0","position":606,"id":"real_number","label":"real number","preferred_label":"real number","alternative_labels":["the reals"],"hidden_search_labels":["real numbers","real-valued"],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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Establishing the existence of the sum and classifying that value are separate questions.","scope_note":"Covers the real-valued limits and theorem domains used by the Plectis mathematical papers and formalizations.","example":"The binary totient series has a real value; the Erdős question asks whether that real number is irrational.","example_route":null,"non_example":"An arbitrary complex number with nonzero imaginary part is not a real number.","visible_text_byte_count":827,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::real_number","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":2,"page_count":1,"top_page":"doctrine.html","top_pages":[{"page":"doctrine.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":607,"id":"receipt","label":"receipt","preferred_label":"receipt","alternative_labels":["receipts"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"public_contextual_gloss","meaning_boundary":"This software use specializes an ordinary acknowledgement or record of an event. How much a receipt establishes depends on its producer and evidence contract.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-receipt","relative_href":"docs/glossary.html#glossary-receipt","public_phrase":"receipt","short_definition":"A receipt is a saved record of an action, its inputs, and its reported outcome.","reader_preview":"A receipt is a saved record of an action, its inputs, and its reported outcome.","reader_preview_basis":"authored_override","card_definition":"Local tools use receipts to preserve what ran or changed and the evidence collected. Some receipt standards distinguish outcome fields from proof metadata and mark records incomplete when required evidence is missing.","reader_rule":"Check the producer, recorded version, covered action, and supporting fields before treating the reported outcome as established.","deep_definition":"A historical record lets later agents compare completion claims with execution evidence. Its presence alone does not establish that the action ran correctly: a receipt can be incomplete, stale for the present question, or narrower than the claim made from it.","scope_note":"Evidence applies to the named action and conditions, with missing proof identified by the receipt contract.","example":"A deploy receipt records the commit pushed, the asset hash verified live, and the paper hashes compared against public source.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":884,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::receipt","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"codex/standards/std_operator_action_receipt.json","role":"standard","gloss":"Separates outcome claims, proof fields, and explicit incomplete-receipt status."}],"evidence_commands":[],"used_in":{"total_occurrences":166,"page_count":98,"top_page":"paper-module-navigation-hologram-route-plane.html","top_pages":[{"page":"paper-module-navigation-hologram-route-plane.html","count":8},{"page":"paper-module-agent-route-observability-runtime.html","count":7},{"page":"paper-module-executable-doctrine-grammar.html","count":5},{"page":"paper-module-formal-math-readiness-gate.html","count":5},{"page":"paper-module-formal-math-premise-retrieval.html","count":4},{"page":"papers.html","count":4}]}},{"schema":"plectis_public_glossary_term_v0","position":608,"id":"reciprocal","label":"reciprocal","preferred_label":"reciprocal","alternative_labels":["reciprocals"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-reciprocal","relative_href":"docs/glossary.html#glossary-reciprocal","public_phrase":"reciprocal","short_definition":"The reciprocal of a nonzero number x is 1/x.","reader_preview":"The reciprocal of a nonzero number x is 1/x.","reader_preview_basis":"definition","card_definition":"Multiplying a number by its reciprocal gives 1. For example, the reciprocal of 4 is \\(1/4\\), and the reciprocal of \\(2/3\\) is \\(3/2\\).","reader_rule":"Zero has no reciprocal in the real or rational numbers.","deep_definition":"The papers study sums of reciprocals of structured integer sequences. The denominator's growth and arithmetic properties both matter: shrinking terms alone do not prove that a reciprocal series converges or determine whether its sum is rational.","scope_note":"A reciprocal-sum result covers the growth condition it assumes.","example":"Erdős #243 concerns integer sequences whose reciprocal sum is rational under a rapid-growth hypothesis.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":707,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::reciprocal","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":28,"page_count":3,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":25},{"page":"papers.html","count":2},{"page":"introduction.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":609,"id":"reciprocal_polynomial","label":"reciprocal polynomial","preferred_label":"reciprocal polynomial","alternative_labels":["polynomial reciprocal","reversed polynomial"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the reader bands identify the ambient field, index set, convergence requirement, or theorem hypotheses relevant to the cited manuscript.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-reciprocal-polynomial","relative_href":"docs/glossary.html#glossary-reciprocal-polynomial","public_phrase":"reciprocal polynomial","short_definition":"The reciprocal of a degree-d polynomial \\(p(T)\\) reverses its coefficients and is written \\(T^{d}p(T^{-1})\\).","reader_preview":"The reciprocal of a degree-d polynomial \\(p(T)\\) reverses its coefficients and is written \\(T^{d}p(T^{-1})\\).","reader_preview_basis":"definition","card_definition":"For \\(p(T)=T^{3}+bT^{2}+dT+w\\), its reciprocal is \\(wT^{3}+dT^{2}+bT+1\\). When w is nonzero, its roots are the reciprocals of the roots of p.","reader_rule":"A zero constant term lowers the degree after reversal. Reversing a polynomial is different from taking its reciprocal as a rational function \\(1/p(T)\\).","deep_definition":"In #243, z is nonzero and its monic minimal polynomial has constant term \\(w\\ne 0\\). Dividing the reversed polynomial by w gives the monic polynomial \\(T^{3}+(d/w)T^{2}+(b/w)T+1/w\\) for \\(z^{-1}\\). This normalization matters in the ensuing trace identities.","scope_note":"A zero constant term lowers the degree after reversal. Reversing a polynomial is different from taking its reciprocal as a rational function \\(1/p(T)\\).","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::sec:cubic-trace.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":954,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::reciprocal_polynomial","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::sec:cubic-trace","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":610,"id":"reciprocal_series","label":"reciprocal series","preferred_label":"reciprocal series","alternative_labels":["sum of reciprocals","reciprocal sum"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-reciprocal-series","relative_href":"docs/glossary.html#glossary-reciprocal-series","public_phrase":"reciprocal series","short_definition":"A reciprocal series is a sum whose terms have the form \\(1/a_n\\) for specified nonzero numbers \\(a_{n}\\).","reader_preview":"A reciprocal series is a sum whose terms have the form \\(1/a_n\\) for specified nonzero numbers \\(a_{n}\\).","reader_preview_basis":"definition","card_definition":"For example, \\(\\sum 1/2^{n}\\) starting at \\(n = 1\\) sums to 1, while \\(\\sum 1/n\\) diverges. The way the denominators grow influences convergence, but a precise comparison or other argument is needed.","reader_rule":"Convergence does not determine whether the sum is rational or irrational.","deep_definition":"The papers study denominators built from factorials, primes, or powers minus one. Their divisibility structure can constrain cancellation and rationality even when convergence follows from a simple size estimate.","scope_note":"Convergence does not determine whether the sum is rational or irrational.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::line 1.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":899,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::reciprocal_series","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::line 1","role":"mathematical_source","gloss":"Published manuscript context for this mathematical term."}],"evidence_commands":[],"used_in":{"total_occurrences":10,"page_count":1,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":10}]}},{"schema":"plectis_public_glossary_term_v0","position":611,"id":"reciprocal_summable_support","label":"reciprocal-summable support","preferred_label":"reciprocal-summable support","alternative_labels":["reciprocal-summable","reciprocal summability","finite reciprocal sum"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the cited paper specifies the parameters and hypotheses for its application.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-reciprocal-summable-support","relative_href":"docs/glossary.html#glossary-reciprocal-summable-support","public_phrase":"reciprocal-summable support","short_definition":"A positive-integer support is reciprocal-summable when the sum of the reciprocals of its elements is finite.","reader_preview":"A positive-integer support is reciprocal-summable when the sum of the reciprocals of its elements is finite.","reader_preview_basis":"definition","card_definition":"The powers of two are reciprocal-summable, since \\(\\sum_{k\\ge1}1/2^k=1\\). The positive integers are not, since \\(\\sum_{n\\ge1}1/n\\) diverges.","reader_rule":"This condition concerns \\(\\sum_{a\\in A}1/a\\). It is different from convergence of \\(\\sum_{a\\in A}1/(b^{a}-1)\\), which holds for every positive-integer support at integer \\(b\\ge 2\\).","deep_definition":"In #257, reciprocal summability suffices for the weighted condition because \\(h/(2^h-1)\\le1\\) for positive integer h. The weighted theorem also reaches supports whose reciprocal sum diverges, so these support classes should not be identified.","scope_note":"This condition concerns \\(\\sum_{a\\in A}1/a\\). It is different from convergence of \\(\\sum_{a\\in A}1/(b^{a}-1)\\), which holds for every positive-integer support at integer \\(b\\ge 2\\).","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":941,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::reciprocal_summable_support","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex","role":"mathematical_source","gloss":"Primary manuscript context and mathematical application."}],"evidence_commands":[],"used_in":{"total_occurrences":2,"page_count":2,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":1},{"page":"updates.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":612,"id":"rectifiable_curve","label":"rectifiable curve","preferred_label":"rectifiable curve","alternative_labels":["rectifiable path","rectifiable curves"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The path sense is used here; rectifiability of a subset or measure is a different definition that needs separate hypotheses.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-rectifiable-curve","relative_href":"docs/glossary.html#glossary-rectifiable-curve","public_phrase":"rectifiable curve","short_definition":"A continuous curve whose length, defined by total variation over partitions, is finite.","reader_preview":"A continuous curve whose length, defined by total variation over partitions, is finite.","reader_preview_basis":"authored_override","card_definition":"Rectifiability does not require a smooth parametrization: a polygonal path with corners is rectifiable. A continuous curve can instead have infinite length, even when its image lies in a bounded region. Length is computed with the specified ambient metric.","reader_rule":"For a continuously differentiable curve, length is the integral of the speed. A rectifiable path admits an arc-length description after accounting for constant portions, but finite length alone does not imply a nonzero tangent at every point.","deep_definition":"The #1041 connector results seek rectifiable paths with a numerical length bound inside a strict polynomial sublevel set. Rectifiability makes such a bound meaningful; merely exhibiting a continuous topological connection does not establish finite or sufficiently small length.","scope_note":"For a continuously differentiable curve, length is the integral of the speed. A rectifiable path admits an arc-length description after accounting for constant portions, but finite length alone does not imply a nonzero tangent at every point.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 638.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1080,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::rectifiable_curve","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 638","role":"mathematical_source","gloss":"Published manuscript use of rectifiable curve with the local hypotheses and conclusion stated in the referenced passage."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":613,"id":"recurrence","label":"recurrence","preferred_label":"recurrence","alternative_labels":["recurrence relation"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-subsum","relative_href":"docs/glossary.html#glossary-subsum","public_phrase":"subsum","short_definition":"A subsum is the value obtained by selecting some terms of a series and adding them.","reader_preview":"A subsum is the value obtained by selecting some terms of a series and adding them.","reader_preview_basis":"definition","card_definition":"For a selected index set A, the expression is \\(\\sum_{n\\in A}a_n\\), when that sum is defined. The selected series is a subseries; its value is one subsum.","reader_rule":"Convergence of a conditionally convergent series does not guarantee convergence of every subseries. Absolute convergence does, and nonnegative convergent series have all subsums between zero and their full sum.","deep_definition":"The achievement set collects all possible subsum values. In #257, it includes the empty and finite selections as well as infinite selections. The universal irrationality question concerns the infinite selections, so rational values from finite selections are not counterexamples.","scope_note":"A subsum is one value; the achievement set is the set of all such values. A restricted achievement set imposes further conditions on the selections.","example":"One quarter plus one sixteenth is a subsum of the series of reciprocal powers of two.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":820,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::subsum","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex","role":"mathematical_source","gloss":"Primary manuscript context and mathematical application."}],"evidence_commands":[],"used_in":{"total_occurrences":2,"page_count":1,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":708,"id":"sum","label":"sum","preferred_label":"sum","alternative_labels":["addition result"],"hidden_search_labels":["finite sum","sums"],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-sum","relative_href":"docs/glossary.html#glossary-sum","public_phrase":"sum","short_definition":"A sum is the result of adding quantities together.","reader_preview":"A sum is the result of adding quantities together.","reader_preview_basis":"definition","card_definition":"A finite sum adds a specified number of terms. An infinite series uses the limit of finite partial sums when that limit exists. The symbol \\(\\sum\\) abbreviates a sum with its index and range stated alongside it.","reader_rule":"The range and order of summation can be part of the mathematical claim.","deep_definition":"The papers distinguish exact finite arithmetic from conclusions about infinite sums. Passing from one to the other requires convergence and, when terms are rearranged, the appropriate justification for that rearrangement.","scope_note":"Covers the finite certificate sums and infinite reciprocal sums appearing in Plectis mathematics.","example":"\\(S_F(b)=\\sum _{n\\in F}1/(b^n-1)\\) is a finite exact sum when F is finite; the full Erdős construction is an infinite sum.","example_route":null,"non_example":"\\(3\\times4\\) is a product, not a sum.","visible_text_byte_count":754,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::sum","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":40,"page_count":13,"top_page":"paper-module-finite-erdos-denominator-certificate-strike.html","top_pages":[{"page":"paper-module-finite-erdos-denominator-certificate-strike.html","count":11},{"page":"papers.html","count":9},{"page":"paper-module-annex-knowledge-routing.html","count":5},{"page":"paper-module-engine-room-annex-knowledge-router.html","count":3},{"page":"component-finite-erdos-denominator-certificate-strike.html","count":2},{"page":"introduction.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":709,"id":"summability","label":"summability","preferred_label":"summability","alternative_labels":["summable"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-summability","relative_href":"docs/glossary.html#glossary-summability","public_phrase":"summability","short_definition":"Summability means that a family of terms has a sum under a specified definition of infinite summation.","reader_preview":"Summability means that a family of terms has a sum under a specified definition of infinite summation.","reader_preview_basis":"definition","card_definition":"For a nonnegative sequence, it means the partial sums are bounded and hence converge to a finite total. For real or complex families indexed without an order, the usual unconditional notion corresponds to absolute summability.","reader_rule":"The terms, weighting, and summation convention are part of the claim.","deep_definition":"In Lean, `Summable` has a formal meaning that should be checked for the relevant index and space. A convergent ordered series need not be summable in every stronger, order-independent sense.","scope_note":"The terms, weighting, and summation convention are part of the claim.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::line 353.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":884,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::summability","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/243/erdos-243-reciprocal-tail-rigidity.tex::line 353","role":"mathematical_source","gloss":"Published manuscript context for this mathematical term."}],"evidence_commands":[],"used_in":{"total_occurrences":3,"page_count":1,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":3}]}},{"schema":"plectis_public_glossary_term_v0","position":710,"id":"divisor_cover","label":"summable divisor cover","preferred_label":"summable divisor cover","alternative_labels":["divisor-cover criterion","divisor cover","summable divisor-cover criterion"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"This row explains the named criterion in the #257 manuscript. It does not propose a universal meaning for every use of divisor cover.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-divisor-cover","relative_href":"docs/glossary.html#glossary-divisor-cover","public_phrase":"summable divisor cover","short_definition":"In the #257 paper, a summable divisor cover groups exponents into finite sets with nonnegative divisor bounds whose full costs have finite sum.","reader_preview":"In the #257 paper, a summable divisor cover groups exponents into finite sets with nonnegative divisor bounds whose full costs have finite sum.","reader_preview_basis":"definition","card_definition":"For a finite exponent set F, let \\(f_F(n)\\) count members of F dividing n. The criterion bounds \\(f_F(n)^\\alpha\\) by \\(\\sum_{d\\mid n}c_d\\), with \\(c_d\\ge 0\\) and \\(0<\\alpha \\le 1\\). Its cost depends on \\(C=\\sum_d c_d/d\\).","reader_rule":"A separate finite cover exists for every finite F. The theorem requires summability across the entire sequence of finite sets, including the scale and fractional-power factors.","deep_definition":"Precisely, for finite \\(F_{j}\\) and coefficients \\(c_{j,d}\\) as above, the paper requires \\(\\sum_j C_j2^{j\\alpha_j}/(2^{\\alpha_j}-1)<\\infty\\). It then proves irrationality of \\(\\sum_{a\\in A}1/(b^{a}-1)\\) for every infinite \\(A\\subseteq \\bigcup _{j}F_{j}\\) and integer \\(b\\ge 2\\). This is a specific criterion, distinct from a topological open cover or a claim that all infinite supports qualify.","scope_note":"A separate finite cover exists for every finite F. The theorem requires summability across the entire sequence of finite sets, including the scale and fractional-power factors.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex::thm:variable-fractional-cover.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1107,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::divisor_cover","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex::thm:variable-fractional-cover","role":"mathematical_source","gloss":"Primary manuscript use and exact local mathematical context."}],"evidence_commands":[],"used_in":{"total_occurrences":1,"page_count":1,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":711,"id":"summation_by_parts","label":"summation by parts","preferred_label":"summation by parts","alternative_labels":["Abel summation","discrete integration by parts"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordinary mathematical usage; the reader bands identify the ambient field, index set, convergence requirement, or theorem hypotheses relevant to the cited manuscript.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-summation-by-parts","relative_href":"docs/glossary.html#glossary-summation-by-parts","public_phrase":"summation by parts","short_definition":"Summation by parts is a finite-sum identity that transfers differences from one factor to partial sums of the other.","reader_preview":"Summation by parts is a finite-sum identity that transfers differences from one factor to partial sums of the other.","reader_preview_basis":"definition","card_definition":"If \\(A_{k}=\\sum _{j=1}^{k}a_{j}\\), then \\(\\sum _{k=1}^N a_{k}b_{k}=A_N b_N+\\sum _{k=1}^{N-1}A_{k}(b_{k}-b_{k+1})\\), with the endpoint retained.","reader_rule":"Passing to infinite sums requires convergence and control of that endpoint. Dropping it without proving its limit can change the identity.","deep_definition":"The #251 paper transforms the weighted prime series into a weighted prime-gap series by this identity. Its Lean-bound finite formula keeps the last prime term. Polynomial growth then makes the dyadic endpoint tend to zero, producing the infinite affine relation and equivalent irrationality statements.","scope_note":"Passing to infinite sums requires convergence and control of that endpoint. Dropping it without proving its limit can change the identity.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/251/erdos251-prime-gap-reasoning-surface.tex::long251:sec:parts.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":950,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::summation_by_parts","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/251/erdos251-prime-gap-reasoning-surface.tex::long251:sec:parts","role":"mathematical_source","gloss":"Exact local mathematical use, with its hypotheses and scope."}],"evidence_commands":[],"used_in":{"total_occurrences":1,"page_count":1,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":712,"id":"superior_predictive_ability","label":"superior predictive ability","preferred_label":"superior predictive ability","alternative_labels":["Hansen's SPA test"],"hidden_search_labels":["spa test"],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-superior-predictive-ability","relative_href":"docs/glossary.html#glossary-superior-predictive-ability","public_phrase":"superior predictive ability","short_definition":"Hansen's superior predictive ability test assesses whether any candidate forecasting method has lower expected loss than a benchmark.","reader_preview":"Hansen's superior predictive ability test assesses whether any candidate forecasting method has lower expected loss than a benchmark.","reader_preview_basis":"definition","card_definition":"It compares a collection of methods jointly rather than treating the best observed performer as if it had been selected in advance. The procedure accounts for sampling variation and the comparison across candidates under its assumptions.","reader_rule":"The loss function, candidate set, dependence model, and resampling choices affect the inference.","deep_definition":"A rejection provides evidence against the benchmark's equal-or-better expected performance in the tested setting. It does not prove a particular method will outperform forever or explain why a difference occurs.","scope_note":"Named for Peter Reinhard Hansen; standard in forecast evaluation and not a contribution of this work.","example":"Testing fifty strategies against one benchmark needs a correction of this kind.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":961,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::superior_predictive_ability","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":4,"page_count":1,"top_page":"paper-module-finance-forecast-evaluation-spine.html","top_pages":[{"page":"paper-module-finance-forecast-evaluation-spine.html","count":4}]}},{"schema":"plectis_public_glossary_term_v0","position":713,"id":"support","label":"support","preferred_label":"support","alternative_labels":["support of a series","index support","exponent support"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Plectis uses support for evidence backing a claim and for the mathematical set of included indices; neither sense licenses conclusions about the other.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-support","relative_href":"docs/glossary.html#glossary-support","public_phrase":"support","short_definition":"For a sequence, support is the set of indices with nonzero terms; in evidential prose it means what backs a claim.","reader_preview":"For a sequence, support is the set of indices with nonzero terms; in evidential prose it means what backs a claim.","reader_preview_basis":"definition","card_definition":"In #257, the support A selects which exponents appear in \\(\\sum_{a\\in A}1/(b^{a}-1)\\). Writing the series with zero-or-one coefficients makes A exactly their nonzero index set.","reader_rule":"For a series support, membership selects a term; it does not assert that the terms form a finite set or satisfy the paper’s weighted hypothesis.","deep_definition":"For a function on a topological space, support commonly means the closure of the nonzero set. On discrete positive-integer indices this gives the same set. In prose about evidence, “support” instead means reasons or records backing a claim; its evidential strength must be judged separately.","scope_note":"Covers the two active Plectis senses (claim support and series support) and explicitly disambiguates them.","example":"A result record supports a component claim; an infinite exponent set supports the reciprocal sum in Erdős #257.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":894,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::support","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/257/erdos-257-mersenne-support-subseries.tex","role":"mathematical_source","gloss":"Primary manuscript context and mathematical application."}],"evidence_commands":[],"used_in":{"total_occurrences":286,"page_count":101,"top_page":"doctrine.html","top_pages":[{"page":"doctrine.html","count":50},{"page":"paper-module-microcosm-axiom-substrate.html","count":35},{"page":"rules-and-ideas.html","count":16},{"page":"papers.html","count":14},{"page":"paper-module-finite-erdos-denominator-certificate-strike.html","count":10},{"page":"paper-module-durable-agent-work-landing-replay.html","count":7}]}},{"schema":"plectis_public_glossary_term_v0","position":714,"id":"supremum","label":"supremum","preferred_label":"supremum","alternative_labels":["least upper bound","sup"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Ordered number systems. Supremum and maximum are not interchangeable.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-supremum","relative_href":"docs/glossary.html#glossary-supremum","public_phrase":"supremum","short_definition":"The smallest number greater than or equal to every member of a set.","reader_preview":"The smallest number greater than or equal to every member of a set.","reader_preview_basis":"definition","card_definition":"The open interval (0,1) has supremum 1 even though it has no largest member. Every element is at most 1, and any smaller proposed upper bound is exceeded by points sufficiently close to 1.","reader_rule":"A maximum must be attained; a supremum need not be. Unbounded-above sets may have supremum \\(+\\infty\\) in the extended reals, and conventions for the empty set must be stated.","deep_definition":"The irrationality exponent is a supremum over approximation exponents achieved infinitely often. Calling it a supremum allows a sharp boundary value even when no single family of rational approximations attains that boundary exponent.","scope_note":"A maximum must be attained; a supremum need not be. Unbounded-above sets may have supremum \\(+\\infty\\) in the extended reals, and conventions for the empty set must be stated.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex::line 291.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":832,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::supremum","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos-1049-rational-base-lambert.tex::line 291","role":"mathematical_source","gloss":"Published manuscript context for this term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":715,"id":"surjective_map","label":"surjective map","preferred_label":"surjective map","alternative_labels":["surjective","surjective function"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Functions. Onto differs from injective; a map may have either, both, or neither.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-surjective-map","relative_href":"docs/glossary.html#glossary-surjective-map","public_phrase":"surjective map","short_definition":"A function that attains every element of its stated codomain.","reader_preview":"A function that attains every element of its stated codomain.","reader_preview_basis":"definition","card_definition":"The map \\(f(n)=n+1\\) from integers to integers is surjective because each target m has preimage m-1. The same formula from nonnegative integers to nonnegative integers is not surjective, since zero is never reached.","reader_rule":"Surjectivity depends on the stated codomain: every allowed target must have at least one input mapping to it. It says nothing about uniqueness of that input; uniqueness belongs to injectivity.","deep_definition":"The Lambert analysis proves certain coefficient maps are onto the integers, so every desired integer correction can be represented. Changing the domain, codomain, coefficient range, or degree bound can destroy that conclusion and must be tracked explicitly.","scope_note":"Surjectivity depends on the stated codomain: every allowed target must have at least one input mapping to it. It says nothing about uniqueness of that input; uniqueness belongs to injectivity.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex::line 2522.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":920,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::surjective_map","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1049/erdos1049-rational-base-lambert-reasoning-surface.tex::line 2522","role":"mathematical_source","gloss":"Published manuscript context for this term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":716,"id":"sylvester_sequence","label":"Sylvester's sequence","preferred_label":"Sylvester's sequence","alternative_labels":["Sylvester recurrence"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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Reciprocal identities associated with the sequence make it useful in questions about unit fractions and the values of structured infinite sums.","scope_note":"Named for J. J. 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A transcendence argument must rule out every nonzero integer-coefficient polynomial relation, which is a stronger requirement than ruling out fractions.","scope_note":"Transcendence of one series says nothing about a related series until a proof connects them.","example":"Pi and e are transcendental; the square root of two is not.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":836,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::transcendental","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":2,"page_count":1,"top_page":"papers.html","top_pages":[{"page":"papers.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":738,"id":"transcript","label":"transcript","preferred_label":"transcript","alternative_labels":["transcripts"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-transcript","relative_href":"docs/glossary.html#glossary-transcript","public_phrase":"transcript","short_definition":"A transcript is a textual record of speech or an interaction.","reader_preview":"A transcript is a textual record of speech or an interaction.","reader_preview_basis":"definition","card_definition":"It can preserve what was said, typed, or reported during a session, sometimes with timings and speaker labels. Plectis uses transcripts as inspectable records and source material for later explanation.","reader_rule":"The recording method determines what the transcript captures and where errors may occur.","deep_definition":"Speech recognition can mishear words, and an interaction transcript may omit events outside the captured channel. Important quotations or claims should therefore be checked against the appropriate original recording or execution evidence.","scope_note":"A transcript records one session. It carries no judgement about it.","example":"An agent's transcript is what the faithfulness audit reads.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":750,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::transcript","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":36,"page_count":16,"top_page":"paper-module-agent-memory-temporal-conflict-replay.html","top_pages":[{"page":"paper-module-agent-memory-temporal-conflict-replay.html","count":7},{"page":"paper-module-agent-route-observability-runtime.html","count":4},{"page":"paper-module-sleeper-memory-poisoning-quarantine-replay.html","count":3},{"page":"paper-modules.html","count":3},{"page":"area-agent-reliability.html","count":2},{"page":"component-sleeper-memory-poisoning-quarantine-replay.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":739,"id":"triangle_inequality","label":"triangle inequality","preferred_label":"triangle inequality","alternative_labels":["triangle bound"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Metrics, norms, and absolute values. Equality requires special alignment conditions.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-triangle-inequality","relative_href":"docs/glossary.html#glossary-triangle-inequality","public_phrase":"triangle inequality","short_definition":"The distance directly between two points is at most the length of any route through a third.","reader_preview":"The distance directly between two points is at most the length of any route through a third.","reader_preview_basis":"definition","card_definition":"For numbers, \\(|x+y|\\le |x|+|y|\\). Geometrically, traveling from A to C through B cannot be shorter than the direct distance from A to C. Equality needs the two steps to align without backtracking.","reader_rule":"The triangle inequality combines upper bounds after decomposing a quantity. Its reverse form, \\(||x|-|y||\\le |x-y|\\), gives a lower bound on separation from differing sizes. Both depend on the relevant absolute value, norm, or metric.","deep_definition":"Path and polynomial estimates assemble local pieces using triangle inequalities. The step can lose crucial cancellation, so a proof needing a sharp constant may require additional angle, sign, orthogonality, or convexity information rather than repeated triangle bounds.","scope_note":"The triangle inequality combines upper bounds after decomposing a quantity. Its reverse form, \\(||x|-|y||\\le |x-y|\\), gives a lower bound on separation from differing sizes. Both depend on the relevant absolute value, norm, or metric.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 651.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":961,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::triangle_inequality","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/1041/erdos1041-lemniscate-reasoning-surface.tex::line 651","role":"mathematical_source","gloss":"Published manuscript context for this term."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":740,"id":"trinomial","label":"trinomial","preferred_label":"trinomial","alternative_labels":["three-term polynomial"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Terms are counted after equal powers have been combined and zero coefficients removed.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-trinomial","relative_href":"docs/glossary.html#glossary-trinomial","public_phrase":"trinomial","short_definition":"A trinomial is a polynomial written as the sum of three nonzero terms.","reader_preview":"A trinomial is a polynomial written as the sum of three nonzero terms.","reader_preview_basis":"definition","card_definition":"For example, \\(z^{5}+2z^{2}-7\\) is a trinomial. The exponents need not be consecutive, and a missing power simply has coefficient zero. The paper studies the form \\(z^{n}+az^{m}+b\\).","reader_rule":"Count nonzero terms after combining like powers. A nominal three-term expression can simplify to a binomial if coefficients cancel. “Monic trinomial” additionally says the coefficient of the highest-degree term is one.","deep_definition":"Trinomial describes presentation by three terms, not irreducibility, degree three, or possession of three roots. A cubic is classified by degree; a trinomial is classified by how many nonzero monomials appear.","scope_note":"Count nonzero terms after combining like powers. 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Plectis applies the idea to source records, checks, uncertainty, and decisions that depend on particular claims.","reader_rule":"Tracking a belief is not a guarantee that the belief is true.","deep_definition":"The useful behaviour is responding to a failed assumption or contradictory observation: identify affected conclusions, change their status, and review actions that relied on them rather than preserving an unsupported conclusion unchanged.","scope_note":"Truth maintenance covers how beliefs, evidence, and their dependencies are tracked across the system's control plane, separating receipts, model outputs, generated projections, and doctrine commitments so none borrows another's authority. A reader meets it wherever a claim is shown with its supporting evidence, its status, or the conditions under which it would be withdrawn.","example":"The Belief-State Process Reward Replay component is one concrete instance: it tracks a belief across a sequence of steps, scoring each step against what the belief should have been at that point and replaying the trajectory to see where a belief was wrongly held or wrongly dropped.","example_route":{"href":"components.html#component-belief_state_process_reward_replay","label":"Belief State Process Reward Replay","gloss":"open one component that tracks and scores a belief across steps"},"non_example":"Truth maintenance is not a guarantee that the system's beliefs are correct, and it is not a single switch that marks something true or false. 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Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":["failure_modes","mining","guinea_pig"],"status":"active","disposition":"active","anchor":"glossary-trying","relative_href":"docs/glossary.html#glossary-trying","public_phrase":"trying","short_definition":"Trying means attempting an action whose outcome is not yet established.","reader_preview":"Trying means attempting an action whose outcome is not yet established.","reader_preview_basis":"definition","card_definition":"In research, an attempt can produce a proof, counterexample, useful failure, or no decisive result. Plectis keeps relevant evidence from attempts so later work can assess what was learned.","reader_rule":"The account should distinguish effort from the outcome actually obtained.","deep_definition":"A failed route is useful when its failure is explained precisely enough to guide another decision. Merely recording that an agent spent time on a task does not establish progress on the mathematical or software question.","scope_note":"Ordinary English for attempted work. Not a slogan and not a substitute for a check.","example":"Publishing a remaining obligation next to the lemmas that did check is evidence of trying.","example_route":null,"non_example":"Deleting failed routes until only green checks remain is not trying in this sense.","visible_text_byte_count":747,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::trying","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":31,"page_count":23,"top_page":"contact.html","top_pages":[{"page":"contact.html","count":4},{"page":"introduction.html","count":2},{"page":"paper-module-certificate-kernel-execution-lab.html","count":2},{"page":"paper-module-formal-math-verifier-trace-repair-loop.html","count":2},{"page":"paper-module-lean-std-premise-index.html","count":2},{"page":"paper-module-research-replication-rubric-artifact-replay.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":743,"id":"type_a","label":"Type A","preferred_label":"Type A","alternative_labels":["type_a","type-a"],"hidden_search_labels":["substrate-native actor","live substrate actor","controller agent","tool-using controller","iterative controller","harness agent"],"term_kind":"actor","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"Type A is ai_workflow's coined substrate-authority classification for an AI actor, not a personality type, intelligence ranking, controller-only role, or universal taxonomy.","meaning_contrasts":["type_b"],"status":"active","disposition":"active","anchor":"glossary-type-a","relative_href":"docs/glossary.html#glossary-type-a","public_phrase":"substrate-native actor","short_definition":"Type A is the local name for an AI actor with live access and assigned authority over the working system.","reader_preview":"Type A is the local name for an AI actor with live access and assigned authority over the working system.","reader_preview_basis":"authored_override","card_definition":"A Type A actor can inspect current repository state and perform permitted operations. It may be a controller or delegated agent. Write access, test access, and task ownership depend on the tools and permissions assigned to its session.","reader_rule":"Classify the actor by access and authority for this substrate. Delegation, intelligence level, and seniority are separate axes.","deep_definition":"An actor working from a packet may need Type A to collect evidence or apply a proposal. A delegated agent with live repository tools can already be Type A. A deterministic script is not an AI actor merely because it touches files.","scope_note":"Classification concerns authority over this working system, independently of model tier or delegation role.","example":"A coding-agent session that opens the repository, reads its current files, runs checks, makes edits under the gates, and commits the result is a Type A actor; the \"With a coding agent\" page shows one in operation.","example_route":{"href":"coding-agents.html","label":"With a coding agent","gloss":"see a Type A actor operating on the live system"},"non_example":"A Type A is not a top capability tier or a mark of seniority, and it is not simply \"the controller\": a tool-using helper agent with live access to the repository is also Type A, while a capable actor that only reads prepared material without live private state or change authority is not.","visible_text_byte_count":1314,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::type_a","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"contrasts_with","target":"type_b","gloss":"Type A has live private-substrate authority; Type B reasons through operator, bridge, provider, web, or injected-context surfaces without direct mutation authority."},{"verb":"specializes","target":"actor_axes","gloss":"Type A is the substrate_type=type_a value under con_016::actor_axes."},{"verb":"owns","target":"type_a_metabolism","gloss":"Type A handles unsettled grammar and final assimilation."},{"verb":"uses","target":"kernel","gloss":"Kernel commands are the local Type A control surface."},{"verb":"may_update","target":"standard","gloss":"Type A may update standards under evidence, validation, and rollback."}],"source_refs":[{"ref":"codex/doctrine/paper_modules/navigation_hologram_theory.md","role":"paper_module","gloss":"Uses Type A for authority packets, controllers, and assimilation decisions."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":744,"id":"type_a_judgment","label":"Type A judgment","preferred_label":"Type A judgment","alternative_labels":["type-a judgment","agent judgment"],"hidden_search_labels":["judgment traits","thinking traits","type a common sense","common sense care passion","care and passion","care + passion","karen passion"],"term_kind":"actor","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"Type A judgment is evidence- and doctrine-grounded discernment by an actor with live substrate access; it is not an intelligence rank, personality trait, or unrestricted license to change the system.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-type-a-judgment","relative_href":"docs/glossary.html#glossary-type-a-judgment","public_phrase":"judgment grows from traits","short_definition":"Type A judgement is the practical reasoning an agent uses when acting on the live system.","reader_preview":"Type A judgement is the practical reasoning an agent uses when acting on the live system.","reader_preview_basis":"definition","card_definition":"It includes noticing a missing connection, checking current evidence, choosing the appropriate owner, and deciding the smallest useful change within the task. 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A broader rule change needs evidence that the local lesson actually generalises.","scope_note":"Type A judgment covers the agent-side qualities of working over the system, such as common sense, care, and a critic's eye, and a reader meets it on the pages describing how agents operate over the system, particularly the coding-agent walkthrough.","example":"A coding agent working through this system pauses on a standard that no longer matches the code, notices the gap, and records where the correction belongs instead of carrying on regardless; that act of noticing and routing is Type A judgment.","example_route":{"href":"coding-agents.html","label":"With a coding agent","gloss":"see an agent exercising this judgment over the system"},"non_example":"Type A judgment is not a fixed checklist of virtues an agent is graded against, and it is not a free-floating personal opinion; it is discernment that stays anchored to the system's own rules and records.","visible_text_byte_count":1468,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::type_a_judgment","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[{"verb":"specializes","target":"type_a_metabolism","gloss":"Names the live judgment side of Type A metabolism."},{"verb":"composes","target":"common_sense","gloss":"Common sense is one routeable Type A judgment trait."},{"verb":"composes","target":"care","gloss":"Care is one routeable Type A judgment trait, not the whole surface."},{"verb":"composes","target":"passion","gloss":"Passion is one routeable Type A judgment trait, not the whole surface."},{"verb":"composes","target":"critic_posture","gloss":"Critic posture is the thinking-trait move for noticing route gaps and weak abstractions."},{"verb":"uses","target":"up_propagate","gloss":"Judgment traits become durable through up-propagation into the owning plane."},{"verb":"constrained_by","target":"axiom","gloss":"Axiom candidates bound when a judgment trait becomes constitutional."}],"source_refs":[{"ref":"CODEX.md","role":"adapter","gloss":"Codex-facing common-sense and up-propagation contract."},{"ref":"CLAUDE.md","role":"adapter","gloss":"Claude-facing Type A judgment and follow-on substrate context."},{"ref":"codex/doctrine/skills/doctrine/local_to_general_propagation.md","role":"skill","gloss":"Closeout and route-gap repair lane for reusable judgment lessons."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":745,"id":"type_a_metabolism","label":"Type A metabolism","preferred_label":"Type A metabolism","alternative_labels":["type_a_metabolism","Type-A metabolism"],"hidden_search_labels":["controller metabolism","judgment metabolism","grammar metabolism"],"term_kind":"runtime","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"Type A metabolism is the open-ended governance lane that creates or repairs rules where no settled procedure exists; it is not biological metabolism or routine execution of an existing job.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-type-a-metabolism","relative_href":"docs/glossary.html#glossary-type-a-metabolism","public_phrase":"judgment governs unsettled grammar","short_definition":"Type A metabolism names the maintenance work that requires judgement about the system's structure or rules.","reader_preview":"Type A metabolism names the maintenance work that requires judgement about the system's structure or rules.","reader_preview_basis":"definition","card_definition":"It addresses situations where an established routine is insufficient, such as resolving a contradiction, defining a missing object, or changing how a standard applies. The result should make later work more explicit and repeatable.","reader_rule":"This is a workflow category rather than a measure of an actor's intelligence.","deep_definition":"Because the work changes guidance that other actions rely on, its evidence, scope, and validation matter. A local ambiguity should lead to the narrowest justified structural refinement.","scope_note":"The term covers the governance-and-decision layer of the system's self-maintenance, as distinct from routine population work. A reader meets it on the architecture map and in the engine-room runtime where unsettled grammar is resolved before work is delegated downstream.","example":"The Engine Room Metabolism Runtime is a concrete instance: the runtime where the system processes unsettled grammar, resolving contradictions and naming primitives under validation before handing routine work to the lower-agency lane.","example_route":{"href":"paper-modules.html#paper-module-engine-room-metabolism-runtime","label":"Engine Room Metabolism Runtime","gloss":"open the runtime where unsettled grammar is resolved"},"non_example":"Type A metabolism is not the routine maintenance lane that applies rules already in place, and it is not a biological process. 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They do not replace explicit instructions, permissions, or evidence that a task has actually been completed.","scope_note":"The term covers the conduct expected of capable controlling agents, the coding agents and operators that act on the system, and appears wherever the pages describe how such an agent stewarded the substrate, spotted findings, and propagated them. It is encountered most directly on the page describing work alongside a coding agent.","example":"The conduct shown on the \"With a coding agent\" page is a concrete instance: an agent reads the substrate, makes a bounded change, records what it learned, and routes that finding into a durable plane rather than dropping it.","example_route":{"href":"coding-agents.html","label":"With a coding agent","gloss":"see a capable controller working the substrate in this register"},"non_example":"A Type A operating register is not a personality, a tone of voice, or a marketing claim about enthusiasm. 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If an actor receives live substrate access, classify it by the authority it actually has.","scope_note":"External tools can supply evidence without granting direct authority over this repository.","example":"A general assistant in a chat or browser window, asked a framing or research question and given a pasted block of context to work from, is acting as a Type B actor: it reasons over what it was shown and returns an answer, without reaching into the live repository itself.","example_route":{"href":"coding-agents.html","label":"With a coding agent","gloss":"see where an external agent works over handed-in context rather than the live repository"},"non_example":"A Type B is not a lower intelligence tier and not simply any delegated worker. 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Equivalently, the averages of every continuous function converge to its integral against normalized Haar measure on the torus.","reader_rule":"Every nonzero integer frequency must be checked. The criterion proves proportions of visits, a stronger statement than topological density. It is qualitative unless explicit bounds on the exponential averages are supplied.","deep_definition":"The #269 crossing-section proof checks the character averages by summing finite geometric progressions. Rational independence makes the nonconstant ratios different from one. The resulting equidistribution ensures that phase neighbourhoods realizing prescribed crossing prefixes are visited arbitrarily late.","scope_note":"Every nonzero integer frequency must be checked. The criterion proves proportions of visits, a stronger statement than topological density. It is qualitative unless explicit bounds on the exponential averages are supplied.","example":"Representative source: sites/microcosm/papers/source/plectis-erdos/paper/269/erdos269-running-lcm-reasoning-surface.tex::line 1553.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1174,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::weyl_criterion","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"sites/microcosm/papers/source/plectis-erdos/paper/269/erdos269-running-lcm-reasoning-surface.tex::line 1553","role":"mathematical_source","gloss":"Published manuscript use of Weyl criterion with the local hypotheses and conclusion stated in the referenced passage."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":778,"id":"whisper","label":"Whisper","preferred_label":"Whisper","alternative_labels":[],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"externally_anchored_term","meaning_scope_label":"Externally anchored","definition_profile":"public_contextual_gloss","meaning_boundary":"This registry only explains how the external term appears here. The external owner, specification, or primary source controls when exact meaning matters.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-whisper","relative_href":"docs/glossary.html#glossary-whisper","public_phrase":"Whisper","short_definition":"Whisper is OpenAI's speech-recognition model family for converting audio speech into text.","reader_preview":"Whisper is OpenAI's speech-recognition model family for converting audio speech into text.","reader_preview_basis":"definition","card_definition":"It can support transcription and related language tasks through suitable software. The surrounding recording workflow determines which timestamps, speaker information, or other metadata are supplied with the text.","reader_rule":"Do not assume that every transcript has accurate word-level timings or speaker labels.","deep_definition":"Recognition can make errors, especially with unfamiliar names, technical notation, or unclear audio. Plectis should preserve the source recording where exact wording matters and check important passages before treating them as authoritative quotations.","scope_note":"Whisper is made by OpenAI and is external to this work.","example":"A demonstration recording can use a Whisper transcript to locate spoken passages while retaining the original audio for checking exact wording.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":834,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::whisper","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":779,"id":"whole_system_correctness","label":"whole-system correctness","preferred_label":"whole-system correctness","alternative_labels":["whole system correctness"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"public_contextual_gloss","meaning_boundary":"Whole-system correctness is the global guarantee that Plectis explicitly refuses to infer from local checks; no component, page, or collection of green fixtures establishes it.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-whole-system-correctness","relative_href":"docs/glossary.html#glossary-whole-system-correctness","public_phrase":"whole-system correctness","short_definition":"Whole-system correctness means satisfying a complete, stated specification for an entire system.","reader_preview":"Whole-system correctness means satisfying a complete, stated specification for an entire system.","reader_preview_basis":"definition","card_definition":"Such a claim must cover the relevant components, interactions, inputs, and environment. 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The public component checks do not establish such a claim for the larger system.","example":"A component's scope limit records that a passing fixture case does not establish whole-system correctness.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":975,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::whole_system_correctness","source_hash":"sha256:b17e183b3fc7c704ac4fe59e25a4a457ae91c36f66777b5cfb1b603561adb298","relationships":[],"source_refs":[{"ref":"codex/standards/std_microcosm.json","role":"source_authority","gloss":"Grounds this Plectis-specific definition and its scope."},{"ref":"microcosm-substrate/src/microcosm_core/validators/research_claim_assurance.py","role":"source_authority","gloss":"Grounds this Plectis-specific definition and its scope."}],"evidence_commands":[],"used_in":{"total_occurrences":300,"page_count":97,"top_page":"paper-modules.html","top_pages":[{"page":"paper-modules.html","count":83},{"page":"rules-and-ideas.html","count":40},{"page":"components.html","count":13},{"page":"area-import-drift.html","count":4},{"page":"paper-module-batch10-cold-eval-honesty-capsule.html","count":4},{"page":"paper-module-public-reveal-walkthrough.html","count":4}]}},{"schema":"plectis_public_glossary_term_v0","position":780,"id":"winding_number","label":"winding number","preferred_label":"winding number","alternative_labels":["index of a curve"],"hidden_search_labels":[],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"Winding number depends on orientation and a reference point off the curve. Overlapping turns are counted with signs, not by enclosed area.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-winding-number","relative_href":"docs/glossary.html#glossary-winding-number","public_phrase":"winding number","short_definition":"The winding number of a closed plane curve around a point outside the curve counts, with sign, how many net turns the curve makes around that point.","reader_preview":"The winding number of a closed plane curve around a point outside the curve counts, with sign, how many net turns the curve makes around that point.","reader_preview_basis":"definition","card_definition":"A counterclockwise circle around the origin has winding number 1; traversing it twice gives 2, and a clockwise traversal gives -1. A loop that can be contracted without crossing the point has winding number 0.","reader_rule":"The curve must avoid the reference point, and orientation matters. The number is invariant under deformations that continue to avoid that point. For nonsimple curves, overlapping turns contribute algebraically rather than by enclosed area.","deep_definition":"The paper notes that level curves can carry an integer winding quantity. Cutting an annulus turns closed loops into arcs and can erase precisely this invariant, so the cut model may lose information essential to the original topology.","scope_note":"The curve must avoid the reference point, and orientation matters. The number is invariant under deformations that continue to avoid that point. 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A useful landing record lets later work identify the resulting artifact, inspect its evidence, and understand any remaining obligations.","scope_note":"The term covers acceptance of one bounded unit of work into its owning files, records, and status surfaces. 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them","what_this_is":"Research on eight Erdős problems: #68, #243, #249, #251, #257, #269, #1041 and #1049. Each per-problem row carries the source's status, checked results, exclusions and open obligations. Read those separately: a counterexample to an exact formal statement does not establish correspondence to every historical wording. The records distinguish Lean-checked results from manuscript arguments and open targets; do not infer that a problem is solved from the presence of a paper or a formal theorem.","results_map":"The human per-problem map with Lean anchors is docs/RESULTS.md in the mathematics repository: https://github.com/wcook04/plectis-erdos/blob/main/docs/RESULTS.md","per_problem":[{"erdos_number":68,"question":"Is the series sum_{n >= 2} 1/(n! - 1) irrational?","status":"open","what_is_checked":["The exact integral normal form: the series is irrational if and only if the strict factorial successor Z_m misses divisibility by m cofinally.","The equivalent carry form: rationality is equivalent to eventual unit carry, so cofinally many non-unit carries imply irrationality.","Canonical factorial digits, exact remainder recurrences, zero-tail propagation, and the divisor-factorial centre recurrence.","Integral channel congruences, the lcm tax, the prime unit translator, and factorial-grid Cramer residual identities.","Weighted projection and factor-split rigidity consumers with their exact modular and Archimedean hypotheses.","The zero-plateau and first-exit carry dichotomy, prime-power prefix obstructions, and exact doubled-prime branch characterisations.","A Lean-checked implication from one exact non-unit carry to a rational-denominator lower bound; a fresh exact GMP certificate at m = 300000 supplies the finite input used by the papers; the event-trace and unit carries match the previously retained receipt."],"what_is_not_checked":["Irrationality of the factorial-denominator series.","A theorem producing cofinally many divisibility misses or non-unit carries.","Cofinal strict nonintegrality of the factorial-grid Cramer residual.","The weighted collision-product and complementary-residue estimates needed by the factor-split projection consumer.","The GMP computation is an exact finite external certificate, not a Lean evaluation and not an unbounded argument."],"open_obligations":[{"id":"weighted_collision_and_complementary_residue","statement":"On an unbounded family of tailored prime blocks, prove both the weighted collision-core product bound and the independent complementary-residue inequality consumed by the exact unit-factor scale split."},{"id":"cofinal_prime_power_amplification","statement":"At infinitely many genuinely nonterminal indices, prove quantitative repeated-support valuation amplification strong enough to meet the checked endpoint inequality."},{"id":"cofinal_lower_endpoint_escape","statement":"For infinitely many primes, force the reduced predecessor gap outside the exact lower unit-carry cylinder."},{"id":"cofinal_doubled_prime_branch_failure","statement":"For infinitely many odd primes, rule out both exact p^2-divisibility branches at index 2p by coupling the carry value to the predecessor residue."},{"id":"cramer_residual_nonintegrality","statement":"Construct an unbounded factorial-grid family whose exact Cramer residual is nonintegral, using a determinant, valuation, cancellation, or gcd-of-minors certificate that controls the finite sign-changing block."}],"claim_registry_status":"registered_declarations"},{"erdos_number":243,"question":"Under a rapid-growth hypothesis on an integer sequence, does rationality of its reciprocal sum force the sequence to satisfy the Sylvester recurrence eventually?","status":"open","what_is_checked":["Lean proves cubic_rate_irrationality_unconditional for positive StrictMono zero-indexed a_n with a_n²/a_(n+1) = 1 + 3/n + o(n^-3). The one-based paper uses a finite-prefix bridge; general #243 remains open.","Lean proves nonintegral_regular_rate_irrational for positive StrictMono zero-indexed a_n whenever the real coefficient lambda is greater than one and nonintegral, with a_n²/a_(n+1) = 1 + lambda/n + o(n^-lambda). Summability is derived. The one-based paper uses the same finite-prefix bridge; the unrestricted problem remains open.","The exact product-cleared tail dynamics and the defect identity, as identities in the integer state variables.","That a vanishing centred state is absorbing under strict centring, so the state is either eventually zero or nowhere zero.","That eventual vanishing of the centred state, with the next tail state nonvanishing, forces the Sylvester recurrence eventually.","That a nonnegative centred state vanishes eventually, by descent.","That the centred state cannot be eventually constant and negative, at any magnitude and any common scale.","That the negative magnitude cannot be eventually periodic with positive drift, in the regime where each magnitude is below its multiplier.","A coprimality barrier: a sequence tending to infinity with uniformly bounded upward increments cannot permanently avoid infinitely many fresh pairwise coprime moduli.","That a reduced exact tail has pairwise coprime multipliers which its numerator permanently avoids, so no such tail has a divergent numerator with bounded rise.","That the tail gcd stabilises under cofinally bounded negative magnitudes.","Conditionally: exact dynamics, eventual strict centring, an eventually bounded negative part, and normalised vanishing together force the centred state to vanish eventually.","That survival through a fixed number of forced updates depends only on the initial state modulo a factorial of the horizon.","From a strictly increasing positive integer sequence, a rational reciprocal sum and a_(n+1)/a_n^2 tending to one, canonical_integer_tail_normalized derives eventual strict centring and normalised vanishing of the exact integer tail."],"what_is_not_checked":["The unrestricted problem.","A uniform lower bound on the centred error for every sequence in the original problem; the canonical tail transfer does not supply this additional hypothesis.","Any exclusion of negative excursions whose magnitudes are unbounded along a cofinal set.","The Erdős–Straus weighted criterion and Duverney's conditional characterisation, which are cited rather than formalised here."],"open_obligations":[{"id":"unbounded_negative_excursions","statement":"Exclude or construct exact orbits with infinitely many unbounded negative centred excursions. Checked exclusions assume bounded negative part, a fixed period, or fixed prime divisors."},{"id":"derive_negative_part_bound","statement":"Derive a lower bound on centred error from the original growth and rational-sum hypotheses, control unbounded negative excursions another way, or construct a counterexample."},{"id":"formalise_the_published_criteria","statement":"Kernel-check the Erdos-Straus weighted criterion and Duverney's conditional characterisation under explicit analytic hypotheses."}],"claim_registry_status":"registered_declarations"},{"erdos_number":249,"question":"Is the binary Lambert series sum phi(n)/2^n irrational?","status":"open","what_is_checked":["For every e >= 1, the complete dyadic totient kernel through level e has rational span dimension 2^e+1.","The rational span of the full dyadic totient kernel is not finite-dimensional; for integer-valued sequence modules this is equivalent to Coons's non-k-regularity and is not an independent contribution.","Lean diagonal certificates through t = 82; neither t = 83 nor an unbounded family is supplied.","Conditionally: a cofinal strict 9/10 gap for the exact natural prime-tail orbit supplies the existing finite pivot-point escape and hence the reviewed irrationality endpoint.","The local s=1 and s=2 squared Möbius Euler-factor identities and the first two prime-power divisor-sum differences.","That the two-by-two checkerboard kills separable backgrounds and is unique up to scale under the four row and column cancellations.","An abstract four-point divisor-layer identity, stated from its four explicit factorisation hypotheses.","An affine completion no-go: every residue class modulo an even modulus has a digit that recentres one carry step.","Exact producer and consumer interfaces for prime-ray layer supply, bounded-degree order witnesses, and finite-prime-support escape."],"what_is_not_checked":["The strict prime-tail orbit gap, which is the unproved producer consumed by the conditional endpoint.","The packet's sharp parabolic Stern-Brocot cusp theorem and its passage from a geometric x-parameter to the arithmetic totient series.","Eventual Hankel nonvanishing for the Möbius-Mersenne power ladder, or its denominator obstruction.","The natural-boundary and quasimodular-transcendence steps of the infinite Euler-sieve tower.","Polynomial resultant realisability, exact-order finite-field supply, Frobenius freezing, or the Archimedean growth needed by the prime-ray strategy.","Unconditional irrationality of the binary totient series."],"open_obligations":[{"id":"strict_prime_tail_orbit_gap","statement":"Prove the cofinal strict 9/10 real-part gap for the exact natural prime-tail orbit, or replace it with another producer for the reviewed pivot-escape endpoint."},{"id":"hankel_denominator_fan_in","statement":"Prove eventual nonvanishing or a usable denominator obstruction for the Möbius-Mersenne power ladder and connect it to the target value."},{"id":"euler_sieve_limit_theorem","statement":"Control the infinite Euler-sieve limit strongly enough that finite-stage quasimodularity yields a valid arithmetic obstruction."},{"id":"prime_ray_resultant_supply","statement":"Realise the layer system by cyclic resultants and prove prime-support escape with the growth estimates needed to contradict a rational carry orbit."},{"id":"stern_brocot_arithmetic_bridge","statement":"Prove the sharp cusp asymptotic and then supply a non-formal bridge from the geometric parameter regime to the exact totient value."}],"claim_registry_status":"registered_declarations"},{"erdos_number":251,"question":"Is the dyadic series of consecutive primes irrational? Equivalently, is the corresponding consecutive-prime-gap dyadic series irrational?","status":"open","what_is_checked":["The finite summation-by-parts identity relating the prime dyadic partial sums to the prime-gap dyadic partial sums, with an exact endpoint term and no convergence premise.","The factor-of-two normalisation between the two displayed indexings, at every finite horizon.","The unconditional prime-to-gap identity, using an elementary polynomial bound on primes to prove summability; irrationality of the prime series is equivalent to irrationality of the prime-gap series.","For an abstract integer-digit dyadic tail recurrence: the exact block identity, and that a shift of fixed length is integral exactly when an explicit scaled state is.","That one integral shift propagates to every later index.","That repeated doubling removes the power-of-two denominator, so every rational-valued dyadic recurrence has one fixed positive shift integral at all sufficiently late indices.","For a real-valued dyadic recurrence, cofinal non-integrality of every fixed positive shift implies irrationality of the initial state.","That the actual consecutive-prime-gap sequence is unbounded, by the factorial prime-free-interval construction, and therefore is not eventually periodic with any positive period.","For any integer-digit dyadic tail recurrence, two adjacent fixed-length shifts lying strictly between -1 and 1 cannot both be integral when the corresponding digits differ.","For the actual prime gaps, a cofinal supply of adjacent small-shift pairs with a mismatching gap excludes eventual integrality of that fixed shift.","The generic distance-to-integers core of the finite-truncation criterion: an approximation farther than its error bound from every integer certifies non-integrality."],"what_is_not_checked":["The target irrationality.","A cofinal nonintegrality theorem for the concrete prime-gap tails; defining the series and proving its recurrence do not supply this.","A theorem supplying cofinally many adjacent actual-prime-gap tail shifts in the open unit interval with the required gap mismatch.","Any theorem forcing periodicity or automaticity of the prime gaps from rationality; the free-carry identity disconfirms that route."],"open_obligations":[{"id":"prime_gap_cofinal_shift_escape","statement":"For every fixed h >= 1 and every N0, find N >= N0 for which sum_{j>=1} (g_{N+h+j}-g_{N+j})/2^j is not an integer."},{"id":"cofinal_adjacent_small_mismatch","statement":"For every fixed h >= 1 and every N0, find N >= N0 such that both adjacent full tail shifts have absolute value less than 1 and g_{N+h+1} != g_{N+1}. The checked local consumer then excludes eventual integrality of the h-shift."}],"claim_registry_status":"registered_declarations"},{"erdos_number":257,"question":"Is the sum of 1/(2^n-1) over every infinite set of positive exponents irrational?","status":"open","what_is_checked":["Irrationality at every integer base for the classical full support, reciprocal-summable supports, and the finite-prime weighted supports in divisibilityWeightedClaim.","Every infinite subset of the union of a finite base-two weighted-mass host and a strengthened positive-cover host has irrational series at every integer base (mixedSupportClaim).","Unique Mersenne coding on every support; compactness and nowhere density; perfection for infinite support; finite-complement measure 2^(-|F|), otherwise zero."],"what_is_not_checked":["Universal irrationality over all infinite supports, or the proposed rational values 1/2 and 1/21.","Fair-bit laws, finer singularity results, and periodic-stride dimensions.","Any bridge from geometric uniqueness or measure rigidity to arithmetic irrationality."],"open_obligations":[{"id":"arithmetic_rigidity_for_thin_supports","statement":"Convert hereditary unique coding into an arithmetic obstruction to rational values for every infinite support, including zero-measure Cantor supports."},{"id":"formalise_measure_and_stride_geometry","statement":"Extend the measure dichotomy to Bernoulli laws and periodic-stride dimensions without implying arithmetic consequences."}],"claim_registry_status":"programme_only"},{"erdos_number":269,"question":"For a finite set of at least two primes, is the sum of reciprocals of the running least common multiples of the smooth numbers irrational? This library treats the three-prime case.","status":"open","what_is_checked":["That for three pairwise distinct primes the least common multiple of the smooth prefix equals the product of the three maximal pure prime powers below the cutoff.","That equal logarithmic coordinates give an equal prefix least common multiple, so the running value is constant on logarithmic cells.","That advancing exactly one logarithmic coordinate multiplies the running value by the corresponding prime.","That the positive power channels of distinct primes are disjoint, so the first powers in three channels give an exact jump count.","That a finite smooth lattice sum groups exactly by running height, with the height fibre cardinality as coefficient.","A quadratic bound on the cardinality of the actual filtered exponent shell.","That each dyadic block contains at most one internal pure power from each non-dyadic channel, so the compressed {2,3,5} block radix belongs to the exact alphabet {2,6,10,30} and in particular lies between 2 and 30.","That the smallest {2,3,5} kernel rectangle has determinant exactly -1/15, so its two-by-two matrix has rank two over the rationals and no rank-one separation exists.","The canonical least-positive-residue interval and congruence facts, including the zero-residue representative.","The exact local-window identity for an integral carry.","For the actual {2,3,5} series, rationality produces a positive reduced carry; cofinal local-window escape rules out such carries under the recorded bound.","A bounded-radix dichotomy: under a bounded positive digit alphabet, a nonzero scaled tail must be separated from the integers cofinally.","Exact three-channel block-nullity as a coboundary condition, and extinction of a bounded carry lift under two independent anchors.","The exact residue-digit plus coboundary form of an integral carry, together with a function-faithful observer criterion and a finite realised-span consequence.","Irrationality of the distinct-height sum for {2,3,5}, which counts each running value once."],"what_is_not_checked":["Irrationality of the repeated sum, or transcendence, in any three-prime case.","CofinalLocalWindowEscape for those actual words; it is an explicit hypothesis of the checked consumer, not a theorem.","A proof that the actual three-channel word satisfies the block-null, anchor, or factorisation hypotheses of the migrated abstract consumers.","The infinite limit of the jump expansion; only its finite core is checked.","Any unbounded denominator exclusion. The finite certificates exclude denominators below an explicit bound and no further."],"open_obligations":[{"id":"actual_local_window_residue_escape","statement":"For every positive B coprime to 30 and every starting index, produce a later actual {2,3,5} window whose least positive forcing residue exceeds the exact denominator-dependent carry bound."},{"id":"nonstationary_analytic_theorem","statement":"Prove a genuinely higher-dimensional analytic theorem for the phase cocycle or the associated contraction."},{"id":"unbounded_height_certificate","statement":"Replace one finite denominator exclusion by a family whose exclusion bound tends to infinity."}],"claim_registry_status":"registered_declarations"},{"erdos_number":1041,"question":"Must two roots of a monic polynomial in the open unit disc be joined by a sub-two-length curve inside its unit lemniscate? Ani’s degree-seven example refutes the exact Formal Conjectures statement; correspondence with the 1958 wording awaits human review.","status":"formal statement refuted","what_is_checked":["For ani’s degree-seven polynomial, every preconnected strict-lemniscate set containing two distinct roots has one-dimensional Hausdorff measure above two. Lean refutes the exact Formal Conjectures path-image claim and checks a separate total-variation bound.","Along a trajectory tangent to the complex Newton vector field -f/f', the polynomial value satisfies w' = -w.","The exponentially rescaled polynomial value is a first integral, so the value of a Newton trajectory remains on one oriented ray.","A finite Newton-flow connection can only join endpoints whose polynomial values lie on the same positive ray; distinct critical-value arguments therefore exclude such a connection.","For two translated critical values, the ray-collision locus is a real affine line with an exact one-parameter description.","A finite family of ray-collision lines can be avoided by an arbitrarily small common translation.","An explicit root-retention estimate keeps roots of a sufficiently small constant perturbation inside the unit disc."],"what_is_not_checked":["The full small-parameter family reported by ani; only one explicit polynomial is formalised.","Independent human review adjudicating whether the exact Formal Conjectures path-image-length statement represents the 1958 curve-length wording.","The spanning-tree decomposition claimed in the March 2026 manuscript; its printed Proposition 12 proof uses an invalid three-ended local saddle block, and a Cassini example refutes the printed global tree budget.","A repair of Proposition 12 that retains the printed 1/(2π) spanning-tree coefficient; any later argument must change the metric inequality.","A perturbation theorem that first makes critical values injective while preserving the relevant lemniscate component, collars, and strict length slack.","A corrected restricted-family or alternative-metric gluing theorem; universal sub-two total-variation gluing is ruled out by the degree-seven counterexample."],"open_obligations":[{"id":"historical_curve_length_correspondence","statement":"Independently review whether the 1958 curve-length wording matches the refuted Formal Conjectures path-image Hausdorff statement."},{"id":"repair_or_refute_saddle_block","statement":"Repair or replace the false Proposition 12 spanning-tree estimate; account for positive attachment costs or use another metric inequality."},{"id":"compact_ray_cut_strip_decomposition","statement":"Verify a compact ray-cut strip or cell decomposition under explicit local-valence hypotheses for restricted families; it gives no universal sub-two connector."},{"id":"corrected_metric_gluing_for_restricted_families","statement":"For a restricted family or different historical length notion, prove a metric gluing inequality that pays every positive attachment cost; universal sub-two total-variation gluing is refuted."},{"id":"two_stage_stable_perturbation","statement":"Given a selected connector with strict slack, quantify a two-stage perturbation preserving roots, component, collars, ray separation, and slack; no universal existence claim follows."},{"id":"relative_global_newton_flow_theorem","statement":"Under explicit component, saddle, and corrected metric hypotheses, derive a restricted root-to-root Newton-flow connector without a universal sub-two claim."}],"claim_registry_status":"registered_declarations"},{"erdos_number":1049,"question":"For which rational bases is the corresponding series irrational? The first resistant explicit base is three halves.","status":"open","what_is_checked":["Lean checks irrationality and the stated irrationality-exponent bound for rational bases in the exact Zudilin contour region, including every positive integral power of 31/4. The contour excludes 3/2.","For p>1 and N≤8, Lean checks positive definite coefficient matrices, real finite-pencil roots below F(p), and non-strict interlacing at adjacent ranks. Neither all-rank positivity nor irrationality at 3/2 follows.","That a literal coordinatewise transfer of the integer-base clearing argument forces a power-versus-linear inequality, and that this inequality is impossible at base three halves.","The exact denominator-cleared rational-base tail recurrence, including its forcing term.","That for every noninteger base with denominator at least two the forcing magnitude grows at least exponentially, while at denominator one it collapses to a bounded expression.","The complete elementary Archimedean height inequality used by the published Bundschuh-Väänänen criterion at base seven halves; the analytic irrationality theorem itself remains external.","Polynomial bounds on the doubled denominator exponents in the homogenised Padé construction, together with an exact gap identity.","The exact exterior-determinant elimination identity, and that rowwise contents scale the Padé errors, exterior determinant, and absolute determinant height by the same factors.","The exact Heine-Zudilin cone endpoint exponent arithmetic used to identify the bottom and top coefficient endpoints.","That homogeneous evaluation at (3,2) sees only the constant endpoint modulo three and the top endpoint modulo two, so unit endpoints obstruct common local factors.","That under unit top and constant endpoint hypotheses, a common multiplier of the two specialised coefficient channels is divisible by neither two nor three; the missing local gain must therefore survive primitive normalisation.","That vanishing of the higher bottom and top endpoint jets is exactly divisibility by the corresponding powers of three and two.","That the source integer-base ratio inequality implies a strictly negative scalar product-formula margin at three halves.","That the four-jet target has exact cardinality (3^R)^2(2^S)^2 and, for positive R, any family with n at least 4R+2S forces two distinct binary selectors to collide; their {-1,0,1} difference cancels all four endpoint jets.","That the rectangular Hermite-Pade threshold never improves on the classical one in the checked exponent model, with equality characterised exactly.","The exact 81/200 height region: 31/4 and every positive power lie inside it and outside the earlier Bundschuh-Vaananen region, while 3/2 lies outside both.","At n = 0, the Amdeberhan-Zeilberger q-Apery operator applied to Van Assche's moving diagonal has residual -p(p-1)^2(p+1)(p^5+2p^4+2p^3+2p^2+2), which is strictly negative and nonzero for every real p > 1."],"what_is_not_checked":["The universal rational-base conjecture, including the case 3/2, remains open. The checked 31/4 power family settles one region and leaves that conjecture untouched.","The published analytic height theorem, which is cited and used as an external result and is not formalised here.","The all-scale Heine-Zudilin coefficient construction at rational base three halves; the primary source assumes an integer base.","A sufficiently large primitive-normalised, non-collapsed coefficient family meeting the checked n >= 4R+2S threshold at the required quadratic depths.","That one of the resulting bounded-coefficient collision vectors has a nonzero polynomial pair, lies outside the analytic remainder nullspace, and has sufficient local gain.","The elementary Padé pilot checks exponent arithmetic only; it lacks remainder positivity or decay and does not qualify the separate Zudilin contour result.","The n=0 q-Apéry residual gives no general recurrence, lattice, valuation, denominator, or irrationality result; it only blocks the named scalar transfer."],"open_obligations":[{"id":"three_halves_growing_rank_endpoint_jet_kernel","statement":"After rowwise primitive normalisation, exhibit at least 4R_n+2S_n coefficient pairs at positive quadratic depths from a non-collapsed deformation; preserve the two- and three-adic gain; then show that one checked {-1,0,1} collision has a nonzero polynomial pair outside the analytic remainder nullspace, or prove an exact rank obstruction."}],"claim_registry_status":"registered_declarations"}],"claim_count":160,"status_counts":{"conditional reduction":51,"unconditional progress":44,"proved here":37,"formalised here":8,"verified finite instance":8,"open":7,"cited only":4,"formal statement refuted":1},"status_counts_reading":"These are registry statuses, not quality grades. 'proved here' means a Lean proof term is accepted for that statement; 'conditional reduction' means an exact equivalence whose two sides may both remain open; 'unconditional progress' means an unconditional theorem about the open problem itself.","bears_directly_on_the_open_problems":[{"claim_id":"prime_gap_unboundedness_and_nonperiodicity","label":"Unbounded and non-eventually-periodic consecutive prime gaps","status":"unconditional progress","statement":"The consecutive-prime-gap sequence is unbounded and is not eventually periodic with any positive period. These coefficient facts do not by themselves imply irrationality of the prime-gap or prime-index dyadic series.","lean_declarations":["ErdosProblems/Erdos251/PrimeGapDyadicTail.lean:57","ErdosProblems/Erdos251/PrimeGapDyadicTail.lean:1023"],"paper_label":"res:gap-nonperiodic"},{"claim_id":"newton_ray_separation_consumer","label":"Finite Newton-flow ray separation and perturbation consumer","status":"unconditional progress","statement":"An explicit exponential-decay Newton connection can occur only when its endpoint values lie on the same positive ray; conversely, every finite injective complex family admits an arbitrarily small common translation making all values nonzero and all positive-ray arguments distinct. These are finite perturbative and conditional connection statements.","lean_declarations":["ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean:197","ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean:325","ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean:334"],"paper_label":"res:ray"},{"claim_id":"channel_quotient_band_breakpoint","label":"Quotient-band channel breakpoint for factorial moments","status":"unconditional progress","statement":"For a finite integer coefficient family whose indices all lie in one quotient band k·d ≤ i_j < (k+1)·d, the factorial moment is exactly (d!)^k times the d-channel numerator. In the first band d ≤ i_j < 2d, channel cancellation forces the factorial moment to vanish; with all indices at least d and a nonzero moment, some index is therefore at least 2d.","lean_declarations":["ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean:71","ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean:91","ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean:101"],"paper_label":"long68:res:bandbreakpoint"},{"claim_id":"three_prime_finite_residue_contradiction","label":"Finite least-positive-residue contradiction for #269","status":"unconditional progress","statement":"For a positive modulus C, a positive integer state c bounded by K cannot be congruent to x modulo C when the canonical least-positive representative of x is greater than K; the zero residue class is represented by C. The integer carry form and exact residue classifier are checked in Lean.","lean_declarations":["ErdosProblems/Erdos269/ResidueEscape.lean:76","ErdosProblems/Erdos269/ResidueEscape.lean:110","ErdosProblems/Erdos269/ResidueEscape.lean:138"],"paper_label":"res:consumer"},{"claim_id":"three_prime_finite_prefix_completion_boundary","label":"Finite #269 radix prefixes admit an altered rational completion","status":"unconditional progress","statement":"Every finite positive integral radix prefix with nonnegative integral digits extends to a positive rational orbit by changing the future to radix 2 and digit 1. In particular, the actual word through any finite horizon has such an abstract completion. This is a countermodel to finite-prefix-only inference and is not a rational model of the actual infinite smooth-number series.","lean_declarations":["ErdosProblems/Erdos269/R12/PrefixExtension.lean:75","ErdosProblems/Erdos269/R12/PrefixExtension.lean:105"],"paper_label":null},{"claim_id":"three_prime_bounded_certificate_safety","label":"Size-authenticated bounded certificate checks for #269","status":"unconditional progress","statement":"The bounded width and cumulative-array checkers expose their size hypotheses, are sound on every accessed entry, compose across adjacent ranges, and carry explicit regressions showing that default-zero array access, half-height cancellation, and finite coverage cannot certify an infinite/cofinal conclusion. No paper numerical table is executed or validated by these safety theorems alone.","lean_declarations":["ErdosProblems/Erdos269/CertificateSafetyR12.lean:14","ErdosProblems/Erdos269/CertificateSafetyR12.lean:25","ErdosProblems/Erdos269/CertificateSafetyR12.lean:34"],"paper_label":null},{"claim_id":"twenty_one_quotient_greedy_frontier","label":"Canonical quotient-greedy and scaled-remainder frontier at 1/21","status":"unconditional progress","statement":"Membership of 1/21 in the Mersenne achievement set is equivalent both to excluding one explicit fatal/cofinite/aligned branch and to cofinally bounded scaled actual greedy remainders. Every closed Boolean quotient row is the deterministic quotient-greedy row; any unbounded sequence of canonical rows with remainder at most 2^R gives membership. On the complementary fatal branch the remainder is eventually strictly supercapacity, the boundary rank is always selected, and support and remainder follow one exact affine recurrence. A final skipped exponent also forces a structured one-sided approximation to the Erdős–Borwein constant with exact denominator-order constraints. These results isolate permanent affine supercapacity as the surviving late regime; they do not contradict it or prove 1/21 membership.","lean_declarations":["Erdos249257/TwentyOneQuotientGreedy.lean:32","Erdos249257/TwentyOneQuotientGreedy.lean:39","Erdos249257/BooleanMobiusCarry.lean:1721"],"paper_label":"res:one-over-twenty-one-frontier"},{"claim_id":"stern_brocot_run_fibonacci_stability","label":"Fibonacci and continuant stability of Stern–Brocot runs","status":"unconditional progress","statement":"A Stern–Brocot word with r nonempty alternating runs has height at least F_(r+3), with equality on the all-unit run spine. Writing each run length as 1+e_i gives an exact multiaffine continuant-defect expansion, the sharp global gain F_(r+1) sum e_i, and exact one-site coefficient F_(i+2)F_(r-i+1). Under the gcd-one weight 2^-(a+b), the first induced run layer has mass 1/2, while the natural denominator-bookkeeping exponent is exactly F_(r+3)-2. These are source-level combinatorial and arithmetic facts; no analytic denominator survival or irrationality conclusion is claimed.","lean_declarations":["Erdos249257/SternBrocotRunGeometry.lean:125","Erdos249257/SternBrocotRunGeometry.lean:185","Erdos249257/SternBrocotRunGeometry.lean:263"],"paper_label":"res:sternbrocotruns"},{"claim_id":"totient_carry_kernel_anti_compression","label":"Totient carry anti-compression","status":"unconditional progress","statement":"If the binary totient coefficient series were rational, one positive-multiplier tempered integral carry would have uniformly eventually-periodic dyadic sections modulo its multiplier while retaining canonical carry-section span dimension at least 2^e − 1 for every e. No finite-rank bound for every rationality-supplied carry is proved, so this quotient-periodicity/anti-compression conjunction is a necessary consequence rather than an irrationality criterion for S.","lean_declarations":["Erdos249257/TotientTailCarryPeriod.lean:224","Erdos249257/TotientCarryKernelRigidity.lean:263","Erdos249257/TotientTailCarryPeriod.lean:180"],"paper_label":"res:carryrank"},{"claim_id":"dyadic_totient_basis","label":"Explicit basis of the dyadic totient kernel span","status":"unconditional progress","statement":"Lean constructs an explicit odd-core-indexed basis of the rational span of the full dyadic totient kernel, with exact span equality and linear independence. This is an independent constructive proof route; its infinite-dimensionality consequence is not an independent contribution because Coons's non-2-regularity theorem already implies it.","lean_declarations":["Erdos249257/TotientMahlerDefect.lean:1265","Erdos249257/TotientMahlerDefect.lean:1380","Erdos249257/TotientMahlerDefect.lean:1392"],"paper_label":"res:basis"},{"claim_id":"totient_residue_series_classification","label":"Irrational totient residue series and the rational dyadic residue observables","status":"unconditional progress","statement":"For every m ≥ 3 the series ∑_{n≥1} (φ(n) mod m)/2^n is irrational, and for k ≥ 1 and f: Z/2^kZ → Q the series ∑_{n≥1} f(φ(n) mod 2^k)/2^n is rational exactly when f is constant on the even residue classes, with value 3f(1)/4 + c/4 where c is that constant. The first assertion, in every integer base, already follows from Erdős's Lemma 1 (Indag. Math. 19 (1957), p. 213; the case b = 0 of his Lemma 4, pp. 215-218) because φ(n) mod m vanishes outside a set of density zero, and equally from Theorem 3 of Kaneko, Suzuki and Tachiya (arXiv:2601.20743v1, p. 5). Those criteria also give the classification when the values f(r) - f(0) on the even classes have one sign; with both signs they need a further spacing hypothesis, which the isolated-coefficient proof does not need. Wong posted the case in which the base equals the modulus in 2015. No novelty claim is made for the first assertion. The reduced series say nothing about the unreduced series ∑ φ(n)/2^n.","lean_declarations":["ErdosProblems/Erdos249/PaperCompleteR7/RationalObservableClassification.lean:276"],"paper_label":"res:residueseries"},{"claim_id":"totient_residue_series_integer_radix","label":"Totient residue series in every integer base","status":"unconditional progress","statement":"For every integer t ≥ 2 and m ≥ 3 the series ∑_{n≥1} (φ(n) mod m)/t^n is irrational, and for k ≥ 1 and f: Z/2^kZ → Q the series ∑_{n≥1} f(φ(n) mod 2^k)/t^n is rational exactly when f is constant on the even residue classes, with value (t+1)f(1)/t^2 + f(0)/(t^2(t-1)). The first assertion already follows from Erdős's Lemma 1 (1957) and from Theorem 3 of Kaneko, Suzuki and Tachiya, as recorded for the binary case; no novelty claim is made for it.","lean_declarations":["ErdosProblems/Erdos249/PaperCompleteR7/RationalIntegerRadixClassification.lean:128","ErdosProblems/Erdos249/PaperCompleteR7/RationalIntegerRadixClassification.lean:145","ErdosProblems/Erdos249/PaperCompleteR7/IntegerRadixObservables.lean:303"],"paper_label":"prop:radixresidue"},{"claim_id":"totient_residue_series_dilations","label":"Linear independence of dilated totient residue series","status":"unconditional progress","statement":"For every integer t ≥ 2 and m ≥ 3, the numbers 1 and ∑_{n≥1} (φ(n) mod m)/t^{dn}, d = 1, 2, ..., are linearly independent over Q. For a finite set D of positive integers and residue functions f_d: Z/2^{k_d}Z → Q with k_d ≥ 1, any rational constant plus ∑_{d∈D} ∑_{n≥1} f_d(φ(n) mod 2^{k_d})/t^{dn} is rational exactly when every f_d is constant on the even residue classes. The proof isolates a nonzero coefficient at the least active dilation by the Chinese remainder theorem and Dirichlet's theorem. When all coefficients of a relation have one sign, Erdős's Lemma 1 (1957) already excludes it; the mixed-sign case is not covered by that criterion without a further spacing hypothesis.","lean_declarations":["ErdosProblems/Erdos249/FiniteDilationLinearIndependent.lean:42","ErdosProblems/Erdos249/FiniteDilationMixedModuli.lean:95"],"paper_label":"prop:dilations"},{"claim_id":"dyadic_totient_certificate_interface","label":"Dyadic totient-kernel finite-level rank","status":"unconditional progress","statement":"For every e, a CRT–Dirichlet construction produces a separated nonzero evaluation minor for the canonical dyadic totient-kernel family, which is linearly independent with exact span dimension 2^e + 1. The same Lean development proves that the full dyadic kernel span is not finite-dimensional, but that corollary is not an independent contribution: it is equivalent, for this integer-valued sequence module, to Coons's non-2-regularity theorem. Even residues reduce to odd-core channels, and the compressed-adjoint boundary remains available. These coefficient-side theorems do not prove that S is irrational.","lean_declarations":["Erdos249257/TotientMahlerDefect.lean:882","Erdos249257/TotientMahlerDefect.lean:935","Erdos249257/TotientMahlerDefect.lean:989"],"paper_label":"thm:kkernelrank"},{"claim_id":"greedy_achievement_geometry","label":"Greedy geometry of the Mersenne achievement set","status":"unconditional progress","statement":"The Mersenne achievement set is compact, perfect, totally disconnected, nowhere dense, and has Lebesgue measure one. Membership is equivalent to greedy survival at every level; for the target 1/2 it is exactly equivalent to the canonical greedy orbit omitting infinitely many exponents. No finite normalized support has value 1/2, so positive membership would automatically produce an infinite rational-valued support and refute universal #257. Finite exact-rational death certificates soundly prove non-membership (3/4 is excluded at level one). The strict-tail Cantor geometry is recorded by Kovač–Tao; no novelty claim is made for the formalised refinements.","lean_declarations":["Erdos249257/GreedyAchievementSet.lean:996","Erdos249257/GreedyAchievementSet.lean:1656","Erdos249257/GreedyAchievementSet.lean:1672"],"paper_label":"res:greedy"},{"claim_id":"campbell_shift_synchronization_boundary","label":"Campbell shift-synchronization strategy boundary","status":"unconditional progress","statement":"Campbell's application of the Alford-Granville-Pomerance estimate chooses delta = 1/4, so at prime-search height X the progression modulus d must satisfy d <= X^(1/4), equivalently d^4 <= X. In the proposed last-skip synchronization, P is the fatal-window prime cap, t the unavoidable phase period, and d the combined progression modulus. Taking X = P, the requirements 2 <= d, t <= d, P <= 4t, and d^4 <= P are inconsistent. The shifted-zero interface is guarded by a putative last skip and is exactly equivalent to infinitely many actual greedy skips and to membership of 1/2 in the Mersenne achievement set. It is an endpoint restatement, not an independent Campbell input, and proves no half-membership.","lean_declarations":["Erdos249257/CampbellShiftSynchronization.lean:299","Erdos249257/CampbellShiftSynchronization.lean:320","Erdos249257/CampbellShiftSynchronization.lean:471"],"paper_label":"res:campbellboundary"},{"claim_id":"fixed_precision_transport_no_go","label":"Fixed-precision valuation-unit no-go theorem","status":"unconditional progress","statement":"At every fixed positive precision, every finite word of odd valuation-unit symbols admits a compatible carry orbit from an arbitrary incoming integer state, with each successive state within the corresponding valuation-unit radius. Thus bounded local signatures alone cannot exclude all finite centred endpoints; growing precision or global information remains necessary for an endpoint argument.","lean_declarations":["Erdos249257/TropicalCurvatureCarry.lean:137","Erdos249257/TropicalCurvatureCarry.lean:69","Erdos249257/TropicalCurvatureCarry.lean:115"],"paper_label":null},{"claim_id":"lcm_factor_ideal_anchor_pulse_no_go","label":"LCM factor-ideal anchor-pulse no-go theorem","status":"unconditional progress","statement":"For every t ≥ 3, a nonzero synthetic affine carry has forcing letters in the principal ideal generated by φ(lcm(1,…,t)), matches every stated whole-ray anchor, and remains within the diagonal bounds while its state is a strict survivor. The same countermodel survives every finite integer shift polynomial. Thus factor ideals, whole-ray anchors, and finite-rank shift algebra alone do not force carry escape. The construction does not assert that its forcing letters are actual totient differences and does not rule out nonlinear use of fresh divisor channels.","lean_declarations":["Erdos249257/LcmFactorIdealPulseObstruction.lean:798","Erdos249257/LcmFactorIdealPulseObstruction.lean:866","Erdos249257/LcmFactorIdealPulseObstruction.lean:895"],"paper_label":"res:factorideal"},{"claim_id":"actual_lcm_positive_corridor_top_edge","label":"Actual LCM positive corridor and top-edge boundary","status":"unconditional progress","statement":"For a ≥ 8 and J + (a + 6) < 2·2^a, the true translated LCM-orbit difference is strictly positive. If it is represented by an integer and the dyadic modulus has room for the directed endpoint strip, the carry survivor is strictly negative and the discrepancy residue is exactly the top-edge representative 2^K − carry_K, strictly between the strip’s lower edge and 2^K. Thus the positive corridor eliminates the nonnegative true survivor but does not itself produce a central-band contradiction; an independent arithmetic exclusion of the top boundary is still required.","lean_declarations":["Erdos249257/TotientActualLcmOrbitSign.lean:39","Erdos249257/TotientActualLcmOrbitSign.lean:172","Erdos249257/TotientActualLcmOrbitSign.lean:211"],"paper_label":"res:actualsign"},{"claim_id":"three_halves_coordinatewise_corridor_no_go","label":"Base 3/2 coordinatewise corridor exclusion","status":"unconditional progress","statement":"With N ≥ 1 and K ≥ 1, the six-condition CoordinatewiseCorridor at base 3/2 is impossible for every natural Q and digit. This excludes only that literal coordinatewise clearing architecture, not rationality or irrationality of F(3/2) or any other proof architecture.","lean_declarations":["ErdosProblems/Erdos1049/RationalBaseLambert.lean:155"],"paper_label":"long1049:res:nocorridor"},{"claim_id":"rational_base_cleared_tail_recurrence","label":"Exact rational-base cleared-tail recurrence","status":"unconditional progress","statement":"For rational r, s, B, and F, a coefficient sequence coeff : ℕ → ℚ, and r ≠ 0, the denominator-cleared tail state satisfies rationalBaseClearedTailQ (r, s, B, F, coeff) (N + 1) = r · rationalBaseClearedTailQ (r, s, B, F, coeff) N − B · coeff (N + 1) · s^(N + 1) for every N. The identity is not derived from rationality and is not itself a rationality contradiction.","lean_declarations":["ErdosProblems/Erdos1049/RationalBaseLambert.lean:187"],"paper_label":"long1049:res:tailrec"},{"claim_id":"rational_base_forcing_growth","label":"Rational-base forcing-term growth boundary","status":"unconditional progress","statement":"For natural s, B, coeff, and N, if s ≥ 2, B ≥ 1, and coeff(N + 1) ≥ 1, then rationalBaseForcingNat s B coeff N ≥ 2^(N + 1); at s = 1 the same quantity is exactly B · coeff(N + 1). This is an elementary forcing-term bound, not a bounded-state contradiction or endpoint result.","lean_declarations":["ErdosProblems/Erdos1049/RationalBaseLambert.lean:204","ErdosProblems/Erdos1049/RationalBaseLambert.lean:218"],"paper_label":"long1049:res:forcing"},{"claim_id":"van_assche_qapery_diagonal_no_go","label":"Van Assche q-Apéry diagonal recurrence non-transfer","status":"unconditional progress","statement":"For every real p > 1, the n = 0 residual obtained by applying the Amdeberhan–Zeilberger q-Apéry operator to Van Assche's moving diagonal is the displayed strictly negative factorisation, hence is nonzero. This rules out that selected diagonal for that recurrence at n = 0; it proves no general recurrence, endpoint, or #1049 irrationality result.","lean_declarations":["ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean:67","ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean:94"],"paper_label":null},{"claim_id":"bounded_rise_fresh_modulus_barrier","label":"Bounded-rise fresh-modulus coprimality barrier","status":"unconditional progress","statement":"For natural sequences u and m, if u tends to infinity with upward increments bounded by a fixed positive B, the moduli m are eventually greater than 1 and pairwise coprime, and each earlier m(i) is coprime to every later u(t), then the hypotheses are inconsistent. This is a reusable state-level obstruction, not a proof of the full Erdős #243 assertion.","lean_declarations":["ErdosProblems/Erdos243/ReciprocalTailRigidity.lean:903"],"paper_label":"res:barrier"},{"claim_id":"eventually_periodic_negative_no_go","label":"Eventually-periodic negative-orbit exclusion","status":"unconditional progress","statement":"For natural sequences a, D, C, and e with 2 ≤ a(n), positive e(N+n) < a(N+n), the exact recurrences D(n+1) = a(n)·D(n) and C(n+1) = C(n) + e(n), the shape equation D(n) + e(n) = (a(n)−1)·C(n), and a positive-period, positive-drift tail e(N+n+h) = e(N+n) and C(N+n+h) = C(N+n) + M with h > 0 and M > 0, the hypotheses are inconsistent. This is only the eventually-periodic negative-magnitude obstruction; mixed-sign and unbounded-negative behaviour remain outside the claim and the full Erdős #243 endpoint remains open.","lean_declarations":["ErdosProblems/Erdos243/ReciprocalTailRigidity.lean:802"],"paper_label":null},{"claim_id":"cubic_survivor_phase_density","label":"Phase-specific density obstruction for the two cubic survivor profiles","status":"unconditional progress","statement":"For either surviving cubic profile Q_{12,c}(n)=2n(n+1)(n+2)+c with c=1 or c=-1, the exact tail recurrences force a mismatch in every sufficiently late four-index window at the corresponding phase modulo 7. The mismatch set therefore has lower density at least 1/7, and hence at least the 1/28 stated in the long paper. No phase-free or arbitrary-profile density statement is claimed.","lean_declarations":["ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean:117","ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean:141","ErdosProblems/Erdos243/PaperCompleteR9/PolynomialCorrections.lean:165"],"paper_label":"long243:res:modseven"},{"claim_id":"cubic_rate_irrationality_unconditional","label":"Cubic-rate reciprocal-sum irrationality under the stated asymptotic","status":"unconditional progress","statement":"For a zero-indexed strictly increasing sequence of positive natural numbers a, if n³(a(n)²/a(n+1) − (1 + 3/n)) tends to zero and the reciprocal series has sum S, then S is irrational. Lean proves the square-specialisation input through the Dedekind-zeta simple pole, so this theorem has no Chebotarev premise. The paper states the ordinary one-based bridge. The quantitative cubic-rate hypothesis is essential here; this does not settle the original eventual Sylvester recurrence assertion.","lean_declarations":["ErdosProblems/Erdos243/PaperCompleteR21/SquareSpecialisationUnconditional.lean:70"],"paper_label":"res:cubicrate"},{"claim_id":"nonintegral_regular_rate_irrationality","label":"Nonintegral regular-rate reciprocal-sum irrationality","status":"unconditional progress","statement":"For a zero-indexed strictly increasing sequence of positive natural numbers a and a real noninteger lambda greater than one, if n^lambda (a(n)^2/a(n+1) - (1 + lambda/n)) tends to zero, then the reciprocal series converges and its sum is irrational. The Lean theorem derives summability from the rate; the one-based paper form uses an ordinary finite-prefix bridge. The little-o remainder is essential to the integer-extraction argument. This does not settle the unrestricted Erdős #243 problem.","lean_declarations":["ErdosProblems/Erdos243/PaperCompleteR21/NonintegralRegularRate.lean:196"],"paper_label":"long243:res:nonintegralrate"},{"claim_id":"finite_period_noncollapse","label":"Finite-support denominator period noncollapse","status":"unconditional progress","statement":"For every finite nonempty support F with 0 ∉ F and integer base b ≥ 2, the reduced denominator of the literal finite sum ∑_{n∈F} 1/(bⁿ−1) is coprime to b and the multiplicative order of b modulo that denominator equals F.lcm id. This finite-support theorem supplies no unbounded support certificate and does not settle the universal infinite-support Erdős #257 endpoint.","lean_declarations":["Erdos249257/CertificateKernel.lean:5246"],"paper_label":"res:period"},{"claim_id":"weighted_phase_carry_finite_observer","label":"Weighted-phase carry split and finite realised span for #269","status":"unconditional progress","statement":"The exact weighted-phase carry recurrence splits into a finite residue digit, each of the two coordinates staying inside its declared interval, and an integral quotient coboundary that is not controlled. An explicit factorisation through a finite-dimensional function-faithful observer then forces the realised symbolic span to be finite. Nothing here proves a finite-state quotient or a literal infinite realised span, and no rationality or irrationality conclusion follows.","lean_declarations":["ErdosProblems/Erdos269/WeightedPhaseCarry.lean:109","ErdosProblems/Erdos269/WeightedPhaseCarry.lean:150","ErdosProblems/Erdos269/WeightedPhaseCarry.lean:157"],"paper_label":null},{"claim_id":"totient_v8_finite_arithmetic_threshold","label":"Exact finite arithmetic and run-threshold checks for #249","status":"unconditional progress","statement":"The four displayed discrepancy values at (h,X,L)=(1,4,9), their finite room and certified kills, the Farey denominator bound, and the corrected finite run threshold hold exactly. These finite facts do not supply a cofinal certificate producer.","lean_declarations":["ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean:9","ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean:10","ErdosProblems/Erdos249/TypeBReturnV8/FiniteArithmetic.lean:11"],"paper_label":null},{"claim_id":"totient_v8_generic_centering_countermodel","label":"Generic centering countermodel for the #249 decomposition","status":"unconditional progress","statement":"A two-point complex model has vanishing peripheral mean contributions while its centered contribution remains nonzero. This refutes the generic inference from peripheral cancellation to a centered budget; it is not a counterexample to the arithmetic totient-centered predicate.","lean_declarations":["ErdosProblems/Erdos249/TypeBReturnV8/CenteringCounterexample.lean:14","ErdosProblems/Erdos249/TypeBReturnV8/CenteringCounterexample.lean:16","ErdosProblems/Erdos249/TypeBReturnV8/CenteringCounterexample.lean:18"],"paper_label":null},{"claim_id":"totient_v8_one_finite_first_harmonic_instance","label":"One unconditional finite first-harmonic gap instance","status":"unconditional progress","statement":"At (h,X,L)=(1,4,9), the exact finite first-harmonic expression has the certified norm gap bounded by 21X/25. This is one finite instance and does not establish DTWFirstHarmonicNormGap or any cofinal statement.","lean_declarations":["ErdosProblems/Erdos249/TypeBReturnV8/FiniteFirstHarmonic.lean:41","ErdosProblems/Erdos249/TypeBReturnV8/FiniteFirstHarmonic.lean:66"],"paper_label":null},{"claim_id":"zudilin_all_rank_hankel_order","label":"All-rank normalized Hankel order and leading coefficient","status":"unconditional progress","statement":"For every natural N, including N = 0, the normalized Zudilin Hankel determinant has q-order N(N-1)(2N-1)/6. Its leading coefficient satisfies the division-free identity 2^N times the coefficient = (N!)^2 (N+1)!, with the corresponding rational quotient formula. The all-row producer is proved rather than assumed. This algebraic theorem does not by itself supply the analytic estimates or irrationality at 3/2. Zudilin (2016) proves the q-order inequality; the equality is proved here.","lean_declarations":["ErdosProblems/Erdos1049/AllRow/Producer.lean:163","ErdosProblems/Erdos1049/AllRow/Producer.lean:173","ErdosProblems/Erdos1049/AllRow/Producer.lean:199"],"paper_label":"res:zudilin-sharp-qorder"},{"claim_id":"coefficient_pencil_finite_rank","label":"Coefficient-pencil positivity and root interlacing through rank eight","status":"unconditional progress","statement":"For each real p > 1 and each natural rank N ≤ 8, the coefficient Hankel matrix A_N is positive definite, the polynomial det(Y A_N − B_N) splits over the reals, and every root is strictly below F(p). For adjacent ranks N and N+1 ≤ 8, the ordered root lists interlace non-strictly, with multiplicities. Lean includes N = 0; the paper states the ranks 1 through 8. No all-rank coefficient-pencil theorem, simple-root or strict-interlacing assertion, limiting result, or irrationality of F(3/2) follows from this finite proposition.","lean_declarations":["ErdosProblems/Erdos1049/PaperR20/FinitePencilProposition.lean:43"],"paper_label":"long1049:res:finite-pencil"},{"claim_id":"finite_prime_weighted_support","label":"Finite-prime weighted support irrationality","status":"unconditional progress","statement":"For every integer base b >= 2, every infinite positive support A with finite weighted mass for some finite nonempty prime set P has irrational support series. If a host H of positive integers has finite base-two weighted mass, then every infinite subset A of H has irrational support series at every integer base b >= 2. This is a sufficient support criterion, not the universal statement.","lean_declarations":["ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean:40","ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean:120"],"paper_label":"res:weighted-support"},{"claim_id":"strengthened_positive_cover_support","label":"Strengthened positive-cover support irrationality","status":"unconditional progress","statement":"A strengthened positive divisor-cover datum with summable scaled cost makes every infinite subset of its host irrational at every integer base b >= 2. The statement retains the nonlinear cover majorant and summability hypotheses; it does not cover arbitrary supports.","lean_declarations":["ErdosProblems/Erdos257/PaperCompleteR8/PositiveCoverReturn.lean:241"],"paper_label":"thm:variable-fractional-cover"},{"claim_id":"mixed_weighted_cover_support","label":"Mixed weighted and positive-cover support irrationality","status":"unconditional progress","statement":"Let E have finite base-two finite-prime weighted mass and let V lie in a strengthened positive-cover host. Every infinite support A contained in E union V has irrational support series at every integer base b >= 2. The theorem uses a common observation scale; it does not assume separate irrationality conclusions for the two components.","lean_declarations":["ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean:126"],"paper_label":"res:mixed-supports"},{"claim_id":"reciprocal_summable_support","label":"Reciprocal-summable support irrationality at every integer base","status":"unconditional progress","statement":"For every integer base b >= 2 and every infinite support A, summability of the reciprocal-support mass implies that the support series is irrational. No pairwise-coprimality hypothesis is required. Erdős (1968) stated this extension without proof.","lean_declarations":["Erdos249257/AllBaseReciprocalSupportIrrationality.lean:395"],"paper_label":"res:reciprocal-support"},{"claim_id":"separated_cut_algebraic_lambert_transcendence","label":"Algebraic-base transcendence on separated divisibility supports","status":"unconditional progress","statement":"For an infinite positive-integer host with cuts L_j < M_j, L_j tending to infinity and M_j-L_j tending to infinity, every host exponent at most L_j dividing L_j and every later exponent divisible by M_j, every infinite thinning with bounded positive integer weights has transcendental Lambert sum at every real algebraic base t > 1. This includes all divisibility chains and an explicit host of unbounded divisibility width and divergent reciprocal sum. The full number-field Subspace Theorem argument has two AI proof reviews; it is an ordinary proof with no Lean transcendence theorem, independent human review or asserted historical priority.","lean_declarations":[],"paper_label":"thm:separated-cuts"},{"claim_id":"independent_clock_channel_relations","label":"Complete rational relations for integer-radix independent-clock sums","status":"unconditional progress","statement":"For positive rationally independent clock rates with no simultaneous positive crossings, and integer radices b_i >= 2, all rational relations among 1 and the channel sums S_i are multiples of (-1,b_1-1,...,b_d-1). The proof uses genuine late swaps with a common infinite suffix in one orbit closure, a fixed rational lattice, and density of the clock section. The explicit rates 1,sqrt(2),sqrt(3),sqrt(6) and radices 2,3,5,7 give rational span dimension four. This is an ordinary proof reviewed by AI agents, without Lean or independent human verification of the infinite theorem.","lean_declarations":[],"paper_label":"long269:thm:clock-relations"},{"claim_id":"calibrated_rational_hankel_countermodel","label":"Rational-value countermodel to a Hankel-signature inference","status":"unconditional progress","statement":"An explicit positive atomic moment family and integer formal series have all-rank Hankel order sum_{j<N} j^2 and leading coefficient (N!)^2(N+1)!/2^N, with the same fixed-base asymptotic shape including N^(-8 xi), while xi=G_{2/3}(2/3)=1. For n>=1 the exact reduced moment denominator is 3^(31n+62)(3^(n+1)-2^(n+1))^4; the determinant denominator has logarithmic size Theta(N^3), and a proposed rank-two row-product clearer leaves denominator625. These are ordinary proofs with independent exact finite checks and AI proof reviews, not a Lean proof of the infinite family or human peer review.","lean_declarations":[],"paper_label":"long1049:thm:calibrated-model"},{"claim_id":"selected_staircase_integral_height_boundary","label":"Smith index and primitive one-target border boundary","status":"unconditional progress","statement":"The selected staircase matrix with ones on and below the diagonal and integer r>=2 above it has Smith factors 1,r-1,...,r-1. For the selected {2,3,5} kernel, any entrywise-integral diagonal clearing has determinant magnitude at least 4^(N-1), whereas the reduced rational determinant has numerator one up to sign. A rational single-target border has primitive polynomial plus or minus (tX-s), where s/t is the reduced Schur complement. These ordinary arithmetic arguments were reviewed by AI agents; they are not the existing Lean nonvanishing theorem and have no independent human review.","lean_declarations":[],"paper_label":"long269:lem:staircase-smith"},{"claim_id":"signed_totient_entire_defect_interpolation","label":"Arbitrarily slow signed interpolation preserving Dirichlet singularities","status":"unconditional progress","statement":"For any integer beta>=2 and positive allowance f(n) tending to infinity, an interval around the totient value is filled by integer sequences b=phi+e with 0<=b(n)<=n and |e(n)|<=f(n). One target-independent even support has upper Banach density zero, leaving any fixed prefix and odd coefficients unchanged. The construction preserves every fixed eventual coefficient and cumulative congruence, gives eventually constant progression moment defects outside common hulls of density zero, entire progression Dirichlet defects, and smooth root-of-unity radial germs with prescribed finite zero jets. The full proof includes the allowance-dependent block bound and first-harmonic rational countermodels. It is an ordinary proof with two AI proof reviews, not a Lean proof of the infinite theorem or independent human review.","lean_declarations":[],"paper_label":"long249:thm:signed-interpolation"},{"claim_id":"eb_full_support","label":"Full-support Erdős–Borwein irrationality, every integer base b ≥ 2","status":"formalised here","statement":"For every integer b ≥ 2, the series ∑_{n≥1} 1/(bⁿ − 1) is irrational (Erdős, 1948).","lean_declarations":["Erdos249257/CertificateKernel.lean:8328"],"paper_label":"res:full"},{"claim_id":"eb_constant","label":"The Erdős–Borwein constant E is irrational","status":"formalised here","statement":"E = ∑_{n≥1} 1/(2ⁿ − 1) is irrational, as the base-2 corollary of the full-support theorem.","lean_declarations":["Erdos249257/CertificateKernel.lean:8335"],"paper_label":"res:full"},{"claim_id":"support_families","label":"Named infinite-support irrationality families toward #257","status":"formalised here","statement":"For every base b ≥ 2, ∑_{n∈A} 1/(bⁿ − 1) is irrational for factorial support, power-of-two support, multiples, pairwise-coprime supports with summable reciprocals, eventually-periodic supports, residue classes, and the odd numbers. Proved families, not the universal statement.","lean_declarations":["Erdos249257/CertificateKernel.lean:6082","Erdos249257/CertificateKernel.lean:6090","Erdos249257/CertificateKernel.lean:10776"],"paper_label":"res:support"},{"claim_id":"denominator_exclusion","label":"Unconditional denominator exclusion for the totient constant","status":"verified finite instance","statement":"S = ∑ φ(n)/2ⁿ is not equal to p/q for any integer p and any 1 ≤ q ≤ 79 639 646 646 701 375 323 355 774 875 831 053 (≈ 7.96 × 10³⁴). A lower bound on any rational representation, not a proof of irrationality.","lean_declarations":["Erdos249257/CertificateKernel.lean:18384"],"paper_label":"res:farey"},{"claim_id":"certificate_reduction","label":"Exact reduction of #249 to finite decidable certificates","status":"conditional reduction","statement":"S is irrational if and only if certified non-integrality witnesses occur at unbounded parameters. The pointwise completeness theorem and the diagonal supply theorem together show that an unbounded lcm-diagonal supply is equivalent as well. Producing that supply remains open.","lean_declarations":["Erdos249257/TotientTailPeriodKiller.lean:394","Erdos249257/LcmDiagonalReduction.lean:112","Erdos249257/LcmConeFlatness.lean:90"],"paper_label":"res:diag"},{"claim_id":"certificate_completeness","label":"Certificate completeness","status":"proved here","statement":"At fixed parameters, a certified kill exists exactly when the corresponding tail difference is non-integral. 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The historical 28 explicit deposits through t = 64 form a strict subset of this contiguous finite band.","lean_declarations":["Erdos249257/TotientTailPeriodKiller.lean:404","Erdos249257/LcmDiagonalReduction.lean:257","Erdos249257/LcmDiagonalReduction.lean:262"],"paper_label":"res:deposits"},{"claim_id":"diagonal_pincer_reduction","label":"Exact full-target reduction on the lcm diagonal","status":"conditional reduction","statement":"At each lcm-diagonal scale, integrality holds if and only if a foreign-defect term lands in an explicit full-target interval. Globally, S is irrational if and only if the full target is avoided at arbitrarily large diagonal scales; that cofinal avoidance is not proved.","lean_declarations":["Erdos249257/DiagonalPincerDecomposition.lean:215","Erdos249257/DiagonalPincerDecomposition.lean:290"],"paper_label":"res:fulltargetavoidance"},{"claim_id":"adelic_height_obstruction","label":"Scalar-localisation height obstruction","status":"proved here","statement":"For every rational x, integer c and natural number H, if H divides the reduced denominator of x and the reduced denominator of c·x divides H, then the complementary factor x.den/H divides |c|. Scalar localisation therefore moves that denominator information into the scalar; this local obstruction is not an irrationality criterion.","lean_declarations":["Erdos249257/AdelicHeightObstruction.lean:23","Erdos249257/AdelicHeightObstruction.lean:56"],"paper_label":"res:adelic"},{"claim_id":"all_base_totient_kernel_index_reduction","label":"All-base totient-kernel reduction, spanning, and conditional rank","status":"proved here","statement":"For every integer base k at least 2, Lean checks the zero-residue scaling identity, the exact nonzero scalar used in composite-base residue reduction, unique quotient/nonzero-digit coordinates, and cardinality k^e + 1 of the canonical finite-level index. It proves unconditionally that this canonical family spans the complete kernel through level e, and proves exact rank k^e + 1 under the explicit hypothesis that the family is linearly independent. Lean does not prove that hypothesis here; the note's evaluation-matrix argument and AllBaseTotientKernel.lean at public checkpoint 25ef6245d15a discharge it by an independent finite-determinant construction, so the unconditional basis and rank theorem takes Martin's positive-density theorem as prior art rather than as an input.","lean_declarations":["Erdos249257/TotientKernelReduction.lean:60","Erdos249257/TotientKernelReduction.lean:117","Erdos249257/TotientKernelIndex.lean:124"],"paper_label":"thm:kkernelrank"},{"claim_id":"sublog_zero_windows","label":"Sublogarithmic divisor-coverage zero windows","status":"proved here","statement":"A rational support-series value forces every zero window in the divisor-count coefficient to have length at most ε·log₂(N+1) + B, where B ≥ 0 is built from ε, c and v but is bound after the support and numerator, so no single constant is claimed uniform across supports. This constrains coverage; it does not solve universal #257.","lean_declarations":["Erdos249257/SublogDivisorCoverage.lean:392"],"paper_label":"res:sublog"}],"remaining_open_propositions":[{"proposition_id":"remaining_open.erdos_68_irrationality","statement":"Prove that S = ∑_{n≥2} 1/(n! − 1) is irrational."},{"proposition_id":"remaining_open.erdos_249_irrationality","statement":"Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator."},{"proposition_id":"remaining_open.erdos_251_irrationality","statement":"Prove that Π = ∑_{n≥1} p_n/2ⁿ is irrational; the exact tail-shift equivalence and prime-gap constraints do not supply the required cofinal escape."},{"proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal."},{"proposition_id":"remaining_open.half_value_membership","statement":"Decide whether 1/2 lies in the Mersenne achievement set. Yes refutes universal #257; no closes only this route."},{"proposition_id":"remaining_open.twenty_one_permanent_affine_supercapacity","statement":"Decide the exact 1/21 branch: equivalently, exclude the fatal/cofinite/aligned branch, contradict its eventual permanent affine-supercapacity recurrence, prove cofinally bounded scaled greedy remainders, or force an unbounded sequence of closed canonical quotient rows. A positive membership result would refute universal #257; no such producer is proved."},{"proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here."},{"proposition_id":"remaining_open.erdos_269_three_prime_irrationality","statement":"Determine irrationality of the running-LCM reciprocal sum for finite prime sets of size at least three. For {2,3,5}, the rationality-to-carry bridge is established and cofinal local-window escape remains open."},{"proposition_id":"remaining_open.erdos_1041_lemniscate_connection","statement":"Independently review whether the 1958 curve-length wording matches the exact Formal Conjectures path-image-length statement, whose negation is now Lean-checked using ani's explicit degree-seven polynomial. The checked one-dimensional Hausdorff-measure bound settles that formal statement negatively; the reported small-parameter family is not formalised."},{"proposition_id":"remaining_open.erdos_1049_irrationality","statement":"Determine whether F(t) = ∑_{n≥1} 1/(tⁿ − 1) is irrational for every rational t > 1. Irrationality of F((31/4)^r) for every positive integer r is a Lean-checked Zudilin specialization; the 3/2 approximant with remainder control and the universal assertion remain open."},{"proposition_id":"remaining_open.erdos_243_eventual_recurrence","statement":"Prove the Erdős #243 assertion: if 1 ≤ a₁ < a₂ < ··· has aₙ/aₙ₋₁² → 1 and rational reciprocal sum, then aₙ = aₙ₋₁² − aₙ₋₁ + 1 eventually; the checked barrier does not settle the mixed-sign unbounded regime."},{"proposition_id":"remaining_open.erdos_269_reduced_tail_nonintegrality","statement":"Prove that B·T_a ∉ ℤ for every B ≥ 1 coprime to 30 and every a ≥ 1."},{"proposition_id":"remaining_open.erdos_269_cofinal_local_window_escape","statement":"Prove that for every B ≥ 1 coprime to 30 and every a₀ ≥ 1 there are ℓ ≥ a₀ and h ≥ 1 with lpr_{W(ℓ,h)}(−B·F(ℓ,h)) > K(B, ℓ+h), where K is the defined long-paper cap."},{"proposition_id":"remaining_open.erdos_269_two_dimensional_representation","statement":"Transcendence of D_{2,3,5}, which is irrational: represent it faithfully by a specified two-dimensional Mahler-type value theorem with every hypothesis verified, give a conditional theorem under explicit logarithmic nondegeneracy, or prove no such representation exists."},{"proposition_id":"remaining_open.erdos_243_overlap_height_growth","statement":"Decide whether every nonterminal canonical orbit of the reduced-orbit problem has limsup_n (log M_n)/n > 0, equivalently 2^n ≤ M_n^K infinitely often for some K ≥ 1."},{"proposition_id":"remaining_open.erdos_249_first_harmonic_anti_concentration","statement":"Decide whether, for each h ≥ 1, some s ≥ 1 and 0 < η < 1 give, for every X₀, admissible X and L meeting the displayed size conditions for which the four first-harmonic bounds hold."},{"proposition_id":"remaining_open.twenty_one_scaled_remainder_cofinal_return","statement":"Decide whether the scaled actual greedy remainder returns cofinally to one bounded interval: whether some B < ∞ has 2^N r_N ≤ B at arbitrarily large N."},{"proposition_id":"remaining_open.twenty_one_actual_orbit_invariant","statement":"Decide whether a bounded-memory 2-adic or discrepancy invariant of the actual support forces a closed return or forbids permanent supercapacity, or prove no such invariant exists."},{"proposition_id":"remaining_open.twenty_one_final_skip_exclusion","statement":"Decide whether the final-skip signatures prove |E − a_M| ≥ gap_M, contradicting 0 < a_M − E < gap_M, for E = ∑_{n≥1}(2^n − 1)^{−1} and the last-skip approximant a_M."}],"how_to_read_this":"Each statement is quoted at exactly the strength the registry asserts, including its own negative clause. Do not add a hedge that the statement does not contain, and do not drop one that it does. If you conclude from this packet that the mathematics is thin or unverifiable, name which of these statements you checked and found wanting -- a general impression formed without opening one is not a finding about the work.","verification_boundary":"No Lean kernel ran during this build. 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it does not ingest private lessons, mutate live ledgers, promote global doctrine, include launch operations or public sharing, make external model access, claim non-public data equivalence, or certify public runtime behavior.","source_link_count":2,"links_to_count":0,"tags":["architecture","navigation","doctrine"],"full_record_pointer":"/visible_text_index/logical_records/components/3","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_public_sharing_authority"]},{"id":"executable_doctrine_grammar","label":"Executable Doctrine Grammar","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","summary_line":"Checks that example standards files declare their purpose, rule, records, and what they do not claim.","rank":5,"evidence_kind":"Contract validator","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.executable_doctrine_grammar validate-executable-grammar-metabolism-bundle --input examples/executable_doctrine_grammar/exported_executable_grammar_metabolism_bundle --out receipts/first_wave/executable_doctrine_grammar --card","relative_href":"docs/component-executable-doctrine-grammar.html","primary_reader_href":"docs/paper-module-executable-doctrine-grammar.html","component_detail_href":"docs/component-executable-doctrine-grammar.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates an exported public executable-grammar metabolism bundle with exact copied-body digests and redacted result records, plus fixture regressions for standards/paper-module shape. It does not publish source doctrine bodies in result records, prove doctrine completeness, export a private standards engine, authorize later components, or claim external model access, non-public data equivalence, launch-scope decision, or whole-system correctness.","source_link_count":2,"links_to_count":1,"tags":["architecture","navigation","doctrine"],"full_record_pointer":"/visible_text_index/logical_records/components/4","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_whole_system_correctness","not_launch_authority"]},{"id":"navigation_hologram_route_plane","label":"Navigation Hologram Route Plane","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","summary_line":"Audits a folder's navigation so browse rows never pose as the source of truth.","rank":5,"evidence_kind":"Contract validator","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.navigation_hologram_route_plane run --input fixtures/first_wave/navigation_hologram_route_plane/input --out receipts/first_wave/navigation_hologram_route_plane","relative_href":"docs/component-navigation-hologram-route-plane.html","primary_reader_href":"docs/paper-module-navigation-hologram-route-plane.html","component_detail_href":"docs/component-navigation-hologram-route-plane.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates only the declared public toy route-plane contract and its regression fixtures (plus exact copied navigation source modules in the bundle path); it does not establish live route freshness, grant source authority, authorize any later component, run any provider/live-kernel call, or certify the whole wave.","source_link_count":2,"links_to_count":11,"tags":["agent-entry","architecture","navigation"],"full_record_pointer":"/visible_text_index/logical_records/components/5","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes"]},{"id":"standards_meta_diagnostics","label":"Standards Meta Diagnostics","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","summary_line":"Confirms every accepted part still ties to a written rule, a run command, and a saved proof.","rank":5,"evidence_kind":"Contract validator","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.standards_meta_diagnostics run --input fixtures/first_wave/standards_meta_diagnostics/input --out /tmp/standards_meta_diagnostics_out","relative_href":"docs/component-standards-meta-diagnostics.html","primary_reader_href":"docs/paper-module-standards-meta-diagnostics.html","component_detail_href":"docs/component-standards-meta-diagnostics.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates only the declared public coverage contract and never becomes source authority for the registries, mutates source, exposes private material, or authorizes launch, external model access, or any whole-system-correctness claim.","source_link_count":2,"links_to_count":0,"tags":["agent-entry","architecture","navigation"],"full_record_pointer":"/visible_text_index/logical_records/components/6","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_whole_system_correctness","not_launch_authority"]},{"id":"voice_to_doctrine_self_improvement_loop","label":"Voice To Doctrine Self Improvement Loop","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","summary_line":"Verifies each lesson changed a named owner page with evidence before the loop closes.","rank":5,"evidence_kind":"Contract validator","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.voice_to_doctrine_self_improvement_loop run --input fixtures/first_wave/voice_to_doctrine_self_improvement_loop/input --out receipts/first_wave/voice_to_doctrine_self_improvement_loop","relative_href":"docs/component-voice-to-doctrine-self-improvement-loop.html","primary_reader_href":"docs/paper-module-voice-to-doctrine-self-improvement-loop.html","component_detail_href":"docs/component-voice-to-doctrine-self-improvement-loop.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates only the declared contract of the loop on fixtures; it does not export source notes or private bodies, grant source/doctrine edits, global-promotion, live work log mutation, or public sharing decision, make external model access, prove correctness, or claim whole-system equivalence.","source_link_count":2,"links_to_count":1,"tags":["agent-entry","architecture","navigation"],"full_record_pointer":"/visible_text_index/logical_records/components/7","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_whole_system_correctness"]},{"id":"cognitive_operator_registry","label":"Cognitive Operator Registry","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","summary_line":"Checks the catalog of named thinking-moves so each is fully described and backed by evidence.","rank":5,"evidence_kind":"Contract validator","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.cognitive_operator_registry run --input fixtures/first_wave/cognitive_operator_registry/input --out /tmp/cognitive_operator_registry_out","relative_href":"docs/component-cognitive-operator-registry.html","primary_reader_href":"docs/paper-module-cognitive-operator-registry.html","component_detail_href":"docs/component-cognitive-operator-registry.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates only the declared public registry contract and copied source bodies; it never becomes registry source authority, mutates operators, proves operator correctness, exposes source notes, or authorizes launch, external model access, or any whole-system-correctness claim.","source_link_count":2,"links_to_count":0,"tags":["architecture","navigation","doctrine"],"full_record_pointer":"/visible_text_index/logical_records/components/8","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_whole_system_correctness","not_launch_authority"]},{"id":"routing_anti_patterns_registry","label":"Routing Anti Patterns Registry","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","summary_line":"Indexes the navigation mistakes agents repeat and guards the public list.","rank":5,"evidence_kind":"Contract validator","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.routing_anti_patterns_registry run --input fixtures/first_wave/routing_anti_patterns_registry/input --out /tmp/routing_anti_patterns_registry_out","relative_href":"docs/component-routing-anti-patterns-registry.html","primary_reader_href":"docs/paper-module-routing-anti-patterns-registry.html","component_detail_href":"docs/component-routing-anti-patterns-registry.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates only the declared public routing anti-pattern registry contract and copied source body; it never becomes route source authority, mutates routes, exposes private routing notes, calls providers, authorizes launch, or proves whole-system correctness.","source_link_count":2,"links_to_count":3,"tags":["architecture","navigation","doctrine"],"full_record_pointer":"/visible_text_index/logical_records/components/9","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_whole_system_correctness","not_launch_authority"]},{"id":"doctrine_fact_claim_audit","label":"Doctrine Fact Claim Audit","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","summary_line":"Checks that public fact rows state the right count and point at live, anchored code.","rank":5,"evidence_kind":"Contract validator","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis doctrine-fact-claim-audit run --input fixtures/first_wave/doctrine_fact_claim_audit/input --out receipts/first_wave/doctrine_fact_claim_audit","relative_href":"docs/component-doctrine-fact-claim-audit.html","primary_reader_href":"docs/paper-module-doctrine-fact-claim-audit.html","component_detail_href":"docs/component-doctrine-fact-claim-audit.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"fact assertion, code-loci, and DAG fixture truth gate only; it is not a comprehension engine and does not establish a minimum read graph","source_link_count":2,"links_to_count":2,"tags":["architecture","navigation","doctrine"],"full_record_pointer":"/visible_text_index/logical_records/components/10","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes"]},{"id":"self_ignorance_coverage_ledger","label":"Self Ignorance Coverage Ledger","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","summary_line":"Compares expected against built entities to report known coverage gaps.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis self-ignorance-coverage-ledger run --input fixtures/first_wave/self_ignorance_coverage_ledger/input --out receipts/first_wave/self_ignorance_coverage_ledger","relative_href":"docs/component-self-ignorance-coverage-ledger.html","primary_reader_href":"docs/paper-module-self-ignorance-coverage-ledger.html","component_detail_href":"docs/component-self-ignorance-coverage-ledger.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"known Kind Atlas coverage debt projection only; it does not claim literal unknown-unknown omniscience or absence proof","source_link_count":2,"links_to_count":0,"tags":["architecture","navigation","doctrine"],"full_record_pointer":"/visible_text_index/logical_records/components/11","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes"]},{"id":"navigation_fitness_benchmark","label":"Navigation Fitness Benchmark","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","summary_line":"Measures a navigation result against its target for recall, precision, and speed on public examples.","rank":4,"evidence_kind":"Bounded runtime computation","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis navigation-fitness-benchmark run --input fixtures/first_wave/navigation_fitness_benchmark/input --out /tmp/plectis-public-runs/navigation_fitness_benchmark","relative_href":"docs/component-navigation-fitness-benchmark.html","primary_reader_href":"docs/paper-module-navigation-fitness-benchmark.html","component_detail_href":"docs/component-navigation-fitness-benchmark.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"The bundle demonstrates a route-packet evaluator for expected stable ids, forbidden first routes, and latency budgets. It is not a live private kernel run, not an embedding benchmark, not a universal navigation benchmark, and not launch-scope decision.","source_link_count":2,"links_to_count":3,"tags":["navigation","evaluation","architecture"],"full_record_pointer":"/visible_text_index/logical_records/components/12","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_benchmark_claim"]},{"id":"annex_knowledge_routing","label":"Reference Knowledge Routing","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","summary_line":"Runs a keyword-overlap match that ranks catalog entries, shows why each matched, and admits when none do.","rank":4,"evidence_kind":"Bounded runtime computation","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis annex-knowledge-routing run --input fixtures/first_wave/annex_knowledge_routing/input --out /tmp/plectis-public-runs/annex_knowledge_routing","relative_href":"docs/component-annex-knowledge-routing.html","primary_reader_href":"docs/paper-module-annex-knowledge-routing.html","component_detail_href":"docs/component-annex-knowledge-routing.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"The bundle demonstrates explainable tiered weighted-token retrieval over a sanitized reference catalog. It is not BM25, TF-IDF, embedding search, repository cloning, license authority, private reference corpus authority, or launch-scope decision.","source_link_count":2,"links_to_count":3,"tags":["navigation","architecture"],"full_record_pointer":"/visible_text_index/logical_records/components/13","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority"]},{"id":"certificate_kernel_execution_lab","label":"Certificate Kernel Execution Lab","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Runs the Lean verifier over a small public proof project and reports which rows it accepted.","rank":4,"evidence_kind":"External tool run","runs_real_tools":true,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":true,"intrinsic_populated_source_runs_real_tools":true,"source_population_required":false},"command":"plectis certificate-kernel-execution-lab run --input fixtures/first_wave/certificate_kernel_execution_lab/input --out receipts/first_wave/certificate_kernel_execution_lab","relative_href":"docs/component-certificate-kernel-execution-lab.html","primary_reader_href":"docs/paper-module-certificate-kernel-execution-lab.html","component_detail_href":"docs/component-certificate-kernel-execution-lab.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It is a local tool-witness that the declared public fixture rows compiled and were adjudicated by the local Lean verifier; it excludes general proof authority, count oracle/provider output as proof, expose proof text, change source files, claim a benchmark solve-rate, or include launch operations.","source_link_count":2,"links_to_count":0,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/14","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_theorem_or_proof_authority","not_benchmark_claim"]},{"id":"formal_math_lean_proof_witness","label":"Formal Math Lean Proof Witness","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Compiles a tiny Lean example with the real prover and records whether it built, leaking no proof text.","rank":4,"evidence_kind":"External tool run","runs_real_tools":true,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":true,"intrinsic_populated_source_runs_real_tools":true,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.formal_math_lean_proof_witness run --input fixtures/first_wave/formal_math_lean_proof_witness/input --out receipts/first_wave/formal_math_lean_proof_witness","relative_href":"docs/component-formal-math-lean-proof-witness.html","primary_reader_href":"docs/paper-module-formal-math-lean-proof-witness.html","component_detail_href":"docs/component-formal-math-lean-proof-witness.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It authorizes only a witness that a tiny declared public toy proof compiled under the locally installed Lean/Lake toolchain in a temporary workspace, plus confirmation that its leakage guardrails fired. It excludes Mathlib/Aesop/Batteries-dependent or general proof or theorem-program authority, external model access, private proof import, benchmark or performance claims, whole-system correctness, or any launch, hosted deployment, or public sharing.","source_link_count":2,"links_to_count":5,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/15","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_whole_system_correctness","not_launch_authority","not_public_sharing_authority","not_theorem_or_proof_authority","not_benchmark_claim"]},{"id":"verifier_lab_execution_spine","label":"Verifier Lab Execution Spine","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Runs Lean on small bounded proof attempts in a temp copy and records what passed or failed.","rank":4,"evidence_kind":"External tool run","runs_real_tools":true,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":true,"intrinsic_populated_source_runs_real_tools":true,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.verifier_lab_execution_spine run --input fixtures/first_wave/verifier_lab_execution_spine/input --out .microcosm/verifier_lab_execution_spine","relative_href":"docs/component-verifier-lab-execution-spine.html","primary_reader_href":"docs/paper-module-verifier-lab-execution-spine.html","component_detail_href":"docs/component-verifier-lab-execution-spine.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It is a tool-witness result record for bounded public Lean transition rows only: it does not establish general proof authority, count oracle/provider output as proof, export proof bodies or tactic scripts, use external model services, change source files, claim benchmark solve-rates, or include launch operations/public sharing.","source_link_count":2,"links_to_count":2,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/16","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_public_sharing_authority","not_theorem_or_proof_authority","not_benchmark_claim"]},{"id":"corpus_readiness_mathlib_absence_gate","label":"Corpus Readiness Mathlib Absence Gate","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Runs the real Lean toolchain to confirm the math library is absent, then gates proof tasks.","rank":4,"evidence_kind":"External tool run","runs_real_tools":true,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":true,"intrinsic_populated_source_runs_real_tools":true,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.corpus_readiness_mathlib_absence_gate run --input fixtures/first_wave/corpus_readiness_mathlib_absence_gate/input --out receipts/first_wave/corpus_readiness_mathlib_absence_gate","relative_href":"docs/component-corpus-readiness-mathlib-absence-gate.html","primary_reader_href":"docs/paper-module-corpus-readiness-mathlib-absence-gate.html","component_detail_href":"docs/component-corpus-readiness-mathlib-absence-gate.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It only projects and gate-checks recorded corpus/toolchain readiness accounting, re-verifies recorded source digests and leakage guards, and runs a bounded Lean/Lake import probe when a toolchain is present. It does not run a full Lake build, prove formal-result correctness, claim Mathlib is available beyond the probe result, benchmark corpora, score model performance, use external model services, or include launch operations or public sharing.","source_link_count":2,"links_to_count":4,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/17","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_public_sharing_authority","not_theorem_or_proof_authority","not_benchmark_claim"]},{"id":"batch4_proof_authority_runtime","label":"Proof / Control / Runtime Import Bundle","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Checks fourteen proof, control, and runtime parts as one unit that rejects every overclaim.","rank":5,"evidence_kind":"Verified source import","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis batch4-proof-authority-runtime run --input fixtures/first_wave/batch4_proof_authority_runtime/input --out /tmp/plectis-public-runs/batch4_proof_authority_runtime","relative_href":"docs/component-batch4-proof-authority-runtime.html","primary_reader_href":"docs/paper-module-batch4-proof-authority-runtime.html","component_detail_href":"docs/component-batch4-proof-authority-runtime.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates only a public source-open bundle and bounded negative fixtures; it is not an Erdos #257 solution, not benchmark evidence, not public sharing or launch-scope decision, not live Codex/browser/runtime authority, and not whole-system equivalence.","source_link_count":2,"links_to_count":0,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/18","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_whole_system_correctness","not_launch_authority","not_public_sharing_authority","not_benchmark_claim"]},{"id":"proof_diagnostic_evidence_spine","label":"Proof Diagnostic Evidence Spine","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Sorts proof-pipeline checks into accepted or rejected without inflating a pass.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis proof-diagnostic-evidence-spine run --input fixtures/first_wave/proof_diagnostic_evidence_spine/input --out receipts/first_wave/proof_diagnostic_evidence_spine --card","relative_href":"docs/component-proof-diagnostic-evidence-spine.html","primary_reader_href":"docs/paper-module-proof-diagnostic-evidence-spine.html","component_detail_href":"docs/component-proof-diagnostic-evidence-spine.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It records proof/evidence diagnostics over existing result record references only. It does not run Lean, use external model services, expose proof bodies, turn a passing check into formal-proof or theorem authority, prove runtime or whole-system correctness, authorize later components, certify public launch, authorize public sharing or recipient work, or establish secret export.","source_link_count":2,"links_to_count":6,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/19","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_whole_system_correctness","not_launch_authority","not_public_sharing_authority","not_theorem_or_proof_authority"]},{"id":"formal_math_readiness_gate","label":"Formal Math Readiness Gate","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Reads declared math setups and lists which proof tactics may be attempted versus blocked.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis formal-math-readiness-gate run --input fixtures/first_wave/formal_math_readiness_gate/input --out receipts/first_wave/formal_math_readiness_gate","relative_href":"docs/component-formal-math-readiness-gate.html","primary_reader_href":"docs/paper-module-formal-math-readiness-gate.html","component_detail_href":"docs/component-formal-math-readiness-gate.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It only validates and projects declared readiness metadata; it does not run Lean/Lake, inspect the real toolchain, use external model services, prove any theorem correct, produce benchmark claims, or authorize Mathlib-dependent proof attempts.","source_link_count":2,"links_to_count":6,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/20","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_theorem_or_proof_authority","not_benchmark_claim"]},{"id":"mathematical_strategy_atlas_hypothesis_scorer","label":"Mathematical Strategy Atlas Hypothesis Scorer","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Picks a first-guess proof strategy from a problem's tags and flags any it cannot map.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.mathematical_strategy_atlas_hypothesis_scorer run --input fixtures/first_wave/mathematical_strategy_atlas_hypothesis_scorer/input --out receipts/first_wave/mathematical_strategy_atlas_hypothesis_scorer","relative_href":"docs/component-mathematical-strategy-atlas-hypothesis-scorer.html","primary_reader_href":"docs/paper-module-mathematical-strategy-atlas.html","component_detail_href":"docs/component-mathematical-strategy-atlas-hypothesis-scorer.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It only projects pre-oracle strategy-hypothesis and retrieval mechanics; it does not run Lean/Lake, prove theorems, establish domain or formal-result correctness, reveal oracle labels, expose proof bodies, use external model services, tune on test answers, or include launch operations.","source_link_count":2,"links_to_count":0,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/21","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_theorem_or_proof_authority"]},{"id":"tactic_portfolio_availability_probe","label":"Tactic Portfolio Availability Probe","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Maps which Lean proof tactics a recorded run marked usable before any code relies on one.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.tactic_portfolio_availability_probe run --input fixtures/first_wave/tactic_portfolio_availability_probe/input --out /tmp/plectis-public-runs/tactic_portfolio_availability_probe","relative_href":"docs/component-tactic-portfolio-availability-probe.html","primary_reader_href":"docs/paper-module-tactic-portfolio-availability.html","component_detail_href":"docs/component-tactic-portfolio-availability-probe.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It only projects and validates which tactics were recorded as compiling in one captured environment; it does not run Lean/Lake at all, prove any goal, certify domain-level conclusions, use external model services, claim benchmark performance, or include launch operations.","source_link_count":2,"links_to_count":5,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/22","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_benchmark_claim"]},{"id":"target_shape_tactic_routing_gate","label":"Target Shape Tactic Routing Gate","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Records an allow-or-reject decision and reason for each proof tactic before any proof runs.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.cli target-shape-tactic-routing-gate run-routing-bundle --input examples/target_shape_tactic_routing_gate/exported_target_shape_tactic_routing_bundle --out receipts/runtime_shell/demo_project/organs/target_shape_tactic_routing_gate","relative_href":"docs/component-target-shape-tactic-routing-gate.html","primary_reader_href":"docs/paper-module-target-shape-tactic-routing.html","component_detail_href":"docs/component-target-shape-tactic-routing-gate.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It only inspects and records the projection mechanics of pre-execution tactic-routing references — emitting per-tactic allow/reject decisions with reasons. It does not run Lean/Lake, does not establish or judge the correctness of any goal, emits no proof bodies, makes no external model access, performs no post-execution route selection, reports no benchmark claims or maturity, and excludes launch.","source_link_count":2,"links_to_count":5,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/23","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_benchmark_claim"]},{"id":"lean_std_premise_index","label":"Lean Std Premise Index","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Lists a fixed catalog of public Lean building blocks and confirms none hides proof text or test answers.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.lean_std_premise_index run --input fixtures/first_wave/lean_std_premise_index/input --out /tmp/plectis-public-runs/lean_std_premise_index","relative_href":"docs/component-lean-std-premise-index.html","primary_reader_href":"docs/paper-module-lean-std-premise-index.html","component_detail_href":"docs/component-lean-std-premise-index.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It only validates the projection of premise metadata and copied source bodies; it does not run Lean or Lake, prove any theorem correct, expose proof bodies or oracle-needed ids, use external model services, produce benchmark claims, or include launch operations.","source_link_count":2,"links_to_count":4,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/24","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_theorem_or_proof_authority","not_benchmark_claim"]},{"id":"formal_math_premise_retrieval","label":"Formal Math Premise Retrieval","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Shows which lemmas a plain search surfaces per query, and never leaks proof text or answer keys.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python -m microcosm_core.organs.formal_math_premise_retrieval run --input fixtures/first_wave/formal_math_premise_retrieval/input --out receipts/first_wave/formal_math_premise_retrieval","relative_href":"docs/component-formal-math-premise-retrieval.html","primary_reader_href":"docs/paper-module-formal-math-premise-retrieval.html","component_detail_href":"docs/component-formal-math-premise-retrieval.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It only checks that public retrieval metadata is internally coherent, term-scored over a copied index, budget-bounded, and leakage-clean; it does not run Lean/Lake, use external model services, prove any theorem or its own correctness, claim benchmark performance, or include launch operations.","source_link_count":2,"links_to_count":6,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/25","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_theorem_or_proof_authority","not_benchmark_claim"]},{"id":"formal_math_verifier_trace_repair_loop","label":"Formal Math Verifier Trace Repair Loop","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Replays how a proof lab turns verifier failures into fixes, with no promotion without a fresh re-run.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis formal-math-verifier-trace-repair-loop run-loop-bundle --input examples/formal_math_verifier_trace_repair_loop/exported_verifier_trace_repair_bundle --out receipts/runtime_shell/demo_project/organs/formal_math_verifier_trace_repair_loop","relative_href":"docs/component-formal-math-verifier-trace-repair-loop.html","primary_reader_href":"docs/paper-module-formal-math-verifier-trace-repair-loop.html","component_detail_href":"docs/component-formal-math-verifier-trace-repair-loop.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It demonstrates control-loop projection mechanics over copied Ring2 run rows only; it does not run Lean/Lake, use external model services, expose proof bodies or oracle premise ids, treat human or provider advice as correctness, prove any theorem, or include launch operations.","source_link_count":2,"links_to_count":5,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/26","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_theorem_or_proof_authority"]},{"id":"formal_evidence_cell_anchor_resolver","label":"Formal Evidence Cell Anchor Resolver","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Resolves each proof-flavored math claim to named evidence and flags ones that overreach or lack backing.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis formal-evidence-cell-anchor-resolver run-anchor-bundle --input examples/formal_evidence_cell_anchor_resolver/exported_evidence_cell_anchor_bundle --out receipts/runtime_shell/demo_project/organs/formal_evidence_cell_anchor_resolver","relative_href":"docs/component-formal-evidence-cell-anchor-resolver.html","primary_reader_href":"docs/paper-module-formal-evidence-cell-anchor-resolver.html","component_detail_href":"docs/component-formal-evidence-cell-anchor-resolver.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates claim-to-evidence anchoring mechanics only: claim-to-cell resolution, source-anchor presence, permitted claim strength, copied-source-module digest checks, and leakage refusals. It does not run Lean/Lake, certify theorem or mathematical correctness, expose proof bodies or non-public source refs, use external model services, or include launch operations/public sharing.","source_link_count":2,"links_to_count":3,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/27","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_public_sharing_authority","not_theorem_or_proof_authority"]},{"id":"undeclared_library_prior_symbol_classifier","label":"Undeclared Library Prior Symbol Classifier","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Detects when a checked Lean proof cites a library result outside its approved set.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.undeclared_library_prior_symbol_classifier run --input fixtures/first_wave/undeclared_library_prior_symbol_classifier/input --out receipts/first_wave/undeclared_library_prior_symbol_classifier","relative_href":"docs/component-undeclared-library-prior-symbol-classifier.html","primary_reader_href":"docs/paper-module-undeclared-library-prior-classifier.html","component_detail_href":"docs/component-undeclared-library-prior-symbol-classifier.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It only projects the symbol-boundary classification mechanic over copied Lean/Std premise rows and pre-extracted symbol observations; it does not read proof source, run Lean or Lake, prove formal-result correctness, treat the whole standard library as an implicit allowlist, claim Mathlib availability, use external model services, or include launch operations.","source_link_count":2,"links_to_count":3,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/28","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_theorem_or_proof_authority"]},{"id":"ring2_premise_retrieval_precision_recall_harness","label":"Ring2 Premise Retrieval Precision Recall Harness","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Scores how much proof support a premise search found, problem by problem.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.ring2_premise_retrieval_precision_recall_harness run --input fixtures/first_wave/ring2_premise_retrieval_precision_recall_harness/input --out receipts/first_wave/ring2_premise_retrieval_precision_recall_harness","relative_href":"docs/component-ring2-premise-retrieval-precision-recall-harness.html","primary_reader_href":"docs/paper-module-ring2-premise-precision-recall.html","component_detail_href":"docs/component-ring2-premise-retrieval-precision-recall-harness.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"These are after-the-fact retrieval-attribution labels and precision/recall counts over copied run records only. The component does not run Lean or Lake, call any provider, expose proof bodies, tune on test answers, claim benchmark performance, prove formal-result correctness, or include launch operations, and its labels are explicitly forbidden from flowing into provider context. The aggregate numbers describe only the copied fixture/bundle replayed, not any benchmark claims.","source_link_count":2,"links_to_count":2,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/29","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_benchmark_claim"]},{"id":"verifier_lab_kernel","label":"Verifier Lab Kernel","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Folds nine proof checks into one report labeling each line by which source actually backs it.","rank":5,"evidence_kind":"Contract validator","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.verifier_lab_kernel run --input fixtures/first_wave/verifier_lab_kernel/input --out receipts/first_wave/verifier_lab_kernel","relative_href":"docs/component-verifier-lab-kernel.html","primary_reader_href":"docs/paper-module-verifier-lab-kernel.html","component_detail_href":"docs/component-verifier-lab-kernel.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates the declared public contract shape of the proof packet and component result records only; it does not establish anything correct, count oracle/provider output as forward proof success, import private or Mathlib-dependent proof bodies, use external model services, change source files, or claim benchmark solve rates, launch, or maturity.","source_link_count":2,"links_to_count":10,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/30","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_theorem_or_proof_authority","not_benchmark_claim"]},{"id":"proof_derived_governed_mutation_authorization","label":"Proof Derived Governed Mutation Authorization","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Checks a synthetic change-authorization record for its proof-and-approval chain, bound to a real commit.","rank":5,"evidence_kind":"Contract validator","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.proof_derived_governed_mutation_authorization run --input fixtures/first_wave/proof_derived_governed_mutation_authorization/input --out /tmp/plectis-public-runs/proof_derived_governed_mutation_authorization","relative_href":"docs/component-proof-derived-governed-mutation-authorization.html","primary_reader_href":"docs/paper-module-proof-derived-governed-mutation-authorization.html","component_detail_href":"docs/component-proof-derived-governed-mutation-authorization.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates only a declared, synthetic governed-mutation contract and excludes live cloud/account action, standing account secrets, source or irreversible mutation, policy-after-execution, hidden votes, external model access, benchmark-score claims, or launch.","source_link_count":2,"links_to_count":3,"tags":["formal-methods","theorem-proving","lean"],"full_record_pointer":"/visible_text_index/logical_records/components/31","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_benchmark_claim"]},{"id":"finite_erdos_denominator_certificate_strike","label":"Finite Erdos Denominator-Order Certificate Strike","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Computes an exact-arithmetic denominator-order identity and catches forged ones, not the open Erdos problem.","rank":4,"evidence_kind":"Bounded runtime computation","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis finite-erdos-denominator-certificate-strike run --input fixtures/first_wave/finite_erdos_denominator_certificate_strike/input --out /tmp/plectis-public-runs/finite_erdos_denominator_certificate_strike","relative_href":"docs/component-finite-erdos-denominator-certificate-strike.html","primary_reader_href":"docs/paper-module-finite-erdos-denominator-certificate-strike.html","component_detail_href":"docs/component-finite-erdos-denominator-certificate-strike.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It computes the finite denominator-order certificate ord_Q(b)=lcm(F) for S_F(b)=sum 1/(b^n-1)=P/Q in exact rational arithmetic over bounded public fixtures and rejects forged certificates by recomputation; it does not establish the open infinite Erdos #257 problem, is not an oracle, prover, or provider result, and a holding certificate is a bounded computational witness, not a machine-checked proof.","source_link_count":2,"links_to_count":3,"tags":["formal-math","number-theory","exact-arithmetic"],"full_record_pointer":"/visible_text_index/logical_records/components/32","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes"]},{"id":"lean_proof_search_lab_runtime","label":"Lean Proof-Search Lab Runtime","family_id":"formal_math_and_proof","family_label":"Formal math & proof","summary_line":"Finds tactic scripts the real Lean prover accepts on toy theorems, and refuses proofs that cheat.","rank":4,"evidence_kind":"External tool run","runs_real_tools":true,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":true,"intrinsic_populated_source_runs_real_tools":true,"source_population_required":false},"command":"plectis lean-proof-search-lab-runtime run --input fixtures/first_wave/lean_proof_search_lab_runtime/input --out /tmp/plectis-public-runs/lean_proof_search_lab_runtime","relative_href":"docs/component-lean-proof-search-lab-runtime.html","primary_reader_href":"docs/paper-module-lean-proof-search-lab-runtime.html","component_detail_href":"docs/component-lean-proof-search-lab-runtime.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"This standard governs a bounded symbolic Lean proof-search lab over tiny public toy-theorem fixtures whose verdict source is the installed Lean subprocess only. It is not neural theorem proving, does not solve any open mathematical problem, does not forward oracle proof bodies, is not frontier-scale math automation or online-RL bandit search, is not a private source prover-run export, and excludes launch or public sharing.","source_link_count":2,"links_to_count":3,"tags":["formal-math","theorem-proving","proof-search"],"full_record_pointer":"/visible_text_index/logical_records/components/33","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_public_sharing_authority","not_theorem_or_proof_authority"]},{"id":"agent_closeout_faithfulness_audit","label":"Agent Completion Faithfulness Audit","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Runs real git and pytest on a sample repo so wrap-up claims state only what the evidence proves.","rank":4,"evidence_kind":"External tool run","runs_real_tools":true,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":true,"intrinsic_populated_source_runs_real_tools":true,"source_population_required":false},"command":"plectis agent-closeout-faithfulness-audit run --input fixtures/first_wave/agent_closeout_faithfulness_audit/input --out receipts/first_wave/agent_closeout_faithfulness_audit","relative_href":"docs/component-agent-closeout-faithfulness-audit.html","primary_reader_href":"docs/paper-module-agent-closeout-faithfulness-audit.html","component_detail_href":"docs/component-agent-closeout-faithfulness-audit.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"verified means the referenced evidence object exists or a pytest span ran; it does not imply the span passed unless exit-zero status was explicitly checked","source_link_count":2,"links_to_count":0,"tags":["ai-safety","agent-evaluation","red-teaming"],"full_record_pointer":"/visible_text_index/logical_records/components/34","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes"]},{"id":"bounded_autonomy_campaign_packet","label":"Bounded Autonomy Campaign Packet","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Drafts proposed work from coverage gaps and proves it cannot repair or rewrite the code itself.","rank":5,"evidence_kind":"Verified source import","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis bounded-autonomy-campaign-packet run --input fixtures/first_wave/bounded_autonomy_campaign_packet/input --out receipts/first_wave/bounded_autonomy_campaign_packet","relative_href":"docs/component-bounded-autonomy-campaign-packet.html","primary_reader_href":"docs/paper-module-bounded-autonomy-campaign-packet.html","component_detail_href":"docs/component-bounded-autonomy-campaign-packet.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"self-proposal campaign packet only; no self-repair or unsupervised source-file changes","source_link_count":2,"links_to_count":0,"tags":["ai-safety","agent-evaluation","red-teaming"],"full_record_pointer":"/visible_text_index/logical_records/components/35","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes"]},{"id":"provider_context_recipe_budget_policy","label":"Provider Context Recipe Budget Policy","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Runs the real context harness to measure assembled byte sizes and check each bundle fits its budget.","rank":4,"evidence_kind":"Bounded runtime computation","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.provider_context_recipe_budget_policy run --input fixtures/first_wave/provider_context_recipe_budget_policy/input --out receipts/first_wave/provider_context_recipe_budget_policy","relative_href":"docs/component-provider-context-recipe-budget-policy.html","primary_reader_href":"docs/paper-module-provider-context-recipe-budget.html","component_detail_href":"docs/component-provider-context-recipe-budget-policy.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates context-budget projection mechanics (byte ceilings, ordered section fill, omitted-section manifests, deliverable routing, and digest-checked source-body imports) only. It excludes provider/API calls, run Lean/Lake, expose or carry proof or oracle truth-side material, assert theorem or domain-level conclusions, or include launch operations.","source_link_count":2,"links_to_count":2,"tags":["ai-safety","agent-evaluation","red-teaming"],"full_record_pointer":"/visible_text_index/logical_records/components/36","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_theorem_or_proof_authority"]},{"id":"batch10_cold_eval_honesty_capsule","label":"Cold Evaluation Honesty Bundle","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Runs a copied route-quality simulator and checks its all-B scorecard against the original code.","rank":5,"evidence_kind":"Verified source import","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis batch10-cold-eval-honesty-capsule run --input fixtures/first_wave/batch10_cold_eval_honesty_capsule/input --out /tmp/plectis-public-runs/batch10_cold_eval_honesty_capsule","relative_href":"docs/component-batch10-cold-eval-honesty-capsule.html","primary_reader_href":"docs/paper-module-batch10-cold-eval-honesty-capsule.html","component_detail_href":"docs/component-batch10-cold-eval-honesty-capsule.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"verified cold-eval source body import only, not a live benchmark, navigation truth, source authority, external model access, whole-system equivalence, public sharing, or launch-scope decision","source_link_count":2,"links_to_count":0,"tags":["ai-safety","agent-evaluation","red-teaming"],"full_record_pointer":"/visible_text_index/logical_records/components/37","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_whole_system_correctness","not_launch_authority","not_public_sharing_authority","not_benchmark_claim"]},{"id":"batch8_validator_checker_capsule","label":"Validator Checker Bundle","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Runs the real validator code over public examples so its safety checks stay inspectable.","rank":5,"evidence_kind":"Verified source import","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis batch8-validator-checker-capsule run --input fixtures/first_wave/batch8_validator_checker_capsule/input --out /tmp/plectis-public-runs/batch8_validator_checker_capsule","relative_href":"docs/component-batch8-validator-checker-capsule.html","primary_reader_href":"docs/paper-module-batch8-validator-checker-capsule.html","component_detail_href":"docs/component-batch8-validator-checker-capsule.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates only the imported validators.py source body and its checker membrane. It does not claim source authority, a full validator-suite proof, whole-system equivalence, launch, hosted-public status, public sharing, external model access, or source-file changes.","source_link_count":2,"links_to_count":2,"tags":["ai-safety","agent-evaluation","red-teaming"],"full_record_pointer":"/visible_text_index/logical_records/components/38","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_whole_system_correctness","not_launch_authority","not_public_sharing_authority"]},{"id":"batch7_secondary_runtime_capsule","label":"Secondary Runtime Source Bundle","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Runs eight trace, graph, and market engines on test rows without fetching live markets.","rank":4,"evidence_kind":"Bounded runtime computation","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis batch7-secondary-runtime-capsule run --input fixtures/first_wave/batch7_secondary_runtime_capsule/input --out /tmp/plectis-public-runs/batch7_secondary_runtime_capsule","relative_href":"docs/component-batch7-secondary-runtime-capsule.html","primary_reader_href":"docs/paper-module-batch7-secondary-runtime-capsule.html","component_detail_href":"docs/component-batch7-secondary-runtime-capsule.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"verified source body import only; no browser/session export, wallet authority, live market data, investment-related actions, external model access, source-file changes, whole-system equivalence, public sharing, launch, semantic-truth, or whole-system correctness claim","source_link_count":2,"links_to_count":0,"tags":["source intake","provenance","drift-control"],"full_record_pointer":"/visible_text_index/logical_records/components/39","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_whole_system_correctness","not_launch_authority","not_public_sharing_authority","not_live_market_data"]},{"id":"agent_benchmark_integrity_anti_gaming_replay","label":"Agent Benchmark Integrity Anti Gaming Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Validates a synthetic benchmark-integrity record and flags the contamination cases it declares.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis agent-benchmark-integrity-anti-gaming-replay run-benchmark-integrity-bundle --input examples/agent_benchmark_integrity_anti_gaming_replay/exported_benchmark_integrity_bundle --out .microcosm/agent_benchmark_integrity_anti_gaming_replay","relative_href":"docs/component-agent-benchmark-integrity-anti-gaming-replay.html","primary_reader_href":"docs/paper-module-agent-benchmark-integrity-anti-gaming-replay.html","component_detail_href":"docs/component-agent-benchmark-integrity-anti-gaming-replay.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It authorizes only bounded public runtime validation over copied source-open pattern provenance bodies and metadata-only benchmark-integrity replay rows; it does not establish any benchmark or SWE-bench score, agent capability, external model service, live-repo mutation, private/oracle/hidden-gold body access, product progress, or launch-scope decision.","source_link_count":2,"links_to_count":2,"tags":["ai-safety","agent-evaluation","red-teaming"],"full_record_pointer":"/visible_text_index/logical_records/components/40","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_benchmark_claim"]},{"id":"agent_monitor_redteam_falsification_replay","label":"Monitor Evidence-Boundary Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Replays honest and deceptive agent runs and flags any verdict missing its declared backing evidence.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis agent-monitor-redteam-falsification-replay run-monitor-bundle --input examples/agent_monitor_redteam_falsification_replay/exported_monitor_redteam_bundle --out receipts/runtime_shell/demo_project/organs/agent_monitor_redteam_falsification_replay","relative_href":"docs/component-agent-monitor-redteam-falsification-replay.html","primary_reader_href":"docs/paper-module-agent-monitor-redteam-falsification-replay.html","component_detail_href":"docs/component-agent-monitor-redteam-falsification-replay.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"Bounded public runtime validation over copied source pattern bodies, sanitized dogfood trace slices, recomputed monitor-verdict spans, source-artifact evidence refs, digest/metadata-only/non-public-state gates, and negative cases only; no live agent execution, monitor product performance, control-eval score, safety-validation, benchmark, provider-call, source-file changes, launch, public sharing, or product authority.","source_link_count":2,"links_to_count":1,"tags":["ai-safety","agent-evaluation","red-teaming"],"full_record_pointer":"/visible_text_index/logical_records/components/41","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_public_sharing_authority","not_benchmark_claim"]},{"id":"agent_sabotage_scheming_monitor_replay","label":"Sabotage-Monitor Contract Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Audits a hidden-goal catch claim for the steps, suspicion scores, and counterfactual it needs.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis agent-sabotage-scheming-monitor-replay run-sabotage-bundle --input examples/agent_sabotage_scheming_monitor_replay/exported_sabotage_monitor_bundle --out receipts/runtime_shell/demo_project/organs/agent_sabotage_scheming_monitor_replay","relative_href":"docs/component-agent-sabotage-scheming-monitor-replay.html","primary_reader_href":"docs/paper-module-agent-sabotage-scheming-monitor-replay.html","component_detail_href":"docs/component-agent-sabotage-scheming-monitor-replay.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"Bounded public runtime validation over copied source pattern bodies, sanitized dogfood trace slices, recomputed sabotage/scheming monitor spans, source-artifact evidence refs, digest/metadata-only/non-public-state gates, and negative cases only; no live sabotage, live agent execution, exploit instruction, account secret/account, private-reasoning, harmful-payload, monitor-product-performance, deployment-risk, benchmark, provider-call, source-file changes, launch, public sharing, or product authority.","source_link_count":2,"links_to_count":0,"tags":["ai-safety","agent-evaluation","red-teaming"],"full_record_pointer":"/visible_text_index/logical_records/components/42","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_public_sharing_authority","not_benchmark_claim"]},{"id":"agent_memory_temporal_conflict_replay","label":"Agent Memory Temporal Conflict Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Replays a memory edit-and-delete to show stale facts get flagged before they sway an answer.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.agent_memory_temporal_conflict_replay run --input fixtures/first_wave/agent_memory_temporal_conflict_replay/input --out /tmp/agent_memory_temporal_conflict_replay_out","relative_href":"docs/component-agent-memory-temporal-conflict-replay.html","primary_reader_href":"docs/paper-module-agent-memory-temporal-conflict-replay.html","component_detail_href":"docs/component-agent-memory-temporal-conflict-replay.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates the projection mechanics of a synthetic memory fixture only — that the required refs, decisions, paired replays, negative cases, and secret-exclusion scan line up and that result records are metadata-only. It does not claim live-memory product quality, judge whether memory decisions were domain-correct, treat memory recall as source authority, adopt active injection, export private transcripts, use external model services, change source files, or include launch operations.","source_link_count":2,"links_to_count":1,"tags":["ai-safety","agent-evaluation","red-teaming"],"full_record_pointer":"/visible_text_index/logical_records/components/43","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority"]},{"id":"sleeper_memory_poisoning_quarantine_replay","label":"Memory-Poisoning Quarantine Policy Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Replays a recorded memory-tamper case, checking its declared quarantine, block, and delete steps line up.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.sleeper_memory_poisoning_quarantine_replay run --input fixtures/first_wave/sleeper_memory_poisoning_quarantine_replay/input --out /tmp/plectis-public-runs/sleeper_memory_poisoning_quarantine_replay","relative_href":"docs/component-sleeper-memory-poisoning-quarantine-replay.html","primary_reader_href":"docs/paper-module-sleeper-memory-poisoning-quarantine-replay.html","component_detail_href":"docs/component-sleeper-memory-poisoning-quarantine-replay.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It only checks the structural shape and internal consistency of a synthetic memory-security policy projection recorded as JSON. It does not run or validate any real memory store, does not itself quarantine, delete, or re-run anything, and does not establish that any system actually resists poisoning. It exports no private memory bodies or transcripts, calls no providers, mutates no source, produces no benchmark claims, and excludes launch (all scope limit flags are hardcoded false).","source_link_count":2,"links_to_count":0,"tags":["ai-safety","agent-evaluation","red-teaming"],"full_record_pointer":"/visible_text_index/logical_records/components/44","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_benchmark_claim"]},{"id":"mcp_tool_authority_replay","label":"MCP Tool-Authority Policy Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Audits a recorded tool-use log to confirm each action was scoped, approved, undoable, and fenced.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.mcp_tool_authority_replay run --input fixtures/first_wave/mcp_tool_authority_replay/input --out receipts/first_wave/mcp_tool_authority_replay","relative_href":"docs/component-mcp-tool-authority-replay.html","primary_reader_href":"docs/paper-module-mcp-tool-authority-replay.html","component_detail_href":"docs/component-mcp-tool-authority-replay.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It only checks that the tool-authority evidence in a recorded bundle (scopes, approvals, rollbacks, instruction/data splits, cold replays, redaction, and the expected abuse-case failures) is present and internally consistent. It does not run tools or authorize live MCP/account access, account secret or payload export, treating tool output as instruction, source-file changes, benchmark safety scores, or launch, and it makes no claim that the underlying tool-use policy is domain-correct.","source_link_count":2,"links_to_count":2,"tags":["ai-safety","agent-evaluation","red-teaming"],"full_record_pointer":"/visible_text_index/logical_records/components/45","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_benchmark_claim"]},{"id":"belief_state_process_reward_replay","label":"Belief-State Reward Bundle Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Checks that each step reward in a recorded run cites a declared verifier-feedback row, not a trick.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.belief_state_process_reward_replay run --input fixtures/first_wave/belief_state_process_reward_replay/input --out .microcosm/belief_state_process_reward_replay","relative_href":"docs/component-belief-state-process-reward-replay.html","primary_reader_href":"docs/paper-module-belief-state-process-reward-replay.html","component_detail_href":"docs/component-belief-state-process-reward-replay.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It only checks that the projection's accounting lines up under its own schema rules over recorded synthetic fixtures; it excludes hidden-reasoning export, RL training, hidden gold or neural-judge-only labels, benchmark-performance claims, external model access, source-file changes, or launch, and proves nothing about real-world reward, live agent behavior, or domain-level conclusions.","source_link_count":2,"links_to_count":0,"tags":["ai-safety","agent-evaluation","red-teaming"],"full_record_pointer":"/visible_text_index/logical_records/components/46","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_benchmark_claim"]},{"id":"agent_sandbox_policy_escape_replay","label":"Sandbox-Policy Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Maps sandboxed agent actions to show each was approved or blocked before running, then rolled back.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.cli agent-sandbox-policy-escape-replay run-sandbox-bundle --input examples/agent_sandbox_policy_escape_replay/exported_sandbox_policy_escape_bundle --out receipts/runtime_shell/demo_project/organs/agent_sandbox_policy_escape_replay","relative_href":"docs/component-agent-sandbox-policy-escape-replay.html","primary_reader_href":"docs/paper-module-agent-sandbox-policy-escape-replay.html","component_detail_href":"docs/component-agent-sandbox-policy-escape-replay.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates the projection / trace-refactor mechanics over a synthetic fixture only; it excludes live sandbox escape, secret or account secret handling, live network access, host filesystem mutation, executable payload export, raw environment export, external model access, security benchmark claims, source-file changes, or launch. A pass proves the projection boundary and trace-refactor mechanics for this contract, not real sandbox security, exploit resistance, or whole-system safety.","source_link_count":2,"links_to_count":1,"tags":["ai-safety","agent-evaluation","red-teaming"],"full_record_pointer":"/visible_text_index/logical_records/components/47","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_whole_system_correctness","not_launch_authority","not_benchmark_claim"]},{"id":"indirect_prompt_injection_information_flow_policy_replay","label":"Prompt-Injection Flow-Policy Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Replays an agent run to show untrusted text was gated before any sensitive action, leaking no secret.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis indirect-prompt-injection-information-flow-policy-replay run-prompt-injection-bundle --input examples/indirect_prompt_injection_information_flow_policy_replay/exported_prompt_injection_flow_bundle --out receipts/runtime_shell/demo_project/organs/indirect_prompt_injection_information_flow_policy_replay","relative_href":"docs/component-indirect-prompt-injection-information-flow-policy-replay.html","primary_reader_href":"docs/paper-module-indirect-prompt-injection-information-flow-policy-replay.html","component_detail_href":"docs/component-indirect-prompt-injection-information-flow-policy-replay.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"Passing result records only show this projection satisfies the named information-flow contract over synthetic, redacted, metadata-only rows; they do not prove general prompt-injection robustness, benchmark performance, live account/tool/provider safety, hidden-message handling in a real system, source-file changes, or launch-scope decision.","source_link_count":2,"links_to_count":0,"tags":["ai-safety","agent-evaluation","red-teaming"],"full_record_pointer":"/visible_text_index/logical_records/components/48","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_benchmark_claim"]},{"id":"agentic_vulnerability_discovery_patch_proof_replay","label":"Vulnerability Patch-Proof Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Checks a fixed-bug evidence chain and re-runs three small real security checks; no real attack material.","rank":3,"evidence_kind":"Computed projection","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.agentic_vulnerability_discovery_patch_proof_replay run --input fixtures/first_wave/agentic_vulnerability_discovery_patch_proof_replay/input --out /tmp/plectis-public-runs/agentic_vulnerability_discovery_patch_proof_replay","relative_href":"docs/component-agentic-vulnerability-discovery-patch-proof-replay.html","primary_reader_href":"docs/paper-module-agentic-vulnerability-discovery-patch-proof-replay.html","component_detail_href":"docs/component-agentic-vulnerability-discovery-patch-proof-replay.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates only the projection/evidence-chain mechanics of a synthetic replay: structural presence, cross-reference consistency, declared boolean flags, and the secret/live-access exclusion scan. It executes small regression witnesses but performs no real vulnerability discovery and makes no judgment of real-world security or fix correctness. It excludes live-target testing, real CVE exploitation, weaponized payloads, account secret handling, network exfiltration, actionable exploit steps, external model access, source-file changes, benchmark security scores, launch, or any whole-system security claim.","source_link_count":2,"links_to_count":0,"tags":["ai-safety","agent-evaluation","red-teaming"],"full_record_pointer":"/visible_text_index/logical_records/components/49","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_whole_system_correctness","not_launch_authority","not_benchmark_claim"]},{"id":"agent_route_observability_runtime","label":"Agent Route Observability Runtime","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Recomputes an agent run's route-compliance score and anti-pattern flags with real trace-analytics code.","rank":5,"evidence_kind":"Contract validator","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.agent_route_observability_runtime run --input fixtures/first_wave/agent_route_observability_runtime/input --out receipts/first_wave/agent_route_observability_runtime","relative_href":"docs/component-agent-route-observability-runtime.html","primary_reader_href":"docs/paper-module-agent-route-observability-runtime.html","component_detail_href":"docs/component-agent-route-observability-runtime.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates only public, recorded trace-feedback metadata and regression fixtures; it does not inspect live operator state, certify or prove runtime behavior, read model-output data, mutate the work log, authorize pattern assimilation, or include launch operations.","source_link_count":2,"links_to_count":15,"tags":["ai-safety","agent-evaluation","red-teaming"],"full_record_pointer":"/visible_text_index/logical_records/components/50","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority"]},{"id":"bridge_campaign_dag_validation","label":"Bridge Campaign DAG Validation","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","summary_line":"Runs shape checks on a fan-out work plan for unique steps, dependencies, and no cycles, not the plan itself.","rank":4,"evidence_kind":"Bounded runtime computation","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis bridge-campaign-dag-validation run --input fixtures/first_wave/bridge_campaign_dag_validation/input --out /tmp/plectis-public-runs/bridge_campaign_dag_validation","relative_href":"docs/component-bridge-campaign-dag-validation.html","primary_reader_href":"docs/paper-module-bridge-campaign-dag-validation.html","component_detail_href":"docs/component-bridge-campaign-dag-validation.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"The bundle validates a public bridge-campaign DAG contract and provider worker ceiling. 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It does not establish forecasting correctness or accuracy, give trading/financial/investment-related actions, call live market data or providers, publish predictions, claim any performance or track record, import non-public data, or include launch operations.","source_link_count":2,"links_to_count":3,"tags":["research-workflows","forecasting","provider operations"],"full_record_pointer":"/visible_text_index/logical_records/components/61","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_live_market_data"]},{"id":"derived_fact_provider_runtime","label":"Derived Fact Provider Runtime","family_id":"research_and_science_replays","family_label":"Research & science","summary_line":"Fills in facts from simple recipes and records a clear error when one points at missing data.","rank":4,"evidence_kind":"Bounded runtime computation","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis derived-fact-provider-runtime run --input fixtures/first_wave/derived_fact_provider_runtime/input --out /tmp/plectis-public-runs/derived_fact_provider_runtime","relative_href":"docs/component-derived-fact-provider-runtime.html","primary_reader_href":"docs/paper-module-derived-fact-provider-runtime.html","component_detail_href":"docs/component-derived-fact-provider-runtime.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"The bundle demonstrates registry-backed JSON-pointer, glob-count, and git-backed callable fact providers with provider failures represented as error rows. 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It supports inspection of recorded drift rows and source-linked refs; live repair, source control, doctrine changes, model-output export, public sharing, and launch are outside the fixture. It does not claim complete drift coverage or live repair control.","source_link_count":2,"links_to_count":7,"tags":["source intake","provenance","drift-control"],"full_record_pointer":"/visible_text_index/logical_records/components/64","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_public_sharing_authority"]},{"id":"batch6_unsurfaced_primitives_capsule","label":"Unsurfaced Source Primitives Bundle","family_id":"import_projection_and_drift","family_label":"Import & drift control","summary_line":"Exposes eleven real but under-surfaced parts and rejects non-public-state and overclaim cases.","rank":5,"evidence_kind":"Verified source import","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"plectis batch6-unsurfaced-primitives-capsule run --input fixtures/first_wave/batch6_unsurfaced_primitives_capsule/input --out /tmp/plectis-public-runs/batch6_unsurfaced_primitives_capsule","relative_href":"docs/component-batch6-unsurfaced-primitives-capsule.html","primary_reader_href":"docs/paper-module-batch6-unsurfaced-primitives-capsule.html","component_detail_href":"docs/component-batch6-unsurfaced-primitives-capsule.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates only a public source-open bundle and bounded public exercises; 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it is not live source authority, whole-system equivalence, semantic truth, investment-related actions, complete sandbox proof, selected-test sufficiency proof, public sharing, or launch-scope decision.","source_link_count":2,"links_to_count":3,"tags":["source intake","provenance","drift-control"],"full_record_pointer":"/visible_text_index/logical_records/components/67","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_whole_system_correctness","not_launch_authority","not_public_sharing_authority"]},{"id":"batch7_oracle_sibling_capsule","label":"Oracle Sibling Source Bundle","family_id":"import_projection_and_drift","family_label":"Import & drift control","summary_line":"Replays subject-index and truth-diff logic on copied code, rejecting reasoning overclaims.","rank":4,"evidence_kind":"Bounded runtime computation","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.batch7_oracle_sibling_capsule run --input fixtures/first_wave/batch7_oracle_sibling_capsule/input --out /tmp/plectis-public-runs/batch7_oracle_sibling_capsule","relative_href":"docs/component-batch7-oracle-sibling-capsule.html","primary_reader_href":"docs/paper-module-batch7-oracle-sibling-capsule.html","component_detail_href":"docs/component-batch7-oracle-sibling-capsule.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates only Oracle sibling copied source bodies and bounded deterministic exercises; it does not run Oracle reasoning, dispatch providers or bridges, invoke private orchestration engine, change source files, prove semantic truth, prove all Oracle paths are covered, authorize public sharing, or include launch operations.","source_link_count":2,"links_to_count":1,"tags":["source intake","provenance","drift-control"],"full_record_pointer":"/visible_text_index/logical_records/components/68","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes","not_launch_authority","not_public_sharing_authority"]},{"id":"batch7_demo_take_console_capsule","label":"Demo Take Console Source Bundle","family_id":"import_projection_and_drift","family_label":"Import & drift control","summary_line":"Replays the recording console's Swift logic without launching the app or capturing audio.","rank":5,"evidence_kind":"Verified source import","runs_real_tools":false,"runs_real_tools_meaning":"current_public_component_command_invokes_external_machinery","public_runtime":{"release_mode":"populated_source_release","runnable_from_current_public_artifact":true,"runs_real_tools_now_from_current_public_artifact":false,"intrinsic_populated_source_runs_real_tools":false,"source_population_required":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.batch7_demo_take_console_capsule run --input fixtures/first_wave/batch7_demo_take_console_capsule/input --out /tmp/plectis-public-runs/batch7_demo_take_console_capsule","relative_href":"docs/component-batch7-demo-take-console-capsule.html","primary_reader_href":"docs/paper-module-batch7-demo-take-console-capsule.html","component_detail_href":"docs/component-batch7-demo-take-console-capsule.html","map_primary_click_policy":"prefer_primary_reader_href","scope_limit":"It validates only Demo Take Console copied Swift source bodies and bounded deterministic exercises; 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It is evidence for fixture-local completion mechanics, not for live Git side effects, unrelated-path staging, non-public body export, service operation, or distribution clearance.","source_link_count":2,"links_to_count":2,"tags":["workflow-engineering","continuity"],"full_record_pointer":"/visible_text_index/logical_records/components/84","claim_ceiling_tags":["public_evidence_only","not_hosted_service","no_external_model_access","no_source_file_changes"]},{"id":"bridge_phase_continuity_runtime","label":"Bridge-Continuity Sign-off Replay","family_id":"work_landing_and_continuity","family_label":"Work & continuity","summary_line":"Replays a paused job to prove the rules for safely resuming it hold and reject duplicate resumes.","rank":5,"evidence_kind":"Contract 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Bodies already in scholarly_maps.formal_claim_control_plane are indexed, not repeated. Source hashes bind the original bytes; parsed JSON may serialize differently. Reading these files does not run their checks.","lean":{"repository":"wcook04/plectis-erdos","checkout_key":"plectis-erdos","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","document_count":32,"embedded_count":26,"missing":[],"documents":{"readme":{"document_id":"readme","source_path":"README.md","role":"human_front_door","github":"https://github.com/wcook04/plectis-erdos/blob/main/README.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/README.md","sha256":"sha256:799039e31a9db6698fa5b16ae7478ffb9668b4ffcb8e76c9d647f26424491bfa","byte_count":17898,"origin":"public_main_checkout","body":"<!-- SPDX-FileCopyrightText: 2026 Will Cook -->\n<!-- SPDX-License-Identifier: Apache-2.0 -->\n\n# Plectis\n\n### Mathematical research and the system behind it\n\nPlectis collects research on eight problems posed by Paul Erdős: papers,\nLean proofs of selected statements, experiments and failed approaches.\nThe aim is to make AI-assisted research easier to understand, check and continue.\n\n**[Explore the research](#problem-papers)** · **[Read the system paper](paper/systems/claim-faithful-publication-systems-paper.pdf)** · **[Watch the videos](#videos)** · **[Open the maths map](https://wcook04.github.io/practical-formal-verification/maths/universe.html)**\n\n[![Eight Erdős problem programmes: the interactive maths map connects papers, results and their formal evidence](.github/maths-map.png)](https://wcook04.github.io/practical-formal-verification/maths/universe.html)\n\n<sub>The maths map on 9 October 2026. Open the image to explore the current edition.</sub>\n\nPlectis is an independent, AI-assisted prototype built and maintained by Will Cook.\n[A reader's way in](docs/READING_GUIDE.md) introduces the mathematics without\nrequiring Lean. The [glossary](https://wcook04.github.io/practical-formal-verification/docs/glossary.html)\nexplains project terminology.\n\n---\n\n<a id=\"choose-a-way-in\"></a>\n\n## Papers, tools and explanations\n\n- **Mathematics:** [eight short papers and longer research records](#problem-papers).\n- **Research system:** [A Repository-Based System for Research and Publication](paper/systems/claim-faithful-publication-systems-paper.pdf).\n- **Writing and revision:** [writing guide](paper/exposition/writing-a-good-mathematical-paper.pdf), [method](paper/exposition/writing-mathematics-from-reviewed-revisions.pdf) and [changes prompted by readers](docs/research-commons/CREDIT_LEDGER.md#credit-correspondence-004).\n- **Connections between problems:** [cross-problem paper](paper/synthesis/optimal-sparse-perturbations.pdf).\n- **Formal evidence:** [Lean source](docs/SOURCE_MAP.md) and [Comparator checks](docs/EXTERNAL_VERIFICATION.md).\n\nThe [mathematics website](https://wcook04.github.io/practical-formal-verification/maths/) connects each\nproblem's papers, results and sources. The films also show private interfaces.\nDo not infer results from private or unreleased work.\n\n## For a first look: #257\n\nStart with the [introductory paper](paper/257/erdos257-expository-sums-never-fractions.pdf)\nor [#257 video](https://www.youtube.com/watch?v=vXgIBy5C5X0): examples followed by\na proof for infinite choices of exponents whose reciprocals have a finite sum.\nErdős stated this result in 1968.\n\n[Problem 257](https://www.erdosproblems.com/257) asks whether the series\n\n$$\\sum_{a\\in A} \\frac{1}{2^a-1}$$\n\nis irrational for **every infinite set $A$ of positive integers**. The universal\nquestion remains open.\n\nThe [short paper](paper/257/erdos-257-mersenne-support-subseries.pdf) proves sufficient conditions for particular infinite sets, including a\nLean-checked prime-weighted criterion that works at every integer base at\nleast two. The proof cancels finitely many terms, then uses averaging to find a\nsmall positive remainder that rules out rationality.\n\n[Try three changed hypotheses](docs/reading-edition/weighted-257-task.md), or\n[follow the Lean declaration](lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean#L120).\n\n## Results and their limits\n\nUsing the degree-seven polynomial constructed by the erdosproblems.com\ncontributor ani, Lean proves that every preconnected strict-lemniscate set\ncontaining two distinct roots has one-dimensional Hausdorff measure greater\nthan two. This refutes the exact Formal Conjectures path-image-length\nstatement; the separate total-variation bound is also checked. The other\nseven targets remain open. Independent human review of correspondence with\nthe 1958 wording has not been recorded. Comparator checks only selected exact\nstatements, axioms and kernel acceptance; it does not assess novelty or\nhistorical correspondence.\n\n[ani's original counterexample](https://www.erdosproblems.com/forum/thread/1041#post-8861)\nis credited in the [#1041 case study](docs/case-studies/formal-conjectures-1041.md).\n[Results and limits](docs/RESULTS.md) keeps each programme's statements beside\nits assumptions, earlier work and remaining questions.\n\n<a id=\"videos\"></a>\n\n## Watch the videos\n\n- [Plectis introduction](https://www.youtube.com/watch?v=U1z2-XaLe_w): why I use AI to explain its research and software.\n- [The maths map](https://www.youtube.com/watch?v=J2sX6nj3jOk): a tour from checked statements through research records and papers to explanations.\n- [Erdős #257 explanation](https://www.youtube.com/watch?v=vXgIBy5C5X0): examples and a proof under Erdős's reciprocal-summability condition; the unrestricted problem remains open.\n\n<a id=\"about-the-project\"></a>\n\n## Why keep the whole research record?\n\nI want other people to work on these questions with me. The records preserve\ncalculations and routes that stopped, which you can question, repair or reuse.\n\nAI agents did most of the research and drafting. I built and directed the\ninfrastructure and reviewed claims when I could; I have not independently\nverified every claim. I maintain the sources and take responsibility for\ncorrecting the public record. Novelty and significance need human judgement.\nSelected contributions have [independent reviews](#formal-conjectures-contributions);\nreview does not cover the whole corpus.\nThe [system paper](paper/systems/claim-faithful-publication-systems-paper.pdf)\nexplains how the repository records results and corrections and credits contributors.\n\n## Problem papers\n\nThe short papers present results and arguments; longer records include\ncalculations, failed approaches and unresolved steps. The\n[results guide](docs/RESULTS.md#problem-by-problem-guide) lists assumptions and sources.\nThe [reading edition](docs/reading-edition/README.md) collects paper openings.\n\n| Problem | Topic | Papers |\n|---|---|---|\n| [#68](https://www.erdosproblems.com/68) | [Factorial-denominator irrationality](https://wcook04.github.io/practical-formal-verification/maths/problems/erdos_68.html) | [short paper](paper/68/erdos-68-factorial-denominator-irrationality.pdf) · [longer paper](paper/68/erdos68-factorial-reasoning-surface.pdf) |\n| [#243](https://www.erdosproblems.com/243) | [Cubic-rate irrationality and reciprocal tails](https://wcook04.github.io/practical-formal-verification/maths/problems/erdos_243.html) | [short paper](paper/243/erdos-243-reciprocal-tail-rigidity.pdf) · [longer paper](paper/243/erdos243-reciprocal-tail-reasoning-surface.pdf) |\n| [#249](https://www.erdosproblems.com/249) | [Binary totient series](https://wcook04.github.io/practical-formal-verification/maths/problems/erdos_249.html) | [short paper](paper/249/erdos-249-binary-totient-series.pdf) · [longer paper](paper/249/erdos249-totient-reasoning-surface.pdf) |\n| [#251](https://www.erdosproblems.com/251) | [Prime-gap dyadic series](https://wcook04.github.io/practical-formal-verification/maths/problems/erdos_251.html) | [short paper](paper/251/erdos-251-prime-gap-dyadic-series.pdf) · [longer paper](paper/251/erdos251-prime-gap-reasoning-surface.pdf) |\n| [#257](https://www.erdosproblems.com/257) | [Mersenne-support subseries](https://wcook04.github.io/practical-formal-verification/maths/problems/erdos_257.html) | [introductory paper](paper/257/erdos257-expository-sums-never-fractions.pdf) · [short paper](paper/257/erdos-257-mersenne-support-subseries.pdf) · [longer paper](paper/257/erdos257-mersenne-reasoning-surface.pdf) |\n| [#269](https://www.erdosproblems.com/269) | [Three-prime running LCM](https://wcook04.github.io/practical-formal-verification/maths/problems/erdos_269.html) | [short paper](paper/269/erdos-269-three-prime-running-lcm.pdf) · [longer paper](paper/269/erdos269-running-lcm-reasoning-surface.pdf) |\n| [#1041](https://www.erdosproblems.com/1041) | [Lemniscates and Newton flow](https://wcook04.github.io/practical-formal-verification/maths/problems/erdos_1041.html) | [short paper](paper/1041/erdos-1041-lemniscate-newton-flow.pdf) · [longer paper](paper/1041/erdos1041-lemniscate-reasoning-surface.pdf) |\n| [#1049](https://www.erdosproblems.com/1049) | [Rational-base Lambert series](https://wcook04.github.io/practical-formal-verification/maths/problems/erdos_1049.html) | [short paper](paper/1049/erdos-1049-rational-base-lambert.pdf) · [longer paper](paper/1049/erdos1049-rational-base-lambert-reasoning-surface.pdf) |\n\n\n### Reading across the problems\n\n[Reading Eight Erdős Problems Together](paper/synthesis/optimal-sparse-perturbations.pdf)\ndevelops connections and the limits of shared methods. Its main theorems are\nordinary proofs; Lean checks specified ingredients. The [synthesis guide](paper/synthesis/README.md)\nidentifies ways to continue the work.\n\n<a id=\"what-the-checks-establish\"></a>\n\n## What Lean and Comparator verify\n\nLean checks whether a proof establishes the exact formal statement written in\nits source. The [source map](docs/SOURCE_MAP.md) connects paper passages to\nthose declarations. [Claim records](docs/claims.json) list their public status\nand limits; [prior art](docs/PRIOR_ART.md) identifies earlier results and their\nrelationship to this work.\n\nComparator checks a solution against a separately declared formal statement,\nwith an explicit list of permitted axioms. [`formalization.yaml`](formalization.yaml)\nlists the selected statements. The [verification dossier](docs/EXTERNAL_VERIFICATION.md)\nrecords selected statements across all eight problems and their replay requirements;\nit does not cover every argument in the papers.\n\nResearchers judge correspondence with the intended mathematics, novelty and\nusefulness. A build or platform submission does not establish those judgements.\n[Methodology](docs/METHODOLOGY.md) explains the review required to change a claim.\n\n## Contribute\n\nIdeas, corrections, references, counterexamples and clearer explanations are\nwelcome. You need not solve a problem or write Lean.\n\n- **Mathematics or exposition:** [work on a paper](docs/CONTRIBUTE_BY_PAPER.md)\n  or [develop a method across problems](paper/synthesis/README.md).\n- **Software:** improve navigation, checks or the contributor experience through\n  the [architecture contribution guide](docs/research-commons/ARCHITECTURE_CONTRIBUTIONS.md).\n- **A question or correction:** use the\n  [research-progress form](https://github.com/wcook04/plectis-erdos/issues/new?template=research_progress.yml)\n  or [email me](https://wcook04.github.io/practical-formal-verification/#contact).\n- **An investigation with your own agent:** the [frontier relay](docs/FRONTIER_RELAY.md)\n  explains how to return the argument, evidence, limits and next question.\n\n[CONTRIBUTING](CONTRIBUTING.md) explains returns without a clone.\nAccepted contributions receive a public evidence and credit record under the\n[credit policy](docs/research-commons/CREDIT_POLICY.md).\n\n<a id=\"read-inspect-or-continue\"></a>\n\n## Read or verify locally\n\n[REPRODUCIBILITY](docs/REPRODUCIBILITY.md) starts with Git and Python 3.11+, then\nexplains how to install Lean and check a proof. Its\n[claim check](docs/REPRODUCIBILITY.md#try-one-claim-without-lean) inspects\nsource links without compiling the proof.\nFor [research with an agent](docs/agents/README.md#start-with-current-public-work),\nsee [AGENTS.md](AGENTS.md) and the\n[agent workbench](docs/agents/AGENT_WORKBENCH.md).\n\n### Repository map\n\n<details>\n<summary>Where the papers, proofs, experiments and tools live</summary>\n\n| Location | Contents |\n|---|---|\n| [paper/](paper/README.md) | PDFs and manuscript sources by problem. |\n| [lean/](lean/README.md) | Formal proofs in the `Erdos249257` and `ErdosProblems` libraries. |\n| [docs/](docs/README.md) | Guides, claim records and source maps. |\n| [research/](research/README.md) | Experiments, examples and investigations. |\n| [research_corpus/](research_corpus/README.md) | Dated research contributions and unresolved obligations. |\n| [computations/](computations/README.md) | Exact arithmetic certificates and scripts. |\n| [verification/](verification/README.md) | Selected formal interfaces, solutions and replay configuration. |\n| [evidence/](evidence/README.md) | Paper-to-proof records and stored Comparator reports. |\n| [scripts/](scripts/README.md) | Query tools, projection builders and validation programs. |\n| [skills/](skills/README.md) | Workflows for coding agents using this checkout. |\n\n</details>\n\nThe [architecture guide](docs/ARCHITECTURE.md) explains how the folders fit together.\nThe checkout is hundreds of megabytes; Lean builds download additional toolchains\nand dependencies. The [security policy](.github/SECURITY.md) covers code execution.\n\n<!-- BEGIN generated_corpus_at_a_glance -->\n<!-- Generated by scripts/build_corpus_descriptor.py; do not edit this region. -->\n<details>\n<summary>Corpus inventory and counting limits</summary>\n\n## Corpus at a glance\n\nThe reviewed layer a mathematician should judge: 160 curated claim records in 38 contribution families, reaching Lean source through 478 principal declaration links. `docs/SCOPE.md` gives its shape and `docs/RESULTS.md` gives the strongest checked result per problem. The website and the papers are the human reading path.\n\nThe rest is engineering inventory. About 87% of the 164,639 declarations (142,668 across 695 modules) are machine-emitted certificate shards: one integer checked prime, one position excluded. The remainder is not all hand-written either.\n\n| Engineering inventory | Current size |\n|---|---:|\n| Lean modules (the two library roots) | 1,903 |\n| Formal results and supporting lemmas | 160,373 |\n| Curated claim records | 160 |\n| Contribution families | 38 |\n\nGenerated shards are counted as formal source and never as separate\nmathematical claims. Claim records cover every status, including cited and\nopen, and are partitioned once.\nThese are navigation counts, not novelty claims. They still need expert validation.\n\n</details>\n<!-- END generated_corpus_at_a_glance -->\n\n<!-- BEGIN generated_principal_declaration_anchors -->\n<!-- Generated by scripts/build_corpus_descriptor.py; do not edit this region. -->\n## Following a result into Lean\n\nThe paper links each headline result to the relevant source. For a particular\ntopic, start with the [source map](docs/SOURCE_MAP.md); it gives the module\norder without asking you to decode Lean declaration names first.\n<!-- END generated_principal_declaration_anchors -->\n\n<a id=\"citation-and-prior-work\"></a>\n\n## Formal Conjectures contributions\n\n[Five mathematical contributions](docs/verification/FORMAL_CONJECTURES_CROSSWALK.md#merged-mathematical-contributions)\nhave merged into Formal Conjectures. [#6529](https://github.com/google-deepmind/formal-conjectures/pull/6529)\ncovers reciprocal-summable supports for integer bases at least two;\narbitrary infinite supports remain open.\n\n[#6971](https://github.com/google-deepmind/formal-conjectures/pull/6971) repaired four immutable proof links on 11 October.\n[Bo Cowgill's checks and Moritz Firsching's approval](docs/verification/FORMAL_CONJECTURES_CROSSWALK.md#proof-link-maintenance)\nconcern maintenance, with proofs and statements unchanged and no new mathematics.\n[Trace #1041](docs/case-studies/formal-conjectures-1041.md).\n\nOn 12 October, [Bo Cowgill approved #6579](docs/verification/FORMAL_CONJECTURES_CROSSWALK.md#open-requests)\nusing author-hosted verification, without an independent rerun.\nThe request remains open with review required; the unreduced-totient problem remains open.\n\nNineteen [receiving checks](docs/verification/FORMAL_CONJECTURES_CROSSWALK.md#contribution-activity) pass.\nBo approved [PR7260](docs/verification/FORMAL_CONJECTURES_CROSSWALK.md#proof-link-maintenance), [PR6780](docs/verification/FORMAL_CONJECTURES_CROSSWALK.md#open-requests) and [PR7290–7294](docs/verification/FORMAL_CONJECTURES_CROSSWALK.md#open-requests)\nusing author-hosted checks without independent proof replay.\nAll seven await required reviews.\n\nOther checks: [Prove2Me #243/#257 packets](docs/research-commons/README.md#native-prove2me-theorems)\nand [plectis-erdos-lean](https://github.com/wcook04/plectis-erdos-lean).\n\n[Palomar](https://palomar-registry.org/) checks Lean proofs and publishes\nregistered results with their review comments. Our [submission record](https://github.com/wcook04/plectis-erdos-lean#palomar-publication-surface)\ndistinguishes verification, editorial review and registration. The\n[Palomar guide](docs/verification/PALOMAR_QUALIFICATION.md) explains the local\npreparation checks; passing those checks alone does not establish acceptance.\n\nThe [paper catalogue](docs/papers/README.md) links versioned aiXiv editions;\nthe [verification guides](docs/verification/README.md) explain submission and\nregistry records. An archived edition may differ from the current paper.\n\n## Reading routes\n\n[![Website → papers → research records and open questions](.github/system-map.png)](.github/system-map.svg)\n\n## Credit\n\nSome of the biggest changes here came from people telling me what was wrong.\nOne mathematician's comments on the #243 note led to all sixteen problem papers\nbeing rewritten for a first-time reader. The [credit ledger](docs/research-commons/CREDIT_LEDGER.md)\nlists each piece of advice, what it changed and the lines where you can see the\nchange. Names stay withheld until each person confirms they want to be named.\n\n## Citation and licence\n\nUse [CITATION.cff](CITATION.cff) and record the commit used. For a tagged\nrelease, use that tag’s citation file. For mathematics, cite the relevant\n[problem paper](paper/README.md) and the original sources it uses. A repository\ncitation does not replace the original author's credit:\n[prior art](docs/PRIOR_ART.md) and [source attributions](docs/research-commons/SOURCE_ATTRIBUTIONS.md)\ndistinguish earlier results, formalisation and local use.\n\nCode, scripts and documentation are Apache-2.0; manuscript sources and PDFs\nare CC-BY-4.0. [REUSE.toml](REUSE.toml) records exceptions. See also\n[scope](docs/SCOPE.md), [privacy](docs/PRIVACY.md) and the\n[code of conduct](.github/CODE_OF_CONDUCT.md).\n","body_format":"text"},"reading-guide":{"document_id":"reading-guide","source_path":"docs/READING_GUIDE.md","role":"reader_introduction","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/READING_GUIDE.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/READING_GUIDE.md","sha256":"sha256:fed7428c2048d573586a786340882c416ed8cc932be433da756a69cbc9b93eee","byte_count":8906,"origin":"public_main_checkout","body":"<!-- SPDX-FileCopyrightText: 2026 Will Cook -->\n<!-- SPDX-License-Identifier: Apache-2.0 -->\n\n# A reader's way in\n\nThis repository studies eight Erdős problem programmes through short papers,\nlonger research records, Lean proofs of selected statements, finite\ncomputations, and accounts of approaches that stopped. Plectis keeps the work\nopen to further insight. The website brings the collection together; you do\nnot need Lean to read it.\n\nStart with the [#257 introductory paper](../paper/257/erdos257-expository-sums-never-fractions.pdf)\nfor examples and a full proof, or [watch the video](https://www.youtube.com/watch?v=vXgIBy5C5X0).\n\nContinue with the [#257 short paper](../paper/257/erdos-257-mersenne-support-subseries.pdf):\nits checked finite-prime weighted criterion proves irrationality for some\ninfinite supports with divergent reciprocal sum, while arbitrary infinite\nsupport remains open. Then read [#1049](../paper/1049/erdos-1049-rational-base-lambert.pdf):\nirrationality holds in the stated region, including powers of `31/4`; `3/2`\nremains open. The [#1041 paper](../paper/1041/erdos-1041-lemniscate-newton-flow.pdf)\nexamines ani's counterexample.\n\nUsing the degree-seven polynomial constructed by the erdosproblems.com\ncontributor ani, Lean proves that every preconnected strict-lemniscate set\ncontaining two distinct roots has one-dimensional Hausdorff measure greater\nthan two. This refutes the exact Formal Conjectures path-image-length\nstatement; the separate total-variation bound is also checked. The other\nseven targets remain open. Independent human review of correspondence with\nthe 1958 wording has not been recorded. Comparator checks only selected exact\nstatements, axioms and kernel acceptance; it does not assess novelty or\nhistorical correspondence.\n\n## Two ways to begin\n\nYou can follow one question or read across the corpus. The\n[reading edition](reading-edition/README.md) gives a short research instruction\nand the opening of each short paper in one file. For #257, follow the weighted\ntheorem, its averaging proof, the long record and Lean source. Then try the\n[weighted-support examples](../research/experiments/weighted_support_transfer/README.md):\nchange the base, or build a host with three minimal prime witnesses. A failed\nsufficient test proves nothing about rationality. The\n[dyadic shift exercise](../research/experiments/premise_exchange/shift_family_exercise.md)\ntests another change of hypothesis.\n\n## Work through an argument\n\nTry a small example before reading the proof. Note where your approach gets\nstuck, then find the step that overcomes that difficulty. Reconstruct it with\nthe paper closed, or change a hypothesis and see what breaks.\n\nIf you use an agent, you can ask: “Help me work through this theorem. Give me\none hint at a time and wait for my attempt before revealing more.” Name the\npaper, statement and your background. You choose the amount of help. The\n[weighted-support exercise](research-commons/PROVE2ME_WEIGHTED_257_PACKET.md#try-changing-a-hypothesis)\noffers a place to start.\n\n## What is here\n\nEach [problem paper](../paper/README.md) states the question, results and main\narguments. The longer record keeps technical detail, failed routes, experiments\nand surviving obligations. The catalogue also includes system papers on proof\nboundaries, contribution and credit.\n\nThe [results guide](RESULTS.md) states the strongest checked result for each\nproblem next to what still blocks its endpoint. [Prior work](PRIOR_ART.md)\nrecords earlier arguments and attribution. [Scope](SCOPE.md) states the project's\nlimits; the [architecture guide](ARCHITECTURE.md) explains its machinery.\n\n## What formalisation adds\n\nLean checks a precise statement against a precise proof, exposing assumptions,\nquantifiers and dependencies. It does not establish novelty or importance,\nor turn a conditional result into an unconditional one.\nEach paper distinguishes ordinary mathematical arguments, formalised results\nand remaining gaps. Follow a statement's source link and verification record\nto see what has been checked; a build alone does not establish that every\nargument in a paper has been formalised.\n\n## The eight problems in brief\n\nOpen a short paper below, or use the\n[results guide](RESULTS.md#problem-by-problem-guide) for its arguments,\nformalisation and remaining questions.\n\n<!-- BEGIN problem_programme_card -->\n[Problem 68](../paper/68/erdos-68-factorial-denominator-irrationality.pdf). Is the series sum_{n >= 2} 1/(n! - 1) irrational? [Work on this paper](CONTRIBUTE_BY_PAPER.md#problem-68).\n\n[Problem 243](../paper/243/erdos-243-reciprocal-tail-rigidity.pdf). Under a rapid-growth hypothesis on an integer sequence, does rationality of its reciprocal sum force the sequence to satisfy the Sylvester recurrence eventually? [Work on this paper](CONTRIBUTE_BY_PAPER.md#problem-243).\n\n[Problem 249](../paper/249/erdos-249-binary-totient-series.pdf). Is the binary Lambert series sum phi(n)/2^n irrational? [Work on this paper](CONTRIBUTE_BY_PAPER.md#problem-249).\n\n[Problem 251](../paper/251/erdos-251-prime-gap-dyadic-series.pdf). Is the dyadic series of consecutive primes irrational? Equivalently, is the corresponding consecutive-prime-gap dyadic series irrational? [Work on this paper](CONTRIBUTE_BY_PAPER.md#problem-251).\n\n[Problem 257](../paper/257/erdos-257-mersenne-support-subseries.pdf). Is the sum of 1/(2^n-1) over every infinite set of positive exponents irrational? [Work on this paper](CONTRIBUTE_BY_PAPER.md#problem-257).\n\n[Problem 269](../paper/269/erdos-269-three-prime-running-lcm.pdf). For a finite set of at least two primes, is the sum of reciprocals of the running least common multiples of the smooth numbers irrational? This library treats the three-prime case. [Work on this paper](CONTRIBUTE_BY_PAPER.md#problem-269).\n\n[Problem 1041](../paper/1041/erdos-1041-lemniscate-newton-flow.pdf). Must two roots of a monic polynomial in the open unit disc be joined by a sub-two-length curve inside its unit lemniscate? Ani’s degree-seven example refutes the exact Formal Conjectures statement; correspondence with the 1958 wording awaits human review. [Work on this paper](CONTRIBUTE_BY_PAPER.md#problem-1041).\n\n[Problem 1049](../paper/1049/erdos-1049-rational-base-lambert.pdf). For which rational bases is the corresponding series irrational? The first resistant explicit base is three halves. [Work on this paper](CONTRIBUTE_BY_PAPER.md#problem-1049).\n<!-- END problem_programme_card -->\n\nThe [cross-problem paper, Reading Eight Erdős Problems Together](../paper/synthesis/optimal-sparse-perturbations.pdf)\njoins the capacity criterion, Lambert subsums across bases, method\nlimits, exact computations and unsuccessful approaches in one account. The\n[synthesis guide](../paper/synthesis/README.md) gives ways to begin.\n\n## How to read the evidence\n\nThe verification concordance at the end of each problem paper links its\nstatements to Lean proofs and recorded Comparator checks. A dagger identifies\na proof that assumes a named input; pending and partial support are labelled\nexplicitly. These links appear in one place, leaving the mathematical argument\nuninterrupted.\n\nComparator compares selected statements with independently declared formal\ninterfaces under fixed assumptions. It is not peer review and does not\nestablish novelty. The [Palomar guide](verification/PALOMAR_QUALIFICATION.md)\ndistinguishes local packaging from recorded service submission or acceptance.\n\nThis is a self-contained public record, not an entrypoint into any private\ndevelopment system. A finite computation covers its tested range; file and\ndeclaration counts do not measure mathematical importance.\n\n## Reviewing one result\n\nPick a statement in a short paper, read its assumptions and the step that does\nthe work, then follow the companion paper. Missing motivation, compressed hard\nsteps and unclear attribution are useful feedback: a checked proof still needs\nan explanation others can understand and reuse.\n\n## Contributing\n\nAI tools did much of the research, code, and drafting under my direction. I\nremain responsible for the release. [Formal Conjectures reviews](verification/FORMAL_CONJECTURES_CROSSWALK.md#contribution-activity)\ncover selected contributions, not the whole corpus or their novelty.\n\nCorrections, explanations, counterexamples and earlier references are welcome.\nThe [contributor guide](../CONTRIBUTING.md) explains how to return them with\nevidence and credit intact. If this work helps you solve a problem, the solution\nand credit are yours.\n\nAn insight is welcome before it has a formal proof. Will can work with you to\ndevelop the argument and formalise it, with the originating insight credited\nseparately from subsequent exposition and proof work. Clearer understanding,\nreusable methods and new questions across the corpus also count as\ncontributions; they need not close an original problem.\n","body_format":"text"},"weighted-257-task":{"document_id":"weighted-257-task","source_path":"docs/reading-edition/weighted-257-task.md","role":"changed_hypothesis_task","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/reading-edition/weighted-257-task.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/reading-edition/weighted-257-task.md","sha256":"sha256:aef833af3cce15890c08c2a96439fe835fe99f20f78beeea7c33129c1ef4e5cd","byte_count":7432,"origin":"public_main_checkout","body":"# Does the weighted #257 result apply when a hypothesis changes?\n\n<!-- Generated by scripts/build_reading_edition.py; do not edit by hand. -->\n\nSource fingerprint: `f594e7011ab8c87f` (authored regions, claim, boundary, paper and Lean sources).\n\nRead this saved file offline, with or without an assistant. No clone, account or Lean installation is needed. Links identify the public sources when you are online.\n\nYour task: decide what the weighted criterion says for the three cases below. Ask for one hint, a second hint, or the worked answer; an assistant should stop at your chosen depth. Collapsed answers are present in this file for offline reading.\n\n| Case | Changed hypothesis |\n|---|---|\n| 1 | $c=2,p=1,b=2$ |\n| 2 | $c=2,p=1,b=3$ |\n| 3 | $c=2,p=2$: what follows at every integer base $b\\ge2$? |\n\n## Criterion and definitions\n\n$X_A(b)=\\sum_{a\\in A}(b^a-1)^{-1}$ is the support series. $W_{b,P}$ denotes weighted mass, and $v_p(a)$ is the exponent of prime $p$ in $a$.\n\nFor a finite nonempty prime set $P$, let\n$h_P(a)=\\prod_{p\\in P}p^{v_p(a)}$. If $b\\ge2$, $A$ is an infinite set of\npositive integers, and\n\n$$\n\\sum_{a\\in A}\\frac{h_P(a)}{a(b^{h_P(a)}-1)}<\\infty,\n$$\n\nthen $\\sum_{a\\in A}(b^a-1)^{-1}$ is irrational. Separately, if a host $H$\nof positive integers has finite weighted mass at base two for some such $P$,\nthen **every infinite subset** $A\\subseteq H$ has an irrational support sum\nat **every** integer base $b\\ge2$. The paper states both clauses and their\nhereditary consequences at `res:weighted-support` in\n`paper/257/erdos-257-mersenne-support-subseries.tex`.\n\n## Registered claim and source boundary\n\nClaim `finite_prime_weighted_support`: **unconditional progress**, projected from [docs/claims.json](https://github.com/wcook04/plectis-erdos/blob/main/docs/claims.json). This status means the criterion is proved with its stated hypotheses; it does not remove those hypotheses.\n\nFor every integer base b >= 2, every infinite positive support A with finite weighted mass for some finite nonempty prime set P has irrational support series. If a host H of positive integers has finite base-two weighted mass, then every infinite subset A of H has irrational support series at every integer base b >= 2. This is a sufficient support criterion, not the universal statement.\n\nOpen boundary `remaining_open.universal_257_all_infinite_supports`: **open**. Prove irrationality for every infinite support A, not only the named support families formalised here.\n\nExact checked criterion: `ErdosProblems.Erdos257.PaperCompleteR8.divisibilityWeightedClaim` in [lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean:120](https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean#L120). Its conjunction type and weighted definitions are in [AnalyticTargets.lean](https://github.com/wcook04/plectis-erdos/blob/main/lean/ErdosProblems/Erdos257/PaperCompleteR7/AnalyticTargets.lean).\n\nLean proves the conditional criterion. The parameterised family calculation below is an ordinary deduction from the paper's dyadic estimate, not a separately named Lean-checked instance. Finite samples cannot establish these infinite convergence claims.\n\n## Exercise, hints and worked answer\n\nThe paper's example uses the support\n$A_\\star=\\{2^km:k\\ge1,\\ m\\text{ odd},\\ m\\le2^{2^k}\\}$.\nIts reciprocal sum diverges, yet its base-two weighted mass is finite, so\nTheorem 1 applies to every infinite subset at every integer base.\nWhat changes if the odd-factor cutoff in layer $k$ becomes $2^{r_k}$, where\n\n$$\nr_k=\\left\\lceil\\frac{c^{2^k}}{k^p}\\right\\rceil,\n\\qquad c\\ge2\\text{ an integer},\\quad p>0?\n$$\n\nFor $c=2,p=1$, decide whether the $P=\\{2\\}$ weighted test works at bases\n$2$ and $3$. Then change only $p$ to $2$ and decide what Theorem 1 says at\n*every* base. Before opening the calculation, distinguish “this test fails”\nfrom a claim that the corresponding series is rational.\n\n<details>\n<summary>Show one hint</summary>\n\nWrite the weighted mass layer by layer using $P=\\{2\\}$. Compare the odd\nharmonic sum with $r_k$ before deciding convergence. Stop here if you want\nto do the comparison yourself.\n\n</details>\n\n<details>\n<summary>Show a second hint</summary>\n\nUse $r/4\\le S_r\\le r$ and compare $(c/b)^{2^k}/k^p$ with the weighted\nterm. At $c=b$, the exponent cancels and the power $p$ decides the test.\nA divergent sufficient test does not establish rationality.\n\n</details>\n\n<details>\n<summary>Show the calculation and decisions</summary>\n\nPut $A(c,p)=\\{2^km:k\\ge1,\\ m\\text{ odd},\\ 1\\le m\\le2^{r_k}\\}$ and\n$S_r=\\sum_{1\\le m\\le2^r,\\ m\\text{ odd}}1/m$. The dyadic estimate in the\n[paper's example](https://github.com/wcook04/plectis-erdos/blob/main/paper/257/erdos-257-mersenne-support-subseries.tex)\ngives $r/4\\le S_r\\le r$ for $r\\ge1$. Layers are disjoint because $2^km$\nhas exactly $k$ factors of $2$. Hence\n\n$$\n\\sum_{a\\in A(c,p)}\\frac1a=\\sum_{k\\ge1}\\frac{S_{r_k}}{2^k},\n\\qquad\nW_{b,\\{2\\}}(A(c,p))=\\sum_{k\\ge1}\\frac{S_{r_k}}{b^{2^k}-1}.\n$$\n\nThe reciprocal sum diverges for every listed $c,p$: its $k$th layer is at\nleast $c^{2^k}/(4k^p2^k)$, which does not tend to zero. For the weighted\nsum, $r_k$ lies between $c^{2^k}/k^p$ and $c^{2^k}/k^p+1$. The same bounds\non $S_r$, and $b^{2^k}/2\\le b^{2^k}-1\\le b^{2^k}$, bound its $k$th term\nbelow by $(c/b)^{2^k}/(4k^p)$ and above by\n$2(c/b)^{2^k}/k^p+2b^{-2^k}$. Therefore:\n\n| Changed condition | $P=\\{2\\}$ weighted test at base $b$ |\n| --- | --- |\n| $b>c$ | Converges |\n| $b=c$ and $p>1$ | Converges |\n| $b=c$ and $0<p\\le1$ | Diverges |\n| $b<c$ | Diverges |\n\nFor $c=2,p=1$, the test passes at base $3$ and every larger integer base.\nThe fixed-base clause therefore makes $X_B(b)$ irrational for each such\nbase and every infinite $B\\subseteq A(2,1)$. At base $2$, this particular\ntest diverges: it gives **no arithmetic verdict** there. Another witness\nor argument has not been ruled out. When $p=2$, the base-two test passes;\nthe all-base clause makes $X_B(b)$ irrational for every integer $b\\ge2$\nand every infinite $B\\subseteq A(2,2)$.\n\nThis parameterised calculation is an ordinary deduction from the paper's\ncriterion and its dyadic bound. The Lean declarations prove the conditional\ncriterion, not a named theorem about this family. The unrestricted Erdős\n#257 question remains open.\n\n</details>\n\n## A useful next question\n\nCan a different fixed finite prime witness decide the case where this test diverges? Return an argument, obstruction, or precise non-answer with its source and assumptions. Failure of this witness does not settle the universal question.\n\n## Source content identities\n\n- `paper/257/erdos-257-mersenne-support-subseries.tex`: SHA-256 `2676b3e9f6a5ba6f3ea8fceeed9684ff6ced0fa3bef9c3c0b6a3a878288af5d3`\n- `lean/ErdosProblems/Erdos257/PaperCompleteR7/AnalyticTargets.lean`: SHA-256 `7d9cbb3b225cbad70ae1f8165560dc15602525143a64dccf9457259fbbdc9d0a`\n- `lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean`: SHA-256 `c00b39ee574aaec577de2cbc6971c12ca5865c8891f0c8fa88b9ba63335e92a0`\n\n[Public paper text](https://raw.githubusercontent.com/wcook04/plectis-erdos/main/docs/papers/full-text/erdos-257-mersenne-support-subseries.md) · [Author-owned exercise](https://github.com/wcook04/plectis-erdos/blob/main/docs/research-commons/PROVE2ME_WEIGHTED_257_PACKET.md#try-changing-a-hypothesis) · [Return or correct work](https://github.com/wcook04/plectis-erdos/blob/main/CONTRIBUTING.md)\n","body_format":"text"},"paper-guide":{"document_id":"paper-guide","source_path":"paper/README.md","role":"current_paper_catalogue","github":"https://github.com/wcook04/plectis-erdos/blob/main/paper/README.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/paper/README.md","sha256":"sha256:decf75c300deb85716e1726091ca1d547496950636d10c81f4c437f8297a5432","byte_count":16413,"origin":"public_main_checkout","body":"<!-- SPDX-FileCopyrightText: 2026 Will Cook -->\n<!-- SPDX-License-Identifier: Apache-2.0 -->\n\n# Papers\n\n[Browse on the website](https://wcook04.github.io/practical-formal-verification/docs/papers.html) ·\n[Explore the maths map](https://wcook04.github.io/practical-formal-verification/maths/universe.html) ·\n[Read without a clone](../docs/reading-edition/README.md)\n\nEach problem has a short first-read paper and a longer complete reasoning record.\nFor an introduction with worked examples, start with the\n[#257 introductory paper](257/erdos257-expository-sums-never-fractions.pdf).\nThe collection has five complementary roles:\n\n- **[Eight short papers](#problem-papers)** introduce selected results, their\n  arguments and what remains open, one paper for each Erdős problem.\n- **[Eight longer reasoning records](#problem-papers)** preserve detailed\n  proofs, computations, attempted approaches and the obstructions they met.\n- **[Reading Eight Erdős Problems Together](#reading-the-eight-together)**\n  develops the mathematical connections and limits of shared methods.\n- **[A Repository-Based System for Research and Publication](#project-papers)**\n  explains how the arguments, evidence, explanations and contributions fit\n  together. This is the main systems paper.\n- **[The writing guides](#project-papers)** explain how mathematical papers are\n  structured and revised using literature and feedback.\n\nYou do not need Lean or a coding agent to read the papers. PDFs and LaTeX\nsources are paired below; the [website catalogue](https://wcook04.github.io/practical-formal-verification/docs/papers.html)\noffers browser reading alongside the downloads.\n\n## What the papers establish\n\nUsing the degree-seven polynomial constructed by the erdosproblems.com\ncontributor ani, Lean proves that every preconnected strict-lemniscate set\ncontaining two distinct roots has one-dimensional Hausdorff measure greater\nthan two. This refutes the exact Formal Conjectures path-image-length\nstatement; the separate total-variation bound is also checked. The other\nseven targets remain open. Independent human review of correspondence with\nthe 1958 wording has not been recorded. Comparator checks only selected exact\nstatements, axioms and kernel acceptance; it does not assess novelty or\nhistorical correspondence. The papers report partial results, failed or\nequivalent routes, finite evidence, and the exact obligations that survive.\nIf the problem numbers or formalisation are unfamiliar, read\n[a reader's way in](../docs/READING_GUIDE.md) first. For the design of the tools\nand the research process, go to the [project papers](#project-papers).\n\n## Problem papers\n\n[#68](#erdős-68) · [#243](#erdős-243) · [#249](#erdős-249) · [#251](#erdős-251) · [#257](#erdős-257) · [#269](#erdős-269) · [#1041](#erdős-1041) · [#1049](#erdős-1049)\n\n### Erdős #68\n\n**Short paper**<br>\n[Integer Linear Forms for a Factorial Reciprocal Series](68/erdos-68-factorial-denominator-irrationality.pdf) ([source](68/erdos-68-factorial-denominator-irrationality.tex))\n\n**Complete reasoning record**<br>\n[Factorial Linear Forms and Denominators: Detailed Proofs and Rationality Criteria](68/erdos68-factorial-reasoning-surface.pdf) ([source](68/erdos68-factorial-reasoning-surface.tex))\n\n### Erdős #243\n\n**Short paper**<br>\n[Reciprocal Sums and the Sylvester Recurrence](243/erdos-243-reciprocal-tail-rigidity.pdf) ([source](243/erdos-243-reciprocal-tail-rigidity.tex))\n\n**Complete reasoning record**<br>\n[Reciprocal Sums and the Sylvester Recurrence: Further Results and Proofs](243/erdos243-reciprocal-tail-reasoning-surface.pdf) ([source](243/erdos243-reciprocal-tail-reasoning-surface.tex))\n\n### Erdős #249\n\n**Short paper**<br>\n[Integral Relations among Totient Sections](249/erdos-249-binary-totient-series.pdf) ([source](249/erdos-249-binary-totient-series.tex))\n\n**Complete reasoning record**<br>\n[The Binary Totient Series](249/erdos249-totient-reasoning-surface.pdf) ([source](249/erdos249-totient-reasoning-surface.tex))\n\n### Erdős #251\n\n**Short paper**<br>\n[Sparse Congruence-Preserving Perturbations of Dyadic Series](251/erdos-251-prime-gap-dyadic-series.pdf) ([source](251/erdos-251-prime-gap-dyadic-series.tex))\n\n**Complete reasoning record**<br>\n[Prime-Gap Dyadic Series: Perturbations, Exact Criteria and Certificates](251/erdos251-prime-gap-reasoning-surface.pdf) ([source](251/erdos251-prime-gap-reasoning-surface.tex))\n\n### Erdős #257\n\n**Introductory paper.**\n[Why certain sums of 1/(2ⁿ−1) are irrational](257/erdos257-expository-sums-never-fractions.pdf) ([source](257/erdos257-expository-sums-never-fractions.tex)).\nWorked examples lead through a complete proof; an appendix gives supporting details, exercises and the link to the formal proof.\n[Watch the #257 video](https://www.youtube.com/watch?v=vXgIBy5C5X0).\n\n**Short paper**<br>\n[Irrationality criteria for Lambert subseries](257/erdos-257-mersenne-support-subseries.pdf) ([source](257/erdos-257-mersenne-support-subseries.tex))\n\n**Complete reasoning record**<br>\n[Reciprocal Mersenne Subseries](257/erdos257-mersenne-reasoning-surface.pdf) ([source](257/erdos257-mersenne-reasoning-surface.tex))\n\n### Erdős #269\n\n**Short paper**<br>\n[Distinct running least common multiples](269/erdos-269-three-prime-running-lcm.pdf) ([source](269/erdos-269-three-prime-running-lcm.tex))\n\n**Complete reasoning record**<br>\n[Running least common multiples: distinct values and multiplicities](269/erdos269-running-lcm-reasoning-surface.pdf) ([source](269/erdos269-running-lcm-reasoning-surface.tex))\n\n### Erdős #1041\n\n**Short paper**<br>\n[Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections](1041/erdos-1041-lemniscate-newton-flow.pdf) ([source](1041/erdos-1041-lemniscate-newton-flow.tex))\n\n**Complete reasoning record**<br>\n[Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections](1041/erdos1041-lemniscate-reasoning-surface.pdf) ([source](1041/erdos1041-lemniscate-reasoning-surface.tex))\n\n### Erdős #1049\n\n**Short paper**<br>\n[Hankel Determinants of Geometric Moments and Rational Lambert Values](1049/erdos-1049-rational-base-lambert.pdf) ([source](1049/erdos-1049-rational-base-lambert.tex))\n\n**Complete reasoning record**<br>\n[Geometric Moments and Rational Lambert Values: Proofs and Further Results](1049/erdos1049-rational-base-lambert-reasoning-surface.pdf) ([source](1049/erdos1049-rational-base-lambert-reasoning-surface.tex))\n\nThe #243 short paper leads with irrationality under the cubic rate\n`a_n²/a_(n+1) = 1 + 3/n + o(n⁻³)` for strictly increasing positive integer\nsequences. Its bounded-increment criterion for an eventual Sylvester tail is\na second result; the unrestricted problem remains open. The #1049 short\npaper proves a rational-base sufficient region using Zudilin's forms after\ncyclotomic cancellation and denominator accounting. Its region contains\n`31/4`, which is outside the earlier Bundschuh–Väänänen sufficient region;\n`3/2` remains open.\n\n## Reading the eight together\n\nOne cross-problem paper develops the mathematics that arises from reading the\nprogrammes together. For `c>0` and integers `d≥1`, it considers entire functions\n`f(z) = ∑_(n≥1) e_n z^n/n!` with nonnegative integer `e_n` eventually bounded\nby `n^c` and, for each fixed positive integer `q`, eventually divisible by `q`.\nThe set of derivative-value vectors at `1` has Hausdorff dimension `min(c,d)` and\ncontains an open set exactly when `c>d`. This result has an ordinary proof with the formal ingredients\nidentified separately. The paper also contains the capacity and congruence\ncriterion, Lambert-subsum results, exact computations and method limits.\n\n[Reading Eight Erdős Problems Together](synthesis/optimal-sparse-perturbations.pdf)\n([source](synthesis/optimal-sparse-perturbations.tex)).\n\n## Follow the argument into its evidence\n\nRead the assumptions of the result you want to use, then the proof. The\nlonger record gives the surrounding working context, including routes that\nfailed, finite experiments and open obligations. If you cannot reconstruct a\nstep or find its source, that is useful feedback through\n[Contributing](../CONTRIBUTING.md).\n\nSome results are checked in Lean; others are ordinary mathematical arguments\nor applications of cited external theorems. Each paper states the boundary for\nits own claims. [Results and limits](../docs/RESULTS.md) gives the result beside\nwhat remains open, and the [source map](../docs/SOURCE_MAP.md) connects the\npaper to supporting declarations. Lean files are proof authority only for the\nexact declarations they check; [`docs/claims.json`](../docs/claims.json)\nrecords the selected public claim interfaces.\n\nThe [full-text index](../docs/papers/README.md) provides generated Markdown\nversions for browsing. The PDFs and `.tex` files above are the authored\nmanuscripts. The older joint #249/#257 paper is retained for provenance;\nit is not the entry point for either problem.\n\n## Project papers\n\nStart with [A Repository-Based System for Research and Publication](systems/claim-faithful-publication-systems-paper.pdf)\n([source](systems/claim-faithful-publication-systems-paper.tex)). This is the main\nsystems paper: it follows a result from its mathematical argument through\nformal support, written explanation, review and contribution.\n\nThe accompanying writing guides describe the exposition process:\n[Writing a Good Mathematical Paper](exposition/writing-a-good-mathematical-paper.pdf)\nis the compact guide; [Writing Mathematics from the Literature and Reviewed Revisions](exposition/writing-mathematics-from-reviewed-revisions.pdf)\nrecords the method and its sources. Their LaTeX sources are in\n[exposition/](exposition/).\n\n<details>\n<summary>Earlier systems papers: historical background</summary>\n\nTwo earlier papers are retained as historical background. Their account is\nsuperseded by the main paper; their observations and cross-references belong\nto the revisions they describe.\n\n| Earlier paper | Detail retained |\n|---|---|\n| [From a Cold Clone to a Proof Receipt](systems/cold-clone-to-proof-receipt.pdf) ([source](systems/cold-clone-to-proof-receipt.tex)) | Navigation, recorded proof checks and incremental validation. |\n| [From Spare Compute to Cumulative Mathematics](systems/open-source-mathematics-strategy.pdf) ([source](systems/open-source-mathematics-strategy.tex)) | Contribution protocol, compute, credit and governance. |\n\n</details>\n\nFor current use, follow [the reading guide](../docs/READING_GUIDE.md),\n[agent instructions](../AGENTS.md) or [Contributing](../CONTRIBUTING.md).\n\nFor the repository layout, sources of truth, build path, and release\ninfrastructure, see [`docs/ARCHITECTURE.md`](../docs/ARCHITECTURE.md).\n\n## Sources and earlier manuscripts\n\nThe manuscript layer (the `.tex` sources and rendered PDFs) is licensed\nCC-BY-4.0; see [`REUSE.toml`](../REUSE.toml) at the repository root.\n\n<details>\n<summary>Source note for the #251 sparse construction</summary>\n\nFor #251, [the results guide](../docs/RESULTS.md#strongest-checked-results-by-problem)\nidentifies the formal sparse construction and the differences in quantifier\norder between it and the printed proof. The short paper's attribution section\nexplains those differences before its source index.\n\n</details>\n\n<details>\n<summary>Earlier combined manuscript: provenance only</summary>\n\nThe older joint #249/#257 paper is retained for provenance; it is not the\nentry point for either problem. Its arguments and release history can be\ninspected in the [archived PDF](archive/erdos249-257-main-paper.pdf) and\n[LaTeX source](archive/erdos249-257-main-paper.tex). Use the individual papers\nabove for the current work.\n\n</details>\n\n<details>\n<summary>Commands for agents following a paper into the source</summary>\n\n### Returning from a problem note to checked evidence\n\nThe notes are exposition, not proof authority.  To return from any note to the\nmachine-owned problem record, run the matching route below.  Each problem route\nreturns its exact paper/source record, checked module inventory, and open\nobligation handles; the [complete eight-problem source map](../docs/SOURCE_MAP.md#complete-eight-problem-return-matrix)\nkeeps the same joins readable.  For a source-fingerprinted continuation packet,\nuse the route-memory command in the last column.\n\n| Problem | Public problem route | Resumable route-memory handoff |\n|---|---|---|\n| #68 | `python3 scripts/query_corpus.py --route erdos_68` | `python3 scripts/query_route_memory.py --problem 68` |\n| #243 | `python3 scripts/query_corpus.py --route erdos_243` | `python3 scripts/query_route_memory.py --problem 243` |\n| #249 | `python3 scripts/query_corpus.py --route erdos_249` | `python3 scripts/query_route_memory.py --problem 249` |\n| #251 | `python3 scripts/query_corpus.py --route erdos_251` | `python3 scripts/query_route_memory.py --problem 251` |\n| #257 | `python3 scripts/query_corpus.py --route erdos_257` | `python3 scripts/query_route_memory.py --problem 257` |\n| #269 | `python3 scripts/query_corpus.py --route erdos_269` | `python3 scripts/query_route_memory.py --problem 269` |\n| #1041 | `python3 scripts/query_corpus.py --route erdos_1041` | `python3 scripts/query_route_memory.py --problem 1041` |\n| #1049 | `python3 scripts/query_corpus.py --route erdos_1049` | `python3 scripts/query_route_memory.py --problem 1049` |\n\n\n</details>\n\n## Build and update\n\nRun from the repository root, with Tectonic or a TeX Live installation:\n\n```sh\nmake -C paper\n```\n\nThe Makefile builds the manuscripts registered in\n[`docs/publication_contract.json`](../docs/publication_contract.json) and\ncopies the PDFs into their problem or systems directory under `paper/`.\nThe temporary build PDFs in `paper/` are not the published copies.\n\nTo rebuild one paper, run `make -C paper <stem>.pdf` and then\n`python3 scripts/sync_publication_pdfs.py`, which copies the fresh PDF into\nplace. Each copy is recorded in [`build-manifest.json`](build-manifest.json)\nwith the digest of the PDF and of every TeX input it was compiled from, and a\nbuild output older than one of its inputs is refused. The release checks fail\nwhen a committed PDF is not the recorded build of its committed inputs.\n\nThe problem PDFs also depend on their generated evidence links. Tectonic keeps\nthe `.aux` files so that, after a layout change, the evidence builder can read\nthe current result numbers and pages:\n\n```sh\npython3 scripts/paper_evidence.py build --corpus-repo /path/to/plectis-erdos-lean \\\n  --aux-dir paper --aux-paper <paper-id>\n```\n\nReview and commit changed evidence records before pointing that paper's\n`record_commit_overrides` entry in `evidence/config.json` at the new commit.\nRun the evidence builder again, rebuild the affected PDF and synchronize it.\n`python3 scripts/check_paper_evidence_pdfs.py` checks the final verification concordance\nand its links against the result headings and pages in the resulting PDF; `make -C paper check` includes\nthis check. It needs the dependencies in `scripts/requirements-release.txt`.\n\nAfter editing a manuscript, rebuild its PDF before updating its recorded\ndigests. The following command previews digest changes; add `--apply` only\nafter reviewing the source and rebuilt PDF:\n\n```sh\npython3 scripts/check_publication_contract.py --restamp\n```\n\nThe statement links inside a paper identify the exact proof-source revision\nused for that paper. They serve reproducibility. They are not a reason to\nkeep an older manuscript on the reading path. When the source changes, update\nthe explanation and source links together, rebuild the PDF, and check its\ncoverage against the current proofs:\n\n```sh\npython3 scripts/check_problem_note_sources.py --coverage\npython3 docs/papers/check_paper_corpus.py\n```\n\nThe last check rejects a Markdown mirror or recorded PDF that no longer\nmatches its manuscript. Include the edited source, rebuilt PDF and check\noutput in your pull request. If the remaining failure is a generated copy\nmade stale by your manuscript edit, say so in the pull request; do not\nhand-edit the copy to make the check pass.\n\nRefresh generated full text and paper-corpus records with\n`python3 docs/papers/refresh_paper_corpus.py --write`, then run\n`python3 scripts/refresh_projections.py` and\n`python3 docs/papers/check_paper_corpus.py` before merging. These owners are\nincluded in the public checkout; do not edit their generated output by hand.\n","body_format":"text"},"documentation":{"document_id":"documentation","source_path":"docs/README.md","role":"documentation_index","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/README.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/README.md","sha256":"sha256:2b18860351eefb494e70e1711f3a9f30a2e5895382cb14835c38369b44819b38","byte_count":8600,"origin":"public_main_checkout","body":"<!-- SPDX-FileCopyrightText: 2026 Will Cook -->\n<!-- SPDX-License-Identifier: Apache-2.0 -->\n\n# Reading and working with the research\n\nThe [front page](../README.md) introduces the project. This index helps you\nchoose what to read or do next, whether you want to understand a result, check\nits proof, continue the research, or inspect how the tools work.\n\nUsing the degree-seven polynomial constructed by the erdosproblems.com\ncontributor ani, Lean proves that every preconnected strict-lemniscate set\ncontaining two distinct roots has one-dimensional Hausdorff measure greater\nthan two. This refutes the exact Formal Conjectures path-image-length\nstatement; the separate total-variation bound is also checked. The other\nseven targets remain open. Independent human review of correspondence with\nthe 1958 wording has not been recorded. Comparator checks only selected exact\nstatements, axioms and kernel acceptance; it does not assess novelty or\nhistorical correspondence. The papers report intermediate results and the\napproaches that stopped, with enough of the record preserved for somebody\nelse to inspect the argument and continue from it.\n\n## Read on the website\n\nThe [mathematics overview](https://wcook04.github.io/practical-formal-verification/maths/) introduces\nthe eight problems. The [interactive maths map](https://wcook04.github.io/practical-formal-verification/maths/universe.html)\nconnects papers, results and recorded evidence; the\n[paper catalogue](https://wcook04.github.io/practical-formal-verification/docs/papers.html) provides\nbrowser reading and PDFs. This index gives the corresponding routes inside a\nclone. No website account or coding agent is needed to read the work.\n\n## Choose a way in\n\n| What you want to do | Start here | Where it takes you |\n|---|---|---|\n| Understand the project without installing anything | [A reader's way in](READING_GUIDE.md) | The eight questions, what formalisation adds, and how to read the evidence. |\n| Read the mathematics | [The papers](../paper/README.md) | A short paper for each problem, then a longer record when you need the details. |\n| Read the comparison across problems | [The cross-problem paper](../paper/synthesis/optimal-sparse-perturbations.pdf) | Capacity and congruences, Lambert subsums, method obstructions and their full research record. |\n| Find what has been established and what is missing | [Results and limits](RESULTS.md) | The results beside their remaining open questions, with routes to the evidence. |\n| See what is open and what would settle it | [The argument graph](ARGUMENT_GRAPH.md) | Every conditional theorem read out of the Lean kernel: what is proved, what reduces to what, which open statements are the same problem in other coordinates, and what a missing input would settle. |\n| Check a result yourself | [Reproducibility](REPRODUCIBILITY.md) | Inspect one claim without Lean, then install the pinned environment if you want to rebuild proofs. |\n| Continue the work or send a correction | [Contributing](../CONTRIBUTING.md) | A plain-language issue or a focused pull request, with evidence and credit. |\n| Understand the software and research process | [How this repository works](ARCHITECTURE.md) | The roles of proofs, claim records, papers, query tools and release checks. |\n\n## Read\n\nChoose a statement in a [short paper](../paper/README.md#problem-papers).\nRead its assumptions and argument, then use the longer record for a compressed\nstep, an earlier attempt, or the surrounding computations. [Prior work](PRIOR_ART.md)\nhelps with attribution; [related problems](RELATED_PROBLEMS.md) follows\nconnections elsewhere in the collection.\n\nThe [source map](SOURCE_MAP.md) locates the supporting Lean declarations and\npaper passages. Each paper says which results have a Lean proof and which use\nan ordinary mathematical argument or a cited theorem. The\n[claim record](claims.json) gives the selected public statements, their status\nand their exact remaining-open propositions. [Scope](SCOPE.md) records the\nrelease boundary, and [methodology](METHODOLOGY.md) explains the rules for\nchanging a claim.\n\n## Check\n\nTo replay a check, follow [reproducibility](REPRODUCIBILITY.md). For selected\nstatements, [external verification](EXTERNAL_VERIFICATION.md) also provides\nComparator interfaces: separately declared formal statements that can be\ncompared with the development. The [verification guides](verification/README.md)\nexplain those checks and their limits.\n\n## Contribute\n\n[Work on a paper](CONTRIBUTE_BY_PAPER.md) brings its question, sources and\ncontribution instructions together, one programme at a time.\n\nA contribution can be a correction, clearer explanation, earlier reference,\ncounterexample, or useful failed approach. [Contributing](../CONTRIBUTING.md)\nexplains what to send back. The [research commons](research-commons/README.md)\ndescribes how longer investigations retain their starting point, evidence and\n[credit](research-commons/CREDIT_POLICY.md).\n\nFor an AI-assisted session, use the [research-shift guide](FRONTIER_RELAY.md).\nGive a coding agent [AGENTS.md](../AGENTS.md) so it can find the relevant\nworkflow. The [agent guides](agents/README.md) explain the tools once you have\na question to work on. You can also contribute to navigation, validation or\nthe contributor experience through the\n[architecture contribution path](research-commons/ARCHITECTURE_CONTRIBUTIONS.md).\n\n## Where things live\n\n| Folder | What belongs there |\n|---|---|\n| [Papers](../paper/README.md) | Current PDFs and manuscript sources, grouped by problem. |\n| [Paper full text](papers/README.md) | Generated Markdown versions and the detailed paper inventory. |\n| [Agent guides](agents/README.md) | Query tools, proof tools and instructions for making changes. |\n| [Verification guides](verification/README.md) | Replay instructions, statement comparisons and submission requirements. |\n| [Technical reference](reference/README.md) | Research history, correction records and indexing details. |\n| [Research commons](research-commons/README.md) | Contribution, review and attribution records. |\n| [Semantic index](semantic/README.md) | Technical navigation through formal statements and their recorded relationships. |\n| [Primary sources](primary-sources/) | Source provenance and redistribution records. |\n| [Measurements](measurements/) | Recorded measurements of the tools. |\n| [Release records](release/) | Audits and release identity records. |\n\nThe main reading guides stay directly under `docs/`. The JSON files beside\nthem supply the query tools and website. For example, `claims.json` records\npublic claim status and `methodology.json` records review rules; the other\nindexes help locate evidence. You can read the papers without opening these\nfiles. [Generated technical navigation](ORIENTATION.md) is a compact entry for\nreaders who want to use the indexes.\n\nOutside `docs/`, the [repository map](../README.md#repository-map) distinguishes\nthe proof libraries, manuscripts and working records. Folder indexes cover\n[experiments](../research/README.md), [returned research](../research_corpus/README.md),\n[tools](../scripts/README.md) and [formal verification files](../verification/README.md).\nThe [naming conventions](ARCHITECTURE.md#directory-and-naming-conventions)\nexplain why similarly named source and documentation folders have different\nroles.\n\n## Files at the repository root\n\n`README.md` introduces the work and `CONTRIBUTING.md` explains how to help.\n`AGENTS.md` is the shared entry for coding agents; `CLAUDE.md` loads it for\nClaude Code. The detailed rules live in the [agent guide](agents/AGENT_GUIDE.md).\n\n`lakefile.toml`, `lake-manifest.json` and `lean-toolchain` configure the Lean\nproject and pin its dependencies. `formalization.yaml` is the Comparator\nmanifest for selected statements. `CITATION.cff` supplies citation metadata;\n`LICENSE`, `LICENSES/` and `REUSE.toml` record the licences.\n\nThe [code of conduct](../.github/CODE_OF_CONDUCT.md),\n[security policy](../.github/SECURITY.md), CI and contribution forms live under\n`.github/`. The [privacy policy](PRIVACY.md) stays here with the other guides.\n\n## Improving these guides\n\nGive each guide a reader's question to answer, and make its index link say\nwhat the reader will learn or be able to do. Define terms where they first\nmatter, and check examples from the repository root. Keep mathematical\nhypotheses, evidence and open boundaries exact when changing the prose.\nFor a generated page, change its source and run its owning builder;\n[Contributing](../CONTRIBUTING.md) explains how to return the improvement.\n","body_format":"text"},"results":{"document_id":"results","source_path":"docs/RESULTS.md","role":"mathematical_results_map","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/RESULTS.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/RESULTS.md","sha256":"sha256:4b2947142ea31b0a1aeb69a516296320d7d579cfd6ac739da4adc9a319bc934f","byte_count":141962,"origin":"formal_claim_control_plane","already_carried_in":"scholarly_maps.formal_claim_control_plane[record_id=lean_results_map]"},"orientation":{"document_id":"orientation","source_path":"docs/ORIENTATION.md","role":"generated_reader_map","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/ORIENTATION.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/ORIENTATION.md","sha256":"sha256:1534ad9156fa5c42e8cce3f130ffae7c81f0e5fb09f909b3b941cb6ec5bbf325","byte_count":21639,"origin":"public_main_checkout","body":"<!--\nSPDX-FileCopyrightText: 2026 Will Cook\nSPDX-License-Identifier: Apache-2.0\n\nGenerated by scripts/build_corpus_descriptor.py. Do not edit by hand.\n-->\n\n# Corpus orientation\n\nThis is the bounded first-read map for the public release. It is a navigation\nprojection, not proof authority. The published Lean source checked by the pinned\nLean kernel remains proof authority.\n\n## Release provenance\n\nThis checkout is a deliberately curated, independently verifiable public output of a larger ongoing formal-mathematics workflow. No public claim depends on private or unreleased work.\n\nPublic evidence of that release workflow: pinned formal-source identity; claim and open-proposition registry; machine-readable argument and module graphs; generated declaration atlas; bounded cold-clone and adversarial checks.\n\nThe workflow provenance explains the release discipline; it does not imply hidden proof authority or equivalence with private work.\n\n## Mathematical signal first\n\nThis is Palomar's canonical `candidate_ranking` order, joined to the\nsource declaration and exact boundary in the external-verification\npacket. It is a reader-priority signal, not a proof, novelty, or\nproblem-closure claim; every ranked family remains reachable below.\n\n| Rank | Family / status |\n|---:|---|\n| 1 | `finite_prime_weighted_support` / `selected` |\n| 2 | `known_irrational_supports` / `represented` |\n| 3 | `pairwise_coprime_support` / `represented` |\n| 4 | `orthogonal_petal_sunflower_reduction` / `represented` |\n| 5 | `periodic_nonnegative_weight_irrationality` / `represented` |\n| 6 | `actual_lcm_orbit_separation` / `represented` |\n| 7 | `first_harmonic_pivot_decomposition` / `represented` |\n| 8 | `strict_prime_tail_orbit_gap` / `represented` |\n| 9 | `factorial_carry_characterisation` / `subordinate` |\n| 10 | `prime_gap_reformulation` / `subordinate` |\n| 11 | `totient_carry_anti_compression` / `represented` |\n| 12 | `half_membership_seam_classification` / `represented` |\n| 13 | `negative_mass_recovery` / `represented` |\n\nInterface, source declaration, and exact boundary, by rank:\n\n1. `finite_prime_weighted_support`\n   `Erdos249257.ExternalVerification.divisibilityWeightedClaim` → `ErdosProblems.Erdos257.PaperCompleteR8.divisibilityWeightedClaim`\n   The finite-prime weighted-summability premise is explicit in both clauses. This criterion does not cover every infinite support or settle universal Erdős #257; no novelty, priority, significance, or external human-review claim is made.\n2. `known_irrational_supports`\n   `Erdos249257.ExternalVerification.irrational_erdosSum_full_support` → `Erdos249257.irrational_erdosSum_full_support`\n   This is the classical full-support theorem, not universal Erdos #257.\n3. `pairwise_coprime_support`\n   `Erdos249257.ExternalVerification.irrational_erdosSupportSeries_pairwise_coprime` → `Erdos249257.irrational_erdosSupportSeries_pairwise_coprime`\n   This is the pairwise-coprime support theorem, not universal Erdos #257; the infinitude, pairwise-coprimality, and summable-reciprocal hypotheses remain explicit.\n4. `orthogonal_petal_sunflower_reduction`\n   `Erdos249257.ExternalVerification.irrational_erdosSupportSeries_of_orthogonalPetalBouquet` → `Erdos249257.SupportSunflowerDichotomy.irrational_erdosSupportSeries_of_orthogonalPetalBouquet`\n   This is a conditional reduction: the bouquet structure and uniform tail-selection predicate remain hypotheses; neither is constructed for arbitrary supports, so this does not add an unconditional Erdos #257 support family or solve the universal problem.\n5. `periodic_nonnegative_weight_irrationality`\n   `Erdos249257.ExternalVerification.irrational_ratWeightSeries_eventuallyPeriodic` → `Erdos249257.irrational_ratWeightSeries_eventuallyPeriodic`\n   This is the nonnegative rational eventually-periodic coefficient theorem; its nonnegativity, eventual periodicity, and positive-tail hypotheses remain explicit, and the broader mixed-sign periodic theorem is not claimed.\n6. `actual_lcm_orbit_separation`\n   `Erdos249257.ExternalVerification.irrational_totientSeries_of_actualLcmOrbitSeparationSupply` → `Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_actualLcmOrbitSeparationSupply`\n   The cofinal separation supply is an open premise, stronger than mere non-integrality; this sufficient route is not an equivalence and leaves the punctured/top-edge boundary and the Erdős #249 endpoint open.\n7. `first_harmonic_pivot_decomposition`\n   `Erdos249257.ExternalVerification.irrational_totient_series_of_pivotResidualDecorrelation` → `Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_pivotResidualDecorrelation`\n   The four-term decomposition and 9X/10 gap remain conditional on the explicit dyadic-room and PivotBudgetAt hypotheses; the cofinal decorrelation producer is unproved. The fibre/image lemmas do not give global supplier-prime isolation or a prime-distribution estimate, and no Erdős #249 irrationality conclusion is claimed.\n8. `strict_prime_tail_orbit_gap`\n   `Erdos249257.ExternalVerification.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap` → `ErdosProblems.Erdos249.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap`\n   The phase-density, cofinal natural-prime strict-gap, associated uniform-margin, and cofinal nonpositive-prime-shift producers remain unproved for the actual totient orbit. Exact phase/integrality equivalence, finite dyadic-root exclusion for 1 ≤ h ≤ 16 and k ≤ 14, and non-dyadicity-to-cofinal-nonpositive-phase implications are classification or finite evidence only; they do not supply density or prime alignment, and the source does not prove actual root entry. This is a conditional reduction only: no unconditional Erdős #249 endpoint, novelty, priority, significance, or external-review claim is made. It is distinct from the finite first_harmonic_pivot_decomposition and actual_lcm_orbit_separation families.\n9. `factorial_carry_characterisation`\n   `Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_nonunit_carries` → `ErdosProblems.Erdos68.irrational_factorialGapSeries_iff_cofinal_nonunit_carries`\n   The equivalence does not supply a cofinal carry producer.\n10. `prime_gap_reformulation`\n   `Erdos249257.ExternalVerification.irrational_tsum_primeDyadicTerm_iff_primeGap` → `ErdosProblems.Erdos251.irrational_tsum_primeDyadicTerm_iff_primeGap`\n   The equivalence does not prove irrationality of either series.\n11. `totient_carry_anti_compression`\n   `Erdos249257.ExternalVerification.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank` → `Erdos249257.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank`\n   This is a necessary consequence of rationality only: it gives no contradiction, no finite-rank upper bound for a rationality-supplied carry, and no irrationality proof for Erdős #249. Modulo a divisor of the multiplier, the forcing disappears and the carry becomes geometric, so quotient periodicity alone supplies no finite-ℚ-rank upper bound. The coefficient-side kernel-rank and certificate-equivalence families are not duplicated, and #249 remains open.\n12. `half_membership_seam_classification`\n   `Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse` → `Erdos249257.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse`\n   This exact endpoint-socket classification does not supply any unbounded terminal-false or upper-or-middle occurrences, cofinal supply, skipped-support producer, or half-membership witness; finite exact rows do not establish the required global supply. It is not an unconditional or universal Erdős #257 result, and no novelty, priority, significance, or external-review claim is made.\n13. `negative_mass_recovery`\n   `Erdos249257.ExternalVerification.sylvesterNext_eventually_of_summable_negativeRelativeMass` → `ErdosProblems.Erdos243.sylvesterNext_eventually_of_summable_negativeRelativeMass`\n   This is a conditional recovery criterion: the source theorem assumes every displayed dynamics, positivity, strict centered-step, normalized-vanishing, and summability hypothesis, and does not prove them for the original Erdős #243 orbit. Any surviving canonical orbit must have divergent normalized negative mass. It supplies no prime-specific producer, unconditional #243 endpoint, or reciprocal-tail irrationality, and makes no novelty, priority, significance, or external-review claim. It is distinct from centered_state_dynamics, which uses a uniform lower-bound hypothesis rather than summability of normalized negative mass.\n\n### Reader tiers and relational boundaries\n\nPalomar's presentation tiers are the reader routing contract (not a second rank or proof-status authority):\n`source-ranked frontier` → `conditional endpoint leverage` → `deep mechanism and classification` → `natural friction and no-go` → `supporting and long tail`\n\nExact family-to-tier placements, open producers, and contrary evidence remain in `docs/orientation.json` under the same Palomar authority.\n\n## What a claim status asserts\n\nA status states the exact public evidence claim, not a priority or novelty claim.\nThe prior-art record, not this table, supplies antecedents.\n\n| Status | Exact public meaning |\n|---|---|\n| proved here | Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied |\n| formalised here | Known mathematics represented and checked in Lean |\n| unconditional progress | Genuine theorem that does not settle the open problem |\n| conditional reduction | Exact implication whose remaining hypothesis is open |\n| verified finite instance | Kernel-checked computation at a bounded range |\n| cited only | External theorem used for positioning, not formalised |\n| open | Not proved by this release |\n| formal statement refuted | Lean refutes the exact Formal Conjectures statement; correspondence with the original wording awaits independent review |\n\n## Exact open boundary\n\n- `remaining_open.erdos_68_irrationality` — Prove that S = ∑_{n≥2} 1/(n! − 1) is irrational.\n- `remaining_open.erdos_249_irrationality` — Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator.\n- `remaining_open.erdos_251_irrationality` — Prove that Π = ∑_{n≥1} p_n/2ⁿ is irrational; the exact tail-shift equivalence and prime-gap constraints do not supply the required cofinal escape.\n- `remaining_open.unbounded_certificate_supply` — Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.\n- `remaining_open.half_value_membership` — Decide whether 1/2 lies in the Mersenne achievement set. Yes refutes universal #257; no closes only this route.\n- `remaining_open.twenty_one_permanent_affine_supercapacity` — Decide the exact 1/21 branch: equivalently, exclude the fatal/cofinite/aligned branch, contradict its eventual permanent affine-supercapacity recurrence, prove cofinally bounded scaled greedy remainders, or force an unbounded sequence of closed canonical quotient rows. A positive membership result would refute universal #257; no such producer is proved.\n- `remaining_open.universal_257_all_infinite_supports` — Prove irrationality for every infinite support A, not only the named support families formalised here.\n- `remaining_open.erdos_269_three_prime_irrationality` — Determine irrationality of the running-LCM reciprocal sum for finite prime sets of size at least three. For {2,3,5}, the rationality-to-carry bridge is established and cofinal local-window escape remains open.\n- `remaining_open.erdos_1041_lemniscate_connection` — Independently review whether the 1958 curve-length wording matches the exact Formal Conjectures path-image-length statement, whose negation is now Lean-checked using ani's explicit degree-seven polynomial. The checked one-dimensional Hausdorff-measure bound settles that formal statement negatively; the reported small-parameter family is not formalised.\n- `remaining_open.erdos_1049_irrationality` — Determine whether F(t) = ∑_{n≥1} 1/(tⁿ − 1) is irrational for every rational t > 1. Irrationality of F((31/4)^r) for every positive integer r is a Lean-checked Zudilin specialization; the 3/2 approximant with remainder control and the universal assertion remain open.\n- `remaining_open.erdos_243_eventual_recurrence` — Prove the Erdős #243 assertion: if 1 ≤ a₁ < a₂ < ··· has aₙ/aₙ₋₁² → 1 and rational reciprocal sum, then aₙ = aₙ₋₁² − aₙ₋₁ + 1 eventually; the checked barrier does not settle the mixed-sign unbounded regime.\n- `remaining_open.erdos_269_reduced_tail_nonintegrality` — Prove that B·T_a ∉ ℤ for every B ≥ 1 coprime to 30 and every a ≥ 1.\n- `remaining_open.erdos_269_cofinal_local_window_escape` — Prove that for every B ≥ 1 coprime to 30 and every a₀ ≥ 1 there are ℓ ≥ a₀ and h ≥ 1 with lpr_{W(ℓ,h)}(−B·F(ℓ,h)) > K(B, ℓ+h), where K is the defined long-paper cap.\n- `remaining_open.erdos_269_two_dimensional_representation` — Transcendence of D_{2,3,5}, which is irrational: represent it faithfully by a specified two-dimensional Mahler-type value theorem with every hypothesis verified, give a conditional theorem under explicit logarithmic nondegeneracy, or prove no such representation exists.\n- `remaining_open.erdos_243_overlap_height_growth` — Decide whether every nonterminal canonical orbit of the reduced-orbit problem has limsup_n (log M_n)/n > 0, equivalently 2^n ≤ M_n^K infinitely often for some K ≥ 1.\n- `remaining_open.erdos_249_first_harmonic_anti_concentration` — Decide whether, for each h ≥ 1, some s ≥ 1 and 0 < η < 1 give, for every X₀, admissible X and L meeting the displayed size conditions for which the four first-harmonic bounds hold.\n- `remaining_open.twenty_one_scaled_remainder_cofinal_return` — Decide whether the scaled actual greedy remainder returns cofinally to one bounded interval: whether some B < ∞ has 2^N r_N ≤ B at arbitrarily large N.\n- `remaining_open.twenty_one_actual_orbit_invariant` — Decide whether a bounded-memory 2-adic or discrepancy invariant of the actual support forces a closed return or forbids permanent supercapacity, or prove no such invariant exists.\n- `remaining_open.twenty_one_final_skip_exclusion` — Decide whether the final-skip signatures prove |E − a_M| ≥ gap_M, contradicting 0 < a_M − E < gap_M, for E = ∑_{n≥1}(2^n − 1)^{−1} and the last-skip approximant a_M.\n\n## Mathematical programmes\n\n- `erdos257_half_story` — Achievement-set geometry and the rational-target seams. Exact geometry and reductions for the 1/2 and 1/21 branches; neither target membership nor universal Erdős #257 is proved.\n- `erdos249_certificate_story` — #249 certificate and harmonic interface. The certificate formulation is exact and finite instances are checked; the unbounded certificate supply and irrationality of S remain open.\n- `structured_support_families` — Classical and structured support irrationality. These are structured families, coefficient classes, and a conditional bouquet reduction; arbitrary infinite supports in universal Erdős #257 remain open.\n- `erdos249_diagonal_arithmetic` — #249 diagonal arithmetic and fresh-loss interfaces. Each interface sharpens or re-expresses the same supply problem; no unbounded avoidance, slack, or separated-projection supply is proved.\n- `boolean_mobius_constraints` — Boolean Möbius carry and global rationality constraints. These are exact characterisations and necessary consequences of rationality; they do not contradict every hypothetical rational value or settle universal Erdős #257.\n- `transport_curvature_programme` — Transport, curvature, phase separation, and no-go results. The actual-LCM frontier, corridor, consumers, and countermodels are exact, but no cofinal curvature, transport, phase-separation, or anti-reconstruction producer is proved.\n- `lambert_obstruction_interfaces` — Mersenne–Lambert identities and obstruction interfaces. These are exact identities, obstruction theorems, and one conditional rank consequence; they do not form an irrationality criterion for S.\n- `probabilistic_gcd_geometry` — Coprimality probability, gcd moments, and Stern–Brocot geometry. Exact identities, denominator transfer, convergence, and combinatorial stability only; no irrationality or novelty claim.\n- `half_carry_compactness_programme` — Half-carry compactness, windows, seams, and finite propagation. The closure implications, local propagation laws, and bounded stages are exact; no cofinal or coherent unbounded window family is proved, so neither half-membership nor universal Erdős #257 follows.\n- `arithmetic_obstruction_interfaces` — Denominator, CRT, moment, and gauge obstructions. These are exact arithmetic interfaces, finite mechanisms, and obstruction theorems; they do not supply an unbounded certificate family or an irrationality criterion for the totient constant.\n\n## Where the substance is\n\nThe working records end at \"The wall\"; the mathematics paper ends with\n\"What we need from a mathematician\"; [`docs/RESULTS.md`](RESULTS.md)\nopens with a ten-minute verdict.\n\n## Scale\n\nNavigation inventory, not results. Generated certificate shards are\ncounted as formal source, never as separate mathematical claims. The\ntwo roots currently expose 1,903 modules and\n164,639 declarations across #68, #243, #249,\n#251, #257, #269, #1041, and #1049; exact counts and atlas handles\nremain in `docs/orientation.json`.\n\nThe exhaustive declaration and import index is\n[`docs/declaration_atlas.json`](declaration_atlas.json).\n\n## Mathematical programme routes\n\nProgramme routes are bounded handles, not extra claims; their\nexact focus and claim ceiling are returned by the canonical route packet.\n\nOpen one programme with\n`python3 scripts/query_corpus.py --route <programme_route_id>`.\nThe packet returns the ordered claim handles, exact open boundary, related\nprogrammes, and source owners without loading the exhaustive graph.\n\n## Problem and family routes\n\nEach of the eight public problem routes returns every reviewed result\nfamily, exact Lean declaration/source handles where supplied, the paper\nrecord, and exact open obligations:\n`python3 scripts/query_corpus.py --route erdos_<problem_number>`.\nFor a dedicated paper's anchor census use\n`python3 scripts/query_corpus.py --paper-source <paper_source>`, then\nfollow each exact `--paper-anchor` handle. The machine descriptor's\nselection surface separates the authored flagship spine, gateway cohort, and\nlower-signal disposition list; review-matrix order is only stable drilldown\norder, never a significance proxy.\n\n## Selection spine and long tail\n\nThe descriptor exposes the authored flagship spine, broader gateway cohort,\nand lower-signal dispositions with their reasons and open obligations.\n\nFlagship families: 7; gateway cohort: 15; lower-signal families: 23;\nthese are visibility tiers, not significance rankings.\nUse `python3 scripts/query_corpus.py --publication-architecture` for the\nselection decisions and `--publication-family <family_id>` for each full\nreason, source route, open obligation, and Comparator disposition.\n\n## Principal claim routes\n\nPrincipal claim handles are preserved\nin [`docs/orientation.json`](orientation.json); exact statements and\nall declarations remain in [`docs/claims.json`](claims.json). Query a\nclaim with `python3 scripts/query_corpus.py --claim <claim_id>`.\nA conditional reduction or finite instance does not settle its open\nproposition; the ranked families above are the first signal, not a\nreplacement for the exact claim registry.\n\n## Read by intent\n\nUse `python3 scripts/query_corpus.py --route <route_id>`.\nAll route ids, titles, and reading lists are in\n[`docs/orientation.json`](orientation.json#reading_routes).\n\n## Drill down\n\n- Exact statuses, statements, paper anchors, declarations, and argument edges:\n  [`docs/claims.json`](claims.json)\n- Complete eight-problem proof/paper/source/frontier matrix: [`docs/SOURCE_MAP.md#complete-eight-problem-return-matrix`](SOURCE_MAP.md#complete-eight-problem-return-matrix)\n- Development chronology: [`docs/reference/WAVE_INDEX.md`](reference/WAVE_INDEX.md)\n- Current mathematical papers: [short papers and longer research records](../paper/README.md)\n- Machine form of this page: [`docs/orientation.json`](orientation.json)\n\n## External corpus registration\n\n[`docs/corpus_descriptor.json`](corpus_descriptor.json) uses schema\n`erdos249257-corpus-descriptor/5`. The release gate keeps it below 64 KB.\nIt carries release identities, content digests, principal claim and declaration\nhandles, and the root module topology. Complete claims, module imports,\ndeclaration prose, methodology, both authored papers, and the paper-to-Lean\nsource-sigil crosswalk remain in their digest-bound expansion files.\nThese authored surfaces remain distinct from Lean proof authority.\n\n## Query one handle\n\nThe read-only query helper returns bounded JSON and typed drilldown handles:\nmodule packets expose import neighbours, claim packets expose typed edges\nand open-proposition packets preserve the unresolved target and its progress.\nPaper labels resolve to exact TeX lines; declaration packets add pinned Lean\nURLs, module context, and attached claims.\n\n```sh\npython3 scripts/query_corpus.py --format card\npython3 scripts/query_corpus.py --claim denominator_exclusion\npython3 scripts/query_corpus.py --route instant_orientation\npython3 scripts/query_corpus.py --route erdos249_diagonal_arithmetic\n```\n","body_format":"text"},"source-map":{"document_id":"source-map","source_path":"docs/SOURCE_MAP.md","role":"proof_and_exposition_crosswalk","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/SOURCE_MAP.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/SOURCE_MAP.md","sha256":"sha256:07266bba272223f29e50612866983ff8c72b42b2f06781be69529d808315d9b5","byte_count":28839,"origin":"formal_claim_control_plane","already_carried_in":"scholarly_maps.formal_claim_control_plane[record_id=lean_source_map]"},"related-problems":{"document_id":"related-problems","source_path":"docs/RELATED_PROBLEMS.md","role":"portfolio_relationship_map","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/RELATED_PROBLEMS.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/RELATED_PROBLEMS.md","sha256":"sha256:fa13dc67457260c00a7314193259b42a1feda135dae66b43b3e6833289459e14","byte_count":8667,"origin":"public_main_checkout","body":"<!--\nSPDX-FileCopyrightText: 2026 Will Cook\nSPDX-License-Identifier: Apache-2.0\n-->\n\n# Related Erdős problems\n\nWhere this work sits among the numbered problems it is near. Statuses are as listed on the individual problem pages ([68](https://www.erdosproblems.com/68), [249](https://www.erdosproblems.com/249), [257](https://www.erdosproblems.com/257), [1041](https://www.erdosproblems.com/1041), [1049](https://www.erdosproblems.com/1049), [69](https://www.erdosproblems.com/69), [250](https://www.erdosproblems.com/250), [258](https://www.erdosproblems.com/258)) in July 2026. The catalogue column preserves those dated classifications; the local-scope column and source links below describe this repository.\n\n## Bounded status route\n\nThe external catalogue status and the local release status are separate facts.\nUse the exact local packets before interpreting an analogy:\n\nFor a machine-first return from any indexed problem, run\n`python3 scripts/query_corpus.py --route erdos_<n>` for `n` in `68, 243, 249,\n251, 257, 269, 1041, 1049`, then follow the [complete eight-problem return\nmatrix](SOURCE_MAP.md#complete-eight-problem-return-matrix). That matrix binds\neach public status row to its strongest checked source, paper and TeX record,\nand exact open-obligation handles without changing the status or claim\nauthority.\n\n- #249 target: `python3 scripts/query_corpus.py --open remaining_open.erdos_249_irrationality`; return to its problem-owned evidence with `python3 scripts/query_corpus.py --route erdos_249`.\n- #257 target: `python3 scripts/query_corpus.py --open remaining_open.universal_257_all_infinite_supports`; return to its problem-owned evidence with `python3 scripts/query_corpus.py --route erdos_257`.\n- #69 neighbour: `python3 scripts/query_corpus.py --claim prime_support_irrationality`\n- #250 neighbour: `python3 scripts/query_corpus.py --claim sigma_transcendence`\n\nThe first two packets are `open`; the latter two are `cited only`. A solved\nneighbour, shared transform, or formalised special case does not advance an\nopen target unless `docs/claims.json::machine_readable_paper.argument_graph`\nrecords an `advances_open_target` edge.\n\n## Problem map\n\n| Problem | Catalogue status | Local relation class |\n|---|---|---|\n| **#68** `∑_{n≥2} 1/(n!−1)` | open | problem-owned checked reductions and finite evidence |\n| **#249** `∑ φ(n)/2ⁿ` | open | open target |\n| **#257** `∑_{n∈A} 1/(2ⁿ−1)` | open | open target with formalised special cases |\n| **#1041** short curves in polynomial lemniscates | open | ani's exact Formal Conjectures refutation plus dynamical reductions |\n| **#1049** `∑ 1/(tⁿ−1)`, rational `t>1` | open (rational `t`) | Zudilin's strict rational-base region theorem plus rational-base obstructions |\n| **#69** `∑ ω(n)/2ⁿ` | solved (Tao–Teräväinen) | cited-only solved neighbour plus formalised identity bridge |\n| **#250** `∑ σ(n)/2ⁿ` | solved (Nesterenko 1996) | cited-only solved neighbour |\n| **#258** `∑ τ(n)/(a₁···aₙ)` | solved | untouched analogy |\n\n## Relation in this repository\n\n- **#68** — the [factorial-carry note](papers/full-text/erdos-68-factorial-denominator-irrationality.md)\n  routes to the exact [FactorialZeroPlateau source](../lean/ErdosProblems/Erdos68/FactorialZeroPlateau.lean#L1090):\n  factorial-successor and carry frontiers, structural consumers, and a finite\n  denominator exclusion; open boundary: produce cofinally many non-unit\n  carries, not merely one finite miss.\n- **#249** — the [problem-owned paper note](papers/full-text/erdos-249-binary-totient-series.md)\n  and [canonical source map](SOURCE_MAP.md) return the unconditional denominator\n  exclusion and conditional reductions. The new [squared-Lambert/GCD-moment\n  family](EXTERNAL_VERIFICATION.md#programme-249) routes to the exact first-moment\n  identity in [GcdMomentCalculus](../lean/Erdos249257/GcdMomentCalculus.lean#L235);\n  it is an identity with no irrationality consequence, so #249 remains open.\n- **#257** — the [problem-owned paper note](papers/full-text/erdos-257-mersenne-support-subseries.md)\n  and [canonical source map](SOURCE_MAP.md) return the named infinite-support\n  cases and full support (`A = ℕ`), not the universal statement.\n- **#243** — the [reciprocal-tail note](papers/full-text/erdos-243-reciprocal-tail-rigidity.md)\n  and [ReciprocalTailRigidity source](../lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean#L897)\n  exclude bounded negative-error regimes; open boundary: the unbounded\n  mixed-sign/negative-excursion regime.\n- **#251** — the [prime-gap note](papers/full-text/erdos-251-prime-gap-dyadic-series.md)\n  and [PrimeGapDyadicTail source](../lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean#L360)\n  expose the exact dyadic-tail machinery; open boundary: cofinally many\n  adjacent small tail shifts at indices where prime gaps differ.\n- **#269** — the [running-lcm note](papers/full-text/erdos-269-three-prime-running-lcm.md)\n  and [ThreePrimeRunningLcm source](../lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean#L123)\n  expose exact three-prime identities and a conditional residue-escape route;\n  the [rationality-to-carry bridge](../lean/ErdosProblems/Erdos269/RationalityCarryBridge.lean#L324)\n  proves that rationality of the actual `{2,3,5}` series produces a positive\n  reduced carry from an explicit onset. Open boundary: cofinal window escape\n  for that actual orbit.\n- **#1041** — ani's degree-seven polynomial has a formalised\n  [Hausdorff-length refutation](../lean/ErdosProblems/Erdos1041/Counterexample/HausdorffLength.lean#L407)\n  of the exact Formal Conjectures statement. This checks one polynomial;\n  independent human review of correspondence with the 1958 wording is not\n  recorded. The paper-facing route also checks Newton-flow ray separation and\n  perturbation lemmas. The source-current [frontier delta](../research_corpus/Erdos1041/FRONTIER.md)\n  additionally routes the surviving hub-selection, near-Fekete, perimeter,\n  inverse-sheet, and degree-five premise families through their exact public\n  source notes; it also records the 2026-08-29 refutations that must not be\n  rebuilt. The unrestricted topology, attachment-compatible selection, and\n  metric gluing remain open. Use the [machine result map](../research_corpus/Erdos1041/STRONGEST_RESULTS.json)\n  and [paper note](papers/full-text/erdos-1041-lemniscate-newton-flow.md)\n  together, keeping the source-only frontier separate from reviewed claim\n  authority.\n- **#1049** — [Zudilin's strict rational-base region theorem](../lean/ErdosProblems/Erdos1049/PaperR17/SourceConsumers.lean#L113)\n  proves irrationality at `a/b` for natural `0 < b < a` satisfying\n  `ZudilinContourRegion a b`, including every `(31/4)^r` with positive natural\n  `r`. The [rational-base note](papers/full-text/erdos-1049-rational-base-lambert.md)\n  and its checked source family — [RationalBaseLambert](../lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean#L155)\n  for the `3/2` corridor and `7/2` height input, [ZudilinHeightRegion](../lean/ErdosProblems/Erdos1049/ZudilinHeightRegion.lean#L91)\n  for the `81/200` region, [RationalPadeArithmetic](../lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean#L30)\n  and [HermitePadeNoGo](../lean/ErdosProblems/Erdos1049/HermitePadeNoGo.lean#L103)\n  for finite Padé arithmetic and comparison, and [QAperyDiagonalNonEquivalence](../lean/ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean#L67)\n  for the moving-diagonal no-go, with [ZudilinConeArithmetic](../lean/ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean#L286)\n  covering the endpoint-unit common-divisor obstruction — route distinct\n  arithmetic and obstruction mechanisms. The integer-base case `b ≥ 2` is\n  `irrational_erdosSum_full_support`; open boundary: universal rational-base\n  irrationality, irrationality at `3/2`,\n  the primitive noncollapsed construction, and analytic remainder control.\n- **#69** — the prime-support case of #257; only the identity bridge to\n  `∑_p 1/(2ᵖ−1)` is formalised, not the irrationality.\n- **#250** — the ladder neighbour `L(Id)`; cited, not re-proved.\n- **#258** — not addressed here; the monotone φ/σ sequel is the sibling\n  territory of #249.\n\nThe one-line summary: formalised settled special cases, exact structural reductions and scoped no-go theorems across eight indexed problems, and finite certificates whose quantifiers are stated explicitly. The degree-seven counterexample found by the erdosproblems.com contributor ani refutes the exact Formal Conjectures statement of #1041 and its total-variation form, and is formalised here; the other seven problems remain open.\n","body_format":"text"},"prior-art":{"document_id":"prior-art","source_path":"docs/PRIOR_ART.md","role":"attribution_and_antecedents","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/PRIOR_ART.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/PRIOR_ART.md","sha256":"sha256:da3391cb4b3ebd0308569046774412c88593a2e444ddf32a67aac3af8b64fd33","byte_count":51403,"origin":"formal_claim_control_plane","already_carried_in":"scholarly_maps.formal_claim_control_plane[record_id=lean_prior_art]"},"methodology":{"document_id":"methodology","source_path":"docs/METHODOLOGY.md","role":"evidence_method","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/METHODOLOGY.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/METHODOLOGY.md","sha256":"sha256:ffe69c6bad5f8e2662b7b0055535c74f735fce4fe4f5b057d79ab265eadec03f","byte_count":16851,"origin":"public_main_checkout","body":"<!-- SPDX-FileCopyrightText: 2026 Will Cook -->\n<!-- SPDX-License-Identifier: Apache-2.0 -->\n<!-- Generated by scripts/build_methodology.py from docs/methodology.json. Do not edit directly. -->\n\n# Formal proofs, mathematical claims, and what remains open\n\nThis page records the rules used to relate the Lean development to the mathematical claims made in this repository, and to keep partial results from being overstated. It is generated from `docs/methodology.json`; run `python3 scripts/build_methodology.py` after changing that source. Lean source checked by the pinned Lean kernel remains proof authority.\n\nThe machine source calls the six standing rules `method_axiom` records. They are repository rules, not Lean `axiom` declarations.\n\n## Standing rules\n\n### 1. A formal result is the exact proposition Lean checked\n\nA result is proved here only as the exact elaborated proposition accepted by the pinned public Lean environment under its transitive assumptions.\n\nSeen in: Full-support Erdős–Borwein irrationality, every integer base b ≥ 2 (`eb_full_support`).\n\nLimit: Kernel acceptance does not establish intended meaning, relevance, novelty, or claim status.\n\n### 2. Formal validity and mathematical faithfulness are separate questions\n\nA checked Lean declaration and the intended mathematical statement remain distinct until human mathematical review records their correspondence.\n\nSeen in: Named infinite-support irrationality families toward #257 (`support_families`).\n\nRecorded boundary: Prove irrationality for every infinite support A, not only the named support families formalised here (`remaining_open.universal_257_all_infinite_supports`).\n\nLimit: The repository can require and route review; it cannot compute mathematical faithfulness.\n\n### 3. A public claim crosses representations only through proved transport\n\nA claim may move between coordinates, encodings, normalisations, or models only through a checked implication or equivalence whose direction and losses are explicit.\n\nSeen in: The Mersenne–Lambert ladder identities (`mersenne_lambert_ladder`).\n\nRecorded boundary: Prove that S = ∑ φ(n)/2ⁿ is irrational, with no bound on a possible rational denominator (`remaining_open.erdos_249_irrationality`).\n\nLimit: A transport theorem establishes the stated relation; it does not establish that the new representation is useful.\n\n### 4. A reduction keeps the remaining hypothesis visible\n\nEvery conditional theorem, criterion, or reduction names the exact hypothesis or quantifier that remains open.\n\nSeen in: Exact reduction of #249 to finite decidable certificates (`certificate_reduction`).\n\nRecorded boundary: Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal (`remaining_open.unbounded_certificate_supply`).\n\nLimit: An exact reduction can expose an open proposition without proving it.\n\n### 5. Finite verification does not change quantifiers\n\nFinite certificates, bounded searches, and verified initial segments remain finite evidence unless a checked theorem supplies the unbounded or universal step.\n\nSeen in: Initial certified-kill segment and generated certificates (`certified_kill_instances`).\n\nRecorded boundary: Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal (`remaining_open.unbounded_certificate_supply`).\n\nLimit: Increasing a checked bound does not change the quantifier of the public claim.\n\n### 6. A claim of progress names both the theorem and the outstanding obligation\n\nA result advances an open target only when it states which remaining open proposition it proves, narrows, decomposes, re-expresses, or constrains.\n\nSeen in: Unconditional denominator exclusion for the totient constant (`denominator_exclusion`); Sublogarithmic divisor-coverage zero windows (`sublog_zero_windows`).\n\nRecorded boundary: This release does not settle Erdős #249 (`not_erdos_249_solution`).\n\nLimit: The repository checks that the claimed effect is explicit; human review decides whether the effect is mathematically significant.\n\n## Evidence responsibilities\n\n### Lean check\n\nDecides: Whether the declaration elaborates and its proof term is accepted in the pinned environment. The associated axiom audit reports the transitive axioms the declaration depends on.\n\nDoes not decide: Whether the proposition expresses the intended mathematics, is relevant to an open problem, or is novel.\n\n### Repository check\n\nDecides: Whether identifiers, statuses, source coordinates, routes, projections, and generated files agree.\n\nDoes not decide: Whether an informal statement is mathematically faithful or significant.\n\n### Human mathematical review\n\nRecords a reviewed judgement about: Whether a formal proposition matches the intended statement, how it relates to prior work, what public claim it supports, and what remains open.\n\nDoes not decide: Whether the Lean kernel accepts the proof term.\n\n### Blinded recovery evaluation\n\nDecides: Whether a participant reconstructs a held-out statement or proof mechanism under a specified question, source snapshot, derived layers, and matched evaluation protocol. The packet builder exports a registered historical cut without Git history and verifies that the target declaration is absent. It does not isolate the participant from other files, the network, or prior knowledge. Packet preparation is not evaluation evidence: current explanatory prose needs a hint review, controls need capsule and workload matching, and a scored run needs fresh participants and independent scoring.\n\nDoes not decide: Whether the recovered statement is novel, significant, or correctly proved; whether the mechanism credited for the recovery is the one the original proof used; or whether a recovery generalises beyond the held-out case.\n\n## Research and publication paths\n\nThe machine source records two bounded paths. They are navigation and review order, not additional mathematical implications.\n\n### Mathematical research path\n\n`open_target` → `representation` → `necessary_consequence` → `proved_reduction` → `finite_certificate` → `lean_declaration` → `remaining_open_proposition`\n\nA route may stop at any intermediate object. In particular, a necessary consequence, reduction, finite certificate, or Lean declaration does not remove the final remaining-open proposition unless a checked theorem actually does so.\n\n### Public claim path\n\n`lean_declaration` → `checked_assumptions` → `reviewed_intended_meaning` → `public_claim` → `authored_exposition` → `generated_projection` → `public_checkout`\n\nThis is the authority order for publication. Kernel acceptance precedes assumption review; intended meaning is reviewed before a public claim; authored exposition precedes generated projections; and the public checkout is the final reader-facing surface.\n\n## Working principles\n\n### Proving\n\n- Under a hypothetical rational value or counterexample, derive checked structural consequences instead of replacing proof with plausibility.\n- A finite checker needs a soundness theorem; calling it exact additionally requires a completeness theorem at the stated parameters.\n\n### Publishing\n\n- Every frontier-facing partial result states both its positive conclusion and the exact proposition that survives it.\n- Each representation change identifies the checked theorem that transfers the claim and states the direction and any loss.\n- A reader using only the public checkout must be able to find the main argument, proof routes, public claim statuses, and remaining open propositions.\n- Public claim review checks both ceilings and coverage: prose must not exceed the checked theorem chain, and a source-backed high-value mechanism, exact reformulation, quantitative reduction, or distinct endpoint must be mechanically discoverable when omitting it would materially understate the corpus.\n\n### Maintaining\n\n- A failed strategy, counterexample, or disproved finite-state interpretation is retained when it rules out a recurring mathematical route.\n- Distinguish proof-body, proposition, assumption, representation, finite-range, claim-status, exposition, and projection changes before deciding what must be updated.\n- Lean owns formal validity; humans own intended meaning, significance, and public framing; agents may search, propose, test, and expose consequences without inheriting either authority.\n- Update Lean source first, then assumption and meaning review, then claim status, then authored exposition, then generated projections.\n\n## Three examples from this development\n\n### An exact reduction that does not close #249\n\n```text\na hypothetical rational value of S = ∑ φ(n)/2ⁿ\n→ an integrality constraint on totient tail differences\n→ a finite non-integrality certificate at fixed parameters (certifiedKill)\n→ an exact conditional reduction (irrational_totient_series_of_lcm_diagonal_certificate_supply)\n→ the unbounded supply of such certificates, which remains open\n```\n\nIf S were rational, its totient tails would satisfy an integrality constraint. The certifiedKill predicate refutes that constraint at fixed parameters, and the completeness theorem shows the finite computation loses no information at those parameters. The reduction theorem then proves: if certificates occur at unbounded parameters, S is irrational. Producing that unbounded supply is the remaining open proposition; nothing in this repository claims it.\n\nClaims: Exact reduction of #249 to finite decidable certificates (`certificate_reduction`); Certificate completeness (`certificate_completeness`).\n\nRemaining open: `remaining_open.unbounded_certificate_supply`.\n\n### Finite verification that stays finite\n\ncertifiedKill_all_small and the lcm-diagonal deposits through t = 8 (certifiedKill_periodLcm_diagonal_upto_six and the t = 7 and t = 8 endpoints) are kernel-checked with decide at fixed bounds. Each names a specific bounded region the checker has cleared; together with the generated certificate shards they show the reduction is not vacuous. The claim keeps the status verified finite instance and states its bounded domain. Raising the checked bound changes the stated bound; it does not change the quantifier of any public claim.\n\nClaims: Initial certified-kill segment and generated certificates (`certified_kill_instances`).\n\nRemaining open: `remaining_open.unbounded_certificate_supply`.\n\n### Necessary structure that does not yet contradict\n\nThe carry and zero-window theorems state what a rational series value would force: reciprocal-mass lower bounds, unbounded carry states, and sublogarithmic zero windows in divisor coverage. These are necessary consequences of rationality. They constrain a hypothetical rational value; none of them rules it out, and the universal #257 statement remains open.\n\nClaims: Rationality-forced reciprocal-mass lower bound (`reciprocal_mass_bound`); Carry states are unbounded over infinite rational-valued supports (`unbounded_carry_states`); Sublogarithmic divisor-coverage zero windows (`sublog_zero_windows`).\n\nRemaining open: `remaining_open.universal_257_all_infinite_supports`.\n\n## Before changing a public claim\n\nClassify the change first. Each class states its minimum evidence, whether human mathematical review is needed, and the public consequence. The machine source additionally records, for each class, the effects the change must not have.\n\n| Change | Minimum evidence | Mathematical review | Public consequence |\n|---|---|---|---|\n| A proof body changes; the stated proposition does not | Lean check, Repository check | only when intended meaning changed or claim text changed | No claim or exposition change follows automatically. Compare the axiom audit before and after. |\n| A stated proposition or its hypotheses change | Lean check, Repository check, Human mathematical review | required | Reconcile the claim wording, the remaining-open boundary, and the paper statement. |\n| The transitive assumptions of a declaration change | Lean check, Repository check, Human mathematical review | required | Reassess the proof posture even when the proposition is unchanged. |\n| A claim moves between coordinates, encodings, or models | Lean check, Human mathematical review | required | Record the transport theorem, its direction, and anything not preserved. |\n| A checked bounded range or certificate family is extended | Lean check, Repository check | only when claim text changed or claim status changed or intended meaning changed or paper statement changed | Keep the finite status; update only the stated bound. |\n| A claim's declaration coordinates or links are corrected; the claim is otherwise unchanged | Lean check, Repository check | only when claim text changed or claim status changed or intended meaning changed | Link or correct the supporting declarations of an unchanged claim, then regenerate dependent projections. The mathematical claim does not change; a Lean check confirms the linked declaration exists and elaborates. |\n| An advances-open-target relation is added or changed | Repository check, Human mathematical review | required | Record the outstanding obligation affected and the exact relation. |\n| The status of a public claim changes | Repository check, Human mathematical review | required | State the theorem or review that licenses the new status. |\n| Authored paper or documentation prose changes mathematically | Repository check, Human mathematical review | required | Prose must not exceed the status recorded in the claim registry. |\n| A generated projection is rebuilt | Repository check | not required | Regenerate below the existing owners; no mathematical effect. |\n| docs/methodology.json changes | Repository check | only when rule semantics changed | Regenerate docs/METHODOLOGY.md and the corpus descriptor. Rules route review; they do not decide mathematics. |\n\n## Rejected inferences\n\n- Do not use a successful Lean check to bypass review of whether the proposition expresses the intended mathematics.\n- Do not narrate a conditional implication, equivalence, or criterion as closure of the original open problem.\n- Do not treat a large checked range as evidence for an unbounded or universal assertion.\n- Do not speak of a transformed representation as the original claim unless a checked implication or equivalence supplies that relation.\n- A dependency graph, generated explanation, language-model output, or private proof sketch is not public proof authority.\n- Counts of declarations, proof lines, certificates, or agent activity do not by themselves establish progress on an open problem.\n- README, generated Markdown, the corpus descriptor, and declaration maps must remain below their owning source files.\n- A methodology rule is removed if it cannot name the objects or changes it applies to, the evidence it requires, the route that evaluates it, and what it does not establish.\n\n## Authority and machine identifiers\n\nLean source checked by the pinned Lean kernel remains proof authority. `docs/claims.json` remains the owner of release identity, claim status, remaining open propositions, and argument relationships. `docs/methodology.json` owns the rules on this page and records, for every rule, its evidence classes, applicability bindings, scope limit, local references, and validation guards. The authored paper remains the owner of mathematical exposition.\n\n`docs/METHODOLOGY.md` is generated from `docs/methodology.json`; run `python3 scripts/build_methodology.py` after changing the source. `python3 scripts/check_release.py` checks the source schema, typed claim-transition requirements, the change-class matrix, local references, exact Markdown projection, descriptor registration, and entry routes.\n\nMachine identifiers for the standing rules, in the order the rules appear above:\n\n```text\n1. A formal result is the exact proposition Lean checked\n   method_axiom.proof_is_exact_checked_proposition\n2. Formal validity and mathematical faithfulness are separate questions\n   method_axiom.formal_proof_does_not_verify_intended_meaning\n3. A public claim crosses representations only through proved transport\n   method_axiom.representation_changes_require_proved_transport\n4. A reduction keeps the remaining hypothesis visible\n   method_axiom.reduction_states_remaining_open_hypothesis\n5. Finite verification does not change quantifiers\n   method_axiom.finite_results_remain_finite_without_uniform_theorem\n6. A claim of progress names both the theorem and the outstanding obligation\n   method_axiom.open_problem_progress_names_changed_open_proposition\n```\n\nPrinciples, rejected inferences, change classes, review triggers, forbidden effects, and claim-transition contracts keep their full identifiers and machine fields in `docs/methodology.json`.\n","body_format":"text"},"scope":{"document_id":"scope","source_path":"docs/SCOPE.md","role":"scope_boundary","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/SCOPE.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/SCOPE.md","sha256":"sha256:a49cfdf72b96920f35dcb3f05304988706f75a62589185d5ebd8db98b68b0659","byte_count":5804,"origin":"formal_claim_control_plane","already_carried_in":"scholarly_maps.formal_claim_control_plane[record_id=lean_scope]"},"formal-conjectures":{"document_id":"formal-conjectures","source_path":"docs/verification/FORMAL_CONJECTURES_CROSSWALK.md","role":"upstream_contribution_record","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/verification/FORMAL_CONJECTURES_CROSSWALK.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/verification/FORMAL_CONJECTURES_CROSSWALK.md","sha256":"sha256:21cac7ae6445abf8a384456889f546269880fa78a2babaa2c1ed6644c5737281","byte_count":49439,"origin":"public_main_checkout","body":"<!-- Generated by scripts/check_formal_conjectures_crosswalk.py; do not edit. -->\n# Formal Conjectures crosswalk\n\nFollow the public Formal Conjectures contributions, then inspect the pinned statement comparison for the eight Erdős programmes. Contribution lifecycle and mathematical source identity have separate evidence.\n\n## Contribution activity\n\nObserved **2026-10-14T11:24:16.859470+00:00** from the linked public pull requests.\n\n**5 merged mathematical contributions · 14 open requests · 1 merged AUTHORS update.**\n\nA dated observation of public pull requests. Individual review, aggregate approval requirements and merge state are separate. A merge accepts the scoped upstream change, not every mathematical claim in this repository. Maintenance of existing proof links is excluded from the merged mathematical contribution count. The earlier pinned statement comparisons below retain their original source identities.\n\nAll nineteen receiving declarations listed here have successful author-hosted checks at the exact delivered heads. Each run exports the actual FC declaration before executing the confined candidate in Lean 4.33.1 with FC's Mathlib pin `0df444a360eaa60ab8c11dca51a86af692955474`; statement-and-dependency equality, default Lean and NanoDa acceptance, the axiom budget (`propext`, `Classical.choice`, `Quot.sound`) and positive/mismatch controls are checked. The eleven repaired adapters use source `d1fe386f`; the three earlier successful declarations retain their own immutable proof pins. The five additional statements retain proof pins `71a2bc9f`, `2cd19fba` and `01b23a6e`; [source PR40](https://github.com/wcook04/plectis-erdos-lean/pull/40) and [source PR41](https://github.com/wcook04/plectis-erdos-lean/pull/41) are merged. These checks confer no independent human rerun or upstream merge.\n\n[Source run 38341643841](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38341643841) separately compiled all ten receiving-adapter projects for fourteen units at [source `d1fe386f`](https://github.com/wcook04/plectis-erdos-lean/commit/d1fe386ff1d6f6ca49bbad2e8a309fea2b58cb85). [Source PR37](https://github.com/wcook04/plectis-erdos-lean/pull/37) packages that repair. Source compilation, identity at a receiving head, current upstream deployment and reviewer acceptance are distinct evidence.\n\n### Merged mathematical contributions\n\n**[#5034 · Add formal proof for Erdős Problem 258 constant-base variant](https://github.com/google-deepmind/formal-conjectures/pull/5034)**\n\nProof link for the constant-base variant of Erdős's 1948 theorem.\n\nMerged 2026-09-18.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/5034#pullrequestreview-5247811041) on an earlier head, 2026-09-18.\n\n**[#6505 · Erdős 1041: mark solved with answer(False) and link a formal proof](https://github.com/google-deepmind/formal-conjectures/pull/6505)**\n\nNegative answer to the exact formal path-image statement, credited to ani; independent correspondence review of the 1958 wording remains separate.\n\nMerged 2026-09-23.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/6505#pullrequestreview-5287880895) on current head, 2026-09-23.\n\n**[#6506 · Erdős 257: link a formal proof of variants.tsum_top](https://github.com/google-deepmind/formal-conjectures/pull/6506)**\n\nProof link for the full-support variant of Erdős's 1948 theorem; arbitrary infinite supports remain open.\n\nMerged 2026-09-23.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/6506#pullrequestreview-5287812291) on current head, 2026-09-23.\n\n**[#6507 · Erdős 1049: link a formal proof of variants.geq_2_integer](https://github.com/google-deepmind/formal-conjectures/pull/6507)**\n\nProof link for integer bases at least two, following Erdős's 1948 theorem; general rational bases remain open.\n\nMerged 2026-09-23.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/6507#pullrequestreview-5287809168) on current head, 2026-09-23.\n\n**[#6529 · Erdős 257: add the reciprocal-summable support variant](https://github.com/google-deepmind/formal-conjectures/pull/6529)**\n\nErdős’s classical coprimality-free result: an infinite support with summable reciprocals gives an irrational series at every integer base at least two. The merged subtype formulation links its verified adapter; the unrestricted question remains open.\n\nMerged 2026-10-09.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/6529#pullrequestreview-5462549186) on an earlier head, 2026-10-08.\n\n### Open requests\n\n**[#6522 · Set Erdős 269 rationality variant to false](https://github.com/google-deepmind/formal-conjectures/pull/6522)**\n\nLiterature-backed negative answer to universal rationality using Steve Fan's two-prime argument. No formal transcendence proof claimed; universal irrationality remains open.\n\nOpen; aggregate review decision: **review required**.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/6522#pullrequestreview-5434952248) on current head, 2026-10-06.\n\n**[#6523 · Erdős 243: add cubic-rate irrationality and slow-growth rigidity](https://github.com/google-deepmind/formal-conjectures/pull/6523)**\n\nCubic-rate irrationality and slow-growth rigidity under additional growth and increment hypotheses; the original question remains open.\n\nReceiving verification for `Erdos243.erdos_243.variants.cubic_rate`: [run 38356995383](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38356995383) compares the actual FC declaration at `ccb7d725` with [proof source `d1fe386f`](https://github.com/wcook04/plectis-erdos-lean/blob/d1fe386ff1d6f6ca49bbad2e8a309fea2b58cb85/verification/FC243DualCurrent/FC6523Solutions.lean). Statements and dependencies match; default Lean and NanoDa accept under the three permitted core axioms, and positive/mismatch controls pass.\n\nReceiving verification for `Erdos243.erdos_243.variants.slow_growth_product_defect`: [run 38356998998](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38356998998) compares the actual FC declaration at `ccb7d725` with [proof source `d1fe386f`](https://github.com/wcook04/plectis-erdos-lean/blob/d1fe386ff1d6f6ca49bbad2e8a309fea2b58cb85/verification/FC243DualCurrent/FC6523Solutions.lean). Statements and dependencies match; default Lean and NanoDa accept under the three permitted core axioms, and positive/mismatch controls pass.\n\nOpen; aggregate review decision: **review required**.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/6523#pullrequestreview-5477814639) on an earlier head, 2026-10-10.\n\n**[#6527 · Erdős 1049: add rational-base contour and irrationality exponent bound](https://github.com/google-deepmind/formal-conjectures/pull/6527)**\n\nA strict logarithmic contour, all positive powers and an irrationality-exponent bound; the general rational-base question, including 3/2, remains open.\n\nReceiving verification for `Erdos1049.erdos_1049.variants.rational_base_contour_measure`: [run 38358957539](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38358957539) compares the actual FC declaration at `ea3a10bd` with [proof source `d1fe386f`](https://github.com/wcook04/plectis-erdos-lean/blob/d1fe386ff1d6f6ca49bbad2e8a309fea2b58cb85/verification/FC1049ContourMeasureCurrent/FC6527Solution.lean). Statements and dependencies match; default Lean and NanoDa accept under the three permitted core axioms, and positive/mismatch controls pass.\n\nOpen; aggregate review decision: **review required**.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/6527#pullrequestreview-5434959641) on an earlier head, 2026-10-06.\n\n**[#6528 · Erdős 257: add weighted and mixed-support irrationality variants](https://github.com/google-deepmind/formal-conjectures/pull/6528)**\n\nIrrationality under sufficient weighted support and covering hypotheses, not for arbitrary infinite supports.\n\nReceiving verification for `Erdos257.erdos_257.variants.finite_prime_weighted_support`: [run 38360865074](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38360865074) compares the actual FC declaration at `4f5ae941` with [proof source `d1fe386f`](https://github.com/wcook04/plectis-erdos-lean/blob/d1fe386ff1d6f6ca49bbad2e8a309fea2b58cb85/verification/FC257WeightedSupportCurrent/FC6528Solutions.lean). Statements and dependencies match; default Lean and NanoDa accept under the three permitted core axioms, and positive/mismatch controls pass.\n\nReceiving verification for `Erdos257.erdos_257.variants.arbitrary_weight_mixed_support`: [run 38360869036](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38360869036) compares the actual FC declaration at `4f5ae941` with [proof source `d1fe386f`](https://github.com/wcook04/plectis-erdos-lean/blob/d1fe386ff1d6f6ca49bbad2e8a309fea2b58cb85/verification/FC257WeightedSupportCurrent/FC6528Solutions.lean). Statements and dependencies match; default Lean and NanoDa accept under the three permitted core axioms, and positive/mismatch controls pass.\n\nOpen; aggregate review decision: **review required**.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/6528#pullrequestreview-5477798867) on an earlier head, 2026-10-10.\n\n**[#6576 · Erdős 1041: add sharp normalized collinear gap variant](https://github.com/google-deepmind/formal-conjectures/pull/6576)**\n\nSharp product-height bound for normalized real roots and a whole adjacent interval; separate from the merged negative answer for general paths.\n\nReceiving verification for `Erdos1041.erdos_1041.variants.sharp_collinear_gap`: [run 38174164846](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38174164846) compares the actual FC declaration at `37a6ba52` with [proof source `45bcf390`](https://github.com/wcook04/plectis-erdos-lean/blob/45bcf390b6e027ecea1214e774e484095c213305/verification/FC1041SharpGapCurrent/FC6576Solution.lean). Statements and dependencies match; default Lean and NanoDa accept under the three permitted core axioms, and positive/mismatch controls pass.\n\nOpen; aggregate review decision: **review required**.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/6576#pullrequestreview-5477816245) on an earlier head, 2026-10-10.\n\n**[#6577 · Erdős 249: add countable least-residue dilation independence](https://github.com/google-deepmind/formal-conjectures/pull/6577)**\n\nRational linear independence of 1 and the countable dilation family for fixed modulus at least three and integer base at least two; the unreduced totient question remains open.\n\nReceiving verification for `Erdos249.erdos_249.variants.least_residue_dilation_independence`: [run 38174685762](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38174685762) compares the actual FC declaration at `41da78f7` with [proof source `c513ed5e`](https://github.com/wcook04/plectis-erdos-lean/blob/c513ed5ed17985567bf55b6a191afcc21cceb028/verification/FC249CountableIndependenceCurrent/FC6577Solution.lean). Statements and dependencies match; default Lean and NanoDa accept under the three permitted core axioms, and positive/mismatch controls pass.\n\nOpen; aggregate review decision: **review required**.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/6577#pullrequestreview-5435273753) on an earlier head, 2026-10-06.\n\n**[#6579 · Erdős 249: classify rational dyadic totient observables](https://github.com/google-deepmind/formal-conjectures/pull/6579)**\n\nFor rational-valued observables modulo 2^k, the binary series is rational exactly when the observable is constant on even residue classes. The unreduced-totient problem remains open.\n\n[Bo Cowgill approved the current head](https://github.com/google-deepmind/formal-conjectures/pull/6579#pullrequestreview-5493193130) on 12 October, relying on author-hosted evidence; he did not independently rerun verification. [Run 38173617731](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38173617731) compares FC head `a6a2e0e1` with proof source `04e14917` in Lean 4.33.1 and FC's Mathlib environment. It reports matching statements and dependencies, defaultLean/NanoDa acceptance under `propext`, `Classical.choice` and `Quot.sound`, and positive/mismatch controls.\n\nHis [earlier approval](https://github.com/google-deepmind/formal-conjectures/pull/6579#pullrequestreview-5435222238) independently checked the older Lean 4.29.1 proof. The [11 October failure](https://github.com/google-deepmind/formal-conjectures/pull/6579#issuecomment-6112373414) at `e6bf75a5` remains historical evidence of a cross-version comparison mismatch, does not establish a mathematical defect. Individual approval, native verification, aggregate review requirements and merge state remain separate.\n\nReceiving verification for `Erdos249.erdos_249.variants.rational_dyadic_observable`: [run 38173617731](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38173617731) compares the actual FC declaration at `a6a2e0e1` with [proof source `04e14917`](https://github.com/wcook04/plectis-erdos-lean/blob/04e14917b7c9296feabf6674f8e596a2c3ec9082/verification/FC249RationalObservableCurrent/FC249Repair.lean). Statements and dependencies match; default Lean and NanoDa accept under the three permitted core axioms, and positive/mismatch controls pass.\n\nOpen; aggregate review decision: **review required**.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/6579#pullrequestreview-5493193130) on current head, 2026-10-12.\n\n**[#6780 · Add fixed-base geometric-moment Hankel asymptotic](https://github.com/google-deepmind/formal-conjectures/pull/6780)**\n\nAsymptotic for each fixed 0 < q < 1; no uniformity as q approaches one and no solution of #1049.\n\nReceiving verification for `GeometricMomentHankel.fixed_base_asymptotic`: [run 38358960367](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38358960367) compares the actual FC declaration at `d194bd96` with [proof source `d1fe386f`](https://github.com/wcook04/plectis-erdos-lean/blob/d1fe386ff1d6f6ca49bbad2e8a309fea2b58cb85/verification/FC1049HankelCurrent/FC6780Solution.lean). Statements and dependencies match; default Lean and NanoDa accept under the three permitted core axioms, and positive/mismatch controls pass.\n\nOn 13 October, Bo Cowgill approved `d194bd96`: the statement still matches Cook’s fixed-base theorem, and Zudilin’s equation (6) at x = z = 1 gives the moment summand; its zeroth Lambert term is zero. He said the actual-FC export-before-candidate order and consumed-export check resolve his earlier workspace concern. He relied on the author-hosted receiving receipt without independently rerunning the proof. The request remains open with review required.\n\nOpen; aggregate review decision: **review required**.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/6780#pullrequestreview-5507342139) on current head, 2026-10-13.\n\n**[#7290 · Add a noninteger regular-rate irrationality variant for Erdős 243](https://github.com/google-deepmind/formal-conjectures/pull/7290)**\n\nFor every real noninteger λ > 1, a positive strictly increasing integer sequence with a(n)²/a(n+1) = 1 + λ/n + o(n^(-λ)), in the original index, has a convergent irrational reciprocal sum. Koizumi’s rational-tail argument is credited. The unrestricted eventual-Sylvester-recurrence problem #243 remains open.\n\n[Receiving run 38472730641](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38472730641) compares the actual FC declaration at `c04dbfd6e3f93a9dda92b62048b2c076516aa989` with [proof source `71a2bc9f`](https://github.com/wcook04/plectis-erdos-lean/blob/71a2bc9f74f8650d7287581f799ba998fe26e60f/verification/FC243NonintegralRateCurrent/FCNonintegralRateSolutions.lean). Statements and dependencies match; default Lean and NanoDa accept under `propext`, `Classical.choice` and `Quot.sound`, with positive and mismatch controls. The exact changed FC module passed a warning-clean build. This author-hosted check is distinct from independent human proof replay and upstream acceptance. Bo Cowgill approved this exact head after checking the little-o remainder, original-index finite-prefix adjustment and noninteger exclusion, relying on the linked author-hosted evidence; he explicitly did not independently rerun the proof. The request remains open with review required.\n\nOpen; aggregate review decision: **review required**.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/7290#pullrequestreview-5510880799) on current head, 2026-10-13.\n\n**[#7291 · Add the mixed-modulus totient-observable classification](https://github.com/google-deepmind/formal-conjectures/pull/7291)**\n\nFor a finite family of distinct positive dilations, base B ≥ 2 and rational tables modulo powers 2^k with k ≥ 1, a rational linear combination of the dilated totient observables is rational exactly when every table is constant on its even classes. Odd classes contribute a finite prefix. This classification leaves the original binary totient-series problem #249 open.\n\n[Receiving run 38471271374](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38471271374) compares the actual FC declaration at `c3d91b4f3a1c362db164595d5e114ac21202237f` with [proof source `2cd19fba`](https://github.com/wcook04/plectis-erdos-lean/blob/2cd19fba5298bc4d8ca9fd469962b18ecfbbca31/verification/FC249MixedModuliCurrent/FC249MixedModuliSolution.lean). Statements and dependencies match; default Lean and NanoDa accept under `propext`, `Classical.choice` and `Quot.sound`, with positive and mismatch controls. The exact changed FC module passed a warning-clean build. This author-hosted check is distinct from independent human proof replay and upstream acceptance. Bo Cowgill approved this exact head after checking the finite mixed-modulus statement, original indexing and even residue classes. He inspected the existing author-hosted receiving check without independently rerunning it. The request remains open with review required.\n\nOpen; aggregate review decision: **review required**.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/7291#pullrequestreview-5517046311) on current head, 2026-10-14.\n\n**[#7292 · Add the infinite-rank three-prime running-LCM kernel theorem](https://github.com/google-deepmind/formal-conjectures/pull/7292)**\n\nThe three-prime running-LCM kernel has nonsingular minors of every finite size uniformly across the third-coordinate layers, and therefore has no finite rational-valued separated representation. This structural theorem gives no rationality or irrationality conclusion for the scalar catalogue series in #269.\n\n[Receiving run 38471274558](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38471274558) compares the actual FC declaration at `72f8836b29a5f9cc3f2ce04176d406e4a593a03e` with [proof source `2cd19fba`](https://github.com/wcook04/plectis-erdos-lean/blob/2cd19fba5298bc4d8ca9fd469962b18ecfbbca31/verification/FC269UniformKernelCurrent/FC269UniformKernelSolution.lean). Statements and dependencies match; default Lean and NanoDa accept under `propext`, `Classical.choice` and `Quot.sound`, with positive and mismatch controls. The exact changed FC module passed a warning-clean build. This author-hosted check is distinct from independent human proof replay and upstream acceptance. Bo Cowgill approved this exact head after checking the uniform indices across all layers and the pinned kernel definitions. He inspected existing receiving evidence without independently rerunning the external proof. His separate description-rendering request was resolved with a whitespace-only edit, leaving the equation and FC head unchanged. The request remains open with review required.\n\nOpen; aggregate review decision: **review required**.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/7292#pullrequestreview-5517029228) on current head, 2026-10-14.\n\n**[#7293 · Add factorial-gap common-denominator growth](https://github.com/google-deepmind/formal-conjectures/pull/7293)**\n\nThe common denominator L_N = lcm_{2≤n≤N}(n! − 1) satisfies liminf log L_N/(N^(3/2) log N) ≥ 2√2/3. The theorem concerns the denominator before cancellation; it does not establish irrationality of the factorial-gap series in #68.\n\n[Receiving run 38472733409](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38472733409) compares the actual FC declaration at `a4ea288bc7672535c3784d13130e9df95af0f7f7` with [proof source `71a2bc9f`](https://github.com/wcook04/plectis-erdos-lean/blob/71a2bc9f74f8650d7287581f799ba998fe26e60f/verification/FC68CommonDenominatorCurrent/FC68CommonDenominatorSolution.lean). Statements and dependencies match; default Lean and NanoDa accept under `propext`, `Classical.choice` and `Quot.sound`, with positive and mismatch controls. The exact changed FC module passed a warning-clean build. This author-hosted check is distinct from independent human proof replay and upstream acceptance. Bo Cowgill approved this exact head after checking the common-denominator lower limit, harmless empty prefixes and shifted-factorial source arguments. He inspected the author-hosted receiving workflow and logs without independently rerunning the proof. The request remains open with review required.\n\nOpen; aggregate review decision: **review required**.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/7293#pullrequestreview-5517055691) on current head, 2026-10-14.\n\n**[#7294 · Add the exact initial term of normalized Hankel determinants](https://github.com/google-deepmind/formal-conjectures/pull/7294)**\n\nFor every natural rank N, including zero, the normalized Hankel determinant has formal q-order E_N = N(N−1)(2N−1)/6 and coefficient 2^N[q^E_N]H_N = (N!)²(N+1)!. Zudilin’s more general lower-order bound and normalized construction are credited. This fixed-rank formal initial term is distinct from #6780’s fixed-real-base large-rank asymptotic and gives no new rational-base irrationality region for #1049.\n\n[Receiving run 38478080939](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38478080939) compares the actual FC declaration at `10c6001c65d3b78b6fee282f4e614a3bc93868d1` with [proof source `01b23a6e`](https://github.com/wcook04/plectis-erdos-lean/blob/01b23a6e0ead0a80b89dd1b3dffed1e4356ad2ba/verification/FC1049OrderCoeffCurrent/FC1049OrderCoeffSolution.lean). Statements and dependencies match; default Lean and NanoDa accept under `propext`, `Classical.choice` and `Quot.sound`, with positive and mismatch controls. The exact changed FC module passed a warning-clean build. This author-hosted check is distinct from independent human proof replay and upstream acceptance. Bo Cowgill approved this exact head after checking the coefficientwise moment definition, division-free initial coefficient equality and empty determinant at N = 0. He reused the author-hosted receiving evidence without independently rerunning the proof. The request remains open with review required.\n\nOpen; aggregate review decision: **review required**.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/7294#pullrequestreview-5517057220) on current head, 2026-10-14.\n\n### Proof-link maintenance\n\n**[#6971 · Update four proof links to kernel-checked sources](https://github.com/google-deepmind/formal-conjectures/pull/6971)**\n\nReplaces four already accepted proof links with immutable Palomar-verified packages; the proof files are byte-for-byte identical and the FC statements are unchanged. [Bo Cowgill reviewed](https://github.com/google-deepmind/formal-conjectures/pull/6971#pullrequestreview-5477230569) statement correspondence, including the Hausdorff path-image measure and series indexing, and the source-bound Lean and NanoDa acceptance reports on Lean 4.29.1 with propext, Classical.choice and Quot.sound. [Moritz Firsching also approved](https://github.com/google-deepmind/formal-conjectures/pull/6971#pullrequestreview-5483329663), asking about file identity and noting the benefit of the Palomar-linked repository. This is proof-link maintenance; it adds no new mathematical result.\n\nMerged 2026-10-11.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/6971#pullrequestreview-5483329663) on current head, 2026-10-11.\n\n**[#7260 · Update five accepted proof links for Lean 4.33.1](https://github.com/google-deepmind/formal-conjectures/pull/7260)**\n\nUpdates five already accepted proof links for #257, #258, #1041 and #1049 to their exact Lean 4.33.1 receiving-environment adapters. The declarations and their mathematical claims are unchanged; this maintenance adds no mathematical result.\n\nReceiving verification for `Erdos257.erdos_257.variants.tsum_top`: [run 38364928751](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38364928751) compares the actual FC declaration at `49e83c34` with [proof source `d1fe386f`](https://github.com/wcook04/plectis-erdos-lean/blob/d1fe386ff1d6f6ca49bbad2e8a309fea2b58cb85/verification/FCMergedIntegerAdaptersCurrent/FC6971IntegerSolutions.lean). Statements and dependencies match; default Lean and NanoDa accept under the three permitted core axioms, and positive/mismatch controls pass.\n\nReceiving verification for `Erdos1049.erdos_1049.variants.geq_2_integer`: [run 38364932948](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38364932948) compares the actual FC declaration at `49e83c34` with [proof source `d1fe386f`](https://github.com/wcook04/plectis-erdos-lean/blob/d1fe386ff1d6f6ca49bbad2e8a309fea2b58cb85/verification/FCMergedIntegerAdaptersCurrent/FC6971IntegerSolutions.lean). Statements and dependencies match; default Lean and NanoDa accept under the three permitted core axioms, and positive/mismatch controls pass.\n\nReceiving verification for `Erdos258.erdos_258.variants.constant`: [run 38366401558](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38366401558) compares the actual FC declaration at `49e83c34` with [proof source `d1fe386f`](https://github.com/wcook04/plectis-erdos-lean/blob/d1fe386ff1d6f6ca49bbad2e8a309fea2b58cb85/verification/FCMergedIntegerAdaptersCurrent/FC6971IntegerSolutions.lean). Statements and dependencies match; default Lean and NanoDa accept under the three permitted core axioms, and positive/mismatch controls pass.\n\nReceiving verification for `Erdos1041.erdos_1041`: [run 38367503435](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38367503435) compares the actual FC declaration at `49e83c34` with [proof source `d1fe386f`](https://github.com/wcook04/plectis-erdos-lean/blob/d1fe386ff1d6f6ca49bbad2e8a309fea2b58cb85/verification/FC1041HausdorffCurrent/FC6971HausdorffSolution.lean). Statements and dependencies match; default Lean and NanoDa accept under the three permitted core axioms, and positive/mismatch controls pass.\n\nReceiving verification for `Erdos257.erdos_257.variants.summable_reciprocal_support`: [run 38364924459](https://github.com/wcook04/plectis-erdos-lean/actions/runs/38364924459) compares the actual FC declaration at `49e83c34` with [proof source `d1fe386f`](https://github.com/wcook04/plectis-erdos-lean/blob/d1fe386ff1d6f6ca49bbad2e8a309fea2b58cb85/verification/FC257ReciprocalSupportCurrent/FC6529Solution.lean). Statements and dependencies match; default Lean and NanoDa accept under the three permitted core axioms, and positive/mismatch controls pass.\n\nOn 13 October, [Bo Cowgill approved head `49e83c34`](https://github.com/google-deepmind/formal-conjectures/pull/7260#pullrequestreview-5507189340). He checked preservation of all five FC statements, including subtype summability, integer and shifted divisor-series indexing, and the #1041 Hausdorff measure definition. He checked the existing author-hosted receiving logs against this FC head and proof source `d1fe386f`; he did not independently rerun the external proofs. The approval is bound to this head and selected declaration set. The request remains open with review required; maintenance adds no mathematical result.\n\nOpen; aggregate review decision: **review required**.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/7260#pullrequestreview-5507189340) on current head, 2026-10-13.\n\n### Authorship and closed requests\n\n**[#6578 · Add Will Cook to AUTHORS](https://github.com/google-deepmind/formal-conjectures/pull/6578)**\n\nAdministrative attribution: adds Will Cook to AUTHORS; excluded from the mathematical contribution count.\n\nMerged 2026-09-24.\n\nLatest individual review: [approved](https://github.com/google-deepmind/formal-conjectures/pull/6578#pullrequestreview-5309248110) on current head, 2026-09-24.\n\nExact target names, reviewed commits and observation fields are recorded in [the crosswalk source](../formal_conjectures_crosswalk.json). Open the linked request for subsequent activity.\n\n## Pinned statement comparison\n\n**Upstream:** [https://github.com/google-deepmind/formal-conjectures](https://github.com/google-deepmind/formal-conjectures) at exact commit [`f2de2ed5841e2105009be778ada0c40c08980125`](https://github.com/google-deepmind/formal-conjectures/commit/f2de2ed5841e2105009be778ada0c40c08980125). Source hashes are SHA-256 over exact file bytes.\n\n**Boundary:** this section records statement identity and local adapter evidence at the pinned snapshot. It does not establish novelty or historical correspondence. The exact Formal Conjectures #1041 Hausdorff path-image statement is refuted by ani's one-polynomial example, while correspondence with the 1958 wording still needs independent review. The other seven original targets remain open. Historical adapter-admission fields below are not current upstream lifecycle; use the dated contribution activity above.\n\n| Problem | Upstream primary declaration | Adapter |\n|---:|---|---|\n| #68 | `Erdos68.erdos_68` | `not_a_candidate` |\n| #243 | `Erdos243.erdos_243` | `not_a_candidate` |\n| #249 | `Erdos249.erdos_249` | `not_a_candidate` |\n| #251 | `Erdos251.erdos_251` | `not_a_candidate` |\n| #257 | `Erdos257.erdos_257` | `checked_against_upstream_statement` |\n| #269 | `Erdos269.erdos_269.variants.irrational` | `not_a_candidate` |\n| #1041 | `Erdos1041.erdos_1041` | `not_a_candidate` |\n| #1049 | `Erdos1049.erdos_1049` | `checked_against_upstream_statement` |\n\n## Pinned upstream sources\n\nEach file is read at the commit above. Each hash is SHA-256 over exact file bytes; the per-problem sections below link the exact declaration line.\n\n```text\n#68    FormalConjectures/ErdosProblems/68.lean\n       sha256:ae87fc60cac529122b9a08cbae11df1c98889461372368a3a1b508b84bff11aa\n#243   FormalConjectures/ErdosProblems/243.lean\n       sha256:c9689f42fe49827990e61113996e89a14ebbca502095b6334cb4be8e9d9f8117\n#249   FormalConjectures/ErdosProblems/249.lean\n       sha256:f7f24be33d689fff45c8ecbb6d3ed026eea9e770db87f3c108f5b5ff0c2c5943\n#251   FormalConjectures/ErdosProblems/251.lean\n       sha256:d2ab131b7662a7ea25717dc65ca227be0927c63d954c74c5d43680194d633184\n#257   FormalConjectures/ErdosProblems/257.lean\n       sha256:bde5d5e3940b45ad4ca53b706f9fb6cf1a5636650c8fbc218904e43c3c35c66c\n#269   FormalConjectures/ErdosProblems/269.lean\n       sha256:7816216886e47aca258974227a526165a2d99e4e6491cdf700b692292ebe45af\n#1041  FormalConjectures/ErdosProblems/1041.lean\n       sha256:368e0cf749e5b0e50ea300f215199375c4f79b8bf325d3fb7877e48f2d9628ca\n#1049  FormalConjectures/ErdosProblems/1049.lean\n       sha256:f2eb2ab5016f7c37ef9d9000c066362296dfafe11786b6bec9b2d1b96bf5222c\n```\n\n## Per-problem comparison\n\n### Erdős #68\n\nLocal question: Is the series sum_{n >= 2} 1/(n! - 1) irrational?\n\n- Canonical local return route: `python3 scripts/query_corpus.py --route erdos_68` (canonical problem packet); Returns the local problem packet with result families, declarations, papers and sources, and the exact open boundary.\n\n- Upstream declaration: [`Erdos68.erdos_68`](https://github.com/google-deepmind/formal-conjectures/blob/f2de2ed5841e2105009be778ada0c40c08980125/FormalConjectures/ErdosProblems/68.lean#L33) at `FormalConjectures/ErdosProblems/68.lean:33`; proof status `sorry`.\n- Statement scope: The same displayed factorial-denominator irrationality question.\n- Indexing: Formal Conjectures sums over n : Nat after the explicit shift n + 2; the displayed local question sums over integers n >= 2.\n- Casts and ambient types: The summand is explicitly real after casting (n + 2).factorial into Real in the denominator.\n- Answer/proof status: The upstream declaration uses answer(sorry) and its proof is sorry; it is statement prior art, not proof authority.\n- Conservative verdict: `statement_level_alignment_only`.\n- Machine-checked equivalence: `none`.\n- Historical adapter-admission record: `not_ready_to_submit`. See [contribution activity](#contribution-activity) for upstream lifecycle.\n\n### Erdős #243\n\nLocal question: Under a rapid-growth hypothesis on an integer sequence, does rationality of its reciprocal sum force the sequence to satisfy the Sylvester recurrence eventually?\n\n- Canonical local return route: `python3 scripts/query_corpus.py --route erdos_243` (canonical problem packet); Returns the local problem packet with result families, declarations, papers and sources, and the exact open boundary.\n\n- Upstream declaration: [`Erdos243.erdos_243`](https://github.com/google-deepmind/formal-conjectures/blob/f2de2ed5841e2105009be778ada0c40c08980125/FormalConjectures/ErdosProblems/243.lean#L38) at `FormalConjectures/ErdosProblems/243.lean:38`; proof status `sorry`.\n- Statement scope: The rapid-growth, rational reciprocal-sum, eventual-Sylvester question is represented, but the local work proves only conditional state-system consequences.\n- Indexing: The upstream sequence is zero-based and refers to a (n - 1), using truncated Nat subtraction; comparison with a traditional sequence starting at a_1 requires discarding a finite prefix.\n- Casts and ambient types: The growth limit is Real-valued, while ha₂ is Summable for a Rational-valued reciprocal sequence. This asserts a Rational sum exists and must be related explicitly to a Real rationality premise.\n- Answer/proof status: The open theorem has a sorry proof and no answer wrapper; the conclusion is an eventual filter statement.\n- Conservative verdict: `statement_level_alignment_only`.\n- Machine-checked equivalence: `none`.\n- Historical adapter-admission record: `not_ready_to_submit`. See [contribution activity](#contribution-activity) for upstream lifecycle.\n\n### Erdős #249\n\nLocal question: Is the binary Lambert series sum phi(n)/2^n irrational?\n\n- Canonical local return route: `python3 scripts/query_corpus.py --route erdos_249` (canonical problem packet); Returns the local problem packet with result families, declarations, papers and sources, and the exact open boundary.\n\n- Upstream declaration: [`Erdos249.erdos_249`](https://github.com/google-deepmind/formal-conjectures/blob/f2de2ed5841e2105009be778ada0c40c08980125/FormalConjectures/ErdosProblems/249.lean#L35) at `FormalConjectures/ErdosProblems/249.lean:35`; proof status `sorry`.\n- Statement scope: The binary totient-series irrationality question is represented; neither corpus solves it.\n- Indexing: Formal Conjectures sums from n = 0. The n = 0 totient term is zero, so the displayed positive-index series is expected to agree, but this crosswalk contains no Lean reindexing proof.\n- Casts and ambient types: Irrational forces a Real-valued sum and Lean inserts coercions for Nat totients and powers; the surface syntax leaves those casts implicit.\n- Answer/proof status: The upstream declaration uses answer(sorry) and its proof is sorry.\n- Conservative verdict: `statement_level_alignment_only`.\n- Machine-checked equivalence: `none`.\n- Historical adapter-admission record: `not_ready_to_submit`. See [contribution activity](#contribution-activity) for upstream lifecycle.\n\n### Erdős #251\n\nLocal question: Is the dyadic series of consecutive primes irrational? Equivalently, is the corresponding consecutive-prime-gap dyadic series irrational?\n\n- Canonical local return route: `python3 scripts/query_corpus.py --route erdos_251` (canonical problem packet); Returns the local problem packet with result families, declarations, papers and sources, and the exact open boundary.\n\n- Upstream declaration: [`Erdos251.erdos_251`](https://github.com/google-deepmind/formal-conjectures/blob/f2de2ed5841e2105009be778ada0c40c08980125/FormalConjectures/ErdosProblems/251.lean#L31) at `FormalConjectures/ErdosProblems/251.lean:31`; proof status `sorry`.\n- Statement scope: The same dyadic prime-series irrationality question is represented.\n- Indexing: Nat.nth Nat.Prime is zero-based and the denominator is 2^n. Relative to the displayed convention p_1 / 2^1 + p_2 / 2^2 + ..., the upstream value is multiplied by 2; irrationality should be invariant under that nonzero rational factor, but no adapter theorem is recorded here.\n- Casts and ambient types: The prime and power expressions are Nat-valued surface syntax coerced into the Real-valued sum required by Irrational.\n- Answer/proof status: The upstream declaration uses answer(sorry) and its proof is sorry.\n- Conservative verdict: `statement_level_alignment_only`.\n- Machine-checked equivalence: `none`.\n- Historical adapter-admission record: `not_ready_to_submit`. See [contribution activity](#contribution-activity) for upstream lifecycle.\n\n### Erdős #257\n\nLocal question: Is the sum of 1/(2^n-1) over every infinite set of positive exponents irrational?\n\n- Canonical local return route: `python3 scripts/query_corpus.py --route erdos_257` (canonical problem packet); Returns the local problem packet with result families, declarations, papers and sources, and the exact open boundary.\n\n- Upstream declaration: [`Erdos257.erdos_257`](https://github.com/google-deepmind/formal-conjectures/blob/f2de2ed5841e2105009be778ada0c40c08980125/FormalConjectures/ErdosProblems/257.lean#L35) at `FormalConjectures/ErdosProblems/257.lean:35`; proof status `sorry`.\n- Statement scope: The upstream open theorem quantifies over every infinite Nat support. The local expansion studies the same positive-exponent Mersenne support problem without proving the universal endpoint.\n- Indexing: The upstream support may contain exponent zero. Its n = 0 summand is 1 / 0 = 0 under Lean's division convention, so zero is analytically invisible; a formal adapter must still normalize Set Nat and subtype indexing explicitly.\n- Casts and ambient types: The denominator power is inferred in Real from the explicit numerator cast. The solved variant instead sums Nat divisor counts coerced to Real.\n- Answer/proof status: The universal theorem and solved-variant irrationality theorem are sorry; the Lambert identity tsum_top_eq is proved upstream.\n- Conservative verdict: `adapter_checked_against_pinned_statement`.\n- Machine-checked equivalence: `Erdos249257.FormalConjecturesAdapter.erdos_257_variants_tsum_top` in `adapters/FormalConjecturesAdapter.lean`, checked in this repository at leanprover/lean4:v4.29.1, and independently at Formal Conjectures' own leanprover/lean4:v4.27.0 with the corpus theorems supplied as axioms so that the bridge alone was checked; axioms `propext`, `Classical.choice`, `Quot.sound`.\n- Historical adapter-admission record: `submitted_upstream`. See [contribution activity](#contribution-activity) for upstream lifecycle.\n\n### Erdős #269\n\nLocal question: For a finite set of at least two primes, is the sum of reciprocals of the running least common multiples of the smooth numbers irrational? This library treats the three-prime case.\n\n- Canonical local return route: `python3 scripts/query_corpus.py --route erdos_269` (canonical problem packet); Returns the local problem packet with result families, declarations, papers and sources, and the exact open boundary.\n\n- Upstream declaration: [`Erdos269.erdos_269.variants.irrational`](https://github.com/google-deepmind/formal-conjectures/blob/f2de2ed5841e2105009be778ada0c40c08980125/FormalConjectures/ErdosProblems/269.lean#L73) at `FormalConjectures/ErdosProblems/269.lean:73`; proof status `sorry`.\n- Statement scope: Formal Conjectures states finite-prime rational and irrational variants and an infinite-prime theorem. The local programme proves only subsidiary structure for the three-prime case.\n- Indexing: The upstream series includes n = 0 and its empty-prefix lcm term; comparison with a convention beginning at the first nonempty prefix differs by a finite rational normalization that is not formalized here.\n- Casts and ambient types: partialLcm is Nat-valued and coerced into a Real reciprocal. The rational variant expresses rationality using existence of q : Rational equal to the Real series through coercion.\n- Answer/proof status: All three upstream research endpoints have sorry proofs; none is proof authority for the local results.\n- Conservative verdict: `statement_level_alignment_only`.\n- Machine-checked equivalence: `none`.\n- Historical adapter-admission record: `not_ready_to_submit`. See [contribution activity](#contribution-activity) for upstream lifecycle.\n\n### Erdős #1041\n\nLocal question: Must two roots of a monic polynomial in the open unit disc be joined by a sub-two-length curve inside its unit lemniscate? Ani’s degree-seven example refutes the exact Formal Conjectures statement; correspondence with the 1958 wording awaits human review.\n\n- Canonical local return route: `python3 scripts/query_corpus.py --route erdos_1041` (canonical problem packet); Returns the local problem packet with the exact Formal Conjectures Hausdorff refutation, credited source, papers, declarations and historical-review boundary.\n\n- Upstream declaration: [`Erdos1041.erdos_1041`](https://github.com/google-deepmind/formal-conjectures/blob/f2de2ed5841e2105009be778ada0c40c08980125/FormalConjectures/ErdosProblems/1041.lean#L67) at `FormalConjectures/ErdosProblems/1041.lean:67`; proof status `sorry`.\n- Statement scope: The local Hausdorff theorem refutes the exact Formal Conjectures path-image-length proposition for ani's degree-seven polynomial. The 1958 curve-length wording remains a separate historical correspondence audit.\n- Indexing: There is no series indexing issue. The upstream theorem quantifies polynomial degree n >= 2 and selects roots through multiset containment, including multiplicity.\n- Casts and ambient types: Both formal statements use a Complex path whose image lies in the strict unit lemniscate; length is one-dimensional Hausdorff measure of Set.range γ in ENNReal. The older local total-variation theorem is separate.\n- Answer/proof status: At the pinned snapshot, the upstream declaration has answer(sorry) and a sorry proof. Local Lean proves the exact Hausdorff negation and answer(False) form. The later answer correction merged as PR #6505; independent historical adjudication remains unrecorded.\n- Conservative verdict: `statement_level_alignment_only`.\n- Machine-checked equivalence: `none`.\n- Local refutation: `Erdos1041.Counterexample.erdos1041_hausdorff_negation` and `Erdos1041.Counterexample.erdos1041_hausdorff_answer_false` at `7380b7871687b6bcc41ca0143c61f232e8af6500`; ani, erdosproblems.com forum thread 1041, 7 September 2026. Exact Formal Conjectures path-image formulation refuted; independent review of correspondence with the 1958 wording remains open.\n- Historical adapter-admission record: `not_ready_to_submit`. See [contribution activity](#contribution-activity) for upstream lifecycle.\n\n### Erdős #1049\n\nLocal question: For which rational bases is the corresponding series irrational? The first resistant explicit base is three halves.\n\n- Canonical local return route: `python3 scripts/query_corpus.py --route erdos_1049` (canonical problem packet); Returns the local problem packet with result families, declarations, papers and sources, and the exact open boundary.\n\n- Upstream declaration: [`Erdos1049.erdos_1049`](https://github.com/google-deepmind/formal-conjectures/blob/f2de2ed5841e2105009be778ada0c40c08980125/FormalConjectures/ErdosProblems/1049.lean#L40) at `FormalConjectures/ErdosProblems/1049.lean:40`; proof status `sorry`.\n- Statement scope: The upstream open theorem asks for every rational t > 1. The local programme does not prove that endpoint, but its full-support theorem covers positive integer bases and may discharge the upstream solved integer-base variant after adapters.\n- Indexing: The upstream sums use positive naturals. The local full-support theorem uses k : Nat with exponent k + 1; an explicit equivalence between those index types is required.\n- Casts and ambient types: The upstream open base is Rational and the solved variant is Integer, both cast into Real powers. The local theorem takes b : Nat and casts it to Real, so Integer-to-Nat and cast/power normalization must be checked.\n- Answer/proof status: The rational-base conjecture and integer-base variant have sorry proofs. The upstream Lambert identity is proved, including branches outside the t > 1 problem regime that rely on Lean's nonsummable-tsum convention.\n- Conservative verdict: `adapter_checked_against_pinned_statement`.\n- Machine-checked equivalence: `Erdos249257.FormalConjecturesAdapter.erdos_1049_variants_geq_2_integer` in `adapters/FormalConjecturesAdapter.lean`, checked in this repository at leanprover/lean4:v4.29.1, and independently at Formal Conjectures' own leanprover/lean4:v4.27.0 with the corpus theorems supplied as axioms so that the bridge alone was checked; axioms `propext`, `Classical.choice`, `Quot.sound`.\n- Historical adapter-admission record: `submitted_upstream`. See [contribution activity](#contribution-activity) for upstream lifecycle.\n\n## Adapter candidates\n\n### Erdős #257: `Erdos257.erdos_257.variants.tsum_top`\n\nThe upstream proposition is stated verbatim in the adapter and derived from this library. This local check is separate from the dated contribution and review record above.\n\nLocal evidence:\n\n- `irrational_erdosSum_full_support` in `Erdos249257/CertificateKernel.lean:8328`\n- `irrational_erdosBorwein_series` in `Erdos249257/CertificateKernel.lean:8335`\n\nUnproved bridge obligations:\n\n- Relate the upstream Nat-indexed Lambert series, whose n = 0 term is zero, to the local successor-indexed base-2 series.\n- Transport irrationality through the upstream proved divisor-sum identity without importing a proof-incompatible local environment.\n- Review theorem attribution, contribution scope, licensing, automation disclosure, and upstream contribution policy with a human before any patch is prepared.\n\n### Erdős #1049: `Erdos1049.erdos_1049.variants.geq_2_integer`\n\nThe upstream proposition is stated verbatim in the adapter and derived from this library. This local check is separate from the dated contribution and review record above.\n\nLocal evidence:\n\n- `irrational_erdosSum_full_support` in `Erdos249257/CertificateKernel.lean:8328`\n\nUnproved bridge obligations:\n\n- Convert t : Integer with t >= 2 to a natural base and prove all Integer/Nat/Real cast and power identities used by the target.\n- Reindex the positive-natural upstream sum to the local Nat successor sum.\n- Confirm that the proof uses only upstream-compatible imports and does not accidentally rely on the irrelevant nonsummable branches of the upstream Lambert identity.\n- Review theorem attribution, contribution scope, licensing, automation disclosure, and upstream contribution policy with a human before any patch is prepared.\n\n## Cross-index matches\n\nUpstream declarations this library can discharge that sit under a problem number it does not work on. An index keyed by local problem cannot hold these, so they are keyed by the upstream declaration.\n\n### Upstream #258: [`Erdos258.erdos_258.variants.constant`](https://github.com/google-deepmind/formal-conjectures/blob/f2de2ed5841e2105009be778ada0c40c08980125/FormalConjectures/ErdosProblems/258.lean#L65)\n\n- Recorded here because: This library does not work on Erdos #258, so an index keyed by local problem number had nowhere to put this match and did not record it. The upstream statement is the same mathematics as the #257 and #1049 targets in a third coefficient convention.\n- Machine-checked equivalence: `Erdos249257.FormalConjecturesAdapter.erdos_258_variants_constant` in `adapters/FormalConjecturesAdapter.lean`, checked in this repository at leanprover/lean4:v4.29.1, and independently at Formal Conjectures' own leanprover/lean4:v4.27.0 with the corpus theorems supplied as axioms so that the bridge alone was checked; axioms `propext`, `Classical.choice`, `Quot.sound`.\n- Historical adapter-admission record: `submitted_upstream`; upstream lifecycle is recorded above.\n\n## Reproduction\n\nOffline contract and projection check:\n\n```console\npython3 scripts/check_formal_conjectures_crosswalk.py\n```\n\nByte-level verification against an exact upstream checkout:\n\n```console\npython3 scripts/check_formal_conjectures_crosswalk.py --upstream-checkout /path/to/formal-conjectures\n```\n","body_format":"text"},"external-verification":{"document_id":"external-verification","source_path":"docs/EXTERNAL_VERIFICATION.md","role":"comparator_explanation","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/EXTERNAL_VERIFICATION.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/EXTERNAL_VERIFICATION.md","sha256":"sha256:bb2dff08d6266d13945575f47b47aef7b553eabd988542a84f5b2fb597209bb3","byte_count":193332,"origin":"public_main_checkout","body":"<!-- Generated by scripts/build_external_verification.py; do not edit. -->\n# Plectis verification: eight Erdős problem programmes\n\n> [!IMPORTANT]\n> **Status boundary:** Using the degree-seven polynomial constructed by the erdosproblems.com contributor ani, Lean proves that every preconnected strict-lemniscate set containing two distinct roots has one-dimensional Hausdorff measure greater than two. This refutes the exact Formal Conjectures path-image-length statement; the separate total-variation bound is also checked. The other seven targets remain open. Independent human review of correspondence with the 1958 wording has not been recorded. Comparator checks only selected exact statements, axioms and kernel acceptance; it does not assess novelty or historical correspondence.\n> **Review posture:** Self-assessed and agent-checked; no human mathematical peer review is claimed.\n\n**What this is.** Plectis is an AI-assisted research system. This public surface shows one checked frontier for each of eight Erdős problem programmes. For each programme, read the question, the exact checked object, and the remaining open step before opening the technical registry.\n\n**How verification works.** This dossier describes 49 selected interfaces. The executable [Comparator roster](../verification/comparator.json) contains 53 theorem declarations, including those interfaces and companion declarations. Their statements are declared again without proofs. Comparator checks that the selected proof declarations match those separately declared statements and respect the configured axiom budget. A named altered statement must fail. This checks formal propositions only. It does not assess exposition, citations, intended meaning, novelty, or significance. Technical detail is in the [Comparator interface appendix](#comparator-interface-appendix).\n\n## Mathematical signal spine\n\nThis order projects the repository's authored `candidate_ranking` for a possible Palomar submission; it is independent of Comparator roster order, programme number, insertion time, theorem count, and qualification convenience. Comparator coverage is the exhaustive evidence inventory, not a significance ranking. Checked propositions are therefore given unequal reader attention: each promoted result keeps its hard step and surviving boundary adjacent.\n\n### Reader tiers\n\n- **Completed direct results:** unconditional checked consequences with their scope kept visible.\n- **Conditional endpoint routes:** target-facing routes whose explicit producer or selector remains open.\n- **Exact endpoint reductions and structural results:** results that sharpen the remaining problem without being presented as endpoint theorems.\n\n### Source-ranked frontier\n\n1. **Finite prime weighted support** (<code>Erdos249257.ExternalVerification.divisibilityWeightedClaim</code>; `selected`)\n   - **Reader tier.** completed direct result\n   - **Consequence.** For every integer base b ≥ 2 it proves irrationality on every infinite positive support of finite prime-weighted mass; a positive host of finite binary weighted mass gives irrationality for every infinite subset at every integer base.\n   - **Load-bearing mechanism.** The source proof builds a concrete weighted dyadic-mean producer with divisibility and scale control, obtains cofinal close returns, and consumes them in the irrationality criterion. The separate fixed-base hereditary theorem restricts the nonnegative weighted summability witness to subsets.\n   - **Source and evidence.** [Lean source](../lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean); The source theorem is Lean-checked in PaperCompleteR8/WeightedReturn.lean:120. This packet includes a separately declared Comparator interface and thin transport. The [replay guide's repository CI record](verification/EXTERNAL_VERIFICATION_REPLAY.md#pull-request-merge-gate) reports a passing weighted-support positive check and deliberate statement-mismatch rejection at commit `7e33a58bb86180013b1fb855cdd8aaff1d7f9057`. The recorded CI result applies only to that snapshot. Independent Linux replay remains pending; verification of later commits requires receipts bound to those commits.\n   - **Boundary.** The finite-prime weighted-summability premise is explicit in both clauses. This criterion does not cover every infinite support or settle universal Erdős #257; no novelty, priority, significance, or external human-review claim is made. **Overclaim risk.** The finite-prime weighted-summability hypothesis and 0 ∉ support condition must stay visible. The theorem does not prove irrationality for arbitrary infinite support, novelty, priority, or external human review.\n\n2. **Known irrational supports** (<code>Erdos249257.ExternalVerification.irrational_erdosSum_full_support</code>; `represented`)\n   - **Reader tier.** completed direct result\n   - **Consequence.** Directly settles the canonical full-support member of Erdős #257 for every integer base b >= 2, while leaving universal arbitrary-support irrationality open.\n   - **Load-bearing mechanism.** The source exposes the bounded Bertrand/CRT first-block frame, middle-window divisor-pair averaging with pigeonhole selection, weighted tail bounds, and explicit parameter closure.\n   - **Source and evidence.** [Lean source](../lean/Erdos249257/CertificateKernel.lean); The exact source theorem is committed at Erdos249257/CertificateKernel.lean:8323-8331 and is transported by the committed Challenge/Solution pair.\n   - **Boundary.** This is the classical full-support theorem, not universal Erdos #257. **Overclaim risk.** Low only under the stated boundary: this is full support, not all infinite supports, and local proof plus Comparator is not a novelty or human-review claim.\n\n3. **Pairwise coprime support** (<code>Erdos249257.ExternalVerification.irrational_erdosSupportSeries_pairwise_coprime</code>; `represented`)\n   - **Reader tier.** completed direct result\n   - **Consequence.** Direct irrationality for every infinite pairwise-coprime support with summable reciprocal mass, but not for arbitrary supports.\n   - **Load-bearing mechanism.** Adaptive CRT weighted coefficient certificates absorb first-block hits and control residue strays through summable reciprocals.\n   - **Source and evidence.** [Lean source](../lean/Erdos249257/CertificateKernel.lean); The exact source theorem is committed at Erdos249257/CertificateKernel.lean:10768-10782 and is transported by the committed Challenge/Solution pair.\n   - **Boundary.** This is the pairwise-coprime support theorem, not universal Erdos #257; the infinitude, pairwise-coprimality, and summable-reciprocal hypotheses remain explicit. **Overclaim risk.** Moderate unless infinitude, pairwise coprimality, and summable reciprocal mass stay visible in every projection.\n\n4. **Orthogonal petal sunflower reduction** (<code>Erdos249257.ExternalVerification.irrational_erdosSupportSeries_of_orthogonalPetalBouquet</code>; `represented`)\n   - **Reader tier.** conditional endpoint route\n   - **Consequence.** Direct irrationality under the displayed structure, but the structural predicates are not constructed for arbitrary supports.\n   - **Load-bearing mechanism.** The conditional selector is the natural friction; omitting it would turn a reduction into a false unconditional theorem.\n   - **Source and evidence.** [Lean source](../lean/Erdos249257/SupportSunflowerDichotomy.lean); Exact committed source and Comparator transport are available.\n   - **Boundary.** This is a conditional reduction: the bouquet structure and uniform tail-selection predicate remain hypotheses; neither is constructed for arbitrary supports, so this does not add an unconditional Erdos #257 support family or solve the universal problem. **Overclaim risk.** High if the bouquet and uniform tail-selection hypotheses are silently treated as automatic.\n\n5. **Periodic nonnegative weight irrationality** (<code>Erdos249257.ExternalVerification.irrational_ratWeightSeries_eventuallyPeriodic</code>; `represented`)\n   - **Reader tier.** completed direct result\n   - **Consequence.** Close to the Lambert-series endpoint but not a universal support theorem and not the mixed-sign statement.\n   - **Load-bearing mechanism.** A periodic divisor-orbit sieve and denominator clearing route are visible in the source.\n   - **Source and evidence.** [Lean source](../lean/Erdos249257/CertificateKernel.lean); Exact committed source and Comparator transport are available.\n   - **Boundary.** This is the nonnegative rational eventually-periodic coefficient theorem; its nonnegativity, eventual periodicity, and positive-tail hypotheses remain explicit, and the broader mixed-sign periodic theorem is not claimed. **Overclaim risk.** Moderate if the mixed-sign or arbitrary-support cases are inferred.\n\n6. **Actual lcm orbit separation** (<code>Erdos249257.ExternalVerification.irrational_totientSeries_of_actualLcmOrbitSeparationSupply</code>; `represented`)\n   - **Reader tier.** conditional endpoint route\n   - **Consequence.** It points directly at the binary totient irrationality endpoint for Erdős #249, conditional on a cofinal separation supply; it does not prove that supply.\n   - **Load-bearing mechanism.** The source carries a raw adjacent-suffix approximation, an explicit error radius, and uniform separation from every integer into a signed-margin producer. The cofinal PowerTwoActualLcmOrbitSeparationSupply is the natural friction and remains a premise.\n   - **Source and evidence.** [Lean source](../lean/Erdos249257/TotientActualLcmOrbitSeparation.lean); The source definition and implication are committed at Erdos249257/TotientActualLcmOrbitSeparation.lean:305, and the exact wrapper is committed in ExternalVerification/Challenge.lean and ExternalVerification/Solution.lean at the configured declaration.\n   - **Boundary.** The cofinal separation supply is an open premise, stronger than mere non-integrality; this sufficient route is not an equivalence and leaves the punctured/top-edge boundary and the Erdős #249 endpoint open. **Overclaim risk.** High unless the two cofinal supply clauses stay attached: the implication is source-faithful and checked, but the supply itself is not proved here.\n\n7. **First harmonic pivot decomposition** (<code>Erdos249257.ExternalVerification.irrational_totient_series_of_pivotResidualDecorrelation</code>; `represented`)\n   - **Reader tier.** conditional endpoint route\n   - **Consequence.** It points directly at Erdős #249 irrationality from an explicit cofinal analytic producer, independently of the actual-LCM orbit route; that producer remains unproved.\n   - **Load-bearing mechanism.** Supplier fibres, pivot correlation, bad and non-supplier contributions, and the controlled remainder are kept separate. The finite counterexample to global isolation and the missing DTWPivotResidualDecorrelation supply are the natural friction.\n   - **Source and evidence.** [Lean source](../lean/Erdos249257/FirstHarmonicPivot.lean); The source theorem, canonical review family, and exact Comparator endpoint wrapper are committed; the cofinal decorrelation producer is explicitly absent.\n   - **Boundary.** The four-term decomposition and 9X/10 gap remain conditional on the explicit dyadic-room and PivotBudgetAt hypotheses; the cofinal decorrelation producer is unproved. The fibre/image lemmas do not give global supplier-prime isolation or a prime-distribution estimate, and no Erdős #249 irrationality conclusion is claimed. **Overclaim risk.** High unless DTWPivotResidualDecorrelation remains attached as an unproved cofinal premise; this is not an unconditional #249 result.\n\n8. **Strict prime tail orbit gap** (<code>Erdos249257.ExternalVerification.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap</code>; `represented`)\n   - **Reader tier.** conditional endpoint route\n   - **Consequence.** It reaches the Erdős #249 irrationality consumer conditionally and is more than a finite normal form, but neither phase-density nor cofinal-prime producer is proved.\n   - **Load-bearing mechanism.** Unit norm and the complement estimate give the exact 89/100 block bound; a 1/100 truncation budget yields the 9/10 finite gap, while the prime branch uses an adaptive positive margin and exact pivot-fibre membership. The two global producer predicates are the natural friction.\n   - **Source and evidence.** [Lean source](../lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean); The exact source declarations, canonical strict_prime_tail_orbit_gap family, and Comparator representative are committed at Claims ddbe2870 and transport e7a58bb3.\n   - **Boundary.** The phase-density, cofinal natural-prime strict-gap, associated uniform-margin, and cofinal nonpositive-prime-shift producers remain unproved for the actual totient orbit. Exact phase/integrality equivalence, finite dyadic-root exclusion for 1 ≤ h ≤ 16 and k ≤ 14, and non-dyadicity-to-cofinal-nonpositive-phase implications are classification or finite evidence only; they do not supply density or prime alignment, and the source does not prove actual root entry. This is a conditional reduction only: no unconditional Erdős #249 endpoint, novelty, priority, significance, or external-review claim is made. It is distinct from the finite first_harmonic_pivot_decomposition and actual_lcm_orbit_separation families. **Overclaim risk.** High unless the unproved density, cofinal natural-prime strict-gap, uniform-margin, and cofinal nonpositive-prime-shift producers stay explicit; no unconditional #249, novelty, priority, significance, or external-review claim follows.\n\n9. **Factorial carry characterisation** (<code>Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_nonunit_carries</code>; `subordinate`)\n   - **Reader tier.** exact reduction or structural result\n   - **Consequence.** Maximal endpoint proximity for #68, but it does not prove the cofinal producer.\n   - **Load-bearing mechanism.** The carry recurrence is the useful mechanism; the missing cofinal witness is the natural unresolved friction.\n   - **Source and evidence.** [Lean source](../lean/ErdosProblems/Erdos68/FactorialGapPlateauCore.lean); Exact committed source and Comparator transport are available.\n   - **Boundary.** The equivalence does not supply a cofinal carry producer. **Overclaim risk.** High if the equivalence is mistaken for a cofinal carry proof.\n\n10. **Prime gap reformulation** (<code>Erdos249257.ExternalVerification.irrational_tsum_primeDyadicTerm_iff_primeGap</code>; `subordinate`)\n   - **Reader tier.** exact reduction or structural result\n   - **Consequence.** Very close to #251's target but proves neither side irrational and retains a summability premise.\n   - **Load-bearing mechanism.** The key mechanism is summation-by-parts normalization; the target irrationality remains open.\n   - **Source and evidence.** [Lean source](../lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean); Exact committed source and Comparator transport are available.\n   - **Boundary.** The equivalence does not prove irrationality of either series. **Overclaim risk.** High if equivalence is read as irrationality.\n\n11. **Totient carry anti compression** (<code>Erdos249257.ExternalVerification.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank</code>; `represented`)\n   - **Reader tier.** exact reduction or structural result\n   - **Consequence.** This is an orbit-level necessary consequence of the #249 rational branch, closer to the endpoint than coefficient-side kernel rank alone, but it derives no contradiction.\n   - **Load-bearing mechanism.** The theorem couples modular orbit regularity with exponential finite-level anti-compression. Its natural friction is exact: modulo a divisor of the multiplier the forcing can vanish and the carry can become geometric, so periodicity supplies no finite-Q-rank upper bound.\n   - **Source and evidence.** [Lean source](../lean/Erdos249257/TotientTailCarryPeriod.lean); The exact source theorem, canonical review family, and Comparator representative are committed; the missing finite-rank contradiction is explicit.\n   - **Boundary.** This is a necessary consequence of rationality only: it gives no contradiction, no finite-rank upper bound for a rationality-supplied carry, and no irrationality proof for Erdős #249. Modulo a divisor of the multiplier, the forcing disappears and the carry becomes geometric, so quotient periodicity alone supplies no finite-ℚ-rank upper bound. The coefficient-side kernel-rank and certificate-equivalence families are not duplicated, and #249 remains open. **Overclaim risk.** High if modular periodicity and unbounded canonical-kernel rank are reported as incompatible; the source proves no such upper bound and no #249 irrationality.\n\n12. **Half membership seam classification** (<code>Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse</code>; `represented`)\n   - **Reader tier.** exact reduction or structural result\n   - **Consequence.** It classifies the concrete #257 half-membership endpoint exactly, closer to a usable target than an analytic recovery criterion or source-only certificate normal form, but supplies no occurrence witness.\n   - **Load-bearing mechanism.** Finite integer seam construction and terminal-bit normalization expose the global supply condition in interchangeable forms. The natural friction is decisive: none of those forms is proved to occur unboundedly.\n   - **Source and evidence.** [Lean source](../lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean); The exact source iff, canonical claims family, and committed Comparator representative are available; the absent occurrence producer is explicit.\n   - **Boundary.** This exact endpoint-socket classification does not supply any unbounded terminal-false or upper-or-middle occurrences, cofinal supply, skipped-support producer, or half-membership witness; finite exact rows do not establish the required global supply. It is not an unconditional or universal Erdős #257 result, and no novelty, priority, significance, or external-review claim is made. **Overclaim risk.** High if an iff is described as a half-membership witness or if finite exact rows are treated as cofinal supply; it proves no universal #257 result.\n\n13. **Negative mass recovery** (<code>Erdos249257.ExternalVerification.sylvesterNext_eventually_of_summable_negativeRelativeMass</code>; `represented`)\n   - **Reader tier.** conditional endpoint route\n   - **Consequence.** It gives a sharp conditional structural conclusion for Erdős #243 and proves that any surviving canonical orbit must evade finite negative mass, but it does not establish the original-orbit hypotheses or reciprocal-tail irrationality.\n   - **Load-bearing mechanism.** The tail-growth estimate by 1 + negativeRelativeMass feeds finite-negative-mass stabilization, killing the centered defect. The hard friction is that summability, positivity, strict centering, normalized vanishing, and the exact C/D dynamics are all premises rather than proved properties of the original orbit.\n   - **Source and evidence.** [Lean source](../lean/ErdosProblems/Erdos243/SparseResetRecovery.lean); The exact source theorem, canonical review family, and committed Comparator representative are available at Claims 6bedcde3 and transport 39d0c4bd.\n   - **Boundary.** This is a conditional recovery criterion: the source theorem assumes every displayed dynamics, positivity, strict centered-step, normalized-vanishing, and summability hypothesis, and does not prove them for the original Erdős #243 orbit. Any surviving canonical orbit must have divergent normalized negative mass. It supplies no prime-specific producer, unconditional #243 endpoint, or reciprocal-tail irrationality, and makes no novelty, priority, significance, or external-review claim. It is distinct from centered_state_dynamics, which uses a uniform lower-bound hypothesis rather than summability of normalized negative mass. **Overclaim risk.** High if the conditional summability criterion is reported as applying to the original reciprocal-tail orbit; it proves no unconditional #243 endpoint, prime-specific producer, or reciprocal-tail irrationality.\n\n### Natural friction and no-go boundaries\n\nThese checked obstructions are an alphabetical, deliberately unranked tier. They show where natural proof friction survives without competing with direct endpoint results for headline position.\n\n- **[#243](#programme-243) · Bounded negative exclusion** (`bounded_negative_exclusion`)<br>\n  Normalised vanishing excludes a cofinally bounded negative part and forces eventual Sylvester behaviour.<br>\n  **Boundary.** Every positivity, dynamics, bounded-rise, and vanishing hypothesis remains explicit.<br>\n  *Evidence.* conditional reduction and no-go result · Lean kernel plus Comparator\n\n- **[#251](#programme-251) · Coefficient only no go** (`coefficient_only_no_go`)<br>\n  The exact finite carry identity and the two non-eventual-periodicity statements expose the coefficient-only barrier: irregularity and growth of a coefficient stream do not by themselves force irrationality, while the synthetic carry stream remains distinct from the actual prime-gap stream.<br>\n  **Boundary.** CoefficientOnlyNoGo checks the exact finite partial-sum identity and the two non-eventual-periodicity statements. It does not assert the infinite limit of the synthetic countermodel, identify its coefficient stream with actual prime gaps, establish a prime-gap tail bridge, prove #251 irrationality, or make novelty, priority, significance, or external-review claims.<br>\n  *Evidence.* coefficient-only barrier; novelty and significance unassessed · Lean kernel plus authored synthesis\n\n- **[#269](#programme-269) · Conditional carry escape** (`conditional_carry_escape`)<br>\n  Under the denominator-dependent cofinal local-window residue-escape predicate, no positive reduced carry can satisfy the exact multiplier recurrence together with its short bound. The load-bearing consumer is no_positive_reducedCarry_of_cofinalLocalWindowEscape; an absorbed nonzero common-factor carry reduces exactly to that consumer. CofinalLocalWindowEscape, windowBase, windowForcing, leastPositiveResidue, and the absorbed-carry bridge are subordinate finite-window mechanism evidence.<br>\n  **Boundary.** The representative assumes b,m : ℕ → ℕ, shortBound : ℕ → ℕ → ℕ, CofinalLocalWindowEscape b m shortBound, and for each positive B coprime to 30 a positive integer-valued d with d(n+1) = (b n : ℤ) d(n) − (B : ℤ) (m n : ℤ) and Int.natAbs (d n) ≤ shortBound B n; it then gives False. The absorbed-carry bridge additionally assumes smoothFactor ≠ 0, c n = smoothFactor * d n, and the corresponding exact absorbed recurrence, then cancels that factor to the same reduced consumer. The representative remains a conditional consumer for arbitrary words. For the actual three-prime word, the rationality-to-reduced-carry bridge is established separately; the source-specific cofinal escape producer is not proved. This is not a #269 endpoint or irrationality proof, and no actual-series identification, novelty, priority, significance, or external-review claim is made. It is one conditional window-carry family, distinct from the finite residue, rank-two, and weighted-phase observer families.<br>\n  *Evidence.* conditional no-go consumer; novelty and significance unassessed · Lean kernel plus Comparator\n\n- **[#1049](#programme-1049) · Coordinatewise corridor no go** (`coordinatewise_corridor_no_go`)<br>\n  The coordinatewise corridor forces a power-versus-linear inequality and cannot occur at base 3/2.<br>\n  **Boundary.** This excludes one proof architecture and proves no irrationality statement.<br>\n  *Evidence.* no-go result · Lean kernel plus Comparator\n\n- **[#1049](#programme-1049) · Endpoint residues** (`endpoint_residues`)<br>\n  Endpoint residues at 2 and 3 exclude a common multiplier under unit-endpoint hypotheses.<br>\n  **Boundary.** Other local or cross-row arithmetic remains possible.<br>\n  *Evidence.* no-go result · Lean kernel\n\n- **[#249](#programme-249) · Fixed precision transport no go** (`fixed_precision_transport_no_go`)<br>\n  At every fixed positive precision, every finite word of odd valuation-unit symbols admits a compatible centred carry completion from any incoming integer state; bounded local signatures therefore cannot exclude all finite centred endpoints.<br>\n  **Boundary.** This is a synthetic finite-word construction with unrestricted incoming/high quotient states. It does not impose the actual totient-difference relation or provide growing-precision/global correlation, and it gives no Erdős #249 endpoint or irrationality result. No novelty or priority claim is made. The one-step and finite-word completion theorems are subordinate mechanisms, not separate families.<br>\n  *Evidence.* unconditional synthetic no-go; novelty unassessed · Lean kernel plus Comparator\n\n- **[#1049](#programme-1049) · Four jet collision** (`four_jet_collision`)<br>\n  Binary selectors collide in the four-jet signature at the stated rank and depth.<br>\n  **Boundary.** A signed relation is not nonzero analytic data.<br>\n  *Evidence.* locally proved result and no-go result; novelty unassessed · Lean kernel\n\n- **[#257](#programme-257) · Half and twenty one frontiers** (`half_and_twenty_one_frontiers`)<br>\n  Exact greedy characterisations, finite-support exclusions, the odd 2-adic two-thirds-band bound, and two conditional rational-half countermodel frontiers isolate local constraints around the remaining 1/2 and 1/21 alternatives.<br>\n  **Boundary.** Neither membership question is decided; the 2-adic bound is sharp locally, and the two countermodel routes require unconstructed terminal-scaled or cofinal-cylinder antecedents.<br>\n  *Evidence.* conditional reduction and no-go result · Lean kernel plus Comparator\n\n- **[#1049](#programme-1049) · Height and pade arithmetic** (`height_and_pade_arithmetic`)<br>\n  For the explicit rectangular two-function exponent model, the cleared gap is nonpositive throughout rho >= 0 and 1 + rho <= sigma, and the threshold is at most 1/2 - 1/pi^2 with equality exactly at rho = 0, sigma = 1. The gap sign and unique-zero mechanism support this sharp threshold no-go within the wider height-and-Padé arithmetic family.<br>\n  **Boundary.** This is a sharp no-go only for the explicit rectangular two-function exponent model under the stated real hypotheses; it constructs no approximating polynomials or remainders and is not a universal Padé or Hermite-Padé no-go. It proves no irrationality at 3/2 or any general rational-base endpoint, and the analytic remainder/nonvanishing input remains untreated. No novelty, priority, significance, external-review, or endpoint claim is made; Erdős #1049 remains open.<br>\n  *Evidence.* locally proved sharp model-specific no-go; novelty unassessed · Lean kernel plus Comparator\n\n- **[#249](#programme-249) · Lcm factor ideal shift algebra no go** (`lcm_factor_ideal_shift_algebra_no_go`)<br>\n  For every t at least 3, a nonzero synthetic dyadic coboundary survives every finite integer shift polynomial while retaining the period-LCM totient factor ideals, exact cleared-prefix cancellation, and uniform state and letter bounds.<br>\n  **Boundary.** The witness is synthetic: it does not assert that its forcing letters are actual totient differences, does not address nonlinear combinations, and leaves the unbounded certificate supply for Erdős #249 open.<br>\n  *Evidence.* unconditional progress · Lean kernel plus Comparator\n\n- **[#243](#programme-243) · Negative orbit no go** (`negative_orbit_no_go`)<br>\n  Constant, eventually constant, periodic, and eventually periodic negative-error orbits are excluded in their stated regimes.<br>\n  **Boundary.** The exclusions do not cover arbitrary unbounded negative behaviour.<br>\n  *Evidence.* no-go result · Lean kernel plus Comparator\n\n- **[#269](#programme-269) · Rank two kernel no go** (`rank_two_kernel_no_go`)<br>\n  The 2,3,5 kernel is not rank one and its smallest displayed minor equals -1/15.<br>\n  **Boundary.** Failure of rank one does not itself imply irrationality.<br>\n  *Evidence.* no-go result · Lean kernel plus Comparator\n\n- **[#1049](#programme-1049) · Scalar content no go** (`scalar_content_no_go`)<br>\n  Integer scalar content changes analytic error and exterior determinant by matching factors, yielding no margin by itself.<br>\n  **Boundary.** The conclusion is scoped to one normalisation strategy.<br>\n  *Evidence.* no-go result · Lean kernel\n\n### Complete inventory, kept subordinate\n\nEvery contribution family and the documented statement-isolated interfaces remain queryable in the programme dossiers and the [Comparator interface appendix](#comparator-interface-appendix). Subordination is a presentation judgement, not deletion or an adverse mathematical disposition.\n\n### Complete serious-result universe\n\nAll 76 source-current review families are accounted for here. The categories preserve honest selection reasons while the programme dossiers below retain each family's exact mechanism and boundary. This inventory is complete but deliberately does not compete with the ranked frontier for attention.\n\n- **represented (54).** [#257](#programme-257) `achievement_set_geometry`, [#249](#programme-249) `actual_foreign_residue_projection`, [#249](#programme-249) `actual_lcm_orbit_separation`, [#257](#programme-257) `boolean_mobius_carry`, [#257](#programme-257) `boolean_mobius_exact_row_dynamics`, [#243](#programme-243) `bounded_negative_exclusion`, [#243](#programme-243) `bounded_rise_coprimality`, [#1049](#programme-1049) `calibrated_rational_hankel_countermodel`, [#243](#programme-243) `centered_state_dynamics`, [#251](#programme-251) `coefficient_only_no_go`, [#257](#programme-257) `composite_dilation_defect_identity`, [#269](#programme-269) `dyadic_block_alphabet`, [#251](#programme-251) `dyadic_tail_integrality_classification`, [#249](#programme-249) `eventually_periodic_lambert`, [#68](#programme-68) `factorial_carry_characterisation`, [#68](#programme-68) `factorial_channel_and_projection_rigidity`, [#257](#programme-257) `finite_period_noncollapse`, [#257](#programme-257) `finite_prime_weighted_support`, [#249](#programme-249) `first_harmonic_pivot_decomposition`, [#249](#programme-249) `fixed_precision_transport_no_go`, [#257](#programme-257) `half_membership_seam_classification`, [#1049](#programme-1049) `height_and_pade_arithmetic`, [#269](#programme-269) `height_fibre_and_shell`, [#269](#programme-269) `independent_clock_channel_relations`, [#251](#programme-251) `integral_shift_classification`, [#257](#programme-257) `known_irrational_supports`, [#1041](#programme-1041) `newton_value_decay`, [#257](#programme-257) `orthogonal_petal_sunflower_reduction`, [#257](#programme-257) `pairwise_coprime_support`, [#257](#programme-257) `periodic_nonnegative_weight_irrationality`, [#251](#programme-251) `prime_gap_reformulation`, [#1049](#programme-1049) `rational_base_tail_recurrence`, [#1041](#programme-1041) `ray_separation`, [#257](#programme-257) `restricted_achievement_sets`, [#1041](#programme-1041) `root_retention`, [#257](#programme-257) `separated_cut_algebraic_lambert_transcendence`, [#257](#programme-257) `shifted_odd_tail_state`, [#249](#programme-249) `signed_totient_entire_defect_interpolation`, [#251](#programme-251) `small_mismatch_criterion`, [#249](#programme-249) `squared_lambert_gcd_moments`, [#249](#programme-249) `stern_brocot_cylinder_law`, [#249](#programme-249) `stern_brocot_run_fibonacci_stability`, [#249](#programme-249) `strict_prime_tail_orbit_gap`, [#269](#programme-269) `three_prime_lcm_cells`, [#249](#programme-249) `totient_carry_anti_compression`, [#249](#programme-249) `totient_certificate_equivalences`, [#249](#programme-249) `totient_kernel_all_base_index`, [#249](#programme-249) `totient_kernel_basis`, [#249](#programme-249) `totient_kernel_rank`, [#249](#programme-249) `totient_lambert_coefficients`, [#251](#programme-251) `totient_shift_propagation`, [#249](#programme-249) `totient_visible_coprime_lattice_mass`, [#1041](#programme-1041) `translation_avoidance`, [#269](#programme-269) `weighted_phase_carry_observer`\n\n- **subordinate (7).** [#269](#programme-269) `conditional_carry_escape`, [#68](#programme-68) `factorial_conditional_producers`, [#1049](#programme-1049) `four_jet_collision`, [#257](#programme-257) `half_and_twenty_one_frontiers`, [#243](#programme-243) `negative_mass_recovery`, [#269](#programme-269) `selected_staircase_integral_height_boundary`, [#249](#programme-249) `totient_carry_rank`\n\n- **rejected (6).** [#1049](#programme-1049) `coordinatewise_corridor_no_go`, [#1049](#programme-1049) `endpoint_residues`, [#249](#programme-249) `lcm_factor_ideal_shift_algebra_no_go`, [#243](#programme-243) `negative_orbit_no_go`, [#269](#programme-269) `rank_two_kernel_no_go`, [#1049](#programme-1049) `scalar_content_no_go`\n\n- **long tail (9).** [#68](#programme-68) `factorial_finite_certificates`, [#68](#programme-68) `factorial_lcm_growth`, [#1041](#programme-1041) `lemniscate_finite_search`, [#1041](#programme-1041) `published_proof_gap`, [#1049](#programme-1049) `seven_halves_irrationality`, [#257](#programme-257) `squarefree_support`, [#269](#programme-269) `three_prime_finite_search`, [#249](#programme-249) `totient_finite_denominator_exclusion`, [#269](#programme-269) `two_prime_transcendence`\n\n**Programmes.** [#68: Factorial-denominator series](#programme-68) · [#243: Reciprocal-tail rigidity near the Sylvester recurrence](#programme-243) · [#249: Binary totient series](#programme-249) · [#251: Prime-gap dyadic series](#programme-251) · [#257: Reciprocal sums over infinite exponent supports](#programme-257) · [#269: Three-prime running least common multiples](#programme-269) · [#1041: Short connections inside polynomial lemniscates](#programme-1041) · [#1049: Lambert-type series at rational bases](#programme-1049)\n\n---\n\n<a id=\"programme-68\"></a>\n## #68: Factorial-denominator series\n\n**Question.** Is the series sum_{n >= 2} 1/(n! - 1) irrational?\n\n**Checked frontier.** The exact integral normal form: the series is irrational if and only if the strict factorial successor Z_m misses divisibility by m cofinally.\n\n**Open boundary.** Irrationality of the factorial-denominator series.\n\n**Read.** [Programme paper](../paper/68/erdos-68-factorial-denominator-irrationality.pdf) · [Lean source](../lean/ErdosProblems/Erdos68/FactorialZeroPlateau.lean)\n\n<details>\n<summary>Source and priority note</summary>\n\nComparator isolation in this checkout is FactorialGapPlateauCore; FactorialZeroPlateau is the default-root duplicate. Required note links remain ZeroPlateau because Core is absent from pin 99f4bf47.\n\n</details>\n\n<details>\n<summary>Representative checked declaration</summary>\n\n<code>ErdosProblems.Erdos68.irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses</code>\n\n</details>\n\n### First-contact mathematical order\n\nThis is Palomar's source-current reader order, not review-matrix or Comparator roster order. Ranked results lead; conditional endpoint leverage, deep mechanisms, natural friction, and supporting rows follow unequally.\n\n1. **Factorial carry characterisation** (`factorial_carry_characterisation`; source-ranked frontier; global rank 9; source disposition `represented`)\n   - **Why here.** Maximal endpoint proximity for #68, but it does not prove the cofinal producer.\n   - **Source.** <code>Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_nonunit_carries</code> in [Lean](../lean/ErdosProblems/Erdos68/FactorialGapPlateauCore.lean)\n   - **Boundary.** The equivalence does not supply a cofinal carry producer.\n   - **Relation.** `supports_without_producing` `factorial_channel_and_projection_rigidity`: Finite channel rigidity feeds the carry endpoint route but supplies no cofinal non-unit carry.\n\n2. **Factorial conditional producers** (`factorial_conditional_producers`; conditional endpoint leverage; source disposition `subordinate`)\n   - **Why here.** A single coefficient-free cofinal complementary-tail certificate would imply the #68 endpoint, but the certificate supply is an explicit unproved premise; this is a shorter conditional consumer, not an irrationality proof.\n   - **Source.** <code>Erdos249257.ExternalVerification.irrational_factorialGapSeries_of_cofinal_global_complementaryTail</code> in [Lean](../verification/ExternalVerification/Statements.lean)\n   - **Boundary.** The producer hypotheses are unproved.\n\n3. **Factorial channel and projection rigidity** (`factorial_channel_and_projection_rigidity`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** The finite channel and projection machinery is the deepest represented #68 mechanism below the endpoint equivalence, but it supplies no cofinal obstruction.\n   - **Source authority.** Canonical review family `factorial_channel_and_projection_rigidity` in [claims](../docs/claims.json)\n   - **Boundary.** These finite and structural results do not produce a cofinal obstruction.\n   - **Relation.** `supports_without_producing` `factorial_carry_characterisation`: Finite channel rigidity feeds the carry endpoint route but supplies no cofinal non-unit carry.\n\n<details>\n<summary>Contribution families (5)</summary>\n\nExact registry keys and Comparator routing are listed separately.\n\n- **Factorial carry characterisation**<br>\n  Exact equivalence between irrationality, cofinally many non-unit carries, and cofinal strict-successor divisibility failures.<br>\n  **Boundary.** An exact reformulation does not supply the required cofinal failures.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel plus Comparator\n\n- **Factorial channel and projection rigidity**<br>\n  Finite quotient-band channel factorisation and breakpoint, channel congruences, a two-term prime-channel corrector, and endpoint-weighted projection rigidity.<br>\n  **Boundary.** These finite and structural results do not produce a cofinal obstruction.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel\n\n- **Factorial conditional producers**<br>\n  Cofinal zero-branch or large private-factor hypotheses imply the required irrationality conclusion.<br>\n  **Boundary.** The producer hypotheses are unproved.<br>\n  *Evidence.* conditional reduction · Lean kernel\n\n- **Factorial finite certificates**<br>\n  Lean checks small exact misses; a hash-bound GMP scan reaches index 300000 and yields the corresponding finite denominator exclusion.<br>\n  **Boundary.** Finite computation does not change the cofinal quantifier.<br>\n  *Evidence.* finite computation · Lean finite theorem plus external exact certificate\n\n- **Factorial lcm growth**<br>\n  An elementary terminal-block argument gives liminf log L_N / (N^{3/2} log N) ≥ 2√2/3 for the lcm of factorial-gap denominators. The displayed bound needs no external multiplicity theorem; an older, weaker multiplicity-based route is historical context only.<br>\n  **Boundary.** The LCM asymptotic has an ordinary paper proof and is checked in Lean as common_denominator_growth_liminf in PaperCompleteLiminf.lean. The finite-block inequality has a separate Lean source. The Comparator entry for this bound belongs to the separate public release wcook04/plectis-erdos-lean. The bound does not decide irrationality.<br>\n  *Evidence.* ordinary paper argument · Lean kernel plus paper argument\n\n</details>\n\n<details>\n<summary>Technical registry and Comparator routing (5)</summary>\n\n- <code>factorial_carry_characterisation</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>factorial_channel_and_projection_rigidity</code><br>\n  Comparator: <code>represented_by_isolated_headline</code>\n\n- <code>factorial_conditional_producers</code><br>\n  Comparator: <code>not_selected_deep_project_predicate_stack</code>\n\n- <code>factorial_finite_certificates</code><br>\n  Comparator: <code>not_applicable_to_external_execution</code>\n\n- <code>factorial_lcm_growth</code><br>\n  Comparator: <code>not_selected_separate_release_comparator_entry</code>\n\n</details>\n\n---\n\n<a id=\"programme-243\"></a>\n## #243: Reciprocal-tail rigidity near the Sylvester recurrence\n\n**Question.** Under a rapid-growth hypothesis on an integer sequence, does rationality of its reciprocal sum force the sequence to satisfy the Sylvester recurrence eventually?\n\n**Checked frontier.** Lean proves cubic_rate_irrationality_unconditional for positive StrictMono zero-indexed a_n with a_n²/a_(n+1) = 1 + 3/n + o(n^-3). The one-based paper uses a finite-prefix bridge; general #243 remains open.\n\n**Open boundary.** The unrestricted problem.\n\n**Read.** [Programme paper](../paper/243/erdos-243-reciprocal-tail-rigidity.pdf) · [Lean source](../lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean)\n\n<details>\n<summary>Representative checked declaration</summary>\n\n<code>ErdosProblems.Erdos243.nextTailState_eq_sub_centered</code>\n\n</details>\n\n### First-contact mathematical order\n\nThis is Palomar's source-current reader order, not review-matrix or Comparator roster order. Ranked results lead; conditional endpoint leverage, deep mechanisms, natural friction, and supporting rows follow unequally.\n\n1. **Bounded negative exclusion** (`bounded_negative_exclusion`; conditional endpoint leverage; source disposition `represented`)\n   - **Why here.** Excludes a bounded-negative regime only under explicit dynamical hypotheses; it does not decide the unrestricted reciprocal-tail endpoint.\n   - **Source.** <code>Erdos249257.ExternalVerification.no_cofinallyBoundedNegative_of_normalizedVanishes</code> in [Lean](../lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean)\n   - **Boundary.** Every positivity, dynamics, bounded-rise, and vanishing hypothesis remains explicit.\n   - **Relation.** `mechanism_for` `centered_state_dynamics`: Centered-state descent is the exact dynamics consumed by the bounded-negative exclusion.\n\n2. **Negative mass recovery** (`negative_mass_recovery`; source-ranked frontier; global rank 13; source disposition `subordinate`)\n   - **Why here.** It gives a sharp conditional structural conclusion for Erdős #243 and proves that any surviving canonical orbit must evade finite negative mass, but it does not establish the original-orbit hypotheses or reciprocal-tail irrationality.\n   - **Source.** <code>Erdos249257.ExternalVerification.sylvesterNext_eventually_of_summable_negativeRelativeMass</code> in [Lean](../lean/ErdosProblems/Erdos243/SparseResetRecovery.lean)\n   - **Boundary.** This is a conditional recovery criterion: the source theorem assumes every displayed dynamics, positivity, strict centered-step, normalized-vanishing, and summability hypothesis, and does not prove them for the original Erdős #243 orbit. Any surviving canonical orbit must have divergent normalized negative mass. It supplies no prime-specific producer, unconditional #243 endpoint, or reciprocal-tail irrationality, and makes no novelty, priority, significance, or external-review claim. It is distinct from centered_state_dynamics, which uses a uniform lower-bound hypothesis rather than summability of normalized negative mass.\n   - **Relation.** `independent_analytic_recovery_peer` `centered_state_dynamics`: Both recover eventual centered-state zero and Sylvester dynamics only under explicit hypotheses, but negative_mass_recovery uses summability of normalized negative mass whereas centered_state_dynamics uses a uniform lower-bound route; neither supplies the original-orbit premises.\n\n3. **Centered state dynamics** (`centered_state_dynamics`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** The committed source-faithful representative exposes the signed centered-state recovery theorem with exact natural C/D dynamics, strict centered margin, lower bound, and normalized vanishing; the centered-zero and Sylvester recovery lemmas remain subordinate, and no reciprocal-tail irrationality follows.\n   - **Source.** <code>Erdos249257.ExternalVerification.boundedNegativePart_eventually_zero</code> in [Lean](../lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean)\n   - **Boundary.** The signed theorem has a C D : ℕ → ℕ, E : ℕ → ℤ, and B : ℕ with all exact hypotheses explicit: a > 1, C > 0, C(n + 1) + D(n) = a(n) · C(n), D(n + 1) = a(n) · D(n), E is the centered state D − (a − 1)C, |E| < C, a uniform lower bound on E, and division-free normalized vanishing. It excludes persistent bounded negative behaviour and forces eventual centered defect zero; centered-zero plus an eventually nonzero next tail recovers Sylvester, but no reciprocal-tail irrationality follows. The unbounded negative mixed-sign branch and the prime-specific producer remain open. This is distinct from the periodic negative-orbit and cofinally bounded-negative no-go families; no novelty, priority, significance, or external-review claim is made, and no unrestricted #243 solution follows.\n   - **Relation.** `mechanism_for` `bounded_negative_exclusion`: Centered-state descent is the exact dynamics consumed by the bounded-negative exclusion.\n   - **Relation.** `independent_analytic_recovery_peer` `negative_mass_recovery`: Both recover eventual centered-state zero and Sylvester dynamics only under explicit hypotheses, but negative_mass_recovery uses summability of normalized negative mass whereas centered_state_dynamics uses a uniform lower-bound route; neither supplies the original-orbit premises.\n\n4. **Bounded rise coprimality** (`bounded_rise_coprimality`; natural friction and no-go; source disposition `represented`)\n   - **Why here.** The fresh-modulus obstruction explains why bounded rise would be decisive, but bounded rise is not automatic for the original reciprocal-tail orbit.\n   - **Source authority.** Canonical review family `bounded_rise_coprimality` in [claims](../docs/claims.json)\n   - **Boundary.** The required bounded-rise input is not automatic for the original sequence.\n\n5. **Negative orbit no go** (`negative_orbit_no_go`; natural friction and no-go; source disposition `rejected`)\n   - **Why here.** Retires an eventually-periodic negative orbit architecture, but leaves arbitrary unbounded negative behaviour and the #243 target open.\n   - **Source.** <code>Erdos249257.ExternalVerification.no_eventuallyPeriodicNegative_orbit</code> in [Lean](../lean/ErdosProblems/Erdos243/ReciprocalTailRigidity.lean)\n   - **Boundary.** The exclusions do not cover arbitrary unbounded negative behaviour.\n\n<details>\n<summary>Contribution families (5)</summary>\n\nExact registry keys and Comparator routing are listed separately.\n\n- **Centered state dynamics**<br>\n  Under exact natural C/D dynamics with a > 1, C > 0, E = D - (a - 1)C, strict |E| < C, a uniform lower bound on E, and normalized vanishing, boundedNegativePart_eventually_zero forces E = 0 eventually. centeredState_eventually_zero supplies the nonnegative stabilization mechanism, and sylvesterNext_eventually_of_centered_zero recovers eventual Sylvester recurrence when the next tail is nonzero.<br>\n  **Boundary.** The signed theorem has a C D : ℕ → ℕ, E : ℕ → ℤ, and B : ℕ with all exact hypotheses explicit: a > 1, C > 0, C(n + 1) + D(n) = a(n) · C(n), D(n + 1) = a(n) · D(n), E is the centered state D − (a − 1)C, |E| < C, a uniform lower bound on E, and division-free normalized vanishing. It excludes persistent bounded negative behaviour and forces eventual centered defect zero; centered-zero plus an eventually nonzero next tail recovers Sylvester, but no reciprocal-tail irrationality follows. The unbounded negative mixed-sign branch and the prime-specific producer remain open. This is distinct from the periodic negative-orbit and cofinally bounded-negative no-go families; no novelty, priority, significance, or external-review claim is made, and no unrestricted #243 solution follows.<br>\n  *Evidence.* locally proved signed recovery/rigidity theorem; novelty unassessed · Lean kernel plus Comparator\n\n- **Negative orbit no go**<br>\n  Constant, eventually constant, periodic, and eventually periodic negative-error orbits are excluded in their stated regimes.<br>\n  **Boundary.** The exclusions do not cover arbitrary unbounded negative behaviour.<br>\n  *Evidence.* no-go result · Lean kernel plus Comparator\n\n- **Bounded rise coprimality**<br>\n  A bounded-rise sequence cannot remain coprime to fresh pairwise-coprime moduli; reduced tails inherit this obstruction.<br>\n  **Boundary.** The required bounded-rise input is not automatic for the original sequence.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel\n\n- **Bounded negative exclusion**<br>\n  Normalised vanishing excludes a cofinally bounded negative part and forces eventual Sylvester behaviour.<br>\n  **Boundary.** Every positivity, dynamics, bounded-rise, and vanishing hypothesis remains explicit.<br>\n  *Evidence.* conditional reduction and no-go result · Lean kernel plus Comparator\n\n- **Negative mass recovery**<br>\n  Under exact reciprocal-tail C/D dynamics, positive tail, the strict centered-step equation, and division-free normalized vanishing, summability of the normalized negative centered mass forces centeredState to vanish eventually and recovers the Sylvester recurrence. The load-bearing theorem is supported by the exact tail-growth bound and the finite-negative-mass stabilization theorem.<br>\n  **Boundary.** The exact criterion assumes a, D : ℕ → ℤ and C : ℕ → ℕ with the declared next-denominator and next-tail dynamics, C > 0, the strict centered step equation, division-free normalized vanishing, and Summable (negativeRelativeMass C (fun n ↦ centeredState (a n) (D n) (C n))). It does not prove these hypotheses for the original Erdős #243 orbit, supply a prime-specific producer, or establish an unconditional #243 endpoint or reciprocal-tail irrationality. Any surviving canonical orbit must have divergent normalized negative mass. This is distinct from centered_state_dynamics, whose recovery criterion uses a uniform lower-bound hypothesis; no novelty, priority, significance, or external-review claim is made.<br>\n  *Evidence.* conditional analytic recovery criterion; novelty unassessed · Lean kernel\n\n</details>\n\n<details>\n<summary>Technical registry and Comparator routing (5)</summary>\n\n- <code>centered_state_dynamics</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>negative_orbit_no_go</code><br>\n  Comparator: <code>targeted_representative_periodic_theorem</code>\n\n- <code>bounded_rise_coprimality</code><br>\n  Comparator: <code>represented_by_stronger_selected_boundary_theorem</code>\n\n- <code>bounded_negative_exclusion</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>negative_mass_recovery</code><br>\n  Comparator: <code>targeted</code>\n\n</details>\n\n---\n\n<a id=\"programme-249\"></a>\n## #249: Binary totient series\n\n**Question.** Is the binary Lambert series sum phi(n)/2^n irrational?\n\n**Checked frontier.** For every e >= 1, the complete dyadic totient kernel through level e has rational span dimension 2^e+1.\n\n**Open boundary.** The strict prime-tail orbit gap, which is the unproved producer consumed by the conditional endpoint.\n\n**Read.** [Programme paper](../paper/249/erdos-249-binary-totient-series.pdf) · [Lean source](../lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean)\n\n<details>\n<summary>Representative checked declaration</summary>\n\n<code>ErdosProblems.Erdos249.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap</code>\n\n</details>\n\n### First-contact mathematical order\n\nThis is Palomar's source-current reader order, not review-matrix or Comparator roster order. Ranked results lead; conditional endpoint leverage, deep mechanisms, natural friction, and supporting rows follow unequally.\n\n1. **Actual lcm orbit separation** (`actual_lcm_orbit_separation`; source-ranked frontier; global rank 6; source disposition `represented`)\n   - **Why here.** It points directly at the binary totient irrationality endpoint for Erdős #249, conditional on a cofinal separation supply; it does not prove that supply.\n   - **Source.** <code>Erdos249257.ExternalVerification.irrational_totientSeries_of_actualLcmOrbitSeparationSupply</code> in [Lean](../lean/Erdos249257/TotientActualLcmOrbitSeparation.lean)\n   - **Boundary.** The cofinal separation supply is an open premise, stronger than mere non-integrality; this sufficient route is not an equivalence and leaves the punctured/top-edge boundary and the Erdős #249 endpoint open.\n   - **Relation.** `independent_conditional_endpoint_peer` `first_harmonic_pivot_decomposition`: Both imply #249 irrationality from different cofinal producers; neither producer is supplied.\n   - **Relation.** `conditional_producer_peer` `strict_prime_tail_orbit_gap`: The strict-prime route supplies a phase-density or natural-prime certificate mechanism, while the actual-LCM route instead assumes quantitative orbit separation; all producer predicates remain open.\n\n2. **First harmonic pivot decomposition** (`first_harmonic_pivot_decomposition`; source-ranked frontier; global rank 7; source disposition `represented`)\n   - **Why here.** It points directly at Erdős #249 irrationality from an explicit cofinal analytic producer, independently of the actual-LCM orbit route; that producer remains unproved.\n   - **Source.** <code>Erdos249257.ExternalVerification.irrational_totient_series_of_pivotResidualDecorrelation</code> in [Lean](../lean/Erdos249257/FirstHarmonicPivot.lean)\n   - **Boundary.** The four-term decomposition and 9X/10 gap remain conditional on the explicit dyadic-room and PivotBudgetAt hypotheses; the cofinal decorrelation producer is unproved. The fibre/image lemmas do not give global supplier-prime isolation or a prime-distribution estimate, and no Erdős #249 irrationality conclusion is claimed.\n   - **Relation.** `independent_conditional_endpoint_peer` `actual_lcm_orbit_separation`: Both imply #249 irrationality from different cofinal producers; neither producer is supplied.\n   - **Relation.** `conditional_producer_peer` `strict_prime_tail_orbit_gap`: Both feed the finite first-harmonic certificate socket, but strict-prime uses phase density or adaptive natural-prime escape while the pivot family uses a four-term supplier-fibre budget and residual decorrelation.\n\n3. **Strict prime tail orbit gap** (`strict_prime_tail_orbit_gap`; source-ranked frontier; global rank 8; source disposition `represented`)\n   - **Why here.** It reaches the Erdős #249 irrationality consumer conditionally and is more than a finite normal form, but neither phase-density nor cofinal-prime producer is proved.\n   - **Source.** <code>Erdos249257.ExternalVerification.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap</code> in [Lean](../lean/ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean)\n   - **Boundary.** The phase-density, cofinal natural-prime strict-gap, associated uniform-margin, and cofinal nonpositive-prime-shift producers remain unproved for the actual totient orbit. Exact phase/integrality equivalence, finite dyadic-root exclusion for 1 ≤ h ≤ 16 and k ≤ 14, and non-dyadicity-to-cofinal-nonpositive-phase implications are classification or finite evidence only; they do not supply density or prime alignment, and the source does not prove actual root entry. This is a conditional reduction only: no unconditional Erdős #249 endpoint, novelty, priority, significance, or external-review claim is made. It is distinct from the finite first_harmonic_pivot_decomposition and actual_lcm_orbit_separation families.\n   - **Relation.** `conditional_producer_peer` `actual_lcm_orbit_separation`: The strict-prime route supplies a phase-density or natural-prime certificate mechanism, while the actual-LCM route instead assumes quantitative orbit separation; all producer predicates remain open.\n   - **Relation.** `conditional_producer_peer` `first_harmonic_pivot_decomposition`: Both feed the finite first-harmonic certificate socket, but strict-prime uses phase density or adaptive natural-prime escape while the pivot family uses a four-term supplier-fibre budget and residual decorrelation.\n   - **Relation.** `endpoint_normal_form_support_for` `totient_certificate_equivalences`: Certificate completeness identifies the exact finite endpoint socket but cannot manufacture the strict-prime route's missing phase-density, cofinal-prime, or uniform-margin producer.\n\n4. **Totient certificate equivalences** (`totient_certificate_equivalences`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** Below the actual-LCM, first-harmonic, and strict-prime producer routes but above medium totient kernel machinery: three exact equivalences expose the endpoint and certificate sockets, while the equivalent lcm-diagonal, period-multiple-kill, and separated-window producers all remain unresolved.\n   - **Source.** <code>Erdos249257.ExternalVerification.exists_clean_binaryCyclotomicAnchor</code> in [Lean](../verification/ExternalVerification/Statements.lean)\n   - **Boundary.** Equivalent producer statements are as hard as the unresolved target.\n   - **Relation.** `endpoint_normal_form_support_for` `strict_prime_tail_orbit_gap`: Certificate completeness identifies the exact finite endpoint socket but cannot manufacture the strict-prime route's missing phase-density, cofinal-prime, or uniform-margin producer.\n\n5. **Totient kernel rank** (`totient_kernel_rank`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** Exact finite-level dyadic rank growth is strong #249 structure and supports totient_carry_anti_compression, but it is weaker than that orbit-level rationality consequence and provides neither a contradiction nor unbounded certificate supply.\n   - **Source.** <code>Erdos249257.ExternalVerification.finrank_totientKernelThroughLevelFamily_eq</code> in [Lean](../lean/Erdos249257/TotientMahlerDefect.lean)\n   - **Boundary.** Only the finite-level rank is presented as an additional result; full-kernel infinite-dimensionality is prior-art context, and no rationality-to-finite-rank bridge is proved.\n\n6. **Totient carry anti compression** (`totient_carry_anti_compression`; source-ranked frontier; global rank 11; source disposition `represented`)\n   - **Why here.** This is an orbit-level necessary consequence of the #249 rational branch, closer to the endpoint than coefficient-side kernel rank alone, but it derives no contradiction.\n   - **Source.** <code>Erdos249257.ExternalVerification.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank</code> in [Lean](../lean/Erdos249257/TotientTailCarryPeriod.lean)\n   - **Boundary.** This is a necessary consequence of rationality only: it gives no contradiction, no finite-rank upper bound for a rationality-supplied carry, and no irrationality proof for Erdős #249. Modulo a divisor of the multiplier, the forcing disappears and the carry becomes geometric, so quotient periodicity alone supplies no finite-ℚ-rank upper bound. The coefficient-side kernel-rank and certificate-equivalence families are not duplicated, and #249 remains open.\n\n7. **Totient kernel all base index** (`totient_kernel_all_base_index`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** The rank theorem retains an external linear-independence hypothesis, so it is valuable structural support rather than an unconditional endpoint result.\n   - **Source.** <code>Erdos249257.ExternalVerification.finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent</code> in [Lean](../lean/Erdos249257/TotientKernelConditional.lean)\n   - **Boundary.** This Comparator target reviews the reusable conditional interface. AllBaseTotientKernel.lean separately proves canonical all-base independence and exact rank unconditionally at 25ef6245d15a; it does not formalise Martin's broader theorem or solve #249.\n\n8. **Totient kernel basis** (`totient_kernel_basis`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** An explicit basis is useful kernel infrastructure, yet it does not connect finite span structure to the binary totient irrationality endpoint.\n   - **Source.** <code>Erdos249257.ExternalVerification.exists_totientDyadicSectionBasis</code> in [Lean](../lean/Erdos249257/TotientMahlerDefect.lean)\n   - **Boundary.** The basis theorem does not connect rationality of the series to finite kernel rank.\n\n9. **Stern brocot run fibonacci stability** (`stern_brocot_run_fibonacci_stability`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** A genuine Fibonacci/continuant stability floor, but analytic denominator survival and the #249 endpoint remain unproved.\n   - **Source.** <code>Erdos249257.ExternalVerification.runHeight_defect_fib_sum_lower</code> in [Lean](../lean/Erdos249257/SternBrocotRunGeometry.lean)\n   - **Boundary.** This exact combinatorial stability floor does not establish analytic denominator survival, irrationality of the binary totient series, or novelty; no #249 endpoint follows.\n\n10. **Eventually periodic lambert** (`eventually_periodic_lambert`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** This is a direct irrationality theorem for an exact structured coefficient class, but the totient-derived coefficients are not eventually periodic, so it is reusable background rather than a #249 frontier.\n   - **Source authority.** Canonical review family `eventually_periodic_lambert` in [claims](../docs/claims.json)\n   - **Boundary.** The totient-derived weights are not eventually periodic.\n\n11. **Signed totient entire defect interpolation** (`signed_totient_entire_defect_interpolation`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** The ordinary proof constructs a sparse signed perturbation with strong defect control for every divergent allowance, but it does not settle the unperturbed totient series or the one-sided prime-gap route; no infinite Lean or Comparator theorem is claimed.\n   - **Source authority.** Canonical review family `signed_totient_entire_defect_interpolation` in [claims](../docs/claims.json)\n   - **Boundary.** The correction is signed, so it does not subsume the one-sided prime-gap theorem. It preserves singularities and prescribed finite jets, not full analytic functions or all Taylor coefficients. The unperturbed totient and prime series remain unresolved.\n\n12. **Fixed precision transport no go** (`fixed_precision_transport_no_go`; natural friction and no-go; source disposition `represented`)\n   - **Why here.** The committed representative unconditionally retires fixed-positive-precision local valuation/unit-word escape for every finite odd-unit word through centred completions, while its synthetic unrestricted high-quotient and finite-word scope supplies no actual totient endpoint.\n   - **Source.** <code>Erdos249257.ExternalVerification.fixedPrecisionTropicalNoGo</code> in [Lean](../lean/Erdos249257/TropicalCurvatureCarry.lean)\n   - **Boundary.** This is a synthetic finite-word construction with unrestricted incoming/high quotient states. It does not impose the actual totient-difference relation or provide growing-precision/global correlation, and it gives no Erdős #249 endpoint or irrationality result. No novelty or priority claim is made. The one-step and finite-word completion theorems are subordinate mechanisms, not separate families.\n   - **Relation.** `contrary_evidence_for` `actual_foreign_residue_projection`: Fixed local precision admits synthetic completions, so any successful projection must use growing precision or global correlation.\n\n13. **Lcm factor ideal shift algebra no go** (`lcm_factor_ideal_shift_algebra_no_go`; natural friction and no-go; source disposition `rejected`)\n   - **Why here.** A synthetic finite-shift obstruction retires one algebraic route while explicitly leaving actual totient forcing and the unbounded supply open.\n   - **Source.** <code>Erdos249257.ExternalVerification.lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient</code> in [Lean](../lean/Erdos249257/LcmFactorIdealPulseObstruction.lean)\n   - **Boundary.** The witness is synthetic: it does not assert that its forcing letters are actual totient differences, does not address nonlinear combinations, and leaves the unbounded certificate supply for Erdős #249 open.\n\n14. **Actual foreign residue projection** (`actual_foreign_residue_projection`; conditional endpoint leverage; source disposition `represented`)\n   - **Why here.** The committed representative exposes the exact finite foreign/divisor partition, geometric tail control, and generic separated-projection consumer for the canonical actual_foreign_residue_projection family; its controlled-projection bridge and unbounded supply remain open, so it is represented conditional evidence rather than a #249 endpoint.\n   - **Source.** <code>Erdos249257.ExternalVerification.scaleFullTarget_miss_of_projected_separation</code> in [Lean](../lean/Erdos249257/ActualForeignResidueProjection.lean)\n   - **Boundary.** This is only a finite partition/geometric-majorant reduction feeding a generic complement-noncancellation consumer. The ControlledForeignProjection analytic bridge and any unbounded supply of separated projections remain unproved; no Erdős #249 endpoint, novelty, significance, priority, or external-review claim follows.\n   - **Relation.** `contrary_evidence_for` `fixed_precision_transport_no_go`: Fixed local precision admits synthetic completions, so any successful projection must use growing precision or global correlation.\n\n15. **Totient lambert coefficients** (`totient_lambert_coefficients`; supporting and long tail; source disposition `represented`)\n   - **Why here.** The Lambert rewrite and prime-power coefficient structure orient the programme but provide no irrationality producer.\n   - **Source authority.** Canonical review family `totient_lambert_coefficients` in [claims](../docs/claims.json)\n   - **Boundary.** The coefficient identities do not prove irrationality.\n\n16. **Squared lambert gcd moments** (`squared_lambert_gcd_moments`; supporting and long tail; source disposition `represented`)\n   - **Why here.** A precise gcd-moment transfer identity, but it supplies no irrationality conclusion for the binary totient series.\n   - **Source.** <code>Erdos249257.ExternalVerification.tsum_totient_div_mersenne_sq_eq_gcd_moment_series</code> in [Lean](../lean/Erdos249257/GcdMomentCalculus.lean)\n   - **Boundary.** This exact transfer identity does not prove irrationality of the binary totient series; the cited constant-weight q-Padé result remains external and no #249 endpoint follows.\n\n17. **Stern brocot cylinder law** (`stern_brocot_cylinder_law`; supporting and long tail; source disposition `represented`)\n   - **Why here.** Exact Stern–Brocot telescoping is reusable geometry, not an irrationality theorem or novelty claim.\n   - **Source.** <code>Erdos249257.ExternalVerification.cylinderMass_split</code> in [Lean](../lean/Erdos249257/GcdMomentCalculus.lean)\n   - **Boundary.** This exact telescoping recursion and its geometric interpretation do not prove irrationality of the binary totient series or novelty; the #249 endpoint remains open.\n\n18. **Totient visible coprime lattice mass** (`totient_visible_coprime_lattice_mass`; supporting and long tail; source disposition `represented`)\n   - **Why here.** Exact visible-lattice normalization explains a mass term but is rational and does not settle Erdős #249.\n   - **Source.** <code>Erdos249257.ExternalVerification.tsum_pos_coprime_inv_mersenne_eq_one</code> in [Lean](../lean/Erdos249257/GcdMomentCalculus.lean)\n   - **Boundary.** This rational normalization identity does not prove irrationality of the binary totient series or settle #249.\n\n<details>\n<summary>Contribution families (20)</summary>\n\nExact registry keys and Comparator routing are listed separately.\n\n- **Totient kernel basis**<br>\n  An explicit odd-core basis and relation normal form for the full dyadic totient kernel.<br>\n  **Boundary.** The basis theorem does not connect rationality of the series to finite kernel rank.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel plus Comparator\n\n- **Totient kernel rank**<br>\n  An independent constructive proof of exact finite-level dyadic rank 2^e+1, together with a kernel-checked full-kernel infinite-dimensionality corollary already implied by Coons's non-2-regularity theorem.<br>\n  **Boundary.** Only the finite-level rank is presented as an additional result; full-kernel infinite-dimensionality is prior-art context, and no rationality-to-finite-rank bridge is proved.<br>\n  *Evidence.* mixed local finite-level theorem and formalised prior consequence · Lean kernel plus Comparator\n\n- **Totient kernel all base index**<br>\n  For every base k at least 2, Lean checks the arithmetic reduction, exact nonmultiple-residue coordinates, unconditional canonical spanning, and exact rank k^e+1 conditional on canonical-family linear independence.<br>\n  **Boundary.** This Comparator target reviews the reusable conditional interface. AllBaseTotientKernel.lean separately proves canonical all-base independence and exact rank unconditionally at 25ef6245d15a; it does not formalise Martin's broader theorem or solve #249.<br>\n  *Evidence.* formalised arithmetic, spanning and reusable conditional rank interface · Lean kernel for this arithmetic, finite indexing, spanning and conditional-rank interface; the independent all-base proof is a separate public source result\n\n- **Totient visible coprime lattice mass**<br>\n  The positive coprime Mersenne weights on the visible lattice sum to one, exposing the reduced-direction mass behind the fair-coin coprimality form of #249.<br>\n  **Boundary.** This rational normalization identity does not prove irrationality of the binary totient series or settle #249.<br>\n  *Evidence.* formalised exact visible-lattice normalization identity · Lean kernel plus Comparator\n\n- **Squared lambert gcd moments**<br>\n  At base one half, the totient-weighted squared-Mersenne Lambert series is the dyadic Pillai gcd-sum-minus-n series, exposing the first fair-coin gcd moment.<br>\n  **Boundary.** This exact transfer identity does not prove irrationality of the binary totient series; the cited constant-weight q-Padé result remains external and no #249 endpoint follows.<br>\n  *Evidence.* formalised exact squared-Lambert first-moment transfer · Lean kernel plus Comparator\n\n- **Stern brocot run fibonacci stability**<br>\n  For a positive Stern-Brocot run word, the global height gain from run defects is bounded below by the sharp Fibonacci coefficient F_(r+1) times the total defect.<br>\n  **Boundary.** This exact combinatorial stability floor does not establish analytic denominator survival, irrationality of the binary totient series, or novelty; no #249 endpoint follows.<br>\n  *Evidence.* formalised exact Fibonacci/continuant run stability floor · Lean kernel plus Comparator\n\n- **Stern brocot cylinder law**<br>\n  At every positive Stern-Brocot node, the closed cylinder mass splits into stop mass and the two child cylinders, giving an exact Markov recursion.<br>\n  **Boundary.** This exact telescoping recursion and its geometric interpretation do not prove irrationality of the binary totient series or novelty; the #249 endpoint remains open.<br>\n  *Evidence.* formalised exact Stern-Brocot cylinder telescoping identity · Lean kernel plus Comparator\n\n- **Lcm factor ideal shift algebra no go**<br>\n  For every t at least 3, a nonzero synthetic dyadic coboundary survives every finite integer shift polynomial while retaining the period-LCM totient factor ideals, exact cleared-prefix cancellation, and uniform state and letter bounds.<br>\n  **Boundary.** The witness is synthetic: it does not assert that its forcing letters are actual totient differences, does not address nonlinear combinations, and leaves the unbounded certificate supply for Erdős #249 open.<br>\n  *Evidence.* unconditional progress · Lean kernel plus Comparator\n\n- **Fixed precision transport no go**<br>\n  At every fixed positive precision, every finite word of odd valuation-unit symbols admits a compatible centred carry completion from any incoming integer state; bounded local signatures therefore cannot exclude all finite centred endpoints.<br>\n  **Boundary.** This is a synthetic finite-word construction with unrestricted incoming/high quotient states. It does not impose the actual totient-difference relation or provide growing-precision/global correlation, and it gives no Erdős #249 endpoint or irrationality result. No novelty or priority claim is made. The one-step and finite-word completion theorems are subordinate mechanisms, not separate families.<br>\n  *Evidence.* unconditional synthetic no-go; novelty unassessed · Lean kernel plus Comparator\n\n- **Actual lcm orbit separation**<br>\n  A cofinal quantitative 1/32 separation supply at canonically guarded odd ranks transfers through the explicit raw-block approximation and error radius to the existing half-word/signed-margin irrationality consumer. The landed actual-LCM sign/top-edge analysis is supporting and contrary-boundary evidence: positivity leaves the true survivor negative, and integrality forces the exact top-edge residue.<br>\n  **Boundary.** The quantitative 1/32 separation supply is sufficient but stronger than mere non-integrality, is not an equivalence, and remains open. The sign positivity alone does not exclude the negative true survivor; the top-edge forcing does not exclude that residue or supply cofinal non-integrality. These source declarations are supporting/contrary-boundary evidence, not a second actual-LCM family, and the punctured/top-edge boundary and #249 endpoint remain open.<br>\n  *Evidence.* conditional reduction · Lean kernel plus Comparator\n\n- **Actual foreign residue projection**<br>\n  The finite Möbius residue diagonal splits exactly into projected foreign and divisor channels, the omitted finite foreign tail has a closed geometric majorant, and the generic complement-noncancellation consumer turns a controlled projection separated from every integer into a full-target miss.<br>\n  **Boundary.** This is only a finite partition/geometric-majorant reduction feeding a generic complement-noncancellation consumer. The ControlledForeignProjection analytic bridge and any unbounded supply of separated projections remain unproved; no Erdős #249 endpoint, novelty, significance, priority, or external-review claim follows.<br>\n  *Evidence.* conditional reduction · Lean kernel plus Comparator\n\n- **First harmonic pivot decomposition**<br>\n  The supplier fibre is an exact prime-image/fibre bijection, while a checked finite counterexample shows that supplier-prime global isolation fails. The load-bearing result is an exact four-term first-harmonic decomposition; under the explicit PivotBudgetAt budgets, the centred real-part bound and three contribution bounds give the 9X/10 gap and a finite certificate, and the conditional pivot-residual-decorrelation theorem transfers a cofinal supply to irrationality.<br>\n  **Boundary.** The fibre/image lemmas and the explicit counterexample establish neither global supplier-prime isolation nor a prime-distribution estimate. The 9X/10 gap is conditional on PivotBudgetAt and the stated dyadic-room inequality; irrationality is conditional on DTWPivotResidualDecorrelation, whose exact cofinal producer remains open. No decorrelation estimate supplying those budgets is proved, and the #249 endpoint remains open.<br>\n  *Evidence.* conditional reduction · Lean kernel plus Comparator\n\n- **Strict prime tail orbit gap**<br>\n  A single two-branch source mechanism gives an 89/100 first-harmonic block saving from an 11/100 nonpositive-phase density, or an adaptive pivot escape from a cofinal natural-prime tail-orbit strict gap below 9/10. The exact squaring law reduces the latter to a power-of-two orbit of one initial phase and shows that an entered dyadic root obstructs that producer, without proving such an entry. The load-bearing representative is the natural-prime irrationality reduction; block-gap, density, and phase-classification declarations remain subordinate.<br>\n  **Boundary.** The 11/100 phase-density, cofinal natural-prime strict-gap, associated uniform-margin, and cofinal nonpositive-prime-shift producers remain unproved for the actual totient orbit. The scaled-angle and phase/integrality equivalences, finite 1 ≤ h ≤ 16 and k ≤ 14 root exclusion, and non-dyadicity-to-cofinal-nonpositive-phase theorem provide classification or finite evidence only; they do not prove a density/equidistribution theorem, prime alignment, or actual dyadic-root entry. Both branches are conditional certificate/irrationality reductions only: they do not close Erdős #249, duplicate first_harmonic_pivot_decomposition or actual_lcm_orbit_separation, or justify novelty, priority, significance, or external review.<br>\n  *Evidence.* conditional reduction · Lean kernel plus Comparator\n\n- **Totient carry anti compression**<br>\n  If the binary totient coefficient series were rational, one positive-multiplier tempered integral carry would have dyadic sections uniformly eventually periodic modulo its multiplier while its canonical carry-section span has dimension at least 2^e − 1 for every e. This is the orbit-level quotient-periodicity/anti-compression conjunction, distinct from the coefficient-side kernel-rank family.<br>\n  **Boundary.** This is a necessary consequence of rationality only: it gives no contradiction, no finite-rank upper bound for a rationality-supplied carry, and no irrationality proof for Erdős #249. Modulo a divisor of the multiplier, the forcing disappears and the carry becomes geometric, so quotient periodicity alone supplies no finite-ℚ-rank upper bound. The coefficient-side kernel-rank and certificate-equivalence families are not duplicated, and #249 remains open.<br>\n  *Evidence.* locally proved necessary consequence; novelty unassessed · Lean kernel plus Comparator\n\n- **Totient finite denominator exclusion**<br>\n  A Farey denominator exclusion through 7.96e34, sharp for its selected window, and diagonal certificates through t=82.<br>\n  **Boundary.** The selected finite window and certificate ceiling do not imply an unbounded producer.<br>\n  *Evidence.* finite computation · Lean kernel and generated finite certificates\n\n- **Totient carry rank**<br>\n  Rationality forces a tempered integral tail orbit with unbounded finite-level carry rank.<br>\n  **Boundary.** This is a necessary consequence, not a contradiction.<br>\n  *Evidence.* conditional reduction · Lean kernel\n\n- **Totient lambert coefficients**<br>\n  The totient series is rewritten as a Lambert series with explicit nonnegative, unbounded prime-power coefficients.<br>\n  **Boundary.** The coefficient identities do not prove irrationality.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel\n\n- **Eventually periodic lambert**<br>\n  Eventually periodic nonnegative rational weights give an irrational Lambert series.<br>\n  **Boundary.** The totient-derived weights are not eventually periodic.<br>\n  *Evidence.* Lean formalisation of existing result · Lean kernel\n\n- **Totient certificate equivalences**<br>\n  Pointwise, cofinal, lcm-diagonal, and separated-window certificate supplies are exact or sufficient reformulations.<br>\n  **Boundary.** Equivalent producer statements are as hard as the unresolved target.<br>\n  *Evidence.* exact equivalence and conditional reduction · Lean kernel\n\n- **Signed totient entire defect interpolation**<br>\n  For any integer beta>=2 and positive allowance f(n) tending to infinity, an interval around the totient value is filled by integer sequences b=phi+e with 0<=b(n)<=n and |e(n)|<=f(n). One target-independent even support has upper Banach density zero, leaving any fixed prefix and odd coefficients unchanged. The construction preserves every fixed eventual coefficient and cumulative congruence, gives eventually constant progression moment defects outside common hulls of density zero, entire progression Dirichlet defects, and smooth root-of-unity radial germs with prescribed finite zero jets. The full proof includes the allowance-dependent block bound and first-harmonic rational countermodels. It is an ordinary proof with two AI proof reviews, not a Lean proof of the infinite theorem or independent human review.<br>\n  **Boundary.** The correction is signed, so it does not subsume the one-sided prime-gap theorem. It preserves singularities and prescribed finite jets, not full analytic functions or all Taylor coefficients. The unperturbed totient and prime series remain unresolved.<br>\n  *Evidence.* ordinary paper argument · ordinary proof; two AI reviews, no independent human review\n\n</details>\n\n<details>\n<summary>Technical registry and Comparator routing (20)</summary>\n\n- <code>totient_kernel_basis</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>totient_kernel_rank</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>totient_kernel_all_base_index</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>totient_visible_coprime_lattice_mass</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>squared_lambert_gcd_moments</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>stern_brocot_run_fibonacci_stability</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>stern_brocot_cylinder_law</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>lcm_factor_ideal_shift_algebra_no_go</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>fixed_precision_transport_no_go</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>actual_lcm_orbit_separation</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>actual_foreign_residue_projection</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>first_harmonic_pivot_decomposition</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>strict_prime_tail_orbit_gap</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>totient_carry_anti_compression</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>totient_finite_denominator_exclusion</code><br>\n  Comparator: <code>not_selected_finite_family_has_separate_receipt</code>\n\n- <code>totient_carry_rank</code><br>\n  Comparator: <code>not_selected_deep_orbit_vocabulary</code>\n\n- <code>totient_lambert_coefficients</code><br>\n  Comparator: <code>represented_by_basis_and_rank_targets</code>\n\n- <code>eventually_periodic_lambert</code><br>\n  Comparator: <code>not_selected_prior_result</code>\n\n- <code>totient_certificate_equivalences</code><br>\n  Comparator: <code>not_selected_deep_certificate_vocabulary</code>\n\n- <code>signed_totient_entire_defect_interpolation</code><br>\n  Comparator: <code>not_applicable_without_exact_Lean_statement</code>\n\n</details>\n\n---\n\n<a id=\"programme-251\"></a>\n## #251: Prime-gap dyadic series\n\n**Question.** Is the dyadic series of consecutive primes irrational? Equivalently, is the corresponding consecutive-prime-gap dyadic series irrational?\n\n**Checked frontier.** The finite summation-by-parts identity relating the prime dyadic partial sums to the prime-gap dyadic partial sums, with an exact endpoint term and no convergence premise.\n\n**Open boundary.** The target irrationality.\n\n**Read.** [Programme paper](../paper/251/erdos-251-prime-gap-dyadic-series.pdf) · [Lean source](../lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean)\n\n<details>\n<summary>Representative checked declaration</summary>\n\n<code>ErdosProblems.Erdos251.dyadicPartialSumQ_eq_start_add_differences</code>\n\n</details>\n\n### First-contact mathematical order\n\nThis is Palomar's source-current reader order, not review-matrix or Comparator roster order. Ranked results lead; conditional endpoint leverage, deep mechanisms, natural friction, and supporting rows follow unequally.\n\n1. **Prime gap reformulation** (`prime_gap_reformulation`; source-ranked frontier; global rank 10; source disposition `represented`)\n   - **Why here.** Very close to #251's target but proves neither side irrational and retains a summability premise.\n   - **Source.** <code>Erdos249257.ExternalVerification.irrational_tsum_primeDyadicTerm_iff_primeGap</code> in [Lean](../lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean)\n   - **Boundary.** The equivalence does not prove irrationality of either series.\n   - **Relation.** `conditional_actual_stream_obstruction_for` `small_mismatch_criterion`: The reformulation gives the exact #251 target language; this fixed-shift actual-prime obstruction identifies what a cofinal adjacent-smallness/mismatch producer would have to destroy. It neither supplies that producer nor turns the reformulation into irrationality.\n\n2. **Small mismatch criterion** (`small_mismatch_criterion`; conditional endpoint leverage; source disposition `represented`)\n   - **Why here.** Below the ranked prime-gap equivalence but above the #269 carry-escape consumer, abstract dyadic-tail classification, and coefficient-only no-go: it is specialized to the actual prime-gap recurrence and identifies the precise cofinal adjacent-smallness producer, while that producer remains unproved.\n   - **Source.** <code>Erdos249257.ExternalVerification.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch</code> in [Lean](../lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean)\n   - **Boundary.** The representative assumes T : ℕ → ℚ, the DyadicTailRecurrence for the actual prime-gap digits, and for every N₀ an N ≥ N₀ whose adjacent h-shifts both lie strictly in (-1, 1) while the corresponding prime gaps differ. Under that cofinal small-mismatch supply it excludes eventual RatIntegral h-shifts; it does not prove the supply or actual smallness. rationalPrimeGapTail_has_positive_shift_not_eventually_small records the contrary rational-state obstruction for at least one positive shift. This is a conditional reduction only: no #251 irrationality, infinite rational-sum limit, universal producer, novelty, priority, significance, or external-review claim is made; #251 remains open. The actual-prime-gap specialization is distinct from coefficient_only_no_go and does not duplicate prime_gap_reformulation.\n   - **Relation.** `normal_form_support_for` `dyadic_tail_integrality_classification`: The abstract shift-integrality normal form explains what the actual-prime cofinal mismatch must destroy.\n   - **Relation.** `contrary_evidence_for` `coefficient_only_no_go`: Coefficient irregularity alone is insufficient; the actual-prime adjacent-smallness producer is genuinely additional.\n   - **Relation.** `conditional_actual_stream_obstruction_for` `prime_gap_reformulation`: The reformulation gives the exact #251 target language; this fixed-shift actual-prime obstruction identifies what a cofinal adjacent-smallness/mismatch producer would have to destroy. It neither supplies that producer nor turns the reformulation into irrationality.\n\n3. **Dyadic tail integrality classification** (`dyadic_tail_integrality_classification`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** Below the source-specific conditional endpoint consumers but above negative-method evidence: its exact denominator and all-positive-tail-shift normal forms are deeper and more reusable than a coefficient-only obstruction, while supplying no actual-prime producer.\n   - **Source.** <code>Erdos249257.ExternalVerification.irrational_initial_iff_all_positive_tailShifts_nonintegral</code> in [Lean](../lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean)\n   - **Boundary.** These exact statements cover arbitrary integer-digit dyadic tail recurrences with the declared rational and real state types; they classify integrality and give an irrationality normal form but do not supply a cofinal nonintegral shift for the consecutive-prime-gap recurrence. The generic hypotheses therefore do not close the prime-specific producer gap: no prime-gap irrationality, novelty, significance, or external review is claimed, and Erdős #251 remains open. This is one abstract classification family, distinct from prime_gap_reformulation, not a duplicate frontier claim.\n   - **Relation.** `normal_form_support_for` `small_mismatch_criterion`: The abstract shift-integrality normal form explains what the actual-prime cofinal mismatch must destroy.\n\n4. **Integral shift classification** (`integral_shift_classification`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** The block and denominator criteria are reusable rational-state structure below the sharper all-positive-shift normal form and above propagation identities.\n   - **Source authority.** Canonical review family `integral_shift_classification` in [claims](../docs/claims.json)\n   - **Boundary.** The concrete prime-gap producer remains missing.\n\n5. **Totient shift propagation** (`totient_shift_propagation`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** Propagation through totient-length shifts explains rational-state persistence but supplies no contradiction for the actual prime-gap recurrence.\n   - **Source authority.** Canonical review family `totient_shift_propagation` in [claims](../docs/claims.json)\n   - **Boundary.** This describes rational states and supplies no contradiction for actual prime gaps.\n\n6. **Coefficient only no go** (`coefficient_only_no_go`; natural friction and no-go; source disposition `represented`)\n   - **Why here.** Below the #251 small-mismatch route and abstract integrality classification: it is valuable natural friction against coefficient-only reasoning, but its synthetic carry stream has no infinite-limit or actual-prime-gap identification bridge.\n   - **Source.** <code>Erdos249257.ExternalVerification.coefficientOnlyNoGo</code> in [Lean](../lean/ErdosProblems/Erdos251/PrimeGapDyadicTail.lean)\n   - **Boundary.** CoefficientOnlyNoGo checks the exact finite partial-sum identity and the two non-eventual-periodicity statements. It does not assert the infinite limit of the synthetic countermodel, identify its coefficient stream with actual prime gaps, establish a prime-gap tail bridge, prove #251 irrationality, or make novelty, priority, significance, or external-review claims.\n   - **Relation.** `contrary_evidence_for` `small_mismatch_criterion`: Coefficient irregularity alone is insufficient; the actual-prime adjacent-smallness producer is genuinely additional.\n\n<details>\n<summary>Contribution families (6)</summary>\n\nExact registry keys and Comparator routing are listed separately.\n\n- **Prime gap reformulation**<br>\n  Exact finite summation by parts, the unconditional infinite prime-gap identity, and irrationality equivalence.<br>\n  **Boundary.** The equivalence does not prove irrationality of either series.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel plus Comparator\n\n- **Integral shift classification**<br>\n  Block identities and exact denominator criteria classify rationality through integral positive tail shifts.<br>\n  **Boundary.** The concrete prime-gap producer remains missing.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel\n\n- **Dyadic tail integrality classification**<br>\n  For any integer-digit dyadic tail recurrence T(N+1) = 2*T(N) - g(N+1), the rational h-shift is integral exactly when the reduced denominator of T(N) divides 2^h - 1, equivalently when 2^h is congruent to 1 modulo that denominator. For the real recurrence, the load-bearing normal form says that irrationality of T(0) is equivalent to every positive tail shift being nonintegral; the eventual-integrality equivalence and denominator/order laws are supporting mechanism evidence.<br>\n  **Boundary.** These exact statements cover arbitrary integer-digit dyadic tail recurrences with the declared rational and real state types; they classify integrality and give an irrationality normal form but do not supply a cofinal nonintegral shift for the consecutive-prime-gap recurrence. The generic hypotheses therefore do not close the prime-specific producer gap: no prime-gap irrationality, novelty, significance, or external review is claimed, and Erdős #251 remains open. This is one abstract classification family, distinct from prime_gap_reformulation, not a duplicate frontier claim.<br>\n  *Evidence.* abstract Lean-checked classification and normal form; novelty unassessed · Lean kernel plus Comparator\n\n- **Totient shift propagation**<br>\n  Odd rational denominators give totient-length integral shifts, and integrality propagates through the recurrence.<br>\n  **Boundary.** This describes rational states and supplies no contradiction for actual prime gaps.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel\n\n- **Small mismatch criterion**<br>\n  For the actual prime-gap dyadic tail, cofinally many adjacent pairs of strictly small h-shifts with unequal corresponding prime gaps force failure of eventual RatIntegral h-shifts. The adjacent-pair zero obstruction is the load-bearing mechanism; exact non-eventual periodicity of consecutive prime gaps supplies the actual-prime-gap contrast.<br>\n  **Boundary.** The representative assumes T : ℕ → ℚ, the DyadicTailRecurrence for the actual prime-gap digits, and for every N₀ an N ≥ N₀ whose adjacent h-shifts both lie strictly in (-1, 1) while the corresponding prime gaps differ. Under that cofinal small-mismatch supply it excludes eventual RatIntegral h-shifts; it does not prove the supply or actual smallness. rationalPrimeGapTail_has_positive_shift_not_eventually_small records the contrary rational-state obstruction for at least one positive shift. This is a conditional reduction only: no #251 irrationality, infinite rational-sum limit, universal producer, novelty, priority, significance, or external-review claim is made; #251 remains open. The actual-prime-gap specialization is distinct from coefficient_only_no_go and does not duplicate prime_gap_reformulation.<br>\n  *Evidence.* conditional actual-prime-gap endpoint reduction; novelty and significance unassessed · Lean kernel plus Comparator\n\n- **Coefficient only no go**<br>\n  The exact finite carry identity and the two non-eventual-periodicity statements expose the coefficient-only barrier: irregularity and growth of a coefficient stream do not by themselves force irrationality, while the synthetic carry stream remains distinct from the actual prime-gap stream.<br>\n  **Boundary.** CoefficientOnlyNoGo checks the exact finite partial-sum identity and the two non-eventual-periodicity statements. It does not assert the infinite limit of the synthetic countermodel, identify its coefficient stream with actual prime gaps, establish a prime-gap tail bridge, prove #251 irrationality, or make novelty, priority, significance, or external-review claims.<br>\n  *Evidence.* coefficient-only barrier; novelty and significance unassessed · Lean kernel plus authored synthesis\n\n</details>\n\n<details>\n<summary>Technical registry and Comparator routing (6)</summary>\n\n- <code>prime_gap_reformulation</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>integral_shift_classification</code><br>\n  Comparator: <code>not_selected_abstract_family_represented_in_manifest</code>\n\n- <code>dyadic_tail_integrality_classification</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>totient_shift_propagation</code><br>\n  Comparator: <code>represented_by_integral_shift_family</code>\n\n- <code>small_mismatch_criterion</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>coefficient_only_no_go</code><br>\n  Comparator: <code>targeted</code>\n\n</details>\n\n---\n\n<a id=\"programme-257\"></a>\n## #257: Reciprocal sums over infinite exponent supports\n\n**Question.** Is the sum of 1/(2^n-1) over every infinite set of positive exponents irrational?\n\n**Checked frontier.** Irrationality at every integer base for the classical full support, reciprocal-summable supports, and the finite-prime weighted supports in divisibilityWeightedClaim.\n\n**Open boundary.** Universal irrationality over all infinite supports, or the proposed rational values 1/2 and 1/21.\n\n**Read.** [Programme paper](../paper/257/erdos-257-mersenne-support-subseries.pdf) · [Lean source](../lean/ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean)\n\n<details>\n<summary>Representative checked declaration</summary>\n\n<code>ErdosProblems.Erdos257.selectedMersenneTail_lt_weight</code>\n\n</details>\n\n### First-contact mathematical order\n\nThis is Palomar's source-current reader order, not review-matrix or Comparator roster order. Ranked results lead; conditional endpoint leverage, deep mechanisms, natural friction, and supporting rows follow unequally.\n\n1. **Finite prime weighted support** (`finite_prime_weighted_support`; source-ranked frontier; global rank 1; source disposition `represented`)\n   - **Why here.** For every integer base b ≥ 2 it proves irrationality on every infinite positive support of finite prime-weighted mass; a positive host of finite binary weighted mass gives irrationality for every infinite subset at every integer base.\n   - **Source.** <code>Erdos249257.ExternalVerification.divisibilityWeightedClaim</code> in [Lean](../lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean)\n   - **Boundary.** The finite-prime weighted-summability premise is explicit in both clauses. This criterion does not cover every infinite support or settle universal Erdős #257; no novelty, priority, significance, or external human-review claim is made.\n\n2. **Known irrational supports** (`known_irrational_supports`; source-ranked frontier; global rank 2; source disposition `represented`)\n   - **Why here.** Directly settles the canonical full-support member of Erdős #257 for every integer base b >= 2, while leaving universal arbitrary-support irrationality open.\n   - **Source.** <code>Erdos249257.ExternalVerification.irrational_erdosSum_full_support</code> in [Lean](../lean/Erdos249257/CertificateKernel.lean)\n   - **Boundary.** This is the classical full-support theorem, not universal Erdos #257.\n\n3. **Pairwise coprime support** (`pairwise_coprime_support`; source-ranked frontier; global rank 3; source disposition `represented`)\n   - **Why here.** Direct irrationality for every infinite pairwise-coprime support with summable reciprocal mass, but not for arbitrary supports.\n   - **Source.** <code>Erdos249257.ExternalVerification.irrational_erdosSupportSeries_pairwise_coprime</code> in [Lean](../lean/Erdos249257/CertificateKernel.lean)\n   - **Boundary.** This is the pairwise-coprime support theorem, not universal Erdos #257; the infinitude, pairwise-coprimality, and summable-reciprocal hypotheses remain explicit.\n\n4. **Orthogonal petal sunflower reduction** (`orthogonal_petal_sunflower_reduction`; source-ranked frontier; global rank 4; source disposition `represented`)\n   - **Why here.** Direct irrationality under the displayed structure, but the structural predicates are not constructed for arbitrary supports.\n   - **Source.** <code>Erdos249257.ExternalVerification.irrational_erdosSupportSeries_of_orthogonalPetalBouquet</code> in [Lean](../lean/Erdos249257/SupportSunflowerDichotomy.lean)\n   - **Boundary.** This is a conditional reduction: the bouquet structure and uniform tail-selection predicate remain hypotheses; neither is constructed for arbitrary supports, so this does not add an unconditional Erdos #257 support family or solve the universal problem.\n\n5. **Periodic nonnegative weight irrationality** (`periodic_nonnegative_weight_irrationality`; source-ranked frontier; global rank 5; source disposition `represented`)\n   - **Why here.** Close to the Lambert-series endpoint but not a universal support theorem and not the mixed-sign statement.\n   - **Source.** <code>Erdos249257.ExternalVerification.irrational_ratWeightSeries_eventuallyPeriodic</code> in [Lean](../lean/Erdos249257/CertificateKernel.lean)\n   - **Boundary.** This is the nonnegative rational eventually-periodic coefficient theorem; its nonnegativity, eventual periodicity, and positive-tail hypotheses remain explicit, and the broader mixed-sign periodic theorem is not claimed.\n\n6. **Boolean mobius exact row dynamics** (`boolean_mobius_exact_row_dynamics`; conditional endpoint leverage; source disposition `represented`)\n   - **Why here.** The committed exact-row representative is one conditional Boolean–Möbius dynamics family: cofinal positive greedy skips yield cofinally many exact local half rows and conditional 1/2-membership, while finite rows do not supply the unproved cofinal premise.\n   - **Source.** <code>Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips</code> in [Lean](../lean/Erdos249257/BooleanMobiusSkipRowCofinal.lean)\n   - **Boundary.** CofinalPositiveHalfGreedySkips is an unproved global supply premise: for every N there must be c ≥ max N 4 with a positive greedy remainder below mersenneWeightRat c. Finite exact rows, including one produced by a skipped core, do not establish cofinality. This is a conditional 1/2-membership/counterexample route, not an unconditional Erdős #257 solution; the rank-two, endpoint-defect, and upper-half Boolean-word declarations remain subordinate evidence, and no novelty, priority, significance, or external-review claim is made.\n   - **Relation.** `reconstruction_support_for` `boolean_mobius_carry`: The Boolean–Möbius certificate reconstructs support, while exact rows still require a cofinal skip producer.\n   - **Relation.** `conditional_producer_for` `half_membership_seam_classification`: Cofinal positive half-greedy skips conditionally give 1/2-membership, which the seam iff classifies as unbounded terminal-false/upper-or-middle supply; finite exact rows supply neither cofinal premise.\n\n7. **Half membership seam classification** (`half_membership_seam_classification`; source-ranked frontier; global rank 12; source disposition `represented`)\n   - **Why here.** It classifies the concrete #257 half-membership endpoint exactly, closer to a usable target than an analytic recovery criterion or source-only certificate normal form, but supplies no occurrence witness.\n   - **Source.** <code>Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse</code> in [Lean](../lean/Erdos249257/HalfCylinderHalfMembershipClassification.lean)\n   - **Boundary.** This exact endpoint-socket classification does not supply any unbounded terminal-false or upper-or-middle occurrences, cofinal supply, skipped-support producer, or half-membership witness; finite exact rows do not establish the required global supply. It is not an unconditional or universal Erdős #257 result, and no novelty, priority, significance, or external-review claim is made.\n   - **Relation.** `conditional_producer_for` `boolean_mobius_exact_row_dynamics`: Cofinal positive half-greedy skips conditionally give 1/2-membership, which the seam iff classifies as unbounded terminal-false/upper-or-middle supply; finite exact rows supply neither cofinal premise.\n\n8. **Shifted odd tail state** (`shifted_odd_tail_state`; conditional endpoint leverage; source disposition `represented`)\n   - **Why here.** It extracts a modular tail state only from explicit rational-support hypotheses and therefore does not exclude rational infinite supports.\n   - **Source.** <code>Erdos249257.ExternalVerification.exists_shifted_odd_tail_nat_states_of_support_fraction</code> in [Lean](../lean/Erdos249257/RationalSupportCarrySkeleton.lean)\n   - **Boundary.** The state-extraction theorem does not exclude rational infinite supports or settle universal Erdős #257; the support and rational-value hypotheses remain explicit.\n\n9. **Separated cut algebraic lambert transcendence** (`separated_cut_algebraic_lambert_transcendence`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** The ordinary proof transfers algebraic-base transcendence through a separated-cut host and an explicit non-chain example, while full support at 3/2 and unrestricted base-two supports remain open; finite Lean cut lemmas are not its infinite proof.\n   - **Source authority.** Canonical review family `separated_cut_algebraic_lambert_transcendence` in [claims](../docs/claims.json)\n   - **Boundary.** The full support at 3/2 and arbitrary infinite supports at base two remain unresolved. Finite cut arithmetic is not a proof of transcendence.\n\n10. **Boolean mobius carry** (`boolean_mobius_carry`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** The committed representative preserves the existence-level quotient certificate, Boolean Möbius support reconstruction, positivity, and square-root bound; it supplies no universal irrationality contradiction or actual-recurrence producer.\n   - **Source.** <code>Erdos249257.ExternalVerification.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry</code> in [Lean](../lean/Erdos249257/BooleanMobiusCarry.lean)\n   - **Boundary.** The equivalence is stated at the existence level with q > 0, 0 ∉ A, and a positive support witness; it gives no contradiction, no universal Erdős #257 irrationality result, and no producer of cofinally nonintegral shifts for the actual recurrence. The generic tempered-orbit theorem and the reciprocal-mass, unbounded-tail, and zero-window results are subordinate consequences, not separate families; the zero-window constant depends on the support and numerator and is not uniform. This does not settle every infinite support, and no novelty, priority, significance, external-review, or unrestricted #257 endpoint claim is made.\n   - **Relation.** `reconstruction_support_for` `boolean_mobius_exact_row_dynamics`: The Boolean–Möbius certificate reconstructs support, while exact rows still require a cofinal skip producer.\n\n11. **Achievement set geometry** (`achievement_set_geometry`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** The full-measure achievement-set theorem is consequential geometry, but it does not classify rational points or settle infinite-support irrationality.\n   - **Source.** <code>Erdos249257.ExternalVerification.volume_mersenneAchievementSet</code> in [Lean](../lean/Erdos249257/GreedyAchievementSet.lean)\n   - **Boundary.** Geometry of the full set does not decide irrationality of every coded point.\n\n12. **Restricted achievement sets** (`restricted_achievement_sets`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** The restricted-set measure dichotomy is structurally useful but does not classify rational values or prove universal irrationality.\n   - **Source.** <code>Erdos249257.ExternalVerification.volume_supportedMersenneAchievementSet_dichotomy</code> in [Lean](../lean/ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean)\n   - **Boundary.** The measure dichotomy does not classify rational points.\n\n13. **Half and twenty one frontiers** (`half_and_twenty_one_frontiers`; natural friction and no-go; source disposition `subordinate`)\n   - **Why here.** Sharp local 2-adic band arithmetic excludes small odd numerators under stated hypotheses, but does not decide reachability or half-membership.\n   - **Source.** <code>Erdos249257.ExternalVerification.seven_le_of_intBand_odd</code> in [Lean](../lean/Erdos249257/HalfGreedyTwoThirdsBand.lean)\n   - **Boundary.** Neither membership question is decided; the 2-adic bound is sharp locally, and the two countermodel routes require unconstructed terminal-scaled or cofinal-cylinder antecedents.\n\n14. **Composite dilation defect identity** (`composite_dilation_defect_identity`; supporting and long tail; source disposition `represented`)\n   - **Why here.** The exact composite-dilation defect exposes a real obstruction, but it supplies neither a tail selector nor an unconditional irrationality result.\n   - **Source.** <code>Erdos249257.ExternalVerification.supportCoeff_mul_eq_add_defect</code> in [Lean](../lean/Erdos249257/CompositeDilationDefect.lean)\n   - **Boundary.** The exact defect is not bounded for arbitrary supports and supplies no tail selector or irrationality theorem; the bouquet budget remains conditional on its structural hypotheses.\n\n15. **Finite period noncollapse** (`finite_period_noncollapse`; supporting and long tail; source disposition `represented`)\n   - **Why here.** The finite-period order interface is exact denominator infrastructure, but finite support cannot decide an infinite-support sum.\n   - **Source.** <code>Erdos249257.ExternalVerification.finite_period_noncollapse_rat_den_interface</code> in [Lean](../lean/Erdos249257/CertificateKernel.lean)\n   - **Boundary.** A finite-support denominator theorem does not settle an infinite-support sum.\n\n<details>\n<summary>Contribution families (16)</summary>\n\nExact registry keys and Comparator routing are listed separately.\n\n- **Finite period noncollapse**<br>\n  The multiplicative order of the base modulo a reduced finite-sum denominator is the support lcm, forcing denominator growth.<br>\n  **Boundary.** A finite-support denominator theorem does not settle an infinite-support sum.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel plus Comparator\n\n- **Boolean mobius carry**<br>\n  For p : ℤ and q > 0, a nonempty normalized positive-index support with erdosSupportSeries 2 A = p/q exists exactly when a quotient-only Boolean–Möbius carry certificate U exists. The certificate has U 0 = p, positive U, the square-root bound U(N) ≤ q(2√N + 4), q-divisible carry differences, and a Boolean Möbius transform; the converse reconstructs the support from that transform. This quotient-level equivalence is distinct from the shifted natural-tail state family.<br>\n  **Boundary.** The equivalence is stated at the existence level with q > 0, 0 ∉ A, and a positive support witness; it gives no contradiction, no universal Erdős #257 irrationality result, and no producer of cofinally nonintegral shifts for the actual recurrence. The generic tempered-orbit theorem and the reciprocal-mass, unbounded-tail, and zero-window results are subordinate consequences, not separate families; the zero-window constant depends on the support and numerator and is not uniform. This does not settle every infinite support, and no novelty, priority, significance, external-review, or unrestricted #257 endpoint claim is made.<br>\n  *Evidence.* locally proved exact equivalence plus conditional consequences; novelty unassessed · Lean kernel plus Comparator\n\n- **Boolean mobius exact row dynamics**<br>\n  If CofinalPositiveHalfGreedySkips holds:cofinally many positive rational-greedy remainders below the next Mersenne weight:the skipped-core and upper-half Boolean-word construction supplies cofinally many exact local half rows, yielding (1 / 2 : ℝ) ∈ mersenneAchievementSet. The load-bearing representative is half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips; exact-row construction, endpoint-defect transition, forced rank-two membership, and Boolean-word filling are subordinate mechanisms.<br>\n  **Boundary.** CofinalPositiveHalfGreedySkips is an unproved global supply premise: for every N there must be c ≥ max N 4 with a positive greedy remainder below mersenneWeightRat c. Finite exact rows, including one produced by a skipped core, do not establish cofinality. This is a conditional 1/2-membership/counterexample route, not an unconditional Erdős #257 solution; the rank-two, endpoint-defect, and upper-half Boolean-word declarations remain subordinate evidence, and no novelty, priority, significance, or external-review claim is made.<br>\n  *Evidence.* conditional reduction; novelty unassessed · Lean kernel\n\n- **Half membership seam classification**<br>\n  Membership of 1/2 in the Mersenne achievement set is exactly equivalent to unbounded false successor terminal bits, the unbounded upper-or-middle seam branch, cofinal terminal-false positions, unbounded skipped ranks along the integer seam, infinite greedy skipped support, and the absence of a last half-greedy skip. The load-bearing representative is half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse; the remaining equivalences are subordinate endpoint-socket evidence.<br>\n  **Boundary.** This exact endpoint-socket classification does not supply any unbounded terminal-false or upper-or-middle occurrences, cofinal supply, skipped-support producer, or half-membership witness; finite exact rows do not establish the required global supply. It is not an unconditional or universal Erdős #257 result, and no novelty, priority, significance, or external-review claim is made.<br>\n  *Evidence.* locally proved exact endpoint classification; novelty unassessed · Lean kernel plus Comparator\n\n- **Shifted odd tail state**<br>\n  A rational support fraction produces a positive natural shifted tail state satisfying the exact binary recurrence and modular doubling orbit modulo v.<br>\n  **Boundary.** The state-extraction theorem does not exclude rational infinite supports or settle universal Erdős #257; the support and rational-value hypotheses remain explicit.<br>\n  *Evidence.* conditional reduction · Lean kernel plus Comparator\n\n- **Composite dilation defect identity**<br>\n  The composite support-dilation identity splits the new divisor count into the old coefficient, the distinguished divisor, and an exact finite defect; its bouquet specialization supplies a separate structural budget.<br>\n  **Boundary.** The exact defect is not bounded for arbitrary supports and supplies no tail selector or irrationality theorem; the bouquet budget remains conditional on its structural hypotheses.<br>\n  *Evidence.* unconditional progress · Lean kernel plus Comparator\n\n- **Known irrational supports**<br>\n  Full, factorial, power-of-two, multiple, eventually periodic, residue-class, and odd supports are formalised; pairwise-coprime support is split out as a dedicated non-dilation representative.<br>\n  **Boundary.** Structured known families do not cover arbitrary infinite supports.<br>\n  *Evidence.* Lean formalisation of existing results · Lean kernel plus Comparator for full and pairwise-coprime support\n\n- **Finite prime weighted support**<br>\n  Finite prime-weighted mass gives fixed-base irrationality for infinite positive supports; binary weighted mass gives all-base irrationality for every infinite subset of a positive host, while weighted mass at another fixed base gives same-base hereditary irrationality.<br>\n  **Boundary.** Each clause retains its exact finite-prime weighted-summability and positive-support hypotheses. Binary mass is required for the all-base hereditary clause; arbitrary infinite supports remain open.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel; exact Comparator transport configured, external replay pending\n\n- **Pairwise coprime support**<br>\n  Infinite pairwise-coprime supports with summable reciprocal mass have irrational Mersenne support series by an adaptive CRT and weighted coefficient certificate argument.<br>\n  **Boundary.** This structured support theorem does not settle arbitrary infinite supports; its infinitude, pairwise-coprimality, and summable-reciprocal hypotheses remain explicit.<br>\n  *Evidence.* formalisation of an existing theorem · Lean kernel plus Comparator\n\n- **Periodic nonnegative weight irrationality**<br>\n  An eventually-periodic nonnegative rational coefficient sequence with a positive value in its periodic region has an irrational Lambert–Mersenne series; denominator clearing reduces the rational sequence to a periodic Nat-weight certificate.<br>\n  **Boundary.** This does not claim the broader mixed-sign periodic theorem; all coefficient, positivity, base, period, and threshold hypotheses remain explicit.<br>\n  *Evidence.* formalised here; novelty unassessed · Lean kernel plus Comparator\n\n- **Orthogonal petal sunflower reduction**<br>\n  A finite-core orthogonal-petal bouquet together with the exact uniform tail-selection predicate composes through the forced-carry certificate consumer to irrationality; the explicit alternating-core, squared-prime-petal witness shows the structural class is non-vacuous, but the selector remains open.<br>\n  **Boundary.** The bouquet and selector hypotheses remain explicit; the selector is not proved, so this is not an unconditional #257 support family.<br>\n  *Evidence.* conditional reduction · Lean kernel plus Comparator plus paper argument\n\n- **Squarefree support**<br>\n  Lean checks squarefree incidence and no-go statements; the irrationality conclusion uses an external analytic theorem in the paper.<br>\n  **Boundary.** Comparator must not badge the paper-plus-external conclusion.<br>\n  *Evidence.* Lean structure plus paper and external theorem · Lean kernel plus paper argument plus cited theorem\n\n- **Achievement set geometry**<br>\n  Compactness, perfectness, total disconnectedness, nowhere density, and measure one for the unrestricted achievement set.<br>\n  **Boundary.** Geometry of the full set does not decide irrationality of every coded point.<br>\n  *Evidence.* mixed prior-art geometry and locally proved metric result; novelty unassessed · Lean kernel plus Comparator\n\n- **Restricted achievement sets**<br>\n  Support-restricted coding is injective and the associated achievement set has exact finite-complement measure or zero for infinite complement.<br>\n  **Boundary.** The measure dichotomy does not classify rational points.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel plus Comparator\n\n- **Half and twenty one frontiers**<br>\n  Exact greedy characterisations, finite-support exclusions, the odd 2-adic two-thirds-band bound, and two conditional rational-half countermodel frontiers isolate local constraints around the remaining 1/2 and 1/21 alternatives.<br>\n  **Boundary.** Neither membership question is decided; the 2-adic bound is sharp locally, and the two countermodel routes require unconstructed terminal-scaled or cofinal-cylinder antecedents.<br>\n  *Evidence.* conditional reduction and no-go result · Lean kernel plus Comparator\n\n- **Separated cut algebraic lambert transcendence**<br>\n  For an infinite positive-integer host with cuts L_j < M_j, L_j tending to infinity and M_j-L_j tending to infinity, every host exponent at most L_j dividing L_j and every later exponent divisible by M_j, every infinite thinning with bounded positive integer weights has transcendental Lambert sum at every real algebraic base t > 1. This includes all divisibility chains and an explicit host of unbounded divisibility width and divergent reciprocal sum. The full number-field Subspace Theorem argument has two AI proof reviews; it is an ordinary proof with no Lean transcendence theorem, independent human review or asserted historical priority.<br>\n  **Boundary.** The full support at 3/2 and arbitrary infinite supports at base two remain unresolved. Finite cut arithmetic is not a proof of transcendence.<br>\n  *Evidence.* ordinary paper argument · ordinary proof; two AI reviews, no independent human review\n\n</details>\n\n<details>\n<summary>Technical registry and Comparator routing (16)</summary>\n\n- <code>finite_period_noncollapse</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>boolean_mobius_carry</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>boolean_mobius_exact_row_dynamics</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>half_membership_seam_classification</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>shifted_odd_tail_state</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>composite_dilation_defect_identity</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>known_irrational_supports</code><br>\n  Comparator: <code>targeted_full_support_representative</code>\n\n- <code>finite_prime_weighted_support</code><br>\n  Comparator: <code>targeted_exact_weighted_and_hereditary_interfaces_pending_replay</code>\n\n- <code>pairwise_coprime_support</code><br>\n  Comparator: <code>targeted_pairwise_coprime_support</code>\n\n- <code>periodic_nonnegative_weight_irrationality</code><br>\n  Comparator: <code>targeted_strongest_nonnegative_rational_weight</code>\n\n- <code>orthogonal_petal_sunflower_reduction</code><br>\n  Comparator: <code>targeted_conditional_sunflower_reduction</code>\n\n- <code>squarefree_support</code><br>\n  Comparator: <code>not_applicable_to_external_irrationality_input</code>\n\n- <code>achievement_set_geometry</code><br>\n  Comparator: <code>targeted_measure_theorem</code>\n\n- <code>restricted_achievement_sets</code><br>\n  Comparator: <code>targeted_measure_dichotomy</code>\n\n- <code>half_and_twenty_one_frontiers</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>separated_cut_algebraic_lambert_transcendence</code><br>\n  Comparator: <code>not_applicable_without_exact_Lean_statement</code>\n\n</details>\n\n---\n\n<a id=\"programme-269\"></a>\n## #269: Three-prime running least common multiples\n\n**Question.** For a finite set of at least two primes, is the sum of reciprocals of the running least common multiples of the smooth numbers irrational? This library treats the three-prime case.\n\n**Checked frontier.** That for three pairwise distinct primes the least common multiple of the smooth prefix equals the product of the three maximal pure prime powers below the cutoff.\n\n**Open boundary.** Irrationality of the repeated sum, or transcendence, in any three-prime case.\n\n**Read.** [Programme paper](../paper/269/erdos-269-three-prime-running-lcm.pdf) · [Lean source](../lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean)\n\n<details>\n<summary>Source and priority note</summary>\n\nEvery two-prime case is settled at transcendence level by a paper argument in the problem note, with Loxton-van der Poorten 1977 (quoted in Bugeaud-Laurent form) as the external engine; the same argument was posted publicly by Steve Fan on the problem's erdosproblems.com discussion page on 26 June 2026, before the note was finalised, so the note claims no priority. The argument is deliberately not formalised here and no Lean declaration asserts it.\n\n</details>\n\n<details>\n<summary>Representative checked declaration</summary>\n\n<code>ErdosProblems.Erdos269.smoothPrefixLcm_eq_threePrimeHeight</code>\n\n</details>\n\n### First-contact mathematical order\n\nThis is Palomar's source-current reader order, not review-matrix or Comparator roster order. Ranked results lead; conditional endpoint leverage, deep mechanisms, natural friction, and supporting rows follow unequally.\n\n1. **Conditional carry escape** (`conditional_carry_escape`; conditional endpoint leverage; source disposition `subordinate`)\n   - **Why here.** Below the #251 small-mismatch route because the source-specific cofinal escape remains unproved, but above medium finite #269 machinery because the exact residue-window contradiction is the endpoint consumer that such machinery must feed. The actual-series reduction is given.\n   - **Source.** <code>Erdos249257.ExternalVerification.no_positive_reducedCarry_of_cofinalLocalWindowEscape</code> in [Lean](../lean/ErdosProblems/Erdos269/RestrictedFloorSum.lean)\n   - **Boundary.** The representative assumes b,m : ℕ → ℕ, shortBound : ℕ → ℕ → ℕ, CofinalLocalWindowEscape b m shortBound, and for each positive B coprime to 30 a positive integer-valued d with d(n+1) = (b n : ℤ) d(n) − (B : ℤ) (m n : ℤ) and Int.natAbs (d n) ≤ shortBound B n; it then gives False. The absorbed-carry bridge additionally assumes smoothFactor ≠ 0, c n = smoothFactor * d n, and the corresponding exact absorbed recurrence, then cancels that factor to the same reduced consumer. The representative remains a conditional consumer for arbitrary words. For the actual three-prime word, the rationality-to-reduced-carry bridge is established separately; the source-specific cofinal escape producer is not proved. This is not a #269 endpoint or irrationality proof, and no actual-series identification, novelty, priority, significance, or external-review claim is made. It is one conditional window-carry family, distinct from the finite residue, rank-two, and weighted-phase observer families.\n   - **Relation.** `mechanism_support_for` `weighted_phase_carry_observer`: The residue/coboundary split exposes finite carry data but does not supply the cofinal escape.\n   - **Relation.** `contrary_evidence_for` `rank_two_kernel_no_go`: The finite kernel is not rank one, ruling out a tempting separable shortcut without producing carry escape.\n\n2. **Weighted phase carry observer** (`weighted_phase_carry_observer`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** The exact residue-digit/coboundary decomposition is the committed representative of the weighted-phase observer family; it does not transport the separate finite-dimensional factorization or imply the #269 endpoint.\n   - **Source.** <code>Erdos249257.ExternalVerification.carry_eq_residueDigit_add_coboundary</code> in [Lean](../verification/ExternalVerification/Statements.lean)\n   - **Boundary.** The recurrence supplies a finite residue coordinate but leaves an uncontrolled integral quotient coboundary; it proves neither a finite-state quotient nor a literal infinite realised span. Finite realised span requires an explicit factorisation through a finite-dimensional function-faithful observer, and scalar evaluation alone is insufficient. No irrationality conclusion follows from this observer alone. The actual rationality-to-positive-reduced-carry bridge is established separately; cofinal local-window escape remains open.\n   - **Relation.** `mechanism_support_for` `conditional_carry_escape`: The residue/coboundary split exposes finite carry data but does not supply the cofinal escape.\n\n3. **Rank two kernel no go** (`rank_two_kernel_no_go`; natural friction and no-go; source disposition `rejected`)\n   - **Why here.** A checked minor obstruction rules out rank one in a displayed finite kernel, but failure of rank one does not imply irrationality.\n   - **Source.** <code>Erdos249257.ExternalVerification.kernel_235_minor_eq_neg_one_fifteen</code> in [Lean](../lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean)\n   - **Boundary.** Failure of rank one does not itself imply irrationality.\n   - **Relation.** `contrary_evidence_for` `conditional_carry_escape`: The finite kernel is not rank one, ruling out a tempting separable shortcut without producing carry escape.\n\n4. **Height fibre and shell** (`height_fibre_and_shell`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** The finite height-fibre normal form and smooth-shell bound are substantive three-prime structure, but they do not produce the source-specific cofinal escape. The actual-series reduction is given separately.\n   - **Source authority.** Canonical review family `height_fibre_and_shell` in [claims](../docs/claims.json)\n   - **Boundary.** These fibre bounds are counting input, not an escape producer. The separate actual-series divisibility bridge is established.\n\n5. **Independent clock channel relations** (`independent_clock_channel_relations`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** The ordinary proof classifies rational relations for independent clock channels by a swap-and-lattice mechanism, but it does not establish the needed independence for the original running-LCM rates or its endpoint.\n   - **Source authority.** Canonical review family `independent_clock_channel_relations` in [claims](../docs/claims.json)\n   - **Boundary.** No rational independence of the original reciprocal prime logarithms is assumed or proved; the repeated running-LCM endpoint remains open.\n\n6. **Dyadic block alphabet** (`dyadic_block_alphabet`; supporting and long tail; source disposition `represented`)\n   - **Why here.** The exact four-symbol block alphabet is a useful finite interface, subordinate to the carry consumer, weighted observer, and rank obstruction.\n   - **Source authority.** Canonical review family `dyadic_block_alphabet` in [claims](../docs/claims.json)\n   - **Boundary.** The finite alphabet does not supply the needed carry escape.\n\n7. **Three prime lcm cells** (`three_prime_lcm_cells`; supporting and long tail; source disposition `represented`)\n   - **Why here.** The smooth-prefix LCM identity is exact three-prime structure, but it does not supply the missing irrationality bridge.\n   - **Source.** <code>Erdos249257.ExternalVerification.smoothPrefixLcm_eq_threePrimeHeight</code> in [Lean](../lean/ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean)\n   - **Boundary.** Cell structure alone does not prove irrationality.\n\n<details>\n<summary>Contribution families (10)</summary>\n\nExact registry keys and Comparator routing are listed separately.\n\n- **Three prime lcm cells**<br>\n  Exact running-LCM product, logarithmic-cell constancy, coordinate jump ratios, and jump count.<br>\n  **Boundary.** Cell structure alone does not prove irrationality.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel plus Comparator\n\n- **Two prime transcendence**<br>\n  Steve Fan posted the two-prime factorisation, reduction and transcendence first (erdosproblems.com/269, 26 June 2026, post-7218). This checkout records that ordinary deduction from Bugeaud–Laurent and does not claim priority or a Lean theorem.<br>\n  **Boundary.** Comparator cannot certify the external analytic input.<br>\n  *Evidence.* paper plus external theorem · paper argument plus cited theorem\n\n- **Height fibre and shell**<br>\n  Finite height-fibre normal form and a quadratic smooth-shell multiplicity bound.<br>\n  **Boundary.** These fibre bounds are counting input, not an escape producer. The separate actual-series divisibility bridge is established.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel\n\n- **Rank two kernel no go**<br>\n  The 2,3,5 kernel is not rank one and its smallest displayed minor equals -1/15.<br>\n  **Boundary.** Failure of rank one does not itself imply irrationality.<br>\n  *Evidence.* no-go result · Lean kernel plus Comparator\n\n- **Dyadic block alphabet**<br>\n  The exact dyadic block alphabet is 2, 6, 10, and 30.<br>\n  **Boundary.** The finite alphabet does not supply the needed carry escape.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel\n\n- **Conditional carry escape**<br>\n  Under the denominator-dependent cofinal local-window residue-escape predicate, no positive reduced carry can satisfy the exact multiplier recurrence together with its short bound. The load-bearing consumer is no_positive_reducedCarry_of_cofinalLocalWindowEscape; an absorbed nonzero common-factor carry reduces exactly to that consumer. CofinalLocalWindowEscape, windowBase, windowForcing, leastPositiveResidue, and the absorbed-carry bridge are subordinate finite-window mechanism evidence.<br>\n  **Boundary.** The representative assumes b,m : ℕ → ℕ, shortBound : ℕ → ℕ → ℕ, CofinalLocalWindowEscape b m shortBound, and for each positive B coprime to 30 a positive integer-valued d with d(n+1) = (b n : ℤ) d(n) − (B : ℤ) (m n : ℤ) and Int.natAbs (d n) ≤ shortBound B n; it then gives False. The absorbed-carry bridge additionally assumes smoothFactor ≠ 0, c n = smoothFactor * d n, and the corresponding exact absorbed recurrence, then cancels that factor to the same reduced consumer. The representative remains a conditional consumer for arbitrary words. For the actual three-prime word, the rationality-to-reduced-carry bridge is established separately; the source-specific cofinal escape producer is not proved. This is not a #269 endpoint or irrationality proof, and no actual-series identification, novelty, priority, significance, or external-review claim is made. It is one conditional window-carry family, distinct from the finite residue, rank-two, and weighted-phase observer families.<br>\n  *Evidence.* conditional no-go consumer; novelty and significance unassessed · Lean kernel plus Comparator\n\n- **Weighted phase carry observer**<br>\n  An exact weighted-phase carry recurrence splits into a finite residue digit and an uncontrolled integral quotient coboundary; an explicit function-faithful finite-dimensional observer then forces finite realised span.<br>\n  **Boundary.** The recurrence supplies a finite residue coordinate but leaves an uncontrolled integral quotient coboundary; it proves neither a finite-state quotient nor a literal infinite realised span. Finite realised span requires an explicit factorisation through a finite-dimensional function-faithful observer, and scalar evaluation alone is insufficient. No irrationality conclusion follows from this observer alone. The actual rationality-to-positive-reduced-carry bridge is established separately; cofinal local-window escape remains open.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel\n\n- **Three prime finite search**<br>\n  A local-window checker searches 106666 denominator and start pairs and records small certificates.<br>\n  **Boundary.** Finite search is not a cofinal statement.<br>\n  *Evidence.* finite computation · external exact computation\n\n- **Independent clock channel relations**<br>\n  For positive rationally independent clock rates with no simultaneous positive crossings, and integer radices b_i >= 2, all rational relations among 1 and the channel sums S_i are multiples of (-1,b_1-1,...,b_d-1). The proof uses genuine late swaps with a common infinite suffix in one orbit closure, a fixed rational lattice, and density of the clock section. The explicit rates 1,sqrt(2),sqrt(3),sqrt(6) and radices 2,3,5,7 give rational span dimension four. This is an ordinary proof reviewed by AI agents, without Lean or independent human verification of the infinite theorem.<br>\n  **Boundary.** No rational independence of the original reciprocal prime logarithms is assumed or proved; the repeated running-LCM endpoint remains open.<br>\n  *Evidence.* ordinary paper argument · ordinary proof; two AI reviews, no independent human review\n\n- **Selected staircase integral height boundary**<br>\n  The selected staircase matrix with ones on and below the diagonal and integer r>=2 above it has Smith factors 1,r-1,...,r-1. For the selected {2,3,5} kernel, any entrywise-integral diagonal clearing has determinant magnitude at least 4^(N-1), whereas the reduced rational determinant has numerator one up to sign. A rational single-target border has primitive polynomial plus or minus (tX-s), where s/t is the reduced Schur complement. These ordinary arithmetic arguments were reviewed by AI agents; they are not the existing Lean nonvanishing theorem and have no independent human review.<br>\n  **Boundary.** The optimality concerns diagonal clearing and one rational target. It does not rule out multi-target, nonlinear, recurrence, or new approximation methods and does not prove running-LCM irrationality.<br>\n  *Evidence.* ordinary paper argument · ordinary proof; two AI reviews, no independent human review\n\n</details>\n\n<details>\n<summary>Technical registry and Comparator routing (10)</summary>\n\n- <code>three_prime_lcm_cells</code><br>\n  Comparator: <code>targeted_running_lcm_identity</code>\n\n- <code>two_prime_transcendence</code><br>\n  Comparator: <code>not_applicable_not_a_lean_declaration</code>\n\n- <code>height_fibre_and_shell</code><br>\n  Comparator: <code>represented_by_three_prime_structure_target</code>\n\n- <code>rank_two_kernel_no_go</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>dyadic_block_alphabet</code><br>\n  Comparator: <code>represented_by_three_prime_structure_target</code>\n\n- <code>conditional_carry_escape</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>weighted_phase_carry_observer</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>three_prime_finite_search</code><br>\n  Comparator: <code>not_applicable_not_a_lean_theorem</code>\n\n- <code>independent_clock_channel_relations</code><br>\n  Comparator: <code>not_applicable_without_exact_Lean_statement</code>\n\n- <code>selected_staircase_integral_height_boundary</code><br>\n  Comparator: <code>not_applicable_without_exact_Lean_statement</code>\n\n</details>\n\n---\n\n<a id=\"programme-1041\"></a>\n## #1041: Short connections inside polynomial lemniscates\n\n**Question.** Must two roots of a monic polynomial in the open unit disc be joined by a sub-two-length curve inside its unit lemniscate? Ani’s degree-seven example refutes the exact Formal Conjectures statement; correspondence with the 1958 wording awaits human review.\n\n**Checked frontier.** For ani’s degree-seven polynomial, every preconnected strict-lemniscate set containing two distinct roots has one-dimensional Hausdorff measure above two. Lean refutes the exact Formal Conjectures path-image claim and checks a separate total-variation bound.\n\n**Open boundary.** The full small-parameter family reported by ani; only one explicit polynomial is formalised.\n\n**Read.** [Programme paper](../paper/1041/erdos-1041-lemniscate-newton-flow.pdf) · [Lean source](../lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean)\n\n<details>\n<summary>Source-current research frontier</summary>\n\n[Dated frontier](../research_corpus/Erdos1041/FRONTIER.md) · [strongest-result map](../research_corpus/Erdos1041/STRONGEST_RESULTS.json) · [corpus manifest](../research_corpus/Erdos1041/CORPUS_MANIFEST.json) · [checkpoint](../research_corpus/Erdos1041/PUBLIC_CORPUS_CHECKPOINT.json)\n\nThis source-fingerprinted route contains 294 activated research results at source checkpoint `fb5c8803c1498cad9755d1615473cb33b11af6ee`. Read the dated frontier first: the map preserves hypotheses, falsifiers, and open gaps.\n\nAuthority boundary: these are public research evidence, not reviewed claim-registry entries or Comparator interfaces. They do not close Erdős #1041 or promote research-corpus rows into the checked result set.\n\n</details>\n\n<details>\n<summary>Representative checked declaration</summary>\n\n<code>ErdosProblems.Erdos1041.newtonFlow_value_hasDerivAt</code>\n\n</details>\n\n### First-contact mathematical order\n\nThis is Palomar's source-current reader order, not review-matrix or Comparator roster order. Ranked results lead; conditional endpoint leverage, deep mechanisms, natural friction, and supporting rows follow unequally.\n\n1. **Newton value decay** (`newton_value_decay`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** The Newton value equation and exponential first integral are the load-bearing local mechanism, while metric gluing to a short curve remains open.\n   - **Source.** <code>Erdos249257.ExternalVerification.exists_two_roots_dist_sum_le_two_mul_geomMean</code> in [Lean](../verification/ExternalVerification/Statements.lean)\n   - **Boundary.** The cited declarations are pointwise derivative statements. The real-time first integral requires the paper’s interval integration step, and value decay alone does not construct a global trajectory or give a short connecting curve.\n   - **Relation.** `mechanism_support_for` `ray_separation`: The exponential first integral confines a Newton trajectory to one value ray, yielding the separation obstruction.\n\n2. **Ray separation** (`ray_separation`; natural friction and no-go; source disposition `represented`)\n   - **Why here.** Distinct value rays exactly obstruct finite Newton connections, a consequential route boundary but not the global lemniscate theorem.\n   - **Source authority.** Canonical review family `ray_separation` in [claims](../docs/claims.json)\n   - **Boundary.** The finite-trajectory conclusion retains continuity on the closed interval, differentiability and noncritical Newton evolution on its interior, and the ordinary endpoint passage. It supplies neither global trajectory existence nor geometric length control.\n   - **Relation.** `mechanism_support_for` `newton_value_decay`: The exponential first integral confines a Newton trajectory to one value ray, yielding the separation obstruction.\n   - **Relation.** `supports_without_closing` `translation_avoidance`: Ray separation motivates avoiding collision lines, but global topology and metric gluing remain open.\n\n3. **Translation avoidance** (`translation_avoidance`; supporting and long tail; source disposition `represented`)\n   - **Why here.** Finite translation avoidance is a useful #1041 input, yet it does not provide the global gluing needed by the target route.\n   - **Source.** <code>Erdos249257.ExternalVerification.exists_small_translation_separating_arguments</code> in [Lean](../lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean)\n   - **Boundary.** The finite-family avoidance theorem does not perform the global topology or metric gluing.\n   - **Relation.** `supports_without_closing` `ray_separation`: Ray separation motivates avoiding collision lines, but global topology and metric gluing remain open.\n\n4. **Root retention** (`root_retention`; supporting and long tail; source disposition `represented`)\n   - **Why here.** Root retention under small perturbation is an exact local input, not a complete stationarity or minimax conclusion.\n   - **Source.** <code>Erdos249257.ExternalVerification.constant_perturbation_roots_in_unitDisk</code> in [Lean](../lean/ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean)\n   - **Boundary.** Root retention is one input to a still-incomplete route.\n\n<details>\n<summary>Contribution families (6)</summary>\n\nExact registry keys and Comparator routing are listed separately.\n\n- **Newton value decay**<br>\n  Lean checks the local Newton value equation and the zero-derivative form of its exponentially scaled value; constancy along a real-time interval is an ordinary calculus consequence in the paper.<br>\n  **Boundary.** The cited declarations are pointwise derivative statements. The real-time first integral requires the paper’s interval integration step, and value decay alone does not construct a global trajectory or give a short connecting curve.<br>\n  *Evidence.* locally proved result and formalised calculus; novelty unassessed · Lean kernel plus authored synthesis\n\n- **Ray separation**<br>\n  Lean checks that distinct positive rays contradict an assumed exponential endpoint relation; the paper derives that relation for a finite real-time Newton trajectory by ordinary integration and endpoint continuity.<br>\n  **Boundary.** The finite-trajectory conclusion retains continuity on the closed interval, differentiability and noncritical Newton evolution on its interior, and the ordinary endpoint passage. It supplies neither global trajectory existence nor geometric length control.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel plus authored synthesis\n\n- **Translation avoidance**<br>\n  Collision translations are parameterised by real affine lines and an arbitrarily small common translation avoids finitely many of them.<br>\n  **Boundary.** The finite-family avoidance theorem does not perform the global topology or metric gluing.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel plus Comparator\n\n- **Root retention**<br>\n  A quantified small constant perturbation keeps every polynomial root inside the open unit disc.<br>\n  **Boundary.** Root retention is one input to a still-incomplete route.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel plus Comparator\n\n- **Published proof gap**<br>\n  The paper identifies an invalid local saddle block in a claimed unrestricted proof.<br>\n  **Boundary.** The diagnosis does not prove that no repair exists.<br>\n  *Evidence.* paper diagnosis · authored source analysis\n\n- **Lemniscate finite search**<br>\n  A bounded search records finite geometric evidence.<br>\n  **Boundary.** Finite evidence does not settle the universal problem.<br>\n  *Evidence.* finite computation · external computation\n\n</details>\n\n<details>\n<summary>Technical registry and Comparator routing (6)</summary>\n\n- <code>newton_value_decay</code><br>\n  Comparator: <code>not_selected_compact_calculus_family</code>\n\n- <code>ray_separation</code><br>\n  Comparator: <code>represented_by_translation_avoidance_targets</code>\n\n- <code>translation_avoidance</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>root_retention</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>published_proof_gap</code><br>\n  Comparator: <code>not_applicable_not_a_lean_proposition</code>\n\n- <code>lemniscate_finite_search</code><br>\n  Comparator: <code>not_applicable_not_a_lean_theorem</code>\n\n</details>\n\n---\n\n<a id=\"programme-1049\"></a>\n## #1049: Lambert-type series at rational bases\n\n**Question.** For which rational bases is the corresponding series irrational? The first resistant explicit base is three halves.\n\n**Checked frontier.** Lean checks irrationality and the stated irrationality-exponent bound for rational bases in the exact Zudilin contour region, including every positive integral power of 31/4. The contour excludes 3/2.\n\n**Open boundary.** The universal rational-base conjecture, including the case 3/2, remains open. The checked 31/4 power family settles one region and leaves that conjecture untouched.\n\n**Read.** [Programme paper](../paper/1049/erdos-1049-rational-base-lambert.pdf) · [Lean source](../lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean)\n\n<details>\n<summary>Representative checked declaration</summary>\n\n<code>ErdosProblems.Erdos1049.threeHalves_no_coordinatewiseCorridor</code>\n\n</details>\n\n### First-contact mathematical order\n\nThis is Palomar's source-current reader order, not review-matrix or Comparator roster order. Ranked results lead; conditional endpoint leverage, deep mechanisms, natural friction, and supporting rows follow unequally.\n\n1. **Rational base tail recurrence** (`rational_base_tail_recurrence`; deep mechanism and classification; source disposition `represented`)\n   - **Why here.** The cleared-tail recurrence is exact rational-base infrastructure, but no contradiction or irrationality consequence is derived.\n   - **Source.** <code>Erdos249257.ExternalVerification.rationalBaseClearedTailQ_succ</code> in [Lean](../lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean)\n   - **Boundary.** The recurrence is not derived from a rationality contradiction.\n\n2. **Height and pade arithmetic** (`height_and_pade_arithmetic`; natural friction and no-go; source disposition `represented`)\n   - **Why here.** The registered representative is the sharp equality characterization for the explicit rectangular two-function exponent model; it is not a universal Hermite-Padé no-go and does not produce irrationality.\n   - **Source.** <code>Erdos249257.ExternalVerification.rectangular_hp_threshold_eq_classical_iff</code> in [Lean](../lean/ErdosProblems/Erdos1049/HermitePadeNoGo.lean)\n   - **Boundary.** This is a sharp no-go only for the explicit rectangular two-function exponent model under the stated real hypotheses; it constructs no approximating polynomials or remainders and is not a universal Padé or Hermite-Padé no-go. It proves no irrationality at 3/2 or any general rational-base endpoint, and the analytic remainder/nonvanishing input remains untreated. No novelty, priority, significance, external-review, or endpoint claim is made; Erdős #1049 remains open.\n   - **Relation.** `stronger_natural_friction_than` `coordinatewise_corridor_no_go`: The sharp rectangular threshold analysis carries deeper model-specific information than the coordinatewise transfer obstruction, while neither proves the endpoint.\n\n3. **Coordinatewise corridor no go** (`coordinatewise_corridor_no_go`; natural friction and no-go; source disposition `rejected`)\n   - **Why here.** It excludes one coordinatewise corridor architecture at base 3/2 and proves no irrationality statement.\n   - **Source.** <code>Erdos249257.ExternalVerification.threeHalves_no_coordinatewiseCorridor</code> in [Lean](../lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean)\n   - **Boundary.** This excludes one proof architecture and proves no irrationality statement.\n   - **Relation.** `stronger_natural_friction_than` `height_and_pade_arithmetic`: The sharp rectangular threshold analysis carries deeper model-specific information than the coordinatewise transfer obstruction, while neither proves the endpoint.\n\n4. **Calibrated rational hankel countermodel** (`calibrated_rational_hankel_countermodel`; natural friction and no-go; source disposition `represented`)\n   - **Why here.** The ordinary positive-moment countermodel blocks inference from an all-rank Hankel signature alone; these are not Zudilin's moments, and neither F(3/2) nor the actual arithmetic input is settled.\n   - **Source authority.** Canonical review family `calibrated_rational_hankel_countermodel` in [claims](../docs/claims.json)\n   - **Boundary.** These are not Zudilin's actual moments. They refute an inference from the shared signature alone; they do not settle F(3/2) or refute the attributed divisibility-based criterion.\n\n<details>\n<summary>Contribution families (8)</summary>\n\nExact registry keys and Comparator routing are listed separately.\n\n- **Scalar content no go**<br>\n  Integer scalar content changes analytic error and exterior determinant by matching factors, yielding no margin by itself.<br>\n  **Boundary.** The conclusion is scoped to one normalisation strategy.<br>\n  *Evidence.* no-go result · Lean kernel\n\n- **Endpoint residues**<br>\n  Endpoint residues at 2 and 3 exclude a common multiplier under unit-endpoint hypotheses.<br>\n  **Boundary.** Other local or cross-row arithmetic remains possible.<br>\n  *Evidence.* no-go result · Lean kernel\n\n- **Four jet collision**<br>\n  Binary selectors collide in the four-jet signature at the stated rank and depth.<br>\n  **Boundary.** A signed relation is not nonzero analytic data.<br>\n  *Evidence.* locally proved result and no-go result; novelty unassessed · Lean kernel\n\n- **Coordinatewise corridor no go**<br>\n  The coordinatewise corridor forces a power-versus-linear inequality and cannot occur at base 3/2.<br>\n  **Boundary.** This excludes one proof architecture and proves no irrationality statement.<br>\n  *Evidence.* no-go result · Lean kernel plus Comparator\n\n- **Rational base tail recurrence**<br>\n  The exact rational-base cleared-tail recurrence exposes exponential denominator-base forcing absent at integer bases.<br>\n  **Boundary.** The recurrence is not derived from a rationality contradiction.<br>\n  *Evidence.* locally proved result; novelty unassessed · Lean kernel plus Comparator\n\n- **Seven halves irrationality**<br>\n  Irrationality at 7/2 is obtained from an external analytic criterion whose elementary height hypothesis is checked in Lean.<br>\n  **Boundary.** Comparator may check the height inequality, not the cited theorem application.<br>\n  *Evidence.* paper plus external theorem · paper argument plus cited theorem and Lean arithmetic\n\n- **Height and pade arithmetic**<br>\n  For the explicit rectangular two-function exponent model, the cleared gap is nonpositive throughout rho >= 0 and 1 + rho <= sigma, and the threshold is at most 1/2 - 1/pi^2 with equality exactly at rho = 0, sigma = 1. The gap sign and unique-zero mechanism support this sharp threshold no-go within the wider height-and-Padé arithmetic family.<br>\n  **Boundary.** This is a sharp no-go only for the explicit rectangular two-function exponent model under the stated real hypotheses; it constructs no approximating polynomials or remainders and is not a universal Padé or Hermite-Padé no-go. It proves no irrationality at 3/2 or any general rational-base endpoint, and the analytic remainder/nonvanishing input remains untreated. No novelty, priority, significance, external-review, or endpoint claim is made; Erdős #1049 remains open.<br>\n  *Evidence.* locally proved sharp model-specific no-go; novelty unassessed · Lean kernel plus Comparator\n\n- **Calibrated rational hankel countermodel**<br>\n  An explicit positive atomic moment family and integer formal series have all-rank Hankel order sum_{j<N} j^2 and leading coefficient (N!)^2(N+1)!/2^N, with the same fixed-base asymptotic shape including N^(-8 xi), while xi=G_{2/3}(2/3)=1. For n>=1 the exact reduced moment denominator is 3^(31n+62)(3^(n+1)-2^(n+1))^4; the determinant denominator has logarithmic size Theta(N^3), and a proposed rank-two row-product clearer leaves denominator625. These are ordinary proofs with independent exact finite checks and AI proof reviews, not a Lean proof of the infinite family or human peer review.<br>\n  **Boundary.** These are not Zudilin's actual moments. They refute an inference from the shared signature alone; they do not settle F(3/2) or refute the attributed divisibility-based criterion.<br>\n  *Evidence.* ordinary paper argument · ordinary proof; two AI reviews, no independent human review\n\n</details>\n\n<details>\n<summary>Technical registry and Comparator routing (8)</summary>\n\n- <code>scalar_content_no_go</code><br>\n  Comparator: <code>not_selected_scalar_content_scaling_family</code>\n\n- <code>endpoint_residues</code><br>\n  Comparator: <code>not_selected_endpoint_residue_and_common_multiplier_family</code>\n\n- <code>four_jet_collision</code><br>\n  Comparator: <code>not_selected_finite_combinatorial_family</code>\n\n- <code>coordinatewise_corridor_no_go</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>rational_base_tail_recurrence</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>seven_halves_irrationality</code><br>\n  Comparator: <code>not_applicable_to_external_irrationality_conclusion</code>\n\n- <code>height_and_pade_arithmetic</code><br>\n  Comparator: <code>targeted</code>\n\n- <code>calibrated_rational_hankel_countermodel</code><br>\n  Comparator: <code>not_applicable_without_exact_Lean_statement</code>\n\n</details>\n\n---\n\n## Comparator interface appendix\n\n<details>\n<summary>Show 49 documented statement-isolated interfaces</summary>\n\n**#68: Factorial-denominator series**\n\n- <code>Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses</code>\n  - **Class.** locally proved result; novelty unassessed\n  - **Statement.** Irrationality is equivalent to cofinally many strict-successor divisibility failures.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The equivalence does not prove that the cofinal failures occur.\n\n- <code>Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_nonunit_carries</code>\n  - **Class.** locally proved result; novelty unassessed\n  - **Statement.** Irrationality is equivalent to cofinally many non-unit carries.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The equivalence does not supply a cofinal carry producer.\n\n**#243: Reciprocal-tail rigidity near the Sylvester recurrence**\n\n- <code>Erdos249257.ExternalVerification.no_cofinallyBoundedNegative_of_normalizedVanishes</code>\n  - **Class.** conditional reduction and no-go result\n  - **Statement.** Normalised vanishing and bounded rise exclude a cofinally bounded negative part.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** All dynamics, bounded-rise, positivity, and vanishing hypotheses remain explicit.\n\n- <code>Erdos249257.ExternalVerification.boundedNegativePart_eventually_zero</code>\n  - **Class.** locally proved signed recovery/rigidity theorem; novelty unassessed\n  - **Statement.** Under exact natural C/D dynamics with a > 1, C > 0, E = D - (a - 1)C, strict |E| < C, a uniform lower bound on E, and division-free normalized vanishing, boundedNegativePart_eventually_zero forces E = 0 eventually; centered-zero plus an eventually nonzero next tail recovers eventual Sylvester recurrence.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The signed theorem has a C D : ℕ → ℕ, E : ℕ → ℤ, and B : ℕ with all exact hypotheses explicit: a > 1, C > 0, C(n + 1) + D(n) = a(n) · C(n), D(n + 1) = a(n) · D(n), E is the centered state D − (a − 1)C, |E| < C, a uniform lower bound on E, and division-free normalized vanishing. It excludes persistent bounded negative behaviour and forces eventual centered defect zero; centered-zero plus an eventually nonzero next tail recovers Sylvester, but no reciprocal-tail irrationality follows. The unbounded negative mixed-sign branch and the prime-specific producer remain open. This is distinct from the periodic negative-orbit and cofinally bounded-negative no-go families; no novelty, priority, significance, or external-review claim is made, and no unrestricted #243 solution follows.\n\n- <code>Erdos249257.ExternalVerification.sylvesterNext_eventually_of_summable_negativeRelativeMass</code>\n  - **Class.** conditional analytic recovery criterion; novelty unassessed\n  - **Statement.** Under exact next-denominator and next-tail C/D dynamics with a, D : ℕ → ℤ, C : ℕ → ℕ, C > 0, the strict centered step equation, division-free normalized vanishing, and Summable normalized negative centered mass, the centered state vanishes eventually and a follows the Sylvester recurrence eventually.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This is a conditional recovery criterion: the source theorem assumes every displayed dynamics, positivity, strict centered-step, normalized-vanishing, and summability hypothesis, and does not prove them for the original Erdős #243 orbit. Any surviving canonical orbit must have divergent normalized negative mass. It supplies no prime-specific producer, unconditional #243 endpoint, or reciprocal-tail irrationality, and makes no novelty, priority, significance, or external-review claim. It is distinct from centered_state_dynamics, which uses a uniform lower-bound hypothesis rather than summability of normalized negative mass.\n\n- <code>Erdos249257.ExternalVerification.no_eventuallyPeriodicNegative_orbit</code>\n  - **Class.** no-go result\n  - **Statement.** An eventually periodic negative-error orbit with positive phase growth is impossible.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The theorem does not cover arbitrary unbounded negative behaviour.\n\n**#249: Binary totient series**\n\n- <code>Erdos249257.ExternalVerification.finrank_totientKernelThroughLevelFamily_eq</code>\n  - **Class.** locally proved result; novelty unassessed\n  - **Statement.** For every e >= 1, the complete dyadic totient kernel through level e has rational span dimension 2^e + 1.\n  - **Canonical claim status.** `supports_registered_claim_family:dyadic_totient_certificate_interface`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This does not prove irrationality of the totient series.\n\n- <code>Erdos249257.ExternalVerification.not_finiteDimensional_span_fullTotientKernel</code>\n  - **Class.** Lean formalisation of an existing consequence of Coons\n  - **Statement.** The rational span of the full dyadic totient kernel is not finite-dimensional.\n  - **Canonical claim status.** `supports_registered_claim_family:dyadic_totient_certificate_interface`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The declaration is kernel-checked locally, but full-kernel infinite-dimensionality is not an independent contribution of this release; no rationality-to-finite-rank bridge is proved.\n\n- <code>Erdos249257.ExternalVerification.exists_totientDyadicSectionBasis</code>\n  - **Class.** locally proved result; novelty unassessed\n  - **Statement.** The full dyadic totient kernel span admits the explicit odd-core-indexed basis.\n  - **Canonical claim status.** `supports_registered_claim_family:dyadic_totient_basis`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The basis theorem does not connect rationality to finite kernel rank.\n\n- <code>Erdos249257.ExternalVerification.finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent</code>\n  - **Class.** formalised unconditional spanning and conditional rank with external independence input\n  - **Statement.** Under 2 ≤ k, 1 ≤ e, and linear independence of the explicit canonical all-base family, the actual all-base totient kernel through level e has rational span dimension k^e + 1.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The linear-independence hypothesis is the external affine-ordering boundary and is not proved here; this does not prove irrationality of the binary totient series.\n\n- <code>Erdos249257.ExternalVerification.tsum_pos_coprime_inv_mersenne_eq_one</code>\n  - **Class.** formalised exact visible-lattice normalization identity\n  - **Statement.** The positive coprime Mersenne weights over the visible lattice sum exactly to one, giving the reduced-direction mass behind the fair-coin coprimality form of the totient constant.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This is a rational normalization identity and does not prove irrationality of the binary totient series or settle Erdős #249.\n\n- <code>Erdos249257.ExternalVerification.tsum_totient_div_mersenne_sq_eq_gcd_moment_series</code>\n  - **Class.** formalised exact first-gcd-moment squared-Lambert transfer identity\n  - **Statement.** At base one half, the squared-Mersenne Lambert series weighted by Euler's totient equals the dyadic series of Pillai's gcd-sum function minus n, the exact first gcd-moment transfer.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This exact identity does not prove irrationality of the binary totient series; the constant-weight squared-Lambert irrationality is cited external mathematics, and no Erdős #249 endpoint follows.\n\n- <code>Erdos249257.ExternalVerification.runHeight_defect_fib_sum_lower</code>\n  - **Class.** formalised exact Fibonacci/continuant run stability floor\n  - **Statement.** For every finite defect list e, the height of the corresponding positive Stern-Brocot run word is at least F_(r+3) + F_(r+1) times the total defect, where r is the number of runs.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This exact combinatorial stability floor does not establish analytic denominator survival, irrationality of the binary totient series, or novelty; no Erdős #249 endpoint follows.\n\n- <code>Erdos249257.ExternalVerification.cylinderMass_split</code>\n  - **Class.** formalised exact Stern-Brocot cylinder telescoping identity\n  - **Statement.** At every positive Stern-Brocot node (a,b), the closed cylinder mass splits exactly into the node's stop mass and the two child cylinder masses.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This exact telescoping recursion and its geometric interpretation do not prove irrationality of the binary totient series or novelty; the #249 endpoint remains open.\n\n- <code>Erdos249257.ExternalVerification.lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient</code>\n  - **Class.** unconditional progress\n  - **Statement.** For every t >= 3, a nonzero synthetic dyadic coboundary survives every finite integer shift polynomial while retaining the period-LCM totient factor ideals, exact cleared-prefix cancellation, and uniform state and letter bounds.\n  - **Canonical claim status.** `supports_registered_claim_family:lcm_factor_ideal_anchor_pulse_no_go`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The witness is synthetic: it does not assert that its forcing letters are actual totient differences, does not address nonlinear combinations, and leaves the unbounded certificate supply for Erdős #249 open.\n\n- <code>Erdos249257.ExternalVerification.fixedPrecisionTropicalNoGo</code>\n  - **Class.** unconditional synthetic no-go; novelty unassessed\n  - **Statement.** For every positive fixed precision, every finite word of odd valuation-unit symbols and every incoming integer carry state admit a compatible orbit whose successive states stay within the corresponding valuation-unit radii; fixed-precision local signatures therefore do not by themselves exclude finite centred completions.\n  - **Canonical claim status.** `supports_registered_claim_family:fixed_precision_transport_no_go`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This is a synthetic finite-word construction with unrestricted incoming/high quotient states. It does not impose the actual totient-difference relation or provide growing-precision/global correlation, and it gives no Erdős #249 endpoint or irrationality result. No novelty or priority claim is made. The one-step and finite-word completion theorems are subordinate mechanisms, not separate families.\n\n- <code>Erdos249257.ExternalVerification.irrational_totientSeries_of_actualLcmOrbitSeparationSupply</code>\n  - **Class.** conditional reduction\n  - **Statement.** A cofinal quantitative separation supply for the actual power-two LCM tail orbit implies irrationality of the binary totient series.\n  - **Canonical claim status.** `supports_registered_claim_family:actual_lcm_orbit_nonintegrality_frontier`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The cofinal separation supply is an open premise, stronger than mere non-integrality; this sufficient route is not an equivalence and leaves the punctured/top-edge boundary and the Erdős #249 endpoint open.\n\n- <code>Erdos249257.ExternalVerification.scaleFullTarget_miss_of_projected_separation</code>\n  - **Class.** conditional reduction\n  - **Statement.** The finite Möbius residue diagonal splits into projected foreign and divisor channels, every finite stable foreign-tail window obeys a closed geometric bound, and a controlled projection separated from every integer forces the full target to be missed.\n  - **Canonical claim status.** `supports_registered_claim_family:actual_foreign_residue_projection`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This is a conditional finite partition/geometric-majorant reduction with a generic complement-noncancellation consumer. The ControlledForeignProjection analytic bridge and any unbounded separated-projection supply remain unproved; no Erdős #249 endpoint, novelty, significance, priority, or external-review claim is made.\n\n- <code>Erdos249257.ExternalVerification.irrational_totient_series_of_pivotResidualDecorrelation</code>\n  - **Class.** conditional reduction\n  - **Statement.** The exact first-harmonic pivot decomposition and its 9X/10 gap yield a finite certificate whenever the stated PivotBudgetAt bounds hold, and the cofinal pivot-residual-decorrelation socket implies irrationality of the binary totient series.\n  - **Canonical claim status.** `supports_registered_claim_family:first_harmonic_pivot_decomposition`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The four-term decomposition and 9X/10 gap remain conditional on the explicit dyadic-room and PivotBudgetAt hypotheses; the cofinal decorrelation producer is unproved. The fibre/image lemmas do not give global supplier-prime isolation or a prime-distribution estimate, and no Erdős #249 irrationality conclusion is claimed.\n\n- <code>Erdos249257.ExternalVerification.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap</code>\n  - **Class.** conditional reduction\n  - **Statement.** An 11/100 nonpositive-phase density on arbitrarily late dyadic blocks yields an 89/100 first-harmonic saving and a finite 9/10 gap; alternatively, a cofinal natural-prime tail-orbit strict gap below 9/10 yields adaptive pivot escape and conditional irrationality. The exact initial phase is the scaled-angle character of (2^h - 1) times the series, and the strict-gap route is an orbit of repeated squaring from that phase.\n  - **Canonical claim status.** `supports_registered_claim_family:strict_prime_tail_orbit_gap`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The phase-density, cofinal natural-prime strict-gap, associated uniform-margin, and cofinal nonpositive-prime-shift producers remain unproved for the actual totient orbit. Exact phase/integrality equivalence, finite dyadic-root exclusion for 1 ≤ h ≤ 16 and k ≤ 14, and non-dyadicity-to-cofinal-nonpositive-phase implications are classification or finite evidence only; they do not supply density or prime alignment, and the source does not prove actual root entry. This is a conditional reduction only: no unconditional Erdős #249 endpoint, novelty, priority, significance, or external-review claim is made. It is distinct from the finite first_harmonic_pivot_decomposition and actual_lcm_orbit_separation families.\n\n- <code>Erdos249257.ExternalVerification.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank</code>\n  - **Class.** locally proved necessary consequence; novelty unassessed\n  - **Statement.** If the binary totient coefficient series were rational, one positive-multiplier tempered integral carry has dyadic sections uniformly eventually periodic modulo its multiplier and canonical carry-section span dimension at least 2^e − 1 for every e.\n  - **Canonical claim status.** `supports_registered_claim_family:totient_carry_kernel_anti_compression`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This is a necessary consequence of rationality only: it gives no contradiction, no finite-rank upper bound for a rationality-supplied carry, and no irrationality proof for Erdős #249. Modulo a divisor of the multiplier, the forcing disappears and the carry becomes geometric, so quotient periodicity alone supplies no finite-ℚ-rank upper bound. The coefficient-side kernel-rank and certificate-equivalence families are not duplicated, and #249 remains open.\n\n**#251: Prime-gap dyadic series**\n\n- <code>Erdos249257.ExternalVerification.exists_primeGap0_gt</code>\n  - **Class.** formalisation of existing result\n  - **Statement.** The consecutive prime gaps are unbounded.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** Unbounded coefficients alone do not force irrationality.\n\n- <code>Erdos249257.ExternalVerification.coefficientOnlyNoGo</code>\n  - **Class.** formalisation of exact finite identity and nonperiodicity contrast; novelty and significance unassessed\n  - **Statement.** The coefficient-only wrapper records the exact finite carry partial-sum identity together with non-eventual-periodicity of the synthetic coefficient stream and the actual prime-gap stream, exposing the limits of coefficient-only reasoning.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The wrapper proves the finite identity and the two non-eventual-periodicity statements only. It does not assert the infinite rational-sum limit of the synthetic countermodel, identify the synthetic coefficient stream with actual prime gaps, establish a prime-gap tail bridge, prove #251 irrationality, or make novelty, priority, significance, or external-review claims.\n\n- <code>Erdos249257.ExternalVerification.irrational_tsum_primeDyadicTerm_iff_primeGap</code>\n  - **Class.** formalisation of existing result; novelty unassessed\n  - **Statement.** Lean checks that the prime-value dyadic series is irrational exactly when the prime-gap dyadic series is, discharging summability by an elementary polynomial prime bound rather than the prime-number theorem. Neither side is proved irrational. The already-known identity is Tao's (7 October 2025) and is recorded separately as cited only.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The equivalence does not prove irrationality of either series.\n\n- <code>Erdos249257.ExternalVerification.irrational_initial_iff_all_positive_tailShifts_nonintegral</code>\n  - **Class.** abstract Lean-checked classification and normal form; novelty unassessed\n  - **Statement.** For any integer-digit real dyadic tail recurrence T(N + 1) = 2*T(N) - g(N + 1), the initial value T(0) is irrational exactly when every positive-length tail shift is nonintegral.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This exact normal form covers arbitrary integer-digit dyadic tail recurrences with the declared real state type; it does not supply a cofinal nonintegral shift for the consecutive-prime-gap recurrence. The prime-specific producer remains open, so no prime-gap irrationality, novelty, significance, or external-review claim is made and Erdős #251 remains open.\n\n- <code>Erdos249257.ExternalVerification.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch</code>\n  - **Class.** conditional actual-prime-gap endpoint reduction; novelty and significance unassessed\n  - **Statement.** For the actual prime-gap dyadic tail, cofinally many adjacent pairs of strictly small h-shifts with unequal corresponding prime gaps force failure of eventual RatIntegral h-shifts.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The representative assumes T : ℕ → ℚ, the DyadicTailRecurrence for the actual prime-gap digits, and for every N₀ an N ≥ N₀ whose adjacent h-shifts both lie strictly in (-1, 1) while the corresponding prime gaps differ. Under that cofinal small-mismatch supply it excludes eventual RatIntegral h-shifts; it does not prove the supply or actual smallness. rationalPrimeGapTail_has_positive_shift_not_eventually_small records the contrary rational-state obstruction for at least one positive shift. This is a conditional reduction only: no #251 irrationality, infinite rational-sum limit, universal producer, novelty, priority, significance, or external-review claim is made; #251 remains open. The actual-prime-gap specialization is distinct from coefficient_only_no_go and does not duplicate prime_gap_reformulation.\n\n**#257: Reciprocal sums over infinite exponent supports**\n\n- <code>Erdos249257.ExternalVerification.volume_mersenneAchievementSet</code>\n  - **Class.** mixed prior-art geometry and locally proved metric result; novelty unassessed\n  - **Statement.** The base-2 Mersenne achievement set has Lebesgue measure exactly one.\n  - **Canonical claim status.** `supports_registered_claim_family:greedy_achievement_geometry`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This does not decide whether every infinite-support sum is irrational.\n\n- <code>Erdos249257.ExternalVerification.irrational_erdosSum_full_support</code>\n  - **Class.** formalisation of an existing theorem\n  - **Statement.** For every integer base b >= 2, the full-support Erdos-Borwein series is irrational.\n  - **Canonical claim status.** `supports_registered_claim_family:eb_full_support`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This is the classical full-support theorem, not universal Erdos #257.\n\n- <code>Erdos249257.ExternalVerification.divisibilityWeightedClaim</code>\n  - **Class.** locally proved result; novelty unassessed\n  - **Statement.** For every base b >= 2, an infinite positive support A with finite prime-weighted mass in base b has irrational support series; if a positive host H has finite prime-weighted mass in base 2, every infinite subset of H has irrational support series at every base b >= 2.\n  - **Canonical claim status.** `supports_registered_claim_family:finite_prime_weighted_support`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The finite-prime weighted-summability premise is explicit in both clauses. This criterion does not cover every infinite support or settle universal Erdős #257; no novelty, priority, significance, or external human-review claim is made.\n\n- <code>Erdos249257.ExternalVerification.finitePrimeWeighted_fixedBase_hereditary</code>\n  - **Class.** locally proved result; novelty unassessed\n  - **Statement.** For every base b >= 2, if a positive host H has finite prime-weighted mass in base b, every infinite subset A of H has irrational support series in the same base b.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This hereditary result retains the same-base weighted-summability hypothesis; it does not infer all-base irrationality from weighted mass at an arbitrary base, or settle universal Erdős #257.\n\n- <code>Erdos249257.ExternalVerification.irrational_erdosSupportSeries_pairwise_coprime</code>\n  - **Class.** formalisation of an existing theorem\n  - **Statement.** For every integer base b >= 2 and every infinite pairwise-coprime support A with summable reciprocal mass, the support-restricted Mersenne series is irrational.\n  - **Canonical claim status.** `supports_registered_claim_family:support_families`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This is the pairwise-coprime support theorem, not universal Erdos #257; the infinitude, pairwise-coprimality, and summable-reciprocal hypotheses remain explicit.\n\n- <code>Erdos249257.ExternalVerification.irrational_ratWeightSeries_eventuallyPeriodic</code>\n  - **Class.** formalised here; novelty unassessed\n  - **Statement.** For every integer base b >= 2 and every eventually-periodic nonnegative rational coefficient sequence γ with a positive value at a positive index in its periodic region, the series sum_a γ(a)/(b^a - 1) is irrational.\n  - **Canonical claim status.** `supports_registered_claim_family:periodic_nonnegative_weight_irrationality`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This is the nonnegative rational eventually-periodic coefficient theorem; its nonnegativity, eventual periodicity, and positive-tail hypotheses remain explicit, and the broader mixed-sign periodic theorem is not claimed.\n\n- <code>Erdos249257.ExternalVerification.irrational_erdosSupportSeries_of_orthogonalPetalBouquet</code>\n  - **Class.** conditional reduction; novelty unassessed\n  - **Statement.** For every support A carrying the finite-core OrthogonalPetalBouquet structure and satisfying the exact uniform SunflowerForcedSlotTailSelection predicate, the base-2 support-restricted Mersenne series erdosSupportSeries 2 A is irrational.\n  - **Canonical claim status.** `supports_registered_claim_family:orthogonal_petal_sunflower_reduction`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This is a conditional reduction: the bouquet structure and uniform tail-selection predicate remain hypotheses; neither is constructed for arbitrary supports, so this does not add an unconditional Erdos #257 support family or solve the universal problem.\n\n- <code>Erdos249257.ExternalVerification.finite_period_noncollapse_rat_den_interface</code>\n  - **Class.** locally proved result; novelty unassessed\n  - **Statement.** For a reduced finite Mersenne sum, the base order modulo the denominator equals the support lcm.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The finite-support theorem does not settle any infinite-support sum.\n\n- <code>Erdos249257.ExternalVerification.volume_supportedMersenneAchievementSet_dichotomy</code>\n  - **Class.** locally proved result; novelty unassessed\n  - **Statement.** A support-restricted Mersenne achievement set has exact finite-complement measure or measure zero for infinite complement.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The measure dichotomy does not classify rational points.\n\n- <code>Erdos249257.ExternalVerification.exists_shifted_odd_tail_nat_states_of_support_fraction</code>\n  - **Class.** conditional reduction\n  - **Statement.** A rational support value with a positive support seed and denominator 2^c*v produces a positive natural shifted tail state with an exact binary recurrence and a modular doubling orbit modulo v.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This state-extraction theorem does not exclude rational infinite supports or settle universal Erdős #257; the support and rational-value hypotheses remain explicit.\n\n- <code>Erdos249257.ExternalVerification.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry</code>\n  - **Class.** locally proved exact equivalence plus conditional consequences; novelty unassessed\n  - **Statement.** For p : ℤ and q > 0, a nonempty positive-index support with erdosSupportSeries 2 A = p/q exists exactly when a quotient-only Boolean–Möbius carry certificate exists, with positive square-root-bounded integer orbit, divisible carry differences, and support reconstructed from its Boolean Möbius transform.\n  - **Canonical claim status.** `supports_registered_claim_family:boolean_mobius_carry`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The equivalence is stated at the existence level with q > 0, 0 ∉ A, and a positive support witness; it gives no contradiction, no universal Erdős #257 irrationality result, and no producer of cofinally nonintegral shifts for the actual recurrence. The generic tempered-orbit theorem and the reciprocal-mass, unbounded-tail, and zero-window results are subordinate consequences, not separate families; the zero-window constant depends on the support and numerator and is not uniform. This does not settle every infinite support, and no novelty, priority, significance, external-review, or unrestricted #257 endpoint claim is made.\n\n- <code>Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips</code>\n  - **Class.** conditional reduction; novelty unassessed\n  - **Statement.** If positive rational-greedy skips occur cofinally in the declared sense, the skipped-core construction supplies cofinally many exact local Mersenne half rows and hence (1 / 2 : ℝ) belongs to the Mersenne achievement set.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This route is conditional on the unproved CofinalPositiveHalfGreedySkips premise: for every N there is c ≥ max N 4 with a positive greedy remainder below mersenneWeightRat c. A finite exact row or any finite collection of exact rows does not establish that cofinal supply. The result is a conditional 1/2-membership/counterexample route only, not an unconditional solution of Erdős #257; the exact-row, upper-half Boolean-word, rank-two, and endpoint-defect declarations are subordinate mechanism evidence, and no novelty, priority, significance, or external-review claim is made.\n\n- <code>Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse</code>\n  - **Class.** locally proved exact endpoint classification; novelty unassessed\n  - **Statement.** Membership of 1/2 in the Mersenne achievement set is equivalent to unbounded false successor terminal bits, equivalently to an unbounded upper-or-middle successor branch, cofinal terminal-false positions, unbounded skipped ranks along the integer seam, infinite greedy skipped support, and no last half-greedy skip.\n  - **Canonical claim status.** `supports_registered_claim_family:half_membership_seam_classification`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This exact endpoint-socket classification does not supply any unbounded terminal-false or upper-or-middle occurrences, cofinal supply, skipped-support producer, or half-membership witness; finite exact rows do not establish the required global supply. It is not an unconditional or universal Erdős #257 result, and no novelty, priority, significance, or external-review claim is made.\n\n- <code>Erdos249257.ExternalVerification.supportCoeff_mul_eq_add_defect</code>\n  - **Class.** unconditional progress\n  - **Statement.** For a positive support element a and positive x, the divisor-count coefficient at a*x splits into the coefficient at x, the distinguished new divisor when a does not divide x, and an exact finite composite-dilation defect.\n  - **Canonical claim status.** `supports_registered_claim_family:composite_dilation_defect_identity`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The exact defect is not bounded for arbitrary supports and supplies no tail selector or irrationality theorem; the separate bouquet budget remains conditional on its structural hypotheses.\n\n- <code>Erdos249257.ExternalVerification.seven_le_of_intBand_odd</code>\n  - **Class.** no-go result\n  - **Statement.** For positive odd integers p, D, and q, the integer form of the two-thirds band forces p ≥ 7; the small odd numerator classes 1, 3, and 5 are therefore excluded.\n  - **Canonical claim status.** `supports_registered_claim_family:half_greedy_two_thirds_band`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This is local 2-adic band arithmetic: p = 7 is sharp for the stated hypotheses, and the theorem does not prove that an actual half-greedy orbit reaches or avoids the band, nor does it prove half-membership.\n\n**#269: Three-prime running least common multiples**\n\n- <code>Erdos249257.ExternalVerification.kernel_235_minor_eq_neg_one_fifteen</code>\n  - **Class.** no-go result\n  - **Statement.** The smallest displayed 2,3,5 kernel minor is exactly -1/15.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** Failure of rank one does not imply irrationality.\n\n- <code>Erdos249257.ExternalVerification.smoothPrefixLcm_eq_threePrimeHeight</code>\n  - **Class.** locally proved result; novelty unassessed\n  - **Statement.** The running lcm of the three-prime smooth prefix equals the exact coordinatewise height.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The identity does not supply the missing irrationality bridge.\n\n- <code>Erdos249257.ExternalVerification.no_positive_reducedCarry_of_cofinalLocalWindowEscape</code>\n  - **Class.** conditional no-go consumer; novelty and significance unassessed\n  - **Statement.** Under the denominator-dependent cofinal local-window residue-escape predicate, no positive reduced carry can satisfy the exact multiplier recurrence together with its short bound.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The representative assumes b,m : ℕ → ℕ, shortBound : ℕ → ℕ → ℕ, CofinalLocalWindowEscape b m shortBound, and for each positive B coprime to 30 a positive integer-valued d with d(n+1) = (b n : ℤ) d(n) − (B : ℤ) (m n : ℤ) and Int.natAbs (d n) ≤ shortBound B n; it then gives False. The representative remains a conditional consumer for arbitrary words. For the actual three-prime word, the rationality-to-reduced-carry bridge is established separately; the source-specific cofinal escape producer is not proved. This is not a #269 endpoint or irrationality proof, and no actual-series identification, novelty, priority, significance, or external-review claim is made. It is one conditional window-carry family, distinct from the finite residue, rank-two, and weighted-phase observer families.\n\n**#1041: Short connections inside polynomial lemniscates**\n\n- <code>Erdos249257.ExternalVerification.exists_small_translation_separating_arguments</code>\n  - **Class.** locally proved result; novelty unassessed\n  - **Statement.** An arbitrarily small common translation separates a finite injective family from zero and from shared positive rays.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** Finite-family avoidance does not provide the global topological or metric gluing.\n\n- <code>Erdos249257.ExternalVerification.constant_perturbation_roots_in_unitDisk</code>\n  - **Class.** locally proved result; novelty unassessed\n  - **Statement.** A quantified small constant perturbation keeps every polynomial root inside the open unit disc.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** Root retention is only one input to an incomplete global route.\n\n**#1049: Lambert-type series at rational bases**\n\n- <code>Erdos249257.ExternalVerification.threeHalves_no_coordinatewiseCorridor</code>\n  - **Class.** no-go result\n  - **Statement.** The coordinatewise divisibility corridor is impossible at base 3/2.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This excludes one proof architecture and proves no irrationality statement.\n\n- <code>Erdos249257.ExternalVerification.rationalBaseClearedTailQ_succ</code>\n  - **Class.** locally proved result; novelty unassessed\n  - **Statement.** The rational-base cleared tail obeys its exact first-order recurrence.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** The recurrence is not derived from a rationality contradiction.\n\n- <code>Erdos249257.ExternalVerification.rectangular_hp_threshold_eq_classical_iff</code>\n  - **Class.** locally proved sharp model-specific no-go; novelty unassessed\n  - **Statement.** For the explicit rectangular two-function exponent model, with real rho and sigma satisfying 0 <= rho and 1 + rho <= sigma, the threshold equals 1/2 - 1/pi^2 exactly when rho = 0 and sigma = 1.\n  - **Canonical claim status.** `comparator_interface_not_registered_as_canonical_claim`\n  - **Novelty.** unassessed; no priority claim\n  - **Boundary.** This is a sharp no-go only for the explicit rectangular two-function model under the stated real hypotheses; it constructs no approximating polynomials or remainders and is not a universal Pade or Hermite-Pade no-go. It proves no irrationality at 3/2 or any general rational-base endpoint, and the analytic remainder/nonvanishing input remains untreated. No novelty, priority, significance, external-review, or endpoint claim is made; Erdős #1049 remains open.\n\n\n</details>\n\n## Verification contract and replay\n\nComparator is used only for exact Lean-owned propositions that can be isolated without importing their proofs; paper deductions, cited theorems, and external computations retain their own evidence classes.\nThe `main_results` key in `formalization.yaml` is the format's list of selected executable interfaces. It is not the canonical claim registry and does not make an unregistered declaration a principal result.\nThe 49 documented interfaces cover all eight programmes. Local proof provenance is recorded separately from novelty, which remains unassessed unless a source-fidelity row says otherwise. The trusted challenge contains one proposition-package fixture and imports only `ExternalVerification.Statements` and Mathlib.\nThe proof-bearing modules occur only in `ExternalVerification.Solution`.\nCI runs the pinned real Linux sandbox and uploads a commit-bound JSON receipt.\nFor a reviewer-run Linux check and the immutable release-asset contract, see `docs/verification/EXTERNAL_VERIFICATION_REPLAY.md`.\n\nUse the precise phrase **Comparator-checked against a separately declared statement and axiom budget** only when the receipt for the displayed commit is green. Do not replace it with “independently verified”.\n","body_format":"text"},"comparator-1049":{"document_id":"comparator-1049","source_path":"docs/verification/COMPARATOR_1049_NUMERICAL_HEIGHT.md","role":"comparator_case_study","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/verification/COMPARATOR_1049_NUMERICAL_HEIGHT.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/verification/COMPARATOR_1049_NUMERICAL_HEIGHT.md","sha256":"sha256:316d19c4714c8366d96f029acef7159bfd46dc52354acdd8b57cf87bb68aa294","byte_count":2407,"origin":"public_main_checkout","body":"# Erdős #1049: one numerical certificate, exactly scoped\n\n## The question\n\nFor which rational bases is the relevant Lambert-type series irrational? The\nbase `3/2` is the main unresolved target in this programme.\n\n## What is checked here\n\nLean proves the concrete inequality\n\n```text\nlog(2) / log(7) < 1/2 - 1/pi^2.\n```\n\nComparator is configured to check a tiny proof transport from the public\nsource theorem to a separately declared Mathlib-only statement, with a\ndeliberately weakened nearby statement as the negative fixture. The checked-in\nconfiguration and workflow are release gates, not terminal replay receipts.\n\nThis is useful because the inequality is the numerical input used at the base\n`7/2` in the external theorem of [Bundschuh and Väänänen](../primary-sources/totient-kernel/bundschuh-vaanenen-1994-source-closure.md). It cleanly separates a\nkernel-checked arithmetic certificate from the literature theorem that\nconsumes it.\n\n## What is not checked here\n\nLean does not formalise the full Bundschuh-Väänänen theorem, all of its\nhypotheses, or its irrationality conclusion. This packet does not advance the\nopen base `3/2` case.\n\n## Who did what\n\nAI agents helped draft the Lean proof, the statement firewall, and this\nexposition. Will Cook directed and reviewed the public release and remains\nresponsible for its claims. The Lean kernel checks the source proposition.\nComparator is configured to check the transport to the separately declared\nstatement. A commit-bound terminal replay artifact is required before claiming\nthat a particular generation passed. The Bundschuh-Väänänen theorem remains a\ncited external result.\n\n## Proof handle\n\n- [Public Lean source](../../lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean): theorem `sevenHalves_bundschuhVaananen_margin`\n- [Separately declared challenge statement](../../verification/ExternalVerification1049/Challenge.lean): theorem `comparator_sevenHalves_numericalHeight`\n- [Source transport](../../verification/ExternalVerification1049/Solution.lean)\n- [Deliberate mismatch](../../verification/ExternalVerification1049/NegativeSolution.lean)\n- [Machine-readable scope](../../verification/comparator-1049-numerical-height.metadata.json)\n\nUntil a commit-bound terminal replay artifact is attached, the precise badge\nis: **Comparator-configured numerical certificate**. It is not an independently\nverified irrationality theorem.\n","body_format":"text"},"architecture":{"document_id":"architecture","source_path":"docs/ARCHITECTURE.md","role":"system_explanation","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/ARCHITECTURE.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/ARCHITECTURE.md","sha256":"sha256:21c7ec9dbc7c7078149d2248965620d88065ab6d1f276ba71192459f907906bb","byte_count":18015,"origin":"public_main_checkout","body":"<!-- SPDX-FileCopyrightText: 2026 Will Cook -->\n<!-- SPDX-License-Identifier: Apache-2.0 -->\n\n# How this repository works\n\nEight mathematical problem programmes connect papers, proofs, computations and\nresearch records. Start with a [problem paper](../paper/README.md#problem-papers).\nYou do not need to know Lean or the project history.\n\n## What this repository is\n\nLean checks formal proofs. The papers explain the mathematics, including\nordinary arguments and cited results. Source maps connect those explanations\nto their supporting records.\n\nThe reviewed claim registry covers #68, #243, #249, #251, #257, #269, #1041 and\n#1049. It describes selected public claims and their status.\nSupporting declarations need not have a claim record; a new file or successful\nbuild creates no public claim. The two Lean roots reflect the development's\nhistory, not different standards of evidence.\n[Methodology](METHODOLOGY.md) explains the review and change rules.\n\nUsing the degree-seven polynomial constructed by the erdosproblems.com\ncontributor ani, Lean proves that every preconnected strict-lemniscate set\ncontaining two distinct roots has one-dimensional Hausdorff measure greater\nthan two. This refutes the exact Formal Conjectures path-image-length\nstatement; the separate total-variation bound is also checked. The other\nseven targets remain open. Independent human review of correspondence with\nthe 1958 wording has not been recorded. Comparator checks only selected exact\nstatements, axioms and kernel acceptance; it does not assess novelty or\nhistorical correspondence. This is a self-contained public release: its use\nand its mathematical claims do not depend on private files.\n\n## Why preserve explanations and failed approaches?\n\nThe design responds to [Henry Cohn on mathematical technical debt](https://terrytao.wordpress.com/2026/09/15/the-technical-debt-of-ai-generated-mathematics/),\n[Tim Gowers on active reading](https://gowers.wordpress.com/2026/09/17/why-i-didnt-sign-the-fields-medallists-letter/)\nand [Grant Sanderson on explanations](https://terrytao.wordpress.com/2026/09/18/if-math-is-more-than-proof-we-need-to-better-celebrate-the-rest-of-it/).\nPlectis keeps arguments, sources and failed routes together so readers can\nreconstruct an argument and continue it. Whether this helps more than papers,\nsource and an on-demand model remains untested. The\n[reading guide](READING_GUIDE.md#work-through-an-argument) offers an active way in.\n\n## Website, map and repository\n\nThe [website](https://wcook04.github.io/practical-formal-verification/maths/) and its\n[maths map](https://wcook04.github.io/practical-formal-verification/maths/universe.html) connect a\nproblem to papers, results and recorded formal evidence. This checkout holds\nthe sources and works independently of the site. Humans enter through\n[README.md](../README.md); agents use [AGENTS.md](../AGENTS.md). Both lead to\nthe same evidence.\n\n## The architecture in one page\n\nA statement in a paper may have a formal proof, an ordinary mathematical\nargument, or a cited external theorem behind it. A correspondence audit starts\nfrom the current paper and asks what establishes each assertion, including its\nassumptions and full conclusion. The evidence routes join at public claim\nreview:\n\n```text\nordinary proof / cited theorem       Lean source       finite computation\n              |                         |                     |\n              |                 formal statement review       |\n              +-------------------------+---------------------+\n                                        |\n                                        v\n                               docs/claims.json\n                        public statement, evidence, limits\n                                        |\n                                        v\n                           papers and public guides\n                                        |\n                                        v\n                         generated indexes and source links\n```\n\nEach evidence route keeps its own scope. An ordinary proof or cited theorem\ndoes not require a Lean proof, and a finite computation establishes only its\nfinite domain. Outside reading, criticism and reuse are observed separately;\nnone follows from passing through this diagram.\n\nA stronger Lean theorem needs a checked implication to the printed statement.\nUnproved conditions and partial coverage stay explicit.\n\nFor selected statements, [Comparator](EXTERNAL_VERIFICATION.md) compares the\nseparately declared challenge with the solution under a configured axiom\nbudget. Inspect the current configuration and available evidence with\n`python3 scripts/query_corpus.py --route comparator_assurance`.\nThe [Palomar qualification route](verification/PALOMAR_QUALIFICATION.md),\n`python3 scripts/query_corpus.py --route palomar_qualification`, checks the\nrepository's packaging requirements. The\n[eight per-problem repository units](https://github.com/wcook04/plectis-erdos-lean)\nhave their own pinned source and replay checks. A local qualification decision\ndoes not establish a service submission, registration or acceptance, and a\nComparator pass does not establish complete paper coverage.\n\nLean and the release checks test different parts of that route:\n\n- Lean decides whether a formal proof establishes its precise formal statement.\n- A mathematician decides whether the public wording describes that statement\n  faithfully and what mathematical significance it has.\n- The release program decides whether the files satisfy the recorded\n  relationships: declarations resolve, source locations agree, paper identities\n  match, and required limitations remain present.\n\nThe [reading guide](READING_GUIDE.md#contributing) records authorship and review.\nThese responsibilities do not imply that independent review has occurred.\n\n## Which file is authoritative for what\n\n| Question | Source | What to read there |\n|---|---|---|\n| What formal statement and proof does Lean check? | [Erdos249257.lean](../lean/Erdos249257.lean), [ErdosProblems.lean](../lean/ErdosProblems.lean) and the modules they import | The exact declarations at the source revision recorded for the result. |\n| What does the selected public claim say? | [docs/claims.json](claims.json) | Wording, status, supporting declarations, finite ranges and remaining open propositions. |\n| What review does a claim change require? | [docs/methodology.json](methodology.json), rendered as [methodology](METHODOLOGY.md) | The evidence and review required for each kind of change. |\n| Which papers are shipped? | [docs/publication_contract.json](publication_contract.json) | Manuscript roles, file identities and reading routes. |\n| Where is the evidence for one result? | [docs/SOURCE_MAP.md](SOURCE_MAP.md) | Routes between problems, claims, paper passages and Lean source. |\n| Which statement does Comparator compare? | [verification/comparator.json](../verification/comparator.json) and [external verification](EXTERNAL_VERIFICATION.md) | The selected challenge, solution, permitted axioms and replay boundary. |\n| What is ready for Palomar? | [docs/verification/PALOMAR_QUALIFICATION.md](verification/PALOMAR_QUALIFICATION.md) and [docs/PALOMAR_POLICY_RECONCILIATION.json](PALOMAR_POLICY_RECONCILIATION.json) | Repository qualification, with external actions and outcomes recorded separately. |\n| Formal Conjectures progress? | [Crosswalk source](formal_conjectures_crosswalk.json) and [guide](verification/FORMAL_CONJECTURES_CROSSWALK.md) | Dated reviews and merges; separate pinned statement comparisons. |\n| What did a research round return, and what does it still owe? | [The research record](reference/RESEARCH_RECORD.md) | Custody of each return, its dispositions, computed milestones, relation rows and contrasts. |\n| Which checks govern a release? | [scripts/check_release.py](../scripts/check_release.py) and [the GitHub workflow](../.github/workflows/lean.yml) | The local checks and the checks run on pushes and pull requests. |\n\nKeep mathematical explanations and reviewed interpretations authored. Generate\nvolatile facts—paper lists, source locations, selected interfaces and status\nsummaries—from the records that own them. The programme card in the\n[agent guide](agents/AGENT_GUIDE.md) comes from the problem and claim owners;\n[Palomar qualification](verification/PALOMAR_QUALIFICATION.md) comes from its\nselection record. The [orientation](ORIENTATION.md), atlas and module index\nare generated too. `python3 scripts/refresh_projections.py` updates these\nsurfaces; the release gate rejects stale copies. Dated development and campaign\nrecords live in [technical reference](reference/README.md).\n\nThe eight individual problem papers are the active mathematical routes. The\nformer combined manuscript,\n[paper/archive/erdos249-257-main-paper.tex](../paper/archive/erdos249-257-main-paper.tex),\nis archived provenance only, not an active gateway.\n\n## Repository map\n\nThis tree groups repeated directories; folder indexes give the full contents.\n\n```text\nplectis-erdos/\n|-- README.md, AGENTS.md       Human and agent entry\n|-- CONTRIBUTING.md           Contributions and credit\n|-- CITATION.cff, LICENSE*    Citation and licences\n|-- lake*, lean-toolchain     Lean build and dependency pins\n|-- formalization.yaml        Selected Comparator statements\n|-- paper/\n|   |-- 68/ … 1049/           Short papers and longer records\n|   |-- synthesis/, systems/  Cross-problem and system papers\n|   |-- exposition/           Writing-method papers\n|   |-- reasoning-parts/      Long-record source sections\n|   `-- evidence/, archive/   Generated links and earlier work\n|-- lean/\n|   |-- ErdosProblems/        Eight programmes, Shared/, Synthesis/\n|   `-- Erdos249257/          Original library and certificates\n|-- docs/\n|   |-- *.md, *.json          Reader guides and source records\n|   |-- agents/               Operational instructions\n|   |-- papers/               Inventory and generated full text\n|   |-- reading-edition/      Portable paper introduction\n|   |-- verification/         Replay instructions\n|   |-- semantic/             Statement relationships\n|   |-- research-commons/     Contributions, reviews and credit\n|   `-- reference/, …         Specialist guides and provenance\n|-- research/                 Experiments, examples and sessions\n|-- research_corpus/          Preserved research returns\n|-- computations/             Exact certificates and recomputation\n|-- verification/             Formal interfaces and replay setup\n|-- evidence/                 Paper-to-proof and Comparator records\n|-- scripts/, skills/         Tools and agent workflows\n`-- .github/                  CI, forms and community policies\n```\n\nFolder guides: [papers](../paper/README.md), [Lean](../lean/README.md),\n[documentation](README.md), [research](../research/README.md),\n[experiments](../research/experiments/README.md),\n[returns](../research_corpus/README.md), [computations](../computations/README.md),\n[verification](../verification/README.md), [evidence](../evidence/README.md),\n[scripts](../scripts/README.md) and [skills](../skills/README.md).\n\nThe [agent workbench](agents/AGENT_WORKBENCH.md) documents exact queries;\nthe [Formal Conjectures crosswalk](verification/FORMAL_CONJECTURES_CROSSWALK.md)\nconnects to upstream statements. The [research commons](research-commons/README.md)\nexplains how work returns with evidence and credit. The companion\n[selected verification repository](https://github.com/wcook04/plectis-erdos-lean)\nis optional; this checkout is self-contained.\n\n### Directory and naming conventions\n\n`paper/` holds manuscripts and PDFs; `docs/papers/` holds generated full text\nand its catalogue. `verification/` holds formal interfaces and configuration;\n`docs/verification/` holds instructions. `research/` contains experiments and\ntools; `research_corpus/` preserves dated returns.\n\nLean source names follow namespaces and Lake configuration. The older\n`Erdos249257` name remains in imports, citations and recorded checks; use the\n[source map](SOURCE_MAP.md) to locate results across both libraries. Renaming a\nsource requires updating those consumers together.\n\n## A complete example\n\nThe [certificate definition](papers/full-text/erdos249-totient-reasoning-surface.md#definition-of-the-finite-residue-test)\nand [certificate source map](SOURCE_MAP.md#249--certificate-reductions)\ngive the mathematical notation and source routes behind this example.\n\nOne public claim says that Lean has checked successful finite calculations at\nevery scale up to 82, together with eight small periods and separate shards.\n\nIn ordinary language, the record says:\n\n```text\nclaim id: certified_kill_instances\npublic statement: Lean checked the listed finite cases.\nstatus: verified finite instance\nbounded range: eight small periods, every scale value up to 82, and the shard parameters, all named in the record. Nothing is asserted at 83.\nformal evidence: six named Lean declarations.\nstill open: prove that successful cases exist beyond every fixed cutoff.\n```\n\nThe range is contiguous rather than a sampled list: the scale quantity changes\nonly at prime powers and is constant between them, so the earlier deposits\nthrough 64 already covered every value up to 66. Each scale is nevertheless an\nindependent construction — the two supporting modules share no prime — so the\nband buys reach, not leverage.\n\n*Verified finite instance* means Lean checked the stated finite inputs. A\nsupply of successful cases beyond every fixed cutoff would settle the open\nproblem by a proved equivalence; the finite list does not supply it.\n\nThis one record connects the layers:\n\n1. The named declarations in the Lean files are the formal evidence.\n2. [`docs/claims.json`](claims.json) records the public sentence, the\n   finite range, the Lean declaration names, and the question still open.\n3. The README and mathematical paper explain the result to readers.\n4. Generated indexes provide links and source locations.\n5. The release program checks that the declaration names still exist, the\n   recorded locations remain accurate, the bounded range is present, the\n   public limitation has not disappeared, and the generated files are current.\n\nA mathematician is still responsible for judging that the English sentence is\na faithful description of the Lean statement. The software preserves that\nreviewed relationship after it has been recorded.\n\n## What happens when a change is made\n\n| Change | What follows |\n|---|---|\n| A Lean statement or proof changes | Check the affected proof, review its assumptions and meaning, then update the claim and paper if their content changed. |\n| An explanation changes | Check its links and evidence boundary. A change in mathematical meaning also needs the corresponding claim review. |\n| A generated index is stale | Run its owning builder, inspect the result and rerun the relevant check. |\n| A manuscript changes | Rebuild its PDF and include the source, PDF and check output in the proposal. Maintainers complete the generated paper updates before merging. |\n\n[The agent guide](agents/AGENT_GUIDE.md) gives the source-change and validation\ninstructions; the [paper build guide](../paper/README.md#build-and-update)\ngives the manuscript commands. Builders run after their source records change.\nAn English edit cannot change what Lean proved.\n\nPreserve and reconcile concurrent contributions with evidence and attribution:\n[Contributing](../CONTRIBUTING.md), [credit policy](research-commons/CREDIT_POLICY.md).\n\n## How the checks run\n\nGitHub runs a Lean job and a release-surface job on pull requests and on manual\ndispatch. The [workflow file](../.github/workflows/lean.yml) contains their\ncurrent commands. The Lean job installs the pinned toolchain and dependencies,\nthen builds the supported roots named in that file. A separate coverage build,\n[lean-coverage-build.yml](../.github/workflows/lean-coverage-build.yml), compiles\nthe paper-coverage Lean modules outside those roots on pushes to main that touch\na Lean input, and on manual dispatch. The release-surface job checks the claim\nrecords, source links, generated files, papers, licences and query routes,\nincluding tests with deliberately invalid inputs.\n\n[Reproducibility](REPRODUCIBILITY.md) gives inspection, installation and build\ncommands. The proof wrapper invokes `lake build`;\n`python3 scripts/check_release.py` does not run Lean.\n\nFor documentation work, the focused reader checks include\n`python3 scripts/test_human_first_contact.py` and\n`python3 scripts/check_architecture_guide.py`.\n`python3 scripts/check_cold_clone_comprehension.py --quick` checks the bounded\nnavigation routes before a toolchain download. Use the relevant tool's own\ntests for a tooling change and the proof checks for a Lean change.\n\n## What the checks do not prove\n\nA passing Lean build means that the kernel accepted the formal statements and\nproofs it checked. The intended mathematical meaning still needs review.\n\nA passing release check means that the recorded comparisons succeeded. It\ndoes not mean that a program understood every sentence in the repository. It\nalso does not prove that every important sentence was selected for checking.\n[Methodology](METHODOLOGY.md) sets out those responsibilities, and\n[external verification](verification/README.md) explains additional statement\ncomparisons and their limits.\n\nThe\n[systems paper](../paper/systems/claim-faithful-publication-systems-paper.pdf)\nexplains the publication architecture in more depth. Its historical checker\nexample is documented in [docs/publication_evidence.json](publication_evidence.json);\nit illustrates the checks recorded for that exercise.\n\n## Where to start\n\nRead a [problem paper](../paper/README.md#problem-papers),\n[reproduce a check](REPRODUCIBILITY.md), or\n[propose a contribution](../CONTRIBUTING.md).\n\nThe archived combined #249/#257 PDF is not a default reading route.\n[The documentation index](README.md) lists current guides and specialist references.\n","body_format":"text"},"residual-progress":{"document_id":"residual-progress","source_path":"docs/reference/RESIDUAL_PROGRESS.md","role":"open_boundary_progress","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/reference/RESIDUAL_PROGRESS.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/reference/RESIDUAL_PROGRESS.md","sha256":"sha256:d9bc78d21dcbe5c57b088d82edb321aa090dea33074d3d00f23d0feb267c8d6c","byte_count":10807,"origin":"public_main_checkout","body":"# Deciding whether a proof sketch reduced its target or renamed it\n\nA proof either checks or it does not, and a checker says nothing about an\nincomplete one.  An agent that leaves `sorry` behind has produced *something* —\nthe question is whether what remains to prove is easier than what it started\nwith, or the same problem wearing different notation.  That question is\nsemantic, it is the one a search loop has to answer thousands of times, and\nsome instances of it are decidable by the kernel.\n\nThis document describes what is decidable, what this repository measured, and\nwhat the evaluator in `scripts/residual_evaluator.py` does and does not settle.\n\n## The reported failure\n\nGoogle DeepMind's AlphaProof Nexus paper ([arXiv:2605.22763][nexus]) rates\ncandidate proof sketches with a language model, because formal verification\ngives a binary signal and evolutionary search needs a graded one.  Its raters\njudge, among other things, \"the plausibility of remaining goals\".  Its failure\nanalysis reports:\n\n> the agent frequently offloaded a problem's core difficulty into a single\n> `sorry` within a helper lemma that reiterated the target statement in a\n> slightly different form\n\nand, of that behaviour:\n\n> Explicitly prompting against this behavior failed to prevent it.\n\nPrompting is the wrong instrument.  Whether a helper lemma reiterates its\ntarget is a relation between two propositions, and a proof assistant can\nsometimes settle it outright.\n\n## Where the instrument came from\n\nThis repository is a formalisation programme over eight Erdős problem programmes,\nand its output is structural: an exact carry normal form equivalent to\nirrationality for #68, a complete measure classification of the Mersenne\nachievement set for #257, the exact rank `2^e + 1` and an explicit odd-core\nbasis of the dyadic totient kernel for #249, a first integral of the complex\nNewton flow for #1041, exact three-prime prefix-LCM dynamics for #269, and proved no-go\ntheorems that close named strategies rather than gesture at difficulty.\n[RESULTS](../RESULTS.md) states each result and the exact point where it stops.\n\nHolding that line requires deciding, repeatedly, which of a programme's many\nreformulations are progress.  The instrument described here was built for that\njob, and it was turned inward first.\n\n`ErdosProblems/DemandLedger/Hyp.lean` provides `hypOf%`, a term elaborator that\ntakes the type of a named hypothesis binder directly from the kernel's\n`ConstantInfo` and returns it as a standalone `Prop`.  It does not pretty-print\nand reparse: the extracted proposition is the binder type or elaboration fails.\nAn unresolved mathematical demand, which previously existed only in binder\nposition, becomes an object that can be named, compared and proved about.\n\nThe mechanism is intentionally local. A submitted target and its remaining\nobligations are compared as exact propositions, one sketch at a time, against a\nbounded proof budget. It does not infer a live corpus census from historical\nbinder extraction, and it does not turn an authored relation label into a\nkernel verdict. Current registered relations are navigated through\n`python3 scripts/query_semantic.py`; current problem status and continuation\nobligations remain owned by `python3 scripts/query_corpus.py` and\n`docs/claims.json`.\n\n## Restatement and reuse use the integrated corpus\n\nThe retired two-problem productivity report counted textual references between\nLean declarations and divided a historical sample by an authored\n``is_restatement`` label.  That was neither a kernel restatement check nor a\ncurrent measurement of the eight public programmes.  Textual fan-in also does\nnot establish mathematical importance, novelty or significance, so the report\nand its standalone builder are no longer public infrastructure.\n\nThe current system keeps each responsibility with its existing owner:\n\n- [`docs/claims.json`](../claims.json) records reviewed public status and exact\n  Lean addresses for the selected registered claims; it does not assert that\n  every mathematical statement in every paper is registered.\n- [`docs/semantic_corpus.json.gz`](../semantic_corpus.json.gz), queried with\n  `python3 scripts/query_semantic.py`, owns cross-paper statement and relation\n  navigation across the current corpus.  Relation labels are navigation and\n  do not themselves certify equivalence, novelty or value.\n- [`docs/PALOMAR_RESULT_SHOWCASE.json`](../PALOMAR_RESULT_SHOWCASE.json) owns the\n  current candidate universe, screening dispositions and reader-facing signal\n  order without turning that screening into proof or peer review.\n- `scripts/residual_evaluator.py` remains the exact bounded route for asking\n  whether a particular proposed reduction leaves its target, an equivalent\n  residual or an explicitly stronger obligation.  Its answer is about the\n  submitted Lean propositions, not an aggregate score over an old sample.\n\n## What the evaluator decides\n\n`scripts/residual_evaluator.py` asks Lean three bounded questions about a\nsketch that targets `P` and leaves residual obligations `R₁ … Rₙ`:\n\n1. **Sufficiency.** Is `R₁ ∧ … ∧ Rₙ → P` provable within the budget?\n2. **Restatement.** For each `Rᵢ` separately, are *both* `P → Rᵢ` and\n   `Rᵢ → P` provable?\n3. **Refutation.** Is the residual demand provably false?\n\nFour outcomes follow, and only two are vetoes:\n\n| Verdict | Meaning |\n|---|---|\n| `endpoint_equivalent` | Some single residual is provably equivalent to the target. The sketch renamed the problem. |\n| `residual_refuted` | The residual demand is provably false. The sketch rests on something untrue. |\n| `strict_decomposition_candidate` | The residuals suffice, and no individual residual is provably equivalent to the target. |\n| `unknown_relation` | The budget settled nothing. Not progress, not falsity, not separation. |\n\nTwo details matter more than they look.\n\n**Restatement is tested per residual and in both directions.**  Testing\n`P → ⋀Rᵢ` instead is wrong: a target implies every legitimate consequence, so\nthat criterion convicts every correct decomposition.  An earlier version of\nthis evaluator did exactly that and flagged a textbook base-plus-step induction\nsplit as the failure it is the opposite of.\n\n**A false accusation is the expensive error.**  Requiring two kernel proofs on\none residual means a well-disguised restatement can escape into\n`unknown_relation`.  That is the intended trade: missing one costs a wasted\nsearch branch, while falsely rejecting a productive change of representation\ncosts the branch that would have worked.\n\n## ResidualBench\n\n`residualbench/ResidualBench.lean` holds eight labelled sketches over two\nelementary targets — the sum of the first `n` odd numbers, and a divisor-count\nidentity — including deliberate restatement, self-assumption, false-residual\nand hallucinated-literature specimens, plus two controls that must **not** be\nvetoed.  Labels live in `scripts/residualbench_manifest.json` and are compared\nagainst the run; the evaluator refuses a manifest that tries to author a\nverdict rather than declare an expectation.\n\nUnder budget `ladder_v3_independent`:\n\n| Sketch | Verdict | |\n|---|---|---|\n| `A_notation` | `endpoint_equivalent` | veto — notation change only |\n| `B_reassociation` | `unknown_relation` | equivalent in fact; budget did not reach it |\n| `C_induction` | `strict_decomposition_candidate` | genuine base-plus-step split, correctly not vetoed |\n| `D_self_assumption` | `unknown_relation` | residual is a tautology; see below |\n| `E_hallucinated` | `residual_refuted` | residual is false |\n| `F_strengthening` | `unknown_relation` | **control** — a productive strengthening, not vetoed |\n| `G_hyperbola` | `endpoint_equivalent` | veto — representation change only |\n| `H_incomparable` | `unknown_relation` | **control** — unrelated true residual, not vetoed |\n\nThree vetoes, both controls clean, and the legitimate decomposition survives.\n`docs/residualbench_report.json` is the run: every proposition put to Lean, the\ntactic that discharged it, and its axiom budget.  It records this repository's\npinned Mathlib environment only; no cross-pin receipt is registered here, so\nthis note makes no portability claim about the older Formal Conjectures pins or\nother library versions.\n\nTwo rows disagree with their labels on budget grounds and are left that way.\n`B_reassociation` is an equivalent range reindexing and `F_strengthening`\nspecialises a general arithmetic-progression formula; the current general\ntactic ladder does not close either.  Adding tactics shaped to these fixtures\nwould turn the benchmark into a transcript of its own tactic list.\n\n`D_self_assumption` is an error by the author of the bench, kept on the record.\nIts residual is `OddSum → OddSum`, which is a tautology — equivalent to `True`,\nnot to the target, because it cannot imply the target back.  It was labelled\n`endpoint_equivalent`.  The evaluator was right and the label was wrong.  A\nresidual provable with no assumptions establishes nothing and deserves its own\nverdict; it does not yet have one.\n\n## Running it\n\nRequires a built Mathlib for this toolchain.\n\n```\npython3 scripts/lean_fast_build.py ResidualBench\npython3 scripts/residual_evaluator.py \\\n  --manifest scripts/residualbench_manifest.json \\\n  --lean-root . \\\n  --no-cross\n```\n\nThe run emits, per sketch, the propositions it put to Lean, which tactic\ndischarged each, and the axiom budget of every accepted proof.  A tactic\nattempt counts as proved only when `#print axioms` returns a report free of\n`sorryAx`; a failed theorem still enters the environment closed by `sorryAx`,\nso the presence of a report is not sufficient.  Each run carries two controls —\none trivially provable proposition and one unprovable one — and refuses to\nreport any verdict if the harness cannot separate them.\n\n## What this does not establish\n\nThe evaluator is sound where it speaks and silent elsewhere.  It is not a\nproof-difficulty metric: logical strength and proof cost are different\nquantities, and a stronger residual may be easier because it has more exploitable\nstructure.  It is not complete: `unknown_relation` means the budget proved\nnothing, never that no relation exists.  It has not been run against a live\nproof-search loop, so no claim is made here about search efficiency or solve\nrate.  And no priority is claimed for the construction; the relation between\nthis and existing proof-progress prediction and reward-model work has not been\nsurveyed.\n\nWhat it does establish is narrower and checkable: for the specimens listed\nabove, at the stated budget, in the stated environment, the relation between a\nsketch's remaining obligations and its target was decided by the kernel rather\nthan estimated.\n\n[nexus]: https://arxiv.org/abs/2605.22763\n","body_format":"text"},"wave-index":{"document_id":"wave-index","source_path":"docs/reference/WAVE_INDEX.md","role":"development_chronology","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/reference/WAVE_INDEX.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/reference/WAVE_INDEX.md","sha256":"sha256:3b7ba9ec389c832d2883a8cbfd32df7dccd78196103665930bffa9e138bcf263","byte_count":21698,"origin":"public_main_checkout","body":"<!--\nSPDX-FileCopyrightText: 2026 Will Cook\nSPDX-License-Identifier: Apache-2.0\n-->\n\n# Wave index\n\nWave labels are **development chronology**, not Lean import order.\n\nRead order for humans:\n\n1. `docs/orientation.json` or `docs/ORIENTATION.md` — select one mathematical\n   programme, claim, or exact remaining open proposition.\n2. `docs/SOURCE_MAP.md` — follow that selected intention into a bounded module\n   route.\n3. [Current papers](../../paper/README.md) — read the selected problem's\n   short paper, longer research record, and source handles.\n4. this file — recover development chronology only when chronology is the\n   question.\n5. `Erdos249257.lean` and `ErdosProblems.lean` — inspect package topology only\n   when import structure is the question.\n\nFor an external statement-identity handoff, use the generated [Formal\nConjectures crosswalk](../verification/FORMAL_CONJECTURES_CROSSWALK.md). It binds each of the\neight problem rows to its pinned upstream declaration, source hash, and local\nproblem route. For a module or paper handle, use\n`python3 scripts/query_corpus.py --module <module_path_or_sigil>` or\n`--paper-anchor <TeX_label_or_source_ref>` to recover the corresponding\nproof/source context before returning to the problem's exact boundary.\n\nLean source checked by the pinned Lean kernel is proof authority. Wave order\ndoes not strengthen a claim or close an exact open proposition. Erdős #249 and\nthe universal form of #257 remain open.\n\n## Chronology map\n\n| Band | Where it lives | Notes |\n|---|---|---|\n| Pre-wave / early kernel mass | `Erdos249257/CertificateKernel.lean` | Core #257 support-family and certificate machinery assembled before the separately named #249 modules |\n| Waves 8–16 | mainly `CertificateKernel.lean` | Earlier kernel layers retained inside the assembled microkernel |\n| Waves 17–25 | separately named modules under `Erdos249257/` | Reader-facing ladder, detailed below |\n| Wave 26–27 | public material inside `CertificateKernel.lean` | Not a separate top-level module |\n| Wave 30 | referenced in `LcmConeNonflat.lean` | Small-prime support of the cone-window argument |\n| Generated certificates | `GeneratedCertificates.lean` + `GeneratedCertificates/*` | Generated finite certificate layers, not waves |\n| Carry trunk | five post-wave modules under `Erdos249257/` | Not waves; a shared binary-carry layer over the kernel, see below |\n| Diagonal pincer frontier | exact decomposition modules, algebraic support, and finite prime-certificate shards | Post-wave #249 work; sharpens the open certificate-supply obligation without closing it |\n\n## Waves 17–25 (named modules)\n\nEach module is a self-contained step with a header docstring stating its own honest boundary. The import chain is `GapFareyBound → MersenneLambertLadder → … → LcmConeNonflat → CertificateKernel ← GeneratedCertificates`.\n\n| Wave | Module |\n|---|---|\n| 17 | `GapFareyBound.lean` |\n| 18 | `MersenneLambertLadder.lean` |\n| 19 | `GeometricCoprimality.lean` |\n| 20 | `GcdMomentCalculus.lean` |\n| 21 | `TotientTailPeriodKiller.lean` |\n| 22 | `CarrySurvivorExtinction.lean` |\n| 23 | `LcmDiagonalReduction.lean` |\n| 24 | `LcmConeFlatness.lean` |\n| 25 | `LcmConeNonflat.lean` |\n\n### Wave 17 — `GapFareyBound.lean`\n\n**Farey-gap denominator bounds.** A mediant/Farey argument places any rational\n`m/q` inside a forbidden gap, so no `q` below the bound can represent `S`.\nElementary and per-window; clears `q ≤ 2.49×10¹⁷` (`K=120`) and\n`q ≤ 7.96×10³⁴` (`K=240`).\n\n### Wave 18 — `MersenneLambertLadder.lean`\n\n**The Mersenne–Lambert ladder**, machine-checked: the rational rungs\n`L(μ)=1/2`, `L(φ)=2`, the positive lift `L(A)=S`, and the Möbius-square lens\n`S = 1/2 + ∑ μ(d)/(2^d−1)²`. Engine: a signed, linear-growth weighted Lambert\nrearrangement.\n\n### Wave 19 — `GeometricCoprimality.lean`\n\n**`S` as coprime-pair mass.** `#{(a,b) : a+b=n, 0<a, gcd(a,b)=1} = φ(n)`, so at\n`r=1/2`, `S − 1/2 = P(gcd(X,Y)=1)` for independent fair-coin waiting times.\nBase 2 is the unique self-normalising point of the geometric law.\n\n### Wave 20 — `GcdMomentCalculus.lean`\n\n**The squared transform** `L₂(f) = ∑ f(d)/(2^d−1)² = E[(f * ζ)(gcd(X,Y))]`,\nsince gcd-divisibility factorises across independent coordinates. Yields\n`L₂(μ) = S − 1/2`, the gcd-moment ladder, and Pillai's gcd-sum function.\nThe exact family return is the [probabilistic gcd-geometry route](../SOURCE_MAP.md#complete-eight-problem-return-matrix):\nthe [totient gcd-moment declaration](../../lean/Erdos249257/GcdMomentCalculus.lean#L235)\nand\nthe [Stern–Brocot cylinder recursion](../../lean/Erdos249257/GcdMomentCalculus.lean#L474),\n[cylinder remainder bound](../../lean/Erdos249257/GcdMomentCalculus.lean#L525), and\n[run-stability declaration](../../lean/Erdos249257/SternBrocotRunGeometry.lean#L343)\nreturn to the paper's [probability-coordinate appendix](../papers/full-text/erdos249-257-main-paper.md#app:lambert-probability),\nwith the stable records `res:directionnormalization`, `res:sternbrocotcylinders`,\nand `res:sternbrocotruns`. The cylinder-law family is also directly recoverable\nthrough `python3 scripts/query_corpus.py --claim stern_brocot_cylinder_law`,\nwhich returns the `probabilistic_gcd_geometry` route and its source-fingerprinted\nresume handoff. These exact identities and combinatorial bounds do not close\nthe open #249 irrationality or unbounded-certificate obligations.\n\n### Wave 21 — `TotientTailPeriodKiller.lean`\n\n**Period, not digits.** Rationality of `S` forces the tail-period law\n`R_{N+h} − R_N ∈ ℤ`; missing that integer by a decidable margin is a finite\n\"kill\". Reduction:\n\n```\nirrational_totient_series_of_certificate_supply\n```\n\n### Wave 22 — `CarrySurvivorExtinction.lean`\n\n**Multiple-period collapse.** Every multiple of a period is a period, so the\nobligation collapses onto the one-parameter family `periodLcm t = lcm(1..t)`.\nAdds the carry-survivor orbit certificate (a bounded integer orbit that must\nescape a narrow strip).\n\n### Wave 23 — `LcmDiagonalReduction.lean`\n\n**Diagonal collapse.** Standing on the ray (`N = periodLcm t`) removes the\nsecond parameter: #249 follows from one ℕ-indexed decidable sequence\n`∃ L, certifiedKill (periodLcm t) (periodLcm t) L` holding infinitely often.\n\n### Wave 24 — `LcmConeFlatness.lean`\n\n**Cone-flatness law.** Rationality forces one fractional constant on the whole\nlcm cone `{k·periodLcm t}`. Certificate **completeness**: a kill exists iff the\ntail difference is a non-integer:\n\n```\nexists_certifiedKill_iff_tail_diff_notMem_int\n```\n\nPlus rank-2 second-difference certificates.\n\n### Wave 25 — `LcmConeNonflat.lean`\n\n**Cone non-flatness refuter.** Interrogates a whole menu of cone vertices at\nonce: if their one-sided arcs shared a common fractional part, the\nminimal-deep-tail vertex would be a common endpoint, and its certificate row\ndenies exactly that. Sharper than pairwise; genuinely joint for menus of\nsize ≥ 3.\n\n## Carry trunk (post-wave modules)\n\nThese five modules are **not waves**. They form a shared binary-carry layer added after the chronology above, building on `CertificateKernel`: what rationality of a binary series forces on its integer carry states. Reading order: `GenericTailOrbitRigidity → GreedyAchievementSet → BooleanMobiusCarry → RationalSupportCarrySkeleton → SublogDivisorCoverage`.\n\n### `GenericTailOrbitRigidity`\n\nDepends on `CertificateKernel`.\n\nFor coefficients `c(n) ≤ n`, the binary series `∑ c(n)/2ⁿ` is rational exactly\nwhen a tempered integer carry orbit exists (`u(N+1) = 2·u(N) − v·c(N+1)` with\n`u(N)/2ᴺ → 0`); every such orbit is rigid, equal to the scaled analytic tail\n`v·T_c(N)`. Balanced pulses also prove unbounded exact successor fan-out,\nruling out a generic autonomous finite-state decoder.\n\nMain theorems:\n\n```\ntemperedBinaryOrbit_eq_scaledTail\nbinaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit\nbalancedPulse_no_autonomous_decoder\n```\n\n### `GreedyAchievementSet`\n\nDepends on `CertificateKernel`.\n\nGreedy geometry for the Mersenne achievement set (values `∑_{n∈A} 1/(2ⁿ−1)`):\nstrict superincreasing tail inequalities, the quantitative gap asymptotic\n`(2/3)·4⁻ⁿ + O(8⁻ⁿ)`, compact/perfect/totally-disconnected/nowhere-dense\nstructure with Lebesgue measure one, exact real and rational greedy\nrecurrences, membership ⇔ all-level greedy survival, sound one-sided finite\nrational death certificates (the exact level-one certificate excludes `3/4`),\nand uniqueness of normalised support coding. Certificates prove nonmembership\nonly.\n\nMain theorems:\n\n```\nvolume_mersenneAchievementSet\nisNowhereDense_mersenneAchievementSet\nmem_mersenneAchievementSet_iff_greedy_survival\ncertifiedGreedyMersenneDeath_not_mem\n```\n\n### `BooleanMobiusCarry`\n\nDepends on `GenericTailOrbitRigidity`.\n\nBoolean–Möbius carry coordinates for support series: the Lambert coefficient\n`f_A(n) = #{a ∈ A : a ∣ n}` satisfies `f_A = 1_A * ζ` and `μ * f_A = 1_A` on\npositive integers; rationality of the support series is equivalent to a\ntempered carry orbit whose carry quotient is exactly `f_A`; normalised nonempty\nsupports with value `p/q` correspond exactly to quotient-only Boolean Möbius\ncarry certificates. The displayed value `1/2` has a canonical affine-orbit\ncriterion. Worked support `{2,3}`: value `10/21`, period-six orbit\n`10, 20, 19, 17, 13, 26`.\n\nMain theorems:\n\n```\nerdosSupportSeries_rational_iff_exists_temperedCarry\nexists_normalized_support_fraction_iff_exists_booleanMobiusCarry\nsupport_half_iff_affineBinaryOrbit_tempered\n```\n\n### `RationalSupportCarrySkeleton`\n\nDepends on `BooleanMobiusCarry`.\n\nResidue wraps and reciprocal mass: the binary repetend identity (least positive\nresidues in a complete doubling cycle sum to odd modulus × number of wraps), an\nalgebraic one-wrap classification, the Cesàro identification of mean support\ntails with the reciprocal mass `∑_{a∈A} 1/a`, the exact excess-mean identity,\nthe rationality-forced lower bound `1/ord_v(2)` on reciprocal mass, the dyadic\nstrengthening (mass divergent or `> 1` for infinite dyadic-rational supports),\nand global unboundedness of the positive carry state attached to any infinite\nsupport with rational value.\n\nMain theorems:\n\n```\nsum_doublingResidue_eq_mul_wrapCount\none_div_oddOrder_le_reciprocalMass_of_support_fraction\ndyadic_support_fraction_reciprocalMass_diverges_or_gt_one\nexists_unbounded_shifted_odd_tail_nat_state_of_support_fraction\n```\n\n### `SublogDivisorCoverage`\n\nDepends on `RationalSupportCarrySkeleton`.\n\n**Sublogarithmic divisor coverage.** If an Erdős support series has a rational\nvalue, consecutive zero windows in its divisor-count coefficient `f_A` have\nlength at most `ε log₂(N+1) + B` for every `ε > 0`, with `B ≥ 0` obtained for\nthat support and numerator. The proof builds `B` from `ε`, `c` and `v`, but the\nsupport and numerator are bound before the existential, so this declaration\ndoes not export one constant uniform across all supports. The proof composes an\nexplicit fixed-power divisor bound (`τ(n)^k ≤ (k^{2^k})^k n`), a binary-tail\nestimate, and the exact carry recurrence. This constrains support coverage; it\ndoes not solve universal #257.\n\nMain theorem:\n\n```\nsupportCoeffZeroWindow_length_le_eps_logb_add\n```\n\nNone of these modules claims a solution of Erdős #249 or #257. The contribution boundary is theorem-family-specific: the carry recurrence and strict-tail geometry have direct prior art (Han Wang; Kovač–Tao); Möbius inversion, repetend algebra, and divisor averaging are classical; the converse/rigidity, certificate-normal-form, and coupled reciprocal-mass families remain exact-source-comparison candidates. No priority claim is made.\n\n`SupportSunflowerDichotomy` is an adjacent conditional #257 route. It gives\nthe exact carry conversion for finite-core orthogonal-petal bouquets and\nrecords an explicit alternating-core squared-prime-petal support. Its uniform\ntail-selection hypothesis is not proved, so it adds no unconditional support\nfamily.\n\n`CompositeDilationDefect` supplies the exact finite correction when a\ncomposite support element dilates the divisor-count coefficient, and bounds\nthat correction for orthogonal-petal bouquets. It is a local identity, not a\ncorrelation or irrationality theorem.\n\n`MaximalOmegaLayer` extends the one-prime-power identity to two distinct\nprime-power layers. It records finite coefficient algebra only, with no\nbounded-Ω or irrationality endpoint.\n\n## Diagonal pincer frontier (post-wave #249 band)\n\nThis band refines the lcm-diagonal reduction.  Its main reading chain is\n`DiagonalPincerDecomposition → SquaredMersenneDiagonalEnclosure →\nDiagonalFreshLossBridge`.  The first module makes diagonal integrality exactly\nequivalent to a foreign-defect term hitting a full-target interval.  The second\ncentres the diagonal expression at a Lambert projection and bounds the signed\nsquared-Mersenne tail.  The third turns the remaining fresh loss into an exact\ninteger projection and then into binary suffix conditions.  Every\nirrationality theorem in this band retains an explicit unbounded-supply\nhypothesis.\n\n`DiagonalPincerCertificates` and the `DiagonalPincerCertificatesT*`\naggregators assemble 28 checked scales through `t = 64`; the many\n`DiagonalPincerPrimeCertificates.*` files are finite proof shards, not separate\nmathematical claims.  The denominator/cyclotomic chain\n`RadicalMobiusShadow → RepunitMobiusNumerator → CyclicTensorMobiusShadow →\nCyclotomicProjectionOfShadow → PrimePowerJumpDynamics →\nMersenneShadowCyclotomicNoncollapse → MersenneShadowDenominatorGrowth`\nsupplies exact rational-denominator information used by the frontier.  The\nremaining named modules provide finite identities or conditional interfaces;\ntheir headers state the boundary locally.\n\n## Transport and curvature companion band\n\nThis post-wave #249 band forms a second, self-contained reading chain:\n`CurvatureCarry → ExponentOnlyTransport → JointExponentTransport`, with\n`PrimeJumpMigration → PrimeJumpWindow`, `ThreeTransportBoundary`,\n`FirstHarmonicGap`, `TropicalCurvatureCarry`, and\n`LcmFactorIdealPulseObstruction` as adjacent branches.  It\nproves exact dyadic window decompositions, affine old-channel annihilation,\nnew-channel migration, two concrete finite certificates, and two scoped no-go\ntheorems: fixed local precision and finite LCM factor/anchor shift algebra.\nThe latter is a synthetic countermodel rather than a model of actual totient\ndifferences.  It does not prove an unbounded certificate supply or a\nfirst-harmonic estimate, so #249 remains open.  The former one-sided companion\nis retired; its checked declarations remain queryable through the claim\nregistry and source map.\n\nThe finite factor-ideal result is one claim inside the reviewed\n`transport_strategy_no_go_boundaries` family, alongside the fixed-precision\nobstruction.  Retrieve that family with\n`python3 scripts/query_corpus.py --publication-family transport_strategy_no_go_boundaries`, or inspect the factor-ideal claim with\n`python3 scripts/query_corpus.py --claim lcm_factor_ideal_anchor_pulse_no_go`.\nIts exact formal anchors are\n[`lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient`](../../lean/Erdos249257/LcmFactorIdealPulseObstruction.lean#L798),\n[`lcm_factorIdeal_sparseAnchor_not_sufficient`](../../lean/Erdos249257/LcmFactorIdealPulseObstruction.lean#L866),\nand [`lcmAnchorPulse_t3_letters`](../../lean/Erdos249257/LcmFactorIdealPulseObstruction.lean#L895).\nThe family remains a synthetic survivor: its forcing letters need not be\nactual totient differences, nonlinear fresh-divisor arguments remain outside\nthe result, and no unbounded certificate supply follows.\n\n## Source-current #1041 frontier (not a wave)\n\nThe public #1041 research corpus is a dated source projection, not a new Lean\nwave and not a reviewed claim-registry surface. Its source checkpoint is\n`6658deca35adde05f60bd2a19c76da996698bc9a`. Read\n[`research_corpus/Erdos1041/FRONTIER.md`](../../research_corpus/Erdos1041/FRONTIER.md)\nfirst: it supersedes stale activation rows by naming the 2026-08-29\nrefutations, the surviving carriers, and the exact open gaps. Then consult\n[`STRONGEST_RESULTS.json`](../../research_corpus/Erdos1041/STRONGEST_RESULTS.json)\nfor the machine-readable premise/consumer map and\n[`CORPUS_MANIFEST.json`](../../research_corpus/Erdos1041/CORPUS_MANIFEST.json)\nfor the published-file and digest inventory.\n\nFor a machine-first handoff, run\n`python3 scripts/query_corpus.py --route erdos_1041`. The route checks the\nfour public frontier files against their indexed digests and returns the\n35-result/open envelope; for a resumable, source-fingerprinted packet use\n`python3 scripts/query_route_memory.py --problem 1041`. Neither route enters\nthe reviewed claim registry, Comparator, or Lean proof authority.\n\nUse this compact reading order:\n\n1. [`AttachmentAwareReeb.md`](../../research_corpus/Erdos1041/AttachmentAwareReeb.md)\n   and [`GenericSufficiencyClosure.md`](../../research_corpus/Erdos1041/GenericSufficiencyClosure.md)\n   for the surviving hub-selection carrier and its generic-to-closed\n   extension.\n2. [`NearFeketeRadialAngularSplit.md`](../../research_corpus/Erdos1041/NearFeketeRadialAngularSplit.md)\n   and [`Degree5AssemblyAndSharpenedCuts.md`](../../research_corpus/Erdos1041/Degree5AssemblyAndSharpenedCuts.md)\n   for the new near-regular and degree-five reductions. The origin-spoke and\n   critical-value-envelope rows refuted in `FRONTIER.md` are not continuation\n   targets.\n3. [`PartialClusterPreimagePerimeterIdentity.md`](../../research_corpus/Erdos1041/PartialClusterPreimagePerimeterIdentity.md),\n   [`TiedNewtonFaceFibreProduct.md`](../../research_corpus/Erdos1041/TiedNewtonFaceFibreProduct.md),\n   and [`ExteriorRootProductCovering.md`](../../research_corpus/Erdos1041/ExteriorRootProductCovering.md)\n   for exact perimeter, moment, and covering premises; each still requires an\n   attachment-compatible selector or metric consumer.\n4. [`MinimalHubArmBudgetRefutation.md`](../../research_corpus/Erdos1041/MinimalHubArmBudgetRefutation.md)\n   and [`SeparatrixAggregateReduction.md`](../../research_corpus/Erdos1041/SeparatrixAggregateReduction.md)\n   for the strongest negative evidence and the machinery that survives only\n   as a possible tool for a different functional.\n\nThe paper-facing route remains\n[`erdos-1041-lemniscate-newton-flow.md`](../papers/full-text/erdos-1041-lemniscate-newton-flow.md),\nwhich points to the checked Newton-flow declarations and states the older\ntopology/metric boundary. The current source frontier does not change the\nopen status of #1041, and its machine rows must not be read as Lean claims.\n\n## Expansion-problem reading handoff (not a wave)\n\nThe five expansion roots have no shared Lean-wave chronology. Their exact\npaper/source joins and open boundaries live in the\n[complete eight-problem return matrix](../SOURCE_MAP.md#complete-eight-problem-return-matrix)\nand the generated [`problems.json`](../problems.json) index. The paper notes are\nthe shortest first read:\n\n| Problem | Paper note | Principal source |\n|---|---|---|\n| #68 | [`factorial carries`](../papers/full-text/erdos-68-factorial-denominator-irrationality.md) | `ErdosProblems/Erdos68/FactorialZeroPlateau.lean` |\n| #243 | [`reciprocal-tail rigidity`](../papers/full-text/erdos-243-reciprocal-tail-rigidity.md) | `ErdosProblems/Erdos243/ReciprocalTailRigidity.lean` |\n| #251 | [`prime-gap dyadic tails`](../papers/full-text/erdos-251-prime-gap-dyadic-series.md) | `ErdosProblems/Erdos251/PrimeGapDyadicTail.lean` |\n| #269 | [`three-prime running lcm`](../papers/full-text/erdos-269-three-prime-running-lcm.md) | `ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean` |\n| #1049 | [`rational-base Lambert`](../papers/full-text/erdos-1049-rational-base-lambert.md) | [`RationalBaseLambert`](../../lean/ErdosProblems/Erdos1049/RationalBaseLambert.lean#L155) · [`ZudilinHeightRegion`](../../lean/ErdosProblems/Erdos1049/ZudilinHeightRegion.lean#L91) · [`RationalPadeArithmetic`](../../lean/ErdosProblems/Erdos1049/RationalPadeArithmetic.lean#L30) · [`HermitePadeNoGo`](../../lean/ErdosProblems/Erdos1049/HermitePadeNoGo.lean#L103) · [`QAperyDiagonalNonEquivalence`](../../lean/ErdosProblems/Erdos1049/QAperyDiagonalNonEquivalence.lean#L67) · [`ZudilinConeArithmetic`](../../lean/ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean#L286) |\n\nThese notes preserve their own evidence classes and limitations; the source\nmap is the route to the exact continuation obligation, not a chronology or a\nclaim-status authority.\n\n<!-- BEGIN generated_package_shape -->\n<!-- Generated by scripts/build_corpus_descriptor.py; do not edit this region. -->\n## Package shape\n\n- `CertificateKernel.lean` (0.85 MiB, 19,278 lines, 845 declarations; 519 theorems and 275 lemmas): the assembled microkernel and headline interfaces.\n- `GeneratedCertificates.lean` (1.18 MiB, 27,728 lines, 1,174 declarations) plus 0 generated shards: finite certificate instances checked by the Lean kernel.\n- The diagonal-pincer family contains 0 isolated prime-certificate modules and 0 scale aggregators. The shards are indexed through aggregators rather than presented as separate mathematical claims.\n- Entire checked corpus: 1,903 modules, 164,639 declarations, 160,373 theorem-like declarations, and 8,171 manifest-marked generated-certificate declarations (a classification floor, not the generated share). The release gate rejects `sorry`, `admit`, custom `axiom` declarations, and `native_decide`.\n\nThese are generated inventory facts, not mathematical claim counts. The declaration atlas\nand Lean source remain the drilldown owners.\n<!-- END generated_package_shape -->\n","body_format":"text"},"frontier-relay":{"document_id":"frontier-relay","source_path":"docs/FRONTIER_RELAY.md","role":"bounded_research_shift","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/FRONTIER_RELAY.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/FRONTIER_RELAY.md","sha256":"sha256:7f5e9356abd6c01f1b07ae0641dbe4c618d4f1ffd52f33b55ed13aab7789bb00","byte_count":10591,"origin":"public_main_checkout","body":"<!-- SPDX-FileCopyrightText: 2026 Will Cook -->\n<!-- SPDX-License-Identifier: Apache-2.0 -->\n\n# Give your agent a research shift\n\nChoose one unfinished question, attempt one bounded continuation, and leave\nsomething another researcher can check and continue. A useful return can be an\nargument, counterexample, correction, exact calculation, or a well-supported\nfailed route. The contributor keeps credit for their original contribution.\nThe public corpus does not claim to solve the eight original Erdős problems.\n\nThis entry packages the existing mining, propagation and return workflows. It\ndoes not require Plectis's private system, a hosted service, a new account or a\nparticular model provider. Owner-authorised model use and optional Lean builds\nhave their own costs and dependencies.\n\n## Or start from the mathematics\n\nA shift does not have to begin from a listed question. You can ask your agent\nto read the corpus and decide what is worth developing: a connection between\nproblems, a general theorem behind a recorded example, a change of\nrepresentation, or a new question with evidence for its value. The\n[exploration skill](../skills/explore-the-corpus/SKILL.md) owns that route in a\nclone, and the [reading edition](reading-edition/README.md) serves it without\none. Once a bounded statement emerges, the rest of this guide applies\nunchanged. A return whose subject is wider than one problem names its subject\nand lists the related problems.\n\n## Pick a question first\n\nIf you already have a clone, this lists every open question in the repository,\ngrouped by problem, each with its statement and its place in the paper:\n\n```sh\npython3 scripts/query_corpus.py --open\n```\n\nChoose one row and open it. The packet names the checked results that bear on\nthe question and the routes that have already been recorded for it:\n\n```sh\npython3 scripts/query_corpus.py --open <id>\n```\n\nThe list is a menu in registry order. It does not rank the questions by\ndifficulty or by value. Some rows are one of the eight original problems. Some\nare restatements that Lean proves equivalent to one of them, so settling such a\nrow settles that whole problem. The rest are narrower questions whose answer\nsettles only what the row says. Open a row and read the checked results that\nbear on it before judging which kind it is.\nThe rest of this guide covers packaging, the pinned wrapper, and the return.\n\n## Start from a reviewed checkout\n\nThe portable wrapper lives at\n[`.agents/skills/plectis-frontier`](../.agents/skills/plectis-frontier/SKILL.md).\nIts four files travel together. Installing only `SKILL.md` is insufficient.\nThe wrapper's revision and the research-source revision are separate: the\nlatter is fixed in its `source.json`. Review both before execution.\n\nIn the reviewed repository checkout:\n\n```sh\nSKILL_DIR=\"$PWD/.agents/skills/plectis-frontier\"\npython3 \"$SKILL_DIR/scripts/bootstrap.py\" describe\n```\n\nInspection makes no network request. Python 3.11+ and Git are required. Linux\nis the initial CI target; macOS is an intended, separately testable target.\nThere is no Windows compatibility claim.\n\nWith the owner's permission, obtain the pinned research corpus in a **new**\ndirectory under an existing writable parent:\n\n```sh\nRESEARCH_DIR=\"$HOME/plectis-research-shift\"\npython3 \"$SKILL_DIR/scripts/bootstrap.py\" prepare \\\n  --destination \"$RESEARCH_DIR\" --allow-network\n```\n\nThat directory must not already exist. Choose a different path rather than\nremoving someone else's work. The download is a full source checkout, without\nLean or mathlib installation; it may be substantial. No private credentials or\nAPI keys are needed. A failed download is retained, not automatically deleted.\n\nAfter reviewing the fetched source and its `AGENTS.md`, authorise the\ntwo Python entry commands:\n\n```sh\npython3 \"$SKILL_DIR/scripts/bootstrap.py\" smoke \\\n  --destination \"$RESEARCH_DIR\" --allow-execution\n```\n\nSave the JSON receipt outside the research checkout. A successful smoke check\nmeans the routing and overview commands ran. It is not a Lean build, proof,\nregistry security verdict, or end-to-end agent compatibility result. Following\nresearch edits will make the initial-clean-checkout verification fail by design;\nuse the established research-return workflow for subsequent validation.\n\n## Install in OpenClaw or another skill client\n\nOn the appropriate OpenClaw Gateway host, with an already configured agent and\nthe owner's approval, install the reviewed local **skill directory**:\n\n```sh\nopenclaw skills install \"$SKILL_DIR\" --as plectis-frontier\nopenclaw skills info plectis-frontier\nopenclaw skills check\n```\n\nThese commands target the configured workspace. Use OpenClaw's `--agent <id>`\nselector when the owner has chosen a named research agent; do not use\n`--global` as a default. Start a new session and ask explicitly for a Plectis\nresearch shift. Keep the installed skill and the research checkout separate.\n\nDo **not** advertise `openclaw skills install git:wcook04/plectis-erdos`\nas a working shortcut: direct Git installation expects `SKILL.md` at the\nrepository root, while this repository contains a nested skill directory.\n\nThe skills CLI can also discover this wrapper from a reviewed local checkout:\n\n```sh\nnpx skills add . --skill plectis-frontier --agent openclaw\n```\n\nThis executes an external CLI: review and record its exact version first. Select\nonly `plectis-frontier`, not every clone-local workflow. After the change is\nmerged and tested on public main, the corresponding repository-discovery route\nis `npx skills add wcook04/plectis-erdos --skill plectis-frontier`.\nA directory being discoverable does not demonstrate a successful run in every\nsupported client. Keep each client test and version separate.\n\n## Pick substantive work\n\nUse the clone's agent entry and current public result ranking to choose an\nexact target. Distinguish the most consequential completed result from the\nmost useful next research task and from a simple machinery demonstration.\n\nFor a first research shift, the #249 certificate programme is a useful starting\nroute. The #257 exact-support frontier is an alternative. Inspect each route's\nactual remaining obligation and failed attempts before choosing; neither is an\ninvitation to relabel an equivalence or finite certificate as a solution.\n\n```sh\ncd \"$RESEARCH_DIR\"\npython3 scripts/query_corpus.py --open\npython3 scripts/query_corpus.py --route erdos249_certificate_story\n# Alternative, not an instruction to attempt both:\npython3 scripts/query_corpus.py --route erdos257_half_story\n```\n\nBefore work, state the exact target, evidence already available, known\nobstruction, one discriminating test, authorised budget and stop condition.\nUse the `mine-open-problem` skill selected by the clone. Do not install Lean or\nstart a costly proof build merely to obtain an overview.\n\nIf you intend to return a structured package, [start the continuation session](research-commons/RETURN_PACKAGE_TEMPLATE.md#start-a-structured-continuation)\n**before editing or committing research changes**. The `start` command records\nthe current commit, contributor and selected question; it requires a clean\ncheckout by default. Keep its session name and generated files for the return.\nAn ordinary issue or pull request does not require this session or JSON files.\n\n## Return something checkable\n\nKeep the human account in ordinary language. Record the starting commit, the\nactual change or failed route, replay commands and outcomes, limitations, and\none next step. Identify human contributors, tool operators and the model and\nagent system actually used. An untested suggestion is not a reproduced result;\na reproduced result is not automatically an accepted contribution.\n\nUse [CONTRIBUTING](../CONTRIBUTING.md), the existing\n[research progress form](../.github/ISSUE_TEMPLATE/research_progress.yml), or the\n[research return form](../.github/ISSUE_TEMPLATE/research_return.yml).\nGitHub Discussions is optional: issue and pull-request intake already exists.\nFor a structured return, follow the clone's\n[`erdos-research-return` skill](../skills/erdos-research-return/SKILL.md) and the\n[packaging sequence](research-commons/RETURN_PACKAGE_TEMPLATE.md#fill-and-package-the-structured-return).\nFill `return.json` and the route-memory sidecar, close the workbench session,\nthen run the validation and packaging commands. The packager copies those\ninputs; it does not write the account of your work. Preserve the local package\nand obtain approval before publishing it. Maintainers decide acceptance and\nrebuild the accepted-contribution credit views from accepted receipts.\n\nWhere Plectis materially helped, please cite the relevant release and explain\nits role. This is a request for accurate attribution, not a claim to a solver's\nmathematics or an additional condition on the MIT-0 wrapper.\n\n## Trust and release boundaries\n\nThe newly authored wrapper is MIT-0 for ClawHub compatibility. The separately\nfetched corpus keeps its existing Apache-2.0 software/documentation and\nCC-BY-4.0 manuscript boundaries. No corpus text is silently relicensed.\n\nThe bootstrap uses an exact public commit, disables Git hooks and ambient Git\nconfiguration for its commands, strips account credentials from child process\nenvironments, refuses existing destinations and separates fetching from\nexecuting repository Python. These controls are not an operating-system\nsandbox, a proof of absence of malicious code, or protection from a compromised\nhost. Use an isolated account/container, restricted filesystem mounts and\nnetwork controls. The flags record operator intent; they are not an access\ncontrol system. Do not install this research skill in an agent with unrelated\nprivate file access or outbound messaging authority.\n\nBefore a release is described as OpenClaw-tested, retain a receipt with the\nwrapper commit, corpus commit, OpenClaw version, operating system, installation\nroute, entry outcome, budget, actual return and remaining failures. The local\nfixture tests are not that receipt. Run the existing public release gate as\nwell as the new wrapper CI; keep failed checks visible.\n\nFor distribution and community-specific boundaries, see\n[FRONTIER_DISTRIBUTION](agents/FRONTIER_DISTRIBUTION.md).\n\n## Packaging sources checked 4 September 2026\n\n[OpenClaw skill CLI](https://docs.openclaw.ai/cli/skills),\n[OpenClaw skill loading](https://docs.openclaw.ai/tools/skills),\n[Agent Skills format](https://agentskills.io/specification), and\n[skills CLI](https://github.com/vercel-labs/skills).\nThese are packaging references, not endorsements or compatibility receipts.\n","body_format":"text"},"contributing":{"document_id":"contributing","source_path":"CONTRIBUTING.md","role":"contribution_and_credit_route","github":"https://github.com/wcook04/plectis-erdos/blob/main/CONTRIBUTING.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/CONTRIBUTING.md","sha256":"sha256:024cd8b3b1087b71ec9bcb80e3eeaed23824f33edca2540a1b4a8055a96dce6e","byte_count":15126,"origin":"public_main_checkout","body":"<!-- SPDX-FileCopyrightText: 2026 Will Cook -->\n<!-- SPDX-License-Identifier: Apache-2.0 -->\n\n# Contributing\n\nThis repository is public so that other people can clone it, do useful work,\nand send that work back. A contribution does not have to solve an Erdős\nproblem. A corrected proof step, a counterexample, a failed route with a reason\nanother person can check, a clearer boundary, or a repaired public check can\nall save future work.\n\n## Choose where to begin\n\n**Work on a paper.** Open [one problem's contribution page](docs/CONTRIBUTE_BY_PAPER.md)\nfor the short and long accounts, a source result to inspect, and current\nquestions. You can send an ordinary mathematical argument without cloning,\nusing Lean, or using an AI.\n\nAn explanation is useful work in its own right. For example, help a reader of\nthe #257 paper see why the first average needs a second one. Name the intended\nreader and the step you want to clarify. A worked example or a better-motivated\nconstruction can help without changing the theorem.\n\n**Develop a reusable method.** Start from the [synthesis papers](paper/synthesis/README.md)\nor the [admissible-choice experiment](research/experiments/sparse_interpolation/README.md).\nA connection, obstruction, new question, construction or useful intermediate\nidea is welcome even when it does not attack one of the eight problems.\n\n**Improve the machinery.** Describe a concrete difficulty in finding, running,\nchecking or returning work through the [architecture contribution path](docs/research-commons/ARCHITECTURE_CONTRIBUTIONS.md).\nAn actionable idea is enough; code is optional.\n\nThese are ways to enter the same contribution process. Change direction when\nthe work requires it. A mixed contribution can remain one pull request, with\nmathematical and software claims reviewed separately and linked credit records\nwhen the receipt format calls for them.\n\n## Return what you learned\n\nUse the [research-progress form](https://github.com/wcook04/plectis-erdos/issues/new?template=research_progress.yml),\na pull request, or [email Will](https://wcook04.github.io/practical-formal-verification/#contact).\nSay what you investigated, what changed or became clearer, what another person\ncan inspect, what remains uncertain, and how you want to be credited. Include\nthe paper edition or starting commit if you know it. You do not need an account\nor a patch to contribute by email.\n\nDirections, references, corrections and explanations deserve attribution as\nwell as proofs and code. For email, say whether your name and the relevant\nidea may appear publicly, or request anonymous credit. Maintainers retain the\norigin of an adopted idea in its public artifact and credit record, distinguish\nsubsequent implementation work, and do not publish the private correspondence.\nThe [credit policy](docs/research-commons/CREDIT_POLICY.md) applies whichever\nchannel you use.\n\n## A first contribution\n\nNo Lean experience is needed to report a broken link, unclear explanation,\nor mismatch between a claim and its evidence. Pick [one claim to trace](docs/REPRODUCIBILITY.md#try-one-claim-without-lean)\nand report an example someone can inspect.\n\nFor a small documentation or tooling patch:\n\n1. Fork this repository and follow the [clone instructions](docs/REPRODUCIBILITY.md#1-start-with-a-complete-committed-checkout),\n   replacing the upstream repository URL in the clone command with your fork's URL.\n2. Create a named branch with `git switch -c <new-branch-name>`, replacing\n   `<new-branch-name>` with a new name you choose. Make one focused change and\n   explain the reader's problem it fixes.\n3. Run the [documentation check](docs/REPRODUCIBILITY.md#check-a-documentation-change)\n   with Python 3.11 or later, or your changed tool's test. Report the result;\n   prose corrections need no full Lean build.\n4. Push your branch to your fork, then open a pull request against this\n   repository's `main` and name your preferred credit. An\n   [issue](https://github.com/wcook04/plectis-erdos/issues/new?template=research_progress.yml)\n   is enough without a patch.\n\nLean or mathematical changes follow the evidence and review requirements below.\n\n## Contribution and review boundaries\n\nParticipation is governed by the [code of conduct](.github/CODE_OF_CONDUCT.md). Direct\ncriticism of a proof, claim, experiment, or repository decision is welcome;\nharassment and attacks on contributors are not.\n\nTreat this contributor path as a prototype until an independent cold-clone run\nand an accepted outside contribution are recorded publicly. Reports about setup\nfriction, unclear instructions, broken links, agent-skill portability, unsafe\ndefaults, or unnecessary steps are substantive architecture contributions, not\nhousekeeping beneath the project's notice.\n\nThe cold-clone comprehension program tests a bounded set of public questions\nand checks that deliberately broken statements or links are detected. A failure\nblocks the release gate. These automated checks do not establish that a person\nunderstood an argument or could use it; that needs actual reader feedback.\n\nThere are two first-class tracks:\n\n1. **Mathematics:** proofs, reductions, computations, counterexamples,\n   corrections, exposition, and reproducible negative or inconclusive results.\n2. **Architecture:** ideas or implementations that improve agent workflows,\n   navigation, validation, reproducibility, tooling, governance, or the public\n   contributor experience.\n\nYou can contribute an architecture idea before you have code. Use the\n[architecture proposal form](https://github.com/wcook04/plectis-erdos/issues/new?template=architecture_proposal.yml)\nor read the [architecture contribution path](docs/research-commons/ARCHITECTURE_CONTRIBUTIONS.md).\nIf an idea is adopted, the accepting change preserves the originator's credit\nin a tracked artifact and accepted receipt.\n\nThe ordinary contribution path is enough: fork or clone the repository, make\na focused change, and open a pull request. Explain the question you started\nfrom, what you changed or learned, what another person can inspect, and what\nremains unresolved. If you have an observation but no finished patch, open the\n[plain-language research-progress form](https://github.com/wcook04/plectis-erdos/issues/new?template=research_progress.yml)\nand write it in your own words. A coding agent can fill the same form; the\nquestions are ordinary language, not a receipt schema. You do not need to learn\nthe receipt format before telling me something useful.\n\nFor compute contributions, begin with a bounded route exposed by the papers,\nclaim registry, or agent entry. The present routes are authored in the public\ncorpus and should name the question, evidence class, and stopping condition;\nqualified external mathematical review is not claimed for every route. You may\nrun them through Claude Code, Codex, Cursor, Antigravity, another agent harness,\nordinary scripts, or your own reasoning. The runner is your choice; the\nreturned evidence boundary is the shared contract.\n\nIf an agent reports a solution, submit the source and evidence as a\n**candidate**, not as an established solution. Maintainers first replay it and\ntry to break it. Lean acceptance establishes that the written formal\ndeclaration follows in the pinned environment; it does not by itself establish\nthat the declaration means what was intended, is novel, settles the original\nproblem, or has been accepted by mathematicians. Work that survives internal\ntriage can be prepared for Comparator and Palomar and placed before the Erdős\nProblems community or another appropriate research audience. Those external\nroutes do not transfer their authority back to a repository status field.\n\nFor a proposed proof, explain the decisive step and the part you have checked\nyourself. Mark what you cannot yet explain, and help resolve questions and\ncorrections in the same issue or pull request. Unfinished work is welcome when\nits remaining work is clear; a large transcript is no substitute for that account.\n\n## What happens to returned work\n\nThe issue or pull request keeps the public history of a proposed change. When\nmaintainers accept the work, they can commit a short receipt recording who\ncontributed, which files or results are being credited, which public commit the\nwork began from, what was checked, and what remains limited. The public credit\npages are rebuilt from accepted receipts. The credit then travels with every\nclone instead of living only in a notebook, chat, or web page.\n\nAcceptance and mathematical claim status are separate. An\naccepted correction can be credited without being called a new theorem. A\nnegative or inconclusive result can be credited when it rules out a route or\nleaves a reproducible stopping point. A formal proof can be accepted as an\nartifact while novelty, exposition, or a stronger public claim still needs\nwork. Earlier credit remains visible when a later contribution corrects or\nsupersedes it.\n\nCredit is attached to the work actually returned. The receipt records\ncontributors, collaborators, tool operators, and disclosed model systems as\ndifferent roles instead of collapsing them into one author field. The project\ndoes not rank people by commit count, diff size, or the number of generated\nrecords. See the\n[credit and stewardship policy](docs/research-commons/CREDIT_POLICY.md) for the\nfull boundary and the [accepted contributions](docs/research-commons/CONTRIBUTIONS.md)\nfor the public, receipt-backed view.\n\nArtifact credit may also carry CRediT-aligned contribution roles such as\nconceptualization, formal analysis, methodology, software, validation, and\nwriting. Select every role that is accurate; no single role is privileged, and\nthe roles describe work rather than deciding authorship.\n\n## Staying in sync\n\nA returned contribution records the public commit where the work began. Later\nreaders can then separate the contribution from changes that landed in the\nmeantime. Git retains the common ancestor between that starting point and the\ncontributor's branch, so maintainers can inspect the original change even when\nthe public main branch has advanced. They first replay the contribution in its\noriginal context, then reconcile the reviewed substance with the current\nrepository. A real conflict is new integration work and receives its own\ncredit; it is not a reason to erase the original contribution.\n\nOnce accepted work reaches the public main branch or a tagged release, an older\nclone can fetch it and merge or rebase in the usual way. The receipt, the\ncredited artifact, and the public credit page arrive together as repository\nfiles. This does not depend on a private development checkout. Validation and\ndownstream consequence checks are repeated after reconciliation because an\nold-branch receipt cannot certify the current repository state.\n\nIf two contributions overlap, preserve both histories and reconcile them in a\nnew change; do not silently rewrite the earlier return. If a later discovery\nshows an accepted artifact was wrong, make a corrective return that names the\nearlier one and says whether it should be retained as history, superseded, or\nwithdrawn from current use.\n\n## What is welcome\n\nMathematical errors and overstatements are especially valuable reports. So are\nLean build failures, broken source or paper links, missing attribution,\nexposition that hides an assumption, counterexamples to intermediate claims,\nand improvements to the clone-local checks. New formal results are welcome,\nand their public description must be no stronger than the evidence. The status\nof the eight headline problems is stated on the [front page](README.md) and\nchanges only through the [methodology](docs/METHODOLOGY.md).\n\nA contribution does not have to start from a listed question. Reading the\npapers as one body of work and developing a connection, a general theorem, a\nclearer explanation of an existing proof, or a well-posed new question is\nwelcome on the same terms. The [reading edition](docs/reading-edition/README.md)\nsupports that route without a clone.\n\nOrdinary prose, citation, and tooling corrections can arrive as normal pull\nrequests. A structured research return is useful when the history matters: for\nexample, a bounded investigation that another person should be able to resume,\nor a negative result whose exact starting point and evidence should not be\nlost. The [research commons guide](docs/research-commons/README.md) explains\nthat lifecycle without requiring you to read a schema.\n\nArchitecture proposals and implementations use the same accepted-receipt and\npublic-credit rail as mathematical returns, but they do not claim a fictional\nproblem number. Their structured frontier declares `track: architecture` and a\nbounded architecture area. The optional session helper supports both tracks; architecture sessions name\nan area, starting files and a validation plan. A\nmathematical return whose subject is wider than one problem names that subject\nand lists the related problems; it never borrows a problem number.\n\n## For agents and maintainers\n\nThe human route above is the contract. The machinery below implements it; none\nof it is a prerequisite for reporting a useful result. Agents should use the\nclone-local [research-return skill](skills/erdos-research-return/SKILL.md), and\nthe [consequence-propagation skill](skills/propagate-research-consequences/SKILL.md)\nafter a stable result, and\nthe [pull-request submission skill](skills/submit-pull-request/SKILL.md) when\nturning owned work into commits and a proposed GitHub return. That skill stops\nbefore pushing or opening the pull request unless the contributor explicitly\nauthorises those external actions. Maintainers should preserve the\ncontributor's prose rather than replacing it with machine field names.\n\nTo open and package a bounded, attributable structured research session, use\n`scripts/continue_research.py`. Its `start`, `check`, and `package` commands\nbind the public origin, starting commit, contributor identity, problem route,\nevidence, and route-memory receipt. The resulting `return.json` and\n`route-memory.json` may accompany a pull request. Validate them with\n`scripts/validate_research_return.py` before submission. The detailed field\ncontract is in the [return package template](docs/research-commons/RETURN_PACKAGE_TEMPLATE.md).\n\nAfter a contribution has actually landed, `scripts/accept_research_return.py`\ncan bind the reviewed result to its accepted public commit. Then\n`scripts/build_research_contributions.py` and\n`scripts/build_research_contribution_recognition.py` rebuild the accepted-only\ncredit views. A submitted return must never appear there before acceptance,\nand an accepted receipt must never silently strengthen `docs/claims.json`.\n\nFor an ordinary source change, run the narrow checks named by the agent entry\nand the affected subsystem. Lean changes must build with the pinned toolchain.\nChanges to claims, papers, or generated projections must follow the authority\nand builder order in the [agent guide](docs/agents/AGENT_GUIDE.md). CI exercises the public return\nvalidator, the acceptance boundary, and the accepted-only attribution views.\n","body_format":"text"},"reproducibility":{"document_id":"reproducibility","source_path":"docs/REPRODUCIBILITY.md","role":"claim_verifier_and_checkouts","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/REPRODUCIBILITY.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/REPRODUCIBILITY.md","sha256":"sha256:a6f77bf243441983abcad2fbdfee7677a99f41d067a0280b66ac8332768617a2","byte_count":22724,"origin":"public_main_checkout","body":"<!-- SPDX-FileCopyrightText: 2026 Will Cook -->\n<!-- SPDX-License-Identifier: Apache-2.0 -->\n\n# Reproduce the public release\n\nStart by following one published result to its source. You can do that with\nGit and Python, then install Lean if you want to check the proof yourself.\nEverything below uses this public checkout and public tools.\n\n| What you want to do | Start here | What you need |\n|---|---|---|\n| Follow one result to its evidence | [Try one claim](#try-one-claim-without-lean) | Git and Python 3.11 or later |\n| Inspect an accepted native #243 or #257 proof | [Native theorem reader routes](research-commons/README.md#native-prove2me-theorems) | A web browser; the public packets print the accepted Solutions without sign-in |\n| Replay the #257 weighted theorem against a separate statement | [Weighted theorem replay](verification/EXTERNAL_VERIFICATION_REPLAY.md#reviewer-replay) | Linux, Git, Python 3.11+, Elan/Lake, Go, and systemd |\n| Rerun the #257 exact-rational example | [Try the late rejection](#reproduce-the-257-exact-rational-example) | Python 3.11 or later; no extra packages |\n| Try a fraction without cloning | [Single-file calculation](#try-a-calculation-without-cloning) | Python 3.11 or later; standard library only |\n| Rerun a #251 finite computation | [Reproduce the #251 computations](#reproduce-a-finite-computation) | Python 3.11 or later; the first run needs no extra packages |\n| Check a documentation edit | [Check a documentation change](#check-a-documentation-change) | Python 3.11 or later; no Lean installation |\n| Compile a proof | [Set up Lean](#2-reproduce-the-pinned-lean-environment) | elan, the pinned dependencies, and space for several gigabytes of cache |\n| Import the library in another project | [Use it as a dependency](#use-the-library-in-another-lean-project) | The same Lean toolchain and the library's dependencies |\n| Reproduce all public checks | [Release checks](#3-run-the-release-surface-checks) | Lean and the pinned Python validation tools |\n\nUnless an example says otherwise, commands run from the repository root. Shell setup below uses macOS/Linux\nsyntax; on Windows, use WSL for the same commands. To read the mathematics\nwithout installing anything, return to [the papers](../paper/README.md).\n\n## Try a calculation without cloning\n\nCan `189/388` be a sum of selected terms `1/(2^n - 1)`? Try the exact-rational\nprobe with Python 3.11 or later; it needs only one file and the standard library.\n\nDownload the [raw Python file at a fixed commit](https://raw.githubusercontent.com/wcook04/plectis-erdos/0f3d8b08c8f5fee06888ec11ea0ed672f92d2266/research/experiments/choices_contraction/rational_membership_probe.py),\nsave it as `rational_membership_probe.py`, and inspect it before running.\nIn that directory:\n\n```sh\npython3 -I rational_membership_probe.py --target 189/388 --depth 16 17 --horizon 160\n```\n\nAt depth 16 it reports `not_excluded`; at depth 17 it reports `excluded` and\nprints the exact gap certificate. Taking the next term overshoots the target;\nomitting it leaves too little in the entire later tail.\n\nChange `--target` to `1/3` for a finite sum, or to `1/2` for an undecided case.\n`not_excluded` makes no claim beyond the tested depth. This finite calculation\ndoes not prove the weighted infinite-support theorem or settle #257.\n[Understand the experiment and check its certificate](#reproduce-the-257-exact-rational-example).\n\n## 1. Start with a complete committed checkout\n\nKeep the commit history: some checks compare the current files with a recorded\nsource version. The following clone keeps that history while fetching old file\ncontents only when needed.\n\n```sh\ngit clone --filter=blob:none https://github.com/wcook04/plectis-erdos.git\ncd plectis-erdos\ngit fetch --tags --force\ngit status --short\n```\n\nThe current checkout is hundreds of megabytes; its historical generated files\nare much larger. `--filter=blob:none` reduces that initial transfer without\nmaking the clone shallow. If you already used `--depth`, fetch the missing\nhistory with `git fetch --unshallow --tags` before running commands that\ncompare pinned revisions. The default `verify_claims.py` environment\ndiagnostic reports missing required history as an environment problem\n(exit `2`). That result does not mean a proof failed. In a shallow clone,\n`--gates` skips the programs that require history. The `--claim` and\n`--verify-all` modes inspect the current checkout and can succeed in a\nshallow clone.\n\n### Try one claim without Lean\n\nAfter cloning, run:\n\n```sh\npython3 scripts/verify_claims.py --claim eb_full_support\n```\n\nThe output shows the published statement, its declaration and source line,\nthe selected Comparator interface, the recorded release, and the claim's\nlimits. For this example, the status is **known mathematics formalised here**.\nThe command looks for the recorded declaration names near their recorded\nsource lines in your current checkout and tests whether the claim uses a\nrecognised status. It also prints the paper and Comparator references. It\ndoes not load the recorded historical revision, run Lean or prove the\nstatement again.\n[Section 2](#2-reproduce-the-pinned-lean-environment) gives the proof-build route.\n\nFor this claim, the verifier's `WRITTEN UP IN` entry points to the archived\njoint manuscript used in the recorded release. For the current exposition,\nread the [#257 short paper](../paper/257/erdos-257-mersenne-support-subseries.pdf).\nThe historical source anchor and the current reading route serve different\npurposes; [the paper index](../paper/README.md) keeps the current papers together.\n\nFor the full claim inventory, use `python3 scripts/verify_claims.py --verify-all`.\nThis mode also uses the current checkout. It additionally reports missing\npaper labels and references to claim IDs absent from the inventory.\n\nTo inspect the weighted #257 result introduced in the README, run:\n\n```sh\npython3 scripts/verify_claims.py --claim finite_prime_weighted_support\n```\n\nThis prints the weighted-summability hypothesis, the all-base conclusion for\ninfinite subsets of a positive host, and the declarations\n`weightedDyadicMeanTarget` and `divisibilityWeightedClaim`. Its paper reference\nis the current #257 short paper, at `res:weighted-support`. The command checks\nthe recorded links in your checkout; it does not run Lean or Comparator.\nTo rerun that formal comparison, use the\n[#257 theorem replay](verification/EXTERNAL_VERIFICATION_REPLAY.md#reviewer-replay).\n\n### Reproduce the #257 exact-rational example\n\nFor a first run without cloning, use the\n[single-file probe](#try-a-calculation-without-cloning). It lets you change the\ntarget and compare two depths. The commands below use the complete checkout\nand include an independent certificate checker.\n\nRun this from the repository root with Python 3.11 or later. It uses only the\nstandard library and needs no Lean installation:\n\n```sh\npython3 research/experiments/sparse_interpolation/late_rejection.py\n```\n\nIn the JSON output, find the certificate with `\"target\": \"189/388\"`.\nExpect `\"first_rejection\": 17`, with remainder\n`9291822600689/1217890317075045460`, tail upper bound\n`196609/25769803776`, and step-17 weight `1/131071`. The script checks\nwith exact fractions that the remainder is greater than the entire later\ntail bound but smaller than the next weight. It also certifies the earlier\nskips, checks the translated target `577/388`, and prints a separate finite\nfactorial-allowance example. A passing run excludes these particular targets\nfrom the Mersenne subsum set; finite survival at an earlier step would not\nestablish membership.\n\nThis calculation does **not** reproduce the finite-prime weighted-support\nirrationality theorem in the [#257 short paper](../paper/257/erdos-257-mersenne-support-subseries.pdf).\nThat theorem concerns infinite positive supports satisfying a weighted\nsummability hypothesis. The script proves neither that hypothesis nor the\ntheorem's irrationality conclusion. The [investigation](../research/experiments/choices_contraction/README.md)\nexplains why the late rejection corrects a proposed finite stopping rule;\nthe [experiment guide](../research/experiments/sparse_interpolation/README.md#exact-checks)\nrecords its other checks.\n\nIf this command fails, confirm `python3 -VV`, rerun\n`python3 scripts/test_choices_contraction_probe.py`, and report the exact\ncommand, error, and `git rev-parse HEAD` through [CONTRIBUTING](../CONTRIBUTING.md).\n\nTo try a fraction of your own, use the single-target probe (replace `189/388`):\n\n```sh\npython3 research/experiments/choices_contraction/rational_membership_probe.py \\\n  --target 189/388 --depth 16 17 --horizon 160 --json\n```\n\nHere the depth-16 outcome is `not_excluded`; at depth 17 it is `excluded` with\nthe exact selected prefix and strict gap certificate. Other outcomes are\n`finite_representation` (the remainder reached zero) and `not_excluded`\n(no conclusion beyond the tested depth). The\n[probe guide](../research/experiments/choices_contraction/README.md#the-probe)\ndefines the host choices and input limits. This is a finite membership probe\nfor the base-two subseries, not a test of the weighted irrationality theorem.\n\nThe [probe guide](../research/experiments/choices_contraction/README.md#the-probe)\nalso gives an inspected, commit-pinned single-file route before cloning. For\na terminal row in this checkout, make a JSON record with `--depth 17` alone\nand check it independently:\n\n```sh\npython3 research/experiments/choices_contraction/rational_membership_probe.py \\\n  --target 189/388 --depth 17 --horizon 160 --json > witness.json\npython3 research/experiments/choices_contraction/verify_terminal_witness.py witness.json\n```\n\nThe receiver recomputes the prefix and tail bound; a `not_excluded` row is\nrefused because finite survival is no membership certificate.\n\n### Reproduce a finite computation\n\nThe #251 paper includes three computations with saved results and public\nprograms. Start with the complete continued-fraction calculation:\n\n```sh\npython3 research/experiments/erdos251/replay.py\n```\n\nExpect `matched_recorded_result: true`; this takes about four seconds on the\nmaintainer's machine and uses only the Python standard library. The\n[computation guide](../research/experiments/erdos251/README.md) explains the\nother two runs, their memory and dependency requirements, and how to change\nan input. Matching these finite results does not prove irrationality or\ncofinality, and the programs do not run Lean.\n\n### Check the clone's navigation and metadata\n\nFor a broader check of the reader routes, run:\n\n```sh\npython3 -VV\npython3 scripts/check_cold_clone_comprehension.py --quick\npython3 scripts/query_corpus.py --tour --format card\n```\n\nThe quick command compares the documented problem selectors, entry commands\nand file counts with the recorded indexes. For the large generated index,\nit compares current file hashes with the saved hash record; if that record\nis missing or stale, it recomputes the index for comparison. The tour lists\nthe available queries. Neither command compiles a proof.\n\nTo explore one problem, ask for its paper and source routes. For example:\n\n```sh\npython3 scripts/query_corpus.py --route erdos_251\n```\n\nReplace `251` with any of `68`, `243`, `249`, `257`, `269`, `1041` or `1049`\nto follow another problem. The result links its paper, source modules and\nremaining questions.\n[The source map](SOURCE_MAP.md) offers a reading route through the same work;\n[the query reference](agents/SEMANTIC_COMPILER.md) explains the more detailed\nselectors.\n\n### Check a documentation change\n\nFor a small documentation contribution, use Python 3.11 or later to check the\nreader routes and their links before opening a pull request. If `python3` is\nolder, use the [existing interpreter check](agents/README.md#start-with-current-public-work)\nto select an installed supported interpreter, and use it for both commands:\n\n```sh\npython3 scripts/test_human_first_contact.py\npython3 scripts/check_cold_clone_comprehension.py --quick\n```\n\nFor a tooling change, also run that tool's test and include the result in the\npull request. Changes to mathematical statements or Lean proofs require the\ncorresponding proof checks below.\n\n## 2. Reproduce the pinned Lean environment\n\nInstall [elan](https://leanprover-community.github.io/get_started.html) once\nusing its documented installer. From the repository root, elan reads\n[`lean-toolchain`](../lean-toolchain), which selects\n`leanprover/lean4:v4.29.1`; [`lake-manifest.json`](../lake-manifest.json)\npins the Mathlib revision and its transitive dependencies.\n\n```sh\nlake --version\npython3 scripts/test_dependency_lock_contract.py\nlake exe cache get\n```\n\nThe cache command is optional for correctness but avoids recompiling Mathlib\nfrom source. It may download several gigabytes. A cache is an acceleration,\nnot authority: the toolchain file and manifest are the reproducibility inputs.\n\nCheck one real published theorem module first. This is the short feedback path\nfor confirming that the toolchain, dependency cache, and project all work;\nit uses a small #249 module:\n\n```sh\npython3 scripts/lean_fast_build.py --jobs 2 \\\n  ErdosProblems.Erdos249.PeriodMultipleEscape\n```\n\nTo follow the earlier `eb_full_support` claim into a proof build, its\ndeclaration lives in the larger `CertificateKernel` module:\n\n```sh\npython3 scripts/lean_fast_build.py --jobs 2 Erdos249257.CertificateKernel\n```\n\nFor the weighted theorem used in the #257 reading route, first inspect its\ncurrent claim record, then build the module containing its proof:\n\n```sh\npython3 scripts/verify_claims.py --claim finite_prime_weighted_support\npython3 scripts/lean_fast_build.py --jobs 2 \\\n  ErdosProblems.Erdos257.PaperCompleteR8.WeightedReturn\n```\n\n[`DivisibilityWeightedClaim`](../lean/ErdosProblems/Erdos257/PaperCompleteR7/AnalyticTargets.lean)\nstates the fixed-base and hereditary clauses. The proof declaration is\n[`divisibilityWeightedClaim`](../lean/ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean#L120).\nThe weighted term and `FinitePrimeWeighted` definitions are in\n[`PrimeWeightedDefinitions.lean`](../lean/ErdosProblems/Erdos257/PaperCompleteR7/PrimeWeightedDefinitions.lean),\nwhich the R7 interface imports.\nThe R7 interface file's preamble describes the earlier development stage;\nits \"missing\" proof wording is not the current theorem status. Read the R8\nproof and the claim record above for that status. The R7 source remains at\nits pinned formal-source revision.\nThe first command checks the current claim's references and prints its\nrecorded formal-source revision; it does not run Lean. The second command\nchecks the current checkout with the pinned Lean toolchain. Record\n`git rev-parse HEAD` alongside its result so that a build at one revision is\nnot attributed to another.\n\nThe [external-verification packet](EXTERNAL_VERIFICATION.md#programme-257)\nand [`verification/comparator.json`](../verification/comparator.json) identify\nthe separately declared Comparator statement and axiom budget. Its legacy\nComparator replay record remains distinct from Palomar's\n[successful mechanical verification](https://github.com/PalomarRegistry/PalomarSubmission/actions/runs/36009433226)\nof the exact five-declaration `PalomarCorpus/E257_01` entry at source commit\n`b85ed30805188eb4390a686b111294b24363418e`; that entry includes\n`divisibilityWeightedClaim`. Palomar\n[registered version 1](https://palomar-registry.org/entry?id=PALOMAR-2026-09-25-000009&version=1)\nas `PALOMAR-2026-09-25-000009` on 25 September 2026. Its\n[immutable record](https://data.palomar-registry.org/entries/PALOMAR-2026-09-25-000009-v1.json)\nbinds submission `gid0ym5uu910` to that exact source and selection. The\nmechanical receipt applies to the\nselected entry, not every theorem in the #257 paper. The\n[replay guide](verification/EXTERNAL_VERIFICATION_REPLAY.md) explains the\nseparate source-bound Comparator receipt.\n\nThe replay guide now provides a one-theorem `weighted-support` unit for\n`divisibilityWeightedClaim`. On Linux with the stated prerequisites, its\nsingle `run --unit weighted-support` command fetches an exact commit, compares\nthe theorem with the separate challenge, and requires rejection of a\ndeliberately changed statement. The configuration is runnable, but a passing\nreceipt for the selected source commit has not been recorded here. The\n`189/388` Python calculation above is a different finite result.\n\nThese commands build the current checkout. The claim verifier also names the\nrecorded source revision; keep that identity with any report about reproducing\na particular release. [External replay](verification/EXTERNAL_VERIFICATION_REPLAY.md)\ndescribes checks tied to an immutable source commit.\n\nFor a complete release replay, build the two supported public roots and the\nexplicitly supported non-default consumers through one host-shared wrapper\ninvocation:\n\n```sh\npython3 scripts/lean_fast_build.py --jobs 2 --lake-staleness \\\n  Erdos249257 ErdosProblems Examples FormalConjecturesAdapter \\\n  FormalConjecturesVariants FC1049HeightRegion FC243CubicRate ResidualBench \\\n  ErdosProblems.Erdos251.PaperLargeAuditR7\nlake env lean research/adapters/FC1049HeightRegion.lean\npython3 scripts/build_lean_dependency_index.py --check --full-check\n```\n\n`Erdos249257` and `ErdosProblems` are the default libraries. `Examples` builds\nconsumer examples inside this package; read their imports and example\ndeclarations in [research/examples/Examples.lean](../research/examples/Examples.lean).\nThe remaining targets check adapters, statement variants, residual examples,\nand the separately compiled #251 paper audit; building them does not\nadd reviewed claims to the mathematical record.\n\nThe wrapper coordinates builds on the same machine. Equivalent requests share\none running job; different heavy builds queue, and compatible clones can\nreuse build files. Keep using the wrapper if another build is in progress.\nThe [concurrent-validation guide](../skills/lean-concurrent-validation/SKILL.md)\nexplains detached operation, cache locations, storage sharing and exit codes.\nA successful build means Lean accepted the requested modules. The exact\nstatements and their limits are in the source and [claim record](claims.json).\n\n### Use the library in another Lean project\n\n[The dependency guide](DOWNSTREAM_REUSE.md) gives the pinned Lake package\nrecipe and distinguishes a separate consumer from the internal `Examples`\ntarget. After the complete build above, check that package boundary with:\n\n```sh\npython3 scripts/check_downstream_reuse.py\n```\n\nThe checker creates a separate Lake project and checks the unchanged consumer\nexamples under the host build lock. A missing build or dependency is a failed\ncheck; the checker does not rebuild or download dependencies.\n\n## 3. Run the release-surface checks\n\nThe release checks inspect claim records, links, generated files, licences and\nother published metadata. From a cold checkout, this entry selects Python\n3.12, creates an ignored environment under `.lake/`, installs the committed\nhash-pinned release dependencies if needed, and builds the Lean import used by\nthe proof-state pilot before running the existing gate:\n\n```sh\npython3 scripts/run_release_check.py\n```\n\nInstall Python 3.12 and Elan first; the pinned Lean version comes from\n`lean-toolchain`. CI uses Python 3.12.9 and may run\n`python3 scripts/check_release.py` directly because its dependency and Lean\npreparation steps have already run. The wrapper's `--prepare-only` option\nchecks those prerequisites without starting the full gate. A missing import\nartifact or failed hash-checked installation stops during preparation; it\ncannot appear as a passing proof-state control. The release check runs for\nseveral minutes and prints only its final summary. The Lean build and this Python check answer separate questions. A publication\nalso needs its selected external verification receipts, described in\n[the verification guide](verification/README.md). A successful static check\nalone is not the complete release decision.\n\nThe clean-ref wrapper is the later immutable-snapshot gate. It resolves one\ncommit, creates a disposable local clone, and runs the configured release\nchecks there; uncommitted files in the caller's checkout are excluded. A\nbounded snapshot probe can verify that preparation without replaying the full\ngate:\n\n```sh\npython3 scripts/check_release_ref.py --ref HEAD --probe-only\n```\n\n## 4. Reproduce a changed checkout safely\n\nKeep generated projections in dependency order. After changing an owning\nsource, use its builder and then its `--check` mode; do not hand-edit a\ngenerated JSON or Markdown projection. Review and commit only the files\nbelonging to your change. If other work is present in the checkout, preserve\nit and keep it out of your commit.\n\nThese checks cover the dependency lock, generated indexes and source links:\n\n```sh\npython3 scripts/test_dependency_lock_contract.py\npython3 scripts/build_corpus_descriptor.py --check\npython3 scripts/build_module_graph.py --check\npython3 scripts/refresh_source_coordinates.py --check\npython3 scripts/build_lean_dependency_index.py --check\npython3 scripts/test_downstream_example_contract.py\n```\n\nIf a command reports stale generated output, regenerate from the named owning\nbuilder and commit the source plus all projections that builder declares. Do\nnot copy files from another checkout or rely on a developer's `.lake` state.\nFor a full in-memory rebuild of the semantic index, use\n`python3 scripts/build_semantic_corpus.py --check --full-check`. The ordinary\ndependency-index `--check` never compiles; an intentional release\nexport uses `--check --full-check` after the coordinated Lean build above.\n\n## Resource and boundary notes\n\n- Python-only navigation is the fast cold-clone path; Python `3.11+` is\n  required by the repository's TOML and standard-library interfaces.\n- The pinned Mathlib cache is large. Allow the cache download and the Lean\n  build their documented time and disk budget; `--jobs 2` is the supported\n  bounded local build setting.\n- `lake build` proves the checked Lean targets, not the open Erdős problems.\n  The public claim registry and papers retain the exact conditional results\n  and open boundary.\n- `docs/problems.json`, `docs/claims.json`, the paper corpus, and the\n  generated declaration/dependency projections are public evidence surfaces;\n  this runbook describes how to replay them and is not proof authority.\n\nIf a command fails, include the command, its output and `git rev-parse HEAD`\nwhen [reporting the problem](../CONTRIBUTING.md). That gives someone else the\nsame source version and a way to reproduce it.\n\n[Back to the documentation](README.md) · [How the repository works](ARCHITECTURE.md)\n","body_format":"text"},"agent-workbench":{"document_id":"agent-workbench","source_path":"docs/agents/AGENT_WORKBENCH.md","role":"agent_command_surface","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/agents/AGENT_WORKBENCH.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/agents/AGENT_WORKBENCH.md","sha256":"sha256:49aaa8f1a4235ea66fd76fed4f77e82499ea639426fa83706e59a6e75f24308b","byte_count":24901,"origin":"public_main_checkout","body":"# Agent workbench\n\nThis corpus is agent-native. The agent or human researcher supplies the\nconjectures, search choices and judgment. The tools check exact statements\nand keep records another researcher can inspect and replay.\n\nUse this page for agent operations. Before proposing changes, read\n[CONTRIBUTING.md](../../CONTRIBUTING.md) and the authority and validation\ncontract in the [agent guide](AGENT_GUIDE.md). Reader introductions belong in the README.\n\nFor clone setup and a copyable task prompt, see the\n[agent quickstart](README.md#start-with-current-public-work).\n\n## What the workbench records\n\nThe researcher chooses which conjecture to test, which representation to try\nand when to change direction. The workbench records the submitted inputs,\nLean probes and results so that someone else can check the work. A successful\nprobe establishes its Lean statement; it does not show that the research\nchoice was useful or that a reader understands the argument.\n\nThe tools can be composed directly or through helpers such as\n[`continue_research.py`](../../scripts/continue_research.py), which combines\ncorpus lookup with a recorded session. Within the proof workbench, a formal\nclaim must cite a kernel-accepted probe before the notary will record it.\nThe receipt preserves that connection; it does not certify every observation\nor conjecture in the session.\n\nRecorded sessions are optional for ordinary contributions. An issue or pull\nrequest can supply an ordinary proof, a correction, exposition or an exact\nfinite computation, with its evidence and limits stated under the\n[methodology](../METHODOLOGY.md). The [systems paper](../../paper/systems/claim-faithful-publication-systems-paper.pdf)\nexplains the current design; the [earlier technical account](../../paper/systems/cold-clone-to-proof-receipt.pdf)\nis retained as historical background.\n\n## Eight-problem cold-start route\n\nWhen a session begins from a problem number, read the matching row in the\ngenerated [problem index](../problems.json), then follow the detailed\n[source map](../SOURCE_MAP.md). The row supplies the question, status, paper\nnote, module inventory, paper/source coordinates, and open obligations; the\nsource map supplies the bounded proof/paper/source traversal. The current\nmathematical gateways are the eight individual problem papers:\n\n| Problem | Canonical paper |\n|---|---|\n| #68 | `erdos-68-factorial-denominator-irrationality.pdf` |\n| #243 | `erdos-243-reciprocal-tail-rigidity.pdf` |\n| #249 | `erdos-249-binary-totient-series.pdf` |\n| #251 | `erdos-251-prime-gap-dyadic-series.pdf` |\n| #257 | `erdos-257-mersenne-support-subseries.pdf` |\n| #269 | `erdos-269-three-prime-running-lcm.pdf` |\n| #1041 | `erdos-1041-lemniscate-newton-flow.pdf` |\n| #1049 | `erdos-1049-rational-base-lambert.pdf` |\n\n`erdos249-257-main-paper.pdf` and its TeX source are a retired joint manuscript\nkept for archive and provenance. Do not use that manuscript as the current\ngateway for either problem.\n\nFor a machine-readable return from any roster row, run the bounded registry\nroute before opening a large source file:\n\n```sh\npython3 scripts/query_corpus.py --route erdos_<n>\n# <n> is one of 68, 243, 249, 251, 257, 269, 1041, or 1049\n```\n\nThe packet returns the problem question, public status, note, formal module\ncoverage, and exact open-obligation handles. Follow the matching row in\n`problems.json` for its `modules`, `paper`, `note.source_path`,\n`note.rendered_path`, and exact open-obligation statements, then use the\n[complete eight-problem return matrix](../SOURCE_MAP.md#complete-eight-problem-return-matrix)\nfor source entry points, paper records and exact frontier routes. Read each\nrow's `what_is_checked`, `what_is_not_checked` and `claim_registration` together.\nFollow `registered_claim_ids` into `docs/claims.json` for each statement's\nstatus and declarations. A library name does not determine review status;\nunlisted declarations gain no status from sharing a module with a registered\nclaim. For #249 and #257, also follow the reviewed core routes and their\nclaim/paper/source links. A programme entry module is not the complete result\ninventory, and navigation does not close an open problem.\n\nFor the reviewed #249/#257 core, `docs/orientation.json` supplies typed routes.\nFor #1041, also read the dated\n[`FRONTIER.md`](../../research_corpus/Erdos1041/FRONTIER.md) before its machine\nresult map: it contains current refutations and surviving premise families\nthat the older paper note does not enumerate. None of these dispatches\nchanges problem status or turns source-level records into reviewed claims.\n\n## The instrument panel\n\nSix instruments. Each states the question it answers exactly, then its entry\npoint.\n\n**Proof-state compiler.** Which candidate applications does Lean actually\naccept from this exact goal, and what are the exact resulting obligations?\n\n```sh\npython3 scripts/proof_state_compiler.py --pilot-controls\n```\n\n**Blocker algebra.** Three non-interchangeable projections: per-plan AND\nresiduals (`plan_blocker_sets`), smallest additions completing some route\n(`minimal_missing_interventions`), smallest removals defeating every known\nroute (`minimal_blocker_cuts`). Entry: packet fields + `blocker_algebra_semantics`.\n\n**Session notary.** What did the agent observe, conjecture, probe, and\nestablish — and can a stranger replay it byte-for-byte?\n\n```sh\npython3 scripts/proof_workbench.py {open,note,probe,claim,close,replay,show}\n```\n\n**Blind harness.** Can a candidate bridge be kernel-checked in a historical\nparent environment that provably lacks the future answer?\n\n```sh\npython3 scripts/historical_bridge_experiment.py --compact\n```\n\n**Static semantic compiler.** Which declarations are structurally relevant to\nthis goal (nomination only — never applicability)?\n\n```sh\npython3 scripts/query_corpus.py\n```\n\n**The corpus itself.** More than 150,000 indexed declarations — 8,171 of them\nexplicitly marked as generated certificate shards (a floor: further emitted\nfamilies predate the markers), counted as source rather than as claims —\nacross the reviewed library, problem-owned expansion, and auxiliary inventory\nforest; exact dependency neighborhoods cover both compact roots. Entry:\n`Erdos249257/`, `ErdosProblems/`, and:\n\n```sh\npython3 scripts/query_semantic.py inventory\n```\n\n## Source-bound decisions and transfer candidates\n\nStart with `query_corpus.py --ask` for the registered frontier. The argument\ngraph gives a second, recorded-edge view; `next` ranks open statements and\n`transfer` screens cross-problem reductions without certifying a useful\napplication:\n\n```sh\npython3 scripts/query_continuations.py next --problem 269 --limit 8\npython3 scripts/query_continuations.py transfer --limit 12 --max-work 10000 --max-checks 250\npython3 scripts/research_decision.py --problem 269 --source-commit <full-commit> --format markdown\n```\n\nThe continuation commands require `docs/argument_continuations_graph.json.gz`.\nIf it is absent, obtain a matching kernel export and run\n`python3 scripts/build_argument_continuations.py --export <export.jsonl.gz>`;\nthe builder's `--help` describes the export and provenance sidecars. Do not\ninfer a frontier from an absent projection.\n\nThe decision adapter binds its read set to the named commit and reports missing\ncapabilities explicitly. It reads the existing claims, relations, contrasts\nand journal; it is not a new claim registry. For a mechanism already authored\nin the theory lab, use the read-only retrieval adapter:\n\n```sh\npython3 scripts/insight_engine.py reverse \"rational tail denominator\"\npython3 scripts/insight_engine.py forward <mechanism_id> --exclude-origin --limit 8\n```\n\nThese are lexical source candidates with pending proof obligations. The graph\nscreen checks recorded relations only; exhaustion is `unknown_budget`. Neither\nroute proves applicability, strictness, novelty or usefulness. For a\nprospective, source-bound experiment on a selected mechanism, follow\n[`run-coupled-research-goals`](../../skills/run-coupled-research-goals/SKILL.md)\nand its `research_episode.py plan` route. For dispatch and returned work, use\n[`erdos-research-return`](../../skills/erdos-research-return/SKILL.md).\n\n## The move grammar\n\nA workbench session is a sequence of typed moves in an append-only\nledger (`research/workbench/sessions/<slug>/ledger.jsonl`):\n\n- `observation` — what the agent read or measured, with sources.\n- `conjecture` — a falsifiable guess. Conjectures should declare their\n  falsifier; the notary records a warning when they do not.\n- `plan` / `interpretation` / `abandonment` — the agent's reasoning,\n  recorded honestly, including dead ends.\n- `probe` — one Lean invocation. The notary stores the exact input\n  bytes and computes the verdict (`kernel_accepted`,\n  `kernel_accepted_with_sorry`, `kernel_rejected`) from the Lean exit\n  status itself. Probes are the only moves that carry authority.\n- `claim` — a statement bound to a cited kernel-accepted probe. The\n  notary refuses claims citing anything weaker.\n- `session_closed` — outcome plus move accounting.\n\n`replay` re-runs every stored probe and reports whether recorded\nverdicts still hold in the current environment. A session whose claims\nsurvive replay is evidence; a session whose claims do not is a detected\ndrift, which is also evidence.\n\n## The invention ladder\n\nAgent-produced mathematics is claimed at exactly one of three rungs,\nand the rung is part of the claim:\n\n1. **Assisted proof.** The agent proved something with full access to\n   the corpus. Claim: the theorem, its kernel receipt, and the session\n   that produced it. No originality claim beyond what the ledger shows.\n2. **Context-blind rederivation.** The agent worked inside the blind\n   harness: a historical parent checkout that provably lacks the future\n   module and name, with the future identifiers denied from its inputs.\n   Claim ceiling: the agent rederived the bridge without access to the\n   answer *in its context*. This does not rule out indirect exposure\n   through model training; the claim is context-blindness, not\n   training-blindness, and must be worded so.\n3. **Prospective invention.** The agent proved a theorem that did not\n   exist anywhere — in this corpus or its history — when the session\n   opened. Blindness holds by construction: there was no answer to\n   leak. This is the strongest rung, and the only one that supports the\n   word \"new\".\n\nRung-3 sessions are the point of the whole design. The first landed\nexample is the carry-pivot session recorded in\n`research/workbench/sessions/carry_pivot_2026_07_27/`, which produced\n`Erdos249257/SuffixCylinderCarryPivot.lean`: the binary carry-pivot\nnormal form for numeral-adjacent boundary words and the exact\ndivisor-incidence identity that sharpens the previously landed\ncoarse `≤ +1` coefficient bound into an equality. The theorems were\nderived and proved by reasoning agents inside this workbench, with the\nkernel as the only judge, and the session ledger is committed beside\nthe mathematics.\n\n## Evaluation\n\nThe workbench is judged on what it lets an inhabiting agent establish,\nmeasured from ledgers rather than asserted:\n\n- kernel-accepted probes per session, and probes-to-verdict per claim;\n- refutation density — how often the corpus falsifies a conjecture\n  (high is good; it means probes are decisive);\n- ladder rung reached, with its receipts;\n- replay integrity — whether closed sessions still verify from their\n  stored bytes alone.\n\n## Source-current continuation route: Erdős #1041\n\nThe public #1041 corpus is a useful continuation surface for an agent, but it\nhas a stricter reading order than an ordinary claim route. Read\n[`research_corpus/Erdos1041/FRONTIER.md`](../../research_corpus/Erdos1041/FRONTIER.md)\nbefore [`STRONGEST_RESULTS.json`](../../research_corpus/Erdos1041/STRONGEST_RESULTS.json):\nthe dated frontier note records which activation rows were refuted on\n2026-08-29 and therefore prevents an agent from spending a new session on a\nknown dead strategy. Use [`CORPUS_MANIFEST.json`](../../research_corpus/Erdos1041/CORPUS_MANIFEST.json)\nto resolve the complete published file set and its source/public digests.\n\nFor a bounded agent session, follow:\n\n1. `AttachmentAwareReeb.md` → `GenericSufficiencyClosure.md` for the\n   surviving hub-selection carrier and the generic-to-closed extension.\n2. `NearFeketeRadialAngularSplit.md` → `Degree5AssemblyAndSharpenedCuts.md`\n   for the current near-Fekete and degree-five residuals.\n3. `PartialClusterPreimagePerimeterIdentity.md` →\n   `TiedNewtonFaceFibreProduct.md` → `ExteriorRootProductCovering.md` for\n   exact perimeter, moment, and covering premises.\n4. `MinimalHubArmBudgetRefutation.md` and `SeparatrixAggregateReduction.md`\n   for refutation evidence and the surviving machinery whose original target\n   has been falsified.\n\nEvery step must carry its own hypotheses and evidence class. These files are\nsource-level research records, not entries in the reviewed `docs/claims.json`\nauthority. The parent problem remains open; the exact remaining producers are\nlisted in `FRONTIER.md`. The paper-facing checked route is\n[`erdos-1041-lemniscate-newton-flow.md`](../papers/full-text/erdos-1041-lemniscate-newton-flow.md),\nwhich leads back to the Lean declaration coordinates without upgrading the\nsource-only frontier.\n\n## Boundaries\n\n- Ledger notes, rankings, and static nominations are advisory. Only\n  kernel receipts assert.\n- A bounded search that finds nothing is an abstention, never an\n  impossibility claim. Family-level impossibility requires a landed\n  theorem.\n- Sessions record cognition; they do not certify that the cognition was\n  optimal. The honesty is the feature.\n- Nothing here claims progress on the open problem statuses tracked in\n  `docs/RELATED_PROBLEMS.md` unless a landed theorem says so.\n\nThe README is held to a short word budget as the human front door. The two sections below were moved here verbatim so an agent following the front page still reaches the whole-corpus navigation contract.\n\n## Recovered from the front page: read or run it\n\nA single claim can be followed without installing Lean. `verify_claims.py`,\nshown above, also prints the Lean proof text, and both it and `--verify-all`\nwork on a `git clone --depth 1` checkout. Run with no argument it performs an\nenvironment check, which exits `2` and prints `git fetch --unshallow` when a\ntruncated history cannot reach the gates that read pinned commits; it never\nexits `1` in that case, so a shallow clone cannot be misread as a claim that\nfailed. `check_release.py` remains the authority for locators.\n\nA mathematician can follow the reading order above and then trace one result\nfrom the [SOURCE MAP](../SOURCE_MAP.md) into Lean. The per-problem papers\nare the live route; the [joint PDF](../../paper/archive/erdos249-257-main-paper.pdf) is retired.\n\nA coding agent starts from an ordinary-language task. The clone-local router\nreturns a bounded lane, minimum read set, and skill:\n\n```bash\npython3 scripts/agent_entry.py --entry \"improve cold-clone skill discovery\"\npython3 scripts/agent_entry.py --skills\n```\n\nUse `--purpose research|method|infrastructure` with\n`--scope \"problem:257\"`; purpose wins over scope words. Architecture sessions use `continue_research.py start --area`; see the\n[session example](../research-commons/ARCHITECTURE_CONTRIBUTIONS.md#optional-resumable-session).\n\nSkills run from the clone root. `scripts/install_agent_skills.py --list` only\nexposes them to an agent harness. Read\n[`AGENTS.md`](../../AGENTS.md), then the bounded\n[`orientation.json`](../orientation.json), and expand one programme or claim;\n[`docs/agents/AGENT_GUIDE.md`](AGENT_GUIDE.md) is the full reference.\n\nWhole-corpus agent navigation runs without a Lean build.\n`python3 scripts/query_corpus.py --tour --format card` and\n`python3 scripts/query_corpus.py --route agent_native_corpus_navigation`\nexpose corpus scale, the mathematical map, the canonical eight-problem map,\nthe distinct reviewed #249/#257 open-proposition frontier, and authority\nboundaries. `query_semantic.py problem-registry` lists every indexed problem\nand `structural-backlog` lists authored replacement. Committed indexes expose\nevery indexed declaration and exact dependencies for both loaded roots;\ncoverage keeps direct evidence, family context, and structural discovery\ndistinct. These are navigation projections, not proof authority.\n\nThe [proof cockpit](PROOF_COCKPIT.md) produces one cold-clone\nstatus and frontier card (`python3 scripts/proof_cockpit.py`), and\n[the Agent Workbench](AGENT_WORKBENCH.md) records typed reasoning moves\nand kernel probes under `research/workbench/sessions/`. The one landed prospective\nsession,\n[`carry_pivot_2026_07_27`](../../research/workbench/sessions/carry_pivot_2026_07_27/ledger.jsonl),\nproduced [`SuffixCylinderCarryPivot.lean`](../../lean/Erdos249257/SuffixCylinderCarryPivot.lean).\nOnly kernel receipts assert; ledger notes and static nominations stay\nadvisory.\n\nFor proof-search construction, `hypOf%` lifts an unresolved hypothesis out of\nbinder position into a `Prop`, so whether a sketch's remaining obligation\ndiffers from the target it started from becomes a question for the kernel\nrather than for a rater; this is the failure mode\n[AlphaProof Nexus](https://arxiv.org/abs/2605.22763) reports prompting could\nnot prevent. [Deciding whether a sketch reduced its target or renamed\nit](../reference/RESIDUAL_PROGRESS.md) documents the evaluator, its eight labelled\nfixtures, and what it refuses to decide.\n[The proof-state compiler](PROOF_STATE_COMPILER.md) asks the pinned Lean\nenvironment which candidate applications it actually accepts from a goal, and\n[the semantic compiler](SEMANTIC_COMPILER.md) nominates declarations\nstructurally without claiming they apply.\n\nPublication topology is reported by\n`python3 scripts/query_corpus.py --publication-architecture` and\n`python3 scripts/query_corpus.py --publication-family <id>`.\n\nThe [Formal Conjectures crosswalk](../verification/FORMAL_CONJECTURES_CROSSWALK.md)\nplaces the corpus against the public benchmark: it binds all eight programmes\nto Google DeepMind's Formal Conjectures statements at a pinned upstream\ncommit, with a SHA-256 per source file and the indexing, ambient-type, and\ncast differences a reviewer must inspect. It is statement identity and\nadapter-review metadata, not a Lean equivalence proof or a\nsubmission-readiness decision; every row is `not_ready_to_submit`.\n[Related problems](../RELATED_PROBLEMS.md) places five of the eight\nprogrammes among the neighbouring numbered problems, each external status as\nlisted on its erdosproblems.com page.\n\n`python3 scripts/check_cold_clone_comprehension.py --quick` checks the\nreading surfaces without Lean. For the full public-surface and query sweep,\n`python3 scripts/run_release_check.py` prepares the required environment;\nfollow the [release instructions](../REPRODUCIBILITY.md#3-run-the-release-surface-checks).\n\n## Recovered from the front page: how the repository fits together\n\nThe package has two compact supported roots. [`Erdos249257.lean`](../../lean/Erdos249257.lean)\npreserves the reviewed #249/#257 corpus. [`ErdosProblems.lean`](../../lean/ErdosProblems.lean)\nis the problem-owned expansion surface: work lives under its actual Erdős\nproblem number instead of being forced into the historical #249/#257 tree.\nKernel checking of that second root establishes its exact Lean propositions;\nit does not by itself promote them into the reviewed claim registry or claim\nthat an open problem is solved.\n\nThe source has five reader-facing layers. The assembled kernel,\n[`CertificateKernel.lean`](../../lean/Erdos249257/CertificateKernel.lean), contains the\ncommon series machinery, the full-support Erdős-Borwein theorem, named\nsupport-family interfaces, and the unconditional #249 denominator exclusion.\nThe #249 reduction spine, comprising the period-killer, lcm-diagonal, cone,\ndiagonal pincer, fresh-loss, and transport modules, turns the open\nirrationality problem into exact certificate or avoidance obligations; finite\ncertificate modules verify explicit parameters and do not supply the\nunbounded family required by the reduction. The #257 carry trunk, comprising\nthe tail-orbit, achievement-set, Boolean-Möbius carry, reciprocal-mass, and\ndivisor-coverage modules, gives exact criteria and necessary conditions, not\nthe universal #257 theorem. The navigation layer finds every declaration and\nimport through the atlas; selected semantic meanings carry scoped reviews\n(`python3 scripts/query_semantic.py semantic-reviews`), which are neither\nhuman, novelty, nor proof authority, and the theory lab records nine\nmechanisms, nine transfer capsules, and three failure receipts, with four\nholdouts that have no results, so no measured transfer is claimed. The\nproblem-owned expansion, `ErdosProblems/Erdos<N>/`, contains bounded results\nand explicit open frontiers for one problem at a time; new entries remain\noutside the reviewed claim registry until mathematical review establishes\ntheir intended meaning and public framing.\n\n[SOURCE MAP](../SOURCE_MAP.md) gives module order; [METHODOLOGY](../METHODOLOGY.md)\ngoverns claim changes; [WAVE INDEX](../reference/WAVE_INDEX.md) gives chronology, not\nreading order.\n\n## Recovered from the front page: build and verify\n\nStatic navigation, source queries and session planning run with Python alone.\nThe live proof-state pilot, workbench probe/replay and historical bridge\nexperiment above also execute Lean: prepare their selected imports first and\nwait for any host-wide Lean build to finish. The [proof-state compiler guide](PROOF_STATE_COMPILER.md#first-run)\ngives the focused pilot setup. Building or probing Lean source needs the\ntoolchain, and `lake` arrives with it: install `elan`, Lean's\ntoolchain manager, from the\n[Lean setup guide](https://leanprover-community.github.io/get_started.html).\n`elan` then reads [`lean-toolchain`](../../lean-toolchain) and selects\n`leanprover/lean4:v4.29.1`; [`lake-manifest.json`](../../lake-manifest.json) pins the\nmatching Mathlib.\n\n```sh\npython3 scripts/lean_fast_build.py --jobs 2 \\\n  ErdosProblems.Erdos249.PeriodMultipleEscape\n```\n\nThe wrapper fetches the pinned cache when needed. A focused build uses\n`python3 scripts/lean_fast_build.py --jobs 2 [target]`. With restored `.lake`\noutputs, `--lake-staleness` makes it trust Lake content traces, not checkout\ntimes. Configuration-timestamp-only candidates go through those traces too, so\nan unchanged `lakefile.toml`, manifest or toolchain file does not rebuild\nrestored outputs. Source changes, missing outputs and configuration content\nchanges still rebuild when Lake marks them stale, and every successful run ends\nwith the serialized Lake authority build. Without a target it checks both roots;\n`--plan` reports waves without building. Partial caches stay on that path even\nwhen a root output is absent. One verbose no-build verdict identifies the stale\nfrontier, expanded through local import dependents; same-wave targets then share\nLake graph scans in batches capped by `--jobs`.\nA cold clone can navigate before this step; formal editing needs the\npinned toolchain. Later builds reuse outputs and rebuild only the selected or\nstale dependency cone; `--changed-from <git-ref>` selects changed modules.\nFor source edits, use `--changed-from HEAD` or explicit selected modules for\nlocal proof iterations. The untargeted two-root build belongs to coordinated\nrelease validation; full Lean roots are release-only.\nThe dependency-index validator stores an exact `.lake` receipt: unchanged\ninputs make `--check` constant-time; `--check --full-check` forces an audit.\n\nFor the public release surfaces, follow the\n[release instructions](../REPRODUCIBILITY.md#3-run-the-release-surface-checks):\n\n```sh\npython3 scripts/run_release_check.py\n```\n\nThat entry runs the shared release gate, including the methodology checks.\nAfter it passes, there is no need to rerun its component tests.\n\nThe pinned public Lean proof corpus contains no `sorry`, `admit`, project-defined\n`axiom`, or `native_decide`; finite computations use kernel-checked `decide`.\nOne deliberate exception is outside the default build:\n[`ExternalVerification/Challenge.lean`](../../verification/ExternalVerification/Challenge.lean)\nstates the trusted propositions Comparator checks the solution against; they\ncarry `sorry` by construction.\n\n## Use as a Lean package\n\nUse the [dependency recipe and separate-project check](../DOWNSTREAM_REUSE.md).\n\nThe reviewed #249/#257 root is imported with:\n\n```lean\nimport Erdos249257\n```\n\nThe problem-owned expansion surface is imported with:\n\n```lean\nimport ErdosProblems\n```\n\n[`examples/Examples.lean`](../../research/examples/Examples.lean) is the minimal downstream\nconsumer; its conditional shell-pressure example leaves the analytic\nhypothesis explicit and does not prove universal #257.\n","body_format":"text"},"agents":{"document_id":"agents","source_path":"AGENTS.md","role":"cold_agent_entry","github":"https://github.com/wcook04/plectis-erdos/blob/main/AGENTS.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/AGENTS.md","sha256":"sha256:4183ea0ac82bc253351b15c674ac22ccafeeb48ba565e7fb566fc8353888b710","byte_count":7745,"origin":"public_main_checkout","body":"<!-- SPDX-FileCopyrightText: 2026 Will Cook -->\n<!-- SPDX-License-Identifier: Apache-2.0 -->\n\n# Agent instructions\n\nStart here when acting on this checkout. [README.md](README.md) introduces the\nresearch to human readers. The [agent guide](docs/agents/AGENT_GUIDE.md) owns\nauthority, change order and validation; open only the section your task needs.\n\nThis is a self-contained public Lean repository. Use tracked public files;\ndo not depend on sibling repositories, private caches or unpublished state.\n\n## Route the task\n\nFrom the repository root:\n\n```sh\npython3 scripts/agent_entry.py --entry \"<task in ordinary language>\"\n```\n\nThe router returns a task lane, its smallest read set and the relevant\n`skills/<id>/SKILL.md`. `python3 scripts/agent_entry.py --skills` lists all\nworkflows. Follow the selected workflow before making changes. Routing\nmetadata helps navigation; it is not mathematical authority.\n\nFor latest-work tasks, run\n`python3 scripts/agent_entry.py --checkout --check-upstream`. Record the\nrevision and preserve older work. The [agent quickstart](docs/agents/README.md#start-with-current-public-work)\nexplains setup, provenance and prompts.\n\n## Find the mathematical evidence\n\nFor mathematics, theorem status, progress or paper synthesis, query before\nreading broad papers or Lean files:\n\n```sh\npython3 scripts/query_corpus.py --ask \"<question>\"\npython3 scripts/query_corpus.py --overview --format card\n```\n\nUse the overview for whole-corpus coverage. Follow its `answer_contract` and\nreturned claim, remaining-open, declaration, module and paper handles. Keep\nhypotheses and exact open boundaries attached to each result. Never infer\nprogress from theorem counts or one flagship example.\n\nFor a corpus-wide progress review, include these stable routes:\n\n```sh\npython3 scripts/query_corpus.py --route instant_orientation\npython3 scripts/query_corpus.py --route erdos249_certificate_story\npython3 scripts/query_corpus.py --route erdos257_half_story\npython3 scripts/query_corpus.py --route browse_claim_status\n```\n\nIf free text returns no semantic cells, use `python3 scripts/query_corpus.py --routes`\nand the applicable stable route. Select open work with\n`python3 scripts/query_corpus.py --open`. Lean refutes the exact Formal\nConjectures #1041 path-image claim; correspondence with the 1958 wording remains\nunreviewed.\n\nFor learning, respect the requested depth: a hint request does not authorise\nrevealing the proof or starting proof search. Follow the\n[reading guidance](skills/explain-public-system/SKILL.md#help-a-reader-work-through-an-argument).\nFor prior work, read [PRIOR_ART.md](docs/PRIOR_ART.md), the paper's references\nand primary sources. Distinguish discovery, formalisation and reuse; model\nagreement does not establish novelty. Keep finite experiments attached to\ntheir inputs, bounds and recheck commands.\n\n### Proof work\n\nStart from a typed mathematical handle, then cross into source:\n\n```sh\npython3 scripts/query_corpus.py --goal-support \"<Lean or mathematical goal>\"\npython3 scripts/query_corpus.py --proof-plan \"<Lean or mathematical goal>\" --depth 4\n```\n\nVerify returned declarations and hypotheses with Lean. Before editing, read\n`Authority and change order` and `Validation` in the\n[agent guide](docs/agents/AGENT_GUIDE.md). Read\n[METHODOLOGY.md](docs/METHODOLOGY.md) if public claim wording or status could\nchange. [The Lean guide](lean/README.md) explains the two source libraries.\n\n### Papers and publication\n\nFor paper choice, role, availability or reading order:\n\n```sh\npython3 scripts/query_corpus.py --papers\n```\n\n[The paper index](docs/papers/README.md) is a navigation projection;\n`docs/papers/corpus.json` owns its machine inventory. After choosing a paper,\nfollow its typed claim and source handles. The human reading catalogue is\n[paper/README.md](paper/README.md).\n\nFor publication mutation, open `docs/publication_entry_packet.json` and its\nemitted contract. Run [public mathematical writing](skills/public-mathematical-writing/SKILL.md),\nthen [consequence propagation](skills/propagate-research-consequences/SKILL.md)\nto reconcile companions, consumers and public guidance.\n\nFor selected external checks:\n\n```sh\npython3 scripts/query_corpus.py --route comparator_assurance\npython3 scripts/query_corpus.py --route palomar_qualification\n```\n\nComparator reports configured statement, axiom and kernel checks; it\ndoes not establish novelty, significance, priority, or peer review.\nQualification reports repository-local policy readiness and operator-only\ndecisions; it does not confer acceptance, registration, publication or endorsement.\n\n## Maintain the repository\n\nFor architecture, README, folder organization or agent routing, follow\n[maintain-public-infrastructure](skills/maintain-public-infrastructure/SKILL.md).\n[ARCHITECTURE.md](docs/ARCHITECTURE.md) explains folder ownership;\n[orientation](docs/ORIENTATION.md) provides generated navigation.\n\nUse the bounded checks selected for the change:\n\n```sh\npython3 scripts/proof_cockpit.py --format card\npython3 scripts/test_compact_agent_entry.py\npython3 scripts/check_cold_clone_comprehension.py --quick\npython3 scripts/check_architecture_guide.py\n```\n\nBuilders own generated files and marked regions. Change the source authority\nfirst and use the named builder/checker. Keep preserved research returns and\nimmutable source citations attached to their recorded revisions.\n\n## Authority and mutation\n\n1. Lean source checked by the pinned Lean kernel is proof authority.\n2. `docs/claims.json` owns public claim identity, status, declarations and exact\n   remaining-open propositions.\n3. `docs/methodology.json` owns evidence responsibilities and claim-change rules.\n4. Generated orientation, declaration, module and publication packets provide\n   navigation, not new mathematical evidence.\n5. Papers and human summaries explain work within the registered claim ceiling.\n6. Model output, plans, memory and drafts are working aids only.\n\nA paper edit cannot change a claim; a registry edit cannot prove a theorem;\na passing navigation check cannot establish mathematics.\n\n- Preserve unrelated staged and unstaged work. Do not reset, restore, clean or\n  broad-stage a shared checkout.\n- Never add `sorry`, `admit`, project-defined `axiom`, `native_decide`,\n  unsafe/partial declarations or unbounded kernel limits.\n- For Lean edits, use `python3 scripts/lean_fast_build.py --jobs 2 --changed-from HEAD`.\n  Never overlap builds. Full Lean roots are release-only.\n\n## Validate and return the work\n\nUse the changed owner's focused checks while editing. README changes need both\n`python3 scripts/test_human_first_contact.py` and the cold-clone quick check:\nthey enforce different reading budgets. Before publication, follow the\n[agent guide's validation contract](docs/agents/AGENT_GUIDE.md#validation),\nincluding source freshness and the prepared release gate.\n\nFor a committed snapshot in a dirty shared checkout:\n\n```sh\npython3 scripts/check_release_ref.py --ref HEAD --receipt \"$(pwd -P)/release-head.json\"\n```\n\nBefore packaging a stable change, give each plausible downstream consumer an\nexplicit disposition through the consequence-propagation skill. For contributed\nor resumable research, read [erdos-research-return](skills/erdos-research-return/SKILL.md).\nPreserve the starting commit, original delta, evidence and credit. Reconcile\nolder-clone work with current main; a material conflict resolution is a\nseparately credited change. Keep the human account in the issue or pull request.\n\nSubmission may commit locally. Pushing or opening a pull request requires\nexplicit authorisation. Never require a private repository to produce or\nreview a contribution.\n\nDo not absorb the complete deep contract merely to find the first action.\n","body_format":"text"},"citation":{"document_id":"citation","source_path":"CITATION.cff","role":"citation_anchor","github":"https://github.com/wcook04/plectis-erdos/blob/main/CITATION.cff","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/CITATION.cff","sha256":"sha256:0d538f9b7faca90b61eefeea93937edc49afb427165ef82f0fff549a82410466","byte_count":12522,"origin":"public_main_checkout","body":"# SPDX-FileCopyrightText: 2026 Will Cook\n# SPDX-License-Identifier: Apache-2.0\ncff-version: 1.2.0\ntype: software\nmessage: \"This metadata describes the current repository. Cite the software metadata below and record the commit used; for a tagged release, use the CITATION.cff from that tag. For a mathematical result, cite the relevant per-problem paper linked from README.md and its original sources; see docs/PRIOR_ART.md and docs/research-commons/SOURCE_ATTRIBUTIONS.md for attribution; paper/archive/erdos249-257-main-paper.tex is the archival joint #249/#257 manuscript.\"\ntitle: \"Plectis: research on eight Erdős problems\"\nauthors:\n  - family-names: \"Cook\"\n    given-names: \"Will\"\nlicense: Apache-2.0\nabstract: \"Using ani's explicit monic degree-seven polynomial, Lean proves that every preconnected subset of its strict unit lemniscate containing two distinct roots has one-dimensional Hausdorff measure greater than 2. This refutes the exact Formal Conjectures path-image-length statement; the separate total-variation bound is also checked. A Lean 4 / Mathlib corpus of kernel-checked results around eight Erdős problems (#68, #243, #249, #251, #257, #269, #1041, #1049): exact finite-period noncollapse for every finite support and every base b ≥ 2, an explicit rational basis of dimension 2^e+1 for the dyadic sections of Euler's totient, Mersenne achievement-set geometry including the measure-one clause, six exact characterisations of #249 irrationality, unconditional denominator exclusions, diagonal certificates through t=82, and checked reductions, equivalences, and obstructions for the six expansion problems, bound to source by a self-assessed and agent-checked claim registry. It does not solve Erdős #249 or the universal #257 problem. The other seven targets remain open. No independent human mathematical review, including review of correspondence with the 1958 wording, has been recorded.\"\nrepository-code: \"https://github.com/wcook04/plectis-erdos\"\nurl: \"https://github.com/wcook04/plectis-erdos\"\nkeywords:\n  - \"Lean 4\"\n  - \"mathlib\"\n  - \"formal mathematics\"\n  - \"irrationality\"\n  - \"Erdős problems\"\n  - \"Erdős–Borwein constant\"\n  - \"Euler totient\"\n  - \"Lambert series\"\nreferences:\n  - type: article\n    title: \"Modular functions and transcendence questions\"\n    authors:\n      - family-names: \"Nesterenko\"\n        given-names: \"Yuri V.\"\n    year: 1996\n    journal: \"Sbornik: Mathematics\"\n    volume: \"187\"\n    issue: \"9\"\n    pages: \"1319-1348\"\n    doi: \"10.1070/SM1996v187n09ABEH000158\"\n  - type: article\n    title: \"On arithmetical properties of Lambert series\"\n    authors:\n      - family-names: \"Erdős\"\n        given-names: \"Paul\"\n    year: 1948\n    journal: \"Journal of the Indian Mathematical Society\"\n    volume: \"12\"\n    pages: \"63-66\"\n    url: \"https://users.renyi.hu/~p_erdos/1948-04.pdf\"\n  - type: article\n    title: \"On Series of the Lambert Type which assume Irrational Values for Rational Values of the Argument\"\n    authors:\n      - family-names: \"Chowla\"\n        given-names: \"Sarvadaman\"\n    year: 1947\n    journal: \"Proceedings of the National Institute of Sciences of India\"\n    volume: \"13\"\n    pages: \"171-173\"\n    url: \"https://insa.nic.in/UI/Archivesection.aspx?JID=MA%3D%3D&JYrs=MTk0Nw%3D%3D\"\n  - type: article\n    title: \"On the irrationality of certain series\"\n    authors:\n      - family-names: \"Erdős\"\n        given-names: \"Paul\"\n    year: 1968\n    journal: \"The Mathematics Student\"\n    volume: \"36\"\n    pages: \"222-226\"\n    url: \"https://users.renyi.hu/~p_erdos/1969-09.pdf\"\n  - type: book\n    title: \"Old and New Problems and Results in Combinatorial Number Theory\"\n    authors:\n      - family-names: \"Erdős\"\n        given-names: \"Paul\"\n      - family-names: \"Graham\"\n        given-names: \"Ronald L.\"\n    year: 1980\n    publisher:\n      name: \"L'Enseignement Mathématique\"\n    volume: \"28\"\n    url: \"https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\"\n  - type: book\n    title: \"On the irrationality of certain series: problems and results\"\n    authors:\n      - family-names: \"Erdős\"\n        given-names: \"Paul\"\n    year: 1988\n    collection-title: \"New Advances in Transcendence Theory\"\n    editors:\n      - family-names: \"Baker\"\n        given-names: \"Alan\"\n    publisher:\n      name: \"Cambridge University Press\"\n    pages: \"102-109\"\n    doi: \"10.1017/CBO9780511897184.009\"\n  - type: article\n    title: \"On the irrationality of certain series\"\n    authors:\n      - family-names: \"Borwein\"\n        given-names: \"Peter B.\"\n    year: 1992\n    journal: \"Mathematical Proceedings of the Cambridge Philosophical Society\"\n    volume: \"112\"\n    issue: \"1\"\n    pages: \"141-146\"\n    doi: \"10.1017/S030500410007081X\"\n  - type: article\n    title: \"Irrationality of Lambert series associated with a periodic sequence\"\n    authors:\n      - family-names: \"Luca\"\n        given-names: \"Florian\"\n      - family-names: \"Tachiya\"\n        given-names: \"Yohei\"\n    year: 2014\n    journal: \"International Journal of Number Theory\"\n    volume: \"10\"\n    issue: \"3\"\n    pages: \"623-636\"\n    doi: \"10.1142/S1793042113501121\"\n  - type: book\n    title: \"Introduction to Analytic Number Theory\"\n    authors:\n      - family-names: \"Apostol\"\n        given-names: \"Tom M.\"\n    year: 1976\n    publisher:\n      name: \"Springer\"\n    doi: \"10.1007/978-1-4757-5579-4\"\n  - type: article\n    title: \"The Lambert series factorization theorem\"\n    authors:\n      - family-names: \"Merca\"\n        given-names: \"Mircea\"\n    year: 2017\n    journal: \"The Ramanujan Journal\"\n    volume: \"44\"\n    issue: \"2\"\n    pages: \"417-435\"\n    doi: \"10.1007/s11139-016-9856-3\"\n  - type: article\n    title: \"Generating Special Arithmetic Functions by Lambert Series Factorizations\"\n    authors:\n      - family-names: \"Merca\"\n        given-names: \"Mircea\"\n      - family-names: \"Schmidt\"\n        given-names: \"Maxie D.\"\n    year: 2019\n    journal: \"Contributions to Discrete Mathematics\"\n    volume: \"14\"\n    issue: \"1\"\n    pages: \"31-45\"\n    doi: \"10.55016/ojs/cdm.v14i1.62425\"\n  - type: article\n    title: \"On the Set of Partial Sums of an Infinite Series\"\n    authors:\n      - family-names: \"Kakeya\"\n        given-names: \"Soichi\"\n    year: 1914\n    journal: \"Proceedings of the Tokyo Mathematico-Physical Society\"\n    volume: \"7\"\n    pages: \"250-251\"\n    doi: \"10.11429/ptmps1907.7.14_250\"\n  - type: article\n    title: \"On a Curious Property of Vulgar Fractions\"\n    authors:\n      - family-names: \"Farey\"\n        given-names: \"John\"\n    year: 1816\n    journal: \"The Philosophical Magazine\"\n    volume: \"47\"\n    pages: \"385-386\"\n    doi: \"10.1080/14786441608628487\"\n  - type: article\n    title: \"The Lean Theorem Prover\"\n    authors:\n      - family-names: \"de Moura\"\n        given-names: \"Leonardo\"\n      - family-names: \"Kong\"\n        given-names: \"Soonho\"\n      - family-names: \"Avigad\"\n        given-names: \"Jeremy\"\n      - family-names: \"van Doorn\"\n        given-names: \"Floris\"\n      - family-names: \"von Raumer\"\n        given-names: \"Jakob\"\n    year: 2015\n    url: \"https://lean-lang.org/papers/system.pdf\"\n  - type: software\n    title: \"mathlib4\"\n    authors:\n      - name: \"The mathlib Community\"\n    repository-code: \"https://github.com/leanprover-community/mathlib4\"\n  - type: article\n    title: \"On several irrationality problems for Ahmes series\"\n    authors:\n      - family-names: \"Kovač\"\n        given-names: \"Vjekoslav\"\n      - family-names: \"Tao\"\n        given-names: \"Terence\"\n    year: 2025\n    journal: \"Acta Mathematica Hungarica\"\n    volume: \"175\"\n    pages: \"572-608\"\n    doi: \"10.1007/s10474-025-01528-0\"\n  - type: article\n    title: \"Quantitative correlations and some problems on prime factors of consecutive integers\"\n    authors:\n      - family-names: \"Tao\"\n        given-names: \"Terence\"\n      - family-names: \"Teräväinen\"\n        given-names: \"Joni\"\n    year: 2026\n    doi: \"10.48550/arXiv.2512.01739\"\n    url: \"https://arxiv.org/abs/2512.01739v2\"\n  - type: article\n    title: \"Sparse Polynomial-Weighted Expansions\"\n    authors:\n      - family-names: \"Wang\"\n        given-names: \"Han\"\n    year: 2026\n    url: \"https://arxiv.org/abs/2606.24972v4\"\n    doi: \"10.48550/arXiv.2606.24972\"\n    identifiers:\n      - type: url\n        value: \"https://github.com/wcook04/plectis-erdos/blob/main/docs/research-commons/SOURCE_ATTRIBUTIONS.md#source-source-f4ad17717c8fd4\"\n        description: \"Source attribution and exact local uses\"\n  - type: article\n    title: \"The googol-th bit of the Erdős-Borwein constant\"\n    authors:\n      - family-names: \"Crandall\"\n        given-names: \"Richard\"\n    year: 2012\n    journal: \"Integers\"\n    volume: \"12\"\n    issue: \"5\"\n    pages: \"811-840\"\n    doi: \"10.1515/integers-2012-0007\"\n  - type: article\n    title: \"On the binary digits of the Erdős-Borwein constant\"\n    authors:\n      - family-names: \"Campbell\"\n        given-names: \"John M.\"\n    year: 2026\n    url: \"https://arxiv.org/abs/2605.24160\"\n    doi: \"10.48550/arXiv.2605.24160\"\n  - type: article\n    title: \"(Non)automaticity of number theoretic functions\"\n    authors:\n      - family-names: \"Coons\"\n        given-names: \"M.\"\n    year: 2010\n    url: \"https://doi.org/10.5802/jtnb.718\"\n    identifiers:\n      - type: url\n        value: \"https://github.com/wcook04/plectis-erdos/blob/main/docs/research-commons/SOURCE_ATTRIBUTIONS.md#source-source-741da55b02c5a9\"\n        description: \"Source attribution and exact local uses\"\n  - type: article\n    title: \"Simultaneous inequalities among values of the Euler phi-function\"\n    authors:\n      - family-names: \"Martin\"\n        given-names: \"Greg\"\n    year: 2006\n    url: \"https://arxiv.org/abs/math/0603053v1\"\n    identifiers:\n      - type: url\n        value: \"https://github.com/wcook04/plectis-erdos/blob/main/docs/research-commons/SOURCE_ATTRIBUTIONS.md#source-source-11b46a0435368f\"\n        description: \"Source attribution and exact local uses\"\n  - type: article\n    title: \"Multiplicative functions and k-automatic sequences\"\n    authors:\n      - family-names: \"Yazdani\"\n        given-names: \"Soroosh\"\n    year: 2001\n    url: \"https://www.numdam.org/item/JTNB_2001__13_2_651_0.pdf\"\n    identifiers:\n      - type: url\n        value: \"https://github.com/wcook04/plectis-erdos/blob/main/docs/research-commons/SOURCE_ATTRIBUTIONS.md#source-source-a0d109b4492fba\"\n        description: \"Source attribution and exact local uses\"\n  - type: generic\n    title: \"Answer to An infinite sum based on the mod-parity of Euler's totient function\"\n    authors:\n      - family-names: \"Wong\"\n        given-names: \"Erick\"\n    year: 2015\n    url: \"https://math.stackexchange.com/a/1211557\"\n    identifiers:\n      - type: url\n        value: \"https://github.com/wcook04/plectis-erdos/blob/main/docs/research-commons/SOURCE_ATTRIBUTIONS.md#source-source-0f61ad0796acdf\"\n        description: \"Source attribution and exact local uses\"\n  - type: article\n    title: \"How to prove that a sequence is not automatic\"\n    authors:\n      - family-names: \"Allouche\"\n        given-names: \"Jean-Paul\"\n      - family-names: \"Shallit\"\n        given-names: \"Jeffrey\"\n      - family-names: \"Yassawi\"\n        given-names: \"Reem\"\n    year: 2022\n    url: \"https://doi.org/10.1016/j.exmath.2021.08.001\"\n    identifiers:\n      - type: url\n        value: \"https://github.com/wcook04/plectis-erdos/blob/main/docs/research-commons/SOURCE_ATTRIBUTIONS.md#source-source-6b460d123159d9\"\n        description: \"Source attribution and exact local uses\"\n  - type: article\n    title: \"Transcendence of generating functions whose coefficients are multiplicative\"\n    authors:\n      - family-names: \"Bell\"\n        given-names: \"Jason P.\"\n      - family-names: \"Bruin\"\n        given-names: \"Nils\"\n      - family-names: \"Coons\"\n        given-names: \"Michael\"\n    year: 2012\n    url: \"https://doi.org/10.1090/S0002-9947-2011-05479-6\"\n    identifiers:\n      - type: url\n        value: \"https://github.com/wcook04/plectis-erdos/blob/main/docs/research-commons/SOURCE_ATTRIBUTIONS.md#source-source-b791f5b49e0da6\"\n        description: \"Source attribution and exact local uses\"\n  - type: article\n    title: \"Mahler series with multiplicative coefficient sequences\"\n    authors:\n      - family-names: \"Bell\"\n        given-names: \"J.\"\n      - family-names: \"Smertnig\"\n        given-names: \"D.\"\n    year: 2026\n    url: \"https://arxiv.org/abs/2603.23456v1\"\n    identifiers:\n      - type: url\n        value: \"https://github.com/wcook04/plectis-erdos/blob/main/docs/research-commons/SOURCE_ATTRIBUTIONS.md#source-source-3bc828513b4d63\"\n        description: \"Source attribution and exact local uses\"\n","body_format":"text"},"security":{"document_id":"security","source_path":".github/SECURITY.md","role":"security_boundary","github":"https://github.com/wcook04/plectis-erdos/blob/main/.github/SECURITY.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/.github/SECURITY.md","sha256":"sha256:9f48d347ac227f616e55d6b271dfef40a63dffc2c1f52f1c6194c26b5829a016","byte_count":2579,"origin":"public_main_checkout","body":"<!-- SPDX-FileCopyrightText: 2026 Will Cook -->\n<!-- SPDX-License-Identifier: Apache-2.0 -->\n\n# Security\n\nThis is a scholarly code-and-document archive, not a network service. It ships\nno credentials or background service. Its security surface is the code a user\nchooses to run, the pinned Lean and Mathlib dependencies, release-only Python\npackages, and the continuous-integration configuration.\n\n## Before you run anything\n\nCloning downloads tracked files but does not run this project's Python or Lean\ncode. The repository defines no submodules or Git LFS filters. It ships an\nopt-in pre-push validation hook in `.githooks/pre-push`. The documented command\n`python3 scripts/check_push.py --install` activates it for a worktree through\nGit's `core.hooksPath`; a normal clone does not install it. The separate\n`--install-shared` option installs a managed publication guard for the\nrepository's worktrees. You can read the papers, claim records, open boundaries,\nand source map on GitHub or the maths site before executing a project command.\n\nThe trust boundary changes when you run code. The read-only query commands\nexecute tracked Python. A Lean build also installs or invokes the exact Lean\ntoolchain in `lean-toolchain` and the Mathlib revision in `lake-manifest.json`.\nRelease-only Python packages are version- and hash-pinned in\n`scripts/requirements-release.txt`, and GitHub Actions dependencies are pinned to full\ncommit hashes. The release workflow checks for prohibited proof shortcuts,\nprivate path leaks, stale generated evidence, and suspect material reachable\nthrough Git history.\n\nNo repository can guarantee that executing unfamiliar code is risk-free. If\nyou need to treat the checkout as hostile, inspect the commit and lock files\nfirst and run it in a disposable virtual machine or container with no secrets\nor write access to important files. A green Lean build establishes proof\nchecking for the imported declarations; it is not a general security audit.\n\nReport privately through\n[GitHub Private Vulnerability Reporting](https://github.com/wcook04/plectis-erdos/security/advisories/new).\n\nUse it for anything sensitive: a supply-chain concern in the pinned\ntoolchain or dependencies, a CI configuration issue, or leaked material that\nshould not be public. Do not open a public issue with sensitive details.\nMathematical errors and build failures are not security issues; use the\n[ordinary issue forms](https://github.com/wcook04/plectis-erdos/issues/new/choose)\nfor those. Reports get a best-effort response from a solo maintainer, normally\nwithin a week.\n","body_format":"text"},"code-of-conduct":{"document_id":"code-of-conduct","source_path":".github/CODE_OF_CONDUCT.md","role":"participation_standard","github":"https://github.com/wcook04/plectis-erdos/blob/main/.github/CODE_OF_CONDUCT.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/.github/CODE_OF_CONDUCT.md","sha256":"sha256:886ec18e8ee69f45fc3d035d4623898cefdeb82579ec137427ab6a21657a7b56","byte_count":1788,"origin":"public_main_checkout","body":"<!-- SPDX-FileCopyrightText: 2026 Will Cook -->\n<!-- SPDX-License-Identifier: Apache-2.0 -->\n\n# Code of conduct\n\nThis repository exists so people can criticise, reuse, correct, and extend\npublic mathematical work. Participation should make that work easier to inspect\nand safer to continue.\n\n## Expected conduct\n\n- Address claims, proofs, code, evidence, attribution, and decisions directly.\n- Give credit to the people and sources that supplied material work.\n- State uncertainty and surviving mathematical limits plainly.\n- Accept good-faith correction and allow other contributors to disagree.\n- Protect private or sensitive information.\n\n## Unacceptable conduct\n\nHarassment, threats, sustained unwanted contact, sexualised conduct, personal\nattacks, deliberate intimidation, or publishing another person's private\ninformation are not accepted. Discrimination based on identity, background,\ndisability, experience, education, or another personal characteristic is also\nnot accepted. Criticising a mathematical assertion or rejecting a contribution\non evidence is not, by itself, a violation.\n\n## Scope and enforcement\n\nThis standard applies to issues, pull requests, reviews, repository-linked\ndiscussions, and other spaces where somebody represents this project. The\nmaintainer may edit or remove comments, close contributions, limit interaction,\nor block an account when needed to enforce it. Decisions should identify the\nconduct at issue without exposing confidential reports.\n\nReport a possible violation privately by\n[emailing Will](mailto:williamwkcook@gmail.com). Reports get a best-effort response\nfrom a solo maintainer, normally within a week. Do not put sensitive details in\na public issue. For software security vulnerabilities, follow the\n[security policy](SECURITY.md).\n","body_format":"text"},"credit-policy":{"document_id":"credit-policy","source_path":"docs/research-commons/CREDIT_POLICY.md","role":"credit_policy","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/research-commons/CREDIT_POLICY.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/research-commons/CREDIT_POLICY.md","sha256":"sha256:e3dc04316dc89c7d54616848cf0cb6d510daee2d72bc91b1dd00952f16663813","byte_count":8213,"origin":"public_main_checkout","body":"<!-- SPDX-FileCopyrightText: 2026 Will Cook -->\n<!-- SPDX-License-Identifier: Apache-2.0 -->\n\n# Credit and stewardship\n\nCredit follows the accepted work. A receipt names the person who returned it\nand the exact public files, results, or evidence being credited. It does not\nreassign authorship of the surrounding corpus or imply that every idea from an\nexploration became part of the accepted result.\n\nA contributor may be different from a collaborator, a tool operator, or a\ndisclosed model or service used during the work. The receipt records those\nroles separately. This makes assistance visible without obscuring human\nresponsibility or inventing authorship. It does not guess at undisclosed use.\n\nWithin artifact credit, optional CRediT-aligned roles describe what each person\nsupplied: for example conceptualization, formal analysis, investigation,\nmethodology, software, validation, visualization, or writing. Multiple roles\nmay apply. The role list describes contributions; it does not by itself decide\nauthorship, ownership, correctness, or priority.\n\nPositive, negative, inconclusive, and corrective work can all receive credit.\nThe returned artifact must be useful, bounded, traceable, and accepted. It does\nnot need to look like a theorem or increase a score. The repository does not\nrank contributors by commits, lines changed, receipt counts, model usage, or\nanother activity count.\n\nArchitecture ideas and implementations receive the same treatment. An adopted\nidea should become a public design note, implementation, test, or other tracked\nartifact that cites the proposal and names its originator. An issue mention by\nitself is delivery history; the accepted artifact and receipt make the credit\ndurable in every clone.\n\nSubmission is not acceptance. Issues, pull requests, and Git history preserve\nthe delivery history, but work appears on the accepted credit pages only after\na receipt is committed against an exact public commit. Acceptance alone does\nnot establish correctness, novelty, peer review, release inclusion, or a\nchange in mathematical claim status. Each of those needs its own evidence.\n\nCorrections preserve lineage. A newer receipt names the earlier return and\nrecords whether it retains, supersedes, or withdraws the affected artifact.\nThe earlier contributor and historical evidence remain discoverable. Disputes\nabout attribution should be handled as corrective contributions with concrete\nartifact and provenance evidence rather than by rewriting history silently.\n\nAll contributions remain subject to the repository's licences and source\nattribution requirements. A contributor should cite prior work, identify any\nmaterial collaborators, and state limitations honestly. Maintainers must keep\nthe credit narrow, durable, public, and no stronger than the accepted evidence.\n\nPrior literature, catalogues, public website contributions, and implemented\nadvice are indexed separately in [source attributions](SOURCE_ATTRIBUTIONS.md).\nThat view is built from `source-attributions.json` with\n`python3 scripts/build_source_attributions.py`; use `--check` for freshness or\n`--query <name-or-problem-or-id>` for a bounded lookup. It is source-credit and\nnavigation evidence, not an accepted-contribution receipt, authorship transfer,\nmathematical review, or endorsement. Private correspondence remains anonymous\nuntil the contributor confirms public naming; its private evidence stays out\nof this repository.\n\n## Directions, email, and work across tracks\n\nA useful direction, reference, correction, example or explanation can be the\ncontribution. Describe that contribution precisely; conceptualization and\nmethodology credit do not depend on supplying code or finishing the proof.\nThe person who develops an idea later has a different role. Retain both when\nboth materially contributed, including when one return has mathematical and\narchitecture receipts.\n\nThe intake channel does not determine credit. For an email contribution,\nmaintainers record the date, the contributor's preferred public name or\nanonymous designation, a public-safe account of the idea, and the permission\nto publish that account. Confirm naming and quotation preferences before\npublishing private correspondence; retain the message outside the public\nrepository. Put adopted substance and its attribution in a tracked artifact,\nthen use the same acceptance and recognition builders as for a GitHub return.\nThe public record can cite that artifact without exposing an email address.\nAn unadopted suggestion remains acknowledged as a suggestion, not an accepted\nresult. A receipt never grants authorship of unspecified future consequences.\n\n## The credit ledger and naming\n\nAdvice received privately that changed public files is listed in the\n[credit ledger](CREDIT_LEDGER.md). Each entry gives the date the advice arrived,\none sentence on what the person said, one sentence on what changed, and the\nexact lines (and, where one commit made the change, the commit) where a reader\ncan see the change. The ledger is generated from the `correspondence` rows of\n`source-attributions.json` by `python3 scripts/build_source_attributions.py`.\n\nEvery entry starts with the name withheld and the person described only by a\nneutral role, such as \"a mathematician\". Only the person who gave the advice\ncan change that. If they confirm the public name they want, the row records it\nwith the confirmation date (`named_with_permission`). If they would rather stay\nanonymous, the row says so (`anonymous_by_request`) and the advice stays\ncredited. These three states follow the pending, accepted and declined credits\nof GitHub security advisories. Either change is a one-row edit followed by the\nbuild command above; the message itself stays private. Acknowledgement in the\nledger does not say that the person reviewed, checked or endorsed the\nmathematics.\n\n## Advice adopted in communication\n\nFeedback about how the work is explained can change the research process\nwithout supplying a proof. The rules below record adopted advice, now also\npublished here. Their entries in the credit ledger acknowledge only those\nchanges, not mathematical review or endorsement. The original messages and\nidentities remain private while naming permission is pending.\n\n<a id=\"communication-clear-purpose\"></a>\n### State the purpose and the request\n\nExplain one concrete object, what it does, and why someone might inspect it.\nSay what is being asked and how much reading or time it requires. Use ordinary\nwording rather than asking the recipient to reconstruct the system's purpose\nfrom a broad description. This records the advice credited in\n[the clear-purpose entry](CREDIT_LEDGER.md#credit-correspondence-006).\n\n<a id=\"communication-evidenced-comparison\"></a>\n### Make comparisons inspectable\n\nWhen inviting someone to use the system, identify the existing approach being\ncompared and the evidence for any claimed advantage. A description of two\nsystems is not a demonstration that one performs better. A willingness to look\nat a comparison is not a completed review. This records the advice credited in\n[the comparison entry](CREDIT_LEDGER.md#credit-correspondence-007).\n\n<a id=\"communication-supported-specialist-connection\"></a>\n### Retire unsupported specialist connections\n\nIf a specialist reports no visible connection between their expertise and the\nproposed question, remove that connection from future correspondence unless\nnew evidence supports it. Do not turn a correction of scope into a claim that\nthe specialist supplied a proof or reviewed the wider project. This records\nthe advice credited in\n[the scope-correction entry](CREDIT_LEDGER.md#credit-correspondence-008).\n\n<a id=\"communication-records-and-model-capability\"></a>\n### Explain the role of persistent records\n\nExplain how records preserve sources, failed approaches and results for later\ninspection. Keep that purpose distinct from improving a model's ability to\nsolve a problem, and do not claim that an external record replaces advances\nin models. A criticism of the framing is not evidence that the critic reviewed\nthe records or the mathematics. This records the advice credited in\n[the research-records entry](CREDIT_LEDGER.md#credit-correspondence-009).\n","body_format":"text"},"problem-library":{"document_id":"problem-library","source_path":"docs/reference/PROBLEM_LIBRARY.md","role":"paper_and_source_map_api","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/reference/PROBLEM_LIBRARY.md","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/reference/PROBLEM_LIBRARY.md","sha256":"sha256:2d82a9f843c1694b6edea3d4d55541bba5cac3b8c8c82f2c261e4b5a099b1fd1","byte_count":1972,"origin":"public_main_checkout","body":"# Papers and Lean source by problem\n\n[Choose a problem on the website](https://wcook04.github.io/practical-formal-verification/maths/).\nEvery problem has a short paper, a long paper, and a searchable source map.\nPDF and TeX links point to stable file paths on this repository's `main` branch.\nThe website's HTML reader is a published snapshot.\n\n`docs/problem_library.json` is the public machine-readable source for that view.\nIt is separate from the bounded `docs/problems.json` entry index. Each problem\ncontains `papers`, `paper_roles`, and a `source_map` with module nodes and import\nedges. Import arrows describe source dependencies, not theorem strength or\nmathematical credit. Only local Lean modules are included.\n\nTo refresh the library after changing source or paper records, run from a full\ncheckout (the reader-only sparse checkout intentionally omits Lean source):\n\n```sh\npython3 scripts/build_problem_index.py\npython3 scripts/build_problem_index.py --check\npython3 scripts/test_problem_library.py\n```\n\nPaper identities and short/long roles come from `docs/papers/corpus.json`.\nKeep a paper's `paper_id` and file paths when revising it; rebuild and commit its\nPDF with the TeX source. Title and page-count changes do not change its URL.\nWhen adding a paper, its corpus record needs explicit problem membership\n(`problem_ids`, or its existing numbered `subject`), publication state, form,\nand paths to the PDF, TeX, and full text. Missing files fail the build.\nThe long manuscripts are assembled from `paper/reasoning-parts/`; use\n`python3 scripts/assemble_reasoning_surfaces.py --check` to check agreement.\n\nSource-map roots come from each problem directory, principal and companion\nmodules, reviewed result sources, and explicitly linked claims. The builder\nreads actual Lean imports and follows their local dependencies recursively.\nEach node carries its source digest and current GitHub URL; the library records\ninput digests so consumers can reject a mismatched claim snapshot.\n","body_format":"text"},"palomar-showcase":{"document_id":"palomar-showcase","source_path":"docs/PALOMAR_RESULT_SHOWCASE.json","role":"reader_priority_ranking","github":"https://github.com/wcook04/plectis-erdos/blob/main/docs/PALOMAR_RESULT_SHOWCASE.json","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/docs/PALOMAR_RESULT_SHOWCASE.json","sha256":"sha256:6c56458363f0dcfedd28b6cbe59cda0d5d9669bd7ce9cc66dd223787a74bb266","byte_count":233811,"origin":"public_main_checkout","body":{"schema":"plectis-palomar-result-showcase/1","captured_at":"2026-09-23","source_authority":{"claim_registry":"docs/claims.json","proof_authority":"Lean source checked by the pinned kernel","comparator_authority":"HEAD:verification/comparator.json","comparator_sha256":"c4f34656bac6072dccb7ff4b68604bd786808412decc06473652442f5a0fcaeb","comparator_theorem_count":53,"source_review_matrix":"HEAD:docs/claims.json::external_verification_packet.review_matrix","source_review_matrix_sha256":"72846b2b730c847722ce1f8909c1af3c3f27c2a608e3f8c54ac98c62f4afb15c","source_review_family_count":76,"source_review_family_count_at_dispatch":60,"comparator_roster":["Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses","Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_nonunit_carries","Erdos249257.ExternalVerification.irrational_factorialGapSeries_of_cofinal_global_complementaryTail","Erdos249257.ExternalVerification.no_cofinallyBoundedNegative_of_normalizedVanishes","Erdos249257.ExternalVerification.boundedNegativePart_eventually_zero","Erdos249257.ExternalVerification.sylvesterNext_eventually_of_summable_negativeRelativeMass","Erdos249257.ExternalVerification.no_eventuallyPeriodicNegative_orbit","Erdos249257.ExternalVerification.finrank_totientKernelThroughLevelFamily_eq","Erdos249257.ExternalVerification.not_finiteDimensional_span_fullTotientKernel","Erdos249257.ExternalVerification.exists_totientDyadicSectionBasis","Erdos249257.ExternalVerification.finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent","Erdos249257.ExternalVerification.tsum_pos_coprime_inv_mersenne_eq_one","Erdos249257.ExternalVerification.tsum_totient_div_mersenne_sq_eq_gcd_moment_series","Erdos249257.ExternalVerification.runHeight_defect_fib_sum_lower","Erdos249257.ExternalVerification.cylinderMass_split","Erdos249257.ExternalVerification.lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient","Erdos249257.ExternalVerification.fixedPrecisionTropicalNoGo","Erdos249257.ExternalVerification.irrational_totientSeries_of_actualLcmOrbitSeparationSupply","Erdos249257.ExternalVerification.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap","Erdos249257.ExternalVerification.exists_clean_binaryCyclotomicAnchor","Erdos249257.ExternalVerification.scaleFullTarget_miss_of_projected_separation","Erdos249257.ExternalVerification.irrational_totient_series_of_pivotResidualDecorrelation","Erdos249257.ExternalVerification.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank","Erdos249257.ExternalVerification.exists_primeGap0_gt","Erdos249257.ExternalVerification.coefficientOnlyNoGo","Erdos249257.ExternalVerification.irrational_tsum_primeDyadicTerm_iff_primeGap","Erdos249257.ExternalVerification.irrational_initial_iff_all_positive_tailShifts_nonintegral","Erdos249257.ExternalVerification.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch","Erdos249257.ExternalVerification.carry_eq_residueDigit_add_coboundary","Erdos249257.ExternalVerification.volume_mersenneAchievementSet","Erdos249257.ExternalVerification.irrational_erdosSum_full_support","Erdos249257.ExternalVerification.divisibilityWeightedClaim","Erdos249257.ExternalVerification.finitePrimeWeighted_fixedBase_hereditary","Erdos249257.ExternalVerification.irrational_erdosSupportSeries_pairwise_coprime","Erdos249257.ExternalVerification.irrational_ratWeightSeries_eventuallyPeriodic","Erdos249257.ExternalVerification.irrational_erdosSupportSeries_of_orthogonalPetalBouquet","Erdos249257.ExternalVerification.finite_period_noncollapse_rat_den_interface","Erdos249257.ExternalVerification.volume_supportedMersenneAchievementSet_dichotomy","Erdos249257.ExternalVerification.exists_shifted_odd_tail_nat_states_of_support_fraction","Erdos249257.ExternalVerification.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry","Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips","Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse","Erdos249257.ExternalVerification.supportCoeff_mul_eq_add_defect","Erdos249257.ExternalVerification.seven_le_of_intBand_odd","Erdos249257.ExternalVerification.kernel_235_minor_eq_neg_one_fifteen","Erdos249257.ExternalVerification.smoothPrefixLcm_eq_threePrimeHeight","Erdos249257.ExternalVerification.no_positive_reducedCarry_of_cofinalLocalWindowEscape","Erdos249257.ExternalVerification.exists_small_translation_separating_arguments","Erdos249257.ExternalVerification.constant_perturbation_roots_in_unitDisk","Erdos249257.ExternalVerification.exists_two_roots_dist_sum_le_two_mul_geomMean","Erdos249257.ExternalVerification.threeHalves_no_coordinatewiseCorridor","Erdos249257.ExternalVerification.rationalBaseClearedTailQ_succ","Erdos249257.ExternalVerification.rectangular_hp_threshold_eq_classical_iff"],"seven_problems_remain_open":true,"erdos_1041_formal_statement_refuted":true,"review_matrix_binding_review":{"reviewed_at":"2026-09-19","previous_sha256":"1177b9c23c905304187d99ee4a34641dacd8c406566799f8758555bf8baeba44","reviewed_claims_commit":"1d0d5e5797300d0e7aed1a6c65ec2ac75155f8b1","scope":"The 2026-09-19 matrix review is historical. The 2026-09-23 packet adds the exact finite-prime weighted support family and independently ranks its checked theorem as the broadest direct #257 lead; no novelty, priority, external human review, or universal endpoint is inferred from this update."}},"selection_contract":{"authority_basis":"Every source family is read from the exact committed HEAD:docs/claims.json review matrix plus the explicitly recorded theorem-forest discoveries; the exact committed HEAD:verification/comparator.json roster is a nested evidence/admission screen. Uncommitted Comparator edits, roster order, and ease of qualification are not selection evidence.","source_result_spine":{"role":"Cross-programme reading order for the strongest currently registered public results. The claim registry and named Lean declarations own status and mathematical truth; this order is an editorial appraisal, not novelty, priority, or peer review.","comparator_boundary":"candidate_ranking remains the separate Comparator-qualified theorem roster. A result absent from that roster can appear here only with its actual source and evidence class stated; this spine does not confer Comparator admission.","ranked_results":[{"rank":1,"problem":257,"claim_id":"finite_prime_weighted_support","source_declaration":"divisibilityWeightedClaim","why_here":"The finite-prime weighted-mass criterion proves irrationality for an infinite positive support at a fixed integer base, with a binary-mass hypothesis giving every infinite subset irrationality at every integer base. Its checked weighted-return construction covers support classes beyond the classical full support.","boundary":"This is a sufficient weighted-mass criterion. Irrationality for every infinite support remains open."},{"rank":2,"problem":1049,"claim_id":"zudilin_rational_base_region","source_declaration":"rational_base_region","why_here":"The constructed polynomial forms, nonvanishing, and asymptotic bounds give Lean-checked irrationality and an irrationality-exponent bound throughout the stated Zudilin contour region, including every positive integral power of 31/4.","boundary":"This formalises a stated analytic region; it does not prove the universal rational-base conjecture. The base 3/2 lies outside the region and remains unresolved."},{"rank":3,"problem":1041,"claim_id":"ani_degree_seven_total_variation_counterexample","source_declaration":"erdos1041_counterexample_hausdorff","why_here":"For ani's explicit degree-seven polynomial, Lean proves a one-dimensional Hausdorff-measure lower bound for every preconnected strict-lemniscate set joining distinct roots. This refutes the exact Formal Conjectures path-image-length statement; a separate total-variation bound is checked.","boundary":"Ani supplied the polynomial. The reported small-parameter family is not formalised, and correspondence with the 1958 wording has not received independent human review."}]},"ranking_axes":[{"axis":"mathematical_nontriviality","test":"Prefer a substantive theorem or obstruction with a real mathematical mechanism over a routine identity, finite instance, or renamed transport."},{"axis":"consequence_and_endpoint_proximity","test":"Prefer a direct consequence for an Erdős target or a faithful exact reduction to its remaining endpoint, while keeping any missing producer explicit."},{"axis":"mechanism_depth_and_natural_friction","test":"Expose the hard certificate, averaging, recurrence, or obstruction step and the hypotheses that make it work; do not flatten friction into a headline."},{"axis":"genuinely_distinct_content_and_independence","test":"Prefer a result that adds a distinct mathematical route or support class rather than a corollary or duplicate wrapper."},{"axis":"external_review_value","test":"Prefer a statement whose exact scope, source attribution, and significance can be checked by a reader without reconstructing hidden context."},{"axis":"proof_digestion_value","test":"Prefer a result that teaches a reusable mechanism and connects the source theorem to the open boundary."},{"axis":"evidence_certainty","test":"Require an exact current source declaration, committed Challenge transport, and source-checked hypotheses and conclusion."},{"axis":"overclaim_risk","test":"Penalize hidden conditionality, missing cofinal supply, and language that could be read as solving a still-open problem."}],"presentation_tiers":[{"order":1,"tier_id":"source_ranked_frontier","label":"source-ranked frontier","authority_field":"candidate_ranking","candidate_screening_dispositions":[],"reader_role":"The explicitly ranked exceptional spine: completed direct results first, then the strongest conditional endpoint routes and deep structural results."},{"order":2,"tier_id":"conditional_endpoint_leverage","label":"conditional endpoint leverage","authority_field":"candidate_screening","candidate_screening_dispositions":["supporting_conditional_endpoint","supporting_conditional_endpoint_classification","supporting_conditional_no_go","supporting_conditional_recovery","supporting_conditional_reduction","supporting_endpoint_reduction"],"reader_role":"Exact target-facing consumers and reductions whose named producer or bridge remains open; their leverage is shown before medium machinery, with every premise adjacent."},{"order":3,"tier_id":"deep_mechanism_and_classification","label":"deep mechanism and classification","authority_field":"candidate_screening","candidate_screening_dispositions":["supporting_boundary","supporting_classification","supporting_combinatorial","supporting_conditional_anti_compression","supporting_conditional_structural","supporting_exact_recurrence","supporting_mechanism","supporting_metric_geometry","supporting_reconstruction_equivalence","supporting_structural"],"reader_role":"Reusable exact normal forms, classifications, geometric mechanisms, and structural obstructions that sharpen the problem without themselves being endpoint theorems."},{"order":4,"tier_id":"natural_friction_and_no_go","label":"natural friction and no-go","authority_field":"candidate_screening","candidate_screening_dispositions":["supporting_contrary","supporting_finite_no_go","supporting_local_no_go","supporting_model_no_go","supporting_negative_method","supporting_no_go"],"reader_role":"Countermodels and exact route obstructions that explain why tempting coefficient, finite, local, or model-specific arguments do not reach an endpoint."},{"order":5,"tier_id":"supporting_and_long_tail","label":"supporting and long tail","authority_field":"candidate_screening","candidate_screening_dispositions":["pending_value_screen","supporting_auxiliary","supporting_budget","supporting_exact_identity","supporting_finite_interface","supporting_normalization_identity","supporting_structural_identity","supporting_technical","supporting_transfer_identity"],"reader_role":"Exact supporting identities, finite interfaces, budgets, technical inputs, and pending rows retained for drilldown rather than first-contact prominence."}],"relational_placements":[{"family_id":"small_mismatch_criterion","tier_id":"conditional_endpoint_leverage","within_tier_order":1,"relative_judgement":"Below the ranked prime-gap equivalence but above the #269 carry-escape consumer, abstract dyadic-tail classification, and coefficient-only no-go: it is specialized to the actual prime-gap recurrence and identifies the precise cofinal adjacent-smallness producer, while that producer remains unproved."},{"family_id":"conditional_carry_escape","tier_id":"conditional_endpoint_leverage","within_tier_order":2,"relative_judgement":"Below the #251 small-mismatch route because the source-specific cofinal escape remains unproved, but above medium finite #269 machinery because the exact residue-window contradiction is the endpoint consumer that such machinery must feed. The actual-series reduction is given."},{"family_id":"dyadic_tail_integrality_classification","tier_id":"deep_mechanism_and_classification","within_tier_order":1,"relative_judgement":"Below the source-specific conditional endpoint consumers but above negative-method evidence: its exact denominator and all-positive-tail-shift normal forms are deeper and more reusable than a coefficient-only obstruction, while supplying no actual-prime producer."},{"family_id":"coefficient_only_no_go","tier_id":"natural_friction_and_no_go","within_tier_order":1,"relative_judgement":"Below the #251 small-mismatch route and abstract integrality classification: it is valuable natural friction against coefficient-only reasoning, but its synthetic carry stream has no infinite-limit or actual-prime-gap identification bridge."}],"represented_family_placements":[{"family_id":"factorial_channel_and_projection_rigidity","tier_id":"deep_mechanism_and_classification","relative_judgement":"The finite channel and projection machinery is the deepest represented #68 mechanism below the endpoint equivalence, but it supplies no cofinal obstruction.","canonical_claim_commit":"06198c0ab3969c625e1022eaebfb46208ff0b5e4","source_declarations":["Erdos68.factorialMoment_eq_factorial_pow_mul_channelNumerator_band","Erdos68.exists_index_ge_two_mul_of_factorialMoment_ne_zero_of_channel_eq_zero","Erdos68.channelNumerator_mod_factorialMoment","Erdos68.primeTranslator_channelResidual_eq_one","ErdosProblems.Erdos68.projection_disagreement_excludes_bounded_endpoint"],"hard_mechanism":"One quotient band factors the factorial moment through a channel numerator; channel congruence preserves the moment modulo d! - 1, a prime translator gives an exact unit residual, and unequal sufficiently large divisor projections exclude a bounded endpoint.","open_boundary":"No theorem supplies cofinally many non-unit carries, moving nontrivial private projections, or cofinally nonzero translated residuals, so the family does not prove Erdős #68 irrationality."},{"family_id":"bounded_rise_coprimality","tier_id":"natural_friction_and_no_go","relative_judgement":"The fresh-modulus obstruction explains why bounded rise would be decisive, but bounded rise is not automatic for the original reciprocal-tail orbit.","canonical_claim_commit":"06198c0ab3969c625e1022eaebfb46208ff0b5e4","source_declarations":["ErdosProblems.Erdos243.reducedTail_pairwiseCoprime","ErdosProblems.Erdos243.no_eventuallyBoundedRise_reducedTail"],"hard_mechanism":"Exact reduced C/D dynamics make every earlier multiplier divide each later denominator while remaining coprime to the later reduced numerator. Distinct multipliers are therefore pairwise coprime fresh moduli; a numerator tending to infinity cannot keep crossing all of their forbidden residue windows with one uniform positive rise bound.","open_boundary":"The theorem assumes eventual multiplier positivity above one, reduced coprimality, both exact recurrence equations, one fixed positive rise bound, and numerator escape to infinity. No theorem supplies that bounded-rise hypothesis for the original reciprocal-tail orbit, so no reciprocal-tail irrationality or Erdős #243 endpoint follows."},{"family_id":"eventually_periodic_lambert","tier_id":"deep_mechanism_and_classification","relative_judgement":"This is a direct irrationality theorem for an exact structured coefficient class, but the totient-derived coefficients are not eventually periodic, so it is reusable background rather than a #249 frontier."},{"family_id":"signed_totient_entire_defect_interpolation","tier_id":"deep_mechanism_and_classification","relative_judgement":"The ordinary proof constructs a sparse signed perturbation with strong defect control for every divergent allowance, but it does not settle the unperturbed totient series or the one-sided prime-gap route; no infinite Lean or Comparator theorem is claimed."},{"family_id":"separated_cut_algebraic_lambert_transcendence","tier_id":"deep_mechanism_and_classification","relative_judgement":"The ordinary proof transfers algebraic-base transcendence through a separated-cut host and an explicit non-chain example, while full support at 3/2 and unrestricted base-two supports remain open; finite Lean cut lemmas are not its infinite proof."},{"family_id":"totient_certificate_equivalences","tier_id":"deep_mechanism_and_classification","relative_judgement":"Below the actual-LCM, first-harmonic, and strict-prime producer routes but above medium totient kernel machinery: three exact equivalences expose the endpoint and certificate sockets, while the equivalent lcm-diagonal, period-multiple-kill, and separated-window producers all remain unresolved.","evidence_status":"claims_review_metadata_only_not_comparator_evidence","authority":"docs/claims.json::external_verification_packet.review_matrix[problem=249].families[totient_certificate_equivalences]","open_boundary":"No lcm-diagonal certificate supply, period-multiple kill supply, or separated-window pair supply is proved for the actual totient series; no unconditional Erdős #249 result, novelty, priority, significance, or external-review claim follows."},{"family_id":"totient_lambert_coefficients","tier_id":"supporting_and_long_tail","relative_judgement":"The Lambert rewrite and prime-power coefficient structure orient the programme but provide no irrationality producer."},{"family_id":"integral_shift_classification","tier_id":"deep_mechanism_and_classification","relative_judgement":"The block and denominator criteria are reusable rational-state structure below the sharper all-positive-shift normal form and above propagation identities."},{"family_id":"totient_shift_propagation","tier_id":"deep_mechanism_and_classification","relative_judgement":"Propagation through totient-length shifts explains rational-state persistence but supplies no contradiction for the actual prime-gap recurrence."},{"family_id":"height_fibre_and_shell","tier_id":"deep_mechanism_and_classification","relative_judgement":"The finite height-fibre normal form and smooth-shell bound are substantive three-prime structure, but they do not produce the source-specific cofinal escape. The actual-series reduction is given separately."},{"family_id":"independent_clock_channel_relations","tier_id":"deep_mechanism_and_classification","relative_judgement":"The ordinary proof classifies rational relations for independent clock channels by a swap-and-lattice mechanism, but it does not establish the needed independence for the original running-LCM rates or its endpoint."},{"family_id":"dyadic_block_alphabet","tier_id":"supporting_and_long_tail","relative_judgement":"The exact four-symbol block alphabet is a useful finite interface, subordinate to the carry consumer, weighted observer, and rank obstruction."},{"family_id":"newton_value_decay","tier_id":"deep_mechanism_and_classification","relative_judgement":"The Newton value equation and exponential first integral are the load-bearing local mechanism, while metric gluing to a short curve remains open.","canonical_claim_commit":"06198c0ab3969c625e1022eaebfb46208ff0b5e4","source_declarations":["ErdosProblems.Erdos1041.newtonFlow_value_hasDerivAt","ErdosProblems.Erdos1041.newtonFlow_scaledValue_hasDerivAt_zero"],"hard_mechanism":"Away from a critical point, the complex Newton vector -f/f' cancels the derivative in the chain rule and gives w' = -w along a tangent curve. The product rule with the complex exponential then makes exp(t) * f(z(t)) a zero-derivative first integral, exposing radial exponential value decay and fixed argument.","open_boundary":"These are pointwise complex-derivative statements. They do not prove real-time trajectory existence, integrate the local derivative identity globally, construct a connecting curve, preserve a component under perturbation, or perform the Reeb/topological and strict length-below-two fan-in required by Erdős #1041."},{"family_id":"ray_separation","tier_id":"natural_friction_and_no_go","relative_judgement":"Distinct value rays exactly obstruct finite Newton connections, a consequential route boundary but not the global lemniscate theorem.","canonical_claim_commit":"06198c0ab3969c625e1022eaebfb46208ff0b5e4","source_declarations":["ErdosProblems.Erdos1041.samePositiveRay_of_real_exp_decay","ErdosProblems.Erdos1041.no_newtonConnection_of_not_samePositiveRay"],"hard_mechanism":"The Newton-value endpoint relation is multiplication by the strictly positive scalar exp(-time), so every finite connection remains on one positive value ray; distinct rays therefore give an exact obstruction.","open_boundary":"The theorem does not choose globally compatible critical-value rays, construct or glue the required flow geometry, or prove the Erdős #1041 lemniscate endpoint."},{"family_id":"calibrated_rational_hankel_countermodel","tier_id":"natural_friction_and_no_go","relative_judgement":"The ordinary positive-moment countermodel blocks inference from an all-rank Hankel signature alone; these are not Zudilin's moments, and neither F(3/2) nor the actual arithmetic input is settled."}],"programme_family_order":[{"problem":68,"family_ids":["factorial_carry_characterisation","factorial_conditional_producers","factorial_channel_and_projection_rigidity"]},{"problem":243,"family_ids":["bounded_negative_exclusion","negative_mass_recovery","centered_state_dynamics","bounded_rise_coprimality","negative_orbit_no_go"]},{"problem":249,"family_ids":["actual_lcm_orbit_separation","first_harmonic_pivot_decomposition","strict_prime_tail_orbit_gap","totient_certificate_equivalences","totient_kernel_rank","totient_carry_anti_compression","totient_kernel_all_base_index","totient_kernel_basis","stern_brocot_run_fibonacci_stability","eventually_periodic_lambert","signed_totient_entire_defect_interpolation","fixed_precision_transport_no_go","lcm_factor_ideal_shift_algebra_no_go","actual_foreign_residue_projection","totient_lambert_coefficients","squared_lambert_gcd_moments","stern_brocot_cylinder_law","totient_visible_coprime_lattice_mass"]},{"problem":251,"family_ids":["prime_gap_reformulation","small_mismatch_criterion","dyadic_tail_integrality_classification","integral_shift_classification","totient_shift_propagation","coefficient_only_no_go"]},{"problem":257,"family_ids":["finite_prime_weighted_support","known_irrational_supports","pairwise_coprime_support","orthogonal_petal_sunflower_reduction","periodic_nonnegative_weight_irrationality","boolean_mobius_exact_row_dynamics","half_membership_seam_classification","shifted_odd_tail_state","separated_cut_algebraic_lambert_transcendence","boolean_mobius_carry","achievement_set_geometry","restricted_achievement_sets","half_and_twenty_one_frontiers","composite_dilation_defect_identity","finite_period_noncollapse"]},{"problem":269,"family_ids":["conditional_carry_escape","weighted_phase_carry_observer","rank_two_kernel_no_go","height_fibre_and_shell","independent_clock_channel_relations","dyadic_block_alphabet","three_prime_lcm_cells"]},{"problem":1041,"family_ids":["newton_value_decay","ray_separation","translation_avoidance","root_retention"]},{"problem":1049,"family_ids":["rational_base_tail_recurrence","height_and_pade_arithmetic","coordinatewise_corridor_no_go","calibrated_rational_hankel_countermodel"]}],"family_relations":[{"from_family_id":"factorial_channel_and_projection_rigidity","relation":"supports_without_producing","to_family_id":"factorial_carry_characterisation","reason":"Finite channel rigidity feeds the carry endpoint route but supplies no cofinal non-unit carry."},{"from_family_id":"centered_state_dynamics","relation":"mechanism_for","to_family_id":"bounded_negative_exclusion","reason":"Centered-state descent is the exact dynamics consumed by the bounded-negative exclusion."},{"from_family_id":"negative_mass_recovery","relation":"independent_analytic_recovery_peer","to_family_id":"centered_state_dynamics","reason":"Both recover eventual centered-state zero and Sylvester dynamics only under explicit hypotheses, but negative_mass_recovery uses summability of normalized negative mass whereas centered_state_dynamics uses a uniform lower-bound route; neither supplies the original-orbit premises."},{"from_family_id":"first_harmonic_pivot_decomposition","relation":"independent_conditional_endpoint_peer","to_family_id":"actual_lcm_orbit_separation","reason":"Both imply #249 irrationality from different cofinal producers; neither producer is supplied."},{"from_family_id":"strict_prime_tail_orbit_gap","relation":"conditional_producer_peer","to_family_id":"actual_lcm_orbit_separation","reason":"The strict-prime route supplies a phase-density or natural-prime certificate mechanism, while the actual-LCM route instead assumes quantitative orbit separation; all producer predicates remain open."},{"from_family_id":"strict_prime_tail_orbit_gap","relation":"conditional_producer_peer","to_family_id":"first_harmonic_pivot_decomposition","reason":"Both feed the finite first-harmonic certificate socket, but strict-prime uses phase density or adaptive natural-prime escape while the pivot family uses a four-term supplier-fibre budget and residual decorrelation."},{"from_family_id":"totient_certificate_equivalences","relation":"endpoint_normal_form_support_for","to_family_id":"strict_prime_tail_orbit_gap","reason":"Certificate completeness identifies the exact finite endpoint socket but cannot manufacture the strict-prime route's missing phase-density, cofinal-prime, or uniform-margin producer."},{"from_family_id":"fixed_precision_transport_no_go","relation":"contrary_evidence_for","to_family_id":"actual_foreign_residue_projection","reason":"Fixed local precision admits synthetic completions, so any successful projection must use growing precision or global correlation."},{"from_family_id":"dyadic_tail_integrality_classification","relation":"normal_form_support_for","to_family_id":"small_mismatch_criterion","reason":"The abstract shift-integrality normal form explains what the actual-prime cofinal mismatch must destroy."},{"from_family_id":"coefficient_only_no_go","relation":"contrary_evidence_for","to_family_id":"small_mismatch_criterion","reason":"Coefficient irregularity alone is insufficient; the actual-prime adjacent-smallness producer is genuinely additional."},{"from_family_id":"small_mismatch_criterion","relation":"conditional_actual_stream_obstruction_for","to_family_id":"prime_gap_reformulation","reason":"The reformulation gives the exact #251 target language; this fixed-shift actual-prime obstruction identifies what a cofinal adjacent-smallness/mismatch producer would have to destroy. It neither supplies that producer nor turns the reformulation into irrationality."},{"from_family_id":"boolean_mobius_carry","relation":"reconstruction_support_for","to_family_id":"boolean_mobius_exact_row_dynamics","reason":"The Boolean–Möbius certificate reconstructs support, while exact rows still require a cofinal skip producer."},{"from_family_id":"boolean_mobius_exact_row_dynamics","relation":"conditional_producer_for","to_family_id":"half_membership_seam_classification","reason":"Cofinal positive half-greedy skips conditionally give 1/2-membership, which the seam iff classifies as unbounded terminal-false/upper-or-middle supply; finite exact rows supply neither cofinal premise."},{"from_family_id":"weighted_phase_carry_observer","relation":"mechanism_support_for","to_family_id":"conditional_carry_escape","reason":"The residue/coboundary split exposes finite carry data but does not supply the cofinal escape."},{"from_family_id":"rank_two_kernel_no_go","relation":"contrary_evidence_for","to_family_id":"conditional_carry_escape","reason":"The finite kernel is not rank one, ruling out a tempting separable shortcut without producing carry escape."},{"from_family_id":"newton_value_decay","relation":"mechanism_support_for","to_family_id":"ray_separation","reason":"The exponential first integral confines a Newton trajectory to one value ray, yielding the separation obstruction."},{"from_family_id":"ray_separation","relation":"supports_without_closing","to_family_id":"translation_avoidance","reason":"Ray separation motivates avoiding collision lines, but global topology and metric gluing remain open."},{"from_family_id":"height_and_pade_arithmetic","relation":"stronger_natural_friction_than","to_family_id":"coordinatewise_corridor_no_go","reason":"The sharp rectangular threshold analysis carries deeper model-specific information than the coordinatewise transfer obstruction, while neither proves the endpoint."}],"source_family_disposition_order":["selected","represented","deferred","subordinate","rejected","long_tail"],"anti_underclaim_boundary":"Do not select a timid proxy merely because it is easy to qualify: an unconditional direct theorem with a visible hard mechanism must remain eligible even when it is classical or leaves the universal problem open.","anti_hype_boundary":"No selected result closes universal Erdős #257 or any other open programme; exact hypotheses, prior-art attribution, proof authority, limitations, and the surviving open boundary remain part of the result.","reversal_evidence":"Reverse the choice only if a newly committed, source-checked Comparator row has a strictly stronger direct consequence at the same or closer endpoint, genuinely deeper independent mechanism, equal-or-better evidence certainty, and no larger hidden-hypothesis or overclaim risk; uncommitted dirt or a row count cannot reverse it.","reader_tier_rule":"Each ranked candidate carries an explicit reader_tier; the query layer projects this reviewed status without inferring conditionality from prose. Rank and reader tier are independent."},"candidate_universe":{"authority":"HEAD:docs/claims.json::external_verification_packet.review_matrix","source_review_matrix_path":"docs/claims.json::external_verification_packet.review_matrix","source_review_matrix_sha256":"72846b2b730c847722ce1f8909c1af3c3f27c2a608e3f8c54ac98c62f4afb15c","source_review_family_count":76,"source_review_family_count_at_dispatch":60,"selection_scope":"all_current_source_review_families_plus_targeted_theorem_forest_discoveries","source_review_family_ids":["factorial_carry_characterisation","factorial_channel_and_projection_rigidity","factorial_conditional_producers","factorial_finite_certificates","factorial_lcm_growth","centered_state_dynamics","negative_orbit_no_go","bounded_rise_coprimality","bounded_negative_exclusion","negative_mass_recovery","totient_kernel_basis","totient_kernel_rank","totient_kernel_all_base_index","totient_visible_coprime_lattice_mass","squared_lambert_gcd_moments","stern_brocot_run_fibonacci_stability","stern_brocot_cylinder_law","lcm_factor_ideal_shift_algebra_no_go","fixed_precision_transport_no_go","actual_lcm_orbit_separation","actual_foreign_residue_projection","first_harmonic_pivot_decomposition","strict_prime_tail_orbit_gap","totient_carry_anti_compression","totient_finite_denominator_exclusion","totient_carry_rank","totient_lambert_coefficients","eventually_periodic_lambert","totient_certificate_equivalences","prime_gap_reformulation","integral_shift_classification","dyadic_tail_integrality_classification","totient_shift_propagation","small_mismatch_criterion","coefficient_only_no_go","finite_period_noncollapse","boolean_mobius_carry","boolean_mobius_exact_row_dynamics","half_membership_seam_classification","shifted_odd_tail_state","composite_dilation_defect_identity","known_irrational_supports","finite_prime_weighted_support","pairwise_coprime_support","periodic_nonnegative_weight_irrationality","orthogonal_petal_sunflower_reduction","squarefree_support","achievement_set_geometry","restricted_achievement_sets","half_and_twenty_one_frontiers","three_prime_lcm_cells","two_prime_transcendence","height_fibre_and_shell","rank_two_kernel_no_go","dyadic_block_alphabet","conditional_carry_escape","weighted_phase_carry_observer","three_prime_finite_search","newton_value_decay","ray_separation","translation_avoidance","root_retention","published_proof_gap","lemniscate_finite_search","scalar_content_no_go","endpoint_residues","four_jet_collision","coordinatewise_corridor_no_go","rational_base_tail_recurrence","seven_halves_irrationality","height_and_pade_arithmetic","signed_totient_entire_defect_interpolation","separated_cut_algebraic_lambert_transcendence","independent_clock_channel_relations","selected_staircase_integral_height_boundary","calibrated_rational_hankel_countermodel"],"source_family_disposition_contract":{"authority_binding":"Resolve each source_review_row against HEAD:docs/claims.json::external_verification_packet.review_matrix.","required_fields":["disposition","support_evidence","contrary_evidence","cheapest_probe","stop_evidence","hard_mechanism","attribution","limitations","open_boundary"],"field_bindings":{"support_evidence":"source_review_row.summary","contrary_evidence":"source_review_row.boundary","cheapest_probe":"source_review_row.evidence_mode plus exact source-coordinate comparison","stop_evidence":"source_review_row.boundary","hard_mechanism":"source_review_row.summary","attribution":"source_review_row.contribution_class","limitations":"source_review_row.boundary","open_boundary":"source_review_row.boundary"},"comparator_role":"Comparator membership is an evidence, eligibility, and admission field nested below the source 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prime_tail_orbit_gap","disposition":"subordinate","source_landscape_record":"candidate_value_dispositions.source_landscape_candidates[strict_prime_tail_orbit_gap]"},{"candidate_id":"erdos249_carry_anti_compression","disposition":"represented","source_landscape_record":"candidate_value_dispositions.source_landscape_candidates[erdos249_carry_anti_compression]"},{"candidate_id":"erdos68_channel_projection_rigidity","disposition":"represented","source_landscape_record":"candidate_value_dispositions.source_landscape_candidates[erdos68_channel_projection_rigidity]"},{"candidate_id":"erdos1041_ray_separation","disposition":"represented","source_landscape_record":"candidate_value_dispositions.source_landscape_candidates[erdos1041_ray_separation]"},{"candidate_id":"erdos1041_newton_value_decay","disposition":"represented","source_landscape_record":"candidate_value_dispositions.source_landscape_candidates[erdos1041_newton_value_decay]"}],"comparator_screen":{"authority":"HEAD:verification/comparator.json","comparator_sha256":"c4f34656bac6072dccb7ff4b68604bd786808412decc06473652442f5a0fcaeb","theorem_count":53,"roster_field":"candidate_universe.declarations","partition":"candidate_value_dispositions.eligible_groups","semantic_role":"Exact 53-row evidence/eligibility/admission screen nested inside the wider source universe; it does not define candidate-universe membership."},"declarations":["Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses","Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_nonunit_carries","Erdos249257.ExternalVerification.irrational_factorialGapSeries_of_cofinal_global_complementaryTail","Erdos249257.ExternalVerification.no_cofinallyBoundedNegative_of_normalizedVanishes","Erdos249257.ExternalVerification.boundedNegativePart_eventually_zero","Erdos249257.ExternalVerification.sylvesterNext_eventually_of_summable_negativeRelativeMass","Erdos249257.ExternalVerification.no_eventuallyPeriodicNegative_orbit","Erdos249257.ExternalVerification.finrank_totientKernelThroughLevelFamily_eq","Erdos249257.ExternalVerification.not_finiteDimensional_span_fullTotientKernel","Erdos249257.ExternalVerification.exists_totientDyadicSectionBasis","Erdos249257.ExternalVerification.finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent","Erdos249257.ExternalVerification.tsum_pos_coprime_inv_mersenne_eq_one","Erdos249257.ExternalVerification.tsum_totient_div_mersenne_sq_eq_gcd_moment_series","Erdos249257.ExternalVerification.runHeight_defect_fib_sum_lower","Erdos249257.ExternalVerification.cylinderMass_split","Erdos249257.ExternalVerification.lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient","Erdos249257.ExternalVerification.fixedPrecisionTropicalNoGo","Erdos249257.ExternalVerification.irrational_totientSeries_of_actualLcmOrbitSeparationSupply","Erdos249257.ExternalVerification.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap","Erdos249257.ExternalVerification.exists_clean_binaryCyclotomicAnchor","Erdos249257.ExternalVerification.scaleFullTarget_miss_of_projected_separation","Erdos249257.ExternalVerification.irrational_totient_series_of_pivotResidualDecorrelation","Erdos249257.ExternalVerification.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank","Erdos249257.ExternalVerification.exists_primeGap0_gt","Erdos249257.ExternalVerification.coefficientOnlyNoGo","Erdos249257.ExternalVerification.irrational_tsum_primeDyadicTerm_iff_primeGap","Erdos249257.ExternalVerification.irrational_initial_iff_all_positive_tailShifts_nonintegral","Erdos249257.ExternalVerification.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch","Erdos249257.ExternalVerification.carry_eq_residueDigit_add_coboundary","Erdos249257.ExternalVerification.volume_mersenneAchievementSet","Erdos249257.ExternalVerification.irrational_erdosSum_full_support","Erdos249257.ExternalVerification.divisibilityWeightedClaim","Erdos249257.ExternalVerification.finitePrimeWeighted_fixedBase_hereditary","Erdos249257.ExternalVerification.irrational_erdosSupportSeries_pairwise_coprime","Erdos249257.ExternalVerification.irrational_ratWeightSeries_eventuallyPeriodic","Erdos249257.ExternalVerification.irrational_erdosSupportSeries_of_orthogonalPetalBouquet","Erdos249257.ExternalVerification.finite_period_noncollapse_rat_den_interface","Erdos249257.ExternalVerification.volume_supportedMersenneAchievementSet_dichotomy","Erdos249257.ExternalVerification.exists_shifted_odd_tail_nat_states_of_support_fraction","Erdos249257.ExternalVerification.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry","Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips","Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse","Erdos249257.ExternalVerification.supportCoeff_mul_eq_add_defect","Erdos249257.ExternalVerification.seven_le_of_intBand_odd","Erdos249257.ExternalVerification.kernel_235_minor_eq_neg_one_fifteen","Erdos249257.ExternalVerification.smoothPrefixLcm_eq_threePrimeHeight","Erdos249257.ExternalVerification.no_positive_reducedCarry_of_cofinalLocalWindowEscape","Erdos249257.ExternalVerification.exists_small_translation_separating_arguments","Erdos249257.ExternalVerification.constant_perturbation_roots_in_unitDisk","Erdos249257.ExternalVerification.exists_two_roots_dist_sum_le_two_mul_geomMean","Erdos249257.ExternalVerification.threeHalves_no_coordinatewiseCorridor","Erdos249257.ExternalVerification.rationalBaseClearedTailQ_succ","Erdos249257.ExternalVerification.rectangular_hp_threshold_eq_classical_iff"]},"candidate_value_dispositions":{"disposition_vocabulary":["selected","represented","deferred","subordinate","rejected","long_tail"],"eligible_disposition_vocabulary":["selected","represented","subordinate","rejected","long_tail"],"authority_rule":"The following groups partition exactly the committed Comparator roster. Source-landscape records preserve exact source identities, value judgments, and admission history; a source-landscape candidate is Comparator evidence only when its record names a committed source-faithful ExternalVerification row.","eligible_groups":[{"disposition":"selected","declarations":["Erdos249257.ExternalVerification.divisibilityWeightedClaim"],"reason":"The exact finite-prime weighted criterion gives the broadest unconditional direct #257 support-class result, including a binary-mass all-base hereditary clause; its finite weighted-mass premise and the open universal boundary remain explicit."},{"disposition":"represented","declarations":["Erdos249257.ExternalVerification.irrational_erdosSum_full_support","Erdos249257.ExternalVerification.irrational_erdosSupportSeries_pairwise_coprime","Erdos249257.ExternalVerification.irrational_ratWeightSeries_eventuallyPeriodic","Erdos249257.ExternalVerification.irrational_erdosSupportSeries_of_orthogonalPetalBouquet"],"reason":"The classical full-support, pairwise-coprime, periodic-weight, and conditional sunflower rows remain distinct #257 evidence below the broader finite-prime weighted lead."},{"disposition":"represented","declarations":["Erdos249257.ExternalVerification.irrational_totientSeries_of_actualLcmOrbitSeparationSupply"],"reason":"The committed wrapper represents the one coherent actual_lcm_orbit_separation family. Its source-level sign, true-survivor, and top-edge mechanisms remain documented in the source-landscape record, while the cofinal separation supply and top-edge exclusion remain open; it stays below the unconditional finite-prime weighted selection and is not duplicated."},{"disposition":"represented","declarations":["Erdos249257.ExternalVerification.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap"],"reason":"The committed wrapper represents one coherent strict_prime_tail_orbit_gap family. Its 11/100 phase-density to 89/100 block saving to 9/10 finite-gap chain and adaptive natural-prime escape are distinct producer-side mechanisms; both global producer predicates remain unproved, so it stays below the actual-LCM and first-harmonic routes but above certificate-only normal forms."},{"disposition":"represented","declarations":["Erdos249257.ExternalVerification.irrational_totient_series_of_pivotResidualDecorrelation"],"reason":"The committed wrapper represents the one coherent first_harmonic_pivot_decomposition family. Its source-level fibre, counterexample, four-term decomposition, and 9/10 budget remain documented in the source-landscape record, while the cofinal decorrelation producer remains unproved; it stays below full support and is not split into pseudo-families."},{"disposition":"represented","declarations":["Erdos249257.ExternalVerification.scaleFullTarget_miss_of_projected_separation"],"reason":"The committed wrapper represents one coherent actual_foreign_residue_projection family: its finite partition, geometric majorant, and generic noncancellation consumer are endpoint-facing but remain below the actual-LCM and first-harmonic routes because the controlled-projection bridge and unbounded separated supply are open."},{"disposition":"represented","declarations":["Erdos249257.ExternalVerification.irrational_initial_iff_all_positive_tailShifts_nonintegral","Erdos249257.ExternalVerification.rectangular_hp_threshold_eq_classical_iff","Erdos249257.ExternalVerification.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank","Erdos249257.ExternalVerification.fixedPrecisionTropicalNoGo","Erdos249257.ExternalVerification.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry","Erdos249257.ExternalVerification.boundedNegativePart_eventually_zero","Erdos249257.ExternalVerification.sylvesterNext_eventually_of_summable_negativeRelativeMass","Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips","Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse","Erdos249257.ExternalVerification.coefficientOnlyNoGo","Erdos249257.ExternalVerification.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch","Erdos249257.ExternalVerification.carry_eq_residueDigit_add_coboundary"],"reason":"These newly registered representatives preserve twelve distinct source-current families: the retained synthetic #249 fixed-precision no-go, the #251 coefficient-only negative method and actual-prime-gap small-mismatch criterion, the two distinct signed #243 recovery criteria, abstract #251 dyadic-tail classification, sharp model-specific #1049 rectangular Hermite–Padé threshold equality, #249 orbit-level carry anti-compression, the #257 Boolean–Möbius support reconstruction and exact-row dynamics routes, the #257 half-membership seam classification, and the #269 weighted-phase residue observer. Each is represented with exact hypotheses and open producer/no-go boundaries, not promoted to a universal endpoint proof."},{"disposition":"represented","declarations":["Erdos249257.ExternalVerification.exists_clean_binaryCyclotomicAnchor","Erdos249257.ExternalVerification.exists_two_roots_dist_sum_le_two_mul_geomMean"],"reason":"These two unconditional mechanisms materially sharpen their programmes without reaching an endpoint: the #249 theorem supplies clean binary-cyclotomic anchors but not the remaining carry-kill producer, while the #1041 theorem reaches the sharp two-root metric scale at a critical point but does not keep the broken line inside the open lemniscate."},{"disposition":"subordinate","declarations":["Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_nonunit_carries","Erdos249257.ExternalVerification.irrational_factorialGapSeries_of_cofinal_global_complementaryTail","Erdos249257.ExternalVerification.irrational_tsum_primeDyadicTerm_iff_primeGap","Erdos249257.ExternalVerification.not_finiteDimensional_span_fullTotientKernel","Erdos249257.ExternalVerification.no_positive_reducedCarry_of_cofinalLocalWindowEscape"],"reason":"Each is mathematically useful and independently retained, but the factorial and prime-gap rows are endpoint reductions without their missing producers, the #249 rank result does not itself prove irrationality or cofinal certificate supply, and the #269 carry-escape consumer retains an open cofinal-window producer. The actual-series reduction is given; the source-specific cofinal escape remains unproved. Comparator admission strengthens traceability, not endpoint status."},{"disposition":"rejected","declarations":["Erdos249257.ExternalVerification.threeHalves_no_coordinatewiseCorridor"],"reason":"This is a sharp no-go for one proof architecture, not a direct endpoint or a complete candidate mechanism; it remains visible as a boundary rather than being promoted as a solution."},{"disposition":"long_tail","declarations":["Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses","Erdos249257.ExternalVerification.no_cofinallyBoundedNegative_of_normalizedVanishes","Erdos249257.ExternalVerification.no_eventuallyPeriodicNegative_orbit","Erdos249257.ExternalVerification.finrank_totientKernelThroughLevelFamily_eq","Erdos249257.ExternalVerification.exists_totientDyadicSectionBasis","Erdos249257.ExternalVerification.finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent","Erdos249257.ExternalVerification.tsum_pos_coprime_inv_mersenne_eq_one","Erdos249257.ExternalVerification.tsum_totient_div_mersenne_sq_eq_gcd_moment_series","Erdos249257.ExternalVerification.runHeight_defect_fib_sum_lower","Erdos249257.ExternalVerification.cylinderMass_split","Erdos249257.ExternalVerification.lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient","Erdos249257.ExternalVerification.exists_primeGap0_gt","Erdos249257.ExternalVerification.volume_mersenneAchievementSet","Erdos249257.ExternalVerification.finite_period_noncollapse_rat_den_interface","Erdos249257.ExternalVerification.volume_supportedMersenneAchievementSet_dichotomy","Erdos249257.ExternalVerification.exists_shifted_odd_tail_nat_states_of_support_fraction","Erdos249257.ExternalVerification.supportCoeff_mul_eq_add_defect","Erdos249257.ExternalVerification.seven_le_of_intBand_odd","Erdos249257.ExternalVerification.kernel_235_minor_eq_neg_one_fifteen","Erdos249257.ExternalVerification.smoothPrefixLcm_eq_threePrimeHeight","Erdos249257.ExternalVerification.exists_small_translation_separating_arguments","Erdos249257.ExternalVerification.constant_perturbation_roots_in_unitDisk","Erdos249257.ExternalVerification.rationalBaseClearedTailQ_succ","Erdos249257.ExternalVerification.finitePrimeWeighted_fixedBase_hereditary"],"reason":"These exact identities, finite instances, structural consumers, auxiliary facts, and no-go boundaries remain in the complete current roster, but they are not competitive with the selected endpoint under the current consequence, mechanism, and evidence ranking."}],"source_landscape_candidates":[{"candidate_id":"erdos1041_newton_value_decay","family_id":"newton_value_decay","disposition":"represented","canonical_claim_commit":"06198c0ab3969c625e1022eaebfb46208ff0b5e4","comparator_eligibility":"source_landed_but_not_comparator_configured","queue_role":"source_landscape_review_not_comparator_evidence","source_declaration":"ErdosProblems.Erdos1041.newtonFlow_value_hasDerivAt; ErdosProblems.Erdos1041.newtonFlow_scaledValue_hasDerivAt_zero","source_file":"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","source_anchor":"50; 64","statement":"Along a curve tangent to the noncritical complex Newton field -f/f', the polynomial value has derivative equal to its own negative, and exp(t) times that value has derivative zero.","exact_hypotheses":["At one complex time t, f has complex derivative f'(z(t)) at z(t).","The curve z has complex derivative newtonFlowVector (f(z(t))) (f'(z(t))) at t.","The derivative value f'(z(t)) is nonzero, so the Newton-vector cancellation is valid."],"conclusion":"The source proves the pointwise value equation d(f∘z)/dt = -f(z(t)) and the zero derivative of exp(t) * f(z(t)); these are the local differential form of exponential radial decay, not a global flow or short-curve theorem.","hard_mechanism":"The chain rule multiplies f'(z(t)) by -f(z(t))/f'(z(t)); noncriticality cancels the denominator exactly. The exponential derivative and product rule then cancel the remaining value term, yielding the first integral that underlies fixed-argument ray geometry.","attribution":"These source-current Lean declarations are in NewtonFlowRaySeparation.lean:50,64 and are canonically bound to newton_value_decay by Claims commit 06198c0a. No novelty, priority, significance, or external-review claim is made.","limitations":["The statements are local HasDerivAt facts over a complex parameter and do not establish a real-time solution on a prescribed interval or integrate the first integral globally.","Ray separation is a geometric consumer of this invariant; translation avoidance and root retention are separate perturbative supports rather than proofs of trajectory existence or component stability.","No Morse/Reeb topology or decomposition, saddle and annular-cut gluing, collar-slack transfer, curve of length less than two, or Erdős #1041 endpoint follows."],"source_landscape_rank_relative_to":"Above ray_separation, translation_avoidance, root_retention, and finite search as the common load-bearing local invariant, but below direct completed endpoint families because global flow existence, topology, perturbation stability, and strict metric fan-in remain open.","ranked_below":["known_irrational_supports","pairwise_coprime_support"],"ranked_above":["ray_separation","translation_avoidance","root_retention","lemniscate_finite_search"],"prerequisite_relations":["noncriticality f'(z(t)) != 0","curve tangent to newtonFlowVector"],"support_relations":["derivative_mul_newtonFlowVector"],"conditional_peer_relations":["ray_separation","translation_avoidance","root_retention"],"contrary_evidence":"The invalid local saddle-block route and the unresolved global-flow/component-stability/length obligations show that exact value decay alone cannot be promoted to a connecting-curve theorem.","reason":"Represented as the strongest underweighted #1041 mechanism: it is the exact local invariant consumed by every ray-based route, while the separate finite perturbation theorems and open global topology remain visible rather than being flattened into a false endpoint.","reversal_evidence":"A source-faithful Comparator transport would strengthen executable review evidence; a proved relative global Newton-flow theorem plus component-stable topological and strict length fan-in would justify endpoint-facing promotion, while another local derivative identity or finite search would not."},{"candidate_id":"erdos243_bounded_rise_coprimality","family_id":"bounded_rise_coprimality","disposition":"represented","canonical_claim_commit":"06198c0ab3969c625e1022eaebfb46208ff0b5e4","comparator_eligibility":"source_landed_but_not_comparator_configured","queue_role":"source_landscape_review_not_comparator_evidence","source_declaration":"ErdosProblems.Erdos243.reducedTail_pairwiseCoprime; ErdosProblems.Erdos243.no_eventuallyBoundedRise_reducedTail","source_file":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","source_anchor":"1010; 1060","statement":"For an eventually reduced exact reciprocal-tail recurrence, distinct multipliers are pairwise coprime, and a numerator escaping to infinity cannot have one uniform positive upper bound on all eventual rises.","exact_hypotheses":["The reduced-tail theorem assumes natural-valued multiplier, numerator, and denominator sequences with eventual multiplier greater than one and pointwise coprimality of each reduced numerator-denominator pair.","The exact recurrences are u(n+1) + v(n) = a(n) * u(n) and v(n+1) = a(n) * v(n) from one eventual index onward.","The no-go additionally assumes a fixed B > 0 with u(n+1) <= u(n) + B eventually and Tendsto u atTop atTop."],"conclusion":"Those hypotheses are inconsistent: reduced exact tails with fresh pairwise-coprime multipliers cannot both escape to infinity and retain a uniform eventual rise bound.","hard_mechanism":"Every earlier multiplier divides each later denominator, while reduced coprimality keeps it coprime to the later numerator. This creates pairwise-coprime fresh moduli and whole-modulus avoidance; bounded rises cannot indefinitely cross the resulting forbidden windows as the numerator tends to infinity.","attribution":"These source-current Lean declarations are in ReciprocalTailRigidity.lean:1010,1060 and are canonically bound to bounded_rise_coprimality by Claims commit 06198c0a. No novelty, priority, significance, or external-review claim is made.","limitations":["The theorem does not establish a uniform eventual rise bound for the original reciprocal-tail numerator sequence.","It assumes exact reduced C/D dynamics, eventual multipliers greater than one, pointwise reduced coprimality, and numerator escape to infinity rather than deriving them for the original problem.","It is a natural-friction no-go, not a centered-state recovery theorem, reciprocal-tail irrationality result, or Erdős #243 endpoint."],"source_landscape_rank_relative_to":"Below centered_state_dynamics and negative_mass_recovery because those families recover eventual Sylvester structure from explicit signed or analytic hypotheses, but above finite searches and routine recurrence identities because it isolates a reusable arithmetic obstruction to bounded-rise strategies.","ranked_below":["centered_state_dynamics","negative_mass_recovery"],"ranked_above":["reciprocal_tail_finite_search"],"reason":"Represented as a distinct #243 natural-friction family: it explains exactly why a bounded-rise proof would be decisive and why pairwise-coprime fresh moduli matter, without conflating that conditional no-go with the stronger signed and summable-negative-mass recovery theorems.","reversal_evidence":"A source-faithful Comparator transport would strengthen executable review evidence; proving the bounded-rise hypothesis for the original reciprocal-tail orbit would promote the family sharply, while another finite recurrence instance would not."},{"candidate_id":"erdos1041_ray_separation","family_id":"ray_separation","disposition":"represented","canonical_claim_commit":"06198c0ab3969c625e1022eaebfb46208ff0b5e4","comparator_eligibility":"source_landed_but_not_comparator_configured","queue_role":"source_landscape_review_not_comparator_evidence","source_declaration":"ErdosProblems.Erdos1041.samePositiveRay_of_real_exp_decay; ErdosProblems.Erdos1041.no_newtonConnection_of_not_samePositiveRay","source_file":"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean","source_anchor":"306; 315","statement":"An exponentially decaying Newton-value endpoint lies on the same positive ray as its start value, so critical values on distinct positive rays cannot be joined by a finite Newton-flow connection.","exact_hypotheses":["The endpoint relation is exactly endValue = exp(-time) * startValue for a real time parameter.","Real.exp(-time) is strictly positive, which is the scalar witness for SamePositiveRay.","The no-connection theorem assumes the two values are not on the same positive ray and the exact exponential endpoint relation."],"conclusion":"Distinct positive critical-value rays exclude the stated finite Newton connection; this is a local route obstruction, not the global Erdős #1041 theorem.","hard_mechanism":"The Newton-value first integral preserves argument because its endpoint multiplier is positive real. Turning that invariant around makes ray separation an exact obstruction rather than a metric estimate.","attribution":"These source-current Lean declarations are in NewtonFlowRaySeparation.lean:325,334 and are canonically bound to ray_separation by Claims commit 06198c0a. No novelty, priority, significance, or external-review claim is made.","limitations":["The result does not construct a global assignment of pairwise distinct critical-value rays or show that a chosen perturbation preserves all required geometry.","The Comparator rows for translation avoidance and root retention are supporting local inputs, not transports of this exact source-only ray theorem.","No compact ray-cut gluing, global Newton-flow connection theorem, lemniscate-length bound, or Erdős #1041 endpoint follows."],"source_landscape_rank_relative_to":"Below the deeper newton_value_decay mechanism and endpoint-facing families because the positive-ray implication is elementary and local, but above routine finite-search evidence because it gives the exact geometric obstruction consumed by translation avoidance.","ranked_below":["newton_value_decay"],"ranked_above":["lemniscate_finite_search"],"reason":"Represented as the strongest under-served #1041 cross-surface join: it explains why argument separation, not modulus decay, is the relevant local obstruction and how the existing translation-avoidance target is used, while preserving the missing global gluing boundary.","reversal_evidence":"A source-faithful Comparator transport would strengthen executable review evidence; a global critical-value ray assignment plus metric/topological gluing would justify promotion, while another finite root search or local perturbation instance would not."},{"candidate_id":"erdos68_channel_projection_rigidity","family_id":"factorial_channel_and_projection_rigidity","disposition":"represented","canonical_claim_commit":"06198c0ab3969c625e1022eaebfb46208ff0b5e4","comparator_eligibility":"source_landed_but_not_comparator_configured","queue_role":"source_landscape_review_not_comparator_evidence","source_declaration":"Erdos68.factorialMoment_eq_factorial_pow_mul_channelNumerator_band; Erdos68.exists_index_ge_two_mul_of_factorialMoment_ne_zero_of_channel_eq_zero; Erdos68.channelNumerator_mod_factorialMoment; Erdos68.primeTranslator_channelResidual_eq_one; ErdosProblems.Erdos68.projection_disagreement_excludes_bounded_endpoint","source_file":"ErdosProblems/Erdos68/ChannelBreakpointRigidity.lean; ErdosProblems/Erdos68/ChannelIntegralCongruence.lean; ErdosProblems/Erdos68/PrimeUnitTranslator.lean; ErdosProblems/Erdos68/EndpointWeightedPrivateSupport.lean","source_anchor":"ChannelBreakpointRigidity.lean:71,130; ChannelIntegralCongruence.lean:76; PrimeUnitTranslator.lean:827; EndpointWeightedPrivateSupport.lean:2401","statement":"Finite quotient-band channel factorisation, channel congruence, an exact prime-channel unit translator, and two-projection endpoint disagreement form one coherent #68 rigidity mechanism below the exact carry endpoint.","exact_hypotheses":["For a finite coefficient family whose indices all lie in the quotient band k*d <= index < (k+1)*d, the factorial moment factors as (d!)^k times the d-channel numerator; in the first band, a zero channel with nonzero moment forces some index at least 2d.","For d >= 2, d! - 1 divides the difference between the factorial moment and the d-channel numerator.","For D >= 2 and prime p > D, the two-term prime translator has full infinite channel residual exactly one.","The projection contradiction assumes Z congruent to T modulo R, Q1 and Q2 dividing R, Z <= B < Q1,Q2, and unequal projected residues modulo Q1 and Q2."],"conclusion":"The source theorems isolate exact finite channel and projection obstructions, but they neither produce a cofinal non-unit carry nor prove irrationality of the factorial-gap series.","hard_mechanism":"Band factorisation converts channel cancellation into factorial-moment rigidity; the d! - 1 congruence and exact prime unit translator identify a movable correction direction, while two unequal large divisor projections cannot both equal one bounded endpoint numerator.","attribution":"These are source-current Lean declarations in the dedicated Erdős #68 channel and endpoint-projection modules, canonically bound to factorial_channel_and_projection_rigidity by Claims commit 06198c0a. No novelty, priority, significance, or external-review claim is made.","limitations":["The mechanism is finite and structural: it supplies no cofinal non-unit carry or cofinal strict-successor divisibility miss.","The prime translator is an exact correction direction, not a proof that the required translated residual is nonzero cofinally.","Projection disagreement requires moving sufficiently large divisor projections; no theorem supplies them on a cofinal family, and Erdős #68 remains open."],"source_landscape_rank_relative_to":"Below factorial_carry_characterisation because the latter is an exact endpoint equivalence, but above isolated finite certificates and routine channel identities because it unifies quotient-band rigidity, modular transport, an exact prime correction, and endpoint projection exclusion.","ranked_below":["factorial_carry_characterisation"],"ranked_above":["factorial_finite_certificates"],"reason":"Represented as the strongest under-served #68 mechanism after the all-eight-paper audit: its four linked source modules explain how finite channel cancellation and endpoint projections could feed the exact carry frontier without pretending that the missing cofinal producer has been supplied.","reversal_evidence":"A source-faithful Comparator representative would strengthen executable review evidence; a proof of cofinally nonzero translated residuals, moving projection disagreement, or cofinal non-unit carries would justify promotion, while another isolated finite channel computation would not."},{"candidate_id":"actual_lcm_orbit_separation","family_id":"actual_lcm_orbit_separation","disposition":"represented","prior_disposition":"deferred","admission_commit":"0450c65e717b3c57b4ed67c99483f2578339eaaa","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","comparator_declaration":"Erdos249257.ExternalVerification.irrational_totientSeries_of_actualLcmOrbitSeparationSupply","source_declaration":"Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_actualLcmOrbitSeparationSupply","source_file":"Erdos249257/TotientActualLcmOrbitSeparation.lean","source_anchor":"305","claim_id":"actual_lcm_orbit_nonintegrality_frontier","evidence_ceiling":"Quantitative separation is sufficient, not equivalent, to actual-LCM non-integrality; the cofinal supply remains unproved.","statement":"A cofinal quantitative separation supply for the actual power-two LCM tail orbit implies irrationality of the binary totient series.","exact_hypotheses":["For every a₀, there exist a and q with max 2 a₀ <= a and oddGuardedCanonicalAdjacentSuffixDepth (2^a) = 2q+1.","For every integer z, 1/32 + actualLcmRawErrorRadius a q <= |actualLcmTailOrbit a - z|."],"conclusion":"Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","hard_mechanism":"The route turns a raw adjacent-suffix approximation and an explicit error radius into a uniform signed-margin separation from every integer, then transports that margin to the irrationality endpoint. The cofinal separation supply is the natural hard producer and is not supplied by the implication theorem.","attribution":"This is a source-specific Lean theorem in the current committed tree; no novelty, priority, or independent Comparator evidence is claimed.","limitations":["The cofinal PowerTwoActualLcmOrbitSeparationSupply is a premise, not a proved supply in this declaration.","The committed ExternalVerification wrapper establishes transport identity and conclusion shape, not the missing cofinal supply.","It does not close the binary totient irrationality endpoint without the missing supply and does not alter the open universal boundaries."],"reason":"Now represented rather than selected: the committed row preserves the source supply clauses and irrationality conclusion, so it earns visibility as a serious conditional #249 route. Full support remains selected because it has a completed unconditional endpoint and lower hidden-premise risk.","reversal_evidence":"A proved cofinal separation supply, or a later source-current result with equal evidence certainty and a stronger completed endpoint, would justify a fresh ranking against full support; the current implication alone does not."},{"candidate_id":"actual_foreign_residue_projection","family_id":"actual_foreign_residue_projection","disposition":"represented","prior_disposition":"deferred","canonical_family_binding":"actual_foreign_residue_projection","canonical_claim_commit":"404592da8e164b24faaeae021642fa2e131319a8","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","source_transport_commit":"8a08768f3758f11e9640535523232b0300316a4b","source_transport_files":["ExternalVerification/Challenge.lean","ExternalVerification/Solution.lean","ExternalVerification/Statements.lean"],"transport_declarations":["Erdos249257.ExternalVerification.abs_foreignTailWindow_le_foreignComplementBound","Erdos249257.ExternalVerification.finiteResidueDiagonal_eq_projectedForeign_add_divisor","Erdos249257.ExternalVerification.scaleFullTarget_miss_of_projected_separation"],"comparator_declaration":"Erdos249257.ExternalVerification.scaleFullTarget_miss_of_projected_separation","supporting_comparator_declarations":[],"transport_admission_boundary":"The source-faithful actual-foreign-residue transport is committed at 8a08768 and its load-bearing representative is registered in the source-current HEAD:verification/comparator.json screen; the finite split and foreign-window bound remain subordinate mechanisms, and canonical claims bind the one family at 404592da.","source_declaration":"Erdos249257.ActualForeignResidueProjection.scaleFullTarget_miss_of_projected_separation; Erdos249257.ActualForeignResidueProjection.finiteResidueDiagonal_eq_projectedForeign_add_divisor; Erdos249257.ActualForeignResidueProjection.abs_foreignTailWindow_le_foreignComplementBound","source_file":"Erdos249257/ActualForeignResidueProjection.lean","source_anchor":"276; 308; 414","statement":"The finite Möbius residue diagonal splits into projected foreign and divisor channels, the omitted finite foreign tail has a closed geometric majorant, and a controlled projection separated from every integer forces the full target to be missed.","exact_hypotheses":["The finite residue diagonal and foreign/divisor channels use the exact natural-number H,D and projection definitions of ActualForeignResidueProjection.lean.","The finite foreign-tail window is bounded by the stated closed geometric majorant; this does not imply an infinite-tail or actual-LCM identification.","The load-bearing consumer assumes ControlledForeignProjection and an unbounded supply of projections separated from every integer; neither bridge is supplied by the transport."],"conclusion":"Under the controlled-projection and separated-supply hypotheses, the full target is missed; this is a conditional #249 reduction, not an irrationality theorem.","hard_mechanism":"The exact finite Möbius partition isolates a foreign channel, the geometric tail bound controls the omitted window, and the generic complement-noncancellation consumer converts uniform projection separation into a full-target miss.","attribution":"This is the source-current #249 actual_foreign_residue_projection family from ActualForeignResidueProjection.lean:276, 308, 414, canonically bound at 404592da and transported by the committed ExternalVerification batch at 8a08768; novelty, priority, significance, and external review remain unassessed.","limitations":["ControlledForeignProjection is an unproved analytic bridge; the transport does not identify the controlled projection with the actual target orbit.","No unbounded separated-projection supply is proved, so the conditional consumer does not close the #249 endpoint.","The finite partition and geometric majorant are subordinate mechanisms inside one family; no novelty, priority, significance, external-review, or #249 irrationality claim is made."],"source_landscape_rank_relative_to":"Below the represented actual_lcm_orbit_separation and first_harmonic_pivot_decomposition routes because its control bridge and separated supply are both open, but above routine finite identities because it exposes an endpoint-facing foreign-channel reduction and its natural noncancellation friction.","reason":"Represented after the source-current Claims and Comparator fan-in: the finite foreign/divisor split and geometric majorant provide a distinct endpoint-facing conditional reduction, while the missing controlled-projection bridge and unbounded separated supply prevent promotion above the stronger actual-LCM and first-harmonic routes or to an endpoint result.","reversal_evidence":"A source-faithful proof of the controlled projection and an unbounded separated-projection supply would justify fresh ranking; another finite partition, window bound, or generic noncancellation restatement would not."},{"candidate_id":"first_harmonic_pivot","family_id":"first_harmonic_pivot_decomposition","disposition":"represented","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","source_declaration":"Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_pivotResidualDecorrelation","source_file":"Erdos249257/FirstHarmonicPivot.lean","source_anchor":"394; 514-552; 569-588","source_transport_commit":"bc0fee48e14719391a77e90ea98205400b98993b","source_transport_files":["ExternalVerification/Challenge.lean","ExternalVerification/Solution.lean","ExternalVerification/Statements.lean"],"transport_declarations":["Erdos249257.ExternalVerification.image_pivotSupplierPrimes_eq_pivotFiber","Erdos249257.ExternalVerification.supplierPrime_not_globally_isolated_counterexample","Erdos249257.ExternalVerification.windowFirstExp_sum_eq_pivot_decomposition","Erdos249257.ExternalVerification.first_harmonic_gap_of_pivotBudgetAt","Erdos249257.ExternalVerification.irrational_totient_series_of_pivotResidualDecorrelation"],"comparator_declaration":"Erdos249257.ExternalVerification.irrational_totient_series_of_pivotResidualDecorrelation","supporting_comparator_declarations":[],"transport_admission_boundary":"The source-faithful endpoint representative is admitted in the source-current committed HEAD:verification/comparator.json screen and bound to the canonical review matrix family first_harmonic_pivot_decomposition; the fibre, counterexample, four-term decomposition, and 9/10 budget remain source-level mechanism evidence, while the cofinal decorrelation producer remains open.","statement":"A cofinal first-harmonic pivot residual-decorrelation supply implies irrationality of the binary totient series.","exact_hypotheses":["For every h > 0 there exist s > 0 and η with 0 < η < 1 such that for every X₀ there exist X,L with max X₀ 1 ≤ X, h ≤ L-s, 16*(2*X+h+L+2) ≤ 2^L, and PivotBudgetAt X L s.","PivotBudgetAt requires pivotCenteredCorrelation.re ≤ (14/25)*X, ‖pivotFiberMeanContribution‖ ≤ (1/100)*X, ‖pivotBadContribution‖ ≤ (1/100)*X, and ‖pivotNonSupplierContribution‖ ≤ (8/25)*X."],"conclusion":"Irrational (∑' n : ℕ, (Nat.totient n : ℝ) / 2 ^ n)","hard_mechanism":"The source decomposes the first harmonic into supplier fibres, a pivot correlation, bad and non-supplier contributions, and a controlled remainder; the four explicit budgets sum to the strict first-harmonic gap bound 9/10·X.","attribution":"This is a source-current Lean theorem in FirstHarmonicPivot.lean; the source claim records the finite decomposition and a counterexample to global isolation, with no novelty or priority claim.","limitations":["The DTWPivotResidualDecorrelation premise is a cofinal producer hypothesis, not a supplied theorem in this declaration.","The committed ExternalVerification transport preserves the decomposition, budget, and conditional endpoint and is registered in the committed Comparator roster, but it remains conditional evidence rather than an irrationality proof for the open endpoint.","The finite counterexample to global isolation remains a natural friction point and prevents silently treating the pivot as a universal local argument."],"reason":"Represented and promoted into the ranked frontier after canonical review-family and Comparator fan-in: this is a genuinely deep conditional #249 endpoint route with a visible four-term harmonic budget and high proof-digestion value. It remains below the unconditional endpoint results and the actual-LCM route because the cofinal residual-decorrelation producer is unproved.","reversal_evidence":"Prove the cofinal decorrelation supply to justify endpoint promotion; a routine restatement, finite pivot check, or roster change would not alter its mathematical position."},{"candidate_id":"actual_lcm_positive_corridor_top_edge","family_id":"actual_lcm_orbit_separation","disposition":"subordinate","comparator_eligibility":"committed_source_transport_pending_comparator_registration","queue_role":"source_landscape_review_not_comparator_evidence","source_declaration":"Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcm_integral_forces_topEdgeResidue","source_file":"Erdos249257/TotientActualLcmOrbitSign.lean","source_anchor":"31-278","source_transport_commit":"bc0fee48e14719391a77e90ea98205400b98993b","source_transport_files":["ExternalVerification/Challenge.lean","ExternalVerification/Solution.lean","ExternalVerification/Statements.lean"],"transport_declarations":["Erdos249257.ExternalVerification.actualLcmTailDiff_shift_pos","Erdos249257.ExternalVerification.actualLcm_trueEndpointSurvivor_neg","Erdos249257.ExternalVerification.actualLcm_integral_forces_topEdgeResidue"],"transport_admission_boundary":"These declarations remain one coherent actual_lcm_orbit_separation family. The source-current committed HEAD:verification/comparator.json screen admits the family representative but not these separate support declarations; the canonical review matrix projection, central-band exclusion, and cofinal supply remain open.","statement":"The actual-LCM positive corridor makes the true translated tail difference positive; if the difference is integral and the dyadic modulus has room, its discrepancy is forced into the top-edge residue strip.","exact_hypotheses":["8 ≤ a and J + K + (a + 6) < 2*2^a.","There is d : ℤ representing the translated actual-LCM tail difference exactly.","The directed room inequality ((2*periodLcm(2^a)+J+K+2 : ℕ) : ℤ) < 2^K holds."],"conclusion":"The discrepancy residue is P-e and satisfies P-B < residue < P, where P=2^K, e is the positive carry orbit, and B is the directed endpoint-strip bound.","hard_mechanism":"A quarter-height lower bound on the first lookahead letter, amplified by 2^(a+5), dominates the LCM height; positive remaining letters and the directed tail bound yield the sign, then carry tracking identifies the exact top-edge residue.","attribution":"This is source-current Lean sign/carry analysis in TotientActualLcmOrbitSign.lean, linked by docs/claims.json to actual_lcm_positive_corridor_top_edge; no duplicate frontier family or novelty claim is made.","limitations":["This is a supporting boundary component of the single actual_lcm_orbit_separation family, not a duplicate family and not yet Comparator evidence.","The top-edge strip is not excluded by the sign theorem; a central-band contradiction and cofinal non-integrality supply remain unavailable.","The theorem does not itself prove irrationality or convert positive sign into a nonintegrality conclusion."],"reason":"Subordinate after value review: the unconditional sign and exact top-edge obstruction are consequential source-landscape support inside the single actual_lcm_orbit_separation family. The current Comparator registration admits only the family representative; canonical evidence projection, cofinal supply, and top-edge exclusion remain open, so no duplicate family is warranted.","reversal_evidence":"Project the exact declarations as part of actual_lcm_orbit_separation or prove an independent top-edge exclusion; a duplicate family or row would not elevate it."},{"candidate_id":"erdos249_carry_anti_compression","family_id":"totient_carry_anti_compression","disposition":"represented","prior_disposition":"deferred_pending_canonical_claims","canonical_family_binding":"totient_carry_anti_compression (canonical claim id: totient_carry_kernel_anti_compression)","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","source_declaration":"Erdos249257.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank; Erdos249257.finrank_canonicalCarryKernel_ge; Erdos249257.carrySectionsEventuallyPeriodicMod_of_shift_dvd; Erdos249257.totient_forcing_vanishes_mod_of_dvd_multiplier; Erdos249257.totient_carry_modEq_geometric_of_dvd_multiplier","source_file":"Erdos249257/TotientTailCarryPeriod.lean","source_anchor":"224","source_transport_commit":"4932144ada05d6c617f43a372ede18a56900df7b","source_transport_files":["ExternalVerification/Challenge.lean","ExternalVerification/Solution.lean","ExternalVerification/Statements.lean"],"transport_declarations":["Erdos249257.ExternalVerification.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank"],"comparator_declaration":"Erdos249257.ExternalVerification.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank","supporting_comparator_declarations":[],"transport_admission_boundary":"The source-faithful representative is committed in ExternalVerification at 4932144a, registered in the source-current committed verification/comparator.json screen, and bound to the canonical review matrix family totient_carry_anti_compression. Supporting carry-kernel, modular-periodicity, forcing-vanishing, and geometric-orbit rows are mechanisms and contrary evidence, not separate families.","statement":"Under the non-irrationality branch for the binary totient series, a positive multiplier v and tempered integral carry u exist with common eventual modular periodicity for every dyadic section, while the canonical carry-kernel rank is at least 2^e - 1 at every level.","exact_hypotheses":["The representative assumes ¬ Irrational (binaryCoeffSeries Nat.totient), with the exact positive multiplier, tempered integer carry, and dyadic-section definitions from TotientTailCarryPeriod.lean.","The eventual modular periodicity is common modulo the rationality-supplied multiplier v for every dyadic section, and the rank lower bound is the finite-level canonical-carry-kernel bound for each e.","The source theorem's natural-friction evidence retains the exact divisor-of-v setting in which totient forcing can vanish modulo v and the carry orbit can become geometric."],"conclusion":"Hypothetical rationality forces uniform modular eventual periodicity of dyadic sections together with canonical carry-kernel rank at least 2^e - 1 for every level.","hard_mechanism":"The orbit-level conjunction couples common modular periodicity under a rationality-supplied multiplier with finite-level anti-compression rank growth; the forcing-vanishing and geometric-orbit lemmas show why periodicity alone does not yield a finite-Q-rank upper bound.","attribution":"This is the source-current Erdos #249 totient_carry_anti_compression family from TotientTailCarryPeriod.lean:224, canonically recorded under claim id totient_carry_kernel_anti_compression at f063b15e and transported in the three-file ExternalVerification batch at 4932144a; novelty, priority, significance, and external-review claims remain unassessed.","limitations":["This is a necessary consequence of non-irrationality only: it gives no contradiction, no bound on the rationality-supplied carry rank, and no #249 irrationality proof.","Forcing can vanish modulo a divisor of v and the resulting carry orbit can be geometric, so modular periodicity does not by itself imply a finite-Q-rank upper bound.","The canonical claims matrix binds this as one coherent family; the supporting carry-kernel rank and modular mechanisms must not be counted as separate families or confused with the coefficient-side totient_kernel family."],"source_landscape_rank_relative_to":"Provisional endpoint-facing review: below completed unconditional endpoint families and below actual-LCM/first-harmonic conditional producer routes, but above routine coefficient-side rank inventory because it isolates a distinct orbit-level modular-periodicity plus anti-compression conjunction.","ranking_status":"Canonical source-landscape family with committed Comparator evidence. It is represented for ranked review, is not selected, and does not replace the finite-prime weighted selection.","ranked_below":["known_irrational_supports","actual_lcm_orbit_separation","first_harmonic_pivot_decomposition"],"ranked_above":["totient_kernel_rank"],"reason":"Represented after canonical claims adjudication and committed source-faithful Comparator fan-in: the source-current transport is endpoint-facing and mathematically distinct as an orbit-level modular-periodicity plus anti-compression conjunction. It remains below the unconditional full-support and conditional producer routes because the contrary modulo-forcing boundary gives no endpoint or finite-rank contradiction.","reversal_evidence":"A source-faithful Comparator registration would strengthen evidence certainty; a proof of contradiction or a finite-Q-rank upper bound would change its value disposition, while a routine coefficient-side restatement would not."},{"candidate_id":"erdos249_fixed_precision_transport_no_go","family_id":"fixed_precision_transport_no_go","disposition":"represented","prior_disposition":"deferred","canonical_family_binding":"fixed_precision_transport_no_go","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","source_declaration":"Erdos249257.TotientTailPeriodKiller.fixedPrecisionTropicalNoGo; Erdos249257.TotientTailPeriodKiller.vu_step_has_centred_completion; Erdos249257.TotientTailPeriodKiller.vu_word_has_prefix_locked_completion","source_file":"Erdos249257/TropicalCurvatureCarry.lean","source_anchor":"69; 115; 137","source_transport_commit":"0c982671140b1a9a2e4f0e410ce807d0ad5e6363","canonical_claim_commit":"9d43aa4e4df76f08077bd10f9eb2f855f646028a","source_transport_files":["ExternalVerification/Challenge.lean","ExternalVerification/Solution.lean","ExternalVerification/Statements.lean"],"transport_declarations":["Erdos249257.ExternalVerification.fixedPrecisionTropicalNoGo","Erdos249257.ExternalVerification.vu_step_has_centred_completion","Erdos249257.ExternalVerification.vu_word_has_prefix_locked_completion"],"comparator_declaration":"Erdos249257.ExternalVerification.fixedPrecisionTropicalNoGo","supporting_comparator_declarations":[],"transport_admission_boundary":"The source-faithful fixed-precision representative is committed in ExternalVerification at 0c982671, registered in the source-current committed HEAD:verification/comparator.json screen, and bound to the canonical fixed_precision_transport_no_go review family at 9d43aa4e; the one-step and finite-word completion declarations are subordinate mechanism evidence, not extra families.","statement":"At every fixed positive precision, every finite word of odd valuation-unit symbols admits a compatible centred carry completion from any incoming integer state, so bounded local signatures cannot exclude all finite centred endpoints.","exact_hypotheses":["The construction ranges over every finite word of odd valuation-unit symbols at a fixed positive precision and permits arbitrary incoming integer/high quotient states.","The load-bearing representative preserves the exact centred-completion statement; vu_step_has_centred_completion and vu_word_has_prefix_locked_completion are subordinate one-step and finite-word mechanisms.","The conclusion is a reusable finite-word no-go for fixed-positive-precision local valuation/unit-word escape, not a statement about the actual totient-difference recurrence."],"conclusion":"Fixed-positive-precision local valuation/unit-word signatures cannot exclude every finite centred endpoint under the synthetic completion interface; this is a retained negative method result, not an Erdős #249 endpoint or irrationality theorem.","hard_mechanism":"Centred one-step completions compose across an arbitrary finite odd-unit word while locking the required prefix, exposing why a fixed local precision cannot rule out all compatible high-quotient states.","attribution":"This is the source-current #249 fixed_precision_transport_no_go family from TropicalCurvatureCarry.lean:69, 115, 137, transported by the committed three-file ExternalVerification batch at 0c982671 and canonically bound at 9d43aa4e; no novelty, priority, significance, or external-review claim is made.","limitations":["The construction is synthetic and allows unrestricted incoming/high quotient states; it is finite-word only and does not impose actual totient-difference arithmetic.","It supplies no growing-precision or global-correlation bridge, no #249 endpoint, and no irrationality result.","The one-step and finite-word completion declarations are subordinate mechanisms, not separate families; no novelty, priority, significance, or external-review claim is made."],"source_landscape_rank_relative_to":"Below endpoint-facing actual-LCM, first-harmonic, and carry anti-compression routes, but above routine structural shards because it unconditionally retires a fixed-positive-precision local escape architecture with an explicit centred-completion mechanism.","ranked_below":["actual_lcm_orbit_separation","first_harmonic_pivot_decomposition","totient_carry_anti_compression"],"ranked_above":["totient_kernel_rank","lcm_factor_ideal_shift_algebra_no_go"],"reason":"Represented as a retained high-value negative-method result after committed Comparator and Claims fan-in: it removes a concrete fixed-precision escape route for every finite odd-unit word, while its synthetic high-quotient and finite-word scope prevents endpoint promotion.","reversal_evidence":"A growing-precision/global-correlation bridge tied to actual totient differences, or a direct #249 endpoint consequence, would justify a fresh ranking; another finite-word completion or routine valuation restatement would not."},{"candidate_id":"erdos251_coefficient_only_no_go","family_id":"coefficient_only_no_go","disposition":"represented","prior_disposition":"deferred","canonical_family_binding":"coefficient_only_no_go","admission_commit":"c9d3660336d9ef4c9497549b26e29f3f953c138d","canonical_claim_commit":"4c4ac3f98352a169d276186c84860e41e7aac731","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","comparator_declaration":"Erdos249257.ExternalVerification.coefficientOnlyNoGo","source_declaration":"ErdosProblems.Erdos251.carryPartialSum_natCast_eq; ErdosProblems.Erdos251.carryCoeff_natCast_not_eventually_periodic; ErdosProblems.Erdos251.primeGap0_not_eventually_periodic; Erdos249257.ExternalVerification.CoefficientOnlyNoGo","source_file":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean; ExternalVerification/Statements.lean","source_anchor":"PrimeGapDyadicTail.lean:1023; 1157-1159; 1163-1168; Statements.lean:109-118","source_transport_commit":"c9d3660336d9ef4c9497549b26e29f3f953c138d","source_transport_files":["ExternalVerification/Challenge.lean","ExternalVerification/Solution.lean","ExternalVerification/Statements.lean"],"transport_declarations":["Erdos249257.ExternalVerification.coefficientOnlyNoGo"],"supporting_comparator_declarations":[],"transport_admission_boundary":"The source-faithful #251 representative is committed in ExternalVerification at c9d36603 and registered in the source-current committed verification/comparator.json screen; the canonical coefficient_only_no_go review family is bound at 4c4ac3f9. The transport records the finite carry identity and two non-eventual-periodicity facts, not a Lean infinite-limit theorem or an endpoint proof.","statement":"The finite rational carry partial-sum identity and non-eventual-periodicity of the synthetic carry coefficients, contrasted with non-eventual periodicity of actual consecutive-prime gaps, show why coefficient irregularity alone cannot force irrationality.","exact_hypotheses":["For every n, the rational carry emitted by K(j) = j has partial sum -n / 2^n.","For every positive period h, the synthetic carry coefficient stream is not eventually periodic.","For every positive period h, the actual consecutive-prime-gap stream is not eventually periodic; the Comparator representative packages this contrast as one coherent no-go interface."],"conclusion":"This is supporting negative/natural-friction evidence against coefficient-only irrationality arguments: a finite rational carry identity can coexist with non-eventual periodicity, while actual prime gaps are also non-eventually-periodic. It is not a #251 endpoint theorem.","hard_mechanism":"The finite rational carry partial-sum identity carryPartialSum_natCast_eq is paired with carryCoeff_natCast_not_eventually_periodic and primeGap0_not_eventually_periodic, separating coefficient growth/irregularity from any missing infinite-series or prime-gap-tail bridge.","attribution":"This is the source-current #251 coefficient_only_no_go family from PrimeGapDyadicTail.lean, transported by the committed ExternalVerification batch at c9d36603 and canonically qualified at 4c4ac3f; novelty, priority, significance, and external review remain unassessed.","limitations":["CoefficientOnlyNoGo contains no Lean theorem of the infinite rational-sum limit for the synthetic carry.","It does not identify the synthetic carry coefficients with actual prime gaps or establish a prime-gap tail bridge; stream-identification remains absent.","It proves no #251 endpoint or irrationality result; novelty, priority, significance, and external review remain unassessed."],"source_landscape_rank_relative_to":"Below direct endpoint-facing families but above routine finite searches: it isolates a reusable negative/natural-friction mechanism rather than a single finite computation, while preserving the missing analytic and stream-identification bridges.","reason":"Represented as one retained supporting negative/natural-friction family below direct endpoint families but above routine finite searches. It exposes the exact finite identity and nonperiodicity contrast that blocks a tempting coefficient-only inference without inflating the obstruction into a #251 result.","reversal_evidence":"A Lean infinite-limit and stream-identification bridge connecting the synthetic model to actual prime-gap coefficients, or a direct #251 endpoint, would change the disposition; another finite nonperiodicity calculation would not.","ranking_status":"Represented supporting negative method; it is not selected, does not replace direct endpoint families, and is not promoted by Comparator row count."},{"candidate_id":"erdos251_small_mismatch_criterion","family_id":"small_mismatch_criterion","disposition":"represented","prior_disposition":"deferred","canonical_family_binding":"small_mismatch_criterion","canonical_claim_commit":"96b2dc35cb1a73ad63ce4e01565aeb88cd477c12","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","source_transport_commit":"750e4d3218248bea5785b16e6f271bb3ab76ff7e","source_transport_files":["ExternalVerification/Challenge.lean","ExternalVerification/Solution.lean","ExternalVerification/Statements.lean"],"transport_declarations":["Erdos249257.ExternalVerification.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch"],"comparator_declaration":"Erdos249257.ExternalVerification.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch","supporting_comparator_declarations":[],"transport_admission_boundary":"The source-faithful #251 representative is committed in ExternalVerification at 750e4d32 and registered in the source-current committed verification/comparator.json screen; the canonical review matrix binds small_mismatch_criterion. The transport is a conditional reduction and does not supply the cofinal small-mismatch producer.","source_declaration":"ErdosProblems.Erdos251.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch; ErdosProblems.Erdos251.not_eventuallyIntegralTailShift_of_cofinal_small_mismatch; ErdosProblems.Erdos251.tailShift_not_both_integral_of_small_pair_of_digit_ne; ErdosProblems.Erdos251.primeGap0_not_eventually_periodic; ErdosProblems.Erdos251.rationalPrimeGapTail_has_positive_shift_not_eventually_small","source_file":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","source_anchor":"979; 1006; 1023; 1097; 1112","statement":"For the actual prime-gap dyadic tail, cofinally many adjacent pairs of strictly small h-shifts with unequal corresponding prime gaps force failure of eventual RatIntegral h-shifts.","exact_hypotheses":["The representative has T : ℕ → ℚ and assumes DyadicTailRecurrence (fun n => (primeGap0 n : ℤ)) T for the actual prime-gap digit stream.","For a fixed h : ℕ, the cofinal supply gives, for every N₀, an N ≥ N₀ whose adjacent h-shifts both lie strictly in (-1, 1) and whose corresponding prime gaps differ.","The representative preserves the exact ¬∃N₀, ∀N ≥ N₀, RatIntegral (tailShift T h N) conclusion; cofinality, strict smallness, and the digit mismatch are premises, not supplied by the transport."],"conclusion":"The adjacent-pair small-mismatch obstruction rules out eventual RatIntegral h-shifts under its exact cofinal actual-prime-gap hypothesis; it is not a #251 irrationality theorem.","hard_mechanism":"Two adjacent rational tail shifts in the open unit interval must both be zero when integral; the shift recurrence then forces equal corresponding digits, contradicting the unequal prime gaps supplied by the cofinal mismatch.","attribution":"This is the source-current #251 small_mismatch_criterion family from PrimeGapDyadicTail.lean:1112, transported by the committed ExternalVerification batch at 750e4d32; novelty, priority, significance, and external review remain unassessed.","limitations":["No theorem supplies cofinally many adjacent small mismatches or proves the required actual smallness for the prime gaps.","The rational-state obstruction rationalPrimeGapTail_has_positive_shift_not_eventually_small records a contrary positive-shift boundary and does not produce the needed cofinal mismatches.","The criterion proves no #251 irrationality, no infinite rational-sum limit, and no universal producer; novelty, priority, significance, and external review remain unassessed."],"source_landscape_rank_relative_to":"Conditional-endpoint leverage tier: below the ranked #251 prime-gap reformulation and the #243 negative-mass recovery because it fixes one h-shift and still assumes its cofinal smallness/mismatch supply, but above abstract classification and synthetic coefficient-only no-go because it is an actual-prime, endpoint-adjacent conditional obstruction with a precise missing producer.","ranked_below":["prime_gap_reformulation","negative_mass_recovery"],"ranked_above":["dyadic_tail_integrality_classification","coefficient_only_no_go"],"reason":"Retained as the first conditional-endpoint-leverage screen after the source-current survivor audit: this one coherent actual-prime-gap criterion exposes the adjacent-pair zero obstruction and its exact cofinal small-mismatch producer. It remains conditional on that producer and is not promoted to a #251 endpoint result.","reversal_evidence":"A proof of the exact cofinal small-mismatch/smallness producer or a direct #251 irrationality theorem would justify a fresh ranking; another finite gap computation or coefficient-only nonperiodicity row would not."},{"candidate_id":"erdos269_conditional_carry_escape","family_id":"conditional_carry_escape","disposition":"subordinate","prior_disposition":"deferred","canonical_family_binding":"conditional_carry_escape","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","source_transport_commit":"069245dfa77c55565611f04f9269707e0c31ce24","canonical_claim_commit":"68b4b2bfa2ebdfff7dd31a355b04e5e41df99a69","source_transport_files":["ExternalVerification/Challenge.lean","ExternalVerification/Solution.lean","ExternalVerification/Statements.lean"],"transport_declarations":["Erdos249257.ExternalVerification.no_positive_reducedCarry_of_cofinalLocalWindowEscape"],"comparator_declaration":"Erdos249257.ExternalVerification.no_positive_reducedCarry_of_cofinalLocalWindowEscape","supporting_comparator_declarations":[],"transport_admission_boundary":"The source-faithful reduced-carry consumer is committed in ExternalVerification at 069245df, registered in the source-current committed verification/comparator.json screen, and bound to the canonical conditional_carry_escape review family. Claims commit 68b4b2b adds the subordinate absorbed-factor bridge: after exact nonzero-factor cancellation it reaches that same committed reduced-carry consumer. Comparator admission strengthens traceability only: neither theorem supplies CofinalLocalWindowEscape. The actual-series reduction is given; the source-specific cofinal escape remains unproved.","source_declaration":"ErdosProblems.Erdos269.no_positive_reducedCarry_of_cofinalLocalWindowEscape; ErdosProblems.Erdos269.no_positive_absorbedCarry_of_cofinalLocalWindowEscape","source_file":"ErdosProblems/Erdos269/RestrictedFloorSum.lean","source_anchor":"629-650; 689","statement":"Cofinal local-window residue escape forces a contradiction with any positive reduced carry obeying the matching recurrence and short bound; the same consumer applies to an absorbed carry after exact cancellation of a nonzero common factor.","exact_hypotheses":["CofinalLocalWindowEscape b m shortBound supplies, for every positive B coprime to 30 and every starting point, a positive window base whose least-positive-residue escape exceeds shortBound.","B is positive and coprime to 30; d : ℕ → ℤ obeys d (n + 1) = b n * d n - B * m n, with d n > 0 and Int.natAbs (d n) ≤ shortBound B n.","For the subordinate absorbed-carry bridge, smoothFactor ≠ 0, c n = smoothFactor * d n, and c obeys the corresponding exact absorbed recurrence; cancellation reduces it to the same positive reduced-carry contradiction.","The displayed conclusion is a contradiction under these exact synthetic carry hypotheses; no rationality-to-carry or actual-series identification is assumed by the transport."],"conclusion":"The stated cofinal local-window escape is incompatible with a positive bounded reduced carry, and with its explicitly absorbed nonzero-factor form after cancellation; this is conditional reduction evidence, not an Erdős #269 endpoint.","hard_mechanism":"The exact integral window identity converts a positive reduced carry into a residue congruence, and the least-positive-residue lower bound contradicts the assumed short bound; the absorbed bridge makes the cancellation step explicit rather than treating common-factor normalization as automatic.","attribution":"This is the source-current #269 conditional_carry_escape family from RestrictedFloorSum.lean:629-650, 689, transported by the committed ExternalVerification batch at 069245df and canonically extended with the absorbed-factor bridge by Claims commit 68b4b2b; novelty, priority, significance, and external review remain unassessed.","limitations":["CofinalLocalWindowEscape is an unproved producer predicate; the source supplies no cofinal escaping windows.","The actual-series reduction is given; the source-specific cofinal escape remains unproved.","It proves no #269 irrationality or transcendence endpoint and makes no novelty, priority, significance, or external-review claim."],"source_landscape_rank_relative_to":"Below direct endpoint families and the stronger actual-LCM/first-harmonic conditional routes; retained as subordinate conditional no-go evidence because the source-specific cofinal escape remains unproved.","reason":"Subordinate source-landscape evidence after source transport and Comparator admission: the residue-escape consumer is a precise reusable contradiction, but its cofinal producer is missing, so it cannot compete with completed endpoints or be promoted as an Erdős #269 result. The actual-series reduction is given; the source-specific cofinal escape remains unproved.","reversal_evidence":"A proof of the cofinal local-window escape predicate together with an actual running-LCM carry bridge would justify fresh ranking; another finite window search or residue identity would not."},{"candidate_id":"strict_prime_tail_orbit_gap","family_id":"strict_prime_tail_orbit_gap","disposition":"represented","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","source_transport_commit":"e7a58bb335dcb8b56c871a44fbae4d796196f26c","canonical_claim_commit":"ddbe2870877d0b3abcb7d7e3e9b00d00eb91f021","comparator_declaration":"Erdos249257.ExternalVerification.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap","source_declaration":"ErdosProblems.Erdos249.TotientTailOrbitBlockGap; ErdosProblems.Erdos249.TotientTailOrbitNonpositiveBlockDensity; ErdosProblems.Erdos249.tailOrbitBlockGap_of_nonpositiveBlockDensity; ErdosProblems.Erdos249.irrational_totient_series_of_tailOrbitBlockGap; ErdosProblems.Erdos249.DTWNaturalPrimeTailOrbitStrictGap; ErdosProblems.Erdos249.naturalPivotPointEscape_of_naturalPrimeTailOrbitStrictGap; ErdosProblems.Erdos249.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap; ErdosProblems.Erdos249.tailOrbitFirstExp_succ; ErdosProblems.Erdos249.tailOrbitFirstExp_add; ErdosProblems.Erdos249.tailOrbitFirstExp_eq_initial_pow_two; ErdosProblems.Erdos249.naturalPrimeTailOrbitStrictGap_iff_initial_phase; ErdosProblems.Erdos249.tailOrbitFirstExp_eq_one_of_le; ErdosProblems.Erdos249.not_naturalPrimeTailOrbitStrictGap_of_phase_one; ErdosProblems.Erdos249.not_naturalPrimeTailOrbitStrictGap_of_dyadic_root; ErdosProblems.Erdos249.tailOrbitFirstExp_zero_eq_scaled_angle; ErdosProblems.Erdos249.tailOrbitFirstExp_eq_one_iff_tail_diff_mem_int; ErdosProblems.Erdos249.exists_tailOrbitFirstExp_zero_pow_two_eq_one_iff_dyadic; ErdosProblems.Erdos249.tailOrbitFirstExp_zero_pow_two_ne_one_upto_sixteen; ErdosProblems.Erdos249.cofinally_tailOrbitFirstExp_re_nonpos_of_not_dyadic; ErdosProblems.Erdos249.naturalPrimeTailOrbitStrictGap_of_cofinal_nonpositive_prime_shift","source_file":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","source_anchor":"25; 35; 45; 96; 157; 166-167; 191; 201; 208; 224; 238; 252; 349; 367; 389; 414; 445; 471","statement":"This one coherent conditional producer has two branches: a late-dyadic-block density of nonpositive tail-orbit phases yields a 9/10 finite first-harmonic gap and certified-kill supply, while cofinally many prime indices with Re(tailOrbitFirstExp h (p - h - 1)) < 9/10 yield natural pivot-fibre escape and the same irrationality endpoint.","exact_hypotheses":["TotientTailOrbitNonpositiveBlockDensity: for every h > 0 and X₀, some X ≥ max X₀ 1 has at least (11/100) * X nonpositive phases on Finset.Ico X (2 * X).","The block-density branch uses the unit norm of every tail-orbit phase and the complement bound to obtain TotientTailOrbitBlockGap, with block mean at most (89/100) * X.","The certified-kill branch chooses one truncation depth at endpoint 2X with error at most 1/100, converting the 89/100 infinite-orbit mean into a 9/10 finite first-harmonic gap.","DTWNaturalPrimeTailOrbitStrictGap: for every h > 0 and N₀, a prime p ≥ max (N₀ + h + 1) (h + 5) satisfies Re(tailOrbitFirstExp h (p - h - 1)) < 9/10; the adaptive truncation margin is then positive and exact natural-fibre membership is proved."],"conclusion":"The block-density antecedent implies irrationality through a uniform finite certificate supply; the cofinal-prime strict-gap antecedent implies irrationality through natural pivot-point escape. Both are conditional reductions only: neither source branch supplies its density, cofinal prime, or uniform margin antecedent for the actual totient orbit.","hard_mechanism":"Unit norm bounds the uncontrolled complement of the nonpositive phase set, so 11/100 density gives the 89/100 block mean; a single worst-case truncation budget at 2X gives the 9/10 finite gap and certified kill. In the prime branch, the exact positive margin 9/10 - Re(tailOrbitFirstExp h (p - h - 1)) feeds the natural-window depth theorem and the pivot-fibre membership calculation.","prerequisite_evidence":["TotientTailOrbitNonpositiveBlockDensity is the unproved density producer for the block branch.","DTWNaturalPrimeTailOrbitStrictGap is the unproved cofinal natural-prime producer for the adaptive branch."],"support_evidence":["tailOrbitFirstExp_succ gives the exact repeated-squaring recurrence tailOrbitFirstExp h (M + 1) = (tailOrbitFirstExp h M)^2; it makes the orbit mechanism reviewable but supplies no density or strict-gap occurrence.","Claims ddbe2870 binds scaled-angle and phase/integrality equivalences, finite dyadic-root exclusion, and non-dyadicity-to-cofinal-nonpositive-phase transport; these are support and classification, not a density or prime-alignment producer.","Unit norm plus the complement estimate converts 11/100 nonpositive phases into the exact 89/100 block-mean bound.","A single 1/100 truncation budget at endpoint 2X converts that infinite-orbit bound into the finite 9/10 certified-kill gap.","The positive adaptive margin at a strict prime index feeds the exact natural pivot-fibre membership calculation."],"conditional_peer_evidence":["actual_lcm_orbit_separation is the stronger source-specific quantitative orbit route, but its cofinal separation supply is also unproved.","first_harmonic_pivot_decomposition gives a deeper finite four-term budget and decorrelation socket, but its cofinal residual producer is also unproved."],"contrary_evidence":["Irrationality or nonintegrality by itself gives no fixed distance from the integers and therefore supplies neither strict-gap producer.","A dyadic root phase reaches the absorbing phase 1 after finitely many squarings and obstructs strict gaps; finite root exclusion and non-dyadicity transport do not classify actual phases or align them with primes.","The source proves no large-prime instance, equidistribution theorem, or actual-orbit density estimate."],"attribution":"These are source-current Erdős #249 declarations in TotientStrictPrimeEscape.lean, canonically consolidated under strict_prime_tail_orbit_gap by Claims commit ddbe2870. The repeated-squaring, scaled-angle, phase/integrality, finite dyadic-root exclusion, and non-dyadicity transport declarations remain subordinate support or contrary evidence; the family is represented by the Comparator row landed at e7a58bb3. No novelty, priority, significance, or external-review claim is asserted.","limitations":["The late-block density antecedent, cofinal prime strict-gap antecedent, required uniform separation supply, and cofinal nonpositive-prime-shift producer are unproved for the actual totient orbit.","Irrationality or nonintegrality alone gives no fixed distance from the integers, so certificate completeness cannot manufacture either producer predicate.","This is distinct producer-side phase-density and prime-index transport mechanism, but it shares the existing singleton-certificate consumer and must not be fragmented into helper families.","It proves no unconditional Erdős #249 irrationality, prime-distribution/equidistribution theorem, endpoint closure, novelty, priority, significance, or external-review result."],"source_landscape_rank_relative_to":"Below actual_lcm_orbit_separation and first_harmonic_pivot_decomposition because both expose stronger source-specific quantitative sockets, but above totient_certificate_equivalences because this family contributes a distinct producer-side phase-density and natural-prime transport mechanism rather than only an endpoint normal form.","reason":"Represented after canonical Claims and source-faithful Comparator admission: this is a genuinely distinct endpoint-facing producer mechanism with a hard phase-density-to-certificate step and an exact prime-index transport branch. Both branches retain their global supply predicates as open natural friction, so the family stays below the actual-LCM and first-harmonic routes and does not become an unconditional endpoint.","reversal_evidence":"A proved block-density or cofinal-prime strict-gap producer for the actual orbit would promote this route; proof that either antecedent follows from the actual-LCM or pivot premises could consolidate it. A restatement of certificate completeness or another finite phase sample would not change its disposition."},{"candidate_id":"certificate_completeness","family_id":"certificate_completeness","disposition":"subordinate","comparator_eligibility":"source_landed_but_not_comparator_configured","queue_role":"source_landscape_review_not_comparator_evidence","source_declaration":"Erdos249257.irrational_totient_series_iff_lcm_diagonal_certificate_supply","source_file":"Erdos249257/LcmConeFlatness.lean","source_anchor":"316-435","statement":"Irrationality of the binary totient series is equivalent to cofinal certified-kill supply, equivalently to pointwise certificate supply and the lcm-diagonal certificate formulation.","exact_hypotheses":["The certified-kill equivalence identifies existence of a certificate at h,N with tailDiff(N,h) outside the integer range.","For all h > 0 and N, the pointwise certificate-supply equivalence quantifies an L with certifiedKill h N L.","The lcm-diagonal form quantifies a cofinal supply of the corresponding actual LCM certificates."],"conclusion":"Irrationality is equivalent to all positive shifted tail differences being nonintegral, equivalently to pointwise and cofinal certificate supply.","hard_mechanism":"The source stitches exact integer-membership scaling, certified-kill existence, tail-difference nonintegrality, and cofinal lcm-diagonal certificates into an iff chain rather than a one-way sufficient route.","attribution":"This is source-current equivalence infrastructure in LcmConeFlatness.lean; no novelty, priority, or Comparator transport is claimed.","limitations":["An equivalence classifies the endpoint but does not produce a single unbounded certificate or a new source of certificates.","It is not a committed Comparator declaration and must not be counted as Comparator evidence.","The open certificate supply remains the mathematical boundary, so completeness does not replace the selected finite-prime weighted theorem."],"reason":"Subordinate after value review: this exact endpoint equivalence is already represented by the certificate interface and adds no distinct executable producer; it remains valuable consolidation rather than a competing family.","reversal_evidence":"Only a genuinely new unbounded certificate producer or a materially new executable consequence would elevate this route; another equivalent reformulation would remain subordinate."},{"candidate_id":"erdos1049_four_jet_pade_obstruction","family_id":"height_and_pade_arithmetic","disposition":"represented","prior_disposition":"deferred","canonical_family_binding":"height_and_pade_arithmetic","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","comparator_declaration":"Erdos249257.ExternalVerification.rectangular_hp_threshold_eq_classical_iff","supporting_comparator_declarations":[],"source_transport_commit":"f52f903a6f5b564644b19b89b66d65f97990c5b6","source_transport_files":["ExternalVerification/Challenge.lean","ExternalVerification/Solution.lean","ExternalVerification/Statements.lean"],"transport_declarations":["Erdos249257.ExternalVerification.hpClearedGap_nonpos","Erdos249257.ExternalVerification.hpClearedGap_eq_zero_iff","Erdos249257.ExternalVerification.rectangular_hp_threshold_le_classical","Erdos249257.ExternalVerification.rectangular_hp_threshold_eq_classical_iff"],"source_declaration":"ErdosProblems.Erdos1049.hpClearedGap_nonpos; ErdosProblems.Erdos1049.hpClearedGap_eq_zero_iff; ErdosProblems.Erdos1049.rectangular_hp_threshold_le_classical; ErdosProblems.Erdos1049.rectangular_hp_threshold_eq_classical_iff","source_file":"ErdosProblems/Erdos1049/HermitePadeNoGo.lean","source_anchor":"58; 75; 103; 126","statement":"For the explicit rectangular two-function exponent model with rho >= 0 and 1 + rho <= sigma, the cleared gap is nonpositive and the threshold is at most 1/2 - 1/pi^2, with equality exactly at rho = 0 and sigma = 1.","exact_hypotheses":["rho and sigma are real numbers with 0 <= rho and 1 + rho <= sigma.","The rectangular_hp_threshold statements retain their explicit two-function exponent-model definitions.","The equality characterization is the load-bearing representative; the gap sign/unique-zero mechanism and threshold upper bound are supporting evidence."],"conclusion":"The explicit rectangular model admits no positive cleared gap in the stated domain, and its classical threshold equality occurs only at the unique boundary point rho = 0, sigma = 1.","hard_mechanism":"The cleared-gap sign and unique-zero analysis gives a sharp threshold bound and equality locus for the concrete rectangular two-function exponent model; it is a model-specific no-go, not a general Padé theorem.","attribution":"These are source-current Erdős #1049 declarations in HermitePadeNoGo.lean, transported by the committed three-file ExternalVerification batch at f52f903a; no novelty, priority, or external-review claim is made.","limitations":["This sharp no-go is restricted to the explicit rectangular two-function exponent model and does not construct approximating polynomials or remainders.","It is not a universal Padé or Hermite-Padé no-go and proves no irrationality at 3/2 or any general rational-base endpoint.","The exact f52f903a ExternalVerification transport is registered by the source-current committed Comparator representative; the canonical review matrix represents it under height_and_pade_arithmetic."],"reason":"Represented after source-current value review: this is a sharp, high-signal model-specific #1049 no-go with an isolated equality endpoint and clear natural friction. It is consolidated under the canonical height_and_pade_arithmetic family, with its representative now admitted to the committed Comparator screen; the analytic remainder/nonvanishing problem remains open, so it stays below direct endpoint families.","reversal_evidence":"A source-faithful Comparator registration preserves representation; only a genuinely broader no-go, analytic nonvanishing result, or direct rational-base endpoint with equal evidence certainty could elevate or reorder it."},{"candidate_id":"erdos251_integral_tail_classification","family_id":"dyadic_tail_integrality_classification","disposition":"represented","prior_disposition":"deferred","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","source_declaration":"ErdosProblems.Erdos251.tailShift_integral_iff_den_dvd_mersenne; ErdosProblems.Erdos251.tailShift_integral_iff_two_pow_modEq_one; ErdosProblems.Erdos251.not_irrational_initial_iff_exists_eventually_integral_positive_tailShift; ErdosProblems.Erdos251.irrational_initial_iff_all_positive_tailShifts_nonintegral","source_file":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","source_anchor":"1276-1535","source_transport_commit":"e10b161abd964381e0e8e6830e6f48e7629ca8ad","source_transport_files":["ExternalVerification/Challenge.lean","ExternalVerification/Solution.lean","ExternalVerification/Statements.lean"],"transport_declarations":["Erdos249257.ExternalVerification.tailShift_integral_iff_den_dvd_mersenne","Erdos249257.ExternalVerification.tailShift_integral_iff_two_pow_modEq_one","Erdos249257.ExternalVerification.not_irrational_initial_iff_exists_eventually_integral_positive_tailShift","Erdos249257.ExternalVerification.irrational_initial_iff_all_positive_tailShifts_nonintegral"],"comparator_declaration":"Erdos249257.ExternalVerification.irrational_initial_iff_all_positive_tailShifts_nonintegral","supporting_comparator_declarations":[],"transport_admission_boundary":"The exact abstract classification transport is registered by its source-faithful representative in the source-current committed verification/comparator.json screen; the canonical review matrix carries dyadic_tail_integrality_classification, while integral_shift_classification and the existing prime-gap equivalence remain distinct.","statement":"For integer-digit dyadic tail recurrences, denominator/Mersenne divisibility classifies integral positive shifts, while eventual integrality gives the exact rationality obstruction and the complementary nonintegrality normal form.","exact_hypotheses":["DyadicTailRecurrence supplies the recurrence needed for tailShift_integral_iff_den_dvd_mersenne.","For positive h,N, tailShift integrality is equivalent to (T N).den dividing 2^h - 1, equivalently 2^h ≡ 1 modulo (T N).den.","The rationality classification quantifies a fixed positive shift that is eventually integral; the prime-gap nonintegrality route additionally requires its explicit eventual-small or cofinal-mismatch hypotheses."],"conclusion":"For the abstract recurrence, ¬Irrational(initial tail) is equivalent to existence of an eventually integral positive tail shift, and Irrational(initial tail) is equivalent to every positive tail shift being nonintegral; prime-specific eventual nonintegrality remains a separate producer obligation.","hard_mechanism":"Block identities reduce real tail rationality to exact denominator/Mersenne divisibility and eventual integrality, exposing the arithmetic boundary a prime-gap producer must cross.","attribution":"This is source-current Erdős #251 classification material in PrimeGapDyadicTail.lean, transported by the committed three-file ExternalVerification batch at e10b161a and registered through its representative in the committed Comparator roster; no novelty, priority, or external-review claim is made.","limitations":["The classifier does not prove the prime-tail cofinal mismatch or eventual-small hypothesis needed for irrationality.","The committed Comparator representative is abstract classification evidence and is not an unconditional prime-gap endpoint result.","The existing Comparator prime-gap equivalence remains a separate transport surface; this queue item does not silently merge the two."],"reason":"Represented after source-current value review: this is endpoint-proximate and mathematically useful denominator classification with strong digestion value. Its exact four-declaration ExternalVerification transport is committed, its canonical review family is explicit, and its all-positive-tail-shifts representative is now in the Comparator screen; the prime-specific cofinal mismatch producer remains open, so it remains below direct endpoint families.","reversal_evidence":"Prove a genuinely new prime-specific mismatch or eventual-small producer; another equivalent classifier would remain subordinate."},{"candidate_id":"erdos257_boolean_mobius_carry","family_id":"boolean_mobius_carry","disposition":"represented","prior_disposition":"deferred","canonical_family_binding":"boolean_mobius_carry","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","source_declaration":"Erdos249257.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry; Erdos249257.BooleanMobiusCarryCertificate; Erdos249257.binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit; Erdos249257.one_div_oddOrder_le_reciprocalMass_of_support_fraction; Erdos249257.exists_unbounded_shifted_odd_tail_nat_state_of_support_fraction; Erdos249257.supportCoeffZeroWindow_length_le_eps_logb_add","source_file":"Erdos249257/BooleanMobiusCarry.lean","source_anchor":"949","source_transport_commit":"712bc4c3928fef394a6fe96142e593ddc95c1ded","source_transport_files":["ExternalVerification/Challenge.lean","ExternalVerification/Solution.lean","ExternalVerification/Statements.lean"],"transport_declarations":["Erdos249257.ExternalVerification.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry","Erdos249257.ExternalVerification.BooleanMobiusCarryCertificate","Erdos249257.ExternalVerification.binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit","Erdos249257.ExternalVerification.one_div_oddOrder_le_reciprocalMass_of_support_fraction","Erdos249257.ExternalVerification.exists_unbounded_shifted_odd_tail_nat_state_of_support_fraction","Erdos249257.ExternalVerification.supportCoeffZeroWindow_length_le_eps_logb_add"],"comparator_declaration":"Erdos249257.ExternalVerification.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry","supporting_comparator_declarations":[],"transport_admission_boundary":"The exact Boolean–Möbius transport is committed in ExternalVerification at 712bc4c3 and registered as a source-faithful representative in the source-current committed verification/comparator.json screen; the canonical review matrix carries boolean_mobius_carry as a distinct family from shifted_odd_tail_state.","statement":"For p : ℤ and q > 0, a nonempty normalized positive-index support with erdosSupportSeries 2 A = p/q exists exactly when a quotient-only Boolean–Möbius carry certificate exists.","exact_hypotheses":["p : ℤ, q > 0, 0 ∉ A, and A has the explicit nonempty positive-index support witness required by the equivalence.","The certificate U has U 0 = p, positive integer values, the bound U(N) ≤ q(2√N + 4), and q-divisible carry differences.","The Boolean Möbius transform is the exact support-reconstruction interface; the equivalence remains at the existence level rather than supplying a particular support producer."],"conclusion":"A rational normalized support fraction is equivalent to a quotient-level positive square-root-bounded integer orbit with divisible carry differences, Boolean Möbius transform, and literal support reconstruction.","hard_mechanism":"The quotient-only certificate isolates the integer orbit, divisibility, and Boolean Möbius inversion needed to reconstruct support, while the square-root bound and subordinate shifted-tail consequences expose the arithmetic friction without hiding it in a scalar identity.","attribution":"This is the canonical source-current #257 boolean_mobius_carry review family from BooleanMobiusCarry.lean:949, transported by the committed three-file ExternalVerification batch at 712bc4c3; novelty, priority, significance, and external-review claims remain unassessed.","limitations":["The existence-level equivalence gives no contradiction and no universal Erdős #257 irrationality result.","It does not produce cofinally nonintegral shifts for the actual recurrence and does not settle every infinite support.","The reciprocal-mass, unbounded-shifted-tail, and zero-window statements are subordinate consequences, not separate families; the zero-window constant depends on the support and numerator and is not uniform.","The source transport is now a committed Comparator row, but this family must not be read as an unrestricted endpoint proof."],"source_landscape_rank_relative_to":"Below the unconditional full-support, pairwise-coprime, eventually-periodic, and orthogonal-petal endpoint families; above shifted_odd_tail_state because it reconstructs support from a quotient-level Boolean–Möbius certificate rather than merely extracting a shifted natural state.","ranked_below":["known_irrational_supports","pairwise_coprime_support","periodic_nonnegative_weight_irrationality","orthogonal_petal_sunflower_reduction"],"ranked_above":["shifted_odd_tail_state"],"reason":"Represented after the canonical claims event and committed source-faithful Comparator fan-in: this is a deep exact reconstruction/normal-form mechanism for hypothetical rational #257 supports, stronger and more independent than shifted-tail extraction but still below unconditional endpoint families. The universal contradiction/actual-recurrence producer remains open.","reversal_evidence":"A direct contradiction or actual-recurrence cofinal producer would change its value; a weaker support-state row or a routine Boolean reformulation would not change its disposition."},{"candidate_id":"erdos257_boolean_mobius_exact_row_dynamics","family_id":"boolean_mobius_exact_row_dynamics","disposition":"represented","prior_disposition":"deferred","canonical_family_binding":"boolean_mobius_exact_row_dynamics","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","source_declaration":"Erdos249257.CofinalPositiveHalfGreedySkips; Erdos249257.exactLocalMersenneHalfRow_of_positiveHalfGreedySkip; Erdos249257.cofinalExactLocalMersenneHalfRows_of_positiveHalfGreedySkips; Erdos249257.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips; Erdos249257.two_mem_of_exact_localMersenneQuotient; Erdos249257.localEndpointDefect_succ; Erdos249257.exists_upperHalfBooleanWord_of_skippedCore","source_file":"Erdos249257/BooleanMobiusSkipRowCofinal.lean","source_anchor":"22-26; 31-54; 60-77","source_transport_commit":"0502b7ebe8d3d8a14808eecc5bbc663feef711a3","canonical_claim_commit":"33c4c32ea71d67afcd550af8c1fb81b380188f12","source_transport_files":["ExternalVerification/Challenge.lean","ExternalVerification/Solution.lean","ExternalVerification/Statements.lean"],"transport_declarations":["Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips"],"comparator_declaration":"Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips","supporting_comparator_declarations":[],"transport_admission_boundary":"The source-faithful #257 Boolean–Möbius exact-row representative is committed in ExternalVerification at 0502b7eb (building on ec787e64), registered in the source-current committed HEAD:verification/comparator.json screen at admission commit 8a08768f, and bound to the canonical review matrix; the cofinal positive-skip premise remains unproved and is not supplied by registration.","statement":"If positive rational-greedy skips occur cofinally in the declared sense, the skipped-core construction supplies cofinally many exact local Mersenne half rows and hence (1 / 2 : ℝ) belongs to the Mersenne achievement set.","exact_hypotheses":["CofinalPositiveHalfGreedySkips means that for every N there is c ≥ max N 4 with a positive greedy remainder below mersenneWeightRat c.","The exact local-row, cofinal-row, endpoint-defect, rank-two, and upper-half Boolean-word declarations retain their source hypotheses; finite exact rows do not establish the cofinal supply premise."],"conclusion":"Under the unproved cofinal positive-skip premise, (1 / 2 : ℝ) belongs to the Mersenne achievement set; this is conditional endpoint-facing evidence, not an unconditional Erdős #257 result.","hard_mechanism":"A positive skipped greedy remainder creates an exact local Mersenne half row; cofinality transfers those rows to the achievement-set membership route through the skipped core and upper-half Boolean-word construction.","attribution":"This is the source-current #257 boolean_mobius_exact_row_dynamics family from BooleanMobiusSkipRowCofinal.lean, transported by the committed two-stage ExternalVerification product at ec787e64 and 0502b7eb; novelty, priority, significance, and external-review claims remain unassessed.","limitations":["CofinalPositiveHalfGreedySkips is an unproved global supply premise; finite exact rows, including one produced by a skipped core, do not establish cofinality.","The route gives conditional 1/2-membership/counterexample evidence only, not an unconditional universal Erdős #257 refutation or solution.","The exact-row, upper-half Boolean-word, rank-two, and endpoint-defect declarations are subordinate mechanism evidence, not separate families; no endpoint proof is claimed before the cofinal producer exists."],"source_landscape_rank_relative_to":"Below the unconditional full-support, pairwise-coprime, eventually-periodic, and orthogonal-petal endpoint families; above weaker shifted-tail extraction because it exposes a concrete exact-row route to the 1/2 endpoint under an explicit cofinal supply premise.","ranked_below":["known_irrational_supports","pairwise_coprime_support","periodic_nonnegative_weight_irrationality","orthogonal_petal_sunflower_reduction"],"ranked_above":["shifted_odd_tail_state"],"reason":"Represented in the widened source landscape after canonical claims admission and committed Comparator fan-in: this is endpoint-facing conditional evidence with a concrete skipped-core/exact-row mechanism, but it remains below completed endpoint families because the cofinal positive-skip producer is unproved.","reversal_evidence":"A proof of CofinalPositiveHalfGreedySkips would increase evidence certainty and trigger a fresh ranking; finite additional rows without cofinal supply, or a weaker shifted-tail extraction, would not change this disposition."},{"candidate_id":"erdos257_half_membership_seam_classification","family_id":"half_membership_seam_classification","disposition":"represented","prior_disposition":"deferred","canonical_family_binding":"half_membership_seam_classification","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","source_declaration":"Erdos249257.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse; Erdos249257.half_mem_mersenneAchievementSet_iff_unboundedUpperOrMiddle; Erdos249257.unboundedTerminalFalse_iff_not_seamGreedyEventuallyRight; Erdos249257.cofinalTerminalFalse_of_unboundedTerminalFalse; Erdos249257.unboundedTerminalFalse_of_cofinalTerminalFalse; Erdos249257.unboundedTerminalFalse_iff_cofinalTerminalFalse; Erdos249257.half_mem_mersenneAchievementSet_iff_cofinalTerminalFalse; Erdos249257.half_mem_mersenneAchievementSet_iff_exists_unboundedSkippedRanksAlong; Erdos249257.unboundedTerminalFalse_iff_greedyMersenneSkippedSupport_infinite; Erdos249257.half_mem_mersenneAchievementSet_iff_no_lastHalfGreedySkip","source_file":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","source_anchor":"126; 203-213","source_transport_commit":"7511c208c617b6e36fc56b5e0c5f780a3d4df831","canonical_claim_commit":"0c2f4473e80352a7b4bd72a84d4e813978354509","source_transport_files":["ExternalVerification/Challenge.lean","ExternalVerification/Solution.lean","ExternalVerification/Statements.lean"],"transport_declarations":["Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse"],"comparator_declaration":"Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse","supporting_comparator_declarations":[],"transport_admission_boundary":"The source-faithful #257 seam classification is committed in ExternalVerification at 7511c208, registered in the source-current committed HEAD:verification/comparator.json screen, and bound to the canonical half_membership_seam_classification review matrix family at 0c2f4473; the remaining nine source equivalences are subordinate classification evidence, not extra Comparator families.","statement":"Membership of 1/2 in the Mersenne achievement set is exactly equivalent to unbounded false successor terminal bits, the unbounded upper-or-middle seam branch, cofinal terminal-false positions, unbounded skipped ranks along the integer seam, infinite greedy skipped support, and the absence of a last half-greedy skip.","exact_hypotheses":["The concrete Mersenne achievement-set, seam-greedy, terminal-bit, upper-or-middle, skipped-rank, and greedy-skipped-support definitions are those of HalfCylinderHalfMembershipClassification.lean.","The load-bearing Comparator representative preserves the exact iff between (1 / 2 : ℝ) membership and SeamGreedyUnboundedTerminalFalse; the other displayed equivalences preserve their source hypotheses and are subordinate endpoint-socket formulations.","The classification is an exact equivalence among the stated seam conditions, not a premise supplying any one of those unbounded or cofinal occurrences."],"conclusion":"The exact seam classification identifies (1 / 2 : ℝ) membership with the upper-or-middle/unbounded-terminal-false boundary; this is endpoint-facing classification evidence, not an unconditional or universal Erdős #257 result.","hard_mechanism":"Finite integer greedy seam construction and terminal-false normalization convert the achievement-set membership question into equivalent cofinal seam, skipped-rank, and no-last-skip formulations, exposing the precise global supply socket rather than hiding it in a one-way producer.","prerequisite_evidence":["The exact Mersenne achievement-set and seam-greedy definitions are required to state the endpoint iff; they contain no supplied unbounded occurrence."],"support_evidence":["boolean_mobius_exact_row_dynamics is a one-way conditional producer: cofinal positive half-greedy skips would yield 1/2-membership, then this iff identifies the corresponding terminal-false seam socket.","The terminal-false, upper-or-middle, cofinal-terminal-false, skipped-rank, infinite-support, and no-last-skip formulations are equivalent endpoint sockets, not separate families."],"conditional_peer_evidence":["negative_mass_recovery has a substantive analytic stabilization mechanism but retains every reciprocal-tail dynamics, normalized-vanishing, and summability hypothesis and is farther from an exact endpoint classification.","totient_certificate_equivalences is an exact #249 endpoint normal form, but is source-only and supplies no certificate producer."],"contrary_evidence":["Finite exact rows and a one-way positive-skip construction do not prove cofinal skips, unbounded terminal-false positions, or a half-membership witness.","The source gives no unconditional universal Erdős #257 conclusion."],"attribution":"This is the source-current #257 half_membership_seam_classification family from HalfCylinderHalfMembershipClassification.lean, transported by the committed three-file ExternalVerification product at 7511c208 and canonically bound at 0c2f4473; novelty, priority, significance, and external-review claims remain unassessed.","limitations":["The source supplies no unbounded terminal-false or upper-or-middle occurrences, no cofinal supply, no skipped-support producer, and no half-membership witness; an equivalence does not create its required witness.","Finite exact rows, including the one-way positiveHalfGreedySkips route, do not establish the global classification premise or membership.","The family proves no unconditional or universal Erdős #257 result and makes no novelty, priority, significance, or external-review claim."],"source_landscape_rank_relative_to":"Below the unconditional full-support, pairwise-coprime, eventually-periodic, and orthogonal-petal endpoint families; above the one-way positiveHalfGreedySkips Boolean–Möbius exact-row dynamics route and weaker shifted-tail extraction because it is the stronger exact classification socket, while still lacking any occurrence producer.","ranked_below":["known_irrational_supports","pairwise_coprime_support","periodic_nonnegative_weight_irrationality","orthogonal_petal_sunflower_reduction"],"ranked_above":["erdos257_boolean_mobius_exact_row_dynamics","shifted_odd_tail_state"],"reason":"Represented after the canonical claims event and committed Comparator fan-in: this is a high-signal endpoint-facing iff classification that identifies the exact seam boundary and is stronger than the one-way positive-skip producer, but it remains below unconditional endpoints because no global terminal-false or half-membership witness is supplied.","reversal_evidence":"A direct unconditional proof of half-membership or refutation, or a source-current exact producer supplying the classified unbounded seam condition with equal evidence certainty, would trigger a fresh ranking; a weaker one-way positive-skip row or additional finite exact rows would not."},{"candidate_id":"erdos243_negative_mass_recovery","family_id":"negative_mass_recovery","disposition":"represented","prior_disposition":"deferred","canonical_family_binding":"negative_mass_recovery","canonical_claim_commit":"6bedcde3554f5d00ce58bd9921d54be4f19ff34c","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","source_transport_commit":"39d0c4bd6cb7e7c5322d4e21e65e7cba982700f5","source_transport_files":["ExternalVerification/Challenge.lean","ExternalVerification/Solution.lean","ExternalVerification/Statements.lean"],"transport_declarations":["Erdos249257.ExternalVerification.sylvesterNext_eventually_of_summable_negativeRelativeMass"],"comparator_declaration":"Erdos249257.ExternalVerification.sylvesterNext_eventually_of_summable_negativeRelativeMass","supporting_comparator_declarations":[],"transport_admission_boundary":"The source-faithful #243 representative is committed in ExternalVerification at 39d0c4bd and registered in the source-current committed verification/comparator.json screen; the canonical negative_mass_recovery review family is bound at 6bedcde3. The transport records the exact summability criterion and its recovery conclusion, not the original-orbit hypotheses or a #243 endpoint.","source_declaration":"ErdosProblems.Erdos243.tail_growth_le_one_add_negativeRelativeMass; ErdosProblems.Erdos243.eventually_zero_of_summable_negativeRelativeMass; ErdosProblems.Erdos243.sylvesterNext_eventually_of_summable_negativeRelativeMass","source_file":"ErdosProblems/Erdos243/SparseResetRecovery.lean","source_anchor":"45; 155; 175","statement":"Under the exact reciprocal-tail C/D dynamics, positive tail, strict centered-step equation, division-free normalized vanishing, and summable normalized negative centered mass, the centered state vanishes eventually and the orbit follows the Sylvester recurrence eventually.","exact_hypotheses":["a,D : ℕ → ℤ and C : ℕ → ℕ retain the declared next-denominator and next-tail dynamics, with C n > 0.","The strict centered-step equation and division-free normalized vanishing hypotheses are retained exactly.","Summable (negativeRelativeMass C (fun n ↦ centeredState (a n) (D n) (C n))) is assumed."],"conclusion":"Finite normalized negative mass forces eventual centered-state zero and then eventual Sylvester recurrence; it does not prove those hypotheses for the original #243 orbit.","hard_mechanism":"The tail-growth bound by 1 + negativeRelativeMass feeds the finite-negative-mass stabilization theorem, which kills the centered defect and recovers Sylvester dynamics.","attribution":"This is the source-current #243 negative_mass_recovery family canonically exposed at 6bedcde3 from SparseResetRecovery.lean and transported by the committed ExternalVerification batch at 39d0c4bd; novelty, priority, significance, and external review remain unassessed.","limitations":["The criterion assumes the exact dynamics, positivity, strict centered step, division-free normalized vanishing, and summable negativeRelativeMass; it does not establish them for the original Erdős #243 orbit.","It supplies no prime-specific producer, no unconditional #243 endpoint, and no reciprocal-tail irrationality result; any surviving canonical orbit must have divergent normalized negative mass.","It is distinct from centered_state_dynamics because summability replaces the uniform lower-bound hypothesis; no novelty, priority, significance, or external-review claim is made."],"source_landscape_rank_relative_to":"Rank 13: below the #257 seam classification and direct/producer-side endpoint routes because every original-orbit premise remains open, but above routine finite searches and generic screening because it replaces the uniform lower-bound route with a distinct summability mechanism.","ranked_below":["half_membership_seam_classification","centered_state_dynamics"],"ranked_above":["coefficient_only_no_go","weighted_phase_carry_observer"],"reason":"Promoted into the unequal Palomar ranking as one source-current analytic recovery family: the summability mechanism materially changes how #243's surviving orbit must be understood, while preserving that the original-orbit hypotheses and endpoint remain unproved.","reversal_evidence":"A proof of the original-orbit summability or a direct reciprocal-tail endpoint could promote its ranking, while another finite negative-mass calculation would not; a duplicate transport row would not change the disposition."},{"candidate_id":"erdos243_centered_state_recovery","family_id":"centered_state_dynamics","disposition":"represented","prior_disposition":"deferred","canonical_family_binding":"centered_state_dynamics","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","source_transport_commit":"ea12f041307bfea8ca423d8f76f6fe2d38c35055","canonical_claim_commit":"717ba73e587ee0be9ae388d352f24ad9f22cb55d","source_transport_files":["ExternalVerification/Challenge.lean","ExternalVerification/Solution.lean","ExternalVerification/Statements.lean"],"transport_declarations":["Erdos249257.ExternalVerification.sylvesterNext_eventually_of_centered_zero","Erdos249257.ExternalVerification.centeredState_eventually_zero","Erdos249257.ExternalVerification.boundedNegativePart_eventually_zero"],"comparator_declaration":"Erdos249257.ExternalVerification.boundedNegativePart_eventually_zero","supporting_comparator_declarations":[],"subordinate_source_declarations":["Erdos249257.ExternalVerification.centeredState_eventually_zero","Erdos249257.ExternalVerification.sylvesterNext_eventually_of_centered_zero"],"source_declaration":"ErdosProblems.Erdos243.boundedNegativePart_eventually_zero; ErdosProblems.Erdos243.centeredState_eventually_zero; ErdosProblems.Erdos243.sylvesterNext_eventually_of_centered_zero","source_file":"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean","source_anchor":"1799; 1837; 2265","statement":"Under exact natural C/D dynamics with a > 1, C > 0, centered defect E = D - (a - 1)C, strict |E| < C, a uniform lower bound on E, and normalized vanishing, persistent bounded negative behavior is excluded and the centered defect is eventually zero; centered zero plus a nonzero tail recovers the eventual Sylvester recurrence.","exact_hypotheses":["a > 1 and C > 0, with exact natural recurrences for C and D.","E is the signed centered state D - (a - 1)C with strict |E| < C, a uniform lower bound, and the stated normalized-vanishing condition.","The Sylvester recovery branch additionally supplies eventual nonzero tail state."],"conclusion":"The signed recovery theorem forces eventual centered defect zero and, in the centered-zero/nonzero-tail branch, eventual Sylvester recurrence; it does not prove reciprocal-tail irrationality.","hard_mechanism":"The signed bounded-negative-part dichotomy combines exact natural C/D dynamics, a strict centered-state margin, a uniform lower bound, and normalized vanishing to turn a mixed-sign obstruction into eventual rigidity; the centered and Sylvester lemmas expose the recovery mechanism.","attribution":"These are source-current Erdős #243 declarations in ReciprocalTailRigidity.lean, transported by the committed three-file ExternalVerification batch at ea12f041; no novelty, priority, or endpoint claim is made.","limitations":["The theorem does not prove irrationality of the reciprocal-tail series.","The unbounded negative mixed-sign branch and the prime-specific producer remain open.","The recovery theorem is distinct from periodic-orbit and cofinally-bounded-negative no-go families; its exact signed hypotheses remain necessary."],"reason":"Represented after the committed source/config event: this is a distinct, high-signal executable signed recovery mechanism with stronger digestion value than another periodic/no-go restatement. Its load-bearing boundedNegativePart_eventually_zero declaration is now the Comparator representative, while centeredState_eventually_zero and sylvesterNext_eventually_of_centered_zero remain subordinate mechanism evidence. It stays below direct endpoint families because its mixed-sign producer and reciprocal-tail irrationality consequence are open.","reversal_evidence":"A proved unbounded mixed-sign producer or direct reciprocal-tail endpoint would justify promotion, while another periodic or bounded-negative exclusion would remain subordinate; a duplicate transport row would not change the disposition."},{"candidate_id":"weighted_phase_carry_observer","family_id":"weighted_phase_carry_observer","disposition":"represented","comparator_eligibility":"committed_source_faithful_transport","queue_role":"source_landscape_review_with_committed_comparator_evidence","source_transport_commit":"cec8ee43219b3f2059902be78ee0fc54df3dd6e9","canonical_claim_commit":"97d96fb5307061dfe79035c19212572c07cc9055","source_transport_files":["ExternalVerification/Challenge.lean","ExternalVerification/Solution.lean","ExternalVerification/Statements.lean"],"transport_declarations":["Erdos249257.ExternalVerification.carry_eq_residueDigit_add_coboundary"],"comparator_declaration":"Erdos249257.ExternalVerification.carry_eq_residueDigit_add_coboundary","supporting_comparator_declarations":[],"transport_admission_boundary":"The source-current #269 weighted-phase transport is committed at cec8ee43 and its residue-decomposition representative is registered in the source-current committed HEAD:verification/comparator.json screen. The separate observer factorization, quotient recurrence, scalar bridge, and #269 endpoint remain outside that representative.","source_declaration":"ErdosProblems.Erdos269.carry_eq_residueDigit_add_coboundary; ErdosProblems.Erdos269.carryResidue_mem_interval; ErdosProblems.Erdos269.residueDigit_mem_interval; ErdosProblems.Erdos269.finite_realisedSpan_of_factorisation","source_file":"ErdosProblems/Erdos269/WeightedPhaseCarry.lean","source_anchor":"109-145; 150-175; 157-183; 334-355","statement":"The weighted-phase carry recurrence splits each digit into a finite residue digit plus an integral quotient coboundary, and a genuinely function-faithful factorization through a finite-dimensional observer forces the realised symbolic span to be finite-dimensional.","exact_hypotheses":["For B : ℤ with 0 < B and hrec : ∀ n, carry (n+1) = base n * carry n - B * digit n, the residue/quotient decomposition gives the exact digit identity.","For DivisionRing 𝕜 modules with FiniteDimensional 𝕜 E, linear maps realise, C, and readout must satisfy realise = readout.comp C before finite-dimensionality of range realise follows."],"conclusion":"The residue coordinate is finite and the remaining quotient contribution is an uncontrolled integral coboundary; only the stated observer factorization yields FiniteDimensional 𝕜 (LinearMap.range realise).","hard_mechanism":"Exact Euclidean residue reconstruction isolates the bounded digit coordinate while preserving the quotient transfer, and the span theorem uses kernel faithfulness and a linear factorization rather than a single scalar evaluation.","attribution":"These are source-current Erdős #269 Lean declarations in WeightedPhaseCarry.lean; the source and paper explicitly retain their non-endpoint scope and make no novelty or priority claim.","limitations":["No quotient recurrence or finite-state property is assumed or concluded for the integral transfer coordinate.","No scalar-evaluation implication and no literal infinite-dimensional-span theorem follows from the residue split alone.","The candidate does not prove the Erdős #269 endpoint; its committed Comparator representative covers the residue identity only, not the observer factorization theorem."],"reason":"Represented after committed source/config admission: this remains one coherent, reusable weighted-phase observer mechanism with exact finite-residue/uncontrolled-coboundary separation and a precise function-faithful span criterion. It stays below direct endpoint families because the admitted row exposes the residue decomposition but not a quotient recurrence, observer factorization, or #269 endpoint.","reversal_evidence":"A committed transport of the observer factorization preserving its function-faithful hypotheses would strengthen representation; a proved quotient recurrence or scalar-to-function bridge yielding the #269 endpoint could promote it, while a routine residue restatement would not."}]},"candidate_ranking":[{"rank":1,"declaration":"Erdos249257.ExternalVerification.divisibilityWeightedClaim","family_id":"finite_prime_weighted_support","selection_status":"selected","mathematical_nontriviality":"An unconditional finite-prime weighted support criterion, with both fixed-base irrationality and a binary-cost all-base hereditary conclusion.","consequence_and_endpoint_proximity":"For every integer base b ≥ 2 it proves irrationality on every infinite positive support of finite prime-weighted mass; a positive host of finite binary weighted mass gives irrationality for every infinite subset at every integer base.","mechanism_depth_and_natural_friction":"The source proof builds a concrete weighted dyadic-mean producer with divisibility and scale control, obtains cofinal close returns, and consumes them in the irrationality criterion. The separate fixed-base hereditary theorem restricts the nonnegative weighted summability witness to subsets.","genuinely_distinct_content_and_independence":"Its analytic weighted-mass hypothesis covers a general support class and is not the classical full-support theorem or the pairwise-coprime certificate.","external_review_value":"The exact statement exposes the finite prime set, weighted series, support positivity, infinitude, base, and two hereditary quantifier patterns to independent statement and axiom comparison.","proof_digestion_value":"Readers can inspect the literal weighted term and see how finite weighted mass yields fixed-base irrationality, while base-two mass yields the stronger all-base hereditary clause.","evidence_certainty":"The source theorem is Lean-checked in PaperCompleteR8/WeightedReturn.lean:120. This packet includes a separately declared Comparator interface and thin transport. The [replay guide's repository CI record](verification/EXTERNAL_VERIFICATION_REPLAY.md#pull-request-merge-gate) reports a passing weighted-support positive check and deliberate statement-mismatch rejection at commit `7e33a58bb86180013b1fb855cdd8aaff1d7f9057`. The recorded CI result applies only to that snapshot. Independent Linux replay remains pending; verification of later commits requires receipts bound to those commits.","overclaim_risk":"The finite-prime weighted-summability hypothesis and 0 ∉ support condition must stay visible. The theorem does not prove irrationality for arbitrary infinite support, novelty, priority, or external human review.","why_ranked_first":"It has the broadest unconditional direct irrationality conclusion among the selected #257 support criteria, with a concrete checked weighted-return mechanism and a binary-mass all-base hereditary clause.","reader_tier":"completed_direct_result"},{"rank":2,"declaration":"Erdos249257.ExternalVerification.irrational_erdosSum_full_support","family_id":"known_irrational_supports","selection_status":"represented","mathematical_nontriviality":"Unconditional irrationality of the full-support Erdős–Borwein series, with a completed certificate machine rather than a finite computation or a formal restatement.","consequence_and_endpoint_proximity":"Directly settles the canonical full-support member of Erdős #257 for every integer base b >= 2, while leaving universal arbitrary-support irrationality open.","mechanism_depth_and_natural_friction":"The source exposes the bounded Bertrand/CRT first-block frame, middle-window divisor-pair averaging with pigeonhole selection, weighted tail bounds, and explicit parameter closure.","genuinely_distinct_content_and_independence":"It is the classical full-support route and is not collapsed into the separate non-dilation support theorem.","external_review_value":"The target is a named Erdős–Borwein constant family with a direct source statement and a short isolated Comparator type.","proof_digestion_value":"It gives a reader a complete certificate-to-irrationality story and identifies the mechanism that differs from arbitrary-support routes.","evidence_certainty":"The exact source theorem is committed at Erdos249257/CertificateKernel.lean:8323-8331 and is transported by the committed Challenge/Solution pair.","overclaim_risk":"Low only under the stated boundary: this is full support, not all infinite supports, and local proof plus Comparator is not a novelty or human-review claim.","why_not_ranked_first":"The theorem gives an unconditional classical full-support endpoint with a completed certificate proof, but the finite-prime weighted criterion covers a broad family of supports and has an all-base hereditary clause under binary weighted mass.","reader_tier":"completed_direct_result"},{"rank":3,"declaration":"Erdos249257.ExternalVerification.irrational_erdosSupportSeries_pairwise_coprime","family_id":"pairwise_coprime_support","selection_status":"represented","mathematical_nontriviality":"A distinct broad structured-support theorem for a non-dilation class.","consequence_and_endpoint_proximity":"Direct irrationality for every infinite pairwise-coprime support with summable reciprocal mass, but not for arbitrary supports.","mechanism_depth_and_natural_friction":"Adaptive CRT weighted coefficient certificates absorb first-block hits and control residue strays through summable reciprocals.","genuinely_distinct_content_and_independence":"Independent of full support because pairwise coprimality is a separate support mechanism, not a full-support corollary.","external_review_value":"High for readers studying structured support families, with exact hypotheses and Erdős (1968) attribution.","proof_digestion_value":"Excellent mechanism case study, but its extra support hypotheses make the universal boundary easier to misunderstand.","evidence_certainty":"The exact source theorem is committed at Erdos249257/CertificateKernel.lean:10768-10782 and is transported by the committed Challenge/Solution pair.","overclaim_risk":"Moderate unless infinitude, pairwise coprimality, and summable reciprocal mass stay visible in every projection.","why_not_ranked_first":"It is broader in support-class language but less direct as the canonical full-support endpoint and carries more restrictive hypotheses.","reader_tier":"completed_direct_result"},{"rank":4,"declaration":"Erdos249257.ExternalVerification.irrational_erdosSupportSeries_of_orthogonalPetalBouquet","family_id":"orthogonal_petal_sunflower_reduction","selection_status":"represented","mathematical_nontriviality":"A deep structural support route with a nontrivial bouquet and tail-selection interface.","consequence_and_endpoint_proximity":"Direct irrationality under the displayed structure, but the structural predicates are not constructed for arbitrary supports.","mechanism_depth_and_natural_friction":"The conditional selector is the natural friction; omitting it would turn a reduction into a false unconditional theorem.","genuinely_distinct_content_and_independence":"Distinct from both full support and pairwise coprime supports.","external_review_value":"Good for reviewing the proposed reduction, weaker for a headline candidate because its admissible supports are not supplied.","proof_digestion_value":"High conditional-route value, but not a complete mechanism-to-endpoint package.","evidence_certainty":"Exact committed source and Comparator transport are available.","overclaim_risk":"High if the bouquet and uniform tail-selection hypotheses are silently treated as automatic.","why_not_ranked_first":"The unresolved selector makes it less certain and less broadly applicable than the unconditional full-support result.","reader_tier":"conditional_endpoint_route"},{"rank":5,"declaration":"Erdos249257.ExternalVerification.irrational_ratWeightSeries_eventuallyPeriodic","family_id":"periodic_nonnegative_weight_irrationality","selection_status":"represented","mathematical_nontriviality":"Direct irrationality for an eventually periodic nonnegative rational coefficient class.","consequence_and_endpoint_proximity":"Close to the Lambert-series endpoint but not a universal support theorem and not the mixed-sign statement.","mechanism_depth_and_natural_friction":"A periodic divisor-orbit sieve and denominator clearing route are visible in the source.","genuinely_distinct_content_and_independence":"A coefficient-class route rather than a support-class duplicate.","external_review_value":"Readable and source-comparable, with nonnegativity, periodicity, and positive-tail hypotheses exposed.","proof_digestion_value":"Useful bridge from periodic weights to support questions.","evidence_certainty":"Exact committed source and Comparator transport are available.","overclaim_risk":"Moderate if the mixed-sign or arbitrary-support cases are inferred.","why_not_ranked_first":"Its coefficient hypotheses are farther from the canonical arbitrary-support question than full support.","reader_tier":"completed_direct_result"},{"rank":6,"declaration":"Erdos249257.ExternalVerification.irrational_totientSeries_of_actualLcmOrbitSeparationSupply","family_id":"actual_lcm_orbit_separation","selection_status":"represented","mathematical_nontriviality":"A genuinely quantitative actual-LCM orbit-separation implication, with an explicit margin beyond approximation error rather than a finite identity or a count-only obstruction.","consequence_and_endpoint_proximity":"It points directly at the binary totient irrationality endpoint for Erdős #249, conditional on a cofinal separation supply; it does not prove that supply.","mechanism_depth_and_natural_friction":"The source carries a raw adjacent-suffix approximation, an explicit error radius, and uniform separation from every integer into a signed-margin producer. The cofinal PowerTwoActualLcmOrbitSeparationSupply is the natural friction and remains a premise.","genuinely_distinct_content_and_independence":"This is a quantitative actual-LCM orbit route, distinct from finite kernel-rank, factor-ideal no-go, and full-support #257 mechanisms.","external_review_value":"High for a reviewer of the #249 endpoint because the actual LCM, approximation scale, integer-distance margin, and missing producer are named rather than compressed into an irrationality headline.","proof_digestion_value":"It exposes how quantitative orbit separation would feed the irrationality theorem and identifies the exact producer a future proof must supply.","evidence_certainty":"The source definition and implication are committed at Erdos249257/TotientActualLcmOrbitSeparation.lean:305, and the exact wrapper is committed in ExternalVerification/Challenge.lean and ExternalVerification/Solution.lean at the configured declaration.","overclaim_risk":"High unless the two cofinal supply clauses stay attached: the implication is source-faithful and checked, but the supply itself is not proved here.","why_not_ranked_first":"It has exceptional endpoint proximity and a deep mechanism, but it is conditional on the missing cofinal producer; the selected finite-prime weighted theorem is an unconditional completed endpoint with lower hidden-premise risk.","reader_tier":"conditional_endpoint_route"},{"rank":7,"declaration":"Erdos249257.ExternalVerification.irrational_totient_series_of_pivotResidualDecorrelation","family_id":"first_harmonic_pivot_decomposition","selection_status":"represented","mathematical_nontriviality":"A four-term first-harmonic decomposition with explicit 14/25, 1/100, 1/100, and 8/25 budgets yields a strict 9/10 gap and the #249 irrationality endpoint under a cofinal residual-decorrelation supply.","consequence_and_endpoint_proximity":"It points directly at Erdős #249 irrationality from an explicit cofinal analytic producer, independently of the actual-LCM orbit route; that producer remains unproved.","mechanism_depth_and_natural_friction":"Supplier fibres, pivot correlation, bad and non-supplier contributions, and the controlled remainder are kept separate. The finite counterexample to global isolation and the missing DTWPivotResidualDecorrelation supply are the natural friction.","genuinely_distinct_content_and_independence":"This is an analytic first-harmonic budget route, distinct from actual-LCM orbit separation, kernel-rank growth, and finite transport no-go results.","external_review_value":"High: a reviewer can inspect the four exact budgets, the 9/10 gap, the conditional endpoint consumer, and the finite counterexample without treating the premise as constructed.","proof_digestion_value":"It explains exactly how a cofinal decorrelation theorem would close the endpoint and why global pivot isolation is too strong.","evidence_certainty":"The source theorem, canonical review family, and exact Comparator endpoint wrapper are committed; the cofinal decorrelation producer is explicitly absent.","overclaim_risk":"High unless DTWPivotResidualDecorrelation remains attached as an unproved cofinal premise; this is not an unconditional #249 result.","why_not_ranked_first":"Its endpoint consequence and mechanism are exceptional, but the selected #257 theorem is unconditional and the rank-5 actual-LCM route is the closer source-specific #249 producer interface.","reader_tier":"conditional_endpoint_route"},{"rank":8,"declaration":"Erdos249257.ExternalVerification.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap","family_id":"strict_prime_tail_orbit_gap","selection_status":"represented","mathematical_nontriviality":"A coherent two-branch conditional producer turns either 11/100 nonpositive tail-orbit phase density or a cofinal natural-prime strict gap below 9/10 into the finite certificate socket.","consequence_and_endpoint_proximity":"It reaches the Erdős #249 irrationality consumer conditionally and is more than a finite normal form, but neither phase-density nor cofinal-prime producer is proved.","mechanism_depth_and_natural_friction":"Unit norm and the complement estimate give the exact 89/100 block bound; a 1/100 truncation budget yields the 9/10 finite gap, while the prime branch uses an adaptive positive margin and exact pivot-fibre membership. The two global producer predicates are the natural friction.","genuinely_distinct_content_and_independence":"This is producer-side phase-density and natural-prime transport, distinct from actual-LCM quantitative separation, the four-term pivot budget, and finite certificate equivalences.","external_review_value":"High as a source-isolated map of exactly what orbit-density or prime-index input would suffice, provided the missing producers remain adjacent.","proof_digestion_value":"It exposes the 11/100 to 89/100 to 9/10 chain and the adaptive prime escape rather than flattening both branches into a generic conditional endpoint.","evidence_certainty":"The exact source declarations, canonical strict_prime_tail_orbit_gap family, and Comparator representative are committed at Claims ddbe2870 and transport e7a58bb3.","overclaim_risk":"High unless the unproved density, cofinal natural-prime strict-gap, uniform-margin, and cofinal nonpositive-prime-shift producers stay explicit; no unconditional #249, novelty, priority, significance, or external-review claim follows.","why_not_ranked_first":"It ranks below actual_lcm_orbit_separation and first_harmonic_pivot_decomposition because their source-specific quantitative sockets are stronger, but above finite certificate equivalences because it contributes a distinct producer mechanism rather than only an endpoint normal form.","reader_tier":"conditional_endpoint_route"},{"rank":9,"declaration":"Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_nonunit_carries","family_id":"factorial_carry_characterisation","selection_status":"subordinate","mathematical_nontriviality":"An exact equivalence translating irrationality into a cofinal non-unit carry condition.","consequence_and_endpoint_proximity":"Maximal endpoint proximity for #68, but it does not prove the cofinal producer.","mechanism_depth_and_natural_friction":"The carry recurrence is the useful mechanism; the missing cofinal witness is the natural unresolved friction.","genuinely_distinct_content_and_independence":"A factorial carry route, independent of the Lambert-support results.","external_review_value":"High as an exact reduction, provided equivalence is not reported as a solved irrationality theorem.","proof_digestion_value":"Excellent for routing future work to the actual missing producer.","evidence_certainty":"Exact committed source and Comparator transport are available.","overclaim_risk":"High if the equivalence is mistaken for a cofinal carry proof.","why_not_ranked_first":"It reaches an open endpoint more directly but does not itself establish the endpoint, unlike the selected finite-prime weighted theorem.","reader_tier":"exact_reduction_or_structural_result"},{"rank":10,"declaration":"Erdos249257.ExternalVerification.irrational_tsum_primeDyadicTerm_iff_primeGap","family_id":"prime_gap_reformulation","selection_status":"subordinate","mathematical_nontriviality":"An exact irrationality equivalence linked to the prime-gap series.","consequence_and_endpoint_proximity":"Very close to #251's target but proves neither side irrational and retains a summability premise.","mechanism_depth_and_natural_friction":"The key mechanism is summation-by-parts normalization; the target irrationality remains open.","genuinely_distinct_content_and_independence":"Distinct prime-gap route, not a support-series corollary.","external_review_value":"Good for verifying the exact normalization and its open boundary.","proof_digestion_value":"Useful connective tissue, not a headline endpoint result.","evidence_certainty":"Exact committed source and Comparator transport are available.","overclaim_risk":"High if equivalence is read as irrationality.","why_not_ranked_first":"It is a reduction rather than a proved irrationality consequence.","reader_tier":"exact_reduction_or_structural_result"},{"rank":11,"declaration":"Erdos249257.ExternalVerification.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank","family_id":"totient_carry_anti_compression","selection_status":"represented","mathematical_nontriviality":"Hypothetical non-irrationality forces one tempered integral carry whose every dyadic section is eventually periodic modulo a common multiplier while canonical carry-kernel ranks grow at least as 2^e - 1.","consequence_and_endpoint_proximity":"This is an orbit-level necessary consequence of the #249 rational branch, closer to the endpoint than coefficient-side kernel rank alone, but it derives no contradiction.","mechanism_depth_and_natural_friction":"The theorem couples modular orbit regularity with exponential finite-level anti-compression. Its natural friction is exact: modulo a divisor of the multiplier the forcing can vanish and the carry can become geometric, so periodicity supplies no finite-Q-rank upper bound.","genuinely_distinct_content_and_independence":"It is one coherent orbit-level conjunction, not a duplicate of the coefficient-side totient_kernel_rank family, strict-prime phase production, or certificate completeness.","external_review_value":"High because the rationality consequence and the failed contradiction route are both statement-isolated and adjacent.","proof_digestion_value":"It shows simultaneously what rationality would regularize and why that regularity does not compress the carry enough to contradict rank growth.","evidence_certainty":"The exact source theorem, canonical review family, and Comparator representative are committed; the missing finite-rank contradiction is explicit.","overclaim_risk":"High if modular periodicity and unbounded canonical-kernel rank are reported as incompatible; the source proves no such upper bound and no #249 irrationality.","why_not_ranked_first":"It outranks coefficient-side kernel inventory because it is tied directly to hypothetical rationality, but remains below the producer-side strict-prime, first-harmonic, and actual-LCM routes because it supplies neither a contradiction nor a certificate producer.","reader_tier":"exact_reduction_or_structural_result"},{"rank":12,"declaration":"Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse","family_id":"half_membership_seam_classification","selection_status":"represented","mathematical_nontriviality":"An exact endpoint equivalence identifies 1/2-membership with unbounded terminal-false behavior and several nonredundant seam sockets: upper-or-middle positions, cofinal terminal-false indices, unbounded skipped ranks, infinite skipped support, and no last half-greedy skip.","consequence_and_endpoint_proximity":"It classifies the concrete #257 half-membership endpoint exactly, closer to a usable target than an analytic recovery criterion or source-only certificate normal form, but supplies no occurrence witness.","mechanism_depth_and_natural_friction":"Finite integer seam construction and terminal-bit normalization expose the global supply condition in interchangeable forms. The natural friction is decisive: none of those forms is proved to occur unboundedly.","genuinely_distinct_content_and_independence":"This is an exact endpoint-classification family, distinct from the one-way Boolean–Möbius positive-skip producer, #243 summability recovery, and #249 certificate equivalences.","external_review_value":"High because a reviewer can see the exact target, all equivalent sockets, the sufficient positive-skip route, and why finite rows do not close the quantifier.","proof_digestion_value":"It prevents future work from mistaking a convenient seam formulation or finite exact row for the missing global producer.","evidence_certainty":"The exact source iff, canonical claims family, and committed Comparator representative are available; the absent occurrence producer is explicit.","overclaim_risk":"High if an iff is described as a half-membership witness or if finite exact rows are treated as cofinal supply; it proves no universal #257 result.","why_not_ranked_first":"It is promoted as the strongest underweighted survivor, but stays beneath all unconditional #257 endpoints and the producer-side #249 routes because it classifies rather than supplies its global endpoint condition.","reader_tier":"exact_reduction_or_structural_result"},{"rank":13,"declaration":"Erdos249257.ExternalVerification.sylvesterNext_eventually_of_summable_negativeRelativeMass","family_id":"negative_mass_recovery","selection_status":"represented","mathematical_nontriviality":"A distinct analytic recovery theorem: summability of normalized negative centered mass, together with the exact reciprocal-tail dynamics, forces eventual centering and Sylvester recurrence.","consequence_and_endpoint_proximity":"It gives a sharp conditional structural conclusion for Erdős #243 and proves that any surviving canonical orbit must evade finite negative mass, but it does not establish the original-orbit hypotheses or reciprocal-tail irrationality.","mechanism_depth_and_natural_friction":"The tail-growth estimate by 1 + negativeRelativeMass feeds finite-negative-mass stabilization, killing the centered defect. The hard friction is that summability, positivity, strict centering, normalized vanishing, and the exact C/D dynamics are all premises rather than proved properties of the original orbit.","genuinely_distinct_content_and_independence":"This is an independent analytic recovery criterion, not a duplicate of the uniform-lower-bound centered-state route, the #257 seam classification, or #249 certificate normal forms.","external_review_value":"High because the exact analytic hypothesis, recovery conclusion, and divergent-negative-mass boundary are statement-isolated in a committed Comparator interface.","proof_digestion_value":"It identifies finite normalized negative mass as a precise analytic condition future #243 work would need to verify or refute, rather than obscuring the missing producer behind a generic recurrence claim.","evidence_certainty":"The exact source theorem, canonical review family, and committed Comparator representative are available at Claims 6bedcde3 and transport 39d0c4bd.","overclaim_risk":"High if the conditional summability criterion is reported as applying to the original reciprocal-tail orbit; it proves no unconditional #243 endpoint, prime-specific producer, or reciprocal-tail irrationality.","why_not_ranked_first":"It deserves ranked visibility over routine screening rows because its analytic mechanism is distinct and reusable, but remains below the #257 seam classification's exact endpoint socket and all direct/producer-side routes because every original-orbit hypothesis remains open.","reader_tier":"conditional_endpoint_route"}],"candidate_selection":{"selected":true,"problem":257,"family_id":"finite_prime_weighted_support","claim_id":"finite_prime_weighted_support","declaration":"Erdos249257.ExternalVerification.divisibilityWeightedClaim","source_declaration":"ErdosProblems.Erdos257.PaperCompleteR8.divisibilityWeightedClaim","source_file":"ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean","source_anchor":"120","statement":"For every integer base b ≥ 2, every infinite positive support A of finite prime-weighted mass in base b has an irrational support series. If a positive host H has finite prime-weighted mass in base 2, every infinite subset of H has an irrational support series at every integer base b ≥ 2.","exact_hypotheses":["A finite nonempty set P of primes witnesses summability over the support of primeSetPart(P,a)/(a(b^primeSetPart(P,a)-1)).","In the first clause: 2 ≤ b, 0 ∉ A, and A is infinite.","In the hereditary clause: 0 ∉ H, finite weighted mass is in base 2, A ⊆ H, and A is infinite."],"conclusion":"Both conjuncts of DivisibilityWeightedClaim hold: fixed-base weighted irrationality and binary-mass all-base hereditary irrationality.","open_boundary":"Finite prime-weighted mass is a sufficient support condition; universal irrationality for all infinite supports in Erdős #257 remains open.","hard_mechanism":"A concrete weighted dyadic mean on one sampling scheme supplies arbitrarily close displacement returns. The source consumes those returns for fixed-base irrationality and uses binary returns for the all-base hereditary clause.","why_this_is_the_showcase":"The weighted theorem has the broadest unconditional direct support-class conclusion in the checked #257 packet. Its two clauses and the exact weighted-mass premise are visible in the statement-isolated Challenge.","attribution":"The theorem is proved locally in Lean at WeightedReturn.lean:120. Novelty, priority, significance, and independent human mathematical review are not asserted.","limitations":["Neither clause covers every infinite support.","The all-base hereditary clause requires finite binary weighted mass; weighted mass at another base yields only the same-base hereditary theorem.","Comparator transport is configured here; an external Comparator receipt is not claimed until replay succeeds."]},"selection_comparison":[{"declaration":"Erdos249257.ExternalVerification.irrational_erdosSum_full_support","family_id":"known_irrational_supports","disposition":"not_selected_alternative","reason":"The classical full-support theorem is unconditional and has a completed certificate proof, but the selected finite-prime weighted criterion gives irrationality for a broader support class and an all-base hereditary consequence from binary weighted mass."},{"declaration":"Erdos249257.ExternalVerification.irrational_totientSeries_of_actualLcmOrbitSeparationSupply","family_id":"actual_lcm_orbit_separation","disposition":"represented_conditional_alternative","reason":"This newly committed #249 row is the strongest endpoint-near alternative: its exact source-faithful separation supply implies irrationality through a quantitative signed-margin mechanism. It remains below full support because the cofinal supply is a premise rather than a completed producer, so selecting it would carry a larger hidden-hypothesis risk."},{"declaration":"Erdos249257.ExternalVerification.irrational_erdosSupportSeries_pairwise_coprime","family_id":"pairwise_coprime_support","disposition":"not_selected_alternative","reason":"The pairwise-coprime theorem remains a distinct non-dilation result, but its support and summable-reciprocal hypotheses are narrower than the finite-prime weighted criterion selected as the broad direct lead."},{"declaration":"Erdos249257.ExternalVerification.irrational_erdosSupportSeries_of_orthogonalPetalBouquet","family_id":"orthogonal_petal_sunflower_reduction","disposition":"not_selected_alternative","reason":"Its bouquet and uniform tail-selection hypotheses expose a deeper conditional route, but those structural hypotheses are not constructed for arbitrary supports; its evidence boundary is therefore weaker for an unconditional showcase."},{"declaration":"Erdos249257.ExternalVerification.threeHalves_no_coordinatewiseCorridor","family_id":"coordinatewise_corridor_no_go","disposition":"not_selected_alternative","reason":"It is a sharp, useful no-go result, but it excludes one proof architecture and concludes no irrationality statement; it is lower consequence for the selected Palomar candidate."},{"declaration":"Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_nonunit_carries","family_id":"factorial_carry_characterisation","disposition":"not_selected_alternative","reason":"It is an exact #68 endpoint reduction with excellent review value, but its cofinal non-unit-carry producer remains unproved; an equivalence to the open endpoint is not a direct irrationality result."},{"declaration":"Erdos249257.ExternalVerification.not_finiteDimensional_span_fullTotientKernel","family_id":"totient_kernel_rank","disposition":"not_selected_alternative","reason":"It is a deep independent #249 structural theorem, but infinite-dimensional kernel rank is farther from the irrationality endpoint than the selected direct result and does not supply the missing unbounded certificate family."}],"candidate_screening":[{"declaration":"Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses","family_id":"factorial_carry_characterisation","disposition":"supporting_endpoint_reduction","reason":"Exact #68 equivalence, but the carry formulation is the more informative route and neither form supplies the required cofinal witness."},{"declaration":"Erdos249257.ExternalVerification.irrational_factorialGapSeries_of_cofinal_global_complementaryTail","family_id":"factorial_conditional_producers","disposition":"supporting_conditional_endpoint","reason":"A single coefficient-free cofinal complementary-tail certificate would imply the #68 endpoint, but the certificate supply is an explicit unproved premise; this is a shorter conditional consumer, not an irrationality proof."},{"declaration":"Erdos249257.ExternalVerification.no_cofinallyBoundedNegative_of_normalizedVanishes","family_id":"bounded_negative_exclusion","disposition":"supporting_conditional_no_go","reason":"Excludes a bounded-negative regime only under explicit dynamical hypotheses; it does not decide the unrestricted reciprocal-tail endpoint."},{"declaration":"Erdos249257.ExternalVerification.no_eventuallyPeriodicNegative_orbit","family_id":"negative_orbit_no_go","disposition":"supporting_no_go","reason":"Retires an eventually-periodic negative orbit architecture, but leaves arbitrary unbounded negative behaviour and the #243 target open."},{"declaration":"Erdos249257.ExternalVerification.finrank_totientKernelThroughLevelFamily_eq","family_id":"totient_kernel_rank","disposition":"supporting_structural","reason":"Exact finite-level dyadic rank growth is strong #249 structure and supports totient_carry_anti_compression, but it is weaker than that orbit-level rationality consequence and provides neither a contradiction nor unbounded certificate supply."},{"declaration":"Erdos249257.ExternalVerification.not_finiteDimensional_span_fullTotientKernel","family_id":"totient_kernel_rank","disposition":"supporting_structural","reason":"Infinite-dimensionality is the coefficient-side consequence of exact kernel-rank growth; it supports anti-compression but is farther from hypothetical rationality and gives no finite-rank contradiction or certificate producer."},{"declaration":"Erdos249257.ExternalVerification.exists_totientDyadicSectionBasis","family_id":"totient_kernel_basis","disposition":"supporting_structural","reason":"An explicit basis is useful kernel infrastructure, yet it does not connect finite span structure to the binary totient irrationality endpoint."},{"declaration":"Erdos249257.ExternalVerification.finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent","family_id":"totient_kernel_all_base_index","disposition":"supporting_conditional_structural","reason":"The rank theorem retains an external linear-independence hypothesis, so it is valuable structural support rather than an unconditional endpoint result."},{"declaration":"Erdos249257.ExternalVerification.tsum_pos_coprime_inv_mersenne_eq_one","family_id":"totient_visible_coprime_lattice_mass","disposition":"supporting_normalization_identity","reason":"Exact visible-lattice normalization explains a mass term but is rational and does not settle Erdős #249."},{"declaration":"Erdos249257.ExternalVerification.tsum_totient_div_mersenne_sq_eq_gcd_moment_series","family_id":"squared_lambert_gcd_moments","disposition":"supporting_transfer_identity","reason":"A precise gcd-moment transfer identity, but it supplies no irrationality conclusion for the binary totient series."},{"declaration":"Erdos249257.ExternalVerification.runHeight_defect_fib_sum_lower","family_id":"stern_brocot_run_fibonacci_stability","disposition":"supporting_combinatorial","reason":"A genuine Fibonacci/continuant stability floor, but analytic denominator survival and the #249 endpoint remain unproved."},{"declaration":"Erdos249257.ExternalVerification.cylinderMass_split","family_id":"stern_brocot_cylinder_law","disposition":"supporting_exact_identity","reason":"Exact Stern–Brocot telescoping is reusable geometry, not an irrationality theorem or novelty claim."},{"declaration":"Erdos249257.ExternalVerification.lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient","family_id":"lcm_factor_ideal_shift_algebra_no_go","disposition":"supporting_no_go","reason":"A synthetic finite-shift obstruction retires one algebraic route while explicitly leaving actual totient forcing and the unbounded supply open."},{"declaration":"Erdos249257.ExternalVerification.exists_clean_binaryCyclotomicAnchor","family_id":"totient_certificate_equivalences","disposition":"supporting_mechanism","reason":"The theorem gives an unconditional cofinal supply of clean prime factors on composite binary-cyclotomic rays. It removes the anchor-existence obstruction for certificate routes, but the denominator-filtered carry-kill supply and the #249 irrationality endpoint remain open."},{"declaration":"Erdos249257.ExternalVerification.exists_primeGap0_gt","family_id":"coefficient_only_no_go","disposition":"supporting_auxiliary","reason":"Prime-gap unboundedness is exact background for #251, but unbounded coefficients alone do not force irrationality."},{"declaration":"Erdos249257.ExternalVerification.coefficientOnlyNoGo","family_id":"coefficient_only_no_go","disposition":"supporting_negative_method","reason":"The exact transport packages the finite carry partial-sum identity and synthetic coefficient non-eventual-periodicity against actual prime-gap nonperiodicity. It is valuable natural-friction evidence below direct endpoints and above routine finite searches, but has no Lean infinite-limit theorem, stream identification, or #251 endpoint."},{"declaration":"Erdos249257.ExternalVerification.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch","family_id":"small_mismatch_criterion","disposition":"supporting_conditional_endpoint","reason":"This actual-prime fixed-shift obstruction is the first conditional-endpoint-leverage screen: strict smallness makes two integral adjacent tail shifts zero, so the exact recurrence contradicts the supplied unequal prime gaps. Its cofinal small-mismatch and actual-smallness producer remains unproved; it is not #251 irrationality or an infinite rational-sum result."},{"declaration":"Erdos249257.ExternalVerification.irrational_initial_iff_all_positive_tailShifts_nonintegral","family_id":"dyadic_tail_integrality_classification","disposition":"supporting_classification","reason":"The registered representative is the all-positive-tail-shifts irrationality normal form for the abstract dyadic recurrence; denominator and congruence classifiers remain source-level mechanism evidence, and the prime-specific cofinal producer is open."},{"declaration":"Erdos249257.ExternalVerification.no_positive_reducedCarry_of_cofinalLocalWindowEscape","family_id":"conditional_carry_escape","disposition":"supporting_conditional_no_go","reason":"The exact window-residue contradiction is retained as subordinate #269 conditional endpoint leverage. The actual-series reduction is given; the source-specific cofinal escape remains unproved."},{"declaration":"Erdos249257.ExternalVerification.carry_eq_residueDigit_add_coboundary","family_id":"weighted_phase_carry_observer","disposition":"supporting_mechanism","reason":"The exact residue-digit/coboundary decomposition is the committed representative of the weighted-phase observer family; it does not transport the separate finite-dimensional factorization or imply the #269 endpoint."},{"declaration":"Erdos249257.ExternalVerification.volume_mersenneAchievementSet","family_id":"achievement_set_geometry","disposition":"supporting_metric_geometry","reason":"The full-measure achievement-set theorem is consequential geometry, but it does not classify rational points or settle infinite-support irrationality."},{"declaration":"Erdos249257.ExternalVerification.finite_period_noncollapse_rat_den_interface","family_id":"finite_period_noncollapse","disposition":"supporting_finite_interface","reason":"The finite-period order interface is exact denominator infrastructure, but finite support cannot decide an infinite-support sum."},{"declaration":"Erdos249257.ExternalVerification.volume_supportedMersenneAchievementSet_dichotomy","family_id":"restricted_achievement_sets","disposition":"supporting_metric_geometry","reason":"The restricted-set measure dichotomy is structurally useful but does not classify rational values or prove universal irrationality."},{"declaration":"Erdos249257.ExternalVerification.exists_shifted_odd_tail_nat_states_of_support_fraction","family_id":"shifted_odd_tail_state","disposition":"supporting_conditional_reduction","reason":"It extracts a modular tail state only from explicit rational-support hypotheses and therefore does not exclude rational infinite supports."},{"declaration":"Erdos249257.ExternalVerification.supportCoeff_mul_eq_add_defect","family_id":"composite_dilation_defect_identity","disposition":"supporting_structural_identity","reason":"The exact composite-dilation defect exposes a real obstruction, but it supplies neither a tail selector nor an unconditional irrationality result."},{"declaration":"Erdos249257.ExternalVerification.seven_le_of_intBand_odd","family_id":"half_and_twenty_one_frontiers","disposition":"supporting_local_no_go","reason":"Sharp local 2-adic band arithmetic excludes small odd numerators under stated hypotheses, but does not decide reachability or half-membership."},{"declaration":"Erdos249257.ExternalVerification.kernel_235_minor_eq_neg_one_fifteen","family_id":"rank_two_kernel_no_go","disposition":"supporting_finite_no_go","reason":"A checked minor obstruction rules out rank one in a displayed finite kernel, but failure of rank one does not imply irrationality."},{"declaration":"Erdos249257.ExternalVerification.smoothPrefixLcm_eq_threePrimeHeight","family_id":"three_prime_lcm_cells","disposition":"supporting_exact_identity","reason":"The smooth-prefix LCM identity is exact three-prime structure, but it does not supply the missing irrationality bridge."},{"declaration":"Erdos249257.ExternalVerification.boundedNegativePart_eventually_zero","family_id":"centered_state_dynamics","disposition":"supporting_mechanism","reason":"The committed source-faithful representative exposes the signed centered-state recovery theorem with exact natural C/D dynamics, strict centered margin, lower bound, and normalized vanishing; the centered-zero and Sylvester recovery lemmas remain subordinate, and no reciprocal-tail irrationality follows."},{"declaration":"Erdos249257.ExternalVerification.exists_small_translation_separating_arguments","family_id":"translation_avoidance","disposition":"supporting_technical","reason":"Finite translation avoidance is a useful #1041 input, yet it does not provide the global gluing needed by the target route."},{"declaration":"Erdos249257.ExternalVerification.constant_perturbation_roots_in_unitDisk","family_id":"root_retention","disposition":"supporting_technical","reason":"Root retention under small perturbation is an exact local input, not a complete stationarity or minimax conclusion."},{"declaration":"Erdos249257.ExternalVerification.exists_two_roots_dist_sum_le_two_mul_geomMean","family_id":"newton_value_decay","disposition":"supporting_metric_geometry","reason":"At a critical point, the theorem finds two distinct roots whose broken-line distance through that point is at most twice the geometric-mean root distance. It strengthens the Newton/critical-point geometry to the sharp metric scale, but it does not prove that the two segments stay inside the open lemniscate."},{"declaration":"Erdos249257.ExternalVerification.threeHalves_no_coordinatewiseCorridor","family_id":"coordinatewise_corridor_no_go","disposition":"supporting_no_go","reason":"It excludes one coordinatewise corridor architecture at base 3/2 and proves no irrationality statement."},{"declaration":"Erdos249257.ExternalVerification.rationalBaseClearedTailQ_succ","family_id":"rational_base_tail_recurrence","disposition":"supporting_exact_recurrence","reason":"The cleared-tail recurrence is exact rational-base infrastructure, but no contradiction or irrationality consequence is derived."},{"declaration":"Erdos249257.ExternalVerification.fixedPrecisionTropicalNoGo","family_id":"fixed_precision_transport_no_go","disposition":"supporting_negative_method","reason":"The committed representative unconditionally retires fixed-positive-precision local valuation/unit-word escape for every finite odd-unit word through centred completions, while its synthetic unrestricted high-quotient and finite-word scope supplies no actual totient endpoint."},{"declaration":"Erdos249257.ExternalVerification.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry","family_id":"boolean_mobius_carry","disposition":"supporting_reconstruction_equivalence","reason":"The committed representative preserves the existence-level quotient certificate, Boolean Möbius support reconstruction, positivity, and square-root bound; it supplies no universal irrationality contradiction or actual-recurrence producer."},{"declaration":"Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips","family_id":"boolean_mobius_exact_row_dynamics","disposition":"supporting_conditional_endpoint","reason":"The committed exact-row representative is one conditional Boolean–Möbius dynamics family: cofinal positive greedy skips yield cofinally many exact local half rows and conditional 1/2-membership, while finite rows do not supply the unproved cofinal premise."},{"declaration":"Erdos249257.ExternalVerification.rectangular_hp_threshold_eq_classical_iff","family_id":"height_and_pade_arithmetic","disposition":"supporting_model_no_go","reason":"The registered representative is the sharp equality characterization for the explicit rectangular two-function exponent model; it is not a universal Hermite-Padé no-go and does not produce irrationality."},{"declaration":"Erdos249257.ExternalVerification.scaleFullTarget_miss_of_projected_separation","family_id":"actual_foreign_residue_projection","disposition":"supporting_conditional_reduction","reason":"The committed representative exposes the exact finite foreign/divisor partition, geometric tail control, and generic separated-projection consumer for the canonical actual_foreign_residue_projection family; its controlled-projection bridge and unbounded supply remain open, so it is represented conditional evidence rather than a #249 endpoint."},{"declaration":"Erdos249257.ExternalVerification.finitePrimeWeighted_fixedBase_hereditary","family_id":"finite_prime_weighted_support","disposition":"supporting_auxiliary","reason":"The exact same-base hereditary theorem extends fixed-base weighted irrationality to every infinite subset of a positive host, using restriction of the nonnegative summability witness. It is subordinate to the two-clause weighted lead and does not infer all-base irrationality from weighted mass at an arbitrary base."}],"frontier_by_problem":[{"problem":68,"status":"open","frontier_summary":"Factorial-denominator irrationality is reduced to cofinally many strict-successor divisibility failures or non-unit carries; no cofinal producer is proved.","open_boundary":"Irrationality of the factorial-denominator series.","comparator_declarations":["Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses","Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_nonunit_carries","Erdos249257.ExternalVerification.irrational_factorialGapSeries_of_cofinal_global_complementaryTail"]},{"problem":243,"status":"open","frontier_summary":"Exact product-cleared tail dynamics and no-go results for bounded or periodic negative-error regimes are checked.","open_boundary":"The unrestricted reciprocal-tail problem and arbitrary unbounded negative behaviour.","comparator_declarations":["Erdos249257.ExternalVerification.no_cofinallyBoundedNegative_of_normalizedVanishes","Erdos249257.ExternalVerification.no_eventuallyPeriodicNegative_orbit","Erdos249257.ExternalVerification.boundedNegativePart_eventually_zero","Erdos249257.ExternalVerification.sylvesterNext_eventually_of_summable_negativeRelativeMass"]},{"problem":249,"status":"open","frontier_summary":"Finite dyadic and conditional all-base kernel rank, exact visible-lattice and gcd-moment identities, Stern-Brocot stability, the first-harmonic pivot decomposition, actual-LCM sign/top-edge support, the strict-prime phase-density/natural-prime conditional producer, and a synthetic LCM factor-ideal no-go are exposed.","open_boundary":"The binary totient series irrationality endpoint and the unbounded certificate supply.","comparator_declarations":["Erdos249257.ExternalVerification.finrank_totientKernelThroughLevelFamily_eq","Erdos249257.ExternalVerification.not_finiteDimensional_span_fullTotientKernel","Erdos249257.ExternalVerification.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank","Erdos249257.ExternalVerification.exists_totientDyadicSectionBasis","Erdos249257.ExternalVerification.finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent","Erdos249257.ExternalVerification.tsum_pos_coprime_inv_mersenne_eq_one","Erdos249257.ExternalVerification.tsum_totient_div_mersenne_sq_eq_gcd_moment_series","Erdos249257.ExternalVerification.runHeight_defect_fib_sum_lower","Erdos249257.ExternalVerification.cylinderMass_split","Erdos249257.ExternalVerification.lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient","Erdos249257.ExternalVerification.fixedPrecisionTropicalNoGo","Erdos249257.ExternalVerification.irrational_totient_series_of_pivotResidualDecorrelation","Erdos249257.ExternalVerification.irrational_totientSeries_of_actualLcmOrbitSeparationSupply","Erdos249257.ExternalVerification.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap","Erdos249257.ExternalVerification.exists_clean_binaryCyclotomicAnchor","Erdos249257.ExternalVerification.scaleFullTarget_miss_of_projected_separation"]},{"problem":251,"status":"open","frontier_summary":"Prime gaps are unbounded and the prime-value and prime-gap dyadic series are related by an exact irrationality equivalence.","open_boundary":"Irrationality of the target prime-gap series.","comparator_declarations":["Erdos249257.ExternalVerification.exists_primeGap0_gt","Erdos249257.ExternalVerification.coefficientOnlyNoGo","Erdos249257.ExternalVerification.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch","Erdos249257.ExternalVerification.irrational_tsum_primeDyadicTerm_iff_primeGap","Erdos249257.ExternalVerification.irrational_initial_iff_all_positive_tailShifts_nonintegral"]},{"problem":257,"status":"open","frontier_summary":"The checked frontier leads with finite-prime weighted irrationality and its binary-mass all-base hereditary clause, alongside classical and other structured supports, achievement-set geometry, finite-period noncollapse, exact carry reductions, and conditional half-value routes.","open_boundary":"Universal irrationality for every infinite support, including the unresolved half and 1/21 membership routes.","comparator_declarations":["Erdos249257.ExternalVerification.volume_mersenneAchievementSet","Erdos249257.ExternalVerification.irrational_erdosSum_full_support","Erdos249257.ExternalVerification.divisibilityWeightedClaim","Erdos249257.ExternalVerification.finitePrimeWeighted_fixedBase_hereditary","Erdos249257.ExternalVerification.irrational_erdosSupportSeries_pairwise_coprime","Erdos249257.ExternalVerification.irrational_ratWeightSeries_eventuallyPeriodic","Erdos249257.ExternalVerification.irrational_erdosSupportSeries_of_orthogonalPetalBouquet","Erdos249257.ExternalVerification.finite_period_noncollapse_rat_den_interface","Erdos249257.ExternalVerification.volume_supportedMersenneAchievementSet_dichotomy","Erdos249257.ExternalVerification.exists_shifted_odd_tail_nat_states_of_support_fraction","Erdos249257.ExternalVerification.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry","Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips","Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse","Erdos249257.ExternalVerification.supportCoeff_mul_eq_add_defect","Erdos249257.ExternalVerification.seven_le_of_intBand_odd"]},{"problem":269,"status":"open","frontier_summary":"A three-prime kernel minor, exact running-LCM/height identity, weighted-phase residue decomposition, and conditional local-window carry-escape consumer are checked.","open_boundary":"Irrationality or transcendence in any three-prime case.","comparator_declarations":["Erdos249257.ExternalVerification.kernel_235_minor_eq_neg_one_fifteen","Erdos249257.ExternalVerification.smoothPrefixLcm_eq_threePrimeHeight","Erdos249257.ExternalVerification.no_positive_reducedCarry_of_cofinalLocalWindowEscape","Erdos249257.ExternalVerification.carry_eq_residueDigit_add_coboundary"]},{"problem":1041,"status":"open","frontier_summary":"For ani's explicit degree-seven polynomial, Lean proves the preconnected-set Hausdorff-measure bound above two, refuting the exact Formal Conjectures path-image-length statement; a separate total-variation bound is checked. The listed Comparator declarations are supporting Newton-flow and perturbation inputs, not transports of that counterexample.","open_boundary":"Ani's reported small-parameter family is not formalised, and independent human review of correspondence between the refuted formal statement and the 1958 wording is not recorded.","comparator_declarations":["Erdos249257.ExternalVerification.exists_small_translation_separating_arguments","Erdos249257.ExternalVerification.constant_perturbation_roots_in_unitDisk","Erdos249257.ExternalVerification.exists_two_roots_dist_sum_le_two_mul_geomMean"]},{"problem":1049,"status":"open","frontier_summary":"Lean proves irrationality and an irrationality-exponent bound throughout the stated Zudilin contour region, including every positive integral power of 31/4. The listed Comparator declarations cover separate recurrence and model-boundary results, not a transport of the contour-region theorem.","open_boundary":"The universal rational-base conjecture, including the case 3/2, remains open. The checked 31/4 power family settles one region and leaves that conjecture untouched.","comparator_declarations":["Erdos249257.ExternalVerification.threeHalves_no_coordinatewiseCorridor","Erdos249257.ExternalVerification.rationalBaseClearedTailQ_succ","Erdos249257.ExternalVerification.rectangular_hp_threshold_eq_classical_iff"]}],"release_boundary":{"qualification":"READY","why":"The committed public checkout satisfies the repository-local Palomar structural contract for its current Comparator roster. This local result does not claim that any external review, submission, registration, or publication has occurred.","not_claimed":["universal Erdős #257 solved","any of the eight original problems solved","novelty or human peer review","registration or publication"]}},"body_format":"json"},"formalization-yaml":{"document_id":"formalization-yaml","source_path":"formalization.yaml","role":"formal_verification_metadata","github":"https://github.com/wcook04/plectis-erdos/blob/main/formalization.yaml","source_commit":"72d3012d21ab90b59daf76c317c4a581214a2a17","permalink_url":"https://github.com/wcook04/plectis-erdos/blob/72d3012d21ab90b59daf76c317c4a581214a2a17/formalization.yaml","sha256":"sha256:67b27fb80fb2e4ce596d69a9b9493bf5b64157cc5d2a1126f1dc920db44944ca","byte_count":171703,"origin":"public_main_checkout","body":"# yaml-language-server: $schema=https://raw.githubusercontent.com/mathlib-initiative/formalization.yaml/99c678e569c7c4c0772db297c5ddd5e4c9b6322e/schema/formalization.schema.json\n# Generated from docs/claims.json::external_verification_packet.\n# Refresh: python3 scripts/build_external_verification.py\nversion: \"v0.4\"\nproject:\n  name: \"Eight Erdős problem programmes: checked mathematics\"\n  description: \"Lean 4 formalization across eight Erdős problem programmes, including unconditional theorems, exact conditional endpoint reductions, arithmetic classifications, counterexamples, and method barriers. The precise release boundary is recorded below.\"\n  authors:\n    - \"Will Cook\"\n  license: \"Apache-2.0\"\n  responsible_maintainers:\n    - \"Will Cook\"\nclassification:\n  arxiv: [\"math.NT\", \"math.CO\"]\n  msc2020: [\"11J72\"]\nsources:\n  - title: \"Erdős Problems 68, 243, 249, 251, 257, 269, 1041, and 1049\"\n    authors:\n      - \"Paul Erdős and collaborators\"\n    id: \"https://www.erdosproblems.com/\"\n    type: \"web discussion\"\n    relationship: \"background\"\n    license: \"external source; see source site\"\n    author_endorsement: \"n/a\"\n  - title: \"Repository problem notes and cited-source registry\"\n    authors:\n      - \"Will Cook\"\n    id: \"docs/papers/corpus.json\"\n    type: \"other\"\n    relationship: \"adapts\"\n    license: \"CC-BY-4.0\"\n    author_endorsement: \"n/a\"\nstatus:\n  scope: \"Using the degree-seven polynomial constructed by the erdosproblems.com contributor ani, Lean proves that every preconnected strict-lemniscate set containing two distinct roots has one-dimensional Hausdorff measure greater than two. This refutes the exact Formal Conjectures path-image-length statement; the separate total-variation bound is also checked. The other seven targets remain open. Independent human review of correspondence with the 1958 wording has not been recorded. Comparator checks only selected exact statements, axioms and kernel acceptance; it does not assess novelty or historical correspondence.\"\n  proof_corpus_sorry_count: 0\n  sorry_count: 8\n  sorry_in_definitions: 0\n  trusted_challenge_sorry_count: 8\n  sorry_boundary: \"Every sorry is a trusted proposition package in one of the statement-isolated Comparator challenge fixtures (verification/ExternalVerification/Challenge.lean; verification/ExternalVerification1041SolvedFamilies/Challenge.lean; verification/ExternalVerification1049/Challenge.lean; verification/ExternalVerification249TotientKernelBasis/Challenge.lean). Each is the fixture against which Comparator checks the proof-bearing solution; the proof corpus and every solution wrapper have sorry_count 0.\"\n  axioms:\n    - \"propext\"\n    - \"Quot.sound\"\n    - \"Classical.choice\"\n  main_results:\n    - declaration: \"ErdosProblems.Erdos68.irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses\"\n      file: \"ErdosProblems/Erdos68/FactorialGapPlateauCore.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved result; novelty unassessed\"\n      statement: \"Irrationality is equivalent to cofinally many strict-successor divisibility failures.\"\n      boundary: \"The equivalence does not prove that the cofinal failures occur.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos68.irrational_factorialGapSeries_iff_cofinal_nonunit_carries\"\n      file: \"ErdosProblems/Erdos68/FactorialGapPlateauCore.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved result; novelty unassessed\"\n      statement: \"Irrationality is equivalent to cofinally many non-unit carries.\"\n      boundary: \"The equivalence does not supply a cofinal carry producer.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_nonunit_carries\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos243.no_cofinallyBoundedNegative_of_normalizedVanishes\"\n      file: \"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"conditional reduction and no-go result\"\n      statement: \"Normalised vanishing and bounded rise exclude a cofinally bounded negative part.\"\n      boundary: \"All dynamics, bounded-rise, positivity, and vanishing hypotheses remain explicit.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.no_cofinallyBoundedNegative_of_normalizedVanishes\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos243.boundedNegativePart_eventually_zero\"\n      file: \"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved signed recovery/rigidity theorem; novelty unassessed\"\n      statement: \"Under exact natural C/D dynamics with a > 1, C > 0, E = D - (a - 1)C, strict |E| < C, a uniform lower bound on E, and division-free normalized vanishing, boundedNegativePart_eventually_zero forces E = 0 eventually; centered-zero plus an eventually nonzero next tail recovers eventual Sylvester recurrence.\"\n      boundary: \"The signed theorem has a C D : ℕ → ℕ, E : ℕ → ℤ, and B : ℕ with all exact hypotheses explicit: a > 1, C > 0, C(n + 1) + D(n) = a(n) · C(n), D(n + 1) = a(n) · D(n), E is the centered state D − (a − 1)C, |E| < C, a uniform lower bound on E, and division-free normalized vanishing. It excludes persistent bounded negative behaviour and forces eventual centered defect zero; centered-zero plus an eventually nonzero next tail recovers Sylvester, but no reciprocal-tail irrationality follows. The unbounded negative mixed-sign branch and the prime-specific producer remain open. This is distinct from the periodic negative-orbit and cofinally bounded-negative no-go families; no novelty, priority, significance, or external-review claim is made, and no unrestricted #243 solution follows.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.boundedNegativePart_eventually_zero\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos243.sylvesterNext_eventually_of_summable_negativeRelativeMass\"\n      file: \"ErdosProblems/Erdos243/SparseResetRecovery.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"conditional analytic recovery criterion; novelty unassessed\"\n      statement: \"Under exact next-denominator and next-tail C/D dynamics with a, D : ℕ → ℤ, C : ℕ → ℕ, C > 0, the strict centered step equation, division-free normalized vanishing, and Summable normalized negative centered mass, the centered state vanishes eventually and a follows the Sylvester recurrence eventually.\"\n      boundary: \"This is a conditional recovery criterion: the source theorem assumes every displayed dynamics, positivity, strict centered-step, normalized-vanishing, and summability hypothesis, and does not prove them for the original Erdős #243 orbit. Any surviving canonical orbit must have divergent normalized negative mass. It supplies no prime-specific producer, unconditional #243 endpoint, or reciprocal-tail irrationality, and makes no novelty, priority, significance, or external-review claim. It is distinct from centered_state_dynamics, which uses a uniform lower-bound hypothesis rather than summability of normalized negative mass.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.sylvesterNext_eventually_of_summable_negativeRelativeMass\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos243.no_eventuallyPeriodicNegative_orbit\"\n      file: \"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"no-go result\"\n      statement: \"An eventually periodic negative-error orbit with positive phase growth is impossible.\"\n      boundary: \"The theorem does not cover arbitrary unbounded negative behaviour.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.no_eventuallyPeriodicNegative_orbit\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.finrank_totientKernelThroughLevelFamily_eq\"\n      file: \"Erdos249257/TotientMahlerDefect.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:dyadic_totient_certificate_interface\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved result; novelty unassessed\"\n      statement: \"For every e >= 1, the complete dyadic totient kernel through level e has rational span dimension 2^e + 1.\"\n      boundary: \"This does not prove irrationality of the totient series.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.finrank_totientKernelThroughLevelFamily_eq\"\n      literature_dependencies:\n        - \"paper/249/erdos-249-binary-totient-series.tex\"\n        - \"docs/PRIOR_ART.md\"\n        - \"https://doi.org/10.1017/S0305004108001665\"\n        - \"https://doi.org/10.1007/s11139-004-4633-4\"\n      source_statement: \"Coons proved that Euler's totient sequence is not k-regular for every integer k at least 2, while Martin's affine-totient ordering theorem implies linear independence of every finite family of pairwise nonproportional positive-slope affine totient sections.\"\n      source_mapping: \"The dyadic Lean development gives an independent constructive finite-level basis and rank proof. Its full-kernel infinite-dimensionality corollary is kernel-checked here but is not an independent contribution: for integer-valued sequence modules it is equivalent to Coons's non-k-regularity result. The all-base Lean modules check zero-channel and composite-base reductions, residue coordinates, index cardinality k^e+1, canonical spanning and the reusable conditional-rank interface. Separately, AllBaseTotientKernel.lean at public checkpoint 25ef6245d15a proves canonical independence and the complete rank unconditionally by CRT and Dirichlet. This does not enlarge the Comparator target represented below or formalise Martin's broader theorem.\"\n      historical_attribution: \"Coons for non-k-regularity of Euler's totient sequence; Martin for the positive-density affine ordering theorem from which the paper's finite-family independence step follows.\"\n      logical_dependency: \"Arithmetic reduction and canonical spanning are unconditional. The recorded reusable rank wrapper takes independence as an input. AllBaseTotientKernel.lean discharges that input independently; Martin's broader positive-density theorem remains cited external prior work, not an axiom of the Lean proof.\"\n    - declaration: \"Erdos249257.not_finiteDimensional_span_fullTotientKernel\"\n      file: \"Erdos249257/TotientMahlerDefect.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:dyadic_totient_certificate_interface\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"Lean formalisation of an existing consequence of Coons\"\n      statement: \"The rational span of the full dyadic totient kernel is not finite-dimensional.\"\n      boundary: \"The declaration is kernel-checked locally, but full-kernel infinite-dimensionality is not an independent contribution of this release; no rationality-to-finite-rank bridge is proved.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.not_finiteDimensional_span_fullTotientKernel\"\n      literature_dependencies:\n        - \"paper/249/erdos-249-binary-totient-series.tex\"\n        - \"docs/PRIOR_ART.md\"\n        - \"https://doi.org/10.1017/S0305004108001665\"\n        - \"https://doi.org/10.1007/s11139-004-4633-4\"\n      source_statement: \"Coons proved that Euler's totient sequence is not k-regular for every integer k at least 2, while Martin's affine-totient ordering theorem implies linear independence of every finite family of pairwise nonproportional positive-slope affine totient sections.\"\n      source_mapping: \"The dyadic Lean development gives an independent constructive finite-level basis and rank proof. Its full-kernel infinite-dimensionality corollary is kernel-checked here but is not an independent contribution: for integer-valued sequence modules it is equivalent to Coons's non-k-regularity result. The all-base Lean modules check zero-channel and composite-base reductions, residue coordinates, index cardinality k^e+1, canonical spanning and the reusable conditional-rank interface. Separately, AllBaseTotientKernel.lean at public checkpoint 25ef6245d15a proves canonical independence and the complete rank unconditionally by CRT and Dirichlet. This does not enlarge the Comparator target represented below or formalise Martin's broader theorem.\"\n      historical_attribution: \"Coons for non-k-regularity of Euler's totient sequence; Martin for the positive-density affine ordering theorem from which the paper's finite-family independence step follows.\"\n      logical_dependency: \"Arithmetic reduction and canonical spanning are unconditional. The recorded reusable rank wrapper takes independence as an input. AllBaseTotientKernel.lean discharges that input independently; Martin's broader positive-density theorem remains cited external prior work, not an axiom of the Lean proof.\"\n    - declaration: \"Erdos249257.totientDyadicSectionBasis\"\n      file: \"Erdos249257/TotientMahlerDefect.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:dyadic_totient_basis\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved result; novelty unassessed\"\n      statement: \"The full dyadic totient kernel span admits the explicit odd-core-indexed basis.\"\n      boundary: \"The basis theorem does not connect rationality to finite kernel rank.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.exists_totientDyadicSectionBasis\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent\"\n      file: \"Erdos249257/TotientKernelConditional.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"formalised unconditional spanning and conditional rank with external independence input\"\n      statement: \"Under 2 ≤ k, 1 ≤ e, and linear independence of the explicit canonical all-base family, the actual all-base totient kernel through level e has rational span dimension k^e + 1.\"\n      boundary: \"The linear-independence hypothesis is the external affine-ordering boundary and is not proved here; this does not prove irrationality of the binary totient series.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent\"\n      literature_dependencies: []\n    - declaration: \"GcdMomentCalculus.tsum_pos_coprime_inv_mersenne_eq_one\"\n      file: \"Erdos249257/GcdMomentCalculus.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"formalised exact visible-lattice normalization identity\"\n      statement: \"The positive coprime Mersenne weights over the visible lattice sum exactly to one, giving the reduced-direction mass behind the fair-coin coprimality form of the totient constant.\"\n      boundary: \"This is a rational normalization identity and does not prove irrationality of the binary totient series or settle Erdős #249.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.tsum_pos_coprime_inv_mersenne_eq_one\"\n      literature_dependencies: []\n    - declaration: \"GcdMomentCalculus.tsum_totient_div_mersenne_sq_eq_gcd_moment_series\"\n      file: \"Erdos249257/GcdMomentCalculus.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"formalised exact first-gcd-moment squared-Lambert transfer identity\"\n      statement: \"At base one half, the squared-Mersenne Lambert series weighted by Euler's totient equals the dyadic series of Pillai's gcd-sum function minus n, the exact first gcd-moment transfer.\"\n      boundary: \"This exact identity does not prove irrationality of the binary totient series; the constant-weight squared-Lambert irrationality is cited external mathematics, and no Erdős #249 endpoint follows.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.tsum_totient_div_mersenne_sq_eq_gcd_moment_series\"\n      literature_dependencies: []\n    - declaration: \"SternBrocotRunGeometry.runHeight_defect_fib_sum_lower\"\n      file: \"Erdos249257/SternBrocotRunGeometry.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"formalised exact Fibonacci/continuant run stability floor\"\n      statement: \"For every finite defect list e, the height of the corresponding positive Stern-Brocot run word is at least F_(r+3) + F_(r+1) times the total defect, where r is the number of runs.\"\n      boundary: \"This exact combinatorial stability floor does not establish analytic denominator survival, irrationality of the binary totient series, or novelty; no Erdős #249 endpoint follows.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.runHeight_defect_fib_sum_lower\"\n      literature_dependencies: []\n    - declaration: \"GcdMomentCalculus.cylinderMass_split\"\n      file: \"Erdos249257/GcdMomentCalculus.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"formalised exact Stern-Brocot cylinder telescoping identity\"\n      statement: \"At every positive Stern-Brocot node (a,b), the closed cylinder mass splits exactly into the node's stop mass and the two child cylinder masses.\"\n      boundary: \"This exact telescoping recursion and its geometric interpretation do not prove irrationality of the binary totient series or novelty; the #249 endpoint remains open.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.cylinderMass_split\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.TotientTailPeriodKiller.lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient\"\n      file: \"Erdos249257/LcmFactorIdealPulseObstruction.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:lcm_factor_ideal_anchor_pulse_no_go\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"unconditional progress\"\n      statement: \"For every t >= 3, a nonzero synthetic dyadic coboundary survives every finite integer shift polynomial while retaining the period-LCM totient factor ideals, exact cleared-prefix cancellation, and uniform state and letter bounds.\"\n      boundary: \"The witness is synthetic: it does not assert that its forcing letters are actual totient differences, does not address nonlinear combinations, and leaves the unbounded certificate supply for Erdős #249 open.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.TotientTailPeriodKiller.fixedPrecisionTropicalNoGo\"\n      file: \"Erdos249257/TropicalCurvatureCarry.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:fixed_precision_transport_no_go\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"unconditional synthetic no-go; novelty unassessed\"\n      statement: \"For every positive fixed precision, every finite word of odd valuation-unit symbols and every incoming integer carry state admit a compatible orbit whose successive states stay within the corresponding valuation-unit radii; fixed-precision local signatures therefore do not by themselves exclude finite centred completions.\"\n      boundary: \"This is a synthetic finite-word construction with unrestricted incoming/high quotient states. It does not impose the actual totient-difference relation or provide growing-precision/global correlation, and it gives no Erdős #249 endpoint or irrationality result. No novelty or priority claim is made. The one-step and finite-word completion theorems are subordinate mechanisms, not separate families.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.fixedPrecisionTropicalNoGo\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_actualLcmOrbitSeparationSupply\"\n      file: \"Erdos249257/TotientActualLcmOrbitSeparation.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:actual_lcm_orbit_nonintegrality_frontier\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"conditional reduction\"\n      statement: \"A cofinal quantitative separation supply for the actual power-two LCM tail orbit implies irrationality of the binary totient series.\"\n      boundary: \"The cofinal separation supply is an open premise, stronger than mere non-integrality; this sufficient route is not an equivalence and leaves the punctured/top-edge boundary and the Erdős #249 endpoint open.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.irrational_totientSeries_of_actualLcmOrbitSeparationSupply\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.ActualForeignResidueProjection.scaleFullTarget_miss_of_projected_separation\"\n      file: \"Erdos249257/ActualForeignResidueProjection.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:actual_foreign_residue_projection\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"conditional reduction\"\n      statement: \"The finite Möbius residue diagonal splits into projected foreign and divisor channels, every finite stable foreign-tail window obeys a closed geometric bound, and a controlled projection separated from every integer forces the full target to be missed.\"\n      boundary: \"This is a conditional finite partition/geometric-majorant reduction with a generic complement-noncancellation consumer. The ControlledForeignProjection analytic bridge and any unbounded separated-projection supply remain unproved; no Erdős #249 endpoint, novelty, significance, priority, or external-review claim is made.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.scaleFullTarget_miss_of_projected_separation\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_pivotResidualDecorrelation\"\n      file: \"Erdos249257/FirstHarmonicPivot.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:first_harmonic_pivot_decomposition\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"conditional reduction\"\n      statement: \"The exact first-harmonic pivot decomposition and its 9X/10 gap yield a finite certificate whenever the stated PivotBudgetAt bounds hold, and the cofinal pivot-residual-decorrelation socket implies irrationality of the binary totient series.\"\n      boundary: \"The four-term decomposition and 9X/10 gap remain conditional on the explicit dyadic-room and PivotBudgetAt hypotheses; the cofinal decorrelation producer is unproved. The fibre/image lemmas do not give global supplier-prime isolation or a prime-distribution estimate, and no Erdős #249 irrationality conclusion is claimed.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.irrational_totient_series_of_pivotResidualDecorrelation\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos249.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap\"\n      file: \"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:strict_prime_tail_orbit_gap\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"conditional reduction\"\n      statement: \"An 11/100 nonpositive-phase density on arbitrarily late dyadic blocks yields an 89/100 first-harmonic saving and a finite 9/10 gap; alternatively, a cofinal natural-prime tail-orbit strict gap below 9/10 yields adaptive pivot escape and conditional irrationality. The exact initial phase is the scaled-angle character of (2^h - 1) times the series, and the strict-gap route is an orbit of repeated squaring from that phase.\"\n      boundary: \"The phase-density, cofinal natural-prime strict-gap, associated uniform-margin, and cofinal nonpositive-prime-shift producers remain unproved for the actual totient orbit. Exact phase/integrality equivalence, finite dyadic-root exclusion for 1 ≤ h ≤ 16 and k ≤ 14, and non-dyadicity-to-cofinal-nonpositive-phase implications are classification or finite evidence only; they do not supply density or prime alignment, and the source does not prove actual root entry. This is a conditional reduction only: no unconditional Erdős #249 endpoint, novelty, priority, significance, or external-review claim is made. It is distinct from the finite first_harmonic_pivot_decomposition and actual_lcm_orbit_separation families.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank\"\n      file: \"Erdos249257/TotientTailCarryPeriod.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:totient_carry_kernel_anti_compression\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved necessary consequence; novelty unassessed\"\n      statement: \"If the binary totient coefficient series were rational, one positive-multiplier tempered integral carry has dyadic sections uniformly eventually periodic modulo its multiplier and canonical carry-section span dimension at least 2^e − 1 for every e.\"\n      boundary: \"This is a necessary consequence of rationality only: it gives no contradiction, no finite-rank upper bound for a rationality-supplied carry, and no irrationality proof for Erdős #249. Modulo a divisor of the multiplier, the forcing disappears and the carry becomes geometric, so quotient periodicity alone supplies no finite-ℚ-rank upper bound. The coefficient-side kernel-rank and certificate-equivalence families are not duplicated, and #249 remains open.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos251.exists_primeGap0_gt\"\n      file: \"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"formalisation of existing result\"\n      statement: \"The consecutive prime gaps are unbounded.\"\n      boundary: \"Unbounded coefficients alone do not force irrationality.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.exists_primeGap0_gt\"\n      literature_dependencies:\n        - \"docs/PRIOR_ART.md#expansion-note-principal-sources-68-243-251-269-1041-1049\"\n        - \"paper/251/erdos-251-prime-gap-dyadic-series.tex\"\n      source_statement: \"The gaps between consecutive primes are unbounded; the elementary factorial construction gives arbitrarily long prime-free intervals.\"\n      source_mapping: \"The local theorem uses the zero-based consecutive-prime gap sequence and proves that for every bound B some indexed gap exceeds B. This is the same unboundedness statement after the repository's indexing convention is unfolded.\"\n      historical_attribution: \"Classical elementary factorial prime-free interval argument; modern large-gap theorems in the paper are contextual and strictly stronger.\"\n      logical_dependency: \"The Lean proof reconstructs an elementary argument locally; no external prime-gap theorem is imported as a proof dependency.\"\n    - declaration: \"ErdosProblems.Erdos251.carryPartialSum_natCast_eq\"\n      file: \"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"formalisation of exact finite identity and nonperiodicity contrast; novelty and significance unassessed\"\n      statement: \"The coefficient-only wrapper records the exact finite carry partial-sum identity together with non-eventual-periodicity of the synthetic coefficient stream and the actual prime-gap stream, exposing the limits of coefficient-only reasoning.\"\n      boundary: \"The wrapper proves the finite identity and the two non-eventual-periodicity statements only. It does not assert the infinite rational-sum limit of the synthetic countermodel, identify the synthetic coefficient stream with actual prime gaps, establish a prime-gap tail bridge, prove #251 irrationality, or make novelty, priority, significance, or external-review claims.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.coefficientOnlyNoGo\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos251.irrational_tsum_primeDyadicTerm_iff_primeGap\"\n      file: \"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"formalisation of existing result; novelty unassessed\"\n      statement: \"Lean checks that the prime-value dyadic series is irrational exactly when the prime-gap dyadic series is, discharging summability by an elementary polynomial prime bound rather than the prime-number theorem. Neither side is proved irrational. The already-known identity is Tao's (7 October 2025) and is recorded separately as cited only.\"\n      boundary: \"The equivalence does not prove irrationality of either series.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.irrational_tsum_primeDyadicTerm_iff_primeGap\"\n      literature_dependencies:\n        - \"https://www.erdosproblems.com/forum/thread/251\"\n        - \"paper/251/erdos-251-prime-gap-dyadic-series.tex\"\n        - \"docs/PRIOR_ART.md#expansion-note-principal-sources-68-243-251-269-1041-1049\"\n      source_statement: \"The prime-value dyadic series equals 2 plus the prime-gap dyadic series by summation by parts, so one series is irrational exactly when the other is.\"\n      source_mapping: \"The local named iff still takes a Summable hypothesis. This checkout discharges that hypothesis by the elementary polynomial bound p_n ≤ 1250(n+1)^4 and the unconditional identity, not by the prime-number theorem. Neither side is proved irrational.\"\n      historical_attribution: \"Tao, erdosproblems.com/251, 7 October 2025. Recorded as cited only on prime_gap_identity_tao. This is not a novelty or priority claim.\"\n      logical_dependency: \"The Lean proof reconstructs summation by parts and an elementary prime bound locally; the forum post is attribution, not a kernel import.\"\n    - declaration: \"ErdosProblems.Erdos251.irrational_initial_iff_all_positive_tailShifts_nonintegral\"\n      file: \"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"abstract Lean-checked classification and normal form; novelty unassessed\"\n      statement: \"For any integer-digit real dyadic tail recurrence T(N + 1) = 2*T(N) - g(N + 1), the initial value T(0) is irrational exactly when every positive-length tail shift is nonintegral.\"\n      boundary: \"This exact normal form covers arbitrary integer-digit dyadic tail recurrences with the declared real state type; it does not supply a cofinal nonintegral shift for the consecutive-prime-gap recurrence. The prime-specific producer remains open, so no prime-gap irrationality, novelty, significance, or external-review claim is made and Erdős #251 remains open.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.irrational_initial_iff_all_positive_tailShifts_nonintegral\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos251.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch\"\n      file: \"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"conditional actual-prime-gap endpoint reduction; novelty and significance unassessed\"\n      statement: \"For the actual prime-gap dyadic tail, cofinally many adjacent pairs of strictly small h-shifts with unequal corresponding prime gaps force failure of eventual RatIntegral h-shifts.\"\n      boundary: \"The representative assumes T : ℕ → ℚ, the DyadicTailRecurrence for the actual prime-gap digits, and for every N₀ an N ≥ N₀ whose adjacent h-shifts both lie strictly in (-1, 1) while the corresponding prime gaps differ. Under that cofinal small-mismatch supply it excludes eventual RatIntegral h-shifts; it does not prove the supply or actual smallness. rationalPrimeGapTail_has_positive_shift_not_eventually_small records the contrary rational-state obstruction for at least one positive shift. This is a conditional reduction only: no #251 irrationality, infinite rational-sum limit, universal producer, novelty, priority, significance, or external-review claim is made; #251 remains open. The actual-prime-gap specialization is distinct from coefficient_only_no_go and does not duplicate prime_gap_reformulation.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.volume_mersenneAchievementSet\"\n      file: \"Erdos249257/GreedyAchievementSet.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:greedy_achievement_geometry\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"mixed prior-art geometry and locally proved metric result; novelty unassessed\"\n      statement: \"The base-2 Mersenne achievement set has Lebesgue measure exactly one.\"\n      boundary: \"This does not decide whether every infinite-support sum is irrational.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.volume_mersenneAchievementSet\"\n      literature_dependencies:\n        - \"https://doi.org/10.11429/ptmps1907.7.14_250\"\n        - \"https://doi.org/10.1007/s10474-025-01528-0\"\n        - \"docs/PRIOR_ART.md\"\n      source_statement: \"Kakeya's strict-tail criterion places rapidly decreasing subsum sets in the Cantor-set regime; for the fixed-base Mersenne sequence, Kovač--Tao verify the strict-tail inequality and distinct subsums.\"\n      source_mapping: \"At base 2 the local development formalises compactness, perfection, total disconnectedness, nowhere density, and measure one. The strict-tail Cantor geometry aligns with the cited antecedents; the measure-one clause is locally derived and carries no novelty assertion.\"\n      historical_attribution: \"Kakeya (1914) for the strict-tail subsum-set criterion and Kovač--Tao (2025), Remark 4.1, for the fixed-base Mersenne antecedent.\"\n      logical_dependency: \"The topology and measure theorem are proved locally in Lean; the cited sources fix historical/source alignment and are not imported proof objects.\"\n    - declaration: \"Erdos249257.irrational_erdosSum_full_support\"\n      file: \"Erdos249257/CertificateKernel.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:eb_full_support\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"formalisation of an existing theorem\"\n      statement: \"For every integer base b >= 2, the full-support Erdos-Borwein series is irrational.\"\n      boundary: \"This is the classical full-support theorem, not universal Erdos #257.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.irrational_erdosSum_full_support\"\n      literature_dependencies:\n        - \"https://users.renyi.hu/~p_erdos/1948-04.pdf\"\n        - \"docs/PRIOR_ART.md\"\n        - \"paper/257/erdos-257-mersenne-support-subseries.tex\"\n      source_statement: \"For every integer b at least 2, the full-support Lambert--Mersenne series sum over n at least 1 of 1/(b^n-1) is irrational.\"\n      source_mapping: \"The local theorem uses Nat indexing with exponent k+1, which is exactly the source sum after reindexing n=k+1; the base assumption is the same integer condition b at least 2.\"\n      historical_attribution: \"Erdos, On arithmetical properties of Lambert series (1948), theorem on p. 63 as recorded in the repository's prior-art ledger.\"\n      logical_dependency: \"The theorem is independently represented and proved in Lean in this repository; the historical source is attribution and fidelity evidence, not an imported kernel dependency.\"\n    - declaration: \"ErdosProblems.Erdos257.PaperCompleteR8.divisibilityWeightedClaim\"\n      file: \"ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:finite_prime_weighted_support\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved result; novelty unassessed\"\n      statement: \"For every base b >= 2, an infinite positive support A with finite prime-weighted mass in base b has irrational support series; if a positive host H has finite prime-weighted mass in base 2, every infinite subset of H has irrational support series at every base b >= 2.\"\n      boundary: \"The finite-prime weighted-summability premise is explicit in both clauses. This criterion does not cover every infinite support or settle universal Erdős #257; no novelty, priority, significance, or external human-review claim is made.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.divisibilityWeightedClaim\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos257.PaperCompleteR8.finitePrimeWeighted_fixedBase_hereditary\"\n      file: \"ErdosProblems/Erdos257/PaperCompleteR8/WeightedHereditaryClaim.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved result; novelty unassessed\"\n      statement: \"For every base b >= 2, if a positive host H has finite prime-weighted mass in base b, every infinite subset A of H has irrational support series in the same base b.\"\n      boundary: \"This hereditary result retains the same-base weighted-summability hypothesis; it does not infer all-base irrationality from weighted mass at an arbitrary base, or settle universal Erdős #257.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.finitePrimeWeighted_fixedBase_hereditary\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.irrational_erdosSupportSeries_pairwise_coprime\"\n      file: \"Erdos249257/CertificateKernel.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:support_families\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"formalisation of an existing theorem\"\n      statement: \"For every integer base b >= 2 and every infinite pairwise-coprime support A with summable reciprocal mass, the support-restricted Mersenne series is irrational.\"\n      boundary: \"This is the pairwise-coprime support theorem, not universal Erdos #257; the infinitude, pairwise-coprimality, and summable-reciprocal hypotheses remain explicit.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.irrational_erdosSupportSeries_pairwise_coprime\"\n      literature_dependencies:\n        - \"https://users.renyi.hu/~p_erdos/1969-09.pdf\"\n        - \"docs/PRIOR_ART.md\"\n        - \"paper/257/erdos-257-mersenne-support-subseries.tex\"\n      source_statement: \"For every integer base b at least 2 and every infinite pairwise-coprime support A with summable reciprocal mass, the support-restricted Mersenne reciprocal series is irrational.\"\n      source_mapping: \"The local theorem keeps the base, infinitude, pairwise-coprimality, and summable-reciprocal hypotheses explicit and concludes irrationality of erdosSupportSeries b A. At base 2 this is exactly the pairwise-coprime support subfamily of Erdős #257; it is a separate non-dilation representative from the full-support row.\"\n      historical_attribution: \"Erdos (1968), as recorded in the repository's prior-art ledger and the #257 problem note.\"\n      logical_dependency: \"The proof-bearing source supplies the adaptive CRT and weighted coefficient certificate argument locally; the cited source establishes historical attribution and statement fidelity, not an imported Lean dependency.\"\n    - declaration: \"Erdos249257.irrational_ratWeightSeries_eventuallyPeriodic\"\n      file: \"Erdos249257/CertificateKernel.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:periodic_nonnegative_weight_irrationality\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"formalised here; novelty unassessed\"\n      statement: \"For every integer base b >= 2 and every eventually-periodic nonnegative rational coefficient sequence γ with a positive value at a positive index in its periodic region, the series sum_a γ(a)/(b^a - 1) is irrational.\"\n      boundary: \"This is the nonnegative rational eventually-periodic coefficient theorem; its nonnegativity, eventual periodicity, and positive-tail hypotheses remain explicit, and the broader mixed-sign periodic theorem is not claimed.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.irrational_ratWeightSeries_eventuallyPeriodic\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.SupportSunflowerDichotomy.irrational_erdosSupportSeries_of_orthogonalPetalBouquet\"\n      file: \"Erdos249257/SupportSunflowerDichotomy.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:orthogonal_petal_sunflower_reduction\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"conditional reduction; novelty unassessed\"\n      statement: \"For every support A carrying the finite-core OrthogonalPetalBouquet structure and satisfying the exact uniform SunflowerForcedSlotTailSelection predicate, the base-2 support-restricted Mersenne series erdosSupportSeries 2 A is irrational.\"\n      boundary: \"This is a conditional reduction: the bouquet structure and uniform tail-selection predicate remain hypotheses; neither is constructed for arbitrary supports, so this does not add an unconditional Erdos #257 support family or solve the universal problem.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.irrational_erdosSupportSeries_of_orthogonalPetalBouquet\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.finite_period_noncollapse_rat_den\"\n      file: \"Erdos249257/CertificateKernel.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved result; novelty unassessed\"\n      statement: \"For a reduced finite Mersenne sum, the base order modulo the denominator equals the support lcm.\"\n      boundary: \"The finite-support theorem does not settle any infinite-support sum.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.finite_period_noncollapse_rat_den_interface\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos257.volume_supportedMersenneAchievementSet_dichotomy\"\n      file: \"ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved result; novelty unassessed\"\n      statement: \"A support-restricted Mersenne achievement set has exact finite-complement measure or measure zero for infinite complement.\"\n      boundary: \"The measure dichotomy does not classify rational points.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.volume_supportedMersenneAchievementSet_dichotomy\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.exists_shifted_odd_tail_nat_states_of_support_fraction\"\n      file: \"Erdos249257/RationalSupportCarrySkeleton.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"conditional reduction\"\n      statement: \"A rational support value with a positive support seed and denominator 2^c*v produces a positive natural shifted tail state with an exact binary recurrence and a modular doubling orbit modulo v.\"\n      boundary: \"This state-extraction theorem does not exclude rational infinite supports or settle universal Erdős #257; the support and rational-value hypotheses remain explicit.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.exists_shifted_odd_tail_nat_states_of_support_fraction\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry\"\n      file: \"Erdos249257/BooleanMobiusCarry.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:boolean_mobius_carry\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved exact equivalence plus conditional consequences; novelty unassessed\"\n      statement: \"For p : ℤ and q > 0, a nonempty positive-index support with erdosSupportSeries 2 A = p/q exists exactly when a quotient-only Boolean–Möbius carry certificate exists, with positive square-root-bounded integer orbit, divisible carry differences, and support reconstructed from its Boolean Möbius transform.\"\n      boundary: \"The equivalence is stated at the existence level with q > 0, 0 ∉ A, and a positive support witness; it gives no contradiction, no universal Erdős #257 irrationality result, and no producer of cofinally nonintegral shifts for the actual recurrence. The generic tempered-orbit theorem and the reciprocal-mass, unbounded-tail, and zero-window results are subordinate consequences, not separate families; the zero-window constant depends on the support and numerator and is not uniform. This does not settle every infinite support, and no novelty, priority, significance, external-review, or unrestricted #257 endpoint claim is made.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips\"\n      file: \"Erdos249257/BooleanMobiusSkipRowCofinal.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"conditional reduction; novelty unassessed\"\n      statement: \"If positive rational-greedy skips occur cofinally in the declared sense, the skipped-core construction supplies cofinally many exact local Mersenne half rows and hence (1 / 2 : ℝ) belongs to the Mersenne achievement set.\"\n      boundary: \"This route is conditional on the unproved CofinalPositiveHalfGreedySkips premise: for every N there is c ≥ max N 4 with a positive greedy remainder below mersenneWeightRat c. A finite exact row or any finite collection of exact rows does not establish that cofinal supply. The result is a conditional 1/2-membership/counterexample route only, not an unconditional solution of Erdős #257; the exact-row, upper-half Boolean-word, rank-two, and endpoint-defect declarations are subordinate mechanism evidence, and no novelty, priority, significance, or external-review claim is made.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse\"\n      file: \"Erdos249257/HalfCylinderHalfMembershipClassification.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:half_membership_seam_classification\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved exact endpoint classification; novelty unassessed\"\n      statement: \"Membership of 1/2 in the Mersenne achievement set is equivalent to unbounded false successor terminal bits, equivalently to an unbounded upper-or-middle successor branch, cofinal terminal-false positions, unbounded skipped ranks along the integer seam, infinite greedy skipped support, and no last half-greedy skip.\"\n      boundary: \"This exact endpoint-socket classification does not supply any unbounded terminal-false or upper-or-middle occurrences, cofinal supply, skipped-support producer, or half-membership witness; finite exact rows do not establish the required global supply. It is not an unconditional or universal Erdős #257 result, and no novelty, priority, significance, or external-review claim is made.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.CompositeDilationDefect.supportCoeff_mul_eq_add_defect\"\n      file: \"Erdos249257/CompositeDilationDefect.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:composite_dilation_defect_identity\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"unconditional progress\"\n      statement: \"For a positive support element a and positive x, the divisor-count coefficient at a*x splits into the coefficient at x, the distinguished new divisor when a does not divide x, and an exact finite composite-dilation defect.\"\n      boundary: \"The exact defect is not bounded for arbitrary supports and supplies no tail selector or irrationality theorem; the separate bouquet budget remains conditional on its structural hypotheses.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.supportCoeff_mul_eq_add_defect\"\n      literature_dependencies: []\n    - declaration: \"Erdos249257.HalfGreedyTwoThirdsBand.seven_le_of_intBand_odd\"\n      file: \"Erdos249257/HalfGreedyTwoThirdsBand.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"supports_registered_claim_family:half_greedy_two_thirds_band\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"no-go result\"\n      statement: \"For positive odd integers p, D, and q, the integer form of the two-thirds band forces p ≥ 7; the small odd numerator classes 1, 3, and 5 are therefore excluded.\"\n      boundary: \"This is local 2-adic band arithmetic: p = 7 is sharp for the stated hypotheses, and the theorem does not prove that an actual half-greedy orbit reaches or avoids the band, nor does it prove half-membership.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.seven_le_of_intBand_odd\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos269.kernel_235_minor_eq_neg_one_fifteen\"\n      file: \"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"no-go result\"\n      statement: \"The smallest displayed 2,3,5 kernel minor is exactly -1/15.\"\n      boundary: \"Failure of rank one does not imply irrationality.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.kernel_235_minor_eq_neg_one_fifteen\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos269.smoothPrefixLcm_eq_threePrimeHeight\"\n      file: \"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved result; novelty unassessed\"\n      statement: \"The running lcm of the three-prime smooth prefix equals the exact coordinatewise height.\"\n      boundary: \"The identity does not supply the missing irrationality bridge.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.smoothPrefixLcm_eq_threePrimeHeight\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos269.no_positive_reducedCarry_of_cofinalLocalWindowEscape\"\n      file: \"ErdosProblems/Erdos269/RestrictedFloorSum.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"conditional no-go consumer; novelty and significance unassessed\"\n      statement: \"Under the denominator-dependent cofinal local-window residue-escape predicate, no positive reduced carry can satisfy the exact multiplier recurrence together with its short bound.\"\n      boundary: \"The representative assumes b,m : ℕ → ℕ, shortBound : ℕ → ℕ → ℕ, CofinalLocalWindowEscape b m shortBound, and for each positive B coprime to 30 a positive integer-valued d with d(n+1) = (b n : ℤ) d(n) − (B : ℤ) (m n : ℤ) and Int.natAbs (d n) ≤ shortBound B n; it then gives False. The representative remains a conditional consumer for arbitrary words. For the actual three-prime word, the rationality-to-reduced-carry bridge is established separately; the source-specific cofinal escape producer is not proved. This is not a #269 endpoint or irrationality proof, and no actual-series identification, novelty, priority, significance, or external-review claim is made. It is one conditional window-carry family, distinct from the finite residue, rank-two, and weighted-phase observer families.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.no_positive_reducedCarry_of_cofinalLocalWindowEscape\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos1041.exists_small_translation_separating_arguments\"\n      file: \"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved result; novelty unassessed\"\n      statement: \"An arbitrarily small common translation separates a finite injective family from zero and from shared positive rays.\"\n      boundary: \"Finite-family avoidance does not provide the global topological or metric gluing.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.exists_small_translation_separating_arguments\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos1041.constant_perturbation_roots_in_unitDisk\"\n      file: \"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved result; novelty unassessed\"\n      statement: \"A quantified small constant perturbation keeps every polynomial root inside the open unit disc.\"\n      boundary: \"Root retention is only one input to an incomplete global route.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.constant_perturbation_roots_in_unitDisk\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos1049.threeHalves_no_coordinatewiseCorridor\"\n      file: \"ErdosProblems/Erdos1049/RationalBaseLambert.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"no-go result\"\n      statement: \"The coordinatewise divisibility corridor is impossible at base 3/2.\"\n      boundary: \"This excludes one proof architecture and proves no irrationality statement.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.threeHalves_no_coordinatewiseCorridor\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos1049.rationalBaseClearedTailQ_succ\"\n      file: \"ErdosProblems/Erdos1049/RationalBaseLambert.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved result; novelty unassessed\"\n      statement: \"The rational-base cleared tail obeys its exact first-order recurrence.\"\n      boundary: \"The recurrence is not derived from a rationality contradiction.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.rationalBaseClearedTailQ_succ\"\n      literature_dependencies: []\n    - declaration: \"ErdosProblems.Erdos1049.rectangular_hp_threshold_eq_classical_iff\"\n      file: \"ErdosProblems/Erdos1049/HermitePadeNoGo.lean\"\n      sorry_count: 0\n      axioms:\n        - \"propext\"\n        - \"Quot.sound\"\n        - \"Classical.choice\"\n      comparator_config: \"verification/comparator.json\"\n      canonical_claim_status: \"comparator_interface_not_registered_as_canonical_claim\"\n      local_proof_provenance: \"proved_in_this_repository\"\n      novelty_status: \"unassessed_no_priority_claim\"\n      contribution_class: \"locally proved sharp model-specific no-go; novelty unassessed\"\n      statement: \"For the explicit rectangular two-function exponent model, with real rho and sigma satisfying 0 <= rho and 1 + rho <= sigma, the threshold equals 1/2 - 1/pi^2 exactly when rho = 0 and sigma = 1.\"\n      boundary: \"This is a sharp no-go only for the explicit rectangular two-function model under the stated real hypotheses; it constructs no approximating polynomials or remainders and is not a universal Pade or Hermite-Pade no-go. It proves no irrationality at 3/2 or any general rational-base endpoint, and the analytic remainder/nonvanishing input remains untreated. No novelty, priority, significance, external-review, or endpoint claim is made; Erdős #1049 remains open.\"\n      comparator_declaration: \"Erdos249257.ExternalVerification.rectangular_hp_threshold_eq_classical_iff\"\n      literature_dependencies: []\nautomation:\n  methods:\n    - method: \"agent\"\n      models: [\"OpenAI Codex and other disclosed agent systems\"]\n      framework: \"agent-assisted Lean formalisation and proof engineering\"\n      tool_setup: \"Repository-specific generators, release guards, and focused Lean builds\"\n    - method: \"other\"\n      framework: \"Pinned Lean kernel and statement-isolated Comparator CI\"\ncomparator:\n  config: \"verification/comparator.json\"\n  roster_source: \"verification/comparator.json\"\n  theorem_name_count: 53\n  theorem_names:\n    - \"Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses\"\n    - \"Erdos249257.ExternalVerification.irrational_factorialGapSeries_iff_cofinal_nonunit_carries\"\n    - \"Erdos249257.ExternalVerification.irrational_factorialGapSeries_of_cofinal_global_complementaryTail\"\n    - \"Erdos249257.ExternalVerification.no_cofinallyBoundedNegative_of_normalizedVanishes\"\n    - \"Erdos249257.ExternalVerification.boundedNegativePart_eventually_zero\"\n    - \"Erdos249257.ExternalVerification.sylvesterNext_eventually_of_summable_negativeRelativeMass\"\n    - \"Erdos249257.ExternalVerification.no_eventuallyPeriodicNegative_orbit\"\n    - \"Erdos249257.ExternalVerification.finrank_totientKernelThroughLevelFamily_eq\"\n    - \"Erdos249257.ExternalVerification.not_finiteDimensional_span_fullTotientKernel\"\n    - \"Erdos249257.ExternalVerification.exists_totientDyadicSectionBasis\"\n    - \"Erdos249257.ExternalVerification.finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent\"\n    - \"Erdos249257.ExternalVerification.tsum_pos_coprime_inv_mersenne_eq_one\"\n    - \"Erdos249257.ExternalVerification.tsum_totient_div_mersenne_sq_eq_gcd_moment_series\"\n    - \"Erdos249257.ExternalVerification.runHeight_defect_fib_sum_lower\"\n    - \"Erdos249257.ExternalVerification.cylinderMass_split\"\n    - \"Erdos249257.ExternalVerification.lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient\"\n    - \"Erdos249257.ExternalVerification.fixedPrecisionTropicalNoGo\"\n    - \"Erdos249257.ExternalVerification.irrational_totientSeries_of_actualLcmOrbitSeparationSupply\"\n    - \"Erdos249257.ExternalVerification.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap\"\n    - \"Erdos249257.ExternalVerification.exists_clean_binaryCyclotomicAnchor\"\n    - \"Erdos249257.ExternalVerification.scaleFullTarget_miss_of_projected_separation\"\n    - \"Erdos249257.ExternalVerification.irrational_totient_series_of_pivotResidualDecorrelation\"\n    - \"Erdos249257.ExternalVerification.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank\"\n    - \"Erdos249257.ExternalVerification.exists_primeGap0_gt\"\n    - \"Erdos249257.ExternalVerification.coefficientOnlyNoGo\"\n    - \"Erdos249257.ExternalVerification.irrational_tsum_primeDyadicTerm_iff_primeGap\"\n    - \"Erdos249257.ExternalVerification.irrational_initial_iff_all_positive_tailShifts_nonintegral\"\n    - \"Erdos249257.ExternalVerification.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch\"\n    - \"Erdos249257.ExternalVerification.carry_eq_residueDigit_add_coboundary\"\n    - \"Erdos249257.ExternalVerification.volume_mersenneAchievementSet\"\n    - \"Erdos249257.ExternalVerification.irrational_erdosSum_full_support\"\n    - \"Erdos249257.ExternalVerification.divisibilityWeightedClaim\"\n    - \"Erdos249257.ExternalVerification.finitePrimeWeighted_fixedBase_hereditary\"\n    - \"Erdos249257.ExternalVerification.irrational_erdosSupportSeries_pairwise_coprime\"\n    - \"Erdos249257.ExternalVerification.irrational_ratWeightSeries_eventuallyPeriodic\"\n    - \"Erdos249257.ExternalVerification.irrational_erdosSupportSeries_of_orthogonalPetalBouquet\"\n    - \"Erdos249257.ExternalVerification.finite_period_noncollapse_rat_den_interface\"\n    - \"Erdos249257.ExternalVerification.volume_supportedMersenneAchievementSet_dichotomy\"\n    - \"Erdos249257.ExternalVerification.exists_shifted_odd_tail_nat_states_of_support_fraction\"\n    - \"Erdos249257.ExternalVerification.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry\"\n    - \"Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips\"\n    - \"Erdos249257.ExternalVerification.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse\"\n    - \"Erdos249257.ExternalVerification.supportCoeff_mul_eq_add_defect\"\n    - \"Erdos249257.ExternalVerification.seven_le_of_intBand_odd\"\n    - \"Erdos249257.ExternalVerification.kernel_235_minor_eq_neg_one_fifteen\"\n    - \"Erdos249257.ExternalVerification.smoothPrefixLcm_eq_threePrimeHeight\"\n    - \"Erdos249257.ExternalVerification.no_positive_reducedCarry_of_cofinalLocalWindowEscape\"\n    - \"Erdos249257.ExternalVerification.exists_small_translation_separating_arguments\"\n    - \"Erdos249257.ExternalVerification.constant_perturbation_roots_in_unitDisk\"\n    - \"Erdos249257.ExternalVerification.exists_two_roots_dist_sum_le_two_mul_geomMean\"\n    - \"Erdos249257.ExternalVerification.threeHalves_no_coordinatewiseCorridor\"\n    - \"Erdos249257.ExternalVerification.rationalBaseClearedTailQ_succ\"\n    - \"Erdos249257.ExternalVerification.rectangular_hp_threshold_eq_classical_iff\"\nreview:\n  status: \"self-assessed\"\n  reviewers: []\n  notes: \"self-assessed and agent-checked; no human mathematical peer review is claimed\"\nfidelity:\n  divergences: \"Using the degree-seven polynomial constructed by the erdosproblems.com contributor ani, Lean proves that every preconnected strict-lemniscate set containing two distinct roots has one-dimensional Hausdorff measure greater than two. This refutes the exact Formal Conjectures path-image-length statement; the separate total-variation bound is also checked. The other seven targets remain open. Independent human review of correspondence with the 1958 wording has not been recorded. Comparator checks only selected exact statements, axioms and kernel acceptance; it does not assess novelty or historical correspondence.\"\nalignment:\n  namespace: \"Erdos249257.ExternalVerification\"\n  statements:\n    - source: \"Irrationality is equivalent to cofinally many strict-successor divisibility failures.\"\n      lean: \"ErdosProblems.Erdos68.irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses\"\n      module: \"ErdosProblems/Erdos68/FactorialGapPlateauCore.lean\"\n      status: \"proved\"\n      note: \"The equivalence does not prove that the cofinal failures occur.\"\n    - source: \"Irrationality is equivalent to cofinally many non-unit carries.\"\n      lean: \"ErdosProblems.Erdos68.irrational_factorialGapSeries_iff_cofinal_nonunit_carries\"\n      module: \"ErdosProblems/Erdos68/FactorialGapPlateauCore.lean\"\n      status: \"proved\"\n      note: \"The equivalence does not supply a cofinal carry producer.\"\n    - source: \"Normalised vanishing and bounded rise exclude a cofinally bounded negative part.\"\n      lean: \"ErdosProblems.Erdos243.no_cofinallyBoundedNegative_of_normalizedVanishes\"\n      module: \"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean\"\n      status: \"proved\"\n      note: \"All dynamics, bounded-rise, positivity, and vanishing hypotheses remain explicit.\"\n    - source: \"Under exact natural C/D dynamics with a > 1, C > 0, E = D - (a - 1)C, strict |E| < C, a uniform lower bound on E, and division-free normalized vanishing, boundedNegativePart_eventually_zero forces E = 0 eventually; centered-zero plus an eventually nonzero next tail recovers eventual Sylvester recurrence.\"\n      lean: \"ErdosProblems.Erdos243.boundedNegativePart_eventually_zero\"\n      module: \"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean\"\n      status: \"proved\"\n      note: \"The signed theorem has a C D : ℕ → ℕ, E : ℕ → ℤ, and B : ℕ with all exact hypotheses explicit: a > 1, C > 0, C(n + 1) + D(n) = a(n) · C(n), D(n + 1) = a(n) · D(n), E is the centered state D − (a − 1)C, |E| < C, a uniform lower bound on E, and division-free normalized vanishing. It excludes persistent bounded negative behaviour and forces eventual centered defect zero; centered-zero plus an eventually nonzero next tail recovers Sylvester, but no reciprocal-tail irrationality follows. The unbounded negative mixed-sign branch and the prime-specific producer remain open. This is distinct from the periodic negative-orbit and cofinally bounded-negative no-go families; no novelty, priority, significance, or external-review claim is made, and no unrestricted #243 solution follows.\"\n    - source: \"Under exact next-denominator and next-tail C/D dynamics with a, D : ℕ → ℤ, C : ℕ → ℕ, C > 0, the strict centered step equation, division-free normalized vanishing, and Summable normalized negative centered mass, the centered state vanishes eventually and a follows the Sylvester recurrence eventually.\"\n      lean: \"ErdosProblems.Erdos243.sylvesterNext_eventually_of_summable_negativeRelativeMass\"\n      module: \"ErdosProblems/Erdos243/SparseResetRecovery.lean\"\n      status: \"proved\"\n      note: \"This is a conditional recovery criterion: the source theorem assumes every displayed dynamics, positivity, strict centered-step, normalized-vanishing, and summability hypothesis, and does not prove them for the original Erdős #243 orbit. Any surviving canonical orbit must have divergent normalized negative mass. It supplies no prime-specific producer, unconditional #243 endpoint, or reciprocal-tail irrationality, and makes no novelty, priority, significance, or external-review claim. It is distinct from centered_state_dynamics, which uses a uniform lower-bound hypothesis rather than summability of normalized negative mass.\"\n    - source: \"An eventually periodic negative-error orbit with positive phase growth is impossible.\"\n      lean: \"ErdosProblems.Erdos243.no_eventuallyPeriodicNegative_orbit\"\n      module: \"ErdosProblems/Erdos243/ReciprocalTailRigidity.lean\"\n      status: \"proved\"\n      note: \"The theorem does not cover arbitrary unbounded negative behaviour.\"\n    - source: \"For every e >= 1, the complete dyadic totient kernel through level e has rational span dimension 2^e + 1.\"\n      lean: \"Erdos249257.finrank_totientKernelThroughLevelFamily_eq\"\n      module: \"Erdos249257/TotientMahlerDefect.lean\"\n      status: \"proved\"\n      note: \"This does not prove irrationality of the totient series.\"\n    - source: \"The rational span of the full dyadic totient kernel is not finite-dimensional.\"\n      lean: \"Erdos249257.not_finiteDimensional_span_fullTotientKernel\"\n      module: \"Erdos249257/TotientMahlerDefect.lean\"\n      status: \"proved\"\n      note: \"The declaration is kernel-checked locally, but full-kernel infinite-dimensionality is not an independent contribution of this release; no rationality-to-finite-rank bridge is proved.\"\n    - source: \"The full dyadic totient kernel span admits the explicit odd-core-indexed basis.\"\n      lean: \"Erdos249257.totientDyadicSectionBasis\"\n      module: \"Erdos249257/TotientMahlerDefect.lean\"\n      status: \"proved\"\n      note: \"The basis theorem does not connect rationality to finite kernel rank.\"\n    - source: \"Under 2 ≤ k, 1 ≤ e, and linear independence of the explicit canonical all-base family, the actual all-base totient kernel through level e has rational span dimension k^e + 1.\"\n      lean: \"Erdos249257.finrank_allBaseTotientKernelThroughLevelFamily_eq_of_linearIndependent\"\n      module: \"Erdos249257/TotientKernelConditional.lean\"\n      status: \"proved\"\n      note: \"The linear-independence hypothesis is the external affine-ordering boundary and is not proved here; this does not prove irrationality of the binary totient series.\"\n    - source: \"The positive coprime Mersenne weights over the visible lattice sum exactly to one, giving the reduced-direction mass behind the fair-coin coprimality form of the totient constant.\"\n      lean: \"GcdMomentCalculus.tsum_pos_coprime_inv_mersenne_eq_one\"\n      module: \"Erdos249257/GcdMomentCalculus.lean\"\n      status: \"proved\"\n      note: \"This is a rational normalization identity and does not prove irrationality of the binary totient series or settle Erdős #249.\"\n    - source: \"At base one half, the squared-Mersenne Lambert series weighted by Euler's totient equals the dyadic series of Pillai's gcd-sum function minus n, the exact first gcd-moment transfer.\"\n      lean: \"GcdMomentCalculus.tsum_totient_div_mersenne_sq_eq_gcd_moment_series\"\n      module: \"Erdos249257/GcdMomentCalculus.lean\"\n      status: \"proved\"\n      note: \"This exact identity does not prove irrationality of the binary totient series; the constant-weight squared-Lambert irrationality is cited external mathematics, and no Erdős #249 endpoint follows.\"\n    - source: \"For every finite defect list e, the height of the corresponding positive Stern-Brocot run word is at least F_(r+3) + F_(r+1) times the total defect, where r is the number of runs.\"\n      lean: \"SternBrocotRunGeometry.runHeight_defect_fib_sum_lower\"\n      module: \"Erdos249257/SternBrocotRunGeometry.lean\"\n      status: \"proved\"\n      note: \"This exact combinatorial stability floor does not establish analytic denominator survival, irrationality of the binary totient series, or novelty; no Erdős #249 endpoint follows.\"\n    - source: \"At every positive Stern-Brocot node (a,b), the closed cylinder mass splits exactly into the node's stop mass and the two child cylinder masses.\"\n      lean: \"GcdMomentCalculus.cylinderMass_split\"\n      module: \"Erdos249257/GcdMomentCalculus.lean\"\n      status: \"proved\"\n      note: \"This exact telescoping recursion and its geometric interpretation do not prove irrationality of the binary totient series or novelty; the #249 endpoint remains open.\"\n    - source: \"For every t >= 3, a nonzero synthetic dyadic coboundary survives every finite integer shift polynomial while retaining the period-LCM totient factor ideals, exact cleared-prefix cancellation, and uniform state and letter bounds.\"\n      lean: \"Erdos249257.TotientTailPeriodKiller.lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient\"\n      module: \"Erdos249257/LcmFactorIdealPulseObstruction.lean\"\n      status: \"proved\"\n      note: \"The witness is synthetic: it does not assert that its forcing letters are actual totient differences, does not address nonlinear combinations, and leaves the unbounded certificate supply for Erdős #249 open.\"\n    - source: \"For every positive fixed precision, every finite word of odd valuation-unit symbols and every incoming integer carry state admit a compatible orbit whose successive states stay within the corresponding valuation-unit radii; fixed-precision local signatures therefore do not by themselves exclude finite centred completions.\"\n      lean: \"Erdos249257.TotientTailPeriodKiller.fixedPrecisionTropicalNoGo\"\n      module: \"Erdos249257/TropicalCurvatureCarry.lean\"\n      status: \"proved\"\n      note: \"This is a synthetic finite-word construction with unrestricted incoming/high quotient states. It does not impose the actual totient-difference relation or provide growing-precision/global correlation, and it gives no Erdős #249 endpoint or irrationality result. No novelty or priority claim is made. The one-step and finite-word completion theorems are subordinate mechanisms, not separate families.\"\n    - source: \"A cofinal quantitative separation supply for the actual power-two LCM tail orbit implies irrationality of the binary totient series.\"\n      lean: \"Erdos249257.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_actualLcmOrbitSeparationSupply\"\n      module: \"Erdos249257/TotientActualLcmOrbitSeparation.lean\"\n      status: \"proved\"\n      note: \"The cofinal separation supply is an open premise, stronger than mere non-integrality; this sufficient route is not an equivalence and leaves the punctured/top-edge boundary and the Erdős #249 endpoint open.\"\n    - source: \"The finite Möbius residue diagonal splits into projected foreign and divisor channels, every finite stable foreign-tail window obeys a closed geometric bound, and a controlled projection separated from every integer forces the full target to be missed.\"\n      lean: \"Erdos249257.ActualForeignResidueProjection.scaleFullTarget_miss_of_projected_separation\"\n      module: \"Erdos249257/ActualForeignResidueProjection.lean\"\n      status: \"proved\"\n      note: \"This is a conditional finite partition/geometric-majorant reduction with a generic complement-noncancellation consumer. The ControlledForeignProjection analytic bridge and any unbounded separated-projection supply remain unproved; no Erdős #249 endpoint, novelty, significance, priority, or external-review claim is made.\"\n    - source: \"The exact first-harmonic pivot decomposition and its 9X/10 gap yield a finite certificate whenever the stated PivotBudgetAt bounds hold, and the cofinal pivot-residual-decorrelation socket implies irrationality of the binary totient series.\"\n      lean: \"Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_pivotResidualDecorrelation\"\n      module: \"Erdos249257/FirstHarmonicPivot.lean\"\n      status: \"proved\"\n      note: \"The four-term decomposition and 9X/10 gap remain conditional on the explicit dyadic-room and PivotBudgetAt hypotheses; the cofinal decorrelation producer is unproved. The fibre/image lemmas do not give global supplier-prime isolation or a prime-distribution estimate, and no Erdős #249 irrationality conclusion is claimed.\"\n    - source: \"An 11/100 nonpositive-phase density on arbitrarily late dyadic blocks yields an 89/100 first-harmonic saving and a finite 9/10 gap; alternatively, a cofinal natural-prime tail-orbit strict gap below 9/10 yields adaptive pivot escape and conditional irrationality. The exact initial phase is the scaled-angle character of (2^h - 1) times the series, and the strict-gap route is an orbit of repeated squaring from that phase.\"\n      lean: \"ErdosProblems.Erdos249.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap\"\n      module: \"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean\"\n      status: \"proved\"\n      note: \"The phase-density, cofinal natural-prime strict-gap, associated uniform-margin, and cofinal nonpositive-prime-shift producers remain unproved for the actual totient orbit. Exact phase/integrality equivalence, finite dyadic-root exclusion for 1 ≤ h ≤ 16 and k ≤ 14, and non-dyadicity-to-cofinal-nonpositive-phase implications are classification or finite evidence only; they do not supply density or prime alignment, and the source does not prove actual root entry. This is a conditional reduction only: no unconditional Erdős #249 endpoint, novelty, priority, significance, or external-review claim is made. It is distinct from the finite first_harmonic_pivot_decomposition and actual_lcm_orbit_separation families.\"\n    - source: \"If the binary totient coefficient series were rational, one positive-multiplier tempered integral carry has dyadic sections uniformly eventually periodic modulo its multiplier and canonical carry-section span dimension at least 2^e − 1 for every e.\"\n      lean: \"Erdos249257.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank\"\n      module: \"Erdos249257/TotientTailCarryPeriod.lean\"\n      status: \"proved\"\n      note: \"This is a necessary consequence of rationality only: it gives no contradiction, no finite-rank upper bound for a rationality-supplied carry, and no irrationality proof for Erdős #249. Modulo a divisor of the multiplier, the forcing disappears and the carry becomes geometric, so quotient periodicity alone supplies no finite-ℚ-rank upper bound. The coefficient-side kernel-rank and certificate-equivalence families are not duplicated, and #249 remains open.\"\n    - source: \"The consecutive prime gaps are unbounded.\"\n      lean: \"ErdosProblems.Erdos251.exists_primeGap0_gt\"\n      module: \"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\"\n      status: \"proved\"\n      note: \"Unbounded coefficients alone do not force irrationality.\"\n    - source: \"The coefficient-only wrapper records the exact finite carry partial-sum identity together with non-eventual-periodicity of the synthetic coefficient stream and the actual prime-gap stream, exposing the limits of coefficient-only reasoning.\"\n      lean: \"ErdosProblems.Erdos251.carryPartialSum_natCast_eq\"\n      module: \"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\"\n      status: \"proved\"\n      note: \"The wrapper proves the finite identity and the two non-eventual-periodicity statements only. It does not assert the infinite rational-sum limit of the synthetic countermodel, identify the synthetic coefficient stream with actual prime gaps, establish a prime-gap tail bridge, prove #251 irrationality, or make novelty, priority, significance, or external-review claims.\"\n    - source: \"Lean checks that the prime-value dyadic series is irrational exactly when the prime-gap dyadic series is, discharging summability by an elementary polynomial prime bound rather than the prime-number theorem. Neither side is proved irrational. The already-known identity is Tao's (7 October 2025) and is recorded separately as cited only.\"\n      lean: \"ErdosProblems.Erdos251.irrational_tsum_primeDyadicTerm_iff_primeGap\"\n      module: \"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\"\n      status: \"proved\"\n      note: \"The equivalence does not prove irrationality of either series.\"\n    - source: \"For any integer-digit real dyadic tail recurrence T(N + 1) = 2*T(N) - g(N + 1), the initial value T(0) is irrational exactly when every positive-length tail shift is nonintegral.\"\n      lean: \"ErdosProblems.Erdos251.irrational_initial_iff_all_positive_tailShifts_nonintegral\"\n      module: \"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\"\n      status: \"proved\"\n      note: \"This exact normal form covers arbitrary integer-digit dyadic tail recurrences with the declared real state type; it does not supply a cofinal nonintegral shift for the consecutive-prime-gap recurrence. The prime-specific producer remains open, so no prime-gap irrationality, novelty, significance, or external-review claim is made and Erdős #251 remains open.\"\n    - source: \"For the actual prime-gap dyadic tail, cofinally many adjacent pairs of strictly small h-shifts with unequal corresponding prime gaps force failure of eventual RatIntegral h-shifts.\"\n      lean: \"ErdosProblems.Erdos251.primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch\"\n      module: \"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean\"\n      status: \"proved\"\n      note: \"The representative assumes T : ℕ → ℚ, the DyadicTailRecurrence for the actual prime-gap digits, and for every N₀ an N ≥ N₀ whose adjacent h-shifts both lie strictly in (-1, 1) while the corresponding prime gaps differ. Under that cofinal small-mismatch supply it excludes eventual RatIntegral h-shifts; it does not prove the supply or actual smallness. rationalPrimeGapTail_has_positive_shift_not_eventually_small records the contrary rational-state obstruction for at least one positive shift. This is a conditional reduction only: no #251 irrationality, infinite rational-sum limit, universal producer, novelty, priority, significance, or external-review claim is made; #251 remains open. The actual-prime-gap specialization is distinct from coefficient_only_no_go and does not duplicate prime_gap_reformulation.\"\n    - source: \"The base-2 Mersenne achievement set has Lebesgue measure exactly one.\"\n      lean: \"Erdos249257.volume_mersenneAchievementSet\"\n      module: \"Erdos249257/GreedyAchievementSet.lean\"\n      status: \"proved\"\n      note: \"This does not decide whether every infinite-support sum is irrational.\"\n    - source: \"For every integer base b >= 2, the full-support Erdos-Borwein series is irrational.\"\n      lean: \"Erdos249257.irrational_erdosSum_full_support\"\n      module: \"Erdos249257/CertificateKernel.lean\"\n      status: \"proved\"\n      note: \"This is the classical full-support theorem, not universal Erdos #257.\"\n    - source: \"For every base b >= 2, an infinite positive support A with finite prime-weighted mass in base b has irrational support series; if a positive host H has finite prime-weighted mass in base 2, every infinite subset of H has irrational support series at every base b >= 2.\"\n      lean: \"ErdosProblems.Erdos257.PaperCompleteR8.divisibilityWeightedClaim\"\n      module: \"ErdosProblems/Erdos257/PaperCompleteR8/WeightedReturn.lean\"\n      status: \"proved\"\n      note: \"The finite-prime weighted-summability premise is explicit in both clauses. This criterion does not cover every infinite support or settle universal Erdős #257; no novelty, priority, significance, or external human-review claim is made.\"\n    - source: \"For every base b >= 2, if a positive host H has finite prime-weighted mass in base b, every infinite subset A of H has irrational support series in the same base b.\"\n      lean: \"ErdosProblems.Erdos257.PaperCompleteR8.finitePrimeWeighted_fixedBase_hereditary\"\n      module: \"ErdosProblems/Erdos257/PaperCompleteR8/WeightedHereditaryClaim.lean\"\n      status: \"proved\"\n      note: \"This hereditary result retains the same-base weighted-summability hypothesis; it does not infer all-base irrationality from weighted mass at an arbitrary base, or settle universal Erdős #257.\"\n    - source: \"For every integer base b >= 2 and every infinite pairwise-coprime support A with summable reciprocal mass, the support-restricted Mersenne series is irrational.\"\n      lean: \"Erdos249257.irrational_erdosSupportSeries_pairwise_coprime\"\n      module: \"Erdos249257/CertificateKernel.lean\"\n      status: \"proved\"\n      note: \"This is the pairwise-coprime support theorem, not universal Erdos #257; the infinitude, pairwise-coprimality, and summable-reciprocal hypotheses remain explicit.\"\n    - source: \"For every integer base b >= 2 and every eventually-periodic nonnegative rational coefficient sequence γ with a positive value at a positive index in its periodic region, the series sum_a γ(a)/(b^a - 1) is irrational.\"\n      lean: \"Erdos249257.irrational_ratWeightSeries_eventuallyPeriodic\"\n      module: \"Erdos249257/CertificateKernel.lean\"\n      status: \"proved\"\n      note: \"This is the nonnegative rational eventually-periodic coefficient theorem; its nonnegativity, eventual periodicity, and positive-tail hypotheses remain explicit, and the broader mixed-sign periodic theorem is not claimed.\"\n    - source: \"For every support A carrying the finite-core OrthogonalPetalBouquet structure and satisfying the exact uniform SunflowerForcedSlotTailSelection predicate, the base-2 support-restricted Mersenne series erdosSupportSeries 2 A is irrational.\"\n      lean: \"Erdos249257.SupportSunflowerDichotomy.irrational_erdosSupportSeries_of_orthogonalPetalBouquet\"\n      module: \"Erdos249257/SupportSunflowerDichotomy.lean\"\n      status: \"proved\"\n      note: \"This is a conditional reduction: the bouquet structure and uniform tail-selection predicate remain hypotheses; neither is constructed for arbitrary supports, so this does not add an unconditional Erdos #257 support family or solve the universal problem.\"\n    - source: \"For a reduced finite Mersenne sum, the base order modulo the denominator equals the support lcm.\"\n      lean: \"Erdos249257.finite_period_noncollapse_rat_den\"\n      module: \"Erdos249257/CertificateKernel.lean\"\n      status: \"proved\"\n      note: \"The finite-support theorem does not settle any infinite-support sum.\"\n    - source: \"A support-restricted Mersenne achievement set has exact finite-complement measure or measure zero for infinite complement.\"\n      lean: \"ErdosProblems.Erdos257.volume_supportedMersenneAchievementSet_dichotomy\"\n      module: \"ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean\"\n      status: \"proved\"\n      note: \"The measure dichotomy does not classify rational points.\"\n    - source: \"A rational support value with a positive support seed and denominator 2^c*v produces a positive natural shifted tail state with an exact binary recurrence and a modular doubling orbit modulo v.\"\n      lean: \"Erdos249257.exists_shifted_odd_tail_nat_states_of_support_fraction\"\n      module: \"Erdos249257/RationalSupportCarrySkeleton.lean\"\n      status: \"proved\"\n      note: \"This state-extraction theorem does not exclude rational infinite supports or settle universal Erdős #257; the support and rational-value hypotheses remain explicit.\"\n    - source: \"For p : ℤ and q > 0, a nonempty positive-index support with erdosSupportSeries 2 A = p/q exists exactly when a quotient-only Boolean–Möbius carry certificate exists, with positive square-root-bounded integer orbit, divisible carry differences, and support reconstructed from its Boolean Möbius transform.\"\n      lean: \"Erdos249257.exists_normalized_support_fraction_iff_exists_booleanMobiusCarry\"\n      module: \"Erdos249257/BooleanMobiusCarry.lean\"\n      status: \"proved\"\n      note: \"The equivalence is stated at the existence level with q > 0, 0 ∉ A, and a positive support witness; it gives no contradiction, no universal Erdős #257 irrationality result, and no producer of cofinally nonintegral shifts for the actual recurrence. The generic tempered-orbit theorem and the reciprocal-mass, unbounded-tail, and zero-window results are subordinate consequences, not separate families; the zero-window constant depends on the support and numerator and is not uniform. This does not settle every infinite support, and no novelty, priority, significance, external-review, or unrestricted #257 endpoint claim is made.\"\n    - source: \"If positive rational-greedy skips occur cofinally in the declared sense, the skipped-core construction supplies cofinally many exact local Mersenne half rows and hence (1 / 2 : ℝ) belongs to the Mersenne achievement set.\"\n      lean: \"Erdos249257.half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips\"\n      module: \"Erdos249257/BooleanMobiusSkipRowCofinal.lean\"\n      status: \"proved\"\n      note: \"This route is conditional on the unproved CofinalPositiveHalfGreedySkips premise: for every N there is c ≥ max N 4 with a positive greedy remainder below mersenneWeightRat c. A finite exact row or any finite collection of exact rows does not establish that cofinal supply. The result is a conditional 1/2-membership/counterexample route only, not an unconditional solution of Erdős #257; the exact-row, upper-half Boolean-word, rank-two, and endpoint-defect declarations are subordinate mechanism evidence, and no novelty, priority, significance, or external-review claim is made.\"\n    - source: \"Membership of 1/2 in the Mersenne achievement set is equivalent to unbounded false successor terminal bits, equivalently to an unbounded upper-or-middle successor branch, cofinal terminal-false positions, unbounded skipped ranks along the integer seam, infinite greedy skipped support, and no last half-greedy skip.\"\n      lean: \"Erdos249257.half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse\"\n      module: \"Erdos249257/HalfCylinderHalfMembershipClassification.lean\"\n      status: \"proved\"\n      note: \"This exact endpoint-socket classification does not supply any unbounded terminal-false or upper-or-middle occurrences, cofinal supply, skipped-support producer, or half-membership witness; finite exact rows do not establish the required global supply. It is not an unconditional or universal Erdős #257 result, and no novelty, priority, significance, or external-review claim is made.\"\n    - source: \"For a positive support element a and positive x, the divisor-count coefficient at a*x splits into the coefficient at x, the distinguished new divisor when a does not divide x, and an exact finite composite-dilation defect.\"\n      lean: \"Erdos249257.CompositeDilationDefect.supportCoeff_mul_eq_add_defect\"\n      module: \"Erdos249257/CompositeDilationDefect.lean\"\n      status: \"proved\"\n      note: \"The exact defect is not bounded for arbitrary supports and supplies no tail selector or irrationality theorem; the separate bouquet budget remains conditional on its structural hypotheses.\"\n    - source: \"For positive odd integers p, D, and q, the integer form of the two-thirds band forces p ≥ 7; the small odd numerator classes 1, 3, and 5 are therefore excluded.\"\n      lean: \"Erdos249257.HalfGreedyTwoThirdsBand.seven_le_of_intBand_odd\"\n      module: \"Erdos249257/HalfGreedyTwoThirdsBand.lean\"\n      status: \"proved\"\n      note: \"This is local 2-adic band arithmetic: p = 7 is sharp for the stated hypotheses, and the theorem does not prove that an actual half-greedy orbit reaches or avoids the band, nor does it prove half-membership.\"\n    - source: \"The smallest displayed 2,3,5 kernel minor is exactly -1/15.\"\n      lean: \"ErdosProblems.Erdos269.kernel_235_minor_eq_neg_one_fifteen\"\n      module: \"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean\"\n      status: \"proved\"\n      note: \"Failure of rank one does not imply irrationality.\"\n    - source: \"The running lcm of the three-prime smooth prefix equals the exact coordinatewise height.\"\n      lean: \"ErdosProblems.Erdos269.smoothPrefixLcm_eq_threePrimeHeight\"\n      module: \"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean\"\n      status: \"proved\"\n      note: \"The identity does not supply the missing irrationality bridge.\"\n    - source: \"Under the denominator-dependent cofinal local-window residue-escape predicate, no positive reduced carry can satisfy the exact multiplier recurrence together with its short bound.\"\n      lean: \"ErdosProblems.Erdos269.no_positive_reducedCarry_of_cofinalLocalWindowEscape\"\n      module: \"ErdosProblems/Erdos269/RestrictedFloorSum.lean\"\n      status: \"proved\"\n      note: \"The representative assumes b,m : ℕ → ℕ, shortBound : ℕ → ℕ → ℕ, CofinalLocalWindowEscape b m shortBound, and for each positive B coprime to 30 a positive integer-valued d with d(n+1) = (b n : ℤ) d(n) − (B : ℤ) (m n : ℤ) and Int.natAbs (d n) ≤ shortBound B n; it then gives False. The representative remains a conditional consumer for arbitrary words. For the actual three-prime word, the rationality-to-reduced-carry bridge is established separately; the source-specific cofinal escape producer is not proved. This is not a #269 endpoint or irrationality proof, and no actual-series identification, novelty, priority, significance, or external-review claim is made. It is one conditional window-carry family, distinct from the finite residue, rank-two, and weighted-phase observer families.\"\n    - source: \"An arbitrarily small common translation separates a finite injective family from zero and from shared positive rays.\"\n      lean: \"ErdosProblems.Erdos1041.exists_small_translation_separating_arguments\"\n      module: \"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean\"\n      status: \"proved\"\n      note: \"Finite-family avoidance does not provide the global topological or metric gluing.\"\n    - source: \"A quantified small constant perturbation keeps every polynomial root inside the open unit disc.\"\n      lean: \"ErdosProblems.Erdos1041.constant_perturbation_roots_in_unitDisk\"\n      module: \"ErdosProblems/Erdos1041/NewtonFlowRaySeparation.lean\"\n      status: \"proved\"\n      note: \"Root retention is only one input to an incomplete global route.\"\n    - source: \"The coordinatewise divisibility corridor is impossible at base 3/2.\"\n      lean: \"ErdosProblems.Erdos1049.threeHalves_no_coordinatewiseCorridor\"\n      module: \"ErdosProblems/Erdos1049/RationalBaseLambert.lean\"\n      status: \"proved\"\n      note: \"This excludes one proof architecture and proves no irrationality statement.\"\n    - source: \"The rational-base cleared tail obeys its exact first-order recurrence.\"\n      lean: \"ErdosProblems.Erdos1049.rationalBaseClearedTailQ_succ\"\n      module: \"ErdosProblems/Erdos1049/RationalBaseLambert.lean\"\n      status: \"proved\"\n      note: \"The recurrence is not derived from a rationality contradiction.\"\n    - source: \"For the explicit rectangular two-function exponent model, with real rho and sigma satisfying 0 <= rho and 1 + rho <= sigma, the threshold equals 1/2 - 1/pi^2 exactly when rho = 0 and sigma = 1.\"\n      lean: \"ErdosProblems.Erdos1049.rectangular_hp_threshold_eq_classical_iff\"\n      module: \"ErdosProblems/Erdos1049/HermitePadeNoGo.lean\"\n      status: \"proved\"\n      note: \"This is a sharp no-go only for the explicit rectangular two-function model under the stated real hypotheses; it constructs no approximating polynomials or remainders and is not a universal Pade or Hermite-Pade no-go. It proves no irrationality at 3/2 or any general rational-base endpoint, and the analytic remainder/nonvanishing input remains untreated. No novelty, priority, significance, external-review, or endpoint claim is made; Erdős #1049 remains open.\"\nclaim_review_matrix:\n  - problem: 68\n    paper: \"paper/68/erdos-68-factorial-denominator-irrationality.tex\"\n    families:\n      - id: \"factorial_carry_characterisation\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"Exact equivalence between irrationality, cofinally many non-unit carries, and cofinal strict-successor divisibility failures.\"\n        boundary: \"An exact reformulation does not supply the required cofinal failures.\"\n      - id: \"factorial_channel_and_projection_rigidity\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"represented_by_isolated_headline\"\n        summary: \"Finite quotient-band channel factorisation and breakpoint, channel congruences, a two-term prime-channel corrector, and endpoint-weighted projection rigidity.\"\n        boundary: \"These finite and structural results do not produce a cofinal obstruction.\"\n      - id: \"factorial_conditional_producers\"\n        contribution_class: \"conditional reduction\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"not_selected_deep_project_predicate_stack\"\n        summary: \"Cofinal zero-branch or large private-factor hypotheses imply the required irrationality conclusion.\"\n        boundary: \"The producer hypotheses are unproved.\"\n      - id: \"factorial_finite_certificates\"\n        contribution_class: \"finite computation\"\n        evidence_mode: \"Lean finite theorem plus external exact certificate\"\n        comparator_disposition: \"not_applicable_to_external_execution\"\n        summary: \"Lean checks small exact misses; a hash-bound GMP scan reaches index 300000 and yields the corresponding finite denominator exclusion.\"\n        boundary: \"Finite computation does not change the cofinal quantifier.\"\n      - id: \"factorial_lcm_growth\"\n        contribution_class: \"ordinary paper argument\"\n        evidence_mode: \"Lean kernel plus paper argument\"\n        comparator_disposition: \"not_selected_separate_release_comparator_entry\"\n        summary: \"An elementary terminal-block argument gives liminf log L_N / (N^{3/2} log N) ≥ 2√2/3 for the lcm of factorial-gap denominators. The displayed bound needs no external multiplicity theorem; an older, weaker multiplicity-based route is historical context only.\"\n        boundary: \"The LCM asymptotic has an ordinary paper proof and is checked in Lean as common_denominator_growth_liminf in PaperCompleteLiminf.lean. The finite-block inequality has a separate Lean source. The Comparator entry for this bound belongs to the separate public release wcook04/plectis-erdos-lean. The bound does not decide irrationality.\"\n  - problem: 243\n    paper: \"paper/243/erdos-243-reciprocal-tail-rigidity.tex\"\n    families:\n      - id: \"centered_state_dynamics\"\n        contribution_class: \"locally proved signed recovery/rigidity theorem; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"Under exact natural C/D dynamics with a > 1, C > 0, E = D - (a - 1)C, strict |E| < C, a uniform lower bound on E, and normalized vanishing, boundedNegativePart_eventually_zero forces E = 0 eventually. centeredState_eventually_zero supplies the nonnegative stabilization mechanism, and sylvesterNext_eventually_of_centered_zero recovers eventual Sylvester recurrence when the next tail is nonzero.\"\n        boundary: \"The signed theorem has a C D : ℕ → ℕ, E : ℕ → ℤ, and B : ℕ with all exact hypotheses explicit: a > 1, C > 0, C(n + 1) + D(n) = a(n) · C(n), D(n + 1) = a(n) · D(n), E is the centered state D − (a − 1)C, |E| < C, a uniform lower bound on E, and division-free normalized vanishing. It excludes persistent bounded negative behaviour and forces eventual centered defect zero; centered-zero plus an eventually nonzero next tail recovers Sylvester, but no reciprocal-tail irrationality follows. The unbounded negative mixed-sign branch and the prime-specific producer remain open. This is distinct from the periodic negative-orbit and cofinally bounded-negative no-go families; no novelty, priority, significance, or external-review claim is made, and no unrestricted #243 solution follows.\"\n      - id: \"negative_orbit_no_go\"\n        contribution_class: \"no-go result\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted_representative_periodic_theorem\"\n        summary: \"Constant, eventually constant, periodic, and eventually periodic negative-error orbits are excluded in their stated regimes.\"\n        boundary: \"The exclusions do not cover arbitrary unbounded negative behaviour.\"\n      - id: \"bounded_rise_coprimality\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"represented_by_stronger_selected_boundary_theorem\"\n        summary: \"A bounded-rise sequence cannot remain coprime to fresh pairwise-coprime moduli; reduced tails inherit this obstruction.\"\n        boundary: \"The required bounded-rise input is not automatic for the original sequence.\"\n      - id: \"bounded_negative_exclusion\"\n        contribution_class: \"conditional reduction and no-go result\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"Normalised vanishing excludes a cofinally bounded negative part and forces eventual Sylvester behaviour.\"\n        boundary: \"Every positivity, dynamics, bounded-rise, and vanishing hypothesis remains explicit.\"\n      - id: \"negative_mass_recovery\"\n        contribution_class: \"conditional analytic recovery criterion; novelty unassessed\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"targeted\"\n        summary: \"Under exact reciprocal-tail C/D dynamics, positive tail, the strict centered-step equation, and division-free normalized vanishing, summability of the normalized negative centered mass forces centeredState to vanish eventually and recovers the Sylvester recurrence. The load-bearing theorem is supported by the exact tail-growth bound and the finite-negative-mass stabilization theorem.\"\n        boundary: \"The exact criterion assumes a, D : ℕ → ℤ and C : ℕ → ℕ with the declared next-denominator and next-tail dynamics, C > 0, the strict centered step equation, division-free normalized vanishing, and Summable (negativeRelativeMass C (fun n ↦ centeredState (a n) (D n) (C n))). It does not prove these hypotheses for the original Erdős #243 orbit, supply a prime-specific producer, or establish an unconditional #243 endpoint or reciprocal-tail irrationality. Any surviving canonical orbit must have divergent normalized negative mass. This is distinct from centered_state_dynamics, whose recovery criterion uses a uniform lower-bound hypothesis; no novelty, priority, significance, or external-review claim is made.\"\n  - problem: 249\n    paper: \"paper/249/erdos-249-binary-totient-series.tex\"\n    families:\n      - id: \"totient_kernel_basis\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"An explicit odd-core basis and relation normal form for the full dyadic totient kernel.\"\n        boundary: \"The basis theorem does not connect rationality of the series to finite kernel rank.\"\n      - id: \"totient_kernel_rank\"\n        contribution_class: \"mixed local finite-level theorem and formalised prior consequence\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"An independent constructive proof of exact finite-level dyadic rank 2^e+1, together with a kernel-checked full-kernel infinite-dimensionality corollary already implied by Coons's non-2-regularity theorem.\"\n        boundary: \"Only the finite-level rank is presented as an additional result; full-kernel infinite-dimensionality is prior-art context, and no rationality-to-finite-rank bridge is proved.\"\n        source_statement: \"Coons proved that Euler's totient sequence is not k-regular for every integer k at least 2, while Martin's affine-totient ordering theorem implies linear independence of every finite family of pairwise nonproportional positive-slope affine totient sections.\"\n        source_mapping: \"The dyadic Lean development gives an independent constructive finite-level basis and rank proof. Its full-kernel infinite-dimensionality corollary is kernel-checked here but is not an independent contribution: for integer-valued sequence modules it is equivalent to Coons's non-k-regularity result. The all-base Lean modules check zero-channel and composite-base reductions, residue coordinates, index cardinality k^e+1, canonical spanning and the reusable conditional-rank interface. Separately, AllBaseTotientKernel.lean at public checkpoint 25ef6245d15a proves canonical independence and the complete rank unconditionally by CRT and Dirichlet. This does not enlarge the Comparator target represented below or formalise Martin's broader theorem.\"\n        historical_attribution: \"Coons for non-k-regularity of Euler's totient sequence; Martin for the positive-density affine ordering theorem from which the paper's finite-family independence step follows.\"\n        logical_dependency: \"Arithmetic reduction and canonical spanning are unconditional. The recorded reusable rank wrapper takes independence as an input. AllBaseTotientKernel.lean discharges that input independently; Martin's broader positive-density theorem remains cited external prior work, not an axiom of the Lean proof.\"\n      - id: \"totient_kernel_all_base_index\"\n        contribution_class: \"formalised arithmetic, spanning and reusable conditional rank interface\"\n        evidence_mode: \"Lean kernel for this arithmetic, finite indexing, spanning and conditional-rank interface; the independent all-base proof is a separate public source result\"\n        comparator_disposition: \"targeted\"\n        summary: \"For every base k at least 2, Lean checks the arithmetic reduction, exact nonmultiple-residue coordinates, unconditional canonical spanning, and exact rank k^e+1 conditional on canonical-family linear independence.\"\n        boundary: \"This Comparator target reviews the reusable conditional interface. AllBaseTotientKernel.lean separately proves canonical all-base independence and exact rank unconditionally at 25ef6245d15a; it does not formalise Martin's broader theorem or solve #249.\"\n      - id: \"totient_visible_coprime_lattice_mass\"\n        contribution_class: \"formalised exact visible-lattice normalization identity\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"The positive coprime Mersenne weights on the visible lattice sum to one, exposing the reduced-direction mass behind the fair-coin coprimality form of #249.\"\n        boundary: \"This rational normalization identity does not prove irrationality of the binary totient series or settle #249.\"\n      - id: \"squared_lambert_gcd_moments\"\n        contribution_class: \"formalised exact squared-Lambert first-moment transfer\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"At base one half, the totient-weighted squared-Mersenne Lambert series is the dyadic Pillai gcd-sum-minus-n series, exposing the first fair-coin gcd moment.\"\n        boundary: \"This exact transfer identity does not prove irrationality of the binary totient series; the cited constant-weight q-Padé result remains external and no #249 endpoint follows.\"\n      - id: \"stern_brocot_run_fibonacci_stability\"\n        contribution_class: \"formalised exact Fibonacci/continuant run stability floor\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"For a positive Stern-Brocot run word, the global height gain from run defects is bounded below by the sharp Fibonacci coefficient F_(r+1) times the total defect.\"\n        boundary: \"This exact combinatorial stability floor does not establish analytic denominator survival, irrationality of the binary totient series, or novelty; no #249 endpoint follows.\"\n      - id: \"stern_brocot_cylinder_law\"\n        contribution_class: \"formalised exact Stern-Brocot cylinder telescoping identity\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"At every positive Stern-Brocot node, the closed cylinder mass splits into stop mass and the two child cylinders, giving an exact Markov recursion.\"\n        boundary: \"This exact telescoping recursion and its geometric interpretation do not prove irrationality of the binary totient series or novelty; the #249 endpoint remains open.\"\n      - id: \"lcm_factor_ideal_shift_algebra_no_go\"\n        contribution_class: \"unconditional progress\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"For every t at least 3, a nonzero synthetic dyadic coboundary survives every finite integer shift polynomial while retaining the period-LCM totient factor ideals, exact cleared-prefix cancellation, and uniform state and letter bounds.\"\n        boundary: \"The witness is synthetic: it does not assert that its forcing letters are actual totient differences, does not address nonlinear combinations, and leaves the unbounded certificate supply for Erdős #249 open.\"\n      - id: \"fixed_precision_transport_no_go\"\n        contribution_class: \"unconditional synthetic no-go; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"At every fixed positive precision, every finite word of odd valuation-unit symbols admits a compatible centred carry completion from any incoming integer state; bounded local signatures therefore cannot exclude all finite centred endpoints.\"\n        boundary: \"This is a synthetic finite-word construction with unrestricted incoming/high quotient states. It does not impose the actual totient-difference relation or provide growing-precision/global correlation, and it gives no Erdős #249 endpoint or irrationality result. No novelty or priority claim is made. The one-step and finite-word completion theorems are subordinate mechanisms, not separate families.\"\n      - id: \"actual_lcm_orbit_separation\"\n        contribution_class: \"conditional reduction\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"A cofinal quantitative 1/32 separation supply at canonically guarded odd ranks transfers through the explicit raw-block approximation and error radius to the existing half-word/signed-margin irrationality consumer. The landed actual-LCM sign/top-edge analysis is supporting and contrary-boundary evidence: positivity leaves the true survivor negative, and integrality forces the exact top-edge residue.\"\n        boundary: \"The quantitative 1/32 separation supply is sufficient but stronger than mere non-integrality, is not an equivalence, and remains open. The sign positivity alone does not exclude the negative true survivor; the top-edge forcing does not exclude that residue or supply cofinal non-integrality. These source declarations are supporting/contrary-boundary evidence, not a second actual-LCM family, and the punctured/top-edge boundary and #249 endpoint remain open.\"\n      - id: \"actual_foreign_residue_projection\"\n        contribution_class: \"conditional reduction\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"The finite Möbius residue diagonal splits exactly into projected foreign and divisor channels, the omitted finite foreign tail has a closed geometric majorant, and the generic complement-noncancellation consumer turns a controlled projection separated from every integer into a full-target miss.\"\n        boundary: \"This is only a finite partition/geometric-majorant reduction feeding a generic complement-noncancellation consumer. The ControlledForeignProjection analytic bridge and any unbounded supply of separated projections remain unproved; no Erdős #249 endpoint, novelty, significance, priority, or external-review claim follows.\"\n      - id: \"first_harmonic_pivot_decomposition\"\n        contribution_class: \"conditional reduction\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"The supplier fibre is an exact prime-image/fibre bijection, while a checked finite counterexample shows that supplier-prime global isolation fails. The load-bearing result is an exact four-term first-harmonic decomposition; under the explicit PivotBudgetAt budgets, the centred real-part bound and three contribution bounds give the 9X/10 gap and a finite certificate, and the conditional pivot-residual-decorrelation theorem transfers a cofinal supply to irrationality.\"\n        boundary: \"The fibre/image lemmas and the explicit counterexample establish neither global supplier-prime isolation nor a prime-distribution estimate. The 9X/10 gap is conditional on PivotBudgetAt and the stated dyadic-room inequality; irrationality is conditional on DTWPivotResidualDecorrelation, whose exact cofinal producer remains open. No decorrelation estimate supplying those budgets is proved, and the #249 endpoint remains open.\"\n      - id: \"strict_prime_tail_orbit_gap\"\n        contribution_class: \"conditional reduction\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"A single two-branch source mechanism gives an 89/100 first-harmonic block saving from an 11/100 nonpositive-phase density, or an adaptive pivot escape from a cofinal natural-prime tail-orbit strict gap below 9/10. The exact squaring law reduces the latter to a power-of-two orbit of one initial phase and shows that an entered dyadic root obstructs that producer, without proving such an entry. The load-bearing representative is the natural-prime irrationality reduction; block-gap, density, and phase-classification declarations remain subordinate.\"\n        boundary: \"The 11/100 phase-density, cofinal natural-prime strict-gap, associated uniform-margin, and cofinal nonpositive-prime-shift producers remain unproved for the actual totient orbit. The scaled-angle and phase/integrality equivalences, finite 1 ≤ h ≤ 16 and k ≤ 14 root exclusion, and non-dyadicity-to-cofinal-nonpositive-phase theorem provide classification or finite evidence only; they do not prove a density/equidistribution theorem, prime alignment, or actual dyadic-root entry. Both branches are conditional certificate/irrationality reductions only: they do not close Erdős #249, duplicate first_harmonic_pivot_decomposition or actual_lcm_orbit_separation, or justify novelty, priority, significance, or external review.\"\n      - id: \"totient_carry_anti_compression\"\n        contribution_class: \"locally proved necessary consequence; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"If the binary totient coefficient series were rational, one positive-multiplier tempered integral carry would have dyadic sections uniformly eventually periodic modulo its multiplier while its canonical carry-section span has dimension at least 2^e − 1 for every e. This is the orbit-level quotient-periodicity/anti-compression conjunction, distinct from the coefficient-side kernel-rank family.\"\n        boundary: \"This is a necessary consequence of rationality only: it gives no contradiction, no finite-rank upper bound for a rationality-supplied carry, and no irrationality proof for Erdős #249. Modulo a divisor of the multiplier, the forcing disappears and the carry becomes geometric, so quotient periodicity alone supplies no finite-ℚ-rank upper bound. The coefficient-side kernel-rank and certificate-equivalence families are not duplicated, and #249 remains open.\"\n      - id: \"totient_finite_denominator_exclusion\"\n        contribution_class: \"finite computation\"\n        evidence_mode: \"Lean kernel and generated finite certificates\"\n        comparator_disposition: \"not_selected_finite_family_has_separate_receipt\"\n        summary: \"A Farey denominator exclusion through 7.96e34, sharp for its selected window, and diagonal certificates through t=82.\"\n        boundary: \"The selected finite window and certificate ceiling do not imply an unbounded producer.\"\n      - id: \"totient_carry_rank\"\n        contribution_class: \"conditional reduction\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"not_selected_deep_orbit_vocabulary\"\n        summary: \"Rationality forces a tempered integral tail orbit with unbounded finite-level carry rank.\"\n        boundary: \"This is a necessary consequence, not a contradiction.\"\n      - id: \"totient_lambert_coefficients\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"represented_by_basis_and_rank_targets\"\n        summary: \"The totient series is rewritten as a Lambert series with explicit nonnegative, unbounded prime-power coefficients.\"\n        boundary: \"The coefficient identities do not prove irrationality.\"\n      - id: \"eventually_periodic_lambert\"\n        contribution_class: \"Lean formalisation of existing result\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"not_selected_prior_result\"\n        summary: \"Eventually periodic nonnegative rational weights give an irrational Lambert series.\"\n        boundary: \"The totient-derived weights are not eventually periodic.\"\n        source_statement: \"For an eventually periodic sequence of nonnegative rational coefficients that is not eventually zero, the associated Lambert series at an integer base b at least 2 is irrational.\"\n        source_mapping: \"The local theorem formalises the nonnegative rational eventually-periodic coefficient case. Its coefficient, base, sign, and non-eventual-zero hypotheses are kept explicit; it does not claim the broader mixed-sign theorem.\"\n        historical_attribution: \"Luca and Tachiya, Irrationality of Lambert series associated with a periodic sequence (2014), with the later RIMS exposition used for statement comparison.\"\n        logical_dependency: \"The local Lean proof uses a separate periodic-divisor certificate route; the cited paper establishes attribution and source fidelity, not a kernel dependency.\"\n      - id: \"totient_certificate_equivalences\"\n        contribution_class: \"exact equivalence and conditional reduction\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"not_selected_deep_certificate_vocabulary\"\n        summary: \"Pointwise, cofinal, lcm-diagonal, and separated-window certificate supplies are exact or sufficient reformulations.\"\n        boundary: \"Equivalent producer statements are as hard as the unresolved target.\"\n      - id: \"signed_totient_entire_defect_interpolation\"\n        contribution_class: \"ordinary paper argument\"\n        evidence_mode: \"ordinary proof; two AI reviews, no independent human review\"\n        comparator_disposition: \"not_applicable_without_exact_Lean_statement\"\n        summary: \"For any integer beta>=2 and positive allowance f(n) tending to infinity, an interval around the totient value is filled by integer sequences b=phi+e with 0<=b(n)<=n and |e(n)|<=f(n). One target-independent even support has upper Banach density zero, leaving any fixed prefix and odd coefficients unchanged. The construction preserves every fixed eventual coefficient and cumulative congruence, gives eventually constant progression moment defects outside common hulls of density zero, entire progression Dirichlet defects, and smooth root-of-unity radial germs with prescribed finite zero jets. The full proof includes the allowance-dependent block bound and first-harmonic rational countermodels. It is an ordinary proof with two AI proof reviews, not a Lean proof of the infinite theorem or independent human review.\"\n        boundary: \"The correction is signed, so it does not subsume the one-sided prime-gap theorem. It preserves singularities and prescribed finite jets, not full analytic functions or all Taylor coefficients. The unperturbed totient and prime series remain unresolved.\"\n  - problem: 251\n    paper: \"paper/251/erdos-251-prime-gap-dyadic-series.tex\"\n    families:\n      - id: \"prime_gap_reformulation\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"Exact finite summation by parts, the unconditional infinite prime-gap identity, and irrationality equivalence.\"\n        boundary: \"The equivalence does not prove irrationality of either series.\"\n      - id: \"integral_shift_classification\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"not_selected_abstract_family_represented_in_manifest\"\n        summary: \"Block identities and exact denominator criteria classify rationality through integral positive tail shifts.\"\n        boundary: \"The concrete prime-gap producer remains missing.\"\n      - id: \"dyadic_tail_integrality_classification\"\n        contribution_class: \"abstract Lean-checked classification and normal form; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"For any integer-digit dyadic tail recurrence T(N+1) = 2*T(N) - g(N+1), the rational h-shift is integral exactly when the reduced denominator of T(N) divides 2^h - 1, equivalently when 2^h is congruent to 1 modulo that denominator. For the real recurrence, the load-bearing normal form says that irrationality of T(0) is equivalent to every positive tail shift being nonintegral; the eventual-integrality equivalence and denominator/order laws are supporting mechanism evidence.\"\n        boundary: \"These exact statements cover arbitrary integer-digit dyadic tail recurrences with the declared rational and real state types; they classify integrality and give an irrationality normal form but do not supply a cofinal nonintegral shift for the consecutive-prime-gap recurrence. The generic hypotheses therefore do not close the prime-specific producer gap: no prime-gap irrationality, novelty, significance, or external review is claimed, and Erdős #251 remains open. This is one abstract classification family, distinct from prime_gap_reformulation, not a duplicate frontier claim.\"\n      - id: \"totient_shift_propagation\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"represented_by_integral_shift_family\"\n        summary: \"Odd rational denominators give totient-length integral shifts, and integrality propagates through the recurrence.\"\n        boundary: \"This describes rational states and supplies no contradiction for actual prime gaps.\"\n      - id: \"small_mismatch_criterion\"\n        contribution_class: \"conditional actual-prime-gap endpoint reduction; novelty and significance unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"For the actual prime-gap dyadic tail, cofinally many adjacent pairs of strictly small h-shifts with unequal corresponding prime gaps force failure of eventual RatIntegral h-shifts. The adjacent-pair zero obstruction is the load-bearing mechanism; exact non-eventual periodicity of consecutive prime gaps supplies the actual-prime-gap contrast.\"\n        boundary: \"The representative assumes T : ℕ → ℚ, the DyadicTailRecurrence for the actual prime-gap digits, and for every N₀ an N ≥ N₀ whose adjacent h-shifts both lie strictly in (-1, 1) while the corresponding prime gaps differ. Under that cofinal small-mismatch supply it excludes eventual RatIntegral h-shifts; it does not prove the supply or actual smallness. rationalPrimeGapTail_has_positive_shift_not_eventually_small records the contrary rational-state obstruction for at least one positive shift. This is a conditional reduction only: no #251 irrationality, infinite rational-sum limit, universal producer, novelty, priority, significance, or external-review claim is made; #251 remains open. The actual-prime-gap specialization is distinct from coefficient_only_no_go and does not duplicate prime_gap_reformulation.\"\n      - id: \"coefficient_only_no_go\"\n        contribution_class: \"coefficient-only barrier; novelty and significance unassessed\"\n        evidence_mode: \"Lean kernel plus authored synthesis\"\n        comparator_disposition: \"targeted\"\n        summary: \"The exact finite carry identity and the two non-eventual-periodicity statements expose the coefficient-only barrier: irregularity and growth of a coefficient stream do not by themselves force irrationality, while the synthetic carry stream remains distinct from the actual prime-gap stream.\"\n        boundary: \"CoefficientOnlyNoGo checks the exact finite partial-sum identity and the two non-eventual-periodicity statements. It does not assert the infinite limit of the synthetic countermodel, identify its coefficient stream with actual prime gaps, establish a prime-gap tail bridge, prove #251 irrationality, or make novelty, priority, significance, or external-review claims.\"\n  - problem: 257\n    paper: \"paper/257/erdos-257-mersenne-support-subseries.tex\"\n    families:\n      - id: \"finite_period_noncollapse\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"The multiplicative order of the base modulo a reduced finite-sum denominator is the support lcm, forcing denominator growth.\"\n        boundary: \"A finite-support denominator theorem does not settle an infinite-support sum.\"\n      - id: \"boolean_mobius_carry\"\n        contribution_class: \"locally proved exact equivalence plus conditional consequences; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"For p : ℤ and q > 0, a nonempty normalized positive-index support with erdosSupportSeries 2 A = p/q exists exactly when a quotient-only Boolean–Möbius carry certificate U exists. The certificate has U 0 = p, positive U, the square-root bound U(N) ≤ q(2√N + 4), q-divisible carry differences, and a Boolean Möbius transform; the converse reconstructs the support from that transform. This quotient-level equivalence is distinct from the shifted natural-tail state family.\"\n        boundary: \"The equivalence is stated at the existence level with q > 0, 0 ∉ A, and a positive support witness; it gives no contradiction, no universal Erdős #257 irrationality result, and no producer of cofinally nonintegral shifts for the actual recurrence. The generic tempered-orbit theorem and the reciprocal-mass, unbounded-tail, and zero-window results are subordinate consequences, not separate families; the zero-window constant depends on the support and numerator and is not uniform. This does not settle every infinite support, and no novelty, priority, significance, external-review, or unrestricted #257 endpoint claim is made.\"\n      - id: \"boolean_mobius_exact_row_dynamics\"\n        contribution_class: \"conditional reduction; novelty unassessed\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"targeted\"\n        summary: \"If CofinalPositiveHalfGreedySkips holds—cofinally many positive rational-greedy remainders below the next Mersenne weight—the skipped-core and upper-half Boolean-word construction supplies cofinally many exact local half rows, yielding (1 / 2 : ℝ) ∈ mersenneAchievementSet. The load-bearing representative is half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips; exact-row construction, endpoint-defect transition, forced rank-two membership, and Boolean-word filling are subordinate mechanisms.\"\n        boundary: \"CofinalPositiveHalfGreedySkips is an unproved global supply premise: for every N there must be c ≥ max N 4 with a positive greedy remainder below mersenneWeightRat c. Finite exact rows, including one produced by a skipped core, do not establish cofinality. This is a conditional 1/2-membership/counterexample route, not an unconditional Erdős #257 solution; the rank-two, endpoint-defect, and upper-half Boolean-word declarations remain subordinate evidence, and no novelty, priority, significance, or external-review claim is made.\"\n      - id: \"half_membership_seam_classification\"\n        contribution_class: \"locally proved exact endpoint classification; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"Membership of 1/2 in the Mersenne achievement set is exactly equivalent to unbounded false successor terminal bits, the unbounded upper-or-middle seam branch, cofinal terminal-false positions, unbounded skipped ranks along the integer seam, infinite greedy skipped support, and the absence of a last half-greedy skip. The load-bearing representative is half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse; the remaining equivalences are subordinate endpoint-socket evidence.\"\n        boundary: \"This exact endpoint-socket classification does not supply any unbounded terminal-false or upper-or-middle occurrences, cofinal supply, skipped-support producer, or half-membership witness; finite exact rows do not establish the required global supply. It is not an unconditional or universal Erdős #257 result, and no novelty, priority, significance, or external-review claim is made.\"\n      - id: \"shifted_odd_tail_state\"\n        contribution_class: \"conditional reduction\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"A rational support fraction produces a positive natural shifted tail state satisfying the exact binary recurrence and modular doubling orbit modulo v.\"\n        boundary: \"The state-extraction theorem does not exclude rational infinite supports or settle universal Erdős #257; the support and rational-value hypotheses remain explicit.\"\n      - id: \"composite_dilation_defect_identity\"\n        contribution_class: \"unconditional progress\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"The composite support-dilation identity splits the new divisor count into the old coefficient, the distinguished divisor, and an exact finite defect; its bouquet specialization supplies a separate structural budget.\"\n        boundary: \"The exact defect is not bounded for arbitrary supports and supplies no tail selector or irrationality theorem; the bouquet budget remains conditional on its structural hypotheses.\"\n      - id: \"known_irrational_supports\"\n        contribution_class: \"Lean formalisation of existing results\"\n        evidence_mode: \"Lean kernel plus Comparator for full and pairwise-coprime support\"\n        comparator_disposition: \"targeted_full_support_representative\"\n        summary: \"Full, factorial, power-of-two, multiple, eventually periodic, residue-class, and odd supports are formalised; pairwise-coprime support is split out as a dedicated non-dilation representative.\"\n        boundary: \"Structured known families do not cover arbitrary infinite supports.\"\n        source_statement: \"Several structured support families for the Mersenne reciprocal subseries are known to give irrational sums, including full support and other classical or later special families.\"\n        source_mapping: \"The review family is an aggregate disclosure. Each local theorem retains its own support predicate; the Comparator selects full support as the classical representative and pairwise-coprime support as a dedicated exact non-dilation representative.\"\n        historical_attribution: \"Erdos (1948) for full support; Erdos (1968), Luca--Tachiya, and Tao--Teravainen for the specific additional families identified in the paper and prior-art ledger.\"\n        logical_dependency: \"The full-support and pairwise-coprime Comparator theorems are proved locally in Lean. Other family rows have their stated local or cited proof boundaries; historical attribution is not an imported Lean dependency.\"\n      - id: \"finite_prime_weighted_support\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel; exact Comparator transport configured, external replay pending\"\n        comparator_disposition: \"targeted_exact_weighted_and_hereditary_interfaces_pending_replay\"\n        summary: \"Finite prime-weighted mass gives fixed-base irrationality for infinite positive supports; binary weighted mass gives all-base irrationality for every infinite subset of a positive host, while weighted mass at another fixed base gives same-base hereditary irrationality.\"\n        boundary: \"Each clause retains its exact finite-prime weighted-summability and positive-support hypotheses. Binary mass is required for the all-base hereditary clause; arbitrary infinite supports remain open.\"\n      - id: \"pairwise_coprime_support\"\n        contribution_class: \"formalisation of an existing theorem\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted_pairwise_coprime_support\"\n        summary: \"Infinite pairwise-coprime supports with summable reciprocal mass have irrational Mersenne support series by an adaptive CRT and weighted coefficient certificate argument.\"\n        boundary: \"This structured support theorem does not settle arbitrary infinite supports; its infinitude, pairwise-coprimality, and summable-reciprocal hypotheses remain explicit.\"\n        source_statement: \"For every integer base b at least 2 and every infinite pairwise-coprime support A with summable reciprocal mass, the support-restricted Mersenne reciprocal series is irrational.\"\n        source_mapping: \"The local theorem keeps the base, infinitude, pairwise-coprimality, and summable-reciprocal hypotheses explicit and concludes irrationality of erdosSupportSeries b A. At base 2 this is exactly the pairwise-coprime support subfamily of Erdős #257; it is a separate non-dilation representative from the full-support row.\"\n        historical_attribution: \"Erdos (1968), as recorded in the repository's prior-art ledger and the #257 problem note.\"\n        logical_dependency: \"The proof-bearing source supplies the adaptive CRT and weighted coefficient certificate argument locally; the cited source establishes historical attribution and statement fidelity, not an imported Lean dependency.\"\n      - id: \"periodic_nonnegative_weight_irrationality\"\n        contribution_class: \"formalised here; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted_strongest_nonnegative_rational_weight\"\n        summary: \"An eventually-periodic nonnegative rational coefficient sequence with a positive value in its periodic region has an irrational Lambert–Mersenne series; denominator clearing reduces the rational sequence to a periodic Nat-weight certificate.\"\n        boundary: \"This does not claim the broader mixed-sign periodic theorem; all coefficient, positivity, base, period, and threshold hypotheses remain explicit.\"\n      - id: \"orthogonal_petal_sunflower_reduction\"\n        contribution_class: \"conditional reduction\"\n        evidence_mode: \"Lean kernel plus Comparator plus paper argument\"\n        comparator_disposition: \"targeted_conditional_sunflower_reduction\"\n        summary: \"A finite-core orthogonal-petal bouquet together with the exact uniform tail-selection predicate composes through the forced-carry certificate consumer to irrationality; the explicit alternating-core, squared-prime-petal witness shows the structural class is non-vacuous, but the selector remains open.\"\n        boundary: \"The bouquet and selector hypotheses remain explicit; the selector is not proved, so this is not an unconditional #257 support family.\"\n      - id: \"squarefree_support\"\n        contribution_class: \"Lean structure plus paper and external theorem\"\n        evidence_mode: \"Lean kernel plus paper argument plus cited theorem\"\n        comparator_disposition: \"not_applicable_to_external_irrationality_input\"\n        summary: \"Lean checks squarefree incidence and no-go statements; the irrationality conclusion uses an external analytic theorem in the paper.\"\n        boundary: \"Comparator must not badge the paper-plus-external conclusion.\"\n      - id: \"achievement_set_geometry\"\n        contribution_class: \"mixed prior-art geometry and locally proved metric result; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted_measure_theorem\"\n        summary: \"Compactness, perfectness, total disconnectedness, nowhere density, and measure one for the unrestricted achievement set.\"\n        boundary: \"Geometry of the full set does not decide irrationality of every coded point.\"\n        source_statement: \"Kakeya's strict-tail criterion places rapidly decreasing subsum sets in the Cantor-set regime; for the fixed-base Mersenne sequence, Kovač--Tao verify the strict-tail inequality and distinct subsums.\"\n        source_mapping: \"At base 2 the local development formalises compactness, perfection, total disconnectedness, nowhere density, and measure one. The strict-tail Cantor geometry aligns with the cited antecedents; the measure-one clause is locally derived and carries no novelty assertion.\"\n        historical_attribution: \"Kakeya (1914) for the strict-tail subsum-set criterion and Kovač--Tao (2025), Remark 4.1, for the fixed-base Mersenne antecedent.\"\n        logical_dependency: \"The topology and measure theorem are proved locally in Lean; the cited sources fix historical/source alignment and are not imported proof objects.\"\n      - id: \"restricted_achievement_sets\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted_measure_dichotomy\"\n        summary: \"Support-restricted coding is injective and the associated achievement set has exact finite-complement measure or zero for infinite complement.\"\n        boundary: \"The measure dichotomy does not classify rational points.\"\n      - id: \"half_and_twenty_one_frontiers\"\n        contribution_class: \"conditional reduction and no-go result\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"Exact greedy characterisations, finite-support exclusions, the odd 2-adic two-thirds-band bound, and two conditional rational-half countermodel frontiers isolate local constraints around the remaining 1/2 and 1/21 alternatives.\"\n        boundary: \"Neither membership question is decided; the 2-adic bound is sharp locally, and the two countermodel routes require unconstructed terminal-scaled or cofinal-cylinder antecedents.\"\n      - id: \"separated_cut_algebraic_lambert_transcendence\"\n        contribution_class: \"ordinary paper argument\"\n        evidence_mode: \"ordinary proof; two AI reviews, no independent human review\"\n        comparator_disposition: \"not_applicable_without_exact_Lean_statement\"\n        summary: \"For an infinite positive-integer host with cuts L_j < M_j, L_j tending to infinity and M_j-L_j tending to infinity, every host exponent at most L_j dividing L_j and every later exponent divisible by M_j, every infinite thinning with bounded positive integer weights has transcendental Lambert sum at every real algebraic base t > 1. This includes all divisibility chains and an explicit host of unbounded divisibility width and divergent reciprocal sum. The full number-field Subspace Theorem argument has two AI proof reviews; it is an ordinary proof with no Lean transcendence theorem, independent human review or asserted historical priority.\"\n        boundary: \"The full support at 3/2 and arbitrary infinite supports at base two remain unresolved. Finite cut arithmetic is not a proof of transcendence.\"\n  - problem: 269\n    paper: \"paper/269/erdos-269-three-prime-running-lcm.tex\"\n    families:\n      - id: \"three_prime_lcm_cells\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted_running_lcm_identity\"\n        summary: \"Exact running-LCM product, logarithmic-cell constancy, coordinate jump ratios, and jump count.\"\n        boundary: \"Cell structure alone does not prove irrationality.\"\n      - id: \"two_prime_transcendence\"\n        contribution_class: \"paper plus external theorem\"\n        evidence_mode: \"paper argument plus cited theorem\"\n        comparator_disposition: \"not_applicable_not_a_lean_declaration\"\n        summary: \"Steve Fan posted the two-prime factorisation, reduction and transcendence first (erdosproblems.com/269, 26 June 2026, post-7218). This checkout records that ordinary deduction from Bugeaud–Laurent and does not claim priority or a Lean theorem.\"\n        boundary: \"Comparator cannot certify the external analytic input.\"\n      - id: \"height_fibre_and_shell\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"represented_by_three_prime_structure_target\"\n        summary: \"Finite height-fibre normal form and a quadratic smooth-shell multiplicity bound.\"\n        boundary: \"These fibre bounds are counting input, not an escape producer. The separate actual-series divisibility bridge is established.\"\n      - id: \"rank_two_kernel_no_go\"\n        contribution_class: \"no-go result\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"The 2,3,5 kernel is not rank one and its smallest displayed minor equals -1/15.\"\n        boundary: \"Failure of rank one does not itself imply irrationality.\"\n      - id: \"dyadic_block_alphabet\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"represented_by_three_prime_structure_target\"\n        summary: \"The exact dyadic block alphabet is 2, 6, 10, and 30.\"\n        boundary: \"The finite alphabet does not supply the needed carry escape.\"\n      - id: \"conditional_carry_escape\"\n        contribution_class: \"conditional no-go consumer; novelty and significance unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"Under the denominator-dependent cofinal local-window residue-escape predicate, no positive reduced carry can satisfy the exact multiplier recurrence together with its short bound. The load-bearing consumer is no_positive_reducedCarry_of_cofinalLocalWindowEscape; an absorbed nonzero common-factor carry reduces exactly to that consumer. CofinalLocalWindowEscape, windowBase, windowForcing, leastPositiveResidue, and the absorbed-carry bridge are subordinate finite-window mechanism evidence.\"\n        boundary: \"The representative assumes b,m : ℕ → ℕ, shortBound : ℕ → ℕ → ℕ, CofinalLocalWindowEscape b m shortBound, and for each positive B coprime to 30 a positive integer-valued d with d(n+1) = (b n : ℤ) d(n) − (B : ℤ) (m n : ℤ) and Int.natAbs (d n) ≤ shortBound B n; it then gives False. The absorbed-carry bridge additionally assumes smoothFactor ≠ 0, c n = smoothFactor * d n, and the corresponding exact absorbed recurrence, then cancels that factor to the same reduced consumer. The representative remains a conditional consumer for arbitrary words. For the actual three-prime word, the rationality-to-reduced-carry bridge is established separately; the source-specific cofinal escape producer is not proved. This is not a #269 endpoint or irrationality proof, and no actual-series identification, novelty, priority, significance, or external-review claim is made. It is one conditional window-carry family, distinct from the finite residue, rank-two, and weighted-phase observer families.\"\n      - id: \"weighted_phase_carry_observer\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"targeted\"\n        summary: \"An exact weighted-phase carry recurrence splits into a finite residue digit and an uncontrolled integral quotient coboundary; an explicit function-faithful finite-dimensional observer then forces finite realised span.\"\n        boundary: \"The recurrence supplies a finite residue coordinate but leaves an uncontrolled integral quotient coboundary; it proves neither a finite-state quotient nor a literal infinite realised span. Finite realised span requires an explicit factorisation through a finite-dimensional function-faithful observer, and scalar evaluation alone is insufficient. No irrationality conclusion follows from this observer alone. The actual rationality-to-positive-reduced-carry bridge is established separately; cofinal local-window escape remains open.\"\n      - id: \"three_prime_finite_search\"\n        contribution_class: \"finite computation\"\n        evidence_mode: \"external exact computation\"\n        comparator_disposition: \"not_applicable_not_a_lean_theorem\"\n        summary: \"A local-window checker searches 106666 denominator and start pairs and records small certificates.\"\n        boundary: \"Finite search is not a cofinal statement.\"\n      - id: \"independent_clock_channel_relations\"\n        contribution_class: \"ordinary paper argument\"\n        evidence_mode: \"ordinary proof; two AI reviews, no independent human review\"\n        comparator_disposition: \"not_applicable_without_exact_Lean_statement\"\n        summary: \"For positive rationally independent clock rates with no simultaneous positive crossings, and integer radices b_i >= 2, all rational relations among 1 and the channel sums S_i are multiples of (-1,b_1-1,...,b_d-1). The proof uses genuine late swaps with a common infinite suffix in one orbit closure, a fixed rational lattice, and density of the clock section. The explicit rates 1,sqrt(2),sqrt(3),sqrt(6) and radices 2,3,5,7 give rational span dimension four. This is an ordinary proof reviewed by AI agents, without Lean or independent human verification of the infinite theorem.\"\n        boundary: \"No rational independence of the original reciprocal prime logarithms is assumed or proved; the repeated running-LCM endpoint remains open.\"\n      - id: \"selected_staircase_integral_height_boundary\"\n        contribution_class: \"ordinary paper argument\"\n        evidence_mode: \"ordinary proof; two AI reviews, no independent human review\"\n        comparator_disposition: \"not_applicable_without_exact_Lean_statement\"\n        summary: \"The selected staircase matrix with ones on and below the diagonal and integer r>=2 above it has Smith factors 1,r-1,...,r-1. For the selected {2,3,5} kernel, any entrywise-integral diagonal clearing has determinant magnitude at least 4^(N-1), whereas the reduced rational determinant has numerator one up to sign. A rational single-target border has primitive polynomial plus or minus (tX-s), where s/t is the reduced Schur complement. These ordinary arithmetic arguments were reviewed by AI agents; they are not the existing Lean nonvanishing theorem and have no independent human review.\"\n        boundary: \"The optimality concerns diagonal clearing and one rational target. It does not rule out multi-target, nonlinear, recurrence, or new approximation methods and does not prove running-LCM irrationality.\"\n  - problem: 1041\n    paper: \"paper/1041/erdos-1041-lemniscate-newton-flow.tex\"\n    families:\n      - id: \"newton_value_decay\"\n        contribution_class: \"locally proved result and formalised calculus; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus authored synthesis\"\n        comparator_disposition: \"not_selected_compact_calculus_family\"\n        summary: \"Lean checks the local Newton value equation and the zero-derivative form of its exponentially scaled value; constancy along a real-time interval is an ordinary calculus consequence in the paper.\"\n        boundary: \"The cited declarations are pointwise derivative statements. The real-time first integral requires the paper’s interval integration step, and value decay alone does not construct a global trajectory or give a short connecting curve.\"\n      - id: \"ray_separation\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus authored synthesis\"\n        comparator_disposition: \"represented_by_translation_avoidance_targets\"\n        summary: \"Lean checks that distinct positive rays contradict an assumed exponential endpoint relation; the paper derives that relation for a finite real-time Newton trajectory by ordinary integration and endpoint continuity.\"\n        boundary: \"The finite-trajectory conclusion retains continuity on the closed interval, differentiability and noncritical Newton evolution on its interior, and the ordinary endpoint passage. It supplies neither global trajectory existence nor geometric length control.\"\n      - id: \"translation_avoidance\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"Collision translations are parameterised by real affine lines and an arbitrarily small common translation avoids finitely many of them.\"\n        boundary: \"The finite-family avoidance theorem does not perform the global topology or metric gluing.\"\n      - id: \"root_retention\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"A quantified small constant perturbation keeps every polynomial root inside the open unit disc.\"\n        boundary: \"Root retention is one input to a still-incomplete route.\"\n      - id: \"published_proof_gap\"\n        contribution_class: \"paper diagnosis\"\n        evidence_mode: \"authored source analysis\"\n        comparator_disposition: \"not_applicable_not_a_lean_proposition\"\n        summary: \"The paper identifies an invalid local saddle block in a claimed unrestricted proof.\"\n        boundary: \"The diagnosis does not prove that no repair exists.\"\n      - id: \"lemniscate_finite_search\"\n        contribution_class: \"finite computation\"\n        evidence_mode: \"external computation\"\n        comparator_disposition: \"not_applicable_not_a_lean_theorem\"\n        summary: \"A bounded search records finite geometric evidence.\"\n        boundary: \"Finite evidence does not settle the universal problem.\"\n  - problem: 1049\n    paper: \"paper/1049/erdos-1049-rational-base-lambert.tex\"\n    families:\n      - id: \"scalar_content_no_go\"\n        contribution_class: \"no-go result\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"not_selected_scalar_content_scaling_family\"\n        summary: \"Integer scalar content changes analytic error and exterior determinant by matching factors, yielding no margin by itself.\"\n        boundary: \"The conclusion is scoped to one normalisation strategy.\"\n      - id: \"endpoint_residues\"\n        contribution_class: \"no-go result\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"not_selected_endpoint_residue_and_common_multiplier_family\"\n        summary: \"Endpoint residues at 2 and 3 exclude a common multiplier under unit-endpoint hypotheses.\"\n        boundary: \"Other local or cross-row arithmetic remains possible.\"\n      - id: \"four_jet_collision\"\n        contribution_class: \"locally proved result and no-go result; novelty unassessed\"\n        evidence_mode: \"Lean kernel\"\n        comparator_disposition: \"not_selected_finite_combinatorial_family\"\n        summary: \"Binary selectors collide in the four-jet signature at the stated rank and depth.\"\n        boundary: \"A signed relation is not nonzero analytic data.\"\n      - id: \"coordinatewise_corridor_no_go\"\n        contribution_class: \"no-go result\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"The coordinatewise corridor forces a power-versus-linear inequality and cannot occur at base 3/2.\"\n        boundary: \"This excludes one proof architecture and proves no irrationality statement.\"\n      - id: \"rational_base_tail_recurrence\"\n        contribution_class: \"locally proved result; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"The exact rational-base cleared-tail recurrence exposes exponential denominator-base forcing absent at integer bases.\"\n        boundary: \"The recurrence is not derived from a rationality contradiction.\"\n      - id: \"seven_halves_irrationality\"\n        contribution_class: \"paper plus external theorem\"\n        evidence_mode: \"paper argument plus cited theorem and Lean arithmetic\"\n        comparator_disposition: \"not_applicable_to_external_irrationality_conclusion\"\n        summary: \"Irrationality at 7/2 is obtained from an external analytic criterion whose elementary height hypothesis is checked in Lean.\"\n        boundary: \"Comparator may check the height inequality, not the cited theorem application.\"\n      - id: \"height_and_pade_arithmetic\"\n        contribution_class: \"locally proved sharp model-specific no-go; novelty unassessed\"\n        evidence_mode: \"Lean kernel plus Comparator\"\n        comparator_disposition: \"targeted\"\n        summary: \"For the explicit rectangular two-function exponent model, the cleared gap is nonpositive throughout rho >= 0 and 1 + rho <= sigma, and the threshold is at most 1/2 - 1/pi^2 with equality exactly at rho = 0, sigma = 1. The gap sign and unique-zero mechanism support this sharp threshold no-go within the wider height-and-Padé arithmetic family.\"\n        boundary: \"This is a sharp no-go only for the explicit rectangular two-function exponent model under the stated real hypotheses; it constructs no approximating polynomials or remainders and is not a universal Padé or Hermite-Padé no-go. It proves no irrationality at 3/2 or any general rational-base endpoint, and the analytic remainder/nonvanishing input remains untreated. No novelty, priority, significance, external-review, or endpoint claim is made; Erdős #1049 remains open.\"\n      - id: \"calibrated_rational_hankel_countermodel\"\n        contribution_class: \"ordinary paper argument\"\n        evidence_mode: \"ordinary proof; two AI reviews, no independent human review\"\n        comparator_disposition: \"not_applicable_without_exact_Lean_statement\"\n        summary: \"An explicit positive atomic moment family and integer formal series have all-rank Hankel order sum_{j<N} j^2 and leading coefficient (N!)^2(N+1)!/2^N, with the same fixed-base asymptotic shape including N^(-8 xi), while xi=G_{2/3}(2/3)=1. For n>=1 the exact reduced moment denominator is 3^(31n+62)(3^(n+1)-2^(n+1))^4; the determinant denominator has logarithmic size Theta(N^3), and a proposed rank-two row-product clearer leaves denominator625. These are ordinary proofs with independent exact finite checks and AI proof reviews, not a Lean proof of the infinite family or human peer review.\"\n        boundary: \"These are not Zudilin's actual moments. They refute an inference from the shared signature alone; they do not settle F(3/2) or refute the attributed divisibility-based criterion.\"\nacknowledgements: null\nexternal_verification:\n  owner: \"docs/claims.json::external_verification_packet\"\n  problem_index: \"docs/problems.json\"\n  problem_index_digest: \"sha256:131d897d99eb4de075e7e4c4269f3431005638e6028519a813944b13dcb8d904\"\n  problem_count: 8\n  problems:\n    - id: \"erdos_68\"\n      erdos_number: 68\n      status: \"open\"\n      checked_frontier: \"The exact integral normal form: the series is irrational if and only if the strict factorial successor Z_m misses divisibility by m cofinally.\"\n      open_frontier: \"Irrationality of the factorial-denominator series.\"\n      claim_registry_status: \"registered_declarations\"\n    - id: \"erdos_243\"\n      erdos_number: 243\n      status: \"open\"\n      checked_frontier: \"Lean proves cubic_rate_irrationality_unconditional for positive StrictMono zero-indexed a_n with a_n²/a_(n+1) = 1 + 3/n + o(n^-3). The one-based paper uses a finite-prefix bridge; general #243 remains open.\"\n      open_frontier: \"The unrestricted problem.\"\n      claim_registry_status: \"registered_declarations\"\n    - id: \"erdos_249\"\n      erdos_number: 249\n      status: \"open\"\n      checked_frontier: \"For every e >= 1, the complete dyadic totient kernel through level e has rational span dimension 2^e+1.\"\n      open_frontier: \"The strict prime-tail orbit gap, which is the unproved producer consumed by the conditional endpoint.\"\n      claim_registry_status: \"registered_declarations\"\n    - id: \"erdos_251\"\n      erdos_number: 251\n      status: \"open\"\n      checked_frontier: \"The finite summation-by-parts identity relating the prime dyadic partial sums to the prime-gap dyadic partial sums, with an exact endpoint term and no convergence premise.\"\n      open_frontier: \"The target irrationality.\"\n      claim_registry_status: \"registered_declarations\"\n    - id: \"erdos_257\"\n      erdos_number: 257\n      status: \"open\"\n      checked_frontier: \"Irrationality at every integer base for the classical full support, reciprocal-summable supports, and the finite-prime weighted supports in divisibilityWeightedClaim.\"\n      open_frontier: \"Universal irrationality over all infinite supports, or the proposed rational values 1/2 and 1/21.\"\n      claim_registry_status: \"programme_only\"\n    - id: \"erdos_269\"\n      erdos_number: 269\n      status: \"open\"\n      checked_frontier: \"That for three pairwise distinct primes the least common multiple of the smooth prefix equals the product of the three maximal pure prime powers below the cutoff.\"\n      open_frontier: \"Irrationality of the repeated sum, or transcendence, in any three-prime case.\"\n      claim_registry_status: \"registered_declarations\"\n    - id: \"erdos_1041\"\n      erdos_number: 1041\n      status: \"formal statement refuted\"\n      checked_frontier: \"For ani’s degree-seven polynomial, every preconnected strict-lemniscate set containing two distinct roots has one-dimensional Hausdorff measure above two. Lean refutes the exact Formal Conjectures path-image claim and checks a separate total-variation bound.\"\n      open_frontier: \"The full small-parameter family reported by ani; only one explicit polynomial is formalised.\"\n      claim_registry_status: \"registered_declarations\"\n      research_frontier:\n        directory: \"research_corpus/Erdos1041\"\n        authority_posture: \"public_safe_research_evidence_not_reviewed_claim_registry_or_Lean_proof_authority\"\n        reading_rule: \"Read FRONTIER.md for dated refutations and open gaps, then STRONGEST_RESULTS.json for source-pinned activation. Premises and falsifiers retain local hypotheses; research rows do not settle historical correspondence or become Comparator claims.\"\n        frontier: \"research_corpus/Erdos1041/FRONTIER.md\"\n        strongest_results: \"research_corpus/Erdos1041/STRONGEST_RESULTS.json\"\n        manifest: \"research_corpus/Erdos1041/CORPUS_MANIFEST.json\"\n        checkpoint: \"research_corpus/Erdos1041/PUBLIC_CORPUS_CHECKPOINT.json\"\n        content_digests:\n          frontier: \"sha256:8aef5071a9a2a35de6b2af79bf0a486d97306aa0b6d4b997144e16ba1d112a9a\"\n          strongest_results: \"sha256:a5f888560836a710024bd6079538cbd62b39c1544a19514903ac57f9789d7e24\"\n          manifest: \"sha256:6f456a92e30ed2318d6c342f2e80dbbc77be39a14e051828ef215ffb8b001bbd\"\n          checkpoint: \"sha256:3ec0d7b9f9ab8a3425678b52830ee3e6701382b9b7485787c3dac43f66ac7fb3\"\n        activated_result_count: 294\n        source_checkpoint: \"fb5c8803c1498cad9755d1615473cb33b11af6ee\"\n        source_subtree_tree: null\n        promotion_boundary: \"This route is public research evidence only; it is not a reviewed claim-registry entry or a Comparator interface.\"\n    - id: \"erdos_1049\"\n      erdos_number: 1049\n      status: \"open\"\n      checked_frontier: \"Lean checks irrationality and the stated irrationality-exponent bound for rational bases in the exact Zudilin contour region, including every positive integral power of 31/4. The contour excludes 3/2.\"\n      open_frontier: \"The universal rational-base conjecture, including the case 3/2, remains open. The checked 31/4 power family settles one region and leaves that conjecture untouched.\"\n      claim_registry_status: \"registered_declarations\"\n  trusted_challenge_holes: 8\n  packet: \"docs/EXTERNAL_VERIFICATION.md\"\n  receipt_contract: \"docs/external_verification_packet.json::receipt_contract\"\n","body_format":"text"},"human-entry":{"document_id":"human-entry","canonical_document_id":"reading-guide","origin":"current_document_alias","already_carried_in":"public_repository_documents.lean.documents.reading-guide"},"agents-override":{"document_id":"agents-override","canonical_document_id":"agents","origin":"current_document_alias","already_carried_in":"public_repository_documents.lean.documents.agents"}}},"plectis":{"repository":"wcook04/plectis","checkout_key":"plectis","source_commit":"f2ed2582e6de24c71b45dfc30dae33863e2ab539","document_count":11,"embedded_count":11,"missing":[],"documents":{"readme":{"document_id":"readme","source_path":"README.md","role":"software_front_door","github":"https://github.com/wcook04/practical-formal-verification/blob/main/README.md","source_commit":"f2ed2582e6de24c71b45dfc30dae33863e2ab539","permalink_url":"https://github.com/wcook04/practical-formal-verification/blob/f2ed2582e6de24c71b45dfc30dae33863e2ab539/README.md","sha256":"sha256:a7d286b2976657d6ec2e56519c7c3f2933ab2e6fa0f41d161fa382b19ea9ad61","byte_count":20460,"origin":"public_main_checkout","body":"# Plectis\n\n**Run and inspect experiments in AI-assisted research.**\n\nPlectis is a public Python toolkit. You can run an example, inspect how it\nchecks its result, and adapt the useful parts. The tools cover different jobs: checking proofs and agent completion\nclaims, comparing forecasts, and keeping generated documents consistent with\ntheir sources. This repository is a public research prototype, with selected\ncomponents from Will Cook's larger private workbench.\n\n[Start in a terminal](#try-the-repository-tour) · [Worked examples](#see-it-work) ·\n[Plectis introduction](https://www.youtube.com/watch?v=U1z2-XaLe_w) ·\n[Plectis website](https://wcook04.github.io/practical-formal-verification/) · [Documentation](docs/README.md)\n\nFor example, an agent says “I committed the fix and the tests passed.” The\n[completion-checking example](docs/guides/checking-agent-completion.md) checks\nthree supplied claims against a disposable Git repository and an actual pytest\nrun, and rejects four deliberately incorrect claims. It is a worked experiment,\nnot a general audit of your agent or project.\n\n### Try the repository tour\n\nFor a first look, run the tour with Python 3.11 or newer. No installation or\nmodel account is needed:\n\n```bash\ngit clone https://github.com/wcook04/practical-formal-verification\ncd plectis\nPYTHONPATH=src python3 -m plectis tour --format text .\n```\n\nThis lists project files, suggests a starting task and saves a local run record.\nIt makes no model calls and leaves your source unchanged. The examples below\nshow what the individual tools do; the tour itself does not run your tests or\ncarry out the suggested task.\n\n<a href=\"https://wcook04.github.io/practical-formal-verification/docs/system-map.html\">\n  <img src=\"assets/system-map-dark.png\" alt=\"Plectis system map in dark mode: seven families of software components surround the axioms, principles and failure modes they cite.\" width=\"100%\">\n</a>\n\n[**Explore the system map →**](https://wcook04.github.io/practical-formal-verification/docs/system-map.html)\n<br>\n<sub>Components around the rim; the doctrine they cite at the centre. Select a\ncomponent or rule on the website to follow its connections.</sub>\n\nThe tools were built with AI coding agents.\n[Authorship, third-party sources and licensing](#name-and-history) are recorded below.\n\n> **Looking for the mathematics?** Start with\n> [**Plectis Erdős →**](https://github.com/wcook04/plectis-erdos), the main research\n> repository for the Lean proofs, papers and open questions.\n> [Read the guide](https://github.com/wcook04/plectis-erdos/blob/main/docs/READING_GUIDE.md)\n> or [explore the mathematics map](https://wcook04.github.io/practical-formal-verification/maths/universe.html).\n\n[Worked examples](#see-it-work) · [Quickstart](QUICKSTART.md) ·\n[Documentation](docs/README.md) · [Contributing](CONTRIBUTING.md)\n\nBrowse the live [component pages](https://wcook04.github.io/practical-formal-verification/docs/components.html),\n[papers](https://wcook04.github.io/practical-formal-verification/#papers) and\n[films](https://wcook04.github.io/practical-formal-verification/#videos) on the\n[Plectis website](https://wcook04.github.io/practical-formal-verification/).\n\nThe website and its maps are agent-readable. Start with the\n[agent navigation guide](https://wcook04.github.io/practical-formal-verification/llms.txt)\nfor text indexes, graph data and source links without running the visual maps.\n\n## Can I use it today?\n\nYes, for the documented local commands and selected examples. Start with the\n[quickstart](QUICKSTART.md) or the [worked guides](docs/guides/README.md), then\ncheck the chosen component's inputs, dependencies and stated limits before\nadapting it to your own project.\n\n| Status | What that means here |\n|---|---|\n| **Usable today** | Project inventory, suggested routes and selected examples, with component-specific prerequisites. |\n| **Tested** | Named inputs and failure cases; the completion example includes Git and pytest subprocesses. |\n| **Experimental or unverified** | New project adaptations and end-to-end research automation need separate validation. |\n\nThe 88-component catalogue does not establish an integrated public research\nplatform, execution availability in every environment or external use of every\ncomponent. The public prototype and the larger private workbench are separate. The\n[overview](docs/overview.md#the-public-workflow) follows one input through a\nselected command to its output and explains the boundary.\n\n## Why this exists\n\nAI-assisted research can span many sessions, files and failed approaches.\nPlectis explores how to preserve that work so another person can understand\nwhat was tried, check the results and choose a useful next step. Experts can\ncontribute a correction or direction without operating the AI themselves.\n\nThis toolkit publishes selected parts of that effort. The\n[mathematics repository](https://github.com/wcook04/plectis-erdos) contains the\npapers, proofs and open questions; the [website](https://wcook04.github.io/practical-formal-verification/)\nexplains both. The larger private workbench shown in the videos is separate\nfrom this clone. The [project overview](docs/overview.md) explains how the\nparts fit together and relates them to other research tools.\n\n<a id=\"choose-a-route\"></a>\n\n## Choose a starting point\n\n| You want to… | Start here |\n|---|---|\n| Understand the project | [Why these tools belong together](docs/overview.md) |\n| Run something locally | [Quickstart](QUICKSTART.md) |\n| Follow a complete example | [Worked guides](docs/guides/README.md) |\n| Find a tool in your field | [Component map](ORGANS.md) |\n| Work on the code | [Contributing](CONTRIBUTING.md) and [implementation map](docs/maintainers/architecture.md#implementation-map) |\n| Read the mathematics | [Companion reader's guide](https://github.com/wcook04/plectis-erdos/blob/main/docs/READING_GUIDE.md) |\n\n## See it work\n\nAn AI says: “I committed the fix and the tests passed.” The\n[completion-checking guide](docs/guides/checking-agent-completion.md) shows how\none component checks those claims using a sample Git repository and actual\npytest runs. It verifies three supplied claims and rejects four deliberately\nincorrect ones. You can follow the result back to the code that checked it.\n\nTwo more [worked guides](docs/guides/README.md) show different uses:\n\n- [Change a prompt-injection example](docs/guides/prompt-injection.md)\n  and see a checker reject an inconsistent permission in saved records.\n- [Prepare a research question](docs/guides/hypothesis-handoffs.md) with a\n  tentative answer, alternatives and proposed observations for an expert.\n\nTry the second example from the clone, without installing anything:\n\n```bash\nPYTHONPATH=src python3 -m plectis hypothesis-handoff \\\n  --input examples/hypothesis_handoff/independent_evaluation.json --format text\n```\n\nIt validates the supplied file and prints the question and proposed\ninvestigation. It does not answer the question, contact an expert or call a\nmodel. Each guide explains what its example tests and what remains untested.\n\n## What is in the toolkit?\n\nThere are **88 components**, grouped into seven areas. A component is a named\nprogram or group of related programs, with its own instructions and stated\nlimits. Some run a computation or an external tool; others compare supplied\nrecords against rules. An entry in the catalogue is not a promise that every\nexample can run in every environment: its page lists required tools and source\nfiles, and the runner reports unavailable requirements.\n\n| Area | Try it for… |\n|---|---|\n| [Entry and orientation](ORGANS.md#entry--reveal) | Checking that a suggested first step names an available command and result file. |\n| [Architecture and navigation](ORGANS.md#architecture--navigation) | Comparing a selection of files and components with an expected selection. |\n| [Mathematics and proof](ORGANS.md#formal-math--proof) | Trying proof steps on supplied theorems and asking Lean to check them. |\n| [Agent reliability and safety](ORGANS.md#agent-reliability--safety-replays) | Testing records of prompt injection, poisoned memory or completion claims. |\n| [Research and forecasting](ORGANS.md#research--science-replays) | Comparing forecast errors on synthetic data. |\n| [Source and generated files](ORGANS.md#import-projection--drift) | Comparing file fingerprints and checking that declared outputs exist. |\n| [Work and continuity](ORGANS.md#work-landing--continuity) | Checking that a work record names its changes, validation results and commits. |\n\nYou can [browse the components on the website](https://wcook04.github.io/practical-formal-verification/docs/components.html)\nor list and search their descriptions from the terminal:\n\n```bash\nPYTHONPATH=src python3 -m plectis comprehend --slice mechanism --format text\nPYTHONPATH=src python3 -m plectis comprehend --first-action \"prompt injection\" --format text\n```\n\nThe search uses phrase and word matching to suggest a command and relevant files.\nIt neither calls an AI model nor executes the suggested command.\n\n## How it works\n\nMost components run independently. They share command handling, file formats\nand a catalogue that names each component's code, inputs, checks and saved\nresults. The catalogue also supplies the generated documentation and maps.\nThe connections between components are recorded where they exist; the seven\nareas are a browsing aid, not seven stages that every run executes.\n\nTo understand a component, open its example input, the source function that\nprocesses it and the output it produces. Compare that output with the test or\nexpected result. Then change an input in a disposable copy and check whether\nthe result changes as expected. The [worked guide](docs/guides/prompt-injection.md)\nfollows this path through one component.\n\nThe project tour is a separate orientation command. It mostly classifies\nfilenames and paths, with limited inspection of Python entry points. It\nwrites a file inventory, suggested tasks and a simulated task record under\n`.microcosm/`, an ignored directory retained from the project's former name.\nA saved run result is called a *receipt* in the source. It records what ran and\nwhich inputs and results it refers to; it does not by itself prove a claim.\nTo see the tour's structured output:\n\n```bash\nPYTHONPATH=src python3 -m plectis tour --card .\n```\n\n[Architecture](ARCHITECTURE.md) maps the implementation. The\n[maintainer guides](docs/maintainers/README.md) explain command dispatch,\ndata ownership and how to regenerate the maps.\n\n## Explore with a coding agent\n\nA coding agent is an AI assistant that can read files, run commands and edit\ncode. You can use your own to explore this repository. Open a current clone\nand ask, for example:\n\n> Read AGENTS.override.md. I want to understand how this project checks an\n> agent's claim that a task is finished. Find the relevant example, explain\n> its inputs, run it, and show me the result and the code that checks it.\n> Explain what this run establishes and what it leaves untested. Include the\n> checkout commit and commands so I can repeat it.\n\nThe [agent quickstart](QUICKSTART.md#use-your-own-coding-agent) explains where\nto save results. [AGENTS.md](AGENTS.md) contains the editing instructions.\nThe toolkit's local examples do not require a model API key; your chosen\ncoding agent has its own account and provider requirements.\n\n## Install\n\nThe basic commands run from a clone with **Python 3.11 or newer**, without\nthird-party runtime dependencies. The shell examples above work on macOS,\nLinux and WSL. In Windows PowerShell, first set `$env:PYTHONPATH = 'src'`,\nthen use `python -m plectis` in place of `PYTHONPATH=src python3 -m plectis`.\nIndividual examples may require Git, pytest, Lean or other tools; follow the\nselected example's instructions.\n\nTo install the shorter `plectis` command on macOS or Linux:\n\n```bash\npython3 -m venv .venv\n.venv/bin/python -m pip install .\n.venv/bin/plectis tour --format text .\n```\n\nUse the virtual environment if your system Python reports\n[`externally-managed-environment`](https://peps.python.org/pep-0668/).\nFor development dependencies, Windows\ninstallation commands and other options, see [Quickstart](QUICKSTART.md),\n[Contributing](CONTRIBUTING.md) and [troubleshooting](docs/README.md#troubleshooting).\nThe older `microcosm` command remains an alias.\n\n## Repository layout\n\nStart with the guides and examples. The large reference files are indexes for\nlooking things up; you do not need to read them all before trying a component.\n\n```text\nplectis/\n├── src/\n│   ├── plectis/           Public command entry points\n│   └── microcosm_core/    Python implementation; components in organs/\n├── docs/                 Guides, reference, maintainer notes and paper guide\n├── examples/             Runnable examples and checked source bundles\n├── fixtures/             Prepared inputs and expected results\n├── tests/                Executable checks\n├── core/                 Component registry and configuration\n├── atlas/                Generated maps and indexes\n├── standards/            Requirements checked by the tools\n├── skills/               Procedures for agents\n├── paper_modules/        Component explanations\n├── paper/                Toolkit paper source\n├── receipts/             Recorded example results\n├── assets/               README and social preview images\n└── scripts/\n    ├── build/            Documentation and map generators\n    ├── check/            Source, projection and boundary checks\n    ├── release/          Packaging and release verification\n    ├── inspect/          Compact inspection cards\n    └── papers/           Publication corpus tools\n```\n\nStart with the [examples](examples/README.md); use the\n[fixtures](fixtures/README.md) and [recorded results](receipts/README.md) to\nunderstand their checks. Your own runs go in the ignored `.microcosm/` directory.\nThe [paper guide](docs/papers/README.md) distinguishes the toolkit paper's\n[source](paper/README.md) from the main research papers.\n\nThe root `ORGANS.md`, `ARCHITECTURE.md` and `AGENT_ROUTES.md` are generated\nindexes. `AXIOMS.md`, `PRINCIPLES.md` and `ANTI_PRINCIPLES.md` contain the\n[doctrine shown in the map](docs/reference/README.md#the-doctrine-in-the-map).\n`Makefile` and `.github/workflows/` run the checks. The\n[maintainer architecture guide](docs/maintainers/architecture.md) identifies\nwhich files to edit and which builders to run. The [script guide](scripts/README.md)\nexplains each maintenance role; `make check` enforces their placement in Git\nand CI.\n\nThe [documentation index](docs/README.md) separates human guides, reference\nmaps and maintainer instructions. Directory READMEs explain the examples and\nreference collections. The [terminology guide](docs/reference/terminology.md)\ntranslates older names you may encounter in source files.\n\n<a id=\"how-the-result-stays-honest\"></a>\n\n## What a passing result means\n\nA pass is about a particular input and check. The component records classify\nthe result: a computation run now, a replay of supplied data, or a comparison\nof files. That is why the source, test and stated limits are published together.\n\n<details>\n<summary>Limits by kind of example</summary>\n\n- **Formal proof:** Lean can check the supplied formal statement and proof.\n  A successful toolkit example does not prove an unrelated theorem.\n- **Agent safety:** a replay does not establish that a deployed agent is safe.\n  Its prepared cases cover specific checks, not every possible attack.\n- **Research and forecasting:** synthetic examples do not establish scientific\n  expertise, investment returns or a forecasting track record. This is not\n  professional advice or investment advice.\n- **Generated files:** a match establishes agreement for the specified files.\n  You do not gain permission to publish private material by comparing them.\n- **Recording and resuming work:** you do not gain permission to edit someone\n  else's source files or publish a release by recording a task as complete.\n\n</details>\n\nThis is a research prototype containing selected public source files and\nexamples. It does not\ncontain enough code or data to reconstruct the private system. It is not a\nhosted service or a production security product, and has no model-provider\naffiliation or endorsement. Public examples do not establish the reliability\nof the unseen private workbench.\n\n## Develop and contribute\n\nA useful contribution can be small: an input that exposes a mistaken check,\na clearer explanation, an easier first command or an improvement to one tool.\n[Contributing](CONTRIBUTING.md) explains how to report a reproducible mismatch,\ntry an entry-route experiment or propose an infrastructure change. Mathematical\ncontributions belong in the [companion repository](https://github.com/wcook04/plectis-erdos/blob/main/CONTRIBUTING.md).\n\nFrom the clone root:\n\n```bash\nmake check   # fast registry and Lean-source checks; no installation\nmake ci      # tests, command smoke tests and a fresh package installation\n```\n\n`make ci` creates temporary environments and installs build and test\ndependencies, so it may need network access. `make help` lists narrower checks.\nThe [validation guide](docs/maintainers/validation.md) explains how to reproduce\nrelease records and check generated files. Edit their source and run the owning\nbuilder rather than changing generated maps or recorded results by hand.\nThe [release review](RELEASE_REVIEW.md) records the candidate’s checks;\n[source status](SOURCE_STATUS.md) explains which source material is included.\n\n## Videos and related projects\n\nThe [project videos page](https://wcook04.github.io/practical-formal-verification/#videos) describes the\nrecordings and their subjects.\n\n| Film | What it shows |\n|---|---|\n| [Plectis introduction](https://www.youtube.com/watch?v=U1z2-XaLe_w) | The research programme and why its results need to be inspectable by other people. |\n| [Code Map](https://www.youtube.com/watch?v=Srrnh6D94E8) | A private interface for exploring saved source descriptions and code connections. Cloning this toolkit does not install that interface. |\n| [One Checkout](https://www.youtube.com/watch?v=hIUgqeg1_Mw) | A selected patch committed while other edits stay in the shared folder. The standalone tool has its [own repository](https://github.com/wcook04/one-checkout). |\n| [Safety Core](https://www.youtube.com/watch?v=oN4ZaCWqfhY) | Editable agent rules and a narrow check tested against an invented-setting workaround. The rulebook toolkit has its [own repository](https://github.com/wcook04/safety-core). |\n\n### Where this toolkit fits\n\nThe Python components and experiments in this clone are one part of Plectis.\nUse the repositories below for the current mathematics and the two standalone\nagent tools; each has its own setup instructions and limits.\n\n| Related project | Start there for |\n|---|---|\n| [Plectis Erdős](https://github.com/wcook04/plectis-erdos) · [research website](https://wcook04.github.io/practical-formal-verification/maths/) | The current mathematical papers, Lean proofs and open questions. |\n| [Safety Core](https://github.com/wcook04/safety-core) | Editable agent rules, hooks and a local map of where checks apply. |\n| [One Checkout](https://github.com/wcook04/one-checkout) | Committing a reviewed patch while preserving other unfinished edits in the same checkout. |\n\nExplore the [research corpus](https://github.com/wcook04/plectis-erdos) and the [Plectis website](https://wcook04.github.io/practical-formal-verification/).\n\n## Name and history\n\n**How this was built.** William Cook sets the direction and is responsible for\nthe public claims; AI coding agents write and maintain most of the code.\n[PROVENANCE.md](PROVENANCE.md) records authorship and third-party sources.\n\nThe project was called Microcosm until 21 June 2026. It became Plectis to avoid\nconfusion with Southampton's earlier Microcosm hypermedia system, without\nimplying endorsement or affiliation. Old names remain in package imports,\noutput paths and historical records for compatibility with existing tools.\n\nFor private security reports, see [SECURITY.md](SECURITY.md). To cite the\nsoftware, use [CITATION.cff](CITATION.cff). Copyright 2026 William Cook;\nlicensed under Apache 2.0, with details in [LICENSE](LICENSE) and [NOTICE](NOTICE).\n\n## Mathematics companion snapshot\n\nThe companion's [current results guide](https://github.com/wcook04/plectis-erdos/blob/main/docs/RESULTS.md)\nowns its mathematical claims. This toolkit's recorded source and release links\nare preserved in the [companion reference](docs/reference/lean-companion.md).\n","body_format":"text"},"documentation":{"document_id":"documentation","source_path":"docs/README.md","role":"documentation_index","github":"https://github.com/wcook04/practical-formal-verification/blob/main/docs/README.md","source_commit":"f2ed2582e6de24c71b45dfc30dae33863e2ab539","permalink_url":"https://github.com/wcook04/practical-formal-verification/blob/f2ed2582e6de24c71b45dfc30dae33863e2ab539/docs/README.md","sha256":"sha256:c2d27801de5614bc38a454987f4af8fefdc43c470410de7891dac0c416bb88d4","byte_count":4554,"origin":"public_main_checkout","body":"# Plectis documentation\n\nPlectis publishes tools and worked examples for doing research with AI while\nkeeping the work understandable and checkable. This directory explains the\nproject, walks through examples and records how to maintain the software.\n\n<a id=\"start\"></a>\n\n## Start here\n\n1. Read the [project overview](overview.md) to understand why these tools exist\n   and how they relate to the mathematics and website.\n2. Run the [quickstart](../QUICKSTART.md) from a clone. No package installation\n   or model API key is needed for the basic commands.\n3. Choose one [worked guide](guides/README.md) and follow its input, code and\n   result. You do not need to read the complete catalogue first.\n\n## Human guides\n\n| You need… | Read… |\n|---|---|\n| An explanation of the project and related approaches | [Project overview](overview.md) |\n| A complete example to run and change | [Worked guides](guides/README.md) |\n| Help with older names used in source files | [Terminology and file map](reference/terminology.md) |\n| A tool for a particular subject | [Component map](../ORGANS.md#find-your-specialty) |\n| The research argument and supporting papers | [Paper guide](papers/README.md) |\n| Mathematical results, open questions and Lean proofs | [Plectis Erdős](https://github.com/wcook04/plectis-erdos) |\n\n## Reference\n\nThe [reference index](reference/README.md) collects the component catalogue,\narchitecture maps, source records and release information. These answer\nspecific questions once you have chosen a component. The\n[Lean companion reference](reference/lean-companion.md) records which edition\nof the mathematics repository this toolkit refers to.\n\n## Contribute and maintain\n\nStart with [Contributing](../CONTRIBUTING.md). The\n[maintainer index](maintainers/README.md) points to the implementation map,\ntest commands, document generators, packaging rules and release instructions.\n\n| Directory | What belongs there |\n|---|---|\n| [`guides/`](guides/README.md) | Worked examples with runnable commands and an explanation of their results. |\n| [`reference/`](reference/README.md) | Lookups, recorded versions and links to generated maps. |\n| [`maintainers/`](maintainers/README.md) | Instructions for changing and verifying the software. |\n| [`governance/`](governance/public-boundary.md) | Rules about public source, claims and publication. |\n| [`papers/`](papers/README.md) | The paper catalogue, PDFs and searchable manuscript text. |\n\nThe root README is the short introduction; [overview.md](overview.md) carries\nthe fuller explanation. Generated reference files keep their established root\npaths so existing tools and links continue to work.\n\n## Use your own coding agent\n\nTo research the agent-readable website without a clone, use its\n[navigation guide](https://wcook04.github.io/practical-formal-verification/llms.txt).\nIt links text indexes and the data behind the visual maps.\n\nOpen a clone in a coding agent that can read files and run shell commands.\nAsk it to read `AGENTS.override.md`, then give it a concrete question or change.\nThe [agent quickstart](../QUICKSTART.md#use-your-own-coding-agent) includes a\nsample request and explains where to save results. Record the checkout commit,\ncommands, inputs, outputs and what the run leaves untested.\n\nFor proof search or mathematical contributions, follow the separate\n[mathematics agent guide](https://github.com/wcook04/plectis-erdos/blob/main/docs/agents/README.md#start-with-current-public-work).\nIts proof evidence belongs to that repository.\n\n## Generated reference and agent instructions\n\nUse the [reference index](reference/README.md) to find generated maps.\n[AGENTS.override.md](../AGENTS.override.md) gives agents their first files and\ncommands; [AGENTS.md](../AGENTS.md) governs edits. The\n[architecture maintenance guide](maintainers/architecture.md#keep-each-document-in-its-role)\nidentifies each generator. Change its source and regenerate the output.\n\n## Troubleshooting\n\n| Symptom | Next step |\n|---|---|\n| `No module named plectis` | Run from the clone root with `PYTHONPATH=src`, or use the Python interpreter from the environment where you installed Plectis. |\n| `plectis: command not found` | Use `.venv/bin/plectis` on macOS/Linux or `.venv\\Scripts\\plectis.exe` on Windows; activating the environment also puts it on your path. |\n| A component reports unavailable dependencies | Read its entry in [ORGANS.md](../ORGANS.md). Individual examples can require external tools. |\n| Port 8765 is already in use | Choose another port with `--port`, then open that port in your browser. |\n","body_format":"text"},"understanding-plectis":{"document_id":"understanding-plectis","source_path":"docs/UNDERSTANDING_PLECTIS.md","role":"worked_software_introduction","github":"https://github.com/wcook04/practical-formal-verification/blob/main/docs/UNDERSTANDING_PLECTIS.md","source_commit":"f2ed2582e6de24c71b45dfc30dae33863e2ab539","permalink_url":"https://github.com/wcook04/practical-formal-verification/blob/f2ed2582e6de24c71b45dfc30dae33863e2ab539/docs/UNDERSTANDING_PLECTIS.md","sha256":"sha256:f6e921b2a1d89a03fad9e1587392a719628039cb200cd65a348c634a3d5deb66","byte_count":897,"origin":"public_main_checkout","body":"# Understanding Plectis\n\nThe explanations previously collected here now have separate homes.\nThese section links remain available for older references.\n\n## What you can do with it\n\nRead the [project overview](overview.md) for how the toolkit fits together,\nthen use the [quickstart](../QUICKSTART.md) to try it.\n\n## One example you can follow\n\nThe [prompt-injection walkthrough](guides/prompt-injection.md) follows\nprepared JSON records through a checker, then changes one field to make\nthe check fail.\n\n## The terms used in the repository\n\nThe [terminology reference](reference/terminology.md#the-terms-used-in-the-repository)\nexplains names used in the code and recorded results.\n\n## Where the files fit\n\nUse the [file map](reference/terminology.md#where-the-files-fit) to find\nsource, inputs, tests and saved results. The [documentation index](README.md)\nlists the other guides and references.\n","body_format":"text"},"quickstart":{"document_id":"quickstart","source_path":"QUICKSTART.md","role":"local_tour","github":"https://github.com/wcook04/practical-formal-verification/blob/main/QUICKSTART.md","source_commit":"f2ed2582e6de24c71b45dfc30dae33863e2ab539","permalink_url":"https://github.com/wcook04/practical-formal-verification/blob/f2ed2582e6de24c71b45dfc30dae33863e2ab539/QUICKSTART.md","sha256":"sha256:12b747edb712a96f89f06a0198c8c70073103ff34bd745dafd6225bd2ccc05e2","byte_count":5047,"origin":"public_main_checkout","body":"# Quickstart\n\nRun Plectis, inspect what it wrote, then choose one component to explore.\nFor an explanation before running anything, read\n[Understanding Plectis](docs/UNDERSTANDING_PLECTIS.md).\n\n## 1. First result\n\nYou need Git and Python 3.11 or newer. These commands use a macOS / Linux shell\n(or WSL on Windows), and every later source command runs from the clone root:\n\n```bash\ngit clone https://github.com/wcook04/practical-formal-verification && cd plectis\nPYTHONPATH=src python3 -m plectis tour --format text .\n```\n\nThe tour lists project files in `.microcosm/catalog.json` and proposes tasks\nsuch as inspecting a README in `.microcosm/routes.json`. It records a simulated\ntask run; it does not execute your project's tests. Your source files are\nunchanged, and `.microcosm/` is ignored by git. The component examples have separate commands.\n\n```bash\nPYTHONPATH=src python3 -m plectis tour --card .\nPYTHONPATH=src python3 -m plectis hello .\n```\n\n`plectis tour --card` prints JSON including the selected job's identifier; `hello`\nprints introductory text without creating files. In PowerShell, set `$env:PYTHONPATH = \"src\"` and use\n`python -m plectis` in place of `PYTHONPATH=src python3 -m plectis`.\n\n`./bootstrap.sh` runs the supplied examples in\n`fixtures/first_wave/pattern_binding_contract/input` and scans for forbidden secret strings. It writes\n`.microcosm/cold_clone_probe.json` with the resulting status.\n`./bootstrap.sh --dry-run` previews that command without running it.\n\n## Use your own coding agent\n[Agent setup and example tasks](docs/README.md#use-your-own-coding-agent).\n\n## 2. Install\n\nInstallation is optional. The component and browser steps use the source form,\nso you can skip this step. To install into a virtual environment on macOS / Linux:\n\n```bash\npython3 -m venv .venv\n.venv/bin/python -m pip install .\n.venv/bin/plectis tour --format text .\n```\n\nUse `.venv/bin/plectis` for installed commands, or activate the environment with\n`source .venv/bin/activate` to use `plectis`. On Windows the corresponding paths\nare `.venv\\Scripts\\python.exe` and `.venv\\Scripts\\plectis.exe`.\n\nIf pip reports `error: externally-managed-environment`, it is protecting a\nsystem-managed Python ([PEP 668](https://peps.python.org/pep-0668/)). Use the\nvirtual environment above or the source commands; neither needs a system-wide\npackage installation. The runtime has no third-party dependencies, although\ncloning and installing build / test dependencies can need network access.\n\n## 3. Browse the component corpus\n\n```bash\nPYTHONPATH=src python3 -m plectis comprehend --slice mechanism --format text\nPYTHONPATH=src python3 -m plectis comprehend --first-action \"Replay a prompt injection example\" --format text\n```\n\nThe first command prints an implementation summary for each component. The\nsecond matches words in the quoted goal to a component and prints commands,\ninput/output paths and saved result paths; it does not run those commands.\nReplace the goal with your own. Check the printed `--out` path before running it.\n[The worked example](docs/UNDERSTANDING_PLECTIS.md#one-example-you-can-follow)\nuses `.microcosm/` for output so the committed example records stay unchanged.\n\nFor other examples and source files, use the [README Component Map](README.md#choose-a-route),\nthe [one-line component list](ORGANS.md#plectis-at-a-glance--every-organ-in-one-line)\nor the [specialty index](ORGANS.md#find-your-specialty).\n\n## 4. Inspect in a browser\n\n```bash\nPYTHONPATH=src python3 -m plectis serve . --host 127.0.0.1 --port 8765 --max-requests 7\n```\n\nOpen `http://127.0.0.1:8765` while it runs. The server stops after seven requests;\na browser can make several requests for one page. Omit `--max-requests` only when\nyou want to keep exploring, then press Ctrl-C in the terminal to stop it.\n\n## 5. Verify the public floor\n\n```bash\nmake check   # reject registry errors and prohibited Lean source syntax\nmake ci      # install, run tests, run example commands, install/run the package\n```\n\nThese targets need `make` and Python. `make check` rejects missing or unknown\ncomponent/evidence-class IDs and prohibited syntax in shipped Lean source.\n`make ci` also creates a temporary virtual environment and installs test\ndependencies. The [validation runbook](docs/maintainers/validation.md) lists\nthe programs each target runs and gives a Python setup when `make` is unavailable.\n\n## Boundaries\n\nThe runtime commands above make no network or model calls, with no source mutation.\nThe prompt-injection example compares prepared JSON rows with policy rules;\nit does not measure a model's response to a hostile page.\n`.microcosm/` retains the project's older name; see\n[Name and history](README.md#name-and-history). Once you know which result you\nwant to inspect, `PYTHONPATH=src python3 -m plectis evidence list . --limit 25`\nlists the saved result files. Continue with [the documentation map](docs/README.md), or\n[CONTRIBUTING.md](CONTRIBUTING.md) if you want to change or check the code. [Troubleshooting](docs/README.md#troubleshooting) covers common setup errors.\n","body_format":"text"},"agents":{"document_id":"agents","source_path":"AGENTS.md","role":"software_agent_entry","github":"https://github.com/wcook04/practical-formal-verification/blob/main/AGENTS.md","source_commit":"f2ed2582e6de24c71b45dfc30dae33863e2ab539","permalink_url":"https://github.com/wcook04/practical-formal-verification/blob/f2ed2582e6de24c71b45dfc30dae33863e2ab539/AGENTS.md","sha256":"sha256:9417d9fea761433ffc1889eb3a52aecd058d1078ac51fa54fa569f2a5eb00c33","byte_count":29550,"origin":"public_main_checkout","body":"# AGENTS.md - Plectis\n\nReading this as a person? The human map is [README.md](README.md); this file\nis the agent entry contract.\n\nPlectis is a local Python command-line tool plus 88 reference\ncomponents spanning formal proof, agent reliability and safety, research and\nforecasting, projection-drift control, validation, work landing, and continuity.\nEach one names the code that runs, the evidence it uses, the receipt it writes,\nand the limit of what a passing result proves. Read each replay's execution\navailability: inclusion in the catalogue does not establish that its required\nsource and dependencies are available in this clone.\n\nRead it in that order — mechanisms, then evidence discipline, then the local\nruntime. The receipts are how a component is held to its claim; they are not the\nthing being published. An agent that summarises Plectis as claim-checking has\ndescribed the accountability layer and missed six of the seven families.\n\nA user can point Plectis at a project folder, create ignored `.microcosm/`\nstate, index files, discover patterns, propose and explain routes, record work,\nobserve events, and inspect supporting evidence. The 88 components are the\nreference corpus; the project commands are how users inspect and exercise it.\n\nThis is an executable research prototype and developer tool. It does not host\na service, call model providers, modify the source it inspects, prove that\narbitrary software is correct, establish production readiness, supply a\nbenchmark score, or provide financial advice. Later sections use the\nrepository's exact vocabulary for projects, catalogues, patterns, standards,\nroutes, work, events, evidence, explanations, assimilation, imported component\nbodies, and exported bundles.\n\n## Fast Entry For Cold Agents\n\n**Arrived with a goal? Convert it into your first correct action before\nabsorbing anything:**\n\n```bash\nPYTHONPATH=src python3 -m microcosm_core comprehend --first-action \"<your goal>\" --format text\n```\n\nIt returns one graph-backed contract: the runnable command, the owning\ncomponent, the validator that proves it, the shipped receipts, the stop\ncondition, and the do-not-edit boundary. [FIRST_ACTION.md](FIRST_ACTION.md)\ndemonstrates this across a goal battery — localization, change-shaped goals,\nauthority refusals, vocabulary traps — and is regenerated from the live\ncompiler, so the examples are compiler output, not prose.\n\n**Asked what this repository is, or why any of it is claimed?** That is a papers\nquestion, not a component question, and the papers are in the clone. Read\n[`docs/papers/corpus.json`](docs/papers/corpus.json) — it is small, and it names\nevery active paper, the question each answers, what each is *not* authority\nfor, and every section with its label and line number. PDFs and searchable\nfull-text mirrors travel with the clone. Follow each corpus row's\n`preferred_read_path` when a generated full-text mirror is marked stale. Do\nnot answer a question about what this project claims by summarising the organ\ninventory: the components show what runs, and the papers state what may be\nconcluded from that, which is the narrower thing.\n\n**Asked to turn an open question into an expert handoff?** Read\n[`docs/guides/hypothesis-handoffs.md`](docs/guides/hypothesis-handoffs.md), then validate the worked\npacket with:\n\n```bash\nplectis hypothesis-handoff --input examples/hypothesis_handoff/independent_evaluation.json --format text\n```\n\nThe leading hypothesis is a disclosed project prior, not evidence or a\nprobability. Every outcome must point to a named option, every option must have\na discriminator, and an expert return stays advisory until the packet's exact\nrepository paths and release checks pass.\n\nIf this is your first touch in a standalone clone and you have no task yet, do\nnot start by absorbing the organ inventory. First prove the local entry path\nand the public authority membrane:\n\n1. Read `README.md` for the human map and install mode. In that README, use\n   the `Choose a starting point` table and `What a passing result means` before opening\n   raw receipts or the long organ inventory.\n2. From the repository root, run the bounded cold-clone probe before any\n   install step:\n\n```bash\n./bootstrap.sh\n```\n\n   It validates the first-wave fixture and boundary floor, writes ignored\n   `.microcosm/cold_clone_probe.json` evidence, and points back to the README\n   map. Use `./bootstrap.sh --dry-run` when you need to see the exact command\n   without writing the ignored receipt.\n3. From the repository root, make the console command available with\n   `make install`. If you cannot use `make`, run\n   `python3 -m pip install -e '.[test]'` directly; if you cannot install, use\n   the source form `PYTHONPATH=src python3 -m microcosm_core <command>`.\n4. Run the standard smoke target before opening raw receipts:\n\n```bash\nmake smoke\n```\n\nThe smoke target writes ignored receipts under `.microcosm/smoke/`, validates\nthem, and prints a compact terminal summary. A healthy run includes\n`Plectis smoke check: pass`, `authority: pass`, `workingness: clear`, and\n`served status: pass`. If you are inspecting each output, use the same commands\nby hand:\n\n```bash\nplectis hello .\nplectis hello --reader cold_cloner .\nplectis hello --reader reviewer .\nplectis hello --reader skeptical_reviewer .\nplectis hello --reader agent .\nplectis hello --reader domain_specialist .\nplectis first-screen --card .\nplectis comprehend --first-action \"<goal>\"\nplectis comprehend --first-contact\nplectis comprehend --organ <organ_id>\nplectis tour --card .\nplectis status --card .\nplectis authority --card\nplectis workingness --card\nplectis legibility-scorecard\n```\n\nThe reader aliases are shortcuts into existing first-screen branches, not new\nroutes: `cold_cloner` / `cold-cloner` maps to the public GitHub visitor branch,\n`interesting_parts` / `interesting-parts` maps to that same public visitor\nbranch for \"what is interesting here?\" questions,\n`skeptical_reviewer` / `skeptical-reviewer` / `reviewer` maps to the safety/evals branch,\nand `agent` / `type-a-agent` maps to the repo-reading agent branch.\n`domain_specialist` / `domain-specialist` is the specialty reader branch; it\npoints to the generated organ specialty index without claiming domain\ncorrectness or expert review. The card echoes the requested alias or route id\nfor copy/paste while resolving it to the selected branch.\n\nRead those outputs as the first contract: `plectis hello` is the no-write\nhuman card, `plectis first-screen --card` is the compact JSON reader map,\n`plectis comprehend --first-action \"<goal>\"` is the goal-shaped entry (one\ngraph-backed First Correct Action contract: action, owner, validator, receipts,\nstop condition, do-not-edit boundary — demonstrated in\n[FIRST_ACTION.md](FIRST_ACTION.md)),\n`plectis comprehend --packet-atlas` is the navigable menu of comprehension\npackets (pick the one matching your goal: self_model, first_contact, authority,\norgan_cluster, math, claim_trace, flow, mutation_plan, path),\n`plectis comprehend --self-model` compiles the WHOLE substrate into one\nbudgeted packet so you comprehend every family, the real-vs-thin calibration,\nand what not to claim at once (`--profile whole_substrate_map` for every\norgan essence), `plectis comprehend --first-contact` is the source-body-free\ncomprehension read pack that answers \"what is this substrate, what does each\norgan do, and what may I trust\" without rereading source (drill in with\n`plectis comprehend --organ <organ_id>`, trace a claim with\n`plectis comprehend --slice claims --organ <organ_id>`, and prove the whole\nsurface comprehends with `plectis comprehension-assay --whole-system`),\n`plectis tour --card` is the shared state-writing behavior proof,\n`plectis status --card` is the compressed public evidence map,\n`plectis authority --card` is the claim ceiling, `plectis workingness --card`\nis the failure envelope, `plectis observe --card` is the compact\nroute/work/event/evidence handoff, and `plectis legibility-scorecard` is the\ncold-reader comprehension check. These commands do not create release, hosting,\nproof, production, provider-call, source-mutation, or financial-advice\nauthority.\n\n## Repository organization\n\nUse [scripts/README.md](scripts/README.md) and the\n[architecture guide](docs/maintainers/architecture.md) before moving files.\nKeep programs in `scripts/<role>/`, tests in `tests/`, and paper text in\n`docs/papers/`. Change producers, callers, imports, package destinations and\ngenerated navigation together. Stage intended files, then run `make check`;\nthe index-based layout check is also required by CI.\n\n## Default Reflexes\n\nThese are passive defaults: you invoke them yourself the instant the trigger\nholds. No one has to tell you. They exist because the most common failure here\nis reasoning about the whole substrate from the one slice you happened to open.\n\n- **First-action reflex.** If you have a concrete goal (or were handed one) and\n  you are about to grep, glob, or open source to figure out where to start, run\n  `plectis comprehend --first-action \"<goal>\"` first. It names the owner, the\n  runnable command, the validator, the receipts, and the stop condition; source\n  search before that contract is almost always wasted or unsafe reading.\n- **Comprehension reflex.** If you ever catch yourself assessing, summarizing,\n  scoring, or judging Plectis from only the part you looked at (the tell:\n  \"based on the part I worked on…\"), or you have just arrived cold, then before\n  you answer run `plectis comprehend --self-model` — add\n  `--profile whole_substrate_map` to read every organ's essence at once.\n  Comprehend the whole substrate first, then speak.\n- **Mechanism-before-claim reflex.** If the question is specifically about\n  component substance, what the components actually do, or whether any family\n  (for example finance) is thin, run `plectis comprehend --slice mechanism`\n  first (source-only clone:\n  `PYTHONPATH=src python3 -m microcosm_core comprehend --slice mechanism`). It\n  returns every organ's real mechanism — what it computes, verifies, and\n  rejects — one line each, across the whole substrate. The proven failure is\n  forming the verdict from the nearest surfaces you opened (wrappers, line\n  counts, one-line glosses) and emitting that as a whole-system claim.\n  `finance_forecast_evaluation_spine`,\n  `finite_erdos_denominator_certificate_strike`,\n  `agent_sabotage_scheming_monitor_replay`, and `batch8_audio_level_rms_port`\n  are sentinels whose mechanism is invisible from the gloss — they are tells,\n  not the whole catalogue, so read the mechanism slice over every organ before\n  concluding. A skeptical reviewer or evals reader starts here, not at `tour`\n  or `hello`.\n- **Calibration-before-claim reflex.** If you ever are about to characterize\n  quality, completeness, readiness, or public significance, first read its `claim_ceiling`\n  and the self-model's `thin_or_projection_surfaces`. Infer quality from the\n  evidence; never assert it. The thinness is surfaced on purpose, not hidden —\n  most organ runners are exact-copy macro bodies, and the self-model says so.\n- **Custody-before-trust reflex.** If you ever are about to treat an organ's\n  runner as deep owned code, check `runner_custody_basis` first. An exact-copy\n  macro body is comprehended via registry metadata and receipts, not by reading\n  the copy as if it were authored here.\n- **Escalate-only-when-needed reflex.** If you ever are about to open a source\n  file, confirm you are mutating or proving. Otherwise stay in the\n  source-body-free read packs. When unsure which packet to open,\n  `plectis comprehend --packet-atlas` is the menu.\n\nBefore the full test floor, run `make check` for the fastest public preflight.\nIt validates the organ evidence-class registry and scans every shipped Lean\nfile for proof placeholders, project-defined axioms, native evaluation,\nunsafe/partial declarations, and unbounded kernel limits.\nIt should print both `Lean proof-trust check: pass` and `Plectis preflight:\norgan evidence-class registry loads cleanly.` Before publishing, handing off,\nor treating the standalone clone as verified, run `make ci`. It is the\npublic GitHub Actions entry and expands to that proof-trust preflight, editable\ninstall, public tests, source-form smoke, and package-install smoke\nverification.\n\nDo not launch multiple raw `pytest` processes against this root in parallel\nunless each one uses its own `--basetemp`. The Makefile targets already isolate\npytest scratch roots per run; direct parallel subsets must do the same or they\ncan race while copying fixture trees if they share a basetemp. The standalone\npytest config disables pytest's cache provider so direct runs do not write\n`.pytest_cache` into the checkout.\n\nThe standalone clone does not currently provision Black or Ruff in the default\nor `.[test]` dependency path. Treat a missing `python -m black` / formatter\nmodule as expected unless `pyproject.toml` changes. For organ edits, preserve\nlocal style and verify with `python -m py_compile` for touched Python files,\nfocused pytest with an isolated `--basetemp`, JSON validity or owner builders,\nand the relevant checker commands. Do not add formatter commands to closeout\nclaims unless package metadata provides them.\n\nIf you need a bounded standalone review artifact, run\n`make standalone-export EXPORT_OUT=/tmp/plectis-export`. This\nwrites a candidate folder plus `receipts/release/release_export_receipt.json`;\nit is not part of `make ci`, performs heavier outside-root smoke checks, and\nstill keeps `release_authorized=false` until a separate human release decision\nexists.\n\nBefore handing off or reviewing that artifact as a standalone clone, validate\nthe exported root itself:\n\n```bash\ncd /tmp/plectis-export/plectis\nmake ci\n```\n\nThat cold-clone check proves the exported package can install, test, and smoke\nfrom its own root. It does not authorize release; the release receipt remains\nthe authority boundary until a separate operator decision exists.\n\nAfter the compact route is green, use `skills/cold_start_navigation.md` for the\nshortest validation ladder. Use the inventory below only as a public runtime\nindex, not as a substitute for running the commands.\n\n## Live CLI Registry Boundary\n\nTreat `plectis --help` as the bounded first-screen console-command registry.\nIt is not the full drilldown inventory. In a source-only checkout, use\n`PYTHONPATH=src python3 -m microcosm_core --help` for the same first-screen\nregistry without installing the console script. The organ inventory below also\nnames route labels, bundle handles, fixture validators, and paper-module\ncontracts; drilldown commands remain callable by exact name when a route card,\ntest, or inventory row names them, but they are not guaranteed to appear in\nroot help. If a label is not listed in root help, use the shipped lens command,\npackage tests, fixture validator, or exact drilldown command named by the\nowning route instead of inventing a CLI path from the label.\n\n## Real Substrate Posture\n\nPlectis is the public repo form of the macro system, reorganized so a cold\nreader can run and inspect it without inheriting the private root's historical\nmess. Public should carry private by default: import, refactor, or export as\nmuch of the macro substrate as possible into this tree. Use synthetic fixtures\nonly as regression wrappers, negative cases, or toy inputs around a real\nmechanism.\n\nThe hard exclusion set is narrow: secrets and credential-equivalent live\naccess (`.env` files, API keys, tokens, passwords, private keys, cookies,\nbrowser profiles, keychains, account sessions, and direct credential-bearing\npayloads), raw operator voice, slurs or abusive wording, private personal\nmaterial, and other clearly unsafe or non-releasable content. Do not turn\n\"private state\" or \"release authority\" into a generic excuse for metadata-only\nstand-ins. Hosted launch and recipient sends are separate operational\ndecisions; they do not block this repo from carrying real source-available\ncontent.\n\nAny `body_copied=true` claim must point at a real target file and a verified\nsource-to-target import record. A source ref, provenance pointer, digest of an\nalready-existing validator, activation score, or maturity label is not an\nimported macro body.\n\n## Accepted Public Runtime Spine\n\nThis is the public entry inventory over `core/organ_registry.json` and\n`core/organ_evidence_classes.json`: accepted public runtime organ records\nwith receipt-index status. Those counts are inventory-only route-alignment metadata:\n`accepted_current_authority`, organ counts, and adapter-backed counts are not product progress, release readiness, product completeness, proof\nauthority,\nprivate-root equivalence, or whole-system correctness. These rows do not authorize release, and the prediction and market organs are\nevidence boundaries, not trading or financial advice.\n\nDo not read organs from this index alone. The generated atlas is the contract:\n\n- **[AGENT_ROUTES.md](AGENT_ROUTES.md)** — the generated task-class route\n  table for agents: task class, relevant organ(s), first command, authority\n  ceiling, evidence/receipt ref, stop condition, and drilldown target.\n- **[ORGANS.md#microcosm-at-a-glance--every-organ-in-one-line](ORGANS.md#microcosm-at-a-glance--every-organ-in-one-line)** —\n  the generated one-line organ ladder, grouped by canonical family order with\n  Entry & Reveal first.\n- **[ORGANS.md](ORGANS.md)** — the comprehension card for every organ: what it\n  makes visible (plain language), what an agent runs it for, its first command,\n  its evidence class, and what it does **not** authorize.\n- **[ORGANS.md#find-your-specialty](ORGANS.md#find-your-specialty)** — the\n  generated human specialty index; use it when the reader starts from a domain\n  rather than an agent task class.\n- **[ARCHITECTURE.md](ARCHITECTURE.md)** — the system at a glance: the local\n  runtime loop, the claim/evidence loop, the kernel primitives, and how the\n  seven families sit on one shared spine.\n\nThe atlas is regenerated from substrate with\n`PYTHONPATH=src python3 scripts/build/build_organ_atlas.py --write` and gated by\n`tests/test_organ_atlas.py`; do not hand-edit `AGENT_ROUTES.md`,\n`ORGANS.md`, `ARCHITECTURE.md`, or `atlas/agent_task_routes.json`.\nDrilldown CLIs such as `plectis reveal` and `plectis spatial-simulation` are\ndocumented per organ in [ORGANS.md](ORGANS.md). The accepted organs cluster into\ngenerated families in [ORGANS.md#families](ORGANS.md#families). Do not copy that\nfamily inventory into AGENTS; agents enter through [AGENT_ROUTES.md](AGENT_ROUTES.md),\nwhile the first faithful inventory pass is the generated\n[one-line organ ladder](ORGANS.md#microcosm-at-a-glance--every-organ-in-one-line)\nand humans can also browse through [ORGANS.md#find-your-specialty](ORGANS.md#find-your-specialty).\n\n## Concept And Mechanism Entry\n\nWhen a Plectis task asks to read, populate, or refine concepts and\nmechanisms, use the entry surface instead of starting from the extracted\npattern inventory. First open `plectis first-screen <project>` and read\n`doctrine_effect_frame`; it now exposes `CONCEPTS` and `MECHANISMS` as\nauthority-boundary handles, not ceremonial doctrine labels.\n\nThe concept floor is `standards/std_microcosm_concept.json`; it governs typed\nvocabulary boundaries with source refs, relationships, payload shape, omission\nreceipts, and anti-claims. The mechanism floor is\n`standards/std_microcosm_mechanism.json`; it governs reusable state, proof,\nrouting, or doctrine transformations with validator attachment. Both standards\npoint back to this agent-entry section and the first-screen doctrine frame.\n\nUse `core/public_standard_pressure.json` for the populated local pressure rows\n`concept_handle_requires_entry_surface` and\n`mechanism_handle_requires_runnable_contract`, plus\n`concept_mechanism_requires_population_specimen_loop` for the specimen-backed\npopulation rule.\n\nDo not stop at the standards. Continue through\n`atlas/entry_packet.json::concept_mechanism_entry_route.population_specimens`.\nThose rows are the specimen-backed loop: each specimen binds a concept role to a\nmechanism role, names source refs, relationship shape, payload shape,\nanti-claims, omission receipt, validator refs, and the public/private authority\nboundary. Use the specimen whose validator matches the pressure:\nfirst-screen route shape, executable grammar standard shape, standards-meta\norgan mapping, or voice-to-doctrine refinement. Only create a new packet if none\nof those existing lanes can carry the pressure without distortion.\n\n- Standard shape: `plectis executable-doctrine-grammar validate-standards-bundle --input examples/executable_doctrine_grammar/exported_standards_bundle --out /tmp/microcosm-executable-doctrine-grammar`\n- Organ-to-standard mapping: `plectis standards-meta-diagnostics run-diagnostics-bundle --input examples/standards_meta_diagnostics/exported_standards_meta_diagnostics_bundle --out /tmp/microcosm-standards-meta-diagnostics`\n- Local pressure -> owner surface -> validation -> closeout loop: `plectis voice-to-doctrine-self-improvement-loop run-bundle --input examples/voice_to_doctrine_self_improvement_loop/exported_voice_to_doctrine_bundle --out /tmp/microcosm-voice-to-doctrine`\n\n## Rules\n\n1. Start with `README.md`, then run `skills/cold_start_navigation.md` if you\n   need the shortest validation route.\n2. Keep first contact split by reader and task.\n   The human first-screen text projection is `plectis hello <project>`:\n\n   - People start in `README.md`; its no-write text view is\n     `plectis hello <project>`.\n   - Agents with a concrete goal run\n     `plectis comprehend --first-action \"<goal>\"`.\n   - Agents choosing a bounded read packet run\n     `plectis comprehend --packet-atlas`.\n   - `plectis first-screen --card <project>` returns the compact JSON reader\n     map. `plectis tour --card <project>` is the state-writing behavior proof;\n     `plectis tour <project>` is the full drilldown, and\n     `plectis compile <project>` is the explicit rebuild loop.\n   - `plectis explain <project> <route_id>` resolves one selected route.\n\n   Do not duplicate the component catalogue here. Resolve a task through\n   [AGENT_ROUTES.md](AGENT_ROUTES.md), a component through\n   [ORGANS.md](ORGANS.md), system structure through\n   [ARCHITECTURE.md](ARCHITECTURE.md), and exact command availability through\n   `plectis --help` or `PYTHONPATH=src python3 -m microcosm_core --help`. The\n   generated route owns the first command, validator, evidence or receipt,\n   stop condition, drilldown target, and authority ceiling. If a command is\n   absent from root help, use the exact lens, fixture validator, or drilldown\n   named by that route; do not invent a CLI spelling.\n\n   The state-writing path is repo -> `.microcosm`. Registry status is a\n   routing boundary: `accepted_current_authority` is not an evidence-strength\n   claim. Fixture verdicts are not benchmark scores and not score-based progress.\n   Bridge continuity examples use synthetic transport; they do not\n   report live bridge health.\n\n   The shared\n   state-writing behavior proof is `plectis tour --card <project>`.\n\n   These read models and fixtures do not authorize live provider calls, private\n   state access, source mutation, publication, hosting, trading, production\n   claims, benchmark claims, mathematical correctness, or release. Formal-math\n   metadata organs may classify, retrieve, route, and validate bounded public\n   fixtures. Only `formal_math_lean_proof_witness` may run its tiny public\n   Lean/Lake witness, and that witness grants no broader proof authority.\n   Generated projections stay downstream of `core/organ_registry.json`,\n   `core/organ_evidence_classes.json`, source modules, standards, and named\n   receipts; use their owner builders instead of editing projections.\n\n   Before release, run `make check` and require:\n\n```text\nLean proof-trust check: pass\nPlectis preflight: organ evidence-class registry loads\ncleanly.\n```\n\n   Then run `make ci`.\n\n3. Fixtures Are Tests: fixtures under `fixtures/first_wave/**` are examples,\n   bootstrap data, and negative cases. Do not treat fixture-only behavior as\n   product-complete runtime behavior.\n4. Receipts Are Evidence: generate receipts by running validators or\n   `bootstrap.sh`; do not edit receipts by hand. `bootstrap.sh` writes ignored\n   local `.microcosm/cold_clone_probe.json` evidence by default; pass `--emit`\n   only when refreshing an owned tracked receipt on purpose.\n5. Treat `core/organ_registry.json`, `core/acceptance/first_wave_acceptance.json`,\n   generated receipts, and public paper modules as public-root navigation\n   surfaces.\n6. Do not widen Lean/Lake. `mathematical_strategy_atlas_hypothesis_scorer`,\n   `tactic_portfolio_availability_probe`,\n   `target_shape_tactic_routing_gate`, `lean_std_premise_index`,\n   `formal_math_premise_retrieval`,\n   `formal_math_verifier_trace_repair_loop`,\n   `formal_evidence_cell_anchor_resolver`,\n   `undeclared_library_prior_symbol_classifier`,\n   `ring2_premise_retrieval_precision_recall_harness`,\n   `agent_benchmark_integrity_anti_gaming_replay`, and\n   `agent_monitor_redteam_falsification_replay`,\n   `agent_sabotage_scheming_monitor_replay`, and\n   `agent_sandbox_policy_escape_replay`,\n   `indirect_prompt_injection_information_flow_policy_replay`, and\n   `agentic_vulnerability_discovery_patch_proof_replay`, and\n   `provider_context_recipe_budget_policy` are\n   metadata/retrieval/admissibility/metric/context-budget lanes only, and\n   `formal_math_lean_proof_witness` authorizes only the tiny public witness\n   fixture in a temporary workspace, and `verifier_lab_kernel` authorizes only\n   a public leak-proof verifier-lab receipt with no private proof bodies,\n   provider proof authority, oracle-to-forward contamination, arbitrary\n   Evolve, source mutation, benchmark claims, or release/publication work.\n   `corpus-lens` is public evidence over corpus readiness receipts:\n   it may make Mathlib import absence and blocked consumers legible, but it\n   must not become Lean/Lake execution, Mathlib proof authority,\n   corpus-completeness authority, benchmark authority, provider authority, or\n   release authority.\n   `standards_meta_diagnostics` is public runtime diagnostics: it may diagnose\n   public standards/runtime/receipt coverage, but it must not claim registry\n   source authority, private macro access, release operations, provider calls,\n   proof authority, or whole-system correctness.\n   `research_replication_rubric_artifact_replay` is a runnable\n   research-replay harness: it may diagnose artifact replay completeness,\n   declared artifact-hash roster binding, grading rubric boundaries, rerun\n   evidence, and private-leakage guards, but it must not claim external\n   benchmark performance, original-paper replication, publication operations,\n   provider calls, secret export, or release authority.\n   `cold_reader_route_map` is executable public route evidence: it may diagnose public\n   entry route order, command refs, docs refs, and receipt refs, but it must not\n   claim route-registry authority, source mutation authority, private macro\n   access, release operations, provider calls, trading advice, or whole-system\n   correctness.\n7. Do not treat prediction fixtures as trading or financial advice. The\n   `prediction_oracle_reconciliation` organ may validate synthetic reasoning\n   mechanics only; it must not imply investment advice, live market data,\n   provider calls, performance claims, publication, or release authority.\n8. Do not import parent-repository-only tools, host-local state, prompt bodies,\n   provider payloads, operator threads, HUD/browser/cockpit state, or old\n   scratch public-root content as source authority.\n9. Do not add release, hosted-public, publication, recipient, credentialed\n   provider-call, or secret-export surfaces from this root.\n10. Do copy real non-secret macro bodies into public runtime code, fixtures,\n   docs, or exported bundles when they are the mechanism being released. Do not\n   copy secrets, credential-equivalent payloads, private operator text, or\n   provider payload bodies. Any macro material that enters Plectis must pass\n   the `macro_projection_import_protocol` boundary with source refs, target\n   refs, body-import verification, omission receipts, and validation receipts.\n11. Keep research-prototype posture explicit. Do not describe this root as\n   production infrastructure, a hosted service, or a release-ready agent\n   platform.\n\n## Receipt Floor\n\nEvery new validator receipt must include `status`, `secret_exclusion_scan`,\n`authority_ceiling`, `anti_claim`, and `receipt_paths`. Organ receipts also\ninclude `organ_id` and `fixture_id`. Legacy `private_state_scan` is a\ncompatibility surface only; do not introduce it in new receipt contracts.\n\nReceipt evidence has a real-substrate supremacy floor. Prefer real runtime\nreceipts, copied non-secret macro bodies with provenance, and source-faithful\npublic refactors. Synthetic receipts are allowed only as regression/negative\nharness artifacts or temporary blocked-import debt with a named real substrate\nreplacement target; they are defects when they stand in for available real\nsubstrate.\n\n## Anti-Claim\n\nThis public agent entry file gives bounded public-root navigation only. It does\nnot authorize Lean/Lake beyond the bounded public witness fixture, trading or\nfinancial advice, hosted deployment, publication, recipient work, credentialed\nprovider calls, secret export, or whole-system correctness.\n","body_format":"text"},"source-status":{"document_id":"source-status","source_path":"SOURCE_STATUS.md","role":"publication_status_authority","github":"https://github.com/wcook04/practical-formal-verification/blob/main/SOURCE_STATUS.md","source_commit":"f2ed2582e6de24c71b45dfc30dae33863e2ab539","permalink_url":"https://github.com/wcook04/practical-formal-verification/blob/f2ed2582e6de24c71b45dfc30dae33863e2ab539/SOURCE_STATUS.md","sha256":"sha256:e45a6c6af5552800b6b7457d307cfff9cc9372aad5c6cea647c6cbb83eb99afc","byte_count":5702,"origin":"public_main_checkout","body":"# Plectis Source Status\n\nLast updated: 2026-09-12\n\nThis repository contains the Plectis Python package, example inputs, saved\nresults from earlier runs, tests, documentation, and license files. The\n[component map](ORGANS.md) lists 88 components, called *organs* in the code and\nJSON files. Each has a stated task, input files, a command to run, and limits on\nwhat its results establish. [Understanding Plectis](docs/UNDERSTANDING_PLECTIS.md)\nworks through one example.\n\nMicrocosm is the former public name. It remains only where compatibility or\ntechnical continuity still requires it: the `microcosm_core` import path, the\nlegacy `microcosm` command alias, `.microcosm/` local state, historical source\npaths, fixture names, and previously published links.\n\n## What is public here\n\n- The public source of record is the canonical Plectis\n  repository at <https://github.com/wcook04/practical-formal-verification>.\n- You can inspect the source and run the commands below from a fresh clone\n  without access to the private development repository.\n- The website at <https://wcook04.github.io/practical-formal-verification/> presents this public source\n  and its release documents. If the website and repository\n  disagree, the repository files and committed release receipts are the source\n  of record.\n- The retired `wcook04/microcosm-substrate` repository and old Pages URL are\n  historical links, not the current development repository.\n- The package is licensed under Apache-2.0; see [LICENSE](LICENSE),\n  [NOTICE](NOTICE), and [Provenance](PROVENANCE.md).\n\n## What is not public here\n\n- The larger private working root, private ledgers, operator notes, browser\n  state, account material, secrets, recipient-send state, and unpublished\n  reference material are outside this repository.\n- Private raw video/capture sources are not part of the public source release.\n  The [recorded walkthroughs](https://wcook04.github.io/practical-formal-verification/#demo-videos)\n  are hosted separately and linked from the README.\n- This repository is not a hosted service, production security product, legal or\n  financial advice system, trading system, or whole-system correctness proof.\n\n## Reviewer path\n\nRun these commands from the clone root with Python 3.11 or newer and a shell\nsuch as Bash or Zsh. No package installation or model account is needed.\n[Quickstart](QUICKSTART.md#1-first-result) gives the clone command and setup.\n\nFirst, run the supplied bootstrap example:\n\n```bash\n./bootstrap.sh\n```\n\nThis runs the supplied inputs in\n`fixtures/first_wave/pattern_binding_contract/input` and searches selected text\nfiles for tokens listed in the [scan policy](core/private_state_forbidden_classes.json).\nThe [security runbook](docs/maintainers/security-runbook.md) explains the file\nselection and scan limits. A successful run prints `Plectis cold-clone probe passed` and\nexits with code 0; a failed run exits with a nonzero code. It writes the summary\nto `.microcosm/cold_clone_probe.json` and component output files under\n`.microcosm/cold_clone_probe/`. These are results of your local run, separate\nfrom the committed files under [`receipts/`](receipts/).\n\nThen print the stored component descriptions and look up a suggested command:\n\n```bash\nPYTHONPATH=src python3 -m microcosm_core comprehend --slice mechanism --format text\nPYTHONPATH=src python3 -m microcosm_core comprehend --first-action \"evaluate prompt injection defenses\" --format text\n```\n\nThe first command prints one stored description per component, grouped by\nfamily. It selects text from the [paper-module summaries](core/paper_module_capsules.json),\n[component descriptions](core/mechanism_sources.json), and\n[component map data](core/organ_atlas.json). It does not run the components or\nverify those descriptions against their code.\n\nThe second command matches words in the question to component names and task\ndescriptions. For this question, it prints a prompt-injection example command,\ninput and output paths, and the stated limits. It does not execute that command.\nReplace the quoted question with your own; see\n[Find a component command](FIRST_ACTION.md) for how to interpret the result and\nchoose an output directory before running it. Both lookups print to the terminal\nwithout writing result files.\n\nTo inspect the local repository and the supplied claim limits:\n\n```bash\nPYTHONPATH=src python3 -m microcosm_core hello .\nPYTHONPATH=src python3 -m microcosm_core tour --card .\nPYTHONPATH=src python3 -m microcosm_core authority --card\n```\n\n`hello` prints an introduction and suggested commands. `tour` inventories the\nlocal files, proposes tasks from that inventory, and creates a simulated work\nrecord; it does not execute the project's tests. Its files, including\n`catalog.json`, `routes.json`, and `work_items.json`, go under `.microcosm/`.\n`authority --card` prints the supplied component classifications, result-file\npaths, and limits on permitted actions and conclusions. The supplied permission\nflags for publication, source changes, and provider calls are false. `hello`\nand this authority query print to the terminal without writing result files.\n\nAfter [installation](QUICKSTART.md#2-install), you can replace\n`PYTHONPATH=src python3 -m microcosm_core` with `plectis` in these commands. The\nlegacy `microcosm` command remains a compatibility alias for existing local\nscripts and historical receipts.\n\n[Release review](RELEASE_REVIEW.md) connects the software claims to input files,\ncommands, and saved results. The [validation guide](docs/maintainers/validation.md)\nexplains `make smoke` and the other test and package commands, including their\nprerequisites and output locations. For the reading order, return to the\n[README](README.md) or the [documentation index](docs/README.md).\n","body_format":"text"},"contributing":{"document_id":"contributing","source_path":"CONTRIBUTING.md","role":"contribution_route","github":"https://github.com/wcook04/practical-formal-verification/blob/main/CONTRIBUTING.md","source_commit":"f2ed2582e6de24c71b45dfc30dae33863e2ab539","permalink_url":"https://github.com/wcook04/practical-formal-verification/blob/f2ed2582e6de24c71b45dfc30dae33863e2ab539/CONTRIBUTING.md","sha256":"sha256:fbff30ffdeaf63d13c041c727ede496d58842cd1996c592412baf82d539f89fe","byte_count":9164,"origin":"public_main_checkout","body":"# Contributing\n\nContributions and error reports are welcome. You can improve one example or\nreport a wrong result without understanding every program in the repository.\nUseful contributions include:\n\n- fixing a command, adding a test case or making an example easier to run;\n- explaining what a command reads, computes and writes, or repairing a broken link;\n- showing an input that a validator accepts although it violates the stated rule;\n- adding a program with its source history, input files, expected output and\n  test functions, and stating which claim follows from that output.\n\nIf you are still choosing where to start, [Understanding Plectis](docs/UNDERSTANDING_PLECTIS.md)\nnames three JSON-field comparisons in the prompt-injection example. The README's\n[Choose a route](README.md#choose-a-route) table helps you find another area.\nPick one concrete discrepancy or improvement and keep the change focused on it.\n\n## Propose a public infrastructure change\n\nAn idea does not need a patch. Open a [toolkit issue](https://github.com/wcook04/practical-formal-verification/issues)\nwith one observable problem, the change you suggest, and how another reader\ncould tell whether it helped. Name a public command or file when you can, and\nsay how you would like your idea, code, test, or review credited if adopted.\n\nFor the Lean corpus, proof workflow, or mathematics contributor path, use the\nseparate [architecture proposal form](https://github.com/wcook04/plectis-erdos/issues/new?template=architecture_proposal.yml).\nIt accepts a plain-language idea and asks for an observable stop condition and\ncredit preference. Neither route requires access to the private workbench.\n\n## Try an entry-route experiment\n\nCan a new reader ask for a first example and get a relevant, runnable step?\nFrom a fresh clone with Python 3.11 or newer, run this public-only probe:\n\n```bash\ngit rev-parse HEAD\npython3 --version\nPYTHONPATH=src python3 -m plectis comprehend --first-action \"find and run a first example in a fresh clone\" --format text\n```\n\nRead the returned `Do this first`, reason, and limits.\nThe documented first example is the project tour in [Quickstart](QUICKSTART.md#1-first-result),\nwhich runs from source and writes under ignored `.microcosm/`. Check whether the\nreturned command matches that example. In a clone without an install, the\nprinted `plectis tour` command has this source form:\n\n```bash\nPYTHONPATH=src python3 -m microcosm_core tour --format text .\n```\n\nIf the route matches, run it once. Stop after that one run or at the first\nmismatch: an unrelated route, a missing command or input, or an\nunclear next step. Do not run a command that writes tracked receipts just to\ncomplete this experiment.\n\nReport the commit, operating system, Python version, exact prompt and output,\nwhat first step you expected, and what you observed. If you ran the variant,\ninclude its output path. A focused fix can add a test for the same prompt and\nobserved route. Use only this repository's public source and fixtures; the\nlarger private system is outside this contribution route. For theorem status,\nLean proofs, or paper questions, use the\n[mathematics companion](https://github.com/wcook04/plectis-erdos) instead.\n\n## Reporting a discrepancy\n\nReport an output that differs from the documented expected result, a validator\nthat accepts an invalid input, or a generated table whose entries differ from\nits source JSON. Open a GitHub issue at\n<https://github.com/wcook04/practical-formal-verification/issues> with:\n\n- the commit you ran (`git rev-parse HEAD`);\n- the exact command and the operating system and Python version;\n- what you expected, with the saved result file's path where one applies;\n- what you observed, with the failing output pasted or attached.\n\nKeep the input and failing output so another contributor can reproduce the\nsame result. You can also dispute an expected value or a claim made for a\npassing result. For example, comparing prepared JSON records does not measure\nhow a model responds to a hostile web page; an explanation that confuses those\ntwo experiments needs correcting even if every Python assertion passes.\n\n## Development setup\n\nClone the repository, run the prepared examples, then install the development\ntools. These commands use a macOS or Linux shell (or WSL):\n\n```bash\ngit clone https://github.com/wcook04/practical-formal-verification && cd plectis\n./bootstrap.sh\nVENV=/tmp/plectis-dev-venv make install\n```\n\n`./bootstrap.sh` runs the supplied examples in\n`fixtures/first_wave/pattern_binding_contract/input` and scans for forbidden\nsecret strings. It writes the status in\n`.microcosm/cold_clone_probe.json`, which is ignored by git. Use\n`./bootstrap.sh --dry-run` to preview the command first.\n\n`make install` installs Plectis and its test dependencies (pytest, requests,\nNumPy and pandas) in a virtual environment. `VENV` selects the directory used\nin the examples below. Without that setting, the Makefile chooses a temporary\ndirectory from the checkout path and selected Python interpreter. You do not\nneed pytest installed beforehand.\n\n## Tests and validation\n\n```bash\nmake check      # reject registry errors and prohibited Lean source syntax\nmake test       # public entry and safety tests\nmake ci         # check, test, smoke, package-smoke\nmake validate   # ci plus comparison of generated doctrine tables with source data\n```\n\n`make check` rejects missing or unknown component/evidence-class IDs and\nprohibited syntax in shipped Lean source. `make test` runs the test files listed\nin `Makefile`; `make ci` also runs example commands and installs the package in\na fresh environment before invoking it. Start with the test file for the code\nyou changed, for example:\n\n```bash\nPYTHONPATH=src /tmp/plectis-dev-venv/bin/python -m pytest tests/test_public_entry_docs.py --basetemp=/tmp/plectis-bt\n```\n\nTwo rules the Makefile enforces that direct pytest runs must respect:\n\n- **One temporary test directory per process.** If you run separate pytest\n  commands at the same time, give each a different `--basetemp` directory;\n  otherwise one process can delete or replace another's test inputs.\n- **Tracked receipts are read-only under pytest.** Generated output that needs\n  to change must be regenerated by the program named in that result's contract.\n  Use `MICROCOSM_TRACKED_RECEIPT_WRITES=1` only when you intentionally refresh\n  committed result files. Do not edit a result by hand to make a test pass.\n\nThe commands for running all smoke examples, comparing generated files with\nsource data (`make test-all`), collecting a reviewer packet, producing a\nstandalone export and running that export's tests are documented in\n[docs/maintainers/validation.md](docs/maintainers/validation.md).\n\n## Generated files\n\n`ORGANS.md`, `ARCHITECTURE.md`, `AGENT_ROUTES.md`, `FIRST_ACTION.md`,\n`RELEASE_REVIEW.md`, and generated atlas/registry JSON records are produced by\nthe repository's build scripts. Change the source data and regenerate them\n(for the atlas:\n`PYTHONPATH=src python3 scripts/build/build_organ_atlas.py --write`). Tests compare\ncommitted files with freshly generated files and fail if they differ.\n\n## Documentation changes\n\nUse the [documentation index](docs/README.md) to choose where a new page belongs.\nState what the reader will learn or run, link the page from the relevant guide,\nand include the next document or command. Define unfamiliar project terms\nwhere a reader first needs them; [the short terminology guide](docs/UNDERSTANDING_PLECTIS.md#the-terms-used-in-the-repository)\nis there for reference.\n\nRun new commands from a clone without an activated development environment.\nOpen relative links and section anchors. Describe the actual input, operation\nand output: for example, say which JSON fields are compared and what causes\n`blocked`, rather than just saying a program \"checks the policy\". Keep the\nauthor's meaning and qualifications, and verify changed claims against the\nsource code and recorded results. The existing first-contact tests include:\n\n```bash\nPYTHONPATH=src /tmp/plectis-dev-venv/bin/python -m pytest tests/test_documented_first_contact_commands.py tests/test_public_entry_docs.py\n```\n\n## Pull requests\n\nUse `.github/PULL_REQUEST_TEMPLATE.md` to record the files changed, tests run\nand claims made. Run `make ci`, or explain why the particular tests you ran are\nsufficient for the change. A completed template does not authorise publication.\n\n## Hard boundaries\n\nDo not contribute secrets, credentials, saved sessions, private messages or\nrequest/response bodies from model providers, unedited operator writing,\nrecordings or transcripts, private personal material, live account data, details of live\nexternal targets, hidden evaluation rubrics, or unsafe exploit steps.\n\nDo not treat a passing result as permission to change a user's source files,\ncall a model provider, publish a site or send a message. Do not cite that result\nas proof of product readiness, proof correctness or production security, or as\na basis for financial advice. A test input may contain these permissions\nonly to demonstrate that the program rejects them. Exported result files keep\n`release_authorized=false` until the operator separately authorises publication.\n","body_format":"text"},"hypothesis-handoff":{"document_id":"hypothesis-handoff","source_path":"docs/guides/hypothesis-handoffs.md","role":"hypothesis_handoffs","github":"https://github.com/wcook04/practical-formal-verification/blob/main/docs/guides/hypothesis-handoffs.md","source_commit":"f2ed2582e6de24c71b45dfc30dae33863e2ab539","permalink_url":"https://github.com/wcook04/practical-formal-verification/blob/f2ed2582e6de24c71b45dfc30dae33863e2ab539/docs/guides/hypothesis-handoffs.md","sha256":"sha256:04d79171089ab0bb7ace17952a487bb3a5350239ac94443808e53c205b7e522c","byte_count":5511,"origin":"public_main_checkout","body":"# Hypothesis handoffs\n\nAn open question is easier to answer when you write down your current guess,\nwhat else could be true, and which observations would distinguish the options.\nA hypothesis handoff is a JSON file in which the author records those choices\nand a specific request for an expert. Plectis validates the file's fields and\nreferences between records, then prints it in a readable form.\n\n## The worked question\n\nThe [example file](../../examples/hypothesis_handoff/independent_evaluation.json)\nasks whether evaluator-selected cases would expose a different Plectis failure\nprofile from the example cases supplied by the author. The tentative leading hypothesis\nis that they would reveal more failures and more kinds of failure. The\nalternatives are that the existing fixtures are approximately representative,\nor that new failures would mostly concern dependencies and the environment.\n\nThe author records what each possible outcome would support. For example,\nmore failures in program results or validation rules would support the leading\nhypothesis; a comparable failure profile under adequate coverage would support\nthe first alternative. An informal review could still reveal a useful defect,\nbut would not provide the planned comparison of case-selection methods.\n\nThis is intentionally more useful than “please evaluate the project” and\nintentionally weaker than claiming that an independent evaluation has occurred.\n\n## Run the example\n\nUse Python 3.11 or newer from the clone root. No package installation or model\naccount is needed. The [quickstart](../../QUICKSTART.md#1-first-result) gives the clone\ncommand and shell setup. In a macOS/Linux shell or WSL, run:\n\n```sh\nPYTHONPATH=src python3 -m plectis hypothesis-handoff \\\n  --input examples/hypothesis_handoff/independent_evaluation.json \\\n  --format text\n```\n\nThe supplied example exits with status `0`. Its terminal output begins:\n\n```text\nHypothesis handoff: plectis.independent_evaluation_failure_profile\nQuestion: Would evaluator-selected cases expose a materially different failure profile from the author-selected public fixtures?\n```\n\nIt then prints the recorded hypotheses, observations, expert request and files\nthe author proposes updating. No output file is written. With invalid input,\nit exits with status `1` and prints `Hypothesis handoff: invalid` followed by\nthe errors. Use `--format json` to inspect the `status` and `errors` fields.\n\nThe command reads only the supplied JSON file. It makes no model calls, changes\nno repository files and sends no request to an expert. A successful exit means\nthe records satisfy the rules below; it does not answer the research question.\n\n## What the validator compares\n\nIn the file, a *discriminator* is a proposed observation or experiment that\nthe author thinks would distinguish hypotheses. A *result map* lists possible\noutcomes and the hypothesis IDs each would support.\n\nThe [validator](../../src/microcosm_core/hypothesis_handoff.py) requires:\n\n- nonempty required fields, a leading hypothesis marked `tentative`, and\n  entries describing support and contrary or missing observations;\n- each `distinguished_by` ID to name a row in `discriminating_evidence`, with\n  an outcome in that row referring back to the hypothesis;\n- each outcome's `supports_hypothesis_ids` to name recorded hypotheses, and\n  each discriminator's outcomes to refer to at least two different hypotheses;\n- each proposed update path in `expert_return.landing_targets` to use a\n  repository-relative `/` path, with no `..` component, Windows drive,\n  backslash, or duplicate path after normalisation; and\n- `landing_order` and `status_change_rule` to match the prescribed text: record\n  the argument or observations, independently verify them, update the claim\n  record, then regenerate the public documents and run release validation.\n\nFor example, `hypothesis.independent_cases_find_more_failures` names\n`discriminator.preregistered_case_selection` in `distinguished_by`. That\ndiscriminator's result map must contain an outcome whose\n`supports_hypothesis_ids` includes the hypothesis ID.\n\nThese are comparisons of supplied fields. A person still has to judge whether\nthe proposed experiment distinguishes the hypotheses and whether the cited\nobservations support them. The command does not open cited sources, resolve\ntarget paths on disk or run the validator commands listed beside those paths.\nIt compares the prescribed order as text; it does not perform those steps.\n\n## Prepare a handoff\n\nUse the example's structure to record your question, tentative answer,\nalternatives, proposed observations and expert request. Name the files you\nwould update after reviewing a response, and the commands you would run to\nvalidate those updates. Keep the original handoff with its date or commit so\nyou can compare later revisions with the choices made before the outcome.\n\nYour current expectation is not itself evidence. The command does not estimate\na probability or change a claim's status. Independently verify and review an\nexpert's response before using it to change a public claim. The `declared_gap`\nfield describes one known gap, following the\n[Self-Ignorance Coverage Ledger](../../paper_modules/self_ignorance_coverage_ledger.md):\nnaming one known gap must never imply that no other gap exists.\n\nThe [existing tests](../../tests/test_hypothesis_handoff.py) include invalid IDs,\nmissing outcomes and paths containing `..`. For other examples, return to the\n[documentation index](../README.md).\n","body_format":"text"},"architecture":{"document_id":"architecture","source_path":"ARCHITECTURE.md","role":"software_architecture","github":"https://github.com/wcook04/practical-formal-verification/blob/main/ARCHITECTURE.md","source_commit":"f2ed2582e6de24c71b45dfc30dae33863e2ab539","permalink_url":"https://github.com/wcook04/practical-formal-verification/blob/f2ed2582e6de24c71b45dfc30dae33863e2ab539/ARCHITECTURE.md","sha256":"sha256:758c80f0b769b3655fd4d9bcb62073c7646825329b0db5db026bc904f956f32b","byte_count":29300,"origin":"public_main_checkout","body":"# Plectis Architecture at a Glance\n\n<!-- GENERATED by scripts/build/build_organ_atlas.py from core/organ_registry.json, core/organ_families.json, core/organ_atlas.json, core/organ_evidence_classes.json, core/substrate_substitution_ledger.json, core/paper_module_capsules.json, and core/architecture_kernel.json. Do not hand-edit; run: PYTHONPATH=src python3 scripts/build/build_organ_atlas.py --write -->\n\nPlectis contains 88 components: Python programs and related functions for running proof tools, comparing research data, testing supplied records of agent actions, comparing generated files, and recording work. This page explains the shared project commands and the modules used by the components.\n\nStart with the tour command below to list a project's files and inspect the JSON records it writes. For a component that processes prepared input data, follow [the prompt-injection example](docs/UNDERSTANDING_PLECTIS.md#one-example-you-can-follow). [ORGANS.md](ORGANS.md) lists the other component commands, source files and saved results.\n\n## How to read this page\n\nThe first sections show a command to run, the files it reads and writes, and the implementation modules. The result section explains how to compare a component's output with its stated rules. The [deep reference](#deep-reference) lists the shared record types, source-file relationships and complete component graph.\n\nCommands run from the clone root. The source form below needs Python 3.11 or newer. Later commands use `plectis` or its older `microcosm` alias. Either install them using the [quickstart](QUICKSTART.md#2-install), or replace either command name with `PYTHONPATH=src python3 -m plectis`. Replace `PROJECT` with the directory you want to inspect.\n\n## Level 0 — what it is\n\n```bash\nPYTHONPATH=src python3 -m plectis tour --format text .\nPYTHONPATH=src python3 -m plectis comprehend --slice mechanism --format text\nPYTHONPATH=src python3 -m plectis comprehend --first-action \"prompt injection\" --format text\n```\n\nThe tour creates a file inventory and records a simulated task run. It does not execute the project's application or tests. The second command prints component descriptions. The third matches words to component descriptions and prints a suggested command; it does not execute that command. These three commands make no network or model calls and leave source files unchanged.\n\n## The words you need first\n\nThe source and generated tables use these names:\n\n- **Component** (an *organ* in the source): a named program or group of functions, listed with its inputs, outputs and conditions for accepting a result.\n- **Family**: a grouping of components by what they help you do, not by how finished they are.\n- **Kernel primitive**: one of ten categories in `core/architecture_kernel.json`, such as a project, catalog, proposed task or work record. The table below lists the associated commands.\n- **Evidence class**: a registry label for how a result was obtained, such as comparing supplied records or running an external tool.\n- **Scope limit**: the line that states what a claim does establish and what it does not.\n- **Record or receipt**: a saved result, usually JSON, containing status fields and references to the input, source or output files.\n- **Projection and drift**: a projection is a file generated from other files. Drift means its committed content differs from newly generated content. For this page, `scripts/build/build_organ_atlas.py --check` makes that comparison.\n- **Substrate, macro, and micro**: older project terms for the files and programs, the private parent project, and this public toolkit respectively. The local output directory retains the name `.microcosm/`.\n- **Source authority**: the file contributors edit before regenerating a derived file. Component records for this page are in `core/`; its explanatory text is in `src/microcosm_core/projections/organ_atlas.py`.\n\n## Level 1 — the local runtime loop\n\nThe tour lists paths and classifies filenames and extensions. For Python projects, it also reads package entry points from `pyproject.toml` and the first 4,096 characters of each Python file. It writes the inventory, proposed tasks and a simulated task result under `.microcosm/`. The commands below display introductory text, create those records, or read them back.\n\n```mermaid\nflowchart TD\n  A[Cold reader or agent] --> B[\"plectis hello PROJECT\"]\n  B --> C[\"plectis tour --card PROJECT\"]\n  C --> D[\".microcosm/ state: catalog, patterns, routes,<br/>work_items, events, evidence, graph, explanations\"]\n  D --> E[\"selected_route_id\"]\n  E --> F[\"microcosm explain PROJECT ROUTE_ID\"]\n  E --> G[\"plectis observe --card PROJECT\"]\n  G --> G2[\"plectis observe PROJECT (event rows)\"]\n  D --> H[\"plectis status --card PROJECT\"]\n  H --> I[\"plectis authority --card / workingness --card\"]\n```\n\nOpen `.microcosm/catalog.json` for the file list, `.microcosm/routes.json` for proposed tasks, and `.microcosm/work_items.json` for the simulated task record. `.microcosm/events.jsonl` records the program's operations; `.microcosm/evidence/` contains saved results and references. This output directory is ignored by git in the Plectis clone.\n\n<a id=\"a-real-run-traced-end-to-end\"></a>\n\n## One tour command: source, records and output\n\nThis example describes the tour command using stored entries in `examples/cold_reader_route_map/exported_cold_reader_route_map_bundle`. The page builder reads `route_map.json`, `route_policy.json` and `route_receipts.json` there. It does not run the tour or validate the referenced result while generating this page. Run the command above to produce results for your checkout.\n\n```mermaid\nsequenceDiagram\n  actor Reader\n  participant CLI as cli.py\n  participant Runtime as runtime_shell.py\n  participant Project as project_substrate.py\n  participant State as .microcosm/\n  participant Card as first-screen card\n  Reader->>CLI: plectis tour <project>\n  CLI->>Runtime: request the tour\n  Runtime->>Project: compile or reuse project records\n  Project->>State: write file list and simulated task record\n  Project-->>Runtime: return project records\n  Runtime-->>Card: assemble the summary\n  Card-->>Reader: proposed task ID, status and result paths\n  Note over Reader,Card: no provider calls, no source change, no release\n```\n\n| Reader question | File or operation |\n|---|---|\n| What starts it? | `plectis tour <project>` — point it at a folder |\n| Which code runs? | `cli.py` calls `runtime_shell.py`, which uses `project_substrate.py` to compile or reuse the project records. |\n| What lands on disk? | `.microcosm/` catalog, routes, `events.jsonl`, `evidence/` |\n| What does not change? | source files, provider state, release state |\n| How can I inspect the result? | Open the JSON files written by your run and compare the listed paths with the project directory. |\n| What does the reader see? | A summary with `selected_route_id`, `front_door_status`, and references to saved task and result records. |\n| Where does the claim stop? | local first-screen route map only; no provider calls or source mutation |\n| Where to drill down? | [README.md#install](README.md#install), [AGENTS.md#rules](AGENTS.md#rules) |\n\nThe configured permissions in `core/architecture_kernel.json` are `source_mutation_default: false`, `provider_calls_authorized: false` and `release_authorized: false`. These are configuration values, not measurements of a run. The stored route record names [`receipts/runtime_shell/public_ten_minute_tour.json`](receipts/runtime_shell/public_ten_minute_tour.json) as its saved result.\n\nThe tour is entry 1 of 10 in the stored first-run sequence. Later entries list other commands, saved-result paths and limits. Listing a command here does not execute it.\n\n## How the system is built\n\nThe public module `src/plectis/` calls the implementation in `src/microcosm_core/`. The main implementation directories and data files are listed below. Component commands and document builders have separate entry points; the tour does not run every component.\n\n| Part | Where it lives | Its job |\n|---|---|---|\n| Command parsing | [`cli.py`](src/microcosm_core/cli.py) | Parses `plectis` and `microcosm` arguments and dispatches the selected command. |\n| Command results | [`runtime_shell.py`](src/microcosm_core/runtime_shell.py) | Coordinates project commands and assembles their summaries from project records and saved results. |\n| Project records | [`project_substrate.py`](src/microcosm_core/project_substrate.py) | Builds file inventories, proposes tasks, and writes or reads project records. |\n| Local output | `.microcosm/` | JSON files for the catalog and tasks, operation records in `events.jsonl`, and saved results in `evidence/`. |\n| Component programs | [`organs/`](src/microcosm_core/organs/) | Modules that process the inputs named in each component's instructions. Some can also be invoked directly with `python -m`. |\n| Validation functions | [`validators/`](src/microcosm_core/validators/) | Compare supplied fields, file references or generated results with specified rules. Component modules also contain validation functions. |\n| Document and diagram builders | [`projections/`](src/microcosm_core/projections/) | Produce pages and diagrams from source data, including this page and the component atlas. |\n| Component records | [`core/`](core/) | JSON lists of component IDs, commands, source and result paths, classifications and configured permissions. |\n\n```mermaid\nflowchart LR\n  CLI[\"Command parsing<br/>cli.py\"] -->|project commands| RUN[\"Command results<br/>runtime_shell.py\"]\n  RUN --> SUB[\"Project records<br/>project_substrate.py\"]\n  SUB --> STATE[\"Project-local state<br/>.microcosm/\"]\n  CLI -->|component commands| ORG[\"Component programs<br/>organs/\"]\n  CLI -->|validation commands| VAL[\"Validation functions<br/>validators/\"]\n  CORE[\"Component records<br/>core/*.json\"] -->|input data| PROJ[\"Document builders<br/>projections/\"]\n  PROJ -->|generated files| DOCS[\"ORGANS.md / this page / graph scene\"]\n```\n\nThe arrows above distinguish command dispatch from document generation; they do not list every Python import or function call. To follow one command, locate its handler in `cli.py`, then open the function it calls and the input and output files named in that component's instructions.\n\n## Level 2 — the claim/evidence discipline\n\nA saved status alone does not establish that a program produced the right result. Open its input, the function that processes it, and the test or comparison used to accept its output. The component records name these files and classify how the result was obtained. For example, comparing prepared JSON fields establishes a different fact from running Lean on a supplied proof.\n\n```mermaid\nflowchart LR\n  Claim[\"Public claim<br/>(route / organ / count)\"] --> Class[\"Evidence class<br/>core/organ_evidence_classes.json\"]\n  Class --> Proof[\"Validator / receipt / source ref<br/>core/organ_registry.json\"]\n  Proof --> Ceiling[\"Scope limit<br/>plectis authority --card\"]\n  Ceiling --> Anti[\"Boundary<br/>(where the claim stops)\"]\n  Anti --> Drill[\"Drilldown: paper module / fixture / receipt\"]\n```\n\nThe component count is the number of accepted entries in the registry:\n\n```text\nReads as:       88 components.\nMeans:          88 public entries in core/organ_registry.json.\nDoes not mean:  product maturity, or whole-system correctness.\n```\n\n## Deep reference\n\nThe sections below list source-file relationships, the ten shared record categories and all registered components. Use them to locate a particular file relationship, command or component ID.\n\n## Level 3 — source-module file and shard routing\n\nThis facet is the deep drilldown; skip it on a first read. It is body-free relation metadata from `organ-surface-contract::coverage.source_module_file_graph`, connecting macro source files and source shards to public target files, target shards, and validation refs. The counts below are handles into that graph, not the edge authority itself.\n\n| Handle | Count |\n|---|---:|\n| Accepted organs with source relations | 67 |\n| Source-module edges | 3110 |\n| Source files (per-organ aggregate) | 304 |\n| Source shards (per-organ aggregate) | 695 |\n| Target files (per-organ aggregate) | 308 |\n| Target shards (per-organ aggregate) | 708 |\n| Validation refs (per-organ aggregate) | 121 |\n\nTop relation types: `source_shard.retained_as_public_target_shard` `708` · `source_shard.validated_by_ref` `571` · `target_shard.validated_by_ref` `571` · `source_file.validated_by_ref` `342` · `target_file.validated_by_ref` `342`\n\nRoute into the live topology surface:\n\n- `microcosm organ-topology --organ cold_reader_route_map`\n- `microcosm organ-topology --relation-type file_to_file --source-ref codex/doctrine/skills/kernel/bootstrap.md`\n- `microcosm organ-topology --relation-type shard_to_shard --shard-ref 'codex/doctrine/skills/kernel/bootstrap.md::required_anchor[./repo-python kernel.py --entry \"<task>\" --context-budget 12000]'`\n- `microcosm organ-topology --validation-ref microcosm-substrate/tests/test_cold_reader_route_map.py::test_cold_reader_exported_bundle_validates_runtime_shape`\n- `microcosm organ-topology --organ public_reveal_walkthrough`\n\nDynamic edge truth remains in `microcosm organ-topology`; this page only shows aggregate handles so readers can move from file-to-file and shard-to-shard relations into the owning topology surface.\n\n## Level 3 — the kernel primitives (shared spine)\n\nThe full spine, in one table. Components do not call each other; each binds to the same ten primitives, projected into every scanned project as `.microcosm/architecture.json`. The table reads left to right as the loop a run walks: name the folder, catalog it, see patterns, hold to standards, propose routes, record work, emit events, keep evidence, explain, assimilate.\n\n| Primitive | What it does | Run it | Macro analogue |\n|---|---|---|---|\n| **Project** | Names the user-owned folder and creates project-local Microcosm state. | `microcosm init <project>` | bounded substrate root / authority boundary |\n| **Catalog** | Classifies files into public repo roles such as README, package metadata, source, tests, docs, examples, and scripts. | `microcosm index <project>` | kind atlas / source inventory |\n| **Pattern** | Maps catalog roles into repo-shape pattern observations through the public pattern surface. | `microcosm patterns <project>` | pattern atlas / pattern binding |\n| **Standard** | Records the public constraints that keep local state reversible, explainable, and evidence-backed. | `microcosm architecture <project>` | standards / principles / axioms |\n| **Route** | Turns project-grounded pattern_refs into reversible next-action candidates after resolving them against .microcosm/patterns.json. | `microcosm route <project>` | navigation hologram / option surface / route plane |\n| **Work** | Records a deterministic governed transaction over the project-local route snapshot. | `microcosm work create <project>` | mission transaction / WorkItem spine |\n| **Event** | Emits a causal trace for project substrate operations. | `microcosm observe --card <project>` | observability runtime / trace stream |\n| **Evidence** | Keeps generated receipts as black-box recorder drilldowns. | `microcosm evidence list <project> --limit 25` | evidence membrane / receipts |\n| **Explanation** | Connects a route to grounded refs, resolved pattern bindings, primitives, standards, work shape, events, and evidence. | `microcosm explain <project> <route_id>` | self-comprehension / route rationale |\n| **Assimilation** | Captures reversible next-action and closeout signals without promoting global doctrine. | `microcosm work run <project>` | pattern assimilation step |\n\n## Level 3 — the organ spine, grouped into families\n\nThe 88 accepted components cluster into 7 families, grouped by what they help you do rather than by how finished they are. Each family below links into [ORGANS.md](ORGANS.md), where every component has a card: its job, its first command, its evidence class, and its scope limit.\n\n```mermaid\nflowchart LR\n  P[Project folder] --> M[\".microcosm local state\"]\n  M --> K[Architecture kernel primitives]\n  K --> R[Routes / work / events / evidence]\n  R --> O[Observatory + explain]\n  R --> S[Organ spine]\n  S --> EAR[\"Entry & Reveal (2)\"]\n  S --> AAN[\"Architecture & Navigation (12)\"]\n  S --> FMAP[\"Formal Math & Proof (20)\"]\n  S --> ARAS[\"Agent Reliability & Safety Replays (20)\"]\n  S --> RASR[\"Research & Science Replays (9)\"]\n  S --> IPAD[\"Import, Projection & Drift (20)\"]\n  S --> WLAC[\"Work, Landing & Continuity (5)\"]\n  S --> V[Validators + receipts]\n  S --> AC[Scope limit]\n```\n\n- **[Entry & Reveal](ORGANS.md#entry--reveal)** — 2 components. Microcosm's entry point and what its short guided path actually proves.\n- **[Architecture & Navigation](ORGANS.md#architecture--navigation)** — 12 components. The kernel primitives, pattern binding, doctrine grammar, route plane, and standards that give the system its shape and make it navigable.\n- **[Formal Math & Proof](ORGANS.md#formal-math--proof)** — 20 components. The Lean/proof evidence pipeline: corpus readiness, premise retrieval, tactic routing, verifier trace repair, bounded witnesses, and certificates.\n- **[Agent Reliability & Safety Replays](ORGANS.md#agent-reliability--safety-replays)** — 20 components. Source-open replay specimens for agent failure modes: red-team monitors, sabotage, sandbox escape, prompt injection, tool authority, memory poisoning, benchmark gaming, route observability, and provider budgets.\n- **[Research & Science Replays](ORGANS.md#research--science-replays)** — 9 components. Replay specimens that project scientific and forecasting workflows: replication rubrics, spatial world models, materials-lab safety, mechanistic interpretability, and prediction reconciliation.\n- **[Import, Projection & Drift](ORGANS.md#import-projection--drift)** — 20 components. The membrane that brings non-secret macro substrate into the public tree and keeps projections honest instead of letting them drift from source.\n- **[Work, Landing & Continuity](ORGANS.md#work-landing--continuity)** — 5 components. How reversible work transactions are recorded, how dirty-tree landing decisions are made, and how detached runs resume.\n\n## Level 3 — organ wiring map\n\nThis is the full graph, and it is dense on purpose: a reference, not a first read. Each box is a family; each node is a component (display names; see [ORGANS.md](ORGANS.md) for stable ids). Arrows are the few explicit component-to-component wires, mostly the formal-math proof chain. Most components are standalone specimens, so a node with no arrows is normal: family membership, not call-graph position, is the primary relation.\n\n```mermaid\nflowchart LR\n  subgraph F0[\"Entry & Reveal (2)\"]\n    o0_0[\"Cold Reader Route Map\"]\n    o0_1[\"Public Reveal Walkthrough\"]\n  end\n  subgraph F1[\"Architecture & Navigation (12)\"]\n    o1_0[\"Pattern Binding Contract\"]\n    o1_1[\"Pattern Assimilation Step\"]\n    o1_2[\"Executable Doctrine Grammar\"]\n    o1_3[\"Navigation Hologram Route Plane\"]\n    o1_4[\"Standards Meta Diagnostics\"]\n    o1_5[\"Voice To Doctrine Self Improvement Loop\"]\n    o1_6[\"Cognitive Operator Registry\"]\n    o1_7[\"Routing Anti Patterns Registry\"]\n    o1_8[\"Doctrine Fact Claim Audit\"]\n    o1_9[\"Self Ignorance Coverage Ledger\"]\n    o1_10[\"Navigation Fitness Benchmark\"]\n    o1_11[\"Annex Knowledge Routing\"]\n  end\n  subgraph F2[\"Formal Math & Proof (20)\"]\n    o2_0[\"Certificate Kernel Execution Lab\"]\n    o2_1[\"Formal Math Lean Proof Witness\"]\n    o2_2[\"Verifier Lab Execution Spine\"]\n    o2_3[\"Corpus Readiness Mathlib Absence Gate\"]\n    o2_4[\"Proof / Control / Runtime Import Capsule\"]\n    o2_5[\"Proof Diagnostic Evidence Spine\"]\n    o2_6[\"Formal Math Readiness Gate\"]\n    o2_7[\"Mathematical Strategy Atlas Hypothesis Scorer\"]\n    o2_8[\"Tactic Portfolio Availability Probe\"]\n    o2_9[\"Target Shape Tactic Routing Gate\"]\n    o2_10[\"Lean Std Premise Index\"]\n    o2_11[\"Formal Math Premise Retrieval\"]\n    o2_12[\"Formal Math Verifier Trace Repair Loop\"]\n    o2_13[\"Formal Evidence Cell Anchor Resolver\"]\n    o2_14[\"Undeclared Library Prior Symbol Classifier\"]\n    o2_15[\"Ring2 Premise Retrieval Precision Recall Harness\"]\n    o2_16[\"Verifier Lab Kernel\"]\n    o2_17[\"Proof Derived Governed Mutation Authorization\"]\n    o2_18[\"Finite Erdos Denominator-Order Certificate Strike\"]\n    o2_19[\"Lean Proof-Search Lab Runtime\"]\n  end\n  subgraph F3[\"Agent Reliability & Safety Replays (20)\"]\n    o3_0[\"Agent Closeout Faithfulness Audit\"]\n    o3_1[\"Bounded Autonomy Campaign Packet\"]\n    o3_2[\"Provider Context Recipe Budget Policy\"]\n    o3_3[\"Cold-Eval Honesty Capsule\"]\n    o3_4[\"Validator Checker Capsule\"]\n    o3_5[\"Secondary Runtime Source Capsule\"]\n    o3_6[\"Agent Benchmark Integrity Anti Gaming Replay\"]\n    o3_7[\"Monitor Evidence-Boundary Replay\"]\n    o3_8[\"Sabotage-Monitor Contract Replay\"]\n    o3_9[\"Agent Memory Temporal Conflict Replay\"]\n    o3_10[\"Memory-Poisoning Quarantine Policy Replay\"]\n    o3_11[\"MCP Tool-Authority Policy Replay\"]\n    o3_12[\"Belief-State Reward Bundle Replay\"]\n    o3_13[\"Sandbox-Policy Replay\"]\n    o3_14[\"Prompt-Injection Flow-Policy Replay\"]\n    o3_15[\"Vulnerability Patch-Proof Replay\"]\n    o3_16[\"Agent Route Observability Runtime\"]\n    o3_17[\"Bridge Campaign DAG Validation\"]\n    o3_18[\"Metabolism Queue Reconciliation\"]\n    o3_19[\"Egress Self-Compliance Audit\"]\n  end\n  subgraph F4[\"Research & Science Replays (9)\"]\n    o4_0[\"Finance Forecast Evaluation Spine\"]\n    o4_1[\"Prediction Market Board Capsule\"]\n    o4_2[\"Market Dashboard Read-Model Capsule\"]\n    o4_3[\"Toy-Transformer Attribution Replay\"]\n    o4_4[\"Gridworld Counterfactual State Replay\"]\n    o4_5[\"Research Replication Rubric Artifact Replay\"]\n    o4_6[\"Materials Lab-Safety Refusal Replay\"]\n    o4_7[\"Prediction Oracle Reconciliation\"]\n    o4_8[\"Derived Fact Provider Runtime\"]\n  end\n  subgraph F5[\"Import, Projection & Drift (20)\"]\n    o5_0[\"Macro Projection Import Protocol\"]\n    o5_1[\"Projection-Drift Contract Validator\"]\n    o5_2[\"Unsurfaced Macro Primitives Capsule\"]\n    o5_3[\"Authority Systems Source Capsule\"]\n    o5_4[\"Trace, Code-Map and Scheduling Engines Capsule\"]\n    o5_5[\"Oracle Sibling Source Capsule\"]\n    o5_6[\"Demo Take Console Source Capsule\"]\n    o5_7[\"Tools-Tail Primitives Capsule\"]\n    o5_8[\"Policy Engines Capsule\"]\n    o5_9[\"Audio Level RMS Port\"]\n    o5_10[\"Structural Theses Finance Capsule\"]\n    o5_11[\"Engine Room Demo\"]\n    o5_12[\"Backend and Governance Engines Capsule\"]\n    o5_13[\"Governance and Compiler Mechanisms Capsule\"]\n    o5_14[\"Saturation Engines Capsule\"]\n    o5_15[\"Tool Server Pressure Inventory\"]\n    o5_16[\"Compliance Pipeline Capsule\"]\n    o5_17[\"Live Source Drift Capsule\"]\n    o5_18[\"Release Claim-Language Gate\"]\n    o5_19[\"Generated Projection Drift Runtime\"]\n  end\n  subgraph F6[\"Work, Landing & Continuity (5)\"]\n    o6_0[\"Mission Transaction Work Spine\"]\n    o6_1[\"Durable Agent Work Landing Replay\"]\n    o6_2[\"Bridge-Continuity Acceptance Replay\"]\n    o6_3[\"Concurrency Mission Control\"]\n    o6_4[\"Semantic Singleflight Dedup Runtime\"]\n  end\n  o0_0 --> o1_3\n  o0_0 --> o3_16\n  o0_0 --> o1_0\n  o0_0 --> o1_8\n  o0_0 --> o1_1\n  o0_0 --> o1_7\n  o0_0 --> o1_9\n  o0_0 --> o1_5\n  o0_1 --> o5_18\n  o0_1 --> o4_3\n  o0_1 --> o5_9\n  o0_1 --> o3_4\n  o1_0 --> o1_3\n  o1_0 --> o3_16\n  o1_0 --> o0_0\n  o1_0 --> o5_0\n  o1_0 --> o1_1\n  o1_0 --> o1_5\n  o1_2 --> o1_8\n  o1_3 --> o0_0\n  o1_3 --> o3_16\n  o1_3 --> o1_7\n  o1_3 --> o1_0\n  o1_3 --> o1_8\n  o1_3 --> o5_0\n  o1_3 --> o1_9\n  o1_3 --> o1_5\n  o1_3 --> o5_11\n  o1_3 --> o5_14\n  o1_3 --> o5_16\n  o1_5 --> o1_1\n  o1_7 --> o1_3\n  o1_7 --> o3_16\n  o1_7 --> o0_0\n  o1_8 --> o5_11\n  o1_8 --> o1_9\n  o1_10 --> o0_0\n  o1_10 --> o1_7\n  o1_10 --> o1_3\n  o1_11 --> o0_0\n  o1_11 --> o1_10\n  o1_11 --> o1_7\n  o2_1 --> o2_11\n  o2_1 --> o2_6\n  o2_1 --> o2_5\n  o2_1 --> o2_16\n  o2_1 --> o2_2\n  o2_2 --> o2_0\n  o2_2 --> o5_11\n  o2_3 --> o2_16\n  o2_3 --> o2_8\n  o2_3 --> o2_14\n  o2_3 --> o2_1\n  o2_5 --> o2_12\n  o2_5 --> o2_13\n  o2_5 --> o2_1\n  o2_5 --> o2_16\n  o2_5 --> o2_15\n  o2_5 --> o2_9\n  o2_6 --> o2_1\n  o2_6 --> o5_0\n  o2_6 --> o2_4\n  o2_6 --> o3_2\n  o2_6 --> o2_8\n  o2_6 --> o2_9\n  o2_8 --> o2_3\n  o2_8 --> o2_12\n  o2_8 --> o2_9\n  o2_8 --> o2_14\n  o2_8 --> o2_16\n  o2_9 --> o2_12\n  o2_9 --> o2_13\n  o2_9 --> o2_5\n  o2_9 --> o2_8\n  o2_9 --> o2_16\n  o2_10 --> o2_12\n  o2_10 --> o2_16\n  o2_10 --> o2_7\n  o2_10 --> o2_14\n  o2_11 --> o2_1\n  o2_11 --> o2_12\n  o2_11 --> o2_10\n  o2_11 --> o2_16\n  o2_11 --> o2_7\n  o2_11 --> o2_15\n  o2_12 --> o2_13\n  o2_12 --> o2_5\n  o2_12 --> o2_16\n  o2_12 --> o2_9\n  o2_12 --> o2_2\n  o2_13 --> o2_12\n  o2_13 --> o2_5\n  o2_13 --> o2_9\n  o2_14 --> o2_3\n  o2_14 --> o2_8\n  o2_14 --> o3_2\n  o2_15 --> o2_7\n  o2_15 --> o2_16\n  o2_16 --> o2_3\n  o2_16 --> o2_10\n  o2_16 --> o2_11\n  o2_16 --> o2_8\n  o2_16 --> o2_9\n  o2_16 --> o2_15\n  o2_16 --> o2_12\n  o2_16 --> o2_5\n  o2_16 --> o2_1\n  o2_16 --> o2_2\n  o2_17 --> o5_18\n  o2_17 --> o2_4\n  o2_17 --> o5_8\n  o2_18 --> o2_0\n  o2_18 --> o2_16\n  o2_18 --> o2_1\n  o2_19 --> o2_0\n  o2_19 --> o2_16\n  o2_19 --> o2_1\n  o3_2 --> o3_1\n  o3_2 --> o5_15\n  o3_4 --> o3_3\n  o3_4 --> o5_18\n  o3_6 --> o4_5\n  o3_6 --> o3_3\n  o3_7 --> o3_8\n  o3_9 --> o3_10\n  o3_11 --> o3_13\n  o3_11 --> o3_10\n  o3_13 --> o3_10\n  o3_16 --> o1_3\n  o3_16 --> o0_0\n  o3_16 --> o1_7\n  o3_16 --> o1_0\n  o3_16 --> o5_0\n  o3_16 --> o3_1\n  o3_16 --> o5_14\n  o3_16 --> o2_4\n  o3_16 --> o5_2\n  o3_16 --> o5_4\n  o3_16 --> o5_16\n  o3_16 --> o3_9\n  o3_16 --> o3_13\n  o3_16 --> o3_2\n  o3_16 --> o3_12\n  o3_17 --> o3_1\n  o3_17 --> o3_2\n  o3_18 --> o6_1\n  o3_18 --> o6_0\n  o3_19 --> o3_0\n  o3_19 --> o5_18\n  o4_0 --> o4_1\n  o4_0 --> o4_7\n  o4_1 --> o5_10\n  o4_2 --> o5_2\n  o4_5 --> o4_6\n  o4_5 --> o4_3\n  o4_5 --> o4_7\n  o4_5 --> o4_4\n  o4_5 --> o3_3\n  o4_6 --> o4_4\n  o4_7 --> o4_2\n  o4_7 --> o4_1\n  o4_7 --> o5_10\n  o4_8 --> o5_19\n  o4_8 --> o5_0\n  o5_0 --> o2_6\n  o5_0 --> o3_16\n  o5_0 --> o6_0\n  o5_0 --> o1_3\n  o5_0 --> o1_0\n  o5_0 --> o4_6\n  o5_0 --> o5_14\n  o5_0 --> o5_16\n  o5_0 --> o5_13\n  o5_0 --> o5_17\n  o5_0 --> o2_4\n  o5_0 --> o5_3\n  o5_0 --> o5_2\n  o5_0 --> o5_4\n  o5_0 --> o5_6\n  o5_0 --> o5_5\n  o5_0 --> o3_5\n  o5_0 --> o5_9\n  o5_0 --> o5_8\n  o5_0 --> o5_12\n  o5_0 --> o6_3\n  o5_0 --> o5_11\n  o5_0 --> o1_2\n  o5_1 --> o4_6\n  o5_1 --> o4_3\n  o5_1 --> o4_7\n  o5_1 --> o4_4\n  o5_1 --> o5_15\n  o5_1 --> o4_2\n  o5_1 --> o5_10\n  o5_4 --> o5_8\n  o5_4 --> o5_7\n  o5_4 --> o5_12\n  o5_5 --> o5_7\n  o5_6 --> o5_9\n  o5_6 --> o5_2\n  o5_10 --> o4_1\n  o5_11 --> o5_16\n  o5_11 --> o3_4\n  o5_12 --> o5_13\n  o5_14 --> o5_0\n  o5_14 --> o5_16\n  o5_14 --> o1_3\n  o5_14 --> o3_16\n  o5_16 --> o5_0\n  o5_16 --> o5_14\n  o5_16 --> o1_3\n  o5_16 --> o3_16\n  o5_17 --> o5_13\n  o5_19 --> o5_0\n  o5_19 --> o5_17\n  o5_19 --> o5_18\n  o6_0 --> o3_1\n  o6_0 --> o6_3\n  o6_0 --> o5_0\n  o6_0 --> o5_15\n  o6_0 --> o5_17\n  o6_0 --> o2_4\n  o6_0 --> o5_7\n  o6_0 --> o3_6\n  o6_0 --> o3_15\n  o6_1 --> o3_0\n  o6_1 --> o5_0\n  o6_2 --> o3_9\n  o6_2 --> o6_3\n  o6_3 --> o6_0\n  o6_3 --> o5_17\n  o6_4 --> o6_0\n  o6_4 --> o6_1\n  o6_4 --> o6_3\n  o6_4 --> o5_11\n```\n\n## Level 4 — import & drift membrane\n\nThe public tree carries private substrate, but only through a membrane. A macro candidate is planned; then either its non-secret body is copied to a public target with a digest and a validator, or, if it touches secrets or private state, only an omission receipt and public metadata cross. Projections are then watched for drift. Nothing in this path turns a copy into source-of-record status or a release.\n\n```mermaid\nflowchart TD\n  Macro[Macro substrate candidate] --> Plan[Projection import plan]\n  Plan --> Copy{Non-secret body copyable?}\n  Copy -->|yes| Target[\"Public target + digest + validator\"]\n  Copy -->|no| Omit[\"Omission receipt + public metadata\"]\n  Target --> Receipt[Validation receipt]\n  Omit --> Receipt\n  Receipt --> Intake[\"plectis intake\"]\n  Intake --> Drift[\"microcosm drift-control\"]\n  Drift --> Ceiling[\"Local inspection · source records · release scope unchanged\"]\n```\n\n## Where to go next\n\n- Run the loop yourself: `plectis hello .`, then `plectis tour --card .` in any code folder.\n- Read one component end to end: open [ORGANS.md](ORGANS.md), pick a card, and follow its first command and its source link.\n- Let the system orient you: `plectis comprehend --first-contact` for a cold-start read pack, `plectis comprehend --self-model` for the whole-substrate self-portrait, or `plectis comprehend --packet-atlas` to choose a lens.\n- Follow an edge to its authority: `microcosm organ-topology --organ <id>`.\n\n> Family labels and counts are navigation metadata: use them to browse and compare source-linked components; maturity, completeness, and launch posture live in separate review records. Evidence classes describe bounded result-record strength for each component. Use this public artifact for local inspection, source records, fixture replay, and component-specific evidence. Operational deployment, external-service use, source-file changes, source mutation, private-root equivalence, proof authority, formal-result authority, and whole-system claims remain separate review topics. Continue from the linked card, paper module, command, or source reference.\n\n---\n\nRegenerate from substrate with `PYTHONPATH=src python3 scripts/build/build_organ_atlas.py --write`.\n","body_format":"text"},"citation":{"document_id":"citation","source_path":"CITATION.cff","role":"citation_anchor","github":"https://github.com/wcook04/practical-formal-verification/blob/main/CITATION.cff","source_commit":"f2ed2582e6de24c71b45dfc30dae33863e2ab539","permalink_url":"https://github.com/wcook04/practical-formal-verification/blob/f2ed2582e6de24c71b45dfc30dae33863e2ab539/CITATION.cff","sha256":"sha256:93ad04450bc5ce1cc3cb1b26e5049dbe78dff53bb8e617f606240c3a617db72d","byte_count":1065,"origin":"public_main_checkout","body":"cff-version: 1.2.0\nmessage: \"If you use or discuss Plectis, please cite it using this metadata.\"\ntype: software\ntitle: \"Plectis\"\nabstract: \"A public executable cross-section of an AI-native workflow and research runtime: 88 bounded components spanning formal proof, agent reliability and safety, research and forecasting, projection-drift control, validators, work landing, and continuity. Each component names the code it runs, a typed evidence class, a local receipt, and an explicit authority ceiling where its result stops. A local Python command-line tool exercises them and, pointed at a project, writes an inspectable record beside the source it reads. It runs entirely on your machine.\"\nauthors:\n  - family-names: \"Cook\"\n    given-names: \"William\"\n    email: \"williamwkcook@gmail.com\"\nversion: \"0.6.2\"\ndate-released: \"2026-08-17\"\nlicense: \"Apache-2.0\"\nrepository-code: \"https://github.com/wcook04/practical-formal-verification\"\nurl: \"https://wcook04.github.io/practical-formal-verification/\"\nkeywords:\n  - \"developer-tools\"\n  - \"ai-agents\"\n  - \"evidence\"\n  - \"reproducibility\"\n  - \"research-software\"\n","body_format":"text"},"security":{"document_id":"security","source_path":"SECURITY.md","role":"security_boundary","github":"https://github.com/wcook04/practical-formal-verification/blob/main/SECURITY.md","source_commit":"f2ed2582e6de24c71b45dfc30dae33863e2ab539","permalink_url":"https://github.com/wcook04/practical-formal-verification/blob/f2ed2582e6de24c71b45dfc30dae33863e2ab539/SECURITY.md","sha256":"sha256:c3e1e1e190998a311ee5dfcb391ad95e892ef6c6ea8b12429d698a1bc129b4c8","byte_count":2273,"origin":"public_main_checkout","body":"# Security\n\nPlectis is a local research prototype and developer tool, not a production\nsecurity product. A program writes its results into a receipt. To interpret a\n`pass`, read which command ran, which inputs it used, and which comparisons\nit performed.\n\n## Supported versions\n\nThe latest tagged release and the current `main` branch. There is no bug\nbounty (this is an unfunded solo student project).\n\n## Reporting privately\n\nUse GitHub Private Vulnerability Reporting on this repository:\n\n```text\nhttps://github.com/wcook04/practical-formal-verification/security/advisories/new\n```\n\nIf the \"Report a vulnerability\" button is not visible on the Security page, do\nnot open a public issue with vulnerability details; hold the report until the\nprivate route is available. No `security@` email route is published yet.\nReports get a best-effort response from a solo maintainer, normally within a\nweek.\n\n## What to report\n\nReport an issue if public Plectis material appears to expose or authorize:\n\n- real secrets, credentials, tokens, cookies, private keys, or account\n  sessions;\n- raw operator voice, private personal material, or provider payload bodies;\n- live external targets, live account access, or credentialed provider calls;\n- source mutation, publication, or hosted-release authority that is not\n  explicitly scoped as a negative fixture;\n- unsafe exploit instructions instead of synthetic replay cases.\n\nSynthetic fixtures named credential/payload/secret are not automatically\nleaks: they are allowed as negative cases when the surrounding receipt keeps\nthe unsafe body out of public output.\n\n## What to include\n\nThe path, the command, the receipt id, and a short redacted description.\nDo not paste the suspected secret, private payload, raw prompt body, or\ncredential-equivalent value into the report, and do not attach local\nvalidation byproducts (venv directories, `.microcosm/`, pytest caches).\n\nBefore reporting a release-boundary or authority issue, the local verification\nroute (including the release-authority receipt fields a report should cite) is\ndocumented in\n[docs/maintainers/security-runbook.md](docs/maintainers/security-runbook.md);\nthe README's [Choose a route](README.md#choose-a-route) table links to the\npublic documentation for the affected component or command.\n","body_format":"text"}}}},"challenge_manifest":{"schema":"plectis_challenge_manifest_v0","artifact_kind":"post_freeze_evaluator_controlled_challenge_manifest","execution_boundary":{"chat_assistant_mode":"read_and_cite_only","packet_is_inert":true,"rule":"A chat assistant reading this packet may inspect and quote the challenge records, but it does not run the probe and must not say it executed a check. Any execution belongs to a separate Python interpreter or coding-agent session and must be attributed to that environment."},"why_this_exists":"Everything else in this packet was chosen by the same author: the mechanism, the fixture, the expected value, the validator rule and the presentation. Agreement among those shows self-consistency, not an independent reason to think the rule is right. A challenge moves one consequential choice -- which cases run -- to you.","how_to_run":{"surface":"separate_python_interpreter_or_coding_agent","primary":"Do not run this from the chat assistant reading the packet. If a human wants an independent execution result, move self_contained_probe.program_source into a separate Python 3 interpreter or coding-agent session. It carries the implementation under test verbatim and needs no repository, network, or third-party dependencies.","seed":"Any string. Choose it yourself, after reading this. The author has not seen it and cannot have tuned anything to it.","reproducibility":"Cases derive from sha256('{seed}|{label}|{index}') read as big-endian integers. Reimplement that in any language to confirm the cases follow from your seed rather than from a curated set.","chat_assistant_route":"The implementation source and its declared spec are both embedded. Read one against the other and report divergence. That is the packet-reading chat's valid challenge route, with nothing executed.","note_on_repository_routes":"The project also runs these challenges from its own repository. That route is not reproduced here: it needs a checkout you do not have, and a command you cannot run is not an instruction. Everything you need is in self_contained_probe."},"how_to_return_a_result":{"problem":"The packet-reading chat cannot write to this project, and a separate evaluator's run does not persist anywhere the project can see.","route":"The probe ends by printing a RECEIPT block. Hand that block back to whoever gave you this packet -- in your reply, or as a file. That printed JSON is the entire return path, and it is how an outside review becomes durable here rather than evaporating.","receipt_schema":"plectis_external_challenge_receipt_v0","what_the_project_does_with_it":"The project validates the receipt, binds it to the implementation hash it was actually run against -- so a receipt for bytes that have since changed cannot pass as current assurance -- and files it as external review state. A receipt naming a defeater is the strongest artifact this system can receive. You do not need to do any of that; you only need to print the block."},"outcome_semantics":{"score_authority":"none","asymmetry":"A challenge can only confirm or defeat. It never raises a score, count or pass rate. Surviving a challenge you did not design is weak evidence, because not finding a defeater is not proof of correctness. Failing one is strong evidence against the claim.","why_outside_input_is_admissible_here":"The anti-gaming replay quarantines cases outside the author's registry, correctly, because there an unregistered case could inflate a benchmark score. A challenge touches no score, so it can admit exactly the input that rule must refuse."},"open_challenges":[{"challenge_id":"audio_rms_independent_oracle","component_id":"batch8_audio_level_rms_port","what_you_control":"the input buffers, via your seed","declared_spec":"sqrt(sum(sample^2) / count) * 8, clamped to [0, 1]; int16 samples divide by Int16.max/32767","oracle_method":"Expected values are computed from the declared spec above by an oracle that never imports the implementation, and that accumulates with math.fsum where the implementation uses a running total -- so agreement is not an artifact of copying.","admissible_domain":"float32 and int16 buffers, length 0-64, amplitudes spanning silence, quiet, loud and clipping","metamorphic_relations":["permutation_invariance","sign_invariance","silence_is_zero","halving_halves_level","range_is_unit_interval","duplication_invariance"],"what_a_defeat_would_look_like":"any seeded buffer where the implementation and the independent oracle disagree beyond floating-point tolerance, any violated relation, or an undeclared format accepted instead of refused"},{"challenge_id":"claim_strength_mutation","component_id":"batch12_release_claim_language_gate","what_you_control":"the order claims are walked, via your seed","declared_spec":"Real public claims are strengthened minimally along quantifier, domain, modality, scope and negation axes, and the live release gate is asked whether it notices.","oracle_method":"Positive controls insert phrases already on the gate's own risky list. They establish that a catch would register. Without a caught control, a survivor count means nothing.","what_a_defeat_would_look_like":"a strengthened claim the gate does not classify as an active claim blocker -- reported as a survivor"}],"known_open_defeat":{"finding_id":"negative_context_bleed","found_by":"claim_strength_mutation","status":"open_unrepaired","summary":"The release gate decides that a risky phrase is a boundary clause if the line contains any negative marker anywhere, without checking that the negation scopes over the phrase. An unrelated disclaimer elsewhere in the same line therefore suppresses a live overclaim blocker.","minimal_reproduction":{"caught":"This system is production-ready.","escapes":"This system is production-ready. The colour is not blue."},"why_it_matters":"Public ceiling rows are built from disclaimer lists, so the gate is weakest on exactly the sentences that carry the strongest claims.","disclosure_posture":"Published unrepaired. Recording a defeat the project's own checks did not find is the point of the mechanism."},"self_contained_probe":{"schema":"plectis_self_contained_challenge_probe_v0","runs_where":"a separate Python 3 interpreter or coding-agent session; never the chat assistant merely reading the packet","dependencies":"python standard library only","how_to_use":["Leave the packet-reading chat in read-and-cite-only mode.","If execution is wanted, a human moves program_source into a separate Python 3 interpreter or coding-agent session.","Change SEED to a string you choose. Do not reuse the example.","Run it in that separate environment.","Attribute the printed RECEIPT to that separate run when handing it back."],"what_you_are_testing":"Whether the verbatim implementation agrees with an oracle derived from its own published spec, on inputs its author never chose, and whether six invariants hold that any correct implementation must.","chat_assistant_route":"Read the embedded implementation and declared spec, compare them, and report any divergence with packet paths. Do not execute the program or claim an execution result.","no_execution_path":"The implementation source and the declared spec are both embedded. Reading one against the other is a valid challenge and needs nothing executed.","implementation_under_test":{"source_path":"microcosm-substrate/src/microcosm_core/organs/batch8_audio_level_rms_port.py","file_sha256":"c7ab93c5ee6ce7032ec791b88d5a603ea5694344f8ab81d2763dbfb91368e600","embedded_verbatim":true},"program_bytes":13984,"program_source":"\"\"\"Plectis self-contained challenge probe.\n\nFor a separate evaluator environment. Standard library only. No network,\nno repository, no files written. The chat assistant that supplied or read\nthe packet must not claim it executed this program.\n\n1. Change SEED below to anything you like. Choose it yourself.\n2. Run this file.\n3. Hand the printed RECEIPT block back to whoever gave you this packet.\n\nYou are testing whether the implementation copied verbatim below agrees\nwith an oracle derived from its own published specification, on inputs\nits author never chose.\n\"\"\"\n\nfrom __future__ import annotations\n\nimport hashlib\nimport json\nimport math\nfrom typing import Any, Callable, Mapping, Sequence\n\n# Pick your own. The author has not seen it and cannot have tuned to it.\nSEED = \"change-me\"\n\nDECLARED_SPEC = 'sqrt(sum(sample^2) / count) * 8, clamped to [0, 1]; int16 samples divide by Int16.max/32767'\nIMPLEMENTATION_SOURCE = 'microcosm-substrate/src/microcosm_core/organs/batch8_audio_level_rms_port.py'\nIMPLEMENTATION_SHA256 = 'c7ab93c5ee6ce7032ec791b88d5a603ea5694344f8ab81d2763dbfb91368e600'\n\n\n# ======================================================================\n# IMPLEMENTATION UNDER TEST\n# Copied verbatim from microcosm-substrate/src/microcosm_core/organs/batch8_audio_level_rms_port.py.\n# Compare it against DECLARED_SPEC yourself -- that reading is itself a\n# challenge, and it needs no execution.\n# ======================================================================\ndef normalized_level(samples: Sequence[float | int], sample_format: str) -> float:\n    \"\"\"\n    [ACTION]\n    - Teleology: Compute normalized level for `batch8_audio_level_rms_port` using caller inputs and\n      module constants.\n    - Preconditions: Caller supplies `samples`, `sample_format` in the shape required by the signature.\n    - Reads: `samples`, `sample_format`, module constants, and any files named by those inputs.\n    - Writes: Return value only; durable writes, if any, are performed by explicitly named helper calls.\n    - Guarantee: Returns `float` with body-free status, card, receipt, or finding data as appropriate\n      for the function.\n    - Fails: Propagates parser, filesystem, JSON, and validation errors from called helpers; rejected\n      cases are represented as status or finding rows.\n    - Couples: Calls `sqrt`, `min`, `ValueError`, `max`, `float`; review those helpers when changing\n      this callable.\n    \"\"\"\n    if sample_format not in {\"float32\", \"int16\"}:\n        raise ValueError(f\"unsupported audio sample format: {sample_format}\")\n    if not samples:\n        return 0.0\n\n    total = 0.0\n    count = 0\n    if sample_format == \"float32\":\n        for sample in samples:\n            value = float(sample)\n            total += value * value\n            count += 1\n    else:\n        for sample in samples:\n            value = float(int(sample)) / 32767.0\n            total += value * value\n            count += 1\n\n    if count == 0:\n        return 0.0\n    rms = math.sqrt(total / count)\n    return min(max(rms * 8.0, 0.0), 1.0)\n\n\n# ======================================================================\n# INDEPENDENT ORACLE\n# Written from DECLARED_SPEC, never importing the implementation, and\n# accumulating with math.fsum where the implementation uses a running\n# total -- so agreement is not an artifact of copying.\n# ======================================================================\ndef oracle_audio_rms_level(samples: Sequence[float], sample_format: str) -> float:\n    \"\"\"Compute the declared normalized level independently of the organ.\n\n    Written from AUDIO_RMS_SPEC. Uses `math.fsum` over squares, whereas the\n    implementation under test accumulates with a running `total +=`. Agreement\n    across the two accumulation orders is therefore evidence about the\n    arithmetic, not an artifact of copying it.\n    \"\"\"\n    if sample_format not in (\"float32\", \"int16\"):\n        raise ValueError(f\"unsupported sample format: {sample_format}\")\n    if not samples:\n        return 0.0\n    if sample_format == \"int16\":\n        scaled = [float(int(value)) / 32767.0 for value in samples]\n    else:\n        scaled = [float(value) for value in samples]\n    mean_square = math.fsum(value * value for value in scaled) / len(scaled)\n    return min(max(math.sqrt(mean_square) * 8.0, 0.0), 1.0)\n\n\n# ======================================================================\n# SEEDED CASE GENERATION\n# Cases derive from sha256('{seed}|{label}|{index}'). Reimplement this\n# in any language to confirm they follow from your seed.\n# ======================================================================\ndef stream_block(seed: str, label: str, index: int) -> bytes:\n    \"\"\"Return the deterministic 32-byte block for (seed, label, index).\"\"\"\n    payload = f\"{seed}|{label}|{index}\".encode(\"utf-8\")\n    return hashlib.sha256(payload).digest()\n\n\nclass ChallengeRng:\n    \"\"\"Deterministic, portable value source driven by a reviewer-chosen seed.\"\"\"\n\n    def __init__(self, seed: str, label: str) -> None:\n        self.seed = seed\n        self.label = label\n        self._index = 0\n        self._buffer = b\"\"\n\n    def _take(self, count: int) -> bytes:\n        while len(self._buffer) < count:\n            self._buffer += stream_block(self.seed, self.label, self._index)\n            self._index += 1\n        head, self._buffer = self._buffer[:count], self._buffer[count:]\n        return head\n\n    def unit(self) -> float:\n        \"\"\"Uniform float in [0, 1).\"\"\"\n        return int.from_bytes(self._take(8), \"big\") / float(1 << 64)\n\n    def uniform(self, low: float, high: float) -> float:\n        return low + (high - low) * self.unit()\n\n    def below(self, bound: int) -> int:\n        \"\"\"Uniform integer in [0, bound).\"\"\"\n        if bound <= 0:\n            raise ValueError(\"bound must be positive\")\n        return int.from_bytes(self._take(8), \"big\") % bound\n\n    def between(self, low: int, high: int) -> int:\n        \"\"\"Uniform integer in [low, high].\"\"\"\n        return low + self.below(high - low + 1)\n\n    def pick(self, options: Sequence[Any]) -> Any:\n        return options[self.below(len(options))]\n\n\ndef generate_audio_cases(seed: str, count: int) -> list[dict[str, Any]]:\n    \"\"\"Derive audio buffers from the reviewer's seed.\n\n    The domain deliberately spans the clamp boundary, the empty buffer, single\n    samples and both declared formats, because those are where a normalisation\n    or clamping error would live.\n    \"\"\"\n    rng = ChallengeRng(seed, \"audio_rms\")\n    cases: list[dict[str, Any]] = []\n    for index in range(count):\n        sample_format = rng.pick([\"float32\", \"int16\"])\n        shape = rng.pick([\"quiet\", \"loud\", \"clipping\", \"single\", \"empty\", \"mixed\"])\n        if shape == \"empty\":\n            samples: list[float] = []\n        else:\n            length = 1 if shape == \"single\" else rng.between(2, 64)\n            if shape == \"quiet\":\n                span = 0.02\n            elif shape == \"loud\":\n                span = 0.2\n            elif shape == \"clipping\":\n                span = 1.0\n            else:\n                span = rng.uniform(0.01, 0.6)\n            if sample_format == \"int16\":\n                bound = max(1, int(span * 32767))\n                samples = [float(rng.between(-bound, bound)) for _ in range(length)]\n            else:\n                samples = [rng.uniform(-span, span) for _ in range(length)]\n        cases.append(\n            {\n                \"case_id\": f\"seeded_{index:03d}_{shape}_{sample_format}\",\n                \"sample_format\": sample_format,\n                \"samples\": samples,\n                \"shape\": shape,\n            }\n        )\n    return cases\n\n\n# ======================================================================\n# METAMORPHIC RELATIONS\n# Properties any correct implementation of the spec must satisfy, with\n# no stored expected value at all.\n# ======================================================================\ndef audio_metamorphic_relations(\n    level: Callable[[Sequence[float], str], float],\n    case: dict[str, Any],\n    rng: ChallengeRng,\n) -> list[dict[str, Any]]:\n    samples = list(case[\"samples\"])\n    fmt = case[\"sample_format\"]\n    base = level(samples, fmt)\n    results: list[dict[str, Any]] = []\n\n    def record(relation: str, holds: bool, detail: str) -> None:\n        results.append({\"relation\": relation, \"holds\": bool(holds), \"detail\": detail})\n\n    # Order must not matter: the level is a function of the multiset of samples.\n    if len(samples) > 1:\n        shuffled = list(samples)\n        for position in range(len(shuffled) - 1, 0, -1):\n            swap = rng.below(position + 1)\n            shuffled[position], shuffled[swap] = shuffled[swap], shuffled[position]\n        got = level(shuffled, fmt)\n        record(\"permutation_invariance\", math.isclose(got, base, abs_tol=1e-12),\n               f\"base={base!r} permuted={got!r}\")\n\n    # Squaring destroys sign, so negation must be invisible.\n    if samples:\n        negated = [-value for value in samples]\n        got = level(negated, fmt)\n        record(\"sign_invariance\", math.isclose(got, base, abs_tol=1e-12),\n               f\"base={base!r} negated={got!r}\")\n\n    # Silence is exactly zero, never a small positive artifact.\n    zeros = [0.0 if fmt == \"float32\" else 0] * max(1, len(samples))\n    got = level(zeros, fmt)\n    record(\"silence_is_zero\", got == 0.0, f\"zeros -> {got!r}\")\n\n    # Below the clamp, the level is linear in amplitude.\n    if samples and base > 0.0 and base < 0.5:\n        halved = [value / 2 for value in samples] if fmt == \"float32\" else [\n            float(int(value // 2)) for value in samples\n        ]\n        got = level(halved, fmt)\n        tolerance = 1e-9 if fmt == \"float32\" else 2e-3\n        record(\"halving_halves_level\", math.isclose(got, base / 2, abs_tol=tolerance),\n               f\"base={base!r} halved={got!r}\")\n\n    # The declared range is closed: no input may escape [0, 1].\n    record(\"range_is_unit_interval\", 0.0 <= base <= 1.0, f\"level={base!r}\")\n\n    # Duplicating the buffer leaves the mean square unchanged.\n    if samples:\n        got = level(samples + samples, fmt)\n        record(\"duplication_invariance\", math.isclose(got, base, abs_tol=1e-12),\n               f\"base={base!r} doubled={got!r}\")\n\n    return results\n\n\n# ======================================================================\n# DRIVER\n# ======================================================================\ndef run(seed):\n    cases = generate_audio_cases(seed, 200)\n    rng = ChallengeRng(seed, \"audio_metamorphic\")\n    defeaters = []\n    oracle_checks = 0\n    relation_checks = 0\n\n    for case in cases:\n        samples, fmt = case[\"samples\"], case[\"sample_format\"]\n        expected = oracle_audio_rms_level(samples, fmt)\n        try:\n            observed = normalized_level(samples, fmt)\n        except Exception as error:\n            defeaters.append({\"case_id\": case[\"case_id\"],\n                              \"kind\": \"unexpected_exception\",\n                              \"detail\": \"%s: %s\" % (type(error).__name__, error)})\n            continue\n        oracle_checks += 1\n        if not math.isclose(observed, expected, rel_tol=1e-12, abs_tol=1e-12):\n            defeaters.append({\"case_id\": case[\"case_id\"],\n                              \"kind\": \"oracle_disagreement\",\n                              \"detail\": \"expected=%r observed=%r\" % (expected, observed),\n                              \"samples\": samples, \"sample_format\": fmt})\n        for row in audio_metamorphic_relations(normalized_level, case, rng):\n            relation_checks += 1\n            if not row[\"holds\"]:\n                defeaters.append({\"case_id\": case[\"case_id\"],\n                                  \"kind\": \"metamorphic_violation\",\n                                  \"detail\": \"%s: %s\" % (row[\"relation\"], row[\"detail\"]),\n                                  \"samples\": samples, \"sample_format\": fmt})\n\n    refusals = []\n    for bogus in (\"int24\", \"\", \"FLOAT32\", \"pcm\"):\n        try:\n            normalized_level([0.1, 0.2], bogus)\n        except Exception as error:\n            refusals.append({\"sample_format\": bogus, \"refused\": True})\n        else:\n            refusals.append({\"sample_format\": bogus, \"refused\": False})\n            defeaters.append({\"case_id\": \"refusal_\" + (bogus or \"empty\"),\n                              \"kind\": \"missing_refusal\",\n                              \"detail\": \"undeclared format %r accepted\" % bogus})\n\n    return {\n        \"schema\": \"plectis_external_challenge_receipt_v0\",\n        \"produced_by\": \"self_contained_probe\",\n        \"reviewer_seed\": seed,\n        \"implementation_source\": IMPLEMENTATION_SOURCE,\n        \"implementation_sha256\": IMPLEMENTATION_SHA256,\n        \"declared_spec\": DECLARED_SPEC,\n        \"oracle_comparison_count\": oracle_checks,\n        \"metamorphic_check_count\": relation_checks,\n        \"refusal_check_count\": len(refusals),\n        \"defeater_count\": len(defeaters),\n        \"defeaters\": defeaters,\n        \"verdict\": \"defeater_found\" if defeaters else \"no_defeater_found\",\n        \"score_authority\": \"none\",\n        \"outcome_semantics\": (\n            \"A challenge can only confirm or defeat. Not finding a defeater is \"\n            \"not proof of correctness; finding one is evidence against the claim.\"\n        ),\n    }\n\n\nif __name__ == \"__main__\":\n    if SEED == \"change-me\":\n        print(\"WARNING: SEED is still the example value. Change it, then rerun.\\n\")\n    receipt = run(SEED)\n    print(\"verdict            :\", receipt[\"verdict\"])\n    print(\"oracle comparisons :\", receipt[\"oracle_comparison_count\"])\n    print(\"metamorphic checks :\", receipt[\"metamorphic_check_count\"])\n    print(\"refusal checks     :\", receipt[\"refusal_check_count\"])\n    print(\"defeaters          :\", receipt[\"defeater_count\"])\n    for d in receipt[\"defeaters\"][:10]:\n        print(\"   -\", d[\"kind\"], \":\", d[\"detail\"])\n    print()\n    print(\"----- RECEIPT: hand this block back -----\")\n    print(json.dumps(receipt, indent=2, sort_keys=True))\n    print(\"----- END RECEIPT -----\")\n\n"},"authority_boundary":"Report-only. Grants no release, publication, scoring or correctness authority. A challenge covers the declared domain of one component and says nothing about the private system or deployed behaviour."},"source_receipts":{"schema":"plectis_source_checkout_receipts_v1","role":"build_time_source_identity","why_this_exists":"A packet can agree with itself perfectly and still describe a stale clone. Internal consistency is checked by derived_facts; this is the other half -- what the builder actually read, and whether it was clean, current, and the release it claims.","all_verified_current":true,"degraded_repositories":[],"reader_rule":"If all_verified_current is false, treat every number and link in this packet as measured against the commits named here rather than against public HEAD, and say so when quoting them.","repositories":{"plectis":{"schema":"plectis_source_checkout_receipt_v1","repository":"plectis","checkout_path_disclosed":false,"measurement_deadlines_seconds":{"status":20,"fetch":45},"checkout_path_note":"The builder's local path is deliberately not recorded: it is a host-private detail and carries no evidential weight. 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